import{s as Ke,o as qe,n as As}from"../chunks/scheduler.9bc65507.js";import{S as Ze,i as Ee,g as m,s as p,r as _,m as c,H as g,A as Fe,h as i,f as a,c as l,j as f,u as z,x as v,n as h,B as d,k as Cs,y as n,a as e,v as M,d as x,t as k,w as T}from"../chunks/index.707bf1b6.js";import{T as ae}from"../chunks/Tip.c2ecdbf4.js";import{D as Os}from"../chunks/Docstring.17db21ae.js";import{C as Ys}from"../chunks/CodeBlock.54a9f38d.js";import{E as Vn}from"../chunks/ExampleCodeBlock.4f515aa9.js";import{H as Us}from"../chunks/Heading.342b1fa6.js";function Be(K){let r,j="Example:",u,y,w;return y=new Ys({props:{code:"ZnJvbSUyMHRyYW5zZm9ybWVycyUyMGltcG9ydCUyMFJ3a3ZDb25maWclMkMlMjBSd2t2TW9kZWwlMEElMEElMjMlMjBJbml0aWFsaXppbmclMjBhJTIwUndrdiUyMGNvbmZpZ3VyYXRpb24lMEFjb25maWd1cmF0aW9uJTIwJTNEJTIwUndrdkNvbmZpZygpJTBBJTBBJTIzJTIwSW5pdGlhbGl6aW5nJTIwYSUyMG1vZGVsJTIwKHdpdGglMjByYW5kb20lMjB3ZWlnaHRzKSUyMGZyb20lMjB0aGUlMjBjb25maWd1cmF0aW9uJTBBbW9kZWwlMjAlM0QlMjBSd2t2TW9kZWwoY29uZmlndXJhdGlvbiklMEElMEElMjMlMjBBY2Nlc3NpbmclMjB0aGUlMjBtb2RlbCUyMGNvbmZpZ3VyYXRpb24lMEFjb25maWd1cmF0aW9uJTIwJTNEJTIwbW9kZWwuY29uZmln",highlighted:`<span class="hljs-meta">&gt;&gt;&gt; </span><span class="hljs-keyword">from</span> transformers <span class="hljs-keyword">import</span> RwkvConfig, RwkvModel

<span class="hljs-meta">&gt;&gt;&gt; </span><span class="hljs-comment"># Initializing a Rwkv configuration</span>
<span class="hljs-meta">&gt;&gt;&gt; </span>configuration = RwkvConfig()

<span class="hljs-meta">&gt;&gt;&gt; </span><span class="hljs-comment"># Initializing a model (with random weights) from the configuration</span>
<span class="hljs-meta">&gt;&gt;&gt; </span>model = RwkvModel(configuration)

<span class="hljs-meta">&gt;&gt;&gt; </span><span class="hljs-comment"># Accessing the model configuration</span>
<span class="hljs-meta">&gt;&gt;&gt; </span>configuration = model.config`,wrap:!1}}),{c(){r=m("p"),r.textContent=j,u=p(),_(y.$$.fragment)},l(o){r=i(o,"P",{"data-svelte-h":!0}),v(r)!=="svelte-11lpom8"&&(r.textContent=j),u=l(o),z(y.$$.fragment,o)},m(o,R){e(o,r,R),e(o,u,R),M(y,o,R),w=!0},p:As,i(o){w||(x(y.$$.fragment,o),w=!0)},o(o){k(y.$$.fragment,o),w=!1},d(o){o&&(a(r),a(u)),T(y,o)}}}function Ge(K){let r,j=`Although the recipe for forward pass needs to be defined within this function, one should call the <code>Module</code>
instance afterwards instead of this since the former takes care of running the pre and post processing steps while
the latter silently ignores them.`;return{c(){r=m("p"),r.innerHTML=j},l(u){r=i(u,"P",{"data-svelte-h":!0}),v(r)!=="svelte-fincs2"&&(r.innerHTML=j)},m(u,y){e(u,r,y)},p:As,d(u){u&&a(r)}}}function He(K){let r,j="Example:",u,y,w;return y=new Ys({props:{code:"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",highlighted:`<span class="hljs-meta">&gt;&gt;&gt; </span><span class="hljs-keyword">from</span> transformers <span class="hljs-keyword">import</span> AutoTokenizer, RwkvModel
<span class="hljs-meta">&gt;&gt;&gt; </span><span class="hljs-keyword">import</span> torch

<span class="hljs-meta">&gt;&gt;&gt; </span>tokenizer = AutoTokenizer.from_pretrained(<span class="hljs-string">&quot;RWKV/rwkv-4-169m-pile&quot;</span>)
<span class="hljs-meta">&gt;&gt;&gt; </span>model = RwkvModel.from_pretrained(<span class="hljs-string">&quot;RWKV/rwkv-4-169m-pile&quot;</span>)

<span class="hljs-meta">&gt;&gt;&gt; </span>inputs = tokenizer(<span class="hljs-string">&quot;Hello, my dog is cute&quot;</span>, return_tensors=<span class="hljs-string">&quot;pt&quot;</span>)
<span class="hljs-meta">&gt;&gt;&gt; </span>outputs = model(**inputs)

<span class="hljs-meta">&gt;&gt;&gt; </span>last_hidden_states = outputs.last_hidden_state`,wrap:!1}}),{c(){r=m("p"),r.textContent=j,u=p(),_(y.$$.fragment)},l(o){r=i(o,"P",{"data-svelte-h":!0}),v(r)!=="svelte-11lpom8"&&(r.textContent=j),u=l(o),z(y.$$.fragment,o)},m(o,R){e(o,r,R),e(o,u,R),M(y,o,R),w=!0},p:As,i(o){w||(x(y.$$.fragment,o),w=!0)},o(o){k(y.$$.fragment,o),w=!1},d(o){o&&(a(r),a(u)),T(y,o)}}}function De(K){let r,j=`Although the recipe for forward pass needs to be defined within this function, one should call the <code>Module</code>
instance afterwards instead of this since the former takes care of running the pre and post processing steps while
the latter silently ignores them.`;return{c(){r=m("p"),r.innerHTML=j},l(u){r=i(u,"P",{"data-svelte-h":!0}),v(r)!=="svelte-fincs2"&&(r.innerHTML=j)},m(u,y){e(u,r,y)},p:As,d(u){u&&a(r)}}}function Xe(K){let r,j="Example:",u,y,w;return y=new Ys({props:{code:"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",highlighted:`<span class="hljs-meta">&gt;&gt;&gt; </span><span class="hljs-keyword">import</span> torch
<span class="hljs-meta">&gt;&gt;&gt; </span><span class="hljs-keyword">from</span> transformers <span class="hljs-keyword">import</span> AutoTokenizer, RwkvForCausalLM

<span class="hljs-meta">&gt;&gt;&gt; </span>tokenizer = AutoTokenizer.from_pretrained(<span class="hljs-string">&quot;RWKV/rwkv-4-169m-pile&quot;</span>)
<span class="hljs-meta">&gt;&gt;&gt; </span>model = RwkvForCausalLM.from_pretrained(<span class="hljs-string">&quot;RWKV/rwkv-4-169m-pile&quot;</span>)

<span class="hljs-meta">&gt;&gt;&gt; </span>inputs = tokenizer(<span class="hljs-string">&quot;Hello, my dog is cute&quot;</span>, return_tensors=<span class="hljs-string">&quot;pt&quot;</span>)
<span class="hljs-meta">&gt;&gt;&gt; </span>outputs = model(**inputs, labels=inputs[<span class="hljs-string">&quot;input_ids&quot;</span>])
<span class="hljs-meta">&gt;&gt;&gt; </span>loss = outputs.loss
<span class="hljs-meta">&gt;&gt;&gt; </span>logits = outputs.logits`,wrap:!1}}),{c(){r=m("p"),r.textContent=j,u=p(),_(y.$$.fragment)},l(o){r=i(o,"P",{"data-svelte-h":!0}),v(r)!=="svelte-11lpom8"&&(r.textContent=j),u=l(o),z(y.$$.fragment,o)},m(o,R){e(o,r,R),e(o,u,R),M(y,o,R),w=!0},p:As,i(o){w||(x(y.$$.fragment,o),w=!0)},o(o){k(y.$$.fragment,o),w=!1},d(o){o&&(a(r),a(u)),T(y,o)}}}function Qe(K){let r,j,u,y,w,o,R,sa,as,Wn='The RWKV model was proposed in <a href="https://github.com/BlinkDL/RWKV-LM" rel="nofollow">this repo</a>',aa,ts,Nn="It suggests a tweak in the traditional Transformer attention to make it linear. This way, the model can be used as recurrent network: passing inputs for timestamp 0 and timestamp 1 together is the same as passing inputs at timestamp 0, then inputs at timestamp 1 along with the state of timestamp 0 (see example below).",ta,ns,Ln="This can be more efficient than a regular Transformer and can deal with sentence of any length (even if the model uses a fixed context length for training).",na,es,In=`This model was contributed by <a href="https://huggingface.co/sgugger" rel="nofollow">sgugger</a>.
The original code can be found <a href="https://github.com/BlinkDL/RWKV-LM" rel="nofollow">here</a>.`,ea,ps,pa,ls,la,ms,Kn="If you want to make sure the model stops generating when <code>&#39;\\n\\n&#39;</code> is detected, we recommend using the following stopping criteria:",ma,is,ia,rs,ra,N,os,It,Js,qn=`This is the configuration class to store the configuration of a <a href="/docs/transformers/main/en/model_doc/rwkv#transformers.RwkvModel">RwkvModel</a>. It is used to instantiate a RWKV
model according to the specified arguments, defining the model architecture. Instantiating a configuration with the
defaults will yield a similar configuration to that of the RWVK-4
<a href="https://huggingface.co/RWKV/rwkv-4-169m-pile" rel="nofollow">RWKV/rwkv-4-169m-pile</a> architecture.`,Kt,Vs,Zn=`Configuration objects inherit from <a href="/docs/transformers/main/en/main_classes/configuration#transformers.PretrainedConfig">PretrainedConfig</a> and can be used to control the model outputs. Read the
documentation from <a href="/docs/transformers/main/en/main_classes/configuration#transformers.PretrainedConfig">PretrainedConfig</a> for more information.`,qt,P,oa,cs,ca,U,hs,Zt,Ws,En="The bare RWKV Model transformer outputting raw hidden-states without any specific head on top.",Et,Ns,Fn=`This model inherits from <a href="/docs/transformers/main/en/main_classes/model#transformers.PreTrainedModel">PreTrainedModel</a>. Check the superclass documentation for the generic methods the
library implements for all its model (such as downloading or saving, resizing the input embeddings, pruning heads
etc.)`,Ft,Ls,Bn=`This model is also a PyTorch <a href="https://pytorch.org/docs/stable/nn.html#torch.nn.Module" rel="nofollow">torch.nn.Module</a> subclass.
Use it as a regular PyTorch Module and refer to the PyTorch documentation for all matter related to general usage
and behavior.`,Bt,q,gs,Gt,Is,Gn='The <a href="/docs/transformers/main/en/model_doc/rwkv#transformers.RwkvModel">RwkvModel</a> forward method, overrides the <code>__call__</code> special method.',Ht,O,Dt,Y,ha,ds,ga,J,us,Xt,Ks,Hn=`The RWKV Model transformer with a language modeling head on top (linear layer with weights tied to the input
embeddings).`,Qt,qs,Dn=`This model inherits from <a href="/docs/transformers/main/en/main_classes/model#transformers.PreTrainedModel">PreTrainedModel</a>. Check the superclass documentation for the generic methods the
library implements for all its model (such as downloading or saving, resizing the input embeddings, pruning heads
etc.)`,St,Zs,Xn=`This model is also a PyTorch <a href="https://pytorch.org/docs/stable/nn.html#torch.nn.Module" rel="nofollow">torch.nn.Module</a> subclass.
Use it as a regular PyTorch Module and refer to the PyTorch documentation for all matter related to general usage
and behavior.`,Pt,Z,ys,Ot,Es,Qn='The <a href="/docs/transformers/main/en/model_doc/rwkv#transformers.RwkvForCausalLM">RwkvForCausalLM</a> forward method, overrides the <code>__call__</code> special method.',Yt,A,At,ss,da,vs,ua,fs,sn,ya,te=`<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>O</mi><mo>=</mo><mrow><mtext>softmax</mtext></mrow><mo stretchy="false">(</mo><mi>Q</mi><msup><mi>K</mi><mi>T</mi></msup><mi mathvariant="normal">/</mi><msqrt><mi>d</mi></msqrt><mo stretchy="false">)</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">O = \\hbox{softmax}(QK^{T} / \\sqrt{d}) V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2311em;vertical-align:-0.25em;"></span><span class="mord">softmax</span><span class="mopen">(</span><span class="mord mathnormal">Q</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span></span><span class="mord">/</span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9811em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathnormal">d</span></span></span><span style="top:-2.9411em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702
c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14
c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54
c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10
s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429
c69,-144,104.5,-217.7,106.5,-221
l0 -0
c5.3,-9.3,12,-14,20,-14
H400000v40H845.2724
s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7
c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z
M834 80h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.0589em;"><span></span></span></span></span></span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span></span></span></span></span>`,va,b,an,fa,ne='<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8778em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">Q</span></span></span></span>',wa,ba,ee='<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span></span></span></span>',_a,za,pe='<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span></span></span></span>',Ma,Fs,Sn="seq_len x hidden_size",tn,xa,le='<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi><msup><mi>K</mi><mi>T</mi></msup></mrow><annotation encoding="application/x-tex">QK^{T}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0358em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">Q</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span></span></span></span></span>',ka,Bs,Pn="seq_len x seq_len",nn,Ta,me='<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span></span></span></span>',ja,Ra,ie='<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>O</mi></mrow><annotation encoding="application/x-tex">O</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">O</span></span></span></span>',$a,Ca,ws,en,Ua,re=`<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>O</mi><mi>i</mi></msub><mo>=</mo><mfrac><mrow><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>i</mi></munderover><msup><mi>e</mi><mrow><msub><mi>Q</mi><mi>i</mi></msub><msubsup><mi>K</mi><mi>j</mi><mi>T</mi></msubsup><mi mathvariant="normal">/</mi><msqrt><mi>d</mi></msqrt></mrow></msup><msub><mi>V</mi><mi>j</mi></msub></mrow><mrow><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>i</mi></munderover><msup><mi>e</mi><mrow><msub><mi>Q</mi><mi>i</mi></msub><msubsup><mi>K</mi><mi>j</mi><mi>T</mi></msubsup><mi mathvariant="normal">/</mi><msqrt><mi>d</mi></msqrt></mrow></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">O_{i} = \\frac{\\sum_{j=1}^{i} e^{Q_{i} K_{j}^{T} / \\sqrt{d}} V_{j}}{\\sum_{j=1}^{i} e^{Q_{i} K_{j}^{T} / \\sqrt{d}}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.3004em;vertical-align:-1.4129em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8875em;"><span style="top:-2.11em;"><span class="pstrut" style="height:3.0871em;"></span><span class="mord"><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9646em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4358em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0871em;"><span style="top:-3.1306em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8329em;"><span style="top:-2.1777em;margin-left:-0.0715em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-2.8448em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4612em;"><span></span></span></span></span></span></span><span class="mord mtight">/</span><span class="mord sqrt mtight"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9378em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mtight" style="padding-left:0.833em;"><span class="mord mathnormal mtight">d</span></span></span><span style="top:-2.8978em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail mtight" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702
c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14
c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54
c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10
s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429
c69,-144,104.5,-217.7,106.5,-221
l0 -0
c5.3,-9.3,12,-14,20,-14
H400000v40H845.2724
s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7
c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z
M834 80h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1022em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.3171em;"><span class="pstrut" style="height:3.0871em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.9129em;"><span class="pstrut" style="height:3.0871em;"></span><span class="mord"><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9646em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4358em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0617em;"><span style="top:-3.1052em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9191em;"><span style="top:-2.214em;margin-left:-0.0715em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span><span style="top:-2.931em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4249em;"><span></span></span></span></span></span></span><span class="mord mtight">/</span><span class="mord sqrt mtight"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9378em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mtight" style="padding-left:0.833em;"><span class="mord mathnormal mtight">d</span></span></span><span style="top:-2.8978em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail mtight" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702
c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14
c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54
c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10
s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429
c69,-144,104.5,-217.7,106.5,-221
l0 -0
c5.3,-9.3,12,-14,20,-14
H400000v40H845.2724
s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7
c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z
M834 80h400000v40h-400000z'/></svg></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.1022em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.4129em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>`,Ja,Q,pn,Va,oe='<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>Q</mi><msup><mi>K</mi><mi>T</mi></msup></mrow><annotation encoding="application/x-tex">QK^{T}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0358em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">Q</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span></span></span></span></span>',Wa,Na,ce='<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>j</mi><mo>&gt;</mo><mi>i</mi></mrow><annotation encoding="application/x-tex">j &gt; i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.854em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.05724em;">j</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6595em;"></span><span class="mord mathnormal">i</span></span></span></span>',La,Ia,bs,ln,Ka,he='<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>O</mi><mi>i</mi></msub><mo>=</mo><mi>σ</mi><mo stretchy="false">(</mo><msub><mi>R</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mfrac><mrow><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>i</mi></munderover><msup><mi>e</mi><mrow><msub><mi>W</mi><mrow><mi>i</mi><mo>−</mo><mi>j</mi></mrow></msub><mo>+</mo><msub><mi>K</mi><mi>j</mi></msub></mrow></msup><msub><mi>V</mi><mi>j</mi></msub></mrow><mrow><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>i</mi></munderover><msup><mi>e</mi><mrow><msub><mi>W</mi><mrow><mi>i</mi><mo>−</mo><mi>j</mi></mrow></msub><mo>+</mo><msub><mi>K</mi><mi>j</mi></msub></mrow></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">O_{i} = \\sigma(R_{i}) \\frac{\\sum_{j=1}^{i} e^{W_{i-j} + K_{j}} V_{j}}{\\sum_{j=1}^{i} e^{W_{i-j} + K_{j}}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:3.0808em;vertical-align:-1.2904em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0077em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7904em;"><span style="top:-2.1454em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9646em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4358em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7834em;"><span style="top:-3.0051em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:-0.1389em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2819em;"><span></span></span></span></span></span></span><span class="mbin mtight">+</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:-0.0715em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2819em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.8258em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9646em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4358em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:-0.1389em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2819em;"><span></span></span></span></span></span></span><span class="mbin mtight">+</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:-0.0715em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2819em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2904em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>',qa,V,mn,Za,ge='<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.00773em;">R</span></span></span></span>',Ea,Fa,de='<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">K</span></span></span></span>',Ba,Ga,ue='<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">V</span></span></span></span>',Ha,Da,ye='<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">W</span></span></span></span>',Xa,Qa,ve='<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>W</mi><mn>0</mn></msub><mo>=</mo><mi>u</mi><mrow><mtext> and </mtext></mrow><msub><mi>W</mi><mi>k</mi></msub><mo>=</mo><mo stretchy="false">(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mi>w</mi><mrow><mtext> for </mtext></mrow><mi>k</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">W_{0} = u \\hbox{  and  } W_{k} = (k-1)w \\hbox{ for } k \\geq 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8444em;vertical-align:-0.15em;"></span><span class="mord mathnormal">u</span><span class="mord"> and </span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361em;"><span style="top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="mord"> for </span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></span>',Sa,$,rn,Pa,fe='<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">u</span></span></span></span>',Oa,Ya,we='<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>w</mi></mrow><annotation encoding="application/x-tex">w</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span></span></span></span>',Aa,Gs,On="time_first",on,Hs,Yn="time_decay",cn,st,be='<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>N</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">N_{i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>',at,tt,_e='<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>D</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">D_{i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>',nt,et,ze='<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>N</mi><mi>i</mi></msub><mo>=</mo><msup><mi>e</mi><mrow><mi>u</mi><mo>+</mo><msub><mi>K</mi><mi>i</mi></msub></mrow></msup><msub><mi>V</mi><mi>i</mi></msub><mo>+</mo><msub><mover accent="true"><mi>N</mi><mo>^</mo></mover><mi>i</mi></msub><mrow><mtext> where </mtext></mrow><msub><mover accent="true"><mi>N</mi><mo>^</mo></mover><mi>i</mi></msub><mo>=</mo><msup><mi>e</mi><msub><mi>K</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub></msup><msub><mi>V</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msup><mi>e</mi><mrow><mi>w</mi><mo>+</mo><msub><mi>K</mi><mrow><mi>i</mi><mo>−</mo><mn>2</mn></mrow></msub></mrow></msup><msub><mi>V</mi><mrow><mi>i</mi><mo>−</mo><mn>2</mn></mrow></msub><mo>⋯</mo><mo>+</mo><msup><mi>e</mi><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mi>w</mi><mo>+</mo><msub><mi>K</mi><mn>1</mn></msub></mrow></msup><msub><mi>V</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">N_{i} = e^{u + K_{i}} V_{i} + \\hat{N}_{i} \\hbox{  where  } \\hat{N}_{i} = e^{K_{i-1}} V_{i-1} + e^{w + K_{i-2}} V_{i-2} \\cdots + e^{(i-2)w + K_{1}} V_{1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0413em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">u</span><span class="mbin mtight">+</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:-0.0715em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0968em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9468em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">^</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"> where </span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9468em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">^</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0997em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:-0.0715em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2025em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0997em;vertical-align:-0.2083em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span><span class="mbin mtight">+</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:-0.0715em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2025em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2083em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.088em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">2</span><span class="mclose mtight">)</span><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span><span class="mbin mtight">+</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:-0.0715em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span>',pt,E,hn,lt,Me='<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>N</mi><mo>^</mo></mover><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\\hat{N}_{i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0968em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9468em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">^</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>',mt,Ds,An="numerator_state",gn,it,xe='<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mover accent="true"><mi>N</mi><mo>^</mo></mover><mn>0</mn></msub><mo>=</mo><mn>0</mn><mrow><mtext> and </mtext></mrow><msub><mover accent="true"><mi>N</mi><mo>^</mo></mover><mrow><mi>j</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msup><mi>e</mi><msub><mi>K</mi><mi>j</mi></msub></msup><msub><mi>V</mi><mi>j</mi></msub><mo>+</mo><msup><mi>e</mi><mi>w</mi></msup><msub><mover accent="true"><mi>N</mi><mo>^</mo></mover><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">\\hat{N}_{0} = 0 \\hbox{  and  } \\hat{N}_{j+1} = e^{K_{j}} V_{j} + e^{w} \\hat{N}_{j}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0968em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9468em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">^</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2329em;vertical-align:-0.2861em;"></span><span class="mord">0 and </span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9468em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">^</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1774em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:-0.0715em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2819em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.2329em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9468em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">^</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span></span>',rt,_s,dn,ot,ke='<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>D</mi><mi>i</mi></msub><mo>=</mo><msup><mi>e</mi><mrow><mi>u</mi><mo>+</mo><msub><mi>K</mi><mi>i</mi></msub></mrow></msup><mo>+</mo><msub><mover accent="true"><mi>D</mi><mo>^</mo></mover><mi>i</mi></msub><mrow><mtext> where </mtext></mrow><msub><mover accent="true"><mi>D</mi><mo>^</mo></mover><mi>i</mi></msub><mo>=</mo><msup><mi>e</mi><msub><mi>K</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub></msup><mo>+</mo><msup><mi>e</mi><mrow><mi>w</mi><mo>+</mo><msub><mi>K</mi><mrow><mi>i</mi><mo>−</mo><mn>2</mn></mrow></msub></mrow></msup><mo>⋯</mo><mo>+</mo><msup><mi>e</mi><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mi>w</mi><mo>+</mo><msub><mi>K</mi><mn>1</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">D_{i} = e^{u + K_{i}} + \\hat{D}_{i} \\hbox{  where  } \\hat{D}_{i} = e^{K_{i-1}} + e^{w + K_{i-2}} \\cdots + e^{(i-2)w + K_{1}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9747em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">u</span><span class="mbin mtight">+</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:-0.0715em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0968em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9468em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1944em;"><span class="mord">^</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"> where </span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9468em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1944em;"><span class="mord">^</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9747em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:-0.0715em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2025em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.9747em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span><span class="mbin mtight">+</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:-0.0715em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2025em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.938em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathnormal mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">2</span><span class="mclose mtight">)</span><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span><span class="mbin mtight">+</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:-0.0715em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>',ct,F,un,ht,Te='<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mover accent="true"><mi>D</mi><mo>^</mo></mover><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\\hat{D}_{i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0968em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9468em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1944em;"><span class="mord">^</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>',gt,Xs,se="denominator_state",yn,dt,je='<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mover accent="true"><mi>D</mi><mo>^</mo></mover><mn>0</mn></msub><mo>=</mo><mn>0</mn><mrow><mtext> and </mtext></mrow><msub><mover accent="true"><mi>D</mi><mo>^</mo></mover><mrow><mi>j</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msup><mi>e</mi><msub><mi>K</mi><mi>j</mi></msub></msup><mo>+</mo><msup><mi>e</mi><mi>w</mi></msup><msub><mover accent="true"><mi>D</mi><mo>^</mo></mover><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">\\hat{D}_{0} = 0 \\hbox{  and  } \\hat{D}_{j+1} = e^{K_{j}} + e^{w} \\hat{D}_{j}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0968em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9468em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1944em;"><span class="mord">^</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2329em;vertical-align:-0.2861em;"></span><span class="mord">0 and </span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9468em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1944em;"><span class="mord">^</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9747em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:-0.0715em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2819em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.2329em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9468em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1944em;"><span class="mord">^</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span></span>',ut,zs,vn,yt,Re='<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><msup><mi>e</mi><msub><mi>x</mi><mi>i</mi></msub></msup><mrow><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msup><mi>e</mi><msub><mi>x</mi><mi>j</mi></msub></msup></mrow></mfrac><mo>=</mo><mfrac><msup><mi>e</mi><mrow><msub><mi>x</mi><mi>i</mi></msub><mo>−</mo><mi>M</mi></mrow></msup><mrow><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msup><mi>e</mi><mrow><msub><mi>x</mi><mi>j</mi></msub><mo>−</mo><mi>M</mi></mrow></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">\\frac{e^{x_{i}}}{\\sum_{j=1}^{n} e^{x_{j}}} = \\frac{e^{x_{i} - M}}{\\sum_{j=1}^{n} e^{x_{j} - M}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.4715em;vertical-align:-1.1301em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3414em;"><span style="top:-2.3057em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8043em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4358em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6065em;"><span style="top:-3.0051em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2819em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6644em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.1301em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.6484em;vertical-align:-1.1301em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5183em;"><span style="top:-2.3057em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8043em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4358em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7834em;"><span style="top:-3.0051em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2819em;"><span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.10903em;">M</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.10903em;">M</span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.1301em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>',vt,B,fn,ft,$e='<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">M</span></span></span></span>',wt,bt,Ce='<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">x_{j}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7167em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span>',_t,zt,Ue='<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mover accent="true"><mi>N</mi><mo>~</mo></mover><mi>i</mi></msub><mo>=</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>M</mi><mi>i</mi></msub></mrow></msup><msub><mover accent="true"><mi>N</mi><mo>^</mo></mover><mi>i</mi></msub><mrow><mtext> and </mtext></mrow><msub><mover accent="true"><mi>D</mi><mo>~</mo></mover><mi>i</mi></msub><mo>=</mo><msup><mi>e</mi><mrow><mo>−</mo><msub><mi>M</mi><mi>i</mi></msub></mrow></msup><msub><mover accent="true"><mi>D</mi><mo>^</mo></mover><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\\tilde{N}_{i} = e^{-M_{i}} \\hat{N}_{i} \\hbox{  and  } \\tilde{D}_{i} = e^{-M_{i}} \\hat{D}_{i}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0702em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9202em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span><span style="top:-3.6023em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0968em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:-0.109em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9468em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">^</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"> and </span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9202em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span><span style="top:-3.6023em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1944em;"><span class="mord">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0968em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:-0.109em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9468em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span><span style="top:-3.2523em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1944em;"><span class="mord">^</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span>',Mt,Ms,wn,xt,Je='<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mover accent="true"><mi>N</mi><mo>~</mo></mover><mn>0</mn></msub><mo>=</mo><mn>0</mn><mrow><mtext> and </mtext></mrow><msub><mover accent="true"><mi>N</mi><mo>~</mo></mover><mrow><mi>j</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msup><mi>e</mi><mrow><msub><mi>K</mi><mi>j</mi></msub><mo>−</mo><mi>q</mi></mrow></msup><msub><mi>V</mi><mi>j</mi></msub><mo>+</mo><msup><mi>e</mi><mrow><mi>w</mi><mo>+</mo><msub><mi>M</mi><mi>j</mi></msub><mo>−</mo><mi>q</mi></mrow></msup><msub><mover accent="true"><mi>N</mi><mo>~</mo></mover><mi>j</mi></msub><mrow><mtext> where </mtext></mrow><mi>q</mi><mo>=</mo><mi>max</mi><mo>⁡</mo><mo stretchy="false">(</mo><msub><mi>K</mi><mi>j</mi></msub><mo separator="true">,</mo><mi>w</mi><mo>+</mo><msub><mi>M</mi><mi>j</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\tilde{N}_{0} = 0 \\hbox{  and  } \\tilde{N}_{j+1} = e^{K_{j} - q} V_{j} + e^{w + M_{j} - q} \\tilde{N}_{j} \\hbox{  where  } q = \\max(K_{j}, w + M_{j})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0702em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9202em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span><span style="top:-3.6023em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2063em;vertical-align:-0.2861em;"></span><span class="mord">0 and </span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9202em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span><span style="top:-3.6023em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1774em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:-0.0715em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2819em;"><span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">q</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.2063em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span><span class="mbin mtight">+</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:-0.109em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2819em;"><span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">q</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9202em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span><span style="top:-3.6023em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mord"> where </span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mop">max</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span>',kt,xs,bn,Tt,Ve='<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mover accent="true"><mi>D</mi><mo>~</mo></mover><mn>0</mn></msub><mo>=</mo><mn>0</mn><mrow><mtext> and </mtext></mrow><msub><mover accent="true"><mi>D</mi><mo>~</mo></mover><mrow><mi>j</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msup><mi>e</mi><mrow><msub><mi>K</mi><mi>j</mi></msub><mo>−</mo><mi>q</mi></mrow></msup><mo>+</mo><msup><mi>e</mi><mrow><mi>w</mi><mo>+</mo><msub><mi>M</mi><mi>j</mi></msub><mo>−</mo><mi>q</mi></mrow></msup><msub><mover accent="true"><mi>D</mi><mo>~</mo></mover><mi>j</mi></msub><mrow><mtext> where </mtext></mrow><mi>q</mi><mo>=</mo><mi>max</mi><mo>⁡</mo><mo stretchy="false">(</mo><msub><mi>K</mi><mi>j</mi></msub><mo separator="true">,</mo><mi>w</mi><mo>+</mo><msub><mi>M</mi><mi>j</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\\tilde{D}_{0} = 0 \\hbox{  and  } \\tilde{D}_{j+1} = e^{K_{j} - q} + e^{w + M_{j} - q} \\tilde{D}_{j} \\hbox{  where  } q = \\max(K_{j}, w + M_{j})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0702em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9202em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span><span style="top:-3.6023em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1944em;"><span class="mord">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.2063em;vertical-align:-0.2861em;"></span><span class="mord">0 and </span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9202em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span><span style="top:-3.6023em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1944em;"><span class="mord">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9747em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:-0.0715em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2819em;"><span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">q</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.2063em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.02691em;">w</span><span class="mbin mtight">+</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:-0.109em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2819em;"><span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">q</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9202em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span><span style="top:-3.6023em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1944em;"><span class="mord">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mord"> where </span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mop">max</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02691em;">w</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0361em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span>',jt,S,_n,Rt,We='<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>M</mi><mrow><mi>j</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mi>q</mi></mrow><annotation encoding="application/x-tex">M_{j+1} = q</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05724em;">j</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span></span></span></span>',$t,Ct,Ne='<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>N</mi><mi>i</mi></msub><mo>=</mo><msup><mi>e</mi><mrow><mi>u</mi><mo>+</mo><msub><mi>K</mi><mi>i</mi></msub><mo>−</mo><mi>q</mi></mrow></msup><msub><mi>V</mi><mi>i</mi></msub><mo>+</mo><msup><mi>e</mi><msub><mi>M</mi><mi>i</mi></msub></msup><msub><mover accent="true"><mi>N</mi><mo>~</mo></mover><mi>i</mi></msub><mrow><mtext> where </mtext></mrow><mi>q</mi><mo>=</mo><mi>max</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>u</mi><mo>+</mo><msub><mi>K</mi><mi>i</mi></msub><mo separator="true">,</mo><msub><mi>M</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N_{i} = e^{u + K_{i} - q} V_{i} + e^{M_{i}} \\tilde{N}_{i} \\hbox{  where  } q = \\max(u + K_{i}, M_{i})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0413em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">u</span><span class="mbin mtight">+</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:-0.0715em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">q</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1146em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:-0.109em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9202em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">N</span></span><span style="top:-3.6023em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1667em;"><span class="mord">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"> where </span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">max</span><span class="mopen">(</span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span>',Ut,ks,zn,Jt,Le='<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>D</mi><mi>i</mi></msub><mo>=</mo><msup><mi>e</mi><mrow><mi>u</mi><mo>+</mo><msub><mi>K</mi><mi>i</mi></msub><mo>−</mo><mi>q</mi></mrow></msup><mo>+</mo><msup><mi>e</mi><msub><mi>M</mi><mi>i</mi></msub></msup><msub><mover accent="true"><mi>D</mi><mo>~</mo></mover><mi>i</mi></msub><mrow><mtext> where </mtext></mrow><mi>q</mi><mo>=</mo><mi>max</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>u</mi><mo>+</mo><msub><mi>K</mi><mi>i</mi></msub><mo separator="true">,</mo><msub><mi>M</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D_{i} = e^{u + K_{i} - q} + e^{M_{i}} \\tilde{D}_{i} \\hbox{  where  } q = \\max(u + K_{i}, M_{i})</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9747em;vertical-align:-0.0833em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">u</span><span class="mbin mtight">+</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:-0.0715em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">q</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1146em;vertical-align:-0.1944em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3281em;"><span style="top:-2.357em;margin-left:-0.109em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9202em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">D</span></span><span style="top:-3.6023em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.1944em;"><span class="mord">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"> where </span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">max</span><span class="mopen">(</span><span class="mord mathnormal">u</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">K</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span>',Vt,Ts,Mn,Wt,Ie='<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>O</mi><mi>i</mi></msub><mo>=</mo><mi>σ</mi><mo stretchy="false">(</mo><msub><mi>R</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mfrac><msub><mi>N</mi><mi>i</mi></msub><msub><mi>D</mi><mi>i</mi></msub></mfrac></mrow><annotation encoding="application/x-tex">O_{i} = \\sigma(R_{i}) \\frac{N_{i}}{D_{i}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">O</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.1963em;vertical-align:-0.836em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0077em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">D</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">N</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.836em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>',Nt,Ss,Lt;return w=new Us({props:{title:"RWKV",local:"rwkv",headingTag:"h1"}}),R=new Us({props:{title:"Overview",local:"overview",headingTag:"h2"}}),ps=new Us({props:{title:"Usage example",local:"usage-example",headingTag:"h2"}}),ls=new Ys({props:{code:"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",highlighted:`<span class="hljs-keyword">import</span> torch
<span class="hljs-keyword">from</span> transformers <span class="hljs-keyword">import</span> AutoTokenizer, RwkvConfig, RwkvModel

model = RwkvModel.from_pretrained(<span class="hljs-string">&quot;sgugger/rwkv-430M-pile&quot;</span>)
tokenizer = AutoTokenizer.from_pretrained(<span class="hljs-string">&quot;sgugger/rwkv-430M-pile&quot;</span>)

inputs = tokenizer(<span class="hljs-string">&quot;This is an example.&quot;</span>, return_tensors=<span class="hljs-string">&quot;pt&quot;</span>)
<span class="hljs-comment"># Feed everything to the model</span>
outputs = model(inputs[<span class="hljs-string">&quot;input_ids&quot;</span>])
output_whole = outputs.last_hidden_state

outputs = model(inputs[<span class="hljs-string">&quot;input_ids&quot;</span>][:, :<span class="hljs-number">2</span>])
output_one = outputs.last_hidden_state

<span class="hljs-comment"># Using the state computed on the first inputs, we will get the same output</span>
outputs = model(inputs[<span class="hljs-string">&quot;input_ids&quot;</span>][:, <span class="hljs-number">2</span>:], state=outputs.state)
output_two = outputs.last_hidden_state

torch.allclose(torch.cat([output_one, output_two], dim=<span class="hljs-number">1</span>), output_whole, atol=<span class="hljs-number">1e-5</span>)`,wrap:!1}}),is=new Ys({props:{code:"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",highlighted:`<span class="hljs-keyword">from</span> transformers <span class="hljs-keyword">import</span> StoppingCriteria

<span class="hljs-keyword">class</span> <span class="hljs-title class_">RwkvStoppingCriteria</span>(<span class="hljs-title class_ inherited__">StoppingCriteria</span>):
    <span class="hljs-keyword">def</span> <span class="hljs-title function_">__init__</span>(<span class="hljs-params">self, eos_sequence = [<span class="hljs-number">187</span>,<span class="hljs-number">187</span>], eos_token_id = <span class="hljs-number">537</span></span>):
        self.eos_sequence = eos_sequence
        self.eos_token_id = eos_token_id

    <span class="hljs-keyword">def</span> <span class="hljs-title function_">__call__</span>(<span class="hljs-params">self, input_ids: torch.LongTensor, scores: torch.FloatTensor, **kwargs</span>) -&gt; <span class="hljs-built_in">bool</span>:
        last_2_ids = input_ids[:,-<span class="hljs-number">2</span>:].tolist()
        <span class="hljs-keyword">return</span> self.eos_sequence <span class="hljs-keyword">in</span> last_2_ids


output = model.generate(inputs[<span class="hljs-string">&quot;input_ids&quot;</span>], max_new_tokens=<span class="hljs-number">64</span>, stopping_criteria = [RwkvStoppingCriteria()])`,wrap:!1}}),rs=new Us({props:{title:"RwkvConfig",local:"transformers.RwkvConfig",headingTag:"h2"}}),os=new Os({props:{name:"class transformers.RwkvConfig",anchor:"transformers.RwkvConfig",parameters:[{name:"vocab_size",val:" = 50277"},{name:"context_length",val:" = 1024"},{name:"hidden_size",val:" = 4096"},{name:"num_hidden_layers",val:" = 32"},{name:"attention_hidden_size",val:" = None"},{name:"intermediate_size",val:" = None"},{name:"layer_norm_epsilon",val:" = 1e-05"},{name:"bos_token_id",val:" = 0"},{name:"eos_token_id",val:" = 0"},{name:"rescale_every",val:" = 6"},{name:"tie_word_embeddings",val:" = False"},{name:"use_cache",val:" = True"},{name:"**kwargs",val:""}],parametersDescription:[{anchor:"transformers.RwkvConfig.vocab_size",description:`<strong>vocab_size</strong> (<code>int</code>, <em>optional</em>, defaults to 50277) &#x2014;
Vocabulary size of the RWKV model. Defines the number of different tokens that can be represented by the
<code>inputs_ids</code> passed when calling <a href="/docs/transformers/main/en/model_doc/rwkv#transformers.RwkvModel">RwkvModel</a>.`,name:"vocab_size"},{anchor:"transformers.RwkvConfig.context_length",description:`<strong>context_length</strong> (<code>int</code>, <em>optional</em>, defaults to 1024) &#x2014;
The maximum sequence length that this model can be be used with in a single forward (using it in RNN mode
lets use any sequence length).`,name:"context_length"},{anchor:"transformers.RwkvConfig.hidden_size",description:`<strong>hidden_size</strong> (<code>int</code>, <em>optional</em>, defaults to 4096) &#x2014;
Dimensionality of the embeddings and hidden states.`,name:"hidden_size"},{anchor:"transformers.RwkvConfig.num_hidden_layers",description:`<strong>num_hidden_layers</strong> (<code>int</code>, <em>optional</em>, defaults to 32) &#x2014;
Number of hidden layers in the model.`,name:"num_hidden_layers"},{anchor:"transformers.RwkvConfig.attention_hidden_size",description:`<strong>attention_hidden_size</strong> (<code>int</code>, <em>optional</em>) &#x2014;
Dimensionality of the attention hidden states. Will default to <code>hidden_size</code> if unset.`,name:"attention_hidden_size"},{anchor:"transformers.RwkvConfig.intermediate_size",description:`<strong>intermediate_size</strong> (<code>int</code>, <em>optional</em>) &#x2014;
Dimensionality of the inner feed-forward layers. Will default to 4 times <code>hidden_size</code> if unset.`,name:"intermediate_size"},{anchor:"transformers.RwkvConfig.layer_norm_epsilon",description:`<strong>layer_norm_epsilon</strong> (<code>float</code>, <em>optional</em>, defaults to 1e-05) &#x2014;
The epsilon to use in the layer normalization layers.`,name:"layer_norm_epsilon"},{anchor:"transformers.RwkvConfig.bos_token_id",description:`<strong>bos_token_id</strong> (<code>int</code>, <em>optional</em>, defaults to 0) &#x2014;
The id of the beginning of sentence token in the vocabulary. Defaults to 0 as RWKV uses the same tokenizer
as GPTNeoX.`,name:"bos_token_id"},{anchor:"transformers.RwkvConfig.eos_token_id",description:`<strong>eos_token_id</strong> (<code>int</code>, <em>optional</em>, defaults to 0) &#x2014;
The id of the end of sentence token in the vocabulary. Defaults to 0 as RWKV uses the same tokenizer as
GPTNeoX.`,name:"eos_token_id"},{anchor:"transformers.RwkvConfig.rescale_every",description:`<strong>rescale_every</strong> (<code>int</code>, <em>optional</em>, defaults to 6) &#x2014;
At inference, the hidden states (and weights of the correponding output layers) are divided by 2 every
<code>rescale_every</code> layer. If set to 0 or a negative number, no rescale is done.`,name:"rescale_every"},{anchor:"transformers.RwkvConfig.tie_word_embeddings",description:`<strong>tie_word_embeddings</strong> (<code>bool</code>, <em>optional</em>, defaults to <code>False</code>) &#x2014;
Whether or not to tie the word embeddings with the input token embeddings.`,name:"tie_word_embeddings"},{anchor:"transformers.RwkvConfig.use_cache",description:`<strong>use_cache</strong> (<code>bool</code>, <em>optional</em>, defaults to <code>True</code>) &#x2014;
Whether or not the model should return the last state.`,name:"use_cache"}],source:"https://github.com/huggingface/transformers/blob/main/src/transformers/models/rwkv/configuration_rwkv.py#L38"}}),P=new Vn({props:{anchor:"transformers.RwkvConfig.example",$$slots:{default:[Be]},$$scope:{ctx:K}}}),cs=new Us({props:{title:"RwkvModel",local:"transformers.RwkvModel",headingTag:"h2"}}),hs=new Os({props:{name:"class transformers.RwkvModel",anchor:"transformers.RwkvModel",parameters:[{name:"config",val:""}],parametersDescription:[{anchor:"transformers.RwkvModel.config",description:`<strong>config</strong> (<a href="/docs/transformers/main/en/model_doc/rwkv#transformers.RwkvConfig">RwkvConfig</a>) &#x2014; Model configuration class with all the parameters of the model.
Initializing with a config file does not load the weights associated with the model, only the
configuration. Check out the <a href="/docs/transformers/main/en/main_classes/model#transformers.PreTrainedModel.from_pretrained">from_pretrained()</a> method to load the model weights.`,name:"config"}],source:"https://github.com/huggingface/transformers/blob/main/src/transformers/models/rwkv/modeling_rwkv.py#L596"}}),gs=new Os({props:{name:"forward",anchor:"transformers.RwkvModel.forward",parameters:[{name:"input_ids",val:": typing.Optional[torch.LongTensor] = None"},{name:"attention_mask",val:": typing.Optional[torch.LongTensor] = None"},{name:"inputs_embeds",val:": typing.Optional[torch.FloatTensor] = None"},{name:"state",val:": typing.Optional[typing.List[torch.FloatTensor]] = None"},{name:"use_cache",val:": typing.Optional[bool] = None"},{name:"output_attentions",val:": typing.Optional[bool] = None"},{name:"output_hidden_states",val:": typing.Optional[bool] = None"},{name:"return_dict",val:": typing.Optional[bool] = None"}],parametersDescription:[{anchor:"transformers.RwkvModel.forward.input_ids",description:`<strong>input_ids</strong> (<code>torch.LongTensor</code> of shape <code>(batch_size, input_ids_length)</code>) &#x2014;
<code>input_ids_length</code> = <code>sequence_length</code> if <code>past_key_values</code> is <code>None</code> else
<code>past_key_values[0][0].shape[-2]</code> (<code>sequence_length</code> of input past key value states). Indices of input
sequence tokens in the vocabulary.</p>
<p>If <code>past_key_values</code> is used, only <code>input_ids</code> that do not have their past calculated should be passed as
<code>input_ids</code>.</p>
<p>Indices can be obtained using <a href="/docs/transformers/main/en/model_doc/auto#transformers.AutoTokenizer">AutoTokenizer</a>. See <a href="/docs/transformers/main/en/main_classes/tokenizer#transformers.PreTrainedTokenizerFast.encode">PreTrainedTokenizer.encode()</a> and
<a href="/docs/transformers/main/en/model_doc/fastspeech2_conformer#transformers.FastSpeech2ConformerTokenizer.__call__">PreTrainedTokenizer.<strong>call</strong>()</a> for details.</p>
<p><a href="../glossary#input-ids">What are input IDs?</a>`,name:"input_ids"},{anchor:"transformers.RwkvModel.forward.attention_mask",description:`<strong>attention_mask</strong> (<code>torch.LongTensor</code> of shape <code>(batch_size, input_ids_length)</code>, <em>optional</em>) &#x2014;
Mask to avoid performing attention on padding token indices. Mask values selected in <code>[0, 1]</code>:</p>
<ul>
<li>1 for tokens that are <strong>not masked</strong>,</li>
<li>0 for tokens that are <strong>masked</strong>.</li>
</ul>
<p>This is currently not used by <code>RwkvModel</code>, but will be supported in the future.</p>
<p><a href="../glossary#attention-mask">What are attention masks?</a>`,name:"attention_mask"},{anchor:"transformers.RwkvModel.forward.inputs_embeds",description:`<strong>inputs_embeds</strong> (<code>torch.FloatTensor</code> of shape <code>(batch_size, sequence_length, hidden_size)</code>, <em>optional</em>) &#x2014;
Optionally, instead of passing <code>input_ids</code> you can choose to directly pass an embedded representation. This
is useful if you want more control over how to convert <code>input_ids</code> indices into associated vectors than the
model&#x2019;s internal embedding lookup matrix.`,name:"inputs_embeds"},{anchor:"transformers.RwkvModel.forward.state",description:`<strong>state</strong> (tuple of five <code>torch.FloatTensor</code> of shape <code>(batch_size, hidden_size, num_hidden_layers)</code>, <em>optional</em>) &#x2014;
If passed along, the model uses the previous state in all the blocks (which will give the output for the
<code>input_ids</code> provided as if the model add <code>state_input_ids + input_ids</code> as context).`,name:"state"},{anchor:"transformers.RwkvModel.forward.use_cache",description:`<strong>use_cache</strong> (<code>bool</code>, <em>optional</em>) &#x2014;
If set to <code>True</code>, the last state is returned and can be used to quickly generate the next logits.`,name:"use_cache"},{anchor:"transformers.RwkvModel.forward.output_attentions",description:`<strong>output_attentions</strong> (<code>bool</code>, <em>optional</em>) &#x2014;
Whether or not to return the attentions tensors of all attention layers. See <code>attentions</code> under returned
tensors for more detail.`,name:"output_attentions"},{anchor:"transformers.RwkvModel.forward.output_hidden_states",description:`<strong>output_hidden_states</strong> (<code>bool</code>, <em>optional</em>) &#x2014;
Whether or not to return the hidden states of all layers. See <code>hidden_states</code> under returned tensors for
more detail.`,name:"output_hidden_states"},{anchor:"transformers.RwkvModel.forward.return_dict",description:`<strong>return_dict</strong> (<code>bool</code>, <em>optional</em>) &#x2014;
Whether or not to return a <a href="/docs/transformers/main/en/main_classes/output#transformers.utils.ModelOutput">ModelOutput</a> instead of a plain tuple.`,name:"return_dict"}],source:"https://github.com/huggingface/transformers/blob/main/src/transformers/models/rwkv/modeling_rwkv.py#L617",returnDescription:`<script context="module">export const metadata = 'undefined';<\/script>


<p>A <code>transformers.models.rwkv.modeling_rwkv.RwkvOutput</code> or a tuple of
<code>torch.FloatTensor</code> (if <code>return_dict=False</code> is passed or when <code>config.return_dict=False</code>) comprising various
elements depending on the configuration (<a
  href="/docs/transformers/main/en/model_doc/rwkv#transformers.RwkvConfig"
>RwkvConfig</a>) and inputs.</p>
<ul>
<li>
<p><strong>last_hidden_state</strong> (<code>torch.FloatTensor</code> of shape <code>(batch_size, sequence_length, hidden_size)</code>) — Sequence of hidden-states at the output of the last layer of the model.</p>
</li>
<li>
<p><strong>state</strong> (list of five <code>torch.FloatTensor</code> of shape <code>(batch_size, hidden_size, num_hidden_layers)</code>) — The state of the model at the last time step. Can be used in a forward method with the next <code>input_ids</code> to
avoid providing the old <code>input_ids</code>.</p>
</li>
<li>
<p><strong>hidden_states</strong> (<code>tuple(torch.FloatTensor)</code>, <em>optional</em>, returned when <code>output_hidden_states=True</code> is passed or when <code>config.output_hidden_states=True</code>) — Tuple of <code>torch.FloatTensor</code> (one for the output of the embeddings, if the model has an embedding layer, +
one for the output of each layer) of shape <code>(batch_size, sequence_length, hidden_size)</code>.</p>
<p>Hidden-states of the model at the output of each layer plus the optional initial embedding outputs.</p>
</li>
<li>
<p><strong>attentions</strong> (<code>tuple(torch.FloatTensor)</code>, <em>optional</em>, returned when <code>output_attentions=True</code> is passed or when <code>config.output_attentions=True</code>) — Tuple of <code>torch.FloatTensor</code> (one for each layer) of shape <code>(batch_size, num_heads, sequence_length, sequence_length)</code>.</p>
<p>Attentions weights after the attention softmax, used to compute the weighted average in the self-attention
heads.</p>
</li>
</ul>
`,returnType:`<script context="module">export const metadata = 'undefined';<\/script>


<p><code>transformers.models.rwkv.modeling_rwkv.RwkvOutput</code> or <code>tuple(torch.FloatTensor)</code></p>
`}}),O=new ae({props:{$$slots:{default:[Ge]},$$scope:{ctx:K}}}),Y=new Vn({props:{anchor:"transformers.RwkvModel.forward.example",$$slots:{default:[He]},$$scope:{ctx:K}}}),ds=new Us({props:{title:"RwkvLMHeadModel",local:"transformers.RwkvForCausalLM",headingTag:"h2"}}),us=new Os({props:{name:"class transformers.RwkvForCausalLM",anchor:"transformers.RwkvForCausalLM",parameters:[{name:"config",val:""}],parametersDescription:[{anchor:"transformers.RwkvForCausalLM.config",description:`<strong>config</strong> (<a href="/docs/transformers/main/en/model_doc/rwkv#transformers.RwkvConfig">RwkvConfig</a>) &#x2014; Model configuration class with all the parameters of the model.
Initializing with a config file does not load the weights associated with the model, only the
configuration. Check out the <a href="/docs/transformers/main/en/main_classes/model#transformers.PreTrainedModel.from_pretrained">from_pretrained()</a> method to load the model weights.`,name:"config"}],source:"https://github.com/huggingface/transformers/blob/main/src/transformers/models/rwkv/modeling_rwkv.py#L764"}}),ys=new Os({props:{name:"forward",anchor:"transformers.RwkvForCausalLM.forward",parameters:[{name:"input_ids",val:": typing.Optional[torch.LongTensor] = None"},{name:"attention_mask",val:": typing.Optional[torch.LongTensor] = None"},{name:"inputs_embeds",val:": typing.Optional[torch.FloatTensor] = None"},{name:"state",val:": typing.Optional[typing.List[torch.FloatTensor]] = None"},{name:"labels",val:": typing.Optional[torch.LongTensor] = None"},{name:"use_cache",val:": typing.Optional[bool] = None"},{name:"output_attentions",val:": typing.Optional[bool] = None"},{name:"output_hidden_states",val:": typing.Optional[bool] = None"},{name:"return_dict",val:": typing.Optional[bool] = None"}],parametersDescription:[{anchor:"transformers.RwkvForCausalLM.forward.input_ids",description:`<strong>input_ids</strong> (<code>torch.LongTensor</code> of shape <code>(batch_size, input_ids_length)</code>) &#x2014;
<code>input_ids_length</code> = <code>sequence_length</code> if <code>past_key_values</code> is <code>None</code> else
<code>past_key_values[0][0].shape[-2]</code> (<code>sequence_length</code> of input past key value states). Indices of input
sequence tokens in the vocabulary.</p>
<p>If <code>past_key_values</code> is used, only <code>input_ids</code> that do not have their past calculated should be passed as
<code>input_ids</code>.</p>
<p>Indices can be obtained using <a href="/docs/transformers/main/en/model_doc/auto#transformers.AutoTokenizer">AutoTokenizer</a>. See <a href="/docs/transformers/main/en/main_classes/tokenizer#transformers.PreTrainedTokenizerFast.encode">PreTrainedTokenizer.encode()</a> and
<a href="/docs/transformers/main/en/model_doc/fastspeech2_conformer#transformers.FastSpeech2ConformerTokenizer.__call__">PreTrainedTokenizer.<strong>call</strong>()</a> for details.</p>
<p><a href="../glossary#input-ids">What are input IDs?</a>`,name:"input_ids"},{anchor:"transformers.RwkvForCausalLM.forward.attention_mask",description:`<strong>attention_mask</strong> (<code>torch.LongTensor</code> of shape <code>(batch_size, input_ids_length)</code>, <em>optional</em>) &#x2014;
Mask to avoid performing attention on padding token indices. Mask values selected in <code>[0, 1]</code>:</p>
<ul>
<li>1 for tokens that are <strong>not masked</strong>,</li>
<li>0 for tokens that are <strong>masked</strong>.</li>
</ul>
<p>This is currently not used by <code>RwkvModel</code>, but will be supported in the future.</p>
<p><a href="../glossary#attention-mask">What are attention masks?</a>`,name:"attention_mask"},{anchor:"transformers.RwkvForCausalLM.forward.inputs_embeds",description:`<strong>inputs_embeds</strong> (<code>torch.FloatTensor</code> of shape <code>(batch_size, sequence_length, hidden_size)</code>, <em>optional</em>) &#x2014;
Optionally, instead of passing <code>input_ids</code> you can choose to directly pass an embedded representation. This
is useful if you want more control over how to convert <code>input_ids</code> indices into associated vectors than the
model&#x2019;s internal embedding lookup matrix.`,name:"inputs_embeds"},{anchor:"transformers.RwkvForCausalLM.forward.state",description:`<strong>state</strong> (tuple of five <code>torch.FloatTensor</code> of shape <code>(batch_size, hidden_size, num_hidden_layers)</code>, <em>optional</em>) &#x2014;
If passed along, the model uses the previous state in all the blocks (which will give the output for the
<code>input_ids</code> provided as if the model add <code>state_input_ids + input_ids</code> as context).`,name:"state"},{anchor:"transformers.RwkvForCausalLM.forward.use_cache",description:`<strong>use_cache</strong> (<code>bool</code>, <em>optional</em>) &#x2014;
If set to <code>True</code>, the last state is returned and can be used to quickly generate the next logits.`,name:"use_cache"},{anchor:"transformers.RwkvForCausalLM.forward.output_attentions",description:`<strong>output_attentions</strong> (<code>bool</code>, <em>optional</em>) &#x2014;
Whether or not to return the attentions tensors of all attention layers. See <code>attentions</code> under returned
tensors for more detail.`,name:"output_attentions"},{anchor:"transformers.RwkvForCausalLM.forward.output_hidden_states",description:`<strong>output_hidden_states</strong> (<code>bool</code>, <em>optional</em>) &#x2014;
Whether or not to return the hidden states of all layers. See <code>hidden_states</code> under returned tensors for
more detail.`,name:"output_hidden_states"},{anchor:"transformers.RwkvForCausalLM.forward.return_dict",description:`<strong>return_dict</strong> (<code>bool</code>, <em>optional</em>) &#x2014;
Whether or not to return a <a href="/docs/transformers/main/en/main_classes/output#transformers.utils.ModelOutput">ModelOutput</a> instead of a plain tuple.`,name:"return_dict"},{anchor:"transformers.RwkvForCausalLM.forward.labels",description:`<strong>labels</strong> (<code>torch.LongTensor</code> of shape <code>(batch_size, sequence_length)</code>, <em>optional</em>) &#x2014;
Labels for language modeling. Note that the labels <strong>are shifted</strong> inside the model, i.e. you can set
<code>labels = input_ids</code> Indices are selected in <code>[-100, 0, ..., config.vocab_size]</code> All labels set to <code>-100</code>
are ignored (masked), the loss is only computed for labels in <code>[0, ..., config.vocab_size]</code>`,name:"labels"}],source:"https://github.com/huggingface/transformers/blob/main/src/transformers/models/rwkv/modeling_rwkv.py#L795",returnDescription:`<script context="module">export const metadata = 'undefined';<\/script>


<p>A <code>transformers.models.rwkv.modeling_rwkv.RwkvCausalLMOutput</code> or a tuple of
<code>torch.FloatTensor</code> (if <code>return_dict=False</code> is passed or when <code>config.return_dict=False</code>) comprising various
elements depending on the configuration (<a
  href="/docs/transformers/main/en/model_doc/rwkv#transformers.RwkvConfig"
>RwkvConfig</a>) and inputs.</p>
<ul>
<li>
<p><strong>loss</strong> (<code>torch.FloatTensor</code> of shape <code>(1,)</code>, <em>optional</em>, returned when <code>labels</code> is provided) — Language modeling loss (for next-token prediction).</p>
</li>
<li>
<p><strong>logits</strong> (<code>torch.FloatTensor</code> of shape <code>(batch_size, sequence_length, config.vocab_size)</code>) — Prediction scores of the language modeling head (scores for each vocabulary token before SoftMax).</p>
</li>
<li>
<p><strong>state</strong> (list of five <code>torch.FloatTensor</code> of shape <code>(batch_size, hidden_size, num_hidden_layers)</code>) — The state of the model at the last time step. Can be used in a forward method with the next <code>input_ids</code> to
avoid providing the old <code>input_ids</code>.</p>
</li>
<li>
<p><strong>hidden_states</strong> (<code>tuple(torch.FloatTensor)</code>, <em>optional</em>, returned when <code>output_hidden_states=True</code> is passed or when <code>config.output_hidden_states=True</code>) — Tuple of <code>torch.FloatTensor</code> (one for the output of the embeddings, if the model has an embedding layer, +
one for the output of each layer) of shape <code>(batch_size, sequence_length, hidden_size)</code>.</p>
<p>Hidden-states of the model at the output of each layer plus the optional initial embedding outputs.</p>
</li>
<li>
<p><strong>attentions</strong> (<code>tuple(torch.FloatTensor)</code>, <em>optional</em>, returned when <code>output_attentions=True</code> is passed or when <code>config.output_attentions=True</code>) — Tuple of <code>torch.FloatTensor</code> (one for each layer) of shape <code>(batch_size, num_heads, sequence_length, sequence_length)</code>.</p>
<p>Attentions weights after the attention softmax, used to compute the weighted average in the self-attention
heads.</p>
</li>
</ul>
`,returnType:`<script context="module">export const metadata = 'undefined';<\/script>


<p><code>transformers.models.rwkv.modeling_rwkv.RwkvCausalLMOutput</code> or <code>tuple(torch.FloatTensor)</code></p>
`}}),A=new ae({props:{$$slots:{default:[De]},$$scope:{ctx:K}}}),ss=new Vn({props:{anchor:"transformers.RwkvForCausalLM.forward.example",$$slots:{default:[Xe]},$$scope:{ctx:K}}}),vs=new Us({props:{title:"Rwkv attention and the recurrent formulas",local:"rwkv-attention-and-the-recurrent-formulas",headingTag:"h2"}}),{c(){r=m("meta"),j=p(),u=m("p"),y=p(),_(w.$$.fragment),o=p(),_(R.$$.fragment),sa=p(),as=m("p"),as.innerHTML=Wn,aa=p(),ts=m("p"),ts.textContent=Nn,ta=p(),ns=m("p"),ns.textContent=Ln,na=p(),es=m("p"),es.innerHTML=In,ea=p(),_(ps.$$.fragment),pa=p(),_(ls.$$.fragment),la=p(),ms=m("p"),ms.innerHTML=Kn,ma=p(),_(is.$$.fragment),ia=p(),_(rs.$$.fragment),ra=p(),N=m("div"),_(os.$$.fragment),It=p(),Js=m("p"),Js.innerHTML=qn,Kt=p(),Vs=m("p"),Vs.innerHTML=Zn,qt=p(),_(P.$$.fragment),oa=p(),_(cs.$$.fragment),ca=p(),U=m("div"),_(hs.$$.fragment),Zt=p(),Ws=m("p"),Ws.textContent=En,Et=p(),Ns=m("p"),Ns.innerHTML=Fn,Ft=p(),Ls=m("p"),Ls.innerHTML=Bn,Bt=p(),q=m("div"),_(gs.$$.fragment),Gt=p(),Is=m("p"),Is.innerHTML=Gn,Ht=p(),_(O.$$.fragment),Dt=p(),_(Y.$$.fragment),ha=p(),_(ds.$$.fragment),ga=p(),J=m("div"),_(us.$$.fragment),Xt=p(),Ks=m("p"),Ks.textContent=Hn,Qt=p(),qs=m("p"),qs.innerHTML=Dn,St=p(),Zs=m("p"),Zs.innerHTML=Xn,Pt=p(),Z=m("div"),_(ys.$$.fragment),Ot=p(),Es=m("p"),Es.innerHTML=Qn,Yt=p(),_(A.$$.fragment),At=p(),_(ss.$$.fragment),da=p(),_(vs.$$.fragment),ua=p(),fs=m("p"),sn=c(`In a traditional auto-regressive Transformer, attention is written as
`),ya=new g(!1),va=p(),b=m("p"),an=c("with"),fa=new g(!1),wa=c(","),ba=new g(!1),_a=c(" and"),za=new g(!1),Ma=c(" are matrices of shape "),Fs=m("code"),Fs.textContent=Sn,tn=c(" named query, key and value (they are actually bigger matrices with a batch dimension and an attention head dimension but we’re only interested in the last two, which is where the matrix product is taken, so for the sake of simplicity we only consider those two). The product"),xa=new g(!1),ka=c(" then has shape "),Bs=m("code"),Bs.textContent=Pn,nn=c(" and we can take the maxtrix product with"),Ta=new g(!1),ja=c(" to get the output"),Ra=new g(!1),$a=c(" of the same shape as the others."),Ca=p(),ws=m("p"),en=c(`Replacing the softmax by its value gives:
`),Ua=new g(!1),Ja=p(),Q=m("p"),pn=c("Note that the entries in"),Va=new g(!1),Wa=c(" corresponding to"),Na=new g(!1),La=c(" are masked (the sum stops at j) because the attention is not allowed to look at future tokens (only past ones)."),Ia=p(),bs=m("p"),ln=c(`In comparison, the RWKV attention is given by
`),Ka=new g(!1),qa=p(),V=m("p"),mn=c("where"),Za=new g(!1),Ea=c(" is a new matrix called receptance by the author,"),Fa=new g(!1),Ba=c(" and"),Ga=new g(!1),Ha=c(" are still the key and value (\\(\\sigma\\) here is the sigmoid function)."),Da=new g(!1),Xa=c(` is a new vector that represents the position of the token and is given by
`),Qa=new g(!1),Sa=p(),$=m("p"),rn=c("with"),Pa=new g(!1),Oa=c(" and"),Ya=new g(!1),Aa=c(" learnable parameters called in the code "),Gs=m("code"),Gs.textContent=On,on=c(" and "),Hs=m("code"),Hs.textContent=Yn,cn=c(" respectively. The numerator and denominator can both be expressed recursively. Naming them"),st=new g(!1),at=c(" and"),tt=new g(!1),nt=c(` we have:
`),et=new g(!1),pt=p(),E=m("p"),hn=c("so"),lt=new g(!1),mt=c(" (called "),Ds=m("code"),Ds.textContent=An,gn=c(` in the code) satistfies
`),it=new g(!1),rt=p(),_s=m("p"),dn=c(`and
`),ot=new g(!1),ct=p(),F=m("p"),un=c("so"),ht=new g(!1),gt=c(" (called "),Xs=m("code"),Xs.textContent=se,yn=c(` in the code) satistfies
`),dt=new g(!1),ut=p(),zs=m("p"),vn=c(`The actual recurrent formula used are a tiny bit more complex, as for numerical stability we don’t want to compute exponentials of big numbers. Usually the softmax is not computed as is, but the exponential of the maximum term is divided of the numerator and denominator:
`),yt=new g(!1),vt=p(),B=m("p"),fn=c("with"),ft=new g(!1),wt=c(" the maximum of all"),bt=new g(!1),_t=c(`. So here on top of saving the numerator state (\\(\\hat{N}\\)) and the denominator state (\\(\\hat{D}\\)) we also keep track of the maximum of all terms encountered in the exponentials. So we actually use
`),zt=new g(!1),Mt=p(),Ms=m("p"),wn=c(`defined by the following recurrent formulas:
`),xt=new g(!1),kt=p(),xs=m("p"),bn=c(`and
`),Tt=new g(!1),jt=p(),S=m("p"),_n=c("and"),Rt=new g(!1),$t=c(`. With those, we can then compute
`),Ct=new g(!1),Ut=p(),ks=m("p"),zn=c(`and
`),Jt=new g(!1),Vt=p(),Ts=m("p"),Mn=c(`which finally gives us
`),Wt=new g(!1),Nt=p(),Ss=m("p"),this.h()},l(s){const t=Fe("svelte-u9bgzb",document.head);r=i(t,"META",{name:!0,content:!0}),t.forEach(a),j=l(s),u=i(s,"P",{}),f(u).forEach(a),y=l(s),z(w.$$.fragment,s),o=l(s),z(R.$$.fragment,s),sa=l(s),as=i(s,"P",{"data-svelte-h":!0}),v(as)!=="svelte-1dglndq"&&(as.innerHTML=Wn),aa=l(s),ts=i(s,"P",{"data-svelte-h":!0}),v(ts)!=="svelte-1iefmuz"&&(ts.textContent=Nn),ta=l(s),ns=i(s,"P",{"data-svelte-h":!0}),v(ns)!=="svelte-1ump7j4"&&(ns.textContent=Ln),na=l(s),es=i(s,"P",{"data-svelte-h":!0}),v(es)!=="svelte-14u6wc5"&&(es.innerHTML=In),ea=l(s),z(ps.$$.fragment,s),pa=l(s),z(ls.$$.fragment,s),la=l(s),ms=i(s,"P",{"data-svelte-h":!0}),v(ms)!=="svelte-1p9yvl5"&&(ms.innerHTML=Kn),ma=l(s),z(is.$$.fragment,s),ia=l(s),z(rs.$$.fragment,s),ra=l(s),N=i(s,"DIV",{class:!0});var G=f(N);z(os.$$.fragment,G),It=l(G),Js=i(G,"P",{"data-svelte-h":!0}),v(Js)!=="svelte-u4a3b2"&&(Js.innerHTML=qn),Kt=l(G),Vs=i(G,"P",{"data-svelte-h":!0}),v(Vs)!=="svelte-o55m63"&&(Vs.innerHTML=Zn),qt=l(G),z(P.$$.fragment,G),G.forEach(a),oa=l(s),z(cs.$$.fragment,s),ca=l(s),U=i(s,"DIV",{class:!0});var L=f(U);z(hs.$$.fragment,L),Zt=l(L),Ws=i(L,"P",{"data-svelte-h":!0}),v(Ws)!=="svelte-1nvylp"&&(Ws.textContent=En),Et=l(L),Ns=i(L,"P",{"data-svelte-h":!0}),v(Ns)!=="svelte-6pahdo"&&(Ns.innerHTML=Fn),Ft=l(L),Ls=i(L,"P",{"data-svelte-h":!0}),v(Ls)!=="svelte-hswkmf"&&(Ls.innerHTML=Bn),Bt=l(L),q=i(L,"DIV",{class:!0});var H=f(q);z(gs.$$.fragment,H),Gt=l(H),Is=i(H,"P",{"data-svelte-h":!0}),v(Is)!=="svelte-118wom5"&&(Is.innerHTML=Gn),Ht=l(H),z(O.$$.fragment,H),Dt=l(H),z(Y.$$.fragment,H),H.forEach(a),L.forEach(a),ha=l(s),z(ds.$$.fragment,s),ga=l(s),J=i(s,"DIV",{class:!0});var I=f(J);z(us.$$.fragment,I),Xt=l(I),Ks=i(I,"P",{"data-svelte-h":!0}),v(Ks)!=="svelte-x9faml"&&(Ks.textContent=Hn),Qt=l(I),qs=i(I,"P",{"data-svelte-h":!0}),v(qs)!=="svelte-6pahdo"&&(qs.innerHTML=Dn),St=l(I),Zs=i(I,"P",{"data-svelte-h":!0}),v(Zs)!=="svelte-hswkmf"&&(Zs.innerHTML=Xn),Pt=l(I),Z=i(I,"DIV",{class:!0});var D=f(Z);z(ys.$$.fragment,D),Ot=l(D),Es=i(D,"P",{"data-svelte-h":!0}),v(Es)!=="svelte-3ehl35"&&(Es.innerHTML=Qn),Yt=l(D),z(A.$$.fragment,D),At=l(D),z(ss.$$.fragment,D),D.forEach(a),I.forEach(a),da=l(s),z(vs.$$.fragment,s),ua=l(s),fs=i(s,"P",{});var xn=f(fs);sn=h(xn,`In a traditional auto-regressive Transformer, attention is written as
`),ya=d(xn,!1),xn.forEach(a),va=l(s),b=i(s,"P",{});var C=f(b);an=h(C,"with"),fa=d(C,!1),wa=h(C,","),ba=d(C,!1),_a=h(C," and"),za=d(C,!1),Ma=h(C," are matrices of shape "),Fs=i(C,"CODE",{"data-svelte-h":!0}),v(Fs)!=="svelte-9fu9w3"&&(Fs.textContent=Sn),tn=h(C," named query, key and value (they are actually bigger matrices with a batch dimension and an attention head dimension but we’re only interested in the last two, which is where the matrix product is taken, so for the sake of simplicity we only consider those two). The product"),xa=d(C,!1),ka=h(C," then has shape "),Bs=i(C,"CODE",{"data-svelte-h":!0}),v(Bs)!=="svelte-v3nqc6"&&(Bs.textContent=Pn),nn=h(C," and we can take the maxtrix product with"),Ta=d(C,!1),ja=h(C," to get the output"),Ra=d(C,!1),$a=h(C," of the same shape as the others."),C.forEach(a),Ca=l(s),ws=i(s,"P",{});var kn=f(ws);en=h(kn,`Replacing the softmax by its value gives:
`),Ua=d(kn,!1),kn.forEach(a),Ja=l(s),Q=i(s,"P",{});var Qs=f(Q);pn=h(Qs,"Note that the entries in"),Va=d(Qs,!1),Wa=h(Qs," corresponding to"),Na=d(Qs,!1),La=h(Qs," are masked (the sum stops at j) because the attention is not allowed to look at future tokens (only past ones)."),Qs.forEach(a),Ia=l(s),bs=i(s,"P",{});var Tn=f(bs);ln=h(Tn,`In comparison, the RWKV attention is given by
`),Ka=d(Tn,!1),Tn.forEach(a),qa=l(s),V=i(s,"P",{});var X=f(V);mn=h(X,"where"),Za=d(X,!1),Ea=h(X," is a new matrix called receptance by the author,"),Fa=d(X,!1),Ba=h(X," and"),Ga=d(X,!1),Ha=h(X," are still the key and value (\\(\\sigma\\) here is the sigmoid function)."),Da=d(X,!1),Xa=h(X,` is a new vector that represents the position of the token and is given by
`),Qa=d(X,!1),X.forEach(a),Sa=l(s),$=i(s,"P",{});var W=f($);rn=h(W,"with"),Pa=d(W,!1),Oa=h(W," and"),Ya=d(W,!1),Aa=h(W," learnable parameters called in the code "),Gs=i(W,"CODE",{"data-svelte-h":!0}),v(Gs)!=="svelte-vzmgec"&&(Gs.textContent=On),on=h(W," and "),Hs=i(W,"CODE",{"data-svelte-h":!0}),v(Hs)!=="svelte-1ng9k7k"&&(Hs.textContent=Yn),cn=h(W," respectively. The numerator and denominator can both be expressed recursively. Naming them"),st=d(W,!1),at=h(W," and"),tt=d(W,!1),nt=h(W,` we have:
`),et=d(W,!1),W.forEach(a),pt=l(s),E=i(s,"P",{});var js=f(E);hn=h(js,"so"),lt=d(js,!1),mt=h(js," (called "),Ds=i(js,"CODE",{"data-svelte-h":!0}),v(Ds)!=="svelte-joi7v1"&&(Ds.textContent=An),gn=h(js,` in the code) satistfies
`),it=d(js,!1),js.forEach(a),rt=l(s),_s=i(s,"P",{});var jn=f(_s);dn=h(jn,`and
`),ot=d(jn,!1),jn.forEach(a),ct=l(s),F=i(s,"P",{});var Rs=f(F);un=h(Rs,"so"),ht=d(Rs,!1),gt=h(Rs," (called "),Xs=i(Rs,"CODE",{"data-svelte-h":!0}),v(Xs)!=="svelte-ep42ss"&&(Xs.textContent=se),yn=h(Rs,` in the code) satistfies
`),dt=d(Rs,!1),Rs.forEach(a),ut=l(s),zs=i(s,"P",{});var Rn=f(zs);vn=h(Rn,`The actual recurrent formula used are a tiny bit more complex, as for numerical stability we don’t want to compute exponentials of big numbers. Usually the softmax is not computed as is, but the exponential of the maximum term is divided of the numerator and denominator:
`),yt=d(Rn,!1),Rn.forEach(a),vt=l(s),B=i(s,"P",{});var $s=f(B);fn=h($s,"with"),ft=d($s,!1),wt=h($s," the maximum of all"),bt=d($s,!1),_t=h($s,`. So here on top of saving the numerator state (\\(\\hat{N}\\)) and the denominator state (\\(\\hat{D}\\)) we also keep track of the maximum of all terms encountered in the exponentials. So we actually use
`),zt=d($s,!1),$s.forEach(a),Mt=l(s),Ms=i(s,"P",{});var $n=f(Ms);wn=h($n,`defined by the following recurrent formulas:
`),xt=d($n,!1),$n.forEach(a),kt=l(s),xs=i(s,"P",{});var Cn=f(xs);bn=h(Cn,`and
`),Tt=d(Cn,!1),Cn.forEach(a),jt=l(s),S=i(s,"P",{});var Ps=f(S);_n=h(Ps,"and"),Rt=d(Ps,!1),$t=h(Ps,`. With those, we can then compute
`),Ct=d(Ps,!1),Ps.forEach(a),Ut=l(s),ks=i(s,"P",{});var Un=f(ks);zn=h(Un,`and
`),Jt=d(Un,!1),Un.forEach(a),Vt=l(s),Ts=i(s,"P",{});var Jn=f(Ts);Mn=h(Jn,`which finally gives us
`),Wt=d(Jn,!1),Jn.forEach(a),Nt=l(s),Ss=i(s,"P",{}),f(Ss).forEach(a),this.h()},h(){Cs(r,"name","hf:doc:metadata"),Cs(r,"content",Se),Cs(N,"class","docstring border-l-2 border-t-2 pl-4 pt-3.5 border-gray-100 rounded-tl-xl mb-6 mt-8"),Cs(q,"class","docstring border-l-2 border-t-2 pl-4 pt-3.5 border-gray-100 rounded-tl-xl mb-6 mt-8"),Cs(U,"class","docstring border-l-2 border-t-2 pl-4 pt-3.5 border-gray-100 rounded-tl-xl mb-6 mt-8"),Cs(Z,"class","docstring border-l-2 border-t-2 pl-4 pt-3.5 border-gray-100 rounded-tl-xl mb-6 mt-8"),Cs(J,"class","docstring border-l-2 border-t-2 pl-4 pt-3.5 border-gray-100 rounded-tl-xl mb-6 mt-8"),ya.a=null,fa.a=wa,ba.a=_a,za.a=Ma,xa.a=ka,Ta.a=ja,Ra.a=$a,Ua.a=null,Va.a=Wa,Na.a=La,Ka.a=null,Za.a=Ea,Fa.a=Ba,Ga.a=Ha,Da.a=Xa,Qa.a=null,Pa.a=Oa,Ya.a=Aa,st.a=at,tt.a=nt,et.a=null,lt.a=mt,it.a=null,ot.a=null,ht.a=gt,dt.a=null,yt.a=null,ft.a=wt,bt.a=_t,zt.a=null,xt.a=null,Tt.a=null,Rt.a=$t,Ct.a=null,Jt.a=null,Wt.a=null},m(s,t){n(document.head,r),e(s,j,t),e(s,u,t),e(s,y,t),M(w,s,t),e(s,o,t),M(R,s,t),e(s,sa,t),e(s,as,t),e(s,aa,t),e(s,ts,t),e(s,ta,t),e(s,ns,t),e(s,na,t),e(s,es,t),e(s,ea,t),M(ps,s,t),e(s,pa,t),M(ls,s,t),e(s,la,t),e(s,ms,t),e(s,ma,t),M(is,s,t),e(s,ia,t),M(rs,s,t),e(s,ra,t),e(s,N,t),M(os,N,null),n(N,It),n(N,Js),n(N,Kt),n(N,Vs),n(N,qt),M(P,N,null),e(s,oa,t),M(cs,s,t),e(s,ca,t),e(s,U,t),M(hs,U,null),n(U,Zt),n(U,Ws),n(U,Et),n(U,Ns),n(U,Ft),n(U,Ls),n(U,Bt),n(U,q),M(gs,q,null),n(q,Gt),n(q,Is),n(q,Ht),M(O,q,null),n(q,Dt),M(Y,q,null),e(s,ha,t),M(ds,s,t),e(s,ga,t),e(s,J,t),M(us,J,null),n(J,Xt),n(J,Ks),n(J,Qt),n(J,qs),n(J,St),n(J,Zs),n(J,Pt),n(J,Z),M(ys,Z,null),n(Z,Ot),n(Z,Es),n(Z,Yt),M(A,Z,null),n(Z,At),M(ss,Z,null),e(s,da,t),M(vs,s,t),e(s,ua,t),e(s,fs,t),n(fs,sn),ya.m(te,fs),e(s,va,t),e(s,b,t),n(b,an),fa.m(ne,b),n(b,wa),ba.m(ee,b),n(b,_a),za.m(pe,b),n(b,Ma),n(b,Fs),n(b,tn),xa.m(le,b),n(b,ka),n(b,Bs),n(b,nn),Ta.m(me,b),n(b,ja),Ra.m(ie,b),n(b,$a),e(s,Ca,t),e(s,ws,t),n(ws,en),Ua.m(re,ws),e(s,Ja,t),e(s,Q,t),n(Q,pn),Va.m(oe,Q),n(Q,Wa),Na.m(ce,Q),n(Q,La),e(s,Ia,t),e(s,bs,t),n(bs,ln),Ka.m(he,bs),e(s,qa,t),e(s,V,t),n(V,mn),Za.m(ge,V),n(V,Ea),Fa.m(de,V),n(V,Ba),Ga.m(ue,V),n(V,Ha),Da.m(ye,V),n(V,Xa),Qa.m(ve,V),e(s,Sa,t),e(s,$,t),n($,rn),Pa.m(fe,$),n($,Oa),Ya.m(we,$),n($,Aa),n($,Gs),n($,on),n($,Hs),n($,cn),st.m(be,$),n($,at),tt.m(_e,$),n($,nt),et.m(ze,$),e(s,pt,t),e(s,E,t),n(E,hn),lt.m(Me,E),n(E,mt),n(E,Ds),n(E,gn),it.m(xe,E),e(s,rt,t),e(s,_s,t),n(_s,dn),ot.m(ke,_s),e(s,ct,t),e(s,F,t),n(F,un),ht.m(Te,F),n(F,gt),n(F,Xs),n(F,yn),dt.m(je,F),e(s,ut,t),e(s,zs,t),n(zs,vn),yt.m(Re,zs),e(s,vt,t),e(s,B,t),n(B,fn),ft.m($e,B),n(B,wt),bt.m(Ce,B),n(B,_t),zt.m(Ue,B),e(s,Mt,t),e(s,Ms,t),n(Ms,wn),xt.m(Je,Ms),e(s,kt,t),e(s,xs,t),n(xs,bn),Tt.m(Ve,xs),e(s,jt,t),e(s,S,t),n(S,_n),Rt.m(We,S),n(S,$t),Ct.m(Ne,S),e(s,Ut,t),e(s,ks,t),n(ks,zn),Jt.m(Le,ks),e(s,Vt,t),e(s,Ts,t),n(Ts,Mn),Wt.m(Ie,Ts),e(s,Nt,t),e(s,Ss,t),Lt=!0},p(s,[t]){const G={};t&2&&(G.$$scope={dirty:t,ctx:s}),P.$set(G);const L={};t&2&&(L.$$scope={dirty:t,ctx:s}),O.$set(L);const H={};t&2&&(H.$$scope={dirty:t,ctx:s}),Y.$set(H);const I={};t&2&&(I.$$scope={dirty:t,ctx:s}),A.$set(I);const D={};t&2&&(D.$$scope={dirty:t,ctx:s}),ss.$set(D)},i(s){Lt||(x(w.$$.fragment,s),x(R.$$.fragment,s),x(ps.$$.fragment,s),x(ls.$$.fragment,s),x(is.$$.fragment,s),x(rs.$$.fragment,s),x(os.$$.fragment,s),x(P.$$.fragment,s),x(cs.$$.fragment,s),x(hs.$$.fragment,s),x(gs.$$.fragment,s),x(O.$$.fragment,s),x(Y.$$.fragment,s),x(ds.$$.fragment,s),x(us.$$.fragment,s),x(ys.$$.fragment,s),x(A.$$.fragment,s),x(ss.$$.fragment,s),x(vs.$$.fragment,s),Lt=!0)},o(s){k(w.$$.fragment,s),k(R.$$.fragment,s),k(ps.$$.fragment,s),k(ls.$$.fragment,s),k(is.$$.fragment,s),k(rs.$$.fragment,s),k(os.$$.fragment,s),k(P.$$.fragment,s),k(cs.$$.fragment,s),k(hs.$$.fragment,s),k(gs.$$.fragment,s),k(O.$$.fragment,s),k(Y.$$.fragment,s),k(ds.$$.fragment,s),k(us.$$.fragment,s),k(ys.$$.fragment,s),k(A.$$.fragment,s),k(ss.$$.fragment,s),k(vs.$$.fragment,s),Lt=!1},d(s){s&&(a(j),a(u),a(y),a(o),a(sa),a(as),a(aa),a(ts),a(ta),a(ns),a(na),a(es),a(ea),a(pa),a(la),a(ms),a(ma),a(ia),a(ra),a(N),a(oa),a(ca),a(U),a(ha),a(ga),a(J),a(da),a(ua),a(fs),a(va),a(b),a(Ca),a(ws),a(Ja),a(Q),a(Ia),a(bs),a(qa),a(V),a(Sa),a($),a(pt),a(E),a(rt),a(_s),a(ct),a(F),a(ut),a(zs),a(vt),a(B),a(Mt),a(Ms),a(kt),a(xs),a(jt),a(S),a(Ut),a(ks),a(Vt),a(Ts),a(Nt),a(Ss)),a(r),T(w,s),T(R,s),T(ps,s),T(ls,s),T(is,s),T(rs,s),T(os),T(P),T(cs,s),T(hs),T(gs),T(O),T(Y),T(ds,s),T(us),T(ys),T(A),T(ss),T(vs,s)}}}const Se='{"title":"RWKV","local":"rwkv","sections":[{"title":"Overview","local":"overview","sections":[],"depth":2},{"title":"Usage example","local":"usage-example","sections":[],"depth":2},{"title":"RwkvConfig","local":"transformers.RwkvConfig","sections":[],"depth":2},{"title":"RwkvModel","local":"transformers.RwkvModel","sections":[],"depth":2},{"title":"RwkvLMHeadModel","local":"transformers.RwkvForCausalLM","sections":[],"depth":2},{"title":"Rwkv attention and the recurrent formulas","local":"rwkv-attention-and-the-recurrent-formulas","sections":[],"depth":2}],"depth":1}';function Pe(K){return qe(()=>{new URLSearchParams(window.location.search).get("fw")}),[]}class ep extends Ze{constructor(r){super(),Ee(this,r,Pe,Qe,Ke,{})}}export{ep as component};
