# Do not Interfere but Cooperate:
A Fully Learnable Code Design for
Multi-Access Channels with Feedback

Emre Ozfatura, Chenghong Bian and Deniz Gündüz  Information Processing and
Communications Lab (IPC-Lab), Imperial College London, London, UK
Email:$\\{$m.ozfatura, c.bian22<EMAIL_ADDRESS>
###### Abstract

Data-driven deep learning based code designs, including low-complexity neural
decoders for existing codes, or end-to-end trainable auto-encoders have
exhibited impressive results, particularly in scenarios for which we do not
have high-performing structured code designs. However, the vast majority of
existing data-driven solutions for channel coding focus on a point-to-point
scenario. In this work, we consider a multiple access channel (MAC) with
feedback and try to understand whether deep learning-based designs are capable
of enabling coordination and cooperation among the encoders as well as
allowing error correction. Simulation results show that the proposed multi-
access block attention feedback (MBAF) code improves the upper bound of the
achievable rate of MAC without feedback in finite block length regime.

###### Index Terms:

mutiple-access channel, channel coding, deep learning, feedback, transformer.

##  I Introduction

The design of deep learning (DL) based solutions for wireless communication
problems has become a rapidly evolving research area. It has been shown that
data-driven solutions are either able to outperform their hand-crafted
counterparts, or exhibit comparable performance with lower computational
complexity. In the context of error correction coding, it has been shown by
various recent studies [1, 2, 3, 4, 5, 6, 7] that DL-assisted designs can be
trained and employed to achieve reliable communication as an alternative
approach to well-known structured codes. These designs can be grouped into two
main categories based on the stage/phase that data-driven design is
incorporated. The first category includes solutions where the decoding of
conventional codes, e.g., LDPC codes and polar codes, is done using deep
neural network (DNN) architectures, e.g., neural belief propagation [4, 8, 9].
The works in the second category, on the other hand, employ DNN-based modules
both at the encoding and decoding sides by introducing a fully learnable auto-
encoder (AE) architecture [1, 10, 11, 5]. The experimental results of
aforementioned works highlight that, DNN-based decoder/AE models either
achieve the state-of-the-art block error rate (BLER) performance in the finite
block length regime, or achieve similar BLER performance with conventional
encoding/decoding schemes with lower complexity.
The use of DNNs is not limited to the feed-forward error correction problem;
recently, it has been shown in [12, 3, 13, 14, 15] that DNN-based solutions
have significant potential when a feedback link from the receiver to the
transmitter is available. Indeed, the feedback channel, highly relevant for
practice, is considered as a challenging model and practical code designs have
been mostly limited to repetition based, e.g., ARQ/HARQ, protocols. On the
other hand, experimental results in [12, 14] demonstrate that DNN-based
feedback code designs can provide significant performance gains in the short
block length regime compared to hand-crafted solutions.
Although the advantages of DNN-based solutions have been shown in point-to-
point systems with/without feedback scenarios, we have a limited understanding
of their role in multi-user scenarios. The broadcast channel scenario has been
recently analyzed in [11]. In this paper, we focus on the multiple access
channel (MAC) with feedback, and analyze the applicability of DNN-based
solutions. It is known that the existence of channel feedback enlarges the
capacity region of the MAC; however, the design of practical codes, especially
for the short block length regime, is still an open problem. To the best of
our knowledge, this is the very first work that studies a fully-learnable
framework for the MAC with feedback scenario. We remark that, in the classical
MAC, since the transmitters cannot cooperate, they generate independent
channel codewords, whereas in the presence of feedback, the transmitters can
cooperate through the feedback channel. Indeed, information theoretic random
coding schemes that achieve rates outside the capacity region of the classical
MAC do enable cooperation by allowing each transmitters to decode each other’s
message (partially) and collaborate for transmission. We believe our results
will open new research directions, since our findings indicate that a fully-
learnable framework is not only capable of forming an error-correction
mechanism, as in[3, 12, 14, 15, 16], but also able to form a co-operation
strategy between the transmitters.

##  II Preliminaries and Problem definition

Figure 1: Visualisation of sequence-to-sequence encoding with block structure
at iteration $\tau$ at the $j$-th user.

###  II-A System Model and Problem Definition

We consider an additive white Gaussian noise (AWGN) MAC with two transmitters.
We assume that feedback from the receiver to both transmitters is available in
the form of perfect channel output. We consider a block feedback scenario:
Communication is performed over $T$ blocks in the forward direction. After the
transmission of each block, the channel output becomes available to the
transmitters. During the $\tau$th communication block, $\tau\in[T]$,
user/transmitter $j\in\left\\{1,2\right\\}$ sends $l_{ff}$ symbols over the
channel, which is referred to as a block of symbols and denoted by
$\mathbf{c}^{(\tau)}_{j}=\left[{c}^{(\tau)}_{j,1},\ldots,{c}^{(\tau)}_{j,l_{ff}}\right]$.
The transmitted symbols are superimposed and exposed to AWGN; that is, the
receiver observes the block of noisy superimposed symbols
$\mathbf{y}^{(\tau)}$:

$\mathbf{y}^{(\tau)}=\mathbf{c}^{(\tau)}_{1}+\mathbf{c}^{(\tau)}_{2}+\mathbf{n}^{(\tau)},$
(1)

where $\mathbf{n}^{(\tau)}\in\mathbb{R}^{l_{ff}}$ is a vector of independent
zero-mean Gaussian random variables with covariance matrix
$\sigma^{2}\mathbf{I}_{l_{ff}}$. Due to the perfect channel output assumption,
$\mathbf{y}^{(\tau)}$ becomes available to the transmitters as well.

User $j$ wants to send a $K$-bit message
$\mathbf{b}_{j}=[b_{j,1},\ldots,b_{j,K}]\in\left\\{0,1\right\\}^{K}$ to the
receiver in $N=Tl$ channel uses, which corresponds to a sum-rate of $R=2K/N$.

Next, we define the notion of knowledge vector, denoted by
$\bm{q}^{(\tau)}_{j}$, which represents all the available information
(history) at the $j$th transmitter at the beginning of block $\tau+1$:

$\bm{q}^{(\tau+1)}_{j}\triangleq\left[\bm{b}_{j},\bm{c}_{j}^{(1)},\ldots,\bm{c}_{j}^{(\tau)},\mathbf{y}^{(\tau)}.\right],$
(2)

While we focus on a passive feedback mechanism in the form of perfect channel
output feedback, more advanced feedback strategies involving active feedback
can be easily incorporated into this framework, following the approach
followed in [14] for the point-to-point scenario.

The objective is to design an encoding function $M^{({\tau})}_{j}$, for each
transmitter $j$, to map its knowledge vector to a block of parity symbols at
each block, i.e.,

$M^{({\tau})}_{j}:\bm{q}^{(\tau)}_{j}\xrightarrow{}\mathbf{c}^{(\tau)}_{j},$
(3)

under an average power constraint imposed for each transmitter:

$\displaystyle\mathbb{E}\left[\frac{1}{N}\sum^{T}_{\tau=1}\langle\mathbf{c}^{(\tau)}_{j},\mathbf{c}^{(\tau)}_{j}\rangle\right]\leq
1,j\in\left\\{1,2\right\\},$ (4)

We also need a decoding function $D$ to be employed at the receiver to
estimate $\mathbf{b}_{1}$ and $\mathbf{b}_{2}$ from the knowledge vector
$\tilde{\bm{q}}=[\mathbf{y}^{(0)},\ldots,\mathbf{y}^{(T)}]$, i.e.,

$D:\tilde{\bm{q}}\xrightarrow{}\hat{\mathbf{b}}_{1},\hat{\mathbf{b}}_{2}\in\left\\{0,1\right\\}^{K}.$
(5)

Hence, the overall objective is to jointly design
$M^{({\tau})}_{1},M^{({\tau})}_{2}$ and $D$. In this paper, inspired by [12,
14], we propose a DNN-based solution built upon the transformer architecture
for the MAC scenario, called the MBAF code. Next, we revisit the fundamental
ideas behind the block attention feedback (BAF) code framework introduced in
[12, 14], and then explain how to adopt it for the MAC scenario. We further
introduce a novel sequential decoding strategy, which decodes the message in
multiple iterations, and includes the prediction in the previous iteration as
an input to the decoder in order to refine it.

###  II-B Block Attention Feedback Code Structure

The key idea behind the BAF code design [12, 14] is to transform the knowledge
vector, both at the receiver and the transmitters, into a sequence of vectors,
and utilize the self-attention mechanism over these sequences to mimic
encoding and decoding operations. Since, from the construction, the self-
attention mechanism explores the similarities and differences between the
elements of a sequence, while taking into account their positions within the
sequence, it is possible to draw an analogy between processing a sequence of
vectors and encoding/decoding process of a bit-stream.

One of the critical aspects highlighted in [12, 14] is how to form these
sequences. The proposed strategy is to divide the $K$ information bits into
$l$ blocks each consisting of $m$ bits, i.e., $K=lm$. As a result, the initial
bit stream is treated as a sequence of $l$ bit blocks, i.e.,
$\bm{b}=\left[\bm{b}_{1},\ldots,\bm{b}_{l}\right]$. In parallel with this
sequence of bit blocks, the knowledge vector $\bm{q}^{(\tau)}$ is also
organized into a sequence of $l$ vectors,
$\mathcal{Q}^{(\tau)}=\left\\{\bm{q}^{(\tau)}_{1},\ldots,\bm{q}^{(\tau)}_{l}\right\\}$,
where $\bm{q}^{(\tau)}_{i}$ denotes the knowledge vector corresponding to the
$i$th block, and initially we have $\bm{q}_{i}^{(1)}=\bm{b}_{i}$. The key
design principle of the BAF code is to process the sequence of knowledge
vectors $\bm{q}^{(\tau)}$ in a parallel manner, using self-attention, and to
generate one encoded symbol for each block. Here, $\bm{c}^{(\tau)}_{i}$
represents the symbol generated corresponding to the $i$th block during
iteration $\tau$. Hence, following the reception of the block of feedback
symbols, the knowledge vector corresponding to each block is also updated
accordingly:

$\bm{q}^{(\tau+1)}_{i}=\left[\bm{b}_{i},\bm{c}_{i}^{(1)},\ldots,\bm{c}_{i}^{(\tau)},{\bm{y}}^{(1)},\ldots,{\bm{y}}^{(\tau)}\right].$
(6)

The iterative process of updating the knowledge vectors by stacking the new
generated symbols and the received feedback symbols, and generating encoded
symbols from the sequence of knowledge vectors is continued until a given
number of interactions $T$ is reached. Once the the communication is
terminated, the decoder decodes the original bit sequence $\bm{b}$ using all
its received symbols $\left[\bm{y}^{(1)},\ldots,\bm{y}^{(T)}\right]$, again by
reshaping each $\bm{y}^{(i)}$ as a sequence of blocks and combining the
sequences over blocks. In the next section, we further explain the
architectural details of the BAF code design, and how the sequences are formed
with a particular focus on the MAC scenario.

Figure 2: Illustration of the overall deep learning assisted communication
framework for MAC with feedback scenario.

##  III Multi-access Block Attention Feedback (MBAF) Codes

###  III-A Overview of the Design

As illustrated in Fig. 2, the entire design consists of three networks, two
parity networks, each located at one of the transmitters and a decoder network
at the receiver. Each network is comprised of similar components: the pre-
processing unit, the feature extractor, the sequence-to-sequence encoder and
the output mapping, which are denoted as $S(\cdot)$, $H_{extract},H_{s2s}$ and
$H_{map}$, respectively. Among them, the pre-processing unit, $S(\cdot)$, is
non-trainable and represents exactly the same operation in all the three
networks. The other components, $H_{extract},H_{s2s}$ and $H_{map}$, however,
are parameterized by DNNs and different networks have different realization of
these modules. To avoid a possible confusion in the following, we use
superscripts ‘$\mathrm{parity}$’ and ‘$\mathrm{decoder}$’ to distinguish the
learnable modules employed by the users and the receiver, respectively. The
details for each of the components are described next.

####  III-A1 Pre-processing unit

$S(\cdot)$ transforms the knowledge vector, $\bm{q}$, into a sequence of
vectors for further processing. To be precise, $S(\cdot)$ at the $j$-th user
partitions $\bm{q}_{j}^{(\tau)}$ into $l$ vectors, i.e.,
$S(\bm{q}_{j}^{(\tau)})=\\{\bm{q}_{j,1}^{(\tau)},\ldots,\bm{q}_{j,l}^{(\tau)}\\}$
and the operations are the same at the decoder network.

####  III-A2 Feature extractor

$H_{\mathrm{extract}}$ adopts a multi-layer perceptron (MLP) architecture with
a ReLU activation function that transforms $d_{in}$-dimensional input vectors
into vectors of dimension $d_{model}$. This component also keeps the output
within a specific range to prevent large noise terms. Note that the neural
network structure for $H_{\mathrm{extract}}$ is identical in all three
networks.

####  III-A3 Sequence-to-sequence encoder

$H_{s2s}$ takes a sequence of $l$ $d_{model}$-dimensional vectors as input and
transforms them into a sequence of $l$ vectors with the same dimension. We use
the transformer architecture as the backbone and denote the output sequence as
$\mathcal{V}=\left\\{\bm{v}_{1},\ldots,\bm{v}_{l}\right\\}$.

$H_{s2s}$ is comprised of $\bar{N}$ identical transformer encoder layers,
where each layer consists of three primary modules: the feed-forward module,
the multi-head attention module, and the layer normalization module as
detailed in [12]. In our implementations, we set $d_{model}=32$, and
$\bar{N}=2$ for all three networks.

Algorithm 1 MAC parity symbol encoding (MIPSE)

1:for $\tau=1,\ldots,T$ do

2: for Transmitter $j\in\\{1,2\\}$ do Simultaneously

3: Update knowledge vector:

4:
$\bm{q}_{j}^{(\tau)}=[\bm{b}_{j},\bm{c}_{j}^{(1)},\ldots,\bm{c}_{j}^{(\tau-1)},\bm{y}^{(1)},\ldots,\bm{y}^{(\tau-1)}]$

5: Pre-process knowledge vector:

6:
$\left\\{\bm{q}^{(\tau)}_{j,1},\ldots,\bm{q}^{(\tau)}_{j,l}\right\\}=S(\bm{q}_{j}^{(\tau)})$

7: Feature extraction:

8: for $i\in[l]$ do
$\bm{f}^{(\tau)}_{j,i}=H^{parity}_{extract}(\bm{q}^{(\tau)}_{j,i})$

9: Attention-based neural-encoding:

10: $\mathcal{V}_{j}^{(\tau)}=H^{parity}_{s2s}(\mathcal{F}_{j}^{(\tau)})$

11: Symbol mapping:

12: for $i\in[l]$ do
$c^{(\tau)}_{j,i}=H^{parity}_{map}(\bm{v}^{(\tau)}_{j,i})$

13: Send block of symbols
$\mathbf{c}_{j}^{(\tau)}=\left[c^{(\tau)}_{j,1},\ldots,c^{(\tau)}_{j,l}\right]$

14: Receiver: broadcast ${\bm{y}}^{(\tau)}$ while $\tau<T$.

####  III-A4 Output mapping

$H_{\mathrm{map}}$ maps the latent representation
$\mathcal{V}=\left\\{\bm{v}_{1},\ldots,\bm{v}_{l}\right\\}$ to the desired
output $\bm{u}_{i}=H_{\mathrm{map}}(\bm{v}_{i})$, for all $i\in[l]$. For all
the three networks, $H_{map}$ is parameterized using an MLP layer.
Specifically, the output parity symbols at the $j$-th user are
${u}_{j,i}=c_{j,i}=H_{map}^{parity}(\bm{v}_{j,i}),i\in[l]$ with $d_{out}=1$
while the decoder network at the receiver maps the latent representation
$\bm{v}_{i}$ to two probability vectors as
$\bm{u}_{i}=\\{\bm{w}_{1,i},\bm{w}_{2,i}\\}=H^{decoder}_{map}(\bm{v}_{i})$
with $d_{out}=2^{m}$, where $\bm{w}_{j,i}$ represents the probability vector
corresponding to the $i$-th bit block of the $j$-th user. Note that the MLP in
$H^{decoder}_{map}$ is followed by an additional softmax layer. Finally, the
decoded bits are estimated from the probability vectors for each user.

Algorithm 2 MAC joint successive parity symbol decoding (MJSPSD)

1:Initialize knowledge vector:

2:$\hat{\bm{q}}^{(0)}=[\bm{y}^{(1)},\ldots,\bm{y}^{(T)}]$

3:Pre-process knowledge vector for decoder network:

4:$\left\\{\hat{\bm{q}}_{1}^{(0)},\ldots,\hat{\bm{q}}_{l}^{(0)}\right\\}=S(\hat{\bm{q}}^{(0)})$,
$\hat{\bm{q}}_{i}^{(0)}=[{\bm{y}}^{(1)}_{i},\ldots,{\bm{y}}^{(T)}_{i}])$

5:Feature extraction:

6:for $i\in[l]$ do
${\bm{f}}_{i}^{(0)}=H^{decoder}_{extract}(\hat{\bm{q}}_{i}^{(0)})$

7:Attention-based decoding:
${\mathcal{V}}^{(0)}=H^{decoder}_{s2s}({\mathcal{F}}^{(0)})$

8:Mapping:

9:for $i\in[l]$ do
$\\{\bm{w}_{1,i}^{(0)},\bm{w}_{2,i}^{(0)}\\}=H^{decoder}_{map}({\bm{v}}_{i}^{(0)})$

10:for $n\in[N_{iter}]$ do

11: Convert class-wise prediction to bit-wise belief:

12: for $i\in[l]$ do
$\\{\bar{\bm{b}}_{1,i}^{(n)},\bar{\bm{b}}_{2,i}^{(n)}\\}=\\{\bm{A}\bm{w}_{1,i}^{(n)},\bm{A}\bm{w}_{2,i}^{(n)}\\}$

13: Enhance the knowledge vectors :

14:
$\hat{\bm{q}}_{i}^{(n)}=[{\bm{y}}^{(1)}_{i},\ldots,{\bm{y}}^{(T)}_{i},\bar{\bm{b}}_{1,i}^{(n)},\bar{\bm{b}}_{2,i}^{(n)}])$

15: Sequence-to-sequence Decoding: Repeat lines 5-9

16:Block-wise classification:

17:for $i\in[l],j\in\\{1,2\\}$ do
$I_{j,i}=arg\,max(\bm{w}_{j,i}^{(N_{iter})})$

18:Blockwise index to bitstream conversion:

19:for $i\in[l],j\in\\{1,2\\}$ do $\hat{\bm{b}}_{j,i}=f_{d2b}(I_{j,i})$

20:Retrieve the bitstream of each user:

21:for $j\in\\{1,2\\}$ do
$\hat{\bm{b}}_{j}=\left[\hat{\bm{b}}_{j,1},\ldots,\hat{\bm{b}}_{j,l}\right]$

###  III-B Algorithmic Flow and Training Procedure

In this subsection, we explain the end-to-end encoding and decoding processes
of the proposed MBAF code, which involves two algorithmic flows: MAC iterative
parity symbol encoding (MIPSE) and MAC joint successive parity symbol decoding
(MJSPSD).

MIPSE, summarized in Algorithm 1, illustrates the encoding process at the
users. As shown in Fig. 1, the exact form of the knowledge vector in (3) is
given as
$\bm{q}_{j}^{(\tau)}=[\bm{b}_{j},\bm{c}_{j}^{(1)},\ldots,\bm{c}_{j}^{(\tau-1)},\tilde{\bm{y}}_{j}^{(1)},\ldots,\tilde{\bm{y}}_{j}^{(\tau-1)}]$,
where $\bm{c}_{j}^{(i)}$ denotes the already transmitted symbol and
$\tilde{\bm{y}}^{(i)}_{j}=\bm{y}^{(i)}-{\bm{c}}^{(i)}_{j}$ is referred to as
the residual feedback symbols for $i\in[1,\tau-1]$. The algorithm iteratively
generates transmit symbols $\bm{c}_{j}^{(\tau)}$ by exploring the feedback
from the receiver, and terminates when $T$ iterations are reached.

After the transmission is completed at the $T$-th communication block, the
receiver employs the MJSPSD algorithm as summarized in Algorithm 2. The
algorithm takes the received signal as input and outputs the predicted bit
sequences $\hat{\bm{b}}_{1},\hat{\bm{b}}_{2}$, for each of the two users. To
improve the error correction ability of the decoder network, we use successive
decoding with $N_{iter}>0$ iterations. The decoding process starts by
generating the latent probability vectors
$\\{\bm{w}_{1,i}^{(0)},\bm{w}_{2,i}^{(0)}\\}$ following the same procedure as
MIPSE, except the outputs $\bm{w}_{j,i}^{(0)}$ represent the probability
vectors with dimension $2^{m}$. We transform these vectors to belief vectors
$\bar{\bm{b}}_{j,i}^{(0)}\in[0,1]^{m}$ by multiplying them with a binary
matrix $\bm{A}$ with dimension $m\times 2^{m}$, which can be expressed as:

$\displaystyle\bar{\bm{b}}_{j,i}^{(0)}$
$\displaystyle=\bm{A}\bm{w}_{j,i}^{(0)},$ (7)

where $\bm{A}$ for $m=3$ is given as:

$\displaystyle\begin{pmatrix}0&0&0&0&1&1&1&1\\\ 0&0&1&1&0&0&1&1\\\
0&1&0&1&0&1&0&1\\\ \end{pmatrix}.$ (8)

Note that the $k$-th entry of $\bar{\bm{b}}_{j,i}^{(0)}$ indicates the
probability of $\bm{b}_{j,i}[k]=1$. For the $n$-th iteration, we update the
knowledge vector by concatenating the received signal $\bm{y}$ with the belief
vectors $\bar{\bm{b}}_{j,i}^{(n)}$, and following the same procedure of MIPSE
to generate refined probability vectors
$\\{\bm{w}_{1,i}^{(n)},\bm{w}_{2,i}^{(n)}\\}$. Finally, after $N_{iter}$
number of iterations, we obtain the index of the largest value in
$\bm{w}_{j,i}^{(N_{iter})}$, denoted by $I_{j,i}$, and transform it into the
bit sequence $\bm{\hat{b}}_{j,i}$, using a decimal-to-binary mapping
$f_{d2b}(\cdot)$. By collecting $\bm{\hat{b}}_{j,i},i\in[l]$ for each user and
concatenating them, the final estimated bit sequences
$\\{\hat{\bm{b}}_{1},\hat{\bm{b}}_{2}\\}$ are obtained.

Next, we will introduce the training methodology for the proposed MBAF code.
The initial step is to randomly generate two bit sequences
$\bm{b}_{j}\in\left\\{0,1\right\\}^{K}$ for each user, then divide each bit
sequence into $l$ blocks, each consisting of $m$ bits. The corresponding label
is assigned to each block and the training data is organized as
$\mathcal{B}_{j}=\left\\{(\bm{b}_{j,1},y_{j,1}),\ldots,(\bm{b}_{j,l},y_{j,l})\right\\},j\in\\{1,2\\}$.
Next, the generated data is fed to the autoencoder, which produces a sequence
of $l$ probability vectors for each user:
$\left\\{\bm{w}_{j,1}^{(N_{iter})},\ldots,\bm{w}_{j,l}^{(N_{iter})}\right\\}$.
The cross-entropy loss is adopted and the total loss is averaged over the
users:

$L_{j}(\bm{W}_{j},Y_{j})=\sum^{l}_{i=1}\sum^{2^{m}-1}_{c=0}-\log{{(\bm{W}_{j}{[i,c]})}}\cdot\mathbbm{1}_{y_{j,i}\neq
c}$ (9)

where $Y_{j}=\left\\{y_{j,1},\ldots,y_{j,l}\right\\}$ is the list of labels
and $\bm{W}_{j}$ is the matrix form of the sequence
$\\{\bm{w}_{j,1}^{(N_{iter})},\ldots,\bm{w}_{j,l}^{(N_{iter})}\\}$ for the
$j$-th user.

TABLE I: Average BLER performance of MBAF code for different sum-rate $R$ values. SNR/Rate | 6/12 | 6/11 | 6/10 | 6/9 | 6/8 | 6/7
---|---|---|---|---|---|---
-1 dB | $2.6\times 10^{-8}$ | $5.9\times 10^{-8}$ | $3.1\times 10^{-5}$ | $3.5\times 10^{-2}$ | - | -
-0.5 dB | - | $1.7\times 10^{-8}$ | $8.8\times 10^{-8}$ | $4.8\times 10^{-5}$ | $8.2\times 10^{-2}$ | -
0 dB | - | - | $6\times 10^{-10}$ | $6.2\times 10^{-7}$ | $2.9\times 10^{-4}$ | -
0.5 dB | - | - | - | $1.7\times 10^{-8}$ | $7.5\times 10^{-6}$ | $5\times 10^{-2}$
1 dB | - | - | - | $2.9\times 10^{-9}$ | $1.7\times 10^{-7}$ | $6.7\times 10^{-3}$
1.5 dB | - | - | - | $7.5\times 10^{-9}$ | $4.48\times 10^{-8}$ | $1.4\times 10^{-4}$

##  IV Numerical Results

To evaluate the performance of the proposed MBAF code, we consider each user
transmitting a bit stream of length $K=51$ bits and set the number of blocks
$l=17$ with block size $m=3$. In all the experiments, we consider the number
of iterations $N_{iter}=2$ for a better BLER performance.
In the training phase, we use a large batch size of $B=8192$ and employ the
AdamW optimizer with an initial learning rate of $0.001$ and a weight decay
parameter $0.01$. The gradient values are clipped with a threshold value of
$0.5$ for stable training. The model is trained using $180K$ batches. Finally,
we follow curriculum learning to train the models for low target signal-to-
noise ratio (SNR) values, where the SNR of the forward channel
($SNR_{ff}\triangleq 1/\sigma^{2}$) decreases linearly from $3$ dB to the
target value in the first $30K$ batches.

We evaluate the BLER performance of the proposed MBAF code for
$SNR_{ff}\in\\{-1,-0.5,0,0.5,1,1.5\\}$ dB, and the number of
interactions/blocks $T=\\{7,\ldots,12\\}$, which corresponds to sum-rate
values $R=\\{6/12,\ldots,6/7\\}$. The overall BLER is defined by the average
BLER of the two users,
$\text{BLER}=\frac{1}{2}\mathbb{E}(\mathbbm{1}_{\hat{\bm{b}}_{1}\neq{\bm{b}}_{1}}+\mathbbm{1}_{\hat{\bm{b}}_{2}\neq{\bm{b}}_{2}})$
and the results for different settings are illustrated in Table I.

As a benchmark, we also consider a time-division multiple access (TDMA)
scheme, where each user, with perfect feedback, transmits its message only in
its assigned time slot. We consider a simplified case where $T$ is an even
number and each user occupies $T/2$ number of interactions; that is, they
equally share the available bandwidth. We use the GBAF code proposed in [12]
to generate the BLER of the TDMA scheme, denoted by GBAF-TD. Note that for a
fair comparison, in GBAF-TD, $SNR_{ff}$ is increased by $3$ dB to ensure that
the total energy consumption is the same. In this simulation, we set
$SNR_{ff}=\\{-1,0,1,2\\}$ dB and $T=8,K=51$. As shown in Table II, the BLER of
the proposed MBAF significantly outperforms the GBAF-TD in all $SNR_{ff}$
values.

As another benchmark, we consider the upper bound on the achievable
performance for a point-to-point channel without feedback in the finite block
length regime [17], which can be obtained by treating the two users as a
‘super user’ that uses twice the power over $T$ interactions. In this
simulation, we consider $K=51,\epsilon=10^{-6}$ for both the proposed MBAF
code and the point-to-point benchmark, where $\epsilon$ is the BLER threshold
which is an indicator of reliable communication. The achievable sum-rate $R$
of the two schemes with respect to different $SNR_{ff}\in[-1,1]$ dB are shown
in Fig. 3. Note that we also provide the MAC sum capacity without feedback,
which is given by $\frac{1}{2}\log_{2}(1+\frac{2P}{\sigma^{2}})$. As shown in
Fig. 3, the proposed MBAF scheme outperforms the achievable rate upper bound
of the point-to-point baseline without feedback in the finite block length,
which confirms the gain brought by exploiting feedback from the receiver. The
achieved rate is below the MAC sum-capacity, but this is expected as we are
considering an extrmeely short blocklength regime.

Figure 3: Illustration of the sum rate, $R$, of the MBAF code with a BLER $\epsilon=10^{-6}$ compared to the theoretical sum rate of the multiple access channel without feedback both in finite block length and infinite block length (capacity). TABLE II: BLER comparison of the GBAF-TD with the proposed MBAF with $K=51,R=6/8$. SNR |  $-1$ dB |  $0$ dB |  $1$ dB
---|---|---|---
GBAF-TD | $8.4\times 10^{-1}$ | $3.6\times 10^{-1}$ | $1.8\times 10^{-3}$
MBAF | $4.0\times 10^{-1}$ | $2.9\times 10^{-4}$ | $1.7\times 10^{-7}$

##  V Conclusion

In this paper, we proposed the MBAF code for reliable communication over a MAC
with perfect channel output feedback. The MBAF code is designed in a purely
data-driven manner. The parity symbols are generated sequentially at each
user, using a block-based communication framework, and employing transformer-
based neural network models. Our results confirm that the users learn not only
to mitigate the adverse effects of the channel noise, but also to cooperate
among each other by utilizing the feedback signal. This is a first step
towards utilizing the transformer architecture to design novel channel codes
in highligh challenging multi-user scenarios.

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Traversable wormhole in logarithmic $f(R)$ gravity by various shape and
redshift functions
J. Sadeghi⋆,†,111Email<EMAIL_ADDRESS>M. Shokri†,222Email:
<EMAIL_ADDRESS>and S. Noori Gashti⋆,333Email:
<EMAIL_ADDRESS>
⋆Department of Physics, Faculty of Basic Sciences, University of Mazandaran
P. O. Box 47416-95447, Babolsar, Iran
†School of Physics, Damghan University, P. O. Box 3671641167, Damghan, Iran

###### Abstract

In this paper, we are going to study the traversable wormholes in the modified
$f(R)$ model. The corresponding model has a polynomial plus a logarithmic term
as $R+\alpha R^{2}+\beta R^{n}+\gamma R^{2}\log(\gamma R)$. Since we deal with
the wormhole geometry, we have two important functions such as the shape
$b(r)$ and the redshift function $\Phi(r)$. So by considering two states for
shape function as $\frac{r}{\exp(r-r_{0})}$ and
${r_{0}\log(r+1)}{\log(r_{0}+1)}$ and redshift function as
$\Phi(r)=\frac{1}{r}$ and $\Phi(r)=c$, we obtain the solution of wormholes
with respect to corresponding modified $f(R)$ model. Also, here we calculate a
series of parameters such as energy density $\rho$, tangential pressure
$P_{t}$ and redial pressure $P_{r}$. In that case, we examine different energy
conditions, such as null energy conditions (NEC), weak energy conditions
(WEC), etc. The satisfactory and un-satisfactory of energy conditions from the
corresponding model can be explained by different figures.
Keywords: Traversable wormholes, Logarithmic $f(R)$, Shape function, Redshift
function, Energy condition

##  1 Introduction

Recently, a series of hypothetical objects have been attended by researchers,
which are called wormholes. These hypothetical object geometry can solve
Einstein equations by tolerating the important condition, namely the violation
of null energy conditions (NEC). As a hypothetical object, the wormhole has a
tubular structure that is asymptotically flat on both sides. This structure
creates a geometry that connects two distinct points of the same space-time or
two different universes. The wormhole structure was first established by
Flamm[1], but his solutions had some problems as to be unstable. Some years
later, Einstein and Rosen examined a similar structure of the wormhole and
introduced a concept called the Einstein-Rosen Bridge[2]. These hypothetical
objects lead to different static and non-static in proportion to the constant
or variable radius of the wormhole throat. Wormholes have always been
investigated from different aspects and conditions[3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28].
Shortly afterward, the Kerr wormholes were studied, which have a quantum
structure and connect different points of space, but their function is always
on the Planck scale. Also, these wormholes were not traversable[29]. In the
continuation of the studies related to wormholes, later Thorne and Morris[3]
specially laid the structure of wormholes, which had concepts such as the
throat. They established the structure of traversable static wormholes. In
their studies, they used general relativity concepts and found a way to
journey, which is a wonderful and laudable work. It is worth noting that the
existence of wormholes requires ”exotic” mater that violate the null energy
condition[30, 31]. As we know, there is a lack of the matter to tolerate the
wormhole geometry. This raised the susceptible question of whether these
modified theories of gravity, which are used in many other concepts such as
the expansion of the universe, could also answer the structure of the
wormhole. Various types of this theory were introduced, and many researchers
studied this concept. Different types of modified theories of gravity, such as
$f(R)$, $f(R,T)$, were the right candidates in the literature for studying the
wormholes. These modified theories of gravity, which replaces the Ricci scalar
$R$ in action, lead to a modification of Einstein’s action. As you know, this
theory presents relatively more complicated equations. As a result, more
comprehensive solutions are being introduced. So far, much work has been done
on effective theories. We can refer the modified gravitational theory for the
investigation of inflation, dark energy, singularities, Big Rip, unify of
inflation and dark energy and including many works related to different types
of these effective theories from cosmological aspects[32, 33, 34, 35, 36, 37,
38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56,
57, 58, 59, 60, 61, 62, 63, 64, 65, 66]. Also, in the last few decades, a lot
of work has been done to obtain the wormhole’s geometry. As you know, the
condition of the wormhole is a violation of energy conditions. Therefore, it
does not appear as a solution to classical gravity, and all of these cases
indicate the possibility of the formation of wormholes at the quantum level.
Therefore, a lot of work has been done in this regard, such as wormholes
derived from the semi-classical equations of the Einstein field and also the
method of effective equations[67, 68, 69, 70]. Several researchers have also
investigated wormholes of this type by using the nonlinear Scalar Ricci in the
gravity action. Also some researchers have investigated wormhole solutions in
a vacuum[71, 72].The structure of traversable wormholes in modified $f(R)$
gravitational model is investigated by [73, 74, 75, 76]. Such studies have
shown that modifications derived from higher curvatures will be responsible
for violating energy conditions. Also, in such theories, different conditions
were used to study the wormhole, for example, by considering the redshift
functions and various shape functions. Here we note that the static wormholes
in higher dimensions have also been investigated[77, 78]. On the other hand,
the studies of wormholes have been performed in energy density and radial
pressure and tangential pressure proportional to the wormhole mouth. Symmetric
spherical wormhole solutions and their stability condition have also been
investigated. They studied traversable wormholes according to the FLRW model
and evaluated the effects of additional dimensions[79, 80]. Cosmic wormholes
have also been studied in the modified theory of gravity. They considered a
dynamic wormhole and investigated such wormhole with different modes as small
wormholes. Such wormholes usually do not require the presence of an exotic
mater near the wormhole throat. Different types of wormholes in the Finslerian
structure of space-time have also been studied. Other types of shape and
redshift functions and their field equations are investigated by[81, 82]. Some
researchers have also studied solutions of wormholes in the modified
$f(R,\phi)$ gravitational model for different types of states. They have also
studied energy conditions without considering the exotic material and examined
it in terms of non-commutative geometry[83, 84]. Novikov[85] also reviewed and
categorized the different types of wormholes. They determined the properties
associated with wormholes and their relations to black holes[86, 87, 88, 89,
90, 91, 92, 93, 94, 95, 96]. Recently, the structure of spherically symmetric
static wormholes has also been studied by many researchers. They found
specific numerical and analytical solutions and shown how different energy
conditions occur under different conditions[97]. All above information give us
motivation to organize this article as follow: In section 2, we briefly
describe the field equations as well as the wormhole geometry. In Section 3,
we express our modified $f(R)$ gravitational model, which is a polynomial plus
a logarithmic term such as$R+\alpha R^{2}+\beta R^{n}+\gamma R^{2}\log\gamma
R$. We apply all the items expressed in section 2 to the gravitational model,
and we examine the different energy conditions by obtaining the required
parameters, i.e., $\rho$, $P_{r}$ and $P_{t}$. It should be noted that we use
the two shape function b(r) and two redshift functions $\Phi(r)$ in our
calculations. Finally we describe the results by plotting some figures. In the
last section we some conclusion and results.

##  2 Field equations and wormhole geometry

In this section, we briefly explain the geometry and field equations of the
wormhole. First, we introduce the wormhole structure with static and symmetric
spherical metrics, which are given by,

$ds^{2}=-e^{2\Phi(r)}dt^{2}+\frac{dr^{2}}{1-\frac{b(r)}{r}}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})$ (1)

In here, we faced two essential functions such as shape function $b(r)$ and
redshift function $\Phi(r)$. In connection with the traversable wormholes,
some points should be considered, of course, the above was mentioned. But, we
will briefly state a few points in the following. In addition to satisfying
Einstein equations, wormhole solutions must have a throat that connects two
different regions and be asymptotically flat. For traversable wormholes, we
should not have an event horizon, and also, the effects of its gravitational
force should be minimal. Shape function $b(r)$ and redshift function $\Phi(r)$
are the function of r and minimum value of $r_{0}$ is the radius of wormhole
throat, which r increase to $\infty$. Of course, we must also pay attention to
one point, and for the wormhole solutions to existing, the shape function
should satisfy the following conditions:

$(1):b(r_{0})=r_{0}$
$(2):\frac{b(r)-b^{\prime}(r)r}{b^{2}}>0$
$(3):b^{\prime}(r_{0})-1\leq 0$
$(4):\frac{b(r)}{r}<1$ ,$\longrightarrow r>r_{0}$
$(5):\frac{b(r)}{r}\longrightarrow 0$, $r\longrightarrow\infty$

As mentioned above, the wormholes were introduced through Morris and Thorne.
In fact, modified $f(R)$ theories of gravity is a generalized theory of
Einstein relativity expressed by substitution (R) in action. Gravitational
action is expressed in the following form according to the modified $f(R)$
theories of gravity and Ricci scalar $R$,

$S=\frac{1}{2\kappa}\int(f(R)+L_{m})\sqrt{-g}d^{4}x$ (2)

where $\kappa=8\pi G$=1, $L_{m}$ is matter lagrangian density, g is determine
of $g_{\mu\nu}$. With respect to equation (2) and $g_{\mu\nu}$ the field
equation is given by,

$FR_{\mu\nu}-\frac{1}{2}fg_{\mu\nu}-\nabla_{\mu}\nabla_{\nu}F+\hbox{$\hbox
to0.0pt{$\sqcup$\hss}\sqcap$}Fg_{\mu\nu}=T^{m}_{\mu\nu}$ (3)

where the $R_{\mu\nu}$ is denote Ricci tensor and $R$ determine scalar
curvature, also$F=\frac{df}{dr}$. Thus the equation (3) is given by,

$FR-2f+3\hbox{$\hbox to0.0pt{$\sqcup$\hss}\sqcap$}F=T$ (4)

where $T=T^{\mu}_{\mu}$. According the equations (3) and (4), the effective
field equations are calculated as follows,

$G_{\mu\nu}\equiv R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=T^{eff}_{\mu\nu},$ (5)

where,

$T^{eff}_{\mu\nu}=T^{c}_{\mu\nu}+\frac{T^{m}_{\mu\nu}}{F}$ (6)

and

$T^{c}_{\mu\nu}=\frac{1}{F}(\nabla_{\mu}\nabla_{\nu}F-\frac{1}{4}g_{\mu\nu}(FR+\hbox{$\hbox
to0.0pt{$\sqcup$\hss}\sqcap$}F+T)$ (7)

where energy momentum tensor, $T_{\mu\nu}=\frac{\partial L_{m}}{\partial
g^{\mu\nu}}$ is given by,

$T_{\mu\nu}=(\rho+P_{t})u_{\mu}u_{\nu}-P_{t}g_{\mu\nu}+(P_{r}-P_{t})\chi_{\mu}\chi_{\nu}$
(8)

where $u^{\mu}u_{\mu}=-1$ and $\chi^{\mu}\chi_{\mu}=1$. $\rho$ is the energy
density, $P_{r}$ and $P_{t}$ are radial pressure and tangential pressure,
respectively. So the Einstein’s field equations with respect to equation (1)
and modified $f(R)$ gravitational model, can obtained by,

$\rho=\frac{Fb^{\prime}(r)}{r^{2}}-(1-\frac{b(r)}{r})F^{\prime}\Phi^{\prime}(r)-H,$
(9)

$P_{r}=-\frac{b(r)F}{r^{3}}+2(1-\frac{b(r)}{r})\frac{\Phi^{\prime}(r)F}{r}-(1-\frac{b(r)}{r})\\{F^{\prime\prime}+\frac{F^{\prime}(rb^{\prime}(r)-b(r))}{2r^{2}(1-\frac{b(r)}{r})}\\}+H,$
(10)

and

$\begin{split}P_{t}=&\frac{F(b(r)-rb^{\prime}(r))}{2r^{3}}-\frac{F^{\prime}}{r}(1-\frac{b(r)}{r})+F(1-\frac{b(r)}{r})(\Phi^{\prime\prime}(r)-\frac{(rb^{\prime}(r)-b(r))\Phi^{\prime}(r)}{2r(r-b)}\\\
&+\Phi^{\prime 2}(r)+\frac{\Phi^{\prime}(r)}{r})+H\end{split}$ (11)

where

$H=\frac{1}{4}(FR+\hbox{$\hbox to0.0pt{$\sqcup$\hss}\sqcap$}F+T)$ (12)

with respect to above equation prime denote the derivative with respect to
radial coordinate $r$. Also the equation of state parameter or radial state
parameter is given by,

$\omega=\frac{P_{r}}{\rho}$ (13)

as well as the mass function and anisotropy parameter are introduced by,

$m=\int_{r_{0}}^{r}4\pi r^{2}\rho dr$ (14)

and

$\Delta=P_{t}-P_{r}$ (15)

$\Delta$ is in terms of radial and tangential pressures, $\Delta$ determines
the attractive $\Delta<0$ and repulsive $\Delta>0$ nature of the geometry.
$\Delta=0$ the geometry contains isotropic pressure. We are introducing the
modified $f(R)$ gravitational model as well as two shape functions b(r) such
as $(\frac{r}{\exp(r-r_{0})})$ and $(\frac{r_{0}\log(r+1)}{\log(r_{0}+1)})$
and two forms for the redshift function, $(\Phi(r)=\frac{1}{r})$ and
$(\Phi(r)=c)$. In next section, we apply all the concepts discussed in this
section to our gravitational model. And we investigate the wormhole solutions
according to the mentioned functions. We also explain different energy
conditions and invetigate the violation and satisfaction of each of the energy
conditions. Also this section, by plotting some figures we interpret the
corresponding results .

##  3 Wormhole solution and logarithmic $f(R)$ model

As mentioned above, the modified gravitational model plays a critical role in
cosmology special in dark energy and cosmic acceleration. This means that the
modified $f(R)$ model will always be accompanied by improvements in
gravitational action. Various modified $f(R)$ models are responsible for
quantum gravity. These modified gravitational models can appear as polynomials
or logarithms, etc. In this article, we will have a polynomial model plus a
logarithmic term. The modified $f(R)$ models without logarithmic modifications
are commonly used to study neutron stars with a strong magnetic field and
other cosmological implications. Logarithmic modifications are also used for
the effects of gluons in non-planar space. For further studying plus mentioned
above, you can see [97, 98, 99, 100, 101, 102, 103, 104, 105]. Therefore, in
this article, we want to discuss the implication of this modified $f(R)$
gravitational model in the study of wormhole geometry . Thus Considering all
the concepts presented in this paper, as well as the geometry and field
equations expressed in the previous section, now we state the modified $f(R)$
gravitational model, which is a polynomial plus a logarithmic term, i.e.,
$R+\alpha R^{2}+\beta R^{n}+\gamma R^{2}\log\gamma R$, which n is a constant
parameter and we check that $0<n<1$ as $n=\frac{1}{2}$. $\alpha$, $\beta$ and
$\gamma$ are constant coefficients. These coefficients solve the dimensional
problem for such a model. Here, for simplicity we assume these values be
positive units. According to the shape function and gravitational redshift
function, we study different energy conditions and evaluate the desired
conditions by plotting some figures. Therefore, we introduce some following
conditions separately,

$(a):b(r)=\frac{r}{\exp(r-r_{0})}$, $\Phi(r)=\frac{1}{r}$
$(b):b(r)=\frac{r}{\exp(r-r_{0})}$, $\Phi(r)=c$
$(c):b(r)=\frac{r_{0}\log(1+r)}{\log(1+r_{0})}$, $\Phi(r)=\frac{1}{r}$
$(d):b(r)=\frac{r_{0}\log(1+r)}{\log(1+r_{0})}$, $\Phi(r)=c$
In the following, we examine each of the above conditions according to the
modified logarithmic gravitational model and the wormhole geometry.

###  3.1 Case a

In this case, we calculate different solutions of the wormhole according to
the modified logarithmic gravitational model, shape function
$b(r)=\frac{r}{\exp(r-r_{0})}$, the redshift function $\Phi(r)=\frac{1}{r}$ as
well as the energy density, radius pressure and tangential pressure which are
obtained by the field equations (9), (10) and (11). So we will have.

$\begin{split}\rho=&\frac{1}{2r^{7}}e^{-3r}(\frac{2^{1+2n}e^{4r-2r_{0}}(e^{r}-e^{r_{0}})(-1+n)^{2}n^{2}(-1+2n)r^{8}(\frac{-e^{-r+r_{0}}(-1+r)}{r^{2}})^{2n}}{(-1+r)^{2}}\\\
&+2^{n}e^{r}(-\frac{e^{-r+r_{0}}(-1+r)}{r^{2}})^{n-1}r^{2}(16e^{2r_{0}}(n-1)n(7+5n)(-1+r)+e^{2r}(-1+n)n(-2+r)r^{2}\\\
&+e^{r+r_{0}}(16(-1+n)n(7+5n)-16(-1+n)n(7+5n)r+2(-1+n)nr^{2}+r^{3}\\\
&+(-1+(-1+n)n)r^{4}))+2e^{r_{0}}(-1+r)(-1520e^{2r_{0}}(-1+r)-e^{2r}r^{2}(-20+10r+r^{3}\\\
&-2e^{r+r_{0}}(760+r(-760+r(10+r+4r^{2}))))+4(2^{3+n}e^{2r}(e^{r}-e^{r_{0}})n(-1+n^{2})(-\frac{e^{-r+r_{0}}(-1+r)}{r^{2}})^{n}r^{4}\\\
&+e^{r}(-1+r)(-544e^{2r_{0}}(-1+r)-2e^{2r}(-2+r)r^{2}-e^{r+r_{0}}(544+r(4+r+r^{2})))))\\\
&\log(-\frac{2e^{-r+r_{0}}(-1+r)}{r^{2}})+348e^{2r_{0}}(e^{r}-e^{r_{0}})(-1+r)^{2}\log(-\frac{2(-1+r)e^{-r+r_{0}}}{r^{2}})^{2}),\\\
\end{split}$ (16)

$\begin{split}&P_{r}=\frac{1}{2r^{7}}(3040e^{-3r+3r_{0}}(-1+r)^{2}-\frac{2^{1+2n}e^{2(r-r_{0})}(-1+n)^{2}n^{2}(-1+2n)r^{8}(-\frac{e^{-r+r_{0}}(-1+r)}{r^{2}})^{2n}}{(-1+r)^{2}}\\\
&-2e^{-r+r_{0}}r^{2}(-12+r(-74+(88+2^{3+n}(-1+n)n(7+5n)(-\frac{e^{-r+r_{0}}(-1+r)}{r^{2}})^{n}-r)r))\\\
&4e^{-2r+2r_{0}}(-1+r)(760+r(-760+r(-6+r(-37+12r))))+4e^{-3r}\log(-\frac{2e^{-r+r_{0}}(-1+r)}{r^{2}})\\\
&(2^{3+n}e^{2r}(-e^{r}+e^{r_{0}})n(-1+n^{2})(-\frac{e^{-r+r_{0}}(-1+r)}{r^{2}})^{n}r^{4}e^{r_{0}}(-1+r)(544e^{2r_{0}}(-1+r)+2e^{2r}r^{2}(2+7r)\\\
&+e^{r+r_{0}}(544+r(-544+r(-4+r(-11+5r))))+96e^{r_{0}}(+e^{r_{0}}-e^{r})(-1+r)\log(-\frac{2(-1+r)e^{-r+r_{0}}}{r^{2}}))))\\\
&+\frac{A+B}{-1+r},\\\
&A=r^{4}(4-4r-2^{n}(-\frac{e^{-r+r_{0}}(-1+r)}{r^{2}})^{n}(-16(-1+n)n(7+5n)\\\
&B=16(-1+n)n(7+5n)r+2nr^{2}+(-1+2(-1+n)n(-1+2n))r^{3}+(1+n-2n^{2})r^{4})),\\\
\end{split}$ (17)

and

$\begin{split}&P_{t}=\frac{1}{4r^{7}}e^{-3r}(6080e^{3r_{0}}(-1+r)^{2}-\frac{4^{n+1}e^{5r-2r_{0}}(-1+n)^{2}n^{2}(-1+2n)r^{8}(\frac{-e^{-r+r_{0}}(-1+r)}{r^{2}})^{2n}}{(-1+r)^{2}}+\\\
&\frac{2^{n+1}e^{4r-r_{0}}nr^{5}(-\frac{e^{-r+r_{0}}(-1+r)}{r^{2}})^{n}(1+r^{2}(-2+n+r+n(-1+2^{1+n}(-1+n)^{2}(-1+2n)(-\frac{e^{-r+r_{0}}(-1+r}{r^{2}})^{n})r))}{(-1+r)^{2}}\\\
&+2e^{2r+r_{0}}r(12+r^{2}(6+r(-22+2^{4+n}(-1+n)n(7+5n)(-\frac{e^{-r+r_{0}}(-1+r}{r^{2}})^{n}+(-1+r)r)))+\\\
&4e^{r+2r_{0}}(-1+r)(1520+r(-1514+r(6+r(19+11r))))+16\log(-\frac{2e^{-r+r_{0}}(-1+r}{r^{2}})(-2^{2+n}e^{2r}\\\
&(e^{r}-e^{r_{0}})n(-1+n^{2})(-\frac{e^{-r+r_{0}}(-1+r}{r^{2}})^{n}r^{4}+e^{r_{0}}(-1+r)(272e^{2r_{0}}(-1+r)-e^{2r}r(1+r+r^{2})\\\
&+e^{r+r_{0}}(272+r(-271+r(1+r)^{2}))+48e^{r_{0}}(-e^{r}-e^{r_{0}})(-1+r)\log(-\frac{2e^{-r+r_{0}}(-1+r}{r^{2}}))))\\\
&+\frac{A+B}{-1+r}\\\
&A=e^{3r}r^{3}(-4+r(4r+2^{n}(-\frac{e^{-r+r_{0}}(-1+r}{r^{2}})^{n}(32(-1+n)n(7+5n)-2n(-113+16n(2+5n))r\\\
&B=2nr^{2}+(2+n(-5+4n))r^{3}+(-2+n(-1+2n))r^{4})))\\\ \end{split}$ (18)

We are calculating the above values, energy density, tangential pressure and
radial pressure according to the field equations mentioned above as well as
the modified logarithmic gravitational model. Here, we can obtain
$\rho+P_{r}$), ($\rho+P_{t}$, $\rho+P_{r}+2P_{t}$, $\frac{P_{r}}{\rho}$,
$P_{t}-P_{r}$, $\rho-|P_{r}|$ and $\rho-|P_{t}|$. Also, we obtain different
energy conditions for the corresponding model and check them carefully.
However, in these calculations, we used the first case of the shape and
redshift function requirements. The figures related to each of these energy
conditions are plotted. You can see the satisfaction and violation conditions
associated with each of the energy conditions. The conditions related to each
energy are expressed in the following form:
(I): The null energy condition (NEC) will satisfy if we have $\rho+P_{r}\geq
0$ and $\rho+P_{t}\geq 0$,
(II): The weak energy condition (WEC) will satisfy if we have $\rho\geq 0$,
$\rho+P_{r}\geq 0$ and $\rho+P_{t}\geq 0$,
(III): The strong energy condition SEC will validate if we have
$\rho+P_{r}\geq 0$ , $\rho+P_{t}\geq 0$ and $\rho+P_{r}+2P_{t}\geq 0$
(IV): The DEC is satisfied by conditions of $\rho>0$, $\rho-|P_{r}|>0$ and
$\rho-|P_{t}|>0$,
and ($\rho+P_{r}+2P_{t}\geq 0$) Here we can write the above equations in terms
of equation of state ($\omega$)($P_{r}=\omega\rho$) and the anisotropy
parameter $\Delta$ ($\Delta=P_{t}-P_{r}$). We can clearly see in the following
figures the satisfaction and violation of corresponding energy conditions,

Figure 1: plot ($\rho$) to ($r$) in (1a), ($\rho+P_{r}$) to ($r$) in (1b),
($\rho+P_{t}$) to ($r$) in (1c), ($\rho-|P_{r}|$) to ($r$) in (1d) and
$(\rho-|P_{t}|$) to ($r$) in (1e)

Figure 2: plot ($\rho+P_{r}+2P_{t}$) to ($r$) in (2a), ($\frac{P_{r}}{\rho}$)
to ($r$) in (2b), ($P_{t}-P_{r}$) to ($r$) in (2c), ($P_{r}$) to ($r$) in (2d)
and ($P_{t}$) to ($r$) in (2e)

###  3.2 Case b

In this case, we calculate the different solutions of the wormhole according
to the modified logarithmic gravitational model, shape function
$b(r)=\frac{r}{\exp(r-r_{0})}$, the redshift function $\Phi(r)=c$ as well as
the energy density, radius pressure, and tangential pressure which are
obtained from the field equations (9), (10) and (11). So we will have.

$\begin{split}\rho=&\frac{1}{2}(2^{2}(-\frac{2e^{-r+r_{0}}(-1+r)}{r^{2}})^{n}-\frac{20e^{-r+r_{0}}(-1+r)}{r^{4}}+\frac{4e^{-2r+2r_{0}}(-1+r)^{2}}{r^{4}}-\frac{2e^{-r+r_{0}}(-1+r)}{r^{2}}\\\
&+\frac{2^{n}(-1+n)n(-\frac{2e^{-r+r_{0}}(-1+r)}{r^{2}})^{-1+n}}{r^{2}}-\frac{8e^{-r+r_{0}}(-1+r)\log(-\frac{e^{2-r+r_{0}}(-1+r)}{r^{2}})}{r^{4}}\\\
&+\frac{4e^{-2r+2r_{0}}(-1+r)^{2}\log(-\frac{2e^{-r+r_{0}}(-1+r)}{r^{2}})}{r^{4}}+\frac{A*B}{r^{3}},\\\
&A=2(1-e^{-r+r_{0}}(-20e^{-r+r_{0}}(-1+r)+2^{n}(-1+n)n(-\frac{e^{-r+r_{0}}(-1+r)}{r^{2}})^{n-1}\\\
&r^{2}8e^{-r+r_{0}}(-1+r)\log(-\frac{2e^{-r+r_{0}}(-1+r)}{r^{2}})\\\
&B=(\frac{1}{-2+2e^{r-r_{0}}}+\frac{2^{n}(-1+n)n(-1+2n)(-\frac{e^{-r+r_{0}}(-1+r)}{r^{2}})^{n-1}}{r^{2}}\\\
&-\frac{4e^{-r+r_{0}}(-1+r)(19+6\log(-\frac{2e^{-r+r_{0}}(-1+r)}{r^{2}})}{r^{4}}\end{split}$
(19)

$\begin{split}P_{r}=&\frac{1}{2r^{7}}e^{-3r}(3040e^{3r_{0}}(-1+r)^{2}+\frac{2^{2n+1}e^{4r-2r_{0}}(-e^{r}+e^{r_{0}})(-1+n)^{2}n^{2}(-1+2n)r^{8}(-\frac{e^{-r+r_{0}}(-1+r)}{r^{2}})^{2n}}{(-1+r)^{2}}\\\
&-2e^{2r+r_{0}}(86+(-86+r))+4e^{r+2r_{0}}(-1+r)(760-760r-37r^{3}+12r^{4})+2^{n}e^{r_{0}}(-1+r)\\\
&(-\frac{e^{-r+r_{0}}(-1+r)}{r^{2}})^{-2+n}(16e^{2r_{0}}(-1+n)n(7+5n)(-1+r)+e^{2r}(-1+n)n(-1+4n)r^{3}e^{r+r_{0}}\\\
&(16(-1+n)n(7+5n)-16(-1+n)n(7+5n)r+(1-2(-1+n)n(-1+2n))r^{3}(-1+n)\\\
&(1+2n)r^{4}))+4(2^{3+n}e^{2r}(-e^{r}+e^{r_{0}})n(-1+n^{2})(-\frac{e^{-r+r_{0}}(-1+r)}{r^{2}})^{n}r^{4}+e^{r_{0}}(-1+r)\\\
&(544e^{2r_{0}}(-1+r)+14e^{2r}r^{3}+e^{r+r_{0}}(544-544r-11r^{3}+5r^{4})))\log(-\frac{2e^{-r+r_{0}}(-1+r)}{r^{2}})\\\
&+384e^{2R_{0}}(-e^{r}+e^{r_{0}})(-1+r^{2})\log(-\frac{e^{-r+r_{0}}(-1+r)}{r^{2}})^{2})\end{split}$
(20)

$\begin{split}P_{t}=&\frac{1}{4r^{7}}e^{-3r}(\frac{4^{1+n}e^{4r-2r_{0}}(-e^{r}+e^{r_{0}}(-1+n)^{2}n^{2}(-1+2n)r^{8}(-\frac{e^{-r+r_{0}}(-1+r)}{r^{2}})^{2n}}{(-1+r)^{2}}+2e^{r_{0}}\\\
&(3040e^{2r_{0}}(-1+r)^{2}+e^{2r}r^{3}(20-20r+r^{3})+2e^{r+r_{0}}(-1+r)(1520-1520r+16r^{3}+11r^{4}))\\\
&+2^{n}e^{r_{0}}(-1+r)(-\frac{e^{-r+r_{0}}(-1+r)}{r^{2}})^{-2+n}(32e^{2r_{0}}(-1+n)n(7+5n)(-1+r)-2e^{r_{0}}(-1+n)nr^{3}\\\
&+e^{r+r_{0}}(32(-1+n)n(7+5n)-32(-1+n)n(7+5n)r+2(-1+n)(-1+2n)r^{3}\\\
&+(-2+n(-1+2n))r^{4}))+8(2^{3+n}e^{2r}(-e^{r}+e^{r_{0}})n(-1+n^{2})(-\frac{e^{-r+r_{0}}(-1+r)}{r^{2}})^{n}r^{4}\\\
&+e^{r_{0}}(-1+r)(544+e^{2r_{0}}(-1+r)(544-544r+3r^{3}+2r^{4})))\log(-\frac{2e^{-r+r_{0}})(-1+r)}{r^{2}})\\\
&+768e^{2r_{0}}(+e^{r_{0}}-e^{r})(-1+r)^{2}\log(-\frac{2e^{-r+r_{0}})(-1+r)}{r^{2}})^{2})\end{split}$
(21)

Also, in this part, we have obtained the above values such as $\rho$, $P_{t}$
and $P_{r}$ concerning the field equations and the modified logarithmic
gravitational model . Just like the previous section, any of the energy
conditions (NEC), (WEC), (DEC), and (SEC) can be obtained. It is a
straightforward task. Also, by using the above values, each of the different
energy conditions for this model can be well obtained, and we check them
carefully. As you know, with respect to mentioned conditions, i.e. (I, II,
III, IV) can be used to examine each of the energy conditions. However, in
these calculations, we used the second case of the conditions related to the
shape and the redshift function that means $b(r)=\frac{r}{\exp(r-r_{0})}$ and
$\Phi(r)=c$. The figures related to each of these energy conditions are
plotted. We can see the satisfaction and violation conditions associated with
each of the energy conditions.

Figure 3: plot ($\rho$) to ($r$) in (3a), ($\rho+P_{r}$) to ($r$) in (3b),
($\rho+P_{t}$) to ($r$) in (3c), ($\rho-|P_{r}|$) to ($r$) in (3d) and
($\rho-|P_{t}|$) to ($r$) in (3e)

Figure 4: plot ($\rho+P_{r}+2P_{t}$) to ($r$) in (4a), ($\frac{P_{r}}{\rho}$)
to ($r$) in (4b), ($P_{t}-P_{r}$) to ($r$) in (4c), ($P_{r}$) to ($r$) in (4d)
and ($P_{t}$) to ($r$) in (4e)

###  3.3 Case c

In this case, we calculate the different solutions of the wormhole according
to the modified logarithmic gravitational model, shape function
$b(r)=\frac{r_{0}\log(1+r)}{\log(1+r_{0})}$, the redshift function
$\Phi(r)=\frac{1}{r}$ as well as the energy density, radius pressure and
tangential pressure which are obtained from the field equations (9), (10) and
(11). So we will have.

$\begin{split}\rho=&\frac{1}{2r^{8}(1+r)^{2}r_{0}^{2}\log(1+r_{0})^{3}}(2^{1+2n}(-1+n)^{2}n^{2}(-1+2n)r^{9}(1+r)^{4}(\frac{r_{0}}{r^{2}(1+r)\log(1+r_{0})})^{2n}\\\
&\log(1+r_{0})^{5}-r_{0}\log(1+r)(2^{n}(-1+n)nr^{4}(1+r)^{2}(\frac{r_{0}}{r^{2}(1+r)\log(1+r_{0})})^{n}\log(1+r_{0})^{2}\\\
&+4r_{0}^{2}(5+2(\frac{r_{0}}{r^{2}(1+r)\log(1+r_{0})})))(-r^{2}(1+r)(2+r)r_{0}\log(1+r_{0})+2^{1+n}(-1+n)n\\\
&(-1+2n)r^{4}(1+r)^{2}(\frac{r_{0}}{r^{2}(1+r)\log(1+r_{0})})^{n}\log(1+R_{0})^{2}+8r_{0}^{2}\\\
&(19+6\log(\frac{2r_{0}}{r^{2}(1+r)\log(1+r_{0})})))+2^{n}r^{5}(1+r)^{2}r_{0}(\frac{r_{0}}{r^{2}(1+r)\log(1+r_{0})})^{n}\\\
&\log(1+r_{0})^{3}((r^{3}+(-1+n)n(112+80n-r^{3}))r_{0}+(-1+n)((-2+r)r^{2}(1+r)\log(1+r_{0})\\\
&+32(1+n)r_{0}\log(\frac{2r_{0}}{r^{2}(1+r)\log(1+r_{0})})))+2rr_{0}^{3}\log(1+r_{0})(r^{2}(1+r)\\\
&\log(1+r_{0})(-20+10r+r^{3}+4(-2+r)\log(\frac{2r_{0}}{r^{2}(1+r)\log(1+r_{0})}))+2r_{0}(760-4r^{3}\\\
&+\log(\frac{r_{0}}{2r^{2}(1+r)\log(1+r_{0})})(544-r^{3}+96\log(\frac{2r_{0}}{r^{2}(1+r)\log(1+r_{0})})))))\end{split}$
(22)

$\begin{split}P_{r}=&\frac{1}{2r^{8}(1+r)^{2}r_{0}\log(1+r_{0})^{3}}(-2^{1+2n}(-1+n)^{2}n^{2}(-1+2n)(1+r)^{3}r^{7}(\frac{r_{0}}{r^{2}(1+r)\log(1+r_{0})})^{-1+2n}\\\
&\log(1+r_{0})^{4}-+2^{n}r^{5}(1+r)^{2}(\frac{r_{0}}{r^{2}(1+r)\log(1+r_{0})})^{n}\log(1+r_{0})^{3}((-1+n)(16n(7+5n)\\\
&-(1+2n)r^{3})r_{0}+nr^{2}(1+r)(2+(-1+n)(-1+4n)r)\log(1+r_{0})+32n(-1+n^{2})r_{0}\\\
&\log(\frac{2r_{0}}{r^{2}(1+r)\log(1+r_{0})}))+4rr_{0}\log(1+r_{0})(4(-190+3r^{3})r_{0}^{2}-r^{2}(1+r)\\\
&\log(1+r_{0})((6+43r)r_{0}+r^{2}(1+r)\log(1+r_{0}))+r_{0}((-544+5r^{3})r_{0}-2r^{2}(1+r)(2+7r)\\\
&\log(1+r_{0}))\log(\frac{2r_{0}}{r^{2}(1+r)\log(1+r_{0})})-96r_{0}^{2}\log(\frac{2r_{0}}{r^{2}(1+r)\log(1+r_{0})})^{2})\\\
&+\log(1+r_{0})(2^{1+2n}(-1+n)^{2}n^{2}(-1+2n)r^{8}(1+r)^{4}(\frac{2r_{0}}{r^{2}(1+r)\log(1+r_{0})})^{2n}\log(1+r_{0})^{4}\\\
&2r_{0}^{2}(1520r_{0}^{2}+r^{2}(1+r)\log(1+r_{0})(2(6+25r)r_{0}+r^{2}(2+r-r^{2})\log(1+r_{0}))+4r_{0}\\\
&(227r_{0}+r^{2}(1+r)(2+3r)\log(1+r_{0}))\log(\frac{2r_{0}}{r^{2}(1+r)\log(1+r_{0})})+192r_{0}^{2}\\\
&\log(\frac{2r_{0}}{r^{2}(1+r)\log(1+r_{0})})^{2})+2^{n}nr^{4}(1+r)^{2}r_{0}(\frac{r_{0}}{r^{2}(1+r)\log(1+r_{0})})^{n}\\\
&\log(1+r_{0})^{2}(r^{2}(1+r)(2+(3+4(-2+n)n)r)\log(1+r_{0})+16r_{0}(-7+2n+5n^{2}+2(-1+n^{2})\\\
&\log(\frac{2r_{0}}{r^{2}(1+r)\log(1+r_{0})})))))\\\ \end{split}$ (23)

$\begin{split}P_{t}=&-\frac{1}{2r^{8}(1+r)^{2}r_{0}\log(1+r_{0})^{3}}(2^{1+2n}(-1+n)^{2}n^{2}(-1+2n)r^{7}(1+r)^{3}\\\
&(\frac{2r_{0}}{r^{2}(1+r)\log(1+r_{0})})^{-1+2n}\log(1+r_{0})^{4}rr_{0}\log(1+r_{0})(-2(-1520+r^{2}(3+11r)\\\
&r_{0}^{2}+r(-12+r(-24+r(-32-21r+r^{3})))r_{0}\log(1+r_{0})-2(r+r^{2})^{3}\log(1+r_{0})\\\
&-4r_{0}((-544+r^{2}+2r^{3})r_{0}+2r(1+r)(1+r+r^{2})\log(1+r_{0}))\log(\frac{2r_{0}}{r^{2}(1+r)\log(1+r_{0})})\\\
&+384r_{0}^{2}\log(\frac{2r_{0}}{r^{2}(1+r)\log(1+r_{0})})^{2})-2^{n-1}r^{5}(1+r)^{2}(\frac{r_{0}}{r^{2}(1+r)\log(1+r_{0})})^{n}\\\
&\log(1+r_{0})^{3}(2nr(1+r)(1+r+(-1+n)r^{2})\log(1+r_{0})+r_{0}(-32(-1+n)n(7+5n)+nr^{2}\\\
&+(-2+n(-1+2n))r^{3}-64n(-1+n^{2})\log(\frac{2r_{0}}{r^{2}(1+r)\log(1+r_{0})})))\\\
&+\frac{1}{2}\log(1+r)(-4^{n+1}(-1+n)^{2}n^{2}(-1+2n)r^{8}(1+r)^{4}(\frac{2r_{0}}{r^{2}(1+r)\log(1+r_{0})})^{2n}\\\
&\log(1+r_{0})^{4}2r_{0}^{2}(-3040r_{0}^{2}+r(1+r)\log(1+r_{0})(6(2+3r+9r^{2})r_{0}-r^{2}(1+r)(-2+(-3+r)r)\\\
&\log(1+r_{0}))+4r_{0}(-544r_{0})+r(1+r)(2+r(3+5r))\log(1+r_{0}))\log(\frac{2r_{0}}{r^{2}(1+r)\log(1+r_{0})})\\\
&-384r_{0}^{2}\log(\frac{2r_{0}}{r^{2}(1+r)\log(1+r_{0})})^{2})+2^{n}nr^{4}(1+r)^{2}r_{0}(\frac{2r_{0}}{r^{2}(1+r)\log(1+r_{0})})^{n}\\\
&\log(1+r_{0})^{2}(r(1+r)(2+r(3-7r+6nr))\log(1+r_{0})+32r_{0}(7-2n-5n^{2}-2(-1+n^{2})\\\
&\log(\frac{2r_{0}}{r^{2}(1+r)\log(1+r_{0})})))))\end{split}$ (24)

In accordance with the previous two-part we investigated the different
important values $\rho$, $P_{t}$ and $P_{r}$. With the help of these values,
we can obtain each of the values of different energy conditions such as (NEC),
(WEC), (DEC), and (SEC) and equation of state $\omega$ as well as anisotropy
parameter $\Delta$. Also, by using the above values, each of the different
energy conditions for this model can be well obtained. According to
conditions, (I, II, III, IV) we examine each of the energy conditions. However
we used the case 3, ie, shape function
$b(r)=\frac{r_{0}\log(1+r)}{\log(1+r_{0})}$ and the redshift function
$\Phi(r)=\frac{1}{r}$. The figures related to each of these energy conditions
are plotted. As we can see in each of the following diagrams, and according to
the conditions that we fully described in the previous part, we see the
satisfaction or violation of the different energy conditions in the following
figures.

Figure 5: plot ($\rho$) to ($r$) in (5a), ($\rho+P_{r}$) to ($r$) in (5b),
plot ($\rho+P_{t}$) to ($r$) in (5c), ($\rho-|P_{r}|$) to ($r$) in (5d) and
($\rho-|P_{t}|$) to ($r$) in (5e)

Figure 6: plot ($\rho+P_{r}+2P_{t}$) to ($r$) in (6a), ($\frac{P_{r}}{\rho}$)
to ($r$) in (6b), ($P_{t}-P_{r}$) to ($r$) in (6c), ($P_{r}$) to ($r$) in (6d)
and ($P_{t}$) to ($r$) in (6e)

###  3.4 Case d

In this case, we calculate the different solutions of the wormhole according
to the modified logarithmic gravitational model, shape function
$b(r)=\frac{r_{0}\log(1+r)}{\log(1+r_{0})}$, the redshift function $\Phi(r)=c$
as well as the energy density, radius pressure, and tangential pressure which
are obtained from the field equations (9), (10) and (11). So we will have.

$\begin{split}\rho=&\frac{1}{2r^{8}(1+r)^{2}r_{0}^{2}\log(1+r_{0})^{3}}(4r^{4}r_{0}^{4}\log(1+r_{0})+2r^{6}(1+r)r_{0}^{3}\log(1+r_{0})^{3}\\\
&2^{n}r^{8}(1+r)^{2}r_{0}^{2}(\frac{r_{0}}{r^{2}(1+r)\log(1+r_{0})})^{n}\log(1+r_{0})^{3}+2^{n}(-1+n)nr^{8}(1+r)^{3}r_{0}\\\
&(\frac{r_{0}}{r^{2}(1+r)\log(1+r_{0})})^{n}\log(1+r_{0})^{4}+4r^{4}r_{0}^{4}\log(1+r_{0})\log(\frac{2r_{0}}{r^{2}(1+r)\log(1+r_{0})})\\\
&+4r^{4}(1+r)r_{0}^{3}\log(1+r_{0})^{2}(5+2\log(\frac{2r_{0}}{r^{2}(1+r)\log(1+r_{0})}))-(2^{n}(-1+n)nr^{4}(1+r^{2})\\\
&(\frac{r_{0}}{r^{2}(1+r)\log(1+r_{0})})^{n}\log(1+r_{0})^{2}+4r_{0}^{2}(5+2\log(\frac{2r_{0}}{r^{2}(1+r)\log(1+r_{0})})))\\\
&(r^{3}r_{0}^{2}(r-(1+r)\log(1+r))\log(1+r_{0})-2^{1+n}(-1+n)n(-1+2n)r^{4}(1+r)^{2}\\\
&(\frac{r_{0}}{r^{2}(1+r)\log(1+r_{0})})^{n}\log(1+r_{0})^{2}(-r_{0}\log(1+r)+r\log(1+r_{0}))\\\
&+8r_{0}^{2}(r_{0}\log(1+r)-r\log(1+r_{0}))(19+6\log(\frac{2r_{0}}{r^{2}(1+r)\log(1+r_{0})}))))\end{split}$
(25)

$\begin{split}&P_{r}=\frac{1}{2r^{8}(1+r)^{2}r_{0}\log(1+r_{0})^{3}}(-2^{1+2n}(-1+n)^{2}n^{2}(-1+2n)r^{7}(1+r)^{3}\log(1+r_{0})^{4}\\\
&(\frac{r_{0}}{r^{2}(1+r)\log(1+r_{0})})^{2n-1}2^{n}(-1+n)r^{5}(1+r)^{2}(\frac{r_{0}}{r^{2}(1+r)\log(1+r_{0})})^{n}\log(1+r_{0})^{3}\\\
&(n(-1+4n)r^{3}(1+r)\log(1+r_{0})+r_{0}(16n(7+5n)-(1+2n)r^{3}+32n(1+n)(\frac{2r_{0}}{r^{2}(1+r)\log(1+r_{0})})))\\\
&+4rr_{0}^{2}\log(1+r_{0})(-r^{3}(1+r)\log(1+r_{0})(43+14\log(\frac{2r_{0}}{r^{2}(1+r)\log(1+r_{0})}))+r_{0}(-760++12r^{3}\\\
&+(-544+5r^{3}-96\log(\frac{2r_{0}}{r^{2}(1+r)\log(1+r_{0})}))(\frac{2r_{0}}{r^{2}(1+r)\log(1+r_{0})})))\\\
&+\log(1+r)(2^{1+2n}(-1+n)^{2}n^{2}(-1+2n)r^{8}(1+r)^{4}(\frac{2r_{0}}{r^{2}(1+r)\log(1+r_{0})})^{2n}\log(1+r_{0})^{4}\\\
&+2r_{0}^{2}(1520r_{0}^{2}+r^{3}(1+r)\log(1+r_{0})(50r_{0}-r^{2}(1+r)\log(1+r_{0}))+4r_{0}(272r_{0}+3r^{3}(1+r)\\\
&\log(1+r_{0}))\log(\frac{2r_{0}}{r^{2}(1+r)\log(1+r_{0})})+192r_{2}^{2}\log(\frac{2r_{0}}{r^{2}(1+r)\log(1+r_{0})})^{2})\\\
&+2^{n}r^{4}n(1+r)^{2}r_{0}(\frac{r_{0}}{r^{2}(1+r)\log(1+r_{0})})^{n}\log(1+r_{0})^{2}((3+4(-2+n)n)r^{3}(1+r)\log(1+r_{0})\\\
&+16r_{0}(-7+2n+5n^{2}+2(-1+n^{2})\log(\frac{2r_{0}}{r^{2}(1+r)\log(1+r_{0})})))))\end{split}$
(26)

$\begin{split}&P_{t}=-\frac{1}{2r^{8}(1+r)^{2}r_{0}\log(1+r_{0})^{3}}(2^{1+2n}(-1+n)^{2}n^{2}(-1+2n)r^{7}(1+r^{3}\log(1+r_{0})^{4}\\\
&\log(\frac{r_{0}}{r^{2}(1+r)\log(1+r_{0})})^{2n-1}+2^{-1+n}r^{5}(1+r)^{2}(\frac{r_{0}}{r^{2}(1+r)\log(1+r_{0})})^{n}\log(1+r_{0})^{3}\\\
&(-2(-1+n)nr^{3}(1+r)\log(1+r_{0})+r_{0}(32(-1+n)n(7+5n)+(2+n-2n^{2})r^{3}+64n(-1+n^{2})\\\
&\log(\frac{2r_{0}}{r^{2}(1+r)\log(1+r_{0})})))+rr_{0}^{2}\log(1+r_{0})(r^{3}(1+r)\log(1+r_{0})(-20+r^{2}\\\
&-8\log(\frac{2r_{0}}{r^{2}(1+r)\log(1+r_{0})}))+2r_{0}(1520-11r^{3}+4\log(\frac{2r_{0}}{r^{2}(1+r)\log(1+r_{0})})\\\
&(272-r^{3}+48\log(\frac{2r_{0}}{r^{2}(1+r)\log(1+r_{0})}))))+\frac{1}{2}\log(1+r)(-4^{1+n}(-1+n)^{2}n^{2}(-1+2n)r^{8}(1+r)^{4}\\\
&(\frac{2r_{0}}{r^{2}(1+r)\log(1+r_{0})})^{2n}\log(1+r_{0})^{4}+2r_{0}^{2}(-3040r_{0}^{2}+r^{3}(1+r)\log(1+r_{0})+4r_{0}(-544r_{0}\\\
&+5r^{3}(1+r)\log(1+r_{0}))\log(\frac{2r_{0}}{r^{2}(1+r)\log(1+r_{0})})-384r_{0}^{2}\log(\frac{2r_{0}}{r^{2}(1+r)\log(1+r_{0})})^{2})\\\
&+2^{n}nr^{4}(1+r)^{2}r_{0}(\frac{2r_{0}}{r^{2}(1+r)\log(1+r_{0})})^{n}\log(1+r_{0})^{2}((-7+6n)r^{3}(1+r)\log(1+r_{0})+32r_{0}\\\
&(7-2n-5n^{2}-2(-1+n^{2})\log(\frac{2r_{0}}{r^{2}(1+r)\log(1+r_{0})})))))\end{split}$
(27)

In this part we used the shape function
$b(r)=\frac{r_{0}\log(1+r)}{\log(1+r_{0})}$ and the redshift function
$\Phi(r)=c$ to study each of the above. WE can obtain the other values as
mentioned the previous part such as $\rho+P_{r}$, $\rho+P_{t}$,
$\rho+P_{r}+2P_{t}$ and etc . And we can evaluate different energy conditions
according to the new conditions. All the arguments in this part are the same
as in the previous parts. Using the obtained values, we expressed each of the
energy conditions by plotting some figures. Also, about the stated conditions
the satisfaction of the energy conditions described above, we can use each of
the different energy conditions for plotting figures, and we analyze the
results with another condition mentioned above.

Figure 7: plot ($\rho$) to ($r$) in (7a), ($\rho+P_{r}$) to ($r$) in (7b),
($\rho+P_{t}$) to ($r$) in (7c), ($\rho-|P_{r}|$) to ($r$) in (7d) and
($\rho-|P_{t}|$) to ($r$) in (7e)

Figure 8: plot ($\rho+P_{r}+2P_{t}$) to ($r$) in (8a), ($\frac{P_{r}}{\rho}$)
to ($r$) in (8b), ($P_{t}-P_{r}$) to ($r$) in (8c), ($P_{r}$) to ($r$) in (8d)
and ($P_{t}$) to ($r$) in (8e)

In this paper, we evaluate a traversable wormhole concerning a modified $f(R)$
logarithmic gravitational model, as well as with respect to different
conditions corresponding to the shape and the redshift function. We
investigated the results associated with each condition in some parts. The
results of this section can be compared with other models. One of the
important points is to examine specific conditions such as the conditions
raised in recent years, the weak gravity conjecture, and compare the results
obtained in each case. Of course, we hope that excellent results will be
obtained in these studies. In the next works, we are looking to review the
results according to specific mentioned conditions. We hope that new windows
of light will be added to our science.

##  4 conclusions

In recent years a series of hypothetical objects have been attended by
researchers, which are called wormholes. The geometry of these hypothetical
objects can solve Einstein equations by tolerating the important condition,
namely the violation of null energy conditions (NEC). The shape function
should satisfy many conditions till would have existed wormhole solutions . In
this paper, our main goal was to study the traversable wormholes from a
modified $f(R)$ gravitational model point of view, which has a polynomial plus
a logarithmic term, i.e., $R+\alpha R^{2}+\beta R^{n}+\gamma R^{2}\log(\gamma
R)$. e studied two very important functions such as the shape function b(r)
and the redshift function $\Phi(r)$, that mean for the shape function such as
$(\frac{r}{\exp(r-r_{0})})$ and $(\frac{r_{0}\log(r+1)}{\log(r_{0}+1)})$ and
for the redshift function, $(\Phi(r)=\frac{1}{r})$ and $(\Phi(r)=c)$. we
obtained the solution of wormholes concerning the Logarithmic $f(R)$ model.
Also, we calculated a series of parameters such as $(\rho)$, $(P_{t})$ and
$(P_{r})$, energy density, tangential pressure and radial pressure
respectively. We examined each of the different energy conditions, such as
null energy conditions (NEC), weak energy conditions(WEC), etc. We described
the results of either the violation or the satisfaction of the energy
conditions completely by plotting some figures.

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# On moments of Cantor and related distributions.

Paweł J. Szabłowski  Department of Mathematics and Information Sciences,
Warsaw University of Technology
ul Koszykowa 75, 00-662 Warsaw, Poland<EMAIL_ADDRESS>
(Date: September 01, 2013)

###### Abstract.

We provide several simple recursive formulae for the moment sequence of
infinite Bernoulli convolution. We relate moments of one infinite Bernoulli
convolution with others having different but related parameters. We give
examples relating Euler numbers to the moments of infinite Bernoulli
convolutions. One of the examples provides moment interpretation of Pell
numbers as well as new identities satisfied by Pell and Lucas numbers.

###### Key words and phrases:

Moments of infinite Bernoulli convolutions, moments of Cantor distribution,
Silver ratio, Pell numbers, Lucas numbers.

###### 2010 Mathematics Subject Classification:

11K41, 11B65, ; Secondary 60E10

##  1\. Introduction

The aim of this note is to add a few simple observations to the analysis of
the distribution of the so called fatigue symmetric walk (term appearing in
[12]). These observations are based on the reformulation of known results
scattered through literature. We however pay more attention to the moment
sequences and less to the properties of distributions that produce these
moment sequences. It seems that the main novelty of the paper lies in the
probabilistic interpretation of Pell and Lucas numbers and easy proofs of some
identities satisfied by these numbers. However in order to place these results
in the proper context we recall definition and basic properties of infinite
Bernoulli convolutions. In deriving properties of these convolutions we recall
some known, important results.

The paper is organized as follows. After recalling definition and basic facts
on the fatigue random walks we concentrate on the moment sequences of infinite
Bernoulli convolutions. We formulate a corollary of the results of the paper
expressed in terms of moment sequence. This corollary formulated in terms of
number sequences provides identities of Pell and Lucas numbers of even order
(Remark 8).

##  2\. Infinite Bernoulli Convolutions

Let $\left\\{X_{n}\right\\}_{n\geq 1}$ be the sequence of i.i.d. random
variables such that $P(X_{1}\allowbreak=\allowbreak 1)\allowbreak=\allowbreak
P(X_{1}\allowbreak=\allowbreak-1)\allowbreak=\allowbreak 1/2.$ Further let
$\left\\{c_{n}\right\\}_{n\geq 1}$ be a sequence of reals such that
$\sum_{n\geq 1}c_{n}^{2}<\infty.$ We define random variable:

$S\allowbreak=\allowbreak\sum_{n\geq 1}c_{n}X_{n}.$

By Kolmogorov 3 series theorem $S$ exists and moreover it is square
integrable. $ES^{2}\allowbreak=\allowbreak\sum_{n\geq 1}c_{n}^{2}.$ Obviously
$ES\allowbreak=\allowbreak 0.$ Let $\varphi(t)$ denote characteristic function
of $S.$ By the standard argument we have $\varphi(t)\allowbreak=\allowbreak
E\exp(it\sum_{s\geq 1}c_{n}X_{n})\allowbreak=\allowbreak E\prod_{n\geq
1}\exp(itc_{n}X_{n})\allowbreak=\allowbreak\prod_{n\geq
1}(\exp(itc_{n})/2+\exp(-itc_{n})/2)\allowbreak=\allowbreak\prod_{n\geq
1}\cos(tc_{n}).$

We will concentrate on the special form of the sequence $c_{n}$ namely we will
assume that $c_{n}\allowbreak=\allowbreak\lambda^{-n}$ for some $\lambda>1.$

It is known (see [13]) that for all $\lambda$ distribution of
$S\allowbreak=\allowbreak S\left(\lambda\right)$ is continuous that is
$P_{S}(\left\\{x\right\\})\allowbreak=\allowbreak 0$ for all $x\in\mathbb{R}.$
Moreover it is also known (see [16], [17]) that if for almost
$\lambda\in(1,2]$ this distribution is absolutely continuous and for almost
all $\lambda\in(1,\sqrt{2}]$ it has square integrable density. Garsia in [7],
Theorem 1.8 showed examples of such $\lambda$ leading to absolutely continuous
distribution. Namely such $\lambda\in(1,2)$ are the roots of monic polynomials
$P$ with integer coefficients such that
$\left|P(0)\right|\allowbreak=\allowbreak 2$ and
$\lambda\prod_{\left|\alpha_{i}\right|>1}\left|\alpha_{i}\right|\allowbreak=\allowbreak
2,$ where $\left\\{\alpha_{i}\right\\}$ are the remaining roots of $P$.

There are known (see [4], [5]) countable instances of $\lambda\in(1,2]$ that
this distribution is singular. We will denote by $\varphi_{\lambda}$ the
characteristic function of $S\left(\lambda\right).$ Following [4] we know that
the values $\lambda$ such that $\varphi_{\lambda}(t)$ does not tend to zero as
$t\longrightarrow\infty$ consequently related distribution is singular (by
Riemann–Lebesgue Lemma) are the so called Pisot or PV- numbers i.e. sole roots
of such monic irreducible polynomials $P$ with integer coefficients having the
property that all other roots have absolute values less than $1.$ We must then
have $P(0)\allowbreak=\allowbreak 1.$ Examples of such numbers are the so
called ’golden ratio’ $\left(1+\sqrt{5}\right)/2$ or the so called ’silver
ratio’ $1+\sqrt{2}.$ Moreover following [14] one knows that PV numbers are the
only numbers $\lambda\in(1,2]$ for which $\varphi_{\lambda}$ does not tend to
zero. Of course singularity of the distribution of $S\left(\lambda\right)$ can
occur for $\lambda$ not being PV numbers.

For $\lambda>2$ it is known that the distribution of $S$ is singular [10].

To simplify notation we will write $\mathop{\mathrm{s}upp}X,$ where $X$ is a
random variable meaning $\mathop{\mathrm{s}upp}P_{X},$ where $P_{X}$ denotes
distribution of $X.$ Similarly $X\ast Y$ denotes random variable whose
distribution is a convolution of distributions of $X$ and $Y.$

We have simple Lemma.

######  Lemma 1.

i)
$\mathop{\mathrm{s}upp}(S\left(\lambda\right))\subset\left[-\frac{1}{\lambda-1},\frac{1}{\lambda-1}\right].$

In particular:

ia) if $\lambda\allowbreak=\allowbreak 2$ then $S\allowbreak\sim\allowbreak
U([-1,1])$ and

ib) if $\lambda\allowbreak=\allowbreak 3$ then $\mathop{\mathrm{s}upp}(S+1/2)$
is equal to Cantor set.

In general if $\lambda\allowbreak$ is a positive integer then
$\mathop{\mathrm{s}upp}(S+1/(\lambda-1))$ consist of all numbers of the form
$\sum_{j\geq 1}r_{j}\lambda^{-j}$ where $r_{j}\in\\{0,2\\}$. Moreover the
distribution of $(S(\lambda)+1/(\lambda-1))$ is ’uniform’ on this a set.

ii) $\forall k\geq 1:$

(2.1)
$\varphi_{\lambda}(\lambda^{k}t)\allowbreak=\allowbreak\varphi_{\lambda}(t)\prod_{j=0}^{k-1}\cos\left(\lambda^{j}t\right).$

iii) $\forall k\geq
1:S\left(\lambda\right)\allowbreak\allowbreak\sim\allowbreak\allowbreak\sum_{i=1}^{k}\lambda^{i-1}S_{i}(\lambda^{k}),$
where $S_{i}\left(\tau\right)$ ( $i\allowbreak\allowbreak=\allowbreak
1,\ldots,k)\allowbreak$ are i.i.d. random variables each having distribution
$S\left(\tau\right).$ Consequently
$\varphi_{\lambda}(t)\allowbreak=\allowbreak\prod_{j=1}^{k}\varphi_{\lambda^{k}}\left(\lambda^{j-1}t\right)$

iv) Let us denote $m_{n}(\lambda)\allowbreak=\allowbreak ES(\lambda)^{n}.$
Then $\forall n\geq 1:m_{2n-1}(\lambda)\allowbreak=\allowbreak 0$ and

$m_{2n}(\lambda)\allowbreak=\allowbreak\frac{1}{\lambda^{2kn}-1}\sum_{j=0}^{n-1}\binom{2n}{2j}m_{2j}(\lambda)W_{2(n-j)}^{(k)}(\lambda),$

with $m_{0}\allowbreak=\allowbreak 1$, where
$W_{n}^{(1)}\allowbreak=\allowbreak 1,$
$W_{n}^{(k)}\left(\lambda\right)\allowbreak=\allowbreak\left.\frac{d^{n}}{dt^{n}}(\prod_{j=1}^{k}\cosh(\lambda^{j-1}t))\right|_{t=0}\allowbreak$
$=\allowbreak\frac{1}{2^{k-1}}\sum_{i_{1}=0,\ldots,i_{k-1}=0}^{1}(1+\sum_{j=1}^{k}(2i_{j}-1)\lambda^{j})^{2n}.$
In particular we have:

(2.2) $\displaystyle m_{2n}(\lambda)$ $\displaystyle=$
$\displaystyle\frac{1}{\lambda^{2n}-1}\sum_{j=0}^{n-1}m_{2j}(\lambda)\binom{2n}{2j},$
(2.3) $\displaystyle m_{2n}\left(\lambda\right)$ $\displaystyle=$
$\displaystyle\frac{1}{\lambda^{4n}-1}\sum_{j=0}^{n-1}\binom{2n}{2j}m_{2j}(\lambda)\sum_{l=0}^{2(n-j)}\binom{2(n-j)}{2l}\lambda^{2l}.$

v) $\forall k\geq
1:m_{2k}(\lambda)\allowbreak\mathbb{=}\allowbreak\frac{-1}{\lambda^{2k}-1}\sum_{j=0}^{k-1}\binom{2k}{2j}\lambda^{2j}E_{2(k-j)}m_{2j}(\lambda),$
where $E_{k}$ denotes $k-$th Euler number.

###### Proof.

i) First of all notice that
$\frac{1}{\lambda-1}\allowbreak=\allowbreak\sum_{n\geq 1}1/\lambda^{n},$ hence
$S+\frac{1}{\lambda-1}\allowbreak=\allowbreak\sum_{n\geq
1}\frac{1}{\lambda^{n}}(X_{n}+1).$ Now since
$P(X_{n}+1\allowbreak=0)\allowbreak=\allowbreak
P(X_{n}+1\allowbreak=\allowbreak 2)\allowbreak=\allowbreak 1/2$ we see that
$\mathop{\mathrm{s}upp}(S+\frac{1}{\lambda-1})\subset[0,\frac{2}{\lambda-1}].$
Notice also that if $\lambda\allowbreak=\allowbreak 2$ then
$S+1\allowbreak=\allowbreak 2\sum_{n\geq 1}\frac{1}{2^{n}}Y_{n},$ where
$P(Y_{n}\allowbreak=\allowbreak 0)\allowbreak=\allowbreak
P(Y_{n}\allowbreak=\allowbreak 1)\allowbreak=\allowbreak 1/2.$ In other words
$(S+1)/2$ is any number chosen from $[0,1]$ with equal chances that is
$(S+1)/2$ has uniform distribution on $[0,1].$

When $\lambda\allowbreak=\allowbreak 3$ we see that $S+1/2$ is a number that
can be written with the help of ${}^{\prime}0^{\prime}$ and
${}^{\prime}2^{\prime}$ in ternary expansion. In other words $S+1/2$ is number
drawn from Cantor set with equal chances. For $\lambda$ integer we argue in
the similar way.

ii) We have $\varphi_{S}(\lambda^{k}t)\allowbreak=\allowbreak\prod_{n\geq
1}\cos(\lambda^{k}t\frac{1}{\lambda^{n}})\allowbreak=\allowbreak\varphi_{S}\left(t\right)\prod_{j=0}^{k-1}\cos\left(\lambda^{j}t\right)$.

iii) Fix integer $k$. Notice that we have:

$\displaystyle S\left(\lambda\right)\allowbreak=\allowbreak\sum_{n\geq
1}\lambda^{-n}X_{n}\allowbreak=\allowbreak$ $\displaystyle\sum_{j\geq
1}\lambda^{-kj}X_{kj}\allowbreak+\allowbreak\newline \sum_{j\geq
1}\lambda^{-kj+1}X_{kj-1}\allowbreak+\allowbreak\ldots\allowbreak+\allowbreak\sum_{j\geq
1}\lambda^{-kj+k-1}X_{kj-k+1}\allowbreak=$
$\displaystyle\allowbreak\sum_{m=1}^{k}\lambda^{m-1}\sum_{j\geq
1}\left(\lambda^{k}\right)^{-j}X_{kj-m+1}.$

Now since by assumption all $X_{i}$ are i.i.d. we deduce that
$S_{i}(\lambda^{k})$ are i.i.d. random variables with distribution defined by
$\varphi_{\lambda^{k}}(t).$ Hence we have
$\varphi_{\lambda}(t)\allowbreak=\allowbreak\prod_{j=1}^{k}\varphi_{\lambda^{k}}(\lambda^{j-1}t).$

iv) First of all we notice that $\varphi_{S}(t)$ is an even function hence all
derivatives of odd order at zero are equal $0$. Secondly let
$\psi_{\lambda}(t)$ denote moment generating function of $S(\lambda).$ It is
easy to notice that
$\psi_{\lambda}(s)\allowbreak=\allowbreak\varphi_{S\left(\lambda\right)}(-is).$
Let us denote
$m_{2n}\left(\lambda\right)\allowbreak=\allowbreak\psi_{\lambda}^{(2n)}(0).$
Basing on the elementary formula

$\cosh(\alpha)\cosh(\beta)\allowbreak=\allowbreak\frac{1}{2}(\cosh(\alpha+\beta)+\cosh\left(\alpha-\beta\right)),$

we can easily obtain by induction the following identity:

$\prod_{j=0}^{k-1}\cosh(\lambda^{j}t)\allowbreak=\allowbreak\frac{1}{2^{k-1}}\sum_{i_{1}=0,\ldots,i_{k-1}=0}^{1}\cosh(t(1+\sum_{j=1}^{k}(2i_{j}-1)\lambda^{j})).$

Since $\left.(\cosh\alpha
t)^{(2n)}\right|_{t=0}\allowbreak=\allowbreak\alpha^{2n}$, we have

$\displaystyle\left.\left(\prod_{j=0}^{k-1}\cosh(\lambda^{j}t)\right)^{(2n)}\allowbreak\right|_{t=0}\allowbreak=$
$\displaystyle\allowbreak\frac{1}{2^{k-1}}\sum_{i_{1}=0,\ldots,i_{k-1}=0}^{1}\left.\left(\cosh(t(1+\sum_{j=1}^{k}(2i_{j}-1)\lambda^{j}))\right)^{(2n)}\right|_{t=0}\allowbreak=W_{2n}^{(k)}\left(\lambda\right).$

Now using Leibnitz formula for differentiation applied to (2.1) we get

$f^{(2n)}(\lambda^{k}t)\lambda^{2kn}=\sum_{j=0}^{2n}\binom{2n}{j}(\prod_{i=0}^{k-1}\cosh\lambda^{i}t)^{\left(j\right)}f^{\left(2n-j\right)}\left(t\right).$

Setting $t=\allowbreak 0$ and using the fact that all derivatives of both $f$
and $\cosh t$ of odd order at zero are zeros we get the desired formula.

v) We use result of [18] that states that for each $N$ inverse of lower
triangular matrix of degree $N\times N$ with $(i,j)$ entry $\binom{2i}{2j}$ is
the lower triangular matrix with $(i,j)-th$ entry equal to
$\binom{2i}{2j}E_{2(i-j)}.$

######  Remark 1.

Formula (2.2) is known in a slightly different form it appeared in [8], [6]
and [1].

######  Remark 2.

Notice that polynomials $\left\\{W_{n}^{(k)}(\lambda)\right\\}_{k,n\geq 1}$
satisfy the following recursive relationship for $k>1:$

$W_{n}^{(k)}\left(\lambda\right)\allowbreak=\allowbreak\sum_{j=0}^{n}\binom{2n}{2j}\lambda^{2j}W_{j}^{\left(k-1\right)}\left(\lambda\right),$

with $W_{n}^{(1)}(\lambda)\allowbreak=\allowbreak 1.$ Hence its generating
functions satisfy $\Theta_{k}(t)$ the following relationship

$\Theta_{k}(t,\lambda)=\Theta_{k-1}(\lambda t,\lambda)\cosh t,$

where we have have denoted:
$\Theta_{k}(t,\lambda)\allowbreak=\allowbreak\sum_{n\geq
0}\frac{t^{2n}}{\left(2n\right)!}W_{n}^{(k)}(\lambda).$

######  Remark 3.

Notice that the above mentioned lemma provides an example of two singular
distributions whose convolution is a uniform distribution. Namely we have
$S(4)\ast 2S(4)\allowbreak=\allowbreak S(2).$ Similarly we have
$S(2)\allowbreak=\allowbreak S\left(8\right)\ast 2S(8)\ast 4S(8)$ or
$S(2)\allowbreak=\allowbreak S(2^{k})\ast\ldots\ast 2^{k-1}S(2^{k}).$ The
first example was already noticed by Kersher and Wintner in [10],(22a).

######  Remark 4.

We can deduce even more from theses examples namely following the result of
Kersher [9], p.451 that characteristic functions $\varphi_{n}(t)$ of
$S\left(n\right)$ (where $n$ is an integer $>2$) do not tend to zero as
$t\longrightarrow\infty.$ Thus since we have
$\varphi_{4}(t)\varphi_{4}(2t)\allowbreak=\allowbreak\sin t/t$ and
$\varphi_{8}(t)\varphi_{8}(2t)\varphi_{8}(4t)\allowbreak=\allowbreak\sin t/t$
we deduce that if $t_{k}\longrightarrow\infty$ is a sequence such that
$\left|\varphi_{4}(t_{k})\right|>\varepsilon>0$ for suitable $\varepsilon$
then $\varphi_{4}(2t_{k})\longrightarrow 0.$ Similarly if
$t_{k}\longrightarrow\infty$ such that
$\left|\varphi_{8}(t_{k})\right|>\varepsilon>0$ then
$\varphi_{8}(2t_{k})\varphi_{8}(4t_{k})\longrightarrow 0$. Similar
observations can be made can be made in more general situation. As it is known
from the papers of Erdős [4], [5] the situation that
$\left|\varphi_{\lambda}(t_{k})\right|>\varepsilon>0$ for some sequence
$t_{k}\longrightarrow\infty$ occurs when $\lambda$ is a Pisot number (briefly
PV-number). On the other hand as it is known roots of Pisot numbers are not
Pisot, hence using above mentioned result of Salem
$\left|\varphi_{\lambda^{1/k}}(t)\right|\longrightarrow 0$ as
$t\longrightarrow\infty,$ where $\lambda$ is some PV number and $k>1$ any
integer. But we have
$\varphi_{\lambda^{1/k}}(t_{n})\allowbreak=\allowbreak\prod_{j=1}^{k}\varphi_{\lambda}\left(\lambda^{(j-1)/k}t_{n}\right)\longrightarrow
0,$ where $t_{n}\longrightarrow\infty$ is such a sequence that
$\left|\varphi_{\lambda}(t_{n})\right|>\varepsilon>0.$

######  Remark 5.

One knows that if $\lambda\allowbreak=\allowbreak q/p$ where $p$ and $q$ are
relatively prime integers and $p>1$ then
$\varphi_{\lambda}(t)\allowbreak=\allowbreak O((\log|t|)^{-\gamma})$ where
$\gamma\allowbreak=\allowbreak\gamma\left(p,q\right)>0$ as
$t\longrightarrow\infty$ (see [9],(3)). Besides we know that then distribution
of $S\left(\lambda\right)$ is singular. Hence from our considerations it
follows that if $\lambda\allowbreak=\allowbreak(q/p)^{1/k}$ for some integer
$k$ then $\varphi_{\lambda}(t)\allowbreak=\allowbreak
O((\log|t|)^{-k\gamma}).$ Is it also singular?

##  3\. Moment sequences

To give connection of certain moment sequences with some known integer
sequences let us remark the following:

######  Remark 6.

i)

$\displaystyle 9^{n}m_{2n}(3)\allowbreak$ $\displaystyle=$
$\displaystyle\allowbreak\allowbreak\sum_{j=0}^{n}\binom{2n}{2j}m_{2j}(3),$
$\displaystyle 81^{n}m_{2n}(3)$ $\displaystyle=$
$\displaystyle\allowbreak\allowbreak\sum_{j=0}^{n}\binom{2n}{2j}m_{2j}(3)(2^{4(n-j)-1}+2^{2\left(n-j\right)-1}).$

ii)

$\displaystyle 5^{n}m_{2n}\left(\sqrt{5}\right)$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{n}\binom{2n}{2j}m_{2j}(\sqrt{5}),$ $\displaystyle
25^{n}m_{2n}\left(\sqrt{5}\right)$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{n}\binom{2n}{2j}m_{2j}(\sqrt{5})4^{n-j}L_{2(n-j)}/2,$

where $L_{n}$ denotes $n-$th Lucas number defined below.

###### Proof.

i) The first assertion is a direct application of (2.2) while in proving the
second one we use (2.3) and the fact that
$\sum_{j=0}^{n}\binom{2n}{2j}9^{j}\allowbreak=\allowbreak 4^{n}(4^{n}+1)/2$ as
shown by [21], (seq. no. A026244). ii) Again the first statement follows (2.2)
while the second follows (2.3) and the fact that
$\sum_{j=0}^{n}\binom{2n}{2j}5^{j}\allowbreak=\allowbreak 4^{n}T_{n}(3/2),$
where $T_{n}$ denotes Chebyshev polynomial of the first kind. ([21], seq. no.
A099140). Further we use the fact that $T_{n}(3/2)\allowbreak=\allowbreak
L_{2n}/2.$ ([21], seq. no. A005248).

We also have the following Lemma.

######  Lemma 2.

$\forall n\geq 1,k\geq 2:$

(3.1)
$m_{2n}(\lambda)\allowbreak=\allowbreak\sum_{i_{1},\ldots,i_{k}=0}^{n}\frac{(2n)!}{(2i_{1})!\ldots(2i_{k})!}\lambda^{2(i_{2}+2i_{3}\ldots(k-1)i_{k})}\prod_{j=1}^{k}m_{2i_{j}}\left(\lambda^{k}\right).$

In particular:

(3.2) $\displaystyle m_{2n}(\lambda)\allowbreak$ $\displaystyle=$
$\displaystyle\allowbreak\sum_{j=0}^{n}\binom{2n}{2j}\lambda^{2j}m_{2j}(\lambda^{2})m_{2n-2j}(\lambda^{2}),$
$\displaystyle m_{2n}(\lambda)\allowbreak$ $\displaystyle=$
$\displaystyle\allowbreak\sum_{\begin{subarray}{c}i,j=0\\\ i+j\leq
n\end{subarray}}\frac{(2n)!}{(2i)!(2j)!(2(n-i-j))!}\times$
$\displaystyle\lambda^{2i}\lambda^{4j}m_{2j}(\lambda^{3})m_{2i}(\lambda^{3})m_{2n-2i-2j}(\lambda^{3}).$

###### Proof.

(3.1) follows directly Lemma 1, iii).

As a corollary we get the following four observations:

######  Corollary 1.

i) $\forall n\geq
1:4^{n}\allowbreak=\allowbreak\sum_{j=0}^{n}\binom{2n+1}{2j+1}$ and
$1\allowbreak=\allowbreak\sum_{j=0}^{n}\binom{2n+1}{2j+1}4^{j}E_{2(n-j)}.$

ii) $S\left(\sqrt{2}\right)\allowbreak$ has density

$g(x)\allowbreak=\allowbreak\left\\{\begin{array}[]{ccc}\sqrt{2}/4&if&\left|x\right|\leq\sqrt{2}-1,\\\
\sqrt{2}(\sqrt{2}+1-\left|x\right|)/8&if&\sqrt{2}-1\leq\left|x\right|\leq\sqrt{2}+1,\\\
0&if&\left|x\right|>1+\sqrt{2}.\end{array}\right.$

iii) Let us denote $\delta_{n}\allowbreak=\allowbreak(\sqrt{2}+1)^{n}$, then

$m_{2n}(\sqrt{2})\allowbreak=\allowbreak\left(\delta_{2n+2}-\delta_{2n+2}^{-1}\right)/(4\sqrt{2}(n+1)(2n+1)).$

###### Proof.

Since for $\lambda\allowbreak=\allowbreak 2$ random variable
$S\allowbreak\sim\allowbreak U[-1,1]$ its moments are equal to
$ES^{2n}\allowbreak=\allowbreak\frac{1}{2n+1}.$ Now we use Lemma 1 iii) and
iv).

ii) From the proof of Lemma 2 it follows that
$S\left(\sqrt{2}\right)\allowbreak\sim\allowbreak
S\left(2\right)\allowbreak+\allowbreak\sqrt{2}S\left(2\right).$ Now keeping in
mind that $S\left(2\right)\allowbreak\sim\allowbreak U(-1,1)$ we deduce that
$g(x)\allowbreak=\allowbreak\frac{\sqrt{2}}{8}\int_{-1}^{1}h(x-t)dt,$ where
$h\left(x\right)\allowbreak=\allowbreak\left\\{\begin{array}[]{ccc}\frac{\sqrt{2}}{4}&if&\left|x\right|\leq\sqrt{2},\\\
0&if&otherwise.\end{array}\right..$ iii) By straightforward calculations we
get $m_{2n}(\sqrt{2})\allowbreak=\allowbreak
2\int_{0}^{\sqrt{2}+1}x^{2n}g(x)dx\allowbreak=\allowbreak\frac{\sqrt{2}}{2}\int_{0}^{\sqrt{2}-1}x^{2n}dx\allowbreak+\allowbreak\frac{\sqrt{2}}{4}\int_{\sqrt{2}-1}^{\sqrt{2}+1}x^{2n}(\sqrt{2}+1-x)dx.$

######  Remark 7.

Let us apply formulae: (3.2), (2.2), (2.3) and observe by direct calculation
that
$2\sum_{j=0}^{n}\binom{2n}{2j}2^{j}\allowbreak=\allowbreak(1+\sqrt{2})^{2n}\allowbreak+\allowbreak(1-\sqrt{2})^{2n}$.
We get the following identities: $\forall n\geq 1:$

$\displaystyle m_{2n}(\sqrt{2})\allowbreak$ $\displaystyle=$
$\displaystyle\allowbreak\sum_{j=0}^{n}\binom{2n}{2j}\frac{2^{j}}{(2n-2j+1)(2j+1)}$
$\displaystyle=$
$\displaystyle\frac{1}{2^{n}-1}\sum_{j=0}^{n-1}\binom{2n}{2j}m_{2j}(\sqrt{2})\allowbreak$
$\displaystyle=$
$\displaystyle\frac{1}{4^{n}-1}\sum_{j=0}^{n-1}\binom{2n}{2j}m_{2j}(\sqrt{2})\tau_{2(n-j)},$

where
$\tau_{n}\allowbreak=\allowbreak(1+(-1)^{n})(\delta_{n}+\delta_{n}^{-1})/4.$

Now let us recall the definition of the so called Pell and Pell–Lucas numbers.
Using sequence $\delta_{n}$ Pell numbers $\left\\{P_{n}\right\\}$ and
Pell–Lucas numbers $\left\\{Q_{n}\right\\}$ are defined

(3.4) $\displaystyle P_{n}\allowbreak$ $\displaystyle=$
$\displaystyle\allowbreak(\delta_{n}+(-1)^{n+1}\delta_{n}^{-1})/(2\sqrt{2}),$
(3.5) $\displaystyle Q_{n}\allowbreak$ $\displaystyle=$
$\displaystyle\allowbreak\delta_{n}+\left(-1\right)^{n}\delta_{n}^{-1},$

where $\delta_{n}$ is defined in 1,iii).

Using these definitions we can rephrase assertions of Corollary 1 and Remark 7
adding to recently discovered ([15], [2]) new identities satisfied by Pell and
Pell-Lucas numbers and of course probabilistic interpretation of Pell numbers.

######  Remark 8.

i) $m_{2n}(\sqrt{2})\allowbreak=\allowbreak\frac{P_{2n+2}}{(2n+2)(2n+1)},$
$\tau_{2n}\allowbreak=\allowbreak Q_{2n}/2.$

ii) $\forall n\geq 1:$

(3.6) $\displaystyle P_{2n+2}\allowbreak$ $\displaystyle=$
$\displaystyle\allowbreak\sum_{j=0}^{n}\binom{2n+2}{2j+1}2^{j},$ (3.7)
$\displaystyle Q_{2n}$ $\displaystyle=$ $\displaystyle
2\sum_{j=0}^{n}\binom{2n}{2j}2^{j},$ (3.8) $\displaystyle
2^{n-1}P_{2n}\allowbreak$ $\displaystyle=$
$\displaystyle\allowbreak\sum_{j=0}^{n}\binom{2n}{2j}P_{2j},$ (3.9)
$\displaystyle 2^{2n-1}P_{2n}\allowbreak$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{n}\binom{2n}{2j}P_{2j}Q_{2(n-j)},$ (3.10)
$\displaystyle\sum_{j=0}^{n}\binom{2n}{2j}(1+\sqrt{2})^{2j}\allowbreak$
$\displaystyle=$ $\displaystyle\allowbreak
2^{n-1}+2^{n-2}Q_{2n}+2^{n-1}\sqrt{2}P_{2n}.$

###### Proof.

Only last statement requires justification. First we find that
$\sum_{j=0}^{n}\binom{2n}{2j}Q_{2j}\allowbreak=\allowbreak 2^{n}(1+Q_{2n}/2)$
using (3.7). Then we use (3.4), (3.5) and (3.8).

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