11footnotetext: 2020 Mathematics Subject Classification. Primary 55S10; 55S05,
46L06, 13A50, 55T15.22footnotetext: Keywords and phrases: Polynomial algebra,
Steenrod algebra, Hit problem, Graded rings.

# On the hit problem for the Steenrod algebra
in some generic degrees and applications

Nguyen Khac Tin  Faculty of Applied Sciences, Ho Chi Minh City University of
Technology and Education
01 Vo Van Ngan, Thu Duc, Ho Chi Minh city, Viet Nam<EMAIL_ADDRESS>
###### Abstract.

Let
$\mathcal{P}_{n}:=H^{*}((\mathbb{R}P^{\infty})^{n})\cong\mathbb{F}_{2}[x_{1},x_{2},\ldots,x_{n}]$
be the polynomial algebra over the prime field of two elements,
$\mathbb{F}_{2}.$ We investigate the Peterson hit problem for the polynomial
algebra $\mathcal{P}_{n},$ viewed as a graded left module over the mod-$2$
Steenrod algebra, $\mathcal{A}.$ For $n>4,$ this problem is still unsolved,
even in the case of $n=5$ with the help of computers.

The purpose of this paper is to continue our study of the hit problem by
developing a result in [18] for $\mathcal{P}_{n}$ in the generic degree
$r(2^{s}-1)+m.2^{s}$ where $r=n=5,\ m=13,$ and $s$ is an arbitrary non-
negative integer. Note that for $s=0,$ and $s=1,$ this problem has been
studied by Phuc [17], and [18], respectively.

As an application of these results, we get the dimension result for the
polynomial algebra in the generic degree
$d=(n-1).(2^{n+u-1}-1)+\ell.2^{n+u-1}$ where $u$ is an arbitrary non-negative
integer, $\ell\in\\{23,67\\},$ and $n=6.$

One of the major applications of hit problem is in surveying a homomorphism
introduced by Singer, which is a homomorphism

$Tr_{n}:\text{Tor}^{\mathcal{A}}_{n,n+d}(\mathbb{F}_{2},\mathbb{F}_{2})\longrightarrow(\mathbb{F}_{2}{\otimes}_{\mathcal{A}}\mathcal{P}_{n})_{d}^{GL(n;\mathbb{F}_{2})}$

from the homology of the Steenrod algebra to the subspace of
$(\mathbb{F}_{2}{\otimes}_{\mathcal{A}}\mathcal{P}_{n})_{d}$ consisting of all
the $GL(n;\mathbb{F}_{2})$-invariant classes. It is a useful tool in
describing the homology groups of the Steenrod algebra,
$\text{Tor}^{\mathcal{A}}_{n,n+d}(\mathbb{F}_{2},\mathbb{F}_{2}).$ The
behavior of the fifth Singer algebraic transfer in degree
$5(2^{s}-1)+13.2^{s}$ was also discussed at the end of this paper.

##  1\. Introduction

Throughout the paper, we denote a prime field with two elements by
$\mathbb{F}_{2}.$ Let $\mathbb{R}\mathcal{P}^{\infty}$ be the infinite
dimensional real projective spaces. Then,
$H^{*}(\mathbb{R}\mathcal{P}^{\infty})\cong\mathbb{F}_{2}[x_{1}],$ and
therefore, the mod-$2$ cohomology algebra of the direct product of $n$ copies
of $\mathbb{R}\mathcal{P}^{\infty}$ is isomorphic to the graded polynomial
algebra $\mathbb{F}_{2}[x_{1},x_{2},\ldots,x_{n}],$ reviewed as an unstable
$\mathcal{A}$-module on $n$ generators $x_{1},x_{2},\ldots,x_{n},$ each of
degree one.

The action of $\mathcal{A}$ on $\mathcal{P}_{n}$ is determined by the formula

$Sq^{i}(x_{j})=\left\\{\begin{array}[]{ll}x_{j},&i=0,\\\ x_{j}^{2},&i=1,\\\
0,&i>1,\end{array}\right.$

and the Cartan formula $Sq^{k}(uv)=\sum_{i=0}^{k}Sq^{i}(u)Sq^{k-i}(v),$ where
$u,v\in\mathcal{P}_{n}$ (see Steenrod and Epstein [24]).

One of central problems of Algebraic Topology is to determine a minimal set of
generators for the ring of invariants $(\mathcal{P}_{n})^{G_{n}},$ where
$G_{n}\subset GL(n;\mathbb{F}_{2}),$ the general linear group of invertible
matrices. This ring is stable under the action of $\mathcal{A}.$ The problem
is so-called hit problem for the polynomial algebra. If we consider
$\mathbb{F}_{2}$ as a trivial $\mathcal{A}$-module, then the hit problem is
equivalent to the problem of finding a basis for the $\mathbb{F}_{2}$-graded
vector space

$\displaystyle\left\\{(\mathbb{F}_{2}{\otimes}_{\mathcal{A}}(\mathcal{P}_{n})^{G_{n}})_{d}\right\\}_{d\geqslant
0}.$ (1.1)

The structure of this space has first been studied by Peterson [16], Wood
[34], Singer [21], Priddy [19], who show its relationship to several classical
problems in cobordism theory, modular respresentation theory, Adams spectral
sequence for the stable homotopy of spheres, stable homotopy type of the
classifying space of finite groups. Then, this problem was investigated by
Hung-Peterson [5], Kameko [11], Janfada [8, 9], Nam [15], Repka-Selick [20],
Silverman [23], Wood [34], Sum [25, 26], Sum-Tin [28, 29], the present writer
[30], [31] and others.

For $G_{n}=\Sigma_{n}$ the symmetric group on $n$ letters, the space (1.1) is
known by Janfada-Wood [6, 7] with $n=3,$ and by Singer in [22] for the dual of
(1.1) with $n\geqslant 0.$ For $G_{n}=GL(n;\mathbb{F}_{2}),$ the problem is
studied by Singer [21] for $n=2,$ and by Hung-Peterson [5] for $n=3,4.$ In
case $G_{n}$ is the trivial group, we have

$\displaystyle\mathbb{F}_{2}{\otimes}_{\mathcal{A}}\mathcal{P}_{n}=\left\\{(\mathbb{F}_{2}{\otimes}_{\mathcal{A}}\mathcal{P}_{n})_{d}\right\\}_{d\geqslant
0}\cong\mathcal{P}_{n}/{\mathcal{A}^{+}}\mathcal{P}_{n},$ (1.2)

as a modular representation of $GL(n;\mathbb{F}_{2}).$ Here,
$(\mathcal{P}_{n})_{d}$ is the subspace of $\mathcal{P}_{n}$ consisting of all
the homogeneous polynomials of degree $d$ in $\mathcal{P}_{n},$ and
$(\mathbb{F}_{2}{\otimes}_{\mathcal{A}}\mathcal{P}_{n})_{d}$ is the subspace
of $\mathbb{F}_{2}{\otimes}_{\mathcal{A}}\mathcal{P}_{n}$ consisting of all
the classes represented by the elements in $(\mathcal{P}_{n})_{d}.$

A polynomial $u$ in $\mathcal{P}_{n}$ is called hit if it can be written as a
finite sum $u=\sum_{i\geqslant 0}Sq^{2^{i}}(f_{i})$ for suitable polynomials
$f_{i}$. That means $u$ belongs to $\mathcal{A}^{+}\mathcal{P}_{n}$, where
$\mathcal{A}^{+}$ denotes the augmentation ideal in $\mathcal{A}$.

For a natural number $d,$ let $\alpha(d)$ be the number of digits 1 in the
binary expansion of $d.$ We define a function
$\mu:\mathbb{N}\longrightarrow\mathbb{N}$ is given by

$\displaystyle\mu(0)=0,{\text{ and }}\mu(d)$
$\displaystyle=\text{min}\\{m\in\mathbb{N}\ :\
d=\sum_{i=1}^{m}(2^{d_{i}}-1),d_{i}>0\\}$
$\displaystyle=\text{min}\\{m\in\mathbb{N}\ :\ \alpha(d+m)\leqslant m\\}.$

In [16], Peterson conjectured that as a module over the Steenrod algebra
$\mathcal{A},$ $\mathcal{P}_{n}$ is generated by monomials in degree $d$ and
satisfying the inequality $\alpha(d+n)\leqslant n$, where $\alpha(d)$ is the
number of digits one in the binary expension of $d,$ and proved it for
$n\leqslant 2,$ in general, it is proved by Wood [34]. This is an extremely
useful tool for determining $\mathcal{A}$-generators for $\mathcal{P}_{n}.$

One of the main tools in the study of the hit problem is Kameko’s squaring
operation

$\widetilde{Sq}^{0}_{*}:=(\widetilde{Sq}^{0}_{*})_{(n;2n+d)}:(\mathbb{F}_{2}{\otimes}_{\mathcal{A}}\mathcal{P}_{n})_{2n+d}\to(\mathbb{F}_{2}{\otimes}_{\mathcal{A}}\mathcal{P}_{n})_{d},$

which is induced by an $\mathbb{F}_{2}$-linear map
$S_{n}:\mathcal{P}_{n}\to\mathcal{P}_{n}$, given by

$S_{n}(x)=\begin{cases}y,&\text{if }x=x_{1}x_{2}\ldots x_{k}y^{2}\\\
0,&\text{otherwise}\end{cases}$

for any monomial $x\in\mathcal{P}_{n}$. Clearly,
$(\widetilde{Sq}^{0}_{*})_{(n;2n+d)}$ is an $\mathbb{F}_{2}$-epimorphism.

From the results of Wood [34], Kameko [10], and Sum [26], the hit problem is
reduced to the case of degree $d$ of the form $d=r(2^{t}-1)+2^{t}m,$ where
$r,m,t$ are non-negative intergers such that $0\leqslant\mu(m)<r\leqslant n.$

Now, the structure of the space (1.2) was completely calculated for
$n\leqslant 4,$ (see Peterson [16] for $n=1,$ and $n=2,$ see Kameko for $n=3$
in his thesis [10], see Sum [26] for $n=4$). For $n\geqslant 5$, it is still
unsolved, even in the case of $n=5$ with the help of computers.

In the present paper, we develop a result in [18] on the hit problem for
$\mathcal{P}_{n}$ in the generic degree $r(2^{s}-1)+m.2^{s},$ where $r=n=5,\
m=13,$ and $s$ is an arbitrary non-negative integer. Remarkably, for $s=0,$
and $s=1,$ this problem has been studied by Phuc [17], and [18], respectively.
Moreover, as an application of the above results, we get the dimension results
for the graded polynomial algebra in the generic degree
$d=(n-1).(2^{n+u-1}-1)+\ell.2^{n+u-1}$ where $u$ is an arbitrary non-negative
integer, $\ell\in\\{23,67\\},$ and $n=6.$

Note that the general linear group $GL(n;\mathbb{F}_{2})$ acts naturally on
$\mathcal{P}_{n}$ by matrix substitution. Since the two actions of
$\mathcal{A}$ and $GL(n;\mathbb{F}_{2})$ upon $\mathcal{P}_{n}$ commute with
each other, there is an action of $GL(n;\mathbb{F}_{2})$ on
$\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{n}.$ From this event, the hit
problem becomes one of the main tools for the studying of the general linear
groups over $\mathbb{F}_{2}.$ Recently, the hit problem and its applications
to representations of general linear groups have been presented in the books
of Walker and Wood [32, 33].

On the other hand, one of the major applications of hit problem is in
surveying a homomorphism introduced by W. M. Singer. It is a useful tool in
describing the cohomology groups of the Steenrod algebra,
$Ext^{n,n+*}_{\mathcal{A}}(\mathbb{F}_{2},\mathbb{F}_{2}).$

Let
$(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{n})_{d}^{GL(n;\mathbb{F}_{2})}$
be the subspace of $(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{n})_{d}$
consisting of all the $GL(n;\mathbb{F}_{2})$-invariant classes of degree $d,$
and
$\mathbb{F}_{2}{\otimes}_{GL(n;\mathbb{F}_{2})}PH_{d}((\mathbb{R}\mathcal{P}^{\infty})^{n})$
be dual to
$(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{n})_{d}^{GL(n;\mathbb{F}_{2})}.$

Singer [21] defined the algebraic transfer, which is a homomorphism

$\psi_{n}:\mathbb{F}_{2}{\otimes}_{GL(n;\mathbb{F}_{2})}PH_{*}((\mathbb{R}\mathcal{P}^{\infty})^{n})\longrightarrow
Ext^{n,n+*}_{\mathcal{A}}(\mathbb{F}_{2},\mathbb{F}_{2}).$

Singer has indicated the importance of the algebraic transfer by showing that
$\psi_{n}$ is a isomorphism with $n=1,2$ and at some other degrees with
$n=3,4$, but he also disproved this for $\psi_{5}$ at degree 9, and then gave
the following conjecture.

######  Conjecture 1.1.

For any $n\geqslant 0,$ the algebraic transfer $\psi_{n}$ is a monomorphism.

It could be seen from the work of Singer the meaning and necessity of the hit
problem. In [2], Boardman confirmed this again by using the modular
representation theory of linear groups to show that $\psi_{3}$ is also an
isomorphism.

For $n\geqslant 4,$ the Singer algebraic transfer was studies by many authors
(See Boardman [2], Bruner-Ha-Hung [3], Minami [14], Sum-Tin [27] and others).
However, Singer’s conjecture is still open for $n\geqslant 4.$

The results of the hit problem are used to study and verify the Singer
conjecture for the algebraic transfer. The behavior of the fifth Singer
algebraic transfer in the degree $d_{s}=5(2^{s}-1)+13.2^{s}$ was also
discussed at the end of this paper.

Next, in Section 2, we recall some needed information on admissible monomials
in $\mathcal{P}_{n}.$ The main results are presented in Section 3. The proofs
of the main results will be presented in Section 4.

##  2\. Preliminaries

In this section, we recall a few useful preliminaries on the admissible
monomials, spike monomials, and the hit monomials from Kameko [10], and Sum
[26], which will be used in the next section.

We will denote by $\mathbb{N}_{n}=\\{1,2,\ldots,n\\}$ and

$\displaystyle
X_{\mathbb{J}}=X_{\\{j_{1},j_{2},\ldots,j_{s}\\}}=\prod_{j\in\mathbb{N}_{n}\setminus\mathbb{J}}x_{j},\
\ \mathbb{J}=\\{j_{1},j_{2},\ldots,j_{s}\\}\subset\mathbb{N}_{n},$

In particular, $X_{\mathbb{N}_{n}}=1,\ X_{\emptyset}=x_{1}x_{2}\ldots x_{n},$
$X_{j}=x_{1}\ldots\hat{x}_{j}\ldots x_{n},\ 1\leqslant j\leqslant n,$ and
$X:=X_{n}\in\mathcal{P}_{n-1}.$

Let $\alpha_{t}(d)$ be the $t$-th coefficient in dyadic expansion of $d.$
Then, $d=\sum_{t\geqslant 0}\alpha_{t}(d).2^{t}$ where
$\alpha_{t}(d)\in\\{0,1\\}.$ Let $x=x_{1}^{a_{1}}x_{2}^{a_{2}}\ldots
x_{n}^{a_{n}}\in\mathcal{P}_{n}$. Denote $\nu_{j}(x)=a_{j},1\leqslant
j\leqslant n$. Set

$\mathbb{J}_{t}(x)=\\{j\in\mathbb{N}_{n}:\alpha_{t}(\nu_{j}(x))=0\\},$

for $t\geqslant 0$. Then, we have $x=\prod_{t\geqslant
0}X_{\mathbb{J}_{t}(x)}^{2^{t}}.$

######  Definition 2.1.

For a monomial $x$ in $\mathcal{P}_{n}$, define two sequences associated with
$x$ by

$\displaystyle\omega(x)=(\omega_{1}(x),\omega_{2}(x),\ldots,\omega_{i}(x),\ldots),\
\ \sigma(x)=(\nu_{1}(x),\nu_{2}(x),\ldots,\nu_{n}(x)),$

where $\omega_{i}(x)=\sum_{1\leqslant j\leqslant
n}\alpha_{i-1}(\nu_{j}(x))=\deg X_{\mathbb{J}_{i-1}(x)},\ i\geqslant 1.$ The
sequence $\omega(x)$ is called the weight vector of $x$.

Let $\omega=(\omega_{1},\omega_{2},\ldots,\omega_{i},\ldots)$ be a sequence of
non-negative integers. The sequence $\omega$ is called the weight vector if
$\omega_{i}=0$ for $i\gg 0$. For a weight vector $\omega$, we define
$\deg\omega=\sum_{i>0}2^{i-1}\omega_{i}$.

The sets of all the weight vectors and the exponent vectors are given the left
lexicographical order.

Denote by $\mathcal{P}_{n}(\omega)$ the subspace of $\mathcal{P}_{n}$ spanned
by all monomials $y$ such that $\deg y=\deg\omega$,
$\omega(y)\leqslant\omega$, and by $\mathcal{P}_{n}^{-}(\omega)$ the subspace
of $\mathcal{P}_{n}$ spanned by all monomials $y\in\mathcal{P}_{n}(\omega)$
such that $\omega(y)<\omega$.

######  Definition 2.2.

Let $\omega$ be a weight vector and $u,\ v$ two polynomials of the same degree
in $\mathcal{P}_{n}.$ We define the equivalence relations $``\equiv"$ and
$``\equiv_{\omega}"$ on $\mathcal{P}_{n}$ by stating that

${\rm(i)}$ $u\equiv v$ if and only if $u-v\in\mathcal{A}^{+}\mathcal{P}_{n}$.

${\rm(ii)}$ $u\equiv_{\omega}v$ if and only if $u,v\in\mathcal{P}_{n}(\omega)$
and
$u-v\in\big{(}\mathcal{A}^{+}\mathcal{P}_{n}\cap\mathcal{P}_{n}(\omega)+\mathcal{P}_{n}^{-}(\omega)\big{)}$
denoting the set of all the $\mathcal{A}$-decomposable elements, in which
$Sq^{2^{r}}:\mathcal{P}_{n-2^{r}}\to\mathcal{P}_{n},$ and $\mathcal{A}^{+}$ is
the kernel of an epimorphism $\mathcal{A}\to\mathbb{F}_{2}$ of graded
$\mathbb{F}_{2}$-algebras. If $u$ is a linear combination of elements in the
images of the Steenrod squaring operations $Sq^{2^{r}}$ for $r>0,$ then $u$ is
called decomposable. Also, $u$ is said to be $\omega$-decomposable if
$u\equiv_{\omega}0.$

Then, we have an $\mathbb{F}_{2}$-qoutient space of $\mathcal{P}_{n}$ by the
equivalence relation $``\equiv_{\omega}"$ as follows

$Q\mathcal{P}_{n}(\omega)=\mathcal{P}_{n}(\omega)/((\mathcal{A}^{+}\mathcal{P}_{n}\cap\mathcal{P}_{n}(\omega))+\mathcal{P}_{n}^{-}(\omega)).$

######  Definition 2.3.

Let $u,\ v$ be monomials of the same degree in $\mathcal{P}_{n}$. We say that
$u<v$ if and only if one of the following holds:

${\rm(i)}$ $\omega(u)<\omega(v)$;

${\rm(ii)}$ $\omega(u)=\omega(v),$ and and $\sigma(u)<\sigma(v).$

######  Definition 2.4.

A monomial $u$ is said to be inadmissible if there exist monomials
$v_{1},v_{2},\ldots,v_{m}$ such that $v_{i}<u$ for $i=1,2,\ldots,m$ and
$u-\sum_{t=1}^{m}v_{i}\in\mathcal{A}^{+}\mathcal{P}_{n}.$

We say $u$ is admissible if it is not inadmissible. From these events, it is
important to remark that the set of all the admissible monomials of degree $d$
in $\mathcal{P}_{n}$ is a minimal set of $\mathcal{A}$-generators for
$\mathcal{P}_{n}$ in degree $d.$ And therefore,
$(\mathbb{F}_{2}{\otimes}_{\mathcal{A}}\mathcal{P}_{n})_{d}$ is an
$\mathbb{F}_{2}$-vector space with a basis consisting of all the classes
represent by the elements in $(\mathcal{P}_{n})_{d}.$

######  Definition 2.5.

Let $u\in\mathcal{P}_{n}.$ We say $u$ is strictly inadmissible if and only if
there exist monomials $v_{1},v_{2},\ldots,v_{m}$ such that $v_{j}<u,$ for
$j=1,2,\ldots,m$ and $u=\sum_{j=1}^{m}v_{j}+\sum_{i=1}^{2^{s}-1}Sq^{i}(f_{i})$
with $s=\max\\{k:\omega_{k}(u)>0\\},$ $f_{i}\in\mathcal{P}_{n}.$

It is easy to check that if $u$ is strictly inadmissible monomial, then it is
inadmissible monomial.

######  Theorem 2.6 (Kameko [10], Sum [26]).

Suppose that $u,v,w\in\mathcal{P}_{n}$ satisfying the conditions
$\omega_{t}(u)=0$ if $t>k>0$, $\omega_{r}(w)\neq 0$ and $\omega_{t}(w)=0$ if
$t>r>0$. Then,

(i) $uw^{2^{k}}$ is inadmissible if $w$ is inadmissible.

(ii) $wv^{2^{r}}$ is strictly inadmissible if $w$ is strictly inadmissible.

######  Definition 2.7.

Let $z=x_{1}^{d_{1}}x_{2}^{d_{2}}\ldots x_{n}^{d_{n}}\in\mathcal{P}_{n}.$ The
monomial $z$ is called a spike if $d_{j}=2^{t_{j}}-1$ for $t_{j}$ a non-
negative integer and $j=1,2,\ldots,n$. Moreover, $z$ is called the minimal
spike, if it is a spike such that $t_{1}>t_{2}>\ldots>t_{r-1}\geqslant
t_{r}>0$ and $t_{j}=0$ for $j>r.$

The following is a Singer’s criterion on the hit monomials in
$\mathcal{P}_{n}.$

######  Theorem 2.8 (Singer [21]).

Assume that $z$ is the minimal spike of degree $d$ in $\mathcal{P}_{n},$ and
$u\in(\mathcal{P}_{n})_{d}$ satisfying the condition $\mu(d)\leqslant n.$ If
$\omega(u)<\omega(z),$ then $u$ is hit.

We will denote by $\mathcal{P}_{n}^{0}$ and $\mathcal{P}_{n}^{+}$ the
$\mathcal{A}$-submodules of $\mathcal{P}_{n}$ spanned all the monomials
$x_{1}^{d_{1}}x_{2}^{d_{2}}\ldots x_{n}^{d_{n}}$ such that $d_{1}\ldots
d_{n}=0,$ and $d_{1}\ldots d_{n}>0,$ respectively. It is easy to see that
$\mathcal{P}_{n}^{0}$ and $\mathcal{P}_{n}^{+}$ are the
$\mathcal{A}$-submodules of $\mathcal{P}_{n}.$ Then, we have a direct summand
decomposition of the $\mathbb{F}_{2}$-vector spaces:

$\mathbb{F}_{2}{\otimes}_{\mathcal{A}}\mathcal{P}_{n}=(\mathbb{F}_{2}{\otimes}_{\mathcal{A}}\mathcal{P}_{n}^{0})\oplus(\mathbb{F}_{2}{\otimes}_{\mathcal{A}}\mathcal{P}_{n}^{+}).$

From now on, let us denote by $\mathcal{C}_{d}^{\otimes n}$ the set of all
admissible monomials of degree $d$ in $\mathcal{P}_{n}.$ For
$f\in\mathcal{P}_{n},$ we denote by $[f]$ the class in
$\mathbb{F}_{2}{\otimes}_{\mathcal{A}}\mathcal{P}_{n}$ represented by $f.$
Denote by $|S|$ the cardinal of a set $S.$

##  3\. Statement of the main results

In this section, we list the main results of this paper, proofs of the main
results will be presented in the next section. First, we study the hit problem
for $\mathcal{P}_{5}$ in the generic degree $k_{s}:=5(2^{s}-1)+13.2^{s}$ with
$s$ an arbitrary non-negative integer.

For $s=0,$ we have $k_{0}=5(2^{0}-1)+13.2^{0}=13.$ Then,
$\dim(\mathbb{F}_{2}{\otimes}_{\mathcal{A}}\mathcal{P}_{5})_{13}=250.$ This
result has been computed in [17] by explicitly determining all admissible
monomials of $\mathcal{P}_{5}$ in degree $13$ (see Phuc [17]).

Moreover, for $s=1,$ $k_{1}=5(2^{1}-1)+13.2^{1}=31,$ the following result has
been shown in [18] by Phuc.

######  Theorem 3.1 (Phuc [18]).

There exist exactly $866$ admissible monomials of degree $31$ in
$\mathcal{P}_{5}.$ Consequently,
$\dim(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{5})_{31}=866.$

One of our main contributions in this article is to study the
$\mathbb{F}_{2}$-graded vector space
$(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{5})_{5(2^{s}-1)+13.2^{s}},$
where $s$ is any positive integer greater than one.

For $s=2,$ we have $k_{2}=5(2^{2}-1)+13.2^{2}=67.$ Since
$\mathcal{P}_{n}=\oplus_{d\geqslant 0}(\mathcal{P}_{n})_{d}$ is the graded
polynomial algebra, and Kameko’s homomorphism

$(\widetilde{Sq}^{0}_{*})_{(5;67)}:(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{5})_{67}\longrightarrow(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{5})_{31}$

is an $\mathbb{F}_{2}$-epimorphism, it follows that

$(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{5})_{67}\cong(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{5}^{0})_{67}\bigoplus\big{(}\text{\rm
Ker}(\widetilde{Sq}^{0}_{*})_{(5;67)}\cap(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{5}^{+})_{67}\big{)}\bigoplus\text{Im}(\widetilde{Sq}^{0}_{*})_{(5;67)}$

Consider the homomorphism $\Phi:\mathcal{P}_{5}\to\mathcal{P}_{5}$ is an
$\mathbb{F}_{2}$-homomorphism determined by
$\Phi(x)=\prod_{i=1}^{5}x_{i}x^{2},$ for $x\in\mathcal{P}_{5}.$ Suppose that
$\mathcal{C}_{31}^{\otimes 5}=\\{a_{i}\in\mathcal{P}_{5}:1\leqslant i\leqslant
866\\}.$ Then, we set

$\mathcal{D}_{Im}^{\otimes 5}=\\{[b_{i}]:\ b_{i}=\Phi(x),\text{ for all
}x\in\mathcal{C}_{31}^{\otimes 5}\\}.$

From this, we easily conclude the following theorem.

######  Theorem 3.2.

$\text{Im}(\widetilde{Sq}^{0}_{*})_{(5;67)}$ is isomorphic to a subspace of
$(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{5})_{67}$ generated by all
the classes of the form $[b_{i}]$, with $1\leqslant i\leqslant 866.$ That
means, the set $\mathcal{D}_{Im}^{\otimes 5}$ is a basis of the
$\mathbb{F}_{2}$-vector space $\text{Im}(\widetilde{Sq}^{0}_{*})_{(5;67)}.$

For any $1\leqslant t\leqslant 5$, consider the homomorphism
$\mathcal{T}_{t}:\mathcal{P}_{4}\to\mathcal{P}_{5}$ of algebras by
substituting

$\mathcal{T}_{t}(x_{i})=\begin{cases}x_{i},&\text{ if }1\leqslant i<t,\\\
x_{i+1},&\text{ if }t\leqslant i<5.\end{cases}$

Then, $\mathcal{T}_{t}$ is a homomorphism of $\mathcal{A}$-modules. Put
$\mathcal{D}_{0}^{\otimes 5}:=\\{c_{i}:\
c_{i}\in\bigcup_{t=1}^{5}\mathcal{T}_{t}(\mathcal{C}_{67}^{\otimes 4})\\}.$

Using the result in Sum [26] (see Theorem 1.4), an easy computation shows that
$|\mathcal{D}_{0}^{\otimes 5}|=460.$ More specifically, we have the following.

######  Theorem 3.3.

The set $\\{[c_{i}]:\ 1\leqslant i\leqslant 460\\}$ is a basis of the
$\mathbb{F}_{2}$-vector space
$(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{5}^{0})_{5(2^{2}-1)+13.2^{2}}.$
Consequently,
$\dim(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{5}^{0})_{5(2^{2}-1)+13.2^{2}}=460.$

######  Remark 3.4.

We recall a result in Mothebe-Kaelo-Ramatebele [13] as follows:

Set $\mathcal{N}_{(n,t)}=\\{I=(i_{1},i_{2},\ldots,i_{t}):1\leqslant
i_{1}<\ldots<i_{t}\leqslant n\\}$, $1\leqslant t<n$. For
$I\in\mathcal{N}_{(n,t)}$, define the homomorphism
$f_{I}:\mathcal{P}_{t}\to\mathcal{P}_{n}$ of algebras by substituting
$f_{I}(x_{\ell})=x_{i_{\ell}}$ with $1\leqslant\ell\leqslant t$. Then, $f_{I}$
is a monomorphism of $\mathcal{A}$-modules. We have a direct summand
decomposition of the $\mathbb{F}_{2}$-vector subspaces:

$\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{n}^{0}=\bigoplus\limits_{1\leqslant
t\leqslant
n-1}\bigoplus\limits_{I\in\mathcal{N}_{(n,t)}}(Qf_{I}(\mathcal{P}_{t}^{+})),$

where
$Qf_{I}(\mathcal{P}_{t}^{+})=\mathbb{F}_{2}\otimes_{\mathcal{A}}f_{I}(\mathcal{P}_{t}^{+})$.

Hence,
$\dim(Qf_{I}(\mathcal{P}_{t}^{+}))_{d}=\dim(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{t}^{+})_{d}$
and $|\mathcal{N}_{(n,t)}|=\binom{n}{t}$. Combining with the results in Wood
[34], one gets

$\dim(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{n}^{0})_{d}=\sum\limits_{\mu(d)\leqslant
t\leqslant
n-1}\binom{n}{t}\dim(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{t}^{+})_{d}.$

Since $\mu(67)=3,$ it folows that if $t<3$ then the spaces
$(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{t}^{+})_{67}$ are trivial.
Moreover, by using the results in Kameko [10] and Sum [26], we have

$\dim(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{t}^{+})_{67}=\begin{cases}14,&\text{if
}\ t=3,\\\ 64,&\text{if }\ t=4.\end{cases}$

From this, we get

$\dim(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{5}^{0})_{67}=\binom{5}{3}\dim(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{3}^{+})_{67}+\binom{5}{4}\dim(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{4}^{+})_{67}=460.$

Next, we explicitly determine the vector space $\text{\rm
Ker}(\widetilde{Sq}^{0}_{*})_{(5;67)}\cap(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{5}^{+})_{67}.$
We will denote by
$Q\mathcal{P}_{n}^{+}(\omega)=Q\mathcal{P}_{n}(\omega)\cap(\mathbb{F}_{2}{\otimes}_{\mathcal{A}}\mathcal{P}_{n}^{+}).$
Then, we have the following theorem.

######  Theorem 3.5.

Let $\widetilde{\omega_{1}}:=(3,4,2,2,2),$
$\widetilde{\omega_{2}}:=(3,4,4,3,1),$ and
$\widetilde{\omega_{3}}:=(3,2,1,1,1).$

(i) Suppose $u$ is an element of $(\mathcal{C}_{67}^{\otimes
5}\cap\mathcal{P}_{5}^{+})$ such that $[u]$ does not belong to $\text{\rm
Im}(\widetilde{Sq}^{0}_{*})_{(5;67)},$ then $\omega(u)$ is one of the
following sequences: $\widetilde{\omega_{m}}$, with $m=1,2,3.$ Moreover, we
have an isomorphism of the $\mathbb{F}_{2}$-vector spaces:

$\text{\rm
Ker}(\widetilde{Sq}^{0}_{*})_{(5;67)}\cap(\mathbb{F}_{2}{\otimes}_{\mathcal{A}}\mathcal{P}_{5}^{+})_{67}\cong\bigoplus_{m=1}^{3}Q\mathcal{P}_{5}^{+}(\widetilde{\omega_{m}}).$

(ii) We have $\dim\text{\rm
Ker}(\widetilde{Sq}^{0}_{*})_{(5;67)}\cap(\mathbb{F}_{2}{\otimes}_{\mathcal{A}}\mathcal{P}_{5}^{+})_{67}=161.$

From the results of Theorems 3.2, 3.3, and 3.5, we obtain the following
corollary.

######  Corollary 3.6.

There exist exactly $1487$ admissible monomials of degree $67$ in
$\mathcal{P}_{5}.$ Consequently,
$\dim(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{5})_{5(2^{2}-1)+13.2^{2}}=1487.$

Consider the degree $k_{s}=5(2^{s}-1)+13.2^{s},$ for any $s\geqslant 3.$ We
set

$\xi(n;d)=\max\\{0,n-\alpha(d+n)-\zeta(d+n)\\},$

where $\zeta(u)$ the greatest integer $v$ such that $u$ is divisible by
$2^{v}.$ We recall a result in [29] the following.

######  Theorem 3.7 (Tin-Sum [29]).

Let $d$ be an arbitrary non-negative integer. Then

$(\widetilde{Sq}^{0}_{*})^{r-t}:(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{n})_{n(2^{r}-1)+2^{r}d}\longrightarrow(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{n})_{n(2^{t}-1)+2^{t}d}$

is an isomorphism of $GL(n;\mathbb{F}_{2})$-modules for every $r\geqslant t$
if and only if $t\geqslant\xi(n;d).$

As an application of Theorem 3.7, we get the following theorem.

######  Theorem 3.8.

The set
$\big{\\{}[x]:x\in\Phi^{s-2}\big{(}\mathcal{C}_{5(2^{2}-1)+13.2^{2}}^{\otimes
5}\big{)}\big{\\}}$ is a basis of the $\mathbb{F}_{2}$-vector space
$(\mathbb{F}_{2}{\otimes}_{\mathcal{A}}\mathcal{P}_{5})_{5(2^{s}-1)+13.2^{s}}$,
for any $s>2.$ This implies that the $\mathbb{F}_{2}$-vector space
$(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{5})_{5(2^{s}-1)+13.2^{s}}$
has dimension $1487,$ for any $s>2.$

Now, we describe the next main results by studying the $\mathbb{F}_{2}$-graded
vector space $\mathbb{F}_{2}{\otimes}_{\mathcal{A}}\mathcal{P}_{n}$ in some
generic degrees of the form ${(n-1)(2^{r+4}-1)+m.2^{r+4}},$ where $r$ is an
arbitrary positive integer, $m\in\\{23,67\\},$ and in the case $n=6.$

As is well known, after explicitly determining
$\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{4},$ Sum [26] has established
an inductive formula by $n$ for the dimension of the vector space
$(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{n})_{d},$ where $d$ is of
general degree (see Theorem 1.3).

From the results in Sum [26], combined with the above results we get the
following theorem.

######  Theorem 3.9.

The vector space
$(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{6})_{5(2^{s+4}-1)+67.2^{s+4}}$
is $93681$-dimensional, for all positive integers $s.$

On the other hand, based on the results in [31] for the hit problem of five
variables, we also obtain the following.

######  Theorem 3.10.

For any integer $r>0$, there exist exactly $78435$ admissible monomials of
degree $5.2^{r+4}-23.2^{r+4}$ in $\mathcal{P}_{6}$. Consequently,
$\dim(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{6})_{5.2^{r+4}-23.2^{r+4}}=78435,$
for all $r>0.$

##  4\. Proofs of main results

###  4.1. Proof of Theorem 3.5

We first prove Part (i) of the theorem. Let us denote by

$Q\mathcal{P}_{5}^{\omega}:=\text{ Span
}\\{[x]\in\mathbb{F}_{2}{\otimes}_{\mathcal{A}}\mathcal{P}_{5}:x\text{ \rm is
admissible and }\omega(x)=\omega\\}.$

Using the results in Walker-Wood [32], we see that

$(\mathbb{F}_{2}{\otimes}_{\mathcal{A}}\mathcal{P}_{5})_{d}=\bigoplus_{\deg\omega=d}Q\mathcal{P}_{5}^{\omega}\cong\bigoplus_{\deg\omega=d}Q\mathcal{P}_{5}(\omega),$
(4.1)

Since $\mathcal{P}_{n}=\oplus_{d\geqslant 0}(\mathcal{P}_{n})_{d}$ is the
graded polynomial algebra, combined with the above results, it follows that
$(\mathbb{F}_{2}{\otimes}_{\mathcal{A}}\mathcal{P}_{5}^{+})_{67}=\bigoplus_{\deg\omega=67}Q\mathcal{P}_{5}^{+}(\omega).$

Suppose that $u$ is an admissible monomial of degree $67$ in
$\mathcal{P}_{5}^{+}$ such that $[u]$ belongs to $\text{\rm
Ker}(\widetilde{Sq}^{0}_{*})_{(5;67)}.$ It is easy to check that
$z=x_{1}^{63}x_{2}^{3}x_{3}$ is the minimal spike of degree $67$ in
$\mathcal{P}_{5}$ and $\omega(z)=\widetilde{\omega_{3}}.$ Using Theorem 2.8,
we obtain $\omega_{1}(u)\geqslant\omega_{1}(z)=3$. Since the degree of $u$ is
odd, it follows that either $\omega_{1}(u)=3$ or $\omega_{1}(u)=5.$

If $\omega_{1}(u)=5,$ then $u=\prod_{i=1}^{5}x_{i}v^{2}$ with $v$ a monomial
in $(\mathcal{P}_{5})_{31}.$ By Theorem 2.6, $v$ is an admissible monomial and
$[v]\not=0$. Thus, $[v]=\text{\rm
Ker}(\widetilde{Sq}^{0}_{*})_{(5;67)}([u])\not=0.$ This contradicts the fact
that $[u]$ belongs to $\text{\rm Ker}(\widetilde{Sq}^{0}_{*})_{(5;67)}.$ From
this, one gets $\omega_{1}(u)=3.$ Then, we have $u=x_{i}x_{j}x_{\ell}y^{2}$
with $1\leqslant i<j<\ell\leqslant 5,$ where $y\in(\mathcal{P}_{5})_{32}.$
Using Theorem 2.6, $y$ is also admissible. By a simple computation shows that
$\omega(y)=(4,4,3,1)$ or $\omega(y)=(4,2,2,2)$ or $\omega(y)=(2,1,1,1,1).$ And
therefore, we get $\omega(u)=\widetilde{\omega_{m}}$, with $m=1,2,3.$

From these above, combined with the aid of (4.1), we have a direct summand
decomposition of the $\mathbb{F}_{2}$-vector spaces:

$\text{\rm
Ker}(\widetilde{Sq}^{0}_{*})_{(5;67)}\cap(\mathbb{F}_{2}{\otimes}_{\mathcal{A}}\mathcal{P}_{5}^{+})_{67}=\bigoplus_{m=1}^{3}Q\mathcal{P}_{5}^{+}(\widetilde{\omega_{m}}).$

Part (i) is proved.

The proof of Part (ii) of the above theorem is too long and computationally
very technical. We sketch its proof as follows:

Consider the set $B_{67}^{\otimes 5}:=\\{X_{\\{i,j\\}}.F^{2}:1\leqslant
i<j\leqslant 5,F\in\mathcal{C}_{32}^{\otimes 5}\\}.$ By using Theorem 2.6, we
see that if $X$ is an admissible monomial of degree $67$ in $\mathcal{P}_{5}$,
then $X\in B_{67}^{\otimes 5}.$ Putting

$\mathcal{C}_{67}^{\otimes 5}(\omega):=\\{X\in B_{67}^{\otimes 5}:X\text{
admissible monomial, and }\omega(X)=\widetilde{\omega_{m}},\text{ for }\
m=1,2,3\\}.$

By direct calculations, we filter out and eliminate the inadmissible monomials
in $B_{67}^{\otimes 5},$ one gets $|\mathcal{C}_{67}^{\otimes
5}(\omega)|=161.$

Note that the result dimension of $\text{\rm
Ker}(\widetilde{Sq}^{0}_{*})_{(5;67)}\cap(\mathbb{F}_{2}{\otimes}_{\mathcal{A}}\mathcal{P}_{5}^{+})_{67}$
has been verified by using a computer calculation program in SAGE by H-V Vu.
We would like to say thank you for his support.

###  4.2. Proof of Theorem 3.8

It is easy to check that for $n=5$ and $d=67$ then $\alpha(d+n)=\alpha(72)=3,$
and $\zeta(d+n)=\zeta(72)=3,$ and therefore $\xi(n;d)=0.$ By Theorem 3.7, we
get an isomorphism of $\mathbb{F}_{2}$-graded vector spaces:

$(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{5})_{5(2^{r}-1)+2^{r}.67}\cong(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{5})_{5(2^{0}-1)+2^{0}.67}$

for all $r\geqslant 0.$

And therefore, we obtain

$(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{5})_{5(2^{s}-1)+13.2^{s}}\cong(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{5})_{5(2^{2}-1)+13.2^{2}}\
\text{for all }s>2.$

That means,
$\dim(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{5})_{5(2^{s}-1)+13.2^{s}}=\dim(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{5})_{5(2^{2}-1)+13.2^{2}}=1487,$
for all $s>2.$

Moreover, for every $s>2,$ the set
$\big{\\{}[x]:x\in\Phi^{s-2}\big{(}\mathcal{C}_{5(2^{2}-1)+13.2^{2}}^{\otimes
5}\big{)}\big{\\}}$ is a basis of the $\mathbb{F}_{2}$-vector space
$(\mathbb{F}_{2}{\otimes}_{\mathcal{A}}\mathcal{P}_{5})_{5(2^{s}-1)+13.2^{s}}.$

###  4.3. Proof of Theorems 3.9 and 3.10

Consider the degree $d=(n-1)(2^{r}-1)+2^{r}m,$ with $r,m$ positive integers
such that $1\leqslant n-3\leqslant\mu(m)\leqslant n-2.$ If $r\geqslant n-1,$
then we have the following result, which was shown in Sum [26].

$\dim(\mathbb{F}_{2}{\otimes}_{\mathcal{A}}\mathcal{P}_{n})_{d}=(2^{n}-1)\dim(\mathbb{F}_{2}{\otimes}_{\mathcal{A}}\mathcal{P}_{n-1})_{m}.$

For $n=6,$ and $m=67,$ we can easily see that
$\mu(67)=3=\alpha(67+\mu(67))=\alpha(70).$ Hence, using the above result, one
gets

$\big{|}\mathcal{C}_{5(2^{r}-1)+67.2^{r}}^{\otimes
6}\big{|}=63.\big{|}\mathcal{C}_{67}^{\otimes 5}\big{|}=93681,\ \text{for any
integer}\ r\geqslant n-1=5.$

So, there exist exactly $93681$ admissible monomials of degree
$5(2^{r}-1)+67.2^{r}$ in $\mathcal{P}_{6},$ for any integer $r>4.$
Consequently,
$\dim(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{6})_{5(2^{s+4}-1)+67.2^{s+4}}=93681,$
for all positive integers $s.$ Theorem 3.9 is proved.

Moreover, Tin [31] showed that
$(\mathbb{F}_{2}{\otimes}_{\mathcal{A}}\mathcal{P}_{5})_{5(2^{r}-1)+9.2^{r}}$
is an $\mathbb{F}_{2}$-vector space of dimension 1245, for any integer $r>0$.
Consequently, $\big{|}\mathcal{C}_{5(2^{r}-1)+9.2^{r}}^{\otimes
5}\big{|}=1245,$ for any integer $r>0.$

Consider the degree $5(2^{u}-1)+23.2^{u}.$ It is easy to check that

$\mu(23)=\alpha(23+\mu(23))=\alpha(26)=n-3=3.$

And therefore, using the result in Sum [26] (see Theorem 1.3), we get

$\big{|}\mathcal{C}_{5(2^{u}-1)+23.2^{u}}^{\otimes
6}\big{|}=(2^{6}-1)\big{|}\mathcal{C}_{23}^{\otimes 5}\big{|},\ \text{for any
integer}\ u\geqslant 5.$

This implies that the $\mathbb{F}_{2}$-graded vector space
$(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{6})_{5(2^{v+4}-1)+23.2^{v+4}}$
has dimension $78435,$ for any integer $v>0.$ Theorem 3.10 is proved.

##  5\. On behavior of the Singer algebraic transfer

We first recall the definition of the Singer algebraic transfer, which is a
homomorphism

$\psi_{n}:\mathbb{F}_{2}{\otimes}_{GL(n;\mathbb{F}_{2})}PH_{*}((\mathbb{R}\mathcal{P}^{\infty})^{n})\longrightarrow
Ext^{n,n+*}_{\mathcal{A}}(\mathbb{F}_{2},\mathbb{F}_{2}).$

Here,
$(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{n})_{d}^{GL(n;\mathbb{F}_{2})}$
is the subspace of $(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{n})_{d}$
consisting of all the $GL(n;\mathbb{F}_{2})$-invariant classes of degree $d,$
and
$\mathbb{F}_{2}{\otimes}_{GL(n;\mathbb{F}_{2})}PH_{d}((\mathbb{R}\mathcal{P}^{\infty})^{n})$
be dual to
$(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{n})_{d}^{GL(n;\mathbb{F}_{2})}.$

Noting that in the dual case, we also have an algebraic homomorphism called
Singer’s algebraic transfer, $Tr_{n}:=(\psi_{n})^{*},$ which is a homomorphism
from the homology of the Steenrod algebra,
$\text{Tor}^{\mathcal{A}}_{n,n+d}(\mathbb{F}_{2},\mathbb{F}_{2}),$ to the
subspace of $(\mathbb{F}_{2}{\otimes}_{\mathcal{A}}\mathcal{P}_{n})_{d}$
consisting of all the $GL(n;\mathbb{F}_{2})$-invariant classes (see Singer
[21]).

In [27], and [31], we based on the the results for the hit problem to study
and verify the Singer conjecture for the algebraic transfer in degrees
$5(2^{s}-1)+2^{s}m,$ for $s$ an arbitrary positive integer, and
$m\in\\{1,2,3\\}$, we obtain the following theorem.

######  Theorem 5.1.

Let $s$ be an arbitrary positive integer. Singer’s conjecture is true for
$n=5$ and the generic degrees $d_{s}=5(2^{s}-1)+2^{s}m,$ where
$m\in\\{1,2,3\\}.$

In the current article, by using Theorem 3.7, we also see that

$(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{5})_{5(2^{s}-1)+13.2^{s}}^{GL(5;\mathbb{F}_{2})}\cong(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{5})_{5(2^{2}-1)+13.2^{2}}^{GL(5;\mathbb{F}_{2})},\
\text{ for all }s>1.$

And therefore, we get

$\mathbb{F}_{2}{\otimes}_{GL(5;\mathbb{F}_{2})}PH_{5(2^{s}-1)+13.2^{s}}((\mathbb{R}\mathcal{P}^{\infty})^{5})\cong\mathbb{F}_{2}{\otimes}_{GL(5;\mathbb{F}_{2})}PH_{5(2^{2}-1)+13.2^{2}}((\mathbb{R}\mathcal{P}^{\infty})^{5}),$

for all $s>1.$ Hence, we need only to compute the dimension of vector space
$\mathbb{F}_{2}{\otimes}_{GL(5;\mathbb{F}_{2})}PH_{5(2^{s}-1)+13.2^{s}}((\mathbb{R}\mathcal{P}^{\infty})^{5})$
for $s=2.$

Noting that for $s=1,$ then
$\dim\big{(}(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{5})_{5(2^{1}-1)+13.2^{1}}^{GL(5;\mathbb{F}_{2})}\big{)}=2$
(see Phuc [18]). Moreover, since Kameko’s homomorphism

$(\widetilde{Sq}^{0}_{*})_{(5;67)}:(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{5})_{5(2^{2}-1)+13.2^{2}}\longrightarrow(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{5})_{5(2^{1}-1)+13.2^{1}}$

is also an $GL(5;\mathbb{F}_{2})$-epimorphism, it follows that

$\dim\big{(}\mathbb{F}_{2}{\otimes}_{GL(5;\mathbb{F}_{2})}PH_{5(2^{s}-1)+13.2^{s}}((\mathbb{R}\mathcal{P}^{\infty})^{5})\big{)}\leqslant\dim(\widetilde{Sq}^{0}_{*})^{GL(5;\mathbb{F}_{2})}_{(5;67)}+2,\text{
for all }s>1.$

Remarkably, for any $s>1,$ all elements of
$\mathbb{F}_{2}{\otimes}_{GL(5;\mathbb{F}_{2})}PH_{5(2^{s}-1)+13.2^{s}}((\mathbb{R}\mathcal{P}^{\infty})^{5})$
are of the form
$\big{(}\xi[\Phi(\upsilon)]+\gamma[\Phi(\beta)]+[f]\big{)}^{*},$ where
$\xi,\gamma\in\mathbb{F}_{2},$ $\upsilon$ and $\beta$ are two generators of
$(\mathbb{F}_{2}\otimes_{\mathcal{A}}\mathcal{P}_{5})_{5(2^{1}-1)+13.2^{1}}^{GL(5;\mathbb{F}_{2})},$
and $f\in(\mathcal{P}_{5})_{5(2^{2}-1)+13.2^{2}}$ such that $[f]$ belongs to
$\text{\rm Ker}(\widetilde{Sq}^{0}_{*})_{(5;67)}.$ By using techniques of the
hit problem, combining the computation of $\text{\rm
Ext}_{\mathcal{A}}^{n,n+*}(\mathbb{Z}_{2},\mathbb{Z}_{2})$ by Lin [12] and
Chen [4], we will describle explicitly all these elments, and verify the
Singer’s conjecture for the fifth algebraic transfer in the near future.

Acknowledgment. I would like to express my warmest thanks to my adviser, Prof.
Nguyen Sum (Sai Gon University) for helpful conversations. I also thank Dr.
Dang Vo Phuc (Khanh Hoa University) for a helpful discussion. This research is
supported by Ho Chi Minh City University of Technology and Education (HCMUTE),
Vietnam.

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# Homogeneous Online Transfer Learning with Online Distribution Discrepancy
Minimization

Yuntao Du Nanjing University,China<EMAIL_ADDRESS><EMAIL_ADDRESS>Nanjing University,China,
<EMAIL_ADDRESS>University,China,
<EMAIL_ADDRESS>University,China, email<EMAIL_ADDRESS>Zhiwen Tan Nanjing<EMAIL_ADDRESS>Nanjing
University,China<EMAIL_ADDRESS>University,China,
<EMAIL_ADDRESS>University,China, email<EMAIL_ADDRESS>Qian Chen Nanjing University,China<EMAIL_ADDRESS>University,China<EMAIL_ADDRESS>University,China, email:
<EMAIL_ADDRESS>
Yi Zhang Nanjing University,China<EMAIL_ADDRESS>University,China, email<EMAIL_ADDRESS>Chongjun Wang Nanjing
University,China, email<EMAIL_ADDRESS>
###### Abstract

Transfer learning has been demonstrated to be successful and essential in
diverse applications, which transfers knowledge from related but different
source domains to the target domain. Online transfer learning(OTL) is a more
challenging problem where the target data arrive in an online manner. Most OTL
methods combine source classifier and target classifier directly by assigning
a weight to each classifier, and adjust the weights constantly. However, these
methods pay little attention to reducing the distribution discrepancy between
domains. In this paper, we propose a novel online transfer learning method
which seeks to find a new feature representation, so that the marginal
distribution and conditional distribution discrepancy can be online reduced
simultaneously. We focus on online transfer learning with multiple source
domains and use the Hedge strategy to leverage knowledge from source domains.
We analyze the theoretical properties of the proposed algorithm and provide an
upper mistake bound. Comprehensive experiments on two real-world datasets show
that our method outperforms state-of-the-art methods by a large margin.

##  1 Introduction

Transfer learning[11] is able to transfer knowledge from labeled source
domains to the target domain. In transfer learning problems, the source domain
and the target domain have different distributions. The distribution
discrepancy between source domain and target domain includes marginal
distribution discrepancy and conditional distribution discrepancy[10, 9, 16].
In view of this challenge, many transfer learning methods are designed to
reduce the distribution discrepancy between domains so that the transfer
performance can be improved [10, 9, 16]. Previous work explored various
techniques for statistics matching, e.g, maximum mean discrepancy(MMD)[10, 9,
16], Correlation Alignment (CORAL)[15] and have achieved remarkable effect.

Most existing transfer learning methods focus on batch learning manners, where
all the source and target data are given in advance. However, this assumption
may not hold in real-world applications, it is expensive to collect sufficient
data at one time. Moreover, in some situations, the instances arrive in a
sequential manner. Recently, Online transfer learning(OTL)[20] has attracted a
lot of attention in the community of machine learning. In this scenario, we
can collect many labeled source data in advance, but we receive the target
data in an online manner, i.e., we receive a target instance at each round.
OTL aims at preforming an online task in the target domain by leveraging
knowledge from some offline source domains. In online learning[7], we need to
train a classification model that can predict for the new arrival instance in
the target domain, then we will receive its true label, and we sequentially
update the existing model based on the loss information between the true label
and the predicted label.

Existing online transfer learning methods focus on how to perform online
learning in the target domain by leveraging knowledge from source domains [20,
5, 18]. They mostly adopt the ensemble learning based strategies, i.e.,
combining the source classifier and the target classifier directly. [20]
proposed to train a classifier with the help of the source domain, and adapted
the classifier to new coming data by constantly adjusting the combination
weights. On the basis of [20], a lot of works focus on multi-source online
transfer learning by using different technologies of multi-source combination,
for example, binary graph[5], selecting source domains adaptively[18]. Though
achieving exciting performance, many previous works have proven that when the
distribution discrepancy between domains is large, directly combining source
classifier and target classifier can not perform well and even lead to
negative transfer[11]. In other words, it is also crucial for online transfer
learning to reduce distribution discrepancy, which is ignored by existing
methods.

Table 1:  Notations and descriptions in this paper Notation | Description | Notation | Description
---|---|---|---
$\mathcal{D}_{S_{i}}$ |  $i$-th source domain data | $\mathcal{D}_{T}^{l},\mathcal{D}_{T}^{u}$ | labeled target domain data, unlabeled target domain data
$n_{S_{i}}$ | # $i$-th source domain data |  $n_{T}^{u}$, $n_{T}^{l}$ | # unlabeled target domain data / labeled target domain data
$n,K$ | # source domains / classes | $\bar{c}_{S_{i}}$ | the mean of the $i$-th source domain data
$A_{i}^{t}$ | transformation matrix of the $i$-th domain at round t | $\bar{c}_{T}$ | the mean of the target domain data
$f_{S_{i}}$ | the classifier of the $i$-th source domain | $\bar{c}_{S_{i}}^{k}$ | the mean of the $k$-th class of the $i$-th source domain data
$f_{T_{i}}$ | the $i$-th classifier of the target domain | $\bar{c}_{T}^{k}$ | the mean of the $k$-th class of the target domain data
$w_{i,t}^{k}$ | the $k$-th weight vector of the $i$-th classifier at round $t$ | $X_{s_{i}}^{p}$ | projected data of the $i$-th source domain data
$F_{t}^{k}$ | the $k$-th component of the final classifier at round t | $x_{t}^{p_{i}}$ | projected data of the $i$-th feature space for $x_{t}$
$u_{i}$ | the weight of the $i$-th source classifier | $T_{w}$ | time window
$v_{i}$ | the weight of the $i$-th target classifier | $\mu$ | regularization parameter

In this paper, we investigate online transfer learning with multiple domains
under homogeneous space, where there are some source domains, and the feature
space of source domains and target domain is the same. We use ensemble
learning based method i.e., we train a classifier in each domain then assign a
weight to each classifier and take advantage of the Hedge strategy[4] to
dynamically adjust the weight of each domain. For each source domain and the
target domain, to reduce domain discrepancy in an online manner, we propose a
novel online method, which jointly minimizes marginal distribution discrepancy
and conditional distribution discrepancy based on MMD in a linear feature
transformation procedure. A linear transformation matrix is used to project
the original source and target data onto a new space for each source domain
and the target domain. This learning process is named as _online stage_.

In the above method, we can just use a random matrix as an initialization
matrix, but it will drop the transfer performance because in the early peroid,
the learning process is performed in an unsuited space. We design an _offline
stage_ to meet this challenge. Unlabeled data are usually easy to collect in
the target domain, with these unlabeled data and source data, we can use some
classical offline MMD-based method, e.g, TCA[10], JDA[9], BDA[16] to get a
good initialization matrix for each source domain and the target domain. After
getting the transformation matrixes, we firstly project the original data to
get the new feature representation. Then, in each source domain, a classifier
is trained for transferring source knowledge.

In general, we propose an algorithm called Homogeneous Online Transfer
Learning with Online Distribution Discrepancy Minimization (HomOTL-ODDM),
which is consisted of the _offline stage_ and the _online stage_. At the
offline stage, an initialization transformation matrix and a classifier is
trained in each source domain. At the online stage, for each source and target
domain, an online classifier is trained in an online manner in new feature
space. The transformation matrixes and the weights of classifiers are also
updated. We theoretically analyze the proposed algorithm and provide an upper
mistake bound. Extensive experiments demonstrate that HomOTL-ODDM outperforms
state-of-the-art methods by a large margin.

To sum up, our contributions are mainly three-fold:

1) We propose a novel method for multiple source online transfer learning by
simultaneously reducing the marginal distribution and the conditional
distribution discrepancy between domains in a linear feature transformation
procedure.

2) We analyze the theoretical properties of the proposed algorithm and provide
an upper mistake bound.

3) We conduct extensive experiments on two real-world data sets and the
results validate the effectiveness of the proposed method.

##  2 Related work

Transfer Learning: The proposed method is mainly related to the feature-based
transfer learning methods. [10, 1, 8] try to correct for the marginal
distribution discrepancy between domains. [10] learns some transfer components
across domains in a Reproducing Kernel Hilbert Space (RKHS) using Maximum Mean
Discrepancy (MMD). A new parametric kernel is used to extract feature, which
can dramatically minimize the distance between domain marginal distribution by
projecting data onto the learned transfer components. [13] tries to correct
for the conditional distribution discrepancy between domains. Based on
measures of divergence between the two domains, it penalizes features with
large divergence, while improving the effectiveness of other less deviant
correlated features. [9] first tries to correct for both the marginal and
conditional distribution discrepancy simultaneously. Instead of treating these
two discrepancy equally, [16] adaptively leverages the importance of the
marginal and conditional distribution discrepancy. However, these methods can
not be used in online manner directly because they are designed for offline
manner. Our method is capable of reducing marginal and conditional
distribution discrepancy simultaneously in an online manner.

Online Learning:One of the most famous approaches is the Perceptron algorithm
[12], which updates the model by adding a new instance with some constant
weight into the current set of support vectors when the instance is
misclassified. Recently, many algorithms have been proposed based on maximum
margin [3, 14, 19]. One example is the Passive-Aggressive (PA) algorithm[3],
which updates the model when a new instance is misclassified or its
classification confidence is smaller than some predefined threshold.
Specially, it can not only be used for binary classification problems but also
for multi-class problems. More extensive surveys for online learning can be
found in [7].

Online Transfer Learning: Unlike conventional transfer learning, online
transfer learning assumes that labeled source data are available in advance
but the target data arrive sequentially. [20] proposes an online transfer
learning algorithm for single source domain under homogeneous and
heterogeneous space, which forms the final classifier by simply ensembling the
source and target classifiers with dynamical weights. On the basis of [20],
[5] proposes an online transfer learning algorithm based on multiple
homogeneous source domain, it assumes that the online target data is
unlabeled. On the premise that the target domain is labeled data, two online
transfer learning algorithms based on multiple homogeneous or heterogeneous
source domains are proposed in [18], which also constitute the final
classifier in an ensemble strategy. [17] researches on the application of
online transfer learning in forecasting advertising volume. Existing methods
have focused on how to transfer knowledge from source domains, however, they
pay little attention to how to reduce the distribution discrepancy between
domains. Instead, our method can reduce both together so that the transfer
performance is improved.

Figure 1:  Structure of HomOTL-ODDM algorithm. It is composed of an the
offline stage and an online stage. At the offline stage, $n$ source
classifiers and initial transformation matrixes are trained in each source
domain for knowledge transfer. At the online stage, the target data is
projected onto $n$ different feature space, and $n$ classifiers are trained in
these new space. Each classifier is assigned a weight and the Hedge strategy
is used to update the weights dynamically. Besides, we design an online method
to update the transform matrixes, which can reduce the marginal and
conditional distribution discrepancy between domains simultaneously.

##  3 problem definition

We begin with the definitions of terminologies, then frequently used notations
are summarized in Table 1.

In multi-source OTL, $n$ source domains
$D_{S}=\\{D_{S_{1}},D_{S_{2}},...,D_{S_{n}}\\}$ and a target domain $T$ are
given. Each source domain $D_{S_{i}}=\\{(x_{j},y_{j}),j=1,2,...,n_{S_{i}}\\}$
contains $n_{S_{i}}$ instances drawn from distribution $P_{S_{i}}(x)$, where
$x_{j}\in\mathcal{X}_{S_{i}}$ with associated labels
$y_{j}\in\mathcal{Y}_{S_{i}}$. Similarly, the target domain consists of
$n_{T}^{u}$ offline unlabeled instances drawn from distribution $P_{T}(x)$,
$D^{u}_{T}=\\{(x_{i},y_{i}),i=1,2,...,n_{T}^{u}\\}$, which are available in
advance as well as $n_{T}^{l}$ online labeled instances
$D^{l}_{T}=\\{(x_{j},y_{j}),j=1,2,...,n_{T}^{l}\\}$, which are sequentially
arriving in the target domain, where $x_{i},x_{j}\in\mathcal{X}_{T}$,
$y_{i},y_{j}\in\mathcal{Y}_{T}$.

In this paper, we focus on homogeneous online transfer learning, where
$\mathcal{X}_{S_{i}}=\mathcal{X}_{T}=R^{m}$,
$\mathcal{Y}_{S_{i}}=\mathcal{Y}_{T}=\\{1,2,...,K\\}$ for $\forall
i=1,2,...,n$. But the marginal distribution and the conditional distribution
between domains are different, i.e., $P_{s_{i}}(x)\neq
P_{t}(x),Q_{s_{i}}(y|x)\neq Q_{t}(y|x)$ for $\forall i=1,2,...,n$. We seek to
find a new feature representation for each source domain and the target domain
by utilizing transformation matrix $A_{i}\in R^{d\times m}$ for the $i$-th
domain, where $d$ is the dimension of the new feature representation.

For an online transfer learning (OTL) task, our goal is to online learn a
prediction function in the target domain from a sequence of labeled examples
$D_{T}^{l}$. Specifically, during the online stage, at the $t$-th trial of
online learning task, the learner receives an instance $x_{t}$. Firstly we use
the current model to predict the label of $x_{t}$, secondly we will receive
its true label, thirdly we sequentially update the current model in the target
domain based on the loss information between the true label and the predicted
label. The key challenge is how to find a good representation that can reduce
the distribution discrepancy between domains as well as how to effectively
transfer knowledge from source domains to the target domain.

###  3.1 Overall Framework

In this paper, we propose a method called Homogeneous Online Transfer Learning
with Online Distribution Discrepancy Minimization (HomOTL-ODDM), which is
composed of an offline stage and an online stage. The structure of the
proposed algorithm is shown in Fig. 1.

At the offline stage, for each source domain, we use JDA algorithm which
utilizes labeled source data and unlabeled target data to get a satisfactory
initial transformation matrix, denoted by $A_{i}$ for the $i$-th source
domain. Then, for the $i$-th source domain, the source data are projected to a
new feature space by $A_{i}$. After that, an existing machine learning method
is used to learn a classifier in new feature space. Generally, there are $n$
transformation matrices and $n$ source classifiers in total.

At the online stage, for $t$-trial target instance $x_{t}$, all the
transformation matrices are used to project $x_{t}$ to the corresponding
feature space, denoted by $x^{p_{1}}_{t},x^{p_{2}}_{t},...,x^{p_{n}}_{t}$. For
the target data in all new feature space, an online learning method, e.g.,
multi-class PA algorithm [3] is used to get the corresponding target
classifiers, denoted by $f_{T_{1}},f_{T_{2}},...,f_{T_{n}}$. Totally, there
are n classifiers in the target domain. We combine source classifiers and
target classifiers by allocating a weight to each classifier to build the
final classifier $F$.

At each round, we will update the target classifiers
$f_{T_{1}},f_{T_{2}},...,f_{T_{n}}$ according to the prediction loss and the
weights of all the classifiers by the Hedge strategy[4]. Besides, to better
reduce the discrepancy between domains as well as improve classification
performance, we propose a method to online update transformation matrices. To
speed up the algorithm, different from multi-class PA algorithm, we set a time
window $T_{w}$ to control the update frequency of transformation matrices
rather than updating transformation matrices in each round.

###  3.2 Multi-class PA algorithm

In this paper, the source classifiers and target classifiers are all trained
with Multi-class PA algorithm[3], we will introduce this algorithm detailly.
At the $t$-th round, the instance $x_{t}$ is associated a different weight
vector with each of the $K$ labels. That is, the multiclass predictor is now
paramterized by $w_{t}^{1},...,w_{t}^{K}$, where $w_{t}^{r}\in R^{d}$. The
output of the predictor is defined as,

$f(x_{t})={((w_{t}^{1}\cdot x_{t}),...,(w_{t}^{K}\cdot x_{t})).}$

The margin loss is used in Multi-class PA algorithm. Generally, at the $t$-th
round we find the pair of indices $r_{t},s_{t}$ which correspond to the
largest violation of the margin constraints,

$\displaystyle r_{t}=y_{t},\quad s_{t}=\arg\max_{s\neq y_{t}}w_{t}^{s}\cdot
x_{t}$ (1)

where $y_{t}$ is the true label. Then, the loss in mutli-class case amounts
to,

$\begin{split}&l(x_{t})=l(w_{t}^{1},...,w_{t}^{K};x_{t},y_{t})=\\\
&\left\\{\begin{array}[]{ll}0&w^{r_{t}}_{t}\cdot x_{t}-w^{s_{t}}_{t}\cdot
x_{t}\geq 1\\\ 1-w^{r_{t}}_{t}\cdot x_{t}+w^{s_{t}}_{t}\cdot
x_{t}&otherwise\end{array}\right.\end{split}$ (2)

Based on these definitions, the new weight vector
${w_{t+1}^{1}},...,w_{t+1}^{K}$ are the solution to the following constrained
optimization problem,

$\begin{split}w_{t+1}^{1},...,w_{t+1}^{K}&=\arg\min_{w^{1},...,w^{K}}\frac{1}{2}\sum_{i=1}^{K}||w^{i}-w_{t}^{i}||_{2}^{2}\quad\\\
&s.t.\quad l(w^{1},...,w^{K};(x_{t},y_{t}))=0\end{split}$ (3)

then we get the resulting update method for multi-class probelm,

$\displaystyle
w_{t+1}^{r_{t}}=w^{r_{t}}_{t}+\tau_{t}x_{t},w_{t+1}^{s_{t}}=w^{s_{t}}_{t}-\tau_{t}x_{t}$
(4) $\displaystyle\tau_{t}=min\\{C,\frac{l(x_{t})}{||w_{t}^{r_{t}}\cdot
x_{t}-w^{s_{t}}_{t}\cdot x_{t}||_{2}}\\}$ (5)

where C is the aggressiveness parameter controlling the influence of the slack
term on the objective function.

###  3.3 Offline Stage

Different from previous online transfer learning methods, we train classifiers
in the new feature space instead of the original feature space. JDA
algorithm[9] is a MMD-based method, which jointly adapts both the marginal
distribution and conditional distribution in a principled dimensionality
reduction procedure, and constructs new feature representation that is
effective and robust for substantial distribution discrepancy. In our method,
JDA algorithm is used to get the initial transformation matrix $A_{i}$ for the
$i$-th source domain with the source data and unlabeled target data. Then, the
original source data is projected to a new feature space, i.e.,
$\mathbf{X_{s_{i}}^{p}={A_{i}}X_{s_{i}}}$. The same transformation process
will also be used in the target domain at the online stage.

To perform knowledge transfer, we train a classifier in each source domain
using the multi-class PA algorithm(section 3.2) with the data in the new
feature space $\mathbf{X^{p}_{s_{i}}}$. And then, we get the classifier
$f_{S_{i}}$ for the $i$-th source domain.

###  3.4 Online Stage

As is shown in Fig. 1, at the $t$-th round, after receiving the target
instance $x_{t}$, all the transformation matrices are used to project $x_{t}$
to the corresponding feature space, i.e.,
$\mathbf{x_{t}^{p_{i}}={A_{i}}x_{t}}$ for the $i$-th transformation matrix.
For the target data in each new feature space, an online learning method,
e.g., multi-class PA algorithm is used to train the corresponding target
classifier, denoted as $f_{T_{i}}$ for the $i$-th feature space. At the $t$-th
round, it is paramterized by $\boldsymbol{w_{i,t}^{1},...,w_{i,t}^{K}}$. To
combine source classifiers and target classifiers, we assign a weight to each
classifier. The weight of the $i$-th source classifier is denoted by $u_{i}$,
and the weight of the $i$-th target classifier is denoted by $v_{i}$. The
initialize value of each weight is $\frac{1}{2n}$.

Then we get the final classifier $F_{t}$ by Equation 6 and make prediction for
$x_{t}$ by Equation 7, where the output of $F_{t}$ is a k-dimensional vector
$\boldsymbol{F_{t}(x_{t}^{p_{i}})}={(F_{t}^{1},F_{t}^{2},...,F_{t}^{K})}$ and
$\widehat{y_{t}}\in\\{1,2...,K\\}$ is the predicted label.

$F_{t}=\sum_{i=1}^{n}u_{i}f_{S_{i}}^{t}(x^{p_{i}}_{t})+v_{i}f_{T_{i}}^{t}(x^{p_{i}}_{t})$
(6) $\widehat{y_{t}}=\arg\max_{k}F_{t}^{k}$ (7)

After prediction, we will receive the true lable $y_{t}$, then a loss is
computed between $y_{t}$ and $\widehat{y_{t}}$ and each target classifier is
updated according to the multi-class PA algorithm. Besides, we except the
weight of each classifier can be adjusted dynamically. Inspired by the Hedge
strategy [4], we suggest the following updating scheme for adjusting the
weights:

$u_{i}=\frac{u_{i}\beta^{z^{i}_{t}}}{\sum_{i=1}^{n}u_{i}\beta^{z^{i}_{t}}+\sum_{i=1}^{n}v_{i}\beta^{r^{i}_{t}}}$
(8)
$v_{i}=\frac{v_{i}\beta^{r^{i}_{t}}}{\sum_{i=1}^{n}u_{i}\beta^{z^{i}_{t}}+\sum_{i=1}^{n}v_{i}\beta^{r^{i}_{t}}}$
(9)

where $\beta\in(0,1)$ is the weight discount parameter,
$z^{i}_{t}=I[\widehat{y}_{s_{i}}\neq y_{t}]$, $\widehat{y}_{s_{i}}$ is the
prediction label of the $i$-th source classifier, similarly,
$r^{i}_{t}=I[\widehat{y}_{t_{i}}\neq y_{t}]$, $\widehat{y}_{t_{i}}$ is the
prediction label of the $i$-th target classifier.

Algorithm 1 HomOTL-ODDM

0: the classifiers of source domains $f_{S_{1}},f_{S_{2}},...,f_{S_{n}}$, the
initial transformation matrices $A_{1},A_{2},...,A_{n}$, initial trade-off
parameter $C$, $\mu$, and the weight discount $\beta\in(0,1)$.
$v_{i}=\frac{1}{2n}$, $u_{i}=\frac{1}{2n}$, $i=1,2,...n$.
$w_{i,0}^{j}=\mathbf{0}$, $i=1,2,...n,j=1,2,...,K$.

1: for $t=1,2,...,T$ do

2: Receive target domain instance ${{x}_{t}}\in{\mathcal{X}_{T}}$.

3: Project $x_{t}$ to corresponding feature space:
$x^{p_{i}}_{t}=A_{i}^{t}x_{t}$

4: Predict $\widehat{y_{t}}$ by Equations 6 and 7.

5: Receive true lable: $y_{t}\in\mathcal{Y}_{t}$.

6: for $i=1,2,...,n$ do

7: Update the weights $v_{i},u_{i}$ by Equations 8 and 9.

8: Calculate $s_{t}$ , $r_{t}$ by equation 1

9: Suffer loss: ${{l}_{i}}(x^{p_{i}}_{t})={[0,1-w^{r_{t}}_{i,t}\cdot
x^{p_{i}}_{t}+w^{s_{t}}_{i,t}\cdot x^{p_{i}}_{t}}]_{+}$

10: if ${{l}_{i}}(x^{p_{i}}_{t})>0$ then

11:
$w_{i,t+1}^{r_{t}}=w^{r_{t}}_{i,t}+\tau_{t}x^{p_{i}}_{t},w_{i,t+1}^{s_{t}}=w^{s_{t}}_{i,t}-\tau_{t}x^{p_{i}}_{t}$

12: $\tau_{t}=min\\{C,\frac{{{l}_{i}}(x^{p_{i}}_{t})}{||w_{i,t}^{r_{t}}\cdot
x^{p_{i}}_{t}-w^{s_{t}}_{i,t}\cdot x^{p_{i}}_{t}||}\\}$

13: end if

14: end for

15: if t mod $T_{w}$ = 0 then

16: Update each transformation matrix $A_{i}$ by Equation 13

17: end if

18: end for

18:
$F_{t}=\sum_{i=1}^{n}u_{i}f_{S_{i}}^{t}(x^{p_{i}}_{t})+v_{i}f_{T_{i}}^{t}(x^{p_{i}}_{t})$.

###  3.5 The Update of Transformation Matrix

At the online stage, besides updating target classifiers and the weights of
each classifier, we also update the transformation matrices. On one hand, the
transformation matrices are got by target unlabeled data, which don’t involve
any label information, so utilizing online labeled target data is helpful for
improving classification performance. On the other hand, as target labeled
data increase, we can use more and more target data so that the measure of
distribution discrepancy between domains can become more accurate. In this
case, the transformation matrices will be closer to the optimal solution and
the distribution discrepancy in new feature space between domains can be
smaller.

Table 2: Experimental dataset details Tasks |  SD size |  TD size |  Dimensions |  Class |  Source domains(SD) |  Target domains(TD)
---|---|---|---|---|---|---
PIE1 |  8222 |  3332 |  1024 |  68 |  PIE2,PIE3,PIE4,PIE5 |  PIE1
PIE2 |  9925 |  1629 |  1024 |  68 |  PIE1,PIE3,PIE4,PIE5 |  PIE2
PIE3 |  9922 |  1632 |  1024 |  68 |  PIE1,PIE2,PIE4,PIE5 |  PIE3
PIE4 |  8225 |  3329 |  1024 |  68 |  PIE1,PIE2,PIE3,PIE5 |  PIE4
PIE5 |  9922 |  1632 |  1024 |  68 |  PIE1,PIE2,PIE3,PIE4 |  PIE5
Amazon |  1575 |  958 |  800 |  10 |  Caltech-256,DSLR,Webcam |  Amazon
Caltech-256 |  1410 |  1123 |  800 |  10 |  Amazon,DSLR,Webcam |  Caltech-256
DSLR |  2376 |  157 |  800 |  10 |  Amazon,Caltech-256,Webcam |  DSLR
Webcam |  2238 |  295 |  800 |  10 |  Amazon,Caltech-256,DSLR |  Webcam

We update each transformation matrix $A_{i}$ of the $i$-th domain by reducing
both the marginal distribution discrepancy and conditional distribution
discrepancy simultaneously. Empirical MMD is adopted as the distance measure
to compare different distributions.The marginal distribution discrepancy is
computed as:

$||{A_{i}\bar{c}_{S_{i}}-A_{i}\bar{c}_{T}}||^{2}_{2}$ (10)

where $\bar{c}_{S_{i}}=\frac{1}{n_{S_{i}}}\sum_{j=1}^{n_{S_{i}}}x_{j}$ is the
mean of the $i$-th source domain data, and
$\bar{c}_{T}=\frac{1}{t}\sum_{j=1}^{t}{x_{j}}$ is the mean of the target data
till round t. By minimizing Equation 10, the marginal distributions between
domains are drawn close under the new feature representation.

As $Q(x|y)$ and $Q(y|x$) can be quite involved according to the sufficient
statistics when sample sizes are large [9], we use the class conditional
distribution $Q(x|y)$ to approximate $Q(y|x)$. The distance between
conditional distribution $Q_{s}(x|y)$ and $Q_{t}(x|y)$ is measured as:

$\sum_{k=1}^{K}||{A_{i}\bar{c}_{S_{i}}^{k}-A_{i}\bar{c}_{T}^{k}}||^{2}_{2}$
(11)

where $\bar{c}_{S_{i}}^{k}=\frac{1}{n^{(c)}}{\sum_{x_{j}\in
D_{S_{i}}^{(c)}}}x_{j}$ represent the mean of examples belonging to class $k$
in the $i$-th source domain, $D_{S_{i}}^{(c)}=\\{x_{j}:x_{j}\in
D_{S_{i}}\wedge y(x_{j})=k\\}$, $n^{(c)}=|D_{S_{i}}^{(c)}|$. Correspondingly,
$\bar{c}_{T}^{k}=\frac{1}{m^{(c)}}{\sum_{x_{j}\in D_{T}^{(c)}}}x_{j}$
represent the mean of the data whose label are $k$ in the target domain,
$D_{T}^{(c)}=\\{x_{j}:x_{j}\in D_{T}\wedge y(x_{j})=k\\}$,
$m^{(c)}=|D_{T}^{(c)}|$.

We aim to simultaneously minimize the discrepancy of both marginal and
conditional distributions while reserving previous information. Thus, we
incorporate Equations 10 and 11, which leads to the following optimization
problem:

${A^{t+1}_{i}}=\arg\min_{A}||{A-A^{t}_{i}}||_{F}^{2}+\mu\sum_{k=0}^{K}||{A(\bar{c}_{S_{i}}^{k}-\bar{c}_{T}^{k})}||^{2}_{2}$
(12)

where $||{A(\bar{c}_{S_{i}}-\bar{c}_{T})}||^{2}_{2}$ is denoted by
$||{A(\bar{c}_{S_{i}}^{0}-\bar{c}_{T}^{0})}||_{2}^{2}$, $\mu$ is the
regularization parameter. Solving the above optimization problem, we will get
the updating method for $A_{i}$:

${A^{t+1}_{i}}={A_{i}^{t}}(I+\mu\sum_{k=0}^{K}{(\bar{c}_{S_{i}}^{k}-{\bar{c}_{T}^{k}})(\bar{c}_{S_{i}}^{k}-{\bar{c}_{T}^{k}})^{T})^{-1}}$
(13)

when
$I+\mu\sum_{k=0}^{K}(\bar{c}_{S_{i}}^{k}-{\bar{c}_{T}^{k}})(\bar{c}_{S_{i}}^{k}-{\bar{c}_{T}^{k}})^{T}$
is reversible, we will update $A_{i}$ using Equation 13, otherwise, we will
use its pseudo-inverse instead.

As we can see, matrix inversion for updating $A_{i}$ would cost a lot of time.
To decrease time cost and enhance the robustness of the proposed method, we
set a time window $T_{w}$ for all transformation matrices such as $T_{w}$= 50
to control the updating frequency. We will update $A_{i}$ once every time
window.

Combing the above update rules of classifiers , weights and transformation
matrices, we propose homogeneous online transfer learning with online domain
adaptation algorithm (HomOTL-ODDM). The pseudocode of the proposed method is
shown in algorithm 1.

###  3.6  Theoretical Analysis

Table 3: Mistakes rates(%) on 9 cross-domain image tasks Task |  PA | PAIO |  HomOTL-I | HomOTL-II |  HomOTLMS |  | HomOTL-ODDM
---
(fixed)
HomOTL-ODDM
PIE1 |  $78.45\pm 0.65$ | $40.10\pm 0.80$ |  $63.08\pm 0.37$ | $64.46\pm 0.34$ |  $67.10\pm 0.71$ | $13.92\pm 0.65$ |  $\mathbf{10.39\pm 1.09}$
PIE2 |  $90.99\pm 0.62$ | $53.82\pm 0.88$ |  $62.15\pm 0.31$ | $62.13\pm 0.31$ |  $69.33\pm 0.75$ | $30.87\pm 0.55$ |  $\mathbf{26.82\pm 1.86}$
PIE3 |  $90.66\pm 0.47$ | $52.49\pm 0.96$ |  $58.93\pm 0.36$ | $58.91\pm 0.33$ |  $69.97\pm 1.22$ | $25.57\pm 1.47$ |  $\mathbf{20.35\pm 2.87}$
PIE4 |  $75.06\pm 0.74$ | $31.49\pm 0.57$ |  $45.22\pm 0.22$ | $45.25\pm 0.21$ |  $60.97\pm 0.86$ | $19.46\pm 0.54$ |  $\mathbf{8.91\pm 0.95}$
PIE5 |  $88.44\pm 0.66$ | $57.53\pm 1.31$ |  $72.04\pm 0.55$ | $72.74\pm 0.57$ |  $74.89\pm 1.33$ | $33.75\pm 1.21$ |  $\mathbf{22.85\pm 1.78}$
Amazon |  $45.66\pm 1.26$ | $\mathbf{39.13\pm 1.29}$ |  $42.89\pm 1.78$ | $41.37\pm 1.04$ |  $39.57\pm 0.96$ | $44.23\pm 1.12$ |  $42.05\pm 1.63$
Caltech-256 |  $58.54\pm 1.23$ | $51.02\pm 1.20$ |  $56.88\pm 1.18$ | $52.64\pm 1.18$ |  $50.94\pm 1.21$ | $51.22\pm 1.36$ |  $\mathbf{50.65\pm 1.15}$
DSLR |  $52.64\pm 3.24$ | $29.68\pm 1.98$ |  $30.27\pm 2.78$ | $31.00\pm 1.38$ |  $24.55\pm 2.19$ | $22.86\pm 2.09$ |  $\mathbf{23.09\pm 3.27}$
Webcam |  $43.54\pm 1.71$ | $34.83\pm 1.65$ |  $34.90\pm 2.43$ | $36.36\pm 1.92$ |  $30.22\pm 1.85$ | $27.86\pm 2.36$ |  $\mathbf{26.17\pm 2.89}$
Average |  $69.33\pm 1.17$ | $43.34\pm 1.18$ |  $51.81\pm 1.11$ | $51.65\pm 0.80$ |  $54.17\pm 1.23$ | $29.97\pm 1.26$ |  $\mathbf{25.69\pm 1.94}$

Similar to [18], we provide the mistake bound of the algorithm HomOTL-ODDM as
follows.

###### Theorem 1

Let us denote $M$ as the number of mistakes made by the algorithm. By choosing
$\beta=\sqrt{M_{min}}/(\sqrt{M_{min}}+\sqrt{\ln 2n})$, we have M bounded by

$M\leq M_{min}+\frac{3}{2}\sqrt{\ln{2n}M_{min}}+\ln{2n}$ (14) $where\quad
M_{min}=min\\{M_{s}^{i},M_{t}^{i}\\},\quad
M_{s}^{i}=\sum_{t=1}^{T}z^{i}_{t},\quad
M_{t}^{i}=\sum_{t=1}^{T}r^{i}_{t}\quad$

######  Proof 3.1.

From the Theorem 2 in Hedge Algorithm [4], if we set all the initial weights
equally to be $1/2n$ for each source and target classifier, we can obtain the
following mistake bound for $M$,

$M\leq\frac{M_{min}\ln(1/\beta)+\ln 2n}{1-\beta}$ (15)

It can be shown that $\ln(1/\beta)\leq(1-\beta^{2})/2\beta$ for
$\beta\in(0,1]$. If we apply this equation to 15, then we can obtain

$\frac{M_{min}\ln(1/\beta)+\ln
2n}{1-\beta}\leq\frac{M_{min}(1+\beta)}{2\beta}+\frac{\ln 2n}{1-\beta}$ (16)

By substituting $\beta=\sqrt{M_{min}}/(\sqrt{M_{min}}+\sqrt{\ln 2n})$ into 16,
we obtain 14.

Theorem 1 provides an upper bound of the mistake for our algorithm. The result
in theorem 1 implies that the learner’s average mistake per round can never be
much larger than that of the best pure strategy. Specially, in HomOTL-ODDM,
the classifiers are trained in new feature space, where the distribution
discrepancy between domains is reduced, so it is expected to get fewer
mistakes in each domain than in original feature space.

##  4 Experiment

In this paper, nine image classification tasks are established on two widely
adopted benchmark datasets PIE and Office+Caltech(refer to Table 2) to
evaluate the proposed method. Codes will be available at
https://github.com/yaoyueduzhen/HomOTL-ODDM.

###  4.1 Datasets

PIE has 68 individuals with 41638 faces of size 32*32. In this paper,
PIE1(C05, left pose), PIE2(C07, upward pose), PIE3(C09, downward pose),
PIE4(C27, frontal pose) and PIE5(C29, right pose) are chosen. Office dataset
is composed by three real-world object domains: Amazon, Webcam and DSLR. It
has 4652 images with 31 object categories. Caltech-256 is a standard database
for object recognition, which has 30607 images and 256 categories.

In the paper, we adopt public Office+Caltech datasets[6]. SURF features are
extracted and quantized into an 800-bin histogram with codebooks computed with
K-means on a subset of images from Amazon. Then the histograms are
standardized by z-score. Specially, we will have four domains: Caltech-256,
Amazon, DSLR and Webcam. We choose one domain as the target domain, the remain
domains are used as source domains. In this way, we can generate four learning
tasks in dataset PIE and five learning tasks in dataset Office+Caltech.

###  4.2 Baseline Methods

To evaluate the performance of the proposed algorithm, we compare our
algorithm with several state-of-the-art methods, and for fair comparison, all
of these methods adopt the mutil-class PA algorithm as classifiers.

  * •

PA [3]: A classical online learning method without exploiting any knowledge
from source domain.

  * •

PAIO [18]: A variant of PA algorithm by initializing PA with a classifier
trained with the whole source domain.

  * •

HomOTL-I [20]: A single-source online transfer learning algorithm, which
adopts a loss-based weight update strategy.

  * •

HomTOL-II [20]: A single-source online transfer learning algorithm, which
adopts a mistake-driven weight update strategy.

  * •

HomOTLMS [18]: A recently proposed mutilple-source online transfer learning
method that combines source classifiers and target classifier adaptively.

  * •

HomOTL-ODDM(fixed): A variant of our proposed method, i.e., we just use the
initial transformation matrices without updating at the online stage

###  4.3 Implementation Details

We adopt the mutil-class PA algorithm to run on the source dataset and adopt
the average classifier as the source classifier in each domain, which
generally enjoys better generalization ability [2]. Futher, we draw 20 times
of random permutation of the instances in the target domain in order to obtain
stable results by averaging over the 20 trials. Considering that HomOTL-I and
HomOTL-II are single-source online transfer learning methods, we combine all
the instances in different source domains as a single source domain for
HomOTL-I and HomOTL-II, with hyper-parameter $C$ = 5 and
$\beta=\sqrt{T}/(\sqrt{T}+\sqrt{ln2})$.

In our method, we set hyper-parameter in JDA algorithm as
$m^{\prime}=100,T=10$ and 1) $\lambda=0.1$ for the PIE datasets, 2)
$\lambda=1$ for the Office+Caltech datasets. Besides, we also set $C$ = 5,
$\beta=\sqrt{T}/(\sqrt{T}+\sqrt{ln2})$, $\mu=1$ and 1) $T_{w}$ = 50 for PIE
datasets, 2) $T_{w}$ = 10 for Office+Caltech datasets. For performance
evaluation, from each of these nine tasks, we randomly sample 30% from the
target domain to form the unlabeled subset $X_{T}^{u}$, and the remaining 70%
in the target domain to form the online labeled subset $X_{T}^{l}$.

In this paper, we evaluate the predictive accuracy of online learning methods
by measuring the standard mistake rate on the labeled subset, which is defined
as,

$mistake\quad rate=\frac{|x:x\in{D_{T}^{l}}\bigwedge\widehat{y_{t}}\neq
y_{t}|}{|x:x\in D_{T}^{l}|}$ (17)

###  4.4 Experiment Results

The classification mistake rates of HomOTL-ODDM and the five baseline methods
on the nine cross-domain image tasks are illustrated in Table 3. Several
observation can be drawn from the experimental results. First of all, PA
algorithm performs the worst in most of the tasks, which indicates that
transferring knowledge from source domains can improve the target performance.
Second, the average classification mistake rate of HomOTL-ODDM is 25.69%
compared to the best baseline method PAIO, i.e., a significant error reduction
of 17.65%. This verifies that HomOTL-ODDM can construct more effective
representation for cross-domain classification tasks. Third, HomOTL-ODDM
outperforms HomOTLMS. A major limitation of HomOTLMS is that the distribution
discrepancy between domains is not explicitly reduced. HomOTL-ODDM avoids this
limitation and achieves much better results. Last but not least, HomOTL-ODDM
achieves much better performance than HomOTL-I and HomOTL-II, which are two
single-source online transfer learning methods, because HomOTL-ODDM not only
reduce distribution discrepancy between domains but also transfer knowledge
from multiple source domains.

Fig. 2 also shows the details of average mistake rates varying over the
learning processes. We only report results on task PIE1 and Amazon, while
similar trends on all other datasets are not shown due to space limitation.
The observations show that the proposed algorithm achieve the best performance
compared with baseline methods.

(a) PIE1

(b) Amazon

Figure 2: Evaluation of online mistake rates varying over learning process

###  4.5 Effectiveness Verification

Effectiveness of Updating transformation Matrix: Table 3 shows the result of
HomOTL-ODDM(fixed) compared with HomOTL-ODDM. As we can see, 1) both methods
perform better than all the baselines, which indicates that both methods can
improve transfer performance by reducing distribution discrepancy between
domains. 2) HomOTL-ODDM gets better results compared with HomOTL-ODDM(fixed),
and the average mistakes rates is reduced by 4.28%, which reveals that
updating transformation matrices using online labeled target data is effective
to improving transfer performance.

Distribution Discrepancy: We evaluate the distribution discrepancy varying
over the learning processes in task PIE1(refer to Fig. 3(a)) and Amazon(refer
to Fig. 3(b)). We compute the aggregate MMD distance of HomOTL-ODDM on their
induced embeddings by Equation 12. Note that, in order to compute the true
distance in both the marginal and conditional distributions between domains,
we have to use the groundtruth labels for target data. However, the
groundtruth labels are only used for verification, not for learning procedure.

Above all, we have three observations. 1) As target data increases, updating
transformation matrices can help extract a more effective and robust
representation, which can reduce the distribution discrepancy between domains
as well as improve the transfer performance. 2) When there is small
discrepancy between domains, e.g., task Amazon, HomOTL-ODDM can achieve the
optimal distance after receiving a small number of examples. 3) Considering
previous classification results, we find that when there is larger discrepancy
between domains, HomOTL-ODDM can achieve better transfer performance
improvement.

(a) PIE

(b) Amazon

Figure 3: Distribution discrepancy varying over the learning processes

###  4.6 Parameter Sensitivity

We conduct sensitivity analysis to validate that HomOTL-ODDM can achieve
optimal performance under a wide range of parameter values.

We run HomOTL-ODDM with varying values of $C$. We only report results on task
PIE1 and Webcam, while similar trends on all other datasets are not shown due
to space limitation. Fig. 4 evaluates the online prediction performance of the
compared algorithms with varied $C$ values across all the tasks. Several
observations can be drawn from the results. First of all, it is clear that the
proposed method is significantly more effective in most cases. Second, among
all the compared algorithms, we observe that HomOTL-ODDM always achieve the
best performance when $C$ is sufficiently large($C>2$), which indicates that a
large $C$ can improve the transfer efficiency. In the experiments, we set $C$
to be 5 for all the algorithms.

(a) PIE1

(b) Webcam

Figure 4: Parameter sensitivity of $C$

We run HomOTL-ODDM with varying values of $T_{w}$. As we can see, the smaller
$T_{w}$ is, the more frequently transformation matrices will be updated, which
will cost a lot of time as well as perform a poor performance. When $T_{w}$
becomes large, the update frequency will decrease, it is possible that it is
not effective to reduce the distribution discrepancy between domains. We plot
average mistakes w.r.t. different values of $T_{w}$ in Fig. 5, which indicates
that $T_{w}\in[30,200]$ can be optimal parameter values.

(a) PIE datasets

(b) Office+Caltech datasets

Figure 5: Parameter sensitivity of $T_{w}$

##  5 Conclusion

In this paper, we propose a method called HomOTL-ODDM. We focus on online
transfer learning with multiple source domains and use the Hedge strategy to
leverage knowledge from source domains. We seek to find a new feature
representation by projecting the original source and target data onto a new
space, where the distribution discrepancy between domains can be online
reduced simultaneously. Furthermore, We provide the mistake bound of the
proposed method and conduct extensive experiments. The experimental results
show that HomOTL-ODDM significantly outperforms several state-of-the-art
methods in a variety of cross-domain problems.

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The random sequence $\\{\beta_{n}\\}_{n\geq 0}$ satisfies
$\beta_{n}\rightarrow 0$ with probability one.

  4. 4.

The admissible set $\Theta$ is a hyperrectangle, i.e., there exist $a$ and $b$
such that $a<b$ and $\Theta=\\{\theta_{n}:a\leq\theta_{n}\leq b\\}$.

  5. 5.

    1. (a)

The function $g_{n}(\cdot)$ is continuous uniformly in $n$. Furthermore, there
exists a function $\bar{g}(\theta)$ such that for all $m>0$, we have
$\lim_{n\rightarrow\infty}\big{\|}\sum_{i=n}^{n+m-1}\epsilon_{i}\big{[}g_{i}(\theta)-\bar{g}(\theta)\big{]}\big{\|}=0$.

    2. (b)

The function $g_{n}(\cdot)$ is continuous uniformly in $n$. Moreover, there
exists an upper semicontinuous set-valued function $G(\theta)$ such that
$\lim_{n,m\rightarrow\infty}\frac{1}{m}\sum_{i=n}^{n+m-1}g_{i}(\theta)\in
G(\theta)$.

###### Theorem 13

Under Assumption 1-4 and 5(a), the asymptotic behavior of the algorithm (48)
is described by the ODE

$\displaystyle\dot{\theta}=\Psi_{\Theta}\big{[}\bar{g}(\theta)\big{]}.$ (49)

###### Theorem 14

Under Assumption 1-5, if there exists a continuously differentiable function
$\omega(\theta)$ such that $\bar{g}(\theta)=\frac{d\omega}{d\theta}$ and
$\omega(\theta)$ is constant on disjoint compact sets $\mathcal{L}_{i}$,
$i=1,\dots,M$, then the parameters $\theta_{n}$ converge with probability one
to $\mathcal{L}_{i}$ for some $i\in\\{1,\dots,M\\}$ as $n\rightarrow\infty$.

###### Theorem 15

Under Assumption 1-4 and 5(b), the limit points are contained in an invariant
set of the differential inclusion

$\displaystyle\dot{\theta}\in\Psi_{\Theta}\big{[}G(\theta)\big{]}.$ (50)

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