自然数集的分拆及其表示函数

令N表示全体非负整数的集合.对给定的集合A C N及n∈N,令R_1(A,n)表示方程n=a+a',a,a'∈A的解的个数.令R_2(A,n)和R_3(A,n)分别表示方程n=a+a',a,a'∈A在条件aa'和a≤a'下解的个数.一个有趣的问题是:给定i∈{1,2,3},确定所有非负整数集合对(A;B),使其表示函数R_i(A,n)及R_i(B,n)最终相等.文章讨论了相关问题.     

PowerNet: Efficient Representations of Polynomials and Smooth Functions by Deep Neural Networks with Rectified Power Units

Deep neural network with rectified linear units (ReLU) is getting more and more popular recently. However, the derivatives of the function represented by a ReLU network are not continuous, which limit the usage of ReLU network to situations only when smoothness is not required. In this paper, we construct deep neural networks with rectified power units (RePU), which can give better approximations for smooth functions. Optimal algorithms are proposed to explicitly build neural networks with sparsely connected RePUs, which we call PowerNets, to represent polynomials with no approximation error. For general smooth functions, we first project the function to their polynomial approximations, then use the proposed algorithms to construct corresponding PowerNets. Thus, the error of best polynomial approximation provides an upper bound of the best RePU network approximation error. For smooth functions in higher dimensional Sobolev spaces, we use fast spectral transforms for tensor-product grid and sparse grid discretization to get polynomial approximations. Our constructive algorithms show clearly a close connection between spectral methods and deep neural networks: PowerNets with $n$ hidden layers can exactly represent polynomials up to degree $s^n$, where $s$ is the power of RePUs. The proposed PowerNets have potential applications in the situations where high-accuracy is desired or smoothness is required.     

Cluster-additive functions on stable translation quivers

Additive functions on translation quivers have played an important role in the representation theory of finite-dimensional algebras, the most prominent ones are the hammock functions introduced by S.?Brenner. When dealing with cluster categories (and cluster-tilted algebras), one should look at a corresponding class of functions defined on stable translation quivers, namely the cluster-additive ones. We conjecture that the cluster-additive functions on a stable translation quiver of Dynkin type $\mathbb{A}_{n}, \mathbb{D}_{n}, \mathbb{E}_{6}, \mathbb {E}_{7}, \mathbb{E}_{8}$ are non-negative linear combinations of cluster-hammock functions (with index set a tilting set). The present paper provides a first study of cluster-additive functions and gives a proof of the conjecture in the case $\mathbb{A}_{n}$ .     

复指数函数系的极小性

A sufficient condition is obtained for the minimality of the complex exponential system E（A, M） = {z^le^λnz： l = 0, 1,,.., mn - 1; n = 1, 2,...} in the Banaeh space La^p consisting of all functions f such that f^-a ∈ LP（N）. Moreover, if the incompleteness holds, each function in the closure of the linear span of exponential system E（A, M） can be extended to an analytic function represented by a Taylor-Dirichlet series.     

Representations of twisted Yangians associated with skew Young diagrams

Let $G_M$ be either the orthogonal group $O_M$ or the symplectic group $Sp_M$ over the complex field; in the latter case the non-negative integer $M$ has to be even. Classically, the irreducible polynomial representations of the group $G_M$ are labeled by partitions $\mu=(\mu_{1},\mu_{2},\,\ldots)$ such that $\mu^{\prime}_1+\mu^{\prime}_2\le M$ in the case $G_M=O_M$, or $2\mu^{\prime}_1\le M$ in the case $G_M=Sp_M$. Here $\mu^{\prime}=(\mu^{\prime}_{1},\mu^{\prime}_{2},\,\ldots)$ is the partition conjugate to $\mu$. Let $W_\mu$ be the irreducible polynomial representation of the group $G_M$ corresponding to $\mu$. Regard $G_N\times G_M$ as a subgroup of $G_{N+M}$. Then take any irreducible polynomial representation $W_\lambda$ of the group $G_{N+M}$. The vector space $W_{\lambda}(\mu)={\rm Hom}_{\,G_M}( W_\mu, W_\lambda)$ comes with a natural action of the group $G_N$. Put $n=\lambda_1-\mu_1+\lambda_2-\mu_2+\ldots\,$. In this article, for any standard Young tableau $\varOmega$ of skew shape $\lm$ we give a realization of $W_{\lambda}(\mu)$ as a subspace in the $n$-fold tensor product $(\mathbb{C}^N)^{\bigotimes n}$, compatible with the action of the group $G_N$. This subspace is determined as the image of a certain linear operator $F_\varOmega (M)$ on $(\mathbb{C}^N)^{\bigotimes n}$, given by an explicit formula. When $M=0$ and $W_{\lambda}(\mu)=W_\lambda$ is an irreducible representation of the group $G_N$, we recover the classical realization of $W_\lambda$ as a subspace in the space of all traceless tensors in $(\mathbb{C}^N)^{\bigotimes n}$. Then the operator $F_\varOmega\(0)$ may be regarded as the analogue for $G_N$ of the Young symmetrizer, corresponding to the standard tableau $\varOmega$ of shape $\lambda$. This symmetrizer is a certain linear operator on $\CNn$$(\mathbb{C}^N)^{\bigotimes n} $ with the image equivalent to the irreducible polynomial representation of the complex general linear group $GL_N$, corresponding to the partition $\lambda$. Even in the case $M=0$, our formula for the operator $F_\varOmega(M)$ is new. Our results are applications of the representation theory of the twisted Yangian, corresponding to the subgroup $G_N$ of $GL_N$. This twisted Yangian is a certain one-sided coideal subalgebra of the Yangian corresponding to $GL_N$. In particular, $F_\varOmega(M)$ is an intertwining operator between certain representations of the twisted Yangian in $(\mathbb{C}^N)^{\bigotimes n}$.     

Derivations and linear functions along rational functions

The main purpose of this paper is to give characterization theorems on derivations as well as on linear functions. Among others the following problem will be investigated: Let ${n \in \mathbb{Z}, f, g\colon\mathbb{R} \to\mathbb{R}}$ be additive functions, ${\left(\begin{array}{cc} a&b\\ c&d \end{array} \right) \in \mathbf{GL}_{2}(\mathbb{Q})}$ be arbitrarily fixed, and let us assume that the mapping $$ \phi(x)=g\left(\frac{ax^{n}+b}{cx^{n}+d}\right)-\frac{x^{n-1}f(x)}{(cx^{n}+d)^{2}} \quad \left(x\in\mathbb{R}, cx^{n}+d\neq 0\right)$$ satisfies some regularity on its domain (e.g. (locally) boundedness, continuity, measurability). Is it true that in this case the above functions can be represented as a sum of a derivation and a linear function? Analogous statements ensuring linearity will also be presented.     

Volterra算子及其伴随算子线性组合的数值域

经典Volterra算子$V$及其伴随算子$V^*$在复空间$L^2[0,1]$中起着关键作用. 关于$V$和$V^*$的线性组合的性质, 我们给出了确保$z_1V+z_2V^*(z_1,z_2\in\mathbb{C})$满足增生性质的等价条件. 本文描述了$(u+iv)I+mV+nV^*(u,v,m,n\in\mathbb{R},m+n\geq0)$数值域的精确表示.     

与正弦函数有关的星形函数逆的Toeplitz行列式

Let S_s*be the class of normalized functions f defined in the open unit■such that the quantity zf’(z)/f(z)lies in an eight-shaped region in the right-half plane and satisfies the condition■.In this paper,we aim to investigate Toeplitz determinants for the inverse of this function classes S_s*associated with sine function.     

Surface Approximation Is Sometimes Easier than Surface Integration

The approximation and integration problems consist of finding anapproximation to a function $f$ or its integral over some fixeddomain $\Sigma$. For the classical version of these problems, wehave partial information about the functions $f$ and completeinformation about the domain $\Sigma$; for example, $\Sigma$ might bea cube or ball in $\reals^d$. When this holds, it is generally thecase that integration is not harder than approximation; moreover,integration can be much easier than approximation. What happens if wehave partial information about $\Sigma$? This paper studies thesurface approximation and surface integration problems, in which$\Sigma=\Sigma_g$ for functions $g$. More specifically, thefunctions $f$ are $r$ times continuously differentiable scalarfunctions of $l$ variables, and the functions $g$ are $s$ times continuouslydifferentiable injective functions of $d$ variables with$l$ components. The class of surfaces considered is generated as images of cubes or balls, or as oriented cellulated regions.Error for the surface approximation problem is measured in the $L_q$-sense.These problems are well defined, provided that $d\le l$, $r\ge 0$, and$s\ge 1$. Information consists of function evaluations of $f$ and $g$.We show that the $\e$-complexity of surfaceapproximation is proportional to $(1/\e)^{1/\mu}$ with $\mu=\mrs/d$.We also show that if $s\ge 2$, then the $\e$-complexity of surfaceintegration is proportional to $(1/\e)^{1/\nu}$ with$\nu=\min\left\{\frac{r}{d},\frac{s-\delta_{s,1}(1-\delta_{d,l})}{\min\{d,l-1\}}\right\}.$(This bound holds as well for several subcases of $s=1$; we conjecturethat it holds for all $r\ge0$, $s\ge1$, and $d\le l$.) Using theseresults, we determine when surface approximation is easier than, aseasy as, or harder than, surface integration; all three possibilitiescan occur. In particular, we find that if $r=s=1$ and $d相似文献   

Projective surfaces with many skew lines

We give an example of a smooth surface of degree that contains pairwise disjoint lines. In particular, our example shows that the degree in Miyaoka's bound is sharp.

     

On Uniform Approximation in R2 of Continuous Functions of Two Variables with Bounded Variation in the Sense of Hardy

Introduce the notation: $\mathbb{Z}$ is the set of integers, $\bar {\mathbb{Z}}={\mathbb{Z}} \cup \{-\infty, +\infty\},{\mathbb{R}}_+^2 =\{x=(x_1,x_2) \in {\mathbb{R}}^2; x_1>0,x_2>0\}$ , $g_{k,m} (x,\alpha,h)= \int\limits_0^1 {g_1 (\frac{(k+u)h_1 - x_1}{\alpha_1})g_2(\frac{(m+u)h_2 - x_2}{\alpha_2})}du$ , where $g_i :\mathbb{R} \to \mathbb{R},x \in \mathbb{R}^2 ,\alpha ,h \in \mathbb{R}_ + ^2 $ . Under certain conditions on the functions g ₁, g ₂, we prove that the system of functions $g_{k,m} (x,\alpha^(n), h^(n)) (k,m \in \bar {\mathbb{Z}})$ , where $\alpha ^{\left( n \right)} ,h^{\left( n \right)} \in \mathbb{R}_ + ^2 $ are arbitrary infinitesimal sequences, is complete in the space C $\mathbb{R}^2 $ of uniformly continuous bounded functions f equipped with the norm $||f|| = \mathop {\sup }\limits_{x \in \mathbb{R}^2 } |f(x)|$ . Starting with the functions g _k,m , it is possible to construct a method for uniform approximating in $\mathbb{R}^2 $ any continuous function of bounded variation in the sense of Hardy. An error estimate is derived in terms of the second order moduli of continuity. Based on the obtained results, we discuss in detail the accuracy of uniform approximation of functions of several variables by linear functions. The error estimates are derived by using second order moduli of continuity. We pay a particular attention to sharpness of constants. Bibliography: 8 titles.     

$c$-半层空间与几乎完全正则空间一点注记

In this paper, we give some characterizations of almost completely regular spaces and c-semistratifiable spaces(CSS) by semi-continuous functions. We mainly show that:(1)Let X be a space. Then the following statements are equivalent:(i) X is almost completely regular.(ii) Every two disjoint subsets of X, one of which is compact and the other is regular closed, are completely separated.(iii) If g, h : X → I, g is compact-like, h is normal lower semicontinuous, and g ≤ h, then there exists a continuous function f : X → I such that g ≤ f ≤ h;and(2) Let X be a space. Then the following statements are equivalent:(a) X is CSS;(b) There is an operator U assigning to a decreasing sequence of compact sets(Fj)j∈N,a decreasing sequence of open sets(U(n,(Fj)))n∈N such that(b1) Fn■U(n,(Fj)) for each n ∈ N;(b2)∩n∈NU(n,(Fj)) =∩n∈NFn;(b3) Given two decreasing sequences of compact sets(Fj)j∈N and(Ej)j∈N such that Fn■Enfor each n ∈ N, then U(n,(Fj))■U(n,(Ej)) for each n ∈ N;(c) There is an operator Φ : LCL(X, I) → USC(X, I) such that, for any h ∈ LCL(X, I),0 Φ(h) h, and 0 Φ(h)(x) h(x) whenever h(x) 0.     

The structure of linear codes of constant weight

In this paper we determine completely the structure of linear codes over of constant weight. Namely, we determine exactly which modules underlie linear codes of constant weight, and we describe the coordinate functionals involved. The weight functions considered are: Hamming weight, Lee weight, two forms of Euclidean weight, and pre-homogeneous weights. We prove a general uniqueness theorem for virtual linear codes of constant weight. Existence is settled on a case by case basis.

     

树的L(3,2,1)-标号问题

An L(3,2,1)-labeling of a graph G is a function f from the vertex set V(G) to the set of all non-negative integers(labels) such that |f(u)-f(v)|≥3 if d(u,v)=1,|f(u)-f(v)≥2 if d(u,v)=2 and |f(u)-f(v)|≥1 if d(u,v)=3.For a non-negative integer k,a k-L(3,2,1)-labeling is an L(3,2,1)-labeling such that no label is greater than k.The L(3,2,1)-labeling number of G,denoted by λ_(3,2,1)(G), is the smallest number k such that G has a k-L(3,2,1)-labeling.In this article,we characterize the L(3,2,1)-labeling numbers of trees with diameter at most 6.     

严格上三角矩阵李代数上的积零导子

设$F$ 为域, $n\geq 3$, $\bf{N}$$(n,\mathbb{F})$ 为域$\mathbb{F}$ 上所有$n\times n$ 阶严格上三角矩阵构成的严格上三角矩阵李代数, 其李运算为$[x,y]=xy-yx$. $\bf{N}$$(n, \mathbb{F})$ 上一线性映射$\varphi$ 称为积零导子,如果由$[x,y]=0, x,y\in \bf{N}$$(n,\mathbb{F})$,总可推出 $[\varphi(x), y]+[x,\varphi(y)]=0$. 本文证明 $\bf{N}$$(n,\mathbb{F})$上一线性映射 $\varphi$ 为积零导子当且仅当 $\varphi$ 为$\bf{N}$$(n,\mathbb{F})$ 上内导子, 对角线导子, 极端导子, 中心导子和标量乘法的和.     

$\alpha$阶星象亚纯函数

令$S(p)$表示单位圆盘$\mathbb{D}$上在$p\in(0,1)$处有一个简单极点的单叶亚纯函数全体.令$\alpha\in[0,1)$,我们用$\Sigma^{*}(p,\omega_{0},\alpha)$表示$f\in S(p)$使得$\hat{\mathbb{C}}\setminus f(\mathbb{D})$是关于不动点$\omega_{0}\neq0$, $\infty$星象的$\alphga$阶区域的函数全体.在本文中,$f\in\Sigma^{*}(p,\omega_{0},\alpha)$的一些解析刻画条件和系数估计被考虑.     

Characterization and recognition of d.c. functions

A function ${f : \Omega \to \mathbb{R}}$ , where Ω is a convex subset of the linear space X, is said to be d.c. (difference of convex) if f =  g ? h with ${g, h : \Omega \to \mathbb{R}}$ convex functions. While d.c. functions find various applications, especially in optimization, the problem to characterize them is not trivial. There exist a few known characterizations involving cyclically monotone set-valued functions. However, since it is not an easy task to check that a given set-valued function is cyclically monotone, simpler characterizations are desired. The guideline characterization in this paper is relatively simple (Theorem 2.1), but useful in various applications. For example, we use it to prove that piecewise affine functions in an arbitrary linear space are d.c. Additionally, we give new proofs to the known results that C ^1,1 functions and lower-C ² functions are d.c. The main goal remains to generalize to higher dimensions a known characterization of d.c. functions in one dimension: A function ${f : \Omega \to \mathbb{R}, \Omega \subset \mathbb{R}}$ open interval, is d.c. if and only if on each compact interval in Ω the function f is absolutely continuous and has a derivative of bounded variation. We obtain a new necessary condition in this direction (Theorem 3.8). We prove an analogous sufficient condition under stronger hypotheses (Theorem 3.11). The proof is based again on the guideline characterization. Finally, we obtain results concerning the characterization of convex and d.c. functions obeying some kind of symmetry.     

On Ginzburg's bivariant Chern classes

The convolution product is an important tool in geometric representation theory. Ginzburg constructed the ``bivariant" Chern class operation from a certain convolution algebra of Lagrangian cycles to the convolution algebra of Borel-Moore homology. In this paper we prove a ``constructible function version" of one of Ginzburg's results; motivated by its proof, we introduce another bivariant algebraic homology theory on smooth morphisms of nonsingular varieties and show that the Ginzburg bivariant Chern class is the unique Grothendieck transformation from the Fulton-MacPherson bivariant theory of constructible functions to this new bivariant algebraic homology theory, modulo a reasonable conjecture. Furthermore, taking a hint from this conjecture, we introduce another bivariant theory of constructible functions, and we show that the Ginzburg bivariant Chern class is the unique Grothendieck transformation from to satisfying the ``normalization condition" and that it becomes the Chern-Schwartz-MacPherson class when restricted to the morphisms to a point.

     

带参数的一阶泛函差分方程的定号周期解

In this paper,the authors obtain the existence of one-signed periodic solutions of the first-order functional difference equation ?u(n) = a(n)u(n)-λb(n)f(u(n-τ(n))),n ∈ Z by using global bifurcation techniques,where a,b:Z → [0,∞) are T-periodic functions with ∑T n=1 a(n) 0,∑T n=1 b(n) 0;τ:Z → Z is T-periodic function,λ 0 is a parameter;f ∈ C(R,R) and there exist two constants s_2 0 s_1 such that f(s_2) = f(0) = f(s_1) = 0,f(s) 0 for s ∈(0,s_1) ∪(s_1,∞),and f(s) 0 for s ∈(-∞,s_2) ∪(s_2,0).     

Non-Negative Integer Matrix Representations of a $\mathbb{Z}_{+}$-Ring

The $\mathbb{Z}_{+}$-ring is an important invariant in the theory of tensor category. In this paper, by using matrix method, we describe all irreducible $\mathbb{Z}_{+}$-modules over a $\mathbb{Z}_{+}$-ring $\mathcal{A}$, where $\mathcal{A}$ is a commutative ring with a $\mathbb{Z}_{+}$-basis{$1$, $x$, $y$, $xy$} and relations: $$ x^{2}=1,\;\;\;\;\; y^{2}=1+x+xy.$$We prove that when the rank of $\mathbb{Z}_{+}$-module $n\geq5$, there does not exist irreducible $\mathbb{Z}_{+}$-modules and when the rank $n\leq4$, there exists finite inequivalent irreducible $\mathbb{Z}_{+}$-modules, the number of which is respectively 1, 3, 3, 2 when the rank runs from 1 to 4.