分类: 数学 >> 离散数学和组合数学 提交时间: 2024-03-27
摘要: Let f(n) be the maximum number of edges in a graph on n vertices in which no two cycles have the same length. Erd¨os raised the problem of determining f(n). Erd¨os conjectured that there exists a positive constant c such that ex(n, C2k) ≥ cn1+1/k. Haj´os conjecture that every simple even graph on n vertices can be decomposed into at most n/2 cycles. We present the problems, conjectures related to these problems and we summarize the know results. We do not think Haj´os conjecture is true.
分类: 数学 >> 离散数学和组合数学 提交时间: 2024-03-27
摘要: The set of all non-increasing nonnegative integers sequence π = (d(v1), d(v2), ..., d(vn)) is denoted by NSn. A sequence π ∈ NSn is said to be graphic if it is the degree sequence of a simple graph G on n vertices, and such a graph G is called a realization of π. The set of all graphic sequences in NSn is denoted by GSn. A graphical sequence π is potentially H-graphical if there is a realization of π containing H as a subgraph, while π is forcibly H-graphical if every realization of π contains H as a subgraph. Let Kk denote a complete graph on k vertices. Let Km −H be the graph obtained from Km by removing the edges set E(H) of the graph H (H is a subgraph of Km). This paper summarizes briefly some recent results on potentially Km −G-graphic sequences and give a useful classification for determining σ(H, n).
分类: 数学 >> 离散数学和组合数学 提交时间: 2024-03-26
摘要: In 1975, P. Erd {o}s proposed the problem of determining the maximum number $f(n)$ of edges in a graph of $n$ vertices in which any two cycles are of different lengths. In this paper, it is proved that $$f(n) geq n+32t-1$$ for $t=27720r+169 , (r geq 1)$ and $n geq frac{6911}{16}t^{2}+ frac{514441}{8}t- frac{3309665}{16}$. Consequently, $ liminf sb {n to infty} {f(n)-n over sqrt n} geq sqrt {2 + {2562 over 6911}}.$
分类: 数学 >> 离散数学和组合数学 提交时间: 2024-03-26
摘要: In 1975,P.Erd {o}sproposedtheproblemofdeterminingthemaximumnumber$f(n)$ofedgesinagraphwith$n$verticesinwhichanytwocyclesareofdifferentlengths.Inthispaper,itisprovedthat$$f(n) geqn+ frac{107}{3}t+ frac{7}{3}$$for$t=1260r+169 , (r geq1)$and$n geq frac{2119}{4}t^{2}+87978t+ frac{15957}{4}$.Consequently,$ liminf sb{n to infty}{f(n)-n over sqrtn} geq sqrt{2+ frac{7654}{19071}},$whichisbetterthanthepreviousbounds$ sqrt2$ Y.Shi,DiscreteMath.71(1988),57-71 ,$ sqrt{2.4}$ C.Lai,Australas.J.Combin.27(2003),101-105 .Theconjecture$ lim_{n rightarrow infty}{f(n)-n over sqrtn}= sqrt{2.4}$isnottrue.
分类: 数学 >> 离散数学和组合数学 提交时间: 2024-03-26
摘要: 设f(n) 是没有等长圈的n个顶点的图的最大可能边数。确定f(n)的问题由Erdos在1975年提出。本文给出了f(n)的下界。
分类: 数学 >> 数学(综合) 提交时间: 2024-03-01
摘要: In this paper, the main aim is to demonstrate the boundedness for commutators of fractional maximal function and sharp maximal function in the context of the p-adic version of Orlicz spaces, where the symbols of the commutators belong to the p-adic version of Lipschitz space, whereby some new characterizations for Λβ(Qnp) spaces are given.
分类: 数学 >> 数学(综合) 提交时间: 2024-03-01
摘要: In this article, the main aim is to introduce the grand variable Herz space over the p-adic fields and demonstrate the boundedness for fractional integral operator, fractional maximal operator in the context of the grand p-adic version of Herz-Morrey spaces with variable exponent, as well as the Lipschitz estimates for the commutators of fractional integral operator, fractional maximal operator, and sharp maximal function on the grand p-adic version of Herz-Morrey spaces with variable exponent.
分类: 数学 >> 几何与拓扑 提交时间: 2024-02-28
摘要: In this paper, we study biharmonic Riemannian submersions $ pi:M^2 times r to (N^2,h)$ from a product manifold onto a surface and obtain some local characterizations of such biharmonic maps. Our results show that when the target surface is flat, then a proper biharmonic Riemannian submersion $ pi:M^2 times r to (N^2,h)$ is locally a projection of a special twisted product, and when the target surface is non-flat, $ pi$ is locally a special map between two warped product spaces with a warping function that solves a single ODE. As a by-product, we also prove that there is a unique proper biharmonic Riemannian submersion $H^2 times r to r^2$ given by the projection of a warped product.
分类: 数学 >> 代数与数论 提交时间: 2024-02-23
摘要: 本文以拓扑Abel群的完备化为基础, 定义了拓扑$k$-代数及其完备化, 并从射影极限的角度对完备化的进行了代数解释.
分类: 数学 >> 应用数学 提交时间: 2024-02-22
摘要: 本文研究Heisenberg群上非退化散度型次椭圆方程弱解的正则性。基于系数矩阵更一般的假设,本文针对齐次与非齐次两种情形,建立弱解在Besov空间中的水平Calderon-Zygmund估计。本文的研究将丰富发展Heisenberg群上非线性Calderon-Zygmund正则性理论。
分类: 数学 >> 离散数学和组合数学 提交时间: 2024-02-18
摘要: Summary: In this paper we consider a variation of the classical Turn-type extremal problems. Let $S$be an $n$-term graphical sequence, and $\sigma(S)$be the sum of the terms in $S$. Let $H$be a graph. The problem is to determine the smallest even $l$such that any $n$-term graphical sequence $S$having $\sigma(S)\geq l$has a realization containing $H$as a subgraph. Denote this value $l$by $\sigma(H,n)$. We show $\sigma(C_{2m+1},n)=m(2n-m-1)+2$, for $m\geq 3$, $n\geq 3m$; $\sigma(C_{2m+2},n)=m(2n-m-1)+4$, for $m\geq 3$, $n\geq 5m-2$.''
分类: 数学 >> 离散数学和组合数学 提交时间: 2024-02-18
摘要: In 1975, P. Erdős proposed the problem of determining the maximum number $f(n)$ of edges in a graph on $n$ vertices in which any two cycles are of different lengths. Let $f^{\ast}(n)$ be the maximum number of edges in a simple graph on $n$ vertices in which any two cycles are of different lengths. Let $M_n$ be the set of simple graphs on $n$ vertices in which any two cycles are of different lengths and with the edges of $f^{\ast}(n)$. Let $mc(n)$ be the maximum cycle length for all $G \in M_n$. In this paper, it is proved that for $n$ sufficiently large, $mc(n)\leq \frac{15}{16}n$. We make the following conjecture: $$\lim_{n \rightarrow \infty} {mc(n)\over n}= 0.$$
分类: 数学 >> 离散数学和组合数学 提交时间: 2024-02-18
摘要: Let $K_{m}-H$ be the graph obtained from $K_{m}$ by removing the edges set $E(H)$ of the graph $H$ ($H$ is a subgraph of $K_{m}$). We use the symbol $Z_4$ to denote $K_4-P_2.$ A sequence $S$ is potentially $K_{m}-H$-graphical if it has a realization containing a $K_{m}-H$ as a subgraph. Let $\sigma(K_{m}-H, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $\sigma(S)\geq \sigma(K_{m}-H, n)$ is potentially $K_{m}-H$-graphical. In this paper, we determine the values of $\sigma (K_{r+1}-U, n)$ for $n\geq 5r+18, r+1 \geq k \geq 7,$ $j \geq 6$ where $U$ is a graph on $k$ vertices and $j$ edges which contains a graph $K_3 \bigcup P_3$ but not contains a cycle on 4 vertices and not contains $Z_4$. There are a number of graphs on $k$ vertices and $j$ edges which contains a graph $(K_{3} \bigcup P_{3})$ but not contains a cycle on 4 vertices and not contains $Z_4$. (for example, $C_3\bigcup C_{i_1} \bigcup C_{i_2} \bigcup >... \bigcup C_{i_p}$ $(i_j\neq 4, j=2,3,..., p, i_1 \geq 5)$, $C_3\bigcup P_{i_1} \bigcup P_{i_2} \bigcup ... \bigcup P_{i_p}$ $(i_1 \geq 3)$, $C_3\bigcup P_{i_1} \bigcup C_{i_2} \bigcup >... \bigcup C_{i_p}$ $(i_j\neq 4, j=2,3,..., p, i_1 \geq 3)$, etc)
分类: 数学 >> 数学(综合) 提交时间: 2024-02-17
摘要: 本文利用拉格朗日反演级数方法,研究了形如级数y=a1x+a2x2+a3x3++anxn+的在x=0的邻域内单调区间,然后针对更一般形式的级数方程,给出了其中一个非零实根的一个计算方法。
分类: 数学 >> 离散数学和组合数学 提交时间: 2024-02-13
摘要: Let $K_{m}-H$ be the graph obtained from $K_{m}$ by removing the edges set $E(H)$ of the graph $H$ ($H$ is a subgraph of $K_{m}$). We use the symbol $Z_4$ to denote $K_4-P_2.$ A sequence $S$ is potentially $K_{m}-H$-graphical if it has a realization containing a $K_{m}-H$ as a subgraph. Let $\sigma(K_{m}-H, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $\sigma(S)\geq \sigma(K_{m}-H, n)$ is potentially $K_{m}-H$-graphical. In this paper, we determine the values of $\sigma (K_{r+1}-Z, n)$ for $n\geq 5r+19, r+1 \geq k \geq 5,$ $j \geq 5$ where $Z$ is a graph on $k$ vertices and $j$ edges which contains a graph $Z_4$ but not contains a cycle on $4$ vertices. We also determine the values of $\sigma (K_{r+1}-Z_4, n)$, $\sigma (K_{r+1}-(K_4-e), n)$, $\sigma (K_{r+1}-K_4, n)$ for $n\geq 5r+16, r\geq 4$.
分类: 数学 >> 离散数学和组合数学 提交时间: 2024-02-13
摘要: Let $K_k$, $C_k$, $T_k$, and $P_{k}$ denote a complete graph on $k$ vertices, a cycle on $k$ vertices, a tree on $k+1$ vertices, and a path on $k+1$ vertices, respectively. Let $K_{m}-H$ be the graph obtained from $K_{m}$ by removing the edges set $E(H)$ of the graph $H$ ($H$ is a subgraph of $K_{m}$). A sequence $S$ is potentially $K_{m}-H$-graphical if it has a realization containing a $K_{m}-H$ as a subgraph. Let $\sigma(K_{m}-H, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $\sigma(S)\geq \sigma(K_{m}-H, n)$ is potentially $K_{m}-H$-graphical. In this paper, we determine the values of $\sigma (K_{r+1}-H, n)$ for $n\geq 4r+10, r\geq 3, r+1 \geq k \geq 4$ where $H$ is a graph on $k$ vertices which contains a tree on $4$ vertices but not contains a cycle on $3$ vertices. We also determine the values of $\sigma (K_{r+1}-P_2, n)$ for $n\geq 4r+8, r\geq 3$.
分类: 数学 >> 离散数学和组合数学 提交时间: 2024-02-10
摘要: A sequence $S$is potentially $K_4-e$graphical if it has a realization containing a $K_4-e$as a subgraph. Let $\sigma(K_4-e,n)$denote the smallest degree sum such that every $n$-term graphical sequence $S$with $\sigma(S)\geq\sigma(K_4-e,n)$is potentially $K_4-e$graphical. Gould, Jacobson, Lehel raised the problem of determining the value of $\sigma(K_4-e,n)$. In this paper, we prove that $\sigma(K_4-e,n)=2[(3n-1)/2]$for $n\geq7$and $n=4,5$, and $\sigma(K_4-e,6)=20$.''
分类: 数学 >> 统计和概率 提交时间: 2024-02-04
摘要: The three frameworks for theories of consciousness taken most seriously by neuroscientists are that consciousness is a biological state of the brain,the global workspace perspective,and the perspective of higher state.Consciousness is discussed from viewpoint of theory of Entropypartition of complex system in present article. Human brains system self-organizably andadaptively implements partition 、 aggregation and integration, and consciousness emerges.The Gibss representation of consciousness is proved and That consciousness originates from quantum mechanical processes of brain activity is explained by means of SW entropy
分类: 数学 >> 统计和概率 提交时间: 2024-01-23
摘要: The nineties of the 20th century, I have proposed and constructed theory of Abstract Neural Automata (ANA) 1 .Aim of present manu
分类: 数学 >> 计算数学 提交时间: 2024-01-04
摘要: This paper presents error analysis of stabilizer free weak Galerkin finite element method (SFWG-FEM) for a second order elliptic equation with low regularity solutions. The standard error analysis of SFWG-FEM requires additional regularity on solutions, such as $H^2$-regularity for the second-order convergence. However, if the solutions are in $H^{1+s}$ with $0< s < 1$, numerical experiments show that the SFWG-FEM is also effective and stable with the $(1+s)$-order convergence rate, so we develop a theoretical analysis for it. We introduce a standard $H^{2}$ finite element approximation for the elliptic problem, and then we apply the SFWG-FEM to approach this smooth approximating finite element solution. Finally, we establish the error analysis for SFWG-FEM with low regularity in both discrete $H^1$-norm and standard $L^2$-norm. The ($P_{k}(T),P_{k-1}(e), P_{k+1}(T) ^d$) elements with dimensions of space $d = 2,3$ are employed and the numerical examples are tested to confirm the theory.