Subjects: Mathematics >> Mathematics (General) submitted time 2024-05-16
Abstract: The 5,000-year-old Chinese traditional culture is the root of the heritage and development of a country and a nation. Promoting the evolution and growth of fine traditional Chinese culture will strengthen cultural confidence for contemporary college students. How to integrate our brilliant traditional Chinese culture into the tedious study to the course of higher mathematics, and to increase the learning interest of contemporary college students for this course, and at the same time to enable the cultural identification for them, all of above are still deserved us to investigate. According to this purpose, we take an ideological and political teaching of an important knowledge point, namely integration by parts, and present a practice of integrating some idols in Chinese traditional culture with the course higher mathematics. In addition, it can be found that our ideological and political teaching significantly increase the learning interest and learning effect of students for this course.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Statistics and Probability Subjects: Statistics >> Mathematical Statistics Subjects: Information Science and Systems Science >> Basic Disciplines of Information Science and Systems Science submitted time 2024-04-11
Abstract: Statistical independence is a core concept in statistics and machine learning. Representing and measuring independence are of fundamental importance in related fields. Copula theory provides the tool for representing statistical independence, while Copula Entropy (CE) presents the tool for measuring statistical independence. This paper first introduces the theory of CE, including its definition, theorem, properties, and estimation method. The theoretical applications of CE to structure learning, association discovery, variable selection, causal discovery, system identification, time lag estimation, domain adaptation, multivariate normality test, two-sample test, and change point detection are reviewed. The relationships between the former four applications and their connection to correlation and causality are discussed. The frameworks based on CE, the kernel method, and distance correlation for measuring statistical independence and conditional independence are compared. The advantage of CE over other independence and conditional independence measures is evaluated. The applications of CE in theoretical physics, astrophysics, geophysics, theoretical chemistry, cheminformatics, materials science, hydrology, climatology, meteorology, environmental science, ecology, animal morphology, agronomy, cognitive neuroscience, motor neuroscience, computational neuroscience, psychology, system biology, bioinformatics, clinical diagnostics, geriatrics, psychiatry, public health, economics, management, sociology, pedagogy, computational linguistics, mass media, law, political science, military science, informatics, energy, food engineering, architecture, civil engineering, transportation, manufacturing, reliability, metallurgy, chemical engineering, aeronautics and astronautics, weapon, automobile, electronics, communication, high performance computing, cybersecurity, remote sensing, and finance are briefly introduced.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Discrete Mathematics and Combinatorics submitted time 2024-03-27
Abstract: 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.
Subjects: Mathematics >> Discrete Mathematics and Combinatorics submitted time 2024-03-27
Abstract: 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).
Subjects: Mathematics >> Discrete Mathematics and Combinatorics submitted time 2024-03-26
Abstract: 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}}.$
Subjects: Mathematics >> Discrete Mathematics and Combinatorics submitted time 2024-03-26
Abstract: In 1975, P. Erd {o}s proposed the problem of determining the
maximum number $f(n)$ of edges in a graph with $n$ vertices in which
any two cycles are of different
lengths. In this paper, it is proved that $$f(n) geq n+ frac{107}{3}t+ frac{7}{3}$$
for $t=1260r+169 , (r geq 1)$
and $n geq frac{2119}{4}t^{2}+87978t+ frac{15957}{4}$. Consequently,
$ liminf sb {n to infty} {f(n)-n over sqrt n} geq sqrt {2 +
frac{7654}{19071}},$ which is better than the previous bounds
$ sqrt 2$ Y. Shi, Discrete Math. 71(1988), 57-71 , $ sqrt {2.4}$
C. Lai, Australas. J. Combin. 27(2003), 101-105 .
The conjecture $ lim_{n rightarrow infty} {f(n)-n over sqrt n}= sqrt {2.4}$ is not true.
Subjects: Mathematics >> Discrete Mathematics and Combinatorics submitted time 2024-03-26
Abstract: In 1975, P. Erd H os proposed the problem of determining the maximum number $f(n)$ of edges in a graph with $n$ vertices in which any two cycles are of different lengths. The sequence $(c_1,c_2, cdots,c_n)$ is the cycle length distribution of a graph $G$ with $n$ vertices, where $c_i$ is the number of cycles of length $i$ in $G$. Let $f(a_1,a_2, cdots,$$a_n)$ denote the maximum possible number of edges in a graph which satisfies $c_i leq a_i$, where $a_i$ is a nonnegative integer. In 1991, Shi posed the problem of determining $f(a_1,a_2, cdots,a_n),$ which extended the problem due to Erd H os. It is clear that $f(n)=f(1,1, cdots,1)$.Let $g(n,m)=f(a_1,a_2, cdots,a_n),$ where $a_i=1$ if $i/m$ is an integer, and $a_i=0$ otherwise.It is clear that $f(n)=g(n,1)$.We prove that $ liminf sb {n to infty} {f(n)-n over sqrt n} geq sqrt {2 + frac{40}{99}},$ which is better than the previous bounds $ sqrt 2$ (Shi, 1988), and $ sqrt {2 + frac{7654}{19071}}$ (Lai, 2017).We show that $ liminf_{n rightarrow infty} {g(n,m)-n over sqrt frac{n}{m}} > sqrt {2.444},$ for all even integers $m$. We make the following conjecture:$ liminf sb {n to infty} {f(n)-n over sqrt n} > sqrt {2.444}.$ par
Subjects: Mathematics >> Mathematics (General) submitted time 2024-03-01
Abstract: 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.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Mathematics (General) submitted time 2024-03-01
Abstract: 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.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Geometry and Topology submitted time 2024-02-28
Abstract: 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.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Algebra and Number Theory submitted time 2024-02-23
Abstract: This article is based on the completion of topological Abel groups, introducts topological $k$-algebras and their completions, and provides an algebraic explanation of the completion by projective limits.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Applied Mathematics submitted time 2024-02-22
Abstract: In this paper, we deal with weak solutions to non-degenerate sub-elliptic equations in the Heisenberg group, and study the regularities of solutions. We establish horizontal Calderon-Zygmund type estimate in Besov spaces with more general assumptions on coefficients for both homogeneous equations and non-homogeneous equations. This study of regularity estimates expands the Calderon-Zygmund theory in the Heisenberg group.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Discrete Mathematics and Combinatorics submitted time 2024-02-18
Abstract: Summary: "In this paper we consider a variation of the classical Turán-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$.''
Subjects: Mathematics >> Discrete Mathematics and Combinatorics submitted time 2024-02-18
Abstract: 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.$$
Subjects: Mathematics >> Discrete Mathematics and Combinatorics submitted time 2024-02-18
Abstract: 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)
Subjects: Mathematics >> Mathematics (General) submitted time 2024-02-17
Abstract: In this paper, we first study the monotone intervals in the neighborhood of x=0 for the series y=a1x+a2x2+a3x3+…+anxn+… by using the Lagrange inversion series method. Then, for the more general series equation, we give a calculation method of the nonzero and minimum real roots.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Discrete Mathematics and Combinatorics submitted time 2024-02-13
Abstract: 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
$n\geq 5r+16, r\geq 4$.
Subjects: Mathematics >> Discrete Mathematics and Combinatorics submitted time 2024-02-13
Abstract: 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
$n\geq 4r+8, r\geq 3$.
Subjects: Mathematics >> Discrete Mathematics and Combinatorics submitted time 2024-02-10
Abstract: "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 $\sigma(K_4-e,6)=20$.''
Subjects: Mathematics >> Statistics and Probability submitted time 2024-02-04
Abstract: 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 Entropy—partition of complex system in present article. Human brain’s system self-organizably and adaptively 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
Peer Review Status:Awaiting Review