Advances in Applied Mathematics最新文献

The authors regret that there was a mistake in [1, Definition 26] with our new basis axiom (nB3). We explain and correct this mistake here.

Let $\left(a\right(n):n\in N_0)$ denote an automatic sequence. Previous research on infinite products involving automatic sequences has mainly dealt with identities for products as in ${\prod }_{n}R{\left(n\right)}^{a\left(n\right)}$ for rational functions $R\left(n\right)$. This inspires the development of techniques for evaluating ${\prod }_{n}f{\left(n\right)}^{a\left(n\right)}$ more generally, for functions $f\left(n\right)$ that are not rational functions. This leads us to apply Euler's product expansion for the Γ-function together with recursive properties of $a\left(n\right)$ to obtain identities as in ${\prod }_{n}f{(n,z)}^{a\left(n\right)}=\Gamma (z+1)$, and this is motivated by how the equivalent series identity ${\sum }_{n}a\left(n\right)\ln f(n,z)=\ln \Gamma (z+1)$ could be applied in relation to the remarkable results due to Gosper on the integration of $\ln \Gamma (z+1)$. We succeed in applying this approach, using Gosper's integration identities, to obtain new infinite products that we evaluate in terms of the Glaisher–Kinkelin constant A and that involve the Thue–Morse sequence, the period-doubling sequence, and the regular paperfolding sequence. A byproduct of our method gives us a way of generalizing a Dirichlet series identity due to Allouche and Sondow, and we also explore applications related to a product evaluation due to Gosper involving A.

In this paper, we extend the chromatic symmetric function X to a chromatic k-multisymmetric function $X_k$, defined for graphs equipped with a partition of their vertex set into k parts. We demonstrate that this new function retains the basic properties and basis expansions of X, and we give a method for systematically deriving new linear relationships for X from previous ones by passing them through $X_k$.

In particular, we show how to take advantage of homogeneous sets of G (those $S\subseteq V\left(G\right)$ such that each vertex of $V\left(G\right)﹨S$ is either adjacent to all of S or is nonadjacent to all of S) to relate the chromatic symmetric function of G to those of simpler graphs. Furthermore, we show how extending this idea to homogeneous pairs $S_1\bigsqcup S_2\subseteq V\left(G\right)$ generalizes the process used by Guay-Paquet to reduce the Stanley-Stembridge conjecture to unit interval graphs.

Let $A=(a_1,a_2,\dots ,a_n)$ be relative prime positive integers with $a_i\ge 2$. The Frobenius number $g\left(A\right)$ is the greatest integer not belonging to the set $\left\{{\sum }_{i=1}^n{a}_i{x}_i\right|x_i\in N\}$. The general Frobenius problem includes the determination of $g\left(A\right)$ and the related Sylvester number $n\left(A\right)$ and Sylvester sum $s\left(A\right)$. We present a combinatorial approach to the Frobenius problem. Basically, we transform the problem into an easier optimization problem. If the new problem can be solved explicitly, then we obtain a formula for $g\left(A\right)$. We illustrate the idea by giving concise proofs and extensions of several existing formulas, as well as new formulas for $g\left(A\right),n\left(A\right),s\left(A\right)$. Moreover, we give a generating function approach to $n\left(A\right),s\left(A\right)$, and even to the more general Sylvester power sum.

A classical result of Erdős and Rademacher (1955) indicates a supersaturation phenomenon. It says that if G is a graph on n vertices with at least $\lfloor n^2/4\rfloor +1$ edges, then G contains at least $\lfloor n/2\rfloor$ triangles. We prove a spectral version of Erdős–Rademacher's theorem. Moreover, Mubayi (2010) [28] extends the result of Erdős and Rademacher from a triangle to any color-critical graph. It is interesting to study the extension of Mubayi from a spectral perspective. However, it is not apparent to measure the increment on the spectral radius of a graph comparing to the traditional edge version (Mubayi's result). In this paper, we provide a way to measure the increment on the spectral radius of a graph and propose a spectral version on the counting problems for color-critical graphs.

We investigate several families of multiple orthogonal polynomials associated with weights for which the moment generating functions are hypergeometric series with slightly varying parameters. The weights are supported on the unit interval, the positive real line, or the unit circle and the multiple orthogonal polynomials are generalizations of the Jacobi, Laguerre or Bessel orthogonal polynomials. We give explicit formulas for the type I and type II multiple orthogonal polynomials and study some of their properties. In particular, we describe the asymptotic distribution of the (scaled) zeros of the type II multiple orthogonal polynomials via the free convolution. Essential to our overall approach is the use of the Mellin transform. Finally, we discuss two applications. First, we show that the multiple orthogonal polynomials appear naturally in the study of the squared singular values of (mixed) products of truncated unitary random matrices and Ginibre matrices. Secondly, we use the multiple orthogonal polynomials to simultaneously approximate certain hypergeometric series and to provide an explicit proof of their $Q$-linear independence.

In 2001, Kanemitsu, Tanigawa, and Yoshimoto studied the following generalized Lambert series,$\sum _{n=1}^{\infty }\frac{n^{N-2h}}{\exp \left(n^{N}x\right)-1},$ for $N\in N$ and $h\in Z$ with some restriction on h. Recently, Dixit and the last author pointed out that this series has already been present in the Lost Notebook of Ramanujan with a more general form. Although, Ramanujan did not provide any transformation identity for it. In the same paper, Dixit and the last author found an elegant generalization of Ramanujan's celebrated identity for $\zeta (2m+1)$ while extending the results of Kanemitsu et al. In a subsequent work, Kanemitsu et al. explored another extended version of the aforementioned series, namely,$\sum _{r=1}^q\sum _{n=1}^{\infty }\frac{\chi \left(r\right)n^{N-2h}\exp (-\frac{r}{q}{n}^{N}x)}{1-\exp (-n^{N}x)},$ where χ denotes a Dirichlet character modulo q, $N\in 2N$ and with some restriction on the variable h. In the current paper, we investigate the above series for any $N\in N$ and $h\in Z$. We obtain a Dirichlet character analogue of Dixit and the last author's identity and there by derive a two variable generalization of Ramanujan's identity for $\zeta (2m+1)$. Moreover, we establish a new identity for $L(1/3,\chi )$ analogous to Ramanujan's famous identity for $\zeta (1/2)$.

In this paper we establish how alphabet size, treewidth and maximum degree of the underlying graph are key parameters which influence the overall computational complexity of finite freezing automata networks. First, we define a general decision problem, called Specification Checking Problem, that captures many classical decision problems such as prediction, nilpotency, predecessor, asynchronous reachability.

Then, we present a fast-parallel algorithm that solves the general model checking problem when the three parameters are bounded, hence showing that the problem is in NC. Moreover, we show that the problem is in XP on the parameters tree-width and maximum degree.

Finally, we show that these problems are hard from two different perspectives. First, the general problem is W[2]-hard when taking either treewidth or alphabet as single parameter and fixing the others. Second, the classical problems are hard in their respective classes when restricted to families of graph with sufficiently large treewidth.

Numerous structural findings of homology manifolds have been derived in various ways in relation to $g_2$-values. The homology 4-manifolds with $g_2\le 5$ are characterized combinatorially in this article. It is well-known that all homology 4-manifolds for $g_2\le 2$ are polytopal spheres. We demonstrate that homology 4-manifolds with $g_2\le 5$ are triangulated spheres and are derived from triangulated 4-spheres with $g_2\le 2$ by a series of connected sum, bistellar 1- and 2-moves, edge contraction, edge expansion, and edge flipping operations. We establish that the above inequality is optimally attainable, i.e., it cannot be extended to $g_2=6$.

We study three combinatorial models for the lower-triangular matrix with entries $t_{n,k}=\left(\begin{array}{c}n\\ k\end{array}\right)n^{n-k}$: two involving rooted trees on the vertex set $[n+1]$, and one involving partial functional digraphs on the vertex set $\left[n\right]$. We show that this matrix is totally positive and that the sequence of its row-generating polynomials is coefficientwise Hankel-totally positive. We then generalize to polynomials $t_{n,k}(y,z)$ that count improper and proper edges, and further to polynomials $t_{n,k}(y,\varphi )$ in infinitely many indeterminates that give a weight y to each improper edge and a weight $m!{\varphi }_m$ for each vertex with m proper children. We show that if the weight sequence ϕ is Toeplitz-totally positive, then the two foregoing total-positivity results continue to hold. Our proofs use production matrices and exponential Riordan arrays.