Advances in Applied Mathematics最新文献

We study divide-and-conquer recurrences of the form$f\left(n\right)=\alpha f(\lfloor \frac{n}{2}\rfloor )+\beta f(\lceil \frac{n}{2}\rceil )+g\left(n\right)(n⩾2),$ with $g\left(n\right)$ and $f\left(1\right)$ given, where $\alpha ,\beta ⩾0$ with $\alpha +\beta >0$; such recurrences appear often in the analysis of computer algorithms, numeration systems, combinatorial sequences, and related areas. We show under an optimum (iff) condition on $g\left(n\right)$ that the solution f always satisfies a simple identity$f\left(n\right)=n^{{\log }_2(\alpha +\beta )}P({\log }_{2}n)-Q\left(n\right),$ where P is a periodic function and $Q\left(n\right)$ is of a smaller order than the dominant term. This form is thus not only an identity but also an asymptotic expansion. Explicit forms for the continuity of the periodic function P are provided, together with a few other smoothness properties. We show how our results can be easily applied to many dozens of concrete examples collected from the literature, and how they can be extended in various directions. Our method of proof is surprisingly simple and elementary, but leads to the strongest types of results for all examples to which our theory applies.

Gamma-positive polynomials frequently appear in finite geometries, algebraic combinatorics and number theory. Sagan and Tirrell (2020) [34] stumbled upon some unimodal sequences, which turn out to be alternating gamma-positive instead of gamma-positive. Motivated by this work, we first show that one can derive alternatingly γ-positive polynomials from γ-positive polynomials. We then prove the alternating γ-positivity and Hurwitz stability of several polynomials associated with the Narayana polynomials of types A and B. In particular, by introducing the definition of colored $2\times n$ Young diagrams, we provide combinatorial interpretations for three identities related to the Narayana numbers of type B. Finally, we present several identities involving the Eulerian polynomials of types A and B.

In this paper, we construct a mixed-base number system over the generalized symmetric group $G(m,1,n)$, which is a complex reflection group with a root system of type $B_n^{\left(m\right)}$. We also establish one-to-one correspondence between all positive integers in the set $\{1,\cdots ,m^{n}n!\}$ and the elements of $G(m,1,n)$ by constructing the subexceedant function in relation to this group. In addition, we provide a new enumeration system for $G(m,1,n)$ by defining the inversion statistic on $G(m,1,n)$. Finally, we prove that the flag-major index is equi-distributed with this inversion statistic on $G(m,1,n)$. Therefore, the flag-major index is a Mahonian statistic on $G(m,1,n)$ with respect to the length function L.

We express the number of anti-invariant subspaces for a linear operator on a finite vector space in terms of the number of its invariant subspaces. When the operator is diagonalizable with distinct eigenvalues, our formula gives a finite-field interpretation for the entries of the q-Hermite Catalan matrix. We also obtain an interesting new proof of Touchard's formula for these entries.

Combinatorial threshold-linear networks (CTLNs) are a special class of recurrent neural networks whose dynamics are tightly controlled by an underlying directed graph. Recurrent networks have long been used as models for associative memory and pattern completion, with stable fixed points playing the role of stored memory patterns in the network. In prior work, we showed that target-free cliques of the graph correspond to stable fixed points of the dynamics, and we conjectured that these are the only stable fixed points possible [19], [8]. In this paper, we prove that the conjecture holds in a variety of special cases, including for networks with very strong inhibition and graphs of size $n\le 4$. We also provide further evidence for the conjecture by showing that sparse graphs and graphs that are nearly cliques can never support stable fixed points. Finally, we translate some results from extremal combinatorics to obtain an upper bound on the number of stable fixed points of CTLNs in cases where the conjecture holds.

Many important enumerative invariants of a matroid can be obtained from its Tutte polynomial, and many more are determined by two stronger invariants, the $G$-invariant and the configuration of the matroid. We show that the same is not true of the most basic connectivity invariants. Specifically, we show that for any positive integer n, there are pairs of matroids that have the same configuration (and so the same $G$-invariant and the same Tutte polynomial) but the difference between their Tutte connectivities exceeds n, and likewise for vertical connectivity and branch-width. The examples that we use to show this, which we construct using an operation that we introduce, are transversal matroids that are also positroids.

We introduce a population model to test the hypothesis that even a single migrant per generation may rescue a dying population. Let $(c_k:k\in N)$ be a sequence of real numbers in $(0,1)$. Let $X_n$ be a size of the population at time $n\ge 0$. Then, $X_{n+1}=X_n-Y_{n+1}+1$, where the conditional distribution of $Y_{n+1}$ given $X_n=k$ is a binomial random variable with parameters $(k,c(k\left)\right)$. We assume that ${\lim }_{k\to \infty }kc\left(k\right)=\rho$ exists. If $\rho <1$ the process is transient with speed $1-\rho$. So for our model a single migrant per generation may rescue a dying population! If $\rho >1$ the process is positive recurrent. In the critical case $\rho =1$ the process is recurrent or transient according to how $kc\left(k\right)$ converges to 1. When $\rho =0$ and under some regularity conditions, the support of the increments is eventually finite.

Extending the classical pop-stack sorting map on the lattice given by the right weak order on $S_n$, Defant defined, for any lattice M, a map ${\mathrm{Pop}}_M:M\to M$ that sends an element $x\in M$ to the meet of x and the elements covered by x. In parallel with the line of studies on the image of the classical pop-stack sorting map, we study ${\mathrm{Pop}}_M\left(M\right)$ when M is the weak order of type $B_n$, the Tamari lattice of type $B_n$, the lattice of order ideals of the root poset of type $A_n$, and the lattice of order ideals of the root poset of type $B_n$. In particular, we settle four conjectures proposed by Defant and Williams on the generating function$\mathrm{Pop}(M;q)=\sum _{b\in {\mathrm{Pop}}_M\left(M\right)}{q}^{\left|U_M\right(b\left)\right|},$ where $U_M\left(b\right)$ is the set of elements of M that cover b.

The Boros-Moll numbers $d_i\left(m\right)$ arise from a quartic integral studied by Boros and Moll. For fixed m, the sequence ${\left\{d_i\right(m\left)\right\}}_{0\le i\le m}$ has been proven to satisfy the Turán inequality, the higher order Turán inequality and 3-log-concavity which are originated from the Laguerre-Pólya class. In this paper, we give sharper bounds for both $d_i(m+1)/d_i\left(m\right)$ and $d_i{\left(m\right)}^2/\left(d_{i-1}\right(m\left)d_{i+1}\right(m\left)\right)$. Applying these bounds, we prove a series of results on the log-behavior, the higher order Turán inequality and the Laguerre inequalities of $d_i\left(m\right)$ for fixed i. In our proofs, we use Mathematica as an auxiliary tool to prove inequalities involving several variables. Moreover, we propose a series of open problems.

A locally uniform random permutation is generated by sampling n points independently from some absolutely continuous distribution ρ on the plane and interpreting them as a permutation by the rule that i maps to j if the ith point from the left is the jth point from below. As n tends to infinity, decreasing subsequences in the permutation will appear as curves in the plane, and by interpreting these as level curves, a union of decreasing subsequences gives rise to a surface. In a recent paper by the author it was shown that, for any $r\ge 0$, under the correct scaling as n tends to infinity, the surface of the largest union of $\lfloor r\sqrt{n}\rfloor$ decreasing subsequences approaches a limit in the sense that it will come close to a maximizer of a specific variational integral (and, under reasonable assumptions, that the maximizer is essentially unique). In the present paper we show that there exists a continuous maximizer, provided that ρ has bounded density and support.

The key ingredient in the proof is a new theorem about real functions of two variables that are increasing in both variables: We show that, for any constant C, any such function can be made continuous without increasing the diameter of its image or decreasing anywhere the product of its partial derivatives clipped by C, that is the minimum of the product and C.