Advances in Applied Mathematics最新文献

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.

We use the Poisson kernel of the continuous q-Hermite polynomials to introduce families of integral operators. One of them is semigroups of linear operators. We describe the eigenvalues and eigenfunctions of one family of operators. The action of the semigroups of operators on the Askey–Wilson polynomials is shown to only change the parameters but preserves the degrees, hence we produce transmutation relations for the Askey–Wilson polynomials. The transmutation relations are then used to derive bilinear generating functions involving the Askey–Wilson polynomials.

The entropy of a semiring ideal of tropical polynomials is introduced. The radical of a semiring ideal consists of all tropical polynomials vanishing on the tropical prevariety determined by the ideal. We prove that the entropy of the radical of a tropical bivariate polynomial with zero coefficients vanishes. Also, we prove that the entropy of a zero-dimensional tropical prevariety vanishes. An example of a non-radical semiring ideal with a positive entropy is exhibited.

We introduce a sheaf-theoretic stability condition for finite acyclic quivers. Our main result establishes that for representations of affine type $\tilde{A}$ quivers, there is a precise relationship between the associated Harder-Narasimhan filtration and the barcode of the periodic zigzag persistence module obtained by unwinding the underlying quiver.