Advances in Applied Mathematics最新文献

We study the enumeration of off-diagonally symmetric domino tilings of odd-order Aztec diamonds in two directions: (1) with one boundary defect, and (2) with maximally-many zeroes on the diagonal. In the first direction, we prove a symmetry property which states that the numbers of off-diagonally symmetric domino tilings of the Aztec diamond of order $2n-1$ are equal when the boundary defect is at the kth position and the $(2n-k)$th position on the boundary, respectively. This symmetry property proves a special case of a recent conjecture by Behrend, Fischer, and Koutschan.

In the second direction, a Pfaffian formula is obtained for the number of “nearly” off-diagonally symmetric domino tilings of odd-order Aztec diamonds, where the entries of the Pfaffian satisfy a simple recurrence relation. The numbers of domino tilings mentioned in the above two directions do not seem to have a simple product formula, but we show that these numbers satisfy simple matrix equations in which the entries of the matrix are given by Delannoy numbers. The proof of these results involves the method of non-intersecting lattice paths and a modification of Stembridge's Pfaffian formula for families of non-intersecting lattice paths. Finally, we propose conjectures concerning the log-concavity and asymptotic behavior of the number of off-diagonally symmetric domino tilings of odd-order Aztec diamonds.

The classical Crofton formula explains how intrinsic volumes of a convex body K in n-dimensional Euclidean space can be obtained from integrating a measurement function at sections of K with invariantly moved affine flats. We generalize this idea by constructing stereologically adapted Crofton formulae for translation invariant Minkowski tensors, expressing a prescribed tensor valuation as an invariant integral of a measurement function of section profiles with flats. The measurement functions are weighed sums of powers of the metric tensor times Minkowski valuations. The weights are determined explicitly from known Crofton formulae using Zeilberger's algorithm. The main result is an exhaustive set of measurement functions where the invariant integration is over flats.

With the main result at hand, a Blaschke-Petkantschin formula allows us to establish new measurement functions valid when the invariant integration over flats is replaced by an invariant integration over subspaces containing a fixed subspace of lower dimension. Likewise, a stereologically adapted Crofton formula valid in the scheme of vertical sections is constructed. Only some special cases of this result have been stated explicitly before, with even the three-dimensional case yielding a new stereological formula. Here, we obtain new vertical section formulae for the surface tensors of even rank.

In this paper we provide an identity between the determinant and other generalized matrix functions, and give a criterion for positive semi-definite matrices to satisfy the permanental dominance conjecture. As a consequence, infinitely many classes of positive semi-definite matrices satisfying the conjecture (does not depend on groups or characters) are provided by generating from any positive semi-definite matrix having no zero in the first column.

As a unification of increasing trees and plane trees, the weakly increasing trees labeled by a multiset was introduced by Lin–Ma–Ma–Zhou (2021). Various intriguing connections and bijections for weakly increasing trees have already been found and the purpose of this paper is to present yet more bijective combinatorics on this unified object. Two of our main contributions are

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  extension of an equidistribution result on plane trees due to Eu–Seo–Shin (2017), regarding levels and degrees of nodes, to weakly increasing trees;

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  a new interpretation of the multiset Schett polynomials in terms of odd left/right chains on weakly increasing binary trees.

Generalizing the idea of elementary simplicial collapses and expansions in classical simple homotopy theory to a stratified setting, we find local combinatorial transformations on stratified simplicial complexes that leave the global stratified homotopy type invariant. In particular, we obtain the notions of stratified formal deformations generalizing J. H. C. Whitehead's formal deformations. We implement the algorithmic execution of such transformations and the computation of intersection homology to illustrate the behavior of stratified simple homotopy equivalences on Vietoris-Rips type complexes associated to point sets sampled near given, possibly singular, spaces.

We construct the s-Gauss probability space by introducing the s-Gaussian density function in $R^n$ for $s\ge 0$, a generalization of the classic Gaussian density function. Based on the s-Gaussian density function, we propose the $(s,k)$-Ehrhard symmetrization which is an extension of the traditional Ehrhard symmetrization for sets in $R^n$. In particular, we establish the s-Gaussian isoperimetric inequality with respect to s-Gaussian measure in $R^2$. Furthermore, we propose and prove the s-Ehrhard-Borell inequalities for $s>0$ when one of the two sets is a Borel set whilst the other being a convex set as well as the case when two sets are convex in $R^1$ with different methods.

There are a number of sporadic coefficient-vanishing results associated with theta series, which suggest certain underlying patterns. By expanding theta powers as linear combinations of products of theta functions, we present two strategies that will provide a unified treatment. Our approaches rely on studying the behavior of products of two theta series under the action of the huffing operator. For this purpose, some explicit criteria are given. We may use the presented methods to not only verify experimentally discovered coefficient-vanishing results, but also to produce a series of general phenomena.

For a subset D of boxes in an $n\times n$ square grid, let ${\chi }_D\left(x\right)$ denote the dual character of the flagged Weyl module associated to D. It is known that ${\chi }_D\left(x\right)$ specifies to a Schubert polynomial (resp., a key polynomial) in the case when D is the Rothe diagram of a permutation (resp., the skyline diagram of a composition). One can naturally define a lower and an upper bound of ${\chi }_D\left(x\right)$. Mészáros, St. Dizier and Tanjaya conjectured that ${\chi }_D\left(x\right)$ attains the upper bound if and only if D avoids a certain single subdiagram. We provide a proof of this conjecture.

We provide a new algebraic solution procedure for the global positioning problem in n dimensions using m satellites. We also give a geometric characterization of the situations in which the problem does not have a unique solution. This characterization shows that such cases can happen in any dimension and with any number of satellites, leading to counterexamples to some open conjectures. We fill a gap in the literature by giving a proof for the long-held belief that when $m\ge n+2$, the solution is unique for almost all user positions. Even better, when $m\ge 2n+2$, almost all satellite configurations will guarantee a unique solution for all user positions. Our uniqueness results provide a basis for predicting the behavior of numerical solutions, as ill-conditioning is expected near the threshold between areas of nonuniqueness and uniqueness. Some of our results are obtained using tools from algebraic geometry.