The Helmholtz equation in heterogeneous media: A priori bounds,well-posedness,and resonances

We consider the exterior Dirichlet problem for the heterogeneous Helmholtz equation, i.e. the equation $\mathrm{?}?(A\mathrm{?}u)+k^{2}nu=?f$ where both A and n are functions of position. We prove new a priori bounds on the solution under conditions on A, n, and the domain that ensure nontrapping of rays; the novelty is that these bounds are explicit in k, A, n, and geometric parameters of the domain. We then show that these a priori bounds hold when A and n are $L^{\infty }$ and satisfy certain monotonicity conditions, and thereby obtain new results both about the well-posedness of such problems and about the resonances of acoustic transmission problems (i.e. A and n discontinuous) where the transmission interfaces are only assumed to be $C^0$ and star-shaped; the novelty of this latter result is that until recently the only known results about resonances of acoustic transmission problems were for $C^{\infty }$ convex interfaces with strictly positive curvature.     

The symbolic defect of an ideal

Let I be a homogeneous ideal of $\mathbb{k}[x_0,\dots ,x_n]$. To compare $I^{(m)}$, the m-th symbolic power of I, with $I^m$, the regular m-th power, we introduce the m-th symbolic defect of I, denoted $\mathrm{sdefect}(I,m)$. Precisely, $\mathrm{sdefect}(I,m)$ is the minimal number of generators of the R-module $I^{(m)}/I^m$, or equivalently, the minimal number of generators one must add to $I^m$ to make $I^{(m)}$. In this paper, we take the first step towards understanding the symbolic defect by considering the case that I is either the defining ideal of a star configuration or the ideal associated to a finite set of points in ${\mathbb{P}}^2$. We are specifically interested in identifying ideals I with $\mathrm{sdefect}(I,2)=1$.     

Schatten properties,nuclearity and minimality of phase shift invariant spaces

We extend Feichtinger's minimality property on the smallest non-trivial time-frequency shift invariant Banach space, to the quasi-Banach case. Analogous properties are deduced for certain matrix spaces.We use these results to prove that the pseudo-differential operator $\mathrm{Op}(a)$ is a Schatten-q operator from $M^{\infty }$ to $M^p$ and r-nuclear operator from $M^{\infty }$ to $M^r$ when $a\in M^r$ for suitable p, q and r in $(0,\infty ]$.     

Stabilization of Boij–Söderberg decompositions of ideal powers

Given an ideal I we investigate the decompositions of Betti diagrams of the graded family of ideals ${\{I^k\}}_k$ formed by taking powers of I. We prove conjectures of Engström from [5] and show that there is a stabilization in the Boij–Söderberg decompositions of $I^k$ for $k>>0$ when I is a homogeneous ideal with generators in a single degree. In particular, the number of terms in the decompositions with positive coefficients remains constant for $k>>0$, the pure diagrams appearing in each decomposition have the same shape, and the coefficients of these diagrams are given by polynomials in k. We also show that a similar result holds for decompositions with arbitrary coefficients arising from other chains of pure diagrams.     

Projective generation of ideals in a certain ring

Let R be an affine domain of dimension $n\ge 3$ over a field of characteristic 0 and $D=R[X,Y]/(XY)$. Let $I?D$ be a local complete intersection ideal of height n such that $\mu (I/I^2)=n$. This paper examines under what condition I is surjective image of a projective D-module of rank n.     

Cancellation of projective modules over polynomial extensions over a two-dimensional ring

Let R be a commutative Noetherian ring of dimension two with $1/2\in R$ and let $A=R[X_1,?,X_n]$. Let P be a projective A-module of rank 2. In this article, we prove that P is cancellative if ${\wedge }^2(P)\oplus A$ is cancellative.     

On invariant fields of vectors and covectors

Let ${\mathbb{F}}_q$ be the finite field of order q. Let G be one of the three groups $\mathrm{GL}(n,{\mathbb{F}}_q)$, $\mathrm{SL}(n,{\mathbb{F}}_q)$ or $\mathrm{U}(n,{\mathbb{F}}_q)$ and let W be the standard n-dimensional representation of G. For non-negative integers m and d we let $mW\oplus d{W}^{?}$ denote the representation of G given by the direct sum of m vectors and d covectors. We exhibit a minimal set of homogeneous invariant polynomials $\{{?}_1,{?}_2,\dots ,{?}_{(m+d)n}\}?{\mathbb{F}}_q{[mW\oplus d{W}^{?}]}^G$ such that ${\mathbb{F}}_q{(mW\oplus d{W}^{?})}^G={\mathbb{F}}_q({?}_1,{?}_2,\dots ,{?}_{(m+d)n})$ for all cases except when $md=0$ and $G=\mathrm{GL}(n,{\mathbb{F}}_q)$ or $\mathrm{SL}(n,{\mathbb{F}}_q)$.     

Representations of elementary abelian p-groups and finite subgroups of fields

Suppose $\mathbb{F}$ is a field of prime characteristic p and E is a finite subgroup of the additive group $(\mathbb{F},+)$. Then E is an elementary abelian p-group. We consider two such subgroups, say E and $E^{\prime }$, to be equivalent if there is an $\alpha \in {\mathbb{F}}^{\times }:=\mathbb{F}?\{0\}$ such that $E=\alpha E^{\prime }$. In this paper we show that rational functions can be used to distinguish equivalence classes of subgroups and, for subgroups of prime rank or rank less than twelve, we give explicit finite sets of separating invariants.     

Characterizations and constructions of n-to-1 mappings over finite fields

n-to-1 mappings have wide applications in many areas, especially in cryptography, finite geometry, coding theory and combinatorial design. In this paper, many classes of n-to-1 mappings over finite fields are studied. First, we provide a characterization of general n-to-1 mappings over ${\mathbb{F}}_{p^m}$ by means of the Walsh transform. Then, we completely determine 3-to-1 polynomials with degree no more than 4 over ${\mathbb{F}}_{p^m}$. Furthermore, we obtain an AGW-like criterion for characterizing some close relationship between the n-to-1 property of a mapping over finite set A and that of another mapping over a subset of A. Finally, we apply the AGW-like criterion into several forms of polynomials and obtain some explicit n-to-1 mappings. Especially, three explicit constructions of the form $x^{r}h\left(x^s\right)$ from the cyclotomic perspective, and several classes of n-to-1 mappings of the form $g(x^{q^k}-x+\delta )+cx$ are provided.     

The Hilbert series and a-invariant of circle invariants

Let V be a finite-dimensional representation of the complex circle ${\mathbb{C}}^{\times }$ determined by a weight vector $\mathit{a}\in {\mathbb{Z}}^n$. We study the Hilbert series ${\mathrm{Hilb}}_{\mathit{a}}(t)$ of the graded algebra $\mathbb{C}{[V]}^{{\mathbb{C}}_{\mathit{a}}^{\times }}$ of polynomial ${\mathbb{C}}^{\times }$-invariants in terms of the weight vector a of the ${\mathbb{C}}^{\times }$-action. In particular, we give explicit formulas for ${\mathrm{Hilb}}_{\mathit{a}}(t)$ as well as the first four coefficients of the Laurent expansion of ${\mathrm{Hilb}}_{\mathit{a}}(t)$ at $t=1$. The naive formulas for these coefficients have removable singularities when weights pairwise coincide. Identifying these cancelations, the Laurent coefficients are expressed using partial Schur polynomials that are independently symmetric in two sets of variables. We similarly give an explicit formula for the a-invariant of $\mathbb{C}{[V]}^{{\mathbb{C}}_{\mathit{a}}^{\times }}$ in the case that this algebra is Gorenstein. As an application, we give methods to identify weight vectors with Gorenstein and non-Gorenstein invariant algebras.