Sampling in a weighted Sobolev space

We show that functions f in some weighted Sobolev space are completely determined by time-frequency samples ${\{f(t_n)\}}_{n\in \mathbb{Z}}\cup {\{\stackrel{?}{f}({\lambda }_k)\}}_{k\in \mathbb{Z}}$ along appropriate slowly increasing sequences ${\{t_n\}}_{n\in \mathbb{Z}}$ and ${\{{\lambda }_n\}}_{n\in \mathbb{Z}}$ tending to ±∞ as $n\to \pm \infty$.     

All double generalized Petersen graphs are Hamiltonian

The double generalized Petersen graph $DP(n,t)$, $n\ge 3$ and $t\in {\mathbb{Z}}_n?\left\{0\right\}$, $2\le 2t<n$, has vertex-set $\{x_i,y_i,u_i,v_i\mid i\in {\mathbb{Z}}_n\}$, edge-set $\{\{x_i,x_{i+1}\},\{y_i,y_{i+1}\},\{u_i,v_{i+t}\},\{v_i,u_{i+t}\},\{x_i,u_i\},\{y_i,v_i\}\mid i\in {\mathbb{Z}}_n\}$. These graphs were first defined by Zhou and Feng as examples of vertex-transitive non-Cayley graphs. Then, Kutnar and Petecki considered the structural properties, Hamiltonicity properties, vertex-coloring and edge-coloring of $DP(n,t)$, and conjectured that all $DP(n,t)$ are Hamiltonian. In this paper, we prove this conjecture.     

On tiling the integers with 4-sets of the same gap sequence

Partitioning a set into similar, if not, identical, parts is a fundamental research topic in combinatorics. The question of partitioning the integers in various ways has been considered throughout history. Given a set $\{x_1,\dots ,x_n\}$ of integers where $x_1<?<x_n$, let the gap sequence of this set be the unordered multiset $\{d_1,\dots ,d_{n?1}\}=\{x_{i+1}?x_i:i\in \{1,\dots ,n?1\}\}$. This paper addresses the following question, which was explicitly asked by Nakamigawa: can the set of integers be partitioned into sets with the same gap sequence? The question is known to be true for any set where the gap sequence has length at most two. This paper provides evidence that the question is true when the gap sequence has length three. Namely, we prove that given positive integers $p$ and $q$, there is a positive integer $r_0$ such that for all $r\ge r_0$, the set of integers can be partitioned into 4-sets with gap sequence $p,q$, $r$.     

On parabolic Kazhdan–Lusztig R-polynomials for the symmetric group

Parabolic R-polynomials were introduced by Deodhar as parabolic analogues of ordinary R-polynomials defined by Kazhdan and Lusztig. In this paper, we are concerned with the computation of parabolic R-polynomials for the symmetric group. Let $S_n$ be the symmetric group on $\{1,2,\dots ,n\}$, and let $S=\{s_i|1\le i\le n?1\}$ be the generating set of $S_n$, where for $1\le i\le n?1$, $s_i$ is the adjacent transposition. For a subset $J?S$, let ${(S_n)}_J$ be the parabolic subgroup generated by J, and let ${(S_n)}^J$ be the set of minimal coset representatives for $S_n/{(S_n)}_J$. For $u\le v\in {(S_n)}^J$ in the Bruhat order and $x\in \{q,?1\}$, let $R_{u,v}^{J,x}(q)$ denote the parabolic R-polynomial indexed by u and v. Brenti found a formula for $R_{u,v}^{J,x}(q)$ when $J=S?\{s_i\}$, and obtained an expression for $R_{u,v}^{J,x}(q)$ when $J=S?\{s_{i?1},s_i\}$. In this paper, we provide a formula for $R_{u,v}^{J,x}(q)$, where $J=S?\{s_{i?2},s_{i?1},s_i\}$ and i appears after $i?1$ in v. It should be noted that the condition that i appears after $i?1$ in v is equivalent to that v is a permutation in ${(S_n)}^{S?\{s_{i?2},s_i\}}$. We also pose a conjecture for $R_{u,v}^{J,x}(q)$, where $J=S?\{s_k,s_{k+1},\dots ,s_i\}$ with $1\le k\le i\le n?1$ and v is a permutation in ${(S_n)}^{S?\{s_k,s_i\}}$.     

Minimal solutions of the isometry equation

For a finite vector space $W$ over ${\mathbb{F}}_q$, there are described all the pairs of multisets $\{V_1,\dots ,V_{q+1}\}$ and $\{U_1,\dots ,U_{q+1}\}$ of subspaces in $W$ such that for all $w\in W$ the equality $\left|\{i\mid w\in V_i\}\right|=\left|\{i\mid w\in U_i\}\right|$ holds.     

Approximation schemes satisfying Shapiro’s Theorem

An approximation scheme is a family of homogeneous subsets $\left(A_n\right)$ of a quasi-Banach space $X$, such that $A_1?A_2?\dots ?X$, $A_n+A_n?A_{K\left(n\right)}$, and $\overline{{\cup }_n{A}_n}=X$. Continuing the line of research originating at the classical paper [8] by Bernstein, we give several characterizations of the approximation schemes with the property that, for every sequence $\left\{{\epsilon }_n\right\}\searrow 0$, there exists $x\in X$ such that $\mathrm{dist}(x,A_n)\ne O\left({\epsilon }_n\right)$ (in this case we say that $(X,\left\{A_n\right\})$ satisfies Shapiro’s Theorem). If $X$ is a Banach space, $x\in X$ as above exists if and only if, for every sequence $\left\{{\delta }_n\right\}\searrow 0$, there exists $y\in X$ such that $\mathrm{dist}(y,A_n)\ge {\delta }_n$. We give numerous examples of approximation schemes satisfying Shapiro’s Theorem.     

Multivariate polynomial interpolation and sampling in Paley–Wiener spaces

In this paper, an equivalence between existence of particular exponential Riesz bases for spaces of multivariate bandlimited functions and existence of certain polynomial interpolants for functions in these spaces is given. Namely, polynomials are constructed which, in the limiting case, interpolate ${\left\{({\tau }_n,f\left({\tau }_n\right))\right\}}_n$ for certain classes of unequally spaced data nodes ${\left\{{\tau }_n\right\}}_n$ and corresponding ${?}_2$ sampled data ${\left\{f\left({\tau }_n\right)\right\}}_n$. Existence of these polynomials allows one to construct a simple sequence of approximants for an arbitrary multivariate bandlimited function $f$ which demonstrates $L_2$ and uniform convergence on ${\mathbb{R}}^d$ to $f$. A simpler computational version of this recovery formula is also given at the cost of replacing $L_2$ and uniform convergence on ${\mathbb{R}}^d$ with $L_2$ and uniform convergence on increasingly large subsets of ${\mathbb{R}}^d$. As a special case, the polynomial interpolants of given ${?}_2$ data converge in the same fashion to the multivariate bandlimited interpolant of that same data. Concrete examples of pertinent Riesz bases and unequally spaced data nodes are also given.     

A new family of exponential iteration methods with quadratic convergence of both diameters and points for enclosing zeros of nonlinear equations

This paper presents a new class of exponential iteration methods with global convergence for finding a simple root $x^1$ of a nonlinear equation $f\left(x\right)=0$ in the interval $[a,b]$. The new methods are shown to be quadratically convergent. Both the sequences of diameters $\{b_n-a_n\}$ and the iterative sequence $\{x_n-x^1\}$ are quadratically convergent to zero. The theoretical analysis and numerical experiments show that new exponential iterative formulae are effective and comparable to those of the well-known Newton and Steffensen methods.     

Primeness property for graded central polynomials of verbally prime algebras

Let F be an infinite field. The primeness property for central polynomials of $M_n(F)$ was established by A. Regev, i.e., if the product of two polynomials in distinct variables is central then each factor is also central. In this paper we consider the analogous property for $M_n(F)$ and determine, within the elementary gradings with commutative neutral component, the ones that satisfy this property, namely the crossed product gradings. Next we consider $M_n(R)$, where R admits a regular grading, with a grading such that $M_n(F)$ is a homogeneous subalgebra and provide sufficient conditions – satisfied by $M_n(E)$ with the trivial grading – to prove that $M_n(R)$ has the primeness property if $M_n(F)$ does. We also prove that the algebras $M_{a,b}(E)$ satisfy this property for ordinary central polynomials. Hence we conclude that, over a field of characteristic zero, every verbally prime algebra has the primeness property.