Approximations in Sobolev spaces by prolate spheroidal wave functions

Recently, there is a growing interest in the spectral approximation by the Prolate Spheroidal Wave Functions (PSWFs) ${\psi }_{n,c},c>0$. This is due to the promising new contributions of these functions in various classical as well as emerging applications from Signal Processing, Geophysics, Numerical Analysis, etc. The PSWFs form a basis with remarkable properties not only for the space of band-limited functions with bandwidth c, but also for the Sobolev space $H^s([?1,1])$. The quality of the spectral approximation and the choice of the parameter c when approximating a function in $H^s([?1,1])$ by its truncated PSWFs series expansion, are the main issues. By considering a function $f\in H^s([?1,1])$ as the restriction to $[?1,1]$ of an almost time-limited and band-limited function, we try to give satisfactory answers to these two issues. Also, we illustrate the different results of this work by some numerical examples.     

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.     

Superconcentration,and randomized Dvoretzky's theorem for spaces with 1-unconditional bases

Let n be a sufficiently large natural number and let B be an origin-symmetric convex body in ${\mathbb{R}}^n$in the ?-position, and such that the space $({\mathbb{R}}^n,{6?6}_B)$ admits a 1-unconditional basis. Then for any $\epsilon \in (0,1/2]$, and for random $c\epsilon \log ?n/\log ?\frac{1}{\epsilon }$-dimensional subspace E distributed according to the rotation-invariant (Haar) measure, the section $B\cap E$ is $(1+\epsilon )$-Euclidean with probability close to one. This shows that the “worst-case” dependence on ε in the randomized Dvoretzky theorem in the ?-position is significantly better than in John's position. It is a previously unexplored feature, which has strong connections with the concept of superconcentration introduced by S. Chatterjee. In fact, our main result follows from the next theorem: Let B be as before and assume additionally that B has a smooth boundary and ${\mathbb{E}}_{{\gamma }_n}{6?6}_B\le n^c{\mathbb{E}}_{{\gamma }_n}{6{\mathrm{grad}}_B(?)6}_2$ for a small universal constant $c>0$, where ${\mathrm{grad}}_B(?)$ is the gradient of ${6?6}_B$ and ${\gamma }_n$ is the standard Gaussian measure in ${\mathbb{R}}^n$. Then for any $p\in [1,c\log ?n]$ the p-th power of the norm ${6?6}_B^p$ is $\frac{C}{\log ?n}$-superconcentrated in the Gauss space.     

On the distance between linear codes

Let V be an n-dimensional vector space over the finite field consisting of q elements and let ${\mathrm{\Gamma }}_k(V)$ be the Grassmann graph formed by k-dimensional subspaces of V, $1<k<n-1$. Denote by $\mathrm{\Gamma }{(n,k)}_q$ the restriction of ${\mathrm{\Gamma }}_k(V)$ to the set of all non-degenerate linear ${[n,k]}_q$ codes. We show that for any two codes the distance in $\mathrm{\Gamma }{(n,k)}_q$ coincides with the distance in ${\mathrm{\Gamma }}_k(V)$ only in the case when $n<{(q+1)}^2+k-2$, i.e. if n is sufficiently large then for some pairs of codes the distances in the graphs ${\mathrm{\Gamma }}_k(V)$ and $\mathrm{\Gamma }{(n,k)}_q$ are distinct. We describe one class of such pairs.     

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 $n\to \pm \infty$.     

On a theorem by Brewer

One of the most frequently referenced monographs on power series rings, “Power Series over Commutative Rings” by James W. Brewer, states in Theorem 21 that if M is a non-SFT maximal ideal of a commutative ring R with identity, then there exists an infinite ascending chain of prime ideals in the power series ring $R?X?$, $Q_0?Q_1???Q_n??$ such that $Q_n\cap R=M$ for each n. Moreover, the height of $M?X?$ is infinite. In this paper, we show that the above theorem is false by presenting two counter examples. The first counter example shows that the height of $M?X?$ can be zero (and hence there is no chain $Q_0?Q_1???Q_n??$ of prime ideals in $R?X?$ satisfying $Q_n\cap R=M$ for each n). In this example, the ring R is one-dimensional. In the second counter example, we prove that even if the height of $M?X?$ is uncountably infinite, there may be no infinite chain $\{Q_n\}$ of prime ideals in $R?X?$ satisfying $Q_n\cap R=M$ for each n.     

Denjoy minimal sets and Birkhoff periodic orbits for non-exact monotone twist maps

A non-exact monotone twist map ${\overline{\phi }}_F$ is a composition of an exact monotone twist map $\overline{\phi }$ with a generating function H and a vertical translation $V_F$ with $V_F((x,y))=(x,y?F)$. We show in this paper that for each $\omega \in \mathbb{R}$, there exists a critical value $F_d(\omega )\ge 0$ depending on H and ω such that for $0\le F\le F_d(\omega )$, the non-exact twist map ${\overline{\phi }}_F$ has an invariant Denjoy minimal set with irrational rotation number ω lying on a Lipschitz graph, or Birkhoff $(p,q)$-periodic orbits for rational $\omega =p/q$. Like the Aubry–Mather theory, we also construct heteroclinic orbits connecting Birkhoff periodic orbits, and show that quasi-periodic orbits in these Denjoy minimal sets can be approximated by periodic orbits. In particular, we demonstrate that at the critical value $F=F_d(\omega )$, the Denjoy minimal set is not uniformly hyperbolic and can be approximated by smooth curves.     

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 ${(S_n)}^{S?\{s_k,s_i\}}$.     

Functional limit theorems for strongly subcritical branching processes in random environment

For a strongly subcritical branching process $(Z_n{)}_{n⩾0}$ in random environment the non-extinction probability at generation n decays at the same exponential rate as the expected generation size and given non-extinction at n the conditional distribution of $Z_n$ has a weak limit. Here we prove conditional functional limit theorems for the generation size process $(Z_k{)}_{0⩽k⩽n}$ as well as for the random environment. We show that given the population survives up to generation n the environmental sequence still evolves in an i.i.d. fashion and that the conditioned generation size process converges in distribution to a positive recurrent Markov chain.     

Current superalgebras and unitary representations

In this paper we determine the projective unitary representations of finite dimensional Lie supergroups whose underlying Lie superalgebra is $\mathfrak{g}=A?\mathfrak{k}$, where $\mathfrak{k}$ is a compact simple Lie superalgebra and A is a supercommutative associative (super)algebra; the crucial case is when $A={\mathrm{\Lambda }}_s(\mathbb{R})$ is a Graßmann algebra. Since we are interested in projective representations, the first step consists in determining the cocycles defining the corresponding central extensions. Our second main result asserts that, if $\mathfrak{k}$ is a simple compact Lie superalgebra with ${\mathfrak{k}}_1\ne \{0\}$, then each (projective) unitary representation of ${\mathrm{\Lambda }}_s(\mathbb{R})?\mathfrak{k}$ factors through a (projective) unitary representation of $\mathfrak{k}$ itself, and these are known by Jakobsen's classification. If ${\mathfrak{k}}_1=\{0\}$, then we likewise reduce the classification problem to semidirect products of compact Lie groups K with a Clifford–Lie supergroup which has been studied by Carmeli, Cassinelli, Toigo and Varadarajan.