Estimates in Beurling-Helson type theorems: Multidimensional case

We consider the spaces A _p ( $\mathbb{T}^m $ ) of functions f on the m-dimensional torus $\mathbb{T}^m $ such that the sequence of Fourier coefficients $\hat f = \{ \hat f(k),k \in \mathbb{Z}^m \} $ belongs to l ^p (? ^m ), 1 ≤ p < 2. The norm on A _p ( $\mathbb{T}^m $ ) is defined by $\left\| f \right\|_{A_p (\mathbb{T}^m )} = \left\| {\hat f} \right\|_{l^p (\mathbb{Z}^m )} $ . We study the rate of growth of the norms $\left\| {e^{i\lambda \phi } } \right\|_{A_p (\mathbb{T}^m )} $ as |λ| → ∞, λ ∈ ?, for C ¹-smooth real functions φ on $\mathbb{T}^m $ (the one-dimensional case was investigated by the author earlier). The lower estimates that we obtain have direct analogs for the spaces A _p (? ^m ).     

Isometric Immersions into {\mathbb{S}^m \times \mathbb{R}} and {\mathbb{H}^m \times \mathbb{R}} with High Codimensions

In this paper, we obtain sufficient and necessary conditions for a simply connected Riemannian manifold (M ⁿ , g) to be isometrically immersed into ${\mathbb{S}^m \times \mathbb{R}}$ and ${\mathbb{H}^m \times \mathbb{R}}$ .     

Vector bundles over iterated suspensions of stunted real projective spaces

Let $X^{k}_{m,n}=\Sigma^{k} (\mathbb{R}\mathbb{P}^{m}/\mathbb{R}\mathbb{P}^{n})$ . In this note we completely determine the values of k, m, n for which the total Stiefel–Whitney class w(ξ)=1 for any vector bundle ξ over  $X^{k}_{m,n}$ .     

Blocks with Defect Group \boldsymbol{Q_{2^n}\times C_{2^m}} and \boldsymbol{SD_{2^n}\times C_{2^m}}

We determine the numerical invariants of blocks with defect group $Q_{2^n}\times C_{2^m}$ and $SD_{2^n}\times C_{2^m}$ , where $Q_{2^n}$ denotes a quaternion group of order 2 ⁿ , $C_{2^m}$ denotes a cyclic group of order 2 ^m , and $SD_{2^n}$ denotes a semidihedral group of order 2 ⁿ . This generalizes Olsson’s results for m?=?0. As a consequence, we prove Brauer’s k(B)-Conjecture, Olsson’s Conjecture, Brauer’s Height-Zero Conjecture, the Alperin–McKay Conjecture, Alperin’s Weight Conjecture and Robinson’s Ordinary Weight Conjecture for these blocks. Moreover, we show that the gluing problem has a unique solution in this case. This paper follows (and uses) (Sambale, J Pure Appl Algebra 216:119–125, 2012; Proc Amer Math Soc, 2012).     

Приближение всплесками и поперечники Фурье классов периодических функций многих переменных. II

Estimates sharp in order for Fourier widths of the classes $ B_{pq}^{sm} (\mathbb{T}^k ) $ and $ L_{pq}^{sm} (\mathbb{T}^k ) $ of Nikol??skii-Besov and Lizorkin-Triebel types, respectively, in the space $ L_r (\mathbb{T}^k ) $ are established for a certain range of the parameters s, p, q, r (here s ?? (0,??) ⁿ , 1 ??p, r, q ???, 1 ?? n ?? k, m = (m ₁, ??,m _n ) ?? ? ⁿ : m ₁ + ?? + m _n = k).     

The Generalization of the Decomposition of Functions by Energy Operators

This work starts with the introduction of a family of differential energy operators. Energy operators $({\varPsi}_{R}^{+}, {\varPsi}_{R}^{-})$ were defined together with a method to decompose the wave equation in a previous work. Here the energy operators are defined following the order of their derivatives $(\varPsi^{-}_{k}, \varPsi^{+}_{k}, k=\{0,\pm 1,\pm 2,\ldots\})$ . The main part of the work demonstrates for any smooth real-valued function f in the Schwartz space $(\mathbf{S}^{-}(\mathbb{R}))$ , the successive derivatives of the n-th power of f ( $n \in \mathbb{Z}$ and n≠0) can be decomposed using only $\varPsi^{+}_{k}$ (Lemma); or if f in a subset of $\mathbf{S}^{-}(\mathbb{R})$ , called $\mathbf{s}^{-}(\mathbb{R})$ , $\varPsi^{+}_{k}$ and $\varPsi^{-}_{k}$ ( $k\in \mathbb{Z}$ ) decompose in a unique way the successive derivatives of the n-th power of f (Theorem). Some properties of the Kernel and the Image of the energy operators are given along with the development. Finally, the paper ends with the application to the energy function.     

Homotopy and nonlinear boundary value problems involving singular {\phi}-Laplacians

Homotopy methods are used to find sufficient conditions for the solvability of nonlinear boundary value problems of the form $$(\phi(u^\prime))^\prime = f(t, u, u^\prime), \quad g(u(\alpha), \phi(u^\prime(\beta))) = 0,$$ where (α, β) = (0, 1), (1, 0), (0, 0) or (1, 1), ${\phi}$ is a homeomorphism from the open ball ${B(a) \subset \mathbb{R}^n}$ onto ${\mathbb{R}^n}$ , f is a Carathéodory function, ${g : \mathbb{R}^n \times \, \mathbb{R}^n \rightarrow \mathbb{R}^m}$ is continuous and m ≤ 2n.     

Grassmann and Weyl embeddings of orthogonal grassmannians

Given a non-singular quadratic form q of maximal Witt index on $V := V(2n+1,\mathbb{F})$ , let Δ be the building of type B _n formed by the subspaces of V totally singular for q and, for 1≤k≤n, let Δ _k be the k-grassmannian of Δ. Let ε _k be the embedding of Δ _k into PG(? ^k V) mapping every point 〈v ₁,v ₂,…,v _k 〉 of Δ _k to the point 〈v ₁∧v ₂∧?∧v _k 〉 of PG(? ^k V). It is known that if $\mathrm{char}(\mathbb{F})\neq2$ then $\mathrm{dim}(\varepsilon_{k})={{2n+1}\choose k}$ . In this paper we give a new very easy proof of this fact. We also prove that if $\mathrm{char}(\mathbb{F}) = 2$ then $\mathrm{dim}(\varepsilon_{k})={{2n+1}\choose k}-{{2n+1}\choose{k-2}}$ . As a consequence, when 1<k<n and $\mathrm{char}(\mathbb{F}) = 2$ the embedding ε _k is not universal. Finally, we prove that if $\mathbb{F}$ is a perfect field of characteristic p>2 or a number field, n>k and k=2 or 3, then ε _k is universal.     

A Congruence Connecting Latin Rectangles and Partial Orthomorphisms

A partial orthomorphism of ${\mathbb{Z}_{n}}$ is an injective map ${\sigma : S \rightarrow \mathbb{Z}_{n}}$ such that ${S \subseteq \mathbb{Z}_{n}}$ and ??(i)?Ci ? ??(j)? j (mod n) for distinct ${i, j \in S}$ . We say ?? has deficit d if ${|S| = n - d}$ . Let ??(n, d) be the number of partial orthomorphisms of ${\mathbb{Z}_{n}}$ of deficit d. Let ??(n, d) be the number of partial orthomorphisms ?? of ${\mathbb{Z}_n}$ of deficit d such that ??(i) ? {0, i} for all ${i \in S}$ . Then ??(n, d) =???(n, d)n ²/d ² when ${1\,\leqslant\,d < n}$ . Let R _{k, n} be the number of reduced k ×?n Latin rectangles. We show that $$R_{k, n} \equiv \chi (p, n - p)\frac{(n - p)!(n - p - 1)!^{2}}{(n - k)!}R_{k-p,\,n-p}\,\,\,\,(\rm {mod}\,p)$$ when p is a prime and ${n\,\geqslant\,k\,\geqslant\,p + 1}$ . In particular, this enables us to calculate some previously unknown congruences for R _{n, n} . We also develop techniques for computing ??(n, d) exactly. We show that for each a there exists??? _a such that, on each congruence class modulo??? _a , ??(n, n-a) is determined by a polynomial of degree 2a in n. We give these polynomials for ${1\,\leqslant\,a\,\leqslant 6}$ , and find an asymptotic formula for ??(n, n-a) as n ?? ??, for arbitrary fixed a.     

Asymptotic behavior of eigenvalues of the two-particle discrete Schrödinger operator

We consider two-particle Schrödinger operator H(k) on a three-dimensional lattice ? ³ (here k is the total quasimomentum of a two-particle system, $k \in \mathbb{T}^3 : = \left( { - \pi ,\pi ]^3 } \right)$ . We show that for any $k \in S = \mathbb{T}^3 \backslash ( - \pi ,\pi )^3$ , there is a potential $\hat v$ such that the two-particle operator H(k) has infinitely many eigenvalues z_n(k) accumulating near the left boundary m(k) of the continuous spectrum. We describe classes of potentials W(j) and W(ij) and manifolds S(j) ? S, i, j ∈ {1, 2, 3}, such that if k ∈ S(3), (k ₂ , k ₃ ) ∈ (?π,π) ² , and $\hat v \in W(3)$ , then the operator H(k) has infinitely many eigenvalues z_n(k) with an asymptotic exponential form as n → ∞ and if k ∈ S(i) ∩ S(j) and $\hat v \in W(ij)$ , then the eigenvalues z_nm(k) of H(k) can be calculated exactly. In both cases, we present the explicit form of the eigenfunctions.     

On binary Kloosterman sums divisible by 3

By counting the coset leaders for cosets of weight 3 of the Melas code we give a new proof for the characterization of Kloosterman sums divisible by 3 for ${\mathbb{F}_{2^m}}$ where m is odd. New results due to Charpin, Helleseth and Zinoviev then provide a connection to a characterization of all ${a\in\mathbb{F}_{2^m}}$ such that ${Tr(a^{1/3})=0}$ ; we prove a generalization to the case ${Tr(a^{1/(2^k-1)})=0}$ . We present an application to constructing caps in PG(n, 2) with many free pairs of points.     

Bases of trigonometric polynomials consisting of shifts of Dirichlet kernels

The system of shifts of Dirichlet kernel on \(\frac{{2k\pi }} {{2n + 1}} \) , k = 0, ± 1, …, ± n, and the system of such shifts of conjugate Dirichlet kernels with \(\frac{1} {2} \) are orthogonal bases in the space of trigonometric polynomials of degree n. The system of shifts of the kernels \(\Sigma _{k = m}^n \) cos kx and \(\Sigma _{k = m}^n \) sin kx on \(\frac{{2k\pi }} {{n - m + 1}} \) , k = 0, 1, …, n?m, is an orthogonal basis in the space of trigonometric polynomials with the components from m ? 1 to n. There is no orthogonal basis of shifts of any function in this space for 0 < m < n.     

Dedekind zeta motives for totally real number fields

Let k be a totally real number field. For every odd n≥3, we construct an element in the category MT(k) of mixed Tate motives over k out of the quotient of a product of hyperbolic spaces by an arithmetic group. By a volume calculation, we prove that its period is a rational multiple of $\pi^{n[k:\mathbb{Q}]}\zeta^{*}_{k}(1-n)$ , where $\zeta^{*}_{k}(1-n)$ denotes the special value of the Dedekind zeta function of k. We deduce that the group $\mathrm {Ext}^{1}_{\mathrm {MT}(k)} (\mathbb{Q}(0),\mathbb{Q}(n))$ is generated by the cohomology of a quadric relative to hyperplanes, and that $\zeta^{*}_{k}(1-n)$ is a determinant of volumes of geodesic hyperbolic simplices defined over k.     

Lipschitz Homotopy Groups of the Heisenberg Groups

Lipschitz and horizontal maps from an n-dimensional space into the (2n + 1)-dimensional Heisenberg group ${\mathbb{H}^n}$ are abundant, while maps from higher-dimensional spaces are much more restricted. DeJarnette-Haj?asz-Lukyanenko-Tyson constructed horizontal maps from S ^k to ${\mathbb{H}^n}$ which factor through n-spheres and showed that these maps have no smooth horizontal fillings. In this paper, however, we build on an example of Kaufman to show that these maps sometimes have Lipschitz fillings. This shows that the Lipschitz and the smooth horizontal homotopy groups of a space may differ. Conversely, we show that any Lipschitz map ${S^k \to \mathbb{H}^1}$ factors through a tree and is thus Lipschitz null-homotopic if ${k \geq 2}$ .     

Randomly weighted sums of dependent subexponential random variables*

We consider the randomly weighted sums $ \sum\nolimits_{k = 1}^n {{\theta_k}{X_k},n \geqslant 1} $ , where $ \left\{ {{X_k},1 \leqslant k \leqslant n} \right\} $ are n real-valued random variables with subexponential distributions, and $ \left\{ {{\theta_k},1 \leqslant k \leqslant n} \right\} $ are other n random variables independent of $ \left\{ {{X_k},1 \leqslant k \leqslant n} \right\} $ and satisfying $ a \leqslant \theta \leqslant b $ for some $ 0 < a \leqslant b < \infty $ and all $ 1 \leqslant k \leqslant n $ . For $ \left\{ {{X_k},1 \leqslant k \leqslant n} \right\} $ satisfying some dependent structures, we prove that $$ {\text{P}}\left( {\mathop {{\max }}\limits_{1 \leqslant m \leqslant n} \sum\limits_{k = 1}^m {{\theta_k}{X_k} > x} } \right)\sim {\text{P}}\left( {\sum\limits_{k = 1}^m {{\theta_k}{X_k} > x} } \right)\sim {\text{P}}\left( {\mathop {{\max }}\limits_{1 \leqslant k \leqslant n} {\theta_k}{X_k} > x} \right)\sim \sum\limits_{k = 1}^m {{\text{P}}\left( {{\theta_k}{X_k} > x} \right)} $$ as x??????.     

Cyclic symmetry of the scaled simplex

Let $\mathcal{Z}_{m}^{k}$ consist of the m ^k alcoves contained in the m-fold dilation of the fundamental alcove of the type A _k affine hyperplane arrangement. As the fundamental alcove has a cyclic symmetry of order k+1, so does $\mathcal{Z}_{m}^{k}$ . By bijectively exchanging the natural poset structure of $\mathcal{Z}_{m}^{k}$ for a natural cyclic action on a set of words, we prove that $(\mathcal{Z}_{m}^{k},\prod_{i=1}^{k} \frac{1-q^{m i}}{1-q^{i}},C_{k+1})$ exhibits the cyclic sieving phenomenon.     

Length of Geodesics and Quantitative Morse Theory on Loop Spaces

Let M ⁿ be a closed Riemannian manifold of diameter d. Our first main result is that for every two (not necessarily distinct) points ${p,q \in M^n}$ and every positive integer k there are at least k distinct geodesics connecting p and q of length ${\leq 4nk^2d}$ . We demonstrate that all homotopy classes of M ⁿ can be represented by spheres swept-out by “short” loops unless the length functional has “many” “deep” local minima of a “small” length on the space ${\Omega_{pq}M^n}$ of paths connecting p and q. For example, one of our results implies that for every positive integer k there are two possibilities: Either the length functional on ${\Omega_{pq} M^n}$ has k distinct non-trivial local minima with length ${\leq 2kd}$ and “depth” ${\geq 2d}$ ; or for every m every map of S ^m into ${\Omega_{pq}M^n}$ is homotopic to a map of S ^m into the subspace ${\Omega_{pq}^{4(k+2)(m+1)d}M^n}$ of ${\Omega_{pq}M^n}$ that consists of all paths of length ${\leq 4(k+2)(m+1)d}$ .     

Lower and upper bounds of Dirichlet eigenvalues for totally characteristic degenerate elliptic operators

Let λkbe the k-th Dirichlet eigenvalue of totally characteristic degenerate elliptic operator-ΔB defined on a stretched cone B0 ■ [0,1) × X with boundary on {x1 = 0}. More precisely,ΔB=(x1αx1)2+ α2x2+ + α2xnis also called the cone Laplacian. In this paper,by using Mellin-Fourier transform,we prove thatλk Cnk2 n for any k 1,where Cn=(nn+2)(2π)2(|B0|Bn)-2n,which gives the lower bounds of the Dirchlet eigenvalues of-ΔB. On the other hand,by using the Rayleigh-Ritz inequality,we deduce the upper bounds ofλk,i.e.,λk+1 1 +4n k2/nλ1. Combining the lower and upper bounds of λk,we can easily obtain the lower bound for the first Dirichlet eigenvalue λ1 Cn(1 +4n)-12n2.