On 1-isometries of affine quadrics over finite fields

Letq be a regular quadratic form on a vector space (V, $\mathbb{F}$ ) and assume $4 \leqslant dim V \leqslant \infty \wedge |\mathbb{F}| \in \mathbb{N}$ . A 1-isometry of the central quadric $\mathcal{F}: = \{ x \in V|q(x) = 1\}$ is a permutation ? of $\mathcal{F}$ such that (*) $$q(x - y) = \nu \Leftrightarrow q(x^\varphi - y^\varphi ) = \nu \forall x,y \in \mathcal{F}$$ holds true for a fixed element ν of $\mathbb{F}$ . For arbitraryν ∈ $\mathbb{F}$ we prove that? is induced (in a certain sense) by a semi-linear bijection $(\sigma ,\varrho ):(V,\mathbb{F}) \to (V,\mathbb{F})$ such thatq oσ =? oq, provided $\mathcal{F}$ contains lines and the exceptional case $(\nu = 2 \Lambda |\mathbb{F}| = 3 \Lambda \dim V = 4 \Lambda |\mathcal{F}| = 24)$ is excluded. In the exceptional case and as well in case of dim V = 3 there are counterexamples. The casesν ≠ 2 and v=2 require different techniques.     

On permutations of central quadrics which preserve an inner distance

Letq be a regular quadratic form on a vector space (V, $\mathbb{F}$ ) and assume dimV ≥ 4 and ¦ $\mathbb{F}$ ¦ ≥ 4. We consider a permutation ? of the central affine quadric $\mathcal{F}$ := {x εV ¦q(x) = 1} such that $$(*)x \cdot y = \mu \Leftrightarrow x^\varphi \cdot y^\varphi = \mu \forall x,y\varepsilon \mathcal{F}$$ holds true, where μ is a fixed element of $\mathbb{F}$ and where “·” is the scalar product associated withq. We prove that ? is induced (in a certain sense) by a semi-linear bijection (σ,?): (V, $\mathbb{F}$ ) → (V, $\mathbb{F}$ ) such thatq o ?o q, provided $\mathcal{F}$ contains lines and the pair (μ, $\mathbb{F}$ ) has additional properties if there ar no planes in $\mathcal{F}$ . The cases μ,≠ 0 and μ = 0 require different techniques.     

Sets Computing the Symmetric Tensor Rank

Let ${\nu_{d} : \mathbb{P}^{r} \rightarrow \mathbb{P}^{N}, N := \left( \begin{array}{ll} r + d \\ \,\,\,\,\,\, r \end{array} \right)- 1,}$ denote the degree d Veronese embedding of ${\mathbb{P}^{r}}$ . For any ${P\, \in \, \mathbb{P}^{N}}$ , the symmetric tensor rank sr(P) is the minimal cardinality of a set ${\mathcal{S} \subset \nu_{d}(\mathbb{P}^{r})}$ spanning P. Let ${\mathcal{S}(P)}$ be the set of all ${A \subset \mathbb{P}^{r}}$ such that ${\nu_{d}(A)}$ computes sr(P). Here we classify all ${P \,\in\, \mathbb{P}^{n}}$ such that sr(P) <  3d/2 and sr(P) is computed by at least two subsets of ${\nu_{d}(\mathbb{P}^{r})}$ . For such tensors ${P\, \in\, \mathbb{P}^{N}}$ , we prove that ${\mathcal{S}(P)}$ has no isolated points.     

On the topology of semi-algebraic functions on closed semi-algebraic sets

We consider a closed semi-algebraic set ${X \subset \mathbb{R}^n}$ and a C ² semi-algebraic function ${f : \mathbb{R}^n \rightarrow\mathbb{R}}$ such that ${f_{\vert X}}$ has a finite number of critical points. We relate the topology of X to the topology of the sets ${X \cap \{ f * \alpha \}}$ , where ${* \in \{\le,=,\ge \}}$ and ${\alpha \in \mathbb{R}}$ , and the indices of the critical points of ${f_{\vert X}}$ and ${-f_{\vert X}}$ . We also relate the topology of X to the topology of the links at infinity of the sets ${X \cap \{ f * \alpha\}}$ and the indices of these critical points. We give applications when ${X=\mathbb{R}^n}$ and when f is a generic linear function.     

Codimension 1 Subvarieties of {\mathcal{M}_g } and Real Gonality of Real Curves

Let ${\mathcal{M}_g }$ be the moduli space of smooth complex projective curves of genus g. Here we prove that the subset of ${\mathcal{M}_g }$ formed by all curves for which some Brill-Noether locus has dimension larger than the expected one has codimension at least two in ${\mathcal{M}_g }$ . As an application we show that if ${X \in \mathcal{M}_g }$ is defined over $\mathbb{R}$ then there exists a low degree pencil ${u:X \to \mathbb{P}^1 }$ defined over $\mathbb{R}.$      

On Uniform Approximation in R2 of Continuous Functions of Two Variables with Bounded Variation in the Sense of Hardy

Introduce the notation: $\mathbb{Z}$ is the set of integers, $\bar {\mathbb{Z}}={\mathbb{Z}} \cup \{-\infty, +\infty\},{\mathbb{R}}_+^2 =\{x=(x_1,x_2) \in {\mathbb{R}}^2; x_1>0,x_2>0\}$ , $g_{k,m} (x,\alpha,h)= \int\limits_0^1 {g_1 (\frac{(k+u)h_1 - x_1}{\alpha_1})g_2(\frac{(m+u)h_2 - x_2}{\alpha_2})}du$ , where $g_i :\mathbb{R} \to \mathbb{R},x \in \mathbb{R}^2 ,\alpha ,h \in \mathbb{R}_ + ^2 $ . Under certain conditions on the functions g ₁, g ₂, we prove that the system of functions $g_{k,m} (x,\alpha^(n), h^(n)) (k,m \in \bar {\mathbb{Z}})$ , where $\alpha ^{\left( n \right)} ,h^{\left( n \right)} \in \mathbb{R}_ + ^2 $ are arbitrary infinitesimal sequences, is complete in the space C $\mathbb{R}^2 $ of uniformly continuous bounded functions f equipped with the norm $||f|| = \mathop {\sup }\limits_{x \in \mathbb{R}^2 } |f(x)|$ . Starting with the functions g _k,m , it is possible to construct a method for uniform approximating in $\mathbb{R}^2 $ any continuous function of bounded variation in the sense of Hardy. An error estimate is derived in terms of the second order moduli of continuity. Based on the obtained results, we discuss in detail the accuracy of uniform approximation of functions of several variables by linear functions. The error estimates are derived by using second order moduli of continuity. We pay a particular attention to sharpness of constants. Bibliography: 8 titles.     

A Criterion for Weak Generalized Localization in the Class L_1 for Multiple Trigonometric Series from the Viewpoint of Isometric Transformations

In this paper, we study the problem of the variation (if any) of the sets of convergence and divergence everywhere or almost everywhere of a multiple Fourier series (integral) of a function $f \in L_p $ , $p \geqslant 1$ , $f(x) = 0$ , on a set of positive measure $\mathfrak{A} \subset \mathbb{T}^N = [ - \pi ,\pi )^N $ , $N \geqslant 2$ , depending on the rotation of the coordinate system, i.e., depending on the element $\tau \in \mathcal{F}$ , where $\mathcal{F}$ is the rotation group about the origin in $\mathbb{R}^N $ . This problem has been reduced to the study of the change in the geometry of the sets $\tau ^{ - 1} (\mathfrak{A}) \cap \mathbb{T}^N $ (where $\tau ^{ - 1} \in \mathcal{F}$ satisfies $\tau ^{ - 1} \cdot \tau = 1$ ) and $\mathbb{T}^N \backslash {\text{supp}}(f \circ \tau )$ depending on the “rotation,” i.e., on $\tau \in \mathcal{F}$ . In the present paper, we consider two settings of this problem (depending on the sense in which the Fourier series of the function $f \circ \tau $ is understood) and give (for both cases) possible solutions of the problem in the class $L_1 (\mathbb{T}^N )$ , $N \geqslant 2$ .     

Metrization of Weighted Graphs

We find a set of necessary and sufficient conditions under which the weight ${w: E \rightarrow \mathbb{R}^{+}}$ on the graph G = (V, E) can be extended to a pseudometric ${d : V \times V \rightarrow \mathbb{R}^{+}}$ . We describe the structure of graphs G for which the set ${\mathfrak{M}_{w}}$ of all such extensions contains a metric whenever w is strictly positive. Ordering ${\mathfrak{M}_{w}}$ by the pointwise order, we have found that the posets $({\mathfrak{M}_{w}, \leqslant)}$ contain the least elements ρ _0,w if and only if G is a complete k-partite graph with ${k \, \geqslant \, 2}$ . In this case the symmetric functions ${f : V \times V \rightarrow \mathbb{R}^{+}}$ , lying between ρ _0,w and the shortest-path pseudometric, belong to ${\mathfrak{M}_{w}}$ for every metrizable w if and only if the cardinality of all parts in the partition of V is at most two.     

Nil-Manifolds Cannot be Immersed as Hypersurfaces in Euclidean Spaces

We prove that the $2n + 1$ -dimensional Heisenberg group H _n and the 4-manifolds $Nil^4 $ and $Nil^3 \times \mathbb{R}$ endowed with an arbitrary left-invariant metric admit no C ³-regular immersions into Euclidean spaces $\mathbb{R}^{2n + 2} $ and $\mathbb{R}^5 $ , respectively.     

The Wiener--Hopf Integral Equation in the Supercritical Case

We consider the scalar homogeneous equation $S(x) = \int_0^\infty {K(x - t)S(t)dt,{\text{ }}x \in \mathbb{R}^ + \equiv (0,\infty ),}$ with symmetric kernel $K:K( - x) = K(x),{\text{ }}x \in \mathbb{R}_1$ satisfying the conditions $0 \leqslant K \in L_1 (\mathbb{R}^ + ) \cap C^{\left( 2 \right)} (\mathbb{R}^ + )$ , $\int_0^\infty {K(t)dt > \frac{1}{2}} $ , $K' \leqslant 0{\text{ }}and 0 \leqslant K'' \downarrow {\text{ }}on \mathbb{R}^ + $ . We prove the existence of a real solution S of the equation given above with asymptotic behavior $S(x) = O(x){\text{ as }}x \to + \infty $ .     

Uniqueness of integrable solutions to {\nabla \zeta=G \zeta, \zeta|_\Gamma = 0} for integrable tensor coefficients G and applications to elasticity

Let ${\Omega \subset \mathbb{R}^{N}}$ be a Lipschitz domain and Γ be a relatively open and non-empty subset of its boundary ${\partial\Omega}$ . We show that the solution to the linear first-order system $$\nabla \zeta = G\zeta, \, \, \zeta|_\Gamma = 0 \quad \quad \quad (1)$$ is unique if ${G \in \textsf{L}^{1}(\Omega; \mathbb{R}^{(N \times N) \times N})}$ and ${\zeta \in \textsf{W}^{1,1}(\Omega; \mathbb{R}^{N})}$ . As a consequence, we prove $$||| \cdot ||| : \textsf{C}_{o}^{\infty}(\Omega, \Gamma; \mathbb{R}^{3}) \rightarrow [0, \infty), \, \, u \mapsto \parallel {\rm sym}(\nabla uP^{-1})\parallel_{\textsf{L}^{2}(\Omega)}$$ to be a norm for ${P \in \textsf{L}^{\infty}(\Omega; \mathbb{R}^{3 \times 3})}$ with Curl ${P \in \textsf{L}^{p}(\Omega; \mathbb{R}^{3 \times 3})}$ , Curl ${P^{-1} \in \textsf{L}^{q}(\Omega; \mathbb{R}^{3 \times 3})}$ for some p, q > 1 with 1/p + 1/q = 1 as well as det ${P \geq c^+ > 0}$ . We also give a new and different proof for the so-called ‘infinitesimal rigid displacement lemma’ in curvilinear coordinates: Let ${\Phi \in \textsf{H}^{1}(\Omega; \mathbb{R}^{3})}$ satisfy sym ${(\nabla\Phi^\top\nabla\Psi) = 0}$ for some ${\Psi \in \textsf{W}^{1,\infty}(\Omega; \mathbb{R}^{3}) \cap \textsf{H}^{2}(\Omega; \mathbb{R}^{3})}$ with det ${\nabla\Psi \geq c^+ > 0}$ . Then, there exist a constant translation vector ${a \in \mathbb{R}^{3}}$ and a constant skew-symmetric matrix ${A \in \mathfrak{so}(3)}$ , such that ${\Phi = A\Psi + a}$ .     

The index of centralizers of elements in classical Lie algebras

The index of a finite-dimensional Lie algebra $\mathfrak{g}$ is the minimum of dimensions of the stabilizers $\mathfrak{g}_\alpha $ over all covectors $\alpha \in \mathfrak{g}^ * $ . Let $\mathfrak{g}$ be a reductive Lie algebra over a field $\mathbb{K}$ of characteristic ≠ = 2. Élashvili conjectured that the index of $\mathfrak{g}_\alpha $ is always equal to the index, or, which is the same, the rank of $\mathfrak{g}$ . In this article, Élashvili’s conjecture is proved for classical Lie algebras. Furthermore, it is shown that if $\mathfrak{g} = \mathfrak{g}\mathfrak{l}_n $ or $\mathfrak{g} = \mathfrak{s}\mathfrak{p}_{2n} $ and $e \in \mathfrak{g}$ is a nilpotent element, then the coadjoint action of $\mathfrak{g}_e $ has a generic stabilizer. For $\mathfrak{g}$ , we give examples of nilpotent elements $e \in \mathfrak{g}$ such that the coadjoint action of $\mathfrak{g}_e $ does not have a generic stabilizer.     

The Roles Played by Order of Convexity or Starlikeness and the Bloch Condition in the Extension of Mappings from the Disk to the Ball

Let ${G: \mathbb {C}^{n-1} \rightarrow \mathbb {C}}$ be holomorphic such that G(0)?=?0 and DG(0)?=?0. When f is a convex (resp. starlike) normalized (f(0)?=?0, f??(0)?=?1) univalent mapping of the unit disk ${\mathbb {D}}$ in ${\mathbb {C}}$ , then the extension of f to the Euclidean unit ball ${\mathbb {B}}$ in ${\mathbb {C}^n}$ given by ${\Phi_G(f)(z)=(f(z_1)+G(\sqrt{f^{\prime}(z_1)} \, \hat{z}),\sqrt{f^{\prime}(z_1)}\, \hat{z})}$ , ${\hat{z}=(z_2,\dots,z_n) \in \mathbb {C}^{n-1}}$ , is known to be convex (resp. starlike) if G is a homogeneous polynomial of degree 2 with sufficiently small norm. Conversely, it is known that G cannot have terms of degree greater than 2 in its expansion about 0 in order for ${\Phi_G(f)}$ to be convex (resp. starlike), in general. We examine whether the restriction that f be either convex or starlike of a certain order ${\alpha \in (0,1]}$ allows, in general, for G to contain terms of degree greater than 2 and still have ${\Phi_G(f)}$ maintain the respective geometric property. Related extension results for convex and starlike Bloch mappings are also given.     

GBRDs over supersolvable groups and solvable groups of order prime to 3

We show that the established necessary conditions for a GBRD ${(v,3,\lambda; \mathbb {G})}$ are sufficient (i) when ${\mathbb {G}}$ is supersolvable and (ii) when ${\mathbb {G}}$ is solvable with ${\vert \mathbb {G} \vert }$ prime to 3.     

Approximation by M. Riesz Kernels in L p for p<1

Let α > 0. We consider the linear span $\mathfrak{X}_\alpha \left( {\mathbb{R}^n } \right)$ of scalar Riesz's kernels $\left\{ {\tfrac{1}{{\left| {x - a} \right|^\alpha }}} \right\}_{a \in \mathbb{R}^n }$ and the linear span $\mathfrak{Y}_\alpha \left( {\mathbb{R}^n } \right)$ of vector Riesz's kernels $\left\{ {\tfrac{1}{{\left| {x - a} \right|^{\alpha + 1} }}\left( {x - a} \right)} \right\}_{a \in \mathbb{R}^n }$ . We study the following problems. (1) When is the intersection $\mathfrak{X}_\alpha \left( {\mathbb{R}^n } \right) \cap L^p \left( {\mathbb{R}^n } \right)$ dense in L^p(?ⁿ)? (2) When is the intersection $\mathfrak{Y}_\alpha \left( {\mathbb{R}^n } \right) \cap L^p \left( {\mathbb{R}^n ,\mathbb{R}^n } \right)$ dense in L^p(?ⁿ, ?ⁿ)? Bibliography: 15 titles.     

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.     

Orthogonality properties of linear combinations of orthogonal polynomials II

Let and be polynomials orthogonal on the unit circle with respect to the measures dσ and dμ, respectively. In this paper we consider the question how the orthogonality measures dσ and dμ are related to each other if the orthogonal polynomials are connected by a relation of the form , for , where . It turns out that the two measures are related by if , where and are known trigonometric polynomials of fixed degree and where the 's are the zeros of on . If the 's and 's are uniformly bounded then (under some additional conditions) much more can be said. Indeed, in this case the measures dσ and dμ have to be of the form and , respectively, where are nonnegative trigonometric polynomials. Finally, the question is considered to which weight functions polynomials of the form where denotes the reciprocal polynomial of , can be orthogonal. This revised version was published online in June 2006 with corrections to the Cover Date.     

Generalized Complex Spherical Harmonics, Frame Functions, and Gleason Theorem

Consider a finite dimensional complex Hilbert space ${\mathcal{H}}$ , with ${dim(\mathcal{H}) \geq 3}$ , define ${\mathbb{S}(\mathcal{H}):= \{x\in \mathcal{H} \:|\: \|x\|=1\}}$ , and let ${\nu_\mathcal{H}}$ be the unique regular Borel positive measure invariant under the action of the unitary operators in ${\mathcal{H}}$ , with ${\nu_\mathcal{H}(\mathbb{S}(\mathcal{H}))=1}$ . We prove that if a complex frame function ${f : \mathbb{S}(\mathcal{H})\to \mathbb{C}}$ satisfies ${f \in \mathbb{L}^2(\mathbb{S}(\mathcal{H}), \nu_\mathcal{H})}$ , then it verifies Gleason’s statement: there is a unique linear operator ${A: \mathcal{H} \to \mathcal{H}}$ such that ${f(u) = \langle u| A u\rangle}$ for every ${u \in \mathbb{S}(\mathcal{H}).\,A}$ is Hermitean when f is real. No boundedness requirement is thus assumed on f a priori.     

Arrangements of Hyperplanes and the Number of Threshold Functions

Two approaches on estimating the number of threshold functions which were recently developed by the author are discussed. Let P(K,n) denote the number of threshold functions in K-valued logic. The first approach establishes that $$P(K,n + 1) \geqslant \frac{1}{2}\left( {\mathop {K^{n - 1} }\limits_{\left\lfloor {n - 4 - 2\frac{n}{{\log _K n}}} \right\rfloor } } \right)P\left( {K,\left\lfloor {{\text{2}}\frac{n}{{\log _K n}} + 3} \right\rfloor } \right).$$ The key argument of investigation is the generalization of the result of Odlyzko on subspaces spanned by random selections of ±1-vectors. Let $E_K = \{ 0,1 \ldots ,K - 1\} $ and let E denote the set of all vectors $w_i ,i = 1, \ldots ,K^n $ , which have the form $(1,a_1 , \ldots ,a_n ),a_i \in E_K $ . Denote by $\Lambda _n (K)$ the number of all collections of different vectors $(w_{i_1 } , \ldots ,w_{i_n } ),2 \leqslant i_1 , \ldots ,i_n \leqslant \mathbb{K}^n $ , such that, for any k, $1 \leqslant k \leqslant n$ , the vector $w_{i_k } $ is minimal among all vectors from the set $E \cap {\text{span}}(w_{i_k } , \ldots ,w_{i_n } )$ . The second approach is based on topology-combinatorical techniques and allows to establish the following inequality $P(K,n) \geqslant 2\Lambda _n (K)$ .     

Ultrapowers ofL 1(μ) and the subsequence splitting principle

We study the ultrapowers $L_1 (\mu )_\mathfrak{U} $ of aL ₁(μ) space, by describing the components of the well-known representation $L_1 (\mu )_\mathfrak{U} = L_1 (\mu _\mathfrak{U} ) \oplus _1 L_1 (\nu _\mathfrak{U} )$ , and we give a representation of the projection from $L_1 (\mu )_\mathfrak{U} $ onto $L_1 (\mu _\mathfrak{U} )$ . Moreover, the subsequence splitting principle forL ₁(μ) motivates the following question: if $\mathfrak{V}$ is an ultrafilter on ? and $[f_i ] \in L_1 (\mu )_\mathfrak{V} $ , is it possible to find a weakly convergent sequence (g _i ) ?L ₁(μ) following $\mathfrak{V}$ and a disjoint sequence (h _i ) ?L ₁(μ) such that [f _i ]=[g _i ]+[h _i ]? If $\mathfrak{V}$ is a selective ultrafilter, we find a positive answer by showing that $f = [f_i ] \in L_1 (\mu )_\mathfrak{V} $ belongs to $L_1 (\mu _{_\mathfrak{V} } )$ if and only if its representatives {f _i } are weakly convergent following $\mathfrak{V}$ and $f \in L_1 (\nu _\mathfrak{V} )$ if and only if it admits a representative consisting of pairwise disjoint functions. As a consequence, we obtain a new proof of the subsequence splitting principle. If $\mathfrak{V}$ is not a p-point then the above characterizations of $L_1 (\nu _{_\mathfrak{V} } )$ and $L_1 (\nu _{_\mathfrak{V} } )$ fail and the answer to the question is negative.