Optimal Set of the Modulus of Continuity in the Sharp Jackson Inequality in the Space $$L_2 $$

To a function $f \in L_2 [ - \pi ,\pi ]$ and a compact set $Q \subset [ - \pi ,\pi ]$ we assign the supremum $\omega (f,Q) = \sup _{t \in Q} ||f( \cdot + t) - f( \cdot )||_{L_2 [ - \pi ,\pi ]} $ , which is an analog of the modulus of continuity. We denote by $K(n,Q)$ the least constant in Jackson's inequality between the best approximation of the function f by trigonometric polynomials of degree $n - 1$ in the space $L_2 [ - \pi ,\pi ]$ and the modulus of continuity $\omega (f,Q)$ . It follows from results due to Chernykh that $K(n,Q) \geqslant 1/\sqrt 2 $ and $K(n,[0,\pi /\pi ]) = 1/\sqrt 2 $ . On the strength of a result of Yudin, we show that if the measure of the set Q is less than $\pi /n$ , then $K(n,Q) >1/\sqrt 2 $ .     

A Characterization of Projective Spaces by a Set of Planes

This note deals with the following question: How many planes of a linear space (P, $\mathfrak{L}$ ) must be known as projective planes to ensure that (P, $\mathfrak{L}$ ) is a projective space? The following answer is given: If for any subset M of a linear space (P, $\mathfrak{L}$ ) the restriction (M, $\mathfrak{L}$ )(M)) is locally complete, and if for every plane E of (M, $\mathfrak{L}$ (M)) the plane $\bar E$ generated by E is a projective plane, then (P, $\mathfrak{L}$ ) is a projective space (cf. 5.6). Or more generally: If for any subset M of P the restriction (M, $\mathfrak{L}$ (M)) is locally complete, and if for any two distinct coplanar lines G1, G2 ∈ $\mathfrak{L}$ (M) the lines $\bar G_1 ,\bar G_2 \varepsilon \mathfrak{L}$ generated by G1, G2 have a nonempty intersection and $\overline {G_1 \cup {\text{ }}G_2 }$ satisfies the exchange condition, then (P, $\mathfrak{L}$ ) is a generalized projective space.     

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.     

Converse Quadrature sum Inequalities for Freud Weights. ii

Let $W: = \exp \left( { - Q} \right)$ , where $Q$ is of smooth polynomial growth at $\infty$ , for example $Q\left( x \right) = \left| x \right|^\beta ,\beta >1$ . We call $W^2 $ a Freud weight. Let $\left\{ {x_{j{\kern 1pt} n} } \right\}_{j = 1}^n $ and $\left\{ {\lambda _{j{\kern 1pt} n} } \right\}_{j = 1}^n $ denote respectively the zeros of the $n$ th orthonormal polynomial $p_n$ for $W^2 $ and the Christoffel numbers of order $n$ . We establish converse quadrature sum inequalities associated with W, such as $$\left\| {\left( {PW} \right)\left( x \right)\left( {1 + \left| x \right|} \right)^r } \right\|_{L_p \left( R \right)} $$ with $C$ independent of $n$ and polynomials P of degree $ < n$ , and suitable restrictions on $r$ , $R$ . We concentrate on the case ${ \geqq 4}$ , as the case ${p < 4}$ was handled earlier. We are able to treat a general class of Freud weights, whereas our earlier treatment dealt essentially with $\left( { - \left| x \right|^\beta } \right),\beta = 2,4,6,....$ Some applications to Lagrange interpolation are presented.     

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 $L_1 (\mathbb{T}^N )$ , $N \geqslant 2$ .     

Regular Semigroups of Endomorphisms of von Neumann Factors

We study a class of $E_0$ -semigroups of endomorphisms of a von Neumann factor $\mathcal{M}$ possessing the following property: an $e_0$ -semigroup of endomorphisms of $\mathcal{B}\left( \mathcal{H} \right)$ , where $\mathcal{H}$ is the standard representation space for $\mathcal{M}$ , and a product system of Hilbert spaces can be associated with each of these $E_0$ -semigroups.     

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.     

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.     

Inverse problem of finding n coefficients of lower derivatives in a parabolic equation

We consider the problem of reconstructing the vector function $\vec b(x) = (b_1 ,...,b_n )$ in the term $(\vec b,\nabla u)$ in a linear parabolic equation. This coefficient inverse problem is considered in a bounded domain Ω ? R ⁿ . To find the above-mentioned function $\vec b(x)$ , in addition to initial and boundary conditions we pose an integral observation of the form $\int_0^T {u(x,t)\vec \omega (t)dt = \vec \chi (x)} $ , where $\vec \omega (t) = (\omega _1 (t),...,\omega _n (t))$ is a given weight vector function. We derive sufficient existence and uniqueness conditions for the generalized solution of the inverse problem. We present an example of input data for which the assumptions of the theorems proved in the paper are necessarily satisfied.     

On Milnor's Invariants of 4 -Component Links

We study the behavior of Milnor's µ-invariants of three- and four-component links with respect to the discriminant determined by $\Delta $ -moves of links. We introduce a new type of $\Delta $ -move, balanced $\Delta $ -moves, or, briefly, $B\Delta $ -moves. Since each four-component link is equivalent to a standard link under a sequence of balanced $\Delta $ -moves, $\Delta $ -moves that involve at most two components, and Reidemeister moves, we manage to define axiomatically µ-invariants of length $3$ for arbitrary semibounding links.     

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.     

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.     

On Total Incomparability of Mixed Tsirelson Spaces

We give criteria of total incomparability for certain classes of mixed Tsirelson spaces. We show that spaces of the form $T\left[ {\left( {\mathcal{M}_k ,\theta _k } \right)_{k = 1}^l } \right]$ with index $i\left( {\mathcal{M}_k } \right)$ finite are either c ₀ or $\ell _p $ saturated for some p and we characterize when any two spaces of such a form are totally incomparable in terms of the index $i\left( {\mathcal{M}_k } \right)$ and the parameter θ _k . Also, we give sufficient conditions of total incomparability for a particular class of spaces of the form $T\left[ {\left( {\mathcal{A}_k ,\theta _k } \right)_{k = 1}^\infty } \right]$ in terms of the asymptotic behaviour of the sequence $\left\| {\sum\limits_{j = 1}^n {e_i } } \right\|$ where (e _i is the canonical basis.     

Lower Bounds for the Riemann Zeta Function on the Critical Line

We establish a relation between the lower bound for the maximum of the modulus of $\zeta (1/2 + iT + s)$ in the disk $|s| \leqslant H$ and the lower bound for the maximum of the modulus of $\zeta (1/2 + iT + it)$ on the closed interval $|t| \leqslant H$ for $0 < H(T) \leqslant {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}$ . We prove a theorem on the lower bound for the maximum of the modulus of $0 < H(T) \leqslant {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}$ on the closed interval $|t| \leqslant H$ for $40 \leqslant H(T) \leqslant \log \log T$ .     

A Nonlinear Loaded Parabolic Equation and a Related Inverse Problem

The solvability of the nonlocal-in-time boundary-value problem for the nonlinear parabolic equation $$u_t - \Delta u + c(\bar u(x,T))u = f(x,t),$$ where $\bar u(x,t) = \alpha (t)u(x,t) + \int_0^t {\beta (\tau )u(x,\tau )d\tau } $ for given functions $\alpha (t)$ and $\beta (t)$ , is studied. Existence and uniqueness theorems for regular solutions are proved; it is shown that the results obtained can be used to study the solvability of coefficient inverse problems.