Strongly Exposed Points of Decomposable Sets in Spaces of Bochner Integrable Functions

A characterization of strongly exposed points of a decomposable bounded closed convex set $\Gamma \subset L_p (T,X)$ , where $1 \leqslant p < \infty $ , in terms of strongly exposed points of values of the set-valued representation $F:T \to 2^X $ of $\Gamma $ is given. As a corollary, necessary conditions characterizing strongly exposed points of the unit ball in $L_p (T,X) $ , where $1 \leqslant p < \infty $ , in terms of strongly exposed points of the unit ball in X are obtained.     

New estimates of continuity in $$M/GI/1/ \infty$$ queues

The paper deals with an estimation of the total variation distance between stationary distributions of waiting time in two queueing systems with equal Poisson inputs and different distributions B and $\widetilde B$ of service time. Assuming equality of two first moments of B and $\widetilde B$ the continuity inequalities are derived in terms of difference pseudomoments of B and $\widetilde B$ . When in addition the third moments of B and $\widetilde B$ coincide then the constant involved in the corresponding inequality has the asymptotics ${\text{O}}\left[ {\left( {1 - \rho } \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}} } \right]$ in the heavy traffic limit $\rho \to 1$ .     

Lower Bounds for the Perron Root of a Sum of Nonnegative Matrices

Let $A^{(l)} (l = 1, \ldots ,k)$ be $n \times n$ nonnegative matrices with right and left Perron vectors $u^{(l)} $ and $v^{(l)} $ , respectively, and let $D^{(l)} $ and $E^{(l)} (l = 1, \ldots ,k)$ be positive-definite diagonal matrices of the same order. Extending known results, under the assumption that $$u^{(1)} \circ v^{(1)} = \ldots = u^{(k)} \circ v^{(k)} \ne 0$$ (where `` $ \circ $ '' denotes the componentwise, i.e., the Hadamard product of vectors) but without requiring that the matrices $A^{(l)} $ be irreducible, for the Perron root of the sum $\sum\nolimits_{l = 1}^k {D^{(l)} A^{(l)} E^{(l)} } $ we derive a lower bound of the form $$\rho \left( {\sum\limits_{l = 1}^k {D^{(l)} A^{(l)} E^{(l)} } } \right) \geqslant \sum\limits_{l = 1}^k {\beta _{l\rho } (A^{(l)} ),{\text{ }}\beta _l >0.} $$ Also we prove that, for arbitrary irreducible nonnegative matrices $A^{{\text{ (}}l{\text{)}}} (l = 1, \ldots ,k),$ , $$\rho \left( {\sum\limits_{l = 1}^k {A^{(l)} } } \right) \geqslant \sum\limits_{l = 1}^k {\alpha _{l\rho } (A^{(l)} ),} $$ where the coefficients ∝₁>0 are specified using an arbitrarily chosen normalized positive vector. The cases of equality in both estimates are analyzed, and some other related results are established. Bibliography: 8 titles.     

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.     

Best Uniform Rational Approximations of Functions by Orthoprojections

Let C[-1,1] be the Banach space of continuous complex functions $f$ on the interval [-1,1] equipped with the standard maximum norm $\left\| f \right\|$ ; let $\omega \left( \cdot \right) = \omega \left( { \cdot ,f} \right)$ be the modulus of continuity of $f$ ; and let $R_n = R_n \left( f \right)$ be the best uniform approximation of $f$ by rational functions (r.f.) whose degrees do not exceed $n = 1, 2, \ldots $ . The space C[-1,1] is also regarded as a pre-Hilbert space with respect to the inner product given by $\left( {f,g} \right) = \left( {1/\pi } \right)\int_{ - 1}^1 {f\left( x \right)g\left( x \right)} \left( {1 - x^2 } \right)^{ - 1/2} dx$ . Let $z_n = \{ z_1 , z_2 , \ldots z_n \} $ be a set of points located outside the interval [-1,1]. By $F\left( { \cdot ,f,z_n } \right)$ we denote an orthoprojection operator acting from the pre-Hilbert space C[-1,1] onto its ( ${n + 1}$ )-dimensional subspace consisting of rational functions whose poles (with multiplicity taken into account) can only be points of the set $z_n $ . In this paper, we show that if $f$ is not a rational function of degree $ \leqslant n$ , then we can find a set of points $z_n = z_n \left( f \right)$ such that $\left\| {f\left( \cdot \right) - F\left( { \cdot ,f,z_n } \right)} \right\| \leqslant 12R_n ln\frac{3}{{\omega ^{ - 1} \left( {R_n /3} \right)}}.$      

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.     

Inscribing a Regular Octahedron in a Three-Dimensional Convex Body with Smooth Boundary

A norm $|| \cdot ||$ and a convex body K with smooth boundary in the standard Euclidean space $\mathbb{R}^3 $ are considered. It is proved that the boundary $\partial $ K of K contains the vertices AA'BB'CC' of a regular octahedron with $||AA\prime || = ||BB\prime || \geqslant ||CC\prime ||$ (or, optionally, $||AA\prime || = ||BB\prime || \leqslant ||CC\prime ||$ ). Bibliography: 4 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 $L_1 (\mathbb{T}^N )$ , $N \geqslant 2$ .     

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.     

A-Systems, Independent Functions, and Sets Bounded in Spaces of Measurable Functions

Let $U \subset L_o ([0,1],\mathcal{M},m)$ be a set of Lebesgue measurable functions. Suppose also that two seminormed spaces of real number sequences are given: $\mathcal{A}$ and $\mathcal{B}$ . We study $\left( {\mathcal{A},\mathcal{B}} \right)$ -sets U defined by the classes $\mathcal{A}$ and $\mathcal{B}$ as follows: $\forall a = (a_n ) \in \mathcal{A}, \forall (f_n (t)) \in u^\mathbb{N} $ (or for sequences similar to $(f_n (t))$ ) $\exists E = E(a) \subset [0,1], mE = 1$ such that $\{ a_n f_n (t)\} 1_E (t)\} \in \mathcal{B}, t \in [0,1]$ . We consider three versions of the definition of $\left( {\mathcal{A},\mathcal{B}} \right)$ -sets, one of which is based on functions independent in the probability sense. The case ${\mathcal{B}}=l_\infty$ is studied in detail. It is shown that $({\mathcal{A}},l_\infty)$ -independent sets are sets bounded or order bounded in some well-known function spaces (L _p , L _p,q , etc.) constructed with respect to the Lebesgue measure. A characterization of such sets in terms of seminormed spaces of number sequences is given. The (l ₁,c _°)- and $(\mathcal{A},l_1 )$ -sets were studied by E. M. Nikishin.     

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 $40 \leqslant H(T) \leqslant \log \log T$ .     

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 $ .     

Integrability of the Majorants of Fourier Series and Divergence of the Fourier Series of Functions with Restrictions on the Integral Modulus of Continuity

We construct an example of a function from the class $H_1^{\omega ^ * } $ , where $\omega ^ * (t) = \sqrt {\log \log (t^{ - 1} )/\log (t^{ - 1} )} $ , $0 < t \leqslant t_0 $ , whose trigonometric Fourier series is divergent almost everywhere. We obtain sharp integrability conditions for the majorants of the partial sums of trigonometric Fourier series in terms of whether the functions in question belong to the classes $H_1^\omega $ .     

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.     

Calderón constants of finite-dimensional couples

The Calderón constant æ( $\bar X$ ) is a numerical invariant of finite-dimensional Banach couple $\bar X = (X_0 ,X_1 )$ measuring its interpolation property with respect to linear operators acting in $\bar X$ . In the paper we prove the duality relation æ( $\bar X$ )≈ æ( $\bar X$ ^*)and calculate the asymptotic behavior of æ( $\bar X$ ) as dim $\bar X \to \infty $ for a few “classical” Banach couples.