An abstract Dirichlet problem in the Hilbert space

In the present paper, we consider an abstract partial differential equation of the form $\frac{{\partial ^2 u}}{{\partial t^2 }} - \frac{{\partial ^2 u}}{{\partial x^2 }} + A\left( {x,t} \right)u = f\left( {x,t} \right)$ , where $\left\{ {A\left( {x,t} \right):\left( {x,t} \right) \in \bar G} \right\}$ is a family of linear closed operators and $\bar G = G \cup \partial G,G$ is a suitable bounded region in the (x, t)-plane with boundary?G. It is assumed thatu is given on the boundary?G. The objective of this paper is to study the considered Dirichlet problem for a wide class of operatorsA(x, t). A Dirichlet problem for non-elliptic partial differential equations of higher orders is also considered.     

Some consequences of the Lindelöf conjecture

Suppose that the Lindelöf conjecture is valid in the following quantitative form: $$|\zeta (\frac{1}{2} + it)| \leqslant c_0 |t|^{\varepsilon (|t|)} $$ , where ε(t) is a monotone decreasing function, $\varepsilon (2t) \geqslant \tfrac{1}{2}\varepsilon (t),\varepsilon (t) \geqslant \tfrac{1}{{\sqrt {log t} }}$ . Then it is proved that for |t|≥T₀ the disk $\{ s:|s - \tfrac{1}{2} - it| \leqslant v\} $ contains at most 20v log |t| zeros of ζ(s) if $\tfrac{1}{2} \geqslant v \geqslant \sqrt {\varepsilon (t)} $ . There exists an absolute constant A such that for |t|≥T₁ the disk $\{ s:|s - \tfrac{1}{2} - it| \leqslant A\varepsilon ^{\tfrac{1}{3}} (t)\} $ contains at least one zero of ζ(s). Bibliography: 2 titles.     

Finite difference method of first boundary problem for quasilinear parabolic systems (IV)

More work is done to study the explicit, weak and strong implicit difference solution for the first boundary problem of quasilinear parabolic system: $$\begin{gathered} u_t = ( - 1)^{M + 1} A(x,t,u, \cdots ,u_x M - 1)u_x 2M + f(x,t,u, \cdots u_x 2M - 1), \hfill \\ (x,t) \in Q_T = \left| {0< x< l,0< t \leqslant T} \right|, \hfill \\ u_x ^k (0,t) = u_x ^k (l,t) = 0 (k = 0,1, \cdots ,M - 1),0< t \leqslant T, \hfill \\ u(x,0) = \varphi (x),0 \leqslant x \leqslant l, \hfill \\ \end{gathered} $$ whereu, ?, andf arem-dimensional vector valued functions, A is anm×m positively definite matrix, and $u_t = \frac{{\partial u}}{{\partial t}},u_x ^k = \frac{{\partial ^k u}}{{\partial x^k }}$ . For this problem, the convergence of iteration for the general difference schemes is proved.     

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

Sensitivity of best recovery in the Sobolev spaces W_\infty ^{r,d} , \widetilde W_\infty ^{r,d}for perturbed sampling

Let $I^d $ be the d‐dimensional cube, $I^d = [0,1]^d $ , and let $F \ni f \mapsto Sf \in L_\infty (I^d ) $ be a linear operator acting on the Sobolev space F, where Fis either $$$$ or $$$$ where $$\left\| f \right\|_F = \sum\limits_{\left| m \right| = r} {\mathop {{\text{esssup}}}\limits_{x \in I^d } \left| {\frac{{\partial f^{\left| m \right|} }} {{\partial x_1^{m_1 } \partial x_2^{m_2 } \cdot \cdot \cdot \partial x_d^{m_d } }}(x)} \right|.} $$ We assume that the problem elements fsatisfy the condition $\sum\nolimits_{\left| m \right| = r} {{\text{esssup}}} _{x \in I^d } \left| {f^{(m)} (x)} \right| \leqslant 1 $ and that Sis continuous with respect to the supremum norm. We study sensitivity of optimal recovery of Sfrom inexact samples of ftaken at npoints forming a uniform grid on $I^d $ . We assume that the inaccuracy in reading the sample vector is measured in the pth norm and bounded by a nonnegative number δ. The sensitivity is defined by the difference between the optimal errors corresponding to the exact and perturbed readings, respectively. Our main result is that this difference is bounded by $\mathcal{A}\delta $ , where $\mathcal{A} $ is a positive constant independent of the number of samples. This indicates that the curse of dimension, which badly affects the optimal errors, does not extend to sensitivity.     

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.     

Removable singular sets for equations of the form\sum {\tfrac{\partial }{{\partial x_i }}a_{ij} (x)\tfrac{{\partial u}}{{\partial x_j }} = f(x,u,\nabla u)}

The following uniformly elliptic equation is considered: $$\sum {\tfrac{\partial }{{\partial x_i }}a_{ij} (x)\tfrac{{\partial u}}{{\partial x_j }} = f(x,u,\nabla u)} , x \in \Omega \subset R^n ,$$ with measurable coefficients. The function f satisfies the condition $$f(x, u, \nabla u) u \geqslant C|u|^{\beta _1 + 1} |\nabla u|^{\beta _1 } , \beta _1 > 0, 0 \leqslant \beta _2 \leqslant 2, \beta _1 + \beta _2 > 1$$ . It is proved that if u(x) is a generalized (in the sense of integral identity) solution in the domain ΩK, where the compactum K has Hausdorff dimension α, and if \(\frac{{2\beta _1 + \beta _2 }}{{\beta _1 + \beta _2 - 1}}< n - \alpha \) , u(x) will be a generalized solution in the domain ω. Moreover, the sufficient removability conditions for the singular set are, in some sense, close to the necessary conditions.     

Regularity of\bar \partial _b , complex on nondegenerate Cauchy-Riemann manifolds, complex on nondegenerate Cauchy-Riemann manifolds

For each point ξ in a CR manifold M of codimension greater than 1, the CR structure of M can be approximated by the CR structure of a nilpotent Lie group G_ξ of step two near ξ. G_ξ varies with ξ. $\square _b $ and $\bar \partial _b $ on M can be approximated by $\square _b $ and $\bar \partial _b $ on the nilpotent Lie group G_ξ We can construct the parametrix of $\square _b $ on M by using the parametrix of $\square _b $ on nilpotent group of step two, and define a quasidistance on M by the approximation. The regularity of $\square _b $ and $\bar \partial _b $ follows from the Harmonic analysis on M.     

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.     

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.     

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??????.     

On the regularity of solutions of model nonlinear elliptic systems with the oblique derivative type boundary condition

Properties of generalized solutions of model nonlinear elliptic systems of second order are studied in the semiball $B_1^ + = B_1 (0) \cap \{ x_n > 0\} \subset $ ? ⁿ , with the oblique derivative type boundary condition on $\Gamma _1 = B_1 (0) \cap \{ x_n = 0\} $ . For solutionsu∈H ¹(B ₁ ⁺ ) of systems of the form $\frac{d}{{dx_\alpha }}a_\alpha ^k (u_x ) = 0, k \leqslant {\rm N}$ , it is proved that the derivatives u_x are Hölder in $B_1^ + \cup \Gamma _1 )\backslash \Sigma $ , where H_n?p(σ)=0,p>2. It is shown for continuous solutions u from H¹(B₁/⁺) of systems $\frac{d}{{dx_\alpha }}a_\alpha ^k (u,u_x ) = 0$ that the derivatives u_x are Hölder on the set $(B_1^ + \cup \Gamma _1 )\backslash \Sigma , dim_\kappa \Sigma \leqslant n - 2$ . Bibliography: 13 titles.     

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 $P(K,n) \geqslant 2\Lambda _n (K)$ .     

Subspace Arrangements of Type B n and D n

Let ${\mathcal{D}}_{n,k} $ be the family of linear subspaces of ?ⁿ given by all equations of the form $\varepsilon _1 x_{i_1 } = \varepsilon _2 x_{i_2 } = \cdot \cdot \cdot \varepsilon _k x_{i_k } ,$ for 1 ≤ < ? ? ? < i _k ≤ i and $\left( {\varepsilon _1 ,...,\varepsilon _k } \right)\varepsilon \left\{ { + 1, - 1} \right\}^k $ Also let ${\mathcal{B}}_{n,k,h} $ be ${\mathcal{D}}_{n,k} $ enlarged by the subspaces $x_{j_1 } = x_{j_2 } = \cdot \cdot \cdot x_{j_h } = 0,$ for 1 ≤. The special cases ${\mathcal{B}}_{n,2,1} $ and ${\mathcal{D}}_{n,2} $ are well known as the reflection hyperplane arrangements corresponding to the Coxeter groups of type B _nand D _n respectively. In this paper we study combinatorial and topological properties of the intersection lattices of these subspace arrangements. Expressions for their Möbius functions and characteristic polynomials are derived. Lexicographic shellability is established in the case of ${\mathcal{B}}_{n,k,h,} 1 \leqslant h < k$ , which allows computation of the homology of its intersection lattice and the cohomology groups of the manifold $\begin{gathered} {\mathcal{D}}_{n,2} \\ M_{n,k,h,} = {\mathbb{R}}^n \backslash \bigcup {{\mathcal{B}}_{n,k,h,} } \\ \end{gathered} $ . For instance, it is shown that $H^d \left( {M_{n,k,k - 1} } \right)$ is torsion-free and is nonzero if and only if d = t(k ? 2) for some $t,0 \leqslant t \leqslant \left[ {{n \mathord{\left/ {\vphantom {n k}} \right. \kern-0em} k}} \right]$ . Torsion-free cohomology follows also for the complement in ?ⁿof the complexification ${\mathcal{B}}_{n,k,h}^C ,1 \leqslant h < k$ .     

Representation of Integers by Positive Quaternary Quadratic Forms

Let $f(x,y,x,w) = x^2 + y^2 + z^2 + Dw^2$ , where $D >1$ is an integer such that $D \ne d^2$ and ${{\sqrt n } \mathord{\left/ {\vphantom {{\sqrt n } {\sqrt D = n^\theta , 0 < \theta < {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}} \right. \kern-0em} {\sqrt D = n^\theta , 0 < \theta < {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}$ . Let $rf(n)$ be the number of representations of n by f. It is proved that $r_f (n) = \pi ^2 \frac{n}{{\sqrt D }}\sigma _f (n) + O\left( {\frac{{n^{1 + \varepsilon - c(\theta )} }}{{\sqrt D }}} \right),$ where $\sigma _f (n)$ is the singular series, $c(\theta ) >0$ , and ε is an arbitrarily small positive constant. Bibliography: 14 titles.     

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 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.     

Construction of characteristic functions for the system of Navier-Stokes-Voigt equations and the BBM equation

For the system of Navier-Stokes-Voigt equations $$\frac{{\partial \vec v}}{{\partial t}} - v\Delta \vec v - \aleph \frac{{\partial \Delta \vec v}}{{\partial t}} + v_\kappa \frac{{\partial \Delta \vec v}}{{\partial x_\kappa }} + grad \rho = 0, div \vec v = 0$$ and the BBM equation $$\frac{{\partial v}}{{\partial t}} + v\frac{{\partial \Delta v}}{{\partial x}} - \frac{{\partial ^3 v}}{{\partial t\partial x^2 }} = 0$$ characteristic functions \(\mathcal{F}\left( {\vec \theta ;t} \right)\) of the measure μ_t(ω)=μ(V _?1 ^t (ω)), describing the evolution in time of the probability measure μ(ω) defined on the set of initial conditions for the first initial boundary-value problem for system (1) or Eq. (2) are constructed and investigated. It is shown that the characteristic functions \(\mathcal{F}\left( {\vec \theta ;t} \right)\) constructed satisfy partial differential equations with an infinite number of independent variables (t; θ₁,θ₂,...) [the statistical equations of E. Hopf for the system (1) or Eq. (2)].     

A note on periodic solutions of a differential equation with two parameters

In [1] (p. 215), the authors Andronov, Leontovich-Andronova, Gordon, and Maier, consider the following equation: $$\left\{ \begin{gathered} \tfrac{{dx}}{{dt}} = y, \hfill \\ \tfrac{{dy}}{{dt}} = x + x^2 - \left( {\varepsilon _1 + \varepsilon _2 x} \right)y, \hfill \\ \end{gathered} \right.$$ whereε ₁ andε ₂ are real constants andε ₁ andε ₂ are not both zero. They proved that there are no non-trivial periodic solutions except possibly for the case $0< \tfrac{{\varepsilon _1 }}{{\varepsilon _2 }}< \tfrac{3}{2}$ . They left that case as an open problem. In this note we prove that there are indeed no non-trivial periodic solutions in the case $0< \tfrac{{\varepsilon _1 }}{{\varepsilon _2 }}< \tfrac{3}{2}$ either. Our method of proof consists essentially of constructing a Dulac function (see [6] and [9]) and using the conception of Duff's rotated vector field (see [4], [7], [8], [10], and [11]).     

Note on Periodic Complementary Sets of Binary Sequences

Denote by $PCS_p^n $ resp. $ACS_p^n $ thecollection consisting of ordered p-tuples of binary sequences(i.e., sequences whose elements are $ \pm 1$ ), each having length n, such that the sum of their periodic resp. aperiodicauto-correlation functions is a delta function. We fill many open cases inthe Bömer and Antweiler diagram [3] of the known cases where $PCS_p^n $ exist for $p \leqslant 12$ and $n \leqslant 50$ . In particular we show that $PCS_2^{34} $ exist, whileit is well known [1] that $ACS_2^{34} $ do not.