Model oblique derivative problem for the heat equation in the Zygmund space <Emphasis Type="Italic">H</Emphasis> <Subscript>1</Subscript>

The third boundary value problem and the oblique derivative problem for the heat equation are considered in model formulations. A difference compatibility condition is introduced for the initial and boundary functions. Under suitable assumptions made about the problem data, the solutions are shown to belong to the parabolic Zygmund space H₁, which is the analogue of the parabolic Hölder space for an integer smoothness exponent.     

Inverse problem with nonlocal observation of finding the coefficient multiplying <Emphasis Type="Italic">u</Emphasis> <Subscript> <Emphasis Type="Italic">t</Emphasis> </Subscript> in the parabolic equation

We study the inverse problem of the reconstruction of the coefficient ?(x, t) = ?⁰(x, t) + r(x) multiplying u_t in a nonstationary parabolic equation. Here ?⁰(x, t) ≥ ?⁰ > 0 is a given function, and r(x) ≥ 0 is an unknown function of the class L_∞(Ω). In addition to the initial and boundary conditions (the data of the direct problem), we pose the problem of nonlocal observation in the form ∫₀^Tu(x, t) dμ(t) = χ(x) with a known measure dμ(t) and a function χ(x). We separately consider the case dμ(t) = ω(t)dt of integral observation with a smooth function ω(t). We obtain sufficient conditions for the existence and uniqueness of the solution of the inverse problem, which have the form of ready-to-verify inequalities. We suggest an iterative procedure for finding the solution and prove its convergence. Examples of particular inverse problems for which the assumptions of our theorems hold are presented.     

The one dimensional parabolic <Emphasis Type="Italic">p</Emphasis>(<Emphasis Type="Italic">x</Emphasis>)-Laplace equation

The Dirichlet problem for the degenerate and singular parabolic p(x)-Laplace equation with one spatial variable is considered. We prove the existence of the unique weak solution such that the derivatives u _t and u _x of a solution u belong to \({L_{\infty}}\). Moreover for the singular case we prove the existence of the strong solution i.e. such that u _t , u _x and u _xx belong to \({L_{\infty}}\).     

Exponentially Many Monochromatic <Emphasis Type="Italic">n</Emphasis>-Matchings in <Emphasis Type="Italic">K</Emphasis><Subscript>3<Emphasis Type="Italic">n</Emphasis>-1</Subscript>

Erdös et al and Gerencsér et al had shown that in any 2-edge-coloring of K _3n-1, there is a n-matching containing edges with the same color(we call such matching monochromatic matching). In this paper we show that for any 2-edge-coloring of K _3n-1 there exists a monochromatic subgraph H of K _3n-1 which contains exponentially many monochromatic n-matchings.     

On Quantum Dynamics on <Emphasis Type="Italic">C</Emphasis>*-Algebras

We study three related extremal problems in the space H of functions analytic in the unit disk such that their boundary values on a part γ₁ of the unit circle Γ belong to the space \(L_{{\psi _1}}^\infty ({\gamma _1})\)of functions essentially bounded on γ₁ with weight ψ₁ and their boundary values on the set γ₀ = Γ γ₁ belong to the space \(L_{{\psi _0}}^\infty ({\gamma _0})\)with weight ψ₀. More exactly, on the class Q of functions from H such that the \(L_{{\psi _0}}^\infty ({\gamma _0})\)-norm of their boundary values on γ₀ does not exceed 1, we solve the problem of optimal recovery of an analytic function on a subset of the unit disk from its boundary values on γ₁ specified approximately with respect to the norm of \(L_{{\psi _1}}^\infty ({\gamma _1})\). We also study the problem of the optimal choice of the set γ₁ for a given fixed value of its measure. The problem of the best approximation of the operator of analytic continuation from a part of the boundary by bounded linear operators is investigated.     

Asymptotic <Emphasis Type="Italic">T</Emphasis><Subscript><Emphasis Type="Italic">u</Emphasis></Subscript>-Toeplitzness of weighted composition operators on <Emphasis Type="Italic">H</Emphasis><Superscript>2</Superscript>

Given a unilateral forward shift S acting on a complex, separable, innite dimensional Hilbert space H, an asymptotically S-Toeplitz operator is a bounded linear operator T on H satisfying that {S* ⁿ TS ⁿ } is convergent with respect to one of the topologies commonly used in the algebra of bounded linear operators on H. In this paper, we study the asymptotic T _u -Toeplitzness of weighted composition operators on the Hardy space H², where u is a nonconstant inner function.     

Theory of (<Emphasis Type="Italic">q</Emphasis> <Subscript>1</Subscript>, <Emphasis Type="Italic">q</Emphasis> <Subscript>2</Subscript>)-quasimetric spaces and coincidence points

We introduce (q₁, q₂)-quasimetric spaces and examine their properties. Covering mappings between (q₁, q₂)-quasimetric spaces are investigated. Sufficient conditions for the existence of a coincidence point of two mappings acting between (q₁, q₂)-quasimetric spaces such that one is a covering mapping and the other satisfies the Lipschitz condition are obtained.     

A new functor from <Emphasis Type="Italic">D</Emphasis><Subscript>5</Subscript>-Mod to <Emphasis Type="Italic">E</Emphasis><Subscript>6</Subscript>-Mod

We find a new representation of the simple Lie algebra of type E ₆ on the polynomial algebra in 16 variables, which gives a fractional representation of the corresponding Lie group on 16-dimensional space. Using this representation and Shen’s idea of mixed product, we construct a new functor from D ₅-Mod to E ₆-Mod. A condition for the functor to map a finite-dimensional irreducible D ₅-module to an infinite-dimensional irreducible E ₆-module is obtained. Our results yield explicit constructions of certain infinite-dimensional irreducible weight E6-modules with finite-dimensional weight subspaces. In our approach, the idea of Kostant’s characteristic identities plays a key role.     

Equivalence of two definitions of a generalized <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> solution to the initial-boundary value problem for the wave equation

In our previous papers, we introduced the notion of a generalized solution to the initial-boundary value problem for the wave equation with a boundary function µ(t) such that the integral ∫ ₀ ^T (T ? t)|µ(t)| ^p dt exists. Here we prove that this solution is a unique solution to the problem in L _p that satisfies the corresponding integral identity.     

Decomposition theorems for<Emphasis Type="Italic">Q</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> spaces

We study the Möbius invariant spacesQ _p andQ _{p, 0} of analytic functions. These scales of spaces include BMOA=Q₁, VMOA=Q_{1, 0} and the Dirichlet space=Q₀. Using the Bergman metric, we establish decomposition theorems for these spaces. We obtain also a fractional derivative characterization for bothQ _p andQ _{p, 0} .     

Coercivity results of a modified <Emphasis Type="Italic">Q</Emphasis><Subscript>1</Subscript>-finite volume element scheme for anisotropic diffusion problems

In this paper, we study a so-called modified Q ₁-finite volume element scheme that is obtained by employing the trapezoidal rule to approximate the line integrals in the classical Q ₁-finite volume element method. A necessary and sufficient condition is obtained for the positive definiteness of a certain element stiffness matrix. Based on this result, a sufficient condition is suggested to guarantee the coercivity of the scheme on arbitrary convex quadrilateral meshes. When the diffusion tensor is an identity matrix, this sufficient condition reduces to a geometric one, covering some standard meshes, such as the traditional h ^1+γ -parallelogram meshes and some trapezoidal meshes. More interesting is that, this sufficient condition has explicit expression, by which one can easily judge on any diffusion tensor and any mesh with any mesh size h > 0. The H ¹ error estimate of the modified Q ₁-finite volume element scheme is obtained without the traditional h ^1+γ -parallelogram assumption. Some numerical experiments are carried out to validate the theoretical analysis.     

Ordinal indices of Small Subspaces of <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

We calculate the ordinal L _p index defined in [3] for Rosenthal’s space X _p , \({\ell_p}\) and \({\ell_2}\). We show that an infinite-dimensional subspace of L _p \({(2 < p < \infty)}\) non-isomorphic to \({\ell_2}\) embeds in \({\ell_p}\) if and only if its ordinal index is the minimal possible. We also give a sufficient condition for a \({\mathcal{L}_p}\) subspace of \({\ell_p \oplus \ell_2}\) to be isomorphic to X _p .     

On systems of linear functional equations of the second kind in <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>

We consider a general system of functional equations of the second kind in L ₂ with a continuous linear operator T satisfying the condition that zero lies in the limit spectrum of the adjoint operator T*. We show that this condition holds for the operators of a wide class containing, in particular, all integral operators. The system under study is reduced by means of a unitary transformation to an equivalent system of linear integral equations of the second kind in L ₂ with Carleman matrix kernel of a special kind. By a linear continuous invertible change, this system is reduced to an equivalent integral equation of the second kind in L ₂ with quasidegenerate Carleman kernel. It is possible to apply various approximate methods of solution for such an equation.     

<Emphasis Type="Italic">M</Emphasis> <Subscript> <Emphasis Type="Italic">p</Emphasis> </Subscript>-supplemented subgroups of finite groups

A subgroup K of G is M_p-supplemented in G if there exists a subgroup B of G such that G = KB and TB < G for every maximal subgroup T of K with |K: T| = p^α. In this paper we prove the following: Let p be a prime divisor of |G| and let H be ap-nilpotent subgroup having a Sylow p-subgroup of G. Suppose that H has a subgroup D with D_p ≠ 1 and |H: D| = p_α. Then G is p-nilpotent if and only if every subgroup T of H with |T| = |D| is M_p-supplemented in G and N_G(T_p)/C_G(T_p) is a p-group.     

Examples of algebraic groups of type <Emphasis Type="Italic">G</Emphasis> <Subscript>2</Subscript> having the same maximal tori

Answering a question of A. Rapinchuk, we construct examples of non-isomorphic semisimple algebraic groups H₁ and H₂ of type G₂ having coherently equivalent systems of maximal k-tori.     

(<Emphasis Type="Italic">q</Emphasis><Subscript>1</Subscript>, <Emphasis Type="Italic">q</Emphasis><Subscript>2</Subscript>)-quasimetrics bi-Lipschitz equivalent to 1-quasimetrics

We prove that the conditions of (q₁, 1)- and (1, q₂)-quasimertricity of a distance function ρ are sufficient for the existence of a quasimetric bi-Lipschitz equivalent to ρ. It follows that the Box-quasimetric defined with the use of basis vector fields of class C₁ whose commutators at most sum their degrees is bi-Lipschitz equivalent to some metric. On the other hand, we show that these conditions are not necessary. We prove the existence of (q₁, q₂)-quasimetrics for which there are no Lipschitz equivalent 1-quasimetrics, which in particular implies another proof of a result by V. Schröder.     

Planar Graded Lattices and the <Emphasis Type="Bold">c</Emphasis><Subscript>1</Subscript>-Median Property

Let L be a lattice of finite length, ξ = (x ₁,…, x _k )∈L ^k , and y ∈ L. The remoteness r(y, ξ) of y from ξ is d(y, x ₁)+?+d(y, x _k ), where d stands for the minimum path length distance in the covering graph of L. Assume, in addition, that L is a graded planar lattice. We prove that whenever r(y, ξ) ≤ r(z, ξ) for all z ∈ L, then y ≤ x ₁∨?∨x _k . In other words, L satisfies the so-called c ₁ -median property.     

The best <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> approximation of the Laplace operator by linear bounded operators in the classes of functions of two and three variables

Close two-sided estimates are obtained for the best approximation in the space L _p (? ^m ), m = 2 and 3, 1 ≤ p ≤ ∞, of the Laplace operator by linear bounded operators in the class of functions for which the second power of the Laplace operator belongs to the space L _p (? ^m ). We estimate the best constant in the corresponding Kolmogorov inequality and the error of the optimal recovery of values of the Laplace operator on functions from this class given with an error. We present an operator whose deviation from the Laplace operator is close to the best.     

Counting extreme <Emphasis Type="Italic">U</Emphasis><Subscript>1</Subscript> matrices and characterizing quadratic doubly stochastic operators

The U ₁ matrix and extreme U ₁ matrix were successfully used to study quadratic doubly stochastic operators by R. Ganikhodzhaev and F. Shahidi [Linear Algebra Appl., 2010, 432: 24–35], where a necessary condition for a U ₁ matrix to be extreme was given. S. Yang and C. Xu [Linear Algebra Appl., 2013, 438: 3905–3912] gave a necessary and sufficient condition for a symmetric nonnegative matrix to be an extreme U ₁ matrix and investigated the structure of extreme U ₁ matrices. In this paper, we count the number of the permutation equivalence classes of the n × n extreme U ₁ matrices and characterize the structure of the quadratic stochastic operators and the quadratic doubly stochastic operators.     

Periodic Groups Saturated with the Groups <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>(<Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

Given an indexing set I and a finite field K_α for each α ∈ I, let ? = {L₂(K_α) | α ∈ I} and \(\mathfrak{N} = \{ SL_2 (K_\alpha )|\alpha \in I\}\). We prove that each periodic group G saturated with groups in \(\Re (\mathfrak{N})\) is isomorphic to L₂(P) (respectively SL₂(P)) for a suitable locally finite field P.