Global Well-posedness of the Stochastic Generalized Kuramoto-Sivashinsky Equation with Multiplicative Noise

Global well-posedness of the initial-boundary value problem for the stochastic generalized Kuramoto- Sivashinsky equation in a bounded domain D with a multiplicative noise is studied. It is shown that under suitable sufficient conditions, for any initial data u₀ ∈ L₂(D × Ω), this problem has a unique global solution u in the space L₂(Ω, C([0, T], L₂(D))) for any T >0, and the solution map u₀ ? u is Lipschitz continuous.     

Construction of Carleman formulas by using mixed problems with parameter-dependent boundary conditions

Let D be an open connected subset of the complex plane C with sufficiently smooth boundary ?D. Perturbing the Cauchy problem for the Cauchy–Riemann system ??u = f in D with boundary data on a closed subset S ? ?D, we obtain a family of mixed problems of the Zaremba-type for the Laplace equation depending on a small parameter ε ∈ (0, 1] in the boundary condition. Despite the fact that the mixed problems include noncoercive boundary conditions on ?D\S, each of them has a unique solution in some appropriate Hilbert space H ⁺(D) densely embedded in the Lebesgue space L ₂(?D) and the Sobolev–Slobodetski? space H ^1/2?δ(D) for every δ > 0. The corresponding family of the solutions {u _ε} converges to a solution to the Cauchy problem in H ⁺(D) (if the latter exists). Moreover, the existence of a solution to the Cauchy problem in H ⁺(D) is equivalent to boundedness of the family {u _ε} in this space. Thus, we propose solvability conditions for the Cauchy problem and an effective method of constructing a solution in the form of Carleman-type formulas.     

On the Completeness of Products of Harmonic Functions and the Uniqueness of the Solution of the Inverse Acoustic Sounding Problem

It is proved that the family of all pairwise products of regular harmonic functions on D and of the Newtonian potentials of points on the line L ? Rⁿ is complete in L₂(D), where D is a bounded domain in Rⁿ, n ≥ 3, such that \(\bar D\) ∩ L = ?. This result is used in the proof of uniqueness theorems for the inverse acoustic sounding problem in R³.     

On the Distribution of Zero Sets of Holomorphic Functions

Let M be a subharmonic function with Riesz measure ν _M in a domain D in the n-dimensional complex Euclidean space ? _n , and let f be a nonzero function that is holomorphic in D, vanishes on a set Z ? D, and satisfies |f| ? expM on D. Then restrictions on the growth of ν _M near the boundary of D imply certain restrictions on the dimensions or the area/volume of Z. We give a quantitative study of this phenomenon in the subharmonic framework.     

To the Spectral Theory of One-Dimensional Matrix Dirac Operators with Point Matrix Interactions

We investigate one-dimensional (2p × 2p)-matrix Dirac operators D_X,α and D_X,β with point matrix interactions on a discrete set X. Several results of [4] are generalized to the case of (p × p)-matrix interactions with p > 1. It is shown that a number of properties of the operators D_X,α and D_X,β (self-adjointness, discreteness of the spectrum, etc.) are identical to the corresponding properties of some Jacobi matrices B_X,α and B_X,β with (p × p)-matrix entries. The relationship found is used to describe these properties as well as conditions of continuity and absolute continuity of the spectra of the operators D_X,α and D_X,β. Also the non-relativistic limit at the velocity of light c → ∞ is studied.     

The local lifting problem for <Emphasis Type="Italic">D</Emphasis><Subscript>4</Subscript>

For a prime p, a cyclic-by-p group G and a G-extension L|K of complete discrete valuation fields of characteristic p with algebraically closed residue field, the local lifting problem asks whether the extension L|K lifts to characteristic zero. In this paper, we characterize D₄-extensions of fields of characteristic two, determine the ramification breaks of (suitable) D₄- extensions of complete discrete valuation fields of characteristic two, and solve the local lifting problem in the affirmative for every D₄-extension of complete discrete valuation fields of characteristic two with algebraically closed residue field; that is, we show that D₄ is a local Oort group for the prime 2.     

Span of a DL-algebra

Unless otherswise specified, all objects are defined over a field k of characteristic 0. Let K be a field. The unessentialness of an extension of the algebra Der K by means of a splittable semisimple Lie algebra is established. Let D _K be the category of differential Lie algebras (DL-algebras) (g;K). In this paper for an extension L/K the functor η:D _K → D _L , defining the tensor product L ? _K of vector spaces and the homomorphism of Lie algebras, is constructed. If the extension L/K is algebraic, then η is unique. The results will be required for strengthening the progress on Gelfand–Kirillov problem and weakened conjecture [1, 2].     

Augmentation quotients for burnside rings of generalized dihedral groups

Let H be a finite abelian group of odd order, D be its generalized dihedral group, i.e., the semidirect product of C₂ acting on H by inverting elements, where C₂ is the cyclic group of order two. Let Ω (D) be the Burnside ring of D, Δ(D) be the augmentation ideal of Ω (D). Denote by Δⁿ(D) and Q_n(D) the nth power of Δ(D) and the nth consecutive quotient group Δⁿ(D)/Δⁿ⁺¹(D), respectively. This paper provides an explicit Z-basis for Δⁿ(D) and determines the isomorphism class of Q_n(D) for each positive integer n.     

Graphs with small diameter determined by their <Emphasis Type="Italic">D</Emphasis>-spectra

Let G be a connected graph with vertex set V(G) = {v₁, v₂,..., v _n }. The distance matrix D(G) = (d _ij )_n×n is the matrix indexed by the vertices of G, where d _ij denotes the distance between the vertices v _i and v _j . Suppose that λ₁(D) ≥ λ₂(D) ≥... ≥ λ _n (D) are the distance spectrum of G. The graph G is said to be determined by its D-spectrum if with respect to the distance matrix D(G), any graph having the same spectrum as G is isomorphic to G. We give the distance characteristic polynomial of some graphs with small diameter, and also prove that these graphs are determined by their D-spectra.     

General Hausdorff functions,and the notion of one-sided measure and dimension

The main facts about Hausdorff and packing measures and dimensions of a Borel set E are revisited, using determining set functions \(\phi_\alpha\colon\mathcal{B}_E\to(0,\infty)\), where \(\mathcal{B}_E\) is the family of all balls centred on E and α is a real parameter. With mild assumptions on φ_α, we verify that the main density results hold, as well as the basic properties of the corresponding box dimension. Given a bounded open set V in ? ^D , these notions are used to introduce the interior and exterior measures and dimensions of any Borel subset of ?V. We stress that these dimensions depend on the choice of φ_α. Two determining functions are considered, φ_α(B)=Vol _D (B∩V)diam(B)^α-D and φ_α(B)=Vol _D (B∩V)^α/D , where Vol _D denotes the D-dimensional volume.     

Estimate of solution stability in a two-dimensional inverse problem for elasticity equations

The problem of determining the density of the medium and one of its elasticity moduli is considered. Properties of the elastic medium and external forces are assumed to be independent of the coordinate x ₃. In this case, the third component of the displacement vector satisfies a scalar equation of the second order, which contains the density ρ of the medium and elasticity modulus μ as coefficients. The parameters ρ and μ are known to be positive and constant everywhere outside some compact domain D ? ?², but they are unknown inside D. The problem of determining these coefficients in D via information, given on the boundary of the domain D for some finite time interval, about a solution of two direct problems is considered. An estimate of the conditional stability of a solution of the inverse problem under consideration is established.     

On a class of differential operators with constant coefficients

A linear differential operator P(D) = P(D ₁, …, D _n ) with constant coefficients is called almost hypoelliptic if all the derivatives D ^α P of the characteristic polynomial P(ξ ₁, …, ξ _n ) can be estimated by P. The paper proves that if P is an almost hypoelliptic operator and f is an infinitely differentiable function, square-summable with a definite exponential weight, then any square summable with the same weight solution u of the equation P(D)u = f is again an infinitely differentiable function and P(ξ) → ∞ as ξ → ∞.     

On Extraction of Smooth Solutions of a Class of Almost Hypoelliptic Equations with Constant Power

A linear differential operator P(x, D) = P(x₁,... x _n , D₁,..., D _n ) = ∑_αγ_α(x)D^α with coefficients γ_α(x) defined in E ⁿ is called formally almost hypoelliptic in E ⁿ if all the derivatives D^ν_ξP(x, ξ) can be estimated by P(x, ξ), and the operator P(x, D) has uniformly constant power in Eⁿ. In the present paper, we prove that if P(x, D) is a formally almost hypoelliptic operator, then all solutions of equation P(x, D)u = 0, which together with some of their derivatives are square integrable with a specified exponential weight, are infinitely differentiable functions.     

A mapping method in inverse Sturm-Liouville problems with singular potentials

In the space L ₂[0, π], the Sturm-Liouville operator L _D(y) = ?y″ + q(x)y with the Dirichlet boundary conditions y(0) = y(π) = 0 is analyzed. The potential q is assumed to be singular; namely, q = σ′, where σ ∈ L ₂[0, π], i.e., q ∈ W ₂ ^?1 [0, π]. The inverse problem of reconstructing the function σ from the spectrum of the operator L _D is solved in the subspace of odd real functions σ(π/2 ? x) = ?σ(π/2 + x). The existence and uniqueness of a solution to this inverse problem is proved. A method is proposed that allows one to solve this problem numerically.     

New discoveries of domination between traffic matrices

In this paper the definition of domination is generalized to the case that the elements of the traffic matrices may have negative values. It is proved that D₃ dominates D₃ + λ(D₂ ? D₁) for any λ ? 0 if D₁ dominates D₂. Let U(D) be the set of all the traffic matrices that are dominated by the traffic matrix D. It is shown that U(D_∞) and U(D_∈) are isomorphic. Besides, similar results are obtained on multi-commodity flow problems. Furthermore, the results are the generalized to integral flows.     

On Lebesgue Measure of Integral Self-Affine Sets

Let A be an expanding integer n×n matrix and D be a finite subset of ? ⁿ . The self-affine set T=T(A,D) is the unique compact set satisfying the equality \(A(T)=\bigcup_{d\in D}(T+d)\). We present an effective algorithm to compute the Lebesgue measure of the self-affine set T, the measure of the intersection T∩(T+u) for u∈? ⁿ , and the measure of the intersection of self-affine sets T(A,D ₁)∩T(A,D ₂) for different sets D ₁, D ₂?? ⁿ .     

Competition Hypergraphs of Products of Digraphs

If D = (V, A) is a digraph, its competition hypergraph \(C {\mathcal H}(D)\) has vertex set V and \(e {\subseteq} V\) is an edge of \(C {\mathcal H} (D)\) iff \(|e| {\geq} 2\) and there is a vertex \(v {\in} V\) , such that e = {w ∈ V|(w, v) ∈ A}. For several products D ₁ ° D ₂ of digraphs D ₁ and D ₂, we investigate the relations between the competition hypergraphs of the factors D ₁, D ₂ and the competition hypergraph of their product D ₁ ° D ₂.     

Generalized competition index of primitive digraphs

For any positive integers k and m, the k-step m-competition graph C _m ^k (D) of a digraph D has the same set of vertices as D and there is an edge between vertices x and y if and only if there are distinct m vertices v₁, v₂, · · ·, v _m in D such that there are directed walks of length k from x to v _i and from y to v _i for all 1 ≤ i ≤ m. The m-competition index of a primitive digraph D is the smallest positive integer k such that C _m ^k (D) is a complete graph. In this paper, we obtained some sharp upper bounds for the m-competition indices of various classes of primitive digraphs.     

The Paranormality of Products and Their Subsets

A topological space is called paranormal if any countable discrete system of closed sets {D_n:n = 1, 2, 3,...} can be expanded to a locally finite system of open sets {U_n:n = 1, 2, 3,...}, i.e., D_n is contained in U_n for all n, and D_m ∩ U_n≠ Ø if and only if D_m = D_n. It is proved that if X is a countably compact space whose cube is hereditarily paranormal, then X is metrizable.     

The weak form of normality

A topological space is said to be paranormal if every countable discrete collection of closed sets {D _n : n < ω} can be expanded to a locally finite collection of open sets {U _n : n < ω}, i.e., D _n ? U _n and D _m ∩ U _n ≠ 0 if and only if D _m = D _n . It is proved that if F: Comp → Comp is a normal functor of degree ≥ 3 and the compact space F(X) is hereditarily paranormal, then the compact space X is metrizable.