Hadamard matrices of order 28 with automorphism groups of order two

We give two examples H₁ and H₂ of Hadamard matrices of order 28 with trivial automorphism groups and show that H₁, H₁^T, H₂ and H₂^T are non-equivalent to each other as Hadamard matrices.     

A discrete “three-particle” Schrödinger operator in the Hubbard model

In the space L ₂(T ^ν ×T ^ν ), where T ^ν is a ν-dimensional torus, we study the spectral properties of the “three-particle” discrete Schrödinger operator ? = H₀ + H₁ + H₂, where H₀ is the operator of multiplication by a function and H₁ and H₂ are partial integral operators. We prove several theorems concerning the essential spectrum of ?. We study the discrete and essential spectra of the Hamiltonians H^t and h arising in the Hubbard model on the three-dimensional lattice.     

Inverse Scattering for a Schroedinger Operator with a Repulsive Potential

We consider a pair of Hamiltonians （H, H0） on L2（R^n）, where H0=p^2 -x^2 is a SchrSdinger operator with a repulsive potential, and H = H0＋V（x）. We show that, under suitable assumptions on the decay of the electric potential, V is uniquely determined by the high energy limit of the scattering operator.     

Hosoya polynomials under gated amalgamations

An induced subgraph H of a graph G is gated if for every vertex x outside H there exists a vertex x^′ inside H such that each vertex y of H is connected with x by a shortest path passing through x^′. The gated amalgam of graphs G₁ and G₂ is obtained from G₁ and G₂ by identifying their isomorphic gated subgraphs H₁ and H₂. Two theorems on Hosoya polynomials of gated amalgams are provided. As their applications, explicit expressions for Hosoya polynomials of hexagonal chains are obtained.     

Extension of isometries on the unit sphere of L p spaces

In this paper we study the isometric extension problem and show that every surjective isometry between the unit spheres of Lp (μ) (1 p ∞, p≠2) and a Banach space E can be extended to a linear isometry from Lp(μ) onto E, which means that if the unit sphere of E is (metrically) isometric to the unit sphere of Lp(μ), then E is linearly isometric to Lp(μ). We also prove that every surjective 1-Lipschitz or anti-1-Lipschitz map between the unit spheres of Lp (μ1, H1) and Lp(μ2,H2) must be an isometry and can be extended to a linear isometry from Lp (μ1,H1) to Lp (μ2,H2), where H1 and H2 are Hilbert spaces.     

Stability theorems for projections of convex sets

LetH ^n?1 denote the set of all (n ? 1)-dimensional linear subspaces of euclideann-dimensional spaceE ⁿ (n≧3), and letJ andK be two compact convex subsets ofE ⁿ. It is well-known thatJ andK are translation equivalent (or homothetic) if for allH ∈H ^n?1 the respective orthogonal projections, sayJ _H, K_H, are translation equivalent (or homothetic). This fact gives rise to the following stability problem: Ifε≧0 is given, and if for everyH ∈H ^n?1 a translate (or homothetic copy) ofK _H is within Hausdorff distanceε ofJ _H, how close isJ to a nearest translate (or homothetic copy) ofK? In the case of translations it is shown that under the above assumptions there is always a translate ofK within Hausdorff distance (1 + 2√2)ε ofJ. Similar results for homothetic transformations are proved and applications regarding the stability of characterizations of centrally symmetric sets and spheres are given. Furthermore, it is shown that these results hold even ifH ^n?1 is replaced by a rather small (but explicitly specified) subset ofH ^n?1.     

Decomposition of bipartite graphs into special subgraphs

Let F and G be two graphs and let H be a subgraph of G. A decomposition of G into subgraphs F₁,F₂,…,F_m is called an F-factorization of G orthogonal to H if F_i≅F and |E(F_i∩H)|=1 for each i=1,2,…,m. Gyárfás and Schelp conjectured that the complete bipartite graph K_4k,4k has a C₄-factorization orthogonal to H provided that H is a k-factor of K_4k,4k. In this paper, we show that (1) the conjecture is true when H satisfies some structural conditions; (2) for any two positive integers r?k, K_kr²,kr² has a K_r,r-factorization orthogonal to H if H is a k-factor of K_kr²,kr²; (3) K_2d²,2d² has a C₄-factorization such that each edge of H belongs to a different C₄ if H is a subgraph of K_2d²,2d² with maximum degree Δ(H)?d.     

The Hosoya polynomial decomposition for catacondensed benzenoid graphs

For a graph G with the vertex set V(G), we denote by d(u,v) the distance between vertices u and v in G, by d(u) the degree of vertex u. The Hosoya polynomial of G is H(G)=∑_{u,v}⊆V(G)x^d(u,v). The partial Hosoya polynomials of G are for positive integer numbers m and n. It is shown that H(G₁)−H(G₂)=x²(x+1)²(H₃₃(G₁)−H₃₃(G₂)),H₂₂(G₁)−H₂₂(G₂)=(x²+x−1)²(H₃₃(G₁)−H₃₃(G₂)) and H₂₃(G₁)−H₂₃(G₂)=2(x²+x−1)(H₃₃(G₁)−H₃₃(G₂)) for arbitrary catacondensed benzenoid graphs G₁ and G₂ with equal number of hexagons. As an application, we give an affine relationship between H(G) with two other distance-based polynomials constructed by Gutman [I. Gutman, Some relations between distance-based polynomials of trees, Bulletin de l’Académie Serbe des Sciences et des Arts (Cl. Math. Natur.) 131 (2005) 1-7].     

Admissible majorants for model subspaces, and arguments of inner functions

Let Θ be an inner function in the upper half-plane ?⁺ and let K _Θ denote the model subspace H ² ? Θ H ² of the Hardy space H ² = H ²(?⁺). A nonnegative function w on the real line is said to be an admissible majorant for K _Θ if there exists a nonzero function f ∈ K _Θ such that {f} ? w a.e. on ?. We prove a refined version of the parametrization formula for K _Θ-admissible majorants and simplify the admissibility criterion (in terms of arg Θ) obtained in [8]. We show that, for every inner function Θ, there exist minimal K _Θ-admissible majorants. The relationship between admissibility and some weighted approximation problems is considered.     

Spectral deformations of one-dimensional Schrödinger operators

We provide a complete spectral characterization of a new method of constructing isospectral (in fact, unitary) deformations of general Schrödinger operatorsH=?d ²/dx ²+V in $H = - d^2 /dx^2 + V in \mathcal{L}^2 (\mathbb{R})$ . Our technique is connected to Dirichlet data, that is, the spectrum of the operatorH ^D onL ²((?∞,x ₀)) ⊕L ²((x ₀, ∞)) with a Dirichlet boundary condition atx ₀. The transformation moves a single eigenvalue ofH ^D and perhaps flips which side ofx ₀ the eigenvalue lives. On the remainder of the spectrum, the transformation is realized by a unitary operator. For cases such asV(x)→∞ as |x|→∞, whereV is uniquely determined by the spectrum ofH and the Dirichlet data, our result implies that the specific Dirichlet data allowed are determined only by the asymptotics asE→∞.     

On ideals in the bidual of the Fourier algebra and related algebras

Let G be a compact nonmetrizable topological group whose local weight b(G) has uncountable cofinality. Let H be an amenable locally compact group, A(G×H) the Fourier algebra of G×H, and UC₂(G×H) the space of uniformly continuous functionals in VN(G×H)=A^∗(G×H). We use weak factorization of operators in the group von Neumann algebra VN(G×H) to prove that there exist at least ²b(G)² left ideals of dimensions at least ²b(G)² in A(G×H)^∗∗ and in UC₂^∗(G×H). We show that every nontrivial right ideal in A(G×H)^∗∗ and in UC₂^∗(G×H) has dimension at least ²b(G)².     

Convergence of H 1-Galerkin mixed finite element method for parabolic problems with reduced regularity on initial data

We study the convergence of H ¹-Galerkin mixed finite element method for parabolic problems in one space dimension. Both semi-discrete and fully discrete schemes are analyzed assuming less regularity on initial data. More precisely, for the spatially discrete scheme, error estimates of order \(\mathcal{O}\) (h ² t ^?1/2) for positive time are established assuming the initial function p ₀ ∈ H ²(Ω) ∩ H ₀ ¹ (Ω). Further, we use energy technique together with parabolic duality argument to derive error estimates of order \(\mathcal{O}\) (h ² t ^?1) when p ₀ is only in H ₀ ¹ (Ω). A discrete-in-time backward Euler method is analyzed and almost optimal order error bounds are established.     

On collision local time of two independent fractional ornstein-uhlenbeck processes

In this article, we study the existence of collision local time of two independent d-dimensional fractional Ornstein-Uhlenbeck processes X_t~(H_1)and _t~(H_2),with different parameters H_i∈(0, 1), i = 1, 2. Under the canonical framework of white noise analysis,we characterize the collision local time as a Hida distribution and obtain its' chaos expansion.     

Multipliers of a wandering subspace for a shift invariant subspace

For a shift-invariant subspace M of the two variable Hardy space H², we consider the associated wandering subspace M₀=M?zM. Then there exists a nonconstant function ? in H^∞ such that ?M₀⊆M₀ if and only if M=qH² for some inner function q.     

The local and global existence of solutions for a generalized Camassa-Holm equation

A fully nonlinear generalization of the Camassa-Holm equation is investigated. Using the pseudoparabolic regularization technique, its local well-posedness in Sobolev space Hs(R) with s3/2 is established via a limiting procedure. Provided that the initial momentum (1-x2)u0 satisfies the sign condition, u0∈Hs(R)(s3/2) and u0∈L1(R),the existence and uniqueness of global solutions for the equation are shown to be true in the space C([0,∞); Hs(R))∩C1([0,∞);Hs-1(R)).     

Albanese and Picard 1-motives

We define, in a purely algebraic way, 1-motives Alb⁺(X), Alb⁻(X), Pic⁺(X), and Pic⁻(X) associated with any algebraic scheme X over an algebraically closed field of characteristic zero. For X over C of dimension n, the Hodge realizations are, respectively, H^{2n − 1} (X, Z(n))/(torsion), H₁ (X, Z)/(torsion), H¹ (X, Z(1)), and H_{2n − 1} (X, Z(1 n))/(torsion).     

On some real hypersurfaces of a complex hyperbolic space

Given a real hypersurface of a complex hyperbolic space #x2102;?H ⁿ ,we construct a principal circle bundle over it which is a Lorentzian hypersurface of the anti-De Sitter space H ₁ ²ⁿ⁺¹ .Relations between the respective second fundamental forms are obtained permitting us to classify a remarkable family of real hypersurfaces of ?H ⁿ .     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> estimates for Riesz transform and their commutators associated with Schrödinger type operator

Let H ₂ = (?Δ)² + V ² be the Schrödinger type operator, where V satisfies reverse Hölder inequality. In this paper, we establish the L ^p boundedness for V ² H ₂ ^?1 , H ₂ ^?1 V ², VH ₂ ^?1/2 and H ₂ ^?1 V ², and that of their commutators. We also prove that H ₂ ^?1 V ²,VH ₂ ^?1/2 are bounded from BMO _L to BMO _L .     

Problems in algebra inspired by universal algebraic geometry

Let Θ be a variety of algebras. In every variety Θ and every algebra H from Θ one can consider algebraic geometry in Θ over H. We also consider a special categorical invariant K _Θ of this geometry. The classical algebraic geometry deals with the variety Θ = Com-P of all associative and commutative algebras over the ground field of constants P. An algebra H in this setting is an extension of the ground field P. Geometry in groups is related to the varieties Grp and Grp-G, where G is a group of constants. The case Grp-F, where F is a free group, is related to Tarski’s problems devoted to logic of a free group. The described general insight on algebraic geometry in different varieties of algebras inspires some new problems in algebra and algebraic geometry. The problems of such kind determine, to a great extent, the content of universal algebraic geometry. For example, a general and natural problem is: When do algebras H ₁ and H ₂ have the same geometry? Or more specifically, what are the conditions on algebras from a given variety Θ that provide the coincidence of their algebraic geometries? We consider two variants of coincidence: 1) K _Θ(H ₁) and K _Θ(H ₂) are isomorphic; 2) these categories are equivalent. This problem is closely connected with the following general algebraic problem. Let Θ⁰ be the category of all algebras W = W(X) free in Θ, where X is finite. Consider the groups of automorphisms Aunt(Θ⁰) for different varieties Θ and also the groups of autoequivalences of Θ⁰. The problem is to describe these groups for different Θ.     

Group actions on Stanley-Reisner rings and invariants of permutation groups

Let H be a group of permutations of x₁ ,…, x_n and let Q^H[x₁ , x₂ ,…, x_n] denote the ring of H-invariant Polynomials in x₁ , x₂ ,…, x_n with rational coefficients. Combinatorial methods for the explicit construction of free bases for Q^H[x₁ , x₂ ,…, x_n] as a module over the symmetric polynomials are developed. The methods are developed by studying the action of the symmetric group on the Stanley-Reisner ring of the subset lattice. Some general results are also obtained by studying the action of a Coxeter group on the Stanley-Reisner ring of the corresponding Coxeter complex. In the case of a Weyl group, a purely combinatorial construction of certain invariants first considered by R. Steinberg (Topology14 (1975), 173–177) is obtained. Some applications to representation theory are also included.