Showing papers in "Open Mathematics in 2007"
TL;DR: In this article, the authors investigated the convergence of the Tucker decomposition with respect to the relative Frobenius norm in higher-order tensors, including inner, outer and Hadamard products, and proposed a mixed CT model based on additive splitting of a tensor as a sum of canonical and Tucker-type representations.
Abstract: This paper investigates best rank-(r
1,..., r
d
) Tucker tensor approximation of higher-order tensors arising from the discretization of linear operators and functions in ℝ
d
. Super-convergence of the best rank-(r
1,..., r
d
) Tucker-type decomposition with respect to the relative Frobenius norm is proven. Dimensionality reduction by the two-level Tucker-to-canonical approximation is discussed. Tensor-product representation of basic multi-linear algebra operations is considered, including inner, outer and Hadamard products. Furthermore, we focus on fast convolution of higher-order tensors represented by the Tucker/canonical models. Optimized versions of the orthogonal alternating least-squares (ALS) algorithm is presented taking into account the different formats of input data. We propose and test numerically the mixed CT-model, which is based on the additive splitting of a tensor as a sum of canonical and Tucker-type representations. It allows to stabilize the ALS iteration in the case of “ill-conditioned” tensors. The best rank-(r
1,..., r
d
) Tucker decomposition is applied to 3D tensors generated by classical potentials, for example
$$\tfrac{1}{{\left| {x - y} \right|}}, e^{ - \alpha \left| {x - y} \right|} , \tfrac{{e^{ - \left| {x - y} \right|} }}{{\left| {x - y} \right|}}$$
and
$$\tfrac{{erf(|x|)}}{{|x|}}$$
with x, y ∈ ℝ
d
. Numerical results for tri-linear decompositions illustrate exponential convergence in the Tucker rank, and robustness of the orthogonal ALS iteration.
91 citations
TL;DR: In this article, a new class of nonlinear variational inclusion problems is presented based on the notion of A-monotonicity, which generalizes H − monotonicity (and in turn generalizes maximal monotonivity).
Abstract: Based on the notion of A — monotonicity, a new class of nonlinear variational inclusion problems is presented. Since A — monotonicity generalizes H — monotonicity (and in turn, generalizes maximal monotonicity), results thus obtained, are general in nature.
40 citations
TL;DR: In this article, the vertex operator (super) algebra is defined for every positive integer m ∈ ℂ ∖ {0, −2} and every nonnegative integer k and the vertex algebra is realized as a subalgebra of a lattice vertex algebra.
Abstract: For every m ∈ ℂ ∖ {0, −2} and every nonnegative integer k we define the vertex operator (super)algebra D
m,k
having two generators and rank
$$\frac{{3m}}{{m + 2}}$$
. If m is a positive integer then D
m,k
can be realized as a subalgebra of a lattice vertex algebra. In this case, we prove that D
m,k
is a regular vertex operator (super) algebra and find the number of inequivalent irreducible modules.
27 citations
TL;DR: In this article, it was shown that approximation is possible in the norm on the space of Schwarzian derivatives of a holomorphic univalent function f in a finitely connected domain D ∋ ∞ with quasiconformal extension to widehat{\mathbb{C}}\) by ϰ(f) ≤ k(f).
Abstract: The Grunsky and Teichmuller norms ϰ(f) and k(f) of a holomorphic univalent function f in a finitely connected domain D ∋ ∞ with quasiconformal extension to \(\widehat{\mathbb{C}}\) are related by ϰ(f) ≤ k(f). In 1985, Jurgen Moser conjectured that any univalent function in the disk Δ* = {z: |z| > 1} can be approximated locally uniformly by functions with ϰ(f) < k(f). This conjecture has been recently proved by R. Kuhnau and the author. In this paper, we prove that approximation is possible in a stronger sense, namely, in the norm on the space of Schwarzian derivatives. Applications of this result to Fredholm eigenvalues are given. We also solve the old Kuhnau problem on an exact lower bound in the inverse inequality estimating k(f) by ϰ(f), and in the related Ahlfors inequality.
22 citations
TL;DR: In this article, a connection between Romanovski polynomials and those that solve the one-dimensional Schrodinger equation with the trigonometric Rosen-Morse and hyperbolic Scarf potential is established.
Abstract: A connection between Romanovski polynomials and those polynomials that solve the one-dimensional Schrodinger equation with the trigonometric Rosen-Morse and hyperbolic Scarf potential is established. The map is constructed by reworking the Rodrigues formula in an elementary and natural way. The generating function is summed in closed form from which recursion relations and addition theorems follow. Relations to some classical polynomials are also given.
18 citations
TL;DR: In this paper, the notion of scattering monodromy for a two degree of freedom hyperbolic oscillator was introduced and applied to determine the Picard-Lefschetz monodrome of the isolated singular point of a quadratic function of two complex variables.
Abstract: We present the notion of scattering monodromy for a two degree of freedom hyperbolic oscillator and apply this idea to determine the Picard-Lefschetz monodromy of the isolated singular point of a quadratic function of two complex variables.
16 citations
TL;DR: In this article, the authors give an alternative description of weak injectivity properties for partially ordered monoids using systems of equations, which generalize the corresponding results for (unordered) acts over monoids proved by Victoria Gould in the 1980s.
Abstract: If S is a partially ordered monoid then a right S-poset is a poset A on which S acts from the right in such a way that the action is compatible both with the order of S and A. By regular weak injectivity properties we mean injectivity properties with respect to all regular monomorphisms (not all monomorphisms) from different types of right ideals of S to S. We give an alternative description of such properties which uses systems of equations. Using these properties we prove several so-called homological classification results which generalize the corresponding results for (unordered) acts over (unordered) monoids proved by Victoria Gould in the 1980’s.
16 citations
TL;DR: In this paper, the complementary polynomials satisfy a hypergeometric-type differential equation themselves, have a three-term recursion among others, and obey Rodrigues formulas.
Abstract: Starting from the Rodrigues representation of polynomial solutions of the general hypergeometric-type differential equation complementary polynomials are constructed using a natural method. Among the key results is a generating function in closed form leading to short and transparent derivations of recursion relations and addition theorem. The complementary polynomials satisfy a hypergeometric-type differential equation themselves, have a three-term recursion among others and obey Rodrigues formulas. Applications to the classical polynomials are given.
15 citations
TL;DR: In this paper, the linear Volterra equation of hyperbolic type was considered and the decay of the associated energy as time goes to infinity was investigated. But the decay was not considered in this paper.
Abstract: This note is concerned with the linear Volterra equation of hyperbolic type
$$\partial _{tt} u(t) - \alpha \Delta u(t) + \int_0^t {\mu (s)\Delta u(t - s)} ds = 0$$
on the whole space ℝN New results concerning the decay of the associated energy as time goes to infinity were established
14 citations
TL;DR: In this paper, the Charlier polynomials Cn(χ) and their zeros asymptotically as n → ∞ were analyzed using the limit relation between Krawtchouk and Charlier, involving some special functions.
Abstract: We analyze the Charlier polynomials Cn(χ) and their zeros asymptotically as n → ∞ We obtain asymptotic approximations, using the limit relation between the Krawtchouk and Charlier polynomials, involving some special functions We give numerical examples showing the accuracy of our formulas
13 citations
TL;DR: In this article, a partial lower semilattice is shown to be a reduct of an expanded Hilbert algebra, and the implication in an implicative partial semilectice is characterised in terms of the?lters of the underlying partial semi-attice.
Abstract: The infimum of elements a and b of a Hilbert algebra are said to be the compatible meet of a and b, if the elements a and b are compatible in a certain strict sense. The subject of the paper will be Hilbert algebras equipped with the compatible meet operation, which normally is partial. A partial lower semilattice is shown to be a reduct of such an expanded Hilbert algebra i ?both algebras have the same ?lters.An expanded Hilbert algebra is actually an implicative partial semilattice (i.e., a relative subalgebra of an implicative semilattice),and conversely.The implication in an implicative partial semilattice is characterised in terms of ?lters of the underlying partial semilattice.
TL;DR: In this paper, basic differential invariants of generic hyperbolic Monge-Ampere equations with respect to contact transformations are constructed and the equivalence problem for these equations is solved.
Abstract: In this paper basic differential invariants of generic hyperbolic Monge-Ampere equations with respect to contact transformations are constructed and the equivalence problem for these equations is solved.
TL;DR: In this article, the authors studied oscillatory and asymptotic properties of solutions of nonlinear differential equations of the third order with quasiderivatives and proved comparison theorems on property A between linear and nonlinear equations.
Abstract: The aim of our paper is to study oscillatory and asymptotic properties of solutions of nonlinear differential equations of the third order with quasiderivatives. We prove comparison theorems on property A between linear and nonlinear equations. Some integral criteria ensuring property A for nonlinear equations are also given. Our assumptions on the nonlinearity of f are restricted to its behavior only in a neighborhood of zero and a neighborhood of infinity.
TL;DR: In this article, the authors studied two classes of light-like submanifolds of codimension two of semi-Riemannian manifolds, according as their radical subspaces are 1-dimensional or 2-dimensional.
Abstract: In this paper we study two classes of lightlike submanifolds of codimension two of semi-Riemannian manifolds, according as their radical subspaces are 1-dimensional or 2-dimensional. For a large variety of both these classes, we prove the existence of integrable canonical screen distributions subject to some reasonable geometric conditions and support the results through examples.
TL;DR: The Kolmogorov distance between the expected spectral distribution of an n × n matrix from the Deformed Gaussian Ensemble and the distribution function of the semi-circle law is of order O(n−2/3+v).
Abstract: It is shown that the Kolmogorov distance between the expected spectral distribution function of an n × n matrix from the Deformed Gaussian Ensemble and the distribution function of the semi-circle law is of order O(n−2/3+v).
TL;DR: In this article, the uniqueness of the solutions of two non-local boundary value problems with continuous and special gluing conditions for parabolic-hyperbolic type equations are proven by the ABC method.
Abstract: In the present paper we study the unique solvability of two non-local boundary value problems with continuous and special gluing conditions for parabolic-hyperbolic type equations. The uniqueness of the solutions of the considered problems are proven by the “abc” method. Existence theorems for the solutions of these problems are proven by the method of integral equations. The obtained results can be used for studying local and non-local boundary-value problems for mixed-hyperbolic type equations with two and three lines of changing type.
TL;DR: In this article, the authors give sufficient conditions such that the spectrum of T is discrete and σ(T) = Λ(T), the set of all the limit points of eigenvalues of the truncated operator T N.
Abstract: Let T be a self-adjoint tridiagonal operator in a Hilbert space H with the orthonormal basis {e n } n=1 ∞ , σ(T) be the spectrum of T and Λ(T) be the set of all the limit points of eigenvalues of the truncated operator T N . We give sufficient conditions such that the spectrum of T is discrete and σ(T) = Λ(T) and we connect this problem with an old problem in analysis.
TL;DR: In this paper, the authors introduce a concept of implication groupoid which is an essential generalization of the implication reduct of intuitionistic logic, i.e. a Hilbert algebra.
Abstract: We introduce a concept of implication groupoid which is an essential generalization of the implication reduct of intuitionistic logic, i.e. a Hilbert algebra. We prove several connections among ideals, deductive systems and congruence kernels which even coincide whenever our implication groupoid is distributive.
TL;DR: In this paper, the first time oscillation of nonlinear delay partial difference equations is discussed by employing frequency measures, and some new oscillatory criteria are established by making use of frequency measures.
Abstract: This paper is concerned with a class of nonlinear delay partial difference equations with variable coefficients, which may change sign. By making use of frequency measures, some new oscillatory criteria are established. This is the first time oscillation of these partial difference equations is discussed by employing frequency measures.
TL;DR: In this article, the authors constructed a representation of the fundamental solution of an elliptic differential equation as a sum of functions, each of which has singularity on a single characteristic.
Abstract: It is known that the fundamental solution to an elliptic differential equation with analytic coefficients exists, is determined up to the kernel of the differential operator, and has singularities on characteristics of the equation in ℂ2. In this paper we construct a representation of fundamental solution as a sum of functions, each of those has singularity on a single characteristic.
TL;DR: In this article, the authors discuss the special diffusive hematopoiesis model with Neumann boundary condition and provide sufficient conditions for the global attractivity and oscillation of the equilibrium for Eq. (*), by using a new theorem.
Abstract: In this paper, we discuss the special diffusive hematopoiesis model
$$\frac{{\partial P(t,x)}}{{\partial t}} = \Delta P(t,x) - \gamma P(t,x) + \frac{{\beta P(t - \tau ,x)}}{{1 + P^n (t - \tau ,x)}}$$
with Neumann boundary condition. Sufficient conditions are provided for the global attractivity and oscillation of the equilibrium for Eq. (*), by using a new theorem we stated and proved. When P(t, χ) does not depend on a spatial variable χ ∈ Ω, these results are also true and extend or complement existing results. Finally, existence and stability of the Hopf bifurcation for Eq. (*) are studied.
TL;DR: In this paper, a hierarchy of Lax type equations on a dual space to the Lie algebra of integro-differential operators with matrix coefficients, extended by evolutions for eigen functions and adjoint eigenfunctions of the corresponding spectral problems, is obtained via some special Backlund transformation.
Abstract: The Hamiltonian representation for a hierarchy of Lax type equations on a dual space to the Lie algebra of integro-differential operators with matrix coefficients, extended by evolutions for eigenfunctions and adjoint eigenfunctions of the corresponding spectral problems, is obtained via some special Backlund transformation. The connection of this hierarchy with integrable by Lax two-dimensional Davey-Stewartson type systems is studied.
TL;DR: The main result of as discussed by the authors is that if G/G/G is divisible by finite, then G itself is also a divisible p-group, and if G is torsion-free then G is actually divisible.
Abstract: Let G be a hypercentral group. Our main result here is that if G/G’ is divisible by finite then G itself is divisible by finite. This extends a recent result of Heng, Duan and Chen [2], who prove in a slightly weaker form the special case where G is also a p-group. If G is torsion-free, then G is actually divisible.
TL;DR: In this paper, it was shown that if G/G′ is 2-divisible, then G is a 2-group and G/V is divisible by finite-of-odd-order, where V is the intersection of the lower central series (continued transfinitely) of O2′ (G) and O 2′ (V).
Abstract: Let G be a hypercyclic group The most substantial results of this paper are the following a) If G/G′ is 2-divisible, then G is 2-divisible b) If G/G′ is a 2′-group, then G is a 2′-group c) If G/G′ is divisible by finite-of-odd-order, then G/V is divisible by finite-of-odd-order, where V is the intersection of the lower central series (continued transfinitely) of O2′ (G)
TL;DR: In this article, strong asymptotic completeness is shown for a pair of Schrodinger type operators on a cylindrical Lipschitz domain, where a key ingredient is a limiting absorption principle valid in a scale of weighted (local) Sobolev spaces with respect to the uniform topology.
Abstract: Strong asymptotic completeness is shown for a pair of Schrodinger type operators on a cylindrical Lipschitz domain. A key ingredient is a limiting absorption principle valid in a scale of weighted (local) Sobolev spaces with respect to the uniform topology. The results are based on a refined version of Mourre’s method within the context of pseudo-selfadjoint operators.
TL;DR: Like in revised simplex method, in this method an auxiliary matrix is used for the computations and the algorithm is suitable for unstable and degenerate problems primarily.
Abstract: The system of inequalities is transformed to the least squares problem on the positive ortant. This problem is solved using orthogonal transformations which are memorized as products. Author’s previous paper presented a method where at each step all the coefficients of the system were transformed. This paper describes a method applicable also to large matrices. Like in revised simplex method, in this method an auxiliary matrix is used for the computations. The algorithm is suitable for unstable and degenerate problems primarily.
TL;DR: In this article, the authors deal with the duality for a multiobjective fractional optimization problem and compare the intermediate dual problem with other similar dual problems known from the literature.
Abstract: The present paper is a continuation of [2] where we deal with the duality for a multiobjective fractional optimization problem. The basic idea in [2] consists in attaching an intermediate multiobjective convex optimization problem to the primal fractional problem, using an approach due to Dinkelbach ([6]), for which we construct then a dual problem expressed in terms of the conjugates of the functions involved. The weak, strong and converse duality statements for the intermediate problems allow us to give dual characterizations for the efficient solutions of the initial fractional problem. The aim of this paper is to compare the intermediate dual problem with other similar dual problems known from the literature. We completely establish the inclusion relations between the image sets of the duals as well as between the sets of maximal elements of the image sets.
TL;DR: In this article, the structure of hom-spaces of all pairs of indecomposable Λ-modules having dimension smaller than or equal to a fixed natural number is described, and their dimensions are calculated in terms of a finite number of finitely generated and generic modules.
Abstract: Let Λ be a finite dimensional algebra over an algebraically closed field k and Λ has tame representation type. In this paper, the structure of Hom-spaces of all pairs of indecomposable Λ-modules having dimension smaller than or equal to a fixed natural number is described, and their dimensions are calculated in terms of a finite number of finitely generated Λ-modules and generic Λ-modules. In particular, such spaces are essentially controlled by those of the corresponding generic modules.
TL;DR: In this paper, the trivial group is shown to be the only injective object in the category of groups in which the trivial groups are injective and the only object in which a trivial group can be a group.
Abstract: In this paper we give a short and simple proof the following theorem of S. Eilenberg and J.C. Moore: the only injective object in the category of groups is the trivial group.
TL;DR: In this article, the convergence in probability of the normalized q-variation of the multiple fractional multiparameter integral processes (MFPIMs) was studied, and the convergence was shown to be independent of the Hurst parameter.
Abstract: We study the convergence in probability of the normalized q-variation of the multiple fractional multiparameter integral processes
$$\begin{gathered} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} _r = (t_1 ,...,t_r ) \to I_r^H (f_r )_{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} _r } : = \int_{[0,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} _r ]} {f_r (s_1 ,...,s_r )dB_{s_1 }^H ...dB_{s_r }^H } , \hfill \\ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} _r = (t_1 ,...,t_r ) \to I_r^{H, - } (f_r )_{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} _r } : = \int_{[0,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} _r ]} {f_r (s_1 ,...,s_r )dS_{s_1 }^H ...dS_{s_r }^H } , \hfill \\ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} _2 = (t_1 ,t_2 ) \to I_r^H (g)_{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} _2 } : = \int_{[0,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{t} _2 ]} {g(s_1 ,s_2 )dB_{s_1 }^{H,1} dB_{s_2 }^{H,2} } , \hfill \\ \end{gathered} $$
where fr, g are continuous deterministic functions, BH (resp. SH) is a fractional (resp. a sub-fractional) Brownian motion with Hurst parameter H > 1/2 and BH,1, BH,1 are independent fractional Brownian motions with Hurst parameter H.