Factorizations of complete multipartite graphs into generalized cubes

For a positive integer d, the usual d‐dimensional cube Q_d is defined to be the graph (K₂)^d, the Cartesian product of d copies of K₂. We define the generalized cube Q(K_k, d) to be the graph (K_k)^d for positive integers d and k. We investigate the decomposition of the complete multipartite graph K into factors that are vertex‐disjoint unions of generalized cubes Q(K_k, d_i), where k is a power of a prime, n and j are positive integers with j ≤ n, and the d_i may be different in different factors. We also use these results to partially settle a problem of Kotzig on Q_d‐factorizations of K_n. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 144–150, 2000     

On spherical representations of quivers and generalized complexes

For any quiverQ we consider spherical representationsV ofQ such that the isomorphism class ofV is a spherical variety. We suggest an approach for classifying such representations for anyQ and obtain a classification forQ being an equioriented Dynkin diagramA _n. In particular, all complexes are spherical representations. We introduce a category of representations that we call generalized complexes and classify spherical generalized complexes. For the quivers that we call crumbly we prove that any spherical generalized complex has a polynomial algebra of covariants on the closure of its isomorphism class.Partially supported by INTAS-OPEN grant 97-1570 and RFFI grant 98-01-00598.     

On regularity of generalized solutions of the third boundary-value problem for an elliptic difference-differential equation

Unlike the case of elliptic differential equations, generalized solutions of elliptic difference-differential equations may be not smooth in a domainQ but remain smooth only in certain subdomainsQ _r Q Conditions are considered which are necessary and sufficient for generalized solutions of the third boundary-value problem to preserve smoothness on the boundary of adjacent subdomainsQ _r .Translated from Ukrainskii Matematicheskii Zhurnal, Vol.47, No. 1, pp. 93–104, January, 1995.The author expresses his deep gratitude to A. D. Myshkis and A. L. Skubachevskii for helpful discussions.     

Smoothness of generalized solutions of the third boundary-value problem for an elliptic differential-difference equation

Unlike the case of elliptic differential equations, generalized solutions of elliptic differential-difference equations may be nonsmooth on an entire domainQ, only preserving smoothness on certain subdomainsQ _r Q. The conditions are considered under which the generalized solutions of the third boundary-value problem remain smooth on the boundaries of the neighboring subdomainsQ _r .Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 8, pp. 1140–1150, August, 1993.The author is grateful to A. D. Myshkis and A. L. Skubachevaskii for their kind attention to the work and valuable advice.     

Some path properties of generalized Lévy sheet

Let X(t) be an N parameter generalized Lévy sheet taking values in ℝ^d with a lower index α, ℜ = {(s, t] = ∏ _i=1 ^N (s _i, t _i], s _i < t _i}, E(x, Q) = {t ∈ Q: X(t) = x}, Q ∈ ℜ be the level set of X at x and X(Q) = {x: ∃t ∈ Q such that X(t) = x} be the image of X on Q. In this paper, the problems of the existence and increment size of the local times for X(t) are studied. In addition, the Hausdorff dimension of E(x, Q) and the upper bound of a uniform dimension for X(Q) are also established.     

Operator time‐splitting techniques combined with quintic B‐spline collocation method for the generalized Rosenau–KdV equation

In this article, the generalized Rosenau–KdV equation is split into two subequations such that one is linear and the other is nonlinear. The resulting subequations with the prescribed initial and boundary conditions are numerically solved by the first order Lie–Trotter and the second‐order Strang time‐splitting techniques combined with the quintic B‐spline collocation by the help of the fourth order Runge–Kutta (RK‐4) method. To show the accuracy and reliability of the proposed techniques, two test problems having exact solutions are considered. The computed error norms L₂ and L_∞ with the conservative properties of the discrete mass Q(t) and energy E(t) are compared with those available in the literature. The convergence orders of both techniques have also been calculated. Moreover, the stability analyses of the numerical schemes are investigated.     

On Generalized Friedrichs and Krein‐von Neumann Extensions and Canonical Systems

Let S be a densely defined and closed symmetric relation in a Hilbert space ℋ︁ with defect numbers (1,1), and let A be some of its canonical selfadjoint extensions. According to Krein's formula, to S and A corresponds a so‐called Q‐function from the Nevanlinna class N . In this note we show to which subclasses N _γ of N the Q‐functions corresponding to S and its canonical selfadjoint extensions belong and specify the Q‐functions of the generalized Friedrichs and Krein‐von Neumann extensions. A result of L. de Branges implies that to each function Q ∈ N there corresponds a unique Hamiltonian H such that Q is the Titchmarsh‐Weyl coefficient of the two‐dimensional canonical system J_y′ = —zHy on [0, ∞) where Weyl's limit point case prevails at ∞. Then the boundary condition y(0) = 0 corresponds to a symmetric relation T_min with defect numbers (1,1) in the Hilbert space L²_H, and Q is equal to the Q‐function with respect to the extension corresponding to the boundary condition y₁(0) = 0. If H satisfies some growth conditions at 0 or ∞, wepresent results on the corresponding Q‐functions and show under which conditions the generalized Friedrichs or Krein‐von Neumann extension exists.     

On the characterization of p ‐adic Colombeau–Egorov generalized functions by their point values

We show that contrary to recent papers by S. Albeverio, A. Yu. Khrennikov and V. Shelkovich, point values do not determine elements of the so‐called p ‐adic Colombeau–Egorov algebra ??(?ⁿ _p ) uniquely. We further show in a more general way that for an Egorov algebra ??(M, R) of generalized functions on a locally compact ultrametric space (M, d) taking values in a nontrivial ring, a point value characterization holds if and only if (M, d) is discrete. Finally, following an idea due to M. Kunzinger and M. Oberguggenberger, a generalized point value characterization of ??(M, R) is given. Elements of ??(?ⁿ _p ) are constructed which differ from the p ‐adic δ ‐distribution considered as an element of ??(?ⁿ _p ), yet coincide on point values with the latter. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Carleman-type theorems for generalized Q-holomorphic functions

In this work, we reduce the boundary condition of Riemann–Hilbert problem for generalized Q-holomorphic functions to the Vekua-type canonical form and obtain an appropriate analogue to the Carleman type representation for generalized Q-holomorphic functions.     

Spreads and ovoids in finite generalized quadrangles

This paper is a survey on the existence and non-existence of ovoids and spreads in the known finite generalized quadrangles. It also contains the following new results. We prove that translation generalized quadrangles of order (s,s ²), satisfying certain properties, have a spread. This applies to three known infinite classes of translation generalized quadrangles. Further a new class of ovoids in the classical generalized quadranglesQ(4, 3 ^e ),e3, is constructed. Then, by the duality betweenQ(4, 3 ^e ) and the classical generalized quadrangleW (3 ^e ), we get line spreads of PG(3, 3 ^e ) and hence translation planes of order 3^2e . These planes appear to be new. Note also that only a few classes of ovoids ofQ(4,q) are known. Next we prove that each generalized quadrangle of order (q ²,q) arising from a flock of a quadratic cone has an ovoid. Finally, we give the following characterization of the classical generalized quadranglesQ(5,q): IfS is a generalized quadrangle of order (q,q ²),q even, having a subquadrangleS isomorphic toQ(4,q) and if inS each ovoid consisting of all points collinear with a given pointx ofS\S is an elliptic quadric, thenS is isomorphic toQ(5,q).     

The generalized centro‐symmetric and least squares generalized centro‐symmetric solutions of the matrix equation AYB + CYTD = E

An n×n real matrix P is said to be a symmetric orthogonal matrix if P = P^?1 = P^T. An n × n real matrix Y is called a generalized centro‐symmetric with respect to P, if Y = PYP. It is obvious that every matrix is also a generalized centro‐symmetric matrix with respect to I. In this work by extending the conjugate gradient approach, two iterative methods are proposed for solving the linear matrix equation and the minimum Frobenius norm residual problem over the generalized centro‐symmetric Y, respectively. By the first (second) algorithm for any initial generalized centro‐symmetric matrix, a generalized centro‐symmetric solution (least squares generalized centro‐symmetric solution) can be obtained within a finite number of iterations in the absence of round‐off errors, and the least Frobenius norm generalized centro‐symmetric solution (the minimal Frobenius norm least squares generalized centro‐symmetric solution) can be derived by choosing a special kind of initial generalized centro‐symmetric matrices. We also obtain the optimal approximation generalized centro‐symmetric solution to a given generalized centro‐symmetric matrix Y₀ in the solution set of the matrix equation (minimum Frobenius norm residual problem). Finally, some numerical examples are presented to support the theoretical results of this paper. Copyright © 2011 John Wiley & Sons, Ltd.     

A factorization of regular generalized Nevanlinna functions

LetQ be a regular operator valued generalized Nevanlinna function with negative index , i.e. . It is shown that then there exists a rational functionB(z), which collects the generalized poles and zeros ofQ that are not of positive type such that the function

On approximation of approximately generalized quadratic functional equation via Lipschitz criteria

Abstract

Let G be an Abelian group with a metric d and E ba a normed space. For any f : G → E we define the generalized quadratic di?erence of the function f by the formula

Q_k f (x, y) := f (x + ky) + f (x ? ky) ? f (x + y) ? f (x ? y) ? 2(k² ? 1)f (y)

for all x, y ∈ G and for any integer k with k ≠ 1, ?1. In this paper, we achieve the general solution of equation Q_k f (x, y) = 0, after it, we show that if Q_k f is Lipschitz, then there exists a quadratic function K : G → E such that f ? K is Lipschitz with the same constant. Moreover, some results concerning the stability of the generalized quadratic functional equation in the Lipschitz norms are presented. In the particular case, if k = 0 we obtain the main result that is in [7].     

On Fractional GJMS Operators

We describe a new interpretation of the fractional GJMS operators as generalized Dirichlet‐to‐Neumann operators associated to weighted GJMS operators on naturally associated smooth metric measure spaces. This gives a geometric interpretation of the Caffarelli‐Silvestre extension for (?Δ)^γ when γ ? (0,1), and both a geometric interpretation and a curved analogue of the higher‐order extension found by R. Yang for (?Δ)^γ when γ > 1. We give three applications of this correspondence. First, we exhibit some energy identities for the fractional GJMS operators in terms of energies in the compactified Poincaré‐Einstein manifold, including an interpretation as a renormalized energy. Second, for γ ? (1,2), we show that if the scalar curvature and the fractional Q‐curvature Q_2γ of the boundary are nonnegative, then the fractional GJMS operator P_2γ is nonnegative. Third, by assuming additionally that Q_2γ is not identically zero, we show that P_2γ satisfies a strong maximum principle.© 2016 Wiley Periodicals, Inc.     

A factorization result for generalized Nevanlinna functions of the classN k

LetQN _k. It is shown that if is a nonreal pole or a real generalized pole of nonpositive type and is a nonreal zero or a real generalized zero of nonpositive type of the functionQ then the function

Regular graphs constructed from the classical generalized quadrangle Q(4, q)

Let c_k be the smallest number of vertices in a regular graph with valency k and girth 8. It is known that c_{k + 1}?2(1 + k + k² + k³) with equality if and only if there exists a finite generalized quadrangle of order k. No such quadrangle is known when k is not a prime power. In this case, small regular graphs of valency k + 1 and girth 8 can be constructed from known generalized quadrangles of order q>k by removing a part of its structure. We investigate the case when q = k + 1 is a prime power, and try to determine the smallest graph under consideration that can be constructed from a generalized quadrangle of order q. This problem appears to be much more difficult than expected. We have general bounds and improve these for the classical generalized quadrangle Q(4, q), q even. © 2010 Wiley Periodicals, Inc. J Combin Designs 19:70‐83, 2010     

Existence of T*(3, 4,v)‐codes

A word of length k over an alphabet Q of size v is a vector of length k with coordinates taken from Q. Let Q^*₄ be the set of all words of length 4 over Q. A T^*(3, 4, v)‐code over Q is a subset C^*? Q^*₄ such that every word of length 3 over Q occurs as a subword in exactly one word of C^*. Levenshtein has proved that a T^*(3, 4, vv)‐code exists for all even v. In this paper, the notion of a generalized candelabra t‐system is introduced and used to show that a T^*(3, 4, v)‐code exists for all odd v. Combining this with Levenshtein's result, the existence problem for a T^*(3,4, v)‐code is solved completely. © 2004 Wiley Periodicals, Inc. J Combin Designs 13: 42–53, 2005.     

Commutators of generalized Hardy operators

LetH_a,b be the commutator generated by the generalized Hardy operator and the CMO function. The (L^p, L^p) boundedness of H_a,b is discussed in this paper. Furthermore, the authors consider the boundedness of H_a,b on the weighted homogeneous Herz spaces (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Certain bilateral generating relations for generalized hypergeometric functions

Recently, we introduced a class of generalized hypergeometric functionsI n:(b _q)/α:(a _p) (x, w) by using a difference operator Δ _x,w , where . In this paper an attempt has been made to obtain some bilateral generating relations associated withI _n ^ga (x, w). Each result is followed by its applications to the classical orthogonal polynomials.