Variational principles for symmetric bilinear forms

Every compact symmetric bilinear form B on a complex Hilbert space produces, via an antilinear representing operator, a real spectrum consisting of a sequence decreasing to zero. We show that the most natural analog of Courant's minimax principle for B detects only the evenly indexed eigenvalues in this spectrum. We explain this phenomenon, analyze the extremal objects, and apply this general framework to the Friedrichs operator of a planar domain and to Toeplitz operators and their compressions. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Isotropic Subspaces of Polylinear Forms

The dimensions of maximal isotropic subspaces of symmetric and skew-symmetric forms of degree d in general position are found.     

Stability of two player game structures

We have extended a two player game-theoretical model proposed by V. Gurvich [To theory of multi-step games, USSR Comput. Math and Math. Phys. 13 (1973)] and H. Moulin [The Strategy of Social Choice, North Holland, Amsterdam, 1983]: All the considered game situations are framed by the same game structure. The structure determines the families of potential decisions of the two players, as well as the subsets of possible outcomes allowed by pairs of such choices. To be a solution of a game, a pair of decisions has to determine a (pure) functional equilibrium of the situational pair of payoff mappings which transforms the realized outcome into real-valued rewards of the players. Accordingly we understand that a structure is stable, if it admits functional equilibria for all possible game situations; and that it is complete, if every situation that only partitions the potential outcomes, is dominated by one of the players. We have generalized and strengthened a theorem by V. Gurvich [Equilibrium in pure strategies, Soviet Math. Dokl. 38 (1989)], proving that a proper structure is stable iff it is complete. Additional results provide game-theoretical insight that focuses the inquiry on the complexity of the stability decision problem; in particular, for coherent structures.These results also have combinatorial importance because every structure is characterized by a pair of hypergraphs [C. Berge, Graphes et Hypergraphes, Dunod, 1970] over a common ground set. The structure is dual (complete/coherent) iff the clutter of one hypergraph equals (includes/is included in) the blocker of the other one. So, for non-void coherent structures, the stability decision problem is equivalent to the much studied subexponential [M.L. Fredman, L. Khachiyan, On the complexity of dualization of monotone disjunctive normal forms, J. Algorithms 21 (1996)] hypergraph duality decision problem.     

Densely continuous forms, pointwise topology and cardinal functions

We consider the space D(X, Y) of densely continuous forms introduced by Hammer and McCoy [5] and investigated also by Holá [6]. We show some additional properties of D(X, Y) and investigate the subspace D*(X) of locally bounded real-valued densely continuous forms equipped with the topology of pointwise convergence τ- _p . The largest part of the paper is devoted to the study of various cardinal functions for (D*(X), τ _p ), in particular: character, pseudocharacter, weight, density, cellularity, diagonal degree, π-weight, π-character, netweight etc. This work was supported by Science and Technology Assistance Agency under the contract No. APVT-51-006904. The authors would like to thank Ľubica Holá for suggestions and comments.     

Weight reduction for mod ? Bianchi modular forms

Let K be an imaginary quadratic field with class number one and ? be a rational prime that splits in K. We prove that mod ?, a system of Hecke eigenvalues occurring in the first cohomology group of some congruence subgroup Γ of SL₂(OK) can be realized, up to twist, in the first cohomology with trivial coefficients after increasing the level of Γ by (?).     

Power and exponential asymptotic forms of correlation functions

Using the Ornstein.Zernike equation, we obtain two asymptotic equations, one describing the exponential asymptotic behavior and the other describing the power asymptotic behavior of the total correlation function h(r). We show that the exponential asymptotic form is applicable only on a bounded distance interval l < r < L. The power asymptotic form is always applicable for r > L and reproduces the form of the interaction potential. In this case, as the density of a rarified gas decreases, L → l, the exponential asymptotic form vanishes, and only the power asymptotic form remains. Conversely, as the critical point is approached, L → ℞, and the applicability domain of the exponential asymptotic form increases without bound. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 3, pp. 454–464, September, 2008.     

The Influence of the Conditions of Sol–Gel Synthesis on the Absorption and Fluorescence Spectra of Porphyrin Molecules in Silicate Matrices

On the basis of the investigation of the electronic absorption and fluorescence spectra of porphin introduced into inorganic and organo–inorganic silicate gel matrices, the participation of the surface hydroxyl groups in the formation of the dication form of porphin is shown. The influence of the gel matrix drying process on the reverse process of deprotonation of the porphin dication has been revealed.     

A fast algorithm for finding small solutions of <Emphasis Type=”Italic”>F(X,Y)=G(X,Y)</Emphasis> over number fields

Summary In the definition of semi-open sets and other kinds of generalized open sets, the role of the starting topology is given to a generalized topology. Some properties of the classes of sets so obtained is examined.     

基于系统空间结构的约当规范形广义特征向量的求解方法

基于系统空间结构的分析 ,给出了求解系统约当规范形广义特征向量的新方法 .与 [1 ]中给出的方法相比 ,本文的方法更简单直接 .最后举例说明了方法的应用 .     

Mathematical Relationships Between Bond-Bending Force Fields

Molecular mechanics softwares adopt various set of force field functions. In some cases, reliable data from one set of force field parameters cannot be used in a software that adopts another set of force field. Using mathematical approach, exact relationships between parameters from three bond-bending force fields, namely the (i) harmonic cosine angle, (ii) polynomial series, and (iii) Fourier series, are herein developed. Parameters from these three potential functions are further related to the approximate form, the harmonic angle function, which is valid for small change in chemical bond angle.