The aim of this paper is to investigate the superstability problem for the pexiderized trigonometric functional equation∑ v∈Φ∫Kf（xkv（y）k^-1）dwK（k）= Φ g（x）h（y）, x, y ∈ G,where G is any topological group...The aim of this paper is to investigate the superstability problem for the pexiderized trigonometric functional equation∑ v∈Φ∫Kf（xkv（y）k^-1）dwK（k）= Φ g（x）h（y）, x, y ∈ G,where G is any topological group, K is a compact subgroup of G, ωK is the normalized Haar measure of K, Φ is a finite group of K-invariant morphisms of G and f, g, h are continuous complex-valued functions.Consequently, we have generalized the results of stability for d＇Alembert＇s and Wilson＇s equations by R. Badora, J. Baker, B. Bouikhalene, P. Gavruta, S. Kabbaj, Pl. Kannappan, G. H.Kim, J.M. Rassias, A. Roukbi, L. Sz′ekelyhidi, D. Zeglami, etc.展开更多
A continuous infinite system of point particles interacting via two-body strong superstable potential is considered in the framework of cell gas (CG) model of classical statistical mechanics. We consider free energy o...A continuous infinite system of point particles interacting via two-body strong superstable potential is considered in the framework of cell gas (CG) model of classical statistical mechanics. We consider free energy of this model as an approximation of the correspondent value of the continuous system. It converges to the free energy of the conventional continuous gas if the parameter of approximation α→0 for any values of an inverse temperature β>0 and volume per particle ν>0.展开更多
The stability problems of the exponential （functional） equation on a restricted domain will be investigated, and the results will be applied to the study of an asymptotic property of that equation. More precisely, t...The stability problems of the exponential （functional） equation on a restricted domain will be investigated, and the results will be applied to the study of an asymptotic property of that equation. More precisely, the following asymptotic property is proved： Let X be a real （or complex） normed space. A mapping f ： X → C is exponential if and only if f（x ＋ y） - f（x）f（y） → 0 as ｜｜x｜｜ ＋ ｜｜y｜｜ → ∞ under some suitable conditions.展开更多
IN model theory, Zil’ ber has proved that any model of nontrivial -categorical theory is verysimilar to a module or field. So some people wanted to prove the famous Vaught’s conjec-ture for superstable theories of m...IN model theory, Zil’ ber has proved that any model of nontrivial -categorical theory is verysimilar to a module or field. So some people wanted to prove the famous Vaught’s conjec-ture for superstable theories of modules. Later, Buechler proved that the Vaught’s conjectureis true for the theories of modules with Morley rank 1. In order to reduce Vaught’s conjec-ture for superstablc theories of modules, Prest suggested discussing the preservation of stabili-ty--theory properties of modules when the underlying ring of the modules changes. Zieglergave some preservation theorems of clementary properties of the theory of modules in ref.展开更多
The aim of this paper is to study the stability problem of the generalized sine functional equations as follows：g（x）f（y）=f（x＋y/2）^2-f（x-y/2）^2 f（x）g（y）=f（x＋y/2）^2-f（x-y/2）^2,g（x）g（y）=f（x＋y/2）...The aim of this paper is to study the stability problem of the generalized sine functional equations as follows：g（x）f（y）=f（x＋y/2）^2-f（x-y/2）^2 f（x）g（y）=f（x＋y/2）^2-f（x-y/2）^2,g（x）g（y）=f（x＋y/2）^-f（x-y/2）^2Namely, we have generalized the Hyers Ulam stability of the （pexiderized） sine functional equation.展开更多
文摘The aim of this paper is to investigate the superstability problem for the pexiderized trigonometric functional equation∑ v∈Φ∫Kf（xkv（y）k^-1）dwK（k）= Φ g（x）h（y）, x, y ∈ G,where G is any topological group, K is a compact subgroup of G, ωK is the normalized Haar measure of K, Φ is a finite group of K-invariant morphisms of G and f, g, h are continuous complex-valued functions.Consequently, we have generalized the results of stability for d＇Alembert＇s and Wilson＇s equations by R. Badora, J. Baker, B. Bouikhalene, P. Gavruta, S. Kabbaj, Pl. Kannappan, G. H.Kim, J.M. Rassias, A. Roukbi, L. Sz′ekelyhidi, D. Zeglami, etc.
文摘A continuous infinite system of point particles interacting via two-body strong superstable potential is considered in the framework of cell gas (CG) model of classical statistical mechanics. We consider free energy of this model as an approximation of the correspondent value of the continuous system. It converges to the free energy of the conventional continuous gas if the parameter of approximation α→0 for any values of an inverse temperature β>0 and volume per particle ν>0.
文摘The stability problems of the exponential （functional） equation on a restricted domain will be investigated, and the results will be applied to the study of an asymptotic property of that equation. More precisely, the following asymptotic property is proved： Let X be a real （or complex） normed space. A mapping f ： X → C is exponential if and only if f（x ＋ y） - f（x）f（y） → 0 as ｜｜x｜｜ ＋ ｜｜y｜｜ → ∞ under some suitable conditions.
文摘IN model theory, Zil’ ber has proved that any model of nontrivial -categorical theory is verysimilar to a module or field. So some people wanted to prove the famous Vaught’s conjec-ture for superstable theories of modules. Later, Buechler proved that the Vaught’s conjectureis true for the theories of modules with Morley rank 1. In order to reduce Vaught’s conjec-ture for superstablc theories of modules, Prest suggested discussing the preservation of stabili-ty--theory properties of modules when the underlying ring of the modules changes. Zieglergave some preservation theorems of clementary properties of the theory of modules in ref.
文摘The aim of this paper is to study the stability problem of the generalized sine functional equations as follows：g（x）f（y）=f（x＋y/2）^2-f（x-y/2）^2 f（x）g（y）=f（x＋y/2）^2-f（x-y/2）^2,g（x）g（y）=f（x＋y/2）^-f（x-y/2）^2Namely, we have generalized the Hyers Ulam stability of the （pexiderized） sine functional equation.
