On some functional equations of Goląb-Schinzel type

Summary LetE be a real Hausdorff topological vector space. We consider the following binary law * on ·E:(, ) * (, ) = (, ^k + ) for(, ), (, ) × E where is a nonnegative real number,k andl are integers.In order to find all subgroupoids of ( ·E, *) which depend faithfully on a set of parameters, we have to solve the following functional equation:f(f(y) ^k x +f(x) ^l y) =f(x)f(y) (x, y E). (1)In this paper, all solutionsf: of (1) which are in the Baire class I and have the Darboux property are obtained. We obtain also all continuous solutionsf: E of (1). The subgroupoids of (^* ·E, *) which dapend faithfully and continuously on a set of parameters are then determined in different cases. We also deduce from this that the only subsemigroup ofL _n ¹ of the form {(F(x ₂,x ₃, ,x _n ),x ₂,x ₃, ,x _n ); (x ₂, ,x _n ) ^{n – 1} }, where the mappingF: ^{n – 1 *} has some regularity property, is {1} × ^{n – 1} .We may noitice that the Gob-Schinzel functional equation is a particular case of equation (1)(k = 0, l = 1, = 1). So we can say that (1) is of Gob—Schinzel type. More generally, whenE is a real algebra, we shall say that a functional equation is of Gob—Schinzel type if it is of the form:f(f(y) ^k x +f(x) ^l y) =F(x,y,f(x),f(y),f(xy)) wherek andl are integers andF is a given function in five variables. In this category of functional equations, we study here the equation:f(f(y) ^k x +f(x) ^l y) =f(xy) (x, y f: ). (4)This paper extends the results obtained by N. Brillouët and J. Dhombres in [3] and completes some results obtained by P. Urban in his Ph.D. thesis [11] (this work has not yet been published).Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth     

Multiplicative Decompositions of Holomorphic Fredholm Functions and ψ*-Algebras

In this article we construct multiplicative decompositions of holomorphic Fredholm operator valued functions on Stein manifolds with values in various algebras of differential and pseudo differential operators which are submultiplicative ψ* - algebras, a concept introduced by the first author. For Fredholm functions T(z) satisfying an obvious topological condition we. Prove (0.1) T(z) = A(z)(I + S(z)), where A(z) is holomorphic and invertible and S(z) is holomorphic with values in an “arbitrarily small” operator ideal. This is a stronger condition on S(z) than in the authors' additive decomposition theorem for meromorphic inverses of holomorphic Fredholm functions [12], where the smallness of S(z) depends on the number of complex variables. The Multiplicative Decomposition theorem (0.1) sharpens the authors' Regularization theorem [11]; in case of the Band algebra L(X) of all bounded linear operators on a Band space, (0.1) has been proved by J. Letterer [20] for one complex variable and by M. 0. Zaidenberg, S. G. Krein, P. A. Kuchment and A. A. Pankov [26] for the Banach ideal of compact operators.     

Some Analytical Properties of γ-Convex Functions in Normed Linear Spaces

For a fixed positive number γ, a real-valued function f defined on a convex subset D of a normed space X is said to be γ-convex if it satisfies the inequality

Hilbert's ϵ-operator in intuitionistic type theories

We investigate Hilbert's ?-calculus in the context of intuitionistic type theories, that is, within certain systems of intuitionistic higher-order logic. We determine the additional deductive strength conferred on an intuitionistic type theory by the adjunction of closed ?-terms. We extend the usual topos semantics for type theories to the ?-operator and prove a completeness theorem. The paper also contains a discussion of the concept of “partially defined” ?-term. MSC: 03B15, 03B20, 03G30.