Positive definite <Emphasis Type="Italic">n</Emphasis>-regular quadratic forms

A positive definite integral quadratic form f is called n-regular if f represents every quadratic form of rank n that is represented by the genus of f. In this paper, we show that for any integer n greater than or equal to 27, every n-regular (even) form f is (even) n-universal, that is, f represents all (even, respectively) positive definite integral quadratic forms of rank n. As an application, we show that the minimal rank of n-regular forms has an exponential lower bound for n as it increases.     

Algebraic form of a theorem of Hahn—Banach type for lattice manifolds

Let E be a vector lattice. A linear functionalf on E is called a lattice homomorphism iff(sup (x, y)) = max (f(x),f(y)) for all x, y E. For lattice homomorphisms a theorem of Hahn—Banach type is valid. In this note we prove an algebraic analog of this theorem.Translated from Matematicheskie Zametki, Vol. 16, No. 4, pp. 595–600, October, 1974.In conclusion the author expresses his thanks to D. A. Raikov for his statement of the problem and his interest in my work.     

On certain functional equations related to Jordan triple (q, f){(\theta, \phi)}-derivations on semiprime rings

The main purpose of this paper is to prove the following result. Let R be a 2-torsion free semiprime ring with symmetric Martindale ring of quotients Q _s and let q{\theta} and f{\phi} be automorphisms of R. Suppose T:R? R{T:R\rightarrow R} is an additive mapping satisfying the relation T(xyx)=T(x)q(y)q(x)-f(x)T(y)q(x)+f(x)f(y)T(x){T(xyx)=T(x)\theta (y)\theta (x)-\phi (x)T(y)\theta (x)+\phi (x)\phi (y)T(x)}, for all pairs x,y ? R{x,y\in R}. In this case T is of the form 2T(x)=qq(x)+f(x)q{2T(x)=q\theta (x)+\phi (x)q}, for all x ? R{x\in R} and some fixed element q ? Q_s{q\in Q_{s}}.     

Generalized Feynman–Kac Semigroups,Associated Quadratic Forms and Asymptotic Properties

In this paper, we study the Feynman–Kac semigroup T _t f(x)=E _x[f(X _t)exp(N _t)],where X _t is a symmetric Levy process and N _t is a continuous additive functional of zero energy which is not necessarily of bounded variation. We identify the corresponding quadratic form and obtain large time asymptotics of the semigroup. The Dirichlet form theory plays an important role in the whole paper.     

Functional Equation Related to Quadratic and Norm Forms

Let T(a ₁, a ₂,..., a _n ) be a norm form of some finite proper extension of ℚ in a certain fixed integral basis, or even some integer form satisfying certain conditions. We are interested in two problems. One is finding all functions f : ℤ → ℂ satisfying the integer functional equation f(T(a ₁, a ₂,..., a _n )) = T(f(a ₁), f(a ₂),..., f(a _n )). Another closely related problem is finding all functions f : ℤ → ℂ such that T(f(a ₁, f(a ₂,..., f(a _n )) depends only on the value of T(a ₁, a ₂,..., a _n ).The second question was studied before for special quadratic forms. We extend these investigations to other types of quadratic forms and, thus, partially solve the second problem for them. The solution of the first problem for one cubic field is also presented. Finally, we give the corresponding conjecture for the first problem and, additionally, several remarks concerning the choice of norm forms.__________Published in Lietuvos Matematikos Rinkinys, Vol. 45, No. 2, pp. 153–172, April–June, 2005.     

Functional equations on semigroups

Summary. We determine the general solution g:S? F g:S\to F of the d'Alembert equation¶¶g(x+y)+g(x+sy)=2g(x)g(y)       (x,y ? S) g(x+y)+g(x+\sigma y)=2g(x)g(y)\qquad (x,y\in S) ,¶the general solution g:S? G g:S\to G of the Jensen equation¶¶g(x+y)+g(x+sy)=2g(x)       (x,y ? S) g(x+y)+g(x+\sigma y)=2g(x)\qquad (x,y\in S) ,¶and the general solution g:S? H g:S\to H of the quadratic equation¶¶g(x+y)+g(x+sy)=2g(x)+2g(y)       (x,y ? S) g(x+y)+g(x+\sigma y)=2g(x)+2g(y)\qquad (x,y\in S) ,¶ where S is a commutative semigroup, F is a quadratically closed commutative field of characteristic different from 2, G is a 2-cancellative abelian group, H is an abelian group uniquely divisible by 2, and s \sigma is an endomorphism of S with s(s(x)) = x \sigma(\sigma(x)) = x .     

Quadratic variation of functionals of two-parameter Wiener process

The quadratic variation of functionals of the two-parameter Wiener process of the form f(W(s, t)) is investigated, where W(s, t) is the standard two-parameter Wiener process and f is a function on the reals. The existence of the quadratic variation is obtained under the condition that f′ is locally absolutely continuous and f^N is locally square integrable.     

Quaternary universal forms over \mathbf{Q}(\sqrt{13})

Let be a real quadratic field over Q with m a square-free positive rational integer and be the integer ring in F. A totally positive definite integral n-ary quadratic form f=f(x ₁,…,x _n )=∑_1≤i,j≤n α _ij x _i x _j ( ) is called universal if f represents all totally positive integers in . Chan, Kim and Raghavan proved that ternary universal forms over F exist if and only if m=2,3,5 and determined all such forms. There exists no ternary universal form over real quadratic fields whose discriminants are greater than 12. In this paper we prove that there are only two quaternary universal forms (up to equivalence) over . For the proof of universality we apply the theory of quadratic lattices.      

Une Remarque sur une Formule Sommatoire liée à la Somme des Chiffres

Let g ≥ 2 be an integer, and let s(n) be the sum of the digits of n in basis g. Let f(n) be a complex valued function defined on positive integers, such that ?_{n ￡ x} f(n)=o(x)\sum_{n\le x} f(n)=o(x) . We propose sufficient conditions on the function f to deduce the equality ?_{n ￡ x} f(s(n))=o(x)\sum_{n\le x} f(s(n))=o(x) . Applications are given, for instance, on the equidistribution mod 1 of the sequence (s(n))^α, where α is a positive real number.     

On finite retract lattices of monounary algebras

For a monounary algebra (A, f) we denote R ^∅(A, f) the system of all retracts (together with the empty set) of (A, f) ordered by inclusion. This system forms a lattice. We prove that if (A, f) is a connected monounary algebra and R ^∅(A, f) is finite, then this lattice contains no diamond. Next distributivity of R ^∅(A, f) is studied. We find a representation of a certain class of finite distributive lattices as retract lattices of monounary algebras.     

Symmetrizations of Markov processes

Two methods for symmetrizing Markov processes are discussed. Letu ^a(x, y) be the potential density of a Lévy process on a compact Abelian groupG. A general condition is given that guarantees thatv(x, y)=u^a(x, y)+u^a(y, x) is the potential density of a symmetric Lévy process onG. The second method arises by considering the linear space of one-potentialsU ¹ f, withf inL ², endowed with the inner product (U ¹ f,U ¹ g)=fU ¹ g+gU ¹ f. If the semigroup ofX(t) is normal, then the completionH of this space is the Dirichlet space of a symmetric processY(t). A set that is semipolar forX(t) is polar forY(t).     

Generalized Hyers-Ulam Stability for a General Mixed Functional Equation in Quasi-��-normed Spaces

In this paper, we establish the general solution and investigate the generalized Hyers-Ulam stability of the following mixed additive and quadratic functional equation

Slopes of integral lattices

We use the dyadic trace to define the concept of slope for integral lattices. We present an introduction to the theory of the slope invariant. The main theorem states that a Siegel modular cusp form f of slope strictly less than the slope of an integral lattice with Gram matrix s satisfies f(sτ)=0 for all τ in the upper half plane. We compute the dyadic trace and the slope of each root lattice and we give applications to Siegel modular cusp forms.     

Bounded solutions of the Golab—Schinzel equation

Summary. Let \Bbb K {\Bbb K} be either the field of reals or the field of complex numbers, X be an F-space (i.e. a Fréchet space) over \Bbb K {\Bbb K} n be a positive integer, and f : X ? \Bbb K f : X \to {\Bbb K} be a solution of the functional equation¶¶f(x + f(x)ⁿ y) = f(x) f(y) f(x + f(x)^n y) = f(x) f(y) .¶We prove that, if there is a real positive a such that the set { x ? X : |f(x)| ? (0, a)} \{ x \in X : |f(x)| \in (0, a)\} contains a subset of second category and with the Baire property, then f is continuous or { x ? X : |f(x)| ? (0, a)} \{ x \in X : |f(x)| \in (0, a)\} for every x ? X x \in X . As a consequence of this we obtain the following fact: Every Baire measurable solution f : X ? \Bbb K f : X \to {\Bbb K} of the equation is continuous or equal zero almost everywhere (i.e., there is a first category set A ì X A \subset X with f(X \A) = { 0 }) f(X \backslash A) = \{ 0 \}) .     

Convex envelopes of products of convex and component-wise concave functions

In this paper, we consider functions of the form f(x,y)=f(x)g(y){\phi(x,y)=f(x)g(y)} over a box, where f(x), x ? \mathbb R{f(x), x\in {\mathbb R}} is a nonnegative monotone convex function with a power or an exponential form, and g(y), y ? \mathbb Rⁿ{g(y), y\in {\mathbb R}^n} is a component-wise concave function which changes sign over the vertices of its domain. We derive closed-form expressions for convex envelopes of various functions in this category. We demonstrate via numerical examples that the proposed envelopes are significantly tighter than popular factorable programming relaxations.     

Some properties of retract lattices of monounary algebras

Necessary and sufficient conditions for a connected monounary algebra (A, f), under which the lattice R ^∅(A, f) of all retracts of (A, f) (together with ∅) is algebraic, are proved. Simultaneously, all connected monounary algebras in which each retract is a union of completely join-irreducible elements of R ^∅(A, f) are characterized. Further, there are described all connected monounary algebras (A, f) such that the lattice R ^∅(A, f) is complemented. In this case R ^∅(A, f) forms a boolean lattice.     

On some composite functional inequalities

The aim of the paper is to deal with the following composite functional inequalities

On the error terms for representation numbers of quadratic forms

Let f be a primitive positive integral binary quadratic form of discriminant −D, and r _f (n) the number of representations of n by f up to automorphisms of f. We first improve the error term E(x) of $ \sum\limits_{n \leqq x} {r_f (n)^\beta } $ \sum\limits_{n \leqq x} {r_f (n)^\beta } for any positive integer β. Next, we give an estimate of ∫₁ ^T |E(x)|² x ^−3/2 dx when β = 1.     

On the functional equation f(x+g(y))-f(y+g(y))=f(x)-f(y) on groups

We solve the equation