Superstability of the generalized orthogonality equation on restricted domains

Chmielinski has proved in the paper [4] the superstability of the generalized orthogonality equation |〈f(x), f(y)〉| = |〈x,y〉|. In this paper, we will extend the result of Chmielinski by proving a theorem: LetD _n be a suitable subset of ℝⁿ. If a function f:D _n → ℝⁿ satisfies the inequality ∥〈f(x), f(y)〉| |〈x,y〉∥ ≤ φ(x,y) for an appropriate control function φ(x, y) and for allx, y ∈ D _n, thenf satisfies the generalized orthogonality equation for anyx, y ∈ D _n.     

Regular left-continuous t-norms

A left-continuous (l.-c.) t-norm ⊙ is called regular if there is an n<ω such that the map x ↦ x⊙a has, for any a∈[0,1], at most n discontinuity points, and if the function mapping every a∈[0,1] to the set behaves in a specifically simple way. The t-norm algebras based on regular l.-c. t-norms generate the variety of MTL-algebras. With each regular l.-c. t-norm, we associate certain characteristic data, which in particular specifies a finite number of constituents, each of which belongs to one out of six different types. The characteristic data determines the t-norm to a high extent; we focus on those t-norms which are actually completely determined by it. Most of the commonly known l.-c. t-norms are included in the discussion. Our main tool of analysis is the translation semigroup of the totally ordered monoid ([0,1];≤,⊙,0,1), which consists of commuting functions from the real unit interval to itself.     

Hyperidentities of lattices and semilattices

Following W. Taylor we define a hyperidentity ∈ to be formally the same as an identity (e.g.,F(G(x, y, z), G(x, y, z))=G(x, y, z)). However, a varietyV is said to satisfy a hyperidentity ∈, if whenever the operation symbols of ∈ are replaced by any choice of polynomials (appropriate forV) of the same arity as the corresponding operation symbols of ∈, then the resulting identity holds inV in the usual sense. For example, if a varietyV of type <2,2> with operation symbols ∨ and ∧ satisfies the hyperidentity given above, then substituting the polynomial (x∨y)∨z for the symbolG, and the polynomialx∧y forF, we see thatV must in particular satisfy the identity ((x∨y)∨z)∧((x∨y)∨z)=((x∨y)∨z). The set of all hyperidentities satisfied by a varietyV, will be denoted byH(V). We shall letH ^m (V) be the set of all hyperidentities hoiding inV with operation symbols of arity at mostm, andH _n (V) will denote the set of all hyperidentities ofV with at mostn distinct variables. In this paper we shall show that ifV is a nontrivial variety of lattices or the variety of all semilattices, then for any integersm andn, there exists a hyperidentity ∈ such that ∈ holds inV, and ∈ is not a consequence ofH ^m (V)∪H _n (V). From this it is deduced that the hyperidentities ofV are not finitely based, partly soling a problem of Taylor [7, Problem 3]. The research of the author was supported by NSERC of Canada. Presented by W. Taylor.     

On the stability of the quadratic functional equation on bounded domains

A result of Skof and Terracini will be generalized; More precisely, we will prove that if a functionf : [-t, t]ⁿ →E satisfies the inequality (1) for some δ > 0 and for allx, y ∈ [-t, t]ⁿ withx + y, x - y ∈ [-t, t]ⁿ, then there exists a quadratic functionq: ℝⁿ →E such that ∥f(x) -q(x)∥ < (2912n² + 1872n + 334)δ for anyx ∈ [-t, t] ⁿ .     

On the boundedness of solutions of nonlinear differential equations in Hilbert spaces

LetH be a complex Hilbert space and letB be the space of all bounded linear operators fromH intoH with the strong operator topology. We will give a boundedness result for the solutions of the differential equationx′=A(t)x+f(t,x) whereA: I=[t ₀, ∞)→B is continuous,f: I×H→H is also continuous and for every bounded setS⊂I×H there exists a constantM(S)>0 such that |f(t,x)−f(t,y)|≤M(S)|x−y|,(t,x), (t,y)∈S.

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Sunto SiaH uno spazio di Hilbert complesso e siaB lo spazio degli operatori lineari limitati daH inH, con la topologia forte. In questo lavoro si prova un risultato di limitatezza per le soluzioni dell'equazione differenzialex′=A(t)x+f(t,x), doveA: I=[t ₀, ∞)→B è continua,f: I×H→H è continua e per ogni insieme limitatoS⊂I×H esiste una costanteM(S)>0 tale che |f(t,x)−f(t,y)|≤M(S)|x−y| per ogni(t,x), (t,y)∈S.

     

Theorie de nevanlinna p-adique

We study meromorphic functions in all ℂp or in a disc of ℂp. Using some properties of the valuation polygon notion, we show p-adic results perfectly analogous to those of Nevanlinna in the complex case. As an application we prove the p-adic analogue of Malmquist-Yosida Theorem: Let m∈ℕ and R(x.y)∈ℂp(x,y). If the differential equation: (dy/dx)^m=R(x,y), m∈ℕ, has a non rational meromorphic solution in all ℂp, then R(x,y) is a polynomial in y of degree ≤2m.      

On functions with the cauchy difference bounded by a functional. III

We are going to discuss special cases of a conditional functional inequality

A class of regular semigroups with regular *- transversals

Let S be a regular semigroup. If there is a subsemigroup S ^* of S and a unary operation * in S satisfying: (1) x ^* ∈ S ^* \cap V_ S ^* (x) for all x∈ S; (2) (x ^* ) ^* =x for all x∈ S ^* ; (3) (x ^* y) ^* =y ^* x ^** and (xy ^* ) ^* =y ^** x ^* for all x,y∈ S, then S ^* is called a regular *- transversal of S ; if (3) is replaced with (xy) ^* =y ^* x ^* for all x,y∈ S, then S ^* is called a strongly regular *- transversal of S. In this paper we consider the class of regular semigroups with a strongly regular *- transversal. It is proved that these semigroups are P - regular semigroups. We characterize the structure of the regular semigroups with a strongly regular *- transversal.     

Lipschitz selections of set-valued mappings and Helly’s theorem

We prove a Helly-type theorem for the family of all m-dimensional convex compact subsets of a Banach space X. The result is formulated in terms of Lipschitz selections of set-valued mappings from a metric space (M, ρ) into this family. Let M be finite and let F be such a mapping satisfying the following condition: for every subset M′ ⊂ M consisting of at most 2^m+1 points, the restriction F|_M′ of F to M′ has a selection f_M′ (i. e., f_M′(x) ∈ F(x) for all x ∈ M′) satisfying the Lipschitz condition ‖ƒ_M′(x) − ƒ_M′(y)‖_X ≤ ρ(x, y), x, y ∈ M′. Then F has a Lipschitz selection ƒ: M → X such that ‖ƒ(x) − ƒ(y)‖_X ≤ γρ(x,y), x, y ∈ M where γ is a constant depending only on m and the cardinality of M. We prove that in general, the upper bound of the number of points in M′, 2^m+1, is sharp. If dim X = 2, then the result is true for arbitrary (not necessarily finite) metric space. We apply this result to Whitney’s extension problem for spaces of smooth functions. In particular, we obtain a constructive necessary and sufficient condition for a function defined on a closed subset of R ² to be the restriction of a function from the Sobolev space W _∞ ² (R ²).A similar result is proved for the space of functions on R ² satisfying the Zygmund condition.     

On Regular Power-Substitution

The necessary and sufficient conditions under which a ring satisfies regular power-substitution are investigated. It is shown that a ring R satisfies regular powersubstitution if and only if a-b in R implies that there exist n ∈ N and a U E GLn （R） such that aU = Ub if and only if for any regular x ∈ R there exist m,n ∈ N and U ∈ GLn（R） such that x^mIn = xmUx^m, where a-b means that there exists x,y, z∈ R such that a =ybx, b = xaz and x= xyx = xzx. It is proved that every directly finite simple ring satisfies regular power-substitution. Some applications for stably free R-modules are also obtained.     

Purely imaginary eigenvalues of operator semigroups

For aC ₀-contraction semigroup (S(t)) _t≥0 of bounded linear operators on a complex Banach spaceX, J. A. Goldstein and B. Nagy [6] have shown that, givenx∈X, S(t)x=e ^iλt x, t≥0, for some λ∈ℝ, provided lim _t→∞ |<S(t)x,x ^* >|=|<x,x ^* >| for allx ^*∈X^*. We present (a) an extension to the case of nonlinear nonexpansive mapsS(t), t≥0, and (b) various generalizations in the linear context.     

On the renewal of a Wiener field on a plane with the use of its values on closed curves

Forw(u, v), (u, v)∉ γ (here,w(x, y), x≥0, y≥0, is a Wiener field and γ is a certain closed curve on a plane), we construct the best mean-square estimate on the basis of the values ofw(x, y) for (x, y)∈ γ. We also calculate the error of this estimate. Donetsk University, Donetsk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 6, pp. 744–752, June, 1999.     

The finite-dimensionalP λ spaces with small λ

It is proved that there is a positive function Φ(∈) defined for sufficiently small ∈>0 such that lim_∈→0 Φ(∈)=0 and for every integerk and everyk-dimensionalP _1+∈ spaceE, d(E, l _∞ ^k )<1+Φ(∈). Author was partially supported by N.S.F. Grant MCS 79-03042. An erratum to this article is available at .     

Every separable metric space is Lipschitz equivalent to a subset ofc 0 +

It is shown that there is a constantK so that, for every separable metric spaceX, there is a mapT:X →c _o satisfyingd(x, y)≦‖Tx−Ty‖≦Kd(x, y) for everyx, y ∈ X. This is a part of the author's Ph.D. Thesis prepared at the Hebrew University of Jerusalem, under the supervision of Professor J. Lindenstrauss.     

On sandwich sets and congruences on regular semigroups

Let S be a regular semigroup and E(S) be the set of its idempotents. We call the sets S(e, f)f and eS(e, f) one-sided sandwich sets and characterize them abstractly where e, f ∈ E(S). For a, a′ ∈ S such that a = aa′a, a′ = a′aa′, we call S(a) = S(a′a, aa′) the sandwich set of a. We characterize regular semigroups S in which all S(e; f) (or all S(a)) are right zero semigroups (respectively are trivial) in several ways including weak versions of compatibility of the natural order. For every a ∈ S, we also define E(a) as the set of all idempotets e such that, for any congruence ϱ on S, aϱa ² implies that aϱe. We study the restrictions on S in order that S(a) or be trivial. For , we define on S by a b if . We establish for which S are or congruences.     

Generalized semi-infinite optimization: A first order optimality condition and examples

We consider a generalized semi-infinite optimization problem (GSIP) of the form (GSIP) min{f(x)‖xεM}, where M={x∈ℝⁿ|h_i(x)=0i=l,...m, G(x,y)⩾0, y∈Y(x)} and all appearing functions are continuously differentiable. Furthermore, we assume that the setY(x) is compact for allx under consideration and the set-valued mappingY(.) is upper semi-continuous. The difference with a standard semi-infinite problem lies in thex-dependence of the index setY. We prove a first order necessary optimality condition of Fritz John type without assuming a constraint qualification or any kind of reduction approach. Moreover, we discuss some geometrical properties of the feasible setM. This work was partially supported by the “Deutsche Forschungsgemeinschaft” through the Graduiertenkolleg “Mathematische Optimierung” at the University of Trier.     

Derivations with engel conditions in prime and semiprime rings

Let R be a prime ring, I a nonzero ideal of R, d a derivation of R and m, n fixed positive integers. (i) If (d[x, y]) ^m = [x, y] _n for all x, y ∈ I, then R is commutative. (ii) If Char R ≠ 2 and [d(x), d(y)] _m = [x, y] ⁿ for all x, y ∈ I, then R is commutative. Moreover, we also examine the case when R is a semiprime ring.     

On polynomiality of the method of analytic centers for fractional problems

We establish polynomial time convergence of the method of analytic centers for the fractional programming problemt→min |x∈G, tB(x)−A(x)∈K, whereG ⊂ ℝ ⁿ is a closed and bounded convex domain,K ⊂ ℝ ^m is a closed convex cone andA(x):G → ℝ ⁿ ,B(x):G→K are regular enough (say, affine) mappings. This research was partly supported by grant #93-012-499 of the Fundamental Studies Foundation of Russian Academy of Sciences     

Regelflächenscharen

Let K be any commutative field and V:=K ⁴. A collection of ruled quadrics in V is called a flock of ruled quadrics if the following holds true. (1) ⋃_{ℱ∈ G} ℱ = V; (2) There is a line S⊂V such that ℱ₁⋂ℱ₂= S for all distinct ℱ₁, ℱ₂∈ . The group ΓL(V) decomposes the set of all those flocks into equivalence classes. Besides that, we consider any cone R in V, say R:= {x∈V|x ₁ x ₃ - x ₂ ² = 0}. Let R denote the set of all regular points of R. Plane sections of R which do not contain the singular point of ℜ are called regular sections. We consider decompositions of R ^* by regular sections and their equivalence classes with respect to the symmetry group ΓL(V)R of the cone ℜ. The main result is as follows. There is a (natural) bijection between the classes of equivalent flocks of rules quadrics and the classes of equivalent decompositions of R ^* by regular sections. A brief discussion of those flocks of ruled quadrics on which the construction of the so-called Betten-Walker planes is based ends the paper. Provided that char K≠3, these planes exist if and only if x∈K→x ³∈K is bijective.