On cyclic biquadratic fields related to a problem of Hasse

In this note we shall prove that there exist infinitely many cyclic biquadratic fieldsK whose integral bases are neither {1, , ², } nor {1, , , ³) for any numbers , inK. Next, we shall construct infinitely many cyclic biquadratic fieldsK which have the index 1, but still have not the integral basis {1, , ², ³) for every inK. Finally we shall give a class of biquadratic fields for a problem of Hasse concerning an integral basis.     

Fredholm integro-differential equation

We consider numerical solution of an integro-differential equation with nonsmooth initspaial values. Unique solvability in Sobolev spaceW ₂ (0, 1), =1,2, is proved. We establish the rate of convergence of the approximate solution to the exact solution in fractional spacesW ₂ ⁺¹ , 01, with approximation order O(h ^++1/2 ) for 01/2 andO(h ⁺¹ |ln h|1/2, for 1/2 #x2264;1.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 64, pp. 8–16, 1988.     

K1 of Twisted Rings of Polynomials

We prove that for an arbitrary endomorphism of a ring R the group K1(R[t]) splits into the direct sum of K1(R) and Ñil (r;). Moreover, for any such R and Ñil (R; ) is isomorphic to Ñil (R ; ) for some ring R with : R R – an isomorphism.     

Bemerkung zu Einem Satz von Baer

Let be an involutorial antiautomorphism of the division ring K with char K 2 and E := {r K | rr = 1}. Refining a result of R. Baer we prove: (card E–1)² < card K =id. This is done by computing card E and the cardinal numbers of the sets consisting of the -symmetric respectively the -skew-symmetric elements.     

Stationary Solutions of the Fractional Kinetic Equation with a Symmetric Power-Law Potential

The properties of stationary solutions of the one-dimensional fractional Einstein--Smoluchowski equation with a potential of the form x ^2m+2, m=1,2,..., and of the Riesz spatial fractional derivative of order , 12, are studied analytically and numerically. We show that for 1<2, the stationary distribution functions have power-law asymptotic approximations decreasing as x ^–(+2m+1) for large values of the argument. We also show that these distributions are bimodal.     

Inner functions and related spaces of pseudocontinuable functions

Let be an inner function, let C, ¦¦=1. Then the harmonic function [(+)]/(–)] is the Poisson integral of a singular measure D. N. Clark's known theorem enables us to identify in a natural manner the space H² H² with the space L² ( ).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 170, pp. 7–33, 1989.     

Lipschitz and darbo conditions for the superposition operator in ideal spaces

Summary In this paper we give necessary and sufficient conditions for the superposition operator Fx(s)=f(s, x(s)) to satisfy a Lipschitz condition Fx₁ - Fx₂kx₁ - x₂ or a Darbo condition (FN)k(N) in ideal spaces of measurable functions, where is the Hausdorff measure of noncompactness. Moreover, we characterize a large class of spaces in which the above mentioned two conditions are equivalent.

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Sunto In questo lavoro diamo delle condizioni necessarie e sufficienti perchè l'operatore di sovrapposizione Fx(s)=f (s, x(s)) soddisfi alla condizione di Lipschitz Fx₁–Fx₂ kx₁–x₂ o quella di Darbo (FN)k(N) in spazi ideali di funzioni misurabili, ove è la misura di non compattezza di Hausdorff. Inoltre, caratterizziamo un'ampia classe di spazi in cui le suddette due condizioni sono equivalenti.

     

On the Suslin Number of Totally-Bounded Left-Topological Groups

For every infinite cardinal , we construct a zero-dimensional totally-bounded left-topological group with Suslin number .     

Semilattices of left reversible semigroups

Let S be a cancellative semigroup which is a semilattice of left reversible semigroups S, . This article studies the relationship between the group of quotients G of S and the groups of quotients G of S, . It is shown that G is the maximum group homomorphic image of an inverse semigroup which is a semilattice of groups G (up to isomorphism).The technique used here which involves the use of Ore's quotients also applies to the study of the maximum group homomorphic image of a semigroup which is a semilattice of inverse semigroups.     

Effect of the interface on the discreteness of the segmental mobility of quartz-filled high-density polyethylene

Conclusion It was shown that in the presence of a filler — finely divided quartz — an interphase layer of bound polymer is formed in HDPE, in which the segmental mobility is more hindered than in the amorphous regions of the unadsorbed polymer. For this reason, a new loss peak is observed in the internal friction spectra which we called the process in conformity with the usual classification. Its intensity increases with an increase in the concentration of filler. The activation energy of the process is equal to 69 kJ/mole.Translated from Mekhanika Kompozitnykh Materialov, No. 4, pp. 600–604, July–August, 1986.     

On the value function in constrained control of Markov chains

It is known that the value function in an unconstrained Markov decision process with finitely many states and actions is a piecewise rational function in the discount factor a, and that the value function can be expressed as a Laurent series expansion about = 1 for close enough to 1. We show in this paper that this property also holds for the value function of Markov decision processes with additional constraints. More precisely, we show by a constructive proof that there are numbers O = _o <₁ <... < _m–1 < _m = 1 such that for everyj = 1, 2, ...,m – 1 either the problem is not feasible for all discount factors in the open interval (_j–1, _j) or the value function is a rational function in a in the closed interval [_j–1, _j]. As a consequence, if the constrained problem is feasible in the neighborhood of = 1, then the value function has a Laurent series expansion about = 1. Our proof technique for the constrained case provides also a new proof for the unconstrained case.     

Concretely Locally Presentable and Locally Generated Categories

We give concrete versions of the characterizations of locally -presentable (resp. -generated) categories as categories of models of (resp. weakly) -ary limit-theories and of (resp. weakly) -ary essentially algebraic theories. By concrete we mean that starting with a category of -structures, the theories obtained are extensions of the original ones, and the equivalences of categories are concrete isomorphisms. -presentable and -generated objects are also investigated from this viewpoint.     

Sur la convergence presque sure,au sens de Cesaro d'ordre α, 0<α<1, de variables aléatoires indepéndantes et identiquement distribuées

Summary We prove the following theorem: «Given 0<1, the (C, )-means of a sequence of i.i.d. random variables X _n converge a.s. iff E|X _n|^1/<.» For 1/2<1 and 0<<1/2 this result is essentially known. We give here a proof of the case =1/2; an important tool is a theorem of Hsu and Robbins [5].     

Sum of Divisors in a Ring of Gaussian Integers

We construct an asymptotic formula for a sum function for _a (), where _a () is the sum of the ath powers of the norms of divisors of the Gaussian integer on an arithmetic progression ₀ (mod ) and in a narrow sector ₁ arg < ₂. For this purpose, we use a representation of _a (n) in the form of a series in the Ramanujan sums.     

The Solution Set to BVP for Some Functional Differential Inclusions

We consider a multivalued BVP x'(t) A(t)x((t))+ F(t,x((t))), Lx= . Under appropriate assumptions on A, L and F, we prove that for sufficiently small the set of solutions to this problem is a nonempty infinite dimensional AR-space (Theorem 4).     

Some isoperimetric norm bounds for the first eigenfunction of wedge-like membranes

Zusammenfassung Das Ziel dieser Arbeit ist die Konstruktion von unteren und oberen Schranken für _D hu dx dy mith=r sin . Dabei istu die erste Eigenfunktion einer auf dem Rande gebundenen Membran, deren Gebiet vollständig im Winkel 0/ liegt. Am Schluss der Arbeit werden zwei einfache Beispiele vorgeschlagen.

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Résumé L'objet de cet article est de construire des bornes inférieures et supérieures pour _D hu dx dy avech=r sin , et oùu est la première fonction propre d'une membrane vibrante fixée sur son contour D, complètement contenu dans l'angle 0/. Comme conclusion, deux applications simples sont présentées.

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This research was supported by the Swiss Nationalfonds.     

A remark on fixed points of functors in topological categories

For a topological category over Set we prove that if a functor T: has a fixed cardinal (i.e. for each object K with card (UK)= we have card (UTK)), then T has a least fixed point, and if T has a successive pair of fixed cardinals and ⁺, then T has a greatest fixed point. This extends results of Adámek and Koubek.Partial financial support of the Grant Agency of the Czech Republic under Grant No. 201/93/0950 is gratefully acknowledged.     

On lower bounds for polynomials in the values of E-functions

The paper considers polynomials in ,f₁(),...,f_s(), where is an algebraic number satisfying certain conditions and f₁(z),...,f_s(z) are some E-functions, algebraically independent over the field of rational functions. Explicit lower bounds in terms of the heights of and the polynomial are obtained for the absolute values of these polynomials. The result is proved by using the method of Siegel and idlovskii.This work was carried out while the author was a research fellow of the Alexander von Humboldt Foundation.     

Traces of functions belonging to Sobolev and Besov spaces and extensions from subsets of the Euclidean space

It is proved that from the existence of the trace operator, 0n and E is a Borel subset of ⁿ.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, AN SSSR, Vol. 157, pp. 137–145, 1987.     

On some properties of multiple conjugate trigonometric series

We have obtained an estimate, in terms of partial and mixed moduli, of the continuity of deviation of the Cesáro (C, ) means ( = (₁,...,_n),_i , ₁ > –1, ) of the sequence of rectangular partial sums ofn-multiple (n>1) conjugate trigonometric series from then-multiple truncated conjugate function. This estimate implies the result on them -convergence (1) of (C, ) means (₁ > 0, ) provided that the essential conditions are imposed on the partial moduli of continuity. Finally, it is shown that them -convergence cannot be replaced by ordinary convergence.