Designs and partial geometries over finite fields

Let be a finite field, and let (, B) be a nontrivial 2-(n, k, 1)-design over . Then each point induces a (k–1)-spread S on /. (, B) is said to be a geometric design if S is a geometric spread on / for each . In this paper, we prove that there are no geometric designs over any finite field .Research partially supported by NSF grant DMS-8703229.     

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.     

On ηα-groups and fields

Let :=. The following are known: two -sets of power are isomorphic. Let >0. Two ordered divisible Abelian groups that are -sets of power are isomorphic, two real closed fields that are -sets of power are isomorphic. The following is shown: (1) there exist 2 nonisomorphic ordered Abelian groups (respectively ordered fields) that are -sets of power ; (2) there exist 2 nonisomorphic ordered divisible Abelian groups (respectively real closed fields) of power all having the same order type; (3) there exist 2 nonisomorphic ordered divisible Abelian groups (respectively real closed fields) that are -sets having the same order type.     

Reflected Brownian motion in a cone: Semimartingale property

Summary In this paper, the object of study is reflected Brownian motion in a cone ind-dimensions (d3) with nonconstant oblique reflection on each radial line emanating from the vertex of the cone. The basic question considered here is When is this process a semimartingale?. Conditions for the existence and uniqueness of the process for which the vertex is an instantaneous state were given by Kwon, which is resolved in terms of a real parameter depending on the cone and the direction of reflection. It is shown that starting from any point of the cone, the process is a semimartingale if < 1, + 0 and not a semimartingale if < < 2.This research is supported by KOSEF grant 941-0100-011-1     

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.     

Monotonicity properties of zeros of generalized Airy functions

Summary We show, among other things, that the positive zeros of a solution ofy +x y=0,y(0)=0 decrease to 1 as increases, 0.

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Sommario Si dimostra, tra l'altro, che gli zeri positivi d'una soiuzione diy +x y=0,y(0)=0 decrescono al limite 1, quando cresce, 0.

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To the memory of Milo Háik

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This research was supported by grants from the Natural Sciences and Engineering Research Council (Canada) and Consiglio Nazionale delle Ricerche (Italy). Some of the work was done while the second-named author was visiting the Department of Mathematics, University of Torino.     

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.     

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.     

OnC 2-diffeomorphisms of the circle which are of type III1

For every irrational number [0, 1) which is not of constant type we construct aC ²-diffeomorphism of the circle with rotation number which is of type III₁. This diffeomorphism can be chosen arbitrarily close to the rotationR . Our methods also allow us to construct, for every Liouville number [0, 1), aC -diffeomorphism of the circle with rotation number which is of type III₁.     

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.

     

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.     

Growth of length of sequential derivation transformed into natural one

We consider the (&, )-fragment of the intuitionistic propositional calculus. It is proved that under the standard transformation of a Gentzen derivation into a natural derivation(), the length of (())2^2·length( ). There is constructed a sequence of Gentzen derivations of length _i, for which the length of (( _i))2^{1/3·length(i}), which shows that the upper bound obtained is not too weak.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 88, pp. 192–196, 1979.     

Zeros in tables of characters for the groups Sn and An

In the representation theory of symmetric groups, for each partition of a natural number n, the partition h() of n is defined so as to obtain a certain set of zeros in the table of characters for S_n. Namely, h() is the greatest (under the lexicographic ordering ) partition among P(n) such that (g) 0. Here, is an irreducible character of S_n, indexed by a partition , and g is a conjugacy class of elements in S_n, indexed by a partition . We point out an extra set of zeros in the table that we are dealing with. For every non self-associated partition P(n), the partition f() of n is defined so that f() is greatest among the partitions of n which are opposite in sign to h() and are such that (g) 0 (Thm. 1). Also, for any self-associated partition of n > 1, we construct a partition () P(n) such that () is greatest among the partitions of n which are distinct from h() and are such that (g) 0 (Thm. 2).Supported by RFBR grant No. 04-01-00463 and by RFBR-BRFBR grant No. 04-01-81001.Translated from Algebra i Logika, Vol. 44, No. 1, pp. 24–43, January–February, 2005.     

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.     

Semi-Classical Analysis of a Dirac Equation Without Adiabatic Decoupling

We study adiabatic decoupling for Dirac equation with some scaling which yields that the mass appears with a coefficient where is the semi-classical parameter and > 0. Therefore, the system presents an avoided crossing. The scale = 1/2 is critical: adiabatic decoupling holds for (0,1/2) while for 1/2, there is energy transfer at leading order between the two modes. We describe this transfer in terms of two-scale Wigner measures by means of Landau-Zener formula which takes into account the change of polarization of the measures after the crossing.     

Isotone solutions of parametric linear complementarity problems

The parametric linear complementarity problem is given by the conditions:q + p + Mz 0, 0,z 0,z ^T (q + p + Mz) = 0. Under the assumption thatM is a P-matrix, Cottle proved that the solution mapz() of the above problem is montonically nondecreasing in the parameter for every nonnegativeq and everyp if and only ifM is a Minkowski matrix. This paper examines whether a similar result holds in various other settings including a nonlinear case.     

Modified (0, 2)-Interpolation on the Roots of Jacobi Polynomials. II (Convergence)

In [4] A. M. Chak, A. Sharma and J. Szabados characterized the Jacobi matrices P(,), (, > –1) for which the (0,2)-interpolation problem is regular. It follows from their result, that if n is odd and = , or if , are both odd integers and n > 1 + ( + )/2, then the (0,2)-interpolation problem is not regular. Recently, the author proved that for , both odd integers, the (0,2)-interpolation problem augmented with boundary (Hermite-type) conditions at the endpoints of the interval [–1,1] is regular. In this paper the convergence of this modified (0,2)-interpolation procedure is studied, if the inner nodal points are the roots of the ultraspherical polynomials with odd integer parameter.     

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.     

Conformal assouad dimension and modulus

Let 1 and let (X, d, ) be an -homogeneous metric measure space with conformal Assouad dimension equal to . Then there exists a weak tangent of (X, d, ) with uniformly big 1-modulus.     

The order of approximation to functions of the Zα. Class by means of positive linear operators

Let C_n (, ) be the upper bound for deviations of periodic functions which form the Zygmund class Z,0 0<<2 from a class of positive linear operators. A study is made of the conditions under which there exists a limit nC_n(, )=C(, ). An explicit expression is given for the functions C(,).Translated from Matematicheskie Zametki, Vol. 4, No. 2, pp. 201–210, August, 1968.