On Mittag-Leffler functions and related distributions

The distribution F () = 1 – E (–), 0 < 1; 0 , where E (x) is the Mittag-Leffler function is studied here with respect to its Laplace transform. Its infinite divisibility and geometric infinite divisibility are proved, along with many other properties. Its relation with stable distribution is established. The Mittag-Leffler process is defined and some of its properties are deduced.     

The measure of a translated ball in uniformly convex spaces

Let (E, ¦·¦) be a uniformly convex Banach space with the modulus of uniform convexity of power type. Let be the convolution of the distribution of a random series inE with independent one-dimensional components and an arbitrary probability measure onE. Under some assumptions about the components and the smoothness of the norm we show that there exists a constant such that |{·<t}–{·+r<t}|r ^q , whereq depends on the properties of the norm. We specify it in the case ofL spaces, >1.     

C ϰ(X) and a property of (db)-spaces

Summary A new class of locally convex linear topological spaces, the (db)-spaees, recently defined by Robertson, Tweddle and Yeomans, interpolates the classes of Baire-like and unordered Baire-like spaces. Saxon and Narayanaswami have given convex metrizable spaces that distinguish among these classes. This paper gives a new characterization of (db)-spaces from which is extracted the class of b-spaces. This class interpolates the classes of -spaces and -spaces of Lehner. Let C(X) be the space of all real-valued continuous functions on the completely regular Sausdorf space X, supplied with the topology of uniform convergence on compact sets of X. It is shown that C(X) is a b-space if and only if it is an -space. A characterization of X for which C(X) is a (db)-space is unknown. Other open questions are stated in the paper.     

Finding the αn-th largest element

We describe an algorithm for selecting the n-th largest element (where 0<<1), from a totally ordered set ofn elements, using at most (1+(1+o(1))H())·n comparisons whereH() is the binary entropy function and theo(1) stands for a function that tends to 0 as tends to 0. For small values of this is almost the best possible as there is a lower bound of about (1+H())·n comparisons. The algorithm obtained beats the global 3n upper bound of Schönhage, Paterson and Pippenger for <1/3.     

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 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.     

Discrete Spectrum of Differential Operators in Spectral Gaps under Nonnegative Perturbations of High Order

Let A be a self-adjoint elliptic second-order differential operator, let (, ) be an inner gap in the spectrum of A, and let B(t) = A + tW ^* W, where W is a differential operator of higher order. Conditions are obtained under which the spectrum of the operator B(t) in the gap (, ) is either discrete, or does not accumulate to the right-hand boundary of the spectral gap, or is finite. The quantity N(, A, W, ), (, ), > 0 (the number of eigenvalues of the operator B(t) passing the point (, ) as t increases from 0 to ) is considered. Estimates of N(, A, W, ) are obtained. For the perturbation W ^* W of a special form, the asymptotics of N(, A, W, ) as + is given. Bibliography: 5 titles.     

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.     

Maximum of the fourth diameter in the family of continua with prescribed capacity

We obtain a complete solution of the problem of the maximum of the fourth diameter in the family of continua with capacity 1. Let E(o, eⁱ, e^–i). 0<i, e^–i; H(=cap E(o, eⁱ, e^–i). Let C() be the common point of three analytic arcs which form E(o, eⁱ, e^–i). One shows that the indicated maximum is realized by the continuum ={z:H(₀)z ²E(o, eⁱ, e^–i)} where ₀, o<₀z eⁱ z+C ( is a real and C is a complex constant). One finds the value of the required maximum. The paper contains a brief exposition of the proof of this result.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 59, pp. 60–79, 1976.     

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.     

A generalized sandwich theorem

It is proved for an arbitrary commutative ring A with identity and any integer n3 that if H is a subgroup of GL_n(A) normalized by E _n(A,q), then there is an ideal of A such that E _n(A,) H GL _n (A, (:q⁴⁰).Furthermore, is uniquely determined up to a certain equivalence relation on the set of ideals of A. The result extends a theorem of Bak, by removing a stability condition he uses on A.     

On the positive definiteness of n ↦ epnα for α close to 2

For >2, let Q ⁺() be the infimum of those q>0 for which the function n e^pn is positive definite on N ₀ for every pq. We shall prove that Q ⁺()0 as 2.     

Points cônes du mouvement brownien plan,le cas critique

Summary LetB=(B _t,t0) be a planar Brownian motion and let >0. For anyt0, the pointz=B _t is called a one-sided cone point with angle if there exist >0 and a wedgeW(,z) with vertexz and angle such thatB _sW(,z) for everys[t, t+]. Burdzy and Shimura have shown independently that one-sided cone points with angle exist when >/2 but not when      

A note on a set-mapping problem of Hajnal and Máté

It is consistent that there exists a set mappingF with <F(, )< for + 2 >w ₂ with no uncountable free sets.Research supported by Hungarian National Research Fund No. 1805 and 1908.     

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.     

On some classes of modules

The aim of this paper is to investigate quasi-corational, comonoform, copolyform and -(co)atomic modules. It is proved that for an ordinal a right R-module M is -atomic if and only if it is -coatomic. And it is also shown that an -atomic module M is quasi-projective if and only if M is quasi-corationally complete. Some other results are developed.     

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.     

Function spacesW 1,p,BV,N α, p characterized by the motions of ℝ n

We characterize the Sobolev spaces W^1,p(ⁿ) (p]1,+]), the space BV(ⁿ) of functions of bounded variation, the Nikol'skij spacesN ^p (R ⁿ =B _p (R ⁿ ) (p [1, + [, ]0,1[).This research was partially supported by M.U.R.S.T.     

Conjugate functions of several variables in the class Lip α

It is known that if a function f of a single variable belongs to the class Lip (, C(T)) (0<<1), then its conjugate function also belongs to the same class; in other words, the class Lip (, C(T)) (0<<1) is invariant with respect to the operator of conjugation acting in it. In the two-dimensional case the class Lip (, C(T²)) (0<<1) is no longer invariant with respect to conjugate functions of two variables. Here a final result elucidating the full character of violation of invariance of the class Lip (, C(T^N)) (0<<1) with respect to the multidimensional conjugation operator acting in it is established.Translated from Matematicheskie Zametki, Vol. 23, No. 3, pp. 361–372, March, 1978.