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.     

Choosing the maximum from a sequence with a discount function

Choosing the maximum value from a sequence ofN independent values is a well known problem often called the candidate problem or secretary problem. This paper treats the above problem with a discount penalty (0<<1) for each additional observation taken. It is shown that asN increases indefinitely, the optimal stopping policy is bounded although the maximum expected payoff goes to zero, and that there exists a sequence 0= ₀<₁<₂<<1, such that the asymptotic optimal stopping rule is the same for all _i–1<_i.     

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.     

On Some Interpolation Rules for Lattice Ordered Groups

Let be an infinite cardinal. In this paper we define an interpolation rule IR() for lattice ordered groups. We denote by C() the class of all lattice ordered groups satisfying IR(), and prove that C() is a radical class.     

Peak sets for analytic Hölder classes

A closed subset E of the unit circumference T is said to be a peak set for the analytic Hölder class A, 0 < < 1 there exists a functionf,fA such that f¦_E1 and ¦f(z)¦<1 for. It is shown that the set E is a peak set of the algebra A if and only if there exists a nonnegative Borel measure on T such that the function coincides almost everywhere with a function of the Hölder class , equal to zero on E. A sufficient condition in order that a closed set E should belong to the family of peak sets is obtained.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 157, pp. 129–136, 1987.     

An observation about a theorem of A. D. Alexandrov concerning Lorentz transformations

By a theorem of A. D. Alexandrov a bijection : R _n ⁿ Rⁿ (n3) satisfiyng d(a,b)=0d ((a),(b))=0 a, b Rⁿ (d distance associated to the Minkowski metric) is a Lorentz transformation up to a dilatation. We prove that the inverse implication in the condition above may be dropped, being a consequence of the direct one.Dedicated to Raphael Artzy on the occasion of his 70^th birthday     

Convergence of the Gram-Charier expansion after the normalizing Box-Cox transformation

Consider an exponential family such that the variance function is given by the power of the mean function. This family is denoted by ED if the variance function is given by ^(2-)/(1-), where is the mean function. When 0<1, it is known that the transformation of ED⁽⁾ to normality is given by the power transformationX ^(1-2)/(3-3), and conversely, the power transformation characterizes ED⁽⁾. Our principal concern will be to show that this power transformation has an another merit, i.e., the density of the transformed variate has an absolutely convergent Gram-Charier expansion.     

A chord-stretching map of a convex loop is an isometry

Let ⁿ be n-dimensional Euclidean space, and let : [0, L] ⁿ and : [0, L] ⁿ be closed rectifiable arcs in ⁿ of the same total length L which are parametrized via their arc length. is said to be a chord-stretched version of if for each 0s tL, |(t)–(s)| |(t)–(s)|. is said to be convex if is simple and if ([0, L]) is the frontier of some plane convex set. Individual work by Professors G. Choquet and G. T. Sallee demonstrated that if were simple then there existed a convex chord-stretched version of . This result led Professor Yang Lu to conjecture that if were convex and were a chord-stretched version of then and would be congruent, i.e. any chord-stretching map of a convex arc is an isometry. Professor Yang Lu has proved this conjecture in the case where and are C ² curves. In this paper we prove the conjecture in general.     

On a certain class of transcendental numbers

We show that the numbers ea are transcendental, where a 0; 1, is irrational, 0, and is algebraic.After this paper was in press the author discovered that Baker (Mathematika, 14(1967), 220–228) proved the transcendental nature of the numbers ea ₁ ¹ ...a_ss where , ₁... are algebraic numbers, 0, and In a₁...,ln a_s, are linearly independent over the field of rationals. The arguments in our paper differ slightly from the arguments of Baker.Translated from Matematicheskie Zametki, Vol. 4, No. 3, pp. 341–348, September, 1968.     

Determinants in net subgroups

Suppose R is a commutative ring with 1, =( _ij ) is a fixedD-net of ideals of R of ordern, and G is the corresponding net subgroup of the general linear group GL (n, R). There is constructed for a homomorphismdet of the subgroup G() into a certain Abelian group (). Let I be the index set {1...,n}. For each subset I let ()= _{ij ji} , wherei, ranges over all indices in and j independently over the indices in the complement I ((I) is the zero ideal). Letdet (a) denote the principal minor of order ||n of the matrixa G () corresponding to the indices in , and let' () be the Cartesian product of the multiplicative groups of the quotient rings R/() over all subsets I. The homomorphismdet is defined as follows: It is proved that if R is a semilocal commutative Bezout ring, then the kernelKer det coincides with the subgroup E() generated by all transvections in G(). For these R is also definedTm det .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 114, pp. 37–49, 1982.     

Bounds for the nonhomogeneous GASPT equation

Summary Weighted a priori bounds for the equation u+(/y)u_y=f(>0), in the halfplane y>0, are proved. If p>1, 0<+p^–1<1+, u has bounded support and y^µu_y0 (as y0₊), then the L^p norms of uu and yD²u are bounded by the L^p norm of yf. A boundary value problem in a rectangle is also studied in the appropriate weighted Sobolev class.Lavoro eseguito nell'ambito dell'Istituto di Analisi Globale ed Applicazioni del C.N.R.Lavoro eseguito nell'ambito del Gruppo Nazionale di Analisi Funzionale ed Applicazioni del C.N.R.     

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.     

Mouvement brownien,cônes et processus stables

Summary LetW=(W _t, t0) denote a two-dimensional Brownian motion starting at 0 and, for 0<<, letC be a wedge in ² with vertex 0 and angle 2. We consider the set of timest's such that the path ofW, up to timet, stays inside the translated wedgeW _t-C. It follows from recent results of Burdzy and Shimura that this set, which we denote byH , contains nonzero times if, and only if, >/4. Here we construct a measure, a local time, supported onH . For /4W, time-changed by the inverse of this local time, is shown to be a two-dimensional stable process with index 2-/2. This results extends Spitzer's construction of the Cauchy process, which is recovered by taking =/2. A formula which describes the behaviour ofW before a timetH is established and applied to the proof of a conjecture of Burdzy. We also obtain a two-dimensional version of the famous theorem of Lévy concerning the maximum process of linear Brownian motion. Precisely, for 0<S _t denote the vertex of the smallest wedge of the typez-C which contains the path ofW up to timet. The processS _t-W_t is shown to be a reflected Brownian motion in the wedgeC , with oblique reflection on the sides. Finally, we investigate various extensions of the previous results to Brownian motion inR ^d, d3. LetC be the cone associated with an open subset of the sphereS ^d-1, and letH be defined asH above. Sufficient conditions are given forH to contain nonzero times, in terms of the first eigenvalue of the Dirichlet Laplacian on .     

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.     

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.     

Nielsen numbers and fixed points of mappings of bouquets of circles

In this note we construct maps of a bouquet of circles into itself, for which the familiar lower bound for the number of fixed points in terms of the Nielsen number is extremely ineffective. Precisely, we prove the following theorem: if n 1 and if f: S¹vS¹-S¹vS¹ is a map such that the homomorphism of the fundamental group induced by it carries the canonical generators and , respectively, into 1 and (gb^–1gb^–1)^n–1, then the Nielsen number of the map f is equal to 0, and any map homotopic to f has not less than 2n–1 fixed points.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 122, pp. 135–136, 1982.     

Remarks on some classes of Fourier coefficients

In this paper equivalent classes of the classes M' and S' _{p r}, p >1, 0,r {0,1,2,...,[]} defined by Sheng [5] are obtained. Then it is shown that the classes of Fourier coefficients S _p, S' _p(case r==0) and S _p(), p>1, defined by . V. Stanojevi, V. B. Stanojevi Sheng and the author of the present note are identical. As a corollary of this result, the L ¹-estimate for cosine series, obtained in [10], is refined.     

On a Class of Convex Sets and Functions

Given a convex subset C of ⁿ, the set-valued mapping C (where 0C is, by convention, the recession cone of C) is increasing on ₊ if and only if C contains the origin, and decreasing on ₊ if and only if C is contained in its recession cone. This simple fact enables us to define a binary operation which combines a concave or convex function on ^m with a convex subset of ⁿ to produce a convex subset of ^n+m. This binary operation is the set theoretic counterpart of a functional operation introduced by the author. In this paper, we present a detailed study of the class of convex subsets which are contained in their recession cones, and we establish some remarkable properties of our binary operation.Mathematics Subject Classifications (2000) 26A51, 26B25, 26E25.     

Application of Hermite-Mahler polynomials to the approximation of the values ofp-adic exponential function

We shall give a further application of Hermite-Mahler polynomials to the consideration ofp-adic exponential function. An effective lower bound is obtained for max {| – | _p ,P(e )| _p }, where is an algebraic number satisfying || _p <p ^–/(p–1), and 0 is ap-adic number with | | _p depending on the degree of the polynomialPZ[y]. The bound obtained implies the transcendence ofe if ap-adic number satisfying 0 < || _p <p ^–/(p–1) is algebraic or can be well approximated by algebraic numbers.This work was carried out while the author was a research fellow of the Alexander von Humboldt Foundation.     

Orderh 2 method for a singular two-point boundary value problem

A method is described based on auniform mesh for the singular two-point boundary value problem:y+(/x)y+f(x, y)=0, 0<x1,y(0)=0,y(1)=A, and it is shown to be orderh ² convergent forall 1.