Lower functions for asymmetric Lévy processes

Summary Let {X _t } be a ¹ process with stationary independent increments and its Lévy measurev be given byv{yy>x}=x ^–L ₁ (x), v{yy<–x}=x ^–L ₂ (x) whereL ₁,L ₂ are slowly varying at 0 and and 0<1. We construct two types of a nondecreasing functionh(t) depending on 0<<1 or =1 such that lim inf a.s. ast 0 andt for some positive finite constantC.This research is partialy supported by a grant from Korea University     

Effilement minimal en une singularité isolée de l'équation de Schrödinger et application au principe de Picard

In this paper, the generalized Schrödinger equation (–)u=0 on the punctured unit disk of ² is investigated. If is rotation free and satisfies the Picard principle at the origin, it is shown that if a setE is minimal thin relatively to an extremal harmonic functionh with zero boundary values at {|x|=1}, there exists a sequence (r _n ) converging to zero such that B(O,r _n ) C E. Lete be the -unit. It is proved that if a measure satisfies _\E e h d<, for a minimal thin, relatively toh , setE then the Picard principle is valid for the measure + .

     

Die Verteilung der schlichten Funktionen in einem Funktionenraum

We consider the set of regular functions . We construct a Borel measure and a class of outer measures _h onH. With these and _h we show that: (HS)=0 and _h (HS)=0, (S is the set of normed univalent functions). From _h (HS)=0 follows—forh=t —that the Hausdorff—Billingsley-dimension ofHS is zero.     

Nonlocal symmetries and the theory of coverings: An addendum to A. M. vinogradov's ‘local symmetries and conservation laws”

For a systemY of partial differential equations, the notion of a covering Y is introduced whereY is infinite prolongation ofY. Then nonlocal symmetries ofY are defined as transformations of which conserve the underlying contact structure. It turns out that generating functions of nonlocal symmetries are integro-differential-type operators.     

Quasi-coherence of Hardy spaces in several complex variables

In this paper, we prove that the Hardy spaceH ^p (), 1p<, over a strictly pseudoconvex domain in ⁿ with smooth boundary is quasi-coherent. More precisely, we show that Toeplitz tuplesT with suitable symbols onH ^p () have property (). This proof is based on a well known exactness result for the tangential Cauchy-Riemann complex.     

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.     

Implicit elliptic boundary-value problems with discontinuous nonlinearities

Let be a bounded domain in ⁿ (n3) having a smooth boundary, let be an essentially bounded real-valued function defined on × ^h, and let be a continuous real-valued function defined on a given subset Y of Y ^h. In this paper, the existence of strong solutions u W ^2,p (, ^h) W _o ^1,p (n/2<p<+) to the implicit elliptic equation (–u)=(x,u), with u=(u₁, u₂, ..., u_h) and u=(u ₁, u ₂, ..., u _h), is established. The abstract framework where the problem is placed is that of set-valued analysis.     

Isometries of symmetric operator spaces associated with AFD factors of typeII and symmetric vector-valued spaces

If is a surjective isometry of the separable symmetric operator spaceE(M, ) associated with the approximately finite-dimensional semifinite factorM and if · _E(M,) is not proportional to · _{L 2}, then there exist a unitary operatorUM and a Jordan automorphismJ ofM such that(x)=UJ(x) for allxME(M, ). We characterize also surjective isometries of vector-valued symmetric spacesF((0, 1), E(M, )).Research supported by the Australian Research Council     

Distribution of the supremum of sums of independent variables with negative drift

Let {_n} be a sequence of identically distributed independent random variables,M₁=<0,M ₁ ² <;S ₀=0,S _n =₁+₂,+...+ _{n, n}1;¯ S=sup {S _n n=0.} The asymptotic behavior ofP(¯ St) as t is studied. If _t ^P (₁x dx=0((t)), thenP(¯ St)– 1/¦¦ _t P (₁x dx=0((t)) (t) is a positive function, having regular behavior at infinity.Translated from Matematicheskie Zametki, Vol. 22, No. 5, pp. 763–770, November, 1977.The author thanks B. A. Rogozin for the formulation of the problem and valuable remarks.     

Asymptotic approximation of crossing probabilities of random sequences

Summary Let {X _i , i1} be a random sequence and {u _ni ,1in, n1} be an array of boundary values. We consider the asymptotic approximation of the probability P _n =P{X _i u _ni ,1in} by . We give sufficient conditions on X _i such that P _n–P _n ^* 0 as n. This generalizes the situation considered in extreme-value theory where the boundary is constant in i. The general theory is applied in particular to Gaussian cases.     

Zur Vollständigkeit der Induktiven Gruppoide der Partiellen Automorphismen von Algebren

For any algebraA let(A) be the set of partial automorphisms (isomorphisms between subalgebras). With the natural multiplication it is an inductive groupoid in the sense of Ehresmann.(A) is complete iff every subset of(A) which is compatible with the semi-ordering has an upper bound. The fact, whether(A) is complete or not, depends on the defining operations ofA. For every direct familyF = (, (A ) ,()_, , of algebras such that all ,, with are one-to-one functions, the direct limit is complete iff all(A) are complete. We give some theorems on the decomposition of inductive groupoids, and employ them in proving the completeness of(A) to variousA.In particular, we obtain that, in case whenG is a finite group,(G) is complete iffG is either cyclic or direct product of a noncyclic group of order 4 and a cyclic group of odd order. For finite acyclic ringsR and finite fieldsK the inductive groupoids(R) and(K) are complete.Further we deal with the question, to what extent algebras are determined by their inductive groupoids. (An algebraA of a classS is defined by(A) iff, for any algebraB of the classS, isomorphism between(A) and(B) implies isomorphism betweenA andB.) Particular attention is paid to finite groups. In general, algebras of classesS are not defined within the classS by their inductive groupoids.

     

Factorization intok-dimensional linear bijections

LetV be a vector space,k withkdimV andS _k{GL(V)|dimV(–1)=k}. ThenS _k generates GL _f (V){GL(V)|V(-1) is finite-dimensional} (with the exception that dimV=2=k and the field is GF2). We study the length problem in GL _f (V) withS _k as set of generators.     

The imaginary zeros of a mixed Bessel function

In the present work we study the existence and monotonicity properties of the imaginary zeros of the mixed Bessel functionM _v(z)=(z²+)J_v(z)+zJ_v(z). Such a function includes as particular cases the functionsJ _v(z)(==0), J_v(z)(=–v²,=1)x andH _v(z)=J_v(z)+zJ_v(z), whereJ _v(z) is the Bessel function of the first kind and of orderv>–1 andJ _v(z), J_v(z) are the first two derivatives ofJ _v(z). Upper and lower bounds found for the imaginary zeros of the functionsJ _v(z), J_v(z) andH _v(z) improve previously known bounds.

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Zusammenfassung Dieser Artikel betrifft die Existenz und Monotonie von Eigenschaften imaginärer Nullen der gemischten BesselfunktionM _v(z)=(z²+)J_v(z)+zJ_v(z). Eine solche Funktion enthält als Spezialfall die FunktionenJ _v(z)(==0), J_v(z)(=–v²,=1) undH _v(z)=J_v(z)+zJ_v(z), woJ _v(z)die Besselfunktion von erster Art und Ordnungv>–1 andJ _v(z), J_v(z) sind die erste und zweite Ableitung vonJ _v(z). Untere und obere Schranken, die für die imaginären Nullen der FunktionenJ _v(z), J_v(z) undH _v(z) gefunden wurden, verbessern früher bekannte Resultate.

     

OnU ϕ-sets of trigonometric integrals

, (t) >0 E(–, +),E<, , ¦f(t)¦ (t) xE, f(t)=0 (–, +).     

Algebraic polynomial bases of space LP

Let (_n) be a system, close to the orthonormal complete system (x _n). An estimate is obtained for the deviation of the system {f_n}, obtained from {_n} by Schmidt's method, from the system {x_n}. This estimate is used to show that, in any L_P(–1,1), withp (1,4/3] [4,), and for any >e¦4 = i,13..., there exists an orthogonal algebraic system (P _n (x)) _n=0 , forming a basis in L_P and such that _n = degP _n (x) n for n>n_o(p,).Translated from Matematicheskie Zametki, Vol. 23, No. 2, pp. 223–230, February, 1978.     

On the boundedness of Weyl sums

Leta be irrational and letf:[0,1] be Riemann-integrable with integral zero. Letf _n (x) denote the Weyl sumf _n (x):= _k=0 ^n–1 f({x k>}),x/[0,1[,n. We prove criteria for the boundedness of the sequence (f _n ) _n1 and discuss the relation of this question to irregularities of the distribution of sequences.     

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.     

Differentiable Connections with Poincar{e} Groups of Local Transformations. {III}

In the canonical smooth fiber bundles :ⁿ⁺¹ⁿ, we study generalized differentiable connections constructed by the author in papers [4] and [5]. Special emphasis is laid on the investigation of the behavior of these connections under local transformations of the classical Poincar{e} groups (1,n) and extended Poincar{e} groups canonically acting in the given connections. We found all first-order nonholonomic affine, ₁, ₂, and _1,2-connections with the groups (1,n) and of local transformations and also constructed classes of the corresponding invariant second-order connections.     

Zur Wachstumsgeschwindigkeit kleiner Flüssigkeitstropfen in einer übersättigten Atmosphäre

Summary The intensity of condensation upon a droplet surrounded by a supercooled mixture of its own vapour and of a gas of arbitrary concentration, and practically motionless relative to this surrounding (Re<1), is represented as a function of the degree of supercooling and of the droplet size. Consideration has been paid to heat and mass transfer and also to the influence of capillarity. The formula obtained is valid for continuons, as well as for free molecular flow conditions.

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Bezeichnungen D m²/sec Diffusionskonstante zwischen den Komponenten der Atmosphäre - g kg/kg Massenanteil der Gaskomponente in der Atmosphäre - j kg/m²sec Diffusionsstrom der Dampfkomponente - m mittlere freie Weglänge der Moleküle in der Atmosphäre - L J/kg Kondensationswärme - kg/sec Kondensationsstärke am Tropfen - p N/m² Gesamtdruck (Summe der Partialdrucke) - P _d N/m² Partialdruck der Dampfkomponente - P _g N/m² Partialdruck der Gaskomponente - P _s N/m² Sättigungsdruck (an ebener Oberfläche) - r m Tropfenradius - r _krit m kritischer Tropfenradius, vergleiche Gleichung (41) - R J/kgK spezifische Gaskonstante - t sec Zeit - T K (absolute) Temperatur - T _s K Sättigungstemperatur (an ebener Oberfläche) - T K Unterkühlung weit vom Tropfen entfernt [T=T _s (P _d , )-T ] - w m/sec radiale Strömungsgeschwindigkeit - x kg/kg Massenanteil der Dampfkomponente in der Atmosphäre - J/m²Ksec Wärmeübergangszahl - sec/m Stoffübergangszahl - m Radialkoordinate - J/m Ksec Wärmeleitzahl - kg/msec dynamische Viskosität - kg/m³ Dichte - N/m Oberflächenspannung des Tropfens Indices usw d Dampfkomponente - g Gaskomponente - r an der Tropfenoberfläche (bei =r) - w Wasser - weit vom Tropfen entfernt (bei =) - j, w Vektorgrössen - – Mittelwert für das atmosphärische Gemisch     

Symmetric balanced ternary designs with ϱ1 = 1 or 2

Summary We prove the following two non-existence theorems for symmetric balanced ternary designs. If ₁ = 1 and 0 (mod 4) then eitherV = + 1 or 4₂ – + 1 is a square and (4₂ – + 1) divides ² – 1. If ₁ = 2 thenV = ((m + 1)/2) ² + 2,K = (m ² + 7)/4 and = ((m – 1)/2)² + 1 wherem 3 (mod 4). An example belonging to the latter series withV = 18 is constructed.