Büschel linearer Abbildungen

If , , are linear mappings out of a projective space (P,G) into a projective space (P', G') and , then is said to belong to the pencil <,<> of linear mappings spanned by and if in the main (x), (x), (x) are collinear for all x P. We give some sufficient conditions for x P and , , such that (x) is uniquely determined by giving, and (z), z P.

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Herrn Prof. Dr.Helmut Karzel zum 60. Geburtstag gewidmet     

Feller Processes Generated by Pseudo-Differential Operators: On the Hausdorff Dimension of Their Sample Paths

Let {X _t} _t0 be a Feller process generated by a pseudo-differential operator whose symbol satisfiesÇⁿ|q(Ç,)|c(1=)()) for some fixed continuous negative definite function (). The Hausdorff dimension of the set {X _t:tE}, E [0, 1] is any analytic set, is a.s. bounded above by dim E. is the Blumenthal–Getoor upper index of the Levy Process associated with ().     

Einschliessungsaussagen für Differentialoperatoren vierter Ordnung und ein Verfahren zur Berechnung von Schrankenfunktionen

For a linear fourth order ordinary differential operator M we study Range Domain Implications (RDI). Let C_o [O,1] be positive; we show under what conditions there exists a C_O[O,1] such that the following RDI holds: Mu(x) (x) (0x1) u(x) (0x1). In particular we provide a numerical procedure to calculate .RDI are used to obtain error estimations and to solve related nonlinear problems.The basic idea to prove RDI is to split M into a product of second order differential operators which are easier to handle. For the general case that there exists no global splitting the concept of a local splitting is introduced.

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The author would like to thank the European Research Office of the United States Army for their kind interest.     

Einschließungsaussagen für gewöhnliche Differentialoperatoren

Summary For differential operatorsM of second order (as defined in (1.1)) we describe a method to prove Range-Domain implications—Mu–u and an algorithm to construct these functions , , , . This method has been especially developed for application to non-inverse-positive differential operators. For example, for non-negativea ₂ and for given functions = we require =C ₀[0, 1] C ₂([0, 1]–T) whereT is some finite set), (M) (t)(t), (t[0, 1]–T) and certain additional conditions for eachtT. Such Range-Domain implications can be used to obtain a numerical error estimation for the solution of a boundary value problemMu=r; further, we use them to guarantee the existence of a solution of nonlinear boundary value problems between the bounds- and .     

Sur les équations fonctionnelles auxq-différences

Summary In this paper, we study the convergence of formal power series solutions of functional equations of the formP _k(x)(^[k](x))=(x), where ^[k] (x) denotes thek-th iterate of the function.We obtain results similar to the results of Malgrange and Ramis for formal solutions of differential equations: if(0) = 0, and(0) =q is a nonzero complex number with absolute value less than one then, if(x)=a(n)x ⁿ is a divergent solution, there exists a positive real numbers such that the power seriesa(n)q ^sn(n+1)2 x ⁿ has a finite and nonzero radius of convergence.

     

Equivalent Sets of Solutions of the Klein–Gordon Equation with a Constant Electric Field

We argue extensively in favor of our earlier choice of the in and out states (among the solutions of a wave equation with one-dimensional potential). In this connection, we study the nonstationary and stationary families of complete sets of solutions of the Klein–Gordon equation with a constant electric field. A nonstationary set _Pv consists of the solutions with the quantum number p _v=p ⁰ v–p₃. It can be obtained from the nonstationary set _P3 with the quantum number p ₃ by a boost along the x ₃ axis (in the direction of the electric field) with the velocity –v. By changing the gauge, we can bring the solutions in all sets to the same potential without changing quantum numbers. Then the transformations of solutions in one set (with the quantum number p _v) to the solutions in another set (with the quantum number p _v) have group properties. The stationary solutions and sets have the same properties as the nonstationary ones and are obtainable from stationary solutions with the quantum number p ⁰ by the same boost. It turns out that each set can be obtained from any other by gauge manipulations. All sets are therefore equivalent, and the classification (i.e., assigning the frequency sign and the in and out indices) in any set is determined by the classification in the set _P3, where it is obvious.     

Construction in a piecewise smooth domain of a function of the class H2 from the value of the conormal derivative

The problem of the construction in a bounded domain ^m with a Lipschitz boundary of a function H²(), for which the conormal derivative on coincides with the normal component of a given vector field u H¹(,C ³), is discussed. The solution of this problem is given for piecewise smooth boundaries in the case m=3.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 163, pp. 17–28, 1987.     

Estimation of a sum along an algebraic curve

Let be an algebraic curve determined over a finite field k = [q]; e,x are subsidiary additive and multiplicative characters of the field k;, are functions in determined over k and satisfying some natural conditions. If P passes through the points of curve , rational over k, then where constant C depends only on the powers of ,,.Translated from Matematicheskie Zametki, Vol. 5, No. 3, pp. 373–380, March, 1969.     

Kennzeichnungen von Minimalschraubflächen Mittels Vertikalisophoten und Horizontalschattengrenzen

In Euclidean space E³, let be a (regular C-) minimal surface without planar points having locally (without loss of generality) the spherical representation n(u,v)=(cos v/cosh u, sin v/cosh u, tanh u), (u,v)G². The corresponding (isothermal) parametrization : x(u,v), (u,v)G can be expressed using agenerating Function (u,v) which satisfies _{uu + vv} – 2_utanh u + =0; the v-curves (coordinate curves u=u₀) in , along each of which the angle between the normal n(u,v) of and the x₃-axis is constant, are thevertical- isophotes of , the u-curves (v=v₀) being their orthogonal trajectories (theorems 1, 2). Considering u-curves and/or v-curves of having additional geometric properties (curves of constant/steepest slope, curves of constant Gaussian curvature, asymptotic curves, lines of curvature or geodesies of ) we prove many newgeometric characterizations of theright helicoid, thecatenoid andScherk's second surface (theorems 3–7). All of these surfaces areminimal hélicoidal surfaces.     

Sûto's method applied to a problem of Färe and Mitchell

Summary A real solution of the functional equation(x + (y – x)) = f(x) + g(y) + h(x)k(y) on a set ² is a 6-tuple (f, g, h, k, , ) of real valued functions such that the equation is identically fulfilled on. Except for cases known before—e.g. when is linear—we present all real solutions in an arbitrary region where the functions have derivatives of second order.     

Integral averages and oscillation of second order sublinear differential equations

New oscillation criteria are given for the second order sublinear differential equation

Approximation of infinitely differentiable periodic functions by interpolation trigonometric polynomials

We establish asymptotically unimprovable interpolation analogs of Lebesgue-type inequalities on the classes of periodic infinitely differentiable functions C C whose elements can be represented in the form of convolutions with fixed generating kernels. We obtain asymptotic equalities for upper bounds of approximations by interpolation trigonometric polynomials on the classes C _, and C H .Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 4, pp. 495–505, April, 2004.     

The existence of some resolvable block designs with divisibility into groups

This paper proves the existence of resolvable block designs with divisibility into groups GD(v; k, m; ₁, ₂) without repeated blocks and with arbitrary parameters such that ₁ = k, (v–1)/(k–1) ₂ v^k–2 (and also ₁ k/2, (v–1)/(2(k–1)) ₂ v^k–2 in case k is even) k 4 andp=1 (mod k–1), k < p for each prime divisor p of number v. As a corollary, the existence of a resolvable BIB-design (v, k, ) without repeated blocks is deduced with X = k (and also with = k/2 in case of even k) k , where a is a natural number if k is a prime power and=1 if k is a composite number.Translated from Matematicheskie Zametki, Vol. 19, No. 4, pp. 623–634, April, 1976.     

The probability of the triangle inequality in probabilistic metric squares

The notion of a random semi-metric space provides an alternate approach to the study of probabilistic metric spaces from the standpoint of random variables instead of distribution functions and permits a new investigation of the triangle inequality. Starting with a probability space (, , P) and an abstract setS, each pair of points,p, q, inS is assigned a random variableX _pq with the interpretation thatX _pq () is the distance betweenp andq at the instant . The probability of the eventJ _pqr = { :X _pr ()X _pq ()+X _qr ()} is studied under distribution function conditions imposed by Menger Spaces (K. Menger, Statistical Metrics, Proc. Nat. Acad. Sci., U.S.A., 28 (1942), 535–537; B. Schweizer and A. Sklar, Statistical Metric Spaces, Pacific J. Math.10 (1960), 313–334). It turns out that for > 0 there are 3 non-negative, identically-distributed random variablesX, Y andZ for whichP(X < Y + Z) < . This and other results show that distribution function triangle inequalities are very weak. Conditional probabilities are introduced to give necessary and sufficient conditions forP(J _pqr ) = 1.     

Some functional inequalities and their Baire category properties

Summary In the paper we consider, from a topological point of view, the set of all continuous functionsf:I I for which the unique continuous solution:I – [0, ) of(f(x)) (x, (x)) and(x, (x)) (f(x)) (x, (x)), respectively, is the zero function. We obtain also some corollaries on the qualitative theory of the functional equation(f(x)) = g(x, (x)). No assumption on the iterative behaviour off is imposed.     

φ-Factors for Function Seminorms

Let (, A, ) be a measure space, a function seminorm on M, the space of measurable functions on , and M the space {f M : (f) < }. Every Borel measurable function : [0, ) [0, ) induces a function : M M by (f)(x) = (|f(x)|). We introduce the concepts of -factor and -invariant space. If is a -subadditive seminorm function, we give, under suitable conditions over , necessary and sufficient conditions in order that M be invariant and prove the existence of -factors for . We also give a characterization of the best -factor for a -subadditive function seminorm when is -finite. All these results generalize those about multiplicativity factors for function seminorms proved earlier.     

Large Deviations of Local Times of Lévy Processes

For X(t) a real-valued symmetric Lévy process, its characteristic function is E(e ^iX(t))=exp(–t()). Assume that is regularly varying at infinity with index 1<2. Let L ^x _t denote the local time of X(t) and L* _t =sup _xR L ^x _t . Estimates are obtained for P(L ⁰ _t y) and P(L* _t y) as y and t fixed.     

An isolation theorem for decomposable forms of purely real algebraic fields of degree n⩾3

Let M be a complete module of a purely algebraic field of degree n3, let be the lattice of this module and let F(X) be its form. By we denote any lattice for which we have = , where is a nondiagonal matrix satisfying the condition ¦-I¦ , I being the identity matrix. The complete collection of such lattices will be denoted by {}. To each lattice we associate in a natural manner the decomposable form F(X). The complete collection of forms, corresponding to the set {}, will be denoted by {F} It is shown that for any given arbitrarily small interval (N–, N+), one can select an such that for each F(X) from {F} there exists an integral vector X₀ such that N– < F(X₀) < N+.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 112, pp. 167–171, 1981.     

Cycles and measure of bifurcation sets for two-dimensional diffeomorphisms

Summary We describe a large class of one-parameter families , {}, , of two-dimensional diffeomorphisms which arestable for <0, exhibit acycle for =0, and thereafter have a bifurcation set of positive but arbitrarily smallrelative measure for in small intervals [0, ]. A main assumption is that the basic sets involved in the cycle havelimit capacities that are not too large.The second author acknowledges hospitality and financial support from IMPA/CNPq during the period this paper was prepared     

On equivalence of rearrangements of the Haar system in dyadic Hardy and BMO spaces

H={h ₁,I } — , . : , I ¦(I)¦=¦I¦, ¦I¦ — I. H H ={h _(I),I} . , , . L ^p .

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Dedicated to Professor B. Szökefalvi-Nagy on his 75th birthday

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This research was supported in part by MTA-NSF Grants INT-8400708 and 8620153.