Contractions with a unilateral shift summand are reflexive

If T is a contraction on the Hilbert space H and there is an invariant subspace K for T such that T/K is similar to a backward shift, then T is reflexive. If similarity is replaced by quasi-similarity, then the same conclusion holds under the additional condition that the defect index of T be finite.This research was partially supported by National Science Council (Republic of China).     

Polynomial Wavelet-Type Expansions on the Sphere

We present a polynomial wavelet-type system on S ^d such that any continuous function can be expanded with respect to these wavelets. The order of the growth of the degrees of the polynomials is optimal. The coefficients in the expansion are the inner products of the function and the corresponding element of a dual wavelet system. The dual wavelets system is also a polynomial system with the same growth of degrees of polynomials. The system is redundant. A construction of a polynomial basis is also presented. In contrast to our wavelet-type system, this basis is not suitable for implementation, because, first, there are no explicit formulas for the coefficient functionals and, second, the growth of the degrees of polynomials is too rapid.     

Regular homomorphisms of generalized projective planes

A generalized projective plane is an incidence structure together with a relation distant on the set of points and also on the set of lines, such that any two distant points A,B (lines a,b) have a unique common line (A,B) (common point (a,b)) and three further axioms hold. Every commutative ring with 1 supplies a model. A homomorphism of into an incidence structure is called regular if the following condition and its dual are valid: A distant B and c IA,B implies c=(A,B). We shall prove the following two theorems. Let be a generalized projective plane satisfying a richness condition called (U). Let M I m. If and are regular homomorphisms of such that X = M X = M for each point X of the line m then A = B A = B for any two points A,B. If is a projective plane over a commutative ring such that (U) holds then the surjective regular homomorphisms of are induced by the ideals of the ring; in particular, the image of under a regular homomorphism is again a projective plane over a ring, and preserves distant.     

A geometric property of extremal surfaces

Let the surface R³ be defined by the equation z = f(x, y), where f(x, y) is a function 3 times continuously differentiable in R². It is proved that if the total (Gaussian) curvature of the surface is nonzero almost everywhere on in the sense of Lebesgue measure in R²), then is extremal, i.e., for almost all (x,y) R² the inequality max (||qx||, ||qy, qf (x, y)) > q^–1/s–. holds for all integral q qo (f), where x is the distance from the real number x to the nearest integer and > 0 is arbitrarily small.Translated from Matematicheskie Zametki, Vol. 23, No. 2, pp. 177–181, February, 1978.In conclusion, the author thanks V. G. Sprindzhuk for suggesting the problem.     

Über die mittelfläche einer isotropen geradenkongruenz

Let be the middle surface of an isotropic rectilinear congruence of class C³ in the real Euclidean space E³. When the spherical image of is parametrized by special isothermal coordinates (u,v) G ², can be described by a generating harmonic function A(u,v). Using such a C-representation of , the basic properties of regularity and curvature of are discussed. Moreover, the cases that be a minimal (regular) surface ₁, or a plane surface ₂ are solved explicitly. In connection with the latter results (which are already well-known from Ribaucour) several new characterizations for being a regular surface ₁ resp. ₂ are given: they are based on special properties (like: being asymptotic lines resp. lines of curvature of ) of those curves c (-Spurlinien) in the tangents of which form in each point Xc a minimal angle with the straight line of passing through X.

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Herrn Prof. Dr. Karl Strubecker zum 80. Geburtstag gewidmet     

Limit theorems for multitype continuous time Markov branching processes

Summary Let X(t)=(X ₁ (t), X ₂ (t), , X _t (t)) be a k-type (2k<) continuous time, supercritical, nonsingular, positively regular Markov branching process. Let M(t)=((m _ij (t))) be the mean matrix where m _ij (t)=E(X _j (t)¦X _r (0)= _ir for r=1, 2, , k) and write M(t)=exp(At). Let be an eigenvector of A corresponding to an eigenvalue . Assuming second moments this paper studies the limit behavior as t of the stochastic process . It is shown that i) if 2 Re >₁, then · X(t)e{–t¦ converges a.s. and in mean square to a random variable. ii) if 2 Re ₁ then [ · X(t)] f(v · X(t)) converges in law to a normal distribution where f(x)=(x) ^–1 if 2 Re <₁ and f(x)=(x log x)^–1 if 2 Re =₁, ₁ the largest real eigenvalue of A and v the corresponding right eigenvector.Research supported in part under contracts N0014-67-A-0112-0015 and NIH USPHS 10452 at Stanford University.     

A proof for Theodorus' theorem by drawing diagrams

Summary In Plato's dialogue Theaetetus, the mathematician Theodorus (ca. 470-400 BC) is said to have proved the irrationality of 3, 5, ..., 17 by drawing diagrams. In this paper such a proof is presented and compared with another, more arithmetical, reconstructed proof by W. Knorr. Moreover a hypothetical early theory of similar rectangles is extracted from Euclid's Elements, which could have served as a basis for the geometrical proof of Theodorus' theorem.Herrn N. Stephanidis in Freundschaft gewidmet     

An error analysis for absolute and relative approximation

We consider the vectorial algorithm for finding best polynomial approximationsp P _n to a given functionf C[a, b], with respect to the norm · _s , defined byp – f _s =w ₁ (p – f)+w ₂ (p – f) A bound for the modulus of continuity of the best vectorial approximation operator is given, and using the floating point calculus of J. H. Wilkinson, a bound for the rounding error in the algorithm is derived. For givenf, these estimates provide an indication of the conditioning of the problem, an estimate of the obtainable accuracy, and a practical method for terminating the iteration.This paper was supported in part by the Canadian NCR A-8108, FCAC 74-09 and G.E.T.M.A.Part of this research was done during the first-named author's visit to theB! Chair of Applied Mathematics, University of Athens, Spring term, 1975.     

The thinnest holding-lattice of a set

In the present note we are going to study the following question proposed by L. Fejes Toth. In the Euclidean plane a lattice is called a holding-lattice of a planar setS if any set congruent toS contains at least one lattice point of . The density of equals (2)^–1, where denotes the area of a fundamental triangle of . A holding-lattice ofS of least possible density is said to be the thinnest holding-lattice ofS. The problem: Find the sets whose thinnest holding-lattice is a regular trianglelattice.To Professor L. Fejes Toth on his 70th Birthday     

Harmonic measure of continua with a fixed diameter

Let be the family of all continua E in , where U={¦z¦ < 1}, let U(E) be the connected component of UE containing the point z=0, let w_E(z₀=w(z₀, E, U(E)) be the harmonic measure of E relative to the domain U(E) at the point z₀ U(E). In the paper one answers affirmatively a question raised by B. Rodkin [K. F. Barth, D. A. Branna, and W. K. Hayman, Research problems in complx analysis, Bull. London Math. Soc.,l6, No. 5, 490–517, 1984]. Namely, one proves that in the family (d₀) of continua E , satisfying the condition diam E=d₀,o<d₀2, one has the inequality arcsin d₀/2, one indicates all the cases for which equality prevails.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 144, pp. 146–148, 1985.     

Асимптоический Анализ Распределения Статистики Диксона

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Склярные дифференциалвные инварианты третвего порядка риамнового простанства трех измерений

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On the exactness of a theorem of G.A. Fomin

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Daugavet type inequalities for operators on L p–spaces

Let T be a regular operator from L ^p L ^p. Then , where T_r denotes the regular norm of T, i.e., T_r=|T| where |T| denotes the modulus operator of a regular operator T. For p=1 every bounded linear operator is regular and T=T_r, so that the above inequality generalizes the Daugavet equation for operators on L ¹–spaces. The main result of this paper (Theorem 9) is a converse of the above result. Let T be a regular linear operator on L _p and denote by T _A the operator T_A. Then for all A with (A)>0 if and only if .     

Hardy spaces of predictable martingales

, P _p Q _p .

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This research was partially supported by the Hungarian Scientific Foundation.     

Orders of one-sided approximations of functions inL p -metric

. L _p , 0<p<, . , f, {E _n (f) _p } ₁ p>0 .

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The author expresses his thanks to S. B. Stekin for the attention he has paid to this work.     

Geometric methods in the study of irregularities of distribution

Let be a signed measure on E ^d with E ^d =0 and ¦¦E^d<. DefineD _s() as sup ¦H¦ whereH is an open halfspace. Using integral and metric geometric techniques results are proved which imply theorems such as the following.Theorem A. Let be supported by a finite pointsetp _i. ThenD _s()>c _d(₁/ ₂)^1/2{ _{i(p i})²}^1/2 where ₁ is the minimum distance between two distinctp _i, and ₂ is the maximum distance. The numberc _d is an absolute dimensional constant. (The number .05 can be chosen forc ₂ in Theorem A.)Theorem B. LetD be a disk of unit area in the planeE ², andp ₁,p ₂,...,p _n be a set of points lying inD. If m if the usual area measure restricted toD, while nP _i=1/n defines an atomic measure _n, then independently of _n,nD _s(m – _n) .0335n ^1/4. Theorem B gives an improved solution to the Roth disk segment problem as described by Beck and Chen. Recent work by Beck shows thatnD _s(m – _n)cn ^1/4(logn)^–7/2.     

Pointwise convergence of expansions with respect to certain product systems

- , - _n A_n-,. , . , ¦_n¦=1 (n=1,2, ), . , — — , .     

A Note on the Place Topology of Projective Planes

It is shown that the place topology induced by a proper epimorphism of a projective plane , which is known to make a Lenz-topological plane, makes even a topological projective plane, if the extended radical of some underlying ternary field is bounded.