On optimality of spline interpolation for recovery of differentiable functions

W _p ^r (1p<) , . - r -1. =1 r-2.     

An infinite dimensional analogue of the Albeverio-Röckner's theorem on Fukushima's conjecture

LetX be the solution of the SDE:dX _t = (X _t)dB _t +b(X _t)dt, with andb C _b (R) such that >0 for some constant , andB a real Brownian motion. Let be the law ofX onE=C([0, 1],R) andk E* – {0}, whereE* is the topological dual space ofE. Consider the classical form: _k (u, v)=u / kv / kd, whereu andv are smooth functions onE. We prove that, if _k is closable for anyk in a dense subset ofE* and if the smooth functions are contained in the domain of the generator of the closure of _k , must be a constant function.     

Cross-cuts in the power set of an infinite set

In the power setP(E) of a setE, the sets of a fixed finite cardinalityk form across-cut, that is, a maximal unordered setC such that ifX, Y E satisfyXY, X someX inC, andY someY inC, thenXZY for someZ inC. ForE=, ₁, and ₂, it is shown with the aid of the continuum hypothesis thatP(E) has cross-cuts consisting of infinite sets with infinite complements, and somewhat stronger results are proved for and ₁.The work reported here has been partially supported by NSERC Grant No. A8054.     

Inner functions and related spaces of pseudocontinuable functions

Let be an inner function, let C, ¦¦=1. Then the harmonic function [(+)]/(–)] is the Poisson integral of a singular measure D. N. Clark's known theorem enables us to identify in a natural manner the space H² H² with the space L² ( ).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 170, pp. 7–33, 1989.     

A family of noetherian rings with their finite length modules under control

We investigate the category mod of finite length modules over the ring =A _k , where is a V-ring, i.e. a ring for which every simple module is injective, k a subfield of its centre and A an elementary k-algebra. Each simple module E _j gives rise to a quasiprogenerator P _j = A E _j . By a result of K. Fuller, P _j induces a category equivalence from which we deduce that mod _j mod EndP _j . As a consequence we can(1) construct for each elementary k-algebra A over a finite field k a nonartinian noetherian ring such that modA mod(2) find twisted versions of algebras of wild representation type such that itself is of finite or tame representation type (in mod)(3) describe for certain rings the minimal almost split morphisms in mod and observe that almost all of these maps are not almost split in Mod.     

Solution of the nonlinear functional equations representing the roll gap relationships in a cold mill

In the development of a roll force model for cold rolling, techniques were developed for solving the system equations which are of general interest. This paper gives a brief introduction to the physical model but concentrates on the solution of the model equations and the simulation. An unusual feature of the model was that the calculated profiles had to satisfy a number of boundary conditions at different points throughout the roll arc. A new method was developed for calculating these profiles and for determining the gradient functions which satisfied the boundary constraints.Nomenclature p() pressure at roll angle - h() gauge - a() roll radius - y() yield stress - g _i () gradient function on iterationi - e() gauge error - (, ) transition function - H(–) Heaviside unit step function at = - (–) unit impulse function at = - H(, ₁, ₂) defined asH( – ₁) –H( – ₂) - angular position from the roll center line - _T angular limits of roll arc represented - _n angular position of the neutral angle - _i angular position ofith strip elastic-plastic boundary - _pi pressure change at the boundaryi - _i , _i , _i constants defined in Appendix A - k ₁,k ₂ elastic region constants - k total number of strip boundaries (elastic-plastic and entry and exit points) - R undeformed work roll radius - R _s roll separation—distance between roll centers - h ₀₁ unstrained gauge in an elastic region - h _in gauge of the strip at the entry to the roll gap - J gauge error cost function - <x, y> inner product ofx andy - x norm ofx - L ₂[0, _T ] the space of Lebesgue square-integrable functions defined on the interval [0, _T ] - JUVY denotes (Dx)() =dx()/d The author would like to acknowledge the help given by Dr. G. F. Bryant, Director, and Mr. M. A. Fuller, Senior Research Engineer, the Industrial Automation Group, Imperial College of Science and Technology, London. He is also grateful to M. J. G. Henderson of the University of Birmingham for his advice and encouragement during the project. He would like to thank the Directors of GEC Electrical Projects Limited for allowing him to undertake the work and also Mr. J. McTaggart and Mr. C. McKenzie (GEC), Professor H. A. Prime of the University of Birmingham, and Dr. G. F. Bryant for arranging the project.     

On term by term dyadic differentiability of Walsh series

. . . . : {ja _j },j=1,2,... — , f(x) , , f ^[1](x) — f .     

Growth of length of sequential derivation transformed into natural one

We consider the (&, )-fragment of the intuitionistic propositional calculus. It is proved that under the standard transformation of a Gentzen derivation into a natural derivation(), the length of (())2^2·length( ). There is constructed a sequence of Gentzen derivations of length _i, for which the length of (( _i))2^{1/3·length(i}), which shows that the upper bound obtained is not too weak.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 88, pp. 192–196, 1979.     

Criteria of algebraic independence of complex numbers over a field of finite transcendence type (II)

In the present paper we establish a criterion of algebraic independence of complex numbers ₁, ..., _n over a field of finite transcendence type using a sequence of nonzero polynomials in several variables with integral coefficients, which satisfy simultaneously certain upper and lower estimates in different orders of magnitude at the point ( ₁, ..., _q , ₁, ..., _n ), where { ₁, ..., _q } is a transcendence basis of over .The Project Supported by the National Natural Science Foundation of China     

Constructing one-parameter transformations on white noise functions in terms of equicontinuous generators

Let be a barreled locally convex space. A continuous operator on is called anequicontinuous generator if { ⁿ /n!;n=0,1,2,...} is an equicontinuous family of operators. For each equicontinuous generator a one-parameter group of operators is constructed by means of power series. There is a one-to-one correspondence between the equicontinuous generators and the locally equicontinuous holomorphic one-parameter groups of operators. If two equicontinuous generators ₁, ₂ satisfy [₁,₂]=₂ for some thena₁+b₂ is also an equicontinuous generator for anya, b. These general results are applied to a study of operators on white noise functions. In particular, a linear combination of the number operator and the Gross Laplacian, which are natural infinite dimensional analogues of a finite dimensional Laplacian, is always an equicontinuous generator. This result contributes to the Cauchy problems in white noise (Gaussian) space.Work supported by Alexander von Humboldt-Stiftung and Japan Society for Promotion of Sciences     

Subsets with Restricted Movement

Let G be a finite permutation group on a set with no fixed points in and let m and k be integers with 0 < m < k. For a finite subset of the movement of is defined as move() = max_gG| ^g \ |. Suppose further that G is not a 2-group and that p is the least odd prime dividing |G| and move() m for all k-element subsets of . Then either || k + m or k (7m – 5) / 2, || (9m – 3)/2. Moreover when || > k + m, then move() m for every subset of .     

On the uniform (C, 1)-summability of Fourier series and integrability of series with respect to multiplicative orthonormal systems

H (G), f(g)H (G) , (, 1)- OHMC G. , OHMC, A. H. . , . , OHMC, lim supp _n=, , ,n .. . , 117 234 . . -      

Radii of convexity and close-to-convexity of certain integral representations

Strict upper bounds are determined for ¦s(z)¦, ¦Re s(z)¦, and ¦Im s(z) ¦ in the class of functions s(z)=a _nzⁿ+a_n+1zⁿ⁺¹+... (n¹) regular in ¦z¦<1 and satisfying the condition ¦u (₁) –u (₂) ¦K¦ ₁- ₂¦, where U()=Re s (eⁱ ), K>0, and ₁ and ₂ are arbitrary real numbers. These bounds are used in the determination of radii of convexity and close-to-convexity of certain integral representations.Translated from Matematicheskie Zametki, Vol. 7, No. 5, pp. 581–592, May, 1970.The author wishes to thank L. A. Aksent'ev for his guidance in this work.     

Spatial dynamics of time periodic solutions for the Ginzburg-Landau equation

We consider the dynamics of the Ginzburg-Landau equation in a small neighborhood of a known pulse solution by studying a Poincaré map,P: _{T T} , where _T is a section which is transverse to the pulse. Due to the fact that the Ginzburg-Landau equation possesses both a rotational symmetry and a spatial symmetry, we are able to conduct a detailed analytical study of this map in neighborhoods arbitrarily close to the pulse solution. Thus, we are able to complement the work of Holmes [8], who conducted an analytical study of the Poincaré map in a punctured neighborhood of the pulse. We find that the Poincaré map contains an invariant set _itT, where is not necessarily a Cantor set of points, such thatP: is homeomorphic to a shift map on (at least) two symbols. Furthermore, we find that for eachm 1 the mapP _itm possesses a fixed point. Since is not necessarily a Cantor set, this is not immediately clear. Finally, we find that when the pulse solution is broken, for eachm1 there exist parameter values such that pulses possessingm maxima appear.On leave at the University of Utah during 1993/94. Supported by the DFG, Habilitationsstipendium Ma 1587/1-1.     

Drachennetze mit geradlinigen Hauptdiagonalen

A C^s-net of curves N (s1) [3] in a regular C^s-2-surface Eⁿ (n2) is called a C^s-kite- net [4] if N and the net N₁ of its angular bisecting curves form a pair of diagonal nets [1] in such a way that each mesh of N-curves possessing two N₁-diagonals shows, with respect to one of these (calledmain diagonal), the same symmetry of angles and lengths as a rectilinear kite in E². Referring to the fact that the main diagonals of any C^s-kite-net N (s2) are geodesics in [5], we ask in this paper for all C^s-kite-nets and, more generally, C^s-D-nets [5] (s1) withstraight main diagonals. This leads, among other results, to a characterization of the skew ruled surfaces in Eⁿ (n3) with constant parameter of distribution and the constant striction /2.

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Herrn Professor Dr. WERNER BURAU zum 70. Geburtstag gewidmet     

Lacunary (0; 0, 1) interpolation on the roots of Jacobi polynomials and their derivatives,respectively. I (Existence,explicit formulae,unicity)

(0; 0, 1) , {x _k <x _k ^* <x _k+1} _k=1 ^n–1 {x _{k k=1} ⁿ }., I, , _n (x)=P _n ^{(, )} (x)–n- , =, n3 . , x ₀=+1 x _n+1= –1. II .

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To the memory of Paul Erds

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The research was supported by the Hungarian National Foundation for Scientific Research under Grant # T 914 244.     

Completion of Countably Seminormed Spaces

Let E be a vector space with a topology generated by countably many increasing seminorms p1 p2 ...and let _{p i} be the completion of E/ker (pi) with respect to pi. If pi are norms compatible in a certain sense, it is known that E is a complete space if and only if E=_{i=1 p i} . In this paper we give a similar characterization of complete spaces in a general case when pi are seminorms, without any additional assumptions. Our characterization coincides with the known one if pi are compatible norms.     

Morera type theorems on the unit disc

, . . : , , , , .     

On the speed of convergence in the central limit theorem of log-likelihood ratio processes

Let , the parameter space, be an open subset ofR ^k ,k1. For each , let the r.v.'sX _n ,n=1, 2,... be defined on the probability space (X, P ) and take values in (S,S,L) whereS is a Borel subset of a Euclidean space andL is the -field of Borel subsets ofS. ForhR _k and a sequence of p.d. normalizing matrices _n = _n ^{k × k} (₀ set _n ^* = ^* = ₀ + _n h, where ₀ is the true value of , such that ^*, . Let _n (^*, _*)( be the log-likelihood ratio of the probability measure with respect to the probability measure , whereP _n is the restriction ofP over _n = (X ₁,X ₂,...,X _n . In this paper we, under a very general dependence setup obtain a rate of convergence of the normalized log-likelihood ratio statistic to Standard Normal Variable. Two examples are taken into account.     

Lattice Polytopes Associated to Certain Demazure Modules of sl n + 1

Let w be an element of the Weyl group of sl _{n + 1}. We prove that for a certain class of elements w (which includes the longest element w₀ of the Weyl group), there exist a lattice polytope R ^l(w) , for each fundamental weight _i of sl _{n + 1}, such that for any dominant weight = _{i = 1} ⁿ a _{i i} , the number of lattice points in the Minkowski sum _w = _{i = 1} ⁿ a _{i i} ^w is equal to the dimension of the Demazure module E _w (). We also define a linear map A ^w : R ^l(w) P _Z R where P denotes the weight lattice, such that char E _w () = e e^–A(x) where the sum runs through the lattice points x of ^w .