A-stable spline-collocation methods of multivalue type

In this paper the general classV of spline-collocation methods presented by Mülthei is investigated. The methods ofV approximate solutions of first order initial value problems. ClassV contains as subclass the methods of so-called multivalue type, and in particular contains the generalized singly-implicit methods treated by Butcher.Any multivalue type representativeU V yields a matrix valued function corresponding toU, which characterizes the region of absolute stability ofU. If a sequence (U()) of multivalue type representatives ofV tending to some singlevalue type representative V is considered, it can easily be seen by the structure of , that the sequence of the greatest eigenvalues of the (.,) tends to the stability function corresponding to . This fact allows one to construct one-parameter families of A-stable methods of multivalue type.     

A Characterisation of the Generalized Quadrangle Q (5, q) Using Cohomology

If a GQ S of order (s, s) is contained in a GQ S of order (s, s ²) as a subquadrangle, then for each point X of S\S the set of points of S collinear with X form an ovoid of S. Thas and Payne proved that if S= (4,q),q even, and is an elliptic quadric for each XS\S,thenS (5,q). In this paper we provide a single proof for the q odd and q even cases by establishing a link between the geometry involved and the first cohomology group of a related simplicial complex.     

Noise reduction by gradual memorization

Zusammenfassung Der vorliegende Beitrag beschreibt ein einfaches, nichtlineares, abgetastetes Filter, das «Gradueller Speicher» genannt wird. Das Filter besteht aus einem Speicher, welches neue Eingangswerte nur graduell, d. h. schrittweise, speichert: Bei jedem neuen Eingangswert wird er um einen konstanten Schritt ±s in Richtung des letzteren nachgestellt. Es wird gezeigt, dass eine solche «Graduelle Speicherung» eine Verminderung von Gaussschem Rauschen um einen Faktor gestattet. (=/s, d. h. ist gleich der Streuung des Eingangsrauschens geteilt durch den Schritt s.) Die Wahrscheinlichkeitsfunktion des Speicherausganges wird berechnet und durch eine Gauss-Funktion angenähert. eine Formel wird hergeleitet, welche die richtige Wahl von s in praktischen Fällen gestattet. Im letzten Abschnitt wird das dynamische Verhalten eines graduellen Speichers betrachtet, die Reaktionszeitt _r , welche der Speicher braucht, um einer plötzlichen Änderung U_o im wahren Eingangssignal nachzufolgen, wird berechnet.t _r wird umso grösser, je grösser die Streuung des Eingangsrauschens ist.

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List of Symbols Real valueX=U ₀+U Input signal (true value + noise) - Real valueU ₀ Normalized valueu ₀=U ₀/s True input signal - Real valueU Normalized valueu=U/s Input noise - Real value s Step of memory - Real valueY=U ₀+R Output signal (true value + noise) - Real valueR Normalized valuer=R/s Output noise - Real value Normalized value Average momentary value of output signal - Real value Normalized value =/s Standard deviation of Gaussian noise at input - Normalized value Standard deviation of Gaussian noise at output - Normalized valueb Parameter of logistical distribution - Real value U ₀ Normalized valuea=U ₀/s Step of true input value - Real valuet Normalized value =t/t Time - Real value t=T Normalized value 1 Sampling interval - Real value Normalized value Average momentary velocity of output signal - Real valueU _tol Prescribed tolerance interval for output signal - Real value Prescribed probability (that the output signal remains within the tolerance interval) - Real valuew(U) Normalized valuew(u) Probability density function (input) - Real value (R) Normalized value (r) Probability (density) function (output) - Normalized valueq _r =1-P _r Cumulative distribution function - Normalized valueK=r/ Momentary value of output noise, expressed in multiples of the standard deviation - Real valueQ Normalized valueQ=/ Noise reduction factor - Real valuet _r Normalized value _r =t _r /t Time of reaction - Normalized valuez Increase in reaction time due to input noise     

Riemann sums for stochastic integrals and L p moduli of continuity

Summary The Brownian stochastic integral for a deterministic function f can be approximated by Riemann sums . Here is a partion of [0,1], and f is obtained by replacing f by its mean value in each interval of . In an earlier paper [3], the author determined which Lipschitz conditions on f imply f db f db a.s. as becomes fine. This is now done for L ^p Lipschitz conditions. The results are stated in terms of Besov spaces. Nonanticipating are considered when p = l. Further, we determine on which L ^p Lipschitz spaces f db defines a bounded linear functional.     

The transfer of the Lyapunov integral criterion to two-dimensional periodic systems of second order

Two possibilities of the transfer of the Lyapunov integral criterion [1, p. 203] (see also [2; 3, p. 177]) for the boundedness of all the solutions of the scalar equation x+(t)x=0 with nonnegative-periodic function(t) to the two-dimensional systems x=A(t)x with piecewise continuous -periodic matrix coefficients are indicated.     

Inertial Gradient-Like Dynamical System Controlled by a Stabilizing Term

Let H be a real Hilbert space and let be a function that we wish to minimize. For any potential and any control function which tends to zero as t+, we study the asymptotic behavior of the trajectories of the following dissipative system:

Popularity of Sets Represented by the Partitions of n

Let us say that a partition of the positive integer n represents a, 0 a n, if there is a submultiset of the multiset of the parts whose sum is a. Erd os and Szalay have proved that almost all partitions of n represent all integers a, 0 a n. If is a finite set of positive integers, let us denote by p~(n, ) the number of partitions of n which represent all integers a, 0 a n, a , n – a but do not represent a for a . For instance, p~(n,) is the number of partitions of n which represent all integers between 0 and n; the result of Erd os and Szalay can be reformulated as p~(n,) p(n), where p(n) is the total number of partitions of n. The aim of this paper is the study of p~(n, ): we shall compare the values of p~(n, ) for small sets and we shall give a close formula for p~(n, ) when is the set of the first k integers.     

Distance-Regular Graphs Related to the Quantum Enveloping Algebra of sl(2)

We investigate a connection between distance-regular graphs and U _q(sl(2)), the quantum universal enveloping algebra of the Lie algebra sl(2). Let be a distance-regular graph with diameter d 3 and valency k 3, and assume is not isomorphic to the d-cube. Fix a vertex x of , and let (x) denote the Terwilliger algebra of with respect to x. Fix any complex number q {0, 1, –1}. Then is generated by certain matrices satisfying the defining relations of U _q(sl(2)) if and only if is bipartite and 2-homogeneous.     

QUADRATIC ESTIMATORS OF QUADRATIC FUNCTIONS WITH PARAMETERS IN NORMAL LINEAR MODELS

ＱＵＡＤＲＡＴＩＣＥＳＴＩＭＡＴＯＲＳＯＦＱＵＡＤＲＡＴＩＣＦＵＮＣＴＩＯＮＳＷＩＴＨＰＡＲＡＭＥＴＥＲＳＩＮＮＯＲＭＡＬＬＩＮＥＡＲＭＯＤＥＬＳ￥ＷＵＱＩＧＵＡＮＧ（吴启光）（ＩｎｓｔｉｔｕｔｅｏｆＳｙｓｔｅｍｅＳｃｉｅｎｃｅ,ｔｈｅＣｈｉｎｅｓｅ...     

On the error term in the mean square formula for the Riemann zeta-function in the critical strip

For 1/2<<1 fixed, letE (T) denote the error term in the asymptotic formula for . We obtain some new bounds forE (T), and an _-result which is the analogue of the strongest _-result in the classical Dirichlet divisor problem.     

Un théorème de Vitali-Hahn-Saks pour les semimartingales

Summary Let (, , P) be a complete probability space; let _t0 be an increasing right-continuous family of -complete sub--fields of ; let be a sequence of semimartingales. Assume that for all positive t and for all bounded predictable processes H, the r.v.'s converge in probability to a limit J(t, H) when n tends to infinity. Then there exists a semimartingale X such that, for all t and H, J(t, H)= .     

Generalized Quasi-Variational Inequalities in Infinite-Dimensional Normed Spaces

In this paper, we deal with the following problem: given a real normed space E with topological dual E*, a closed convex set XE, two multifunctions :X2^X and , find such that We extend to the above problem a result established by Ricceri for the case (x)X, where in particular the multifunction is required only to satisfy the following very general assumption: each set (x) is nonempty, convex, and weakly-star compact, and for each yX–:X the set is compactly closed. Our result also gives a partial affirmative answer to a conjecture raised by Ricceri himself.     

Finely holomorphic functions and quasi-analytic classes

We study the set of functions in quasi-analytic classes and the set of finely holomorphic functions. We show that no one of these two sets is contained in the other.LetI denote the set of complex functionsf: for which there exists a quasi-analytic classC{M _n} containingf. Let denote the set of complex functionsf: for which there exist a fine domainU containing the real line and a function finely holomorphic onU satisfyingf(x)= (x) for allx . The power of unique continuation is incomparable in these two cases (I\ is non-empty, \I is non-empty).Research supported by the grant No. 201/93/2174 of Czech Grant Agency and by the grant No. 354 of Charles University.     

The type set for some measures on \mathbb{R}^{2n} with n-dimensional support

Let be realhomogeneous functions in ofdegree and let bethe Borel measure on given by

Tensor ideals in the category of tilting modules

We study the tensor category of tilting modules over a quantum groupU _q with divided powers. The setX ₊ of dominant weights is a union of closed alcoves numbered by the elementswW ^f of a certain subset of affine Weyl groupW. G. Lusztig and N. Xi defined a partition ofW ^f into canonical right cells and the right order _R on the set of cells. For a cellAW ^f we consider a full subcategory formed by direct sums of tilting modulesQ() with highest weights . We prove that is a tensor ideal in , generalizing H. Andersen's theorem about the ideal of negligible modules which in our notations is nothing else then . The proof is an application of a recent result by W. Soergel who has computed the characters of tilting modules.This material is based upon work supported by the U.S. Civilian Research and Development Foundation under Award No. RM1-265.     

Some Characterization of Locally Nonconical Convex Sets

A closed convex set Q in a local convex topological Hausdorff spaces X is called locally nonconical (LNC) if for every x, y Q there exists an open neighbourhood U of x such that . A set Q is local cylindric (LC) if for x, y Q, x y, z (x, y) there exists an open neighbourhood U of z such that U Q (equivalently: bd(Q) U) is a union of open segments parallel to [x, y]. In this paper we prove that these two notions are equivalent. The properties LNC and LC were investigated in [3], where the implication LNC LC was proved in general, while the inverse implication was proved in case of Hilbert spaces.     

Convex independent sets and 7-holes in restricted planar point sets

For a finite setA of points in the plane, letq(A) denote the ratio of the maximum distance of any pair of points ofA to the minimum distance of any pair of points ofA. Fork>0 letc (k) denote the largest integerc such that any setA ofk points in general position in the plane, satisfying for fixed , contains at leastc convex independent points. We determine the exact asymptotic behavior ofc (k), proving that there are two positive constants=(), such thatk ^1/3c (k)k ^1/3. To establish the upper bound ofc (k) we construct a set, which also solves (affirmatively) the problem of Alonet al. [1] about the existence of a setA ofk points in general position without a 7-hole (i.e., vertices of a convex 7-gon containing no other points fromA), satisfying . The construction uses Horton sets, which generalize sets without 7-holes constructed by Horton and which have some interesting properties.     

The Milnor–Totaro Theorem for Space Polygons

If denotes the curvature and the torsion of a closed, generic, and oriented polygonal space curve X in , then we show that _X (² + ²) ds = _X ds + _X | | ds > 4 if is positive. We also show that _X (² + ²) ds 2n if no four consecutive vertices lie in a plane and X has linking number n with a straight line. These extend theorems of Milnor and Totaro.     

A Group Algebra for Inductive Limit Groups. Continuity Problems of the Canonical Commutation Relations

Given an inductive limit group where each is locally compact, and a continuous two-cocycle , we construct a C^*-algebra group algebra is imbedded in its multiplier algebra , and the representations of are identified with the strong operator continuous of G. If any of these representations are faithful, the above imbedding is faithful. When G is locally compact, is precisely , the twisted group algebra of G, and for these reasons we regard in the general case as a twisted group algebra for G. Applying this construction to the CCR-algebra over an infinite dimensional symplectic space (S,\,B),we realise the regular representations as the representation space of the C^*-algebra , and show that pointwise continuous symplectic group actions on (S,\, B) produce pointwise continuous actions on , though not on the CCR-algebra. We also develop the theory to accommodate and classify 'partially regular' representations, i.e. representations which are strong operator continuous on some subgroup H of G (of suitable type) but not necessarily on G, given that such representations occur in constrained quantum systems.