Free Algebras over Biregularizations of Varieties

Let : F N be a type of algebras, where F is a set of fundamental operation symbols and N is the set of nonnegative integers. An identity of type is called biregular if the sets of variables in and are identical and the sets of fundamental operation symbols in and are identical. If K is a variety of type , we denote by K_b the variety of type defined by all biregular identities from Id(K). K_b will be called the biregularization of K. In this paper we give a representation of free algebras over K_b by means of free algebras over K.     

Allgemeine Zerlegungstheorie

The notion -disjoint decomposition is introduced as a common generalization of the elementary geometric decomposition of polyhedras and the disjoint decomposition of sets: Let Rø be a set, G a subgroup of the symmetric group S_R and 2^R with ø and ¦¦>1; then A, B 2^R are called -disjoint if ø. is the only set of contained in A B, and then A B A B is called the disjoint union. If fullfills the axioms (0, I, II, III) of §2, then can be provided in a natural way with certain binary relations. With respect to the decomposition equivalence can be turned in a partially ordered commutative semigroup with identity (cf. Satz §5).

---

---

Dedicated to Prof.R.Artzy on the occasion of his 80. birthday     

Radon—Nikodym theorems for multimeasures and transition multimeasures

— . , — . , .

---

---

Research supported by N. S. F. Grant DMS-8802688.     

Semimodular lattices with isomorphic graphs

Two discrete modular lattice and have isomorphic graphs if and only if is of the form A × and is of the form A × for some lattices A and and . We prove that for discrete semimodular lattices and this latter condition holds if and only if and have isomorphic graphs and the isomorphism preserves the order on all cover-preserving sublattices of which are isomorphic to the seven-element, semimodular, nonmodular lattice (see Figure 1). This answers in the affirmative a question posed by J. Jakubik.     

Аппроксимация субга рмонических функций

The following statement is proved. Letu be a subharmonic function in the region and _u the associated measure. Then there exists a functionf holomorphic in and such that if _f is the associated measure of the function in ¦f¦, then ¦u(z)–ln¦f(z)¦ A¦ln s¦+B¦ln diam¦+ s(¦lns¦+1)+C. hold at every point z for which the setsD(z, t)={w: ¦w–z¦},t(0,s) lie in and satisfy(D(z, t))t both for= _u and for= _f . In the case where is an unbounded region, In diam should be replaced by ln ¦z¦. The constants, , do not depend on andu.

---

---

. . .     

A Selection Theorem for Strongly Regular Multivalued Mappings

We prove the following generalization of a theorem of Ferry concerning selections of strongly regular multivalued maps onto the class of paracompact spaces: Let : X (Z, ) be a map of a paracompact space X into a metric space (Z, ) whose values (x) are complete subspaces of Z and absolute extensors (AE), for every x X. Suppose that can be represented as = , where : X Y is a continuous singlevalued map of X onto some finite-dimensional paracompact space Y and : Y (Z, ) is a strongly regular map. Then for every closed subset A X and every selection r : A Z of the map |A : A Z, there exists an extension : X Z of r such that is a selection of the map . We also prove a local version of this theorem.     

Asymptotic behavior of the dwell time distribution for a random walk on a positive semi-axis

Let ₁, ₂, ... be a sequence of independent identically distributed random variables with zero means. We consider the functional _n = _k=o ⁿ (S _k ) where S₁=0, S_k= _i=1 ^k _i (k1) and(x)=1 for x0,(x) = 0 for x<0. It is readily seen that _n is the time spent by the random walk S_n, n0, on the positive semi-axis after n steps. For the simplest walk the asymptotics of the distribution P (_n = k) for n and k, as well as for k = O(n) and k/n<1, was studied in [1]. In this paper we obtain the asymptotic expansions in powers of n^–1 of the probabilities P(h_n = nx) and P(nx_{1 n} nx₂) for 0<₁, x = k/n ₂<1, 0<₁x₁2₂<1.Translated from Matematicheskie Zametki, Vol. 15, No. 4, pp. 613–620, April, 1974.The author wishes to thank B. A. Rogozin for valuable discussions in the course of his work.     

Eine Verallgemeinerung des Satzes vom abgeschlossenen Wertebereich in lokalkonvexen Räumen

This paper extends Kato's proof [5] of Banach's closed range theorem to locally convex spaces. Thus we consider a locally convex space (E,) and pairs (M,N) of closed subspaces. We call such a pair -open, if and only if there exists a directed, total system of seminorms generating the topology induced by a on M+N, such that the minimal gap _p(M,N)>O for each p. Our main result is a generalisation of the closed range theorem and it consists of statements on relationships between the following properties: (a) M+N -closed, (b) M+N (E,E)-closed, (c) M+N (E,E)-closed, (d) (M,N) -open, (e) (M,N) (E,E)-open, (f) (M,N) (E,E)-open, (g) (M,N) (E,E)-open, (h) M+N=(MN), (i) M+N=(MN).By specialising the space (E,) and the subspaces M,N, our generalisation includes the closed range theorems of Dieudonné and Schwartz [4], Browder [1] and Mochizuki [12]. It is shown that these theorems not only hold for closed linear operators but even for closed linear relations. We are therefore able to obtain closed domain theorems which extend Brown's examinations in Banach-spaces [2] to locally convex spaces.

---

---

Herrn Gottfried Köthe zum 70. Geburtstag am 25.12.1975 gewidmet     

Counting Bumps

The number of modes of a density f can be estimated by counting the number of 0-downcrossings of an estimate of the derivative f, but this often results in an overestimate because random fluctuations of the estimate in the neighbourhood of points where f is nearly constant will induce spurious counts. Instead of counting the number of 0-downcrossings, we count the number of "significant" modes by counting the number of downcrossings of an interval [-, ]. We obtain consistent estimates and confidence intervals for the number of "significant" modes. By letting converge slowly to zero, we get consistent estimates of the number of modes. The same approach can be used to estimate the number of critical points of any derivative of a density function, and in particular the number of inflection points.     

Produktsätze für Verfahren zur Limitierung von Doppelfolgen

{p _mn } - ₀₀>0, (1, 1) (1.1) (1.2). {s _mn } J _p - ( bJ _p -lims _mn =), (1.3) 0<x,y<1 p _s (, )/p(x, y) x, y 1-. {r _mn } - , (1.5) 0<, <1. N _rp - , (1.6). , bJ _p -lims _mn = bJ _q -lim(N _rps) _mn =. J _p - . , .     

Cutting disjoint disks by straight lines

Fork>0 letf(k) denote the minimum integerf such that, for any family ofk pairwise disjoint congruent disks in the plane, there is a direction such that any line having direction intersects at mostf of the disks. We determine the exact asymptotic behavior off(k) by proving that there are two positive constantsd ₁,d ₂ such thatd ₁k logkf(k)d ₂k logk. This result has been motivated by problems dealing with the separation of convex sets by straight lines.The work of the first author was supported in part by the Allon Fellowship, by the Bat Sheva de Rothschild Foundation, by the Fund for Basic Research administered by the Israel Academy of Sciences, and by the Center for Absorbtion in Science. Work by the second author was supported by the Technion V. P.R. Fund, Grant No. 100-0679. The third author's work was supported by the Natural Sciences and Engineering Research Council, Canada, and the joint project Combinatorial Optimization of the Natural Science and Engineering Research Council (NSERC), Canada, and the German Research Association (Deutsche Forschungsgemeinschaft, SFB 303).     

(q, δ)-Numeration Systems with Missing Digits

We consider the (q, ) numeration system, with basis q2 and the set of digits {, +1,,q+–1} where –(q–1)0. We study properties of numbers where some digits do not occur. This is analogous to the Cantor set {0.a₁a₂a_i{0,2}}. We compute an asymptotic equivalent of the nth moment of the Cantor (q, D)-distribution which can be described as the numbers 0. w₁w₂ with w_iD{,,q+–1}, and each such letter can occur with the same probability 1/CardD. Furthermore, we consider n random strings according to the distribution and the expected minimum of them. We find a recursion which we solve asymptotically.This author was supported by the CNRS/NRF-project no 10959. Part of this work was done during the first authors visit to the John Knopfmacher Centre for Applicable Analysis and Number Theory at the University of the Witwatersrand, Johannesburg, South Africa.This author was supported by the CNRS/NRF-project no 10959.     

Fortsetzungssätze und Moduln über Semifields

. . — . — —.

---

---

Herrn Professor Dr. Frank Terpe zum 60. Geburtstag gewidmet     

Canonical and Boundary Representations on a Hyperboloid of One Sheet

For the hyperboloid of one sheet X=G/H, G=SO₀(1,2), H=SO₀(1,1), canonical representations R _,, C, =0,1, are defined as the restrictions to G of representations of the overgroup =SO₀(2,2) associated with a cone. They act on the torus containing two copies of X as open G-orbits. We study boundary representations generated by R _,. For some , they contain Jordan blocks. The decomposition of R _, into irreducible constituents includes a finite number (depending on ) of irreducible parts of the boundary representations.     

Maximal Partial Spreads and Flocks

Denote by a flock of a quadratic cone of PG(3,q) by S() the spread of PG(3,q) associated with and by l the common line of the base reguli. Suppose that there are two lines not transversal to a base regulus which share the same lines of S() Then we prove that is either linear or a Kantor-Knuth semifield flock. Using this property we can extend the result of J3 on derivable flocks proving that, if a set of q + 1 lines of S() defines a derivable net different from a base regulus-net, then is either linear or a Kantor-Knuth semifield flock. Moreover if l is not a component of the derivable net, then is linear.     

Dilation-analytic wave operators for three particles

Let H(0) be a dilation-analytic three-particle Schrödinger operator with analytic continuation H() (>0). Let a be zero or the energy of a two-particle bound state. Let- (a) be the Laplace operator representing the kinetic energy of the relative motion of fragments scattered in channel a. By recent results, wave operators W (±, a, ) with conjugates W (±, a, ) exist such that W (±, a, ) W (±, a, ) is a projection P (a, ) commuting with H () while [H ()-a]W (±, a, ) equals-W(±, a, ) (a) e²ⁱ. This paper shows that the wave operators transform dilation-analytic functions of particle coordinates into dilation-analytic functions. Specifically, if the left shoulder of the spectrum of P (a,) H () does not sweep across eigenvalues of H() when , then W(-, a, ) and W (+, a, ) are dilation analytic in [, ]. If the right shoulder does not sweep across eigenvalues, W(+, a, ) and W(-, a, ) are dilation analytic in [,]. A semisimple eigenvalue of H () embedded in the spectrum of P (a, ) H () does not prevent the wave operators from being dilation analytic in an interval [, ] with as an interior point.This work was supported in part by the National Science Foundation under grant DMS-8301096.     

Moduli of smoothness and best approximation in the spacesL p , 0<p<1

L ^p , 0<<1, .     

On the bestL p multipoint local approximation

P (f) — , f L ^p - , k . f 02k–2 P (f) 0.