Taut Distance-Regular Graphs of Odd Diameter

Let denote a bipartite distance-regular graph with diameter D 4, valency k 3, and distinct eigenvalues ₀ > ₁ > ··· > _D. Let M denote the Bose-Mesner algebra of . For 0 i D, let E _i denote the primitive idempotent of M associated with _i . We refer to E ₀ and E _D as the trivial idempotents of M. Let E, F denote primitive idempotents of M. We say the pair E, F is taut whenever (i) E, F are nontrivial, and (ii) the entry-wise product E F is a linear combination of two distinct primitive idempotents of M. We show the pair E, F is taut if and only if there exist real scalars , such that _{i + 1 i + 1} – _{i – 1 i – 1} = _i ( _{i + 1} – _{i – 1}) + _i ( _{i + 1} – _{i – 1}) + (1 i D – 1)where ₀, ₁, ..., _D and ₀, ₁, ..., _D denote the cosine sequences of E, F, respectively. We define to be taut whenever has at least one taut pair of primitive idempotents but is not 2-homogeneous in the sense of Nomura and Curtin. Assume is taut and D is odd, and assume the pair E, F is taut. We show

A Proof of the Jungnickel-Tonchev Conjecture on Quasi-Multiple Quasi-Symmetric Designs

A proof of the following conjecture of Jungnickel and Tonchev on quasi-multiple quasi-symmetric designs is given: Let D be a design whose parameter set (v,b,r,k,) equals (v,sv,sk,k, s) for some positive integer s and for some integers v,k, that satisfy (v-1) = k(k-1) (that is, these integers satisfy the parametric feasibility conditions for a symmetric (v,k,)-design). Further assume that D is a quasi-symmetric design, that is D has at most two block intersection numbers. If (k, (s-1)) = 1, then the only way D can be constructed is by taking multiple copies of a symmetric (v,k, )-design.     

Limiting behavior of certain mixtures of distributions

U — [0, 1] Y — . X=[1–U ^1/v /Y], U Y.     

Über den algorithmischen Nachweis von Ungleichungen I

We shall develop a method to prove inequalities in a unified manner. The idea is as follows: It is quite often possible to find a continuous functional : ⁿ , such that the left- and the right-hand side of a given inequality can be written in the form (u)(v) for suitable points,v=v(u). If one now constructs a map ^{n n} , which is functional increasing (i.e. for each x ⁿ (which is not a fixed point of ) the inequality (x)<((x)) should hold) one specially gets the chain (u)( u))( ²(u))... ⁿ (u)). Under quite general conditions one finds that the sequence { ⁿ (u)} _n converges tov=v(u). As a consequence one obtains the inequality (u)(v).     

Generalization of some results concerning Walsh series and the dyadic derivative

, , , . , . , , x(0,1),x2^–j ,j=1,2,..., 2 ⁿ . , ka _k 0 k k. , (0, 1) , , , , . , .     

On cubature formulas on the basis of lattices

Q (.. , L). Q . P(S_r(2)) — 2 (S _r(2) (r — ). , M(P(S _r(m=sup{t(·)t(·)₁:t P(S _r(2)),t 0}. , /4+(1)M(P(S _r(2)))/r ²15/17+(1)(r+). (Q), Q L.     

On a Singular Semilinear Elliptic Boundary Value Problem and the Boundary Harnack Principle

On a bounded C ²-domain we consider the singular boundary-value problem 1/2u=f(u) in D, u _D =, where d3, f:(0,)(0,) is a locally Hölder continuous function such that f(u) as u0 at the rate u ^–, for some (0,1), and is a non-negative continuous function satisfying certain growth assumptions. We show existence of solutions bounded below by a positive harmonic function, which are smooth in D and continuous in . Such solutions are shown to satisfy a boundary Harnack principle.     

On the sums of a family of Kapteyn series

The series ₁ n ^r–1 J _n (n)J _n (n) (r 0, 0 < 1) arise in studying the emission and absorption of radiation by a charged particle on a Kepler orbit. For the first few even,r, the sums are obtained in closed form, and for oddr they are given in terms of a certain definite integral. The integral is expressed as a power series in for ||<1, and, near =1, an asymptotic expansion in powers of (1–²)^1/2 may be obtained.

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Résumé La série ₁ n ^r–1 J _n (n)J _n (n) (r 0, 0 < 1) se trouve par l'émission et l'absorption du rayonnement d'une particule chargée sur l'orbite Keplerien. Pour les plus petites valuers paires der, les sommes s'obtienment en forme compacte, et pour les valuers impaires, elles se déterminent d'après une intégrale definie. Pour ||<1, cette intégrale se développe dans une série de puissances de , et dans le voisinage de =1, on obtient une série asymptotique et puissances de (1–²)^1/2.

     

Inequalities in Products of Minors of Totally Nonnegative Matrices

Let _I,I be the minor of a matrix which corresponds to row set I and column set I. We give a characterization of the inequalities of the form _{I,I K,K J,J L,L} which hold for all totally nonnegative matrices. This generalizes a recent result of Fallat, Gekhtman, and Johnson.     

Infinite families of non-embeddable quasi-residual hadamard designs

LetD be a quasi-residual Hadamard design with =(2m + 1)2ⁿ–1, wherem andn are positive integers. IfD contains a pair of blocks intersecting in m2ⁿ⁺¹ points together with a third block intersecting each of the first two blocks in (m + 1)2ⁿ points thenD is non-embeddable. Using this result together with a recursive construction for quasi-residual Hadamard designs the existence of a previously unknown infinite family of non-embeddable quasi-residual Hadamard designs with =5(2ⁿ)–1 is established. An additional infinite family of non-embeddable quasi-residual Hadamard designs is given. This family has = 2ⁿ–1 with each design in the family having a pair of blocks meeting in (3 + 3)/4 points and a third block meeting each of the first two blocks in (5 + 5)/8 points.     

Majorization versus power majorization

, . , [1]. 42.     

On unconditionally convergent basis expansions in the space of continuous functions

[0,1], - H .

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This paper was written during the author's scholarship at the State University of Odessa in the USSR.     

Maximal inequalities,convexity inequality,and their duality. II

, , . . . [1], , . , , ., , L logL. , , . . . . [5]. , .     

Cohomology of solvable lie algebras and solvmanifolds

The cohomology H^* (G/,) of the de Rham complex *(G/) of a compact solvmanifold G/ with deformed differential d = d + , where is a closed 1 -form, is studied. Such cohomologies naturally arise in Morse-Novikov theory. It is shown that, for any completely solvable Lie group G containing a cocompact lattice G, the cohomology H^*(G/, ) is isomorphic to the cohomology H^*( ) of the tangent Lie algebra of the group G with coefficients in the one-dimensional representation : defined by () = (). Moreover, the cohomology H ^*(G/,) is nontrivial if and only if -[] belongs to a finite subset of H ¹(G/,) defined in terms of the Lie algebra .Translated from Matematicheskie Zametki, vol. 77, no. 1, 2005, pp. 67–79.Original Russian Text Copyright © 2005 by D. V. Millionshchikov.This revised version was published online in April 2005 with a corrected issue number.     

Exact estimates for partially monotone approximation

f(x) — , - [–1,1], (f, ) — , as— f, . . (- ) (x _i,x _{i+ 1}) (i=0, 1, ...,s–1; =–1,x _s,=1), f(x) . , n=0,1,... _n() , [– 1,1] signf(x) sign _n(x) 0, ¦f(x)– _n(x)¦ C(s) (f, 1/n+1, C(s) s. , - , « » .     

Surfaces and the second homology of a group

LetG be a group andK(G, 1) an Eilenberg—MacLane space, i.e. ₁(K(G,1))G, _i (K(G,1))=0,i1. We give a purely algebraic proof that the second homology groupH ₂(G)=H ₂(G,)H ₂(K(G,1)) is isomorphic to the group of stable equivalence classes of continuous mapsFK(G,1) inducing surjections on fundamental groups (resp. surjections, whereF{F _g=closed orientable surface of genusg,g}. As a corollary we obtain an algebraic proof of the well-known isomorphismH ₂(G)₂(K(G,1)) (2-dimensional bordism group).     

On the angular derivative and univalence

f . , , — , A f f(). , , f() 0 . , , ,A , f . , f() - f() . , , . (1976) ( ¦f(z)¦<1) . . (1969) ( ).     

Isomorphismen von rechtseiträumen

Spaces called rectangular spaces were introduced in [5] as incidence spaces (P,G) whose set of linesG is equipped with an equivalence relation and whose set of point pairs P² is equipped with a congruence relation , such that a number of compatibility conditions are satisfied. In this paper we consider isomorphisms, automorphisms, and motions on the rectangular spaces treated in [5]. By an isomorphism of two rectangular spaces (P,G, , ) and (P,G, , ) we mean a bijection of the point setP onto P which maps parallel lines onto parallel lines and congruent points onto congruent points. In the following, we consider only rectangular spaces of characteristic 2 or of dimension two. According to [5] these spaces can be embedded into euclidean spaces. In case (P,G, , ) is a finite dimensional rectangular space, then every congruence preserving bijection ofP onto P is in fact an isomorphism from (P,G, , ) onto (P,G, , ) (see (2.4)). We then concern ourselves with the extension of isomorphisms. Our most important result is the theorem which states that any isomorphism of two rectangular spaces can be uniquely extended to an isomorphism of the associated euclidean spaces (see (3.2)). As a consequence the automorphisms of a rectangular space (P,G, , ) are precisely the restrictions (onP) of the automorphisms of the associated euclidean space which fixP as a whole (see (3.3)). Finally we consider the motions of a rectangular space (P,G, , ). By a motion of(P. G,, ) we mean a bijection ofP which maps lines onto lines, preserves parallelism and satisfies the condition((x), (y)) (x,y) for allx, y P. We show that every motion of a rectangular space can be extended to a motion of the associated euclidean space (see (4.2)). Thus the motions of a rectangular space (P,G, , ) are seen to be the restrictions of the motions of the associated euclidean space which mapP into itself (see (4.3)). This yields an explicit representation of the motions of any rectangular plane (see (4.4)).

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Herrn Professor Burau zum 85. Geburtstag gewidmet     

On an inequality of Longnecker and Serfling

— [2], [5], [7], [8], , , . , . , .     

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.