Extremal Eigenvalues of the Laplacian in a Conformal Class of Metrics: The `Conformal Spectrum'

Let M be a compact connected manifold of dimension n endowed witha conformal class C ofRiemannian metrics of volume one. For any integer k 0, we consider the conformal invariant _k ^c (C) defined as the supremum of the k-th eigenvalue _k (g) of the Laplace–Beltrami operator _g , where g runs over C.First, we give a sharp universal lower bound for _k ^c (C) extending to all k a result obtained by Friedlander andNadirashvili for k = 1. Then, we show that the sequence \{ _k ^c (C)\}, that we call `conformal spectrum',is strictly increasing and satisfies, k 0, _k+1 ^c (C) ^n/2– _k ^c (C) ^n/2 n ^n/2 _n , where _n is the volume of the n-dimensionalstandard sphere.When M is an orientable surface of genus , we also considerthe supremum _k ^top()of _k (g) over theset of all the area one Riemannian metrics on M, and study thebehavior of _k ^top() in terms of .     

On a new class of multiplicative pseudo-random number generators

In the computing literature, there are few detailed analytical studies of the global statistical characteristics of a class of multiplicative pseudo-random number generators.We comment briefly on normal numbers and study analytically the approximately uniform discrete distribution or (j,)-normality in the sense of Besicovitch for complete periods of fractional parts {x _{0 1} ⁱ /p} on [0, 1] fori=0, 1,..., (p–1)p^–1–1, i.e. in current terminology, generators given byx _{n+1 1} x _n mod p wheren=0, 1,..., (p–1)p ^–1–1,p is any odd prime, (x ₀,p)=1, ₁ is a primitive root modp ², and 1 is any positive integer.We derive the expectationsE(X, ),E(X ², ),E(X _nX_n+k); the varianceV(X, ), and the serial correlation coefficient k. By means of Dedekind sums and some results of H. Rademacher, we investigate the asymptotic properties of k for various lagsk and integers 1 and give numerical illustrations. For the frequently used case =1, we find comparable results to estimates of Coveyou and Jansson as well as a mathematical demonstration of a so-called rule of thumb related to the choice of ₁ for small k.Due to the number of parameters in this class of generators, it may be possible to obtain increased control over the statistical behavior of these pseudo-random sequences both analytically as well as computationally.     

Elementarteiler von Inzidenzmatrizen symmetrischer Blockpläne

By a study of the integral code generated by the rows of the incidence matrix and its extention the following results are obtained: Let d ₁,...,d _V(d ₁|d ₂,d ₂|d ₃...) be the elementary divisors of the incidence matrix of a symmetric (v,n+, ) design. Then d _v=(n+)n/g.c.d. (n, ). Moreover, if p is a prime such that p|n, p and if x _p denotes the p-part of x, then (d _id_v+2–i) _p =n _p for 2iv. For projective planes it can be shown that d ₁=···=d _3n–2=1, hence and . The paper also contains some results about elementary divisors of incidence matrices G satisfying the condition G ^tG=nI+ J.

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Elementarteiler von Inzidenzmatrizen symmetrischer Blockpläne

     

On the convergence of Hunter's quadrature rule for Cauchy principal value integrals

Hunter's (n+1)-point quadrature rule for the approximate evaluation of the Cauchy principal value integralf ¹ _–1 (w(x)f(x)/(x – ))dx, –1<<1, is based on approximatingf by the polynomial which interpolatesf at the point and then zeros of the orthogonal polynomialp _n generated by the weight functionw. Sufficient conditions are given to ensure the convergence of a suitably chosen subsequence of the quadrature rules to the integral, whenf is Hölder continuous on [–1,1].     

An extreme value theory for long head runs

Summary For an infinite sequence of independent coin tosses withP(Heads)=p(0,1), the longest run of consecutive heads in the firstn tosses is a natural object of study. We show that the probabilistic behavior of the length of the longest pure head run is closely approximated by that of the greatest integer function of the maximum ofn(1-p) i.i.d. exponential random variables. These results are extended to the case of the longest head run interrupted byk tails. The mean length of this run is shown to be log(n)+klog(n)+(k+1)log(1–p)–log(k!)+k+/–1/2+ r₁(n)+ o(1) where log=log_1/p , =0.577 ... is the Euler-Mascheroni constant, =ln(1/p), andr ₁(n) is small. The variance is ₂/6²+1/12 +r ₂(n)+ o(1), wherer ₂(n) is again small. Upper and lower class results for these run lengths are also obtained and extensions discussed.This work was supported by a grant from the System Development Foundation     

The Müntz-Szasz problem

A condition is obtained on the placement of point _n (in some sense, the final point) with which completeness of the system of functionsexp (– _n x), Re_n>0, in spaces L^p, 1p<2. is equivalent to divergence of the series re_n(1+¦_n¦²)^–1.Translated from Matematicheskie Zametki, Vol. 23, No. 1, pp. 91–103, January, 1978.Deceased.     

The Lê varieties,I

Summary For a complex polynomial,f:( ⁿ⁺¹ ,0) (, 0), with a singular set of complex, dimensions at the origin, we define a sequence of varieties—the Lê varieties, _f ^(k) , off at 0. The multiplicities of these varieties, _f ^(k) , generalize the Milnor number for an isolated singularity. In particular, we show that ifsn-2, the Milnor, fibre off is obtained fromB ²ⁿ by successively attaching _f ^{(n – k)} k-handles, wheren-skn Ifs=n-1, the Milnor fibre off is obtained from a2n-manifold with the homotopy type of a bouquet of _f ^{(n – 1)} circles by successively attaching _f ^{(n – k)} k-handles, where 2kn.The author is a National Science Foundation, Postdoctoral Research Fellow supported by grant # DMS-8807216     

Divisible designs and groups

We study (s, k, ₁, ₂)-translation divisible designs with ₁0 in the singular and semi-regular case. Precisely, we describe singular (s, k, ₁, ₂)-TDD's by quasi-partitions of suitable quotient groups or subgroups of their translation groups. For semi-regular (s, k, ₁, ₂)-TDD's (and, more general, for the case ₂>₁) we prove that their translation groups are either Frobenius groups or p-groups of exponent p. Some examples are given for the singular, semi-regular and regular case.     

On the immersion of metrics close to immersible metrics

For a C^r,-immersion z:X E, r 2, 0 < < 1, of an n-dimensional (n 1) simply-connected C^r+2,-manifold X into Euclidean space E, the metric I(z) induced by z has a neighborhood in C^r,-topology in which every metric from a given subbundle of metrics is C^r,-immersible into E. In particular, it is proved that metric ds ₀ ² of the Riemannian product of p spheres of dimensions ₁, , _p 2 has a neighborhood in C^2,-topology from which any conformally equivalent metric to ds ₀ ² , is immersible into E with dimE = ₁ + + _p + p. The proofs are based on the investigation of a varied system of Gauss—Codazzi—Ricci equations for an infinitely small deformation of surface z(X) in E with a prescribed variation of the metric.Translated from Ukrainskii Geometricheskii Sbornik, No. 35, pp. 49–67, 1992.     

Quasi-symmetric 3-designs with triangle-free graph

The following result is proved: Let D be a quasi-symmetric 3-design with intersection numbers x, y(0x<y<k). D has no three distinct blocks such that any two of them intersect in x points if and only if D is a Hadamard 3-design, or D has a parameter set (v, k, ) where v=(+2)(²+4+2)+1, k=²+3+2 and =1,2,..., or D is a complement of one of these designs.     

On polynomial EkP matrices

In this paper the relation betweenEP--matrices andE ^k P--matrices over an arbitrary filedF is studied. Further, conditions for the product ofE ^k P--matrices to be anE ^k P--matrix and for the reverse order law to hold for the polynomial Moore-Penrose inverse of the product ofE ^k P--matrices are determined     

The existence of some resolvable block designs with divisibility into groups

This paper proves the existence of resolvable block designs with divisibility into groups GD(v; k, m; ₁, ₂) without repeated blocks and with arbitrary parameters such that ₁ = k, (v–1)/(k–1) ₂ v^k–2 (and also ₁ k/2, (v–1)/(2(k–1)) ₂ v^k–2 in case k is even) k 4 andp=1 (mod k–1), k < p for each prime divisor p of number v. As a corollary, the existence of a resolvable BIB-design (v, k, ) without repeated blocks is deduced with X = k (and also with = k/2 in case of even k) k , where a is a natural number if k is a prime power and=1 if k is a composite number.Translated from Matematicheskie Zametki, Vol. 19, No. 4, pp. 623–634, April, 1976.     

Problem of the maximum of the product of the conformal radii of nonoverlapping domains

Let c_k, k=1,...,4, be arbitrary distinct points of . LetD be the family of all systems of simply connected domains in. By R(D_k, c_k) we denote the conformal radius of the domain D_k, relative to the point c_k. We prove that in the familyD one has the sharp inequality, (1) where a=(+i)/(–1), being the cross-ratio of the points c₁,c₂, c₃, c₄: E(–1, 1,a) is the continuum of least capacity containing the points –1,1,a. An explicit expression for capE(–1,1,a.) in terms of elliptic Jacobi functions has been obtained earlier by the author [Tr. Mat. Inst. Akad. Nauk SSSR,94, 47–65, 1968]. On the basis of the well-known properties of continua of least capacity, one shows that the largest value of the right-hand side of (1) is attained for a=± i3 and it is equal to 4^–8/3·3². One gives all the configurations for which equality prevails in the obtained estimates.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 100, pp. 131–145, 1980.     

A technique for constructing divisible difference sets

An (m, n, k, ₁,₂) divisible difference set in a groupG of ordermn relative to a subgroupN of ordern is ak-subsetD ofG such that the list {xy^–1:x, y D} contains exactly ₁ copies of each nonidentity element ofN and exactly ₂ copies of each element ofG N. It is called semi-regular ifk > ₁ and k²=mn₂. We develop a method for constructing a divisible difference set as a product of a difference set and a relative difference set or a difference set and a subset ofG which we call a relative divisible difference set. The method results in several parametrically new families of semi-regular divisible difference sets.     

Necessary conditions for the existence of divisible designs

BOSE and CONNOR [2] proved that a symmetric regular divisible design with w classes of sizes g and joining numbers ₁ and ₂ must satisfy for every prime p the arithmetic condition (d₁, (–1)^sw)_p(d₂,(–l)^tg^w)_p=1, where d₁=k²–v₂, d₂= k–₁ s=(w-1)(w-2)/2, t=(v-w)(v-w-1)/2 and (*,*) is the Hilbert symbol. We show that if in addition _{1 2} and the design is fully symmetric divisible then (d₁, (–1)^s w)_p=(d₂, (–1)^tg^w)=1. Our assumption is by a result of CONNOR [5] fulfilled, if d₁ and ₁–₂ are relatively prime. Thus, we can exclude parameters not accessible to the Bose-Connor-Theorem. Our result can be derived from a theorem of RAGHAVARAO [9], and we give the precise assumptions of this theorem. We also discuss arithmetic restrictions for divisible designs which satisfy diverse other rules for the intersection numbers and generalize a result of DEMBOWSKI [6; 2.1.11].Dedicated to Professor Benz on occasion of his sixtieth birthday     

Modular andp-adic cyclic codes

This paper presents some basic theorems giving the structure of cyclic codes of lengthn over the ring of integers modulop ^a and over thep-adic numbers, wherep is a prime not dividingn. An especially interesting example is the 2-adic cyclic code of length 7 with generator polynomialX ³ +X ² +(–1)X–1, where satisfies ² - + 2 = 0. This is the 2-adic generalization of both the binary Hamming code and the quaternary octacode (the latter being equivalent to the Nordstrom-Robinson code). Other examples include the 2-adic Golay code of length 24 and the 3-adic Golay code of length 12.     

Skew-symmetric association schemes with two classes and strongly regular graphs of type L 2n−1(4n−1)

A construction of a pair of strongly regular graphs ⁿ and ⁿ of type L _2n–1(4n–1) from a pair of skew-symmetric association schemes W, W of order 4n–1 is presented. Examples of graphs with the same parameters as ⁿ and ⁿ, i.e., of type L _2n–1(4n–1), were known only if 4n–1=p ³, where p is a prime. The first new graph appearing in the series has parameters (v, k, )=(225, 98, 45). A 4-vertex condition for relations of a skew-symmetric association scheme (very similar to one for the strongly regular graphs) is introduced and is proved to hold in any case. This has allowed us to check the 4-vertex condition for ⁿ and ⁿ, thus to prove that ⁿ and ⁿ are not rank three graphs if n>2.     

Building Symmetric Designs With Building Sets

We introduce a uniform technique for constructing a family of symmetric designs with parameters (v(q ^m+1-1)/(q-1), kq ^m ,q ^m), where m is any positive integer, (v, k, ) are parameters of an abelian difference set, and q = k ²/(k - ) is a prime power. We utilize the Davis and Jedwab approach to constructing difference sets to show that our construction works whenever (v, k, ) are parameters of a McFarland difference set or its complement, a Spence difference set or its complement, a Davis–Jedwab difference set or its complement, or a Hadamard difference set of order 9 · 4 ^d , thus obtaining seven infinite families of symmetric designs.     

On the integersn for which Ω(n)=k (II)

Let (n) be the number of prime divisors ofn, counted with multiplicity. We denote byS(x, k) the set of thenx for which (n)=k, and byV _p(n) the exponent of the primep in the factorization ofn. In a previous paper we proved a result which implies that, ify=x/2 ^k tends to infinity withk>2loglogx where >1, then the distribution of the numbers on the setS(x, k) converges to the normal distribution of Gauss. Here, besides a slight improvement of that result, we give, for the moment of orderq of the above mentioned distribution, a formula which holds uniformly for 2loglogxklog (x/3)/log2 where 1<<3/2.