A Higher Level Bailey Lemma: Proof and Application

In a recent letter, new representations were proposed for the pair of sequences (,), as defined formally by Bailey in his famous lemma. Here we extend and prove this result, providing pairs (,) labelled by the Lie algebra AN – 1, two nonnegative integers and k and a partition , whose parts do not exceed N – 1. Our results give rise to what we call a higher level Bailey lemma. As an application it is shown how this lemma can be applied to yield general q-series identities, which generalize some well-known results of Andrews and Bressoud.     

On the solution of a generalized Schrödinger-type equation

Lu(t)+(u,F)g(t)=f(t), tS. , ( F, g). .

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The authors wish to thank Professor Yu. A. Rozanov for his help and discussions.     

Склярные дифференциалвные инварианты третвего порядка риамнового простанства трех измерений

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A conditional dilatation equation

Summary The following theorem holds true. Theorem. Let X be a normed real vector space of dimension 3 and let k > 0 be a fixed real number. Suppose that f: X X and g: X × X are functions satisfying x – y = k f(x) – f(y) = g(x, y)(x – y) for all x, y X. Then there exist elements and t X such that f(x) = x + t for all x X and such that g(x, y) = for all x, y X with x – y = k.     

Mittelwert der Eulerschen φ-Funktion und des Quadrates der Dirichletschen Teilerfunktion in algebraischen Zahlkörpern

LetK be an algebraic number field, and for every integer K let () andd(), respectively, denote the number of relatively prime residue classes and the number of divisors of the principal ideal (). Asymptotic equalities are proved for the sums () and d ²(), where runs through certain finite sets of integers ofK.     

Number of quasimodes of “bouncing-ball” type

A new two-scale expansion is proposed for eigenfunctions of bouncing-ball type and the corresponding eigenvalues of the Laplace operator with the Dirichlet condition in a domain of the plane. The eigenfunctions are concentrated in a neighborhood of a stable diameter of the domain and are numbered with two indices (p, q) where p is the number of longitudinal nodes and q the number of nodes in a direction orthogonal to the diameter. The validity of the asymptotic expansions is guaranteed for 0 q const p^–1 and >0 as p +.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 117, pp. 172–182, 1981.     

Boolean Term Orders and the Root System Bn

A Boolean term order is a total order on subsets of [n] ={1,..., n} such that for all [n], , and for all with ( ) = . Boolean term orders arise in several different areas of mathematics, including Gröbner basis theory for the exterior algebra, and comparative probability.The main result of this paper is that Boolean term orders correspond to one-element extensions of the oriented matroid M(B_n), where B_n is the root system {e_i : 1 i n} {e_i ± e_j : 1 i < j n}. This establishes Boolean term orders in the framework of the Baues problem, in the sense of (Reiner, 1998). We also define a notion of coherence for a Boolean term order, and a flip relation between different term orders. Other results include examples of noncoherent term orders, including an example exhibiting flip deficiency, and enumeration of Boolean term orders for small values of n.     

Positive Solutions for a Singular Nonlinear Problem on a Bounded Domain in R 2

For a bounded regular Jordan domain in R ², we introduce and study a new class of functions K() related on its Green function G. We exploit the properties of this class to prove the existence and the uniqueness of a positive solution for the singular nonlinear elliptic equation u+(x,u)=0, in D(), with u=0 on and uC(), where is a nonnegative Borel measurable function in ×(0,) that belongs to a convex cone which contains, in particular, all functions (x,t)=q(x)t ^–,>0 with nonnegative functions qK(). Some estimates on the solution are also given.     

Complex whitham deformations in problems with “integrable instability”

The focusing nonlinear Schrödinger equation with finite-density boundary conditions as |x| is considered. The asymptotic behavior of the solution ast is investigated by means of the complex theory of deformations of Whitham.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 104, No. 3, pp. 393–419, September, 1995.     

Testing for Unit Roots in a Nearly Nonstationary Spatial Autoregressive Process

The limiting distribution of the normalized periodogram ordinate is used to test for unit roots in the first-order autoregressive model _st= _s-1,t+_s,t-1- _s-1,t-1+_st. Moreover, for the sequence _n = e ^c/n , _n = e ^d/n of local Pitman-type alternatives, the limiting distribution of the normalized periodogram ordinate is shown to be a linear combination of two independent chi-square random variables whose coefficients depend on c and d. This result is used to tabulate the asymptotic power of a test for various values of c and d. A comparison is made between the periodogram test and a spatial domain test.     

Distribution of the supremum of sums of independent variables with negative drift

Let {_n} be a sequence of identically distributed independent random variables,M₁=<0,M ₁ ² <;S ₀=0,S _n =₁+₂,+...+ _{n, n}1;¯ S=sup {S _n n=0.} The asymptotic behavior ofP(¯ St) as t is studied. If _t ^P (₁x dx=0((t)), thenP(¯ St)– 1/¦¦ _t P (₁x dx=0((t)) (t) is a positive function, having regular behavior at infinity.Translated from Matematicheskie Zametki, Vol. 22, No. 5, pp. 763–770, November, 1977.The author thanks B. A. Rogozin for the formulation of the problem and valuable remarks.     

Stochastic models of homogeneity

The probabilityp()=P{–<) is used as a characteristic of homogeneity of the system (, ) of independent identically distributed random variables. We consider some estimation methods of this characteristic and a method of classification of observations minimizing the between-class dissimilarity measure.Translated from Statisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 120–123, 1986.     

On the Classification of Flag-Transitive c.c*- Geometries

Let ( G) be a flag-transitive c.c*-geometry whose point-stabilizer is not an affine group. We list all known examples and show that, if (, G) is a minimal unknown example, then G is an almost simple group and the commutator subgroup G is a simple group of Lie type.     

Tangential Limits of Potentials on Homogeneous Trees

Let T be a homogeneous tree of homogeneity q+1. Let denote the boundary of T, consisting of all infinite geodesics b=[b ₀,b ₁,b ₂,] beginning at the root, 0. For each b, 1, and a0 we define the approach region _,a (b) to be the set of all vertices t such that, for some j, t is a descendant of b _j and the geodesic distance of t to b _j is at most (–1)j+a. If >1, we view these as tangential approach regions to b with degree of tangency . We consider potentials Gf on T for which the Riesz mass f satisfies the growth condition _T f ^p (t)q ^–|t|<, where p>1 and 0<<1, or p=1 and 0<1. For 11/, we show that Gf(s) has limit zero as s approaches a boundary point b within _,a (b) except for a subset E of of -dimensional Hausdorff measure 0, where H (E)=sup_>0inf _i q ^{–|t i|}:E a subset of the boundary points passing through t _i for some i,|t _i |>log _q (1/).     

Net subgroups of Chevalley groups. II. Gauss decomposition

This paper is a continuation of RZhMat 1980, 5A439, where there was introduced the subgroup () of the Chevalley group G(,R) of type over a commutative ring R that corresponds to a net , i.e., to a set =(),, of ideals of R such that ₊ whenever ,,+ . It is proved that if the ring R is semilocal, then () coincides with the group ₀ considered earlier in RZhMat 1976, 10A151; 1977, 10A301; 1978, 6A476. For this purpose there is constructed a decomposition of () into a product of unipotent subgroups and a torus. Analogous results are obtained for sub-radical nets over an arbitrary commutative ring.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 114, pp. 62–76, 1982.In conclusion, the authors would like to thank Z. I. Borevich for his interest in this paper.     

A quantitative refinement of Rado's theorem

The fundamental result of the paper is the following. Theorem: Let be a k-quasiconformal Jordan curve and let be another Jordan curve (not necessarily quasiconformal). Assume that f maps conformallyext ontoext , f()=, f()>0. We assume that there exists a homeomorphism between and such that Then there exist numbers =(k)>0 and A=A(k), such that f(())– A, .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 157, pp. 103–112, 1987.     

Small Ball Probabilities Around Random Centers of Gaussian Measures and Applications to Quantization

Let be a centered Gaussian measure on a separable Hilbert space (E, ). We are concerned with the logarithmic small ball probabilities around a -distributed center X. It turns out that the asymptotic behavior of –log (B(X,)) is a.s. equivalent to that of a deterministic function _R (). These new insights will be used to derive the precise asymptotics of a random quantization problem which was introduced in a former article by Dereich, Fehringer, Matoussi, and Scheutzow.⁽⁸⁾     

Estimating Equations with Nuisance Parameters: Theory and Applications

In a variety of statistical problems the estimate _n of a parameter is defined as the root of a generalized estimating equation G_n(_nn)=0 where _n is an estimate of a nuisance parameter . We give sufficient conditions for the asymptotic normality of #x0398;_n defined in this way and derive their asymptotic distribution. A circumstance under which the asymptotic distribution of #x0398;_n will not be influenced by that of _n) is noted. As an example, we consider a covariance structure analysis in which both the population mean and the population fourth-order moment are nuisance parameters. Applications to pseudo maximum likelihood, generalized least squares with estimated weights, and M-estimation with an estimated scale parameter are discussed briefly.     

A parallel projection method for overdetermined nonlinear systems of equations

We consider overdetermined nonlinear systems of equationsF(x)=0, whereF: ^{n m} ,mn. For this type of systems we define weighted least square distance (WLSD) solutions, which represent an alternative to classical least squares solutions and to other solutions based on residual normas. We introduce a generalization of the classical method of Cimmino for linear systems and we prove local convergence results. We introduce a practical strategy for improving the global convergence properties of the method. Finally, numerical experiments are presented.Work supported by FAPESP (Grant 90/3724/6), FINEP, CNPq and FAEP-UNICAMP.     

On generalized gamma functions related to the Laurent coefficients of the Riemann zeta function

Summary We study a class of generalized gamma functions _k (z) which relate to the generalized Euler constants _k (basically the Laurent coefficients of(s)) as (z) does to the Euler constant. A new series expansion for _k is derived, and the constant term in the asymptotic expansion for log _k (z) is studied in detail. These and related constants are numerically computed for 1 k 15.