Distribution of norms of primitive hyperbolic classes

We prove that the asymptotic formula for the number of primitive hyperbolic classes includes no summands corresponding to the discriminants d of indefinite binary quadratic forms such that h(d) < cd/ log² d for a certain constant c > 0. A similar result is obtained for prime discriminants. Bibliography: 7 titles.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 237, 1997, pp. 40–45.     

Upper Bounds and Asymptotics for the q-Binomial Coefficients

In this article, we a derive an upper bound and an asymptotic formula for the q -binomial, or Gaussian, coefficients. The q -binomial coefficients, that are defined by the expression are a generalization of the binomial coefficients, to which they reduce as q tends toward 1. In this article, we give an expression that captures the asymptotic behavior of these coefficients using the saddle point method and compare it with an upper bound for them that we derive using elementary means. We then consider as a case study the case q =1+ z / m , z <0, that was actually encountered by the authors before in an application stemming from probability and complexity theory. We show that, in this case, the asymptotic expression and the expression for the upper bound differ only in a polynomial factor; whereas, the exponential factors are the same for both expressions. In addition, we present some numerical calculations using MAPLE (a computer program for performing symbolic and numerical computations), that show that both expressions are close to the actual value of the coefficients, even for moderate values of m .     

Number of points of a translated lattice in a domain on a multidimensional ellipsoid

We prove an asymptotic formula for the number of representations of the number m by n-ary quadratic form f which lie in a given residue class (mod a) and in a given domain on the surface . The parameters of the problem are unconstrained. For n=3 the asymptotic formula is conditional (or incomplete).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 151, pp. 176–183, 1986.     

Grafting hyperbolic metrics and Eisenstein series

The family hyperbolic metric for the plumbing variety {zw = t} and the non holomorphic Eisenstein series are combined to provide an explicit expansion for the hyperbolic metrics for degenerating families of Riemann surfaces. Applications include an asymptotic expansion for the Weil–Petersson metric and a local form of symplectic reduction.     

A note on the zeros of the zeta-function of Riemann

Archiv der Mathematik - We consider the qth root number function for the symmetric group. Our aim is to develop an asymptotic formula for the multiplicities of the qth root number function as q...     

Ensembles de classe (0, 1, q,q + 1) de PG (d,q)

We determine all sets Q of points of any finite dimensional protective space P such that each line intersecting Q in more than one point, either is contained in Q or contains exactly one point not on Q. If P is a finite protective space of order q, these sets are the so called sets of class (0, 1, q, q + 1).     

Counting domains in {p,q} tessellations

For any given regular {p,q} tessellation in the hyperbolic plane, we compute the number of vertices and tiles to be found as we distance from a given point, enabling a complete characterization of the asymptotic behavior.     

A lattice point problem on the regular tree

Huber (1956) [8] considered the following problem on the hyperbolic plane H. Consider a strictly hyperbolic subgroup of automorphisms on H with compact quotient, and choose a conjugacy class in this group. Count the number of vertices inside an increasing ball, which are images of a fixed point x∈H under automorphisms in the chosen conjugacy class, and describe the asymptotic behaviour of this number as the size of the ball goes to infinity. We use a well-known analogy between the hyperbolic plane and the regular tree to solve this problem on the regular tree.     

Total curvature of complete surfaces in hyperbolic space

We prove a Gauss-Bonnet formula for the extrinsic curvature of complete surfaces in hyperbolic space under some assumptions on the asymptotic behavior. The result is given in terms of the measure of geodesics intersecting the surface non-trivially, and of a conformal invariant of the curve at infinity.     

On Asymptotics for the Signless Noncentral q-Stirling Numbers of the First Kind

In this paper, we derive asymptotic formulas for the signless noncentral q -Stirling numbers of the first kind and for the corresponding series. The signless noncentral q -Stirling numbers of the first kind appear as coefficients of a polynomial of q -number [ t ] _q , expressing the noncentral ascending q -factorial of t of order m and noncentrality parameter k . In this paper, we have two main purposes. The first is to give an expression by which we obtain the asymptotic behavior of these coefficients, using the saddle point method . The second main purpose is to derive an asymptotic expression for the signless noncentral q -Stirling of the first kind series by using the singularity analysis method. We then apply our first formula to provide asymptotic expressions for probability functions of the number of successes in m trials and of the number of trials until the occurrence of the n th success in sequences of Bernoulli trials with varying success probability which are both written in terms of the signless noncentral q -Stirling numbers of the first kind. In addition, we present some numerical calculations using the computer program MAPLE indicating that our expressions are close to the actual values of the signless noncentral q -Stirling numbers of the first kind and of the corresponding series even for moderate values of m .     

关于表自然数因子个数的函数

设ｇ（Ｎ）是满足ｇ（０）＝０的任一实值数论函数．当是ｎ的标准分解式时，定义和ｆ（１）＝０．本文给出了和的渐近公式，此处Ω（ｎ）表ｎ的全部素因子的个数．     

Dedekind zeta-functions and Dedekind sums

In this paper we use Dedekind zeta functions of two real quadratic number fields at -1 to denote Dedekind sums of high rank. Our formula is different from that of Siegel’s. As an application, we get a polynomial representation of ζK(-1): ζK(-1) = 1/45(26n³ -41n± 9),n = ±2(mod 5), where K = Q(√5q), prime q = 4n² + 1, and the class number of quadratic number field K2 = Q(vq) is 1.     

Asymptotics of the Distribution of Eigenvalues of a Selfadjoint Second Order Hyperbolic Differential Operator on the Two-Dimensional Torus

We study the asymptotic properties of the discrete spectrum for general selfadjoint second order hyperbolic operators on the two-dimensional torus. For a broad class of operators with sufficiently smooth coefficients and the principal part coinciding with the wave operator in the light cone coordinates we prove the discreteness of the spectrum and obtain an asymptotic formula for the distribution of eigenvalues. In some cases we can indicate the first two asymptotic terms. We discuss the relations of these questions to analytic number theory and mathematical physics.     

Tree-graded spaces and asymptotic cones of groups

We introduce a concept of tree-graded metric space and we use it to show quasi-isometry invariance of certain classes of relatively hyperbolic groups, to obtain a characterization of relatively hyperbolic groups in terms of their asymptotic cones, to find geometric properties of Cayley graphs of relatively hyperbolic groups, and to construct the first example of a finitely generated group with a continuum of non-π₁-equivalent asymptotic cones. Note that by a result of Kramer, Shelah, Tent and Thomas, continuum is the maximal possible number of different asymptotic cones of a finitely generated group, provided that the Continuum Hypothesis is true.     

New dynamical invariants on hyperbolic manifolds

The rotation measure is an asymptotic dynamical invariant assigned to a typical point of a flow in a fiber bundle over a hyperbolic manifold. The total mass of the rotation measure is the average speed of the orbit and its “direction” is the ergodic invariant probability measure of the hyperbolic geodesic flow which best captures the asymptotic dynamics of the given point. The rotation measure exists almost everywhere and is constant for an ergodic measure of the given flow and so it may be viewed as assigning an ergodic measure of the geodesic flow to one of the given flow. It generalizes the usual notion of homology rotation vector by encoding homotopy information.     

Asymptotic number of solutions of some systems of diophantine inequalities

The problem of finding the asymptotic number of solutions of the system of inequalities $$\begin{gathered} \left\| {\alpha _i q} \right\|< q^{ - \sigma _i } (i = 1,...,n), \sigma _i > 0, \hfill \\ \sigma = \sum\nolimits_{i = 1}^n {\sigma _i< c(\alpha _1 ,...,\alpha _n ), q = 1,...,N,} \hfill \\ \end{gathered}$$ is solved under the assumption that for real numbers α₁,..., α_n, starting from some Q=max(q₁...,q_n) the inequality holds for any real λ≥0.     

On J. Bolyai's parallel construction

Given a model of Hilbert's incidence, order and congruence axioms (an H-plane) in which the Line-Circle Principle and the hypothesis of the acute angle hold, J. Bolyai's parallel construction is shown to yield the two parallels through the given point that have a common perpendicular at infinity with the given line through the ideal points at which they meet the given line; these need not be asymptotic parallels, for the plane need not be hyperbolic. Two new characterizations of hyperbolic planes are given, based on W. Pejas' classification of H-planes. The role of Archimedes' axiom is clarified.     

Holditch Theorem and Steiner Formula for the Planar Hyperbolic Motions

The Steiner formula and the Holditch Theorem for one-parameter closed planar Euclidean motions [1, 7] were expressed by H.R. Müller [9] under the one-parameter closed planar motions in the complex sense. In this paper, in analogy with complex motions as given by Müller [9], the Steiner formula and the mixture area formula are obtained under one parameter hyperbolic motions. Also Holditch theorems were expressed in the hyperbolic sense. The classical Holditch Theorem: If the endpoints A, B of a segment of fixed length are rotated once on an oval, then a given point X of this segment, with , describes a closed, not necessarily convex, curve. The area of the ring-shaped domain bounded by the two curves is πab, [1, 7].     

On Neyman–Pearson minimax detection of Poisson process intensity

The problem of the minimax testing of the Poisson process intensity \({\mathbf{s}}\) is considered. For a given intensity \({\mathbf{p}}\) and a set \(\mathcal{Q}\), the minimax testing of the simple hypothesis \(H_{0}: {\mathbf{s}} = {\mathbf{p}}\) against the composite alternative \(H_{1}: {\mathbf{s}} = {\mathbf{q}},\,{\mathbf{q}} \in \mathcal{Q}\) is investigated. The case, when the 1-st kind error probability \(\alpha \) is fixed and we are interested in the minimal possible 2-nd kind error probability \(\beta ({\mathbf{p}},\mathcal{Q})\), is considered. What is the maximal set \(\mathcal{Q}\), which can be replaced by an intensity \({\mathbf{q}} \in \mathcal{Q}\) without any loss of testing performance? In the asymptotic case (\(T\rightarrow \infty \)) that maximal set \(\mathcal{Q}\) is described.

     

The number of graphs and a random graph with a given degree sequence

We consider the set of all graphs on n labeled vertices with prescribed degrees D = (d₁,…,d_n). For a wide class of tame degree sequences D we obtain a computationally efficient asymptotic formula approximating the number of graphs within a relative error which approaches 0 as n grows. As a corollary, we prove that the structure of a random graph with a given tame degree sequence D is well described by a certain maximum entropy matrix computed from D. We also establish an asymptotic formula for the number of bipartite graphs with prescribed degrees of vertices, or, equivalently, for the number of 0‐1 matrices with prescribed row and column sums. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013