On type numbers of split orders of definite quaternion algebras

In this paper, we shall give a new relation between the arithmetic of quaternion algebras and modular forms; we shall express the type numberT _{q, N} of a split order of type (q, N) as the sums of dimensions of some subspaces of the space of cusp forms of weight 2 with respect to Γ₀(qN) which are common eigenspaces of Atkin-Lehner's involutions.     

Traces of Eichler—Brandt matrices and type numbers of quaternion orders

LetA be a totally definite quaternion algebra over a totally real algebraic number fieldF andM be the ring of algebraic integers ofF. For anyM-orderG ofA we derive formulas for the massm(G) and the type numbert(G) of G and for the trace of the Eichler-Brandt matrixB(G, J) ofG and any integral idealJ ofM in terms of genus invariants ofG and of invariants ofF andJ. Applications to class numbers of quaternion orders and of ternary quadratic forms are indicated.     

An embedding theorem for Eichler orders

Let B be a quaternion algebra over number field K. Assume that B satisfies the Eichler condition (i.e., there is at least one archimedean place which is unramified in B). Let Ω be an order in a quadratic extension L of K. The Eichler orders of B which admit an embedding of Ω are determined. This is a generalization of Chinburg and Friedman's embedding theorem for maximal orders.     

On embedding expanders into ℓ p spaces

In this note we show that the minimum distortion required to embed alln-point metric spaces into the Banach space ℓ _p is between (c ₁/p) logn and (c ₂/p) logn, wherec ₂>c ₁>0 are absolute constants and 1≤p<logn. The lower bound is obtained by a generalization of a method of Linial et al. [LLR95], by showing that constant-degree expanders (considered as metric spaces) cannot be embedded any better. Research supported by Czech Republic Grant GAČR 201/94/2167 and Charles University grants No. 351 and 361.     

A circle representation using complex and quaternion numbers

In this paper, we describe a simple representation of a circular arc in the space using quaternions. Using this representation, we obtain a subdivision of the arc, describe circular splines, and give a few applications with circular surfaces. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 2, pp. 298–310, April–June, 2006.     

On embedding trees into uniformly convex Banach spaces

We investigate the minimum value ofD =D(n) such that anyn-point tree metric space (T, ρ) can beD-embedded into a given Banach space (X, ∥·∥); that is, there exists a mappingf :T →X with 1/D ρ(x,y) ≤ ∥f(x) −f(y)∥ ≤ρ(x,y) for anyx,y εT. Bourgain showed thatD(n) grows to infinity for any superreflexiveX (and this characterized super-reflexivity), and forX =ℓ _p, 1 <p < ∞, he proved a quantitative lower bound of const·(log logn)^min(1/2,1/p). We give another, completely elementary proof of this lower bound, and we prove that it is tight (up to the value of the constant). In particular, we show that anyn-point tree metric space can beD-embedded into a Euclidean space, with no restriction on the dimension, withD =O(√log logn). This paper contains results from my thesis [Mat89] from 1989. Since the subject of bi-Lipschitz embeddings is becoming increasingly popular, in 1997 I finally decided to publish this English version. Supported by Czech Republic Grant GAČR 0194 and by Charles University grants No. 193, 194.     

On quaternion algebras

It is proved that the distributiveness of the right ideals lattice for a quaternion algebra over a commutative ring A is equivalent to the following property: the equation x²+y²+z²=0 is uniquely solvable in the field A/M for any maximal ideals M of A, the lattice of the ideals of A being distributive. Bibliography: 5 titles. Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 17, pp. 209–214, 1994.     

An embedding theorem for orders in central simple algebras

For an n-dimensional central simple algebra defined by a generalized Hilbert symbol over a number field, we compute the number of conjugacy classes of maximal orders that contain a conjugate of the order spanned by the canonical generators of the algebra. This has applications to the study of certain finite subgroups of arithmetic orbifold groups.     

On the set of natural numbers which only yield orders of Abelian groups

Let A denote the set of all natural numbers n such that every group of order n is Abelian. Let C denote the set of all natural numbers n such that every group of order n is cyclic. We prove that Σ_{n ≤ x,n?A?C}1 has roughly the order of magnitude x(log log x)^?1.     

On the embedding of constant-weight codes into perfect codes

We show that each q-ary constant-weight code of weight 3, minimum distance 4, and length m embeds in a q-ary 1-perfect code of length n = (q ^m ? 1)/(q ? 1).     

On the embedding of topological domains into quotient fields

Necessary and sufficient conditions to embed a Hausdorff left Ore domain A in a topological field Q(A) of quotients are given in terms of conditions on an invariant uniformity topologizing the multiplicative semigroup A* of A. The categorical aspects of such an embedding are discussed, and a new proof of S. Warner's theorem on the embedding of a left Ore domain as an open subring of a topological field of quotients is given.This work comprises part of the author's doctoral dissertation submitted to SUNY at Buffalo. The author wishes to thank Professor Dov Tamari for suggesting this problem.     

On the embedding of BV space into Besov-Orlicz space

We give a sufficient (and, in the case of a compact domain, a necessary) condition for the embedding of Sobolev space of functions with integrable gradient into Besov-Orlicz spaces to be bounded. The condition has a form of a simple integral inequality involving Young and weight functions. We provide an example with Matuszewska-Orlicz indices of involved Orlicz norm equal to one. The main tool is the molecular decomposition of functions from a BV space.     

Indecomposable totally positive numbers in real quadratic orders

In a totally real number field, every totally positive integral number is a finite sum of (additively) indecomposable totally positive integral numbers, and up to multiplication by totally positive units, there exist only finitely many indecomposables. In the paper it is shown that in quadratic fields all these numbers can be listed in a very efficient way by using the so-called intermediate convergents of a certain quadratic irrationality. The method can be viewed as a simple extension of the standard method of calculating the fundamental unit by using continued fractions. As an application it is shown that for instance in Z|√d| a number is decomposable if its norm is >d. It is remarkable that this bound does not depend on the size of the fundamental unit.     

Factorial functions and stirling numbers of fractional orders

The aim of this paper is to study the binomial coefficients ( _n ^x ), the factorial polynomials [x]_n and [x]ⁿ, the Stirling numbers of first and second kind, namely s(n,k) and S(n,k), in the case that n ∈ ? is replaced by real α ∈ ?. In the course of the paper, the Vandermonde convolution formula is presented in an infinite series frame, the binomial coefficient function ( _a ^x ), α ∈ ?, is sampled in terms of the binomial coefficients ( _k ^x ) for k ∈ ?_o, Bell numbers of fractional orders are introduced. Emphasis is placed on the fractional order Stirling numbers s(α,k) and S(α,k), first studied here. Some applications of the S(α,k) are given.