A New Way to Represent Links. One-Dimensional Formalism and Untangling Technology

An alternative link representation different from planar diagrams is discussed. Isotopy classes of unordered nonoriented links are realized as central elements of a monoid presented explicitly by a finite number of generators and relations. The group presented by two generators and three relations [[a,b],a ² ba ^–2]=[[a,b],b ² ab ^–2]=[[a,b],[a ^–1,b ^–1]]=1, where [x,y]=xyx ^–1 y ^–1, is proved to have a commutator subgroup isomorphic to the braid group on infinitely many strands. A new partial algorithm for unknot recognition is constructed. Experiments show that the algorithm allows the untangling of unknots whose planar diagram has hundreds of crossings. Here 'untangling' means 'finding an isotopy to the circle'.     

On Algebras of Skew Polynomials Generated by Quadratic Homogeneous Relations

We consider the algebras, with two generators a and b, generated by the quadratic relations ba = α² + βab + γb², where the coefficients α, β, and γ belong to an arbitrary field F of characteristic 0. We find conditions for such an algebra to be expressed as a skew polynomial algebra with generator b over the polynomial ring F [a]. These conditions are equivalent to the existence of the Poincaree-Birkhoff-Witt basis, i. e., basis of the form {a^m, bⁿ}. Bibliography: 16 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 301, 2003, pp. 144–171.     

Unique representation of primitive factors of 2 n −1,n ODD,in certain quadratic forms

Primitive factorsq of 2 ⁿ –1,n odd, are either of the form 8m–1 or 8m+1. In the first case there exists a unique representation ofq in the quadratic forma ²–2b ², (a, b)=1,a andb odd,b<[q/2], and in the latter a unique representation ofq in the quadratic formc ²+2 ^j d², (c, d)=1,c andd odd,j even and 6. Thus the uniqueness ofq=a ²–2b ² orc ²+2 ^j d² exhibits a proof of the primality ofq.A program that determinesa, b, c, andd for anyq not exceeding 16 decimal digits is described, and as an example the 13-digit prime 4432676798593 (a primitive factor of 2⁴⁹–1) is uniquely represented by 1374273²+2¹⁴·12461².     

Equations in a group with a length function

A simple counterexample to Lyndon's conjecture is given. There is defined a group of continuous functions with a regular Archimedean length function. Analogues of Wicks' theorem on the commutator and of Lyndon's theorem on the solutions of the equation a²b²=c² are proved for a wider class of groups with a regular Archimedean length function.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, Nos. 7 and 8, pp. 935–942, July–August, 1991.     

Regular splittings and the discrete Neumann problem

Summary Iterative methods are discussed for approximating a solution to a singular but consistent square linear systemAx=b. The methods are based upon splittingA=M–N withM nonsingular. Monotonicity and the concept of regular splittings, introduced by Varga, are used to determine some necessary and some sufficient conditions in order that the iterationx ⁱ⁺¹=M^–1Nxⁱ+M^–1b converge to a solution to the linear system. Finally, applications are given to solving the discrete Neumann problem by iteration which are based upon the inherent monotonicity in the formulation.This research was supported by the U. S. Army Research Office-Durham under contract no. DAHCO4 74 C 0019.     

On Chebyshev–Markov Rational Functions over Several Intervals

Chebyshev–Markov rational functions are the solutions of the following extremal problem

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withKbeing a compact subset of andω_n(x) being a fixed real polynomial of degree less thann, positive onK. A parametric representation of Chebyshev–Markov rational functions is found forK=[b₁, b₂]…[b_2p−1, b_2p], −∞<b₁b₂<…<b_2p−1b_2p<+∞ in terms of Schottky–Burnside automorphic functions.     

The method of geometric progression for finding a minimum of unimodal functions

An asymptotically optimal method for finding a minimal point x^* for a function f(x) C², unimodal on an interval [a_k,b_k], is proposed. This method partitions the intervals _k = b_k – a_k, k = 1, 2, ..., and locates x^* by means of a decreasing geometric progression.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 70, 104–108, 1990.     

Zur Kennzeichnung von Lorentztransformationen in Endlichen Ebenen

Given a commutative field K we define d(A,B):= (a₁–b₁)²–(a₂–b₂)² for A=(a₁,a₂), B=(b₁,b₂) K². Given moreover a fixed k K^O, W. Benz has asked for all mappings : K²K² such that d(A,B)=k implies. This paper gives an answer if K=GF(p), p=5,7,11: must be a bijective collineation in case p = 7,11; there are non-injective mappings in case p=5. To obtain some of these results we have mads use of a computer.

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Prof. R. Artzy zum 70. Geburtstag gewidmet.     

Infinite p-groups containing exactly p2 solutions of the equation xP=1

We study arbitrary infinite 2-groups with three involutions and infinite locally finite p-groups (p2), containing p²–1 elements of order p. For odd p the groupG=a, where A is a direct product of two quasicyclic 3-groups ¦b¦=9, b³A, and subgroup A is generated by the elements of the commutator ladder of element b, is a unique infinite non-Abelian locally finite p-group whose equation xP=1 has p² solutions.Translated from Matematicheskie Zametki, Vol. 20, No. 1, pp. 11–18, July, 1976.     

Coefficients of univalent functions which assume no pair of values W and −W

In this paper we study the behavior of the coefficients of functions univalent in the disk ¦z¦< 1 and assuming there are no pair of values Wand –W. In particular, we establish the asymptotic behavior of b_n (n); for the coefficients we obtain the estimate ¦b_n¦ < 2.34 exp {l/4n} (n = 2,3, ...) and for each function of the class indicated we prove, subject to a certain condition, the relationship b_n+1¦–¦b_n=O(n^–1/2).Translated from Matematicheskie Zametki, Vol. 11, No. 1, pp. 3–14, January, 1972.     

Rational Surfaces with Many Nodes

We describe smooth rational projective algebraic surfaces over an algebraically closed field of characteristic different from 2 which contain n b ₂ –2 disjoint smooth rational curves with self-intersection –2, where b ₂ is the second Betti number. In the last section this is applied to the study of minimal complex surfaces of general type with p _g = 0 and K² = 8, 9 which admit an automorphism of order 2.     

Deux remarques sur les flots riemanniens

Let M be a connected oriented closed n-manifold. A riemannian flow on M is an oriented one dimensional foliation which admits a bundle-like metric.We give a caracterization of isometric flows as riemannian flows whose basic cohomology H _b ^n–1 (M, ) is non trivial in degree (n–1). A second caracterization involves the triviality of the central sheaf.We show also that has a section if and only if H _b ^n–1 (M, ) has a non trivial image in H^n–1(M).     

Signal Analytic Proofs of Two Basic Results on Lattice Expansions

We present new and short proofs of two theorems in the theory of lattice expansions. These proofs are based on a necessary and sufficient condition, found by Wexler and Raz, for biorthogonality. The first theorem is the Lyubarskii–Seip–Wallstén theorem for lattices, according to which the set of Gaussians 2^1/4 exp(-π(t - na)² + 2πimbt), n, m , constitutes a frame when a > 0,b > 0,ab < 1. In addition, we display dual functions for this case. The second theorem is the result that a set g_na,mb(t) = g(t - na) exp(2πimbt), n, m of time–frequency translates of a g L²( ) cannot be a frame when a > 0,b > 0,ab > 1.     

Effect of loading on intensity of fracture and deformation of polymers

An expression is proposed for the activation energy U = U₀ –a + b², by means of which it is possible qualitatively to describe the various cases of the kinetics of deformation and rupture in polymers.Scientific-Research Institute of Plastics, Moscow. Translated from Mekhanika Polimerov, No. 2, pp. 366–368, March–April, 1972.     

Error-free computation with rational numbers

A method is described for doing error-free computation when the operands are rational numbers. A rational operanda/b is mapped onto the integer ¦a·b ^–1¦ _p and the arithmetic is performed inGF(p). A method is given for taking an integer result and finding its rational equivalent (the one which corresponds to the correct rational result).     

The birth and death processes with zero as their absorbing barrier

LetE=(0, 1,...), Q ^b=(q_ij), i, j=0, 1, ..., whereq _{i, i–1}=a_i, q_{i, i+1}=b_i, q_ii=–(a_i+b_i), q_ij=0, when|i–j|>1. a ₀=0, b₀=b>0, a_i, b_i>0 (i>0). Lettingb=0 inQ ^b, we get the matrixQ ⁰.The time homogeneous Markov processX ^b ={x ^b (t,w), 0t< ^b (w)} (X ⁰={x⁰(t,w), 0t<⁰(w)}), withQ ^b (Q ⁰, respectively) as its density matrix and withE as its state space, is calledQ ^b (Q ⁰, respectively) process in this paper.Q ^b andQ ⁰ processes are all called the birth and death processes, with zero being the reflecting barrier ofQ ^b processes, the absorbing barrier ofQ ⁰ processes.AllQ ^b processes have been constructed by both probability and analytical methods (Wang [2], Yang [1]). In this paper, theQ ⁰ processes are imbedded intoQ ^b processes and all theQ ⁰ processes are directly constructed from theQ ^b processes. The main results are:Letb>0 be arbitrarily fixed, then there is a one to one correspondence between theQ ⁰ processes and theQ ^b processes (in the sense of transition probability).TheQ ⁰ process is unique iffR ^*=. SupposingR<, then:IfX ⁰={x⁰(t,w), 0t<⁰(w)} is a non-minimalQ ⁰ process, then its eigensequence (p, q, r _n, n–1) satisfies § 4(7).Conversely, let a non-negative number sequence (p, q, r _n, n–1) satisfying § 4(7) be arbitrarily given, then there exists a unique non-minimalQ ⁰ processX ⁰ with eigensequence (p, q, r _n, n–1). The Laplace transform of the transition probability (p _ij ⁰ (t)) ofX ⁰ is determined by § 4(15). X ⁰ is honest iffr _–1=0.X ⁰ satisfies the forward equation iffp=0.     

On the identric and logarithmic means

Summary Leta, b > 0 be positive real numbers. The identric meanI(a, b) of a andb is defined byI = I(a, b) = (1/e)(b ^b /a ^a ) ^1/(b–a) , fora b, I(a, a) = a; while the logarithmic meanL(a, b) ofa andb isL = L(a, b) = (b – a)/(logb – loga), fora b, L(a, a) = a. Let us denote the arithmetic mean ofa andb byA = A(a, b) = (a + b)/2 and the geometric mean byG =G(a, b) = . In this paper we obtain some improvements of known results and new inequalities containing the identric and logarithmic means. The material is divided into six parts. Section 1 contains a review of the most important results which are known for the above means. In Section 2 we prove an inequality which leads to some improvements of known inequalities. Section 3 gives an application of monotonic functions having a logarithmically convex (or concave) inverse function. Section 4 works with the logarithm ofI(a, b), while Section 5 is based on the integral representation of means and related integral inequalities. Finally, Section 6 suggests a new mean and certain generalizations of the identric and logarithmic means.     

Simple Curves on Surfaces

We study simple closed geodesics on a hyperbolic surface of genus g with b geodesic boundary components and c cusps. We show that the number of such geodesics of length at most L is of order L ^6g+2b+2c–6. This answers a long-standing open question.     

Lower Bounds for the Difference ax n − by m

In this work we give completely explicit lower bounds for |ax ⁿ – by ^m | depending only on a, b, n, m and a, b, n, x, respectively.     

The set of values of b for which there exist block designs with b blocks

Let B denote the set of values of b for which there exists a block design with b blocks and for k3, let B_k denote the subset of B determined by the designs with block size k. We present some information about B and the sets B_k. In particular, we discuss, for certain integers h, the question as to whether there exist integers k and k^′ such that the equation b^′=b+h has infinitely many solutions b,b^′ satisfying bB_k and b^′B_k^′. The study is restricted to the case λ=1.