Reciprocal polynomials with all zeros on the unit circle

Let f(x)=a _d x ^d +a _d−1 x ^d−1+⋅⋅⋅+a ₀∈ℝ[x] be a reciprocal polynomial of degree d. We prove that if the coefficient vector (a _d ,a _d−1,…,a ₀) or (a _d−1,a _d−2,…,a ₁) is close enough, in the l ¹-distance, to the constant vector (b,b,…,b)∈ℝ ^d+1 or ℝ ^d−1, then all of its zeros have moduli 1.     

On the difference between the number of prime divisors at consecutive integers

Suppose thatg(n) is equal to the number of divisors ofn, counting multiplicity, or the number of divisors ofn, a≠0 is an integer, andN(x,b)=|{n∶n≤x, g(n+a)−g(n)=b orb+1}|. In the paper we prove that sup _b N(x,b)≤C(a)x)(log log 10 _x )^−1/2 and that there exists a constantC(a,μ)>0 such that, given an integerb |b|≤μ(log logx)^1/2,x≥x _o, the inequalityN(x,b)≥C(a,μ)x(log logx(^−1/2) is valid. Translated fromMatematicheskie Zametki, Vol. 66, No. 4, pp. 579–595, October, 1999.     

Groups of typeF a,b,−c

For integersa, b andc, the groupF ^a,b,−c is defined to be the group 〈R, S : R ²=RS ^aRS^bRS^−c=1〉. In this paper we identify certain subgroups of the group of affine linear transformations of finite fields of orderp ⁿ (for certainp andn) as groups of typeF ^a,b,−c for certain (not unique) choices ofa, b andc.     

On Ramanujan asymptotic expansions and inequalities for hypergeometric functions

In this paper we first discuss refinement of the Ramunujan asymptotic expansion for the classical hypergeometric functionsF(a,b;c;x), c ≤a + b, near the singularityx = 1. Further, we obtain monotonous properties of the quotient of two hypergeometric functions and inequalities for certain combinations of them. Finally, we also solve an open problem of finding conditions ona, b > 0 such that 2F(−a,b;a +b;r ²) < (2−r ²)F(a,b;a +b;r ²) holds for all r∈(0,1).     

Metacyclic groups and modular forms

In the paper we find the metacyclic groups of the form 〈a, b:a ^m=e, b ^s=e, b ⁻¹ ab=a ^r〉, wherem=3, 4, 5, 7, 11, 23, such that the modular forms associated with all elements of these groups by some faithful representation are multiplicative η-products. Translated fromMatematicheskie Zametki, Vol. 67, No. 2, pp. 163–173, February, 2000.     

On a difference equation arising in a learning-theory model

An analysis is presented of the equationf(x+a)−f(x)=e ^−x {f(x)−f(x−b)}. Herea andb denote arbitrary positive constants, and a solution is sought which satisfies the following conditions:f(−∞)=0,f(+∞)=1, 0≦f(x)≦1. Existence and uniqueness of solution are established, and then an analytical form of the solution is obtained by use of bilateral Laplace transform. Research supported by the National Science Foundation, Grant GP-2558.     

The Number of Lattice Points Above the Catenary

For fixed c > 1 and for arbitrary and independent a,b ≧ 1 let Z ²|b( cosh(x/a)−c) ≦ y < 0}. We investigate the asymptotic behaviour of R(a,b) for a,b → ∞. In the special case b = o(a ^5/6) the lattice rest has true order of magnitude . This revised version was published online in June 2006 with corrections to the Cover Date.     

Dehn Twists and Products of Mapping Classes of Riemann Surfaces with One Puncture

Let S be a Riemann surface that contains one puncture x. Let ℐ be the collection of simple closed geodesics on S, and let ℱ denote the set of mapping classes on S isotopic to the identity on S ∪ {x}. Denote by t _c the positive Dehn twist about a curve c ∈ ℐ. In this paper, the author studies the products of forms (t _b ^−m ∘ t _a ⁿ ) ∘ f ^k , where a, b ∈ ℐ and f ∈ ℱ. It is easy to see that if a = b or a, b are boundary components of an x-punctured cylinder on S, then one may find an element f ∈ ℱ such that the sequence (t _b ^−m ∘ t _n ^a ) ∘ f ^k contains infinitely many powers of Dehn twists. The author shows that the converse statement remains true, that is, if the sequence (t _b ^−m ∘ t _a ⁿ ) ∘ f ^k contains infinitely many powers of Dehn twists, then a, b must be the boundary components of an x-punctured cylinder on S and f is a power of the spin map t _b ⁻¹ ∘ t _a .     

Dense inverse subsemigroups of a topological inverse semigroup

In this paper we study dense inverse subsemigroups of topological inverse semigroups. We construct a topological inverse semigroup from a semilattice. Finally, we give two examples of the closure of B _{( −∞, ∞ )}¹, a topological inverse semigroup obtained by starting with the real numbers as a semilattice with the operation a ∨ b=sup{a,b}. The author would like to thank to the referee for useful suggestions.     

Hilbert-Kunz function of monomial ideals and binomial hypersurfaces

The aim of this note is to determine the Hilbert-Kunz functions of rings defined by monomial ideals and of rings defined by a single binomial equationX ^a−X^b with gcd(X ^a, X^b)=1.     

On the embedding problem with non-Abelian kernel of orderp 4. V

A survey of solvability conditions for the embedding problem of number fields, in which the kernel is a non-Abelian group of order p⁴, is completed. As a kernel, the two 2-groups with two generators a, b and with the following relations are considered: a ⁸ =1, b ² =1, [a,b]=a⁻² in the first group, and a ⁸ =1, b ² =a ⁴ , [a,b]=a⁻² in the second. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 211, 1994, pp. 127–132. Translated by V. V. Ishkhanov.     

On the Terai-Jésmanowicz conjecture

In this paper, we prove that if a, b and c are pairwise coprime positive integers such that a^2＋b^2=c^r,a〉b,a≡3 （mod4）,b≡2 （mod4） and c-1 is not a square, thena a^x＋b^y=c^z has only the positive integer solution （x, y, z） = （2, 2, r）.
Let m and r be positive integers with 2｜m and 2 r, define the integers Ur, Vr by （m ＋√-1）^r=Vr＋Ur√-1. If a = ｜Ur｜,b=｜Vr｜,c = m^2＋1 with m ≡ 2 （mod 4）,a ≡ 3 （mod 4）, and if r 〈 m/√1.5log3（m^2＋1）-1, then a^x ＋ b^y = c^z has only the positive integer solution （x,y, z） = （2, 2, r）. The argument here is elementary.     

On the compactification of concave ends

Let X be a complex manifold which admits a proper strictly plurisubharmonic function ρ : X →]a, b[, where −∞ ≤ a < b ≤ ∞. It is well-known that the Hausdorffness of H ¹(X) is necessary for the existence of a Stein completion of X. We show that this condition is also sufficient.     

A note on Weil index

Let F be a non-archimedean local field of characteristic 0 and(?)a nontrivial additive character.Weil first defined the Weil indexγ(a,(?))(a∈F~*)in his famous paper,from which we know thatγ(a,(?))γ(b,(?))=γ(ab,(?))γ(1,(?))(a,b)andγ(a,(?))~4 =(-1,-1),where(a,b)is the Hilbert symbol for F.The Weil index plays an important role in the theory of theta series and in the general representation theory.In this paper,we establish an identity relating the Weil indexγ(a,(?))and the Gauss sum.     

On the Drazin index of regular elements

It is known that the existence of the group inverse a ^# of a ring element a is equivalent to the invertibility of a ² a ⁻ + 1 − aa ⁻, independently of the choice of the von Neumann inverse a ⁻ of a. In this paper, we relate the Drazin index of a to the Drazin index of a ² a ⁻ + 1 − aa ⁻. We give an alternative characterization when considering matrices over an algebraically closed field. We close with some questions and remarks.      

Almost poised basic hypergeometric series

Given a basic hypergeometric series with numerator parametersa ₁,a ₂, ...,a _r and denominator parametersb ₂, ...,b _r, we say it isalmost poised ifb _i, =a ₁ q ^δ,i a _i,δ_i = 0, 1 or 2, for 2 ≤i ≤r. Identities are given for almost poised series withr = 3 andr = 5 when a₁, =q ⁻²ⁿ. Partially supported by N.S.F. Grant No. DMS-8521580.     

Penultimate limiting forms in extreme value theory

Summary Let {X _n}_n≧1 be a sequence of independent, identically distributed random variables. If the distribution function (d.f.) ofM _n=max (X ₁,…,X _n), suitably normalized with attraction coefficients {α_n}_n≧1(α_n>0) and {b _n}_n≧1, converges to a non-degenerate d.f.G(x), asn→∞, it is of interest to study the rate of convergence to that limit law and if the convergence is slow, to find other d.f.'s which better approximate the d.f. of(M _n−b_n)/a_n thanG(x), for moderaten. We thus consider differences of the formF ⁿ(a_nx+b_n)−G(x), whereG(x) is a type I d.f. of largest values, i.e.,G(x)≡Λ(x)=exp (-exp(−x)), and show that for a broad class of d.f.'sF in the domain of attraction of Λ, there is a penultimate form of approximation which is a type II [Ф _α(x)=exp (−x^−α), x>0] or a type III [Ψ _α(x)= exp (−(−x)^α), x<0] d.f. of largest values, much closer toF ⁿ(a_nx+b_n) than the ultimate itself.     

Derived length and products of conjugacy classes

Let G be a supersolvable group and A be a conjugacy class of G. Observe that for some integer η(AA ⁻¹) > 0, AA ⁻¹ = {ab ⁻¹: a, b ∈ A} is the union of η(AA ⁻¹) distinct conjugacy classes of G. Set C _G (A) = {g ∈ G: a ^g = a for all a ∈ A. Then the derived length of G/C _G (A) is less or equal than 2η(AA ⁻¹) − 1.     

Algebras with bracket

An algebra with bracket is an associative algebra A equipped with a bilinear operation [−,−] satisfying [a · b, c] = [a, c]·b+a · [b, c]. Our main result claims that the operad corresponding to algebras with bracket is Koszul.     

Recursive method for building Arnold’s triangle X p (a,b)

We give a recursive method for building X _p (a,b) for each prime p. Arnold’s triangle is composed of positive integers: for a>1 and 0<b<a, X _p (a,b) is the degree of the highest power of p dividing the difference of the binomial coefficients C _pa ^pb −C _a ^b .