Properties of certain semigroups and their potential as platforms for cryptosystems

In this paper, we study some properties of semigroups with presentation 〈a,b ; a ^p =b ^r ,a ^q =b ^s 〉. We will also study their potential as platforms for the Diffie-Hellman key exchange protocol.     

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.     

On the semigroup nilpotency and the Lie nilpotency of associative algebras

To each associative ringR we can assign the adjoint Lie ringR ⁽⁻⁾ (with the operation(a,b)=ab−ba) and two semigroups, the multiplicative semigroupM(R) and the associated semigroupA(R) (with the operationaob=ab+a+b). It is clear that a Lie ringR ⁽⁻⁾ is commutative if and only if the semigroupM(R) (orA(R)) is commutative. In the present paper we try to generalize this observation to the case in whichR ⁽⁻⁾ is a nilpotent Lie ring. It is proved that ifR is an associative algebra with identity element over an infinite fieldF, then the algebraR ⁽⁻⁾ is nilpotent of lengthc if and only if the semigroupM(R) (orA(R)) is nilpotent of lengthc (in the sense of A. I. Mal'tsev or B. Neumann and T. Taylor). For the case in whichR is an algebra without identity element overF, this assertion remains valid forA(R), but fails forM(R). Another similar results are obtained. Translated fromMatematicheskie Zametki, Vol. 62, No. 4, pp. 510–519, October, 1997. Translated by A. I. Shtern     

On the algebraic independence in the selberg class

We prove that a functionF of the Selberg class ℐ is ab-th power in ℐ, i.e.,F=H ^b for someHσ ℐ, if and only ifb divides the order of every zero ofF and of everyp-componentF _p. This implies that the equationF ^a=G^b with (a, b)=1 has the unique solutionF=H ^b andG=H ^a in ℐ. As a consequence, we prove that ifF andG are distinct primitive elements of ℐ, then the transcendence degree of ℂ[F,G] over ℂ is two.     

Adjoint bi-continuous semigroups and semigroups on the space of measures

For a given bi-continuous semigroup (T(t)) _t⩾0 on a Banach space X we define its adjoint on an appropriate closed subspace X° of the norm dual X′. Under some abstract conditions this adjoint semigroup is again bi-continuous with respect to the weak topology σ(X°,X). We give the following application: For Ω a Polish space we consider operator semigroups on the space C_b(Ω) of bounded, continuous functions (endowed with the compact-open topology) and on the space M(Ω) of bounded Baire measures (endowed with the weak*-topology). We show that bi-continuous semigroups on M(Ω) are precisely those that are adjoints of bi-continuous semigroups on C_b(Ω). We also prove that the class of bi-continuous semigroups on C_b(ω) with respect to the compact-open topology coincides with the class of equicontinuous semigroups with respect to the strict topology. In general, if is not a Polish space this is not the case.     

On the Hamilton-Waterloo Problem

 The Hamilton-Waterloo problem asks for a 2-factorisation of K _v in which r of the 2-factors consist of cycles of lengths a ₁,a ₂,…,a _t and the remaining s 2-factors consist of cycles of lengths b ₁,b ₂,…,b _u (where necessarily ∑ _i=1 ^t a _i =∑ _j=1 ^u b _j =v). In this paper we consider the Hamilton-Waterloo problem in the case a _i =m, 1≤i≤t and b _j =n, 1≤j≤u. We obtain some general constructions, and apply these to obtain results for (m,n)∈{(4,6),(4,8),(4,16),(8,16),(3,5),(3,15),(5,15)}. Received: July 5, 2000     

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.     

Contractive presentations: A family of inverse monoids and semigroups with finiteR-classes

There has recently been considerable interest in inverse monoids which are presented by generators and relations. In this work the author employs graphical techniques to investigate the word problem for presentations of inverse monoids which generalize the case in which all relations in a presentation are of the formw=w ² . The work also investigates free objects in finitely based varieties of inverse semigroups, where the free objects have similar presentations. A fundamental charecteristic of the monoids (semigroups) investigated is: ifF is a free inverse monoid andM=F/θ, then form∈F, theR-class ofmθ has no more elements than theR-class ofm.     

A Conjecture Concerning the Pure Exponential Diophantine Equation a^x ＋ b^y = c^z

Let a, b, c, r be fixed positive integers such that a^2 ＋ b^2 = c^r, min（a, b, c, r） 〉 1 and 2 r. In this paper we prove that if a ≡ 2 （mod 4）, b ≡ 3 （mod 4）, c 〉 3.10^37 and r 〉 7200, then the equation a^x ＋ b^y = c^z only has the solution （x, y, z） = （2, 2, r）.     

On the Existence of Specific Stars in Planar Graphs

Given a graph G, a (k;a,b,c)-star in G is a subgraph isomorphic to a star K_1,3 with a central vertex of degree k and three leaves of degrees a, b and c in G. The main result of the paper is: Every planar graph G of minimum degree at least 3 contains a (k;a,b,c)-star with a≤ b≤ c and (i) k = 3, a≤ 10, or (ii) k = 4, a = 4, 4≤ b≤ 10, or (iii) k = 4, a = 5, 5≤ b≤ 9, or (iv) k = 4, 6≤ a≤ 7, 6≤ b≤ 8, or (v) k = 5, 4≤ a≤ 5, 5≤ b≤ 6 and 5≤ c≤ 7, or (vi) k = 5 and a = b = c = 6.     

Attainability of the minimal exponential growth rate for free products of finite cyclic groups

We consider free products of two finite cyclic groups of orders 2 and n, where n is a prime power. For any such group ℤ₂ * ℤ _n = 〈a, b | a ² = b ⁿ = 1〉, we prove that the minimal growth rate α _n is attained on the set of generators {a, b} and explicitly write out an integer polynomial whose maximal root is α _n . In the cases of n = 3, 4, this result was obtained earlier by A. Mann. We also show that under sufficiently general conditions, the minimal growth rates of a group G and of its central extension [(G)\tilde]\tilde G coincide and that the attainability of one implies the attainability of the other. As a corollary, the attainability is proved for some cyclic extensions of the above-mentioned free products, in particular, for groups 〈a, b | a ² = b ⁿ 〉, which are groups of torus knots for odd n.     

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.     

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.     

Homogenization of some parabolic operators with several time scales

The main focus in this paper is on homogenization of the parabolic problem ∂ _t uɛ − ∇ · (a(x/ɛ,t/ɛ,t/ɛ ^r )∇u ^ɛ ) = f. Under certain assumptions on a, there exists a G-limit b, which we characterize by means of multiscale techniques for r > 0, r ≠ 1. Also, an interpretation of asymptotic expansions in the context of two-scale convergence is made.     

A Generalized Radon Transform on the Plane

A new generalized Radon transform R _α, β on the plane for functions even in each variable is defined which has natural connections with the bivariate Hankel transform, the generalized biaxially symmetric potential operator Δ _α, β , and the Jacobi polynomials P_k^(b, a)(t)P_{k}^{(\beta,\,\alpha)}(t). The transform R _α, β and its dual R_a, b^*R_{\alpha,\,\beta}^{\ast} are studied in a systematic way, and in particular, the generalized Fuglede formula and some inversion formulas for R _α, β for functions in L_a, b^p(\mathbbR²₊)L_{\alpha,\,\beta}^{p}(\mathbb{R}^{2}_{+}) are obtained in terms of the bivariate Hankel–Riesz potential. Moreover, the transform R _α, β is used to represent the solutions of the partial differential equations Lu:=?_j=1^ma_jD_a, b^ju=fLu:=\sum_{j=1}^{m}a_{j}\Delta_{\alpha,\,\beta}^{j}u=f with constant coefficients a _j and the Cauchy problem for the generalized wave equation associated with the operator Δ _α, β . Another application is that, by an invariant property of R _α, β , a new product formula for the Jacobi polynomials of the type P_k^(b, a)(s)C_2k^a+b+1(t)=còòP_k^(b, a)P_{k}^{(\beta,\,\alpha)}(s)C_{2k}^{\alpha+\beta+1}(t)=c\int\!\!\int P_{k}^{(\beta,\,\alpha)} is obtained.     

Bi-Parametric Potentials, Relevant Function Spaces and Wavelet-Like Transforms

We introduce new potential type operators J^a_b = (E+(-D)^b/2)^-a/bJ^{\alpha}_{\beta} = (E+(-\Delta)^{\beta/2})^{-\alpha/\beta}, (α > 0, β > 0), and bi-parametric scale of function spaces H^a_{b, p}(\mathbbRⁿ)H^{\alpha}_{\beta , p}({\mathbb{R}}^n) associated with J^α_β. These potentials generalize the classical Bessel potentials (for β = 2), and Flett potentials (for β = 1). A characterization of the spaces H^a_{b, p}(\mathbbRⁿ)H^{\alpha}_{\beta, p}({\mathbb{R}}^n) is given with the aid of a special wavelet–like transform associated with a β-semigroup, which generalizes the well-known Gauss-Weierstrass semigroup (for β = 2) and the Poisson one (for β = 1).     

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.     

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.     

Proportionally modular diophantine inequalities and their multiplicity

Let I be an interval of positive rational numbers. Then the set S （I） = T ∩ N, where T is the submonoid of （Q0＋, ＋） generated by T, is a numerical semigroup. These numerical semigroups are called proportionally modular and can be characterized as the set of integer solutions of a Diophantine inequality of the form ax rood b 〈 cx. In this paper we are interested in the study of the maximal intervals I subject to the condition that S （I） has a given multiplicity. We also characterize the numerical semigroups associated with these maximal intervals.