Residual finiteness of outer automorphism groups of certain 1-relator groups

We prove that certain 1-relator groups have Property E. Using this fact, we characterize all conjugacy separable 1-relator groups of the form a,b;(a-αbβaαbγ)t , t 1, having residually finite outer automorphism groups.     

The Lie algebra of skew-symmetric elements and its application in the theory of Jordan algebras

We prove that the Lie algebra of skew-symmetric elements of the free associative algebra of rank 2 with respect to the standard involution is generated as a module by the elements [a, b] and [a, b]³, where a and b are Jordan polynomials. Using this result we prove that the Lie algebra of Jordan derivations of the free Jordan algebra of rank 2 is generated as a characteristic F-module by two derivations. We show that the Jordan commutator s-identities follow from the Glennie-Shestakov s-identity.     

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.     

Latin bitrades,dissections of equilateral triangles,and abelian groups

Let T=(T^*, T^?) be a spherical latin bitrade. With each a=(a₁, a₂, a₃)∈T^* associate a set of linear equations Eq(T, a) of the form b₁+b₂=b₃, where b=(b₁, b₂, b₃) runs through T^*\{a}. Assume a₁=0=a₂ and a₃=1. Then Eq(T,a) has in rational numbers a unique solution $b_{i}=\bar{b}_{i}Let T=(T^*, T^?) be a spherical latin bitrade. With each a=(a₁, a₂, a₃)∈T^* associate a set of linear equations Eq(T, a) of the form b₁+b₂=b₃, where b=(b₁, b₂, b₃) runs through T^*\{a}. Assume a₁=0=a₂ and a₃=1. Then Eq(T,a) has in rational numbers a unique solution $b_{i}=\bar{b}_{i}$. Suppose that $\bar{b}_{i}\not= \bar{c}_{i}$ for all b, c∈T^* such that $\bar{b}_{i}\not= \bar{c}_{i}$ and i∈{1, 2, 3}. We prove that then T^? can be interpreted as a dissection of an equilateral triangle. We also consider group modifications of latin bitrades and show that the methods for generating the dissections can be used for a proof that T^* can be embedded into the operational table of a finite abelian group, for every spherical latin bitrade T. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 1–24, 2010     

Numerical Semigroups on Compound Sequences

We generalize the geometric sequence {a^p, a^p?1b, a^p?2b²,…, b^p} to allow the p copies of a (resp. b) to all be different. We call the sequence {a₁a₂a₃… a_p, b₁a₂a₃…a_p, b₁b₂a₃…a_p,…, b₁b₂b₃…b_p} a compound sequence. We consider numerical semigroups whose minimal set of generators form a compound sequence, and compute various semigroup and arithmetical invariants, including the Frobenius number, Apéry sets, Betti elements, and catenary degree. We compute bounds on the delta set and the tame degree.     

On the Diophantine System a^2 ＋ b^2 = c^3 and a^x ＋ b^y = c^z for b is an Odd Prime

Let a, b and c be fixed coprime positive integers. In this paper we prove that if a^2 ＋ b^2 = c^3 and b is an odd prime, then the equation a^x ＋ b^y = c^z has only the positive integer solution （x, y, z） = （2,2,3）.     

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）.     

A note on the diophantine equation x 2 + b y = c z

Let a, b, c, r be positive integers such that a ² + b ² = c ^r , min(a, b, c, r) > 1, gcd(a, b) = 1, a is even and r is odd. In this paper we prove that if b ≡ 3 (mod 4) and either b or c is an odd prime power, then the equation x ² + b ^y = c ^z has only the positive integer solution (x, y, z) = (a, 2, r) with min(y, z) > 1.     

Third power associative,antiflexible rings satisfying (a,b,ac) = a(a,b,c)

In this paper, we study third power associative, antiflexible rings satisfying the identity (a,b,ac)?=?a(a,b,c). We prove that third power associative, antiflexible rings satisfying the identity (a,b,ac)?=?a(a,b,c) with characteristic ≠2,3 are associative of degree 5. As a consequence of this result, we prove that a third power associative semiprime antiflexible ring satisfying the identity (a,b,ac)?=?a(a,b,c) is associative.     

Fourier Series with Linear Splines on an Interval

We construct orthonormal bases of linear splines on a finite interval [a, b] and then we study the Fourier series associated to these orthonormal bases. For continuous functions defined on [a, b], we prove that the associated Fourier series converges pointwisely on (a, b) and also uniformly on [a, b], if it convergences pointwisely at a and b.     

Reverse Order Law for the Group Inverse in Semigroups and Rings

In this paper, we provide equivalent conditions for the two-sided reverse order law for the group inverse (ab)^# = b ^# a ^# and (ba)^# = a ^# b ^#, in semigroups and rings. Moreover, we prove that, under finiteness conditions, these conditions are also equivalent with the one-sided reverse order law (ab)^# = b ^# a ^#.     

On the number of solutions to systems of Pell equations

We prove that, for positive integers a, b, c and d with c≠d, a>1, b>1, the number of simultaneous solutions in positive integers to ax²−cz²=1, by²−dz²=1 is at most two. This result is the best possible one. We prove a similar result for the system of equations x²−ay²=1, z²−bx²=1.     

Semigroups of linear endomorphisms closed under conjugation

Let a,b be singular endomorphisms of a. finite dimensional vector space V and denote by S _a the semigroup generated by all the elements g ^-1ag, where g?Aut(V).The aim of this paper is to prove that b?S _a if and only if rank(b) ≥ rank(a).     

Decomposition of Complete Bipartite Even Graphs into Closed Trails

We prove that any complete bipartite graph K _a,b , where a, b are even integers, can be decomposed into closed trails with prescribed even lengths.     

Games of Chains and Cutsets in the Boolean Lattice II

Let 2 ^[n] denote the poset of all subsets of [n]={1,2,...,n} ordered by inclusion. Following Gutterman and Shahriari (Order 14, 1998, 321–325) we consider a game G _n (a,b,c). This is a game for two players. First, Player I constructs a independent maximal chains in 2 ^[n]. Player II will extend the collection to a+b independent maximal chains by finding another b independent maximal chains in 2 ^[n]. Finally, Player I will attempt to extend the collection further to a+b+c such chains. The last Player who is able to complete her move wins. In this paper, we complete the analysis of G _n (a,b,c) by considering its most difficult instance: when c=2 and a+b+2=n. We prove, the rather surprising result, that, for n7, Player I wins G _n (a,n–a–2,2) if and only if a3. As a consequence we get results about extending collections of independent maximal chains, and about cutsets (collections of subsets that intersect every maximal chain) of minimum possible width (the size of largest anti-chain).     

On Solvable R* Groups of Finite Rank

Abstract

For an element a of a group G,let S(a) denote the semigroup generated by all conjugates of a in G. We prove that if G is solvable of finite rank and 1 ? S(a) for all 1 ≠ a ∈ G,then ?a ^G ?/?b ^G ? is a periodic group for every b ∈ S(a). Conversely if every two generator subgroup of a finitely generated torsion-free solvable group G has this property then G has finite rank,and if every finitely generated subgroup has this property then every partial order on G can be extended to a total order.     

Exceptional 0-Alia algebras

Analgebra (A, ∘) with the identity [a, b]∘c + [b, c]∘a + [c, a]∘b = 0, where [a, b] = a∘b−b∘a, is called 0-Alia. We prove that the algebra (ℂ[x], ∘) with multiplication a ∘ b = ∂2(2a∂(b)+∂(a)b) is a simple, exceptional 0-Alia algebra. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 60, Algebra, 2008.     

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.     

On the Consistency of the Quantum-Like Representation Algorithm for Hyperbolic Interference

Recently quantum-like representation algorithm (QLRA) was introduced by A. Khrennikov [20]–[28] to solve the so-called “inverse Born’s rule problem”: to construct a representation of probabilistic data by a complex or hyperbolic probability amplitude or more general complex together with hyperbolic which matches Born’s rule or its generalizations. The outcome from QLRA is coupled to the formula of total probability with an additional term corresponding to trigonometric, hyperbolic or hyper-trigonometric interference. The consistency of QLRA for probabilistic data corresponding to trigonometric interference was recently proved [29]. We complete the proof of the consistency of QLRA to cover hyperbolic interference as well. We will also discuss hyper trigonometric interference. The problem of consistency of QLRA arises, because formally the output of QLRA depends on the order of conditioning. For two observables (e.g., physical or biological) a and b, b|a- and a|b-conditional probabilities produce two representations, say in Hilbert spaces H ^b|a and H ^a|b (in this paper over the hyperbolic algebra). We prove that under “natural assumptions” these two representations are unitary equivalent (in the sense of hyperbolic Hilbert space).