Two finite embedding theorems for partial 3-quasigroups

In this paper it is shown that a finite partial (x, x, y) = y 3-quasigroup can be embedded in a finite (x, x, y) = y 3-quasigroup. This result coupled with the technique of proof is then used to show that a finite partial almost Steiner 3-quasigroup {(x, x, y) = y, (x, y, z) = (x, z, y) = (y, x, z)} can be embedded in a finite almost Steiner 3-quasigroup. Almost Steiner 3-quasigroups are of more than passing interest because just like Steiner 3-quasigroups ( = Steiner quadruple systems) all of their derived quasigroups are Steiner quasigroups.     

Duality for a Class of Multiobjective Control Problems

In this paper, a class of multiobjective control problems is considered, where the objective and constraint functions involved are f(t, x(t), ?(t), y(t), z(t)) with x(t) ∈ Rⁿ, y(t) ∈ Rⁿ, and z(t) ∈ R^m, where x(t) and z(t) are the control variables and y(t) is the state variable. Under the assumption of invexity and its generalization, duality theorems are proved through a parametric approach to related properly efficient solutions of the primal and dual problems.     

Unit groups of group algebras of some small groups

Let FG be a group algebra of a group G over a field F and U (FG) the unit group of FG. It is a classical question to determine the structure of the unit group of the group algebra of a finite group over a finite field. In this article, the structure of the unit group of the group algebra of the non-abelian group G with order 21 over any finite field of characteristic 3 is established. We also characterize the structure of the unit group of FA ₄ over any finite field of characteristic 3 and the structure of the unit group of FQ ₁₂ over any finite field of characteristic 2, where Q ₁₂ = 〈x, y; x ⁶ = 1, y ² = x ³, x ^y = x ^?1〉.     

The existence and uniqueness of strong kings in tournaments

A king x in a tournament T is a player who beats any other player y directly (i.e., x→y) or indirectly through a third player z (i.e., x→z and z→y). For x,y∈V(T), let b(x,y) denote the number of third players through which x beats y indirectly. Then, a king x is strong if the following condition is fulfilled: b(x,y)>b(y,x) whenever y→x. In this paper, a result shows that for a tournament on n players there exist exactly k strong kings, 1?k?n, with the following exceptions: k=n-1 when n is odd and k=n when n is even. Moreover, we completely determine the uniqueness of tournaments.     

A bilinear version of Orlicz-Pettis theorem

Given three Banach spaces X, Y and Z and a bounded bilinear map , a sequence x=n(xn)⊆X is called B-absolutely summable if is finite for any y∈Y. Connections of this space with are presented. A sequence x=n(xn)⊆X is called B-unconditionally summable if is finite for any y∈Y and z^∗∈Z^∗ and for any M⊆N there exists xM∈X for which _∑n∈M〈B(xn,y),z^∗〉=〈B(xM,y),z^∗〉 for all y∈Y and z^∗∈Z^∗. A bilinear version of Orlicz-Pettis theorem is given in this setting and some applications are presented.     

On difference sets in high exponent 2-groups

We investigate the existence of difference sets in particular 2-groups. Being aware of the famous necessary conditions derived from Turyn’s and Ma’s theorems, we develop a new method to cover necessary conditions for the existence of (2^2d+2,2^2d+1?2 ^d ,2^2d ?2 ^d ) difference sets, for some large classes of 2-groups. If a 2-group G possesses a normal cyclic subgroup 〈x〉 of order greater than 2 ^d+3+p , where the outer elements act on the cyclic subgroup similarly as in the dihedral, semidihedral, quaternion or modular groups and 2 ^p describes the size of G′∩〈x〉 or C _G (x)′∩〈x〉, then there is no difference set in such a group. Technically, we use a simple fact on how sums of 2 ⁿ -roots of unity can be annulated and use it to characterize properties of norm invariance (prescribed norm). This approach gives necessary conditions when a linear combination of 2 ⁿ -roots of unity remains unchanged under homomorphism actions in the sense of the norm.     

Self-converse Mendelsohn designs with block size 4t + 2

A Mendelsohn design MD(v, k, λ) is a pair (X, B) where X is a v-set together with a collection B of cyclic k-tuples from X such that each ordered pair from X, as adjacent entries, is contained in exactly λk-tuples of B. An MD(v, k, λ) is said to be self-converse, denoted by SCMD(v, k, λ) = (X, B, f), if there is an isomorphic mapping from (X, B) to (X, B⁻¹), where B⁻¹ = {B⁻¹ = 〈x_k, x_k−1, … x₂, x₁〉; B = 〈x₁, … ,x_k〉 ∈ B.}. The existence of SCMD(v, 3, λ) and SCMD(v, 4, 1) has been settled by us. In this article, we will investigate the existence of SCMD(v, 4t + 2, 1). In particular, when 2t + 1 is a prime power, the existence of SCMD(v, 4t + 2, 1) has been completely solved, which extends the existence results for MD(v, k, 1) as well. © 1999 John Wiley & Sons, Inc. J. Combin Designs 7: 283–310, 1999     

A new off-step high order approximation for the solution of three-space dimensional nonlinear wave equations

In this paper, we propose a new high accuracy numerical method of O(k² + k²h² + h⁴) based on off-step discretization for the solution of 3-space dimensional non-linear wave equation of the form u_tt = A(x,y,z,t)u_xx + B(x,y,z,t)u_yy + C(x,y,z,t)u_zz + g(x,y,z,t,u,u_x,u_y,u_z,u_t), 0 < x,y,z < 1,t > 0 subject to given appropriate initial and Dirichlet boundary conditions, where k > 0 and h > 0 are mesh sizes in time and space directions respectively. We use only seven evaluations of the function g as compared to nine evaluations of the same function discussed in  and . We describe the derivation procedure in details of the algorithm. The proposed numerical algorithm is directly applicable to wave equation in polar coordinates and we do not require any fictitious points to discretize the differential equation. The proposed method when applied to a telegraphic equation is also shown to be unconditionally stable. Comparative numerical results are provided to justify the usefulness of the proposed method.     

The k-factor conjecture is true

A sequence 〈d_i〉, 1≤i≤n, is called graphical if there exists a graph whose i^th vertex has degree d_i for all i. It is shown that the sequences 〈d_i〉 and 〈d_i-k〉 are graphical only if there exists a graph G whose degree sequence is 〈d_i〉 and which has a regular subgraph with k lines at each vertex. Similar theorems have been obtained for digraphs. These theorems resolve comjectures given by A.R. Rao and S.B. Rao, and by B. Grünbaum.     

A generalization of the Bernoulli model occurring in order statistics. I.

Let X=(x₁, ..., x_n) and Y=(y₁, ..., y_m) be independent samples from populations G_x and G_y, x⁽¹⁾ ,... x⁽ⁿ⁾ be ordered statistics constructed from the sample X. A model of trials associated with the occurrence of dependent events A_k={y_k (x⁽ⁱ⁾}, x^(j), i < j, k=1, 2, ..., m, where x⁽ⁱ⁾, x^(j) are order statistics, is considered. This model is a generalization of the Bernoulli model. Distribution of frequencies of occurrences of events A_k and the limit theorems which describe asymptotic properties of these frequencies are investigated.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 4, pp. 518–528, April, 1990.     

A subgroup involvement of the Fibonacci length

For a non-Abelian 2-generated finite group G=〈a,b〉, the Fibonacci length of G with respect to A={a,b}, denoted by LEN _A (G), is defined to be the period of the sequence x ₁=a,x ₂=b,x ₃=x ₁ x ₂,…,x _n+1=x _n?1 x _n ,… of the elements of G. For a finite cyclic group C _n =〈a〉, LEN _A (C _n ) is defined in a similar way where A={1,a} and it is known that LEN _A (C _n )=k(n), the well-known Wall number of n. Over all of the interesting numerical results on the Fibonacci length of finite groups which have been obtained by many authors since 1990, an intrinsic property has been studied in this paper. Indeed, by studying the family of minimal non-Abelian p-groups it will be shown that for every group G of this family, there exists a suitable generating set A′ for the derived subgroup G′ such that LEN _A′(G′)|LEN _A (G) where, A is the original generating set of G.     

An Erdös-Ko-Rado theorem for the subcubes of a cube

LetP be that partially ordered set whose elements are vectors x=(x ₁, ...,x _n ) withx _i ε {0, ...,k} (i=1, ...,n) and in which the order is given byx≦y iffx _i =y _i orx _i =0 for alli. LetN _i (P)={x εP : |{j:x _j ≠ 0}|=i}. A subsetF ofP is called an Erdös-Ko-Rado family, if for allx, y εF it holdsx ≮y, x ≯ y, and there exists az εN ₁(P) such thatz≦x andz≦y. Let ? be the set of all vectorsf=(f ₀, ...,f _n ) for which there is an Erdös-Ko-Rado familyF inP such that |N _i (P) ∩F|=f _i (i=0, ...,n) and let 〈?〉 be its convex closure in the (n+1)-dimensional Euclidean space. It is proved that fork≧2 (0, ..., 0) and \(\left( {0,...,0,\overbrace {i - component}^{\left( {\begin{array}{*{20}c} {n - 1} \\ {i - 1} \\ \end{array} } \right)}k^{i - 1} ,0,...,0} \right)\) (i=1, ...,n) are the vertices of 〈?〉.     

Oscillatory and asymptotic behavior of solutions of differential equations with deviating arguments

The oscillatory and asymptotic behavior of solutions of a class of nth order nonlinear differential equations, with deviating arguments, of the form (E, δ) L_nx(t) + δq(t) f(x[g₁(t)],…, x[g_m(t)]) = 0, where δ = ± 1 and L₀x(t) = x(t), L_kx(t) = a_k(t)(L_{k ? 1}x(t))^., $\text{k = 1, 2,…, n (}{}^{\mathrm{.}}\text{=}\text{d}\text{dt}\text{)}$, is examined. A classification of solutions of (E, δ) with respect to their behavior as t → ∞ and their oscillatory character is obtained. The comparisons of (E, 1) and (E, ?1) with first and second order equations of the form y^.(t) + c₁(t) f(y[g₁(t)],…, y[g_m(t)]) = 0 and (a_{n ? 1}(t)z^.(t))^. ? c₂(t) f(z[g₁(t)],…, z[g_m(t)]) = 0, respectively, are presented. The obtained results unify, extend and improve some of the results by Graef, Grammatikopoulos and Spikes, Philos and Staikos.     

On positive solutions of perturbed nonlinear differential equations

Some results are given concerning positive solutions of equations of the form x⁽ⁿ⁾ + P(t) G(x) = Q(t, x).Let class I (II) consist of all n-times differentiable functions x(t), such that x(t)>0 and x^{(n ? 1)}(t) ? 0 (x^{(n ? 1)}(t) ? 0) for all large t. Two theorems are given guaranteeing the nonexistence of solutions in class I and II, respectively, and three theorems ensure the convergence to zero of positive solutions. A recent result of Hammett concerning the second-order case is extended to the general case.     

Achievement and avoidance games for generating abelian groups

For any finite groupG, the DO GENERATE game is played by two players Alpha and Beta as follows. Alpha moves first and choosesx ₁ ∈G. Thek-th play consists of a choice ofx _k ∈G ?S _k ?1 whereS _n ={itx ₁,...,x _n }. LetG _n = 〈S _n 〉. The game ends whenG _n =G. The player who movesx _n wins. In the corresponding avoidance game, DON'T GENERATE, the last player to move loses. Of course neither game can end in a draw. For an arbitrary group, it is an unsolved problem to determine whether Alpha or Beta wins either game. However these two questions are answered here for abelian groups.     

Transitive orientations in bull-reducible Berge graphs

A bull is a graph with five vertices r,y,x,z,s and five edges ry, yx, yz, xz, zs. A graph G is bull-reducible if every vertex of G lies in at most one bull of G. We prove that every bull-reducible Berge graph G that contains no antihole is weakly chordal, or has a homogeneous set, or is transitively orientable. This yields a fast polynomial time algorithm to color the vertices of such a graph exactly.     

Sur l'équation diophantienne

We announce several new theorems on the Diophantine equation (xⁿ−1)/(x−1)=y^q, which include the fact that any integer greater than 2 and with all digits equal to 1 in base ten cannot be a pure power. We then apply these results to solve this equation for any integer x of the form x=z¹ with 2 ≤ z ≤ 10000 and t ≥ 1.     

Stability interval for explicit difference schemes for multi-dimensional second-order hyperbolic equations with significant first-order space derivative terms

In this piece of work, we introduce a new idea and obtain stability interval for explicit difference schemes of O(k²+h²) for one, two and three space dimensional second-order hyperbolic equations u_tt=a(x,t)u_xx+α(x,t)u_x-2η²(x,t)u,u_tt=a(x,y,t)u_xx+b(x,y,t)u_yy+α(x,y,t)u_x+β(x,y,t)u_y-2η²(x,y,t)u, and u_tt=a(x,y,z,t)u_xx+b(x,y,z,t)u_yy+c(x,y,z,t)u_zz+α(x,y,z,t)u_x+β(x,y,z,t)u_y+γ(x,y,z,t)u_z-2η²(x,y,z,t)u,0<x,y,z<1,t>0 subject to appropriate initial and Dirichlet boundary conditions, where h>0 and k>0 are grid sizes in space and time coordinates, respectively. A new idea is also introduced to obtain explicit difference schemes of O(k²) in order to obtain numerical solution of u at first time step in a different manner.     

Note on Locating Pairs of Vertices on Hamiltonian Cycles

Given a fixed positive integer k ≥ 2, let G be a simple graph of order n ≥ 6k. It is proved that if the minimum degree of G is at least n/2 + 1, then for every pair of vertices x and y, there exists a Hamiltonian cycle such that the distance between x and y along that cycle is precisely k.