Self-orthogonal cyclicn-quasigroups

A quasigroup (Q,) satisfying the identityx(yx) =y (or the equivalent identity (xy)x =y) is called semisymmetric. Ann-quasigroup (Q, A) satisfying the identityA(A(x ₁, ...,x _n ),x ₁, ...,x _n–1) =x _n is called cyclic. So, cyclicn-quasigroups are a generalization of semisymmetric quasigroups. In this paper, self-orthogonal cyclicn-quasigroups (SOCnQs) are considered. Some constructions ofSOCnQs are described and the spectrum of suchn-quasigroups investigated.     

An application of L-functions of imaginary quadratic fields

Let F₁(x, y),…, F_2h+1(x, y) be the representatives of equivalent classes of positive definite binary quadratic forms of discriminant ?q (q is a prime such that q ≡ 3 mod 4) with integer coefficients, then the number of integer solutions of F_i(x, y) = n (i = 1,…, 2h + 1) can be calculated for each natural number n using L-functions of imaginary quadratic field $\text{Q}$((?q)^1/2).     

Self-orthogonal semisymmetric quasigroups

A quasigroup satisfying the 2-variable identity x(yx) = y is called semisymmetric. It is observed that the transpose of a semisymmetric quasigroup is also semisymmetric. Consequently, the existence of a self-orthogonal semisymmetric quasigroup (SOSQ) gives rise to a pair of orthogonal semisymmetric quasigroups. In this paper, the spectrum of SOSQs is investigated and it is found that the spectrum contains all positive integers n = 1 (mod 3), except n = 10.     

On dynamic programming equations for utility indifference pricing under delta constraints

In this paper we study the problem of utility indifference pricing in a constrained financial market, using a utility function defined over the positive real line. We present a convex risk measure −v(•:y) satisfying q(x,F)=x+v(F:u₀(x)), where u₀(x) is the maximal expected utility of a small investor with the initial wealth x, and q(x,F) is a utility indifference buy price for a European contingent claim with a discounted payoff F. We provide a dynamic programming equation associated with the risk measure (−v), and characterize v as a viscosity solution of this equation.     

Some theorems on the existence,uniqueness, and nonexistence of limit cycles for quadratic systems

Given a quadratic system (QS) with a focus or a center at the origin we write it in the form ẋ = y + P₂(x, y), ẏ = −x + dy + Q₂(x, y) where P₂ and Q₂ are homogeneous polynomials of degree 2. If we define F(x, y) = (x − dy) P₂(x, y) + yQ₂(x, y) and g(x, y) = xQ₂(x, y) − yP₂(x, y) we give results of existence, nonexistence, and uniqueness of limit cycles if F(x, y) g(x, y) does not change of sign. Then, by using these results plus the properties on the evolution of the limit cycles of the semicomplete families of rotated vector fields we can study some particular families of QS, i.e., the QS with a unique finite singularity and the bounded QS with either one or two finite singularities.     

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.     

First-order optimality conditions in set-valued optimization

A a set-valued optimization problem min _C F(x), x ∈X ₀, is considered, where X ₀ ⊂ X, X and Y are normed spaces, F: X ₀ ⊂ Y is a set-valued function and C ⊂ Y is a closed cone. The solutions of the set-valued problem are defined as pairs (x ⁰,y ⁰), y ⁰ ∈F(x ⁰), and are called minimizers. The notions of w-minimizers (weakly efficient points), p-minimizers (properly efficient points) and i-minimizers (isolated minimizers) are introduced and characterized through the so called oriented distance. The relation between p-minimizers and i-minimizers under Lipschitz type conditions is investigated. The main purpose of the paper is to derive in terms of the Dini directional derivative first order necessary conditions and sufficient conditions a pair (x ⁰, y ⁰) to be a w-minimizer, and similarly to be a i-minimizer. The i-minimizers seem to be a new concept in set-valued optimization. For the case of w-minimizers some comparison with existing results is done.     

Modular metric spaces, I: Basic concepts

The notion of a modular is introduced as follows. A (metric) modular on a set X is a function w:(0,∞)×X×X→[0,∞] satisfying, for all x,y,z∈X, the following three properties: x=y if and only if w(λ,x,y)=0 for all λ>0; w(λ,x,y)=w(λ,y,x) for all λ>0; w(λ+μ,x,y)≤w(λ,x,z)+w(μ,y,z) for all λ,μ>0. We show that, given x₀∈X, the set X_w={x∈X:lim_λ→∞w(λ,x,x₀)=0} is a metric space with metric , called a modular space. The modular w is said to be convex if (λ,x,y)?λw(λ,x,y) is also a modular on X. In this case X_w coincides with the set of all x∈X such that w(λ,x,x₀)<∞ for some λ=λ(x)>0 and is metrizable by . Moreover, if or , then ; otherwise, the reverse inequalities hold. We develop the theory of metric spaces, generated by modulars, and extend the results by H. Nakano, J. Musielak, W. Orlicz, Ph. Turpin and others for modulars on linear spaces.     

Duality theorems for perturbed convex optimization

We show that if F, X are two locally convex spaces and $\text{h: F →}\text{R}\text{?}\text{, ?: F × X → R}$ are two convex functionals satisfying h(y) = ?(y, x₀) (y?F) for some x₀?X, then, under suitable assumptions, the computation of inf h(F) can be reduced to the computation of inf ?(H) on certain hyperplanes H of F × X. We give some applications.     

Characterization of projective graphs

We denote the distance between vertices x and y of a graph by d(x, y), and p_ij(x, y) = ∥ {z : d(x, z) = i, d(y, z) = j} ∥. The (s, q, d)-projective graph is the graph having the s-dimensional subspaces of a d-dimensional vector space over GF(q) as vertex set, and two vertices x, y adjacent iff $\text{dim}\text{(x ? y) = s ? 1}$. These graphs are regular graphs. Also, there exist integers λ and μ > 4 so that μ is a perfect square, p₁₁(x, y) = λ whenever d(x, y) = 1, and p₁₁(x, y) = μ whenever d(x, y) = 2. The (s, q, d)-projective graphs where $\text{2d}\text{3}\text{≤ s < d ? 2}$ and $\text{(s, q, d) ≠ (}\text{2d}\text{3}\text{, 2, d)}$, are characterized by the above conditions together with the property that there exists an integer r satisfying certain inequalities.     

The increase of sums and products dependent on (y 1, …,y n ) by rearrangement of this set

LetF(u, v) be a symmetric real function defined forα<u, v<β and assume thatG(u, v, w)=F(u, v)+F(u, w)−F(v, w) is decreasing inv andw foru≦min (u, v). For any set (y)=(y ₁, …,y _n ),α<y _i <β, given except in arrangement Σ _i ^=1/n F(y _i ,y _i+1) wherey _n+1=y ₁) is maximal if (and under some additional assumptions only if) (y) is arranged in circular symmetrical order. Examples are given and an additional result is proved on the productΠ _i ^=1/n [(y2i−1y2i) ^m +α ₁(y _2i−1 y _2i ) ^m−1+ … +a _m ] wherea _k ≧0 and where the set (y)=(y ₁, ..,y _n ),y _i ≧0 is given except in arrangement. The problems considered here arose in connection with a theorem by A. Lehman [1] and a lemma of Duffin and Schaeffer [2]. This paper is part of the author’s Master of Science dissertation at the Technion-Israel Institute of Technology. The author wishes to thank Professor B. Schwarz and Professor E. Jabotinsky for their help in the preparation of this paper.     

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 〈?〉.     

A general class of fuzzy operators,the demorgan class of fuzzy operators and fuzziness measures induced by fuzzy operators

In this paper we examine operators which can be derived from the general solution of functional equations on associativity. We define the characteristics of those functions f(x) which are necessary for the production of operators. We shall show, that with the help of the negation operator for every such function f(x) a function g(x) can be given, from which a disjunctive operator can be derived, and for the three operators the DeMorgan identity is fulfilled. For the fulfillment of the DeMorgan identity the necessary and sufficient conditions are given.We shall also show that an f_λ(x) can be constructed for every f(x), so that for the derived k_λ(x,y) and d_λ(x,y) lim_λ→∞k_λ(x,y) and lim_λ→∞d_λ(x,y) = max(x,y).As Yager's operator is not reducible, for every λ there exists an α, for which, in case x < α and y<α, k_λ(x,y) = 0.We shall give an f(x) which has the characteristics of Yager's operator, and which is strictly monotone.Finally we shall show, that with the help of all those f(x), which are necessary when constructing a k(x,y), an F(x) can be constructed which has the properties of the measures of fuzziness introduced by A. De Luca and S. Termini. Some classical fuzziness measures are obtained as special cases of our system.     

Oscillation of partial difference equations with continuous variables

In this paper, we consider the partial difference equation with continuous variables of the form P₁z(x + a, y + b) + p₂z (x + a, y) + p₃z (x, y + b) − p₄z (x, y) + P (x, y) z (x − τ, y − σ) = 0, where P ϵ C(R⁺ × R⁺, R⁺ − {0}), a, b, τ, σ are real numbers and p_i (i = 1, 2, 3, 4) are nonnegative constants. Some sufficient conditions for all solutions of this equation to be oscillatory are obtained.     

Generalized (Jordan) left derivations on rings associated with an element of rings

In this paper, we introduce a new notion of generalized (Jordan) left derivation on rings as follows: let R be a ring, an additive mapping F : R → R is called a generalized (resp. Jordan) left derivation if there exists an element w ∈ R such that F(xy) = xF(y) + yF(x) + yxw (resp. F(x ²) = 2xF(x) + x ² w) for all x, y ∈ R. Then, some related properties and results on generalized (Jordan) left derivation of square closed Lie ideals are obtained.     

A new inequality for symmetric functions

Let x_i ≥ 0, y_i ≥ 0 for i = 1,…, n; and let a_j(x) be the elementary symmetric function of n variables given by a_j(x) = ∑_{1 ≤ ii < … <ij ≤ n}x_ii … x_ij. Define the partical ordering x <y if a_j(x) ≤ a_j(y), j = 1,… n. We show that $\text{x}\text{\$}\text{?}\text{y ? x}{}^{\mathrm{\alpha }}\text{\$}\text{?}\text{y}{}^{\mathrm{\alpha }}\text{, 0}\text{\$}\text{?}\text{α ≤ 1}$, where {x^α}_i = x^α_i. We also give a necessary and sufficient condition on a function f(t) such that x <y ? f(x) <f(y). Both results depend crucially on the following: If x <y there exists a piecewise differentiable path z(t), with z_i(t) ≥ 0, such that z(0) = x, z(1) = y, and z(s) <z(t) if 0 ≤ s ≤ t ≤ 1.     

Commutativity of rings with derivations

I. N. Herstein [10] proved that a prime ring of characteristic not two with a nonzero derivation d satisfying d(x)d(y) = d(y)d(x) for all x, y must be commutative, and H. E. Bell and M. N. Daif [8] showed that a prime ring of arbitrary characteristic with nonzero derivation d satisfying d(xy) = d(yx) for all x, y in some nonzero ideal must also be commutative. For semiprime rings, we show that an inner derivation satisfying the condition of Bell and Daif on a nonzero ideal must be zero on that ideal, and for rings with identity, we generalize all three results to conditions on derivations of powers and powers of derivations. For example, let R be a prime ring with identity and nonzero derivation d, and let m and n be positive integers such that, when charR is finite, m ∨ n < charR. If d(x ^m y ⁿ ) = d(y ⁿ x ^m ) for all x, y ∈ R, then R is commutative. If, in addition, charR≠ 2 and the identity is in the image of an ideal I under d, then d(x) ^m d(y) ⁿ = d(y) ⁿ d(x) ^m for all x, y ∈ I also implies that R is commutative.     

A duality relation for entrance and exit laws for Markov processes

Markov processes X_t on (X, F_X) and Y_t on (Y, F_Y) are said to be dual with respect to the function f(x, y) if E_xf(X_t, y) = E_yf(x, Y_t for all x ? X, y ? Y, t ? 0. It is shown that this duality reverses the role of entrance and exit laws for the processes, and that two previously published results of the authors are dual in precisely this sense. The duality relation for the function f(x, y) = 1_{x<y} is established for one-dimensional diffusions, and several new results on entrance and exit laws for diffusions, birth-death processes, and discrete time birth-death chains are obtained.