Modular <Emphasis Type="Italic">W</Emphasis> and <Emphasis Type="Italic">H</Emphasis> Type Lie Algebras

The modular Witt algebra W(p, n) and H(p, 2n) are defined on the polynomial rings Z_p[x₁,...,x_n] and Z_p[X₁,...,x_n, y₁,...,y_n] respectively. We generalize Z_p[x₁,...,xn] and Zp[x₁,...,xn, y₁,...,y_n], so we get the generalized W-type and H-type modular Lie algebras. We find all the derivations of W(p, 1).AMS Subject Classification: Primary 17B40, 17B56.     

On the existence of certain measures appearing in distance geometry

Abstract. We prove the following result: Let X be a compact connected Hausdorff space and f be a continuous function on X x X. There exists some regular Borel probability measure m\mu on X such that the value of¶¶ ò\limit _X f(x,y)dm(y)\int\limit _X f(x,y)d\mu (y) is independent of the choice of x in X if and only if the following assertion holds: For each positive integer n and for all (not necessarily distinct) x₁,x₂,...,x_n,y₁,y₂,...,y_n in X, there exists an x in X such that¶¶ ?_i=1ⁿ f(x_i,x)=?_i=1ⁿ f(y_i,x).\sum\limits _{i=1}^n f(x_i,x)=\sum\limits _{i=1}^n f(y_i,x).     

Modern thermodynamics as a branch of mathematics (mathematical physics)

In the paper, it is proved that, if f(x₁,..., x_n)g(y₁,..., y_m) is a multilinear central polynomial for a verbally prime T-ideal Γ over a field of arbitrary characteristic, then both polynomials f(x₁,..., x_n) and g(y₁,..., y_m) are central for Γ.     

Distribution of lattice points on surfaces of second order

Let f₁(x₁,..., xl₁) and f₂(y₁,...,yl₂) be positive definite primitive quadratic forms in l₁ and l₂ variables, respectively. We obtain new results in the well-known problem on the number of lattice points on the cone f₁(x₁,...,xl₁)=f₂(y₁,...,yl₂), in the domain f₁(x₁,...,xl₁)≦N for N»∞. Our technical tool is the Rankin-Selberg convolution. In several special cases the results can be sharpened by other methods. In addition, new facts concerning the uniform distribution of lattice points on ellipsoids in l variables, l odd, l≧5 are obtained.     

An Engel condition with generalized derivations on multilinear polynomials

Let R be a prime ring with extended centroid C, g a nonzero generalized derivation of R, f (x ₁,..., x _n) a multilinear polynomial over C, I a nonzero right ideal of R. If [g(f(r ₁,..., r _n)), f(r ₁,..., r _n)] = 0, for all r ₁, ..., r _n ∈ I, then either g(x) = ax, with (a − γ)I = 0 and a suitable γ ∈ C or there exists an idempotent element e ∈ soc(RC) such that IC = eRC and one of the following holds:

The Embedding Theorem for Cantor Varieties

Let m and n be fixed integers, with 1 m < n. A Cantor variety C _m,n is a variety of algebras with m n-ary and n m-ary basic operations which is defined in a signature ={g₁,...,g_m,f₁,...,f_n} by the identities f_ig₁x₁,...,x_n),...,g_mx₁,...,x_n) = x_i, _i=1,...,n, g_jf₁x₁,...,x_m),...,f_nx₁,...,x_m)) = x_j, _j=1,...,m. We prove the following: (a) every partial C _m,n-algebra A is isomorphically embeddable in the algebra G= A; S(A) of C _m,n; (b) for every finitely presented algebra G= A; S in C _m,n, the word problem is decidable; (c) for finitely presented algebras in _{C m}, the occurrence problem is decidable; (d) _{C m,n} has a hereditarily undecidable elementary theory.     

A functional equation arising in multi-agent statistical decision theory

Using recent results of Járai we show that the measurable solutions of the functional equationf(x ₁ y ₁,...,x _n y _n )f((1?x ₁)(1?y ₁),..., (1?x _n )(1?y _n ))=f(x ₁(1?y ₁),...,x _n (1 ? (y _n ))f(y ₁(1?x ₁),...,y _n(1 ?x _n )), wheref: (0, 1) ⁿ → (0, ∞) and 0<x _i ,y _i <1,i=1,...,n, are of the form $$f(x_1 ,...,x_n ) = c \exp \left( {\sum\limits_{i = 1}^n {a_i (x_1 - x_1^2 ))} \prod\limits_{i = 1}^n {x_i^{b_1 } ,} } \right.$$ wherec>0,a ₁,...,a _n andb ₁,..., _b are arbitrary real constants. This result enables one to characterize certain independence-preserving methods of aggregating probability distributions over four alternatives.     

A result on rearrangements

Circular symmetry is defined for ordered sets ofn real numbers: (y)=(y ₁,...,y _n). Letf(x) be non-decreasing and convex forx≧0 and let (y) be given except in arrangement. The Σ _i ⁼¹ⁿ f(|y _i−y _i+1|) (wherey _n+1=y ₁) is minimal if (and under some additional assumptions only if) (y) is arranged in circular symmetrical order. Sponsored by the Mathematics Research Center, United States Army under Contract No. DA-11-022-ORD-2059, University of Wisconsin, Madison.     

Multilinear polynomials and cocentralizing conditions in prime rings

Let R be a noncommutative prime ring of characteristic different from 2, let Z(R) be its center, let U be the Utumi quotient ring of R, let C be the extended centroid of R, and let f(x ₁,..., x _n ) be a noncentral multilinear polynomial over C in n noncommuting variables. Denote by f(R) the set of all evaluations of f(x ₁, …, xn) on R. If F and G are generalized derivations of R such that [[F(x), x], [G(y), y]] ∈ Z(R) for any x, y ∈ f(R), then one of the following holds: . Decomposer equations:

f(f^*(x)f(y))=f(y),f(f(x)f_*(y))=f(x)

.Strong decomposer equations:

f(f^*(x)y)=f(y),f(xf_*(y))=f(x)

.Canceler equations:

f(f(x)y)=f(xy),f(xf(y))=f(xy),f(xf(y)z)=f(xyz)

, where f^*(x) f(x) = f (x) f_* (x) = x. In this paper we solve them and introduce the general solution of the decomposer and strong decomposer equations in the sets with a binary operation and semigroups respectively and also associative equations in arbitrary groups. Moreover we state some equivalent equations to them and study the relations between the above equations. Finally we prove that the associative equations and the system of strong decomposer and canceler equations do not have any nontrivial solutions in the simple groups.     

Multivariate discriminant and iterated resultant

In this paper,we study the relationship between iterated resultant and multivariate discriminant.We show that,for generic form f(x_n) with even degree d,if the polynomial is squarefreed after each iteration,the multivariate discriminant △(f) is a factor of the squarefreed iterated resultant.In fact,we find a factor Hp(f,[x_1,...,x_n]) of the squarefreed iterated resultant,and prove that the multivariate discriminant △(f) is a factor of Hp(f,[x_1,...,x_n]).Moreover,we conjecture that Hp(f,[x_1,...,x_n]) = △(f) holds for generic form/,and show that it is true for generic trivariate form f(x,y,z).     

Boundedness and Stability for Higher Order Difference Equations*

Sufficient conditions are given under which the higher order difference equation x _n+1= f(x _n,x _n-1,...,x_n-k ), n=0,1,2,... generates an order preserving discrete dynamical system with respect to the discrete exponential ordering. It is shown that under the above monotonicity assumption the boundedness of all solutions as well as the local and global stability of an equilibrium hold if and only if they hold for the much simpler first order equation x _n+1=h(x _n ), where h(x)=f(x,x,…,x). As an application, a second order nonlinear difference equation from macroeconomics and a discrete analogue of a model of haematopoiesis are discussed.     

Classes of orderings of measures and related correlation inequalities II. Multivariate reverse rule distributions

A function f(x) defined on = ₁ × ₂ × … × _n where each _i is totally ordered satisfying f(x y) f(x y) ≥ f(x) f(y), where the lattice operations and refer to the usual ordering on , is said to be multivariate totally positive of order 2 (MTP₂). A random vector Z = (Z₁, Z₂,…, Z_n) of n-real components is MTP₂ if its density is MTP₂. Classes of examples include independent random variables, absolute value multinormal whose covariance matrix Σ satisfies −DΣ⁻¹D with nonnegative off-diagonal elements for some diagonal matrix D, characteristic roots of random Wishart matrices, multivariate logistic, gamma and F distributions, and others. Composition and marginal operations preserve the MTP₂ properties. The MTP₂ property facilitate the characterization of bounds for confidence sets, the calculation of coverage probabilities, securing estimates of multivariate ranking, in establishing a hierarchy of correlation inequalities, and in studying monotone Markov processes. Extensions on the theory of MTP₂ kernels are presented and amplified by a wide variety of applications.     

Extension theorems for functional equations with bisymmetric operations

Summary. In this paper we deal with the extension of the following functional equation¶¶ f (x) = M (f (m₁(x, y)), ..., f (m_k(x, y)))        (x, y ? K) f (x) = M \bigl(f (m_{1}(x, y)), \dots, f (m_{k}(x, y))\bigr) \qquad (x, y \in K) , (*)¶ where M is a k-variable operation on the image space Y, m₁,..., m_k are binary operations on X, K ì X K \subset X is closed under the operations m₁,..., m_k, and f : K ? Y f : K \rightarrow Y is considered as an unknown function.¶ The main result of this paper states that if the operations m₁,..., m_k, M satisfy certain commutativity relations and f satisfies (*) then there exists a unique extension of f to the (m₁,..., m_k)-affine hull K^* of K, such that (*) holds over K^*. (The set K^* is defined as the smallest subset of X that contains K and is (m₁,..., m_k)-affine, i.e., if x ? X x \in X , and there exists y ? K^* y \in K^* such that m₁(x, y), ?, m_k(x, y) ? K^* m_{1}(x, y), \ldots, m_{k}(x, y) \in K^* then x ? K^* x \in K^* ). As applications, extension theorems for functional equations on Abelian semigroups, convex sets, and symmetric convex sets are obtained.     

The 2-circle and 2-disk problems on trees

Our purpose here is to consider on a homogeneous tree two Pompeiutype problems which classically have been studied on the plane and on other geometric manifolds. We obtain results which have remarkably the same flavor as classical theorems. Given a homogeneous tree, letd(x, y) be the distance between verticesx andy, and letf be a function on the vertices. For each vertexx and nonnegative integern let Σ _n f(x) be the sum Σ _{d(x, y)=n} f(y) and letB _n f(x)=Σ _{d(x, y)≦n} f(y). The purpose is to study to what extent Σ _n f andB _n f determinef. Since these operators are linear, this is really the study of their kernels. It is easy to find nonzero examples for which Σ _n f orB _n f vanish for one value ofn. What we do here is to study the problem for two values ofn, the 2-circle and the 2-disk problems (in the cases of Σ _n andB _n respectively). We show for which pairs of values there can exist non-zero examples and we classify these examples. We employ the combinatorial techniques useful for studying trees and free groups together with some number theory.     

On nonconvex optimization problems with separated nonconvex variables

A mathematical programming problem is said to have separated nonconvex variables when the variables can be divided into two groups: x=(x ₁,...,x _n ) and y=( y ₁,...,y _n ), such that the objective function and any constraint function is a sum of a convex function of (x, y) jointly and a nonconvex function of x alone. A method is proposed for solving a class of such problems which includes Lipschitz optimization, reverse convex programming problems and also more general nonconvex optimization problems.     

Linear stability of stiff differential equation solvers

When a linear multistep method is used to solve a stiff differential equationy(x)=f(y(x)), producing an approximationy _n toy(x _n ), it is preferable to approximate the valuey(x _n ) in subsequent formulae by a value which exactly satisfies the corrector equation used, rather than by the valuef(y _n ). We prove that the resulting method is stable if the underlying corrector equation is absolutely stable, provided that the residuals obtained in solving successive nonlinear equations remain uniformly bounded.     

Nonhomogeneous recursions and Generalised Continued Fractions

Consider the (n+1)st order nonhomogeneous recursionX _k+n+1=b _k X _k+n +a _k ⁽ⁿ⁾ X _k+n-1+...+a _k ⁽¹⁾ X _k +X _k .Leth be a particular solution, andf ⁽¹⁾,...,f ⁽ⁿ⁾,g independent solutions of the associated homogeneous equation. It is supposed thatg dominatesf ⁽¹⁾,...,f ⁽ⁿ⁾ andh. If we want to calculate a solutiony which is dominated byg, but dominatesf ⁽¹⁾,...,f ⁽ⁿ⁾, then forward and backward recursion are numerically unstable. A stable algorithm is derived if we use results constituting a link between Generalised Continued Fractions and Recursion Relations.     

On symmetric basic sequences in lorentz sequence spaces II

It is shown that if {y _n} is a block of type I of a symmetric basis {x _n} in a Banach spaceX, then {y _n} is equivalent to {x _n} if and only if the closed linear span [y _n] of {y _n} is complemented inX. The result is used to study the symmetric basic sequences of the dual space of a Lorentz sequence spaced(a, p). Let {x _n,f _n} be the unit vector basis ofd(a, p), for 1≤p<+∞. It is shown that every infinite-dimensional subspace ofd(a, p) (respectively, [f _n] has a complemented subspace isomorphic tol _p (respectively,l _q, 1/p+1/q=1 when 1<p<+∞ andc ₀ whenp=1) and numerous other results on complemented subspaces ofd(a, p) and [f _n] are obtained. We also obtain necessary and sufficient conditions such that [f _n] have exactly two non-equivalent symmetric basic sequences. Finally, we exhibit a Banach spaceX with symmetric basis {x _n} such that every symmetric block basic sequence of {x _n} spans a complemented subspace inX butX is not isomorphic to eitherc ₀ orl _p, 1≤p<+∞.     

A Generalized Cubic Functional Equation

In this paper, we determine the general solution of the functional equation f1 （2x ＋ y） ＋ f2（2x - y） = f3（x ＋ y） ＋ f4（x - y） ＋ f5（x） without assuming any regularity condition on the unknown functions f1,f2,f3, f4, f5 ： R→R. The general solution of this equation is obtained by finding the general solution of the functional equations f（2x ＋ y） ＋ f（2x - y） = g（x ＋ y） ＋ g（x - y） ＋ h（x） and f（2x ＋ y） - f（2x - y） = g（x ＋ y） - g（x - y）. The method used for solving these functional equations is elementary but exploits an important result due to Hosszfi. The solution of this functional equation can also be determined in certain type of groups using two important results due to Szekelyhidi.     

Decomposer and associative functional equations

In the previous researches [2,3] b-integer and b-decimal parts of real numbers were introduced and studied by M.H. Hooshmand. The b-parts real functions have many interesting number theoretic explanations, analytic and algebraic properties, and satisfy the functional equation f (f(x) + y - f(y)) = f(x). These functions have led him to a more general topic in semigroups and groups (even in an arbitrary set with a binary operation [4] and the following functional equations have been introduced: Associative equations:

f(xf(yz))=f(f(xy)z),f(xf(yz))=f(f(xy)z)=f(xyz)