Commutativity for a Certain Class of Rings

We first establish the commutativity for the semiprime ring satisfying [x ⁿ , y]x ^r = ±y ^s[x, y ^m]y ^t for all x, y in R, where m, n, r, s, and t are fixed non-negative integers, and further, we investigate the commutativity of rings with unity under some additional hypotheses. Moreover, it is also shown that the above result is true for s-unital rings. Also, we provide some counterexamples which show that the hypotheses of our theorems are not altogether superfluous. The results of this paper generalize some of the well-known commutativity theorems for rings which are right s-unital.     

Derivations with engel conditions in prime and semiprime rings

Let R be a prime ring, I a nonzero ideal of R, d a derivation of R and m, n fixed positive integers. (i) If (d[x, y]) ^m = [x, y] _n for all x, y ∈ I, then R is commutative. (ii) If Char R ≠ 2 and [d(x), d(y)] _m = [x, y] ⁿ for all x, y ∈ I, then R is commutative. Moreover, we also examine the case when R is a semiprime ring.     

Commutativity of One Sided S- Unital Rings

In the present paper we extend some commutativity theorems for rings as follows: Let m > 1, n and k be fixed non- negative integers, and let R be a left or right s- unital ring satisfying the polynomial identity [xⁿ]y ? y^mx^k,x] = 0. Then R is commutative. Under appropriate conditions the commutativity of R has also been proved for the case m = 1.     

On structure and commutativity of certain periodic near rings

In the present paper we first establish decomposition theorems for near rings satisfying either of the properties xy = x^my^pxⁿ or xy = y^mx^pyⁿ, where m≥1, n≥1, p≥1 are positive integers depending on the pair of near ring elements x,y; and further, we investigate commutativity of such near rings. Moreover, it is also shown that under some additional hypotheses, such nearrings turn out to be commutative rings.     

Engel type identities with derivations for right ideals in prime rings

Let R be a prime ring of char R≠2, d a non-zero derivation of R and ρ a non-zero right ideal of R such that [[d(x),d(y)]_n [y,x]_m] = 0 for all x,y ∈ ρ or [[d(x),d(y)]_n d[y,x]_m] = 0 for all x,y ∈ ρ, n, m ≥ 0 are fixed integers. If [ρ,ρ]ρ ≠ 0, then d(ρ)ρ = 0.     

A Class of Simple Derivations of k[x,y]

If k is a field of characteristic zero, c ∈ k?{0}, s, t ≥ 1, and r ≥ 0 are integers, then it is shown that the k-derivation y ^r ? _x  + (y ^s x ^t  + c)? _y of the polynomial algebra k[x, y] is simple.     

Some Simple Derivations of k[x,y]

It is shown that the derivation y ^r ? _x  + (xy ^s  + g)? _y , where 0 ≤ r < s are integers, is a simple derivation of k[x, y], the polynomial ring in two variables over a field k of characteristic zero.     

Onn-th commutators with nilpotent or regular values in rings

Then-th commutator for a,b in a ringR is defined inductively as follows: [a,b]₁=[a,b]=ab−ba and[a,b] _n=[[a,b]₋₁,b]. We characterize the ringsR without non-zero nil right ideals in which[a,b] _nis nilpotent or regular for alla,b∈R. We also examine the case whereR is a semiprime ring with involution in which[t ₁, t₂]_nis nilpotent or regular for all tracest ₁,t₂∈R.     

A convexity-preservingC 2 parametric rational cubic interpolation

Summary AC ² parametric rational cubic interpolantr(t)=x(t) i+y(t) j,t[t ₁,t _n] to data S={(x_j, y_j)|j=1,...,n} is defined in terms of non-negative tension parameters _j ,j=1,...,n–1. LetP be the polygonal line defined by the directed line segments joining the points (x _j ,y _j ),t=1,...,n. Sufficient conditions are derived which ensure thatr(t) is a strictly convex function on strictly left/right winding polygonal line segmentsP. It is then proved that there always exist _j ,j=1,...,n–1 for whichr(t) preserves the local left/righ winding properties of any polygonal lineP. An example application is discussed.This research was supported in part by the natural Sciences and Engineering Research Council of Canada.     

A new continuation method for complementarity problems with uniformP-functions

The complementarity problem with a nonlinear continuous mappingf from the nonnegative orthantR ₊ ⁿ ofR ⁿ intoR ⁿ can be written as the system of equationsF(x, y) = 0 and(x, y) R ₊ ²ⁿ , whereF denotes the mapping from the nonnegative orthantR ₊ ²ⁿ ofR ²ⁿ intoR ₊ ⁿ × Rⁿ defined byF(x, y) = (x ₁y₁,,x_ny_n, f₁(x) – y₁,, f_n(x) – y_n) for every(x, y) R ₊ ²ⁿ . Under the assumption thatf is a uniformP-function, this paper establishes that the mappingF is a homeomorphism ofR ₊ ²ⁿ ontoR ₊ ⁿ × Rⁿ. This result provides a theoretical basis for a new continuation method of tracing the solution curve of the one parameter family of systems of equationsF(x, y) = tF(x ⁰, y⁰) and(x, y) R ₊ ²ⁿ from an arbitrary initial point(x ⁰, y⁰) R ₊ ²ⁿ witht = 1 until the parametert attains 0. This approach is an extension of the one used in the polynomially bounded algorithm recently given by Kojima, Mizuno and Yoshise for solving linear complementarity problems with positive semi-definite matrices.     

On diophantine equations of the form {({\boldsymbol x}-{\boldsymbol a}_{\bf 1})({\boldsymbol x}-{\boldsymbol a}_{\bf 2}) \ldots ({\boldsymbol x}-{\boldsymbol a}_{\boldsymbol k}) + {\boldsymbol r} = {\boldsymbol y}^{ \boldsymbol n}}

Erd?s and Selfridge [3] proved that a product of consecutive integers can never be a perfect power. That is, the equation x(x?+?1)(x?+?2)...(x?+?(m???1))?=?y ⁿ has no solutions in positive integers x,m,n where m, n?>?1 and y?∈?Q. We consider the equation $$ (x-a_1)(x-a_2) \ldots (x-a_k) + r = y^n $$ where 0?≤?a ₁?<?a ₂?<???<?a _k are integers and, with r?∈?Q, n?≥?3 and we prove a finiteness theorem for the number of solutions x in Z, y in Q. Following that, we show that, more interestingly, for every nonzero integer n?>?2 and for any nonzero integer r which is not a perfect n-th power for which the equation admits solutions, k is bounded by an effective bound.     

The combinatorics of a three-line circulant determinant

We study the polynomial , where ω is a primitivepth root of unity. This polynomial arises in CR geometry [1]. We show that it is the determinant of thep×p circulant matrix whose first row is (1, −x,0,…,0,−y,0,…,0), the −y being in positionq+1. Therefore, the coefficients of this polynomial Φ are integers that count certain classes of permutations. We show that all of the permutations that contribute to a fixed monomialx ^ry^s in Φ have the same sign, and we determine that sign. We prove that a monomialx ^ry^s appears in Φ if and only ifp dividesr+sq. Finally, we show that the size of the largest coefficient of the monomials in Φ grows exponentially withp, by proving that the permanent of the circulant whose first row is (1, 1, 0, …, 0, 1, 0, …, 0) is the sum of the absolute values of the monomials in the polynomial Φ. Supported by NSF Postdoctoral research grants.     

Inequalities for Means in Two Variables

We present various new inequalities involving the logarithmic mean L(x,y)=(x-y)/(logx-logy) L(x,y)=(x-y)/(\log{x}-\log{y}) , the identric mean I(x,y)=(1/e)(x^x/y^y)^1/(x-y) I(x,y)=(1/e)(x^x/y^y)^{1/(x-y)} , and the classical arithmetic and geometric means, A(x,y)=(x+y)/2 A(x,y)=(x+y)/2 and G(x,y)=?{xy} G(x,y)=\sqrt{xy} . In particular, we prove the following conjecture, which was published in 1986 in this journal. If M_r(x,y) = (x^r/2+y^r/2)^1/r(r 1 0) M_r(x,y)= (x^r/2+y^r/2)^{1/r}(r\neq{0}) denotes the power mean of order r, then $ M_c(x,y)<\frac{1}{2}(L(x,y)+I(x,y)) {(x,y>0,\, x\neq{y})} $ M_c(x,y)<\frac{1}{2}(L(x,y)+I(x,y)) {(x,y>0,\, x\neq{y})} with the best possible parameter c=(log2)/(1+log2) c=(\log{2})/(1+\log{2}) .     

Extensions of the measurable choice theorem by means of forcing

Using the method of forcing of set theory, we prove the following two theorems on the existence of measurable choice functions: LetT be the closed unit interval [0,1] and letm be the usual Lebesgue measure defined on the Borel subsets ofT. Theorem1. LetS⊂T×T be a Borel set such that for alltεT,S _t ^def={x|(t,x)εS} is countable and non-empty. Then there exists a countable series of Lebesgue-measurable functionsf _n: T→T such thatS _t={f_n(t)|nεω} for alltε[0,1],W _x={y|(x,y)εW} is uncountable. Then there exists a functionh:[0,1]×[0,1]→W with the following properties: (a) for each xε[0,1], the functionh(x,·) is one-one and ontoW _x and is Borel measurable; (b) for eachy, h(·, y) is Lebesgue measurable; (c) the functionh is Lebesgue measurable.     

Center conditions III: Parametric and model center problems

We consider an Abel equation (*)y’=p(x)y ² +q(x)y ³ withp(x), q(x) polynomials inx. A center condition for (*) (closely related to the classical center condition for polynomial vector fields on the plane) is thaty ₀=y(0)≡y(1) for any solutiony(x) of (*). Folowing [7], we consider a parametric version of this condition: an equation (**)y’=p(x)y ² +εq(x)y ³ p, q as above, ε ∈ ℂ, is said to have a parametric center, if for any ɛ and for any solutiony(ɛ,x) of (**)y(ɛ, 0)≡y(ɛ, 1).. We give another proof of the fact, shown in [6], that the parametric center condition implies vanishing of all the momentsm _k (1), wherem _k (x)=∫ ₀ ^{x pk} (t)q(t)(dt),P(x)=∫ ₀ ^x p(t)dt. We investigate the structure of zeroes ofm _k (x) and generalize a “canonical representation” ofm _k (x) given in [7]. On this base we prove in some additional cases a composition conjecture, stated in [6, 7] for a parametric center problem. The research of the first and the third author was supported by the Israel Science Foundation, Grant No. 101/95-1 and by the Minerva Foundation.     

ON THE NORMALITY OF MONOMIAL IDEALS OF MIXED PRODUCTS

Let R = K[x, y] be a polynomial ring in two disjoint sets of variables x, y over a field K. We study ideals of mixed products L = I_kJ_r + I_sJ_t such that k + r = s + t, where I_k (resp. J_r ) denotes the ideal of R generated by the square-free monomials of degree k (resp. r) in the x (resp. y ) variables. Our main result is a characterization of when a given ideal L of mixed products is normal.

     

Visualizing the Relationship between Two Time Series by Hierarchical Smoothing Models

Abstract

Let x(t_i), y(t_i) be two time series such that y(t_i) = μ(t_i, x) + ε_i, where μ is a smooth function and ε_i is a zero mean stationary process. Which model may be assumed for μ depends on the subject specific context. This article was motivated by questions raised in the context of musical performance theory. The general problem is to understand the relationship between the symbolic structure of a music score and its performance. Musical structure typically consists of a hierarchy of global and local structures. This motivates the definition of hierarchical smoothing models (or HISMOOTH models) that are characterized by a hierarchy of bandwidths b ₁ > b ₂ > … > b_M and a vector of coefficients β ∈ R^M. The expected value μ(t_i x) = E[y(t_i)‖x] is equal to a weighted sum of smoothed versions of x. The “errors” ε_i are modeled by a Gaussian process that may exhibit long memory. More generally, we may observe a collection of time series y_r (r = 1, …, N) that are related to a common time series x by y_r(t_i) = μ _r(t_i, x) + ε_{r, i} where ε _r are independent error processes. For repeated time series, HISMOOTH models lead to a visual and formal classification into clusters that can be interpreted in terms of the relationship to x. An analysis of tempo curves from 28 performances of Schumann's “Träumerei” op. 15/7 illustrates the method. In particular, similarities and differences of “melodic styles” can be identified.     

On the stability of the quadratic functional equation on bounded domains

A result of Skof and Terracini will be generalized; More precisely, we will prove that if a functionf : [-t, t]ⁿ →E satisfies the inequality (1) for some δ > 0 and for allx, y ∈ [-t, t]ⁿ withx + y, x - y ∈ [-t, t]ⁿ, then there exists a quadratic functionq: ℝⁿ →E such that ∥f(x) -q(x)∥ < (2912n² + 1872n + 334)δ for anyx ∈ [-t, t] ⁿ .     

Edge-isoperimetric inequalities in the grid

The grid graph is the graph on [k] ⁿ ={0,...,k–1} ⁿ in whichx=(x _i ) ₁ ⁿ is joined toy=(y _i ) ₁ ⁿ if for somei we have |x _i –y _i |=1 andx _j =y _j for allji. In this paper we give a lower bound for the number of edges between a subset of [k] ⁿ of given cardinality and its complement. The bound we obtain is essentially best possible. In particular, we show that ifA[k] ⁿ satisfiesk ⁿ /4|A|3k ⁿ /4 then there are at leastk ^n–1 edges betweenA and its complement.Our result is apparently the first example of an isoperimetric inequality for which the extremal sets do not form a nested family.We also give a best possible upper bound for the number of edges spanned by a subset of [k] ⁿ of given cardinality. In particular, forr=1,...,k we show that ifA[k] ⁿ satisfies |A|r ⁿ then the subgraph of [k] ⁿ induced byA has average degree at most 2n(1–1/r).Research partially supported by NSF Grant DMS-8806097     

Simplicity of Some Derivations of k[x,y]

If k is a field of characteristic zero, c ∈ k?0, and 1 ≤ s ≤ r are integers such that either r ? s + 1 divides s or s divides r ? s + 1, then it is shown that the derivation y ^r ? _x  + (xy ^s  + c)? _y of the polynomial ring k[x, y] is simple.