Decomposition tree of a lexicographic product of binary structures

Given a set S and a positive integer k, a binary structure is a function . The set S is denoted by V(B) and the integer k is denoted by . With each subset X of V(B) associate the binary substructure B[X] of B induced by X defined by B[X](x,y)=B(x,y) for any x≠y∈X. A subset X of V(B) is a clan of B if for any x,y∈X and v∈V(B)?X, B(x,v)=B(y,v) and B(v,x)=B(v,y). A subset X of V(B) is a hyperclan of B if X is a clan of B satisfying: for every clan Y of B, if X∩Y≠0?, then X⊆Y or Y⊆X. With each binary structure B associate the family Π(B) of the maximal proper and nonempty hyperclans under inclusion of B. The decomposition tree of a binary structure B is constituted by the hyperclans X of B such that Π(B[X])≠0? and by the elements of Π(B[X]). Given binary structures B and C such that , the lexicographic product B⌊C⌋ of C by B is defined on V(B)×V(C) as follows. For any (x,y)≠(x^′,y^′)∈V(B)×V(C), B⌊C⌋((x,x^′),(y,y^′))=B(x,y) if x≠y and B⌊C⌋((x,x^′),(y,y^′))=C(x^′,y^′) if x=y. The decomposition tree of the lexicographic product B⌊C⌋ is described from the decomposition trees of B and C.     

Varieties of strange plane curves

Abstract

Let A be a commutative ring with identity, let X, Y be indeterminates and let F(X,Y), G(X, Y) ∈ A[X, Y] be homogeneous. Then the pair F(X, Y), G(X, Y) is said to be radical preserving with respect to A if Rad((F(x, y), G(x, y))R) = Rad((x,y)R) for each A-algebra R and each pair of elements x, y in R. It is shown that infinite sequences of pairwise radical preserving polynomials can be obtained by homogenizing cyclotomic polynomials, and that under suitable conditions on a ?-graded ring A these can be used to produce an infinite set of homogeneous prime ideals between two given homogeneous prime ideals P ? Q of A such that ht(Q/P) = 2.     

Mappings Preserving Submodules of Hilbert C*-Modules

A Hilbert module over a C*-algebra B is a right B-module X,equipped with an inner product ·, · which is linearover B in the second factor, such that X is a Banach space withthe norm ||x||:=||x, x||^1/2. (We refer to [8] for the basictheory of Hilbert modules; the basic example for us will beX=B with the inner product x, y=x*y.) We denote by B(X) thealgebra of all bounded linear operators on X, and we denoteby L(X) the C*-algebra of all adjointable operators. (In thebasic example X=B, L(X) is just the multiplier algebra of B.)Let A be a C*-subalgebra of L(X), so that X is an A-B-bimodule.We always assume that A is nondegenerate in the sense that [AX]=X,where [AX] denotes the closed linear span of AX. Denote by A_X the algebra of all mappings on X of the form (1.1) where m is an integer and a_iA, b_iB for all i. Mappings of form(1.1) will be called elementary, and this paper is concernedwith the question of which mappings on X can be approximatedby elementary mappings in the point norm topology.     

An eigenvalue problem for the numerical range of a bounded linear operator

LetX be a complex Lebesgue space with a unique duality mapJ fromX toX ^*, the conjugate space ofX. LetA be a bounded linear operator onX. In this paper we obtain a non-linear eigenvalue problem for (A)=sup{Re: W(A} whereW(A)={J(x)A(x)) : x=1}, under the assumption that (A) and the convex hull ofW(A) for some linear operatorsA onl _p , 2<p<.     

The Density of Linear Symplectic Cocycles with Simple Lyapunov Spectrum in YIC（X, SL（2, R））

Let （X, G（X）, m） be a probability space with a-algebra G（X） and probability measure m. The set V in G is called P-admissible, provided that for any positive integer n and positive-measure set Vn∈ contained in V, there exists a Zn∈G such that Zn belong to Vn and 0 〈 m（Zn） 〈 1/n. Let T be an ergodic automorphism of （X, G） preserving m, and A belong to the space of linear measurable symplectic cocycles     

On the vector space of 0-configurations

Let α be a rational-valued set-function on then-element sexX i.e. α(B) εQ for everyB ⫅X. We say that α defines a 0-configuration with respect toA⫅2 ^x if for everyA εA we have α(B)=0. The 0-configurations form a vector space of dimension 2 ⁿ − |A| (Theorem 1). Let 0 ≦t<k ≦n and letA={A ⫅X: |A| ≦t}. We show that in this case the 0-configurations satisfying α(B)=0 for |B|>k form a vector space of dimension , we exhibit a basis for this space (Theorem 4). Also a result of Frankl, Wilson [3] is strengthened (Theorem 6).     

Rationally convex sets on the unit sphere in ℂ2

Let X be a rationally convex compact subset of the unit sphere S in ?², of three-dimensional measure zero. Denote by R(X) the uniform closure on X of the space of functions P/Q, where P and Q are polynomials and Q≠0 on X. When does R(X)=C(X)? Our work makes use of the kernel function for the $\bar{\delta}_{b}Let X be a rationally convex compact subset of the unit sphere S in ℂ², of three-dimensional measure zero. Denote by R(X) the uniform closure on X of the space of functions P/Q, where P and Q are polynomials and Q≠0 on X. When does R(X)=C(X)? Our work makes use of the kernel function for the operator on S, introduced by Henkin in [5] and builds on results obtained in Anderson–Izzo–Wermer [3]. We define a real-valued function ε _X on the open unit ball intB, with ε _X (z,w) tending to 0 as (z,w) tends to X. We give a growth condition on ε _X (z,w) as (z,w) approaches X, and show that this condition is sufficient for R(X)=C(X) (Theorem 1.1). In Section 4, we consider a class of sets X which are limits of a family of Levi-flat hypersurfaces in intB. For each compact set Y in ℂ², we denote the rationally convex hull of Y by . A general reference is Rudin [8] or Aleksandrov [1].     

Counting rational points on hypersurfaces of low dimension

Let N(X,B) be the number of rational points of height at most B on a variety X⊂Pn defined over Q. We establish new upper bounds for N(X,B) for hypersurfaces of dimension at most four. We also study N(X^′,B) for the open complements X^′ of all lines or all planes on such hypersurfaces. One of the goals is to show that for smooth hypersurfaces in P⁵ defined by a form F(x₀,…,x₅) of degree d?9. This improves upon a classical upper estimate of Hua for the form .     

Counting Cocomponents of a Topological Space

Dualizing the statement about a number of components of a topological space X, we say that, for a natural number n, the space X has at most n cocomponents if every continuous map f:X ⁿ⁺¹X factorizes through X ⁿ , i.e., f depends on at most n coordinates. We construct metrizable spaces X ₁,X ₂,X ₃ such that(1) X ₁ does not have finitely many cocomponents but every continuous map X ₁ X ₁ depends only on finitely many coordinates;(2) X ₂=A×B for rigid spaces A,B and there is a continuous map X ₂ X ₂ depending on all coordinates;(3) X ₃(n) has precisely n cocomponents but it cannot be expressed as A×B with |A|>1 and |B|>1.     

The generalized singular value decomposition and the general (A-λB)-problem

In the study of the canonical structure of matrix pencilsA-B, the column and row nullities ofA andB and the possible common nullspaces give information about the Kronecker structure ofA-B. It is shown how to extract the significant information concerning these nullspaces from the generalized singular value decomposition (GSVD) of the matrix pair (A, B), These properties are the basis for the RGSVD-algorithm that will be published elsewhere [9]. An algorithm for the numerical computation of the transformation matrices (V, X), used in an equivalence transformationV ^H (A-B)X, is presented and discussed. A generalizedQZ-decomposition (GQZD) of a matrix pair (A, B) is also formulated, giving a unitary equivalent transformationV ^H (A-B)Q. If during the computationsX gets too illconditioned we switch to the GQZD, sacrificing a simple diagonal structure of the transformedB-part in order to maintain stability.Dedicated to Germund Dahlquist, on the occasion of his 60th birthday.     

A Finite Algorithm for Solving Infinite Dimensional Optimization Problems

We consider the general optimization problem (P) of selecting a continuous function x over a -compact Hausdorff space T to a metric space A, from a feasible region X of such functions, so as to minimize a functional c on X. We require that X consist of a closed equicontinuous family of functions lying in the product (over T) of compact subsets Y _t of A. (An important special case is the optimal control problem of finding a continuous time control function x that minimizes its associated discounted cost c(x) over the infinite horizon.) Relative to the uniform-on-compacta topology on the function space C(T,A) of continuous functions from T to A, the feasible region X is compact. Thus optimal solutions x ^* to (P) exist under the assumption that c is continuous. We wish to approximate such an x ^* by optimal solutions to a net {P _i }, iI, of approximating problems of the form min_{xX i} c i(x) for each iI, where (1) the net of sets {X _i } _I converges to X in the sense of Kuratowski and (2) the net {c _i } _I of functions converges to c uniformly on X. We show that for large i, any optimal solution x ^* _i to the approximating problem (P _i ) arbitrarily well approximates some optimal solution x ^* to (P). It follows that if (P) is well-posed, i.e., limsupX _i ^* is a singleton {x ^*}, then any net {x _i ^*} _I of (P _i )-optimal solutions converges in C(T,A) to x ^*. For this case, we construct a finite algorithm with the following property: given any prespecified error and any compact subset Q of T, our algorithm computes an i in I and an associated x _i ^* in X _i ^* which is within of x ^* on Q. We illustrate the theory and algorithm with a problem in continuous time production control over an infinite horizon.     

Existence of log canonical flips and a special LMMP

Let (X/Z,B+A) be a Q-factorial dlt pair where B,A??0 are Q-divisors and K _X +B+A?? _Q 0/Z. We prove that any LMMP/Z on K _X +B with scaling of an ample/Z divisor terminates with a good log minimal model or a Mori fibre space. We show that a more general statement follows from the ACC for lc thresholds. An immediate corollary of these results is that log flips exist for log canonical pairs.     

Cohomology of toric bundles

Let p: E B be a principal bundle with fibre and structure group the torus T ( ^*)ⁿ over a topological space B. Let X be a nonsingular projective T-toric variety. One has the X-bundle : E(X) B where E(X) = E × _T X, ([e,x]) = p(e). This is a Zariski locally trivial fibre bundle in case p: E B is algebraic. The purpose of this note is to describe (i) the singular cohomology ring of E(X) as an H ^* (B;)-algebra, (ii) the topological K-ring of K ^* (E(X)) as a K ^* (B)-algebra when B is compact. When p : E B is algebraic over an irreducible, nonsingular, noetherian scheme over , we describe (iii) the Chow ring of A ^* (E(X)) as an A ^* (B)-algebra, and (iv) the Grothendieck ring $\mathcal K$⁰ (E (X)) of algebraic vector bundles on E (X) as a $\mathcal K$⁰(B)-algebra.     

Commuting traces and commutativity preserving maps on triangular algebras

Let A be a triangular algebra. The problem of describing the form of a bilinear map satisfying B(x,x)x=xB(x,x) for all xA is considered. As an application, commutativity preserving maps and Lie isomorphisms of certain triangular algebras (e.g., upper triangular matrix algebras and nest algebras) are determined.     

Indexes and Special Discretization Methods for Linear partial Differential Algebraic Equations

Linear partial differential algebraic equations (PDAEs) of the form Au _t(t, x) + Bu _xx(t, x) + Cu(t, x) = f(t, x) are studied where at least one of the matrices A, B R ^n×n is singular. For these systems we introduce a uniform differential time index and a differential space index. We show that in contrast to problems with regular matrices A and B the initial conditions and/or boundary conditions for problems with singular matrices A and B have to fulfill certain consistency conditions. Furthermore, two numerical methods for solving PDAEs are considered. In two theorems it is shown that there is a strong dependence of the order of convergence on these indexes. We present examples for the calculation of the order of convergence and give results of numerical calculations for several aspects encountered in the numerical solution of PDAEs.     

On intertwining operators

LetB(H) denote the algebra of operators on the Hilbert spaceH into itself. GivenA,BB(H), defineC (A, B) andR (A, B):B(H)B(H) byC (A, B) X=AX–XB andR(A, B) X=AXB–X. Our purpose in this note is a twofold one. we show firstly that ifA andB ^*B (H) are dominant operators such that the pure part ofB has non-trivial kernel, thenC ⁿ (A, B) X=0, n some natural number, implies thatC (A, B)X=C(A ^*,B ^*)X=0. Secondly, it is shown that ifA andB ^* are contractions withC ₀ completely non-unitary parts, thenR ⁿ (A, B) X=0 for some natural numbern implies thatR (A, B) X=R (A ^*,B ^*)X=C (A, B ^*)X=C (A ^*,B) X=0. In the particular case in whichX is of the Hilbert—Schmidt class, it is shown that his result extends to all contractionsA andB.     

Monotone optimal decision rules and their computation

For a given objective functionw(x, a) onX × A, a maximizinga=(x) has to be determined for eachx in the totally ordered setX. We give conditions onw such that there is a monotone which can be computed recursively ifA is finite.     

Random symmetric polynomials

Let {Q⁽ⁿ⁾(x₁,...,x_n)} be a sequence of symmetric polynomials having a fixed degree equal to k. Let {X_n1,...,X_nn}, n k, be some sequence of series of random variables (r.v.). We form the sequence of r.v. Y_n=Q⁽ⁿ⁾(X_n1, ... X_nn), n k One obtains limit theorems for the sequence Y_n, under very general assumptions.Translated from Veroyatnostnye Raspredeleniya i Matematicheskaya Statistika, pp. 170–188, 1986.     

Classification Des Ideaux a Droite De A1(k)

Les études récentes sur les idéaux àdroite de A₁(k), la première algèbre de Weyl surun corps algébriquement clos et de caractéristiquenulle k, nous montrent que : pour tout idéal I 0 àdroite de A₁(k), il existe x Q = frac(A₁(k)), et V V telsque : I = xD(R, V) o V est l'ensemble des sous-espaces primairementdécomposables de k[t] = R, et D(R, V), l'idéalà droite {d A₁(k/d(R V}. Dans cet article nous montreronsprincipalement que: pour tout 0 I idéal à droitede A₁(k, !n N, (x, ) Q^* x Aut_k(A₁(k)) : I = x(D(R, O(X_n))),où X_n est la courbe d'algèbre des fonctions régulières: O(X_n = k+tⁿ⁺¹k[t]. La forme des idéaux décriteci-dessus permet de voir dans une hypothèse de Letzteret Makar-Limanov, pour deux courbes algébriques affinesX et X' on a : D(XD(X') co dim D(X = co dim D(X'). Recent studies on right ideals of the first Weyl algebra A₁(k)over an algebraic closed field k with characteristic zero showthat: for each right ideal I 0 of A₁(k), there exist x Q =fracA₁(k)) and a primary decomposable sub-space V of k[t] suchthat I=xD(R,V), where D(R,V) : = {d A₁(k)/d(R) V} is a rightideal of A₁(k). In this paper, we show that for all right idealsI 0 of A₁(k), !n N, (x, ) Q^* x Aut_k(A₁(k)) : I = x(D(R, O(X_n))),where X_n denotes the affine algebraic curve with ring of regularfunctions O(X_n=k+tⁿ⁺¹k[t]. With ideals as described above, onecan easily see, under a hypothesis given by Letzter and Makar-Limanov,that for two affine algebraic curves X and X', D(X)D(X') codim D(X) = co dim D(X'). 2000 Mathematics Subject Classification16S32.