On the dualities between categories of k-completely regular modules

A semigroup S is called collapsing if there exists a positive integer n such that for every subset of n elements in S at least two distinct words of length n on these letters are equal in S. Let U(A) denote the group of units of an associative algebra A over an infinite field of characteristic p > 0. We show that if A is unitally generated by its nilpotent elements then the following conditions are equivalent: U(A) is collapsing; U(A) satisfies some semigroup identity; U(A) satisfies an Engel identity; A satisfies an Engel identity when viewed as a Lie algebra; and, A satisfies a Morse identity. The characteristic zero analogue of this result was proved by the author in a previous paper.     

A Matrix Setting for the q-Schur Algebra

The Schur algebra S(n, r) has a basis (described in [6, 2.3])consisting of certain elements _i,j, where i, jI(n, r), the setof all ordered r-tuples of elements from the set n={1, 2, ...,n}. The multiplication of two such basis elements is given bya formula known as Schur's product rule. In recent years, aq-analogue S_q(n, r) of the Schur algebra has been investigatedby a number of authors, particularly Dipper and James [3, 4].The main result of the present paper, Theorem 3.6, shows howto embed the q-Schur algebra in the rth tensor power T^r(M_n)of the nxn matrix ring. This embedding allows products in theq-Schur algebra to be computed in a straightforward manner,and gives a method for generalising results on S(n, r) to S_q(n,r). In particular we shall make use of this embedding in subsequentwork to prove a straightening formula in S_q(n, r) which generalisesthe straightening formula for codeterminants due to Woodcock[12]. We shall be working mainly with three types of algebra: thequantized enveloping algebra U(gl_n) corresponding to the Liealgebra gl_n, the q-Schur algebra S_q(n, r), and the Hecke algebra,H(A_r–1). It is often convenient, in the case of the q-Schuralgebra and the Hecke algebra, to introduce a square root ofthe usual parameter q which will be denoted by v, as in [5].This corresponds to the parameter v in U(gl_n). We shall denotethis ‘extended’ version of the q-Schur algebra byS_v(n, r), and we shall usually refer to it as the v-Schur algebra.All three algebras are associative and have multiplicative identities,and the base field will be the field of rational functions,Q(v), unless otherwise stated. The symbols n and r shall bereserved for the integers given in the definitions of thesethree algebras.     

A Lie Theoretic Approach to Thompson's Theorems on Singular Values-Diagonal Elements and Some Related Results

Thompson's famous theorems on singular values–diagonalelements of the orbit of an nxn matrix A under the action (1)U(n) U(n) where A is complex, (2) SO(n) SO(n), where A isreal, (3) O(n) O(n) where A is real are fully examined. Coupledwith Kostant's result, the real semi-simple Lie algebra so_n,n yields (2) and hence (3) and the sufficient part (the hardpart) of (1). In other words, the curious subtracted term(s)are well explained. Although the diagonal elements correspondingto (1) do not form a convex set in Cⁿ, the projection of thediagonal elements into Rⁿ (or iRⁿ) is convex and the characterizationof the projection is related to weak majorization. An elementaryproof is given for this hidden convexity result. Equivalentstatements in terms of the Hadamard product are also given.The real simple Lie algebra su_{n, n} shows that such a convexityresult fits into the framework of Kostant's result. Convexityproperties and torus relations are studied. Thompson's resultson the convex hull of matrices (complex or real) with prescribedsingular values, as well as Hermitian matrices (real symmetricmatrices) with prescribed eigenvalues, are generalized in thecontext of Lie theory. Also considered are the real simple Liealgebras so_{p, q} and so_{p, q}, p < q, which yield the rectangularcases. It is proved that the real part and the imaginary partof the diagonal elements of complex symmetric matrices withprescribed singular values are identical to a convex set inRⁿ and the characterization is related to weak majorization.The convex hull of complex symmetric matrices and the convexhull of complex skew symmetric matrices with prescribed singularvalues are given. Some questions are asked.     

Rigid representations of a double quiver of type A, and Richardson elements in seaweed Lie algebras

In this paper, we show that there is always an open adjointorbit in the nilpotent radical of a seaweed Lie algebra in gl_n(k),thus answering positively in this gl_n(k) case to a questionraised independently by Michel Duflo and Dmitri Panyushev. Theproof gives an explicit construction, using -filtered modulesof quasi-hereditary algebras arising from quotients of the doubleof quivers of type A. An example of a seaweed Lie algebra ina simple Lie algebra of type E₈ not admitting an open orbitin its nilpotent radical is given.     

On Ratio Inequalities for Heat Content

Let U be a domain, convex in x and symmetric about the y-axis,which is contained in a centered and oriented rectangle S. Itis proved that H_t(U⁺)/H_t(U)H_t(S⁺)/H_t(S) where H_t stands forheat content, that is, the remaining heat in the domain at timet if it initially has uniform temperature 1, with Dirichletboundary conditions, where A⁺=A{(x,y):x>0}. It is also shownthat the analog of this inequality holds for some other Schrödingeroperators.     

Recognition of finite groups by a set of orders of their elements

For G a finite group, ω(G) denotes the set of orders of elements in G. If ω is a subset of the set of natural numbers, h(ω) stands for the number of nonisomorphic groups G such that ω(G)=ω. We say that G is recognizable (by ω(G)) if h(ω(G))=1. G is almost recognizable (resp., nonrecognizable) if h(ω(G)) is finite (resp., infinite). It is shown that almost simple groups PGL_n(q) are nonrecognizable for infinitely many pairs (n, q). It is also proved that a simple group S₄(7) is recognizable, whereas A₁₀, U₃(5), U₃(7), U₄(2), and U₅(2) are not. From this, the following theorem is derived. Let G be a finite simple group such that every prime divisor of its order is at most 11. Then one of the following holds: (i) G is isomorphic to A₅, A₇, A₈, A₉, A₁₁, A₁₂, L₂(q), q=7, 8, 11, 49, L₃(4), S₄(7), U₄(3), U₆(2), M₁₁, M₁₂, M₂₂, HS, or M^cL, and G is recognizable by the set ω(G); (ii) G is isomorphic to A₆, A₁₀, U₃(3), U₄(2), U₅(2), U₃(5), or J₂, and G is nonrecognizable; (iii) G is isomorphic to S₆(2) or O ₈ ⁺ (2), and h(ω(G))=2. Supported by RFFR grant No. 96-01-01893. Translated fromAlgebra i Logika, Vol. 37, No. 6, pp. 651–666, November–December, 1998.     

Pointwise bounded approximation and Dirichlet algebras

Those open sets U of S² for which A(U) is pointwise boundedly dense in H^∞(U) are characterized in terms of analytic capacity. It is also shown that the real parts of the functions in A(U) are uniformly dense in C_R(∂U) if and only if each component of U is simply connected and A(U) is pointwise boundedly dense in H^∞(U).     

Reduction of CalderON Commutators

Let A₁,..., A_n be Lipschitz functions on R such that A'₁,...,A'_nVMO. We show that on any bounded interval, the Calderóncommutator associated with the kernel (A₁(x)–A₁(y)) ...(A_n(x) – A_n(y))/(x–y) ⁿ¹ is a compact perturbationof , where H is the Hilberttransform. 1991 Mathematics Subject Classification 47B38, 47B47,47G10, 45E99.     

Indecomposable flat cotorsion modules

An additive functor from the category of flat right R-modulesto the category of abelian groups is continuous if it is isomorphicto a functor of the form–_R M, where M is a left R-module.It is shown that for any simple subfunctor A of– M thereis a unique indecomposable flat cotorsion module U_R for whichA(U)0. It is also proved that every subfunctor of a continuousfunctor contains a simple subfunctor. This implies that everyflat right R-module may be purely embedded into a product ofindecomposable flat cotorsion modules. If CE(R) is the cotorsion envelope of R_R and S= End;_R CE(R),then a local ring monomorphism is constructed from R/J(R) toS/J(S). This local morphism of rings is used to associate asemiperfect ring to any semilocal ring. It also proved thatif R is a semilocal ring and M a simple left R-module, thenthe functor–_R M on the category of flat right R-modulesis uniform, and therefore contains a unique simple subfunctor.     

Implicit finite difference schemes for a linear model of well-reservoir coupling

Standard reservoir models usually consider wells as Dirac measuresover an interval length. Moreover, the well-reservoir couplingis taken into account under quite simplified assumptions. Mostrecently, however, attention has been drawn to the fact thatin some situations, such as those related to non-vertical wells,these simplifications do not allow us to model some relevantmechanisms of the coupled flow. Therefore, more complex alternativemodels have been proposed recently in the oil reservoir simulationliterature. A linearized version for the well-reservoir couplingcan be written, in an appropriate functional setting, in theform U'(t) + AU(t) = F(t). In this work we discuss implicitin time discretizations of this equation, of the form { Uⁿ⁺¹_h - Uⁿ_h÷+A_hUⁿ⁺¹_h = Fⁿ⁼¹_h, U⁰_h=U_0.h We propose two different approximations, corresponding to first-and second-order spatial truncation errors, and we establishthe convergence of both approximations.     

Lie superalgebras whose enveloping algebras satisfy a non-matrix polynomial identity

Let L be a Lie superalgebra with its enveloping algebra U(L) over a field F. A polynomial identity is called non-matrix if it is not satisfied by the algebra of 2×2 matrices over F. We characterize L when U(L) satisfies a non-matrix polynomial identity. We also characterize L when U(L) is Lie solvable, Lie nilpotent, or Lie super-nilpotent.     

Type 2 Semi-Algebras of Continuous Functions

A semi-algebra of continuous functions is a cone A of continuousreal functions on a compact Hausdorff space X such that A containsthe products of its elements. A cone A is said to be of typen if fA implies fⁿ(1 + f)^–1 A. Uniformly closed semi-algebrasof types 0 and 1 have long been characterized in a manner analogousto the Stone–Weierstrass theorem, but, except for thecase when A is generated by a single function, little has beenknown about type 2. Here, progress is reported on two problems.The first is the characterization of those continuous linearfunctionals on C(X) that determine semi-algebras of type 2.The second is the determination of the type of the tensor productof two type 1 semi-algebras. 1991 Mathematics Subject Classification:46J10.     

Generation of classical groups over finite fields

Let U_n(V) and Sp_n(V) denote the unitary group and the symplectic group of the n dimensional vector space V over a finite field of characteristic not 2, respectively. Assume that the hyperbolic rank of U_n(V) is at least one. Then U_n(V) is generated by 4 elements and Sp_n(V) by 3 elements. Further, U_2m+1(V) is generated by 3 elements and Sp_4m(V) by 2 elements.     

Generalized euler-type partition identities

Let A and S be subsets of the natural numbers. Let A′(n) be the number of partitions of n where each part appears exactly m times for some m?A. Let S(n) be the number of partitions of n into parts which are elements of S.     

Smoothness Properties of the Unit Ball in a JB*-Triple

An element a of norm one in a JB^*-triple A is said to be smoothif there exists a unique element x in the unit ball A₁^* of thedual A^* of A at which a attains its norm, and is said to beFréchet-smooth if, in addition, any sequence (x_n) ofelements in A₁^* for which (x_n(a)) converges to one necessarilyconverges in norm to x. The sequence (a²ⁿ⁺¹) of odd powers ofa converges in the weak^*-topology to a tripotent u(a) in theJBW^*-envelope A^** of A. It is shown that a is smooth if andonly if u(a) is a minimal tripotent in A^** and a is Fréchet-smoothif and only if, in addition, u(a) lies in A.     

Hypersurfaces in a Unit Sphere Sn+1(1) with Constant Scalar Curvature

The paper considers n-dimensional hypersurfaces with constantscalar curvature of a unit sphere S^n–1(1). The hypersurfaceS^k(c₁)xS^n–k(c₂) in a unit sphere Sⁿ⁺¹(1) is characterized,and it is shown that there exist many compact hypersurfaceswith constant scalar curvature in a unit sphere Sⁿ⁺¹(1) whichare not congruent to each other in it. In particular, it isproved that if M is an n-dimensional (n > 3) complete locallyconformally flat hypersurface with constant scalar curvaturen(n–1)r in a unit sphere Sⁿ⁺¹(1), then r > 1–2/n,and (1) when r (n–2)/(n–1), if then M is isometric to S¹xS^n–1(c),where S is the squared norm of the second fundamental form ofM; (2) there are no complete hypersurfaces in Sⁿ⁺¹(1) with constantscalar curvature n(n–1)r and with two distinct principalcurvatures, one of which is simple, such that r = (n–2)/(n–1)and      

A Convergence Acceleration Method based on a good Estimation of the Absolute value of the Error

Suppose a sequence (S_n)_n, converges to a limit S. In this paperwe construct, from a good estimate (D_n)_n of (|S_n–S|),an acceleration method for (S_n)_n based on: (a) an automaticselection between two transformations if (D_n)_n converges logarithmically;(b) composite sequence transformations if (D_n)_n converges linearly.     

SCALED ASYMPTOTICS FOR SOME q-SERIES

This work, investigates the asymptotics for Euler’s q-exponentialE_q(z), Ramanujan’s function A_q(z), Jackson’s q-Besselfunction J_v⁽²⁾ (z; q), the Stieltjes–Wigert orthogonalpolynomials S_n(x; q) and q-Laguerre polynomials L_n⁽⁾ (x; q)as q approaches 1.     

A continuous function space with a Faber basis

Let be compact with #S=∞ and let C(S) be the set of all real continuous functions on S. We ask for an algebraic polynomial sequence (P_n)_n=0^∞ with deg P_n=n such that every fC(S) has a unique representation f=∑_i=0^∞ α_iP_i and call such a basis Faber basis. In the special case of , 0<q<1, we prove the existence of such a basis. A special orthonormal Faber basis is given by the so-called little q-Legendre polynomials. Moreover, these polynomials state an example with A(S_q)≠U(S_q)=C(S_q), where A(S_q) is the so-called Wiener algebra and U(S_q) is the set of all fC(S_q) which are uniquely represented by its Fourier series.     

Almost homomorphisms between unital <Emphasis Type="Italic">C</Emphasis>*-algebras: A fixed point approach

Let A , B be two unital C*-algebras. By using fixed pint methods, we prove that every almost unital almost linear mapping h : A → B which satisfies h(2 n uy) = h(2 n u)h(y) for all u ∈ U(A), all y ∈ A, and all n = 0, 1, 2, … , is a homomorphism. Also, we establish the generalized Hyers-Ulam-Rassias stability of *-homomorphisms on unital C*-algebras.