Showing papers in "Colloquium Mathematicum in 2001"

Computation of some examples of Brown's spectral measure in free probability

Abstract: We use free probability techniques for computing spectra and Brown measures of some non hermitian operators in finite von Neumann algebras. Examples include u_n+u_oo where u_n and u_oo are the generators of Z_n and Z respectively, in the free product group Z_n*Z, or elliptic elements, of the form S_a+iS_b where S_a and S_b are free semi-circular elements of variance a and b. We give some pictorial evidence for connections with spectra of random matrices.

Lyapunov functions and $L^{p}$-estimates for a class of reaction-diffusion systems

Abstract: In this paper we give a sufficient condition for the existence of a Lyapunov function for the system at = ∇(k(a, c)∇a− h(a, c)∇c), x ∈ Ω, t > 0 εct = kc∆c− f(c)c + g(a, c), x ∈ Ω, t > 0 } (1) for Ω ⊂ IR completed with either a = c = 0, or ∂a ∂n = ∂c ∂n = 0, or k(a, c) ∂a ∂n = h(a, c) ∂c ∂n , c = 0 on ∂Ω× {t > 0}. Furthermore we study the asymptotic behaviour of the solution and give some uniform L−estimates.

Estimates for homological dimension of configuration spaces of graphs

Abstract: We show that the homological dimension of a configuration space of a graph Γ is estimated from above by the number b of vertices in Γ whose valence is greater than 2. We show that this estimate is optimal for the n-point configuration space of Γ if n ≥ 2b. 0. Introduction. Let Γ be a finite graph and n a natural number. The marked n-point configuration space of Γ is a subspace CnΓ in the nth cartesian power of Γ defined by CnΓ := {(x1, . . . , xn) ∈ Γ n : xi 6= xj for i 6= j}. Consider the natural free action of the symmetric group Sn on the space CnΓ defined by σ(x1, . . . , xn) = (xσ(1), . . . , xσ(n)) and put CnΓ := CnΓ/Sn. The space CnΓ is called the (unmarked) n-point configuration space of Γ . This paper reports on partial progress towards understanding the homology of configuration spaces of graphs, or even more generally of compact polyhedra. For another recent result in that direction, see [G]. We call a vertex v of Γ branched if it is adjacent to at least three edges. We denote by b = b(Γ ) the number of branched vertices in Γ . The main result of this paper is the following. 0.1. Theorem. Let Γ be a finite graph and n a natural number. (1) There exists a cube complex KnΓ of dimension min(b(Γ ), n) which embeds as a deformation retract into the configuration space CnΓ . (2) The fundamental group π1(CnΓ ) contains a subgroup isomorphic to the free abelian group Z with k = min(b(Γ ), [n/2]), where [x] denotes the integer part of x. 2000 Mathematics Subject Classification: Primary 55M10; Secondary 20J05, 51F99. The author was supported by the Polish State Committee for Scientific Research (KBN) grant 2 P03A 023 14.

Quasi-Einstein hypersurfaces in semi-Riemannian space forms

Abstract: We investigate curvature properties of hypersurfaces of a semi-Rieman- nian space form satisfyingR·C = LQ(S,C), which is a curvature condition of pseu- dosymmetry type. We prove that under some additional assumptions the ambient space of such hypersurfaces must be semi-Euclidean and that they a re quasi-Einstein Ricci- semisymmetricmanifolds.

Blowup rates for nonlinear heat equations with gradient terms and for parabolic inequalities

Abstract: Consider the nonlinear heat equation (E): ut − ∆u = |u| u + b|∇u| . We prove that for a large class of radial, positive, nonglobal solutions of (E), one has the blowup estimates C1(T − t) −1/(p−1) ≤ ‖u(t)‖∞ ≤ C2(T − t) . Also, as an application of our method, we obtain the same upper estimate if u only satisfies the nonlinear parabolic inequality ut − uxx ≥ u . More general inequalities of the form ut − uxx ≥ f(u) with, for instance, f(u) = (1 + u) log (1 + u) are also treated. Our results show that for solutions of the parabolic inequality, one has essentially the same estimates as for solutions of the ordinary differential inequality v̇ ≥ f(v).

Generalized Hardy spaces on tube domains over cones

Abstract: We define a class of spaces H μ, 0 < p < ∞, of holomorphic functions on the tube, with a norm of Hardy type: ‖F‖ H p μ = sup y∈Ω \\

Generalized canonical algebras and standard stable tubes

Abstract: We introduce a new wide class of finite-dimensional algebras which admit families of standard stable tubes (in the sense of Ringel [17]). In particular, we prove that there are many algebras of arbitrary nonzero (finite or infinite) global dimension whose Auslander–Reiten quivers admit faithful standard stable tubes. Introduction. Throughout the paper K will denote a fixed algebraically closed field. By an algebra we mean a finite-dimensional K-algebra (associative, with an identity), which we moreover assume to be basic. An algebra A can be written as a bound quiver algebra A ∼= KQ/I, where Q = QA is the Gabriel quiver of A and I is an admissible ideal in the path algebra KQ of Q. Equivalently, we will consider A as a K-category whose class of objects is the set of vertices of QA. For an algebra A, we denote by modA the category of finite-dimensional (over K) right A-modules, by rad(modA) the Jacobson radical of modA and by rad(modA) the infinite radical of modA. Recall that rad(modA) is generated by nonisomorphisms between indecomposable objects in modA, and rad(modA) is the intersection of all finite powers rad(modA), i ≥ 1, of rad(modA). By anA-module we mean an object of modA. For each vertex i of QA, we denote by SA(i) the simple A-module at i, and by PA(i) (respectively, IA(i)) the projective cover (respectively, injective envelope) of SA(i) in modA. Moreover, we denote by D the standard duality HomK(−,K) on modA. We shall denote by ΓA the Auslander–Reiten quiver of A and by τA and τ A the Auslander–Reiten translations DTr and TrD in ΓA, respectively. We do not distinguish between an indecomposable A-module and the vertex of ΓA corresponding to it. By a component of ΓA we mean a connected component of ΓA. For a family C of components in ΓA, we denote by suppA C the support of C and by annA C the annihilator of C. Recall that suppA C is the full subcategory of A given by all objects i such that SA(i) is a 2000 Mathematics Subject Classification: 16G10, 16G70, 18G05. Supported by the Foundation for Polish Science.

An analogue of hardy's theorem for the heisenberg group

Abstract: We observe that the classical theorem of Hardy on Fourier transform pairs can be reformulated in terms of the heat kernel associated with the Laplacian on the Euclidean space. This leads to an interesting version of Hardy's theorem for the sublaplacian on the Heisenberg group. We also consider certain Rockland operators on the Heisenberg groupand Schrodinger operators on R n relatedto them.

Harmonic analysis for spinors on real hyperbolic spaces

Abstract: We develop the $L^2$ harmonic analysis for (Dirac) spinors on the real hyperbolic space $H^n(\R)$ and give the analogue of the classical notions and results known for functions and differential forms: we investigate the Poisson transform, the spherical function theory, the spherical Fourier transform and the Fourier transform. Very explicit expressions and statements are obtained by reduction to Jacobi analysis on $L^2(\R)$. As applications, we describe the exact spectrum of the Dirac operator, study the Abel transform and derive explicit expressions for the heat kernel associated with the spinor Laplacian

On the Neumann problem for an elliptic system of equations involving the critical Sobolev exponent

Abstract: We consider the Neumann problem for an elliptic system of two equations involving the critical Sobolev nonlinearity. Our main objective is to study the effect of the coefficient of the critical Sobolev nonlinearity on the existence and nonexistence of least energy solutions. As a by-product we obtain a new weighted Sobolev inequality.

On supports of dynamical laminations and biaccessible points in polynomial Julia sets

Abstract: We use Beurling estimates and Zdunik’s theorem to prove that the support of a lamination of the circle corresponding to a connected polynomial Julia set has zero length, unless f is conjugate to a Chebyshev polynomial. Equivalently, except for the Chebyshev case, the biaccessible points in the connected polynomial Julia set have zero harmonic measure. A connected, locally connected, full, compact subset K of the complex plane C can be topologically described by a lamination of a circle, which tells how to pinch the circle to obtain K (see, e.g., [Dou]). A lamination is an equivalence relation on the unit circle T, identifying points ζ and ζ ′ if they are mapped to one point in K by the Riemann uniformization map of the complement of K. To obtain a topological model of the compact set K, we glue together points of the unit circle, belonging to one equivalence class. The support of a lamination is defined as the union of all non-trivial (containing 2 or more points) equivalence classes, i.e. it includes those points which are identified with some other points. The laminations so defined are topologically fully characterized (among all equivalence relations on T, see [Dou]) by the following properties: (1) the graph {(ζ, ζ ) : ζ ∼ ζ ′} is a closed set in T× T, (2) the convex hulls of different equivalence classes are disjoint, (3) each equivalence class is totally disconnected. There are also analytical properties (e.g. the logarithmic capacity of each equivalence class is zero) which are not fully understood, and it is a difficult open question how to characterize laminations analytically among all equivalence relations on T. It also makes sense to consider laminations corresponding to not necessarily locally connected compacta, but those lamina2000 Mathematics Subject Classification: Primary 37F20; Secondary 30C85, 30D05.

An addendum and corrigendum to "Almost free splitters" (Colloq. Math. 81 (1999), 193-221)

Abstract: LetRbe a subring of the rationalnumbers Q. We recall from (3) that an R-moduleGisasplitterifExt 1 (G,G)=0.InthisnotewecorrectthestatementofMain Theorem 1.5 in (3) and discuss the existence of non-free splitters of cardinality@1under the negationofthe special continuum hypothesisCH.

A combinatorial construction of sets with good quotients by an action of a reductive group

Abstract: The aim of this paper is to construct open sets with good quotients by an action of a reductive group starting with a given family of sets with good quotients. In particular, in the case of a smooth projective variety X with Pic(X) = Z, we show that all open sets with good quotients that embed in a toric variety can be obtained from the family of open sets with projective good quotients. Our method applies in particular to the case of Grassmannians. Introduction. Let X be an algebraic variety with an action of a reductive group G. One of the main problems of geometric invariant theory is to describe all open G-invariant subsets U ⊂ X such that there exists a good quotient U → U/G. In the case of X = P this problem was solved in [5]. We give in 1.10 and 2.5 a new formulation of the results obtained in [5] for G = T a torus and show that open T -invariant sets with good quotients are unions of collections of intersections of subsets with projective quotients. Moreover, we emphasize the fact that, for any T -maximal subset of P, the quotient space embeds in a toric variety. In the present paper we investigate two possible ways of generalizing the results of [5]. First we assume that we are given a set of G-invariant open subvarieties of X with good quotients by the action of G, and we describe a procedure for constructing a larger class of G-invariant subsets of X that admit good quotients (Theorem 2.6). The new sets obtained by our method are unions of “good collections” of intersections of old ones. In this way we generalize the results of [5] concerning existence of good quotients to the case of an arbitrary reductive group G and any variety X. Then we consider the action of a reductive group G on a projective smooth variety with Pic(X) = Z and we study the problem of describing all open G-invariant subsets of X admitting quotients that embed in a toric 2000 Mathematics Subject Classification: Primary 14L24, 14L30.

Subcategories of the derived category and cotilting complexes

Abstract: We show that there is a one-to-one correspondence between basic cotilting complexes and certain contravariantly finite subcategories of the bounded derived category of an artin algebra. This is a triangulated version of a result by Auslander and Reiten. We use this to find an existence criterion for complements to exceptional complexes. Introduction. Homologically finite subcategories were introduced by Auslander and Smalo (3), and they have proved to be important in the study of artin algebras. Homologically finite subcategories of the category of finitely generated modules have been studied by several authors. In (1), Auslander and Reiten showed that there is a correspondence between cer- tain contravariantly finite subcategories and basic cotilting modules. In this paper we consider some subcategories of the bounded derived category of an artin algebra that are associated with cotilting complexes. In the first sec- tion we give definitions and basic results that we use in the second section, where we show that there is a correspondence between cotilting complexes and certain contravariantly finite subcategories of the derived category. The third section is devoted to examples. In the fourth section we use the cor- respondence to prove an existence criterion for complements of exceptional complexes. 1. Subcategories of the derived category. Let Λ be an artin algebra. Let mod Λ be the category of finitely generated left Λ-modules, and let D = D b (mod Λ) be the bounded derived category. This is a triangulated category. We denote the shift functor by (1), and its inverse by (−1). We let I(Λ) be the full subcategory of mod Λ formed by the injective objects, and, similarly, P(Λ) stands for the projectives. Then D b (mod Λ) is equivalent to K +,b (I(Λ)). We consider this an identification, and let K b (I(Λ)) denote the coperfect complexes. By a subcategory, we will always mean a full additive

Left-right projective bimodules and stable equivalences of Morita type

Abstract: We study a connection between left-right projective bimodules and stable equivalences of Morita type for finite-dimensional associative algebras over a field. Some properties of the category of all finite-dimensional left-right projective bimodules for selfinjective algebras are also given. Introduction. Let K be a fixed field. In the representation theory of finite-dimensional associative K-algebras with identity, stable equivalences of Morita type seem to be of particular relevance. They play a substantial role in the representation theory of finite groups (see [7, 12, 15]). It was observed by Rickard that if two self-injectiveK-algebras are derived equivalent then they are stably equivalent of Morita type [14]. Rickard [14] showed that for two derived equivalent K-algebras their Hochschild cohomology algebras are isomorphic. Happel also proved it but did not publish. This result was generalized in [13] to self-injective Kalgebras A and B which are stably equivalent of Morita type. The main trick used in the proof of this generalization rests heavily on an easy observation that there are two very small subcategories (they consist only of one Ω-orbit) which are stably equivalent; one of them is contained in the category of finite-dimensional A-bimodules, the other in the category of finite-dimensional B-bimodules. An important property of these subcategories is that they consist only of bimodules which are projective as left modules and as right modules. Following Butler and King [8] we call such bimodules left-right projective. They were first studied by Auslander and Reiten [5]. It is natural to ask whether a stable equivalence of Morita type between two finite-dimensional K-algebras A and B induces a stable equivalence between their categories of left-right projective bimodules. One of the main aims of this paper is to give an affirmative answer to the last question. We show even more: existence of a stable equivalence between two finite-dimensional K-algebras is equivalent to existence of a special stable equivalence between the categories of left-right projective bimodules. 2000 Mathematics Subject Classification: 16D20, 16G20. Supported by Polish KBN Grant 2 PO3A 012 14.

Infinitesimal unipotent group schemes of complexity 1

Abstract: We classify the uniserial infinitesimal unipotent commutative groups of finite representation type over algebraically closed fields. As an application we provide detailed information on the structure of those infinitesimal groups whose distribution algebras have a representation-finite principal block.

Product preserving gauge bundle functors on vector bundles

Abstract: A complete description is given of all product preserving gauge bundle functorsF on vector bundles in terms of pairs (A,V ) consisting of a Weil algebraA and anA-moduleV with dimR(V )<1. Some applications of this result are presented.

Additive functions on trees

Abstract: The motivation of considering positive additive functions on trees was the characterization of extended Dynkin graphs (see I. Reiten [R]) and the application of additive functions in the representation theory (see H. Lenzing and I. Reiten [LR] and T. H¨ubner [H]). We consider graphs equipped with functions of integer values, i.e.valued graphs (see also [DR]). Methods are given for the construction of additive functions on valued trees (in particular on Euclidean graphs) and for the characterization of their structure. We introduce the concept of almost additive functions, which are additive on each vertex of a graph except for one (called exceptional vertex). On (valued) trees (with fixed exceptional vertex) the almost additive functions are unique up to rational multiples. For valued trees a necessary and sufficient condition is given for the existence of positive almost additive functions.

Abstract parabolic problems in ordered Banach spaces

Abstract: Parabolic Problems in Ordered Banach Spaces Alexandre N. Carvalho* Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo-Campus de São Carlos, Caixa Postal 668, 13560-970 São Carlos SP, Brazil E-mail: andcarva@icmsc.sc.usp.br Jan W. Cholewa and Tomasz DÃlotko† Institute of Mathematics, Silesian University, 40-007 Katowice, Poland E-mail: jcholewa@ux2.math.us.edu.pl and tdlotko@ux2.math.us.edu.pl We consider abstract parabolic problems in ordered Banach spaces and give conditions under which they possess global attractors. Our approach goes via comparison of solutions. Within this approach abstract comparison principles are obtained and bounds on the attractors are given by order intervals in Banach spaces. These results are applied to ordinary differential equations and to parabolic equations for which the main part is given by a sum of fractional powers of sectorial operators having increasing resolvent and integral operators having positive kernels. March, 2001 ICMC-USP AMS-Subject Classification 2000: Primary 35K90, Secondary 35S15, 35B40, 35B41.

A classification of two-peak sincere posets of finite prinjective type and their sincere prinjective representations

Abstract: Assume thatK is an arbitraryfield Let (I,�) be a two-peak poset of finiteprinjectivetypeandletKIbetheincidencealgebraofIWestudysincereposetsI andsincereprinjectivemodulesoverKIThecompletesetofallsinceretwo-peakposetsof finiteprinjectivetypeisgiveninTheorem31Moreover,foreachsuchposetI,acomplete setofrepresentativesofisomorphismclassesofsincereindecomposableprinjectivemodules overKIispresented in Tables 81

Local-global principle for annihilation of general local cohomology

Abstract: Let A be a Noetherian ring, let M be a finitely generated A-module and letbe a system of ideals of A. We prove that, for any ideal a in �, if, for every prime ideal p of A, there exists an integer k(p), depending on p, such that a k(p) kills the general local cohomology module H j �p (Mp) for every integer j less than a fixed integer n, where �p :={ap : a2 �}, then there exists an integer k such that a k H j (M) = 0 for every j < n.

On the set representation of an orthomodular poset

Abstract: LetP be an orthomodular poset and letB be a Boolean subalgebra ofP. A mappings :P→h 0,1iis said to be a centrally additiveB-state if it is order preserving, satisfies s(a ' ) = 1−s(a), is additive on couples that contain a central element, and restricts to a state onB. It is shown that, for any Boolean subalgebraB ofP,P has an abundance of two-valued centrally additiveB-states. This answers positively a question raised in (13, Open question, p. 13). As a consequence one obtains a somewhat better set representation of orthomodular posets and a better extension theorem than in (2, 12, 13). Further improvement in the Boolean vein is hardly possible as the concluding example shows. Our notation is standard. We use OMP to abbreviate orthomodular poset, OML to abbreviate orthomodular lattice, Z to denote the centre of an orthomodular poset,⊂ for set inclusion, andh 0,1ifor the real unit interval. We remind the reader that a subset B of an orthomodular poset P is called a Boolean subalgebra of P if B is closed under orthocomplementation and finite orthogonal joins and B forms a Boolean algebra under these inherited operations. It is well known that any two elements of B also have a join (resp., a meet) in P and that the join (resp., the meet) taken in B coincides with the join (resp., the meet) taken in P. For general background on or- thomodular posets the reader should consult (11), on orthomodular lattices (1, 6), and for various results related to set representations of orthomodular posets (2, 5, 7, 8, 9, 13, 14).

On an estimate for the linearized compressible Navier-Stokes equations in the $L_{p}$-framework

Abstract: An Lp-estimate with a constant independent of time for solutions of the linearized compressible Navier–Stokes system in the whole space (under the assumption that solutions have compact supports in space) is obtained.

Vertical variation of harmonic functions in upper half-spaces

Abstract: Some results of Bourgain on the radial variation of harmonic functions in the disk are extended to the setting of harmonic functions in upper half-spaces.