The Finite Subsets of Z2 Having the Extension Property

The paper considers finite subsets Z^d which possess the extensionproperty, namely that every collection {c_k}_k– of complexnumbers which is positive definite with respect to is the restrictionof the Fourier coefficients of some positive measure on T^d.All finite subsets of Z² which possess the extension propertyare described.     

Faithful Representations of Free Products

In 1940 Nisnevi published the following theorem [3]. Let (G) be a family of groups indexed by some set and (F) a family of fields of the same characteristic p0. Iffor each the group G has a faithful representation of degreen over F then the free product* G has a faithful representationof degree n+1 over some field of characteristic p. In [6] Wehrfritzextended this idea. If (G) GL(n, F) is a family of subgroupsfor which there exists ZGL(n, F) such that for all the intersectionGF.1_n=Z, then the free product of the groups *_ZG with Z amalgamatedvia the identity map is isomorphic to a linear group of degreen over some purely transcendental extension of F. Initially, the purpose of this paper was to generalize theseresults from the linear to the skew-linear case, that is, togroups isomorphic to subgroups of GL(n, D) where the D are divisionrings. In fact, many of the results can be generalized to ringswhich, although not necessarily commutative, contain no zero-divisors.We have the following.     

Asymptotics for Fractional Nonlinear Heat Equations

The Cauchy problem is studied for the nonlinear equations withfractional power of the negative Laplacian where (0,2), with critical = /n and sub-critical (0,/n)powers of the nonlinearity. Let u₀ L^1,a L C, u₀(x) 0 in Rⁿ, = . The case of not small initial data is of interest. It is proved that the Cauchy problemhas a unique global solution u C([0,); L L^1,a C) and the largetime asymptotics are obtained.     

Piecewise Absolutely Continuous Cocycles Over Irrational Rotations

For an irrational rotation of the circle group T=R/Z and apiecewise absolutely continuous function f:TR, the unitary operatorVh(x)=e^2if(x)h(x+) on L²(T) is studied. It is shown that iff has a single discontinuity with non-integer jump then V is-weakly mixing for some with 0<||<1. In particular Vhas continuous singular spectrum. The property of -weak mixing(with possible change of the value of , 0<||<1) holdsfor all irrational rotations and, given , is stable under perturbationsof f by functions with sufficiently small O(1/n)-norm. On theother hand, there exists a piecewise linear function f withtwo non-integer jumps such that the spectrum of V is continuoussingular for one value of and Lebesgue for another.     

Geometry of Critical Loci

Let :(Z,z)(U,0) be the germ of a finite (that is, proper with finite fibres)complex analytic morphism from a complex analytic normal surfaceonto an open neighbourhood U of the origin 0 in the complexplane C². Let u and v be coordinates of C² defined on U. Weshall call the triple (, u, v) the initial data. Let stand for the discriminant locus of the germ , that is,the image by of the critical locus of . Let ()_A be the branches of the discriminant locus at O whichare not the coordinate axes. For each A, we define a rational number d by where I(–, –) denotes the intersection number at0 of complex analytic curves in C². The set of rational numbersd, for A, is a finite subset D of the set of rational numbersQ. We shall call D the set of discriminantal ratios of the initialdata (, u, v). The interesting situation is when one of thetwo coordinates (u, v) is tangent to some branch of , otherwiseD = {1}. The definition of D depends not only on the choiceof the two coordinates, but also on their ordering. In this paper we prove that the set D is a topological invariantof the initial data (, u, v) (in a sense that we shall definebelow) and we give several ways to compute it. These resultsare first steps in the understanding of the geometry of thediscriminant locus. We shall also see the relation with thegeometry of the critical locus.     

Concavity and B-Concavity of Solutions of Quasilinear Filtration Equations

Spatial concavity properties of non-negative weak solutionsof the filtration equations with absorption u_t = ((u))_xx–(u)in Q = Rx(0, ), '0, 0 are studied. Under certain assumptionson the coefficients , it is proved that concavity of the pressurefunction is a consequence of a ‘weak’ convexityof travelling-wave solutions of the form V(x, t) = (x–t+a).It is established that the global structure of a so-called properset B = {V} of such particular solutions determines a propertyof B-concavity for more general solutions which is preservedin time. For the filtration equation u_t = ((u))_xx a semiconcavityestimate for the pressure, v_xx(t+)^–1'(), due to the B-concavityof the solution to the subset B of the explicit self-similarsolutions (x/t+)) is proved. The analysis is based on the intersection comparison based onthe Sturmian argument of the general solution u(x, t) with subsetsB of particular solutions. Also studied are other aspects ofthe B-concavity/convexity with respect to different subsetsof explicit solutions.     

BOUNDARY CONCENTRATION IN RADIAL SOLUTIONS TO A SYSTEM OF SEMILINEAR ELLIPTIC EQUATIONS

We study concentration phenomena for the system in the unit ball B₁ of ³ with Dirichlet boundaryconditions. Here , , > 0 and p > 1. We prove the existenceof positive radial solutions (, ) such that concentrates ata distance (/2)|log | away from the boundary B₁ as the parameter tends to 0. The approach is based on a combination of Lyapunov–Schmidtreduction procedure together with a variational method.     

Real Interpolation and Two Variants of Gehring's Lemma

Let be a fixed open cube in Rⁿ. For r[1, ) and [0, ) we define where Q is a cube in Rⁿ (with sides parallel to the coordinateaxes) and _Q stands for the characteristic function of the cubeQ. A well-known result of Gehring [5] states that if (1.1) for some p(1, ) and c(0, ), then there exist q(p, ) and C=C(p,q, n, c)(0, ) such that for all cubes Q, where |Q| denotes the n-dimensional Lebesguemeasure of Q. In particular, a function fL¹() satisfying (1.1)belongs to L^q(). In [9] it was shown that Gehring's result is a particular caseof a more general principle from the real method of interpolation.Roughly speaking, this principle states that if a certain reversedinequality between K-functionals holds at one point of an interpolationscale, then it holds at other nearby points of this scale. Usingan extension of Holmstedt's reiteration formulae of [4] andresults of [8] on weighted inequalities for monotone functions,we prove here two variants of this principle involving extrapolationspaces of an ordered pair of (quasi-) Banach spaces. As an applicationwe prove the following Gehring-type lemmas.     

The Structure of Biserial Algebras

By an algebra we mean an associative k-algebra with identity,where k is an algebraically closed field. All algebras are assumedto be finite dimensional over k (except the path algebra kQ).An algebra is said to be biserial if every indecomposable projectiveleft or right -module P contains uniserial submodules U andV such that U+V=Rad(P) and UV is either zero or simple. (Recallthat a module is uniserial if it has a unique composition series,and the radical Rad(M) of a module M is the intersection ofits maximal submodules.) Biserial algebras arose as a naturalgeneralization of Nakayama's generalized uniserial algebras[2]. The condition first appeared in the work of Tachikawa [6,Proposition 2.7], and it was formalized by Fuller [1]. Examplesinclude blocks of group algebras with cyclic defect group; finitedimensional quotients of the algebras (1)–(4) and (7)–(9)in Ringel's list of tame local algebras [4]; the special biserialalgebras of [5, 8] and the regularly biserial algebras of [3].An algebra is basic if /Rad() is a product of copies of k.This paper contains a natural alternative characterization ofbasic biserial algebras, the concept of a bisected presentation.Using this characterization we can prove a number of resultsabout biserial algebras which were inaccessible before. In particularwe can describe basic biserial algebras by means of quiverswith relations.     

Tensor Products of C*-Algebras Over Abelian Subalgebras

Suppose that A is a C*-algebra and C is a unital abelian C*-subalgebrawhich is isomorphic to a unital subalgebra of the centre ofM(A), the multiplier algebra of A. Letting = , so that we maywrite C = C(), we call A a C()-algebra (following Blanchard[7]). Suppose that B is another C()-algebra, then we form A_CB, the algebraic tensor product of A with B over C as follows:A B is the algebraic tensor product over C, I_C = {ⁿ_i–1(f_i 1–1f_i)x|f_iC, xAB} is the ideal in AB generated by f1–1f|fC,and A _CB = AB/I_C. Then A_CB is an involutive algebra over C,and we shall be interested in deciding when A_CB is a pre-C*-algebra;that is, when is there a C*-norm on A_C B? There is a C*-semi-norm,which we denote by ||·||_C-min, which is minimal in thesense that it is dominated by any semi-norm whose kernel containsthe kernel of ||·||_C-min. Moreover, if A _C B has a C*-norm,then ||·||_C-min is a C*-norm on A_C B. The problem isto decide when ||·||_C-min is a norm. It was shown byBlanchard [7, Proposition 3.1] that when A and B are continuousfields and C is separable, then ||·||_C-min is a norm.In this paper we show that ||·||_C-min is a norm whenC is a von Neumann algebra, and then we examine some consequences.     

Auslander algebras and initial seeds for cluster algebras

Let Q be a Dynkin quiver and the corresponding set of positiveroots. For the preprojective algebra associated to Q, a rigid-module I_Q is produced with r = || pairwise non-isomorphic indecomposabledirect summands by pushing the injective modules of the Auslanderalgebra of k Q to . If N is a maximal unipotent subgroup ofa complex simply connected simple Lie group of type |Q|, thenthe coordinate ring [N] is an upper cluster algebra. It is shownthat the elements of the dual semicanonical basis which correspondto the indecomposable direct summands of I_Q coincide with certaingeneralized minors which form an initial cluster for [N] andthat the corresponding exchange matrix of this cluster can beread from the Gabriel quiver of End(I_Q). Finally, the fact thatthe categories of injective modules over and over its covering are triangulated is exploited in order to show several interesting identities in the respectivestable module categories.     

Constructing {kappa}-like Models of Arithmetic

A model (M, <, ...) is -like if M has cardinality but, forall M, the cardinality of {x M : x < a} is strictly lessthan . In this paper we shall give constructions of -like modelsof arithmetic satisfying an arbitrarily large finite part ofPA but not PA itself, for various singular cardinals . The mainresults are: (1) for each countable nonstandard M ₂–Th(PA)with arbitrarily large initial segments satisfying PA and eachuncountable of cofinality there is a cofinal extension K ofM which is -like; also hierarchical variants of this resultfor _n–Th(PA); and (2) for every n 1, every singular and every M B_n+exp+¬ I_n there is a -like model K elementarilyequivalent to M.     

The Bestvina-Brady Construction Revisited: Geometric Computation of {sum}-Invariants for Right-Angled Artin Groups

The starting point of our investigation is the remarkable paper[2] in which Bestvina and Brady gave an example of an infinitelyrelated group of type FP₂. The result about right-angled Artingroups behind their example is best interpreted by means ofthe Bieri–Strebel–Neumann–Renz -invariants. For a group G the invariants ⁿ(G) and ⁿ(G, Z) are sets of non-trivialhomomorphisms :GR. They contain full information about finitenessproperties of subgroups of G with abelian factor groups. Themain result of [2] determines for the canonical homomorphism, taking each generator of the right-angled Artin group G to1, the maximal n with ⁿ(G), respectively ⁿ(G, Z). In [6] Meier, Meinert and VanWyk completed the picture by computingthe full -invariants of right-angled Artin groups using as wellthe result of Bestvina and Brady as algebraic techniques from-theory. Here we offer a new account of their result which istotally geometric. In fact, we return to the Bestvina–Bradyconstruction and simplify their argument considerably by bringinga more general notion of links into play. At the end of thefirst section we re-prove their main result. By re-computingthe full -invariants, we show in the second section that thesimplification even adds some power to the method. The criterionwe give provides new insight on the geometric nature of the‘n-domination’ condition employed in [6].     

Support Varieties of Non-Restricted Modules over Lie Algebras of Reductive Groups

Let G be a connected semisimple group over an algebraicallyclosed field K of characteristic p>0, and g=Lie (G). Fixa linear function g* and let Z_g() denote the stabilizer of in g. Set N_p(g)={xg|x^[p]=0}. Let C(g) denote the category offinite-dimensional g-modules with p-character . In [7], Friedlanderand Parshall attached to each MOb(C(g)) a Zariski closed, conicalsubset V_g(M)N_p(g) called the support variety of M. Suppose thatG is simply connected and p is not special for G, that is, p2if G has a component of type B_n, C_n or F₄, and p3 if G has acomponent of type G₂. It is proved in this paper that, for anynonzero MOb(C(g)), the support variety V_g(M) is contained inN_p(g)Z_g(). This allows one to simplify the proof of the Kac–Weisfeilerconjecture given in [18].     

On Simultaneously Badly Approximable Numbers

For any pair i,j 0 with i+j=1 let Bad(i,j) denote the set ofpairs (,ß) R² for which max{||q||^1/i||qß|^1/j}>c/qfor all q N. Here c=c(,ß) is a positive constant.If i=0 the set Bad(0, 1) is identified with RxBad where Badis the set of badly approximable numbers. That is, Bad(0, 1)consists of pairs (, ß) with R and ß Bad If j=0 the roles of and ß are reversed. It isproved that the set Bad(1,0)Bad (0,1) Bad(i,j) has Hausdorffdimension 2, that is, full dimension. The method easily generalizesto give analogous statements in higher dimensions.     

Extremal Subgroups in Chevalley Groups

Throughout this paper G(k) denotes a Chevalley group of rankn defined over the field k, where n3. Let be the root systemassociated with G(k) and let ={₁, ₂, ..., _n} be a set of fundamentalroots of , with ⁺ being the set of positive roots of with respectto . For and ⁺, let n() be the coefficient of in the expressionof as a sum of fundamental roots; so =n(). Also we recall thatht(), the height of , is given by ht()=n(). The highest rootin ⁺ will be denoted by . We additionally assume that the Dynkindiagram of G(k) is connected.     

Positive Eigenfunctions of a Schrodinger Operator

The paper considers the eigenvalue problem where , and for some bounded open set R^N. Given >0, does there exist a value of >0 for which theproblem has a positive solution? It is shown that this occursif and only if lies in a certain interval (,₁) and that inthis case the value of is unique, =(). The properties of thefunction () are also discussed.     

Base Sizes and Regular Orbits for Coprime Affine Permutation Groups

Let G be a permutation group on a finite set . A sequence B=(₁,..., _b) of points in is called a base if its pointwise stabilizerin G is the identity. Bases are of fundamental importance incomputational algorithms for permutation groups. For both practicaland theoretical reasons, one is interested in the minimal basesize for (G, ), For a nonredundant base B, the elementary inequality2^|B||G|||^|B| holds; in particular, |B|log|G|/log||. In the casewhen G is primitive on , Pyber [8, p. 207] has conjectured thatthe minimal base size is less than Clog|G|/log|| for some (large)universal constant C. It appears that the hardest case of Pyber's conjecture is thatof primitive affine groups. Let H=GV be a primitive affine group;here the point stabilizer G acts faithfully and irreduciblyon the elementary abelian regular normal subgroup V of H, andwe may assume that =V. For positive integers m, let mV denotethe direct sum of m copies of V. If (v₁, ..., v_m)mV belongsto a regular G-orbit, then (0, v₁, ..., v_m) is a base for theprimitive affine group H. Conversely, a base (₁, ..., _b) forH which contains 0V= gives rise to a regular G-orbit on (b–1)V. Thus Pyber's conjecture for affine groups can be viewed asa regular orbit problem for G-modules, and it is therefore aspecial case of an important problem in group representationtheory. For a related result on regular orbits for quasisimplegroups, see [4, Theorem 6].     

Set Theory is Interpretable in the Automorphism Group of an Infinitely Generated Free Group

In [6] S. Shelah showed that in the endomorphism semi-groupof an infinitely generated algebra which is free in a varietyone can interpret some set theory. It follows from his resultsthat, for an algebra F which is free of infinite rank in avariety of algebras in a language L, if > |L|, then thefirst-order theory of the endomorphism semi-group of F, Th(End(F)),syntactically interprets Th(,L₂), the second-order theory ofthe cardinal . This means that for any second-order sentence of empty language there exists *, a first-order sentence ofsemi-group language, such that for any infinite cardinal >|L|, Th(,L₂)*Th(End(F)) In his paper Shelah notes that it is natural to study a similarproblem for automorphism groups instead of endomorphism semi-groups;a priori the expressive power of the first-order logic for automorphismgroups is less than the one for endomorphism semi-groups. Forinstance, according to Shelah's results on permutation groups[4, 5], one cannot interpret set theory by means of first-orderlogic in the permutation group of an infinite set, the automorphismgroup of an algebra in empty language. On the other hand, onecan do this in the endomorphism semi-group of such an algebra. In [7, 8] the author found a solution for the case of the varietyof vector spaces over a fixed field. If V is a vector spaceof an infinite dimension over a division ring D, then the theoryTh(, L₂) is interpretable in the first-order theory of GL(V),the automorphism group of V. When a field D is countable anddefinable up to isomorphism by a second-order sentence, thenthe theories Th(GL(V)) and Th(, L₂) are mutually syntacticallyinterpretable. In the general case, the formulation is a bitmore complicated. The main result of this paper states that a similar result holdsfor the variety of all groups.     

Necessary and Sufficient Conditions for Exponential Stability and Ultimate Boundedness of Systems Governed by Stochastic Partial Differential Equations

Consider the following infinite dimensional stochastic evolutionequation over some Hilbert space H with norm |·|: It is proved that under certain mild assumptions, the strongsolution X_t(x₀)VHV*, t 0, is mean square exponentially stableif and only if there exists a Lyapunov functional (·,·):HxR₊R¹ which satisfies the following conditions: (i)c₁|x|²–k₁e^–µ₁t(x,t)c₂|x|²+k₂+k₂e^–µ₂t; (ii) L(x,t)–c₃(x,t)+k₃e^–µ₃t, xV, t0; where L is the infinitesimal generator of the Markov processX_t and c_i, k_i, µ_i, i = 1, 2, 3, are positive constants.As a by-product, the characterization of exponential ultimateboundedness of the strong solution is established as the nulldecay rates (that is, µ_i = 0) are considered.