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.     

Some Properties of Free Groups of Some Soluble Varieties of Groups

Let F be a free group, and let _n(F) be the nth term of the lowercentral series of F. It is proved that F/[_j(F), _i(F), _k(F)]and F/[_j(F), _i(F), _k(F), _l(F)] are torsion free and residuallynilpotent for certain values of i, j, k and i, j, k, l, respectively.In the process of proving this, it is proved that the analogousLie rings are torsion free.     

The Consistency of Holt's Conjectures on Cohomological Dimension of Locally Finite Groups

Let G be a locally finite group of cardinality _n where n isa natural number. Let (G) be the set of primes p for which Ghas an element of order p. In [5], Holt conjectures that ifk is a finite field with char k (G) then (1) G has cohomological dimension n+1 over k; (2) Hⁿ⁺¹(G, kG) has cardinality 2^_n; (3) Hⁱ(G, kG) = 0 for 0 i n.     

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.     

Lp Lq Estimates for Parabolic Systems in Non-Divergence Form with VMO Coefficients

Consider a parabolic NxN-system of order m on ⁿ with top-ordercoefficients a VMOL. Let 1 < p, q < and let be a Muckenhouptweight. It is proved that systems of this kind possess a uniquesolution u satisfying whereAu = _||m a Du and J = [0,). In particular, choosing = 1, therealization of A in L^p(ⁿ)^N has maximal L^p – L^q regularity.     

Construction of p-1 Irreducible Modules with Fundamental Highest Weight for the Symplectic Group in Characteristic p

Let K be a field and let V be a vector space of dimension 2mover K. Let V denote the exterior algebra of V and ^kV its kthexterior power for 0k2m. Let f be a non-degenerate alternatingbilinear form defined on VxV. The symplectic group Sp_2m(K) isthe group of all isometries of f and it acts as a group of vectorspace automorphisms on ^kV. In the case that K is algebraicallyclosed and 1km, it is known that ^kV contains a composition factorcorresponding to the fundamental weight _k of a root system oftype C_m. We shall refer to the irreducible module for Sp_2m(K)given by this composition factor as a fundamental module.     

The canonical volume of 3-folds of general type with {chi} <= 0

We prove that the canonical volume K³ 1/30 for all 3-foldsof general type with ()0. This bound is sharp.     

Movement and Separation of Subsets of Points Under Group Actions

Let G be a permutation group on a set , and let m and k be integerswhere 0<m<k. For a subset of , if the cardinalities ofthe sets ^g\, for gG, are finite and bounded, then is said tohave bounded movement, and the movement of is defined as move()=max_gG|^g\|. If there is a k-element subset such that move()m, it is shown that some G-orbit has length at most (k²–m)/(k–m).When combined with a result of P. M. Neumann, this result hasthe following consequence: if some infinite subset has boundedmovement at most m, then either is a G-invariant subset withat most m points added or removed, or nontrivially meets aG-orbit of length at most m²+m+1. Also, if move ()m for allk-element subsets and if G has no fixed points in , then either||k+m (and in this case all permutation groups on have thisproperty), or ||5m–2. These results generalise earlierresults about the separation of finite sets under group actionsby B. J. Birch, R. G. Burns, S. O. Macdonald and P. M. Neumann,and groups in which all subsets have bounded movement (by theauthor).     

Sub-Laplacians of Holomorphic Lp-Type on Exponential Solvable Groups

Let L denote a right-invariant sub-Laplacian on an exponential,hence solvable Lie group G, endowed with a left-invariant Haarmeasure. Depending on the structure of G, and possibly alsothat of L, L may admit differentiable L^p-functional calculi,or may be of holomorphic L^p-type for a given p 2. ‘HolomorphicL^p-type’ means that every L^p-spectral multiplier for Lis necessarily holomorphic in a complex neighbourhood of somenon-isolated point of the L²-spectrum of L. This can in factonly arise if the group algebra L¹(G) is non-symmetric. Assume that p 2. For a point in the dual g^* of the Lie algebrag of G, denote by ()=Ad^*(G) the corresponding coadjoint orbit.It is proved that every sub-Laplacian on G is of holomorphicL^p-type, provided that there exists a point g^* satisfying Boidol'scondition (which is equivalent to the non-symmetry of L¹(G)),such that the restriction of () to the nilradical of g is closed.This work improves on results in previous work by Christ andMüller and Ludwig and Müller in twofold ways: on theone hand, no restriction is imposed on the structure of theexponential group G, and on the other hand, for the case p>1,the conditions need to hold for a single coadjoint orbit only,and not for an open set of orbits. It seems likely that the condition that the restriction of ()to the nilradical of g is closed could be replaced by the weakercondition that the orbit () itself is closed. This would thenprove one implication of a conjecture by Ludwig and Müller,according to which there exists a sub-Laplacian of holomorphicL¹ (or, more generally, L^p) type on G if and only if there existsa point g^* whose orbit is closed and which satisfies Boidol'scondition.     

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.     

Antisupercyclic Operators and Orbits of the Volterra Operator

We say that a bounded linear operator T acting on a Banach spaceB is antisupercyclic if for any x B either Tⁿx = 0 for somepositive integer n or the sequence {Tⁿx/||Tⁿx||} weakly convergesto zero in B. Antisupercyclicity of T means that the angle criterionof supercyclicity is not satisfied for T in the strongest possibleway. Normal antisupercyclic operators and antisupercyclic bilateralweighted shifts are characterized. As for the Volterra operator V, it is proved that if 1 p and any f L_p [0,1] then the limit lim_n (n!||Vⁿf||_p)^1/n doesexist and equals 1 – inf supp (f). Upon using this asymptoticformula it is proved that the operator V acting on the Banachspace L_p[0,1] is antisupercyclic for any p (1,). The same statementfor p = 1 or p = is false. The analogous results are provedfor operators when the real part of z C is positive.     

The Rationals have an AZ-Enumeration

Let M be an -categorical structure (that is, M is countableand Th(M) is -categorical). A nice enumeration of M is a totalordering of M having order-type and satisfying the following.Whenever a_i, i<, is a sequence of elements from M, thereexist some i<j< and an automorphism of M such that (a_i)= a_j and whenever ba_i, then (b)a_j. Such enumerations were introduced by Ahlbrandt and Ziegler in[1] where they showed that any Grassmannian of an infinite-dimensionalprojective space over a finite field (or of a disintegratedset) admits a nice enumeration; this combinatorial propertyplayed an essential role in their proof that almost stronglyminimal totally categorical structures are quasi-finitely axiomatisable. Recall that if M is -categorical and is a k-tuple of distinctelements from M (with tp() non-algebraic), then the GrassmannianGr(M; ) is defined as follows. The domain of Gr(M; ) is theset of realisations of tp() in M^k, modulo the equivalence relationxEy if x and y are equal as sets. This is a 0-definable subsetof M^eq, and now the relations on Gr(M; ) are by definition preciselythose which are 0-definable in the structure M^eq. (In particular,Gr(M; ) is also -categorical.) Notice that it is by no means clear that if M admits a niceenumeration, then so do Grassmannians of M. However, there isa strengthening of the notion of nice enumeration for whichthis is the case.     

Turan's Inequalities for Trigonometric Polynomials

We present here a technique for establishing inequalities ofthe form in the set of alltrigonometric polynomials of order n which have only real zeros.The function is assumed to be convex and increasing on [0,). As a corollary of the main result we get Turan's inequalities with the exact constantc(n, k, q) for each 1 q , n and k.     

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.     

On the Boundedness and Compactness of a Class of Integral Operators

Let > 0. The operator of the form is considered, where the real weight function v(x) is locallyintegrable on R⁺ := (0, ). In case v(x) = 1 the operator coincideswith the Riemann–Liouville fractional integral, L^p L^qestimates of which with power weights are well known. This workgives L^p L^qboundedness and compactness criteria for the operatorT in the case 0 < p, q < , p > max(1/, 1).     

Bounding the number of stable homotopy types of a parametrized family of semi-algebraic sets defined by quadratic inequalities

We prove a nearly optimal bound on the number of stable homotopytypes occurring in a k-parameter semi-algebraic family of setsin R, each defined in terms of m quadratic inequalities. Ourbound is exponential in k and m, but polynomial in . More precisely,we prove the following. Let R be a real closed field and let = {P₁, ... , P_m} R[Y₁, ... ,Y,X₁, ... ,X_k], with deg_Y(P_i) 2, deg_X(P_i) d, 1 i m. Let S R^+k be a semi-algebraic set,defined by a Boolean formula without negations, with atoms ofthe form P 0, P 0, P . Let : R^+k R^k be the projection onthe last k coordinates. Then the number of stable homotopy typesamongst the fibers S_x = ^–1(x) S is bounded by (2^mkd)^O(mk).     

The Convergence of a Class of Quasimonotone Reaction-Diffusion Systems

It is proved that every solution of the Neumann initial-boundaryproblem converges to some equilibrium, if the system satisfies (i) F_i/u_j 0 for all 1 i j n, (ii) F(u * g(s)) h(s) * F(u) wheneveru and 0 s 1, where x *y = (x₁y₁, ..., x_ny_n) and g, h : [0, 1] [0, 1]ⁿ are continuousfunctions satisfying g_i(0) = h_i(0) = 0, g_i(1) = h_i(1) = 1, 0< g_i(s); h_i(s) < 1 for all s (0, 1) and i = 1, 2, ...,n, and (iii) the solution of the corresponding ordinary differentialequation system is bounded in . We also study the convergence of the solution of the Lotka–Volterrasystem where r_i > 0, 0, and a_ij 0 for i j.     

A Strong Law for the Largest Nearest-Neighbour Link between Random Points

Suppose that X₁, X₂, X₃, ... are independent random points inR^d with common density f, having compact support with smoothboundary , with f| continuous. Let Rⁿ_{i, k} denote the distancefrom X_i to its kth nearest neighbour amongst the first n points,and let M_{n, k} = max_in Rⁿ_{i, k}. Let denote the volume of theunit ball. Then as n , , almost surely If instead the points lie in a compact smooth d-dimensionalRiemannian manifold K, then nM^d_{n, k}/log n (min_Kf)^–1,almost surely.     

On a Comparison Principle and the Critical Fujita Exponents for Solutions of Semilinear Parabolic Inequalities

The paper examines blow-up phenomena for the inequality u_t–Lu–|u|^q–1u_t–L–||^q–1 (*) in the half-space x Rⁿ, n 1, where L is a linear second-order partial differential operatorin divergence form. The paper studies weak solutions of (*) that belong only locallyto the corresponding Sobolev spaces in the half-space x Rⁿ. It also requires no conditionsfor the behavior of solutions of (*) on the hyperplane t = 0. The existence of critical blow-up exponents is obtained forsolutions of (*) as a special case of a comparison principlefor the corresponding solutions of (*). For example, the well-knownFujita result is a consequence of the comparison principle. The approach developed in the paper is directly applicable tothe study of analogous problems involving nonlinear differentialoperators. Its elliptic analogue has been recently developedby the authors.