TURAN'S EXTREMAL PROBLEM FOR POSITIVE DEFINITE FUNCTIONS ON GROUPS

We study the following question: given an open set , symmetricabout 0, and a continuous, integrable, positive definite functionf, supported in and with f(0) = 1, how large can f be? Thisproblem has been studied so far mostly for convex domains inEuclidean space. In this paper we study the question in arbitrarylocally compact abelian groups and for more general domains.Our emphasis is on finite groups as well as Euclidean spacesand ^d. We exhibit upper bounds for f assuming geometric propertiesof of two types: (a) packing properties of and (b) spectralproperties of . Several examples and applications of the maintheorems are shown. In particular, we recover and extend severalknown results concerning convex domains in Euclidean space.Also, we investigate the question of estimating f over possiblydispersed sets solely in dependence of the given measure m :=||of . In this respect we show that in and the integral is maximalfor intervals.     

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).     

Minimal Geodesics on Manifolds with Discontinuous Metrics

The paper describes some qualitative properties of minimizerson a manifold M endowed with a discontinuous metric. The discontinuityoccurs on a hypersurface disconnecting M. Denote by ₁ and₂ the open subsets of M such that M\ =₁₂. Assume that and are endowed with metrics ·, · ₍₁₎ and ·,·₍₂₎, respectively, such that (i=1, 2) is convex or concave. The existence of a minimizerof the length functional on curves joining two given pointsof M is proved. The qualitative properties obtained allows therefraction law in a very general situation to be described.     

A Torsion-Free Milnor-Moore Theorem

Let X be the space of Moore loops on a finite, q-connected,n-dimensional CW complex X, and let R Q be a subring containing1/2. Let (R) be the least non-invertible prime in R. For a gradedR-module M of finite type, let FM = M/Torsion M. We show thatthe inclusion P FH_*(X;R) of the sub-Lie algebra of primitiveelements induces an isomorphism of Hopf algebras provided that (R) n/q. Furthermore, the Hurewiczhomomorphism induces an embedding of F(_*(X) R) in P, with P/F(_*(X)R)torsion. As a corollary, if X is elliptic, then FH_*(X;R) isa finitely generated R-algebra.     

Compact Embeddings of Besov Spaces in Exponential Orlicz Spaces

Let 1 < p < , 0 < v < p', let be a bounded domainin Rⁿ, and denote by id the limiting compact embedding of theBesov space (Rⁿ) into the exponentialOrlicz space L_exp(t^v)(), mapping a function f onto its restrictionf|. In 1993 Triebel established, among others, two-sided estimatesfor the entropy numbers of id, which are even asymptoticallyoptimal for ‘small’ . The aim of the paper is toimprove the upper bounds in the case of ‘large’, where Triebel's estimates are not yet sharp, thus making afurther step towards the conjectured correct asymptotic behaviour.     

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.     

Identity Theorems for Functions of Bounded Characteristic

Suppose that f(z) is a meromorphic function of bounded characteristicin the unit disk :|z|<1. Then we shall say that f(z)N. Itfollows (for example from [3, Lemma 6.7, p. 174 and the following])that where h₁(z), h₂(z) are holomorphic in and have positive realpart there, while ₁(z), ₂(z) are Blaschke products, that is, where p is a positive integer or zero, 0<|a_j|<1, c isa constant and (1–|a_j|)<. We note in particular that, if c0, so that f(z)0, (1.1) so that f(z)=0 only at the points a_j. Suppose now that z_j isa sequence of distinct points in such that |z_j|1 as j and (1–|z_j|)=. (1.2) If f(z_j)=0 for each j and fN, then f(z)0. N. Danikas [1] has shown that the same conclusion obtains iff(z_j)0 sufficiently rapidly as j. Let _j, _j be sequences of positivenumbers such that _j< and _j as j. Danikas then defines and proves Theorem A.     

An Access Theorem for Continuous Functions

Let f be a continuous function on an open subset of R² suchthat for every x there exists a continuous map : [–1,1] with (0) = x and f increasing on [–1, 1]. Thenfor every there exists a continuous map : [0, 1) suchthat (0) = y, f is increasing on [0; 1), and for every compactsubset K of , max{t : (t) K} < 1. This result gives an answerto a question posed by M. Ortel. Furthermore, an example showsthat this result is not valid 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.     

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.     

Properties of Removable Singularities for Hardy Spaces of Analytic Functions

Removable singularities for Hardy spaces H^p() = {f Hol(): |f|^p u in for some harmonic u}, 0 < p < are studied. A setE = is a weakly removable singularity for H^p(\E) if H^p(\E) Hol(), and a strongly removable singularity for H^p(\E) if H^p(\E)= H^p(). The two types of singularities coincide for compactE, and weak removability is independent of the domain . The paper looks at differences between weak and strong removability,the domain dependence of strong removability, and when removabilityis preserved under unions. In particular, a domain and a setE that is weakly removable for all H^p, but not strongly removablefor any H^p(\E), 0 < p < , are found. It is easy to show that if E is weakly removable for H^p(\E)and q > p, then E is also weakly removable for H^q(\E). Itis shown that the corresponding implication for strong removabilityholds if and only if q/p is an integer. Finally, the theory of Hardy space capacities is extended, anda comparison is made with the similar situation for weightedBergman spaces.     

A New Notion of Transitivity for Groups and Sets of Permutations

Let = {1, 2, ..., n} where n 2. The shape of an ordered setpartition P = (P₁, ..., P_k) of is the integer partition =(₁, ..., _k) defined by _i = |P_i|. Let G be a group of permutationsacting on . For a fixed partition of n, we say that G is -transitiveif G has only one orbit when acting on partitions P of shape. A corresponding definition can also be given when G is justa set. For example, if = (n – t, 1, ..., 1), then a -transitivegroup is the same as a t-transitive permutation group, and if = (n – t, t), then we recover the t-homogeneous permutationgroups. We use the character theory of the symmetric group S_n to establishsome structural results regarding -transitive groups and sets.In particular, we are able to generalize a celebrated resultof Livingstone and Wagner [Math. Z. 90 (1965) 393–403]about t-homogeneous groups. We survey the relevant examplescoming from groups. While it is known that a finite group ofpermutations can be at most 5-transitive unless it containsthe alternating group, we show that it is possible to constructa nontrivial t-transitive set of permutations for each positiveinteger t. We also show how these ideas lead to a combinatorialbasis for the Bose–Mesner algebra of the association schemeof the symmetric group and a design system attached to thisassociation scheme.     

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.     

Finite Element Approximation of a Rigid Punch Indenting a Membrane

Optimal order H¹ and L error bounds are obtained for a continuouspiecewise linear finite element approximation of an obstacleproblem, where the obstacle's height as well as the contactzone, _c, are a priori unknown. The problem models the indentationof a membrane by a rigid punch. For R², given ,g R⁺ and an obstacle defined over E we consider the minimization of |v|²_1,+gµover (v, µ) H¹₀() x R subject to v+µ on E. In additionwe show under certain nondegeneracy conditions that dist (_c,^h_c)Ch ln 1/h, where ^h_c is the finite element approximation to_c. Finally we show that the resulting algebraic problem canbe solved using a projected SOR algorithm.     

Immanant Inequalities, Induced Characters, and Rank Two Partitions

If = {₁, ₂, ..., _s}, where _{1 2} ... _s > 0, is a partitionof n then denotes the associated irreducible character of S_n,the symmetric group on {1, 2, ..., n}, and, if cCS_n, the groupalgebra generated by C and S_n, then d_c(·) denotes thegeneralized matrix function associated with c. If c₁, c₂ CS_nthen we write c₁ c₂ in case (A) (A) for each n x n positivesemi-definite Hermitian matrix A. If cCS_n and c(e) 0, wheree denotes the identity in S_n, then or denotes (c(e))^–1 c. The main result, an estimate for the norms of tensors of a certainanti-symmetry type, implies that if = {₁, ₂, ..., _s, 1^t} isa partition of n such that s > 1 and _s = 2, and ' denotes{₁, ₂, ..., _s-1, 1^t} then (, _{2}) where denotes characterinduction from S_n–2 x S₂ to S_n. This in turn implies thatif = {₁, ₂, ..., _s, 1^t} with s > 1, _s = 2, and ßdenotes {₁ + 2, ₂, ..., _s-1, 1^t} then _ß which,in conjunction with other known results, provides many new inequalitiesamong immanants. In particular it implies that the permanentfunction dominates all normalized immanants whose associatedpartitions are of rank 2, a result which has proved elusivefor some years. We also consider the non-relationship problem for immanants– that is the problem of identifying pairs, (,ß)such that _ß and _ß are both false.     

On the Behaviour of the First Eigenfunction of the p-Laplacian Near its Critical Points

In this paper, the behaviour of the positive eigenfunction of in u_| = 0, p > 1, isstudied near its critical points. Under some convexity and symmetryassumptions on , is seen to have a unique critical point atx = 0; also, the behaviour of both and is determined nearby.Positive solutions u to some general problems –_pu = f(u)in , u_| = 0, are also considered, with some convexity restrictionson u. 2000 Mathematics Subject Classification 35B05 (primary),35J65, 35J70 (secondary).     

Essential norms and localization moduli of Sobolev embeddings for general domains

We introduce new measures of non-compactness for the embeddingoperator E_p,q():L_p¹() L_q() and describe their relations withthe essential norm of E_{p, q}(), ‘local’ isoperimetricand isocapacitary constants. An explicit formula for the essentialnorm of E_{p, q}() is obtained for domains with a power cusp onthe boundary and bounded C¹ domains. The Neumann problem fora particular Schrödinger operator is discussed on domainswith a power cusp.     

Calderon-Zygmund Operators on Mixed Lebesgue Spaces and Applications to Null Forms

The boundedness of Calderón–Zygmund operators isproved in the scale of the mixed Lebesgue spaces. As a consequence,the boundedness of the bilinear null forms Q_{i j} (u,) =_i u_j - _j u_i , Q₀(u,)=u_{t t} -_x u· _x on various space–timemixed Sobolev–Lebesgue spaces is shown.     

Some Remarks on the Cone of Completely Positive Maps between Von Neumann Algebras

The close relationship between the notions of positive formsand representations for a C*-algebra A is one of the most basicfacts in the subject. In particular the weak containment ofrepresentations is well understood in terms of positive forms:given a representation of A in a Hilbert space H and a positiveform on A, its associated representation is weakly containedin (that is, ker ker ) if and only if belongs to the weak*closure of the cone of all finite sums of coefficients of .Among the results on the subject, let us recall the followingones. Suppose that A is concretely represented in H. Then everypositive form on A is the weak* limit of forms of the typex ^k_{i=1 i}, x_i with the _i in H; moreover if A is a von Neumannsubalgebra of (H) and is normal, there exists a sequence (_i)_{i 1} in H such that (x) = _{i 1 i}, x_i for all x.     

Normal and Non-Normal Points of Self-Similar Sets and Divergence Points of Self-Similar Measures

Let K and µ be the self-similar set and the self-similarmeasure associated with an IFS (iterated function system) withprobabilities (S_i, p_i)_i=1,...,N satisfying the open set condition.Let ={1,...,N}^N denote the full shift space and let : K denotethe natural projection. The (symbolic) local dimension of µat is defined by lim_n (log µK_|n/log diam K_|n), where for = (₁, ₂,...) . A point for which the limit lim_n (log µK_|n/log diam K_|n) doesnot exist is called a divergence point. In almost all of theliterature the limit lim_n (log µK_|n/log diam K_|n) is assumedto exist, and almost nothing is known about the set of divergencepoints. In the paper a detailed analysis is performed of theset of divergence points and it is shown that it has a surprisinglyrich structure. For a sequence (_n)_n, let A(_n) denote the setof accumulation points of (_n)_n. For an arbitrary subset I ofR, the Hausdorff and packing dimension of the set and related sets is computed. An interesting and surprisingcorollary to this result is that the set of divergence pointsis extremely ‘visible’; it can be partitioned intoan uncountable family of pairwise disjoint sets each with fulldimension. In order to prove the above statements the theory of normaland non-normal points of a self-similar set is formulated anddeveloped in detail. This theory extends the notion of normaland non-normal numbers to the setting of self-similar sets andhas numerous applications to the study of the local propertiesof self-similar measures including a detailed study of the setof divergence points.