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.     

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.     

An extension of the recursively enumerable Turing degrees

Consider the countable semilattice _T consisting of the recursivelyenumerable Turing degrees. Although _T is known to be structurallyrich, a major source of frustration is that no specific, naturaldegrees in _T have been discovered, except the bottom and topdegrees, 0 and 0'. In order to overcome this difficulty, weembed _T into a larger degree structure which is better behaved.Namely, consider the countable distributive lattice _w consistingof the weak degrees (also known as Muchnik degrees) of massproblems associated with non-empty ₀¹ subsets of 2. It is knownthat _w contains a bottom degree 0 and a top degree 1 and isstructurally rich. Moreover, _w contains many specific, naturaldegrees other than 0 and 1. In particular, we show that in _wone has 0 < d < r₁ < f(r₂, 1) < 1. Here, d is theweak degree of the diagonally non-recursive functions, and r_nis the weak degree of the n-random reals. It is known that r₁can be characterized as the maximum weak degree of a ₀¹ subsetof 2 of positive measure. We now show thatf(r₂, 1) can be characterizedas the maximum weak degree of a ₀¹ subset of 2, the Turing upwardclosure of which is of positive measure. We exhibit a naturalembedding of _T into _w which is one-to-one, preserves the semilatticestructure of _T, carries 0 to 0, and carries 0' to 1. Identifying_T with its image in _w, we show that all of the degrees in _Texcept 0 and 1 are incomparable with the specific degrees d,r₁, andf(r₂, 1) in _w.     

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].     

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.     

Optimal Cardinals for Metrizable Barrelled Spaces

We seek the smallest or largest cardinals for which certainbasic results hold, as did Mazur when he proved that c is thesmallest infinite-dimensionality for a Fréchet space.As with Mazur, we make no axiomatic assumptions outside theusual ZFC model. We discover three instances in which the optimalcardinal is the dominating number and three in which it isthe bounding number b, apparently giving the first locally convexspace characterizations of these venerable and easily describedcardinals. Here are two samples: it is known that for any non-normablemetrizable locally convex space E, the minimal size _b(E) fora fundamental system of bounded sets must satisfy _{1 b}(E) c;we prove that _b(E) = . Again, it is known that if E is a non-normablemetrizable barrelled space of minimal dimension, then ₁ dim(E) c; we prove that dim(E) = b. The most important individualresult is the reconstruction of Tweddle's space without useof the Continuum Hypothesis (₁ = c). The reconstruction is vitalin the characterizations of b and in subsequent papers answeringopen questions about countable enlargements.     

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.     

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.     

Metrical Theorems on Prime Values of the Integer Parts of Real Sequences II

Let [ ] denote the integer part. Among other results in [3]we gave a complete solution to the following problem. PROBLEM. Given an increasing sequence a_n R⁺, n = 1, 2, ...,where a_n as n , are there infinitely many primes in the sequence[a_n] for almost all ?     

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.     

Recurrence, Dimension and Entropy

Let (_A, T) be a topologically mixing subshift of finite typeon an alphabet consisting of m symbols and let :_A R^d be a continuousfunction. Denote by (x) the ergodic limit when the limit exists. Possible ergodic limits arejust mean values dµ for all T-invariant measures. Forany possible ergodic limit , the following variational formulais proved: where h_µ denotes the entropy of µ and h_top denotestopological entropy. It is also proved that unless all pointshave the same ergodic limit, then the set of points whose ergodiclimit does not exist has the same topological entropy as thewhole space _A     

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.     

The Isolated Points of G{rho} and the w*-Strongly Exposed Points of P{rho}(G)0

Let G be a separable locally compact group and let be its dualspace with Fell's topology. It is well known that the set P(G)of continuous positive-definite functions on G can be identifiedwith the set of positive linear functionals on the group C^*-algebraC^*(G). We show that if is discrete in , then there exists anonzero positive-definite function associated with such that is a w^*-strongly exposed point of P(G)₀, where P(G)₀={f P(G):f(e)1. Conversely, if some nonzero positive-definite function associatedwith is a w^*-strongly exposed point of P(G)₀, then is isolatedin . Consequently, G is compact if and only if, for every ,there exists a nonzero positive-definite function associatedwith that is a w^*-strongly exposed point of P(G)₀. If, in addition,G is unimodular and , then is isolated in if and only if somenonzero positive-definite function associated with is a w^*-stronglyexposed point of P(G)₀, where is the left regular representationof G and is the reduced dual space of G. We prove that if B(G)has the Radon–Nikodym property, then the set of isolatedpoints of (so square-integrable if G is unimodular) is densein . It is also proved that if G is a separable SIN-group, thenG is amenable if and only if there exists a closed point in. In particular, for a countable discrete non-amenable groupG (for example the free group F₂ on two generators), there isno closed point in its reduced dual space .     

Sandwiching C0-Semigroups

Let T = {T(t)}_t0 be a C₀-semigroup on a Banach space X. Thefollowing results are proved. (i) If X is separable, there exist separable Hilbert spacesX₀ and X₁, continuous dense embeddings j₀:X₀ X and j₁:X X₁,and C₀-semigroups T₀ and T₁ on X₀ and X₁ respectively, suchthat j₀ T₀(t) = T(t) j₀ and T₁(t) j₁ = j₁ T(t) for all t 0. (ii) If T is -reflexive, there exist reflexive Banach spacesX₀ and X₁ , continuous dense embeddings j:D(A²) X₀, j₀:X₀ X, j₁:X X₁, and C₀-semigroups T₀ and T₁ on X₀ and X₁ respectively,such that T₀(t) j = j T(t), j₀ T₀(t) = T(t) j₀ and T(t) j₁ = j₁ T(t) for all t 0, and such that (A₀) = (A) = (A₁),where A_k is the generator of T_k, k = 0, Ø, 1.     

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.     

Contractions et Hyperdistributions a Spectre de Carleson

Let =(_n)_n1 be a log concave sequence such that lim inf_n+n/n^c>0for some c>0 and ((log _n)/n)_n1 is nonincreasing for some<1/2. We show that, if T is a contraction on the Hilbertspace with spectrum a Carleson set, and if ||T^–n||=O(_n)as n tends to + with _n11/(n log _n)=+, then T is unitary. Onthe other hand, if _n11/(n log _n)<+, then there exists a (non-unitary)contraction T on the Hilbert space such that the spectrum ofT is a Carleson set, ||T^–n||=O(_n) as n tends to +, andlim sup_n+||T^–n||=+.     

Herman Rings and Arnold Disks

For (,a) C* x C, let f_,a be the rational map defined by f_,a(z)= z² (az+1)/(z+a). If R/Z is a Brjuno number, we let D bethe set of parameters (,a) such that f_,a has a fixed Hermanring with rotation number (we consider that (e²ⁱ,0) D). Resultsobtained by McMullen and Sullivan imply that, for any g D, theconnected component of D(C* x (C/{0,1})) that contains g isisomorphic to a punctured disk. We show that there is a holomorphic injection F:DD such thatF(0) = (e²ⁱ ,0) and , where r is the conformal radius at 0 of the Siegel disk of the quadraticpolynomial z e²ⁱ z(1+z). As a consequence, we show that for a (0,1/3), if f_l,a has afixed Herman ring with rotation number and if m_a is the modulusof the Herman ring, then, as a0, we have e m_a=(r/a) + O(a). We finally explain how to adapt the results to the complex standardfamily z e^(a/2)(z-1/z).     

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.