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 Positive Multipeak Solutions of a Nonlinear Elliptic Problem

In this paper we continue our investigation in [5, 7, 8] onmultipeak solutions to the problem –²u+u=Q(x)|u|^q–2u, xR^N, uH¹(R^N) (1.1) where = ^N_i=1²/x²_i is the Laplace operator in R^N, 2 < q < for N = 1, 2, 2 < q < 2N/(N–2) for N3, and Q(x)is a bounded positive continuous function on R^N satisfying thefollowing conditions. (Q₁) Q has a strict local minimum at some point x₀R^N, that is,for some > 0 Q(x)>Q(x₀) for all 0 < |x–x₀| < . (Q₂) There are constants C, > 0 such that |Q(x)–Q(y)|C|x–y| for all |x–x₀| , |y–y₀| . Our aim here is to show that corresponding to each strict localminimum point x₀ of Q(x) in R^N, and for each positive integerk, (1.1) has a positive solution with k-peaks concentratingnear x₀, provided is sufficiently small, that is, a solutionwith k-maximum points converging to x₀, while vanishing as 0 everywhere else in R^N.     

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.     

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.     

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.     

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].     

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.     

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.     

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.     

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.     

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.     

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 ?     

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.     

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].     

A Class of Infinite Dimensional Simple Lie Algebras

Let A be an abelian group, F be a field of characteristic 0,and , ß be linearly independent additive maps fromA to F, and let ker()\{0}. Then there is a Lie algebra L = L(A,, ß, ) = _xA Fe_x under the product [e_x, e_y]]=(x–y)e_x+y+(ß) (x, y) e_x+y–. If, further, ß() = 1, and ß(A) = Z, thereis a subalgebra L⁺:=L(A⁺, , ß, ) = _xA⁺ Fe_x, whereA⁺ = {xA|ß(x)0}. The necessary and sufficient conditionsare given for L' = [L, L] and L⁺ to be simple, and all semi-simpleelements in L' and L⁺ are determined. It is shown that L' andL⁺ cannot be isomorphic to any other known Lie algebras andL' is not isomorphic to any L⁺, and all isomorphisms betweentwo L' and all isomorphisms between two L⁺ are explicitly described.     

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.     

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).     

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.     

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.     

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 .