On the Compactness and Approximation Numbers of Hardy-Type Integral Operators in Lorentz Spaces

We characterize the mapping properties such as the boundedness,compactness and measure of noncompactness for those real weightfunctions , , u0, v0, for which the Hardy-type integral operatorof the form acts from to , when the parameters are restricted to the range 1 < max (r,s) min (p, q) < and the kernel k(x, y) 0 satisfies theOinarov condition (see (2) below). For the case k(x, y) = 1,we obtain lower and upper estimates of the approximation numbers,extending the result of [5].     

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.     

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.     

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.     

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.     

On Borel Sets in Function Spaces with the Weak Topology

It is proved that the duality map ,:(, weak)x(()*, weak*)R isnot Borel. More generally, the evaluation e:(C)(K),x KR, e(f,x) = f(x), is not Borel for any function space C(K) on a compactF-space. It is also shown that a non-coincidence of norm-Boreland weak-Borel sets in a function space does not imply thatthe duality map is non-Borel.     

On the Total Coloring of Graphs Embeddable in Surfaces

The paper shows that any graph G with the maximum degree (G) 8, which is embeddable in a surface of Euler characteristic() 0, is totally ((G)+2)-colorable. In general, it is shownthat any graph G which is embeddable in a surface and satisfiesthe maximum degree (G) (20/9) (3–())+1 is totally ((G)+2)-colorable.     

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.     

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.     

On the Edge Reconstruction of Graphs Embedded in Surfaces II

In this paper, we prove the following theorems. (i) Let G bea graph of minimum degree 5. If G is embeddable in a surface and satisfies (–5)|V(G)|+6()0, then G is edge reconstructible.(ii) Any graph of minimum degree 4 that triangulates a surfaceis edge reconstructible. (iii) Any graph which triangulatesa surface of characteristic 0 is edge reconstructible.     

Systems of Diagonal Equations Over p-Adic Fields

Let K be a p-adic field, and consider the system F = (F₁,...,F_R)of diagonal equations (1) with coefficients in K. It is an interesting problem in numbertheory to determine when such a system possesses a nontrivialK-rational solution. In particular, we define *(k, R, K) tobe the smallest natural number such that any system of R equationsof degree k in N variables with coefficients in K has a nontrivialK-rational solution provided only that N*(k, R, K). For example,when k = 1, ordinary linear algebra tells us that *(1, R, K)= R + 1 for any field K. We also define *(k, R) to be the smallestinteger N such that *(k, R, Q_p) N for all primes p.     

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.     

5-Torsion Points on Curves of Genus 2

Let C be a smooth proper curve of genus 2 over an algebraicallyclosed field k. Fix a Weierstrass point in C(k) and identifyC with its image in its Jacobian J under the Albanese embeddingthat uses as base point. For any integer N1, we write J_N forthe group of points in J(k) of order dividing N and for the subset of J_N of points oforder N. It follows from the Riemann–Roch theorem thatC(k)J₂ consists of the Weierstrass points of C and that C(k) and C(k) are empty (see [3]). The purpose of this paper is to study curvesC with C(k) non-empty.     

On the Length of Snakes in Powers of Complete Graphs

The conjecture stated in an earlier paper by the author thatthere is a constant (independent from both n and k) such that n^d–1 holds for everyn 2 and d 2, where is the length of the longest snake (cycle without chords) in the Cartesianproduct of d copies of thecomplete graph K_n, is proved.     

Logistic Type Equations on RN by a Squeezing Method Involving Boundary Blow-Up Solutions

We study, on the entire space R^N(N 1), the diffusive logisticequation u_t–u=u–u^p, u0 (1.1) and its generalizations. Here p > 1 is a constant. Problem(1.1) plays an important role in understanding various populationmodels and some other problems in applied mathematics. When = 1 and p = 2, it is also known as the Fisher equation andKPP equation, due to the pioneering works of Fisher [8] andKolmogoroff, Petrovsky and Piscounoff [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.     

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.     

Elliptic Genera for Z/k-Manifolds I

Using Z/k-manifolds we give a geometric interpretation of thering homomorphism from cobordismwith Z/k coefficients to elliptic cohomology with Z/k coefficients,induced by the map of spectra MSO Ell associated to the naturaltransformation of cohomology theories MSO* Ell*.     

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.