Multivalued evolution equations with nonlocal initial conditions in Banach spaces

We prove the existence of integral solutions to the nonlocal Cauchy problem in a Banach space X, where is m-accretive and such that –A generates a compact semigroup, has nonempty, closed and convex values, and is strongly-weakly upper semicontinuous with respect to its second variable, and . The case when A depends on time is also considered.      

A Real Analyticity Result for Symmetric Functions of the Eigenvalues of a Domain-Dependent Neumann Problem for the Laplace Operator

Let Ω be an open connected subset of for which the imbedding of the Sobolev space W ^1,2(Ω) into the space L ²(Ω) is compact. We consider the Neumann eigenvalue problem for the Laplace operator in the open subset (Ω) of , where is a Lipschitz continuous homeomorphism of Ω onto (Ω). Then we prove a result of real analytic dependence for symmetric functions of the eigenvalues upon variation of . This paper represents an extension of a part of the work performed by P.D. Lamberti in his PhD Thesis at the University of Padova under the guidance of M. Lanza de Cristoforis.     

A singular Moser-Trudinger embedding and its applications

Let Ω be a bounded domain in , we prove the singular Moser-Trudinger embedding: if and only if where and . We will also study the corresponding critical exponent problem.     

Proof Of A Conjecture Of Erdős On Triangles In Set-Systems

A triangle is a family of three sets A,B,C such that A∩B, B∩C, C∩A are each nonempty, and . Let be a family of r-element subsets of an n-element set, containing no triangle. Our main result implies that for r ≥ 3 and n ≥ 3r/2, we have This settles a longstanding conjecture of Erdős [7], by improving on earlier results of Bermond, Chvátal, Frankl, and Füredi. We also show that equality holds if and only if consists of all r-element subsets containing a fixed element. Analogous results are obtained for nonuniform families.     

Strongly 1-Bounded Von Neumann Algebras

Suppose F is a finite tuple of selfadjoint elements in a tracial von Neumann algebra M. For α > 0, F is α-bounded if where is the free packing α-entropy of F introduced in [J3]. M is said to be strongly 1-bounded if M has a 1-bounded finite tuple of selfadjoint generators F such that there exists an with . It is shown that if M is strongly 1-bounded, then any finite tuple of selfadjoint generators G for M is 1-bounded and δ₀(G) ≤ 1; consequently, a strongly 1-bounded von Neumann algebra is not isomorphic to an interpolated free group factor and δ₀ is an invariant for these algebras. Examples of strongly 1-bounded von Neumann algebras include (separable) II ₁-factors which have property Γ, have Cartan subalgebras, are non-prime, or the group von Neumann algebras of . If M and N are strongly 1-bounded and M ∩ N is diffuse, then the von Neumann algebra generated by M and N is strongly 1-bounded. In particular, a free product of two strongly 1-bounded von Neumann algebras with amalgamation over a common, diffuse von Neumann subalgebra is strongly 1-bounded. It is also shown that a II ₁-factor generated by the normalizer of a strongly 1-bounded von Neumann subalgebra is strongly 1-bounded. Received: November 2005, Revision: March 2006, Accepted: March 2006     

Some results on the qualitative properties of positive solutions of quasilinear elliptic equations

We consider the Dirichlet problem in Ω with zero Dirichlet boundary conditions. We prove local summability properties of and we exploit these results to give geometric characterizations of the critical set . We extend to the case of changing sign nonlinearities some results known in the case f(s) > 0 for s > 0. Berardino Sciunzi: Supported by MURST, Project “Metodi Variazionali ed Equazioni Differenziali Non Lineari”     

Periodic parabolic problems with concave and convex nonlinearities

Let be a smooth bounded domain, let a, b be two functions that are possibly discontinuous and unbounded with a ≥ 0 in and b > 0 in a set of positive measure and let 0 < p < 1 < q. We prove that there exists some 0 < Λ < ∞ such that the nonlinear Dirichlet periodic parabolic problem in has a positive solution for all 0 < λ < Λ and that there is no positive solution if λ > Λ. In some cases we also show the existence of a minimal solution for all 0 < λ < Λ and that the solution u _λ can be chosen such that λ → u _λ is differentiable and increasing. We also give some upper and lower estimates for such a Λ. All results remain true for the analogous elliptic problems. Partially supported by CONICET, Secyt-UNC, ANPCYT and Agencia Cordoba Ciencia     

Property ∏σ and tensor products of operator algebras

In this paper, we introduce Property ∏σ of operator algebras and prove that nest subalgebras and the finite-width CSL subalgebras of arbitrary von Neumann algebras have Property ∏σ.Finally, we show that the tensor product formula alg ML1-(×)algNL2 = algM-(×)N(L1 (×) L2) holds for any two finite-width CSLs L1 and L2 in arbitrary von Neumann algebras M and N, respectively.     

Explicit relation between the solutions of the heat and the Hermite heat equation

There are lots of results on the solutions of the heat equation but much less on those of the Hermite heat equation due to that its coefficients are not constant and even not bounded. In this paper, we find an explicit relation between the solutions of these two equations, thus all known results on the heat equation can be transferred to results on the Hermite heat equation, which should be a completely new idea to study the Hermite equation. Some examples are given to show that known results on the Hermite equation are obtained easily by this method, even improved. There is also a new uniqueness theorem with a very general condition for the Hermite equation, which answers a question in a paper in Proc. Japan Acad. (2005). Supported partially by 973 project (2004CB318000)     

Reality properties of conjugacy classes in algebraic groups

Let be a (not necessarily semi-finite) σ-finite von Neumann algebra. We prove that there exists a finite von Neumann algebra so that for every 1 < p < 2, the Haagerup L ^p -space associated with embeds isomorphically into . We also provide a proof of the following non-commutative generalization of a classical result of Rosenthal: if is a semi-finite von Neumann algebra then every reflexive subspace of embeds isomorphically into L ^r ( ) for some r > 1. Dedicated to Professor H. P. Rosenthal on the occasion of his sixty-fifth birthday Research partially supported by NSF grant DMS-0456781.     

The Cauchy-Neumann problem for a viscous Hamilton-Jacobi equation

We study the weak solvability of viscous Hamilton-Jacobi equation: \,0,\,x\,\in\,\Omega,$" align="middle" border="0"> with Neumann boundary condition and irregular initial data ₀. The domain is a bounded open set and p > 0. The last part deals with the case a convex set and the initial data ₀ = in a open set D such that and      

Supersaturation For Ramsey-Turán Problems

For an l-graph , the Turán number is the maximum number of edges in an n-vertex l-graph containing no copy of . The limit is known to exist [8]. The Ramsey–Turán density is defined similarly to except that we restrict to only those with independence number o(n). A result of Erdős and Sós [3] states that as long as for every edge E of there is another edge E′of for which |E∩E′|≥2. Therefore a natural question is whether there exists for which . Another variant proposed in [3] requires the stronger condition that every set of vertices of of size at least εn (0<ε<1) has density bounded below by some threshold. By definition, for every . However, even is not known for very many l-graphs when l>2. We prove the existence of a phenomenon similar to supersaturation for Turán problems for hypergraphs. As a consequence, we construct, for each l≥3, infinitely many l-graphs for which . We also prove that the 3-graph with triples 12a, 12b, 12c, 13a, 13b, 13c, 23a, 23b, 23c, abc, satisfies . The existence of a hypergraph satisfying was conjectured by Erdős and Sós [3], proved by Frankl and R?dl [6], and later by Sidorenko [14]. Our short proof is based on different ideas and is simpler than these earlier proofs. * Research supported in part by the National Science Foundation under grants DMS-9970325 and DMS-0400812, and an Alfred P. Sloan Research Fellowship. † Research supported in part by the National Science Foundation under grants DMS-0071261 and DMS-0300529.     

A characterization of free generating sets in MV-algebras

MV-algebras are a generalization of Boolean algebras. As is well known, a free generating set for a Boolean algebra is characterized by the following simple algebraic condition: whenever A and B are finite disjoint subsets of X then . Our aim in this note is to give a similar characterization of free generating sets in MV-algebras. Received January 30, 2005; accepted in final form March 13, 2007.     

Sharp Sobolev inequalities of second order

Let (M, g) be a smooth compact Riemannian manifold of dimension n≥5, and ₂ ² (M) be the Sobolev space consisting of functions in L²(M) whose derivatives up to the order two are also in L²(M). Thanks to the Sobolev embedding theorem, there exist positive constants A and B such that for any U ∈ H ₂ ² (M),

Minimal number of idempotent generators for certain algebras

It is known [KRS] that for each finitely generated Banach algebra there exists a numberN such that for eachn>N the matrix algebras can be generated by three idempotents. In this paper we show that the same statement is true for direct sums and , where is a finitely generated free algebra, i.e. polynomials in several non-commuting variables. These results are new even for algebras because the numberN we obtain here improves known estimates (see for example [R]). We show that the algebra can be generated by two idempotents if and only ifn _j =2 for eachj and is singly generated. Also we give an example of a free singly generated algebra for which can not be generated by two idempotents. But% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf% gDOfdaryqr1ngBPrginfgDObYtUvgaiuaacuWFSeIqgaacaaaa!409A!\[{\tilde {\cal B}}\] can be generated by three idempotents for each singly generated free algebra .     

Multiplicity of standing wave solutions of nonlinear Schrödinger equations with electromagnetic fields

In this paper, we are concerned with the multiplicity of standing wave solutions of nonlinear Schr?dinger equations with electromagnetic fields

Explicit Bounded-Degree Unique-Neighbor Concentrators

Here we solve an open problem considered by various researchers by presenting the first explicit constructions of an infinite family of bounded-degree ‘unique-neighbor’ concentrators Γ; i.e., there are strictly positive constants α and ε, such that all Γ = (X,Y,E(Γ)) ∈ satisfy the following properties. The output-set Y has cardinality times that of the input-set X, and for each subset S of X with no more than α|X| vertices, there are at least ε|S| vertices in Y that are adjacent in Γ to exactly one vertex in S. Also, the construction of is simple to specify, and each has fewer than edges. We then modify to obtain explicit unique-neighbor concentrators of maximum degree 3. * Supported by NSF grant CCR98210-58 and ARO grant DAAH04-96-1-0013.     

On the Schwarz reflection principle for monogenic functions

Let ∑ be either an oriented hyperplane or the unit sphere in , let be open and connected and let be an open and connected domain in such that . If in is a null solution of the Dirac operator (also called a monogenic function in ) which is continuously extendable to , then conditions upon are given enabling the monogenic extension of across . In such a way Schwarz reflection type principles for monogenic functions are established in the Spin (1) and Spin cases. The Spin (1) case includes the classical Schwarz reflection principle for holomorphic functions in the plane. The Spin case deals with so-called “half boundary value problems” for the Dirac operator. Received: 2 February 2006     

Hermitean Clifford-Hermite Polynomials

Orthogonal Clifford analysis in flat m–dimensional Euclidean space focusses on monogenic functions, i.e. null solutions of the rotation invariant vector valued Dirac operator , where ( ) forms an orthogonal basis for the quadratic space underlying the construction of the Clifford algebra . When allowing for complex constants and taking the dimension to be even: m = 2n, the same set of generators produces the complex Clifford algebra , which we equip with a Hermitean Clifford conjugation and a Hermitean inner product. Hermitean Clifford analysis then focusses on the simultaneous null solutions of two mutually conjugate Hermitean Dirac operators, naturally arising in the present context and being invariant under the action of a realization of the unitary group U (n). In this so–called Hermitean setting Clifford–Hermite polynomials are constructed, starting from a Rodrigues formula involving both Dirac operators mentioned. Due to the specific features of the Hermitean setting, four different types of polynomials are obtained, two types of even degree and two types of odd degree. We investigate their properties: recurrence relations, structure, explicit form and orthogonality w.r.t. a deliberately chosen weight function. They also give rise to the definition of the Hermitean Clifford–Hermite functions, and may be used to develop a Hermitean continuous wavelet transform, see [4].     

Lie and Jordan Ideals in Reflexive Algebras

A CDCSL algebra is a reflexive operator algebra with completely distributive and commutative subspace lattice. In this paper, we show, for a weakly closed linear subspace of a CDCSL algebra , that is a Lie ideal if and only if for all invertibles A in , and that is a Jordan ideal if and only if it is an associative ideal.