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     

Singular Integrals and Potential Theory

In this paper I consider a class of non-standard singular integrals motivated by potential theoretic and probabilistic considerations. The probabilistic applications, which are by far the most interesting part of this circle of ideas, are only outlined in Section 1.5: They give the best approximation of the solution of the classical Dirichlet problem in a Lipschitz domain by the corresponding solution by finite differences. The potential theoretic estimate needed for this gives rise to a natural duality between the L _p functions on the boundary ∂Ω and a class of functions A on Ω that was first considered by Dahlberg. The actual duality is given by ∫_Ω S f(x)A(x)dx = (f, A) where S f(x) = ∫_∂Ω |x − y|¹⁻ⁿ f(y)dy is the Newtonian potential. We can identify the upper half Lipschitz space with in the obvious way and express for an appropriate kernel K. It is the boundedness properties of the above (for , ) that is the essential part of this work. This relates with more classical (but still “rough”) singular integrals that have been considered by Christ and Journé. Lecture held in the Seminario Matematico e Fisico on March 14, 2005 Received: April 2007     

Quasi-reflection algebras and cyclotomic associators

We develop a cyclotomic analogue of the theory of associators. Using a trigonometric version of the universal KZ equations, we prove the formality of a morphism , where B _n ¹ is a braid group of type B. The formality isomorphism depends algebraically on a series Ψ_KZ, the “KZ pseudotwist”. We study the scheme of pseudotwists and show that it is a torsor under a group GTM(N, k), mapping to Drinfeld’s group GT(k), and whose Lie algebra is isomorphic to its associated graded (N, k). We prove that Ihara’s subgroup GTK of the Grothendieck–Teichmüller group, defined using distribution relations, in fact coincides with it. We show that the subscheme of pseudotwists satisfying distribution relations is a subtorsor. We study the corresponding analogue (N, k) of (N, k); it is a graded Lie algebra with an action of , and we give a lower bound for the character of its space of generators.      

Optimal constant in a new estimate for the degree

In this paper, we prove the estimate

Metrics in the sphere of a C*-module

Given a unital C*-algebra and a right C*-module over , we consider the problem of finding short smooth curves in the sphere = {x ∈ : 〈x, x〉 = 1}. Curves in are measured considering the Finsler metric which consists of the norm of at each tangent space of . The initial value problem is solved, for the case when is a von Neumann algebra and is selfdual: for any element x ₀ ∈ and any tangent vector ν at x ₀, there exists a curve γ(t) = e ^tZ (x ₀), Z ∈ , Z* = −Z and ∥Z∥ ≤ π, such that γ(0) = x ₀ and (0) = ν, which is minimizing along its path for t ∈ [0, 1]. The existence of such Z is linked to the extension problem of selfadjoint operators. Such minimal curves need not be unique. Also we consider the boundary value problem: given x ₀, x ₁ ∈ , find a curve of minimal length which joins them. We give several partial answers to this question. For instance, let us denote by f ₀ the selfadjoint projection I − x ₀ ⊗ x ₀, if the algebra f ₀ f ₀ is finite dimensional, then there exists a curve γ joining x ₀ and x ₁, which is minimizing along its path.      

Local existence and finite time blow-up of solutions in the 2-D Keller-Segel system

We consider the 2-D Keller-Segel system (KS) for γ > 0. We first construct a mild solution of (KS) for every . The local existence time is characterized for with 1 < q _* < 2. Next, we prove the finite time blow-up of strong solution under the assumption and , where g(s) is an increasing function of s > 1 with an explicit representation. As an application of our mild solutions, an exact blow-up rate near the maximal existence time is obtained.      

The solutions of the<Emphasis Type="Italic">n</Emphasis>-dimensional Bessel diamond operator and the Fourier-Bessel transform of their convolution

In this article, the operator is introduced and named as the Bessel diamond operator iteratedk times and is defined by

Rational invariants of certain classical groups over finite fields

Let be a finite field with q elements, where q is a prime power. Let G be a subgroup of the general linear group over and be the rational function field over . We seek to understand the structure of the rational invariant subfield . In this paper, we prove that is rational (or, purely transcendental) by giving an explicit set of generators when G is the symplectic group. In particular, the set of generators we gave satisfies the Dickson property.      

Borel hierarchies in infinite products of Polish spaces

Let H be a product of countably infinite number of copies of an uncountable Polish space X. Let Σ_ξ be the class of Borel sets of additive class ξ for the product of copies of the discrete topology on X (the Polish topology on X), and let . We prove in the Lévy-Solovay model that

On the relative Wall-Witt groups

LetR* be a simplicial involutive ring. According to certain involutions onK(R*) and _ε L R _∗, there are 1/2-local splittings and . It is known [2] that _ε L _\ga ^α R _∗, the Wall-Witt group. SupposeI→R S is a split extension of discrete involutive rings withI ²=0, andI is a freeS-bimodule. Then we have and . The trace map Tr: Prim _n ∧*M(I ⊗ ℚ)→ ₀ ρ _n ;I ⊗ ℚ) is an isomorphism. We prove in Lemma 1 that the trace map Tr is ℤ/2-equivariant. In Theorem 2 we show that under a certain assumption the rational relative Wall-Witt group vanishes. Theorem 2 can be extended to a more general case (Theorem 3) by employing Goodwillie’s reduction technique [3]. This work was partially supported by KOSEF under Grant 923-0100-010-1.     

Sesquitransitive and Localizing Operator Algebras

An algebra of operators on a Banach space X is said to be transitive if X has no nontrivial closed subspaces invariant under every member of the algebra. In this paper we investigate a number of conditions which guarantee that a transitive algebra of operators is “large” in various senses. Among these are the conditions of algebras being localizing or sesquitransitive. An algebra is localizing if there exists a closed ball B ∌ 0 such that for every sequence (x _n ) in B there exists a subsequence and a bounded sequence (A _k ) in the algebra such that converges to a non-zero vector. An algebra is sesquitransitive if for every non-zero z ∈ X there exists C > 0 such that for every x linearly independent of z, for every non-zero y ∈ X, and every there exists A in the algebra such that and ||Az|| ≤ C||z||. We give an algebraic version of this definition as well, and extend Jacobson’s density theorem to algebraically sesquitransitive rings. The second and the third authors were supported by NSERC.     

Strong surjectivity of mappings of some 3-complexes into $$ M_{Q_8 } $$

Let K be a CW-complex of dimension 3 such that H ³(K;ℤ) = 0 and the orbit space of the 3-sphere with respect to the action of the quaternion group Q ₈ determined by the inclusion Q ₈ ⊆ . Given a point a ∈ , we show that there is no map f:K → which is strongly surjective, i.e., such that MR[f,a]=min{#(g ⁻¹(a))|g ∈ [f]} ≠ 0.      

Additive preservers of idempotence and Jordan homorphisms between rings of square matrices

Suppose R is an idempotence-diagonalizable ring. Let n and m be two arbitrary positive integers with n ≥ 3. We denote by Mn（R） the ring of all n x n matrices over R. Let （Jn（R）） be the additive subgroup of Mn（R） generated additively by all idempotent matrices. Let JJ = （Jn（R）） or Mn（R）. We describe the additive preservers of idempotence from JJ to Mm（R） when 2 is a unit of R. Thereby, we also characterize the Jordan （respectively, ring and ring anti-） homomorphisms from Mn （R） to Mm （R） when 2 is a unit of R.     

On a map between two K3 surfaces associated to a net of quadrics

We study the geometry of the birational map between an intersection of a net of quadrics in that contains a line and the double sextic branched along the discriminant of the net. We show that the branch locus of a smooth double sextic S ₆ is discriminant of a net of quadrics in such that S ₆ is isomorphic to the intersection of this net iff a certain configuration of rational curves on S₆ is weakly even. Received: 14 September 2005 Suported by the DFG Schwerpunktprogramm ‘Global methods in complex geometry’. The first named author is partially supported by the KBN Grant No. 1 P03A 008 28. The second named author is partially supported by the KBN Grant No. 2 P03A 016 25.     

Derivations of the even part of the odd hamiltonian superalgebra in modular case

In this paper we mainly study the derivations for even part of the finite-dimensional odd Hamiltonian superalgebra HO over a field of prime characteristic. We first give the generating set of the even part g of HO. Then we compute the derivations from g into the even part m of the generalized Witt superalgebra. Finally, we determine the derivation algebra and outer derivation algebra of and the dimension formulas. In particular, the first cohomology groups H^1（g;m） and H^1（g;g） are determined.     

A superalgebraic interpretation of the quantization maps of Weil algebras

Let G be a Lie group whose Lie algebra g is quadratic. In the paper ＂the non-commutative Weil algebra＂, Alekseev and Meinrenken constructed an explicit G-differential space homomorphism ￡, called the quantization map, between the Well algebra Wg = S（g^＊） χ∧A（g^＊） and Wg= U（g） χ Cl（g） （which they call the noncommutative Weil algebra） for g. They showed that ￡ induces an algebra isomorphism between the basic cohomology rings Hbas^＊（Wg） and Hbas^＊（Wg）. In this paper, we will interpret the quantization map .~ as the super Duflo map between the symmetric algebra S（Tg[1]） and the universal enveloping algebra U（Tg[1]） of a super Lie algebra T9[1] which is canonically associated with the quadratic Lie algebra g. The basic cohomology rings Hbas^＊（Wg） and Hbas^＊（Wg） correspond exactly to S（Tg[1]）^inv and U（Tg[1]）^inv, respectively. So what they proved is equivalent to the fact that the super Duflo map commutes with the adjoint action of the super Lie algebra, and that the super Duflo map is an algebra homomorphism when restricted to the space of invariants.     

The Hopf algebra Rep U_{q} \widehat{\frak g \frak l}_\infty

We define the Hopf algebra structure on the Grothendieck group of finite-dimensional polynomial representations of in the limitN→∞. The resulting Hopf algebra Rep is a tensor product of its Hopf subalgebras Rep_a ,a ∈ ℂ^×/q^2ℤ. Whenq is generic (resp.,q ² is a primitive root of unity of orderl), we construct an isomorphism between the Hopf algebra Rep _a and the algebra of regular functions on the prounipotent proalgebraic group (resp., ). Whenq is a root of unity, this isomorphism identifies the Hopf subalgebra of Rep _a spanned by the modules obtained by pullback with respect to the Frobenius homomorphism with the algebra generated by the coefficients of the determinant of an element of considered as anl×l matrix over the Taylor series. This gives us an explicit formula for the Frobenius pullbacks of the fundamental representations. In addition, we construct a natural action of the Hall algebra associated to the infinite linear quiver (resp., the cyclic quiver withl vertices) on Rep _a and describe the span of tensor products of evaluation representations taken at fixed points as a module over this Hall algebra.     

Weighted Non-Trivial Multiply Intersecting Families

Let n and r be positive integers. Suppose that a family satisfies F₁∩···∩F_r ≠∅ for all F₁, . . .,F_r ∈ and . We prove that there exists ε=ε(r) >0 such that holds for 1/2≤w≤1/2+ε if r≥13.     

（Z2）^k-Actions with Fixed Point Set of Constant Codimension 2^k＋ 2

Let J.^r ,k denote the ideal in MO. of cobordism classes containing a representative that admits (Z2)^k-actions with a fixed point set of constant codimension r. In this paper we determine J,k^2k 2 and Ja,3^2^3 1.     

On quasiconformal deformations of the universal hyperbolic solenoid

We investigate the Teichmüller metric and the complex structure on the Teichmüller space (H _∞) of the universal hyperbolic solenoid H _∞. In particular, a version of the Reich-Strebel inequality for H _∞ is obtained. As a consequence, we show that the Teichmüller type Beltrami coefficients determine unique geodesics in (H _∞), and we compute the infinitesimal form of the Teichmüller metric. In addition, we show that a Beltrami coefficient is Teichmüller extremal if and only if it is infinitesimally extremal. Finally, we show that the Kobayashi metric on (H _∞) equals the Teichmüller metric.