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Mathematical Logic: Proof Theory, Constructive Mathematics
Authors: Buss Samuel R. , Kohlenbach Ulrich , Schwichtenberg Helmut
Published: December 2009

The workshop
Mathematical Logic: Proof Theory, Constructive Mathematics
was held April 6-12, 2008 and had the following main aims:

To promote
the interaction of proof theory with core areas of mathematics. This, in particular, refers to uses of proof theoretic techniques (most notably various forms of functional and realizability interpretations) to unwind prima facie ineffective proofs in mathematics. For instance a number of talks presented such applications in the areas of fixed point theory, ergodic theory, topological dynamics and symbolic dynamics resulting in new theorems, surprising to the experts in the respective fields (Avigad, Briseid, Gerhardy, Leuştean, Simpson). Another talk addressed uses of analytic number theory in connection with new unprovability results and phase transitions in provability (Bovykin).

To further develop
new directions in both the foundations as well as applied forms of constructive mathematics. There were a number of talks focusing on new formal frameworks for constructive mathematics and their proof theoretic properties (Aczel, Iemhoff, J. Moschovakis, Palmgren, Streicher), but also new constructive (and partly computer generated) developments in specific areas such as algebra, analysis, combinatorics and quantum theory (Bauer, Delzell, Lombardi, Paule, Spitters).

To explore
further the connections between proof theory and computational complexity (e.g. in connection with the study of systems of so-called bounded arithmetic, Beckmann, Pollett, Thapen). Another topic in this area was the study of implicit computational complexity and the design of functional programming languages corresponding to specific complexity classes (Aehlig, Hofmann, Matthes). Yet another approach to the issue of the intrinsic complexity for general classes of algorithmic problems was developed in a talk by Y. Moschovakis.

In order to provide the participants with an overview over new developments in the study of functional and realizability interpretations (as mentioned above) two invited lecture series were given.

The first one (given by Fernando Ferreira and Paulo Oliva) focussed on novel (so-called monotone and bounded) variants of Gödel's functional interpretation (first published exactly 50 years ago) and its decomposition into an interpretation of linear logic combined with Girard's embedding of intuitionistic logic into linear logic. The second course was delivered by Ulrich Berger and presented a number of proof theoretic approaches for analyzing proofs based on dependent choice (such as Spector's bar recursive solution, the realizability solution by modified bar recursion due to Berger and Oliva and a novel approach based on open induction and its relation to previous work by Berardi, Bezem and Coquand). Moreover, Berger reported on recent experiments (with Schwichtenberg) concerning the automated extraction (based on the MinLog tool) of efficient algorithms from proofs in constructive analysis.
Groups and Geometries
Authors: Liebeck Martin W. , Mühlherr Bernhard , Stroth Gernot
Published: December 2009

The workshop
Groups and Geometries
was one of a series of Oberwolfach workshops on this topic which has taken place every 3 years for some time. It focused on algebraic and finite groups, their interactions with the geometry of buildings, and applications.

A particular highlight of the meeting was a celebration of the recent award of the Abel Prize jointly to John Thompson and Jacques Tits, two of the great pioneers of modern group theory and geometry, and both leading participants at many Oberwolfach meetings. The celebration took the form of two special lectures, given on the Tuesday evening. In the first, Bernd Fischer gave some personal recollections about Thompson, and spoke about his enormous influence on the development of finite group theory. The second lecture was given by Richard Weiss, who described some of the revolutionary innovations of Tits in the geometrical aspects of group theory, and also told some stories illustrating how Tits inspired people through his warmth and sense of humour.

There were 45 participants and 26 talks. These were on three main inter-related themes: buildings and their relationship with algebraic groups; structure of finite groups; and applications. Particularly pleasing was the participation of a substantial number (more than 10) of young researchers, several of whom – Alice Devillers, Silvia Onofrei, Rebecca Waldecker, Pierre-Emmanuel Caprace, Ralf Gramlich, Harald Helfgott and Nikolay Nikolov – gave talks. This demonstrates the attractiveness of the field.

The conference showed that the theories of buildings, algebraic groups, and finite simple groups and their geometries are very active areas with a great deal of interaction between them and also with other areas. People from these different areas were brought together, and their interaction was indeed very lively. The conference stands in the tradition of very successful meetings on
Groups and Geometries
at Oberwolfach.
Nonstandard Finite Element Methods
Authors: Brenner Susanne C. , Carstensen Carsten , Monk Peter
Published: December 2009

The workshop
Nonstandard Finite Element Methods
, organized by Susanne C. Brenner (Baton Rouge), Carsten Carstensen (Berlin) and Peter Monk (Newark) was held August 10 – 16, 2008. This meeting was well attended with over 40 participants with broad geographic representation.

Although Courant is often credited with the discovery of finite elements [2, 3], the first practical use of such methods in engineering seems to date back to the work of Argyris and Clough et. al. (apparently Clough coined the name “finite elements”) in the 1950s [1, 4]. By the 1970s the finite element method was firmly established in engineering practice and the basic theory of conforming elements for elliptic problems was well understood [2]. Engineers and mathematicians have since expanded the use of finite elements to a wide variety of new applications and have improved the theoretical underpinnings of the method. In addition, efficient computational algorithms have been developed, such as the multigrid method, that allow for the application of finite elements to sophisticated three dimensional problems.

Within current research, we can distinguish classical or standard finite element methods, and newer nonstandard finite elements The former include conforming elements applied to “standard” Galerkin discretizations of problems. Usually accuracy is obtained by mesh refinement. As the range of applications has increased, and to alleviate perceived shortcomings in standard conforming elements, numerical analysts and engineers have sought more general finite element methods based, for example, on mixed formulations, and non-conforming or even discontinuous Galerkin schemes. These methods may be superior to standard finite elements in particular applications because of enhancements to stability, robustness or conservation properties. We refer to them as non-standard finite element methods.

This Oberwolfach workshop was devoted to non-standard finite element methods and their analyses. Inevitability such a snapshot of the subject is biased, but we have tried to include non-standard methods in the broadest sense including mixed, discontinuous Galerkin, generalized, partition of unity and mortar FEM methods. All these schemes have in common that stability and convergence is not obvious and requires mathematical analysis. This is even more true for developing fast solvers, a posteriori error estimation and adaptive mesh design.

Through the week there were 29 presentations on various non-standard topics. These provided an overview of current research directions, new developments and open problems.
$\mathrm{C}^*$-Algebras
Authors: Anantharaman-Delaroche Claire , Echterhoff Siegfried , Haagerup Uffe , Voiculescu Dan-Virgil
Published: December 2009

A

\mathrm{C}^*

-algebra is an involutive Banach algebra

A

which satisfies the

\mathrm{C}^*

-condition

\|a^*a\|=\|a\|^2

for all

a\in A

. The theory of

\mathrm{C}^*

-algebras goes back to work of Murray and von Neumann, who first studied a special variant now known as von Neumann algebras. The theory developed rapidly after some ground breaking work of Gelfand and Naimark in 1943 in which they showed that

The aim of the workshop

\mathrm{C}^*

-algebras

, organized by Claire Anantharaman–Delaroche, Siegfried Echterhoff, Uffe Haagerup, and Dan Voiculescu, is to bring together leading researchers from basically all areas related to the field. This gives a unique opportunity to maintain a broad view on the subject and to create new cooperations between researchers with different background. Among the 42 participants was a good number of young researchers, some of them already on the top of the field. There have been 27 lectures presented at the workshop with topics ranging from classification of

\mathrm{C}^*

-algebras, group actions on

\mathrm{C}^*

-algebras, orbit equivalence of dynamical systems, Jones' theory of subfactors, fundamental groups of II

_1

-factors,

L^2

-invariants, duality for quantum groups and quantum groupoids, Index theorems,

\mathrm{C}^*

-algebras related to number theory, free probability, the relation between

\mathrm{C}^*

-algebras and Harmonic Analysis, and others.

Among the most exciting recent developments in the field we mention the breathtaking advances of Popa and Vaes on the fundamental groups of II

_1

-factors in the theory of von Neumann algebras, with beautiful applications to the classification of equivalence relations and dynamical systems. Other milestones are the progress of Kirchberg and coworkers on the classification of actions on

\mathcal O_2

and the beautiful results of Dadarlat on continuous bundles of

\mathrm{C}^*

-algebras. Important progress has been achieved also in the classification theory of simple and non-simple

\mathrm{C}^*

-algebras by Elliot, Rørdam, Toms, and Winter–with some further progress being made during the workshop. There were many other exciting contributions to this very successful workshop, which can be seen from the reports presented in this issue.

It is a pleasure for the organizers to thank all participants of the workshop for their beautiful lectures and fruitful discussions. We also want to use this opportunity to thank the Mathematisches Forschungs-institut Oberwolfach for providing a very stimulating environment and strong support for organizing this conference. Special thanks also go to the very competent and helpful staff of the institute.
Arbeitsgemeinschaft: Ricci Flow and the Poincaré Conjecture
Authors: Ecker Klaus , Wilking Burkhard
Published: December 2009

The Ricci Flow, introduced by Richard Hamilton [Ha1], is a geometric evolution equation which deforms the metric on a Riemannian manifold smoothly in the direction of its Ricci curvature. More precisely, the evolution equation for the family of metrics

(g_{ij})

on a manifold

M

is given by

\frac{∂}{∂t} g_{ij} = - 2 R_{ij}

where

R_{ij}

denotes the Ricci tensor corresponding to the metric. Written in suitable local coordinates this equation has the form of a nonlinear heat type equation for the metric symbols. Because of this one might naively expect that the equation will try to evolve the geometry on

M

to one which looks the same at every point on the manifold, a homogeneous geometry. This intuition is correct in dimension two, where the Ricciflow can be used [Ha2], [Ch] to conformally deform any metric on a closed surface to one of constant curvature, which provides a new proof of the famous uniformization theorem.

In higher dimensions, however, the geometry will in general become singular in finite time, i.e. the norm of some of the sectional curvatures will tend to infinity at certain points on the manifold. In three dimensions, if the initial metric has positive Ricci curvature and

M

is closed and simply connected, Hamilton [Ha1] showed that, after suitable rescaling of the evolving metric, such as to keep the volume of the manifold constant, the metric tends smoothly to the metric on the standard

S^3

.

Soon after that, Hamilton [Ha3] set up a programme which had the aim of settling Thurston's geometrization conjecture using Ricci Flow. This conjecture asks whether any closed 3-manifold can be decomposed along 2-spheres and incompressible tori in such a way that after capping of the 2-sphere boundaries by 3-balls, the resulting finitely many geodesically complete pieces would each admit one out of a list of eight homogeneous geometries formulated by Thurston [Th]. In particular, this would prove the famous Poincaré conjecture, that any simply connected orientable closed 3-manifold had to be topologically equivalent to

S^3

.

Hamilton himself, but also many others, completed many of the crucial steps in this programme (see [CLN]) but several severe technical difficulties remained unsettled for at least one decade. In 2002, Perelman [P1]–[P3] introduced a number of completely novel ideas and techniques that eventually led to the resolution of the geometrization and hence also the Poincaré conjecture.
Mini-Workshop: Group Actions on Curves: Reduction and Lifting
Authors: Bouw Irene , Mézard Ariane , Wewers Stefan
Published: December 2009

The mini-workshop
Group actions on curves: reduction and lifting
, organized by Irene I. Bouw (Ulm), Ariane Mézard (Versailles) and Stefan Wewers (Hannover) was held November 16th–November 22nd, 2008. There were 15 participants, 6 of whom were PhD-students or recent PhDs. There were 15 talks and 2 discussion sessions. There was a joint evening session with the mini-workshop
Symmetric Varieties and Involutions of Algebraic Groups
.

The talks discussed very recent progress on the following closely related topics:

The theme of the first discussion session was the local lifting problem. Several participants gave an informal presentation of possible approaches for solving it. Afterwards, we discussed how the different approaches fit together. The main emphasis was the connection between differential data and torsors under group schemes.

The theme of the second discussion session was the structure of deformation spaces. We discussed several questions that came out of the talks. Among other things, given a family of lifts of a cover of formal germs of curves from positive characteristic to characteristic zero, is there a reasonable notion of ‘moduli’ of such lifts, and how does it relate to the position of the branch points? We formulated several conjectural statements and discussed their relevance and possible proofs.
Random Trees
Authors: Baake Ellen , Dawson Donald A. , Greven Andreas , den Hollander Frank
Published: December 2009

The meeting was devoted to random trees, a central concept in mathematics that provides a key way of thinking about relationships between objects such as particles in a ﬂuid, individuals in a population, or labels in a search algorithm. Particular emphasis was put on the role of random trees in physics, biology, and computer science.
Mini-Workshop: Category Theory and Related Fields: History and Prospects
Authors: Krömer Ralf , McLarty Colin , Wright Michael
Published: December 2009

he workshop concerned various topics in the history of category theory and related ﬁelds, paying attention to some extent also to open questions, present and possible future development.
Mini-Workshop: Support Varieties
Authors: Erdmann Karin , Krause Henning
Published: December 2009

The notion of support is a fundamental concept which provides a geometric approach for studying various algebraic structures. The prototype for this has been Quillen’s description of the algebraic variety corresponding to the cohomology ring of a ﬁnite group, based on which Carlson introduced support varieties for modular representations. This has made it possible to apply methods of algebraic geometry to obtain representation theoretic information. Their work has inspired the development of analogous theories in various contexts, notably modules over commutative complete intersection rings, and over cocommutative Hopf algebras. The aim of this workshop has been to bring together experts from these ﬁelds and to stimulate interaction and exchange of ideas.
Topological and Variational Methods for Partial Differential Equations
Authors: Bartsch Thomas , Dancer E. Norman
Published: December 2009

In recent years, there has been a major impact of topological and variational methods on the study of nonlinear elliptic and parabolic partial differential equations. In particular, surprising results for classical open problems have been obtained with new techniques far beyond the classical approaches. The purpose of the meeting was to provide a forum for these developments and to bring together experts with complementary background.
Computational Multiscale Methods
Authors: Carstensen Carsten , Engquist Björn
Published: December 2009

Computational Multiscale Methods play an important role in many modern computer simulations in material sciences with different time scales and different scales in space. Besides various computational challenges, the meeting brought together various applications from many disciplines and scientists from various scientiﬁc communities.
On the Cauchy problem for a one-dimensional compressible viscous polytropic ideal gas
Authors: Qin Yuming , Wu Yumei , Liu Fagui
Published: December 2009

In this paper, we first prove the regularity and continuous dependence on initial data for

H^i

-solutions

(i=1,2,4)

for large initial data and then show the large-time behavior of

H^i(i=2,4)

-global solutions for small initial data to the Cauchy problem for the compressible Navier–Stokes equations of a one-dimensional viscous polytropic ideal gas. Moreover, we also obtain the large-time behavior of “small” classical solutions
in the norm of classical solutions
for this model.
Euler constants for the ring of $S$-integers of a function field
Authors: Car Mireille
Published: December 2009

The Euler constant

\gamma

may be defined as the limit for

n

tending to

+\infty

, of the difference

\sum\limits_{j=1}^n\frac1{j}-\log n

. Alternatively, it may be defined as the limit at 1 of the difference

\sum\limits_{n=1}^{\infty}\frac1{j^s}-\frac 1{s-1}

,

s

being a complex number in the half-plane

\Re(s)>1

. Mertens theorem states that for

x

real number tending to +

\infty

,

\prod\limits_{p\leq x}(1-\frac{1}p)\sim \frac{e^{-\gamma}}{\log x}

, the product being over prime numbers

\leq x

. We prove analog results for the ring of

S

-integers of a function field. However, in the function field case, the three approaches lead to different constants.
A class of high order TVD schemes for systems of conservation laws
Authors: Hashem Zahran Yousef
Published: December 2009

The purpose of this paper is to develop improved finite difference schemes. We firstly propose to use the fourth order TVD flux presented in [21] as the building block in ENO schemes for scalar problems. A way to extend this scheme and the scheme [20] to general systems of nonlinear hyperbolic conservation laws, in one and two dimensions, is discussed in details. A semi-discrete version of the two schemes are presented. The performance of the schemes is assessed by solving test problems for the Euler equations of gas dynamics in one and two dimensions. We use exact solutions and other results to validate the results.
The Tutte polynomial of a morphism of matroids 4. Computational complexity
Authors: Las Vergnas Michel
Published: December 2009

We determine the easy points of the 3-variable Tutte polynomial of a matroid perspective. It turns out that all but one of the sporadic easy points of the 3-variable Tutte polynomial proceed from the 8 sporadic easy points determined in the seminal paper of Jaeger–Vertigan–Welsh on the computational complexity of the Tutte polynomial of a matroid. The exceptional easy point, namely

(-1,-1,-1)

, can be evaluated with polynomial complexity for binary matroid perspectives by a previous result of the author.
A generalisation of Cartwright’s Theorem: nonautonomous differential equations case
Authors: Berboucha Ahmed
Published: December 2009

In this article we show that the almost-periodic solutions of a large class of nonautonomous delay differential equations are quasi-periodic. This result is a generalisation of a theorem proved by Cartwright for ordinary differential equations.
A free boundary problem for a torrential flow
Authors: Yahia-Djouadi R. Ait , Hernane-Boukari D. , Teniou D.
Published: December 2009

A theoretical method based on the hodograph transformation is presented to solve the problem of an irrotational and steady flow of an inviscid and incompressible fluid, over a two-dimensional obstacle lying on the bottom of a channel. The suggested method for the solution of the fully non-linear problem is presented for a super-critical flow (Froude number Fr > 1). The results obtained are based on those established in [6] and [7].
On periodic solutions of a nonlinear singular differential equation
Authors: Sidi Ammi Moulay R.
Published: December 2009

We find sufficient conditions on the rotation number of solutions to a nonlinear singular problem which guarantee the existence of T-periodic solutions.
Quadratic forms for the Liouville equation $w_{tt}+ λ^2a(t)w = 0$ with applications to Kirchhoff equation
Authors: Manfrin Renato
Published: December 2009

We introduce a set of quadratic forms for the solutions of the Liouville equation

w_{tt}+ λ^2a(t)w = 0

. From these forms we derive estimates for the wave equation

u_{tt} − a(t)Δu = 0

and then prove the global solvability for the Kirchhoff equation in suitable classes of not necessarily smooth or small initial data.
Existence of a renormalized solution for a class of nonlinear parabolic equations in Orlicz spaces
Authors: Azroul Elhoussine , Redwane Hicham , Rhoudaf Mohamed
Published: December 2009

An existence result for a solution of a class of nonlinear parabolic equations in Orlicz spaces is established. The data belongs to L1, no growth assumption is made on the nonlinearities and the N-function does not satisfy the Δ2 condition.
There is no odd perfect polynomial over $\mathbb F_2$ with four prime factors
Authors: Gallardo Luis H. , Rahavandrainy Olivier
Published: December 2009

A perfect polynomial over the binary field

\mathbb F_2

is a polynomial

A ∈ \mathbb F_2[x]

that equals the sum of all its divisors. If

\operatorname{gcd}(A,x^2 + x) = 1

then we say that

A

is odd. It is believed that odd perfect polynomials do not exist. In this article we prove this for odd perfect polynomials

A

with four prime divisors, i.e., polynomials of the form

A = P^aQ^bR^cS^d

where

P, Q, R, S

are distinct irreducible polynomials of degree

> 

1 over

\mathbb F_2

and

a, b, c, d

are positive integers.
Holomorphic extension theorem for tempered ultrahyperfunctions
Authors: Franco Daniel H. T.
Published: December 2009

In this article we are concerned with the space of tempered ultrahyperfunctions corresponding to a proper open convex cone. A holomorphic extension theorem (the version of the celebrated Edge-of-the-Wedge Theorem) will be given for this setting. As application, a version is also given of the principle of determination of an analytic function by its values on a non-empty open real set. The article finishes with the generalization of holomorphic extension theorem à la Martineau.
Nonlinear stability properties of periodic travelling wave solutions of the classical Korteweg–de Vries and Boussinesq equations
Authors: Arruda Lynnyngs Kelly
Published: December 2009

This article is concerned with nonlinear stability properties of periodic travelling wave solutions of the classical Korteweg de Vries and Boussinesq equations. Periodic travelling wave solutions with a fixed fundamental period L will be constructed by using Jacobi’s elliptic functions. It will be shown that these solutions, called
cnoidal waves
, are nonlinearly stable in the respective energy space by periodic disturbances with period L.
On splitting perfect polynomials over $\mathbb F_{p^2}$
Authors: Gallardo Luis H. , Rahavandrainy Olivier
Published: December 2009

We study some properties of the exponents of the terms appearing in the splitting perfect polynomials over

\mathbb F_{p^2}

, where p is a prime number. This generalizes the work of Beard et al. over

\mathbb F_{p^2}

.
Existence of solutions to a one-dimensional model of bounded piezoelectric material
Authors: Hamroun Djamila
Published: December 2009

We study the dynamic of piezoelectric bodies with switchable domains, described by a coupling of an hyperbolic equation and a parabolic one. We consider the one dimensional case and prove local existence of solutions.
Nonlinear reaction diffusion systems of degenerate parabolic type
Authors: Aliziane Tarik , Hadjadj Lila , Moulay Mohamed Said
Published: December 2009

In this paper we study the following parabolic problem

\begin{cases}
∂t(u_i) − Δ(|u_i|^{σ_i}u_i) = g_i(u) + \overrightarrow{b_i}∇(|ui|^{m_i − 1}u_i) | &\text{in } ]0,∞[×Ω,\\
u_i =0 &\text{on }]0,∞[×∂Ω,\\
u_i(0,.) = u_{i0} &\text{in }Ω,
\end{cases}

where

Ω

is a bounded domain with smooth boundary and

i = 1, 2,…,d

. Our aim is to study existence of globally bounded weak solutions or blow-up, depending on the relations between the parameters that appear in the problem.
Free energies and Fichera’s quasi-static problem for materials with fading memory
Authors: Amendola Giovambattista , Fabrizio Mauro
Published: December 2009

In this work we are concerned with the Fichera problem related to the quasi-static equation of viscoelasticity. By the use of the space derived from the minimum free energy, connected with a viscoelastic material, we prove a uniqueness and existence theorem in a new large space of data.
Approximating the inverse matrix of the G-limit through changes of variables in the plane
Authors: Moscariello Gioconda , Sbordone Carlo , Murat François
Published: December 2009

Let

A_j

be a sequence of coercive symmetric matrices of

L^\infty(\mathbb{R}^2)^{2\times 2}

with

det \, A_j=1

which

G

-converges to

A

. We prove that there exists a sequence of

K

-quasiconformal mappings

F_j

which converge locally uniformly to a

K

-quasiconformal mapping

F

such that

A_j^{-1}\circ F_j^{-1}

G

-converges to

A^{-1}\circ F^{-1}

. The result is specific to the two dimensional case but a similar result holds in dimension

1

.
Closed curves in $\mathbb{R}^{3}$ with prescribed curvature and torsion in perturbative cases. Part 1: Necessary condition and study of the unperturbed problem
Authors: Caldiroli Paolo , Guida Michela
Published: December 2009

We study the problem of

(\kappa,\tau)

-loops, namely closed curves in the three-dimensional Euclidean space, with prescribed curvature

\kappa

and torsion

\tau

. We state a necessary condition for the existence of a bounded sequence of

(\kappa_{n},\tau_{n})

-loops when the functions

\kappa_{n}

and

\tau_{n}

converge to the constants 1 and 0, respectively. Moreover we prove some Fredholm-type properties for the “unperturbed” problem, with

\kappa\equiv 1

and

\tau\equiv 0

.
Positivity of the Bondi mass in Bondi's radiating spacetimes
Authors: Huang Wen-ling , Yau Shing-Tung , Zhang Xiao
Published: December 2009

We find two conditions related to the
news functions
of the Bondi's radiating vacuum spacetimes. We provide a complete proof of the positivity of the Bondi mass by using Schoen–Yau's method under one condition and by using Witten's method under another condition.
Dissipative stochastic equations in Hilbert space with time dependent coefficients
Authors: Da Prato Giuseppe , Röckner Michael
Published: December 2009

We prove existence and, under an additional assumption, uniqueness of an evolution system of measures

(\nu_t)_{t\in \R}

for a stochastic differential equation with time dependent dissipative coefficients. We prove that if

P_{s,t}

denotes the corresponding transition evolution operator, then

P_{s,t}\varphi

behaves asymptotically as

t\to +\infty

like a limit curve (which is independent of

s

) for any continuous and bounded “observable”

\varphi

.
Necessary optimality conditions for geodesics in weighted Wasserstein spaces
Authors: Ambrosio Luigi , Santambrogio Filippo
Published: December 2009

The geodesic problem in Wasserstein spaces with a metric perturbed by a conformal factor is considered, and necessary optimality conditions are estabilished in a case where this conformal factor favours the spreading of the probability measure along the curve. These conditions have the form of a system of PDEs of the kind of the compressible Euler equations. Moreover, self-similar solutions to this system are discussed.
Multiple solutions of superlinear elliptic equations
Authors: Rabinowitz Paul H. , Su Jiabao , Wang Zhi-Qiang
Published: December 2009

In this paper we give some multiplicity results on existence of nontrivial solutions for superlinear elliptic equations with a saddle structure near

0

. We make use of a combination of bifurcation theory and minimax methods.
Transversality of covariant mappings admitting a potential
Authors: Lesimple Marc , Valent Tullio
Published: December 2009

It is shown that for any mapping between Banach spaces which is covariant under a non linear action of a Lie group and have a symmetric first derivative with respect to some duality, the image of any point close to the origin belongs to the polar of the orbit of that point, under the commutator subgroup of the symmetry group.
The Grötzsch problem in higher dimensions
Authors: Adamowicz Tomasz
Published: December 2009

An extension of the Grötzsch Problem to higher dimensions is considered. The problem is formulated and proven for a subclass of polyconvex energy integrals and counterexamples in general case are given. A conjecture about the generalized distortion functions is stated.
Viscosity solutions of the Monge–Ampère equation with the right hand side in $L^p$
Authors: Amadori Anna Lisa , Brandolini Barbara , Trombetti Cristina
Published: December 2009

In this paper we compare various notions of solutions of Monge–Ampère equations with discontinuous functions in the right hand side. Precisely, we show that the weak solutions defined by Trudinger can be obtained by the vanishing viscosity approximation method. Moreover, we investigate existence and uniqueness of

L^p

-viscosity solutions.
A transcendence criterion for infinite products
Authors: Corvaja Pietro , Hancl Jaroslav
Published: December 2009

We prove a transcendence criterion for certain infinite products of algebraic numbers. Namely, for an increasing sequence of positive integers

a_n

and an algebraic number

\alpha>1

, we consider the convergent infinite product

\prod_{n}([\alpha^{a_n}]/\alpha^{a_n})

, where

[\cdot]

stands for the integral part. We prove (Thm. 1) that its value is transcendental, under certain hypothesis; Thm. 3 will show that such hypothesis are in a sense unavoidable.
Geometry and balance of hyperstresses
Authors: Mariano Paolo Maria
Published: December 2009

Surface hyperstress is introduced along a layer with vanishing thichness, a layer across which two second-grade elastic Cauchy bodies are glued. Surface balances are obtained by making use of the relative power of actions and of invariance requirements for the Lagrangian. The action of bulk hyperstress on vacancies, linear inclusions and cracks is also evaluated.
Conservative principle for differential forms
Authors: Masamune Jun
Published: December 2009

Motivated by recent development of conservative principle for differential forms, we will study the sufficient conditions of a manifold to be conservative.
Intrinsic Harnack inequalities for quasi-linear singular parabolic partial differential equations
Authors: DiBenedetto Emmanuele , Gianazza Ugo , Vespri Vincenzo
Published: December 2009

Intrinsic Harnack estimates for non-negative solutions of singular, quasi-linear, parabolic equations, are established, including the prototype

p

-Laplacean equation (1.4) below. For

p

in the super-critical range

\frac{2N}{N+1}<p<2

, the Harnack inequality is shown to hold in a parabolic form, both forward and backward in time, and in a elliptic form at fixed time. These estimates fail for the heat equation (

p\to2

). It is shown by counterexamples, that they fail for

p

in the sub-critical range

1<p\le \frac{2N}{N+1}

. Thus the indicated super-critical range is optimal for a Harnack estimate to hold. The novel proofs are based on measure theoretical arguments, as opposed to comparison principles and are sufficiently flexible to hold for a large class of singular parabolic equation including the porous medium equation and its quasi-linear versions.
Correlation inequalities for spin glass in one dimension
Authors: Contucci Pierluigi , Unguendoli Francesco
Published: December 2009

We prove two inequalities for the direct and truncated correlation functions for the nearest-neighboor one-dimensional Edwards-Anderson model with symmetric quenched disorder. The second inequality has the opposite sign of the GKS inequality of type II. In the non symmetric case with positive average we show that while the direct correlation keeps its sign the truncated one changes sign when crossing a suitable line in the parameter space. That line separates the regions satisfying the GKS second inequality and the one proved here.
A criterion for the reality of the spectrum of $PT$-symmetric Schrödinger operators with complex-valued periodic potentials
Authors: Caliceti Emanuela , Graffi Sandro
Published: December 2009

Consider in

L^2(\R)

the Schrödinger operator family

H(g):=-d^2_x+V_g(x)

depending on the real parameter

g

, where

V_g(x)

is a complex-valued but

PT

symmetric periodic potential. An explicit condition on

V

is obtained which ensures that the spectrum of

H(g)

is purely real and band shaped; furthermore, a further condition is obtained which ensures that the spectrum contains at least a pair of complex analytic arcs.
Uniqueness of signed measures solving the continuity equation for Osgood vector fields
Authors: Ambrosio Luigi , Bernard Patrick
Published: December 2009

Nonnegative measure-valued solutions of the continuity equation are uniquely determined by their initial condition, if the characteristic ODE associated to the velocity field has a unique solution. In this paper we give a partial extension of this result to signed measure-valued solutions, under a quantitative two-sided Osgood condition on the velocity field. Our results extend those obtained for log-Lipschitz vector fields in [6].
A Calderon–Zygmund theory for inﬁnite energy minima of some integral functionals
Authors: Boccardo Lucio
Published: December 2009

A Calderon–Zygmund theory in Lebesgue and Marcinkiewicz spaces for inﬁnite energy minima of some integral functionals is proved.
The relative power and its invariance
Authors: Mariano Paolo Maria
Published: December 2009

The relative power of actions in Cauchy bodies su¤ering mutations due to defect evolution is introduced. It is shown that its invariance under the action of the Euclidean group over the ambient space and the material space allows one to obtain (i) the balance of standard and conﬁgurational actions and (ii) the identiﬁcation of conﬁgurational ingredients from a unique source.
Orlicz–Sobolev regularity of mappings with subexponentially integrable distortion
Authors: Clop Albert , Koskela Pekka
Published: December 2009

We study regularity properties of mappings of ﬁnite distortion. We show that some sort of self-improvement phenomena hold also when only subexponential integrability is assumed for the distortion function. We extend to this setting results by Faraco, Koskela and Zhong [9] and Bildhauer, Fuchs and Zhong [6].
Recognizing the Farey–Stern–Brocot AF algebra
Authors: Mundici Daniele
Published: December 2009

In his 2008 paper published in the Canadian Journal of Mathematics, F. Boca investigates an AF algebra

\mathfrak A

, whose Bratteli diagram arises from the Farey–Stern–Brocot sequence. It turns out that

\mathfrak A

coincides with the AF algebra

\mathfrak M_1

introduced in 1988 by the present author in a paper published in Advances in Mathematics. We give a procedure to recognize

\mathfrak A

among all ﬁnitely presented AF algebras whose Murray–von Neumann order of projections is a lattice. Further: (i)

\mathfrak A

is a *-subalgebra of Glimm universal algebra; (ii) tracial states of

\mathfrak A

correspond to Borel probability measures on the unit real interval; (iii) all primitive ideals of

\mathfrak A

are essential; (iv) the automorphism group of

\mathfrak A

has exactly two connected components.
Stability-Instability criteria for nonautonomous systems
Authors: Rionero Salvatore
Published: December 2009

Nonautonomous binary systems of O.D.Es are considered. Apart from a critical case, it is shown that a temporal uniform validity of the Hurwitz conditions appear to be a basic condition to require for guaranteing the stability. Stability-instability criteria are obtained. Applications to the equation d2/d_t_2
x
+
p
(
t
) d/d_t_
x
+
q
(
t
)
x
= 0 and in particular to the Hill equation, are furnished. The Hill equation associated to the (linear) stability of the nonautonomous Lotka–Volterra system is considered.
On Brooks–Jewett, Vitali–Hahn–Saks and Nikodým convergence theorems for quasi-triangular functions
Authors: Cavaliere Paola , de Lucia Paolo
Published: December 2009

Connections are established between Brooks–Jewett, Vitali–Hahn–Saks and Nikodym type convergence theorems for quasi-triangular functions on Boolean rings satisfying the Subsequential Completeness Property and valued into Hausdorff topological spaces.
The Generalized Libera Transform on Hardy, Bergman and Bloch Spaces on the Unit Polydisc
Authors: Stevic Stevo
Published: December 2009

We consider the boundedness of the Libera transform Lz0 on Hardy, Bergman and a-Bloch spaces of analytic functions on the unit polydisk.
Morozov's Discrepancy Principle under General Source Conditions
Authors: Nair M. Thamban , Schock E. , Tautenhahn Ulrich
Published: December 2009

In this paper we study linear ill-posed problems

Ax = y

in a Hilbert space setting where instead of exact data

y

noisy data

y^\delta

are given satisfying

\Vert y - y^\delta\Vert ≤ \delta

with known noise level

\delta

. Regularized approximations are obtained by a general regularization scheme where the regularization parameter is chosen from Morozov's discrepancy principle. Assuming the unknown solution belongs to some general source set

M

we prove that the regularized approximation provides order optimal error bounds on the set

M

. Our results cover the special case of finitely smoothing operators

A

and extend recent results for infinitely smoothing operators.
Complements on Growth Envelopes of Spaces with Generalized Smoothness in the Sub-Critical Case
Authors: Bricchi M. , Moura Susana D.
Published: December 2009

We describe the growth envelope of Besov and Triebel-Lizorkin spaces

B_{p,q}^\sigma (\mathbb R^n)

and

F_{p,q}^\sigma (\mathbb R^n)

with generalized smoothness, i.e.\ instead of the usual scalar regularity index

\sigma \in \mathbb R

we consider now the more general case of a sequence

\sigma = \{\sigma_j\}_{j \in\mathbb N_0}

. We take under consideration the range of the parameters

\sigma, p,q

which, in analogy to the classical terminology, we call sub-critical.
Approximation by Superpositions of a Sigmoidal Function
Authors: Lewicki Grzegorz , Marino G.
Published: December 2009

We generalize a result of B. Gao and Y. Xu [J. Math. Anal. Appl. 178 (1993) 221–226] concerning the approximation of functions of bounded variation by linear combinations of a fixed sigmoidal function to the class of functions of bounded f-variation. Also, in the case of one variable, a proposition of A. R. Barron [IEEE Trans. Inf. Theory 36 (1993) 930–945] is improved. Our proofs are similar to that of Gao and Xu [loc. cit.].
Erratum to: $C^{1, \alpha}$ Local Regularity for the Solutions of the $p$-Laplacian on the Heisenberg Group for $2 \leq p < 1 + \sqrt{5}$
Authors: Marchi Silvana
Published: December 2009

The paper mentioned in the title [Z. Anal. Anwend. 20 (2001) 617–636] dealt with some equations approximating the p-Laplacian on the Heisenberg group. We proved the existence of the second derivatives of their solutions for

2 \leq p < 1 + \sqrt{5}

. But the differentiation of the approximating equations is incorrect. So the regularity for the solutions of the

p

-Laplacian obtained from it as a by-product must be considered as not proved.
On a Class of Inclusions in Ordered Spaces
Authors: Huy Nguyen Bich , Dung Dao Bao , Khanh Nguyen Huu
Published: December 2009

Let W be a non-empty set, X an ordered topological space, L from W to X a single-valued operator and N a set-valued operator assigning to each element of W a nonempty subset of X. Under approximate assumptions on the monotonicity of L and N we prove existence results for inclusions of the form Lx in Nx. An application of the obtained results to implicit elliptic equations of the form Lu = f(x, u, Lu) is given.
A Class of Linear Integral Equations and Systems with Sum and Difference Kernel
Authors: von Wolfersdorf Lothar
Published: December 2009

By means of Fourier transform and Cauchy integral techniques a complete investigation of a class of linear integral equations and corresponding systems of equations of cross-correlation type in the Lebesgue spaces L^1 and L^2 is performed. Integral equations of first and second kind are reduced to explicitly solvable Riemann-Hilbert problems for a holomorphic function in the upper half-plane and the system of equations to conjugacy problems for a sectionally holomorphic function, where in the case of a finite interval also the analytic continuation of the solutions to the lower half-plane can be carried out in explicit way. Further, a resolvent representation of the solution to the integral equation and its adjoint equation is derived.
Bounds for the Best Constant in an Improved Hardy-Sobolev Inequality
Authors: Chaudhuri Nirmalendu
Published: December 2009

We show that the best constant

C

in the improved Hardy-Sobolev inequality of Adimurthi, N. Chaudhuri and M. Ramaswamy [Proc. Amer. Math. Soc. 130 (2002) 489–505] for

2 \le p < n

is bounded by

{p - 1 \over p^2} \big({n - p \over p}\big)^{p-2} \le C \le {p - 1 \over 2} \Big({n - p \over p}\Big)^{p-2}.
Existence and Relaxation for Finite-Dimensional Optimal Control Problems Driven by Maximal Monotone Operators
Authors: Papageorgiou Nikolaos S. , Papalini Francesca
Published: December 2009

We study the optimal control of a class of nonlinear finite-dimensional optimal control problems driven by a maximal monotone operator which is not necessarily everywhere defined. So our model problem incorporates systems monitored by variational inequalities. First we prove an existence theorem using the reduction method of Berkovitz and Cesari. This requires a convexity hypothesis. When this convexity condition is not satisfied, we have to pass to an augmented, convexified problem known as the "relaxed problem". We present four relaxation methods. The first uses Young measures, the second uses multi-valued dynamics, the third is based on Caratheodory's theorem for convex sets in RN and the fourth uses lower semicontinuity arguments and Γ-limits. We show that they are equivalent and admissible, which roughly speaking means that the corresponding relaxed problem is in a sense the "closure" of the original one.
Extinction and Asymptotic Behavior of Solutions to a System Arising in Biology
Authors: Kouche M. , Tatar Nasser-edine
Published: December 2009

A generalization to n species of a system by L. Bass, A. J. Bracken, K. Holmaker and B. Jefferies [J. Austral. Math. Soc. (Ser. B) 29 (1987) 156–194], which describes the self-organization of liver zones in a liver capillary in the case of two species, is proposed. We establish some hypotheses on the coefficient parameters of the system under which a part of the species is driven to extinction while the remaining ones are attracted by the non-trivial stationary solution.
Asymptotic Behavior of a Vibrating Membrane Model
Authors: Boella M.
Published: December 2009

We study the asymptotic behaviour of the solutions of an approximated model of the vibrating membrane. The results we give concern three different approximations; for each the boundedness of the energy and the presence of an absorbing set are proved, provided that the forcing term remains bounded in a suitable norm. The absorbing set is found via a known technique, while the boundedness of the energy is obtained by "ad hoc" calculations. Moreover, in one approximation, in the autonomous case, the existence of an attractor is proved.
Hölder-Zygmund Regularity in Algebras of Generalized Functions
Authors: Hörmann Günther
Published: December 2009

We introduce an intrinsic notion of Hölder-Zygmund regularity for Colombeau generalized functions. In case of embedded distributions belonging to some Zygmund-Hölder space this is shown to be consistent. The definition is motivated by the well-known use of Littlewood-Paley decompositions in characterizing Hölder-Zygmund regularity for distributions. It is based on a simple interplay of differentiated convolution-mollification with wavelet transforms, which directly translates wavelet estimates into properties of the regularizations. Thus we obtain a scale of new subspaces of the Colombeau algebra. We investigate their basic properties and indicate first applications to differential equations whose coefficients are non-smooth but belong to some Hölder-Zygmund class (distributional or generalized). In applications problems of this kind occur, for example, in seismology when Earth's geological properties of fractal nature have to be taken into account while the initial data typically involve strong singularities.
A Global Bifurcation Theorem for Convex-Valued Differential Inclusions
Authors: Domachowski S. , Gulgowski J.
Published: December 2009

We prove a global bifurcation theorem for convex-valued completely continuous maps. Basing on this theorem we prove an existence theorem for convex-valued differential inclusions with Sturm–Liouville boundary conditions

u''(t) \in \varphi(t,u(t),u'(t))\ \ \hbox{for a.e.}\ \ t \in (a,b)

l(u) = 0

The assumptions refer to the appropriate asymptotic behaviour of

\varphi(t,x,y)

for

|x| + |y|

close to

0

and to

+\infty

, and they are independent from the well known Bernstein-type conditions. In the last section we give a set of examples of

\varphi

satisfying the assumptions of the given theorem but not satisfying the Bernstein conditions.
A Class of Partial Integro-Differential Equations with Correlation-Convolution Integral II
Authors: von Wolfersdorf Lothar
Published: December 2009

Three classes of nonlinear integral equations and first order integro-differential equations in two variables are dealt with where the quadratic nonlinearity is given by the correlation-convolution integral. In the case of the quarter plane the equations are reduced to boundary value problems for holomorphic functions which can be solved in closed form. In this way existence and constructive formulas for the solutions to the equations are derived.
Asymptotic Behavior of the Solutions of an Elliptic-Parabolic System Arising in Flow in Porous Media
Authors: Amirat Y. , Ziani A.
Published: December 2009

We study the asymptotic behavior, with respect to high Peclet numbers, of the solutions of the nonlinear elliptic-parabolic system governing the displacement of one incompressible fluid by another, completely miscible with the first, in a porous medium. Using compensated compactness techniques, we obtain the existence of a global weak solution to the nonlinear degenerate elliptic-parabolic system modelling the flow when the molecular diffusion effects are neglected.