bluuebunny/crossref_metadata_2025 huggingface dataset:
Hopf algebroids and secondary characteristic classes
Authors: Kaminker Jerome , Tang Xiang
Published: December 2009

We study a Hopf algebroid,

\mathcal H

, naturally associated to the groupoid

U^δ_n ⋉ U_n

. We show that classes in the Hopf cyclic cohomology of

\mathcal H

can be used to define secondary characteristic classes of trivialized flat

U_n

-bundles. For example, there is a cyclic class which corresponds to the universal transgressed Chern character and which gives rise to the continuous part of the

ρ

-invariant of Atiyah–Patodi–Singer. Moreover, these cyclic classes are shown to extend to pair with the K-theory of the associated C

^*

-algebra. This point of view gives leads to homotopy invariance results for certain characteristic numbers. In particular, we define a subgroup of the cohomology of a group analogous to the Gelfand–Fuchs classes described by Connes [3] and show that the higher signatures associated to them are homotopy invariant.
On spectral triples in quantum gravity II
Authors: Aastrup Johannes , Grimstrup Jesper Møller , Nest Ryszard
Published: December 2009

A semifinite spectral triple for an algebra canonically associated to canonical quantum gravity is constructed. The algebra is generated by based loops in a triangulation and its barycentric subdivisions. The underlying space can be seen as a gauge fixing of the unconstrained state space of Loop Quantum Gravity. This article is the second of two papers on the subject.
Noncommutative field theory on rank one symmetric spaces
Authors: Bieliavsky Pierre , Gurau Razvan G. , Rivasseau Vincent
Published: December 2009

Quantum field theory has been shown recently renormalizable on flat Moyal space and better behaved than on ordinary space-time. Some models at least should be completely finite, even beyond perturbation theory. In this article a first step is taken to extend such theories to non-flat backgrounds such as solvable symmetric spaces.
Localization over complex-analytic groupoids and conformal renormalization
Authors: Perrot Denis
Published: December 2009

We present a higher index theorem for actions of a discrete group on the complex plane by local conformal mappings. The novelty is the use of the local anomaly formula established in a previous paper, which represents the bivariant Chern character of a quasihomomorphism as the chiral anomaly associated to a renormalized noncommutative chiral field theory. In the present situation the geometry is non-metric and the corresponding field theory can be renormalized in a purely conformal way, exploiting the complex-analytic structure of the groupoid only. The index formula is automatically localized at the automorphism subset of the groupoid and involves a cap-product with the sum of two different cyclic cocycles over the groupoid algebra. The first cocycle is a trace involving a generalization of the Lefschetz numbers to higher-order fixed points. The second cocycle is a noncommutative Todd class, constructed from the modular automorphism group of the algebra.
Fusion rules for quantum reflection groups
Authors: Banica Teodor , Vergnioux Roland
Published: December 2009

We find the fusion rules for the quantum analogues of the complex reflection groups

H^s_n = ℤ_s ≀ S_n

. The irreducible representations can be indexed by the elements of the free monoid

ℕ^{∗s}

, and their tensor products are given by formulae which remind the Clebsch–Gordan rules (which appear at

s = 1

).
Bost–Connes type systems for function fields
Authors: Jacob Benoît
Published: December 2009

We describe a construction which associates to any function field

k

and any place

∞

of

k

a

\mathrm C^*

-dynamical system

(C_{k,∞},σ_t)

that is analogous to the Bost–Connes system associated to

ℚ

and its archimedean place. Our construction relies on Hayes’ explicit class field theory in terms of sign-normalized rank one Drinfel’d modules. We show that

C_{k,∞}

has a faithful continuous action of

\mathrm{Gal} (K/k)

, where

K

is a certain field constructed by Hayes such that

k^{\mathrm{ab},∞} ⊂ K ⊂ k^{\mathrm{ab}}

. Here

k^{\mathrm{ab},∞}

is the maximal abelian extension of

k

that is totally split at

∞

. We classify the extremal KMS

{}_β

states of

(C_{k,∞},σ_t)

at any temperature

0 < 1/β < ∞

and show that a phase transition with spontaneous symmetry breaking occurs at temperature

1/β = 1

. At high temperature

1/β ≥ 1

, there is a unique KMS

{}_β

state, of type III

{}_{q^{−β}}

, where

q

is the cardinal of the constant subfield of

k

. At low temperature

1/β < 1

, the space of extremal KMS

{}_β

states is principal homogeneous under

\mathrm{Gal} (K/k)

. Each such state is of type I

_∞

and the partition function is the Dedekind zeta function

ζ_{k,∞}

. Moreover, we construct a

\ast

-subalgebra

\mathcal H

, we give a presentation of

\mathcal H

and of

C_{k,∞}

, and we show that the values of the low-temperature extremal KMS

_β

states at certain elements of

\mathcal H

are related to special values of partial zeta functions.
Représentations non unitaires, morphisme de Baum–Connes et complétions inconditionnelles
Authors: Gomez Aparicio Maria Paula
Published: December 2009

We show that the Baum–Connes morphism twisted by a non-unitary representation, defined in [GA08], is an isomorphism for a large class of groups satisfying the Baum–Connes conjecture. Such class contains all the real semi-simple Lie groups, all hyperbolic groups and many infinite discrete groups having Kazhdan’s property (T). We define a tensorisation by a non-unitary finite dimensional representation on the left-hand side of the Baum–Connes morphism and we show that its analogue in K-theory must be defined on the K-theory of the twisted group algebras introduced in [GA07b].
On the classification of certain fusion categories
Authors: Jordan David , Larson Eric
Published: December 2009

We advance the classification of fusion categories in two directions. Firstly, we completely classify integral fusion categories – and consequently, semi-simple Hopf algebras – of dimension

pq^2

, where

p

and

q

are distinct primes. This case is especially interesting because it is the simplest class of dimensions where not all integral fusion categories are group-theoretical. Secondly, we classify a certain family of

ℤ/3ℤ

-graded fusion categories, which are generalizations of the

ℤ/2ℤ

-graded Tambara–Yamagami categories. Our proofs are based on the recently developed theory of extensions of fusion categories.
A period map for generalized deformations
Authors: Fiorenza Domenico , Manetti Marco
Published: December 2009

For every compact Kähler manifold we give a canonical extension of Griffith’s period map to generalized deformations, intended as solutions of Maurer–Cartan equation in the algebra of polyvector fields. Our construction involves the notion of Cartan homotopy and a canonical

L_∞

-structure on mapping cones of morphisms of differential graded Lie algebras.
Dirac operators on all Podleś quantum spheres
Authors: D'Andrea Francesco , Dąbrowski Ludwik , Landi Giovanni , Wagner Elmar
Published: December 2009

We construct spectral triples on all Podleś quantum spheres S2qt. These noncommutative geometries are equivariant for a left action of

\mathcal{U}

q(su(2)) and are regular, even and of metric dimension 2. They are all isospectral to the undeformed round geometry of the sphere S2. There is also an equivariant real structure for which both the commutant property and the first order condition for the Dirac operators are valid up to infinitesimals of arbitrary order.
Poincaré–Birkhoff–Witt deformations of Calabi–Yau algebras
Authors: Berger Roland , Taillefer Rachel
Published: December 2009

Recently, Bocklandt proved a conjecture by Van den Bergh in its graded version, stating that a graded quiver algebra A (with relations) which is Calabi–Yau of dimension 3 is defined from a homogeneous potential W. In this paper, we prove that if we add to W any potential of smaller degree, we get a Poincaré–Birkhoff–Witt deformation of A. Such PBW deformations are Calabi–Yau and are characterised among all the PBW deformations of A. Various examples are presented.
Any generating set of an arbitrary property T von Neumann algebra has free entropy dimension ≤ 1
Authors: Jung Kenley , Shlyakhtenko Dimitri L.
Published: December 2009

Suppose that

N

is a diffuse, property T von Neumann algebra and

X

is an arbitrary finite generating set of selfadjoint elements for

N

. By using rigidity/deformation arguments applied to representations of

N

in ultraproducts of full matrix algebras, we deduce that the microstate spaces of

X

are asymptotically discrete up to unitary conjugacy. We use this description to show that the free entropy dimension of

X

,

\delta_0(X)

, is less than or equal to

1

. It follows that when

N

embeds into the ultraproduct of the hyperfinite

\mathrm{II}_1

factor, then

\delta_0(X)=1

and otherwise,

\delta_0(X)=−\infty

. This generalizes the earlier results of Voiculescu, and Ge, Shen pertaining to

\mathrm{SL}_n(\mathbb Z)

as well as the results of Connes, Shlyakhtenko pertaining to group generators of arbitrary property T algebras.
Base change and $K$-theory for $\mathrm{GL}(n)$
Authors: Mendes Sergio , Plymen Roger
Published: December 2009

Let F be a nonarchimedean local field and let

G = \mathrm{GL}(n) = \mathrm{GL}(n, F)

. Let

E/F

be a finite Galois extension. We investigate base change

E/F

at two levels: at the level of algebraic varieties, and at the level of

K

-theory. We put special emphasis on the representations with Iwahori fixed vectors, and the tempered spectrum of

\mathrm{GL}(1)

and

\mathrm{GL}(2)

. In this context, the prominent arithmetic invariant is the residue degree

f(E/F)

.
Geometric Numerical Integration
Authors: Hairer Ernst , Hochbruck Marlis , Iserles Arieh , Lubich Christian
Published: December 2009

The subject of this workshop was numerical methods that preserve geometric properties of the flow of an ordinary or partial differential equation: symplectic and multisymplectic integrators for Hamiltonian systems, symmetric integrators for reversible systems, methods preserving first integrals and numerical methods on manifolds, inclusive of Lie group methods and integrators for constrained mechanical systems, and methods for problems with highly oscillatory solutions. The unifying theme was structure preservation: not just the “how?” but also “why?”, “where?” and “what for?”.

The motivation for developing structure-preserving algorithms for special classes of problems originates independently in such diverse areas of research as astronomy, molecular dynamics, mechanics, control theory, theoretical physics and numerical analysis, with important contributions from other areas of both applied and pure mathematics. Moreover, it turns out that preservation of geometric properties of the flow not only produces an improved qualitative behaviour, but also allows for a significantly more accurate long-time integration than with general-purpose methods.

In addition to the construction of geometric integrators, an important aspect of geometric integration is the light it sheds on the relationship between geometric properties of a numerical method and favourable error propagation in long-time integration. A very successful organising principle is backward error analysis, whereby the numerical one-step map is interpreted as (almost) the flow of a modified differential equation. In this way, geometric properties of the numerical integrator translate seamlessly into structure preservation on the level of the modified equation. Much insight and rigourous error estimates over long time intervals can then be obtained by combining backward error analysis with the KAM theory and related perturbation theories for Hamiltonian and reversible systems. While this approach has been very successful for ordinary differential equations, much less is currently known about highly oscillatory systems and geometric integrators for partial differential equations.

Geometric numerical integration has been an active interdisciplinary research area since the last decade. Although the subject is in a lively phase of intensive development, the results so far are substantive and they impact on a wide range of application areas and on our understanding of core issues in computational mathematics. This is evidenced by the monographs [1, 2].
Explicit Methods in Number Theory
Authors: Cohen Henri , Lenstra, Jr. Hendrik W. , Zagier Don B.
Published: December 2009

The workshop
Explicit Methods in Number Theory
was organised by Henri Cohen (Talence), Hendrik W. Lenstra (Leiden), and Don B. Zagier (Bonn), with the assistance of Karim Belabas (Talence), and it took place July 15–21, 2007. Four previous workshops on the topic had been held in 1999, 2001, 2003, and 2005. The goal of the meeting was to present new methods and results on concrete aspects of number theory. In several cases, this included algorithmic and experimental work, but the emphasis was on the implications for number theory.

There were three ‘mini-series’ highlighting important recent developments: one of three hours, by Jordan Ellenberg and Akshay Venkatesh, on asymptotics for field extensions and class numbers; one of two hours by Mark Watkins, on random matrices and

L

-functions; and one of three hours by Noam Elkies, on the construction of elliptic curves of high Mordell–Weil rank.

Some of the other themes were:

As always in Oberwolfach, the atmosphere was lively and active, providing an ideal environment for the exchange of ideas and productive discussions. The meeting was well-attended, with 51 participants from a variety of backgrounds, including a large number of younger researchers. There were 32 talks of various lengths, and ample time was allotted to informal collaboration.
Notes on periodic solutions of discrete steady state systems
Authors: Wang Gen-Qiang , Cheng Sui Sun
Published: December 2009

A three term discrete system is considered and periodic solutions are found by means of a mountian pass theorem in the critical point theory.
The influence of domain geometry in the boundary behavior of large solutions of degenerate elliptic problems
Authors: Grillot Michèle , Grillot Philippe
Published: December 2009

In this paper we study the asymptotic boundary behavior of large solutions of the equation

\Delta u=d^{\alpha}u^p

in a regular bounded domain

\Omega

in

\R^N

,

N\geq 2

, where

d(x)

denotes the distance from

x

to

\partial\Omega

,

p>1

and

\alpha>0

. We precise the expansion which depends on the mean curvature of the boundary.
Compléments sur les opérateurs précompacts et faiblements compacts positifs
Complement on positive precompact and weakly compact operators
Authors: Aqzzouz Belmesnaoui , Nouira Redouane
Published: December 2009

Nous établissons deux conditions nécessaires concernant le problème de domination, l'une pour les opérateurs précompacts positifs et l'autre pour les opérateurs faiblement compacts positifs.

We prove two necessary conditions about the domination problem, one for positive precompact operators and the other for positive weakly compact operators.
Simplicity of Lyapunov spectra: a sufficient criterion
Authors: Avila Artur , Viana Marcelo
Published: December 2009

We exhibit an explicit sufficient condition for the Lyapunov exponents of a linear cocycle over a Markov map to have multiplicity
1. This builds on work of Guivarc'h–Raugi and Gol'dsheid–Margulis, who considered products of random matrices, and of Bonatti–Viana, who dealt with the case when the base dynamics is a subshift of finite type. Here the Markov structure may have infinitely many symbols and the ambient space needs not be compact. As an application, in another paper we prove the Zorich–Kontsevich conjecture on the Lyapunov spectrum of the Teichmüller flow in the space of translation surfaces.
Remarks on the Navier–Stokes evolution equations under slip type boundary conditions with linear friction
Authors: Beirão da Veiga Hugo
Published: December 2009

In general terms, we show that the regularity results proved in [2] (see also [7]) for strong solutions to the
linear stationary
Stokes system under slip type boundary conditions allow to extend to these boundary conditions many of the well known results for solutions to the
nonlinear evolution
Navier–Stokes equations under the classical non-slip boundary condition.
Complete reducibility of systems of equations with respect to R
Authors: Almeida Jorge , Costa José Carlos , Zeitoun Marc
Published: December 2009

It is shown that the pseudovariety

\mathsf R

of all finite

\mathcal R

-trivial semigroups is completely reducible with respect to the canonical signature. Informally, if the variables in a finite system of equations with rational constraints may be evaluated by pseudowords so that each value belongs to the closure of the corresponding rational constraint and the system is verified in

\mathsf R

, then there is some such evaluation which is “regular”, that is one in which, additionally, the pseudowords only involve multiplications and

\omega

-powers.
A revision of J.-L. Lions’ notion of sentinels
Authors: Nakoulima Ousseynou
Published: December 2009

We revisit the notion of sentinel introduced by J.-L. Lions [12] considering the case where the observation of the system and control function of the sentinel have their supports in two different open sets. This point of view leads to problems of null-controllability with or without constraints on the control.  This article focuses on the case of parabolic equations although similar developments can be done for other PDE’s.  The main tool used is an observability inequality of Carleman type which is “adapted” to the constraints.
Existence and iteration of positive solution for a multi-point boundary value problem with a $p$-Laplacian operator
Authors: Ma De-Xiang , Chen Xue-Gang
Published: December 2009

In this paper we obtain the existence of a positive solution and establish a corresponding iterative scheme for the following boundary value problem:

\begin{cases}
Φp(u'))' + q(t) f(t,u) = 0,\quad 0 < t < 1,\\
u(0) = \sum_{i=1}^{q−1} γ+iu(δ_i),\   u(1)= \sum_{i=1}^{m−1} η_iu(ξ_i).
\end{cases}

The main tool is the monotone iterative technique. Here the coefficient

q(t)

may be singular at

t = 0,1

.
Uniqueness of meromorphic functions sharing two sets with finite weight
Authors: Banerjee Abhijit
Published: December 2009

In this paper we consider a problem initially posed by Yi [15]. We prove two uniqueness theorems on meromorphic functions which improve and extend results of Lahiri [9] and Fang and Lahiri [1]. We provide an example that shows that a condition in one of our results is sharp.
Triple positive solution to the one-dimensional $p$-Laplacian equation
Authors: Zhang Xuemei , Feng Meiqiang , Ge Weigao
Published: December 2009

We obtain sufficient conditions for the existence of at least three positive solutions to the second-order nonlinear delay differential equation with one-dimensional

p

-Laplacian

\begin{cases}
(Φ_p(x'(t)))' + w(t) f(t, x(t), x(t − τ), x'(t)) = 0, t ∈ (0,1), & τ > 0,\\
x(t) = 0, &−τ ≤ t ≤ 0,\\
x(1) = 0,
\end{cases}

where

Φp(s)

is the

p

-Laplacian operator, i.e.,

Φ_p(s) = |s|^{p−2}s

,

p > 1

,

(Φ_p)^{−1} = Φ_q

,

\frac{1}{p} + \frac{1}{q} = 1

. The arguments are based upon a new fixed point theorem in a cone introduced by Avery and Peterson.
$τ(p;q)$-summing mappings and the domination theorem
Authors: Mujica Ximena
Published: December 2009

We extend the concept of

τ

-summing operators presented by Pietsch in his monograph on Operator Ideals to multilinear mappings and polynomials. We also present a domination theorem for

τ(p)

-summing mappings and polynomials, showing their relation with

p

-semi-integral mappings and polynomials.
Every strict sum of cubes in $\mathbb F_4[t]$ is a strict sum of 6 cubes
Authors: Gallardo Luis H.
Published: December 2009

It is easy to see that an element

P(t) ∈ \mathbb F_4[t]

is a strict sum of cubes if and only if

P(t) ∈ M(4)

where

\begin{align*}
M(4) = \{ P(t) ∈ \mathbb F_4[t] \ |\ &P(r) ∈ \{0,1\} \text{ for all }r ∈ \mathbb F_4 \text{ and such that} \\
&\text{either 3 does not divide }\mathrm{deg}(P(t)), \\
&\text{or 3 does divide }\mathrm{deg}(P(t)) \text{ and }P(t) \text{ is monic} \}.
\end{align*}

We say that

P(t)

is a “strict” sum of cubes

A_1(t)^3 +···+ A_g(t)^3

if

\mathrm{deg}(A_i^3) < \mathrm{deg}(P) + 3

for each

i

, and we define

g(3,\mathbb F_4[t])

as the least

g

such that every element of

M(4)

is a strict sum of

g

cubes. The main result is that

g(3,\mathbb F_4[t]) ≤ 6.

This improves an earlier result of the author that

g(3,\mathbb F_4[t]) ≤ 9

.
Bivariate classical and $q$-series transformations
Authors: Chu Wenchang , Jia Cangzhi
Published: December 2009

By means of bivariate inverse series relations, we review several bivariate classical hypergeometric series transformation formulae and establish their

q

-analogues.
Wedderburn quasialgebras
Authors: Albuquerque Helena , Elduque Alberto , Pérez-Izquierdo José María
Published: December 2009

The Wedderburn–Artin Theorem for G-graded quasialgebras is proved. This provides new examples of quasialgebras.
The Fourier–Borel transform between spaces of entire functions of a given type and order
Authors: Vieira Fávaro Vinícius
Published: December 2009

In this article we study the Fourier–Borel transform between the dual of

\mathrm{Exp}^k_{Ñ,(s;( r,q)),A}(E)

, the space of entire functions on

E

of

(s;(r,q))

-quasi-nuclear order

k

and

(s;(r,q))

-quasi-nuclear type strictly less than

A

, and the space

\mathrm{Exp}^{k'}_{(s',m(r';q')),0,(θ(k)A)^{−1}}(E')

of entire functions on

E'

of

(s',m(r';q'))

-summing order

k

and

(s',m(r';q'))

-summing type less than or equal to

(θ(k)A)^{−1}

. This mapping identifies algebraically and topologically these two spaces. On the dual space it is considered the strong topology. This generalizes results of Matos [4] and Martineau [3].
Irréductibilité et spécialisation des polynômes
Authors: Najib Salah
Published: December 2009

À partir de la décomposition en base b(X) (un polynôme non constant), nous donnons un critère d'irréductibilité des polynômes en deux variables. Puis nous étudions un problème de spécialisation qui a un lien avec le théorème d'irréductibilité de Hilbert.

Starting from the decomposition in base b(X) (a nonconstant polynomial), we give an irreducibility criterion of the bivariate polynomials. Moreover we study a problem of specialization which has a link with Hilbert’s irreducibility theorem.
Symmetry and bifurcation of periodic solutions in Neumann boundary value problems
Authors: Castro Sofia B. S. D.
Published: December 2009

We study a vector-valued reaction-diffusion equation with Neumann boundary conditions

(u: [0,π] → ℝ^2

). Unlike what is observed for scalar equations, where no heteroclinic connections involving periodic solutions occur, we find that steady-state/Hopf and Hopf/Hopf mode interactions produce heteroclinic solutions connecting at least one solution of standing wave type. This is achieved by restricting a problem with periodic boundary conditions and equivariant under

\mathrm{O}(2)

symmetry to a suitable fixed-point space.

For completeness, we include a description of the solutions for Hopf bifurcation and mode interactions involving Hopf bifurcation, namely, steady-state/Hopf and Hopf/Hopf.
A remark on multiple solutions for a nonlinear eigenvalue problem
Authors: Wang Gen-Qiang , Cheng Sui Sun
Published: December 2009

Multiple solutions are shown to exist for the system

Bu = λF(u)

, where

λ > 0

,

B

is a positive definite matrix and

F

is ‘superquadratic’ or ‘subquadratic’. We make use of Clark's theorem in critical point theory to obtain our results and provide examples to show that the results are sharp.
Existence and uniqueness of periodic solutions for a delay differential equation with piecewise constant arguments
Authors: Liang Haihua , Wang Gen-Qiang
Published: December 2009

We establish existence and uniqueness criteria for the periodic solutions of a first order differential equation with piecewise constant arguments.
Stability criteria for certain third order nonlinear delay differential equations
Authors: Tunç Cemil
Published: December 2009

In this paper we study the asymptotic stability of the trivial solution of third order nonlinear delay differential equations of the form

x'''(t) + f(x(t), x'(t))x''(t) + g(x(t − r), x'(t − r)) + h(x(t − r)) = 0,

where

r > 0

is a constant delay. In proving our result we make use of Lyapunov’s second method by constructing a Lyapunov functional.
The counting hierarchy in binary notation
Authors: Ferreira Gilda
Published: December 2009

We present a new recursion-theoretic characterization of

\mathsf{FCH}

, the
hierarchy of counting functions
, in binary notation. Afterwards we introduce a theory of
bounded arithmetic
,

\mathsf{TCA}

, that can be seen as a reformulation, in the binary setting, of Jan Johannsen and Chris Pollett's system

\mathsf D^0_2

. Using the previous inductive characterization of

\mathsf{FCH}

, we show that a strategy similar to the one applied to

\mathsf{D}^0_2

can be used in order to characterize

\mathsf{FCH}

as the class of functions provably total in

\mathsf{TCA}

.
Multilinear functionals of Schatten class type and approximation numbers
Authors: Mendes Cristiane de Andrade
Published: December 2009

In this article we investigate the relationship between multilinear functionals of Schatten class type and approximation numbers, and obtain some results that are similar to the those we know in the linear theory.
An existence theorem for a perturbed singular elliptic problem
Authors: Anello Giovanni
Published: December 2009

We prove the existence of a positive solution for singular elliptic problems of the type

− Δu = α(x)u^{−β} + λf(x, u)

in

Ω

,

u_{| ∂Ω} = 0

, where

β

is any positive real number.
Examples of optimal solutions of infinite horizon variational problems arising in continuum mechanics
Authors: Zaslavski Alexander J.
Published: December 2009

In this article we construct several important examples of optimal solutions of infinite horizon second order variational problems. The problems are related to a model in thermodynamics introduced in Coleman, Marcus and Mizel,
Arch. Rational Mech. Anal.
117
(1992), 321–347.
On the existence of positive solutions for a nonhomogeneous elliptic system
Authors: dos Santos Ederson Moreira
Published: December 2009

The main goal of this work is to prove the existence and in some cases the uniqueness of positive solutions for some nonhomogeneous systems of Lane–Emden type. The nonhomogeneous part must satisfy a suitable positiveness condition, but it may change sign.
Asymptotic distribution of certain statistics relevant to the fitting of max-semistable models
Authors: Canto e Castro Luísa , Dias Sandra
Published: December 2009

According to the results in Canto e Castro et al. [1], max-semistable distribution functions can be characterized by a parameter r > 1, by the extreme value index γ and by a real function w defined in [0,log r]. The estimation of the parameters r and γ based on ratios of differences of order statistics, or appropriate functions of these sequences, was treated in Dias and Canto e Castro [4]. In this work we study the asymptotic distribution of these sequences of statistics.
Parabolic problems with dynamical boundary conditions: eigenvalue expansions and blow up
Authors: Bandle Catherine , von Below Joachim , Reichel Wolfgang
Published: December 2009

An existence theory for local solutions of a parabolic problem under a dynamical boundary condition

\sigma u_t+u_n=0

is developed and a spectral representation formula is derived. It relies on the spectral theory of an associated elliptic problem with the eigenvalue parameter both in the equation and the boundary condition. The well-posedness of the parabolic problem holds in some natural space only if the number of negative eigenvalues is finite. This depends on the parameter

\sigma

in the boundary condition. If

\sigma\geq 0

the parabolic problem is always well-posed. For

\sigma<0

it is well-posed only if the space dimension is

1

and ill-posed in space dimension

\geq 2

. By means of the theory of compact operators the spectrum is analyzed and some qualitative properties of the eigenfunctions are derived. An interesting phenomenon is the “parameter-resonance”, where for a specific parameter-value

\sigma_0

two eigenvalues of the elliptic problem cross. Examples are added for which the eigenvalues and eigenfunctions can be computed explicitly. The last part of the paper deals with blow up of solutions of the parabolic problem with nonlinear positive sources.
Surface classification and local and global fundamentals groups, I
Authors: Catanese Fabrizio
Published: December 2009

Given a smooth complex surface

S

, and a compact connected global normal crossings divisor

D = \bigcup_i D_i

, we consider the local fundamental group

\pi_1 (T \setminus D)

, where

T

is a good tubular neighbourhood of

D

. One has an exact sequence

1 \to \mathcal K \to \Gamma : = \pi_1 (T - D) \rightarrow \Pi : = \pi_1 (D) \to 1

, and the kernel

\mathcal K

is normally generated by geometric loops

\gamma_i

around the curve

D_i

. Among the main results, which are strong generalizations of a well known theorem of Mumford, is the nontriviality of

\gamma_i

in

\Gamma = \pi_1 (T - D)

, provided all the curves

D_i

of genus zero have selfintersection

D_i^2 \leq -2

(in particular this holds if the canonical divisor

K_S

is nef on

D

), and under the technical assumption that the dual graph of

D

is a tree.
A curious identity and the volume of the root spherical simplex
Authors: De Concini Corrado , Procesi Claudio
Published: December 2009

We show a curious identity on root systems which gives the evaluation of the volume of the spherical simpleces cut by the cone generated by simple roots. In the appendix John Stembridge gives a conceptual proof of our identity.
Étale and crystalline beta and gamma functions via Fontaine's periods
Authors: Baldassarri Francesco
Published: December 2009

We compare the Ihara–Anderson theory of the

p

-adic étale beta function, which describes the Galois action on

p

-adic étale homology for the tower of Fermat curves over

\mathbb{Q}

of degree a power of

p

, with the crystalline theory of Dwork–Coleman, based on the calculation of the Frobenius action on

p

-adic de Rham cohomology of the same curves. The two constructions are easily related via a ramified extension of Fontaine's period ring

\mathbb B_{\mathrm{crys}} =
\mathbb B_{\mathrm{crys},p}

contained in

\mathbb B_{\mathrm{dR}} =
\mathbb B_{\mathrm{dR},p}

,
namely

\mathbb B_p :=
\mathbb B_{\mathrm{crys},p} \otimes_{\mathbb Q_p^{\mathrm{ur}}}\bar{\mathbb Q}_p
\subset \mathbb B_{\mathrm{dR},p}

.
We propose, but do not carry out, a similar comparison for the

p

-adic étale gamma function of Anderson and the Morita–Dwork–Coleman

p

-adic crystalline gamma function.
Steklov eigenvalues for the $\infty$-Laplacian
Authors: García-Azorero Jesús , Manfredi Juan J. , Peral Ireneo , Rossi Julio D.
Published: December 2009

We study the Steklov eigenvalue problem for the

\infty

-laplacian. To this end we consider the limit as

p \to \infty

of solutions of

-\Delta_p u_p =0

in a domain

\Omega

with

|\nabla u_p|^{p-2} \partial u_p / \partial \nu = \lambda |u|^{p-2} u

on

\partial\Omega

. We obtain a limit problem that is satisfied in the viscosity sense and a geometric characterization of the second eigenvalue.
Local clustering of the non-zero set of functions in $W^{1,1}(E)$
Authors: DiBenedetto Emmanuele , Gianazza Ugo , Vespri Vincenzo
Published: December 2009

We extend to the

p=1

case a measure theoretic result previously proved by DiBenedetto and Vespri for functions that belong to

u\in W^{1,p}(K_\rho(x_0))

where

K_\rho(x_0))

is a

N

-dimensional cube of edge

\rho

centered at

x_0

. It basically states that if the set where

u

is bounded away from zero occupies a sizable portion of

K_\rho

, then the set where

u

is positive clusters about at least one point of

K_\rho

.
Periodic solutions of nonlinear wave equations for asymptotically full measure sets of frequencies
Authors: Baldi Pietro , Berti Massimiliano
Published: December 2009

We prove existence and multiplicity of small amplitude periodic solutions of completely resonant nonlinear wave equations with Dirichlet boundary conditions for asymptotically full measure sets of frequencies, extending the results of [7] for new types of nonlinearities.
Configuration spaces of tori
Authors: Feler Yoel
Published: December 2009

The

n

-point configuration spaces

{\mathcal E}^n({\mathbb T}^2)=\{(q_1,...,q_n) \in ({\mathbb T}^2)^n\,|\ q_i\ne q_j \ \forall\,i\ne j\}

and

{\mathcal C}^n({\mathbb T}^2) =\{Q\subset {\mathbb T}^2\,|\ \# Q=n\}

of a complex torus

{\mathbb T}^2

are complex manifolds. We prove that for

n>4

any holomorphic self-map

F

of

{\mathcal C}^n({\mathbb T}^2)

either carries the whole of

{\mathcal C}^n({\mathbb T}^2)

into an orbit of the diagonal

\operatorname{Aut}({\mathbb T}^2)

action in

{\mathcal C}^n({\mathbb T}^2)

or is of the form

F(Q)=T(Q)Q

, where

T\colon{\mathcal C}^n({\mathbb T}^2)\to\operatorname{Aut}({\mathbb T}^2)

is a holomorphic map. We also prove that for

n>4

any endomorphism of the torus braid group

B_n({\mathbb T}^2)=\pi_1({\mathcal C}^n({\mathbb T}^2))

with a non-abelian image preserves the pure torus braid group

P_n({\mathbb T}^2) =\pi_1({\mathcal E}^n({\mathbb T}^2))

.
Linear isometries of some function algebras
Authors: Vesentini Edoardo
Published: December 2009

Linear isometries of a class of logmodular algebras which are generated by unitary functions are represented by Holszty\'{n}ski type weighted composition operators. The description of these operators leads - among other things - to a characterization of the linear isometries of the disc algebra and of the Hardy space of all bounded holomorphic functions on the open unit disc of

\mathbb{C}

. Spectral properties of these isometries are also investigated.
On weak solutions for $p$-Laplacian equations with weights
Authors: Pucci Patrizia , Servadei Raffaella
Published: December 2009

Here we summarize the results obtained in the forthcoming papers [32, 33], in which we prove existence and non-existence theorems of weak solutions of quasilinear singular elliptic equations with weights. We also establish regularity and qualitative properties of the solutions.
Remarkable identities related to the Riemann zeta function
Authors: Di Cerbo Gabriele
Published: December 2009

We derive some formulas and identities for the Riemann zeta function related to the celebrated Ramanujan's formula.
Characteristic varieties and constructible sheaves
Authors: Dimca Alexandru
Published: December 2009

We explore the relation between the positive dimensional irreducible components of the characteristic varieties of rank one local systems on a smooth surface and the associated (rational or irrational) pencils. Our study, which may viewed as a continuation of D. Arapura's paper [1], yields new geometric insight into the translated components relating them to the multiplicities of curves in the associated pencil, in a close analogy to the compact situation treated by A. Beauville [3]. The new point of view is the key role played by the constructible sheaves naturally arising from local systems.
Some existence results for the Toda system on closed surfaces
Authors: Malchiodi Andrea , Ndiaye Cheikh Birahim
Published: December 2009

Given a compact closed surface

\Sigma

, we consider the generalized Toda system of equations on

\Sigma

:

- \Delta u_i = \sum_{j=1}^2 \rho_j a_{ij} \left( \frac{h_j e^{u_j}}{\int_\Sigma h_j e^{u_j} dV_g} - 1 \right)

for

i = 1, 2

, where

\rho_1, \rho_2

are real parameters and

h_1, h_2

are smooth positive functions. Exploiting the variational structure of the problem and using a new minimax scheme we prove existence of solutions for generic values of

\rho_1

and for

\rho_2 < 4 \pi

.
Spectral analysis of transfer operators associated to Farey fractions
Authors: Bonanno Claudio , Graffi Sandro , Isola Stefano
Published: December 2009

The spectrum of a one-parameter family of signed transfer operators associated to the Farey map is studied in detail. We show that when acting on a suitable Hilbert space of analytic functions they are self-adjoint and exhibit absolutely continuous spectrum and no non-zero point spectrum. Polynomial eigenfunctions when the parameter is a negative half-integer are also discussed.
A metric on shape space with explicit geodesics
Authors: Younes Laurent , Michor Peter W. , Shah Jayant M. , Mumford David B.
Published: December 2009

This paper studies a specific metric on plane curves that has the property of being isometric to classical manifold (sphere, complex projective, Stiefel, Grassmann) modulo change of parametrization, each of these classical manifolds being associated to specific qualifications of the space of curves (closed-open, modulo rotation etc\ldots) Using these isometries, we are able to explicitely describe the geodesics, first in the parametric case, then by modding out the paremetrization and considering horizontal vectors. We also compute the sectional curvature for these spaces, and show, in particular, that the space of closed curves modulo rotation and change of parameter has positive curvature. Experimental results that explicitly compute minimizing geodesics between two closed curves are finally provided.
A note on de Concini and Procesi's curious identity
Authors: Denham Graham
Published: December 2009

We give a short, case-free and combinatorial proof of de Concini and Procesi's formula for the volume of the simplicial cone spanned by the simple roots of any finite root system. The argument presented here also extends their formula to include the non-crystallographic root systems.
Entire solutions of autonomous equations on $\mathbb R^n$ with nontrivial asymptotics
Authors: Malchiodi Andrea
Published: December 2009

We prove existence of a new type of solutions for the semilinear equation

- \Delta u + u = u^p

on

\R^n

, with

1 < p < \frac{n+2}{n-2}

. These solutions are positive, bounded, decay exponentially to zero away from three half-lines with a common origin, and at infinity are asymptotically periodic.
Rational points in periodic analytic sets and the Manin–Mumford conjecture
Authors: Pila Jonathan , Zannier Umberto
Published: December 2009

We present a new proof of the Manin–Mumford conjecture about torsion points on algebraic subvarieties of abelian varieties. Our principle, which admits other applications, is to view torsion points as rational points on a complex torus and then compare (i) upper bounds for the number of rational points on a transcendental analytic variety (Bombieri–Pila–Wilkie) and (ii) lower bounds for the degree of a torsion point (Masser), after taking conjugates. In order to be able to deal with (i), we discuss (Thm. 2.1) the semi-algebraic curves contained in an analytic variety supposed invariant for translations by a full lattice, which is a topic with some independent motivation.
Transversality of equivariant mappings with closed associated differential forms on $G$-manifolds
Authors: Lesimple Marc , Valent Tullio
Published: December 2009

It is proved that any equivariant mapping between

G

-manifolds (

G

a Lie group), with a closed associated differential form, is transversal to the orbit of any point in its image up to a

G

-invariant subspace of the tangent space.
Axial impact on a semi-infinite elastic rod
Authors: Villaggio Piero
Published: December 2009

An approximate theory for treating the axial collision of a rigid mass against an elastic rod was proposed by Saint–Venant more than 150 years ago. The method works only for short bars, under the assumption that the elastic axial displacement instantaneously propagates from one end to the other after the impact. Moreover, this hypothesis is unrealistic for long rods.

Here we suggest an extension of the method which is able to treat the axial impact on the initial cross-section of a semi-infinite rod.
Conformal metrics on $ℝ^{2m}$ with constant $Q$-curvature
Authors: Martinazzi Luca
Published: December 2009

We study the conformal metrics on

\mathbb R^{2m}

with constant

Q

-curvature

Q\in\R{}

having finite volume, particularly in the case

Q\leq 0

. We show that when

Q<0

such metrics exist in

\R{2m}

if and only if

m>1

. Moreover we study their asymptotic behavior at infinity, in analogy with the case

Q>0

, which we treated in a recent paper. When

Q=0

, we show that such metrics have the form

e^{2p}g_{\R{2m}}

, where

p

is a polynomial such that

2\leq \deg p\leq 2m-2

and

\sup_{\R{2m}}p<+\infty

. In dimension

4

, such metrics correspond to the polynomials

p

of degree

2

with

\lim_{|x|\to+\infty}p(x)=-\infty

.
On the notion of ergodicity for finite quantum systems
Authors: Degli Espositi Mirko , Graffi Sandro , Isola Stefano
Published: December 2009

We show that the orginal definition of ergodicity of Boltzmann can be directly applied to finite quantum systems, such as those arising from the quantization of classical systems on a compact phase space. It yields a notion of quantum ergodicity strictly stronger than the Von Neumann one. As an example, we remark that the quantized hyperbolic symplectomorphisms (a particular case is the quantized Arnold cat) are ergodic in this sense.
A remark on the semiclassical Fefferman–Phong inequality for certain systems of PDEs
Authors: Parmeggiani Alberto
Published: December 2009

We give here an extension of the semiclassical version of the Fefferman–Phong inequality to certain
N × N
systems of PDEs. In addition, we give an example of system for which the semiclassical Fefferman–Phong inequality cannot hold.
Existence and multiplicity results for a weighted $p$-Laplace equation involving Hardy potentials and critical nonlinearities
Authors: Musina Roberta
Published: December 2009

We study a class of elliptic problems involving weighted
p
-Laplace operators, critical growths and Hardy potentials. The main motivation lies in some Hardy–Sobolev type inequalities that were proved by Caffarelli–Kohn–Nirenberg in 1984.
Plücker formulae for curves in high dimensions
Authors: Wall C.T.C.
Published: December 2009

The classical relations of Plücker between the invariants and singularities of a plane curve can be expressed as two linear relations and two involving quadratic terms. The linear relations were generalised to curves in n-space already in the nineteenth century, but true generalisations of the others were obtained only in 3-space. In this article, using the classical method of correspondences, we obtain formulae in n-space corresponding to the original ones in the plane.
Editorial note
Authors: Ambrosetti Antonio
Published: December 2009

Editorial Note

On the occasion of the thirtieth anniversary of the death of Guido Stampacchia (1922-1978), the International Conference “Recent trends in nonlinear partial differential equations” has been organised by some of his former colleagues and students at the Accademia dei Lincei in Roma, November 6, 2008.

Stampacchia was the President of the Unione Matematica Italiana in the period of 1967-1973, and a very active member of the Accademia dei Lincei. One of the leaders of the Italian school of partial differential equations, his contributions to differential equations, calculus of variations and variational inequalities are well known. His popularity in the Italian mathematical community explains very well the participation of so many researchers in the Conference.

After the opening addresses by Enrico Magenes and David Kinderlehrer, plenary lectures were given by Louis Nirenberg, Haïm Brezis, Lucio Boccardo, Vieri Benci, Henry Berestycki and Luis Caffarelli. Some of the speakers have gladly agreed to write up their lectures and Rendiconti Lincei - Matematica e Applicazioni is proud to publish them. The paper by Brezis already appeared in vol. XIX (2008), No 4, pp. 335-338 while others are collected in the present issue.

The Editor-in-ChiefAntonio Ambrosetti
Surfaces minimizing nonlocal energies
Authors: Caffarelli Luis A.
Published: December 2009

In this lecture, we discuss what we understand by a non local diffusion equation and explain the particular case of surface evolution by non local mean curvature and the corresponding minimal surfaces.
The level 1 case of Serre’s conjecture revisited
Authors: Dieulefait Luis Victor
Published: December 2009

We prove existence of conjugate Galois representations, and we use it to derive a simple method of weight reduction. As a consequence, an alternative proof of the level 1 case of Serre’s conjecture follows.
A sharp isoperimetric inequality in the plane involving Hausdorff distance
Authors: Alvino Angelo , Ferone Vincenzo , Nitsch Carlo
Published: December 2009

We show that among all the convex bounded domain in

ℝ^2

having an assigned asymmetry index related to Hausdorff distance, there exists only one convex set (up to a similarity) which minimizes the isoperimetric deﬁcit. We also show how to construct this set. The result can be read as a sharp improvement of the isoperimetric inequality for convex planar domain.
Deep foundations
Authors: Villaggio Piero
Published: December 2009

The stress transmission between a rigid foundation and the ground below is traditionally formulated into mathematical terms as the elastic problem of ﬁnding the stress state in a half-plane loaded by a rigid indentor. But this model is not realistic since foundations are not built on the surface of the ground but below its level, at the bottom of an excavation. We here suggest a solution for a notched elastic half-plane loaded by a rigid punch applied at the throat of the cavity.
T1 Theorems on Besov and Triebel-Lizorkin Spaces on Spaces of Homogeneous Type and Their Applications
Authors: Yang Dachun
Published: December 2009

The author first establishes the reduced T1 theorems for Besov and Triebel-Lizorkin spaces on spaces of homogeneous type. Using these T1 theorems, the author proves that an operator of Bessel potential type can be used as the lifting operator of these spaces.
Homogenization Structures and Applications I
Authors: Nguetseng Gabriel
Published: December 2009

We lay the foundations of a mathematical theory of homogenization structures and show how the latter arises in the homogenization of partial differential equations. We find out that the concept of a homogenization structure turns out to be exactly the right tool that is needed to systematically extend homogenization theory beyond the classical periodic setting. This permits to work out various outstanding nonperiodic homogenization problems that were out of reach till then for lack of an appropriate mathematical framework. The classical Gelfand representation theory is one of our main tools and our basic approach is an adaptation of the two-scale convergence method.
Exponential Growth for a Fractionally Damped Wave Equation
Authors: Kirane Mokhtar , Tatar Nasser-edine
Published: December 2009

We consider a nonlinear wave equation with an internal damping represented by a fractional time derivative and with a polynomial source. It is proved that the solution is unbounded and grows up exponentially in the Lp-norm for sufficiently large initial data. To this end we use some techniques based on Fourier transforms and some inequalities such as the Hardy-Littlewood inequality.
On the Set of Reachable States in the Problem of Controllability of Rotating Timishenko Beams
Authors: Krabs Werner , Sklyar Grigory M. , Wozniak J.
Published: December 2009

This work continues the authors' previous investigation on the controllability problem of a slowly rotating Timoshenko beam. We obtain conditions of exact controllability under the assumption that the parameter g appearing in the model equation is rational. Our result rests on a generalization of the theorem by Ullrich on the Riesz basis property of exponential divided differences.