bluuebunny/crossref_metadata_2025 huggingface dataset:
Continuum limits of atomistic energies allowing smooth and sharp interfaces in 1D elasticity
Authors: Mora-Corral Carlos
Published: December 2009

We present two atomistic models for the energy of a one-dimensional elastic crystal. We assume that the macroscopic displacement equals the microscopic one. The energy of the ﬁrst model is given by a two-body interaction potential, and we assume that the atoms follow a continuous and piecewise smooth macroscopic (continuum) deformation. We calculate the ﬁrst terms of the Taylor expansion (with respect to the parameter representing the interatomic distance) of the atomistic energy, and deduce that the coefﬁcients of that Taylor expansion represent, respectively, an elastic energy, a sharp-interface energy, and a smooth-interface energy. The second atomistic model is a variant of the ﬁrst one, and its Taylor expansion predicts, in addition, a new term that accounts for the repulsion force between two sharp interfaces.
On the convexity of some free boundaries
Authors: Greco Antonio , Kawohl Bernd
Published: December 2009

We demonstrate that a method of Colesanti and Salani, which compares solutions of elliptic differential equations to their quasiconcave envelopes, can be extended to derive convexity of free boundaries. As examples we present the so-called dam problem, a free boundary problem modelling pollution and a Bernoulli problem. Moreover, we prove strict convexity of the wet region in the dam problem in arbitrary dimensions.
Second order numerical scheme for motion of polygonal curves with constant area speed
Authors: Beneš Michal , Kimura Masato , Yazaki Shigetoshi
Published: December 2009

We study polygonal analogues of several moving boundary problems and their time discretization which preserves the constant area speed property. We establish various polygonal analogues of geometric formulas for moving boundaries and make use of the geometric formulas for our numerical scheme to analyse general constant area speed motion of polygons. Accuracy and efﬁciency of our numerical scheme are checked through numerical simulations for several polygonal motions such as motion by curvature and area-preserving advected ﬂow etc.
Null structure and almost optimal local regularity for the Dirac-Klein-Gordon system
Authors: D'Ancona Piero , Foschi Damiano , Selberg Sigmund
Published: December 2009

We prove almost optimal local well-posedness for the coupled Dirac-Klein-Gordon (DKG) system of equations in

1+3

dimensions. The proof relies on the null structure of the system, combined with bilinear spacetime estimates of Klainerman-Machedon type. It has been known for some time that the Klein-Gordon part of the system has a null structure; here we uncover an additional null structure in the Dirac equation, which cannot be seen directly, but appears after a duality argument.
Corrigendum: A construction of the Deligne–Mumford orbifold (J. Eur. Math. Soc. 8, 611–699 (2006))
Authors: Robbin Joel W. , Salamon Dietmar A.
Published: December 2009

We correct an error in [3, Lemma 8.2]. As stated, the lemma only holds for surfaces of genus greater than

1

or in the case

α=0

. When the genus is

0

or

1

and in addition

α≠0

, equation (2) in [3] (in the present corrigendum this is equation (2)) is only a necessary condition for the integrability of

J

but is not sufficient. In [3] Lemma 8.2 is only used twice. On page 637 it is used in the trivial case

α=0

. On page 642 only the “only if” direction is used and the proof of that direction is correct in [3]. In this note we prove a corrected version of [3, Lemma 8.2].
Single-point blow-up on the boundary where the zero Dirichlet boundary condition is imposed
Authors: Fila Marek , Winkler Michael
Published: December 2009

We consider a reaction-diffusion-convection equation on the halfline

(0,\infty)

with the zero Dirichlet boundary condition at

x=0

. We find a positive selfsimilar solution

u

which blows up in a finite time

T

at

x=0

while

u(x,T)

remains bounded for

x>0

.
Asymptotic behavior of a stochastic combustion growth process
Authors: Ramírez Alejandro F. , Sidoravicius Vladas
Published: December 2009

We study a continuous time growth process on the

d

-dimensional hyper-cubic lattice

\Bbb Z^d

, which admits a phenomenological interpretation as the combustion reaction

A+B\to 2A

, where

A

represent heat particles and

B

inert particles. This process can be described as an interacting particle system in the following way: at time

0

a simple symmetric continuous time random walk of total jump rate one begins to move from the origin of the hyper-cubic lattice; then, as soon as any random walk visits a site previously unvisited by any other random walk, it creates a new independent simple symmetric random walk starting from that site. Let us call

P_d

the law of such a process and

S^0_d(t)

the set of visited sites at time

t

. In this article we prove that there exists a bounded, non-empty, convex set

C_d\subset{\Bbb R}^d

, such that for every

\epsilon>0

,

P_d

-a.s. eventually in

t

, the set

S_d^0(t)

is within an

\epsilon t

neighborhood of the set

[C_dt]

, where for

A\subset{\Bbb R}^d

we define

[A]:=A\cap {\Bbb Z}^d

. Furthermore, answering questions posed by M. Bramson and R. Durrett, we prove that the empirical density of particles converges weakly to a product Poisson measure of parameter one, and moreover, for

d

large enough, we establish that the set

C_d

is not a ball under the Euclidean norm.
Blow ups of complex solutions of the 3D Navier–Stokes system and renormalization group method
Authors: Li Dong , Sinai Yakov G.
Published: December 2009

We consider complex-valued solutions of the three-dimensional Navier-Stokes system without external forcing on

R^3

. We show that there exists an open set in the space of

10

-parameter families of initial conditions such that for each family from this set there are values of parameters for which the solution develops blow up in finite time.
Invariance of tautological equations I: conjectures and applications
Authors: Lee Y.-P.
Published: December 2009

The main goal of this paper is to introduce a set of conjectures on the relations in the tautological rings. In particular, this framework gives an efficient algorithm to calculate all tautological equations using only finite dimensional linear algebra. Other applications include the proofs of
Witten's conjecture
on the relations between higher spin curves and Gelfand–Dickey hierarchy and
Virasoro conjecture
for target manifolds with conformal semisimple quantum cohomology, both for genus up to two.
Symmetry of solutions of semilinear elliptic problems
Authors: Van Schaftingen Jean , Willem Michel
Published: December 2009

We study symmetry properties of least energy positive or nodal solutions of semilinear elliptic problems with Dirichlet or Neumann boundary conditions. The proof is based on symmetrizations in the spirit of Bartsch, Weth and Willem (J. Anal. Math., 2005) together with a strong maximum principle for quasi-continuous functions of Ancona and an intermediate-value property for such functions.
Random Witten Laplacians: traces of semigroups, $L^2$-Betti numbers and index
Authors: Albeverio Sergio , Daletskii Alexei , Kalyuzhnyi Alexander
Published: December 2009

Random Witten Laplacians on infinite coverings of compact manifolds are considered. The probabilistic representations of the corresponding heat kernels are given. The finiteness of the von Neumann traces of the corresponding semigroups is proved, and the short-time asymptotics of the corresponding super-trace is computed. Examples associated with Gibbs measures on configuration spaces and product manifolds are considered.
A second order SDE for the Langevin process reflected at a completely inelastic boundary
Authors: Bertoin Jean
Published: December 2009

It was shown in [2] that a Langevin process can be reflected at an energy absorbing boundary. Here, we establish that the law of this reflecting process can be characterized as the unique weak solution to a certain second order stochastic differential equation with constraints, which is in sharp contrast with a deterministic analog.
Exact boundary controllability of a nonlinear KdV equation with critical lengths
Authors: Coron Jean-Michel , Crépeau Emmanuelle
Published: December 2009

We study the boundary controllability of a nonlinear Korteweg-de Vries equation with the Dirichlet boundary condition on an interval with a critical length for which it has been shown by Rosier that the linearized control system around the origin is not controllable. We prove that the nonlinear term gives the local controllability around the origin.
The moduli space of commutative algebras of finite rank
Authors: Poonen Bjorn
Published: December 2009

The moduli space of rank-

n

commutative algebras equipped with an ordered basis is an affine scheme

\mathfrak B_n

of finite type over

\mathbb Z

, with geometrically connected fibers. It is smooth if and only if

n \le 3

. It is reducible if

n \ge 8

(and the converse holds, at least if we remove the fibers above

2

and

3

). The relative dimension of

\mathfrak B_n

is

\frac{2}{27} n^3 + O(n^{8/3})

. The subscheme parameterizing étale algebras is isomorphic to

\operatorname{GL}_n/S_n

, which is of dimension only

n^2

. For

n \ge 8

, there exist algebras that are not limits of étale algebras. The dimension calculations lead also to new asymptotic formulas for the number of commutative rings of order

p^n

and the dimension of the Hilbert scheme of

n

points in

d

-space for

d \ge n/2

.
Scales, fields, and a problem of Hurewicz
Authors: Tsaban Boaz , Zdomskyy Lyubomyr
Published: December 2009

Menger's basis property is a generalization of

\sigma

-comp\-actness and admits an elegant combinatorial interpretation. We introduce a general combinatorial method to construct non

\sigma

-compact sets of reals with Menger's property. Special instances of these constructions give known counterexamples to conjectures of Menger and Hurewicz. We obtain the first explicit solution to the Hurewicz 1927 problem, that was previously solved by Chaber and Pol on a dichotomic basis. The constructed sets generate nontrivial subfields of the real line with strong combinatorial properties, and most of our results can be stated in a Ramsey-theoretic manner. Since we believe that this paper is of interest to a diverse mathematical audience, we have made a special effort to make it self-contained and accessible.
Expansion and random walks in $\mathrm{SL}_d(\mathbb{Z}/p^n\mathbb{ℤ})$: I
Authors: Bourgain Jean , Gamburd Alex
Published: December 2009

We prove that Cayley graphs of

\mathrm{SL}_2(ℤ/p^nℤ)

are expanders with respect to the projection of any fixed elements in

\mathrm{SL}_2(ℤ)

generating a Zariski-dense subgroup.
Deformations of Lie brackets: cohomological aspects
Authors: Crainic Marius , Moerdijk Ieke
Published: December 2009

We introduce a new cohomology for Lie algebroids, and prove that it provides a differential graded Lie algebra which “controls” deformations of the structure bracket of the algebroid.
Convergence of a non-local eikonal equation to anisotropic mean curvature motion. Application to dislocations dynamics
Authors: Da Lio Francesca , Forcadel Nicolas , Monneau Régis
Published: December 2009

In this paper we prove the convergence at a large scale of a non-local first order equation to an anisotropic mean curvature motion. This first order equation is an eikonal-type equation with a velocity depending in a non-local way on the solution itself, that arises in the theory of dislocations dynamics. We show that if an anisotropic mean curvature motion is approximated by this type of equations then it is always of variational type, whereas the converse is true only in dimension two.
On the Ladyzhenskaya–Smagorinsky turbulence model of the Navier–Stokes equations in smooth domains. The regularity problem
Authors: Beirão da Veiga Hugo
Published: December 2009

We establish regularity results
up to the boundary
for solutions to generalized Stokes and Navier–Stokes systems of equations in the stationary and in the evolutive cases. Generalized here means the presence of a shear dependent viscosity. We treat here the case

p\geq 2

. Actually, we are interested in proving regularity results in

L^q(\Omega)

spaces for all the second order derivatives of the velocity and all the first order derivatives of the pressure. The main aim of the present paper is to extend our previous scheme, introduced in references [3] and [4] for the flat-boundary case, to the case of curvilinear boundaries.
Single-point blow-up for a semilinear parabolic system
Authors: Souplet Philippe
Published: December 2009

We consider positive solutions of a semilinear parabolic system coupled by power nonlinearities, in a ball or in the whole space. Relatively little is known on the blow-up set for parabolic systems and, up to now, no result was available for this basic system except for the very special case of equal powers. Here we prove single-point blow-up for a large class of radial decreasing solutions. This in particular solves a problem left open in a paper of A. Friedman and Y. Giga (1987). We also obtain lower pointwise estimates for the final blow-up profiles.
On the stabilization problem for nonholonomic distributions
Authors: Rifford Ludovic , Trélat Emmanuel
Published: December 2009

Let

M

be a smooth connected and complete manifold of dimension

n

, and

\Delta

be a smooth nonholonomic distribution of rank

m\leq n

on

M

. We prove that, if there exists a smooth Riemannian metric on

\Delta

for which no nontrivial singular path is minimizing, then there exists a smooth repulsive stabilizing section of

\Delta

on

M

. Moreover, in dimension three, the assumption of the absence of singular minimizing horizontal paths can be dropped in the Martinet case. The proofs are based on the study, using specific results of nonsmooth analysis, of an optimal control problem of Bolza type, for which we prove that the corresponding value function is semiconcave and is a viscosity solution of a Hamilton-Jacobi equation, and establish fine properties of optimal trajectories.
Strong spectral gaps for compact quotients of products of PSL(2,ℝ)
Authors: Kelmer Dubi , Sarnak Peter
Published: December 2009

The existence of a strong spectral gap for quotients

\Gamma\backslash G

of noncompact connected semisimple Lie groups is crucial in many applications. For congruence lattices there are uniform and very good bounds for the spectral gap coming from the known bounds towards the Ramanujan–Selberg Conjectures. If

G

has no compact factors then for general lattices a spectral gap can still be established, however, there is no uniformity and no effective bounds are known. This note is concerned with the spectral gap for an irreducible co-compact lattice

\Gamma

in

G=\operatorname{PSL}(2,\mathbb R)^d

for

d\geq 2

which is the simplest and most basic case where the congruence subgroup property is not known. The method used here gives effective bounds for the spectral gap in this setting.
Polarizations of Prym varieties for Weyl groups via abelianization
Authors: Lange Herbert , Pauly Christian
Published: December 2009

Let

\pi: Z \to X

be a Galois covering of smooth projective curves with Galois group the Weyl group of a simple and simply connected Lie group

G

. For any dominant weight

\lambda

consider the curve

Y = Z/\operatorname{Stab}(\lambda)

. The Kanev correspondence defines an abelian subvariety

P_\lambda

of the Jacobian of

Y

. We compute the type of the polarization of the restriction of the canonical principal polarization of

\operatorname{Jac}(Y)

to

P_\lambda

in some cases. In particular, in the case of the group

E_8

we obtain families of Prym-Tyurin varieties. The main idea is the use of an abelianization map of the Donagi–Prym variety to the moduli stack of principal

G

-bundles on the curve

X

.
Face enumeration—from spheres to manifolds
Authors: Swartz Ed
Published: December 2009

We prove a number of new restrictions on the enumerative properties of homology manifolds and semi-Eulerian complexes and posets. These include a determination of the affine span of the fine

h

-vector of balanced semi-Eulerian complexes and the toric

h

-vector of semi-Eulerian posets.

The lower bounds on simplicial homology manifolds, when combined with higher dimensional analogues of Walkup’s 3-dimensional constructions [47], allow us to give a complete characterization of the

f

-vectors of arbitrary simplicial triangulations of

S^1 \times S^3

,

ℂP^2, K3

surfaces, and

(S^2 \times S^2) \# (S^2 \times S^2)

. We also establish a principle which leads to a conjecture for homology manifolds which is almost logically equivalent to the

g

-conjecture for homology spheres. Lastly, we show that with sufficiently many vertices, every triangulable homology manifold without boundary of dimension three or greater can be triangulated in a 2-neighborly fashion.
Relative integral functors for singular fibrations and singular partners
Authors: Hernández Ruipérez Daniel , López Martín Ana Cristina , Sancho de Salas Fernando
Published: December 2009

We study relative integral functors for singular schemes and characterise those which preserve boundedness and those which have integral right adjoints. We prove that a relative integral functor is an equivalence if and only if its restriction to every fibre is an equivalence. This allows us to construct a non-trivial auto-equivalence of the derived category of an arbitrary genus one fibration with no conditions on either the base or the total space and getting rid of the usual assumption of irreducibility of the fibres. We also extend to Cohen–Macaulay schemes the criterion of Bondal and Orlov for an integral functor to be fully faithful in characteristic zero and give a different criterion which is valid in arbitrary characteristic. Finally, we prove that for projective schemes both the Cohen–Macaulay and the Gorenstein conditions are invariant under Fourier–Mukai functors.
On Kahane's ultraflat polynomials
Authors: Bombieri Enrico , Bourgain Jean
Published: December 2009

This paper is devoted to the construction of polynomials of almost constant modulus on the unit circle, with coefficients of constant absolute value. In particular, one obtains a much improved estimate for the error term. A major part of this paper deals also with the long-standing problem of the effective construction of ultraflat polynomials.
Critical points via Γ-convergence: general theory and applications
Authors: Jerrard Robert L. , Sternberg Peter
Published: December 2009

It is well-known that Γ-convergence of functionals provides a tool for studying global and local minimizers. Here we present a general result establishing the existence of critical points of a Γ-converging sequence of functionals provided the associated Γ-limit possesses a nondegenerate critical point, subject to certain mild additional hypotheses. We then go on to prove a theorem that describes suitable nondegenerate critical points for functionals, involving the arclength of a limiting singular set, that arise as Γ-limits in a number of problems. Finally, we apply the general theory to prove some new results, and give new proofs of some known results, establishing the existence of critical points of the 2d Modica–Mortola (Allen–Cahn) energy and 3d Ginzburg–Landau energy with and without magnetic ﬁeld, and various generalizations, all in a uniﬁed framework.
Hypoellipticity, fundamental solution and Liouville type theorem for matrix-valued differential operators in Carnot groups
Authors: Baldi Annalisa , Franchi Bruno , Tesi Maria Carla
Published: December 2009

Let

\mathcal L

be a non-negative self-adjoint

N\times N

matrix-valued operator of order

a\leq Q

on a Carnot group

\mathbb G

. Here

Q

is the homogeneous dimension of

\mathbb G

. The aim of this paper is to investigate the relationship between hypoellipticity and maximal hypoellipticity (i.e. sharp

L^p

estimates in appropriate Sobolev spaces),

L^p

-maximal hypoellipticity (i.e. sharp

L^p

estimates in appropriate Sobolev spaces for {

1 < p < \infty

}), and what we call maximal subellipticity of

\mathcal L

(which is basically a sharp higher order energy estimate).
Construction of Kähler surfaces with constant scalar curvature
Authors: Rollin Yann , Singer Michael
Published: December 2009

We present new constructions of Kähler metrics with constant scalar curvature on complex surfaces, in particular on certain del Pezzo surfaces. Some higher-dimensional examples are provided as well.
Expansion and random walks in $\mathrm{SL}_d(\mathbb{Z}/p^n \mathbb{Z})$: II
Authors: Bourgain Jean , Gamburd Alex
Published: December 2009

We prove that Cayley graphs of

\mathrm{SL}_d(ℤ/p^nℤ)

are expanders with respect to the projection of any fixed elements in

\mathrm{SL}_d(ℤ)

generating a Zariski dense subgroup.
On the structure of Hardy–Sobolev–Maz'ya inequalities
Authors: Filippas Stathis , Tertikas Achilles , Tidblom Jesper
Published: December 2009

We establish new improvements of the optimal Hardy inequality in the half-space. We ﬁrst add all possible linear combinations of Hardy type terms, thus revealing the structure of this type of inequalities and obtaining best constants. We then add the critical Sobolev term and obtain necessary and sufﬁcient conditions for the validity of Hardy–Sobolev–Maz’ya type inequalities.
Confirmation of Matheron's conjecture on the covariogram of a planar convex body
Authors: Averkov Gennadiy , Bianchi Gabriele
Published: December 2009

The covariogram

g_K

of a convex body

K

in

\mathbb E^d

is the function which associates to each

x\in \mathbb E^d

the volume of the intersection of

K

with

K+x

. In 1986 G. Matheron conjectured that for

d = 2

the covariogram

g_K

determines

K

within the class of all planar convex bodies, up to translations and reﬂections in a point. This problem is equivalent to some problems in stochastic geometry and probability as well as to a particular case of the phase retrieval problem in Fourier analysis. It is also relevant for the inverse problem of determining the atomic structure of a quasicrystal from its X-ray diffraction image. In this paper we conﬁrm Matheron’s conjecture completely. This problem is equivalent to some problems in stochastic geometry and probability as well as to a particular case of the phase retrieval problem in Fourier analysis. It is also relevant for the inverse problem of determining the atomic structure of a quasicrystal from its X-ray diffraction image. In this paper we confirm Matheron's conjecture completely.
The cubic nonlinear Schrödinger equation in two dimensions with radial data
Authors: Killip Rowan , Tao Terence , Vișan Monica
Published: December 2009

We establish global well-posedness and scattering for solutions to the mass-critical nonlinear Schrödinger equation

iu_t + \Delta u= ±|u|^2 u

for large spherically symmetric

L_x^2(ℝ^2)

initial data; in the focusing case we require, of course, that the mass is strictly less than that of the ground state. As a consequence, we deduce that in the focusing case, any spherically symmetric blowup solution must concentrate at least the mass of the ground state at the blowup time.

We also establish some partial results towards the analogous claims in other dimensions and without the assumption of spherical symmetry.
Metrical theory for $α$-Rosen fractions
Authors: Dajani Karma , Kraaikamp Cor , Steiner Wolfgang
Published: December 2009

The Rosen fractions form an inﬁnite family which generalizes the nearest-integer continued fractions. In this paper we introduce a new class of continued fractions related to the Rosen fractions, the
α
-Rosen fractions. The metrical properties of these
α
-Rosen fractions are studied.

We ﬁnd planar natural extensions for the associated interval maps, and show that their domains of deﬁnition are closely related to the domains of the ‘classical’ Rosen fractions. This uniﬁes and generalizes results of diophantine approximation from the literature.
Geometry of the theta divisor of a compactiﬁed jacobian
Authors: Caporaso Lucia
Published: December 2009

The object of this paper is the theta divisor of the compactiﬁed jacobian of a nodal curve. We determine its irreducible components and give it a geometric interpretation. A characterization of hyperelliptic irreducible stable curves is appended as an application.
Formal power series rings over a $\pi$-domain
Authors: Kang Byung Gyun , Oh Dong Yeol
Published: December 2009

Let

R

be an integral domain,

Χ

be a set of indeterminates over

R

, and

R[[\mathcal X]]_3

be the full ring of formal power series in

\mathcal X]

over

R

. We show that the Picard group of

R[[\mathcal X]]_3

is isomorphic to the Picard group of

R

. An integral domain is called a

π

-domain if every principal ideal is a product of prime ideals. An integral domain is a

π

-domain if and only if it is a Krull domain that is locally a unique factorization domain. We show that

R[[\mathcal X]]_3

is a

π

-domain if

R[[Χ_1 , . . . , Χ_n]]

is a

π

-domain for every

n ≥ 1

. In particular,

R[[\mathcal X]]_3

is a

π

-domain if

R

is a Noetherian regular domain. We extend these results to rings with zero-divisors. A commutative ring

R

with identity is called a

π

-ring if every principal ideal is a product of prime ideals. We show that

R[[\mathcal X]]_3

is a

π

-ring if

R

is a Noetherian regular ring.
Pluripotential theory, semigroups and boundary behavior of infinitesimal generators in strongly convex domains
Authors: Bracci Filippo , Contreras Manuel D. , Díaz-Madrigal Santiago
Published: December 2009

We characterize infinitesimal generators of semigroups of holomorphic self-maps of strongly convex domains using the pluricomplex Green function and the pluricomplex Poisson kernel. Moreover, we study boundary regular fixed points of semigroups. Among other things, we characterize boundary regular fixed points both in terms of the boundary behavior of infinitesimal generators and in terms of pluripotential theory.
On the Lenstra constant associated to the Rosen continued fractions
Authors: Nakada Hitoshi
Published: December 2009

The purpose of this paper is to describe the relation between the Legendre and the Lenstra constants. Indeed we show that they are equal whenever the Legendre constant exists; in particular, this holds for both Rosen continued fractions and α-continued fractions. We also give the explicit value of the entropy of the Rosen map with respect to the absolutely continuous invariant probability measure.
Simply connected minimal symplectic 4-manifolds with signature less than $−1$
Authors: Akhmedov Anar , Baldridge Scott , Baykur R. İnanç , Kirk Paul , Park B. Doug
Published: December 2009

For each pair

(e,σ)

of integers satisfying

2e + 3σ ≥ 0

,

σ ≤ −2

, and

e + σ ≡ 0

(mod

4

), with four exceptions, we construct a minimal, simply connected symplectic 4-manifold with Euler characteristic

e

and signature

σ

. We also produce simply connected, minimal symplectic 4-manifolds with signature zero (resp. signature

−1

) with Euler characteristic

4k

(resp.

4k + 1

) for all

k ≥ 46

(resp.

k ≥ 49

).
Lyapunov inequalities for Neumann boundary conditions at higher eigenvalues
Authors: Cañada Antonio , Villegas Salvador
Published: December 2009

This paper is devoted to the study of Lyapunov-type inequality for Neumann boundary conditions at higher eigenvalues. Our main result is derived from a detailed analysis about the number and distribution of zeros of nontrivial solutions and their first derivatives, together with the use of suitable minimization problems. This method of proof allows to obtain new information on Lyapunov constants. For instance, we prove that as in the classical result by Lyapunov, the best constant is not attained. Additionally, we exploit the relation between Neumann boundary conditions and disfocality to provide new nonresonance conditions at higher eigenvalues.
Descriptive Kakutani equivalence
Authors: Miller Benjamin D. , Rosendal Christian
Published: December 2009

We consider a descriptive set-theoretic analog of Kakutani equivalence for Borel automorphisms of Polish spaces. Answering a question of Nadkarni, we show that up to this notion, there are exactly two aperiodic Borel automorphisms of uncountable Polish spaces. Using this, we classify all Borel

ℝ

-flows up to

C^∞

-time-change isomorphism. We then extend the notion of descriptive Kakutani equivalence to all (not necessarily injective) Borel functions, and provide a variety of results leading towards a complete classification. The main technical tools are a series of Glimm–Effros and Dougherty–Jackson–Kechris-style embedding theorems.
Estimates for differential operators of vector analysis involving $L^1$-norm
Authors: Maz'ya Vladimir G.
Published: December 2009

New Hardy and Sobolev type inequalities involving

L^1

-norms of scalar and vector valued functions in

ℝ^n

are obtained. The work is related to some problems stated in the recent paper by Bourgain and Brezis [BB2].
Measures of maximal entropy for random $\beta$-expansions
Authors: Dajani Karma , de Vries Martijn
Published: December 2009

Let

\beta >1

be a non-integer. We consider

\beta

-expansions of the form

\sum_{i=1}^{\infty} \frac{d_i}{\beta^i}

, where the digits

(d_i)_{i \geq 1}

are generated by means of a Borel map

K_{\beta}

defined on

\{0,1\}^{\N}\times \left[ 0, \lfloor \beta \rfloor /(\beta -1)\right]

. We show that

K_{\beta}

has a unique mixing measure

\nu_{\beta}

of maximal entropy with marginal measure an infinite convolution of Bernoulli measures. Furthermore, under the measure

\nu_{\beta}

the digits

(d_i)_{i \geq 1}

form a uniform Bernoulli process. In case 1 has a finite greedy expansion with positive coefficients, the measure of maximal entropy is Markov. We also discuss the uniqueness of

\beta

-expansions.
Branching processes, and random-cluster measures on trees
Authors: Grimmett Geoffrey R. , Janson Svante
Published: December 2009

Random-cluster measures on infinite regular trees are studied in conjunction with a general type of ‘boundary condition’, namely an equivalence relation on the set of infinite paths of the tree. The uniqueness and non-uniqueness of random-cluster measures are explored for certain classes of equivalence relations. In proving uniqueness, the following problem concerning branching processes is encountered and answered. Consider bond percolation on the family-tree

T

of a branching process. What is the probability that every infinite path of

T

, beginning at its root, contains some vertex which is itself the root of an infinite open sub-tree?
The Cauchy Problem for a Strongly Degenerate Quasilinear Equation
Authors: Andreu Fuensanta , Caselles Vicent , Mazón José M.
Published: December 2009

We prove existence and uniqueness of entropy solutions for the Cauchy problem for the quasilinear parabolic equation

u_t = \operatorname{div} \, \mathbf a(u,Du)

, where

\mathbf a(z,\xi) = \nabla_\xi f(z,\xi)

, and

f

is a convex function of

\xi

with linear growth as

\Vert \xi\Vert \to\infty

, satisfying other additional assumptions. In particular, this class includes a relativistic heat equation and a flux limited diffusion equation used in the theory of radiation hydrodynamics.
On the Newton partially flat minimal resistance body type problems
Authors: Comte Myriam , Díaz Jesús Ildefonso
Published: December 2009

We study the flat region of stationary points of the functional

\int_{\Omega }F(\left\vert \nabla u(x)\right\vert )dx

under the constraint

u\leq M,

where

\Omega

is a bounded domain of

{\mathbb{R}}^{2}

. Here

F(s)

is a function which is concave for

s

small and convex for

s

large, and

M>0

is a given constant. The problem generalizes the classical minimal resistance body problems considered by Newton. We construct a family of partially flat radial solutions to the associated stationary problem when

\Omega

is a ball. We analyze also some other qualitative properties. Moreover, we show the uniqueness of a radial solution minimizing the above mentioned functional. Finally, we consider nonsymmetric domains

\Omega

and provide sufficient conditions which insure that a stationary solution has a flat part.
Moser-Trudinger and logarithmic HLS inequalities for systems
Authors: Shafrir Itai , Wolansky Gershon
Published: December 2009

We prove several optimal Moser-Trudinger and logarithmic Hardy-Littlewood-Sobolev inequalities for systems in two dimensions. These include inequalities on the sphere

S2

, on a bounded domain

\Omega\subset\R2

and on all of

\R2

. In some cases we also address the question of existence of minimizers.
The p-Laplace eigenvalue problem as p goes to infinity in a Finsler metric
Authors: Belloni M. , Kawohl Bernd , Juutinen Petri
Published: December 2009

We consider the

p

-Laplacian operator on a domain equipped with a Finsler metric. We recall relevant properties of its first eigenfunction for finite

p

and investigate the limit problem as

p\to\infty

.
A note on a critical problem with natural growth in the gradient
Authors: Abdellaoui Boumediene , Peral Alonso Ireneo
Published: December 2009

The paper analyzes the influence on the meaning of natural growth in the gradient, of a perturbation by a Hardy potential in some elliptic equations. We obtain a linear differential operator that, in a natural way, is the corresponding gradient for the perturbed elliptic problem.

The main results are: i) Optimal summability of the data to have weak solutions; ii) Optimal linear operator associated, and, iii) Multiplicity and characterization of all solutions in terms of some measures. The results also are new for the Laplace operator perturbed for an inverse-square potential.
General results on the eigenvalues of operators with gaps, arising from both ends of the gaps. Application to Dirac operators
Authors: Dolbeault Jean , Esteban Maria J. , Séré Éric
Published: December 2009

This paper is concerned with an extension and reinterpretation of previous results on the variational characterization of eigenvalues in gaps of the essential spectrum of self-adjoint operators. We state two general abstract results on the existence of eigenvalues in the gap and a continuation principle. Then these results are applied to Dirac operators in order to characterize simultaneously eigenvalues corresponding to electronic and positronic bound states.
On the number of positive solutions of singularly perturbed 1D NLS
Authors: Felmer Patricio A. , Martínez Salomé , Tanaka Kazunaga
Published: December 2009

We study singularly perturbed 1D nonlinear Schrödinger equations (1.1). When

V(x)

has multiple critical points, (1.1) has a wide variety of positive solutions for small

\varepsilon

and the number of positive solutions increases to

\infty

as

\varepsilon\to 0

. We give an estimate of the number of positive solutions whose growth order depends on the number of local maxima of

V(x)

. Envelope functions or equivalently adiabatic profiles of high frequency solutions play an important role in the proof.
Sharp estimates for the Ambrosetti–Hess problem and consequences
Authors: Gámez José L. , Ruiz-Hidalgo Juan F.
Published: December 2009

Motivated by [3], we define the “Ambrosetti–Hess problem” to be the problem of bifurcation from infinity and study of the local behavior of continua of solutions of nonlinear elliptic eigenvalue problems. Although the works in this way underline the asymptotic properties on the nonlinearity, here we point out that this local behavior is determined by the global shape of the nonlinearity.
A fully nonlinear version of the Yamabe problem on manifolds with boundary
Authors: Li YanYan , Li Aobing
Published: December 2009

We propose a fully nonlinear version of the Yamabe problem on manifolds with boundary. The boundary condition for the conformal metric is the mean curvature. We establish some Liouville type theorems and Harnack type inequalities.
Multiplicity and stability of closed geodesics on Finsler 2-spheres
Authors: Long Yiming
Published: December 2009

A survey of recent progress on the multiplicity and stability problems of closed geodesics on Finsler

2

-spheres is given.
Uniqueness and stability of ground states for some nonlinear Schrödinger equations
Authors: Stuart Charles A.
Published: December 2009

We discuss the orbital stability of standing waves of a class of nonlinear Schrödinger equations in one space dimension. The crucial feature for our treatment is the presence of a non-constant linear potential that is even and decreasing away from the origin in space. This enables us to establish the orbital stability of all ground states over the whole range of frequencies for which such solutions exist.
Path Functionals over Wasserstein Spaces
Authors: Brancolini Alessio , Buttazzo Giuseppe , Santambrogio Filippo
Published: December 2009

Given a metric space

X

we consider a general class of functionals which measure the cost of a path in

X

joining two given points

x_0

and

x_1

, providing abstract existence results for optimal paths. The results are then applied to the case when

X

is a Wasserstein space of probabilities on a given set

\Omega

and the cost of a path depends on the value of classical functionals over measures. Conditions to link arbitrary extremal measures

\mu_0

and

\mu_1

by means of finite cost paths are given.
On local CR-transformations of Levi-degenerate group orbits in compact Hermitian symmetric spaces
Authors: Kaup W. , Zaitsev Dmitri
Published: December 2009

We present a large class of homogeneous

2

-nondegenerate CR-manifolds

M

, both of hypersurface type and of arbitrarily high CR-codimension, with the following property: Every CR-equivalence between domains

U,V

in

M

extends to a global real-analytic CR-automorphism of

M

. We show that this class contains

G

-orbits in Hermitian symmetric spaces

Z

of compact type, where

G

is a real form of the complex Lie group

\mathsf{Aut}(Z)^{0}

and

G

has an open orbit that is a bounded symmetric domain of tube type.
Random walk in random environment with asymptotically zero perturbation
Authors: Menshikov M.V. , Wade Andrew R.
Published: December 2009

We give criteria for ergodicity, transience and null recurrence for the random walk in random environment on

\Z^+=\{0,1,2,\ldots\}

, with reflection at the origin, where the random environment is subject to a vanishing perturbation. Our results complement existing criteria for random walks in random environments and for Markov chains with asymptotically zero drift, and are significantly different to these previously studied cases. Our method is based on a martingale technique - the method of Lyapunov functions.
A construction of the Deligne-Mumford orbifold
Authors: Robbin Joel W. , Salamon Dietmar A.
Published: December 2009

The Deligne–Mumford moduli space is the space

\overline{\mathcal{M}}_{g,n}

of isomorphism classes of stable nodal Riemann surfaces of arithmetic genus

g

with

n

marked points. A marked nodal Riemann surface is stable if and only if its isomorphism group is finite. We introduce the notion of a universal unfolding of a marked nodal Riemann surface and show that it exists if and only if the surface is stable. A natural construction based on the existence of universal unfoldings endows the Deligne–Mumford moduli space with an orbifold structure. We include a proof of compactness. Our proofs use the methods of differential geometry rather than algebraic geometry.
On covering and quasi-unsplit families of curves
Authors: Bonavero Laurent , Casagrande Cinzia , Druel Stéphane
Published: December 2009

Given a covering family

V

of effective 1-cycles on a complex projective variety

X

, we find conditions allowing to construct a geometric quotient

q~: X \to Y

, with

q

regular on the whole of

X

, such that every fiber of

q

is an equivalence class for the equivalence relation naturally defined by

V

. Among others, we show that on a normal and

\mathbb Q

-factorial projective variety

X

with

\dim(X) \leq 4

, every covering and quasi-unsplit family

V

of rational curves generates a geometric extremal ray of the Mori cone

\overline{\rm NE}(X)

of classes of effective 1-cycles and that the associated Mori contraction yields a geometric quotient for

V

provided

X

has canonical singularities.
Invariant densities for random $\beta$-expansions
Authors: Dajani Karma , de Vries Martijn
Published: December 2009

Let

\beta >1

be a non-integer. We consider expansions of the form

\sum_{i=1}^{\infty} \frac{d_i}{\beta^i}

, where the digits

(d_i)_{i \geq 1}

are generated by means of a Borel map

K_{\beta}

defined on

\{0,1\}^{\N}\times \left[ 0, \lfloor \beta \rfloor /(\beta -1)\right]

. We show existence and uniqueness of an absolutely continuous

K_{\beta}

-invariant probability measure w.r.t.

m_p \otimes \lambda

, where

m_p

is the Bernoulli measure on

\{0,1\}^{\N}

with parameter

p

( 0 < p < 1)

and

\lambda

is the normalized Lebesgue measure on

[0 ,\lfloor \beta \rfloor /(\beta -1)]

. Furthermore, this measure is of the form

m_p \otimes \mu_{\beta,p}

, where

\mu_{\beta,p}

is equivalent with

\lambda

. We establish the fact that the measure of maximal entropy and

m_p \otimes \lambda

are mutually singular. In case

1

has a finite greedy expansion with positive coefficients, the measure

m_p \otimes \mu_{\beta,p}

is Markov. In the last section we answer a question concerning the number of universal expansions, a notion introduced in [EK].
Necessary and sufficient conditions for the chain rule in $W_{\mathrm{loc}}^{1,1}(\mathbb{R}^N; \mathbb{R})$ and $BV_{\mathrm{loc}}(\mathbb{R}^N; \mathbb{R}^d)$
Authors: Leoni Giovanni , Morini Massimiliano
Published: December 2009

In this paper we prove necessary and sufficient conditions for the validity of the classical chain rule in Sobolev spaces and in the space of functions of bounded variation.
Vortex collisions and energy-dissipation rates in the Ginzburg–Landau heat flow. Part II: The dynamics
Authors: Serfaty Sylvia
Published: December 2009

We deduce from the first part of this paper [S1] estimates on the energy-dissipation rates for solutions of the Ginzburg–Landau heat-flow, which allow to study various phenomena occurring in this flow, among which vortex-collisions; allowing in particular to extend the dynamical law of vortices passed collisions.
One cannot hear the winding number
Authors: Bourgain Jean , Kozma Gady
Published: December 2009

We construct an example of two continuous maps f and g of the circle to itself with the same absolute value of the Fourier transform but with different winding numbers, answering a question of Brezis.
Maximizers for the Strichartz Inequality
Authors: Foschi Damiano
Published: December 2009

We compute explicitly the best constants and, by solving some functional equations, we find all maximizers for homogeneous Strichartz estimates for the Schrödinger equation and for the wave equation in the cases when the Lebesgue exponent is an even integer.
Hamiltonicity of cubic Cayley graphs
Authors: Glover Henry H. , Marusic Dragan
Published: December 2009

Following a problem posed by Lovász in 1969, it is believed that every finite connected vertex-transitive graph has a Hamilton path. This is shown here to be true for cubic Cayley graphs arising from finite groups having a

(2,s,3)

-presentation, that is, for groups

G=\langle a,b\mid a^2=1, b^s=1, (ab)^3=1, \dots \rangle

generated by an involution

a

and an element

b

of order

s\geq3

such that their product

ab

has order

3

. More precisely, it is shown that the Cayley graph

X=\operatorname{Cay}(G,\{a,b,b^{-1}\})

has a Hamilton cycle when

|G|

(and thus

s

) is congruent to

2

modulo

4

, and has a long cycle missing only two adjacent vertices (and thus necessarily a Hamilton path) when

|G|

is congruent to

0

modulo

4

.
On the number of components of the symplectic representatives of the canonical class
Authors: Vidussi Stefano
Published: December 2009

In this paper we show that there exists a family of simply connected, symplectic 4-manifolds such that the (Poincar\'e dual of the) canonical class admits both connected and disconnected symplectic representatives. This answers a question of Fintushel and Stern.
A general Fredholm theory I: a splicing-based differential geometry
Authors: Hofer Helmut W. , Wysocki Kris , Zehnder Eduard
Published: December 2009

This is the first paper in a series introducing a generalized Fredholm theory in a new class of smooth spaces called polyfolds. These spaces, in general, are locally not homeomorphic to open sets in Banach spaces. The present paper describes some of the differential geometry of this new class of spaces. The theory will be illustrated in upcoming papers by applications to Floer Theory, Gromov-Witten Theory, and Symplectic Field Theory.
Quantum field theory on projective modules
Authors: Gayral Victor , Jureit Jan-Hendrik , Krajewski Thomas , Wulkenhaar Raimar
Published: December 2009

We propose a general formulation of perturbative quantum field theory on (finitely generated) projective modules over noncommutative algebras. This is the analogue of scalar field theories with non-trivial topology in the noncommutative realm. We treat in detail the case of Heisenberg modules over noncommutative tori and show how these models can be understood as large rectangular

p × q

matrix models, in the limit

p/q → θ

, where

θ

is a possibly irrational number. We find out that the model is highly sensitive to the number theoretical aspect of

θ

and suffers from an UV/IR-mixing. We give a way to cure the entanglement and prove 1-loop renormalisability.
Hopf algebra of non-commutative field theory
Authors: Tanasă Adrian , Vignes-Tourneret Fabien
Published: December 2009

We construct here the Hopf algebra structure underlying the process of renormalization of non-commutative quantum field theory.
The Chern character of semifinite spectral triples
Authors: Carey Alan L. , Phillips John , Rennie Adam , Sukochev Fyodor A.
Published: December 2009

In previous work we generalised both the odd and even local index formula of Connes and Moscovici to the case of spectral triples for a *-subalgebra

\mathcal{A}

of a general semifinite von Neumann algebra. Our proofs are novel even in the setting of the original theorem and rely on the introduction of a function valued cocycle (called the resolvent cocycle) which is ‘almost’ a (b,B)-cocycle in the cyclic cohomology of

\mathcal{A}

. In this paper we show that this resolvent cocycle ‘almost’ represents the Chern character and assuming analytic continuation properties for zeta functions we show that the associated residue cocycle, which appears in our statement of the local index theorem does represent the Chern character.
The universal Hopf-cyclic theory
Authors: Kaygun Atabey
Published: December 2009

We define a Hopf cyclic (co)homology theory in an arbitrary symmetric strict monoidal category. Thus we unify all different types of Hopf cyclic (co)homologies under one single universal theory. We recover Hopf cyclic (co)homology of module algebras, comodule algebras and module coalgebras along with Hopf–Hochschild (co)homology of module algebras, and describe the missing theory for comodule coalgebras.
Cohomology of Yang–Mills algebras
Authors: Movshev Michael
Published: December 2009

In this paper we compute cyclic and Hochschild homology of the universal envelope

U(YM)

of the Yang–Mills Lie algebra

YM

. We also compute Hochschild cohomology with coefficients in

U(YM)

, considered as a bimodule over itself.

The result of the calculations depends on the number of generators

n

of

YM

. The semidirect product

\mathfrak{so}(n) ⋉ ℂ^n

acts by derivations upon

U(YM)

. One of the important consequences of our results is that if

n ≥ 3

then the Lie algebra of outer derivations of

U(YM)

coincides with

\mathfrak{so}(n) ⋉ ℂ^n

.
Quantization of projective homogeneous spaces and duality principle
Authors: Ciccoli Nicola , Fioresi Rita , Gavarini Fabio
Published: December 2009

We introduce a general recipe to construct quantum projective homogeneous spaces, with a particular interest for the examples of the quantum Grassmannians and the quantum generalized flag varieties. Using this construction, we extend the quantum duality principle to quantum projective homogeneous spaces.