Portugaliae Mathematica

In the same way that a contact manifold determines and is determined by a symplectic cone, a Sasaki manifold determines and is determined by a suitable Kahler cone. Kahler-Sasaki geometry is the geometry of these cones. This paper presents a symplectic action-angle coordinates approach to toric Kahler geometry and how it was recently generalized, by Burns-Guillemin-Lerman and Martelli-Sparks-Yau, to toric Kahler-Sasaki geometry. It also describes, as an application, how this approach can be used to relate a recent new family of Sasaki-Einstein metrics constructed by Gauntlett-Martelli-Sparks-Waldram in 2004, to an old family of extremal Kahler metrics constructed by Calabi in 1982. Comment: 20 pages, 1 figure, added references and improved exposition. To appear in Portugaliae Mathematica

There is a standard "word length" metric canonically associated to any set of generators for a group. In particular, for any integers a and b greater than 1, the additive group of integers has generating sets {a^i}_{i=0}^{\infty} and {b^j}_{j=0}^{\infty} with associated metrics d_A and d_B, respectively. It is proved that these metrics are bi-Lipschitz equivalent if and only if there exist positive integers m and n such that a^m = b^n.

For two general polytopal complexes the set of face-wise affine maps between them is shown to be a polytopal complex in an algorithmic way. The resulting algorithm for the affine hom-complex is analyzed in detail. There is also a natural tensor product of polytopal complexes, which is the left adjoint functor for Hom(-,-). This extends the corresponding facts from single polytopes, systematic study of which was initiated in [5,11]. Explicit examples of computations of the resulting structures are included. In the special case of simplicial complexes, the affine hom-complex is a functorial subcomplex of Kozlov's combinatorial hom-complex [13], which generalizes Lovasz' well-known construction [14] for graphs.

We construct a family of irreducible representations of the quantum plane and of the quantum Weyl algebra over an arbitrary field, assuming the deformation parameter is not a root of unity. We determine when two representations in this family are isomorphic, and when they are weight representations, in the sense of Bavula.

We show that the path construction integration of Lie algebroids by Lie groupoids is an actual equivalence from the category of integrable Lie algebroids and complete Lie algebroid comorphisms to the category of source 1-connected Lie groupoids and Lie groupoid comorphisms. This allows us to construct an actual symplectization functor in Poisson geometry. We include examples to show that the integrability of comorphisms and Poisson maps may not hold in the absence of a completeness assumption.

We prove the uniqueness of solutions of the Maxwell-Schr"odinger system with given asymptotic behaviour at infinity in time. The assumptions include suitable restrictions on the growth of solutions for large time and on the accuracy of their asymptotics, but no restriction on their size. The result applies to the solutions with prescribed asymptotics constructed in a previous paper.

We classify the most common local forms of smooth maps from a smooth manifold L to the plane. The word "local" can refer to locations in the source L, but also to locations in the target. The first point of view leads us to a classification of certain germs of maps, which we review here although it is very well known. The second point of view leads us to a classification of certain multigerms of maps.

The tropical variety defined by linear equations with constant coefficients is the Bergman fan of the corresponding matroid. Building on a self-contained introduction to matroid polytopes, we present a geometric construction of the Bergman fan, and we discuss its relationship with the simplicial complex of nested sets in the lattice of flats. The Bergman complex is triangulated by the nested set complex, and the two complexes coincide if and only if every connected flat remains connected after contracting along any subflat. This sharpens a result of Ardila-Klivans who showed that the Bergman complex is triangulated by the order complex of the lattice of flats. The nested sets specify the De Concini-Procesi compactification of the complement of a hyperplane arrangement, while the Bergman fan specifies the tropical compactification. These two compactifications are almost equal, and we highlight the subtle differences.

We present type preserving bijections between noncrossing and nonnesting partitions for all classical reflection groups, answering a question of Athanasiadis and Reiner. The bijections for the abstract Coxeter types B, C and D are new in the literature. To find them we define, for every type, sets of statistics that are in bijection with noncrossing and nonnesting partitions, and this correspondence is established by means of elementary methods in all cases. The statistics can be then seen to be counted by the generalized Catalan numbers Cat(W) when W is a classical reflection group. In particular, the statistics of type A appear as a new explicit example of objects that are counted by the classical Catalan numbers. © 2009 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France.

This short note is a sequel to our previous papers on the asymptotic behavior of Bloch wave packet solutions of the wave equation in periodic media. The purpose is to prove that the group velocity for these Bloch wave packets is bounded by the maximal speed of propagation for the original wave equation.

Variational approaches have been used successfully as a strategy to take advantage from real data measurements. In several applications, this approach gives a means to increase the accuracy of numerical simulations. In the particular case of fluid dynamics, it leads to optimal control problems with non standard cost functionals which, when constraint to the Navier-Stokes equations, require a non-standard theoretical frame to ensure the existence of solution. In this work, we prove the existence of solution for a class of such type of optimal control problems. Before doing that, we ensure the existence and uniqueness of solution for the 3D stationary Navier-Stokes equations, with mixed-boundary conditions, a particular type of boundary conditions very common in applications to biomedical problems.

The Krohn-Rhodes complexity of the Brauer type semigroups Bn and An is computed. In three-quarters of the cases the result is the 'expected' one: the complexity coincides with the (essential) J-depth of the respective semigroup. The exception (and perhaps the most interesting case) is the annular semigroup A2n of even degree in which case the complexity is the J-depth minus 1. For the 'rook' versions PBn and PAn it is shown that c(PBn) = c(Bn) and c(PA2n-1) = c(A2n-1) for all n≥1. The computation of c(PA2n) is left as an open problem.

We provide an example of a trivalent, 3-connected graph G such that, for any choice of metric on G, the resulting metric graph is Brill-Noether special.

For any C 1 -expanding map f of the circle we study the equilibrium states for the potential ψ=-log|f ' |. We formulate a C 1 generalization of Pesin’s Entropy Formula that holds for all the Sinai-Ruelle-Bowen (SRB) measures if they exist, and for all the (necessarily existing) SRB-like measures. In the C 1 -generic case Pesin’s Entropy Formula holds for a unique SRB measure which is not absolutely continuous with respect to Lebesgue. The result also stands in the non-generic case for which no SRB measure exists.

We prove interpolation estimates between Morrey-Campanato spaces and Sobolev spaces. These estimates give in particular concentration-compactness inequalities in the translation-invariant and in the translation- and dilation-invariant case. They also give in particular interpolation estimates between Sobolev spaces and functions of bounded mean oscillation. The proofs rely on Sobolev integral representation formulae and maximal function theory. Fractional Sobolev spaces are also covered.

Quantization problems suggest that the category of symplectic manifolds and symplectomorphisms be augmented by the inclusion of canonical relations as morphisms. These relations compose well when a transversality condition is satisfied, but the failure of the most general compositions to be smooth manifolds means that the canonical relations do not comprise the morphisms of a category. We discuss several existing and potential remedies to the nontransversality problem. Some of these involve restriction to classes of lagrangian submanifolds for which the transversality property automatically holds. Others involve allowing lagrangian "objects" more general than submanifolds. Comment: 16 pages, for proceedings of Geometry Summer School, Lisbon, July 2009

We consider the Cauchy problem for nonlinear Schrodinger equations in the presence of a smooth, possibly unbounded, potential. No assumption is made on the sign of the potential. If the potential grows at most linearly at infinity, we construct solutions in Sobolev spaces (without weight), locally in time. Under some natural assumptions, we prove that the $H^1$-solutions are global in time. On the other hand, if the potential has a super-linear growth, then the Sobolev regularity of positive order is lost instantly, not matter how large it is, unless the initial datum decays sufficiently fast at infinity.

For a local Lie group M we define odd order cohomology classes. The first class is an obstruction to globalizability of the local Lie group. The third class coincides with Godbillon-Vey class in a particular case. These classes are secondary as they emerge when curvature vanishes. Comment: To appear in Portugaliae Mathematicae

The aim of this paper is to continue the study of sg-compact spaces, a topological notion much stronger than hereditary compactness. We investigate the relations between sg-compact and $C_2$-spaces and the interrelations to hereditarily sg-closed sets.

In 2013, Abo and Wan studied the analogue of Waring's problem for systems of skew-symmetric forms and identified several defective systems. Of particular interest is when a certain secant variety of a Segre-Grassmann variety is expected to fill the natural ambient space, but is actually a hypersurface. Algorithms implemented in Bertini are used to determine the degrees of several of these hypersurfaces, and representation-theoretic descriptions of their equations are given. We answer Problem 6.5 [Abo-Wan2013], and confirm their speculation that each member of an infinite family of hypersurfaces is minimally defined by a (known) determinantal equation. While led by numerical evidence, we provide non-numerical proofs for all of our results.

We consider variational inequality solutions with prescribed gradient constraints for first order linear boundary value problems. For operators with coefficients only in $L^2$, we show the existence and uniqueness of the solution by using a combination of parabolic regularization with a penalization in the nonlinear diffusion coefficient. We also prove the continuous dependence of the solution with respect to the data, as well as, in a coercive case, the asymptotic stabilization as time $t\rightarrow+\infty$ towards the stationary solution. In a particular situation, motivated by the transported sandpile problem, we give sufficient conditions for the equivalence of the first order problem with gradient constraint with a two obstacles problem, the obstacles being the signed distances to the boundary. This equivalence, in special conditions, illustrates also the possible stabilization of the solution in finite time.

We prove strong convergence of a semi-discrete finite difference method for the KdV and modified KdV equations. We extend existing results to non-smooth data (namely, in $L^2$), without size restrictions. Our approach uses a fourth order (in space) stabilization term and a special conservative discretization of the nonlinear term. Convergence follows from a smoothing effect and energy estimates. We illustrate our results with numerical experiments, including a numerical investigation of an open problem related to uniqueness posed by Y. Tsutsumi.

Following Geroch, Traschen, Mars and Senovilla, we consider Lorentzian manifolds with distributional curvature tensor. Such manifolds represent spacetimes of general relativity that possibly contain gravitational waves, shock waves, and other singular patterns. We aim here at providing a comprehensive and geometric (i.e., coordinate-free) framework. First, we determine the minimal assumptions required on the metric tensor in order to give a rigorous meaning to the spacetime curvature within the framework of distribution theory. This leads us to a direct derivation of the jump relations associated with singular parts of connection and curvature operators. Second, we investigate the induced geometry on a hypersurface with general signature, and we determine the minimal assumptions required to define, in the sense of distributions, the curvature tensors and the second fundamental form of the hypersurface and to establish the Gauss-Codazzi equations.

Elements of the tropical vertex group are formal families of symplectomorphisms of the 2-dimensional algebraic torus. Commutators in the group are related to Euler characteristics of the moduli spaces of quiver representations and the Gromov-Witten theory of toric surfaces. After a short survey of the subject (based on lectures of Pandharipande at the 2009 Geometry summer school in Lisbon), we prove new results about the rays and symmetries of scattering diagrams of commutators (including previous conjectures by Gross-Siebert and Kontsevich). Where possible, we present both the quiver and Gromov-Witten perspectives. Comment: 43 pages

In our paper we give a characterization of (set-valued) maps which are minimal usco and minimal cusco simultaneously. Let X be a topological space and Y be a Banach space. We show that there is a bijection between the space MU(X, Y) of minimal usco maps from X to Y and the space MC(X, Y) of minimal cusco maps from X to Y, and we study this bijection with respect to various topologies on underlying spaces. Let X be a Baire space and Y be a Banach space. Then (MU(X, Y), tau(U)) and (MC(X, Y), tau(U)) are homeomorphic, where tau(U) is the topology of uniform convergence.

We study evolution equations governed by an averaging operator on a directed tree, showing existence and uniqueness of solutions. In addition we find conditions of the initial condition that allows us to find the asymptotic decay rate of the solutions as $t\to \infty$. It turns out that this decay rate is not uniform, it strongly depends on how the initial condition goes to zero as one goes down in the tree.

We obtain global and local theorems on the existence of invariant manifolds for perturbations of non autonomous linear difference equations assuming a very general form of dichotomic behavior for the linear equation. The results obtained include situations where the behavior is far from hyperbolic. We also give several new examples and show that our result includes as particular cases several previous theorems.

Here we prove that the homological dimension of the category of sheaves on a topological space satisfying some suitable conditions is finite. In particular, we find conditions to bound the homological dimension of o-minimal and subanalytic sheaves. Comment: 10 pages, uses xy-pic

We consider nonclassical entropy solutions to scalar conservation laws with concave-convex flux functions, whose set of left- and right-hand admissible states across undercompressive shocks is selected by a kinetic function \phi. We introduce a new definition for the (generalized) strength of classical and nonclassical shocks, allowing us to propose a generalized notion of total variation functional. Relying only upon the natural assumption that the composite function \phi o \phi is uniformly contracting, we prove that the generalized total variation of front-tracking approximations is non-increasing in time, and we conclude with the existence of nonclassical solutions to the initial-value problem. We also propose two definitions of generalized interaction potentials which are adapted to handle nonclassical entropy solutions and we investigate their monotonicity properties. In particular, we exhibit an interaction functional which is globally non-increasing along a splitting-merging interaction pattern.

It is shown that the nonstandard representatives of Schwartz-distributions, as introduced by K.D. Stroyan and W.A.J. Luxemburg in their book 'Introduction to the theory of infinitesimals', are locally equal to a finite-order derivative of a finite-valued and S-continuous function. By 'equality', we mean a pointwise equality, not an equality in a distributional sense. This proves a conjecture by M. Oberguggenberger in [Z. Anal. Anwend. 10 (1991), 263-264]. Moreover, the representatives of the zero-distribution are locally equal to a finite-order derivative of a function assuming only infinitesimal values. These results also unify the nonstandard theory of distributions by K.D. Stroyan and W.A.J. Luxemburg with the theory by R.F. Hoskins and J. Sousa Pinto in [Portugaliae Mathematica 48(2), 195-216].

Consider a Hamiltonian system of type \[ -\Delta u=H_{v}(u,v),\ -\Delta v=H_{u}(u,v) \ \ \text{ in } \Omega, \qquad u,v=0 \text{ on } \partial \Omega \] where $H$ is a power-type nonlinearity, for instance $H(u,v)= |u|^p/p+|v|^q/q$, having subcritical growth, and $\Omega$ is a bounded domain of $\mathbb{R}^N$, $N\geq 1$. The aim of this paper is to give an overview of the several variational frameworks that can be used to treat such a system. Within each approach, we address existence of solutions, and in particular of ground state solutions. Some of the available frameworks are more adequate to derive certain qualitative properties; we illustrate this in the second half of this survey, where we also review some of the most recent literature dealing mainly with symmetry, concentration, and multiplicity results. This paper contains some original results as well as new proofs and approaches to known facts.

Necessary and sufficient conditions are given for the endomorphism monoid of a profinite semigroup to be profinite. A similar result is established for the automorphism group.

In this paper we study the effect of a homoclinic tangency in the variation of the topological entropy. We prove that a diffeomorphism with a homoclinic tangency associated to a basic hyperbolic set with maximal entropy is a point of entropy variation in the $C^{\infty}$-topology. We also prove results about variation of entropy in other topologies and when the tangency does not correspond to a basic set with maximal entropy. We also show an example of discontinuity of the entropy among $C^{\infty}$ diffeomorphisms of three dimensional manifolds.

This paper deals with well-known notion of $PF$-rings, that is, rings in which principal ideals are flat. We give a new characterization of $PF$-rings. Also, we provide a necessary and sufficient condition for $R\bowtie I$ (resp., $R/I$ when $R$ is a Dedekind domain or $I$ is a primary ideal) to be $PF$-ring. The article includes a brief discussion of the scope and precision of our results.

In this paper we are concerned with the space of tempered ultrahyperfunctions corresponding to a proper open convex cone. A holomorphic extension theorem (the version of the celebrated edge of the wedge theorem) will be given for this setting. As application, a version is also given of the principle of determination of an analytic function by its values on a non-empty open real set. The paper finishes with the generalization of holomorphic extension theorem \`a la Martineau.

We give a self contained and elementary description of normal forms for symplectic matrices, based on geometrical considerations. The normal forms in question are expressed in terms of elementary Jordan matrices and integers with values in $\{-1,0,1\}$ related to signatures of quadratic forms naturally associated to the symplectic matrix.

We introduce a generalization of Glimm's random choice method, which provides us with an approximation of entropy solutions to quasilinear hyperbolic system of balance laws. The flux-function and the source term of the equations may depend on the unknown as well as on the time and space variables. The method is based on local approximate solutions of the generalized Riemann problem, which form building blocks in our scheme and allow us to take into account naturally the effects of the flux and source terms. To establish the nonlinear stability of these approximations, we investigate nonlinear interactions between generalized wave patterns. This analysis leads us to a global existence result for quasilinear hyperbolic systems with source-term, and applies, for instance, to the compressible Euler equations in general geometries and to hyperbolic systems posed on a Lorentzian manifold.

We give a brief exposition of the 2d TQFT that captures the structure of the GL Verlinde numbers, following Witten.

This is a concise introduction to the theory of Lie groupoids, with emphasis in their role as models for orbispaces. After some preliminaries, we review the foundations on Lie groupoids, and we carefully study equivalences and proper groupoids. Orbispaces are geometric objects which have manifolds and orbifolds as special instances, and can be presented as the transverse geometry of a Lie groupoid. Two Lie groupoids are equivalent if they are presenting the same orbispace, and proper groupoids are presentations of separated orbispaces, which by the linearization theorem are locally modeled by linear actions of compact groups. We discuss all these notions in detail. Our treatment diverges from the expositions already in the literature, looking for a complementary insight over this rich theory that is still in development.

We study natural variations of the G2 structure {\sigma}_0 \in {\Lambda}^3_+ existing on the unit tangent sphere bundle SM of any oriented Riemannian 4-manifold M. We find a circle of structures for which the induced metric is the usual one, the so-called Sasaki metric, and prove how the original structure has a preferred role in the theory. We deduce the equations of calibration and cocalibration, as well as those of W3 pure type and nearly-parallel type.

We calculate the hyperelliptic Hodge integral lambda_g lambda_{g-1} / (1 - psi) for use in arXiv:math/0702219. The proof uses the WDVV equations for the genus zero Gromov--Witten invariants of P(1,1,2).

Motivated by recent appearance of multivalued structures in categorification, tropical geometry and other areas, we study basic properties of abstract multisemigroups. We give many new and old examples and general constructions for multisemigroups. Special attention is paid to simple and nilpotent multisemigroups. We also show that "almost all" randomly chosen multivalued binary operations define multisemigroups.

We introduce a class of finite semigroups obtained by considering Rees quotients of numerical semigroups. Several natural questions concerning this class, as well as particular subclasses obtained by considering some special ideals, are answered while others remain open. We exhibit nice presentations for these semigroups and prove that the Rees quotients by ideals of N, the positive integers under addition, constitute a set of generators for the pseudovariety of commutative and nilpotent semigroups.

For a map $f:I \rightarrow I$, a point $x \in I$ is periodic with period $p \in \mathbb{N}$ if $f^p(x)=x$ and $f^j(x)\not=x$ for all $0<j<p$. When $f$ is continuous and $I$ is an interval, a theorem due to Sharkovskii (\cite{BC}) states that there is an order in $\mathbb{N}$, say $\lhd$, such that, if $f$ has a periodic point of period $p$ and $p \lhd q$, then $f$ also has a periodic point of period $q$. In this work, we will see how an extension of this order $\lhd$ to an ultrapower of the integer numbers yields a Sharkovskii-type result for non-wandering points of $f$.

We study the tropicalizations of Severi varieties, which we call tropical Severi varieties. In this paper, we give a partial answer to the following question, ``describe the tropical Severi varieties explicitly.'' We obtain a description of tropical Severi varieties in terms of regular subdivisions of polygons. As an intermediate step, we construct explicit parameter spaces of curves. These parameter spaces are much simpler objects than the corresponding Severi variety and they are closely related to flat degenerations of the Severi variety, which in turn describes the tropical Severi variety. As an application, we understand G.Mikhalkin's correspondence theorem for the degrees of Severi varieties in terms of tropical intersection theory. In particular, this provides a proof of the independence of point-configurations in the enumeration of tropical nodal curves.

We consider a stationary free boundary problem for the Navier-Stokes equations governing effluence of a viscous incompressible liquid out of unbounded non-expanding at infinity, in general, non-symmetric strip-like domainOmega Gamma outside which the liquid forms a sector-like jet with free (unknown) boundary and with the limiting opening ` 2 (0; =2). Conditions at the free boundary take account of the capillary forces but external forces are absent. The total flux of the liquid through arbitrary cross-section ofOmega Gamma is prescribed and assumed to be small. Under this condition, we prove the existence of an isolated solution of the problem which is found in a certain weighted Holder space of functions. x1. Introduction In the paper [1] there was considered a symmetric viscous flow through an aperture in an infinite straight line. In the present paper we remove the condition of symmetry. We assume that the domainOmega ae R 2 filled with the liquid consists of two parts,O...

In this paper we generalize the theorems of exact controllability for the linear wave equation with a distributed control to the semilinear case, showing that, given T large enough, for every initial state in a su#ciently small neighbourhood of the origin in a certain function space, there exists a distributed control, supported on a part of a domain, driving the system to rest. Also, if the control is allowed to support on the entire domain, then we prove that the system is globally exactly controllable at any time T . Key Words: Distributed controllability; semilinear wave equation. AMS subject classification: 93B05, 35B37, 35L05. # Supported by the Killam Postdoctoral Fellowship. 1 1 Introduction The main purpose of this paper is to generalize the theorems of exact controllability for the linear wave equation to the following semilinear case # # # # # y ## -#y + f(y) = h in Q, y(x, 0) = y 0 , y # (x, 0) = y 1 in# , y = 0 on #, (1.1) In (1.1),# is a bounded domain (none...

In this article we prove approximation formulae for a class of unitary evolution operators $U (t,s)_{s,t \epsilon [0,T]}$ associated with linear non-autonomous evolution equations of Schrödinger type de…ned in a Hilbert space $\mathcal{H}$. An important feature of the equations we consider is that both the corresponding self-adjoint generators and their domains may depend explicitly on time, whereas the associated quadratic form domains may not. Furthermore the evolution operators we are interested in satisfy the equations in a weak sense. Under such conditions the approximation formulae we prove for $U(t,s)$ involve weak operator limits of products of suitable approximating functions taking values in $\mathcal{L}(\mathcal{H})$, the algebra of all linear bounded operators on $\mathcal{H}$. Our results may be relevant to the numerical analysis of $U(t,s)$ and we illustrate them by considering two evolution problems in quantum mechanics.

In this paper, we consider two classes $A$ and $B$ of the family of integral functions defined b Dirichlet series of finite order. We establish some results concerning the integral functions represented by Dirichlet series of finite order $\rho_1$ and $\rho_2$ with respect to variables $s_1$ and $s_2$.