Singularities of the Hamiltonian vectorfield in nonautonomous variational problems

Variational problems with n degrees of freedom give rise (by Pontriaguine maximum principle) to a Hamiltonian vectorfield in , that presents singularities (nonsmoothness points) when the Lagrangian is not convex. For one degree of freedom nonautonomous problems of the calculus of variations where the Hamiltonian vectorfield in depends explicitly on the time, we consider the associated autonomous vectorfield in and classify its singularities up to an equivalence that takes into account the special role played by the time coordinate, i.e., that respects the foliation of into planes of constant time.     

Holomorphic foliations desingularized by punctual blow-ups

Given be a germ of codimension-one singular holomorphic foliation at the origin . We assume that can be desingularized by a certain sequence of punctual blow-ups producing only simple singularities (Definition 1). This case is studied in analogy with the case of Kleinian singularities of complex surfaces. It is proved that is given by a simple poles closed meromorphic 1-form provided that, along the reduction process, the simple singularities exhibit a hyperbolic transverse type (Theorem 3). In the non-hyperbolic case, we prove the existence of a formal integrating factor if we interdict the existence of holomorphic first integrals for the transverse types (Theorem 4). The proof relies strongly on a result of Deligne regarding the fundamental group of the complement of algebraic curves in the complex projective plane.     

Solitary charged waves interacting with the electrostatic field

In this paper we study the semiclassical limit for the following system of Schrödinger-Maxwell equations in the unit ball B₁ of R³:

     

Quadratic addition rules for quantum integers

For every positive integer n, the quantum integer [n]_q is the polynomial [n]_q=1+q+q²+?+q^n-1. A quadratic addition rule for quantum integers consists of sequences of polynomials , , and such that for all m and n. This paper gives a complete classification of quadratic addition rules, and also considers sequences of polynomials that satisfy the associated functional equation .     

On codimension one foliations with Morse singularities on three-manifolds

A closed, connected oriented three-manifold supporting a codimension one oriented smooth foliation with Morse singularities having more centers than saddles and without saddle connections is diffeomorphic to the three-sphere. The use of the Reeb Stability theorem in place of the Poincaré-Bendixson theorem paves the way to a three-dimensional version, for foliations with singularities of Morse type, of a classical result of Haefliger. Finally, we give an example of a codimension one C^∞ foliation in the closed ball , with only one singularity which is of saddle type 2-2 and transverse to the boundary S³=∂B⁴.     

Various types of stochastic integrals with respect to fractional Brownian sheet and their applications

In this paper, we develop a stochastic calculus related to a fractional Brownian sheet as in the case of the standard Brownian sheet. Let be a fractional Brownian sheet with Hurst parameters H=(H₁,H₂), and (²[0,1],B(²[0,1]),μ) a measure space. By using the techniques of stochastic calculus of variations, we introduce stochastic line integrals along all sufficiently smooth curves γ in ²[0,1], and four types of stochastic surface integrals: , i=1,2, , , , . As an application of these stochastic integrals, we prove an Itô formula for fractional Brownian sheet with Hurst parameters H₁,H₂∈(1/4,1). Our proof is based on the repeated applications of Itô formula for one-parameter Gaussian process.     

Exact solutions and branching of singularities for some hyperbolic equations in two variables

Exact global propagators are constructed for the singular hyperbolic operators in two variables , λ a real parameter, and for the degenerate hyperbolic operators . Qualitative phenomena such as uniqueness in the Cauchy problem and branching of singularities vary with λ, as shown earlier by Treves and by Taniguchi and Tozaki.     

A moduli scheme of embedded curve singularities

The main purpose of this paper is to prove the existence of the moduli space parameterizing the embedded curve singularities of with an admissible Hilbert polynomial p and to study its basic properties.     

Quartic, octic residues and Lucas sequences

Let be a prime and a,b∈Z with a²+b²≠p. Suppose p=x²+(a²+b²)y² for some integers x and y. In the paper we develop the calculation technique of quartic Jacobi symbols and use it to determine . As applications we obtain the congruences for modulo p and the criteria for (if ), where {Un} is the Lucas sequence given by U₀=0, U₁=1 and Un+1=bUn+k²Un−1(n?1). We also pose many conjectures concerning , or .     

U(1)-invariant special Lagrangian 3-folds. III. Properties of singular solutions

This is the last of three papers studying special Lagrangian 3-submanifolds (SLV 3-folds) N in invariant under the U(1)-action e^iθ:(z₁,z₂,z₃)?(e^iθz₁,e^iθz₂,z₃), using analytic methods. If N is such a 3-fold then |z₁|²−|z₂|²=2a on N for some . Locally, N can be written as a kind of graph of functions satisfying a nonlinear Cauchy-Riemann equation depending on a, so that u+iv is like a holomorphic function of x+iy.The first paper studied the case a nonzero, and proved existence and uniqueness for solutions of two Dirichlet problems derived from the nonlinear Cauchy-Riemann equation. This yields existence and uniqueness of a large class of nonsingular U(1)-invariant SL 3-folds in , with boundary conditions. The second paper extended these results to weak solutions of the Dirichlet problems when a=0, giving existence and uniqueness of many singular U(1)-invariant SL 3-folds in , with boundary conditions.This third paper studies the singularities of these SL 3-folds. We show that under mild conditions the singularities are isolated, and have a multiplicityn>0, and one of two types. Examples are constructed with every multiplicity and type. We also prove the existence of large families of U(1)-invariant special Lagrangian fibrations of open sets in , including singular fibres.     

Continuity and Schatten-von Neumann p-class membership of Hankel operators with anti-holomorphic symbols on (generalized) Fock spaces

In this paper we investigate Hankel operators with anti-holomorphic symbols ∈L²(C,m|z|), where are general Fock spaces. We will show that is not continuous if the corresponding symbol is not a polynomial . For polynomial symbols we will give necessary and sufficient conditions for continuity and compactness in terms of N and m. For monomials we will give a complete characterization of the Schatten-von Neumann p-class membership for p>0. Namely in case 2k<m the Hankel operators are in the Schatten-von Neumann p-class iff p>2m/(m−2k); and in case 2k?m they are not in the Schatten-von Neumann p-class.     

Polynomial approximation, local polynomial convexity, and degenerate CR singularities

We begin with the following question: given a closed disc and a complex-valued function , is the uniform algebra on generated by z and F equal to ? When F∈C¹(D), this question is complicated by the presence of points in the surface that have complex tangents. Such points are called CR singularities. Let p∈S be a CR singularity at which the order of contact of the tangent plane with S is greater than 2; i.e. a degenerate CR singularity. We provide sufficient conditions for S to be locally polynomially convex at the degenerate singularity p. This is useful because it is essential to know whether S is locally polynomially convex at a CR singularity in order to answer the initial question. To this end, we also present a general theorem on the uniform algebra generated by z and F, which we use in our investigations. This result may be of independent interest because it is applicable even to non-smooth, complex-valued F.     

Fourier analysis on domains in compact groups

Let Ω be a measurable subset of a compact group G of positive Haar measure. Let be a non-negative function defined on the dual space and let L²(μ) be the corresponding Hilbert space which consists of elements (ξπ)_π∈suppμ satisfying , where ξπ is a linear operator on the representation space of π, and is equipped with the inner product: . We show that the Fourier transform gives an isometric isomorphism from L²(Ω) onto L²(μ) if and only if the restrictions to Ω of all matrix coordinate functions , π∈suppμ, constitute an orthonormal basis for L²(Ω). Finally compact connected Lie groups case is studied.     

Spectral properties of a Schrödinger equation with a class of complex potentials and a general boundary condition

In this paper we investigate the spectrum and the spectral singularities of an operator L generalized in by the differential expression

     

Large deviations for martingales and derivatives

Fix a sequence of positive integers (mn) and a sequence of positive real numbers (wn). Two closely related sequences of linear operators (Tn) are considered. One sequence has given by the Lebesgue derivatives . The other sequence has given by the dyadic martingale when (l−1)/n²?x<l/n² for l=1,…,n². We prove both positive and negative results concerning the convergence of .     

Monoids with sub-log-exponential free spectra

A classic result from the 1960s states that the asymptotic growth of the free spectrum of a finite group is sub-log-exponential if and only if is nilpotent. Thus a monoid is sub-log-exponential implies , the pseudovariety of semigroups with nilpotent subgroups. Unfortunately, little more is known about the boundary between the sub-log-exponential and log-exponential monoids.The pseudovariety consists of those finite semigroups satisfying (x^ωy^ω)^ω(y^ωx^ω)^ω(x^ωy^ω)^ω≈(x^ωy^ω)^ω. Here it is shown that a monoid is sub-log-exponential implies . A quick application: a regular sub-log-exponential monoid is orthodox. It is conjectured that a finite monoid is sub-log-exponential if and only if it is , the finite monoids in having nilpotent subgroups. The forward direction of the conjecture is proved; moreover, the conjecture is proved for when is completely (0)-simple. In particular, the six-element Brandt monoid (the Perkins semigroup) is sub-log-exponential.