On the Hilbert function of Artinian local complete intersections of codimension three

In singularity theory or algebraic geometry, it is natural to investigate possible Hilbert functions for special algebras A such as local complete intersections or more generally Gorenstein algebras. The sequences that occur as the Hilbert functions of standard graded complete intersections are well understood classically thanks to Macaulay and Stanley. Very little is known in the local case except in codimension two. In this paper we characterise the Hilbert functions of quadratic Artinian complete intersections of codimension three. Interestingly we prove that a Hilbert function is admissible for such a Gorenstein ring if and only if is admissible for such a complete intersection. We provide an effective construction of a local complete intersection for a given Hilbert function. We prove that the symmetric decomposition of such a complete intersection ideal is determined by its Hilbert function.     

F-rationality of certain Rees algebras and counterexamples to “Boutot's Theorem” for F-rational rings

F-rational rings are defined for rings of characteristic p > 0 using the Frobenius endomorphism and corresponds to rational singularities in characteristic 0. We study F-rationality of certain Rees algebras and prove that every Cohen-Macaulay local ring with isolated singularity and negative a-invariant has a Rees algebra which is F-rational. As a consequence, we find that “Boutot's Theorem” asserting that a pure subring of a rational singularity is a rational singularity is not true for a F-rational ring.     

On the oscillation of Hill's equations under periodic forcing

We use the Floquet theory of the Hill's equation to prove the conjecture that all solutions of the second order forced linear differential equation y^″+c(sint)y=cost, are oscillatory on [0,∞) for all c≠0.     

Landau-Ginzburg/Calabi-Yau correspondence,global mirror symmetry and Orlov equivalence

We show that the Gromov-Witten theory of Calabi-Yau hypersurfaces matches, in genus zero and after an analytic continuation, the quantum singularity theory (FJRW theory) recently introduced by Fan, Jarvis and Ruan following a proposal of Witten. Moreover, on both sides, we highlight two remarkable integral local systems arising from the common formalism of $\widehat {\Gamma }$ -integral structures applied to the derived category of the hypersurface {W=0} and to the category of graded matrix factorizations of W. In this setup, we prove that the analytic continuation matches Orlov equivalence between the two above categories.     

A singular approach to discontinuous vector fields on the plane

In this paper we deal with discontinuous vector fields on R² and we prove that the analysis of their local behavior around a typical singularity can be treated via singular perturbation. The regularization process developed by Sotomayor and Teixeira is crucial for the development of this work.     

四维Minkowski空间中类空超曲面的双曲高斯映射的奇点

介绍了四维Minkowski空间中类空超曲面的局部理论,定义了类空超曲面上的双曲高斯映射,双曲高度函数及距离平方函数,给出了一些定理的详细证明.介绍了一种证明高度函数是Morse族的新方法并应用Arnold等建立的Lagrange奇点理论对类空超曲面的双曲高斯映射的奇点进行了分类.     

A splitting criterion for an isolated singularity at x = 0 in a family of even hypersurfaces

Assume given a family of even local analytic hypersurfaces, whose central fiber has an isolated singularity at x =?0 which is not an ordinary double point. We prove that if the family is sufficiently general, for instance if the general fiber is smooth and the general singular fiber has only ordinary double points, then the singularity at x = 0 “splits in codimension one”, i.e., the local discriminant divisor has an irreducible component, over which a general fiber has more than one singularity specializing to the original one. As a corollary, we deduce the result by Grushevsky and Salvati Manni (Singularities of the theta divisor at points of order two, IMRN, 2007, Proposition 8) that on a principally polarized abelian variety (A, Θ) with dim(A) = g ≥ 4, a singularity of even multiplicity on Θ, isolated or not, at a point of order two and not an ordinary double point, must be a limit of two distinct ordinary double points {x, ?x} on nearby theta divisors.     

On the Cauchy problem for second order strictly hyperbolic equations with non–regular coefficients

In this paper we shall consider some necessary and sufficient conditions for well–posedness of second order hyperbolic equations with non–regular coefficients with respect to time. We will derive some optimal regularities for well–posedness from the intensity of singularity to the coefficients by WKB representation of the solution and some counter examples which are constructed by using ideas of Floquet theory.     

Analytic Solutions and Singularity Formation for the Peakon b-Family Equations

This paper deals with the well-posedness of the b-family equation in analytic function spaces. Using the Abstract Cauchy-Kowalewski theorem we prove that the b-family equation admits, locally in time, a unique analytic solution. Moreover, if the initial data is real analytic and it belongs to H ^s with s>3/2, and the momentum density u ₀?u _0,xx does not change sign, we prove that the solution stays analytic globally in time, for b≥1. Using pseudospectral numerical methods, we study, also, the singularity formation for the b-family equations with the singularity tracking method. This method allows us to follow the process of the singularity formation in the complex plane as the singularity approaches the real axis, estimating the rate of decay of the Fourier spectrum.     

On the GLY Conjecture of upper estimate of positive integral points in real right-angled simplices

The GLY (Granville-Lin-Yau) Conjecture is a generalization of Lin, Xu and Yau's results. An important application of GLY is its use in characterizing an affine hypersurface in Cn as a cone over a nonsingular projective variety. In addition, the Rough Upper Estimate Conjecture in GLY, recently proved by Yau and Zhang, implies the Durfee Conjecture in singularity theory. This paper develops a unified approach to prove the Sharp Upper Estimate Conjecture for general n. Using this unified approach, we prove that the Sharp Upper Estimate Conjecture is true for n=4,5,6. After giving a counter-example to show that the Sharp Upper Estimate Conjecture is not true for n=7, we propose a Modified GLY Conjecture. For each fixed n, our unified approach can be used to prove this Modified GLY Conjecture.     

On the instability of elliptic traveling wave solutions of the modified Camassa–Holm equation

This paper is concerned with the orbital instability for a specific class of periodic traveling wave solutions with the mean zero property and large spatial period related to the modified Camassa–Holm equation. These solutions, called snoidal waves, are written in terms of the Jacobi elliptic functions. To prove our result we use the abstract method of Grillakis, Shatah and Strauss [23], the Floquet theory for periodic eigenvalue problems and the n-gaps potentials theory of Dubrovin, Matveev and Novikov [19].     

Finite Gorenstein representation type implies simple singularity

Let R be a commutative noetherian local ring and consider the set of isomorphism classes of indecomposable totally reflexive R-modules. We prove that if this set is finite, then either it has exactly one element, represented by the rank 1 free module, or R is Gorenstein and an isolated singularity (if R is complete, then it is even a simple hypersurface singularity). The crux of our proof is to argue that if the residue field has a totally reflexive cover, then R is Gorenstein or every totally reflexive R-module is free.     

Periodic orbits of radially symmetric Keplerian-like systems: A topological degree approach

We are concerned with non-autonomous radially symmetric systems with a singularity, which are T-periodic in time. By the use of topological degree theory, we prove the existence of large-amplitude periodic solutions whose minimal period is an integer multiple of T. Precise estimates are then given in the case of Keplerian-like systems, showing some resemblance between the orbits of those solutions and the circular orbits of the corresponding classical autonomous system.     

Nonexistence of self-similar singularities in the viscous magnetohydrodynamics with zero resistivity

We are concerned on the possibility of finite time singularity in a partially viscous magnetohydrodynamic equations in Rn, n=2,3, namely the MHD with positive viscosity and zero resistivity. In the special case of zero magnetic field the system reduces to the Navier-Stokes equations in Rn. In this paper we exclude the scenario of finite time singularity in the form of self-similarity, under suitable integrability conditions on the velocity and the magnetic field. We also prove the nonexistence of asymptotically self-similar singularity. This provides us information on the behavior of solutions near possible singularity of general type as described in Corollary 1.1.     

Isosingular Sets and Deflation

This article introduces the concept of isosingular sets, which are irreducible algebraic subsets of the set of solutions to a system of polynomial equations constructed by taking the closure of points with a common singularity structure. The definition of these sets depends on deflation, a procedure that uses differentiation to regularize solutions. A weak form of deflation has proven useful in regularizing algebraic sets, making them amenable to treatment by the algorithms of numerical algebraic geometry. We introduce a strong form of deflation and define deflation sequences, which, in a different context, are the sequences arising in Thom–Boardman singularity theory. We then define isosingular sets in terms of deflation sequences. We also define the isosingular local dimension and examine the properties of isosingular sets. While isosingular sets are of theoretical interest as constructs for describing singularity structures of algebraic sets, they also expand the kinds of algebraic set that can be investigated with methods from numerical algebraic geometry.     

Global analysis of a p-Ginzburg-Landau energy with radial structure

This paper is concerned with the asymptotic behavior of a p-Ginzburg-Landau functional with radial structure as parameter goes to zero in the case of p≠2. By analyzing the functional globally, we show that the singularity of p-Ginzburg-Landau energy concentrates on the origin. By the fact the singularity can be balanced by some infinitesimal weight, we prove that an energy with a proper weight is globally bounded.     

Lifshitz tails for 2-dimensional random Schrödinger operators

The purpose of this paper is to prove that for a 2-dimensional continuous Anderson model, at an open band edge, the density of states exhibits a Lifshitz behavior. The Lifshitz exponent need not bed/2 = 1 but is determined by the behaviour of the Floquet eigenvalues of a well chosen background periodic Schrödinger operator.     

Limit holonomy and extension properties of Sobolev and Yang-Mills bundles

We consider the local behavior of Sobolev connections in a neighborhood of a singularity of codimension 2 and determine sufficient conditions for existence and local constancy of the limit holonomy of such connection. We prove that every Sobolev connection on an mdimensional manifold with locally L^m/2-integrable curvature and trivial limit holonomy extends through singularity of codimension 2. Additionally, if the connection satisfies the Yang-Mills-Higgs equation, the extension also satisfies the equation.     

Local existence with physical vacuum boundary condition to Euler equations with damping

In this paper, we consider the local existence of solutions to Euler equations with linear damping under the assumption of physical vacuum boundary condition. By using the transformation introduced in Lin and Yang (Methods Appl. Anal. 7 (3) (2000) 495) to capture the singularity of the boundary, we prove a local existence theorem on a perturbation of a planar wave solution by using Littlewood-Paley theory and justifies the transformation introduced in Liu and Yang (2000) in a rigorous setting.     

Unobstructed Lagrangian deformations

We prove that deformations of a Lagrangian singularity are unobstructed if the usual (flat) deformations are unobstructed and if a cohomological vanishing condition is satisfied. This gives another application to deformation theory of the Lagrangian de Rham complex introduced in Sevenheck and van Straten (Math. Ann. 327 (1) (2003) 79–102). To prove our theorem, we use the T¹-lifting criterion due to Ran, Kawamata and others. To cite this article: C. Sevenheck, C. R. Acad. Sci. Paris, Ser. I 338 (2004).