Results 1  10
of
21
On Hamiltonian perturbations of hyperbolic systems of conservation laws
, 2004
"... We study the general structure of formal perturbative solutions to the Hamiltonian perturbations of spatially onedimensional systems of hyperbolic PDEs vt + [φ(v)]x = 0. Under certain genericity assumptions it is proved that any bihamiltonian perturbation can be eliminated in all orders of the pert ..."
Abstract

Cited by 33 (5 self)
 Add to MetaCart
We study the general structure of formal perturbative solutions to the Hamiltonian perturbations of spatially onedimensional systems of hyperbolic PDEs vt + [φ(v)]x = 0. Under certain genericity assumptions it is proved that any bihamiltonian perturbation can be eliminated in all orders of the perturbative expansion by a change of coordinates on the infinite jet space depending rationally on the derivatives. The main tools is in constructing of the socalled quasiMiura transformation of jet coordinates eliminating an arbitrary deformation of a semisimple bihamiltonian structure of hydrodynamic type (the quasitriviality theorem). We also describe, following [35], the invariants of such bihamiltonian structures with respect to the group of Miuratype transformations depending
ALGEBROGEOMETRIC ASPECTS OF HEINESTIELTJES THEORY
, 2008
"... The goal of the paper is to develop a HeineStieltjes theory for univariate linear differential operators of higher order. Namely, for a given linear ordinary differential operator d(z) = Pk di i=1 Qi(z) dzi with polynomial coefficients set r = maxi=1,...,k(deg Qi(z) − i). If d(z) satisfies the co ..."
Abstract

Cited by 9 (2 self)
 Add to MetaCart
The goal of the paper is to develop a HeineStieltjes theory for univariate linear differential operators of higher order. Namely, for a given linear ordinary differential operator d(z) = Pk di i=1 Qi(z) dzi with polynomial coefficients set r = maxi=1,...,k(deg Qi(z) − i). If d(z) satisfies the conditions: i) r ≥ 0 and ii) deg Qk(z) = k + r we call it a nondegenerate higher Lamé operator. Following the classical approach of E. Heine and T. Stieltjes, see [18], [41] we study the multiparameter spectral problem of finding all polynomials V (z) of degree at most r such that the equation: d(z)S(z) + V (z)S(z) = 0 has for a given positive integer n a polynomial solution S(z) of degree n. We show that under some mild nondegeneracy assumptions there exist exactly n+r ´ such polynomials Vn,i(z) whose corresponding eigenpolynomials Sn,i(z)
Deformations Of Polynomials, Boundary Singularities And Monodromy
 Mosc. Math. J
"... We study the topology of polynomial functions by deforming them generically. We explain how the nonconservation of the total "quantity" of singularity in the neighbourhood of infinity is related to the variation of topology in certain families of boundary singularities along the hyperplane at infin ..."
Abstract

Cited by 9 (2 self)
 Add to MetaCart
We study the topology of polynomial functions by deforming them generically. We explain how the nonconservation of the total "quantity" of singularity in the neighbourhood of infinity is related to the variation of topology in certain families of boundary singularities along the hyperplane at infinity. 1.
Cluster tilting for onedimensional hypersurface singularities
 Adv. Math
"... Abstract. In this article we study CohenMacaulay modules over onedimensional hypersurface singularities and the relationship with representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete d ..."
Abstract

Cited by 8 (7 self)
 Add to MetaCart
Abstract. In this article we study CohenMacaulay modules over onedimensional hypersurface singularities and the relationship with representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete description by homological method using higher almost split sequences and results from birational geometry. We obtain a large class of 2CY tilted algebras which are finite dimensional symmetric and satisfies τ 2 = id. In particular, we compute 2CY tilted algebras for simple/minimally elliptic curve singuralities.
Global asymptotics for multiple integrals with boundaries
 Duke Math. J
"... Dedicated to Frédéric Pham on the occasion of his 65th birthday. Under convenient geometric assumptions, the saddlepoint method for multidimensional Laplace integrals is extended to the case where the contours of integration have boundaries. The asymptotics are studied in the case of nondegenerate ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
Dedicated to Frédéric Pham on the occasion of his 65th birthday. Under convenient geometric assumptions, the saddlepoint method for multidimensional Laplace integrals is extended to the case where the contours of integration have boundaries. The asymptotics are studied in the case of nondegenerate and of degenerate isolated critical points. The incidence of the Stokes phenomenon is related to the monodromy of the homology via generalized PicardLefschetz formulae and is quantified in terms of geometric indices of intersection. Exact remainder terms and the hyperasymptotics are then derived. A direct consequence is a numerical algorithm to determine the Stokes constants and indices of intersections. Examples are provided. 1.
Birational automorphisms of nodal quartic threefolds arXiv:0803.4348
, 2008
"... Abstract. It is wellknown that a nonsingular minimal cubic surface is birationally rigid; the group of its birational selfmaps is generated by biregular selfmaps and birational involutions such that all relations between the latter are implied by standard relations between reflections on an ellipti ..."
Abstract

Cited by 4 (3 self)
 Add to MetaCart
Abstract. It is wellknown that a nonsingular minimal cubic surface is birationally rigid; the group of its birational selfmaps is generated by biregular selfmaps and birational involutions such that all relations between the latter are implied by standard relations between reflections on an elliptic curve. It is also known that a factorial nodal quartic threefold is birationally rigid and its group of birational selfmaps is generated by biregular ones and certain birational involutions. We prove that all relations between these involutions are implied by standard relations on elliptic curves, complete the proof of birational rigidity over a nonclosed field and describe the situations when some of the birational involutions in question become regular (and, in particular, complete the proof of the initial theorem on birational rigidity, since some details were not established in the original paper of M. Mella). 1.
Asymptotic integration and dispersion for hyperbolic equations, with applications to Kirchhoff equations, in preparation
"... Abstract. The aim of this paper is to establish time decay properties and dispersive estimates for strictly hyperbolic equations with homogeneous symbols and with timedependent coefficients whose derivatives belong to L 1 (R). For this purpose, the method of asymptotic integration is developed for ..."
Abstract

Cited by 4 (3 self)
 Add to MetaCart
Abstract. The aim of this paper is to establish time decay properties and dispersive estimates for strictly hyperbolic equations with homogeneous symbols and with timedependent coefficients whose derivatives belong to L 1 (R). For this purpose, the method of asymptotic integration is developed for such equations and representation formulae for solutions are obtained. These formulae are analysed further to obtain time decay of L p –L q norms of propagators for the corresponding Cauchy problems. It turns out that the decay rates can be expressed in terms of certain geometric indices of the limiting equation and we carry out the thorough analysis of this relation. This provides a comprehensive view on asymptotic properties of solutions to timeperturbations of hyperbolic equations with constant coefficients. The formulae are then applied to the global solvability of Kirchhoff equations of higher order with small data. Moreover, we also obtain the time decay rate of the L p –L q estimates for nonlinear equations of these kinds, so the time wellposedness of the corresponding equations with additional semilinearity can be treated by standard Strichartz estimates. 1.
Geometry of planar logfronts
, 2008
"... Given two polynomials P(z, w) and Q(z, w), one can study solutions of the system { ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Given two polynomials P(z, w) and Q(z, w), one can study solutions of the system {
ON EIGENVALUES OF RECTANGULAR MATRICES
"... Abstract. Given a (k+1)tuple A, B1,..., Bk of (m×n)matrices with m ≤ n we call the set of all ktuples of complex numbers {λ1,..., λk} such that the linear combination A + λ1B1 + λ2B2 +... + λkBk has rank smaller than m the eigenvalue locus of the latter pencil. Motivated primarily by applications ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Abstract. Given a (k+1)tuple A, B1,..., Bk of (m×n)matrices with m ≤ n we call the set of all ktuples of complex numbers {λ1,..., λk} such that the linear combination A + λ1B1 + λ2B2 +... + λkBk has rank smaller than m the eigenvalue locus of the latter pencil. Motivated primarily by applications to multiparameter generalizations of the HeineStieltjes spectral problem, see [He] and [Vol], we study a number of properties of the eigenvalue locus in the most important case k = n − m + 1. Introduction and Main Results In recent years there appeared a number of publications discussing the eigenvalues of pencils of nonsquare matrices and their approximations, see, e.g., [BEGM], [CG], [TW] and references therein. But to the best of our knowledge the following natural problem either has been overlooked by specialists in linear algebra or is
Morse theory for manifolds with boundary, eprint arXiv:1207.3066
, 2012
"... Abstract. We develop Morse theory for manifolds with boundary. Beside standard and expected facts like the handle cancellation theorem and the Morse lemma for manifolds with boundary, we prove that under suitable connectedness assumptions a critical point in the interior of a Morse function can be m ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Abstract. We develop Morse theory for manifolds with boundary. Beside standard and expected facts like the handle cancellation theorem and the Morse lemma for manifolds with boundary, we prove that under suitable connectedness assumptions a critical point in the interior of a Morse function can be moved to the boundary, where it splits into a pair of boundary critical points. As an application, we prove that every cobordism of connected manifolds with boundary splits as a union of left product cobordisms and right product cobordisms. 1.