structure of α stratified modules for finite dimensional lie algebras i for complex simple finite dimensional lie algebras we study the structure of generalized verma modules which are torsion free with respect to some subalgebra isomorphic tosl 2 we obtain a criterion of irreducibility and establish necessary and sufficient conditions for the existence of a non trivial homomorphism between such modules generalizing the bgg theorem for verma modules

existence and instability of steady states for a triangular cross diffusion system a computer assisted proof abstract in this paper we present and apply a computer assisted method to study steady states of a triangular cross diffusion system our approach consist in an a posteriori validation procedure that is based on using a fixed point argument around a numerically computed solution in the spirit of the newton kantorovich theorem it allows to prove the existence of various non homogeneous steady states for different parameter values in some situations we obtain as many as 13 coexisting steady states we also apply the a posteriori validation procedure to study the linear stability of the obtained steady states proving that many of them are in fact unstable

some unexpected results on the brillouin singular equation fold bifurcation of periodic solutions abstract in this paper we find some new patterns regarding the periodic solvability of the brillouin electron beam focusing equation x 1 cos t x 1 x in particular we prove that there exists 0 248 for which a 2 periodic solution exists for every 0 and the bifurcation diagram with respect to displays a fold for this result significantly contributes to the discussion about the well known conjecture asserting that the brillouin equation admits a periodic solution for every 0 1 4 leading to doubt about its truthfulness for the first time moreover we prove multiplicity of periodic solutions for a range of values of near the technique used relies on rigorous computation and can be extended to some generalizations of the brillouin equation with right hand side equal to 1 x p we briefly discuss the cases p 2 and p 3

efficient rigorous numerics for higher dimensional pdes via one dimensional estimates we present an efficient rigorous computational method which is an extension of the work analytic estimates and rigorous continuation for equilibria of higher dimensional pdes m gameiro and j p lessard j differential equations 249 2010 pp 2237 2268 the idea is to generate sharp one dimensional estimates using interval arithmetic which are then used to produce high dimensional estimates these estimates are used to construct the radii polynomials which provide an efficient way of determining a domain on which the contraction mapping theorem is applicable computing the equilibria using a finite dimensional projection the method verifies that the numerically produced equilibrium for the projection can be used to explicitly define a set which contains a unique equilibrium for the pde a new construction of the polynomials is presented where the nonlinearities are bounded by products of one dimensional estimates as opposed to using fft with large inputs it is demonstrated that with this approac

a numerical verification method for solutions of boundary value problems with local uniqueness by banach s fixed point theorem in this paper we propose a method to prove the existence and the local uniqueness of solutions to infinite dimensional fixed point equations using computers choosing a set which possibly includes a solution we transform it by an approximate linearization of the operator appearing in the equation then we calculate the radii of the transformed set in order to check sufficient conditions for banach s fixed point theorem this method is applied to elliptic problems and numerical examples are given

an alternative approach to norm bound computation for inverses of linear operators in hilbert spaces abstract in the present paper we propose a computer assisted procedure to prove the invertibility of a linear operator in a hilbert space and to compute a verified norm bound of its inverse a number of the authors have previously proposed two verification approaches that are based on projection and constructive a priori error estimates the approach of the present paper is expected to bridge the gap between the two previous procedures in actual numerical verifications several verification examples that confirm the actual effectiveness of the proposed procedure are reported

rigorous computation of smooth branches of equilibria for the three dimensional cahn hilliard equation in this paper we propose a new general method to compute rigorously global smooth branches of equilibria of higher dimensional partial differential equations the theoretical framework is based on a combination of the theory introduced in global smooth solution curves using rigorous branch following van den berg et al math comput 79 271 1565 1584 2010 and in analytic estimates and rigorous continuation for equilibria of higher dimensional pdes gameiro and lessard j diff equ 249 9 2237 2268 2010 using this method one can obtain proofs of existence of global smooth solution curves of equilibria for large continuous parameter ranges and about local uniqueness of the solutions on the curve as an application we compute several smooth branches of equilibria for the three dimensional cahn hilliard equation

computational fixed point theory for differential delay equations with multiple time lags abstract we introduce a general computational fixed point method to prove existence of periodic solutions of differential delay equations with multiple time lags the idea of such a method is to compute numerical approximations of periodic solutions using newton s method applied on a finite dimensional projection to derive a set of analytic estimates to bound the truncation error term and finally to use this explicit information to verify computationally the hypotheses of a contraction mapping theorem in a given banach space the fixed point so obtained gives us the desired periodic solution we provide two applications the first one is a proof of coexistence of three periodic solutions for a given delay equation with two time lags and the second one provides rigorous computations of several nontrivial periodic solutions for a delay equation with three time lags

existence of secondary bifurcations or isolas for pdes abstract in this paper we introduce a method to conclude about the existence of secondary bifurcations or isolas of steady state solutions for parameter dependent nonlinear partial differential equations the technique combines the global bifurcation theorem knowledge about the non existence of nontrivial steady state solutions at the zero parameter value and explicit information about the coexistence of multiple nontrivial steady states at a positive parameter value we apply the method to the two dimensional swift hohenberg equation

generalized verma modules induced from sl 2 c and associated verma modules abstract with each generalized verma module induced from a well embedded parabolic subalgebra of a lie algebra with triangular decomposition we associate a verma module over the same algebra in a natural way in the case when the semisimple part of the levi factor of the parabolic subalgebra is isomorphic to sl 2 c and the generalized verma module is induced from an infinite dimensional simple module we prove that the associated verma module is simple if and only if the original generalized verma module is simple

multiple solutions for a semilinear boundary value problem a computational multiplicity proof abstract we prove the first genuine partial differential equation result on a conjecture concerning the number of solutions of second order elliptic boundary value problems with a nonlinearity which grows superlinearly at the proof makes massive use of computer assistance after approximate solutions have been computed by a numerical mountain pass algorithm combined with a newton iteration to improve accuracy a fixed point argument is used to show the existence of exact solutions close to the approximations

automatic differentiation for fourier series and the radii polynomial approach abstract in this work we develop a computer assisted technique for proving existence of periodic solutions of nonlinear differential equations with non polynomial nonlinearities we exploit ideas from the theory of automatic differentiation in order to formulate an augmented polynomial system we compute a numerical fourier expansion of the periodic orbit for the augmented system and prove the existence of a true solution nearby using an a posteriori validation scheme the radii polynomial approach the problems considered here are given in terms of locally analytic vector fields i e the field is analytic in a neighborhood of the periodic orbit hence the computer assisted proofs are formulated in a banach space of sequences satisfying a geometric decay condition in order to illustrate the use and utility of these ideas we implement a number of computer assisted existence proofs for periodic orbits of the planar circular restricted three body problem pcrtbp

analytic estimates and rigorous continuation for equilibria of higher dimensional pdes abstract in this paper we extend the ideas of the so called validated continuation technique to the context of rigorously proving the existence of equilibria for partial differential equations defined on higher dimensional spatial domains for that effect we present a new set of general analytic estimates these estimates are valid for any dimension and are used together with rigorous computations to construct a finite number of radii polynomials these polynomials provide a computationally efficient method to prove via a contraction argument the existence and local uniqueness of solutions for a rather large class of nonlinear problems we apply this technique to prove existence and local uniqueness of equilibrium solutions for the cahn hilliard and the swift hohenberg equations defined on two and three dimensional spatial domains

on the determinant of shapovalov form for generalized verma modules abstract we define a generalization of the shapovalov form for contragradient lie algebras and compute its determinant for generalized verma modules induced from a well embeddedsl 2 c subalgebra as a corollary we obtain a generalization of the bgg theorem for generalized verma modules

recent advances about the uniqueness of the slowly oscillating periodic solutions of wright s equation abstract an old conjecture in delay equations states that wright s equation y t y t 1 1 y t r has a unique slowly oscillating periodic solution sops for every parameter value 2 we reformulate this conjecture and we use a method called validated continuation to rigorously compute a global continuous branch of sops of wright s equation using this method we show that a part of this branch does not have any fold point partially answering the new reformulated conjecture

quadratic geometry of numbers we give upper bounds for zeros of quadratic forms for example we prove that for any nondegenerate quadratic form w xl xn with rational integer coefficients which vanishes on a d dimensional rational subspace d 0 there exist sublattices ro r1 rn d of zn of rank d on which w vanishes with the following properties rank ro n ri d 1 rank ro u rl u u rn d n and det ro n d detrl det rn d 4 f n d 2 where f is the maximum modulus of the coefficients of w

rigorous numerics for nonlinear differential equations using chebyshev series a computational method based on chebyshev series to rigorously compute solutions of initial and boundary value problems of analytic nonlinear vector fields is proposed the idea is to recast solutions as fixed points of an operator defined on a banach space of rapidly decaying chebyshev coefficients and to use the so called radii polynomials to show the existence of a unique fixed point near an approximate solution as applications solutions of initial value problems in the lorenz equations and symmetric connecting orbits in the gray scott equation are rigorously computed the symmetric connecting orbits are obtained by solving a boundary value problem with one of the boundary values in the stable manifold

constructive a priori error estimates for a full discrete approximation of the heat equation in this paper we consider the constructive a priori error estimates for a full discrete numerical solution of the heat equation our method is based on the finite element galerkin method with an interpolation in time that uses the fundamental solution for semidiscretization in space the present estimates play an essential role in the numerical verification method of exact solutions for the nonlinear parabolic equations this implies that by utilizing the present results we could get the guaranteed a posteriori error estimates for various kinds of nonlinear evolutional problems our results can also be considered as an explicit optimal estimate with the limited regularity of solutions which should be unknown up to the present

a combinatorial description of blocks in o p λ associated with sl 2 induction abstract we study the category o p where is an admissible category of dense weight sl 2 modules we give a combinatorial description of projectively stratified algebras arising from o p and prove a double centralizer property moreover we determine the characters of tilting modules in o p and prove that the finite dimensional algebra associated with the principal block of our o p is its own ringel dual

global smooth solution curves using rigorous branch following in this paper we present a new method for rigorously computing smooth branches of zeros of nonlinear operators f l 1 b 1 l 2 b 2 where b 1 and b 2 are banach spaces the method is first introduced for parameter continuation and then generalized to pseudo arclength continuation examples in the context of ordinary partial and delay differential equations are given

blow up profile for solutions of a fourth order nonlinear equation abstract it is well known that the nontrivial solutions of the equation u r u r f u r 0 blow up in finite time under suitable hypotheses on the initial data and f these solutions blow up with large oscillations knowledge of the blow up profile of these solutions is of great importance for instance in studying the dynamics of suspension bridges the equation is also commonly referred to as extended fisher kolmogorov equation or swift hohenberg equation in this paper we provide details of the blow up profile the key idea is to relate this blow up profile to the existence of periodic solutions for an auxiliary equation

rigorous numerics for piecewise smooth systems in this paper a rigorous computational method to compute solutions of piecewise smooth systems using a functional analytic approach based on chebyshev series is introduced a general theory based on the radii polynomial approach is proposed to compute crossing periodic orbits for continuous and discontinuous filippov piecewise smooth systems explicit analytic estimates to carry the computer assisted proofs are presented the method is applied to prove existence of crossing periodic orbits in a model nonlinear filippov system and in the chua s circuit system a general formulation to compute rigorously crossing connecting orbits for piecewise smooth systems is also introduced

lie algebra modules with finite dimensional weight spaces i let g denote a reductive lie algebra over an algebraically closed field of characteristic zero and let 4 denote a cartan subalgebra of g in this paper we study finitely generated g modules that decompose into direct sums of finite dimensional 4 weight spaces we show that the classification of irreducible modules in this category can be reduced to the classification of a certain class of irreducible modules those we call torsion free modules we also show that if g is a simple lie algebra that admits a torsion free module then g is of type a or c

rigorous numerics for ill posed pdes periodic orbits in the boussinesq equation in this paper we develop computer assisted techniques for the analysis of periodic orbits of ill posed partial differential equations as a case study our proposed method is applied to the boussinesq equation which has been investigated extensively because of its role in the theory of shallow water waves the idea is to use the symmetry of the solutions and a newton kantorovich type argument the radii polynomial approach to obtain rigorous proofs of existence of the periodic orbits in a weighted l1 banach space of space time fourier coefficients with exponential decay we present several computer assisted proofs of the existence of periodic orbits at different parameter values

validated continuation for equilibria of pdes one of the most efficient methods for determining the equilibria of a continuous parameterized family of differential equations is to use predictor corrector continuation techniques in the case of partial differential equations this procedure must be applied to some finite dimensional approximation which of course raises the question of the validity of the output we introduce a new technique that combines the information obtained from the predictor corrector steps with ideas from rigorous computations and verifies that the numerically produced equilibrium for the finite dimensional system can be used to explicitly define a set which contains a unique equilibrium for the infinite dimensional partial differential equation using the cahn hilliard and swift hohenberg equations as models we demonstrate that the cost of this new validated continuation is less than twice the cost of the standard continuation method alone