the infinite dimensional evans function abstract we introduce generalized operator valued jost solutions of first order ill posed differential equations on hilbert spaces we then construct an infinite dimensional evans function for abstract differential equations as a 2 modified fredholm determinant of the operator obtained by adding the values at zero of the generalized operator valued jost solutions next we prove a formula that connects the 2 modified evans determinant and the 2 modified determinant related to the birman schwinger type operator associated to the ill posed equation using this formula we construct a holomorphic infinite dimensional evans function for second order differential operators on infinite cylinders whose zeros are the eigenvalues of the differential operators

the gap lemma and geometric criteria for instability of viscous shock profiles an obstacle in the use of evans function theory for stability analysis of traveling waves occurs when the spectrum of the linearized operator about the wave accumulates at the imaginary axis since the evans function has in general been constructed only away from the essential spectrum a notable case in which this difficulty occurs is in the stability analysis of viscous shock profiles here we prove a general theorem the gap lemma concerning the analytic continuation of the evans function associated with the point spectrum of a traveling wave into the essential spectrum of the wave this allows geometric stability theory to be applied in many cases where it could not be applied previously we demonstrate the power of this method by analyzing the stability of certain undercompressive viscous shock waves a necessary geometric condition for stability is determined in terms of the sign of a certain melnikov integral of the associated viscous profile this sign can easily be evaluated numerically we also compute it analytically for solutions of several important classes of systems in particular we show for a wide class of systems that homoclinic solitary waves are linearly unstable confirming these as the first known examples of unstable viscous shock waves we also show that strong heteroclinic undercompressive waves are sometimes unstable similar stability conditions are also derived for lax and overcompressive shocks and for nn

infinite dimensional evans function theory for elliptic eigenvalue problems in a channel abstract an infinite dimensional evans function theory is developed for the elliptic eigenvalue problem associated with the stability of travelling solitary waves in a channel also a bundle is constructed over the complex domain so that its first chern number gives the number of eigenvalues inside the domain

not residually finite groups of intermediate growth commensurability and non geometricity abstract we construct a series of not residually finite groups of intermediate growth in particular we construct groups of intermediate growth that are not commensurable up to finite kernels with any residually finite group also we show that certain properties of groups among those solvability of the word problem and the property of having torsion of unbounded order are not weakly geometric the argument is based in particular on studying commensurability up to finite kernels

center manifolds for a class of degenerate evolution equations and existence of small amplitude kinetic shocks abstract we construct center manifolds for a class of degenerate evolution equations including the steady boltzmann equation and related kinetic models establishing in the process existence and behavior of small amplitude kinetic shock and boundary layers notably for boltzmann s equation we show that elements of the center manifold decay in velocity at near maxwellian rate in accord with the formal chapman enskog picture of near equilibrium flow as evolution along the manifold of maxwellian states or grad moment approximation via hermite polynomials in velocity our analysis is from a classical dynamical systems point of view with a number of interesting modifications to accommodate ill posedness of the underlying evolution equation

groups with infinite mod p schur multiplier abstract the non finiteness of the mod p schur multiplier of a finitely generated group g with trivial center implies the existence of uncountably many non residually finite non isomorphic central extension groups with kernel c p see thm a this phenomenon is related to the comparison of the cohomology of the profinite completion g of a group g and the cohomology of the group g itself see thm b according to j p serre 8 a group g is good if the cohomologies of g and g are naturally isomorphic on finite coefficients it is shown that non uniform arithmetic lattices of algebraic rank 1 groups over local fields of positive characteristic p are not good see thm c

l valued propositional logic l vpl in this paper two types of gradational l type lattice valued propositional logic l v p l with truth values in a lattice implication algebra are introduced and some fundamental questions of them such as semantics syntax and compactness are investigated and where the l type a soundness theorem a b completeness theorem a b consistent theorem and a b q deduction theorem are also proved

multiary α resolution principle for a lattice valued logic this paper focuses on resolution based automated reasoning theory in a lattice valued logic system with truth values that are defined in a lattice valued logical algebraic structure lattice implication algebras lias which essentially aims to extend the classical logic to handle automated deduction under an uncertain environment concretely we investigate a generalization of the known conjunctive normal form cnf in the classical logic representation that we call the generalized conjunctive normal form gcnf which aims to characterize the constants and implication connectives that are essentially different from the ones in classical logic we then extend the established resolution principle at a certain truth value level called resolution in this lattice valued logic to a more general form i e from binary resolution to multiary resolution the extension to multiary resolution starts from lattice valued propositional logic lp x while its theorems of both soundness and completeness are proved multiary resolution principle is then further established in the corresponding lattice valued first order logic lf x along with its soundness theorem lifting lemma and completeness theorem meanwhile an important result that multiary resolution principle in lf x can be equivalently transformed into that in lp x to some extent is obtained all these works will theoretically support the establishment of an automated reasoning algorithm and its implementation with further applications into automated deduction and decision making problems under uncertainty

reverse norms and l exponential decay for a class of degenerate evolution systems arising in kinetic theory abstract we consider the question of exponential decay to equilibrium of solutions of an abstract class of degenerate evolution equations on a hilbert space modeling the steady boltzmann and other kinetic equations specifically we provide conditions suitable for construction of a stable manifold in a particular reverse l norm and examine when these do and do not hold

the dichotomy theorem for evolution bi families we prove that the operator g the closure of the rst order dierential operator d dt d t on l2 r x is fredholm if and only if the not well posed equation u 0 t d t u t t 2 r has exponential dichotomies on r and r and the ranges of the dichotomy projections form a fredholm pair moreover the index of this pair is equal to the fredholm index of g here x is a hilbert space d t a b t a is the generator of a bi semigroup b is a bounded piecewise strongly continuous operator valued function also we prove some perturbations results and consider various examples of not well posed problems

the symplectic evans matrix and the instability of solitary waves and fronts hamiltonian evolution equations which are equivariant with respect to the action of a lie group are models for physical phenomena such as oceanographic flows optical fibres and atmospheric flows and such systems often have a wide variety of solitary wave or front solutions in this paper we present a new symplectic framework for analyzing the spectral problem associated with the linearization about such solitary waves and fronts at the heart of the analysis is a multi symplectic formulation of hamiltonian partial differential equations where a distinct symplectic structure is assigned for the time and space directions with a third symplectic structure with two form denoted by omega associated with a coordinate frame moving at the speed of the wave this leads to a geometric decomposition and symplectification of the evans function formulation for the linearization about solitary waves and fronts we introduce the concept of the symplectic evans matrix a matrix consisting of restricted omega symplectic forms by applying hodge duality to the exterior algebra formulation of the evans function we find that the zeros of the evans function correspond to zeros of the determinant of the symplectic evans matrix based on this formulation we prove several new properties of the evans function restricting the spectral parameter lambda to the real axis we obtain rigorous results on the derivatives of the evans function near the origin based solely on the abstract geometry of the equations and results for the large lambda behaviour which use primarily the symplectic structure but also extend to the non symplectic case the lie group symmetry affects the evans function by generating zero eigenvalues of large multiplicity in the so called systems at infinity we present a new geometric theory which describes precisely how these zero eigenvalues behave under perturbation by combining all these results a new rigorous sufficient condition for instability of solitary waves and fronts is obtained the theory applies to a large class of solitary waves and fronts including waves which are biasymptotic to a nonconstant manifold of states as x tends to infinity to illustrate the theory it is applied to three examples a boussinesq model from oceanography a class of nonlinear schrodinger equations from optics and a nonlinear klein gordon equation from atmospheric dynamics

homoclinic orbits in reversible systems and their applications in mechanics fluids and optics abstract this survey article reviews the theory and application of homoclinic orbits to equilibria in reversible continuous time dynamical systems where the homoclinic orbit and the equilibrium are both reversible the focus is on even order reversible systems in four or more dimensions local theory generic argument and global existence theories are examined for each qualitatively distinct linearisation several recent results such as coalescence caused by non transversality and the reversible orbit flip bifurcation are covered a number of open problems are highlighted applications are reviewed to systems arising from a variety of disciplines with the aid of numerical methods three examples are presented in detail one of which is infinite dimensional

evans functions jost functions and fredholm determinants the principal results of this paper consist of an intrinsic definition of the evans function in terms of newly introduced generalized matrix valued jost solutions for general first order matrix valued differential equations on the real line and a proof of the fact that the evans function a finite dimensional determinant by construction coincides with a modified fredholm determinant associated with a birman schwinger type integral operator up to an explicitly computable nonvanishing factor

the congruence kernel of an arithmetic lattice in a rank one algebraic group over a local field let k be a global field and let kv be the completion of k with respect to v a non archimedean place of k let g be a connected simply connected algebraic group over k which is absolutely almost simple of kv rank 1 let g g kv let g be an arithmetic lattice in g and let c c g be its congruence kernel lubotzky has shown that c is infi nite confirming an earlier conjecture of serre here we provide complete solution of the congruence subgroup problem for g by determining the structure of c it is shown that c is a free profinite product one of whose factors is f o the free profinite group on countably many generators the most surprising conclusion from our results is that the structure of c depends only on the characteristic of k the structure of c is already known for a number of special cases perhaps the most important of these is the non uniform example g sl2 o s where o s is the ring of s integers in k with s f vg which plays a central role in the theory of drinfeld modules the proof makes use of a decomposition theorem of lubotzky arising from the action of g on the bruhat tits tree associated with g

filter based resolution principle for lattice valued propositional logic lp x as one of most powerful approaches in automated reasoning resolution principle has been introduced to non classical logics such as many valued logic however most of the existing works are limited to the chain type truth value fields lattice valued logic is a kind of important non classical logic which can be applied to describe and handle incomparability by the incomparable elements in its truth value field in this paper a filter based resolution principle for the lattice valued propositional logic lp x based on lattice implication algebra is presented where filter of the truth value field being a lattice implication algebra is taken as the criterion for measuring the satisfiability of a lattice valued logical formula the notions and properties of lattice implication algebra filter of lattice implication algebra and the lattice valued propositional logic lp x are given firstly the definitions and structures of two kinds of lattice valued logical formulae i e the simple generalized clauses and complex generalized clauses are presented then finally the filter based resolution principle is given and after that the soundness theorem and weak completeness theorems for the presented approach are proved

exponential dichotomies for solitary wave solutions of semilinear elliptic equations on infinite cylinders in applications solitary wave solutions of semilinear elliptic equations u g u u 0 x y r in infinite cylinders frequently arise as travelling waves of parabolic equations as such their bifurcations are an interesting issue interpreting elliptic equations on infinite cylinders as dynamical systems inxhas proved very useful still there are major obstacles in obtaining for instance bifurcation results similar to those for ordinary differential equations in this article persistence and continuation of exponential dichotomies for linear elliptic equations is proved with this technique at hands lyapunov schmidt reduction near solitary waves can be applied as an example existence of shift dynamics near solitary waves is shown if a perturbation h x u u periodic inxis added

fuzzy implicative and boolean filters of r 0 algebras the notions of fuzzy filters fuzzy implicative filters and fuzzy boolean filters are introduced in r0 algebras respectively some of their properties are investigated the relations between fuzzy implicative filters and fuzzy boolean filters are discussed the characterizations of fuzzy implicative filters are studied and the following is proved to be equivalent 1 a fuzzy filter f is implicative 2 f is a fuzzy boolean filter 3 x y z l f x z f x z y f y z 4 x y l f x y f x y y 5 x l n 1 f nx f x

an evans function approach to spectral stability of internal solitary waves in stratified fluids frequently encountered in nature internal solitary waves in strat ified fluids are well observed and well studied from the experimental the the oretical and the numerical perspective from the mathematical point of view these waves are exact solutions of the 2d euler equations for incompressible inviscid fluids contrasting with a rich theory for their existence and the de velopment of methods for computing these waves their stability analysis has hardly received attention at a rigorous mathematical level this paper proposes a new approach to the investigation of stability of in ternal solitary waves in a continuously stratified fluidic medium and carries out the following four steps of this approach i to formulate the eigenvalue problem as an infinite dimensional spatial dynamical system ii to intro duce finite dimensional truncations of the spatial dynamics description iii to demonstrate that each truncation of any order permits a well defined evans function iv to prove absence of small zeros of the evans function in the small amplitude limit the latter notably implies the low frequency spectral stability of small amplitude internal solitary waves to arbitrarily high trunca tion order

an infinite dimensional evans function theory for elliptic boundary value problems abstract an infinite dimensional evans function e and a stability index theorem are developed for the elliptic eigenvalue problem in a bounded domain r m the number of zero points of the evans function in a bounded simply connected complex domain d is shown to be equal to the number of eigenvalues of the corresponding elliptic operator in d when the domain is star shaped an associated unstable bundle e d based on d is constructed and the first chern number of e d also gives the number of eigenvalues of the elliptic operator inside d

determination of α resolution in lattice valued first order logic lf x key issues for resolution based automated reasoning in lattice valued first order logic lf x are investigated with truth values in a lattice valued logical algebraic structure lattice implication algebra lia the determination of resolution at a certain truth value level called a resolution in lf x is proved to be equivalently transformed into the determination of a resolution in lattice valued propositional logic lp x based on lia the determination of a resolution of any quasi regular generalized literals and constants under various cases in lp x is further analyzed specified and subsequently verified hence the determination of a resolution in lf x can be accordingly solved to a very broad extent which not only lays a foundation for the practical implementation of automated reasoning algorithms in lf x but also provides a key support for a resolution based automated reasoning approaches and algorithms in lia based linguistic truth valued logics

evans functions for periodic waves on infinite cylindrical domains abstract using galerkin approximations an evans function for spatially periodic waves on infinite cylindrical domains is constructed it is also shown that the evans function can be used to define a parity index for periodic waves that detects whether the wave admits an odd number of real unstable eigenvalues this parity index depends only on local information for the existence problem of the wave in particular it uses information about the linear dispersion relation near zero and the orientability of the unstable and stable manifolds along the nonlinear wave the results are applied to small amplitude wave trains for a scalar equation on an infinite strip

cohomological goodness and the profinite completion of bianchi groups the concept of cohomological goodness was introduced by j p serre in his book on galois cohomology 31 this property relates the cohomology groups of a group to those of its profinite completion we develop properties of goodness and establish goodness for certain important groups we prove for example that the bianchi groups i e the groups psl 2 o where o is the ring of integers in an imaginary quadratic number field are good as an application of our improved understanding of goodness we are able to show that certain natural central extensions of fuchsian groups are residually finite this result contrasts with examples of p deligne 5 who shows that the analogous central extensions of sp 4 z do not have this property

variations on a theme of jost and pais abstract we explore the extent to which a variant of a celebrated formula due to jost and pais which reduces the fredholm perturbation determinant associated with the schrodinger operator on a half line to a simple wronski determinant of appropriate distributional solutions of the underlying schrodinger equation generalizes to higher dimensions in this multi dimensional extension the half line is replaced by an open set r n n n n 2 where has a compact nonempty boundary satisfying certain regularity conditions our variant involves ratios of perturbation determinants corresponding to dirichlet and neumann boundary conditions on and invokes the corresponding dirichlet to neumann map as a result we succeed in reducing a certain ratio of modified fredholm perturbation determinants associated with operators in l 2 d n x n n to modified fredholm determinants associated with operators in l 2 d n 1 n 2 applications involving the birman schwinger principle and eigenvalue counting functions are discussed