some koszul rings from geometry abstract we give examples of koszul rings that arise naturally in algebraic geometry in the first part we prove a general result on koszul property associated to an ample line bundle on a projective variety specifically we show how koszul property of multiples of a base point free ample line bundle depends on its castelnuovo mumford regularity in the second part we give examples of koszul rings that come from adjoint line bundles on minimal irregular surfaces of general type

nonlinear three point third order boundary value problems abstract in this paper we consider three point boundary value problems for third order differential equations we provide simple sufficient conditions in order to obtain a priori bounds on solutions we then rely on the topological transversality theorem of granas or on schaefer s theorem to prove existence of solutions

gaussian beams and legendre polynomials as invariants of the time reversal operator for a large rigid cylinder the dort method french acronym for decomposition of the time reversal operator is an active remote sensing technique using an array of antennas for the detection and localization of scatterers this method is based on the singular value decomposition of the interelement response matrix in this paper an analytical expression of the singular vectors due to the reflection from a large rigid cylinder is provided depending on the array aperture two asymptotic regimes are described it is shown that the singular vectors correspond to hermite gaussian modes for large apertures and legendre polynomials for small ones using perturbation theory the corresponding singular values are deduced theoretical predictions are in good agreement with experimental results

singular perturbations for third order nonlinear multi point boundary value problem abstract this paper is devoted to study the following third order multi point singularly perturbed boundary value problem x t f t x t x t x t 0 0 t 1 0 1 x 0 0 ax 0 bx 0 i 1 n 2 i x i a cx 1 dx 1 i 1 n 2 i x i b where a b c d 0 a b r a b 0 c d 0 i 0 i 0 i 1 2 n 2 0 1 2 n 2 1 and 0 1 2 n 2 1 the existence uniqueness and asymptotic estimates of solutions of the boundary value problem are give by using priori estimates differential inequalities technique and leray schauder degree theory

existence of solutions of boundary value problem for 3rd order nonlinear system by appropriate bounding function pair and modified functions using the theory of differential inequality and the theory of topological degree this paper presents the existence criterions of solutions for the system of 3rd order differential equations under nonlinear boundary conditions and extends many known results

extremality and comparison results for discontinuous implicit third order functional initial boundary value problems extremality results are derived for implicit third order functional initial boundary value problems differential equations initial conditions and boundary conditions of problems may involve discontinuous nonlinearities

pbw deformations of koszul algebras over a nonsemisimple ring let b be a generalized koszul algebra over a finite dimensional algebra s we construct a bimodule koszul resolution of b when the projective dimension of s b equals two using this we prove a poincare birkhoff witt pbw type theorem for a deformation of a generalized koszul algebra when the projective dimension of s b is greater than two we construct bimodule koszul resolutions for generalized smash product algebras obtained from braidings between finite dimensional algebras and koszul algebras and then prove the pbw type theorem the results obtained can be applied to standard koszul artin schelter gorenstein algebras in the sense of minamoto and mori adv math 226 4061 4095 2011

characterization of subwavelength elastic cylinders with the decomposition of the time reversal operator theory and experiment the decomposition of the time reversal operator provides information on the scattering medium it has been shown chambers and gautesen j acoust soc am 109 2616 2624 2001 that a small spherical scatterer is in general associated with four eigenvalues and eigenvectors of the time reversal operator in this paper the 2d problem of scattering by an elastic cylinder imbedded in water measured by a linear array of transducers is considered in this case the array response matrix has three nonzero singular values experimental results are obtained with linear arrays of transducers and for wires of different diameters smaller that the wavelength it is shown how the singular value distribution and the singular vectors depend on the elastic velocities cl ct the density of each wire and on the density 0 and velocity c0 of the surrounding fluid these results offer a new perspective towards solution of the inverse problem by determining more than scattering contrast using conventional array proce

bounding functions methods for fully nonlinear boundary value problems abstract using the bounding functions method and the theory of topological degree this paper presents the existence criterion of solution for third order bvp with nonlinear boundary conditions and extends the existing results

uniqueness and existence results for a third order nonlinear multi point boundary value problem this paper provides sufficient conditions for the existence and uniqueness of solutions to a nonlinear multi point boundary value problem for third order differentia equation the nonlinear term f in the differential equation under consideration may depend on second order derivative the emphasis here is the boundary conditions are nonlinear and this is where the main novelty of this work lies by applying the upper and lower solutions method as well as degree theory the existence and uniqueness results of the problem are established some examples are given to demonstrate the main results

graded decomposition numbers for the blob algebra abstract we give a closed formula for the graded decomposition numbers of the blob algebra over a field of characteristic zero at a root of unity

analysis of the time reversal operator for a small spherical scatterer in an electromagnetic field the time reversal operator tro for a planar array of crossed dipole elements illuminating a small conducting and or dielectric sphere is investigated in order to determine the general properties of an electromagnetic time reversing array system the behavior of such a system for a given frequency is analyzed by studying the eigenvalues and eigenvectors of the tro each eigenvector specifies a set of complex driving currents for the array elements that produce received voltages that are proportional to the conjugates of the drive currents the proportionality constant is equal to the square root of the associated eigenvalue and is the same for all elements the eigenvalues and eigenvectors can be determined by performing a singular value decomposition svd on the multistatic data matrix of the array the eigenvalues of the tro are the squares of the singular values and the eigenvectors are identical to the singular vectors we have shown that the maximum number of singular vectors associated with the sphere is equal to the number of orthogonal orientations of the dipole moments induced in the sphere when irradiated by the array so there is a maximum of six for a conducting sphere but only three are significant when the conductivity is small and the sphere may be considered being just a dielectric numerical results are presented for linear and circular arrays to show the general behavior of the system

solvability for a third order discontinuous fully equation with nonlinear functional boundary conditions abstract we prove an existence and location result for the third order functional nonlinear boundary value problem u t f t u u t u t for t a b 0 l 0 u u u t 0 0 l 1 u u u a u a 0 l 2 u u u b u b with t 0 a b given f i c i r 2 r is a l 1 caratheodory function and l 0 l 1 l 2 are continuous functions depending functionally on u and u the arguments make use of an a priori estimate on u lower and upper solutions method and degree theory applications to a multipoint problem and to a beam equation will be presented

eigenvalue for a singular third order three point boundary value problem in this paper the existence of positive solutions for a singular third order three point boundary value problem is investigated by using krasnoselskii s fixed point theorem several sufficient conditions for the existence of positive solutions and the eigenvalue intervals on which there exists a positive solution are obtained

modular perverse sheaves on flag varieties iii positivity conditions we further develop the general theory of the mixed modular derived category introduced by the authors in a previous paper in this series we then use it to study positivity and q koszulity phenomena on flag varieties

twin monotone positive solutions to a singular nonlinear third order differential equation abstract this paper discusses the existence of at least one or two nondecreasing positive solutions for the following singular nonlinear third order differential equation x t t f t x t 0 a t b x a x a x b 0 where 0 is a parameter c a b r f c a b 0 r t may be singular at t a b and f t s may be singular at s 0 green s function and the fixed point theorem of cone expansion and compression type are utilized to illustrate the results four nontrivial examples are included

existence and uniqueness of solutions for nth order nonlinear two point boundary value problems in this paper we present some general results of existence and uniqueness of solutions for nth order nonlinear two point boundary value problems by using shooting method together with maximum principle and kneser hukahara continuum theorem as applications we give certain concrete sufficient conditions for the existence and uniqueness of solutions of nonlinear two point boundary value problems of nth order lipschitz equations

a generalized koszul theory and its relation to the classical theory abstract let a i 0 a i be a graded locally finite k algebra where a 0 is a finite dimensional algebra whose finitistic dimension is 0 in this paper we develop a generalized koszul theory preserving many classical results and show an explicit correspondence between this generalized theory and the classical theory applications in representations of certain categories and extension algebras of standard modules of standardly stratified algebras are described

existence and uniqueness results for third order nonlinear differential systems abstract sufficient conditions are given for the existence of solutions to third order boundary value problems for nonlinear differential systems some appropriate conditions are given to guarantee that the nagumo condition is satisfied by constructing appropriate a lower solution upper solution pair a concept that is defined in this paper the uniqueness result of the problem is also established the emphasis here is that the differential systems has nonlinear dependence on all over order derivatives and the boundary conditions are nonlinear

the prolate spheroidal wave functions as invariants of the time reversal operator for an extended scatterer in the fraunhofer approximation the decomposition of the time reversal operator known by the french acronym dort is widely used to detect locate and focus on scatterers in various domains such as underwater acoustics medical ultrasound and nondestructive evaluation in the case of point scatterers the theory is well understood the number of nonzero eigenvalues is equal to the number of scatterers and the eigenvectors correspond to the scatterers green s function in the case of extended objects however the formalism is not as simple it is shown here that in the fraunhofer approximation analytical solutions can be found and that the solutions are functions called prolate spheroidal wave functions these functions have been studied in information theory as a basis of band limited and time limited signals they also arise in optics the theoretical solutions are compared to simulation results most importantly the intuition that for an extended objects the number of nonzero eigenvalues is proportional to the number of resol

existence of solutions for a class of third order nonlinear boundary value problems abstract in this paper we are concerned with the following third order ordinary differential equation x t f t x t x t x t 0 0 with the nonlinear boundary conditions x 0 0 g x 0 x 0 a h x 1 x 1 b where a b r f 0 1 r 3 r is continuous g h r 2 r are continuous the existence result is given by using a priori estimate nagumo condition upper and lower solutions and leray schauder degree and we give an example to demonstrate our result

bgg algebras and the bgg reciprocity principle the bgg reciprocity principle is a famous result in the representation theary of complex semisimple lie algebras proved by bernstein gelfand and gelfand in bgg 21 it may be viewed as an analogue for the category 6 of the brauer reciprocity principle in the modular representation theory of finite groups and of humphreys reciprocity principle in the representation theory of restricted lie algebras indeed the result of humphreys served as motivation for the bgg result similar results have since been proved in a number of other settings by various approaches for all these results one works in a suitable category of modules in which each simple module l has a projective cover p in addition there is a family of intermediate modules mi such that each p has a filtration with m s as quotients then the appropriate reciprocity principle states that m occurs as a filtration quotient in p as often as l occurs as a composition factor in mi in the case of the category 0 the simple modules are the simple highest weight modules the category is introduced as a setting for projective covers to exist and the intermediate modules are the verma modules the proof of bgg reciprocity in bgg 21 is fairly straightforward once the basic facts about the category 0 are developed the crucial point is the construction of enough projective modules in 0 once these are constructed the existence of projective covers the existence of suitable filtrations on these and the reciprocity principle are ail proved with reference to the construction however another proof of reciprocity exists which is more natural in the sense that it makes no further reference to the construction of projective modules this proof can be used in some of the other settings as well although in any given setting no drastic simplifications result i believe it is still worthwhile to draw attention to this more conceptual approach the properties of the category 0 which are needed for the proof may be

on a common generalization of koszul duality and tilting equivalence abstract we propose a new definition of koszulity for graded algebras where the degree zero part has finite global dimension but is not necessarily semi simple the standard koszul duality theorems hold in this setting

existence result for a third order ode with nonlinear boundary conditions in presence of a sign type nagumo control abstract in this work we provide an existence and location result for the third order nonlinear differential equation u t f t u t u t u t where f a b r 3 r is a continuous function and two types of boundary conditions u a a u b u b 0 u a b or u a a u a u a 0 u b c with r 2 r continuous functions monotonous in the second variable and a b c r we assume that f satisfies a sign type nagumo condition which allows an asymmetric unbounded behaviour on the nonlinearity the arguments used concern leray schauder degree theory and lower and upper solutions technique

koszul duality patterns in representation theory the aim of this paper is to work out a concrete example as well as to provide the general pattern of applications of koszul duality to repre sentation theory the paper consists of three parts relatively independent of each other the first part gives a reasonably selfcontained introduction to koszul rings and koszul duality koszul rings are certain z graded rings with particularly nice homological properties which involve a kind of duality thus to a koszul ring one associates naturally the dual koszul ring the second part is devoted to an application to representation theory of semisimple lie algebras we show license or copyright restrictions may apply to redistribution see http www ams org journal terms of use koszul duality patterns 527 that the block of the bernstein gelfand gelfand category o that corresponds to any fixed central character is governed by the koszul ring moreover the dual of that ring governs a certain subcategory of the category o again this generalizes the selfduality theorem conjectured by beilinson and ginsburg in 1986 and proved by soergel in 1990 in the third part we study certain cate gories of mixed perverse sheaves on a variety stratified by affine linear spaces we provide a general criterion for such a category to be governed by a koszul ring in the flag variety case this reduces to the setup of part two in the more general case of affine flag manifolds and affine grassmannians the criterion should yield interesting results about representations of quantum groups and affine lie algebras department of mathematics massachusetts institute of technology cambridge massachusetts 02139 e mail address sasha math mit edu department of mathematics the university of chicago chicago illinois 60637 e mail address ginzburg math uchicago edu max planck institut fur mathematik gottfried claren strase 26 d 53 bonn 3 germany current address mathematisches institut universitat freiburg albertstrase 23b d 79104 freiburg germany e mail address soergel sun1 mathematik uni freiburg de license or copyright restrictions may apply to redistribution see http www ams org journal terms of use

n cohomology of simple highest weight modules on walls and purity letg p b be respectively a complex connected linear algebraic semisimple group a parabolic subgroup and a borel subgroup the first main result is the following theorem let be a pure complex ong b smooth with respect to bruhat cells then its restriction to anyp orbit is pure as well of the same weight as a consequence we are able to compute then cohomology of simple highest weight modules on walls

tilting modules of finite projective dimension tilting modules are well known and are useful in the representation theory of artinian algebras 2 4 6 11 the purpose of this paper is to define tilting modules of finite projective dimension and prove some fundamental results a unital left a module a t over a ring a is said to be a tilting module of projective dimension 0 is an integer if it satisfies the following three conditions 1 a t has a projective resolution

quasi hereditary algebras and generalized koszul duality abstract we present an easily applicable sufficient condition for standard koszul algebras to be koszul with respect to if a quasi hereditary algebra is koszul with respect to then and the yoneda extension algebra of are koszul dual in a sense explained below implying in particular that their bounded derived categories of finitely generated graded modules are equivalent we also prove that the extension algebra of is koszul in the classical sense

analysis of the response matrix for an extended target in this paper we study the response matrix obtained from the interelement response of an active array of transducers that can send out signals and record reflected signals in particular we analyze the eigenvalues and eigenvectors of the response matrix corresponding to the acoustic field reflected by an extended target the size of which is comparable to the wavelength we show that the eigenvalues are not well separated for a single extended target in general however when both the size of the target and the size of the active array are small compared to the distance from the array to the target it is shown that the eigenvalues are well separated and that the leading eigenvalues and eigenvectors can be characterized in terms of the location and dimension of the target numerical experiments are presented to verify the analysis

on modules with trivial self extensions throughout this note all rings will be self basic connected artin algebras over a fixed commutative local artin ring k and all modules will be finitely generated homomorphisms will be written on the opposite side of the scalars by mod a resp a mod we denote the category of all right resp left a modules for an artin algebra a the ordinary duality functor is denoted by d mod a 2 a mod i e d hom f with the minimal injective cogenerator i over k we are interested in the bimodules bta with the following properties

existence and uniqueness of solutions for third order nonlinear boundary value problems in this paper we present some general results of existence and uniqueness of solutions of nonlinear two point boundary value problems for third order nonlinear differential equations by using the shooting method as applications we give certain concrete sufficient conditions for the existence and uniqueness

symplectic reflection algebras calogero moser space and deformed harish chandra homomorphism to any finite group sp v of automorphisms of a symplectic vector space v we associate a new multi parameter deformation h of the algebra v smash product of with the polynomial algebra on v the parameter runs over points of r where r number of conjugacy classes of symplectic reflections in the algebra h called a symplectic reflection algebra is related to the coordinate ring of a poisson deformation of the quotient singularity v this leads to a symplectic analogue of mckay correspondence which is most complete in case of wreath products if is the weyl group of a root system in a vector space and v then the algebras h are certain rational degenerations of the double affine hecke algebra introduced earlier by cherednik

equivariant sheaves on flag varieties we show that the borel equivariant derived category of sheaves on the flag variety of a complex reductive group is equivalent to the perfect derived category of differential graded modules over the extension algebra of the direct sum of the simple equivariant perverse sheaves this proves a conjecture of soergel and lunts in the case of flag varieties

positive solutions of singular third order three point boundary value problems abstract the positive solutions of a class of singular third order three point boundary value problems are considered by using the guo krasnosel skii fixed point theorem of cone expansion compression type in this class of problems the nonlinear term is allowed to be singular main results show that this class of problems can have n positive solutions provided that the conditions on the nonlinear term on some bounded sets are appropriate

analysis of the time reversal operator for scatterers of finite size recently it was shown that the time reversal operator for a single small spherical scatterer could have up to four distinguishable eigenstates chambers and gautesen j acoust soc am 109 2616 2624 2001 in this paper that analysis is generalized for scatterers of arbitrary shape and larger size it is shown that the time reversal operator may have an indefinitely large number of distinguishable eigenstates with the exact number depending on the nature of the scatterer and the geometry of the time reversal mirror in addition the case of a multiple number of well separated scatterers is investigated with the result that the total spectrum is the direct combination of the eigenstates associated with each scatterer as an example the singular value spectrum of the time reversal operator for a linear array is calculated explicitly for bubbles and hard rubber spheres of finite size both resonance peaks and apparent crossing points can be observed in the spectrum as the size of the scatterer increases

the weyl extension algebra of gl2 f p abstract we compute the yoneda extension algebra of the collection of weyl modules for g l 2 over an algebraically closed field of characteristic p 0 by developing a theory of generalised koszul duality for certain 2 functors one of which controls the rational representation theory of g l 2 over such a field

on monotone solution of the third order differential equation the existence and uniqueness of monotone solution of linear differential equation of the third order are proved using ky fan fixed point theorem the existence of such solution of nonlinear equation of the third order is obtained

spatio temporal invariants of the time reversal operator the decomposition of the time reversal operator known by the french acronym dort is a technique to extract point scatterers monochromatic green s functions from a medium it is used to detect locate and focus on scatterers in various domains such as underwater acoustics medical ultrasound and nondestructive evaluation a limitation of the method arises from its single frequency nature when the signals used in acoustics are often broadband reconstruction of the broadband green s functions from the single frequency green s functions can be very difficult when numerous scatterers are present in the medium moreover the method does not take advantage of the axial resolution associated with broadband signals time domain methods are investigated here as an answer to these problems it is shown that the time reversal operator in the time domain takes the form of a tensor the properties of the invariants are discussed it is shown they do not have all the expected properties another method is proposed that requires a priori information on the medium

fourth order problems with fully nonlinear boundary conditions differential inequality method bounding function method and topological degree are applied to obtain the existence criterions of at least one solution for the general fourth order differential equations under nonlinear boundary conditions and many existing results are complemented

existence of positive solutions for nonlinear third order three point boundary value problems abstract this paper is concerned with the following nonlinear third order three point boundary value problem u t a t f u t 0 t 0 1 u 0 u 0 0 u 1 u where 0 1 and 1 1 first the green s function for the associated linear boundary value problem is constructed and then some useful properties of the green s function are obtained by a new method finally existence results for at least one positive solution for the above problem are established when f is superlinear or sublinear

a fundamentally simple approach to a singular boundary value problem abstract existence results by means of a simple approach for the 3rd order differential equation x t t f t x t x t x t 0 t 1 satisfying the three point boundary value conditions x 0 x x 1 0 are given where 0 1 is a fixed point in contrast to the usual case 1 2 1 f t x y z 0 for any t 0 1 x 0 y z r and the map t is positive on 0 1 and suitable singular at the end points furthermore the obtained solution x t is positive and concave on 0 1 in addition without any monotonicity assumption on the nonlinearity we prove the existence of a sequence of such solutions with lim n x n 0 our principal tools are very simple applications on the plane of the well known from combinatorial topology sperner s lemma in comparison with other classical approaches like fixed point or degree theorems

green s function estimation in speckle using the decomposition of the time reversal operator application to aberration correction in medical imaging the fdort method french acronym for decomposition of the time reversal operator using focused beams is a time reversal based method that can detect point scatterers in a heterogeneous medium and extract their green s function it is particularly useful when focusing in a heterogeneous medium this paper generalizes the theory of the fdort method to random media speckle and shows that it is possible to extract green s functions from the speckle signal using this method therefore it is possible to achieve a good focusing even if no point scatterers are present moreover a link is made between fdort and the van cittert zernike theorem it is deduced from this interpretation that the normalized first eigenvalue of the focused time reversal operator is a well known focusing criterion the concept of an equivalent virtual object is introduced that allows the random problem to be replaced by an equivalent deterministic problem and leads to an intuitive understanding of fdort in speckle applications to aberration correction are presented the reduction of the variance of the green s function estimate is discussed finally it is shown that the method works well in the presence of strong interfering scatterers

singular perturbations of third order nonlinear differential equations with full nonlinear boundary conditions in this paper we discuss singular perturbations of third order nonlinear ordinary differential equations with full nonlinear boundary conditions the emphasis here is that the nonlinear term depends on the first second order derivatives and the boundary conditions are full nonlinear that is where the main novelty of this work lies by applying the upper and lower solutions method as well as analysis technique the existence uniqueness results for the singularly perturbed boundary value problem are established and asymptotic estimates of solutions is also obtained

the method of lower and upper solutions for second third fourth and higher order boundary value problems abstract in this paper we develop the monotone method in the presence of lower and upper solutions for the problem u n t t u t u i a u i b i r i 0 n 1 where is a caratheodory function we obtain necessary and sufficient conditions in to guarantee the existence of solutions between a lower solution and an upper solution for n 2 if n 3 either or and n 4 if furthermore we obtain sufficient conditions in for n 2 k 6 when