the primal dual active set strategy as a semismooth newton method this paper addresses complementarity problems motivated by constrained optimal control problems it is shown that the primal dual active set strategy which is known to be extremely efficient for this class of problems and a specific semismooth newton method lead to identical algorithms the notion of slant differentiability is recalled and it is argued that the max function is slantly differentiable in lp spaces when appropriately combined with a two norm concept this leads to new local convergence results of the primal dual active set strategy global unconditional convergence results are obtained by means of appropriate merit functions

representation dimension and finitely generated cohomology abstract we consider selfinjective artin algebras whose cohomology groups are finitely generated over a central ring of cohomology operators for such an algebra we show that the representation dimension is strictly greater than the maximal complexity occurring among its modules this provides a unified approach to computing lower bounds for the representation dimension of group algebras exterior algebras and artin complete intersections we also obtain new examples of classes of algebras with arbitrarily large representation dimension

complexity and tor on a complete intersection let r m be a complete intersection that is a local ring whose m adic completion is the quotient of a regular local ring by a regular sequence supposemis a finitely generatedr module it is known that the even and odd betti sequences ofmare eventually given by polynomials of the same degreen the complexity ofmis the nonnegative integern 1 we use this notion of complexity to study the vanishing of torri m n for finitely generated modulesmandnover a complete intersectionr we prove several theorems dealing with rigidity of tor which are generalizations and in certain situations improvements of known results the main idea of these rigidity theorems is that the number of consecutive vanishing tors required in the hypothesis of a rigidity theorem depends more on the minimum of the complexities ofmandnrather than on the codimension ofr we give examples showing that this dependence is sharp we also show that ifm rnhas finite length then for sufficiently high indices two consecutive vanishing tors force the vanishing of all higher tors

syzygies and tensor product of modules we give an application of the new intersection theorem and prove the following let r be a local complete intersection ring of codimension c and let m and n be nonzero finitely generated r modules assume n is a nonnegative integer and that the tensor product m otimes r n is an n c th syzygy of some finitely generated r module if mathrm tor r 0 m n 0 then both m and n are n th syzygies of some finitely generated r modules

the varieties and the cohomology ring of a module let g be a finite group and let k be a field of characteristic p 0 recently several studies have focused on the homological invariants of finitely generated kg modules in l alperin proposed the study of the complexity of a module the complexity is related to the degree of the polynomial rate of growth of the terms in a projective resolution of the module see definition 2 4 in 2 alperin and evens proved that the complexity of a kg module m is equal to the maximum of the complexities of the restrictions of m to elementary abelian p subgroups of g one of the roots of the alperin evens theorem is quillen s dimension theorem see 21 or 22 which with some difficulty can be interpreted as saying the same thing for the special case of the trivial kg module alperin and evens 3 and avrunin 4 have further studied the annihilator in ext k k of the cohomology of m and have produced theorems of a similar nature these are related to the work in 9 on the structure of the cohomology ring ext m m the purpose of this paper is to investigate the relationship between these homological invariants and the structure of the module we concentrate on the case in which g is an elementary abelian p group however using such results as those mentioned above we can apply the present work to more general groups even as the earlier results indicate that much of the structure and cohomology of a module is revealed in the restrictions to elementary abelian p subgroups we show here that in the case of an elementary abelian group much of this information can be found by looking at the restrictions to cyclic p subgroups of the group of units of kg some of the results in this paper were announced in s let g xl xn be an elementary abelian group of order p suppose that k is algebraically closed to any finitely generated kg module m we

an all at once approach for the optimal control of the unsteady burgers equation we apply an all at once method for the optimal control of the unsteady burgers equation the nonlinear burgers equation is discretized in time using the semi implicit discretization and the resulting first order optimality conditions are solved iteratively by newton s method the discretize then optimize approach is used because it leads to a symmetric indefinite saddle point problem numerical results for the distributed unconstrained and control constrained problems illustrate the performance of the all at once approach with semi implicit time discretization

ergodicity of the stochastic fractional reaction diffusion equation abstract a stochastic reaction diffusion equation with fractional dissipation is considered the smaller the exponent of the equation is the weaker the dissipation of the equation is the equation is discussed in detail when the exponent changes the aim is to prove the well posedness existence and uniqueness of an invariant measure as well as strong law of large numbers and convergence to equilibrium without the analytic property of the semigroup some methods are used to overcome the difficulties to get the energy estimates the results in this paper can be applied to the classic reaction diffusion equation with wiener noise

equivariant isomorphisms of ext and tor modules abstract in this article we establish equivariant isomorphisms of ext and tor modules over different relative complete intersections more precisely for a commutative ring q this paper investigates how ext q f m n and tor q f m n change when one varies f among all koszul regular sequences of a fixed length such that f m 0 and f n 0 of notable interest is how the theory of perturbations is used to establish isomorphisms of certain dg modules

g dimension complete intersection dimension complexity and relative homological algebra abstract it is shown that the g dimension and the complete intersection dimension are relative projective dimensions relative auslander buchsbaum formulas are discussed new cohomology theories called complexity cohomology are constructed the new theories play the same role in identifying rings and modules with prescribed complexity as tate vogel cohomology does in identifying modules of finite projective dimension

strong solutions of stochastic equations with singular time dependent drift we prove existence and uniqueness of strong solutions to stochastic equations in domains with unit diffusion and singular time dependent drift b up to an explosion time we only assume local l q l p integrability of b in g with d p 2 q 1 we also prove strong feller properties in this case if b is the gradient in x of a nonnegative function blowing up as g x g we prove that the conditions 2d t k 2d t ke 0 2 imply that the explosion time is infinite and the distributions of the solution have sub gaussian tails

support varieties an axiomatic approach we provide an axiomatic approach for studying support varieties of objects in a triangulated category via the action of a tensor triangulated category where the tensor product is not necessarily symmetric this is illustrated by examples taken in particular from the representation theory of finite dimensional algebras

cohomology of twisted tensor products abstract it is well known that the cohomology of a tensor product is essentially the tensor product of the cohomologies we look at twisted tensor products and investigate to which extent this is still true we give an explicit description of the ext algebra of the tensor product of two modules and under certain additional conditions describe an essential part of the hochschild cohomology ring of a twisted tensor product as an application we characterize precisely when the cohomology groups over a quantum complete intersection are finitely generated over the hochschild cohomology ring moreover both for quantum complete intersections and in related cases we obtain a lower bound for the representation dimension of the algebra

gradient estimates and applications for sdes in hilbert space with multiplicative noise and dini continuous drift abstract consider the stochastic evolution equation in a separable hilbert space h with a nice multiplicative noise and a locally dini continuous drift we prove that for any initial data the equation has a unique possibly explosive mild solution under a reasonable condition ensuring the non explosion of the solution the strong feller property of the associated markov semigroup is proved gradient estimates and log harnack inequalities are derived for the associated semigroup under certain global conditions which are new even in finite dimensions

modules with reducible complexity abstract for a commutative noetherian local ring we define and study the class of modules having reducible complexity a class containing all modules of finite complete intersection dimension various properties of this class of modules are given together with results on the vanishing of homology and cohomology

well posedness of semilinear stochastic wave equations with holder continuous coefficients abstract we prove that semilinear stochastic abstract wave equations including wave and plate equations are well posed in the strong sense with an holder continuous drift coefficient if 2 3 1 the uniqueness may fail for the corresponding deterministic pde and well posedness is restored by adding an external random forcing of white noise type this shows a kind of regularization by noise for the semilinear wave equation to prove the result we introduce an approach based on backward stochastic differential equations we also establish regularizing properties of the transition semigroup associated to the stochastic wave equation by using control theoretic results

on the growth of the betti sequence of the canonical module we study the growth of the betti sequence of the canonical module of a cohen macaulay local ring it is an open question whether this sequence grows exponentially whenever the ring is not gorenstein we answer the question of exponential growth affirmatively for a large class of rings and prove that the growth is in general not extremal as an application of growth we give criteria for a cohen macaulay ring possessing a canonical module to be gorenstein

1 torsion of finite modules over semiperfect rings abstract we initiate the study of 1 torsion of finite modules over two sided noetherian semiperfect rings in particular we give a criterion for determining when the 1 torsion submodule contains minimal generators of the module we also provide an explicit construction for a projective cover of the submodule generated by the torsion elements in the top of the module some of the obtained results hold without the noetherian assumption we also give several applications to local algebra

the variety of an indecomposable module is connected let g be a finite group and let k be an algebraically closed field of characteristic p 0 the connections between the modular representation theory of finite groups and cohomology theory is currently a topic of great interest in particular cohomology theory has been used to attach to each finitely generated kgmodule m a homogeneous affine variety v m if we let 17 m be the corresponding projective variety then our main result is the following

pathwise uniqueness for a class of sde in hilbert spaces and applications abstract an abstract evolution equation in hilbert spaces is considered in the deterministic case it includes several examples of non uniqueness it is proved that pathwise uniqueness is restored by means of a suitably non degenerate additive noise the proof is based on the associated infinite dimensional kolmogorov equation

quillen stratification for modules let g be a finite group and k a fixed algebraically closed field of characteristic p o if p is odd let h be the subring of h g k consisting of elements of even degree following 20 221 we take h h g k if p 2 though one could just as well use the subring of elements of even degree for all p h is a finitely generated commutative k algebra 13 and we let v denote its associated affine variety max h if m is any finitely generated kg module then the cohomology variety v m of m may be defined as the support in v of the h module h g m if g is a p group and in general as the largest support of h g l m where l is any kg module 4 91 a module l with each irreducible kc module as a direct summand will serve d quillen 20 221 proved a number of beautiful results relating k to the varieties v associated with the various elementary abelian p subgroups e of g culminating in his stratification theorem 20 221 this theorem gives a piecewise description of v almost explicitly in terms of the subgroups e and their normalizers in g a well known corollary is that dim v max dim v where e

well posedness of the transport equation by stochastic perturbation we consider the linear transport equation with a globally holder continuous and bounded vector field with an integrability condition on the divergence while uniqueness may fail for the deterministic pde we prove that a multiplicative stochastic perturbation of brownian type is enough to render the equation well posed this seems to be the first explicit example of a pde of fluid dynamics that becomes well posed under the influence of a multiplicative noise the key tool is a differentiable stochastic flow constructed and analyzed by means of a special transformation of the drift of ito tanaka type

homology of perfect complexes abstract it is proved that the sum of the loewy lengths of the homology modules of a finite free complex f over a local ring r is bounded below by a number depending only on r this result uncovers in the structure of modules of finite projective dimension obstructions to realizing r as a closed fiber of some flat local homomorphism other applications include as special cases uniform proofs of known results on free actions of elementary abelian groups and of tori on finite cw complexes the arguments use numerical invariants of objects in general triangulated categories introduced here and called levels they allow one to track through changes of triangulated categories homological invariants like projective dimension as well as structural invariants like loewy length an intermediate result sharpens with a new proof the new intersection theorem for commutative algebras over fields under additional hypotheses on the ring r stronger estimates are proved for loewy lengths of modules of finite projective dimension

homological algebra modulo a regular sequence with special attention to codimension two abstract let m be a finite module over a ring r obtained from a commutative ring q by factoring out an ideal generated by a regular sequence the homological properties m over r and over q are intimately related their links are analyzed here from the point of view of differential graded homological algebra over a koszul complex that resolves r over q one outcome of this approach is a transparent derivation of some central results of the theory another is a new insight into codimension two phenomena yielding an explicit finitistic construction of the generally infinite minimal r free resolution of m it leads to theorems on the structure and classification of finite modules over codimension two local complete intersections that are exact counterparts of eisenbud s results for modules over hypersurfaces

growth of hilbert coefficients of syzygy modules abstract let a m be a local complete intersection ring of dimension d and let i be an m primary ideal let m be a maximal cohen macaulay a module for i 0 1 d let e i i m denote the i th hilbert coefficient of m with respect to i we prove that for i 0 1 2 the function j e i i syz j a m is of quasi polynomial type with period 2 let g i m be the associated graded module of m with respect to i if g i a is cohen macaulay and dim a 2 we also prove that the functions j depth g i syz 2 j i a m are eventually constant for i 0 1 let i m lim l depth g i l m finally we prove that if dim a 2 and g i a is cohen macaulay then the functions j i syz 2 j i a m are eventually constant for i 0 1

huneke ulrich almost complete intersections of cohen macaulay type two abstract let r m k be a commutative noetherian local ring n 2 an integer x a 2n 1 2n 1 alternating matrix with entries from m y a 1 2n 1 matrix with entries from m i the ideal i i1 yx and a the quotient ring r i assume that the grade of i is at least 2n in this case i is a perfect ideal of grade equal to 2n and i is minimally generated by 2n 1 elements we prove that the minimal resolution of a by free r modules is a dg algebra furthermore we identify the algebra torr a k and prove that if r is regular and char k 0 or n 2 2 char k then the poincare series pma z i 0 dimk torai m k zi is a rational function for every finitely generated a module m as a consequence we deduce that if the projective dimension of m is infinite then eventually the betti numbers of m form an increasing sequence with strong exponential growth

solving optimal control problems for the unsteady burgers equation in comsol multiphysics the optimal control of unsteady burgers equation without constraints and with control constraints are solved using the high level modelling and simulation package comsol multiphysics using the first order optimality conditions projection and semi smooth newton methods are applied for solving the optimality system the optimality system is solved numerically using the classical iterative approach by integrating the state equation forward in time and the adjoint equation backward in time using the gradient method and considering the optimality system in the space time cylinder as an elliptic equation and solving it adaptively the equivalence of the optimality system to the elliptic partial differential equation pde is shown by transforming the burgers equation by the cole hopf transformation to a linear diffusion type equation numerical results obtained with adaptive and nonadaptive elliptic solvers of comsol multiphysics are presented both for the unconstrained and the control constrained case

asymptotic behavior of ext functors for modules of finite complete intersection dimension let r be a local ring and let m and n be finitely generated r modules such that m has finite complete intersection dimension in this paper we define and study under certain conditions a pairing using the modules rm ext r i m n which generalizes buchweitz s notion of the herbrand difference we exploit this pairing to examine the number of consecutive vanishing of rm ext r i m n needed to ensure that rm ext r i m n 0 for all i gg 0 our results recover and improve on most of the known bounds in the literature especially when r has dimension at most two

free resolutions and change of rings abstract projective resolutions of modules over a ring r are constructed starting from appropriate projective resolutions over a ring q mapping to r it is shown that such resolutions may be chosen to be minimal in codimension 2 but not in codimension 3 this is used to obtain minimal resolutions for essentially all modules over local or graded rings r with codimension 2 explicit resolutions are given for cyclic modules over multigraded rings and necessary and sufficient conditions are obtained for their minimality

vanishing theorems for complete intersections the starting point for this work is auslander s seminal paper a the main focus of his paper was to understand when the tensor product m rn of finitely generated modules m and n over a regular local ring r is torsion free this condition forces the vanishing of a certain tor module associated to m and n which in turn by auslander s famous rigidity theorem implies that tori m n 0 for all i 1 the vanishing of tor carries a great deal of information for example it implies the depth formula depth m depth n dim r depth m r n from this formula one can deduce for example the highly non trivial fact that if m r n is torsion free and non zero then m and n must both be torsion free in hw huneke and wiegand generalized some of auslander s results to hypersurfaces though with some unavoidable extra hypotheses in particular they proved a rigidity theorem hw 2 4 and the following vanishing theorem

degenerate sdes with singular drift and applications to heisenberg groups abstract by using the ultracontractivity of a reference diffusion semigroup krylov s estimate is established for a class of degenerate sdes with singular drifts which leads to existence and pathwise uniqueness by means of zvonkin s transformation the main result is applied to singular sdes on generalized heisenberg groups

distributed optimal control of time dependent diffusion convection reaction equations using space time discretization we apply two different strategies for solving unsteady distributed optimal control problems governed by diffusion convection reaction equations in the first approach the optimality system is transformed into a biharmonic equation in the space time domain the system is then discretized in space and time simultaneously and solved by an equation based finite element package i e comsol multiphysics the second approach is a classical gradient based optimization method to solve the state and adjoint equations and the optimality condition iteratively the convection dominated state and adjoint equations are stabilized using the streamline upwind petrov galerkin supg method numerical results show favorable accuracy and efficiency of the two strategies for unstabilized and stabilized numerical solutions

the derived category of a locally complete intersection ring abstract in this paper we answer a question of dwyer greenlees and iyengar by proving a local ring r is a complete intersection if and only if every complex of r modules with finitely generated homology is proxy small moreover we establish that a commutative noetherian ring r is locally a complete intersection if and only if every complex of r modules with finitely generated homology is virtually small

jacobian criteria for complete intersections the graded case letp be a positively graded polynomial ring over a fieldk of characteristic zero leti be a homogeneous ideal ofp and setr p i the paper investigates the homological properties of somer modules canonically associated withr among them the module r k of kahler differentials and the conormal modulei i 2 it is shown that a subexponential bound on the betti numbers of any of these modules implies thati is generated by ap regular sequence in particular the finiteness of the projective dimension of the conormal module impliesr is a complete intersection similarly the finiteness of the projective dimension of the differential module impliesr is a reduced complete intersection this provides strong converses to some well known properties of complete intersections and establishes special cases of conjectures of vasconcelos the proofs of these results make extensive use of differential graded homological algebra the crucial step is to show that any homomorphism of complexes from the minimal cotangent complexl r k of andre and quillen into the minimal free resolution of the irrelevant maximal ideal m ofr which extends the euler map r k is a split embedding of gradedr modules

existence of strong solutions for ito s stochastic equations via approximations given strong uniqueness for an ito s stochastic equation with discontinuous coefficients we prove that its solution can be constructed on any probability space by using for example euler s polygonal approximations stochastic equations in d and in domains in d are considered

complexity and elementary abelian p groups one of the main differences between ordinary and modular representation theory of finite groups is the existence of nonprojective modules in the modular representation algebra however given a finitely generated module m over the group algebra kg where k is a field of characteristic p 0 there exists a uniquely determined up to kg isomorphism projective module p called the projective cover of m and an epimorphism f

boundedness versus periodicity over commutative local rings over commutative graded local artinian rings examples are constructed of periodic modules of arbitrary minimal period and modules with bounded betti numbers which are not eventually periodic they provide counterexamples to a conjecture of d eisenbud that every module with bounded betti numbers over a commutative local ring is eventually periodic of period 2 it is proved however that the conjecture holds over rings of small length

on the stochastic burgers equation in the real line this is the publisher s version also available electronically from http projecteuclid org euclid aop 1022677386 abstract

on the regularization effect of space time white noise on quasi linear parabolic partial differential equations we show that an existence and uniqueness and a comparison theorem hold if we add a space time white noise to a quasi linear parabolic equation in one space dimension even if the nonlinearity is only measurable and not even locally bounded

support varieties and cohomology over complete intersections quillen s geometric approach to the cohomology of finite groups 30 has revolutionized modular representation theory the ideology and techniques involved have been generalized and extended to representations of various hopf algebras culminating in the recent work of friedlander and suslin 17 on finite group schemes in this paper we develop geometric methods for the study of finite modules over a local complete intersection r if r is such a ring and m is its maximal ideal then the m adic completion r has the form q f where f is a regular sequence and q is a regular local ring that can be taken to be a ring of formal power series over a field or a discrete valuation ring the least number of equations needed to cut out r from a regular local ring equals the codimension of r where codim r r m dim r and r m denotes the minimal number of generators of an r module m in 5 a cone that is a homogeneous algebraic set vr m is attached to each finite r module m and used to study its minimal free resolution here we prove

stable cohomology over local rings abstract for a commutative noetherian ring r with residue field k stable cohomology modules ext r n k k have been defined for each n z but their meaning has remained elusive it is proved that the k rank of any ext r n k k characterizes important properties of r such as being regular complete intersection or gorenstein these numerical characterizations are based on results concerning the structure of z graded k algebra carried by stable cohomology it is shown that in many cases it is determined by absolute cohomology through a canonical homomorphism of algebras ext r k k ext r k k some techniques developed in the paper are applicable to the study of stable cohomology functors over general associative rings

sobolev differentiable flows of sdes with local sobolev and super linear growth coefficients by establishing a characterization for sobolev differentiability of random fields we prove the weak differentiability of solutions to stochastic differential equations with local sobolev and super linear growth coefficients with respect to the starting point moreover we also study the strong feller property and the irreducibility to the associated diffusion semigroup

three themes of syzygies we present three exciting themes of syzygies where major progress was made recently boij soderberg theory stillman s question and syzygies over complete intersections

tor and torsion on a complete intersection abstract let r m be a complete intersection that is a ring whose m adic completion is the quotient of a regular local ring by a regular sequence supposemandnare finitely generatedr modules we give a necessary condition for the vanishing of torri m n for alli 0 in terms of the intersection of certain affine algebraic sets associated tomandn we apply this condition to the study of torsion in tensor products for example we show that ifris a domain andmis anr module of infinite projective dimension then there exist infinitely manynfor which the tensor product ofmwith one of itsnth syzygy modules has torsion we also give a sufficient condition for the vanishing torri m n for alli 0 in terms of the ability to leftmandnto disjoint complete intersections of smaller codimension we use this condition to construct tensor products of non free modules which are maximal cohen macaulay