weak strong uniqueness of dissipative measure valued solutions for polyconvex elastodynamics for the equations of elastodynamics with polyconvex stored energy and some related simpler systems we define a notion of a dissipative measure valued solution and show that such a solution agrees with a classical solution with the same initial data when such a classical solution exists as an application of the method we give a short proof of strong convergence in the continuum limit of a lattice approximation of one dimensional elastodynamics in the presence of a classical solution also for a system of conservation laws endowed with a positive and convex entropy we show that dissipative measure valued solutions attain their initial data in a strong sense after time averaging

a differential algebraic riccati equation for applications in flow control we investigate existence and structure of solutions to quadratic control problems with semiexplicit differential algebraic constraints as they appear in the modeling of flows we introduce a decoupled representation of the state and the formal adjoint equations to identify the conditions for the existence of solutions we derive a differential algebraic riccati equation formulated in the original coefficients for the efficient numerical computation of the optimal solution

quasiconvex elastodynamics weak strong uniqueness for measure valued solutions a weak strong uniqueness result is proved for measure valued solutions to the system of conservation laws arising in elastodynamics the main novelty brought forward by the present work is that the underlying stored energy function of the material is assumed strongly quasiconvex the proof employs tools from the calculus of variations to establish general convexity type bounds on quasiconvex functions and recasts them in order to adapt the relative entropy method to quasiconvex elastodynamics

a regularity theorem for minimizers of quasiconvex integrals we prove c1 partial regularity for minimizers of functionals with quasiconvex integrand f x u du depending on vector valued functions u the integrand is required to be twice continuously differentiable in du and no assumption on the growth of the derivatives of f is made a polynomial growth is required only on f itself

strongly absolute stability of lur e type differential algebraic systems in this paper we consider lur e type differential algebraic systems lds and introduce the concept of strongly absolute stability such a notion is a generalization of absolute stability for lur e type standard state space systems lss by a lur e type lyapunov function we derive an lmi based stability criterion for lds to be strongly absolutely stable using extended strictly positive realness espr we present the frequency domain interpretation of the obtained criterion by which we simplify the criterion and show that the criterion is a generalization of the well known popov criterion finally we illustrate the effectiveness of the main results by a numerical example

a variational approximation scheme for three dimensional elastodynamics with polyconvex energy we construct a variational approximation scheme for the equations of three dimensional elastodynamics with polyconvex stored energy the scheme is motivated by some recently discovered geometric identities qin 18 for the null lagrangians the determinant and cofactor matrix and by an associated embedding of the equations of elastodynamics into an enlarged system which is endowed with a convex entropy the scheme decreases the energy and its solvability is reduced to the solution of a constrained convex minimization problem we prove that the approximating process admits regular weak solutions which in the limit produce a measure valued solution for polyconvex elastodynamics that satisfies the classical weak form of the geometric identities this latter property is related to the weak continuity properties of minors of jacobian matrices here exploited in a time dependent setting

w1 p quasiconvexity and variational problems for multiple integrals abstract variational problems for the multiple integral i u g u x dx where r m and u r n are studied a new condition on g called w1 p quasiconvexity is introduced which generalizes in a natural way the quasiconvexity condition of c b morrey it being shown in particular to be necessary for sequential weak lower semicontinuity of i in w 1 p r n and for the existence of minimizers for certain related integrals counterexamples are given concerning the weak continuity properties of jacobians in w 1 p r n p n m an existence theorem for nonlinear elastostatics is proved under optimal growth hypotheses

covers and envelopes in grothendieck categories flat covers of complexes with applications abstract in the general setting of grothendieck categories with enough projectives we prove theorems that make possible to restrict the study of the problem of the existence of f covers and envelopes to the study of some properties of the class f we then prove the existence of flat covers and cotorsion envelopes of complexes giving some examples this generalizes the earlier work j algebra201 1998 86 102 and finishes the problem of the existence of flat covers of complexes

oscillations and concentrations in weak solutions of the incompressible fluid equations the authors introduce a new concept of measure valued solution for the 3 d incompressible euler equations in order to incorporate the complex phenomena present in limits of approximate solutions of these equations one application of the concepts developed here is the following important result a sequence of leray hopf weak solutions of the navier stokes equations converges in the high reynolds number limit to a measure valued solution of 3 d euler defined for all positive times the authors present several explicit examples of solution sequences for 3 d incompressible euler with uniformly bounded local kinetic energy which exhibit complex phenomena involving both persistence of oscillations and development of concentrations an extensions of the concept of young measure is developed to incorporate these complex phenomena in the measure valued solutions constructed here

explicit characterization oflp young measures the full explicit description of young measures attainable by bounded sequences from the lebesgue spacelp rm was known so far for the casep only in the paper a suitable condition characterizing such measures is isolated also for the case 1 p a generalization of thelp young measures constructed recently by diperna and majda can be thus described also but in special cases only

homological dimensions and regular rings abstract a question of avramov and foxby concerning injective dimension of complexes is settled in the affirmative for the class of noetherian rings a key step in the proof is to recast the problem on hand into one about the homotopy category of complexes of injective modules analogous results for flat dimension and projective dimension are also established

cohomology theories based on flats abstract let a be an associative ring with identity k flat a the homotopy category of flat modules and k p flat a the full subcategory of pure complexes the quotient category k flat a k p flat a called here the pure derived category of flats was introduced by neeman in this category flat resolutions are unique up to homotopy and so can be used to compute cohomology we develop theories of tate and complete cohomology in the pure derived category of flats these theories extend naturally to sheaves over semi separated noetherian schemes where there are not always enough projectives but we do have enough flats as applications we characterize rings with finite sfli and schemes which are locally gorenstein

optimal control of causal differential algebraic systems abstract existence theory of filippov roxin type as well as a maximum principle for optimal control problem governed by nonlinear differential algebraic equations of index 1 or 2 or 3 are formulated and proved the index 3 case is illustrated on mechanical descriptor systems arising in robotics

gorenstein injective complexes of modules over noetherian rings abstract a complex c is called gorenstein injective if there exists an exact sequence of complexes i 1 i 0 i 1 such that each i i is injective c ker i 0 i 1 and the sequence remains exact when hom e is applied to it for any injective complex e we show that over a left noetherian ring r a complex c of left r modules is gorenstein injective if and only if c m is gorenstein injective in r mod for all m z also gorenstein injective dimensions of complexes are considered

optimal control problems for singular systems singular systems of the form e xdot x u t are considered where e is a square matrix and may be singular it is assumed that for any admissible initial state x t 0 any control u t u yields one and only one continuous state x t and there is one and only one continuous adjoint state t the formulae for functional variation are derived the necessary condition for optimality the maximum principle is obtained the boundary conditions for the adjoint equations of the singular systems are given and the necessary and sufficient condition for optimality of linear singular systems is derived

on pure acyclic complexes abstract in this paper we study the pure acyclic complexes of modules we obtain several characterizations of these complexes extending results that are known for the pure acyclic complexes of flat modules in particular we extend neeman s characterization 18 of the pure acyclic complexes of flat modules in terms of complexes of projective modules by considering more generally complexes of pure projective modules as a consequence we obtain simson s result 22 on the pure projectivity of pure periodic modules

weak strong uniqueness for measure valued solutions we prove the weak strong uniqueness for measure valued solutions of the incompressible euler equations these were introduced by diperna and majda in their landmark paper commun math phys 108 4 667 689 1987 where in particular global existence to any l2 initial data was proven whether measure valued solutions agree with classical solutions if the latter exist has apparently remained open

on the relation between k flatness and k projectivity abstract krause 19 has proved that the homotopy category k r pureproj of pure projective modules over an associative ring r is compactly generated and equivalent to the verdier quotient k r t r where t r is the triangulated subcategory of the homotopy category k r consisting of the pure acyclic complexes we provide an alternative proof of this result when restricted to the homotopy category of projective modules this equivalence reduces to that obtained by neeman 26 as an application of this description of k r pureproj we show that the k flat complexes defined by spaltenstein 29 are up to homotopy equivalence the filtered colimits of k projective complexes of pure projective modules

rank one convexity does not imply quasiconvexity we consider variational integrals defined for sufficiently regular functions u r m here is a bounded open subset of r n du x denotes the gradient matrix of u at x and f is a continuous function on the space of all real m n matrices m m n one of the important problems in the calculus of variations is to characterise the functions f for which the integral i is lower semicontinuous in this connection the following notions were introduced see 3 9 10

some adjoints in homotopy categories let r be a ring in a previous paper 11 we found a new description for the category k r proj it is equivalent to the verdier quotient k r flat s for some suitable s c k r flat in this article we show that the quotient map from k r flat to k r flat s always has a right adjoint this gives a new fully faithful embedding of k r proj into k r flat its virtue is that it generalizes to nonaffine schemes

flat covers of complexes abstract in this article we define and study flat complexes over any ring also we prove that any complex over a commutative noetherian ring with finite krull dimension has a flat cover and a dg flat cover

minimal semi flat cotorsion replacements and cosupport abstract over a commutative noetherian ring r of finite krull dimension we show that every complex of flat cotorsion r modules decomposes as a direct sum of a minimal complex and a contractible complex moreover we define the notion of a semi flat cotorsion complex as a special type of semi flat complex and provide functorial ways to construct a quasi isomorphism from a semi flat complex to a semi flat cotorsion complex consequently every r complex can be replaced by a minimal semi flat cotorsion complex in the derived category over r furthermore we describe structure of semi flat cotorsion replacements by which we recover classic theorems for finitistic dimensions in addition we improve some results on cosupport and give a cautionary example we also explain that semi flat cotorsion replacements always exist and can be used to describe the derived category over any associative ring

necessary conditions for optimal control problems involving nonlinear differential algebraic equations abstract dynamic models which take the form of a coupled set of differential and algebraic equations daes are widely used in process systems engineering necessary conditions of optimality for optimal control problems involving such models are derived a strong maximum principle is obtained under a convexity hypothesis on the velocity set an example illustrates that the strong maximal principle may be violated when this hypothesis is dropped for problems involving nonconvex velocity sets however a weak maximum principle is valid

auslander type conditions and cotorsion pairs we study the properties of rings satisfying auslander type conditions if an artin algebra satisfies the auslander condition that is is an gorenstein artin algebra then we construct two kinds of subcategories which form functorially finite cotorsion pairs

gorenstein flat covers of modules over gorenstein rings abstract a left and right noetherian ring r is called gorenstein if both r r and r r have finite injective dimensions these rings were studied by bass for the commutative case and iwanaga for the noncommutative case in this paper we define gorenstein flat modules over a gorenstein ring these modules are a generalization of flat modules after discussing the properties of modules over a gorenstein ring we prove that every module over a gorenstein ring has a gorenstein flat cover or a minimal right approximation by a gorenstein flat module in auslander and reiten s terminology

triangulated equivalence between a homotopy category and a triangulated quotient category abstract given two complete hereditary cotorsion pairs q r and q r in a bicomplete abelian category g such that q q and q r q r becker showed that there exists a hereditary abelian model structure m q w r on g where w is a thick subcategory of g we prove that the homotopy category ho m of m is triangulated equivalent to the triangulated quotient category d b g q r k b q r where d b g q r is the subcategory of d b g consisting of all homology bounded complexes with both finite q dimension and r dimension and k b q r is the bounded homotopy category of q r core objects applications are given in the category of modules it is shown that the homotopy category of the gorenstein flat resp ding projective and gorenstein ac projective model structure on the category of modules established by gillespie and his coauthors can be realized as a certain triangulated quotient category

boundary regularity and sufficient conditions for strong local minimizers abstract in this paper we present a new proof of the sufficiency theorem for strong local minimizers concerning c 1 extremals at which the second variation is strictly positive the results are presented in the quasiconvex setting in accordance with the original statement by grabovsky and mengesha 31 the strategy that we follow relies on a decomposition theorem that allows to split a sequence of variations into its oscillating and its concentrating parts as well as on a sufficiency result according to which smooth extremals are spatially local minimizers furthermore we prove partial regularity up to the boundary for strong local minimizers in the non homogeneous case and a full regularity result for lipschitz extremals with gradient of vanishing mean oscillation as a consequence we also establish a sufficiency result for this class of extremals the regularity results are established via a blow up argument

the homotopy category of flat modules and grothendieck duality let r be a ring we prove that the homotopy category k r proj is always aleph 1 compactly generated and depending on the ring r it may or may not be compactly generated we use this to give a description of k r proj as a quotient of k r flat the remarkable fact is that this new description of k r proj generalizes to non affine schemes this will appear in murfet s thesis

structural properties of stress relaxation and convergence from viscoelasticity to polyconvex elastodynamics we consider a model of stress relaxation approximating the equations of elastodynamics necessary and sufficient conditions are derived for the model to be equipped with a global free energy and to have positive entropy production the resulting class allows for both convex and non convex equilibrium potentials for convex equilibrium potentials we prove a strong dissipation estimate and two relative energy estimates for the relative entropy of the relaxation process and the modulated relative energy both give convergence results to smooth solutions for polyconvex equilibrium potentials an augmenting of the system of polyconvex elastodynamics and the null lagrangian structure suggest an appropriate notion of relative energy we prove convergence of viscosity approximations to polyconvex elastodynamics in the regime where the limit solution remains smooth a modulated relative energy is also obtained for the polyconvex case which shows stability of relaxation approximations

covers and envelopes over gr gorenstein rings abstract in this paper we study the existence of gorenstein injective envelopes and gorenstein projective and flat covers in the category of graded modules and we relate them with the corresponding envelopes and covers in the category of modules