a test of static equilibrium models and rates of return to quasi fixed factors with an application to the bell system abstract the paper provides a statistical test to assess the adequacy of the static equilibrium framework which can be used prior to specifying a fully dynamic equilibrium model we develop and apply this test to the production structure and factor demand equations based on the theory of restricted cost functions this diagnostic test can also be used in other areas of applied economics such as consumer demand and labor supply studies where the analyst faces the choice between static and dynamic models of economic behavior the framework is used to calculate rates of return to quasi fixed factors to measure the extent of over and underinvestment in such factors to adjust measured productivity growth for departures from static equilibrium and to provide measures of marginal tobin s q for individual assets that could be used to model investment functions for those assets

rigidity for anosov actions of higher rank lattices this is the simplest example of a large class of analytic standard actions of lattices in semisimple lie groups on locally homogeneous spaces a basic problem is to understand the differentiable actions near to such a standard action in terms of their geometry and dynamics cf 13 50 51 a cr action so if x x x of a group f on a compact manifold x is said to be anosov if there exists at least one element yh e f such that p yh is an anosov diffeomorphism of x we begin in this paper to study the anosov differentiable actions of lattices including many standard algebraic examples and especially to study their stability properties our main theme is that either the cr rigidity or the cr deformation rigidity of an anosov action for 1 r cr or even for the real analytic case can be shown just by studying the behavior of the periodic orbits for the action there are two notions of structural stability that appear in this paper rigidity and deformation rigidity a cl perturbation of a cr action so is simply another cr action spl such that for a finite set of generators 81 ad of f the cr diffeomorphisms p 8i and pl 8i are cl close for all i an action so is said to be cr rigid or topologically rigid if r 0 if every sufficiently small c perturbation of so is cr conjugate to so for 0 r oo or r w in the case of real analytic actions a cl deformation of an action so is a continuous path of cr actions pt defined for some 0 t e with spo sd an action sd is said to be cr deform

semi global exponential stability of augmented primal dual gradient dynamics for constrained convex optimization abstract primal dual gradient dynamics that find saddle points of a lagrangian have been widely employed for handling constrained optimization problems building on existing methods we extend the augmented primal dual gradient dynamics aug pdgd to incorporate general convex and nonlinear inequality constraints and we establish its semi global exponential stability when the objective function is strongly convex we also provide an example of a strongly convex quadratic program of which the aug pdgd fails to achieve global exponential stability numerical simulation also suggests that the exponential convergence rate could depend on the initial distance to the kkt point

the topological trace formula the topological trace formula is a computation of the lefschetz number of a hecke correspondence c acting on the weighted cohomology groups defined in ghm of a locally symmetric space x it expresses this lefschetz number as a sum of contributions from fixed point components of c on the reductive borel serre compactifi cation of x the proof uses the lefschetz fixed point formula of gm2

stability and robustness for saddle point dynamics through monotone mappings abstract motivated by recent interest in saddle point dynamics where given a convex in x and concave in y function trajectories follow the steepest descent in x and the steepest ascent in y and where convergence of trajectories to saddle points is desired this note revisits the maximal monotone mapping approach to saddle point dynamics mildly improves one convergence result and proposes new results on the robustness of pointwise asymptotic stability of the set of saddle points the results apply to nonsmooth convex concave functions and constraints and as a special case to projected saddle point dynamics

hodge cohomology of gravitational instantons we study the space of l2 harmonic forms on complete manifolds with metrics of fibred boundary or fibred cusp type these metrics generalize the geometric structures at infinity of several different well known classes of metrics including asymptotically locally euclidean manifolds the known types of gravitational instantons and also poincare metrics on q rank 1 ends of locally symmetric spaces and on the complements of smooth divisors in kahler manifolds the answer in all cases is given in terms of intersection cohomology of a stratified compactification of the manifold the l2 signature formula implied by our result is closely related to the one proved by dai 25 and more generally by vaillant 67 and identifies dai s invariant directly in terms of intersection cohomology of differing perversities this work is also closely related to a recent paper of carron 12 and the forthcoming paper of cheeger and dai 17 we apply our results to a number of examples gravitational instantons among them arising in predictions about l2 harmonic forms in duality theories in string theory

tests on the validity of static equilibrium models abstract this paper provides a framework to test the validity of static cost minimizing equilibrium assumptions that form the basis for much of the empirical literature on industrial production the point of departure in our model is to allow the observed technology to be at a short run equilibrium where firms minimize variable costs while being constrained by the utilization levels of quasi fixed factors the long run equilibrium is then inferred by minimizing total costs with the optimal adjustment of quasi fixed factor levels we use results from the optimization problem to form tests with respect to quantities and prices in the quantity space version departures between the actual and the optimal long run levels of quasi fixed factors are tested for statistical significance a significant non zero departure implies the rejection of a static equilibrium specification in the price space version the test is cast as a comparison of the market price and the long run shadow value of a quasi fixed factor although the two versions would give identical results in the non stochastic case the rejection powers of these two tests are found to depend on the particular functional form chosen to represent the production process i e cost function in an application based on aggregate u s manufacturing where capital is taken to be quasi fixed we were able to reject the static equilibrium specification these results cast doubt on the validity of a number of previous empirical studies

simplicial differential forms with poles shadow forms consider a simplicial complex k whitney associated to each simplicial cochain 4 a differential form v 4 on k whitney s map v from simplicial cochains to differential forms is a map of cochain complexes which induces the derham isomorphism we call the forms v 4 whitney forms they have been useful in several contexts most notably integral geometry and rational homotopy theory in this paper we introduce a generalization of whitney forms called shadow forms we associate to certain simplicial chains g with respect to the first derived subdivision k of k a shadow form w on k let us illustrate the construction of shadow forms when k is a single n simplex which we now denote by a in this case w g is defined for k simplices g of ax which do not lie in the boundary of a or for linear combinations of these for example if a is the 2 simplex then either of the 1 simplices or 2 pictured below has an associated shadow form

infinitesimal rigidity for the action of sl n ℤ on n let r sl n z or any subgroup of finite index then the action of r on tn by automorphisms is infinitesimally rigid for n 7 i e the cohomology h i r vec tn 0 where vec t n denotes the module of c vector fields on t

smooth classification of cartan actions of higher rank semisimple lie groups and their lattices let g be a connected semisimple lie group without compact factors whose real rank is at least 2 and let g be an irreducible lattice we provide a c classification for volume preserving cartan actions of an d g also if g has real rank at least 3 we provide a c classification for volume preserving multiplicity free trellised anosov actions on compact manifolds

online learning of feasible strategies in unknown environments define an environment as a set of convex constraint functions that vary arbitrarily over time and consider a cost function that is also convex and arbitrarily varying agents that operate in this environment intend to select actions that are feasible for all times while minimizing the cost s time average such action is said optimal and can be computed offline if the cost and the environment are known a priori an online policy is one that depends causally on the cost and the environment to compare online policies to the optimal offline action define the fit of a trajectory as a vector that integrates the constraint violations over time and its regret as the cost difference with the optimal action accumulated over time fit measures the extent to which an online policy succeeds in learning feasible actions while regret measures its success in learning optimal actions this paper proposes the use of online policies computed from a saddle point controller which are shown to have fit and regret that are either bounded or grow at a sublinear rate these properties provide an indication that the controller finds trajectories that are feasible and optimal in a relaxed sense concepts are illustrated throughout with the problem of a shepherd that wants to stay close to all sheep in a herd numerical experiments show that the saddle point controller allows the shepherd to do so

distributed linear programming with event triggered communication we consider a network of agents whose objective is for the aggregate of their states to converge to a solution of a linear program in standard form each agent has limited information about the problem data and can communicate with other agents at discrete time instants of their choosing our main contribution is the synthesis of a distributed dynamics and a set of state based rules termed triggers that individual agents use to determine when to opportunistically broadcast their state to neighboring agents to ensure asymptotic convergence to a solution of the linear program our technical approach to the algorithm design and analysis overcomes a number of challenges including establishing convergence in the absence of a common smooth lyapunov function ensuring that the triggers are detectable by agents using only local information accounting for asynchronism in the state broadcasts and ruling out various causes of arbitrarily fast state broadcasting various simulations illustrate our results

distributed gradient algorithm for constrained optimization with application to load sharing in power systems abstract in this paper a distributed constrained optimization problem is discussed to achieve the optimal point of the sum of agents local objective functions while satisfying local constraints here neither the local objective function nor local constraint functions of each agent can be shared with other agents to solve the problem a novel distributed continuous time algorithm is proposed by using the kkt condition combined with the lagrangian multiplier method and the convergence is proved with the help of lyapunov functions and an invariance principle for hybrid systems furthermore this distributed algorithm is applied to optimal load sharing control problem in power systems both theoretical and numerical results show that the optimal load sharing can be achieved within both generation and delivering constraints in a distributed way

the impact of stochastic noisy feedback on distributed network utility maximization the implementation of distributed network utility maximization num algorithms hinges heavily on information feedback through message passing among network elements in practical systems the feedback is often obtained using error prone measurement mechanisms and suffers from random errors in this paper we investigate the impact of noisy feedback on distributed num we first study the distributed num algorithms based on the lagrangian dual method and focus on the primal dual p d algorithm which is a single time scale algorithm in the sense that the primal and dual parameters are updated simultaneously assuming strong duality we study both cases when the stochastic gradients are unbiased or biased and develop a general theory on the stochastic stability of the p d algorithms in the presence of noisy feedback when the gradient estimators are unbiased we establish via a combination of tools in martingale theory and convex analysis that the iterates generated by distributed p d algorithms converge with probability one to the optimal point under standard technical conditions in contrast when the gradient estimators are biased we show that the iterates converge to a contraction region around the optimal point provided that the biased terms are asymptotically bounded by a scaled version of the true gradients we also investigate the rate of convergence for the unbiased case and find that in general the limit process of the interpolated process corresponding to the normalized iterate sequence is a stationary reflected linear diffusion process not necessarily a gaussian diffusion process we apply the above general theory to investigate stability of cross layer rate control for joint congestion control and random access next we study the impact of noisy feedback on distributed two time scale num algorithms based on primal decomposition we establish via the mean ode method the convergence of the stochastic two time scale algorithm under mild conditions for the cases where the gradient estimators in both time scales are unbiased numerical examples are used to illustrate the finding that compared to the single time scale counterpart the two time scale algorithm although having lower complexity is less robust to noisy feedback

parametric tests for static and dynamic equilibrium econometric estimates of the characteristics of production are highly sensitive to the choice of dynamic specification in this paper we propose a new approach for testing whether inputs are in static equilibrium the procedure which is based on the use of a generalized cost function tests for divergences between the actual and static equilibrium quantities of all inputs simultaneously and symmetrically we apply it to two widely used data sets one at a high level of aggregation and the other at the firm level and strongly reject static equilibrium in both cases

distributed convergence to nash equilibria in two network zero sum games this paper considers a class of strategic scenarios in which two networks of agents have opposing objectives with regard to the optimization of a common objective function in the resulting zero sum game individual agents collaborate with neighbors in their respective network and have only partial knowledge of the state of the agents in the other network for the case when the interaction topology of each network is undirected we synthesize a distributed saddle point strategy and establish its convergence to the nash equilibrium for the class of strictly concave convex and locally lipschitz objective functions we also show that this dynamics does not converge in general if the topologies are directed this justifies the introduction in the directed case of a generalization of this distributed dynamics which we show converges to the nash equilibrium for the class of strictly concave convex differentiable functions with globally lipschitz gradients the technical approach combines tools from algebraic graph theory nonsmooth analysis set valued dynamical systems and game theory

on the reductive borel serre compactification l p cohomology of arithmetic groups for large p the l 2 cohomology of an arithmetic quotient of an hermitian symmetric space i e of a locally symmetric variety is known to have the topological interpretation as the intersection homology of its baily borel satake compactification in this article we observe that even without the hermitian hypothesis the l p cohomology of an arithmetic quotient for p finite and sufficiently large is isomorphic to the ordinary cohomology of its reductive borel serre compactification we use this to generalize a theorem of mumford concerning homogeneous vector bundles their invariant chern forms and the canonical extensions of the bundles here though we are referring to canonical extensions to the reductive borel serre compactification of any arithmetic quotient to achieve that we give a systematic discussion of vector bundles and chern classes on stratified spaces

relative artin motives and the reductive borel serre compactification of a locally symmetric variety we introduce the notion of artin motives and cohomological motives over a scheme x given a cohomological motive m over x we consider the universal artin motive mapping to m and denote it omega 0 x m we use this to define a motive mathbb e x over x which is an invariant of the singularities of x the first half of the paper is devoted to the study of the functors omega 0 x and the computation of the motives mathbb e x

dynamical properties of hybrid automata hybrid automata provide a language for modeling and analyzing digital and analogue computations in real time systems hybrid automata are studied here from a dynamical systems perspective necessary and sufficient conditions for existence and uniqueness of solutions are derived and a class of hybrid automata whose solutions depend continuously on the initial state is characterized the results on existence uniqueness and continuity serve as a starting point for stability analysis lyapunov s theorem on stability via linearization and lasalle s invariance principle are generalized to hybrid automata

toroidal and reductive borel serre compactifications of locally symmetric spaces by hermitian locally symmetric space we mean an arithmetic quotient of a bounded symmetric domain both the toroidal and the reductive borel serre compactifications of such a space come equipped with canonical mappings to the baily borel satake compactification in this article we show that there is a mapping from the toroidal compactification to the reductive borel serre compactification whose composition with the projection to the baily borel compactification agrees with the canonical projection up to an arbitrarily small homotopy we also consider arithmetic quotients of a self adjoint homogeneous cone there is a canonical mapping from the reductive borel serre compactification to the standard compactification of such a locally symmetric cone we show that this projection when restricted to the closure of a polyhedral cone has contractible fibers

lefschetz properties for noncompact arithmetic ball quotients we prove a lefschetz property for the restriction of the cohomology of noncompact arithmetic ball quotients to ball quotients of smaller dimension

stability of primal dual gradient dynamics and applications to network optimization this paper considers dynamic laws that seek a saddle point of a function of two vector variables by moving each in the direction of the corresponding partial gradient this method has old roots in the classical work of arrow hurwicz and uzawa on convex optimization and has seen renewed interest with its recent application to resource allocation in communication networks this paper brings other tools to bear on this problem in particular krasovskii s method to find lyapunov functions and recently obtained extensions of the lasalle invariance principle for hybrid systems these methods are used to obtain stability proofs of these primal dual laws in different scenarios and applications to cross layer network optimization are exhibited

asymptotic convergence of constrained primal dual dynamics this paper studies the asymptotic convergence properties of the primal dual dynamics designed for solving constrained concave optimization problems using classical notions from stability analysis we motivate the need for this study by providing an example that rules out the possibility of employing the invariance principle for hybrid automata to study asymptotic convergence we understand the solutions of the primal dual dynamics in the caratheodory sense and characterize their existence uniqueness and continuity with respect to the initial condition we use the invariance principle for discontinuous caratheodory systems to establish that the primal dual optimizers are globally asymptotically stable under the primal dual dynamics and that each solution of the dynamics converges to an optimizer

on the reductive borel serre compactification ii excentric quotients and least common modifications let x be a locally symmetric variety i e the quotient of a bounded symmetric domain by a say neat arithmetically defined group of isometries let overline x rm exc and x rm tor exc denote its excentric borel serre and toroidal compactifications respectively we determine their least common modification and use it to prove a conjecture of goresky and tai concerning canonical extensions of homogeneous vector bundles in the process we see that overline x rm exc and x rm tor exc are homotopy equivalent

l2 index and the selberg trace formula abstract a method is developed for computing the l 2 index of a locally symmetric elliptic differential operator d acting on a locally symmetric manifold m g k with g semisimple of real rank one and of finite co volume based on applying the selberg trace formula to the difference of the two heat kernels associated to d the applications include an extension of the osborne warner multiplicity formula to certain non integrable discrete series derived from the l 2 spinor formula and showing the existence in some cases of non invariant l 2 cohomology classes in the middle dimension via the l 2 signature formula

equilibrium independent dissipativity with quadratic supply rates equilibrium independent dissipativity eid is a recently introduced system property that requires a system to be dissipative with respect to any forced equilibrium configuration this paper is a detailed examination of eid with quadratic supply rates for a common class of nonlinear control affine systems we provide an algebraic characterization of eid for such systems in the spirit of the hill moylan lemma where the usual stability condition is replaced by an incremental stability condition based on this characterization we state results concerning internal stability feedback stability and absolute stability of eid systems finally we study eid for discrete time systems providing the relevant definitions and an analogous hill moylan type characterization results for both continuous time and discrete time systems are illustrated through examples on physical systems and convex optimization algorithms

sur la l 2 cohomologie des varietes a courbure negative we give a topological interpretation of the space of l2 harmonic forms on finite volume manifolds with sufficiently pinched negative curvature we give examples showing that this interpretation fails if the curvature is not sufficiently pinched and that our result is sharp with respect to the pinching constants the method consists first in comparing l2 cohomology with weighted l2 cohomology thanks to previuos works done by t ohsawa and then in identifying these weighted spaces

intersection homology ii in 19 20 we introduced topological invariants ih x called intersection homology groups for the study of singular spaces x these groups depend on the choice of a perversity p a perversity is a function from 2 3 to the non negative integers such that both c and c 2 c are positive and increasing functions of c 2 1 the group ihr is defined for spaces x called pseudomanifolds a pseudomanifold of dimension n is a space that admits a stratification x x n x n z x n 3 x 1 x 0

the role of convexity in saddle point dynamics lyapunov function and robustness this paper studies the projected saddle point dynamics associated to a convex concave function which we term saddle function the dynamics consists of gradient descent of the saddle function in variables corresponding to convexity and projected gradient ascent in variables corresponding to concavity we examine the role that the local and or global nature of the convexity concavity properties of the saddle function plays in guaranteeing convergence and robustness of the dynamics under the assumption that the saddle function is twice continuously differentiable we provide a novel characterization of the omega limit set of the trajectories of this dynamics in terms of the diagonal blocks of the hessian using this characterization we establish global asymptotic convergence of the dynamics under local strong convexity concavity of the saddle function when strong convexity concavity holds globally we establish three results first we identify a lyapunov function that decreases strictly along the trajectory for the projected saddle point dynamics when the saddle function corresponds to the lagrangian of a general constrained convex optimization problem second for the particular case when the saddle function is the lagrangian of an equality constrained optimization problem we show input to state stability iss of the saddle point dynamics by providing an iss lyapunov function third we use the latter result to design an opportunistic state triggered implementation of the dynamics various examples illustrate our results

on livsic s theorem superrigidity and anosov actions of semisimple lie groups we prove a generalization of livsic s theorem on the vanishing of the cohomology of certain types of dynamical systems as a consequence we strengthen a result due to zimmer concerning algebraic hulls of anosov actions of semisimple lie groups combining this with topological superrigidity we find a holder geometric structure for multiplicity free anosov actions

the cauchy riemann equations on product domains we establish the l 2 theory for the cauchy riemann equations on product domains provided that the cauchy riemann operator has closed range on each factor we deduce regularity of the canonical solution on p 1 forms in special sobolev spaces represented as tensor products of sobolev spaces on the factors of the product this leads to regularity results for smooth data

foliated stratified spaces and a de rham complex describing intersection space cohomology the method of intersection spaces associates cell complexes depending on a perversity to certain types of stratified pseudomanifolds in such a way that poincare duality holds between the ordinary rational cohomology groups of the cell complexes associated to complementary perversities the cohomology of these intersection spaces defines a cohomology theory hi for singular spaces which is not isomorphic to intersection cohomology ih mirror symmetry tends to interchange ih and hi the theory ih can be tied to type iia string theory while hi can be tied to iib theory for pseudomanifolds with stratification depth 1 and flat link bundles the present paper provides a de rham theoretic description of the theory hi by a complex of global smooth differential forms on the top stratum we prove that the wedge product of forms introduces a perversity internal cup product on hi for every perversity flat link bundles arise for example in foliated stratified spaces and in reductive borel serre compactifications of locally symmetric spaces a precise topological definition of the notion of a stratified foliation is given

hodge decompositions and dolbeault complexes on normal surfaces give the smooth subset of a normal singular complex projective surface the metric induced from the ambient projective space the l2 cohomology of this incomplete manifold is isomorphic to the surface s intersection cohomology which has a natural hodge decomposition this paper identifies dolbeault complexes whose 0 closed and 0 coclosed forms represent the classes of pure type in the corresponding hodge decomposition of l2 cohomology

unitary spectrum and the fundamental group for actions of semisimple lie groups suppose a locally compact group g acts on a separable metrizable space m preserving a finite measure then there is an associated unitary representation of g on lz m o the square integrable functions orthogonal to the constants this representation will in general be far from irreducible and in fact when g is not compact need not have any irreducible subrepresentations thus in the decomposition of lz g o into irreducible constituents one not only requires a direct integral rather than a direct sum but the associated spectral measure on the unitary dual g say for type i groups will except in special circumstances have no atoms the point of this paper is to give natural and general conditions mostly involving just rq m under which there will be an infinite atomic spectrum i e the spectral measure has infinitely many atoms the basic example of actions in which the spectrum is purely atomic i e when the representation is a direct sum of irreducible representations is the case of m g f where f c g is a cocompact discrete subgroup in this case assuming 9 g to be simply connected we have 7tl m f the first results of this paper show that when g is a simple lie group with r rank g 2 we can deduce that any action of g on a compact manifold m with fundamental group isomorphic to such a f and satisfying a natural hypothesis of either a geometric or ergodic theoretic nature must also have an infinite atomic spectrum in fact under certain natural hypotheses the same is true as long as there is a surjection of hi m onto f to formulate this more precisely let f c ad g be a cocompact lattice and let pr be the unitary representation of ad g on l2 ad g f o we recall that an action of a group on a manifold or more generally on a standard space in the sense of 4 is called engaging if there is no loss of ergodicity in passing to finite covers see i 4 for discussion

fixed time stable gradient flows applications to continuous time optimization this paper proposes novel gradient flow schemes that yield convergence to the optimal point of a convex optimization problem within a fixed time from any given initial condition for unconstrained optimization constrained optimization and min max problems it is shown that the solution of the modified gradient flow dynamics exists and is unique under certain regularity conditions on the objective function while fixed time convergence to the optimal point is shown via lyapunov based analysis the application of the modified gradient flow to unconstrained optimization problems is studied under the assumption of gradient dominance a relaxation of strong convexity then a modified newton s method is presented that exhibits fixed time convergence under some mild conditions on the objective function building upon this method a novel technique for solving convex optimization problems with linear equality constraints that yields convergence to the optimal point in fixed time is developed finally the general min max problem is considered and a modified scheme to obtain the optimal solution of saddle point dynamics in fixed time is developed

intersection numbers of hecke cycles on hilbert modular varieties let o be the ring of integers of a totally real number field e and set g res e q gl 2 fix an ideal c 0 for each ideal m 0 let t m denote the mth hecke operator associated to the standard compact open subgroup u 0 c of g a f setting x 0 c g q g a k u 0 c where k is a certain subgroup of g r we use t m to define a hecke cycle z m ih 2 e q x 0 c x 0 c here ih denotes intersection homology we use zucker s conjecture proven by looijenga and independently by saper and stern to obtain a formula relating the intersection number z m z n to the trace of t m o t n considered as an endomorphism of the space of hilbert cusp forms on u 0 c

ex post tests for short and long run optimization abstract the purpose of this paper is to develop tests of long run equilibrium models where the long run is defined as the state where the observed levels of output and capital are consistent with optimizing behavior we use the implied relationship between a restricted translog cost function and derived demand equations to provide a series of nested tests which can be interpreted as ex post tests for short and long run optimization behavior a set of parameter restrictions on the translog system are specified for testing whether observable levels of capital and or output of a firm correspond to the long run optimal level our tests of the validity of long run equilibrium specifications are based on data for 28 german industries for the period 1960 1981 for most of the industries the set of restrictions implied by long run equilibrium had to be rejected

exponential convergence of distributed primal dual convex optimization algorithm without strong convexity abstract this paper establishes exponential convergence rates for a class of primal dual gradient algorithms in distributed optimization without strong convexity the convergence analysis is based on a carefully constructed lyapunov function by evaluating metric subregularity of the primal dual gradient map we present a general criterion under which the algorithm achieves exponential convergence to facilitate practical applications of this criterion several simplified sufficient conditions are derived we also prove that although these results are developed for the continuous time algorithms they carry over in a parallel manner to the discrete time algorithms constructed by using euler s approximation method

the proximal augmented lagrangian method for nonsmooth composite optimization we study a class of optimization problems in which the objective function is given by the sum of a differentiable but possibly nonconvex component and a nondifferentiable convex regularization term we introduce an auxiliary variable to separate the objective function components and utilize the moreau envelope of the regularization term to derive the proximal augmented lagrangian a continuously differentiable function obtained by constraining the augmented lagrangian to the manifold that corresponds to the explicit minimization over the variable in the nonsmooth term the continuous differentiability of this function with respect to both primal and dual variables allows us to leverage the method of multipliers mm to compute optimal primal dual pairs by solving a sequence of differentiable problems the mm algorithm is applicable to a broader class of problems than proximal gradient methods and it has stronger convergence guarantees and a more refined step size update rules than the alternating direction method of multipliers admm these features make it an attractive option for solving structured optimal control problems we also develop an algorithm based on the primal descent dual ascent gradient method and prove global exponential asymptotic stability when the differentiable component of the objective function is strongly convex and the regularization term is convex finally we identify classes of problems for which the primal dual gradient flow dynamics are convenient for distributed implementation and compare contrast our framework to the existing approaches

cohomologie l 2 des varietes qale we give a topological interpretation of the space of l 2 harmonis forms of some qale manifolds introduced by d joyce we introduce a analytical criterium which make possible the used of mayer vietoris sequence

weighted cohomology of arithmetic groups let g be a semisimple algebraic group defined over the rational numbers k a maximal compact subgroup ofg g r and g q a neat arithmetic subgroup letx g k be the locally symmetric space associated to and e the local system on x constructed out of a finite dimensional irreducible algebraic representation e of g fix a maximally q split torus s in g s is assumed to be nontrivial so that x is noncompact let a s r 0 and by a slight abuse of notation let x a denote the group of rational characters of s choose a minimal rational parabolic subgroup p0 s the choice determines a notion of positivity in x a and a set of positive roots among the roots of s in g let 0 be the half sum of the positive roots weighted cohomology is an invariant of introduced by m goresky g harder and r macpherson 12 in the study 15 16 12 14 of the trace of hecke operators in the cohomology of for each p x a q there are weight profiles p and p upper and lower p see 1 5 and groups w ph i e and w h i e the notion of profile here differs from but is equivalent to that of 12 see remark 1 2 below if p q 2 0 then the profiles p and q are dual and the corresponding groups are in poincare duality for very positive p one gets the compactly supported cohomology h i c x e and dually for very negative p the full cohomology h i x e the definition of the weighted cohomology groups is somewhat involved so i will attempt to give some idea of the motivation behind it detailed definitions are recalled in section 1 associated to p and p are complexes of sheaves wpc e and wpc e on the reductive borel serre compactification x of x let p be a rational parabolic subgroup np its unipotent radical sp the split centre of p np and xp the p boundary component in x the idea behind the construction in 12 is as follows the stalk cohomology of the direct image ri e by i x x at x xp is