on the cartan matrix of mackey algebras let k be a field of characteristic p 0 and let g be a finite group the first result of this paper is an explicit formula for the determinant of the cartan matrix of the mackey algebra k g of g over k the second one is a formula for the rank of the cartan matrix of the cohomological mackey algebra co k g of g over k and a characterization of the groups g for which this matrix is nonsingular the third result is a generalization of this rank formula and characterization to blocks of co k g in particular if b is a block of kg the cartan matrix of the corresponding block co k b of co k g is nonsingular if and only if b is nilpotent with cyclic defect groups

the classification of endo trivial modules this paper settles the classification of all endo trivial modules for a finite p group by treating the remaining case of a p group having maximal elementary abelian subgroups of rank 2 it is shown that the modules constructed by alperin using relative syzygies provide the complete list of endo trivial modules in this case

equivalences between blocks of cohomological mackey algebras let g be a finite group and k mathcal o k be a p modular system which is large enough let r mathcal o or k there is a bijection between the blocks of the group algebra rg and the central primitive idempotents the blocks of the so called cohomological mackey algebra co mu r g here we introduce the notion of permeable derived equivalence and we prove that a permeable derived equivalence between two blocks of group algebras implies the existence of a derived equivalence between the corresponding blocks of cohomological mackey algebras in particular in the context of broue s abelian defect group conjecture if two blocks are splendidly derived equivalent then the corresponding blocks of cohomological mackey algebras are derived equivalent

non autonomous second order hamiltonian systems abstract we study the existence of periodic solutions for a second order non autonomous dynamical system containing variable kinetic energy terms our assumptions balance the interaction between the kinetic energy and the potential energy with neither one dominating the other we study sublinear problems and the existence of non constant solutions

entanglement quantum entropy and mutual information the operational structure of quantum couplings and entanglements is studied and classified for semi finite von neumann algebras we show that the classical quantum correspondences such as quantum encodings can be treated as diagonal semi classical d couplings and the entanglements characterized by truly quantum q couplings can be regarded as truly quantum encodings the relative entropy of the d compound and entangled states leads to two different types of entropy for a given quantum state the von neumann entropy which is achieved as the maximum of mutual information over all d entanglements and the dimensional entropy which is achieved at the standard entanglement true quantum entanglement coinciding with a d entanglement only in the case of pure marginal states the d and q information of a quantum noisy channel are respectively defined via the input d and q encodings and the q capacity of a quantum noiseless channel is found to be the logarithm of the dimensionality of the input algebra the quantum capacity may double the classical capacity achieved as the supremum over all d couplings or encodings bounded by the logarithm of the dimensionality of a maximal abelian subalgebra

periodic solutions of delay differential systems via hamiltonian systems abstract with a new definition of the j m index theory developed in liu 2006 we give another application of this index theory in delay differential systems via hamiltonian systems where we do not need the potential function to be c 2 continuous

symplectic topology and hamiltonian dynamics on etudie des applications symplectiques non lineaires capacites symplectiques construction d une capacite symplectique problemes de plongement problemes de rigidite

absolute continuity for operator valued completely positive maps on c algebras motivated by applicability to quantum operations quantum information and quantum probability we investigate the notion of absolute continuity for operator valued completely positive maps on c algebras previously introduced by parthasarathy in athens conference on applied probability and time series analysis i springer verlag berlin 1996 pp 34 54 we obtain an intrinsic definition of absolute continuity we show that the lebesgue decomposition defined by parthasarathy is the maximal one among all other lebesgue type decompositions and that this maximal lebesgue decomposition does not depend on the jointly dominating completely positive map we obtain more flexible formulas for calculating the maximal lebesgue decomposition and we point out the nonuniqueness of the lebesgue decomposition as well as a sufficient condition for uniqueness in addition we consider radon nikodym derivatives for absolutely continuous completely positive maps that in general are unbounded positive self adjoint ope

a continuity theorem for stinespring s dilation abstract we show a continuity theorem for stinespring s dilation two completely positive maps between arbitrary c algebras are close in cb norm if and only if we can find corresponding dilations that are close in operator norm the proof establishes the equivalence of the cb norm distance and the bures distance for completely positive maps we briefly discuss applications to quantum information theory

the dade group of a p group let p be a prime number this paper solves the question of the structure of the group d p of endo permutation modules over an arbitrary finite p group p that was open after dade s original papers in 1978 19 20 and it gives a proof of the conjectures proposed in 4 and 10 this leads to a presentation of d p by explicit generators and relations generalizing the presentation obtained by dade when p is abelian

non convex self dual lagrangians new variational principles of symmetric boundary value problems abstract we study the concept and the calculus of non convex self dual nc sd lagrangians and their derived vector fields which are associated to many partial differential equations and evolution systems they indeed provide new representations and formulations for the superposition of convex functions and symmetric operators they yield new variational resolutions for large class of hamiltonian partial differential equations with variety of linear and nonlinear boundary conditions including many of the standard ones this approach seems to offer several useful advantages it associates to a boundary value problem several potential functions which can often be used with relative ease compared to other methods such as the use of euler lagrange functions these potential functions are quite flexible and can be adapted to easily deal with both nonlinear and homogeneous boundary value problems additionally in most cases the solutions generated using this new method have greater regularity than the solutions obtained using the standard euler lagrange function perhaps most remarkable however are the permanence properties of nc sd lagrangians their calculus is relatively manageable and their applications are quite broad

endo p permutation modules this paper introduces the notion of endo p permutation kg module i give a characterization of indecomposable endo p permutation modules with vertex p via their source modules which are exactly the indecomposable endo permutation modules whose classes in the dade group d p are g stable in particular when the normalizer of p controls p fusion i give a classification of sources of endo p permutation modules using the recent classification of the dade group

a perturbation theory near convex hamiltonian systems soit he x p une famille de fonctions sur r n r n dependant du parametre e r d pour e petit on considere des solutions periodiques aux equations de hamilton dx i dt he x p p i dp i dt he x p x i 1 i et on cherche a les relier aux solutions periodiques du systeme non perturbe dx i dt h 0 x p p i dp i dt h 0 x p x i 1 i

discord of response the presence of quantum correlations in a quantum state is related to the state s response to local unitary perturbations such a response is quantified by the distance between the unperturbed and perturbed states minimized with respect to suitably identified sets of local unitary operations in order to be a bona fide measure of quantum correlations the distance function must be chosen among those that are contractive under completely positive and trace preserving cptp maps the most relevant instances of such physically well behaved metrics include the trace the bures and the hellinger distance to each of these metrics one can associate the corresponding discord of response namely the trace or hellinger or bures minimum distance from the set of unitarily perturbed states all these three discords of response satisfy the basic axioms for a proper measure of quantum correlations in the present work we focus in particular on the bures distance which enjoys the unique property of being both riemannian and contractive under cptp maps and admits important operational interpretations in terms of state distinguishability we compute analytically the bures discord of response for two qubit states with maximally mixed marginals and we compare it with the corresponding bures geometric discord namely the geometric measure of quantum correlations defined as the bures distance from the set of classical quantum states finally we investigate and identify the maximally quantum correlated two qubit states according to the bures discord of response these states exhibit a remarkable nonlinear dependence on the global state purity

equivalences between blocks of p local mackey algebras let g be a finite group and k o k be a p modular system let r o or k there is a bijection between the blocks of the group algebra and the blocks of the so called p local mackey algebra mu 1 r g let b be a block of rg with abelian defect group d let b be its brauer correspondant in n g d it is conjectured by broue that the blocks rgb and rng d b are derived equivalent here we look at equivalences between the corresponding blocks of p local mackey algebras we prove that an analogue of the broue s conjecture is true for the p local mackey algebras in the following cases for the principal blocks of p nilpotent groups and for blocks with defect 1

foncteurs d ensembles munis d une double action resume sommaire 1 introduction 2 notations generalites 3 ensembles munis d une double action 3 1 notations 3 2 formule de mackey 3 3 decomposition d un morphisme 3 4 sous categories 3 4 1 ensembles p q libres 3 4 2 groupes reductifs 4 foncteurs simples 5 une theorie de vortex et de source 6 une seconde theorie de vortex et source 7 le foncteur de burnside 7 1 un homomorphisme deb g dans end g 7 2 les sous foncteurs du foncteur de burnside en caracteristique 0 7 2 1 premier exemple 7 2 2 les constantesmg n 7 2 3 lesb groupes 7 2 4 les sous foncteurs deb 7 2 5 exemples deb groupes 8 exemples de foncteurs simples 8 1 le cas des foncteurs de mackey globaux 8 2 les foncteurssh k 8 3 certains foncteurs simples en caracteristique 0 9 l algebreegg end g egg 9 1 une base naturelle 9 2 multiplication 9 3 semi simplicite 10 le foncteur des representations complexes 10 1 caracteres 10 2 sous foncteurs et decomposition 10 3 semi simplicite

periodic solutions of a class of hamiltonian systems with singularities this paper deals with a class of time periodic hamiltonian systems obtained by a time dependent perturbation from an autonomous system with a singularity at q 0 in configuration space it is shown that t being the period of the perturbation nondegenerate families of t periodic orbits in the unperturbed problem branch off into a certain number of t periodic orbits for the perturbed problem

conley type index applied to hamiltonian inclusions in the paper we develop the theory of a cohomological index of the conley type detecting invariant sets of a multivalued dynamical system generated by semilinear differential inclusion in an infinite dimensional hilbert space an application to the existence of periodic orbits to asymptotically linear hamiltonian inclusions is presented

on a theorem by ekeland and lasry concerning the number of periodic hamiltonian trajectories this note deals with the existence of iz distinct modes of vibration for hamihonian systems with n degrees of freedom on a prescribed energy surface this problem has been studied in a recent remarkable paper by ekeland and lasry 5 who have given an important contribution to the complete extension in the large of the local results of liapunov ri weinstein 111 and moser s the purpose of the present paper is to give a new proof of the result of 5 jointly with a slight improvement

on morita and derived equivalences for cohomological mackey algebras by results of the second author a source algebra equivalence between two p blocks of finite groups induces an equivalence between the categories of cohomological mackey functors associated with these blocks and a splendid derived equivalence between two blocks induces a derived equivalence between the corresponding categories ofcohomological mackey functors the main result of this paper proves a partial converse an equivalence resp rickard equivalence between the categories of cohomological mackey functors of two blocks of finite groups induces a permeable morita resp derived equivalence between the two block algebras

applications of clark duality to periodic solutions with minimal period for discrete hamiltonian systems by making use of clark duality perturbation technique and dual least action principle some results on the existence of subharmonic solutions with minimal period to second order subquadratic discrete hamiltonian systems are obtained

a new variational principle and duality for periodic solutions of hamilton s equations we are dealing with a classical boundary value problem concerning periodic solutions of hamiltonian equations p f k f 4th p f i f h x t p t 1 where h r x r x r r is a given function called the hamiltonian in mechanics 1 and also in mathematics 171 with a given hamiltonian the lagrangian l defined by the legendre transform l 4 x a sup k p wc x pi d 2

on the new variational principles and duality for periodic solutions of lagrange equations with superlinear nonlinearities periodic problem 1 1 was studied in the 1980s by many authors forthe sublinear as well as for the superlinear case see e g 6 however we believe that our paper may contribute some new look at this problem this is because we propose to study 1 1 by duality methods in a waythat is to some extent analogous to the methods developed for 1 1 insublinear cases 2 3 6 8 some cases of 1 1 for superlinear v

operational distance and fidelity for quantum channels we define and study a fidelity criterion for quantum channels which we term the minimax fidelity through a noncommutative generalization of maximal hellinger distance between two positive kernels in classical probability theory like other known fidelities for quantum channels the minimax fidelity is well defined for channels between finite dimensional algebras but it also applies to a certain class of channels between infinite dimensional algebras explicitly those channels that possess an operator valued radon nikodym density with respect to the trace in the sense of belavkin staszewski and induces a metric on the set of quantum channels that is topologically equivalent to the cb norm distance between channels precisely in the same way as the bures metric on the density operators associated with statistical states of quantum mechanical systems derived from the well known fidelity generalized transition probability of uhlmann is topologically equivalent to the trace norm distance

periodic solutions with prescribed minimal period for convex autonomous hamiltonian systems clarke has shown that the problem of findingt periodic solutions for a convex hamiltonian system is equivalent to the problem of finding critical points to a certain functional dual to the classical action functional in this paper we relate the morse index of the critical point to the minimal period of the correspondingt periodic solution in particular we show that if the critical point is obtained by the ambrosetti rabinowitz mountain pass theorem the corresponding solution has minimal periodt that is it cannot bet k periodic withk integer k 2 as a consequence we prove that if the hamiltonian is flat near an equilibrium and superquadratic near infinity then for anyt 0 the corresponding hamiltonian system has a periodic solution with minimal periodt

affirmative answer to a question of feit as he points out if p is abelian the affirmative answer follows from dade s classification 4 our purpose here is to show that the general case reduces to the abelian one which answers affirmatively feit s question more generally in 4 dade determines the cap group associated with a finite abelian p group showing in particular that this group is finitely generated our main result here cf theorem 2 2 below implies that finite generation is still true in the non abelian case cf corollary 2 4 below this result and the main steps of a proof were already announced in 17 section81 and here we are just discharging ourselves of an old debt actually as we said in 7 section81 our result would follow from a complete classification of endo permutation modules

the dade group of mackey functors for p groups abstract we introduce endo permutation mackey functors and the mackey dade group of a p group and study their basic properties we exhibit relations between the mackey dade group of a finite p group p and the dade groups of the weyl groups of all subgroups of p as a result we show that the mackey dade group tensored with q over z is the kernel of the linearization map for mackey functors introduced by the author

the classification of torsion endo trivial modules the main purpose of this paper is to show that if g is a finite p group which is not cyclic quaternion or semi dihedral then the group of endo trivial g modules is detected on restriction to elementary abelian p subgroups of rank 2 consequently the torsion subgroup of the group of endo trivial modules is trivial for any finite p group except in the known cases of cyclic quaternion and semi dihedral groups this requires a large amount of group cohomology recent bounds for the number of bocksteins in serre s theorem carlson s recent theorem expressing serre s theorem in terms of modules and finally the theory of varieties attached to modules a detection theorem for the torsion subgroup of the dade group of all endo permutation modules is deduced from the main theorem the structure of this torsion subgroup can then be fully described when p is odd using the results of bouc thevenaz

a remark on the dade group and the burnside group abstract the object of this note is to show that the formula for tensor induction of relative syzygies in the dade group stated in s bouc j reine angew math 523 2000 113 can be viewed as a special case of a functorial homomorphism from the dual b of the burnside group to the subgroup d of the dade group generated by relative syzygies it follows that there exists a short exact sequence of functors 0 r q b d d tors 0 where r q is the functor of rational representations this may be viewed as an improvement from q to z of theorem d of invent math 139 2000 275

disguising quantum channels by mixing and channel distance trade off we consider the reverse problem of the distinguishability of two quantum channels which we call the disguising problem given two quantum channels the goal here is to make the two channels identical by mixing with some other channels with minimal mixing probabilities this quantifies how much one channel can disguise as the other in addition the possibility to trade off between the two mixing probabilities allows one channel to be more preserved less mixed at the expense of the other we derive lower and upper bounds of the trade off curve and apply them to a few example channels optimal trade off is obtained in one example we relate the disguising problem and the distinguishability problem by showing that the former can lower and upper bound the diamond norm we also show that the disguising problem gives an upper bound on the key generation rate in quantum cryptography

existence of multiple normal mode trajectories on convex energy surfaces of even classical hamiltonian systems hamiltonian systems of n degrees of freedom for which the hamiltonian is a function that is even both in its joint n coordinate variables as well as in its joint n momentum variables are discussed for such systems the number of distinct trajectories which correspond to particular periodic solutions normal modes with the same energy is investigated to that end a constrained dual action principle is introduced applying min max methods to this variational problem several results are obtained among which the existence of at least n distinct trajectories if specific conditions are satisfied

the information disturbance tradeoff and the continuity of stinespring s representation stinespring s dilation theorem is the basic structure theorem for quantum channels it states that any quantum channel arises from a unitary evolution on a larger system here we prove a continuity theorem for stinespring s dilation if two quantum channels are close in cb norm then it is always possible to find unitary implementations which are close in operator norm with dimension independent bounds this result generalizes uhlmann s theorem from states to channels and allows to derive a formulation of the information disturbance tradeoff in terms of quantum channels as well as a continuity estimate for the no broadcasting theorem we briefly discuss further implications for quantum cryptography thermalization processes and the black hole information loss puzzle

equivariant morse theory for starshaped hamiltonian systems soit une hypersurface en forme d etoile dans r 2n le probleme de trouver des caracteristiques fermees de peut se reduire classiquement a un probleme variationnel ceci conduit a etudier une fonctionnelle s 1 equivariante sur un espace de hilbert

existence of periodic solutions for nonlinear differential equations abstract in this paper we shall study the existence of a periodic solution to the nonlinear differential equationxtt axt a xt a ax b t x f t x t 0 in some complex hilbert space using duality and variational methods

the relative morse index theory for infinite dimensional hamiltonian systems with applications abstract we established an index theory for a kind of infinite dimensional linear hamiltonian system as applications we considered the existence and multiplicity of periodic solution for some nonlinear infinite dimensional hamiltonian systems

connected components of the category of elementary abelian p subgroups we determine the maximal number of conjugacy classes of maximal elementary abelian subgroups of rank 2 in a finite p group g for an odd prime p namely it is p if g has rank at least 3 and it is p 1 if g has rank 2 more precisely if g has rank 2 there are exactly 1 2 p 1 or possibly 3 classes for some 3 groups of maximal nilpotency class

splendid equivalences derived categories and permutation modules we introduce the notion of a splendid equivalence between blocks of finite groups which is a strengthening of the notion of an equivalence of derived categories such an equivalence is always induced by a two sided tilting complex whose terms are summands of permutation modules a feature that is compatible with the belief that these complexes should be related to the adic cohomology of deligne lusztig varities in the case of reductive groups over finite fields we show that a splendid equivalence has good consequences over and above those that follow from a derived equivalence in particular it gives for principal blocks a structural explanation of brou s concept of an isotypy a compatible family of perfect isometries we give several examples including examples of morita equivalences that are splendid in an unexpectedly non trivial way

strong fusion control and stable equivalences abstract this article is dedicated to the proof of the following theorem let g be a finite group p be a prime number and e be a p block of g assume that the centraliser c g p of an e subpair p e p strongly controls the fusion of the block e and that a defect group of e is either abelian or for odd p has a non cyclic centre then there exists a stable equivalence of morita type between the block algebras o g e and o c g p e p where o is a complete discrete valuation ring of residual characteristic p this stable equivalence is constructed by gluing together a family of local morita equivalences which are induced by bimodules with fusion stable endo permutation sources broue had previously obtained a similar result for principal blocks in relation with the search for a modular proof of the odd z p theorem thus our theorem points towards a block theoretic analogue of the z p theorem which we state in terms of fusion control and morita equivalences

the dual variational principle and elliptic problems with discontinuous nonlinearities on etudie des problemes aux valeurs limites elliptiques du type u f u p x x u 0 x ou r n est un domaine borne et f peut avoir des discontinuites on cherche des solutions en utilisant le principe d action duale de clarke

radon nikodym derivatives of quantum operations given a completely positive cp map t there is a theorem of the radon nikodym type w b arveson acta math 123 141 1969 v p belavkin and p staszewski rep math phys 24 49 1986 that completely characterizes all cp maps s such that t s is also a cp map this theorem is reviewed and several alternative formulations are given along the way we then use the radon nikodym formalism to study the structure of order intervals of quantum operations as well as a certain one to one correspondence between cp maps and positive operators already fruitfully exploited in many quantum information theoretic treatments we also comment on how the radon nikodym theorem can be used to derive norm estimates for differences of cp maps in general and of quantum operations in particular

biset functors and genetic sections for p groups abstract in this note i show that if f is a biset functor defined over finite p groups then for each finite p group p there is a direct summand of f p admitting a natural direct sum decomposition indexed by the irreducible rational representations of p or equivalently by the equivalence classes of origins in p or also by equivalence classes of genetic sections of p this leads to a description of the torsion part of the group of relative syzygies in the dade group of p and to a conjecture on the structure of the torsion part of the whole dade group of p

borel smith functions and the dade group we show that there is an exact sequence of biset functors over p groups 0 cb jb d 0 where cb is the biset functor for the group of borel smith functions b is the dual of the burnside ring functor d is the functor for the subgroup of the dade group generated by relative syzygies and the natural transformation is the transformation recently introduced by the first author in s bouc a remark on the dade group and the burnside group j algebra 279 2004 180 190 we also show that the kernel of mod 2 reduction of is naturally equivalent to the functor b of units of the burnside ring and obtain exact sequences involving the torsion part of d mod 2 reduction of cb and b

cryptographic distinguishability measures for quantum mechanical states this paper mostly expository in nature surveys four measures of distinguishability for quantum mechanical states this is done from the point of view of the cryptographer with a particular eye on applications in quantum cryptography each of the measures considered is rooted in an analogous classical measure of distinguishability for probability distributions namely the probability of an identification error the kolmogorov distance the bhattacharyya coefficient and the shannon 1948 distinguishability as defined through mutual information these measures have a long history of use in statistical pattern recognition and classical cryptography we obtain several inequalities that relate the quantum distinguishability measures to each other one of which may be crucial for proving the security of quantum cryptographic key distribution in another vein these measures and their connecting inequalities are used to define a single notion of cryptographic exponential indistinguishability for two families of quantum states this is a tool that may prove useful in the analysis of various quantum cryptographic protocols

generalized gradients of lipschitz functionals abstract the purpose of this article is threefold i to present in a unified fashion the theory of generalized gradients whose elements are at present scattered in various sources ii to give an account of the ways in which the theory has been applied iii to prove new results concerning generalized gradients of summation functionals pointwise maxima and integral functionals on subspaces of l these last mentioned formulas are obtained with an eye to future applications in the calculus of variations and optimal control their proofs can be regarded as applications of the existing theory of subgradients of convex functionals as developed by rockafellar ioffe and levin valadier and others

periodic solutions of hamilton s equations and local minima of the dual action the dual action is a functional whose extremals lead to solutions of hamilton s equations up to now extremals of the dual action have been obtained either through its global minimization or through application of crit ical point theory a new methodology is introduced in which local minima of the dual action are found to exist applications are then made to the existence of hamiltonian trajectories having prescribed period 1 introduction a classical and very interesting boundary value problem is concerned with periodic solutions of hamilton s equations 1 1 p i vxtf x i p i x t vph x t p t where the function h rn x rn r is called the hamiltonian any such hamil tonian trajectory x p is such that h x t p t is constant this constant is called the energy associated with x p if x 0 x t p 0 p t for some t 0 then x p is said to have period t one can ask for what energies and periods are their associated hamiltonian trajectories this and related issues have both a long history and an active present we refer to desolneux and moulis 16 mancini 20 and rabinowitz 23 for surveys and a more extensive bibliography than would be appropriate here in this article attention will be focused upon the case in which the hamiltonian is a convex function a situation which has drawn perhaps the most attention of late we begin by reviewing briefly the basis for much of this recent work a dual action principle introduced by clarke in 1978 6 let g denote the conjugate of h in the sense of convex analysis i e the function

an example to stability for the minimal of a sequence of dc functions homogenization for a class of nonlinear sturm liouville problems combining the clarke ekeland dual least action principle and the epi convergence we state an existence result and study the asymptotic behaviour for the periodic solution of a nonlinear sturm liouville problem deriving from a convex subquadratic potential when the data are perturbed in a suitable sense the result appears like a stability result for the minimizers of a sequence of dc functions

complexity and cohomology of cohomological mackey functors abstract let k be a field of characteristic p 0 call a finite group g a poco group over k if any finitely generated cohomological mackey functor for g over k has polynomial growth the main result of this paper is that g is a poco group over k if and only if the sylow p subgroups of g are cyclic when p 2 or have sectional rank at most 2 when p 2 a major step in the proof is the case where g is an elementary abelian p group in particular when p 2 all the extension groups between simple functors can be determined completely using a presentation of the graded algebra of self extensions of the simple functor s 1 g by explicit generators and relations

duality for non convex variational principles abstract this paper treats the construction of dual variational principles for non convex problems using lagrangian methods besides the well known saddle point theory there are dualities involving minimizing the lagrangian in both variables and dualities where one first maximizes and then minimizes the lagrangian in each variable when the primal problem has certain structure an associated canonical lagrangian is defined which leads to the appropriate dual variational principle and to a hamiltonian description the properties of and correspondence between the critical points are analyzed the last five sections are devoted to examples of these dual principles

the group of endo permutation modules the group d p of all endo permutation modules for a finite p group p is a finitely generated abelian group we prove that its torsion free rank is equal to the number of conjugacy classes of non cyclic subgroups of p we also obtain partial results on its torsion subgroup we determine next the structure of q otimes d viewed as a functor which turns out to be a simple functor s e q indexed by the elementary group e of order p 2 and the trivial out e module q finally we describe a rather strange exact sequence relating q otimes d p q otimes b p and q otimes r p where b p is the burnside ring and r p is the grothendieck ring of qp modules

clarke duality for hamiltonian systems with nonstandard growth abstract we consider the existence of periodic solutions to hamiltonian systems with growth conditions involving g function we introduce the notion of symplectic g function and provide relation for the growth of hamiltonian in terms of certain constant c g associated to symplectic g function g we discuss an optimality of this constant for some special cases we also provide applications to the laplacian type systems

periodic solutions to hamiltonian inclusions abstract we prove the existence of periodic trajectories of hamiltonian inclusions which reduce to the usual hamiltonian equations in the presence of smoothness this is accomplished via a direct variational principle involving a new action integral in an extended sense

uniform continuity bounds for information characteristics of quantum channels depending on input dimension and on input energy we obtain continuity bounds for basic information characteristics of quantum channels depending on their input dimension if it is finite and on the input energy bound if the input dimension is infinite we pay special attention to the case in which a multimode quantum oscillator is an input system first we prove continuity bounds for the output conditional mutual information for a single channel and for copies of a channel then we obtain estimates for the variation of the output holevo quantity with respect to simultaneous variations of a channel and of an input ensemble as a result tight and close to tight continuity bounds for basic capacities of quantum channels which depend on the input dimension are obtained they complement the leung smith continuity bounds which depend on the output dimension finally we obtain tight and close to tight continuity bounds for basic capacities of infinite dimensional energy constrained channels with respect to the energy constrained bures distance generating the strong convergence of quantum channels

periodic and subharmonic solutions for fourth order nonlinear difference equations abstract by using the critical point theory some new criteria are obtained for the existence and multiplicity of periodic and subharmonic solutions to fourth order nonlinear difference equations the main approach used in our paper is a variational technique and the linking theorem our results generalize and improve the existing ones

nonlinear oscillations and boundary value problems for hamiltonian systems we prove the existence of solutions of various boundary value problems for nonautonomous hamiltonian systems with forcing terms begin gathered dot x t h p t x t p t g t hfill dot p t h x t x t p t f t hfill end gathered

the functor of rational representations for p groups abstract let k be a field of characteristic p 0 let c p k be the category whose objects are the finite p groups morphisms are the k linear combinations of bisets and composition of morphisms is obtained by k linear extension from the usual product of bisets let f p k denote the category of k linear functors from c p k to the category of k vector spaces this paper investigates the structure of the functor kr q mapping a p group p to kr q p k z r q p as an object of f p k where r q p is the grothendieck group of the category of finite dimensional q p modules the main result is an explicit description of the lattice of all subfunctors of kr q in particular for p odd it is shown that kr q is a uniserial object in f p k for p 2 the lattice of subfunctors of kr q can be described as the lattice of closed subsets of a graph whose vertices are the 2 groups of normal 2 rank 1 in both cases a composition series of kr q is obtained which leads to a formula giving the dimension of the evaluations sq k p of the simple objects sq k of f p k associated to p groups q of normal p rank 1 different from cp this formula can be phrased in terms of rational representations of p but also in terms of the geometry of the lattice of subgroups of p using the notions of basic subgroups and origins for example if p 2 the dimension of s 1 k p is equal to the number of absolutely irreducible q p modules