non isochronicity of the center at the origin in polynomial hamiltonian systems with even degree nonlinearities abstract in 2002 x jarque and j villadelprat proved that no center in a planar polynomial hamiltonian system of degree 4 is isochronous and raised a question is there a planar polynomial hamiltonian system of even degree which has an isochronous center in this paper we give a criterion for non isochronicity of the center at the origin of planar polynomial hamiltonian systems moreover the orders of weak centers are determined our results answer a weak version of the question proving that there is no planar polynomial hamiltonian system with only even degree nonlinearities having an isochronous center at the origin

conformal invariance in two dimensional percolation the word percolation borrowed from the latin refers to the seeping or oozing of a liquid through a porous medium usually to be strained in this and related senses it has been in use since the seventeenth century it was introduced more recently into mathematics by s r broadbent and j m hammersley bh and is a branch of probability theory that is especially close to statistical mechanics broadbent and hammersley distinguish between two types of spreading of a fluid through a medium or between two aspects of the probabilistic models of such processes diffusion processes in which the random mechanism is ascribed to the fluid and percolation processes in which it is ascribed to the medium a percolation process typically depends on one or more probabilistic parameters for example if molecules of a gas are absorbed at the surface of a porous solid as in a gas mask then their ability to penetrate the solid depends on the sizes of the pores in it and their positions both conceived to be distributed in some random manner a simple mathematical model of such a process is often defined by taking the pores to be distributed in some regular manner that could be determined by a periodic graph and to be open thus very large or closed thus smaller than the molecules with probabilities p and 1 p as p increases the probability of deeper penetration of the gas into the interior of the solid grows

on polynomial hamiltonian planar vector fields abstract we study the dynamical behaviour of polynomial hamiltonian planar vector fields particularly we analyze the structure of finite and infinite critical points and we obtain the best upper bound of the number of centers and of the number of saddles that a system of this type can exhibit depending on its degree

the period function for hamiltonian systems with homogeneous nonlinearities abstract the paper deals with hamiltonian systems with homogeneous nonlinearities we prove that such systems have no isochronous centers that the period annulus of any of its centres is either bounded or the whole plane and that the period function associated to the origin has at most one critical point

new exact exponents for two dimensional self avoiding walks the author establishes using standard coulomb gas methods the values of a new set of geometrical exponents that have been previously conjectured these exponents give for any p the number of configurations of p two dimensional self avoiding walks of the same length l which are attached by their ends

on the period function for a family of complex differential equations abstract we consider planar differential equations of the form z f z g z being f z and g z holomorphic functions and prove that if g z is not constant then for any continuum of period orbits the period function has at most one isolated critical period which is a minimum among other implications the paper extends a well known result for meromorphic equations z h z that says that any continuum of periodic orbits has a constant period function

the logarithmic triplet theory with boundary the boundary theory for the c 2 triplet model is investigated in detail in particular we show that there are four different boundary conditions that preserve the triplet algebra and check the consistency of the corresponding boundary operators by constructing their ope coefficients explicitly we also compute the correlation functions of two bulk fields in the presence of a boundary and verify that they are consistent with factorization

fusion of irreducible modules in cal wlm p p prime based on symmetry principles we derive a fusion algebra generated from repeated fusions of the irreducible modules appearing in the extended logarithmic minimal model in addition to the irreducible modules themselves closure of the commutative and associative fusion algebra requires the participation of a variety of reducible yet indecomposable modules we conjecture that this fusion algebra is the same as the one obtained by application of the nahm gaberdiel kausch algorithm and find that it reproduces the known such results for and for p 1 this fusion algebra does not contain a unit requiring that the spectrum of modules is invariant under a natural notion of conjugation however introduces additional p 1 p 1 reducible yet indecomposable rank 1 modules among which the identity is found still yielding a well defined fusion algebra in this greater fusion algebra the aforementioned symmetries are generated by fusions with the three irreducible modules of conformal weights kp 1 1 k 1 2 3 we also identify polynomial fusion rings associated with our fusion algebras

the monotonicity of period function for codimension four quadratic system q4 abstract in this paper we study the codimension four quadratic system q 4 z iz 4 z 2 2 z 2 b ic z 2 where b and c are real constants i 2 1 z x iy b ic 2 it is proved that the period function of periodic trajectories for q 4 is monotone the content of this paper can be viewed as a contribution to the proof of chicone s conjecture cf mr 94h 58072

radford drinfeld and cardy boundary states in the 1 p logarithmic conformal field models we introduce p 1 pseudocharacters in the space of 1 p model vacuum torus amplitudes to complete the distinguished basis in the 2p dimensional fusion algebra to a basis in the whole 3p 1 dimensional space of torus amplitudes and the structure constants in this basis are not necessarily non negative integer numbers we obtain a generalized verlinde formula that gives these structure constants in the context of theories with boundaries we identify the space of vacuum torus amplitudes with the space of ishibashi states then we propose 3p 1 boundary states satisfying the cardy condition

non isochronicity of the center in polynomial hamiltonian systems abstract in 2002 jarque and villadelprat proved that planar polynomial hamiltonian systems of degree 4 have no isochronous centers and raised an open question for general planar polynomial hamiltonian systems of even degree recently it was proved that a planar polynomial hamiltonian system is non isochronous if a quantity denoted by m 2 m 2 can be computed such that m 2 m 2 0 as a corollary of this criterion the open question was answered for those systems with only even degree nonlinearities in this paper we consider the case of m 2 m 2 0 and give a new criterion for non isochronicity applying the new criterion we also answer the open question for some cases in which some terms of odd degree are included

kazhdan lusztig dual quantum group for logarithimic extensions of virasoro minimal models we derive and study a quantum group gp q that is kazhdan lusztig dual to the w algebra wp q of the logarithmic p q conformal field theory model the algebra wp q is generated by two currents w z and w z of dimension 2p 1 2q 1 and the energy momentum tensor t z the two currents generate a vertex operator ideal r with the property that the quotient wp q r is the vertex operator algebra of the p q virasoro minimal model the number 2pq of irreducible gp q representations is the same as the number of irreducible wp q representations on which r acts nontrivially we find the center of gp q and show that the modular group representation on it is equivalent to the modular group representation on the wp q characters and pseudocharacters the factorization of the gp q ribbon element leads to a factorization of the modular group representation on the center we also find the gp q grothendieck ring which is presumably the logarithmic fusion of the p q model

vertex operator super algebras and lcft we review some of the developments in logarithmic conformal field theory from the vertex algebra point of view several important examples of vertex operator super algebras of the triplet type are discussed including their representation theory particular emphasis is put on c2 cofiniteness of these vertex algebras a description of zhu s algebras and the construction of logarithmic modules

on the period function of reversible quadratic centers with their orbits inside quartics abstract this paper is concerned with the monotonicity of the period function for a class of reversible quadratic centers with their orbits inside quartics it is proved that such a system has a period function with at most one critical point

optimal distribution of the internal null control for the one dimensional heat equation abstract this paper addresses the problem of optimizing the distribution of the support of the internal null control of minimal l 2 norm for the 1 d heat equation a measure constraint is imposed on the support but no topological assumption such as the number of connected components therefore the problem typically lacks of solution in the class of characteristic functions and needs of relaxation we show that the relaxed formulation is obtained by replacing the set of characteristic functions by its convex envelope the proof requires that the observability constant related to the control problem be uniform with respect to the support property which is obtained by the control transmutation method the optimality conditions of the relaxed problem as well as the case where the number of connected components is fixed a priori are also discussed several numerical experiments complete the study and suggest the ill posedness of the problem in contrast to the wave situation

reachability of maximal number of critical periods without independence abstract the reachability of the maximal number of critical periods bifurcated from a weak center is judged with independence of period quantities since the independence is a sufficient condition in this paper we give a method to judge the reachability without the independence our main idea is to consider how the number of zeros for derivative of the period function changes as a curve in the parameter space scans near a specific point and choose an appropriate curve to find the largest number of zeros we apply this method to two polynomial differential systems having zero dimensional isochronicity variety and nonzero dimensional one separately to show that the maximal number is reached and obtain greater numbers than those obtained by classical independence separately with our method the two examples give the possibility that more critical periods arise from an isochronous center than a finite order weak center an opposite fact to many known results

properties of the period function for some hamiltonian systems and homogeneous solutions of a semilinear elliptic equation abstract in this work we study the period function t of solutions to the conservative equation x t f x t 0 we present conditions on f that imply the monotonicity and convexity of t as a consequence we obtain the criterium established by c chicone and find conditions easier to apply we also get a condition obtained by cima gasull and manosas about monotonicity and following some of their calculations present results on the period function of hamiltonian systems where h x y f x n 1 y n using the monotonicity of t we count the homogeneous solutions to the semilinear elliptic equation u u 1 in two dimensions

conformal algebras of two dimensional disordered systems we discuss the structure of two dimensional conformal field theories at a central charge c 0 describing critical disordered systems polymers and percolation we construct a novel extension of the c 0 virasoro algebra characterized by a number b measuring the effective number of massless degrees of freedom and by a logarithmic partner of the stress tensor it is argued to be present at a generic random critical point lacking super kac moody or other higher symmetries and is a tool to describe and classify such theories interestingly this algebra is not only consistent with but indeed naturally accommodates in general an underlying global supersymmetry polymers and percolation realize this algebra unexpectedly we find that the c 0 kac table of the degenerate fields contains two distinct theories with b 5 6 and b 5 8 which we conjecture to correspond to percolation and polymers respectively a given kac table field can be degenerate only in one of them remarkably we also find this algebra and thereby an ensuing hidden supersymmetry realized at general replica averaged critical points for which we derive an explicit formula for b

quadratic hamiltonian vector fields abstract let h be a cubic polynomial in two variables over r then h defines a quadratic hamiltonian vector field h y h x the purpose of this paper is to prove that there are exactly 28 non equivalent topologic phase portraits of quadratic hamiltonian vector fields

exact critical exponents for two dimensional dense polymers using conformal invariance and coulomb gas results the author gives the exact value in two dimensions of the eta exponent of l dense polymers attached by their extremities eta l l2 4 8 the value in two dimensions of the gamma exponent of a dense branched polymer of fixed topology with nl l leg vertices l or 1 is then deduced to be gamma sigma l or 1 nl 2 l l 18 32 these values correspond to a conformally invariant theory with central charge c 2

modular group representations and fusion in logarithmic conformal field theories and in the quantum group center the sl 2 representation on the center of the restricted quantum group open image in new window at the primitive 2pth root of unity is shown to be equivalent to the sl 2 representation on the extended characters of the logarithmic 1 p conformal field theory model the multiplicative jordan decomposition of the open image in new window ribbon element determines the decomposition of into a pointwise product of two commuting sl 2 representations one of which restricts to the grothendieck ring this restriction is equivalent to the sl 2 representation on the 1 p characters related to the fusion algebra via a nonsemisimple verlinde formula the grothendieck ring of open image in new window at the primitive 2pth root of unity is shown to coincide with the fusion algebra of the 1 p logarithmic conformal field theory model as a by product we derive q binomial identities implied by the fusion algebra realized in the center of open image in new window

isochronicity for several classes of hamiltonian systems abstract in this paper we study isochronous centers of analytic hamiltonian systems giving special attention to the polynomial case we first revisit the potential systems and we show the connection between isochronicity and involutions we then study a more general system namely the ones associated to hamiltonians of the form h x y a x b x y c x y 2 as an application we classify the cubic polynomial hamiltonian isochronous centers and we give examples of nontrivial and nonglobal polynomial hamiltonian isochronous centers

polynomial fusion rings of w extended logarithmic minimal models the countably infinite number of virasoro representations of the logarithmic minimal model lm p p can be reorganized into a finite number of w representations with respect to the extended virasoro algebra symmetry w using a lattice implementation of fusion we recently determined the fusion algebra of these representations and found that it closes albeit without an identity for p 1 here we provide a fusion matrix realization of this fusion algebra and identify a fusion ring isomorphic to it we also consider various extensions of it and quotients thereof and introduce and analyze commutative diagrams with morphisms between the involved fusion algebras and the corresponding quotient polynomial fusion rings one particular extension is reminiscent of the fundamental fusion algebra of lm p p and offers a natural way of introducing the missing identity for p 1 working out explicit fusion matrices is facilitated by a further enlargement based on a pair of mutual moore penrose inverses intertwining bet

fusion matrices generalized verlinde formulas and partition functions in cal wlm 1 p the infinite series of logarithmic minimal models is considered in the extended picture where they are denoted by as in the rational models the fusion algebra of is described by a simple graph fusion algebra the corresponding fusion matrices are mutually commuting but in general not diagonalizable nevertheless they can be simultaneously brought to jordan form by a similarity transformation the spectral decomposition of the fusion matrices is completed by a set of refined similarity matrices converting the fusion matrices into jordan canonical form consisting of jordan blocks of rank 1 2 or 3 the various similarity transformations and jordan forms are determined from the modular data this gives rise to a generalized verlinde formula for the fusion matrices its relation to the partition functions in the model is discussed in a general framework by application of a particular structure matrix and its moore penrose inverse this verlinde formula reduces to the generalized verlinde formula for the associated grothendieck ring

on the number of critical periods for planar polynomial systems abstract in this paper we get some lower bounds for the number of critical periods of families of centers which are perturbations of the linear one we give a method which lets us prove that there are planar polynomial centers of degree l with at least 2 l 2 2 critical periods as well as study concrete families of potential reversible and lienard centers this last case is studied in more detail and we prove that the number of critical periods obtained with our approach does not increases with the order of the perturbation

virasoro representations and fusion for general augmented minimal models in this paper we present explicit results for the fusion of irreducible and higher rank representations in two logarithmically conformal models the augmented c2 3 0 model as well as the augmented yang lee model at c2 5 22 5 we analyse their spectrum of representations which is consistent with the symmetry and associativity of the fusion algebra we also describe the first few higher rank representations in detail in particular we present the first examples of consistent rank 3 indecomposable representations and describe their embedding structure knowing these two generic models we also conjecture the general representation content and fusion rules for general augmented cp q models

w extended fusion algebra of critical percolation two dimensional critical percolation is the member lm 2 3 of the infinite series of yang baxter integrable logarithmic minimal models lm p p we consider the continuum scaling limit of this lattice model as a rational logarithmic conformal field theory with extended w w2 3 symmetry and use a lattice approach on a strip to study the fundamental fusion rules in this extended picture we find that the representation content of the ensuing closed fusion algebra contains 26 w indecomposable representations with eight rank1 representations 14 rank 2 representations and four rank 3 representations weidentifytheserepresentationswithsuitablelimitsofyang baxterintegrable boundary conditions on the lattice and obtain their associated w extended characters the latter decompose as finite non negative sums of w irreducible characters of which 13 are required implementation of fusion on the lattice allows us to read off the fusion rules governing the fusion algebra of the 26 representations and to construct an explicit cayley table the closure of these representations among themselves under fusion is remarkable confirmation of the proposed extended symmetry

integrable boundary conditions and w extended fusion in the logarithmic minimal models lm 1 p we consider the logarithmic minimal models lm 1 p as rational logarithmic conformal field theories with extended w symmetry to make contact with the extended picture starting from the lattice we identify 4p 2 boundary conditions as specific limits of integrable boundary conditions of the underlying yang baxter integrable lattice models specifically we identify 2p integrable boundary conditions to match the 2p known irreducible w representations these 2p extended representations naturally decompose into infinite sums of the irreducible virasoro representations r s a further 2p 2 reducible yet indecomposable w representations of rank 2 are generated by fusion and these decompose as infinite sums of indecomposable rank 2 virasoro representations the fusion rules in the extended picture are deduced from the known fusion rules for the virasoro representations of lm 1 p and are found to be in agreement with previous works the closure of the fusion algebra on a finite number of representations in the extended picture is remarkable confirmation of the consistency of the lattice approach comment 15 pages

the period function for quadratic integrable systems with cubic orbits this paper is concerned with the monotonicity of the period function for families of quadratic systems with centers whose orbits lie on cubic planar curves it is proved that each such system has a period function with at most one critical point

area preserving normalizations for centers of planar hamiltonian systems abstract it is well known that a nondegenerate center of an analytic hamiltonian planar system can be brought to normal form by means of an analytic canonical change of coordinates this normal form that we denote by cnf does not depend on the coordinate transformation in this paper we give an elementary proof of these facts and we show some interesting applications of the machinery that we develop in order to prove them for instance we describe the space of coordinate transformations that bring a hamiltonian nondegenerate center to its cnf and we prove that they are all canonical when the center is non isochronous we also show that two hamiltonian systems with a nondegenerate center are canonically conjugated if and only if both centers have the same period function

bifurcation of critical periods for plane vector fields on etudie un probleme de bifurcation dans des familles de champs de vecteurs analytiques plans qui ont un centre non degenere a l origine pour toutes les valeurs d un parametre r n on definit la fonction periode p on determine les points critiques de cette fonction pres du centre avec comme parametre de bifurcation

integrable and conformal boundary conditions for widehat s ell 2 a d e lattice models and unitary minimal conformal field theories integrable boundary conditions are studied for critical a d e and general graph based lattice models of statistical mechanics in particular using techniques associated with the temperley lieb algebra and fusion a set of boundary boltzmann weights which satisfies the boundary yang baxter equation is obtained for each boundary condition when appropriately specialized these boundary weights each of which depends on three spins decompose into more natural two spin edge weights the specialized boundary conditions for the a d e cases are naturally in one to one correspondence with the conformal boundary conditions of widehat s ell 2 unitary minimal conformal field theories supported by this and further evidence we conclude that in the continuum scaling limit the integrable boundary conditions provide realizations of the complete set of conformal boundary conditions in the corresponding field theories

on urabe s criteria of isochronicity we give a short proof of urabe s criteria for the isochronicity of periodical solutions of the equation we show that apart from the harmonic oscillator there exists a large family of isochronous potentials which must all be non polynomial and not symmetric an even function of the coordinate x

first derivative of the period function with applications abstract given a centre of a planar differential system we extend the use of the lie bracket to the determination of the monotonicity character of the period function as far as we know there are no general methods to study this function and the use of commutators and lie bracket was restricted to prove isochronicity we give several examples and a special method which simplifies the computations when a first integral is known

isochronicity and linearizability of planar polynomial hamiltonian systems in this paper we study isochronicity and linearizability of planar polynomial hamiltonian systems first we prove a theorem which supports a negative answer to the following open question stated by jarque and villadelprat in 15 do there exist planar polynomial hamiltonian systems of even degree having an isochronous center additionally we obtain some conditions for linearizability of complex cubic hamiltonian systems

attainable subspaces and the bang bang property of time optimal controls for heat equations in this paper we study two subjects on internally controlled heat equations with time varying potentials the attainable subspaces and the bang bang property for some time optimal control problems we present some equivalent characterizations on the attainable subspaces and provide a sufficient condition to ensure the bang bang property both the above mentioned characterizations and the sufficient condition are closely related to some function spaces consisting of some solutions to the adjoint equations it seems for us that the existing ways to derive the bang bang property for heat equations with time invariant potentials see for instance h o fattorini infinite dimensional linear control systems the time optimal and norm optimal problems elsevier amsterdam 2005 f gozzi and p loreti siam j control optim 37 1999 pp 1195 1221 v mizel and t seidman siam j control optim 35 1997 pp 1204 1216 and g wang siam j control optim 47 2008 pp 1701 1720 do not work

fusion algebras of logarithmic minimal models we present explicit conjectures for the chiral fusion algebras of the logarithmic minimal models lm p p considering virasoro representations with no enlarged or extended symmetry algebra the generators of fusion are countably infinite in number but the ensuing fusion rules are quasi rational in the sense that the fusion of a finite number of representations decomposes into a finite direct sum of representations the fusion rules are commutative associative and exhibit an s 2 structure but require so called kac representations which are typically reducible yet indecomposable representations of rank 1 in particular the identity of the fundamental fusion algebra p 1 is a reducible yet indecomposable kac representation of rank 1 we make detailed comparisons of our fusion rules with the results of gaberdiel and kausch for p 1 and with eberle and flohr for p p 2 5 corresponding to the logarithmic yang lee model in the latter case we confirm the appearance of indecomposable representations of rank 3 we also find that closure of a fundamental fusion algebra is achieved without the introduction of indecomposable representations of rank higher than 3 the conjectured fusion rules are supported within our lattice approach by extensive numerical studies of the associated integrable lattice models details of our lattice findings and numerical results will be presented elsewhere the agreement of our fusion rules with the previous fusion rules lends considerable support for the identification of the logarithmic minimal models lm p p with the augmented cp p minimal models defined algebraically

nonsemisimple fusion algebras and the verlinde formula we find a nonsemisimple fusion algebra open image in new window associated with each 1 p virasoro model we present a nonsemisimple generalization of the verlinde formula which allows us to derive open image in new window from modular transformations of characters

logarithmic torus amplitudes abstractfor the example of the logarithmic triplet theory at c 2 the chiral vacuumtorus amplitudes are analysed it is found that the space of these torus amplitudes isspanned by the characters of the irreducible representations as well as a function thatcan be associated to the logarithmic extension of the vacuum representation a fewimplications and generalisations of this result are discussed 1 introduction during the last twenty years much has been understood about the struc ture of rational conformal eld theories rational conformal eld theories are characterisedby the property that they have only nitely many irreducible highest weight representationsof the chiral algebra or vertex operator algebra and that every highest weight represen tation is completely decomposable into irreducible representations the structure of thesetheories is well understood in particular the characters of the irreducible representationstransform into one another under modular transformations 1 see also 2 and the modulars matrix determines the fusion rules via the verlinde formula 3 a general proof for thishas only recently been given in 4 on the other hand it is clear that rational conformal eld theories are rather special andit is therefore important to understand the structure of more general classes of conformal eld theories one such class are the rational logarithmic theories that possess only nitelymany indecomposable representations but for which not all highest weight representationsare completely decomposable the name logarithmic comes from the fact that their chiralcorrelation functions typically have logarithmic branch cuts the rst example of a non rational logarithmic conformal eld theory was found in 5 see also 6 and the rstrational example that shall also concern us in this paper was constructed in 7 for somerecent reviews see 8 9 10 from a physics point of view logarithmic conformal eldtheories appear naturally in various models of statistical physics for example in the theory

a modular invariant bulk theory for the c 0 triplet model a proposal for the bulk space of the logarithmic cal w 2 3 triplet model at central charge zero is made the construction is based on the idea that one may reconstruct the bulk theory of a rational conformal field theory from its boundary theory the resulting bulk space is a quotient of the direct sum of projective representations which is isomorphic as a vector space to the direct sum of tensor products of the irreducible representations with their projective covers as a consistency check of our analysis we show that the partition function of the bulk theory is modular invariant and that the boundary state analysis is compatible with the proposed annulus partition functions of this model

global behaviour of the period function of the sum of two quasi homogeneous vector fields abstract we study the global behaviour of the period function on the period annulus of degenerate centres for two families of planar polynomial vector fields these families are the quasi homogeneous vector fields and the vector fields given by the sum of two quasi homogeneous hamiltonian ones in the first case we prove that the period function is globally decreasing extending previous results that deal either with the hamiltonian quasi homogeneous case or with the general homogeneous situation in the second family and after adding some more additional hypotheses we show that the period function of the origin is either decreasing or has at most one critical period and that both possibilities may happen this result also extends some previous results that deal with the situation where both vector fields are homogeneous and the origin is a non degenerate centre

an explicit expression of the first liapunov and period constants with applications in this paper we study systems in the plane having a critical point with pure imaginary eigenvalues and we search for effective conditions to discern whether this critical point is a focus or a center in the case of it being a center we look for additional conditions in order to be isochronous we stress that the essential differences between the techniques used in this work and the more usual ones are basically two the elimination of the integration constants when we consider primitives of functions see also remark 3 2 and the fact that we maintain the complex notation in the whole study thanks to these aspects we reach with relative ease an expression of the first three liapunov constants v3 v5 andv7 and of the first two period ones p2andp4 for a general system as far as we know this is the first time that a general and compact expression ofv7has been given moreover the use of a computer algebra system is only needed in the computation ofv7andp4 these results are applied to give a classification of centers and isochronous centers for certain families of differential equations

w extended kac representations and integrable boundary conditions in the logarithmic minimal models wlm 1 p we construct new yang baxter integrable boundary conditions in the lattice approach to the logarithmic minimal model giving rise to reducible yet indecomposable representations of rank 1 in the continuum scaling limit we interpret these extended kac representations as finitely generated extended feigin fuchs modules over the triplet w algebra the extended fusion rules of these representations are inferred from the recently conjectured virasoro fusion rules of the kac representations in the underlying logarithmic minimal model we also introduce the modules contragredient to the extended kac modules and work out the correspondingly extended fusion algebra our results are in accordance with the kazhdan lusztig dual of tensor products of modules over the restricted quantum universal enveloping algebra at q e i p finally polynomial fusion rings isomorphic with the various fusion algebras are determined and the corresponding grothendieck ring of characters is identified

optimal actuator location of minimum norm controls for heat equation with general controlled domain abstract in this paper we study optimal actuator location of the minimum norm controls for a multi dimensional heat equation with control defined in the space l 2 0 t the actuator domain is time varying in the sense that it is only required to have a prescribed lebesgue measure for any moment we select an optimal actuator location so that the optimal control takes its minimal norm over all possible actuator domains we build a framework of finding the nash equilibrium so that we can develop a sufficient and necessary condition to characterize the optimal relaxed solutions for both actuator location and corresponding optimal control of the open loop system the existence and uniqueness of the optimal classical solutions are therefore concluded as a result we synthesize both optimal actuator location and corresponding optimal control into a time varying feedbacks

logarithmic conformal field theory beyond an introduction this article aims to review a selection of central topics and examples in logarithmic conformal field theory it begins with the remarkable observation of cardy that the horizontal crossing probability of critical percolation may be computed analytically within the formalism of boundary conformal field theory cardy s derivation relies on certain implicit assumptions which are shown to lead inexorably to indecomposable modules and logarithmic singularities in correlators for this a short introduction to the fusion algorithm of nahm gaberdiel and kausch is provided while the percolation logarithmic conformal field theory is still not completely understood there are several examples for which the formalism familiar from rational conformal field theory including bulk partition functions correlation functions modular transformations fusion rules and the verlinde formula has been successfully generalized this is illustrated for three examples the singlet model related to the triplet model symplectic fermions and the fermionic bc ghost system the fractional level wess zumino witten model based on at related to the bosonic ghost system and the wess zumino witten model for the lie supergroup related to at and 1 the bershadsky polyakov algebra and the feigin semikhatov algebras these examples have been chosen because they represent the most accessible and most useful members of the three best understood families of logarithmic conformal field theories the logarithmic minimal models the fractional level wess zumino witten models and the wess zumino witten models on lie supergroups excluding in this review the emphasis lies on the representation theory of the underlying chiral algebra and the modular data pertaining to the characters of the representations each of the archetypal logarithmic conformal field theories is studied here by first determining its irreducible spectrum which turns out to be continuous as well as a selection of natural reducible but indecomposable modules this is followed by a detailed description of how to obtain character formulae for each irreducible a derivation of the action of the modular group on the characters and an application of the verlinde formula to compute the grothendieck fusion rules in each case the genuine fusion rules are known so comparisons can be made and favourable conclusions drawn in addition each example admits an infinite set of simple currents hence extended symmetry algebras may be constructed and a series of bulk modular invariants computed the spectrum of such an extended theory is typically discrete and this is how the triplet model arises for example moreover simple current technology admits a derivation of the extended algebra fusion rules from those of its continuous parent theory finally each example is concluded by a brief description of the computation of some bulk correlators a discussion of the structure of the bulk state space and remarks concerning more advanced developments and generalizations the final part gives a very short account of the theory of staggered modules the simplest class of representations that are responsible for the logarithmic singularities that distinguish logarithmic theories from their rational cousins these modules are discussed in a generality suitable to encompass all the examples met in this review and some of the very basic structure theory is proven then the important quantities known as logarithmic couplings are reviewed for virasoro staggered modules and their role as fundamentally important parameters akin to the three point constants of rational conformal field theory is discussed an appendix is also provided in order to introduce some of the necessary but perhaps unfamiliar language of homological algebra

nonexistence of isochronous centers in planar polynomial hamiltonian systems of degree four abstract in this paper we study centers of planar polynomial hamiltonian systems and we are interested in the isochronous ones we prove that every center of a polynomial hamiltonian system of degree four that is with its homogeneous part of degree four not identically zero is nonisochronous the proof uses the geometric properties of the period annulus and it requires the study of the hamiltonian systems associated to a hamiltonian function of the form h x y a x b x y c x y 2 d x y 3