a topos foundation for theories of physics iv categories of systems this paper is the fourth in a series whose goal is to develop a fundamentally new way of building theories of physics the motivation comes from a desire to address certain deep issues that arise in the quantum theory of gravity our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system classical physics arises when the topos is the category of sets other types of theory employ a different topos the previous papers in this series are concerned with implementing this program for a single system in the present paper we turn to considering a collection of systems in particular we are interested in the relation between the topos representation for a composite system and the representations for its constituents we also study this problem for the disjoint sum of two systems our approach to these matters is to construct a category of systems and to find a topos representation of the entire category

linear preservers of weak majorization on l1 i when i is an infinite set abstract the necessary and sufficient conditions that a bounded linear map on the banach space l 1 i may be considered as a linear preserver of weak majorization on l 1 i where i is an arbitrary infinite set are given also we prove that the set of all linear preservers of weak majorization is closed under the norm topology

une borne superieure pour l entropie topologique d une application rationnelle let x be a complex projective manifold and let f be a dominating rational map from x onto x we show that the topological entropy h f of f is bounded from above by the logarithm of its maximal dynamical degree

exponential stability of nondissipative linear system in banach space and application to unbounded bilinear systems abstract we consider linear control systems of the form d z t d t a z t b z t where is a real parameter a is the infinitesimal generator of a linear c 0 semigroup of contractions s 0 t on a banach space x and the linear perturbation b is a relatively bounded operator with respect to a we provide necessary and sufficient conditions for exponential stability of the above system our stability results are then applied to establish uniform exponential stabilization of linear and bilinear systems with unbounded control operators applications to hyperbolic like equations are considered

dynamics of bimeromorphic maps of surfaces we classify bimeromorphic self maps f x inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i of compact kahler surfaces x in terms of their actions f h 1 1 x inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i on cohomology we observe that the growth rate of f n is invariant under bimeromorphic conjugacy and that by conjugating one can always arrange that f n f n we show that the sequence f n can be bounded grow linearly grow quadratically or grow exponentially in the first three cases we show that after conjugating f is an automorphism virtually isotopic to the identity f preserves a rational fibration or f preserves an elliptic fibration respectively in the last case we show that there is a unique up to scaling expanding eigenvector for f that is nef and that f is bimeromorphically conjugate to an automorphism if and only if 2 0 we go on in this case to construct a dynamically natural positive current representing and we study the growth rate of periodic orbits of f we conclude by illustrating our results with a particular family of examples

sur les groupes de transformations birationnelles des surfaces ni agissant par transformations bi rationnelles sur les surfaces complexes compactes kahl eriennes nous mon trons a que le groupe des transformations birationnelles d une surface satisfait l alternative de tits b que les actions birationnelles de groupes de kazhdan sur les surfaces sont toutes birationnellement conjugu eesa des actions homographiques sur le plan projectif et c que si f et g sont deux transformations birationnelles de surfaces qui commutent alors ou bien f pr eserve un pinceau de courbes ou bien l un des it er es g m de g m 0 concide avec un it er e f n de f n 2 z let s be any compact complex kahler surface and bir s the group of birational transformations of s we study the structure of nitely genera ted subgroups of bir s and prove three main results a bir s satises the tits alternative b if a group with kazhdan property t acts on a compact kahler surface by birational transformations then the action is conjugate to an action by linear projective transformations on the projec tive plane and c if f is an element of bir s which does not preserve any pencil of curves and if g commutes with f then a positive iterate of g coincides with an iterate of f the second statement provides a positive answer to zimmer s conjecture for birational actions on surfaces

the damped string problem revisited abstract we revisit the damped string equation on a compact interval with a variety of boundary conditions and derive an infinite sequence of trace formulas associated with it employing methods familiar from supersymmetric quantum mechanics we also derive completeness and riesz basis results with parentheses for the associated root functions under less smoothness assumptions on the coefficients than usual using operator theoretic methods rather than detailed eigenvalue and root function asymptotics only

energy decay rate of wave equations with indefinite damping abstract we consider the one dimensional wave equation with an indefinite sign damping and a zero order potential term using a shooting method we establish the asymptotic expansion of eigenvalues and eigenvectors of the damped wave equation for a large class of coefficients in addition if the damping coefficient is more positive than negative we prove that the energy of system decays uniformly exponentially to zero this sharp result generalizes a previous work of freitas and zuazua 1996

the generally covariant locality principle a new paradigm for local quantum field theory a new approach to the model independent description of quantum field theories will be introduced in the present work the main feature of this new approach is to incorporate in a local sense the principle of general covariance of general relativity thus giving rise to the concept of a locally covariant quantum field theory such locally covariant quantum field theories will be described mathematically in terms of covariant functors between the categories on one side of globally hyperbolic spacetimes with isometric embeddings as morphisms and on the other side of algebras with unital injective monomorphisms as morphisms moreover locally covariant quantum fields can be described in this framework as natural transformations between certain functors the usual haag kastler framework of nets of operator algebras over a fixed spacetime background manifold together with covariant automorphic actions of the isometry group of the background spacetime can be re gained from this new approach as a special case examples of this new approach are also outlined in case that a locally covariant quantum field theory obeys the time slice axiom one can naturally associate to it certain automorphic actions called relative cauchy evolutions which describe the dynamical reaction of the quantum field theory to a local change of spacetime background metrics the functional derivative of a relative cauchy evolution with respect to the spacetime metric is found to be a divergence free quantity which has as will be demonstrated in an example the significance of an energy momentum tensor up to addition of scalar functions for the locally covariant quantum field theory furthermore we discuss the functorial properties of state spaces of locally covariant quantum field theories that entail the validity of the principle of local definiteness

linear maps preserving group majorization abstract a necessary and sufficient condition for a linear map to preserve group majorizations is given the condition is applied to prove some preservation results

super potentials densities of currents and number of periodic points for holomorphic maps abstract we prove that if a positive closed current is bounded by another one with bounded continuous or holder continuous super potentials then it inherits the same property there are two different methods to define wedge products of positive closed currents of arbitrary bi degree on compact kahler manifolds using super potentials and densities when the first method applies we show that the second method also applies and gives the same result as an application we obtain a sharp upper bound for the number of isolated periodic points of holomorphic maps on compact kahler manifolds whose actions on cohomology are simple a similar result still holds for a large class of holomorphic correspondences

self adjoint operators as functions i lattices galois connections and the spectral order observables of a quantum system described by self adjoint operators in a von neumann algebra or affiliated with it in the unbounded case form a conditionally complete lattice when equipped with the spectral order using this order theoretic structure we develop a new perspective on quantum observables

spectral order of operators and range projections we study the effect algebra i e the positive part of the unit ball of an operator algebra and its relation to the projection lattice from the perspective of the spectral order a spectral orthomorphism is a map between effect algebras which preserves the spectral order and orthogonality of elements we show that if the spectral orthomorphism preserves the multiples of the unit then it is a restriction of a jordan homomorphism between the corresponding algebras this is an optimal extension of the dye s theorem on orthomorphisms of the projection lattices to larger structures containing the projections moreover results on automatic countable additivity of spectral homomorphisms are proved further we study the order properties of the range projection map assigning to each positive contraction in a jbw algebra its range projection it is proved that this map preserves infima of positive contractions in the spectral respectively standard order if and only if the projection lattice of the algebra in question is a modular lattice

a theorem of tits type for automorphism groups of projective varieties in arbitrary characteristic we prove a theorem of tits type for automorphism groups of projective varieties over an algebraically closed field of arbitrary characteristic which was first conjectured by keum oguiso and zhang for complex projective varieties

exponential stability of an abstract nondissipative linear system in this paper we consider an abstract linear system with perturbation of the form frac dy dt ay varepsilon by on a hilbert space cal h where a is skew adjoint b is bounded and varepsilon is a positive parameter motivated by a work of freitas and zuazua on the one dimensional wave equation with indefinite viscous damping p freitas and e zuazua j differential equations 132 1996 pp 338 352 we obtain a sufficient condition for exponential stability of the above system when b is not a dissipative operator we also obtain a hautus type criterion for exact controllability of system a g where g is a bounded linear operator from another hilbert space to cal h our result about the stability is then applied to establish the exponential stability of several elastic systems with indefinite viscous damping as well as the exponential stabilization of the elastic systems with noncolocated observation and control

a topos foundation for theories of physics i formal languages for physics this paper is the first in a series whose goal is to develop a fundamentally new way of constructing theories of physics the motivation comes from a desire to address certain deep issues that arise when contemplating quantum theories of space and time our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system classical physics arises when the topos is the category of sets other types of theory employ a different topos in this paper we discuss two different types of language that can be attached to a system s the first is a propositional language pl s the second is a higher order typed language l s both languages provide deductive systems with an intuitionistic logic the reason for introducing pl s is that as shown in paper ii of the series it is the easiest way of understanding and expanding on the earlier work on topos theory and quantum physics however the main thrust o

a spillover phenomenon in the optimal location of actuators in this paper we are interested in finding the optimal location and shape of the actuators in a stabilization problem namely we consider the one dimensional wave equation damped by an internal feedback supported on a subdomain omega of given length the criterion we want to optimize represents the rate of decay of the total energy of the system it theoretically involves all the eigenmodes of the operator from an engineering point of view it seems more realistic to consider only a finite number of modes say the n first ones in that context we are able to prove existence and uniqueness of an optimal domain omega n it is the better possible location for the actuators we characterize this optimal domain and we point out the following strange phenomenon at least for small lengths the optimal domain omega n which is the better one for the n first modes is actually the worse one for the n 1th mode this looks like the well known spillover phenomenon in control theory at last we will give some possible extension and open problems in higher dimension

a comparison of two topos theoretic approaches to quantum theory the aim of this paper is to compare the two topos theoretic approaches to quantum mechanics that may be found in the literature to date the first approach which we will call the contravariant approach was originally proposed by isham and butterfield and was later extended by doring and isham the second approach which we will call the covariant approach was developed by heunen landsman and spitters

minimum positive entropy of complex enriques surface automorphisms we determine the minimum positive entropy of complex enriques surface automorphisms this together with mcmullen s work completes the determination of the minimum positive entropy of complex surface automorphisms in each class of the enriques kodaira classification of complex surfaces

spectral order for unbounded operators abstract the spectral order a notion originated by olson for bounded operators is investigated here in the context of unbounded operators dissimilarities between bounded and unbounded cases are pointed out new criteria for two operators to be comparable are supplied a way of reducing the study of the spectral order to the case of bounded operators is proposed connections with essential selfadjointness are established integral inequalities for monotonically increasing functions are characterized in terms of distribution functions some illustrative examples are furnished

a topos foundation for theories of physics ii daseinisation and the liberation of quantum theory this paper is the second in a series whose goal is to develop a fundamentally new way of constructing theories of physics the motivation comes from a desire to address certain deep issues that arise when contemplating quantum theories of space and time our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system classical physics arises when the topos is the category of sets other types of theory employ a different topos in this paper we study in depth the topos representation of the propositional language pl s for the case of quantum theory in doing so we make a direct link with and clarify the earlier work on applying topos theory to quantum physics the key step is a process we term daseinisation by which a projection operator is mapped to a subobject of the spectral presheaf the topos quantum analog of a classical state space in the second part of the paper we change gear

a topos for algebraic quantum theory the aim of this paper is to relate algebraic quantum mechanics to topos theory so as to construct new foundations for quantum logic and quantum spaces moti vated by bohr s idea that the empirical content of quantum physics is accessible only through classical physics we show how a noncommutative c algebra of observables a induces a topos t a in which the amalgamation of all of its commutative subalgebras comprises a single commutative c algebra a according to the constructive gelfand duality theorem of banaschewski and mulvey the latter has an internal spectrum a in t a which in our approach plays the role of the quantum phase space of the sys tem thus we associate a locale which is the topos theoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a heyting algebra to a c algebra which is the noncommutative notion of a space in this setting states on a become probability measures more precisely valuations on and self adjoint elements of a define continuous functions more precisely locale maps fromto scott s interval domain noting that open subsets of a correspond to propositions about the system the pairing map that assigns a generalized truth value to a state and a proposition assumes an extremely simple categorical form formulated in this way the quantum theory defined by a is essentially turned into a classical theory internal to the topos t a these results were inspired by the topos theoretic approach to quantum physics pro posed by butterfield and isham as recently generalized by doring and isham

stability results for the wave equation with indefinite damping abstract the one dimensional wave equation with damping of indefinite sign in a bounded interval with dirichlet boundary conditions is considered it is proved that solutions decay uniformly exponentially to zero provided the damping potential is in thebv class has positive average is small enough and satisfies a finite number of further constraints guaranteeing that the derivative of the real part of the eigenvalues is negative when the damping vanishes this sharp result completes a previous one by the first author showing that an indefinite sign damping always produces unstable solutions if it is large enough and it answers by the afirmative to a conjecture concerning small damping terms the method of proof relies on the developments by s cox and the second author on the high frequency asymptotic expansion of the spectrum for damped wave equations and on the characterization of the decay rate in terms of the spectral abscissa

stabilization of the wave equations with potential and indefinite damping by means of global carleman type estimate we study the stabilization problem of the wave equations with potential and indefinite damping the energy decay rate of the system is given explicitly also we obtain an upper bound estimate on the negative damping to guarantee the energy of the system decays exponentially

majorization doubly stochastic matrices and comparison of eigenvalues abstract 1 basic properties of majorization 2 isotone maps and algebraic operations 3 double sub and superstochasticity 4 doubly stochastic matrices 5 doubly stochastic matrices with minimum permanent 6 comparison of eigenvalues 7 doubly stochastic maps

multidimensional spectral order for selfadjoint operators abstract the notion of the spectral order is extended to finite families of pairwise commuting bounded and unbounded selfadjoint operators in hilbert space it is shown that the multidimensional spectral order is preserved by transformations represented by spectral integrals of separately increasing borel functions on r in particular the dimensional spectral order is the restriction of product of spectral orders for selfadjoint operators in the context of positive tuples of pairwise commuting selfadjoint operator the relation a b holds if and only if a b for every tuples of nonnegative integer numbers

stabilization of high eigenfrequencies of a beam equation with generalized viscous damping in this paper using the one dimensional vibrating beam equation with generalized viscous damping as a model of vibration of flexible robot arms it is shown that for such a system the high eigenmodes decay at a uniform rate the proof is obtained by perturbation theory of linear operators y h luo acta math sinica 32 1991 pp 556 563 and asymptotic estimates of eigenvectors based on an earlier work of g b birkhoff and m h stone feedback control for this class of system is investigated and a finite dimensional controller is presented for an exponentially stable closed loop system our method may be used to study nondissipative systems

vigier s theorem for the spectral order and its applications abstract the paper mainly deals with suprema and infima of self adjoint operators in a von neumann algebra m with respect to the spectral order let m s a be the self adjoint part of m and let be the spectral order on m s a we show that a decreasing net in m s a with a lower bound has the infimum equal to the strong operator limit the similar statement is proved for an increasing net bounded above in m s a this version of vigier s theorem for the spectral order is used to describe suprema and infima of nonempty bounded sets of self adjoint operators in terms of the strong operator limit and operator means as an application of our results on suprema and infima we study the order topology on m s a with respect to the spectral order we show that it is finer than the restriction of the mackey topology

classical and quantum probabilities as truth values we show how probabilities can be treated as truth values in suitable sheaf topoi the scheme developed in this paper is very general and applies both in classical and quantum physics on the quantum side the results naturally tie in with the topos approach to quantum theory that has been developed in the last 14 years by the authors and others earlier results on the representation of arbitrary quantum states are complemented with a purely logical perspective

some eigenvalue inequalities for positive semidefinite matrix power products abstract we study the eigenvalues of positive semidefinite matrix power products and obtain some inequalities most of which are in terms of majorization in particular for a b 0 0 we prove log 1 a b log 1 a b the result is a generalization of some work of marcus lieb thirring le couteur bushell and trustrum

normal subgroups in the cremona group let k be an algebraically closed field we show that the cremona group of all birational transformations of the projective plane mathbb p mathbf k 2 is not a simple group the strategy makes use of hyperbolic geometry geometric group theory and algebraic geometry to produce elements in the cremona group that generate non trivial normal subgroups

bresse system with indefinite damping abstract we consider the bresse system in a bounded domain 0 l r 1 the system has an indefinite damping mechanism i e with a damping function a a x possibly changing sign presented only in the equation for the rotation angle we shall prove that the system is still exponentially stable under the same conditions as in the positive constant damping case and provided a 1 l 0 l a x d x 0 and a a l 2 for small enough in the arguments we shall also give a new proof of exponential stability for the constant case a a

timoshenko systems with indefinite damping abstract we consider the timoshenko system in a bounded domain 0 l r 1 the system has an indefinite damping mechanism i e with a damping function a a x possibly changing sign present only in the equation for the rotation angle we shall prove that the system is still exponentially stable under the same conditions as in the positive constant damping case and provided a 0 l a x d x 0 and a a l 2 for small enough the decay rate will be described explicitly in the arguments we shall also give a new proof of exponential stability for the constant case a a moreover we give a precise description of the decay rate and demonstrate that the system has the spectrum determined growth sdg property i e the type of the induced semigroup coincides with the spectral bound for its generator

eigenvalue asymptotics inverse problems and a trace formula for the linear damped wave equation abstract we determine the general form of the asymptotics for dirichlet eigenvalues of the one dimensional linear damped wave operator as a consequence we obtain that given a spectrum corresponding to a constant damping term this determines the damping term in a unique fashion we also derive a trace formula for this problem

a note on the lyapunov equation some results on the sensitivity of the solution of the stable lyapunov equation are shown to follow easily from well known theorems of functional analysis

exponential stability of wave equations with potential and indefinite damping abstract first we consider the linear wave equation u t t u x x a x u t b x u 0 on a bounded interval 0 l r the damping function a is allowed to change its sign if a 1 l 0 l a x d x is positive and the spectrum of the operator x x b is negative exponential stability is proved for small a a l 2 explicit estimates of the decay rate are given in terms of a and the largest eigenvalue of x x b second we show the existence of a global small smooth solution of the corresponding nonlinear wave equation u t t u x x a x u t b x u 0 if additionally the negative part of a is small enough compared with

exponential stability for wave equations with non dissipative damping abstract we consider the nonlinear wave equation u t t u x x a x u t 0 in a bounded interval 0 l r 1 the function a is allowed to change sign but has to satisfy a 1 l 0 l a x d x 0 for this non dissipative situation we prove the exponential stability of the corresponding linearized system for i possibly large a l with small a a l 2 and ii a class of pairs a l with possibly negative moment 0 l a x sin 2 x l d x estimates for the decay rate are also given in terms of a moreover we show the global existence of smooth small solutions to the corresponding nonlinear system if additionally the negative part of a is small enough

a topos foundation for theories of physics iii the representation of physical quantities with arrows this paper is the third in a series whose goal is to develop a fundamentally new way of viewing theories of physics our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system in paper ii we studied the topos representations of the propositional language pl s for the case of quantum theory and in the present paper we do the same thing for the more extensive local language l s one of the main achievements is to find a topos representation for self adjoint operators this involves showing that for any physical quantity a there is an arrow o a r where r is the quantity value object for this theory the construction of o a is an extension of the daseinisation of projection operators that was discussed in paper ii the object r is a monoid object only in the topos setsv h op of the theory and to enhance the applicability of the formalism we apply to r a topos a

majorizations and inequalities in matrix theory abstract in matrix theory majorization plays a significant role for instance majorization relations among eigenvalues and singular values of matrices produce a lot of norm inequalities and even matrix inequalities this survey article is intended as a review of recent results in matrix theory related to majorization

extremal majorizing and anti majorizing matrices in this note we study the convex set of all real n x n matrices which either increase or decrease mi xednlnrss when applied to any monotonously ordered probability n vector and we compute a complete list of its extremals in both cases 0 i998 published by

optimal design of the damping set for the stabilization of the wave equation abstract we consider the problem of optimizing the shape and position of the damping set for the internal stabilization of the linear wave equation in r n n 1 2 in a first theoretical part we reformulate the problem into an equivalent non convex vector variational one using a characterization of divergence free vector fields then by means of gradient young measures we obtain a relaxed formulation of the problem in which the original cost density is replaced by its constrained quasi convexification this implies that the new relaxed problem is well posed in the sense that there exists a minimizer and in addition the infimum of the original problem coincides with the minimum of the relaxed one in a second numerical part we address the resolution of the relaxed problem using a first order gradient descent method we present some numerical experiments which highlight the influence of the over damping phenomena and show that for large values of the damping potential the original problem has no minimizer we then propose a penalization technique to recover the minimizing sequences of the original problem from the optimal solution of the relaxed one

the solution of van der waerden s problem for permanents in this paper a solution to van der waerden s problem is presented it is hoped that the solution of this problem will stimulate new and vigorous interest in the theory of permanents and related topics permanents are ideal objects of study having both practical applications and abstract mathematical problems problems involving permanents tend to be easily stated but only solvable using deep methods the theory of permanents as an independent branch of mathematics which has developed parallel to the theory of determinants is far from exhausted many difficult questions must still be resolved in particular the problem of calculation and estimation of permanents for specific classes of matrices is of prime importance the solution of van der waerden s problem makes use of the penetrating force of the method of mixed volumes from combinatorial analysis even hilbert noted that there were connections to be uncovered between the theory of mixed volumes and other areas of mathematics the method of mixed volumes is ideally suited to solving extremal problems and problems of uniqueness and obtaining deep new inequalities it is reasonable to assume that in the future the method of mixed volumes will stand with that of generating functions as one of the basic analytic tools of computational combinatorial analysis

eigenvalue asymptotics for the damped wave equation on metric graphs abstract we consider the linear damped wave equation on finite metric graphs and analyse its spectral properties with an emphasis on the asymptotic behaviour of eigenvalues in the case of equilateral graphs and standard coupling conditions we show that there is only a finite number of high frequency abscissas whose location is solely determined by the averages of the damping terms on each edge we further describe some of the possible behaviour when the edge lengths are no longer necessarily equal but remain commensurate

periodic points of birational transformations on projective surfaces we give a classification of birational transformations on smooth projective surfaces which have a zariski dense set of noncritical periodic points in particular we show that if the first dynamical degree is greater than one the union of all noncritical periodic orbits is zariski dense

energy and invariant measures for birational surface maps given a birational self map of a compact complex surface it is useful to find an invariant measure that relates the dynamics of the map to its action on cohomology under a very weak hypothesis on the map we show how to construct such a measure the main point in the construction is to make sense of the wedge product of two positive closed 1 1 currents we are able to do this in our case because local potentials for each current have finite energy with respect to the other our methods also suffice to show that the resulting measure is mixing does not charge curves and has nonzero lyapunov exponents

topos models for physics and topos theory what is the role of topos theory in the topos models for quantum theory as used by isham butterfield doring heunen landsman spitters and others in other words what is the interplay between physical motivation for the models and the mathematical framework used in these models concretely we show that the presheaf topos model of butterfield isham and doring resembles classical physics when viewed from the internal language of the presheaf topos similar to the copresheaf topos model of heunen landsman and spitters both the presheaf and copresheaf models provide a quantum logic in the form of a complete heyting algebra although these algebras are natural from a topos theoretic stance we seek a physical interpretation for the logical operations finally we investigate dynamics in particular we describe how an automorphism on the operator algebra induces a homeomorphism or isomorphism of locales on the associated state spaces of the topos models and how elementary propositions and truth values transform under the action of this homeomorphism also with dynamics the focus is on the internal perspective of the topos

notes on egoritsjev s proof of the van der waerden conjecture abstract a solution to the van der waerden conjecture on the minimum of the permanent of doubly stochastic matrices has recently been found by g p ergoritsjev in russian because of the wide interest in this conjecture it seems useful to make the solution more accessible professor van lint has kindly consented to have his notes including a direct proof of an inequality available in the literature only in a more general form published in this journal ed r a b

degree growth of meromorphic surface maps we study the degree growth of iterates of meromorphic self maps of compact kahler surfaces using cohomology classes on the riemann zariski space we show that the degrees grow similarly to those of mappings that are algebraically stable on some bimeromorphic model 0 introduction let x be a compact kahler surface and let f x x be a dominant meromorphic mapping fix a kahler class on x normalized by x 1 and define the degree of f with respect to to be the positive real number deg f f x f x where x denotes the intersection form on h 1 1 r x when x p2 and is the class of a line this coincides with the usual algebraic degree of f one can show that deg f n m 2 deg f deg f for all m n hence the limit 1 lim n deg f exists we refer to it as the asymptotic degree of f it follows from standard arguments see proposition 3 1 that 1 does not depend on the choice of that 1 is invariant under bimeromorphic conjugacy and that 1 2 where 2 is the topological degree of f

invariant curves for birational surface maps we classify invariant curves for birational surface maps that are expanding on cohomology when the expansion is exponential the arithmetic genus of an invariant curve is at most one this implies severe constraints on both the type and number of irreducible components of the curve in the case of an invariant curve with genus equal to one we show that there is an associated invariant meromorphic two form

asymptotic stability for bresse systems abstract in this paper consider a vibrating system of bresse type in a one dimensional bounded domain with a nonlinear damping acting in the equation of the shear angle displacement and nonlinear localized damping in other equations we show the asymptotic stability without imposing conditions about the equal speed wave propagation using a method developed by lasiecka and tataru 9

instability results for the damped wave equation in unbounded domains abstract we extend some previous results for the damped wave equation in bounded domains in r d to the unbounded case in particular we show that if the damping term is of the form a with bounded a taking on negative values on a set of positive measure then there will always exist unbounded solutions for sufficiently large positive in order to prove these results we generalize some existing results on the asymptotic behaviour of eigencurves of one parameter families of schrodinger operators to the unbounded case which we believe to be of interest in their own right

optimal shape and position of the actuators for the stabilization of a string abstract the energy in a string subject to constant viscous damping k on a subset of length l 0 decays exponentially in time we consider the problem of optimizing the decay rate for the which are the unions of at most n intervals this rate is given by the spectral abscissa of the linear operator associated to the wave equation we are interested in small values of k therefore we consider the derivative of the spectral abscissa at k 0 we prove that except for the case l 1 2 when the number of intervals is not fixed a priori an optimal domain does not exist we study numerically the case of one or two intervals using a genetic algorithm these numerical results are not intuitive in particular the optimal position of one interval is never at the middle of the string