equations of motion for variational electrodynamics abstract we extend the variational problem of wheeler feynman electrodynamics by generalizing the electromagnetic functional to a local space of absolutely continuous trajectories possessing a derivative velocities of bounded variation we show here that the gateaux derivative of the generalized functional defines two partial lagrangians for variations in our generalized local space one for each particle we prove that the critical point conditions of the generalized variational problem are i the euler lagrange equations must hold lebesgue almost everywhere and ii the momentum of each partial lagrangian and the legendre transform of each partial lagrangian must be absolutely continuous functions generalizing the weierstrass erdmann conditions

minimizers with discontinuous velocities for the electromagnetic variational method the electromagnetic two body problem has neutral differential delay equations of motion that for generic boundary data can have solutions with discontinuous derivatives if one wants to use these neutral differential delay equations with arbitrary boundary data solutions with discontinuous derivatives must be expected and allowed surprisingly wheeler feynman electrodynamics has a boundary value variational method for which minimizer trajectories with discontinuous derivatives are also expected as we show here the variational method defines continuous trajectories with piecewise defined velocities and accelerations and electromagnetic fields defined by the euler lagrange equations on trajectory points here we use the piecewise defined minimizers with the lienard wierchert formulas to define generalized electromagnetic fields almost everywhere but on sets of points of zero measure where the advanced retarded velocities and or accelerations are discontinuous along with this generalization we formulate the generalized absorber hypothesis that the far fields vanish asymptotically almost everywhere and show that localized orbits with far fields vanishing almost everywhere must have discontinuous velocities on sewing chains of breaking points we give the general solution for localized orbits with vanishing far fields by solving a linear neutral differential delay equation for these far fields we discuss the physics of orbits with discontinuous derivatives stressing the differences to the variational methods of classical mechanics and the existence of a spinorial four current associated with the generalized variational electrodynamics

improper regular conditional distributions improper regular conditional distributions rcd s given a field a have the following anomalous property for sets a a pr a a is not always equal to the indicator of a such a property makes the conditional probability puzzling as a representation of uncertainty when rcd s exist and the field a is countably generated then almost surely the rcd is proper we give sufficient conditions for an red to be improper in a maximal sense and show that these conditions apply to the tail field and the field of symmetric events

on equilibria of non compact generalized games abstract a fixed point theorem is first proved from which a theorem on the existence of maximal elements for lc majorized correspondences defined on a non compact set is obtained next an equilibrium theorem for non compact qualitative games with lc majorized correspondences defined on a non compact strategy set is proved and is applied to prove an equilibrium theorem for non compact generalized games with lc majorized correspondences defined on a non compact strategy set our results generalize several previously published corresponding results

singular hopf bifurcation in a differential equation with large state dependent delay we study the onset of sustained oscillations in a classical state dependent delay sdd differential equation inspired by control theory owing to the large delays considered the hopf bifurcation is singular and the oscillations rapidly acquire a sawtooth profile past the instability threshold using asymptotic techniques we explicitly capture the gradual change from nearly sinusoidal to sawtooth oscillations the dependence of the delay on the solution can be either linear or nonlinear with at least quadratic dependence in the former case an asymptotic connection is made with the rayleigh oscillator in the latter van der pol s equation is derived for the small amplitude oscillations sdd differential equations are currently the subject of intense research in order to establish or amend general theorems valid for constant delay differential equation but explicit analytical construction of solutions are rare this paper illustrates the use of singular perturbation techniques and the unusual way in which solvability conditions can arise for sdd problems with large delays

electromagnetic two body problem recurrent dynamics in the presence of state dependent delay we study the electromagnetic two body problem of classical electrodynamics as a prototype dynamical system with state dependent delays the equations of motion are analysed with reference to motion along a straight line in the presence of an electrostatic field we consider the general electromagnetic equations of motion for point charges with advanced and retarded interactions and study two limits a retarded only interactions dirac electrodynamics and b half retarded plus half advanced interactions wheeler feynman electrodynamics a fixed point is created where the electrostatic field balances the coulombian attraction and we use local analysis near this fixed point to derive necessary conditions for a hopf bifurcation in case a we study a hopf bifurcation about an unphysical fixed point and find that it is subcritical in case b there is a hopf bifurcation about a physical fixed point and we study several families of periodic orbits near this point the bifurcating periodic orbits are illustrated and simulated numerically by introducing a surrogate dynamical system into the numerical analysis which transforms future data into past data by exploiting the periodicity thus obtaining systems with only delays

a new approach to partial synchronization in globally coupled rotators we develop a formalism to analyse the behaviour of pulse coupled identical phase oscillators with specific attention devoted to the onset of partial synchronization the method which allows describing the dynamics both at the microscopic and macroscopic level is introduced in a general context but then the application to the dynamics of leaky integrate and fire lif neurons is analysed as a result we derive a set of delayed equations describing exactly the lif behaviour in the thermodynamic limit we also investigate the weak coupling regime by means of a perturbative analysis which reveals that the evolution rule reduces to a set of ordinary differential equations robustness and generality of the partial synchronization regime is finally tested both by adding noise and considering different force fields

finitely maxitive conditional possibilities bayesian like inference disintegrability and conglomerability the aim of the paper is to study bayesian like inference processes involving coherent finitely maxitive t conditional possibilities assessed on infinite sets of conditional events coherence of an assessment consisting of an arbitrary possibilistic prior and an arbitrary possibilistic likelihood function is proved thus a closed form expression for the envelopes of the relevant joint and posterior possibilities is given when t is the minimum or a strict t norm the notions of disintegrability and conglomerability are also studied and their relevance in the infinite version of the possibilistic bayes formula is highlighted

generalized vector quasi equilibrium problems with applications in this paper we consider the generalized vector quasi equilibrium problem with or without involving condensing maps and prove the existence of its solution by using known fixed point and maximal element theorems as applications of our results we derive some existence results for a solution to the vector quasi optimization problem for nondifferentiable functions and vector quasi saddle point problem

on the approximation of upper semi continuous correspondences and the equilibriums of generalized games in section 1 we explain some of the definitions and terminology that we use in section 2 we prove several theorems concerning the approximation of upper semi continuous correspondences having for range a locally convex space theorems 1 and 2 and corollary 1 generalize certain approximation theorems by g haddad 16 pp 1352213541 g haddad and j m lasry 17 pp 299 3001 and j p aubin and a cellina 2 pp 86891 see also f s de blasi lo and j m lasry and r robert 21 although some of the details of the proofs of theorems 1 and 2 are new the basic ideas are taken from the above mentioned papers theorem 3 shows that the correspondences in the approximating families can be chosen so that they are regular see section 1 this theorem see the remark following its proof contains a classical result of m hukuhara 18 pp 56573 theorem 3 is proved using theorem 2 and propositions 1 and 4 variants of propositions 2 3 and 4 were given in 193 in section 3 we give among others theorem 5 which concerns the existence of equilibriums of generalized games abstract economies the main purpose of this result is to replace the continuity hypothesis in the w shafer h sonnenschein equilibrium theorem for generalized games by an upper semi continuity one the proof of theorem 5 is based on the results on the approximation of upper semi continuous correspondences obtained in section 2 theorem 5 us used in section 4 in section 4 we establish theorems 6 and 7 these theorems show that certain statements concerning the equilibrium of generalized games are equivalent to certain statements concerning minimax inequalities of k fan type theorems 1 2 3 5 6 and 7 are the main results of this paper 267 0022 247x 88 3 00

the extent of non conglomerability of finitely additive probabilities an arbitrary finitely additive probability can be decomposed uniquely into a convex combination of a countably additive probability and a purely finitely additive pfa one the coefficient of the pfa probability is an upper bound on the extent to which conglomerability may fail in a finitely additive probability with that decomposition if the probability is defined on a field the bound is sharp hence non conglomerability or equivalently non disintegrability characterizes finitely as opposed to countably additive probability nonetheless there exists a pfa probability which is simultaneously conglomerable over an arbitrary finite set of partitions neither conglomerability nor non conglomerability in a given partition is closed under convex combinations but the convex combination of pfa ultrafilter probabilities each of which cannot be made conglomerable in a common margin is singular with respect to any finitely additive probability that is conglomerable in that margin

finite element boundary value integration of wheeler feynman electrodynamics the electromagnetic two body problem is solved as a boundary value problem associated to an action functional we show that the functional is frechet differentiable and that its conditions for criticality are the mixed type neutral differential delay equations with state dependent delay of wheeler feynman electrodynamics we construct a finite element method that finds c 1 smooth solutions when suitable past and future positions of the particles are given as boundary data the numerical trajectories satisfy a variational problem defined in a finite dimensional hermite functional space of c 1 piecewise polynomials the numerical variational problem is solved using a combination of newton s method intercalated with boundary adjustments to ensure that the velocity of the solution is continuous with the boundary data we recover the known circular orbits and compute several other novel trajectories of the wheeler feynman electrodynamics we also discuss the local convexity of the functional close to the new found trajectories and the possibility of solutions with less regularity

variational principle for the wheeler feynman electrodynamics we adapt the formally defined fokker action into a variational principle for the electromagnetic two body problem we introduce properly defined boundary conditions to construct a functional of a finite orbital segment into the reals the boundary conditions for the variational principle are an end point along each trajectory plus the respective segment of trajectory for the other particle inside the light cone of each end point we show that the conditions for an extremum of our functional are the mixed type neutral equations with implicit state dependent delay of the electromagnetic two body problem we put the functional on a natural banach space and show that the functional is frechet differentiable we develop a method to calculate the second variation for c2 orbital perturbations in general and in particular about circular orbits of large enough radii we prove that our functional has a local minimum at circular orbits of large enough radii at variance with the limiting kepler action that has a

a further generalization of yannelis prabhakar s continuous selection theorem and its applications abstract in this article we first prove one improved version of the yannelis prabhakar continuous selection theorem and next as its applications a fixed point theorem in noncompact product spaces a nonempty intersection theorem some existence theorems of solutions for the generalized quasi variational inequalities and some equilibrium existence theorems for the abstract economies are given

collective fixed points and maximal elements with applications to abstract economies abstract in this paper we first establish collective fixed points theorems for a family of multivalued maps with or without assuming that the product of these multivalued maps is condensing as an application of our collective fixed points theorem we derive the coincidence theorem for two families of multivalued maps defined on product spaces then we give some existence results for maximal elements for a family of ls majorized multivalued maps whose product is condensing we also prove some existence results for maximal elements for a family of multivalued maps which are not ls majorized but their product is condensing as applications of our results some existence results for equilibria of abstract economies are also derived the results of this paper are more general than those given in the literature

analysis of a model representing stage structured population growth with state dependent time delay a stage structured model of population growth is proposed where the time to ma turity is itself state dependent it is shown that under appropriate assumptions all solutions are positive and bounded criteria for the existence of positive equilibria and further conditions for the uniqueness of the equilibria are given the stability of the equilibria are also discussed in addition an attracting region is determined for solutions such that this region collapses to the unique positive equilibrium in the state independent case

equilibria of non compact generalized games with l majorized preference correspondences an existence theorem of maximal elements in a non compact set for l majorized correspondences is obtained next an existence theorem of an equilibrium in a qualitative game is proved and is then applied to achieve an existence theorem of equilibrium in a non compact abstract economy with l majorized preference correspondences these results are either very closely related to or generalizations of those recent results of borglin keiding yannelis prabhakar toussaint and tulcea

fixed point and maximal element theorems with applications to abstract economies and minimax inequalities in this paper we establish some fixed point theorems for a family of multivalued maps under mild conditions by using our fixed point theorems we derive some maximal element theorems for a particular family of multivalued maps namely the condensing multivalued maps as applications of our results we prove some general equilibrium existence theorems in the generalized abstract economies with preference correspondences further applications of our results are also given to minimax inequalities for a family of functions

geometric integration of the electromagnetic two body problem the equations of motion of the two body problem of dirac s electrodynamics of point charges consist of a delay equation for the proton and a delay equation for the electron these equations involve the third derivative of the charges s position and have runaway solutions which make forward numerical integration troublesome dirac s equations of motion are algebraic delay equations involving a degenerate linear form of the past accelerations a fredholm alternative yields a system of second order delay equations of motion plus a constraint on the initial segment of trajectory here we use the fredholm constraint as a geometric tool to derive covariant second order equations of motion in position for backward time integration we also extend the backward integration scheme to include a generalized version of dirac s theory that includes two delays

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0 1 laws for regular conditional distributions is c g see 2 3 and theorem 1 of 4 in other terms not onlyeverywhere properness is to be weakened into condition 2 but the latterholds if and only if a is c g under p if a fails to be c g under p various weaker notions of r c d have beeninvestigated roughly speaking in such notions is asked to be everywhereproper but additivity and or measurability are relaxed see 1 and refer ences therein however not very much is known on r c d s regarded in theusual sense when a is not c g under p one exception is 11 in particu lar when 2 fails one question is whether some of its consequences are stillavailable in this paper among these consequences we focus on 3 h 0 1 for all a