absolute continuity and existence of solutions to dynamic inclusions in time scales we provide some properties for absolutely continuous functions in time scales then we consider a class of dynamical inclusions in time scales and extend to this class a convergence result of a sequence of almost inclusion trajectories to a limit which is actually a trajectory of the inclusion in question we also introduce the so called euler solution to dynamical systems in time scales and prove its existence a combination of the existence of euler solutions with the compactness type result described above ensures the existence of an actual trajectory for the dynamical inclusion when the setvalued vector field is nonempty compact convex and has closed graph

approximation algorithms for the capacitated multi item lot sizing problem via flow cover inequalities we study the classical capacitated multi item lot sizing problem with hard capacities there are n items each of which has a specified sequence of demands over a finite planning horizon of t discrete periods the demands are known in advance but can vary from period to period all demands must be satisfied on time each order incurs a time dependent fixed ordering cost regardless of the combination of items or the number of units ordered but the total number of units ordered cannot exceed a given capacity c on the other hand carrying inventory from period to period incurs holding costs the goal is to find a feasible solution with minimum overall ordering and holding costs we show that the problem is strongly np hard and then propose a novel facility location type lp relaxation that is based on an exponentially large subset of the well known flow cover inequalities the proposed lp can be solved to optimality in polynomial time via an efficient separation procedure for this subset of inequalities moreover the optimal solution of the lp can be rounded to a feasible integer solution with cost that is at most twice the optimal cost this provides a 2 approximation algorithm which is the first constant approximation algorithm for the problem we also describe an interesting on the fly variant of the algorithm that does not require solving the lp a priori with all the flow cover inequalities as a by product we obtain the first theoretical proof regarding the strength of flow cover inequalities in capacitated inventory models

approximation algorithms for supply chain planning and logistics problems with market choice we propose generalizations of a broad class of traditional supply chain planning and logistics models that we call supply chain planning and logistics problems with market choice instead of the traditional setting we are given a set of markets each specified by a sequence of demands and associated with a revenue decisions are made in two stages in the first stage one chooses a subset of markets and rejects the others once that market choice is made one needs to construct a minimum cost production plan set of facilities to satisfy all of the demands of all the selected markets the goal is to minimize the overall lost revenues of rejected markets and the production facility opening and connection costs these models capture important aspects of demand shaping within supply chain planning and logistics models we introduce a general algorithmic framework that leverages existing approximation results for the traditional models to obtain corresponding results for their counterpart models with market choice more specifically any lp based approximation for the traditional model can be leveraged to a frac 1 1 e 1 alpha approximation algorithm for the counterpart model with market choice our techniques are also potentially applicable to other covering problems

hukuhara differentiability of interval valued functions and interval differential equations on time scales abstract using the concept of the generalized hukuhara difference in this paper we introduce and study the differentiability and the integrability for the interval valued functions on time scales some illustrative examples to interval differential equations on time scales are presented

primal dual algorithms for deterministic inventory problems we consider several classical models in deterministic inventory theory the single item lot sizing problem the joint replenishment problem and the multistage assembly problem these inventory models have been studied extensively and play a fundamental role in broader planning issues such as the management of supply chains for each of these problems we wish to balance the cost of maintaining surplus inventory for future demand against the cost of replenishing inventory more frequently for example in the joint replenishment problem demand for several commodities is specified over a discrete finite planning horizon the cost of maintaining inventory is linear in the number of units held but the cost incurred for ordering a commodity is independent of the size of the order furthermore there is an additional fixed cost incurred each time a nonempty subset of commodities is ordered the goal is to find a policy that satisfies all demands on time and minimizes the overall holding and ordering cost we shall give a novel primal dual framework for designing algorithms for these models that significantly improve known results in several ways the performance guarantees for the quality of the solutions improve on or match previously known results the performance guarantees hold under much more general assumptions about the structure of the costs and the algorithms and their analysis are significantly simpler than previous known results finally our primal dual framework departs from the structure of previously studied primal dual approximation algorithms in significant ways and we believe that our approach may find applications in other settings more specifically we provide 2 approximation algorithms for the joint replenishment problem and for the assembly problem and solve the single item lot sizing problem to optimality the results for the joint replenishment and the lot sizing problems also hold for their generalizations with back orders allowed as a byproduct of our work we prove known and new upper bounds on the integrality gap of some linear programming lp relaxations of the abovementioned problems

on nirenberg type conditions for higher order systems on time scales we study the existence of periodic solutions for a nonlinear system of nth order differential equations on time scales assuming a suitable nirenberg type condition we prove the existence of at least one solution of the problem using mawhin s coincidence degree

homoclinic orbits and periodic solutions for a class of hamiltonian systems on time scales abstract in this paper we are concerned with a second order non autonomous hamiltonian system on time scales t u t v u t u t f t t t under certain conditions the existence and multiplicity of periodic solutions are obtained for this hamiltonian system on time scales by using the saddle point theory the least action principle as well as the three critical point theorem in addition the existence of homoclinic orbit is obtained as a limit of 2 k t periodic solutions of a given sequence of hamiltonian system on time scales by means of the mountain pass theorem and the standard minimizing argument

a dynamic matrix exponential via a matrix cylinder transformation abstract in this work we develop a new matrix exponential on time scales via a cylinder transformation with a component wise locally integrable square matrix subscript our resulting matrix function can be written in terms of the matrix exponential of a lebesgue integral added to a logarithmic sum in terms of the gaps of a general time scale under strict commutativity conditions we show our dynamic matrix exponential is equivalent to the one in the standard literature finally we demonstrate that our matrix exponential satisfies a nonlinear dynamic integral equation

integrable equations on time scales integrable systems are usually given in terms of functions of continuous variables on r in terms of functions of discrete variables on z and recently in terms of functions of q variables on kq we formulate the gel fand dikii gd formalism on time scales by using the delta differentiation operator and find more general integrable nonlinear evolutionary equations in particular they yield integrable equations over integers difference equations and over q numbers q difference equations we formulate the gd formalism also in terms of shift operators for all regular discrete time scales we give a method allowing to construct the recursion operators for integrable systems on time scales finally we give a trace formula on time scales and then construct infinitely many conserved quantities casimirs of the integrable systems on time scales

a threshold of ln n for approximating set cover given a collection f of subsets of s 1 n setcover is the problem of selecting as few as possiblesubsets from f such that their union covers s and maxk cover is the problem of selecting k subsets from f such that their union has maximum cardinality both these problems arenp hard we prove that 1 o 1 ln n is a threshold below which setcover cannot be approximated efficiently unless np has slightlysuperpolynomial time algorithms this closes the gap up to low orderterms between the ratio of approximation achievable by the greedyalogorithm which is 1 o 1 lnn and provious results of lund and yanakakis that showed hardness ofapproximation within a ratio of log 2 n 2s0 72 ln n for max k cover we show an approximationthreshold of 1 1 e up tolow order terms under assumption that p np

noether s theorem on time scales abstract we show that for any variational symmetry of the problem of the calculus of variations on time scales there exists a conserved quantity along the respective euler lagrange extremals

fast approximation algorithms for the one warehouse multi retailer problem under general cost structures and capacity constraints we consider a well studied multi echelon deterministic inventory control problem known in the literature as the one warehouse multi retailer owmr problem we propose a simple and fast 2 approximation algorithm for this np hard problem by recombining the solutions of single echelon relaxations at the warehouse and at the retailers we then show that our approach remains valid under quite general assumptions on the cost structures and under capacity constraints at some retailers in particular we present the first approximation algorithms for the owmr problem with nonlinear holding costs truckload discount on procurement costs or with capacity constraints at some retailers in all cases the procedure is purely combinatorial and can be implemented to run in low polynomial time

reflection positivity and monotonicity we prove general reflection positivity results on riemannian manifolds for both scalar fields and dirac fields and comment on applications to quantum field theory as another application we prove the inequality cd cn between dirichlet and neumann covariance operators on a manifold with a reflection

hamilton jacobi theory over time scales and applications to linear quadratic problems abstract in this paper we first derive the verification theorem for nonlinear optimal control problems over time scales that is we show that the value function is the only solution of the hamilton jacobi equation in which the minimum is attained at an optimal feedback controller applications to the linear quadratic regulator problem lqr problem gives a feedback optimal controller form in terms of the solution of a generalized time scale riccati equation and that every optimal solution of the lqr problem must take that form a connection of the newly obtained riccati equation with the traditional one is established problems with shift in the state variable are also considered as an important tool for the latter theory we obtain a new formula for the chain rule on time scales finally the corresponding lqr problem with shift in the state variable is analyzed and the results are related to previous ones

higher order self adjoint boundary value problems on time scales in this study higher order self adjoint differential expressions on time scales and their associated self adjoint boundary conditions are discussed the symmetry property of the corresponding green s functions is shown together with specific formulas of green s functions for select time scales

hamiltonian systems on time scales linear and nonlinear hamiltonian systems are studied on time scales t we unify symplectic flow properties of discrete and continuous hamiltonian systems a chain rule which unifies discrete and continuous settings is presented for our so called alpha derivatives on generalized time scales this chain rule allows transformation of linear hamiltonian systems on time scales under simultaneous change of independent and dependent variables thus extending the change of dependent variables recently obtained by dosl and hilscher we also give the legendre transformation for nonlinear euler lagrange equations on time scales to hamiltonian systems on time scales

singular value conditions for stability of dynamic switched systems abstract in this paper the stability of a certain class of time varying systems evolving over nonuniformly spaced discrete domains is analyzed switched systems used here in the context of dynamic equations over time scale domains arise naturally in applications when a continuous time system is discretized via a sample and hold method with multiple sample rates the stability of switched systems is typically deduced by appealing to certain interrelated properties of the subsystems such as pairwise commutativity 24 simultaneous diagonalization 7 simultaneous triangularizability 8 or other lie algebraic conditions 1 which imply the existence of a common quadratic lyapunov function a novel approach is used here to determine the existence of quadratic lyapunov functions which does not rely on how the subsystems interact with each other this new method instead examines the role they each play in the aggregate system by way of singular value conditions results implying switched system stability and instability are developed under the two primary methods of examining such systems how the system behaves during arbitrary switching and how it behaves under the influence of a particular switching signal several examples illuminating the results are provided throughout the text as well as important insight discussing certain bounds on these results as they apply to stability theory in general

the laplace transform on time scales revisited abstract in this work we reexamine the time scale laplace transform as defined by bohner and peterson m bohner a peterson dynamic equations on time scales an introduction with applications birkhauser boston 2001 m bohner a peterson laplace transform and z transform unification and extension methods appl anal 9 1 2002 155 162 in particular we give conditions on the class of functions which have a transform develop an inversion formula for the transform and further we provide a convolution for the transform the notion of convolution leads to considering its algebraic structure in particular the existence of an identity element motivating the development of the dirac delta functional on time scales applications and examples of these concepts are given

uniqueness and existence results for initial value problems on time scales through a reciprocal problem and applications this paper is devoted to deriving the existence uniqueness and approximation of solutions to a first order initial value problem on an interval of an arbitrary time scale taking values in a different time scale we obtain the results considering dynamic equations whose nonlinear part is nonnegative and satisfies a type of inverse caratheodory s conditions and discontinuity several examples are given to illustrate the applicability of these results

realizations of nonlinear control systems on time scales nonlinear partially defined systems on an arbitrary unbounded time scale are studied they include continuous time and discrete time systems the main problem is to find necessary and sufficient conditions for an abstract input output map to have a realization as a nonlinear system of a specific class on the time scale the obtained results extend criteria of realizability of continuous time polynomial systems a simple construction of a realization is provided

lot sizing with constant batches formulation and valid inequalities we consider the classical lot sizing problem with constant production capacities lcc and a variant in which the capacity in each period is an integer multiple of some basic batch size lcb we first show that the classical dynamic program for lcc simplifies for lcb leading to an on2 min n c algorithm where n is the number of periods and c the batch size using this new algorithm we show how to formulate both problems as linear programs with on3 variables and constraints a class of so called k l s i inequalities are described for lcb which capture both the dynamic nature of the problem as well as the capacity aspects for lcb we prove that these inequalities are the only facet defining inequalities of a certain form for lcc we show that these inequalities include all the known classes of valid inequalities finally we discuss several open questions and possible extensions

calculus of variations on time scales weak local piecewise crd1 solutions with variable endpoints abstract a nonlinear calculus of variations problem on time scales with variable endpoints is considered the space of functions employed is that of piecewise rd continuously differentiable functions c1prd for this problem the euler lagrange equation the transversality condition and the accessory problem are derived as necessary conditions for weak local optimality assuming the coercivity of the second variation a corresponding second order sufficiency criterion is established

line integrals and green s formula on time scales in this paper we study curves parametrized by a time scale parameter introduce line delta and nabla integrals along time scale curves and obtain an analog of green s formula in the time scale setting

factorization method on time scales abstract we present an approach to the factorization method for second order difference equations on time scales we construct hilbert spaces of functions on the time scale and show how to construct a chain of intertwined first order difference operators a k in the sense that a k and a k act as ladder operators generating new solutions of eigenproblem of a k a k

approximation algorithms for metric facility location and k median problems using the primal dual schema and lagrangian relaxation we present approximation algorithms for the metric uncapacitated facility location problem and the metric k median problem achieving guarantees of 3 and 6 respectively the distinguishing feature of our algorithms is their low running time o m log m and o m log m l log n respectively where n and m are the total number of vertices and edges in the underlying complete bipartite graph on cities and facilities the main algorithmic ideas are a new extension of the primal dual schema and the use of lagrangian relaxation to derive approximation algorithms

on stabilisability of nonlinear systems on time scales in this article stabilisability of nonlinear finite dimensional control systems on arbitrary time scales is studied the classical results on stabilisation of nonlinear continuous time and discrete time systems are extended to systems on arbitrary time scales with bounded graininess function it is shown that uniform exponential stability of the linear approximation of a nonlinear system implies uniform exponential stability of the nonlinear system then this result is used to show a similar implication for uniform exponential stabilisability

mimetic methods on measure chains we introduce the divergence and the gradient for functions defined on a measure chain and this includes as special cases both continuous derivatives and discrete forward differences it is shown that in one dimension subject to dirichlet boundary conditions the divergence and the gradient are negative adjoints of each other and that the divergence of the gradient is negative semidefinite these are well known results in tile continuous theory and hence mimic those properties also for the case of a general measure chain 2001 elsevier science ltd all rights reserved

approximation algorithms for inventory problems with submodular or routing costs we consider the following two deterministic inventory optimization problems with non stationary demands submodular joint replenishment problem this involves multiple item types and a single retailer who faces demands over a finite planning horizon of t periods in each time period any subset of item types can be ordered incurring a joint ordering cost which is submodular moreover items can be held in inventory while incurring a holding cost the objective is to find a sequence of orders that satisfies all demands and minimizes the total ordering and holding costs inventory routing problem this involves a single depot that stocks items and multiple retailer locations facing demands over a finite planning horizon of t periods in each time period any subset of locations can be visited using a vehicle originating from the depot there is also cost incurred for holding items at any retailer the objective here is to satisfy all demands while minimizing the sum of routing and holding costs we present a unified approach that yields mathcal o left frac log t log log t right ologtloglogt factor approximation algorithms for both problems when the holding costs are polynomial functions a special case is the classic linear holding cost model wherein this is the first sub logarithmic approximation ratio for either problem

a constant approximation algorithm for the one warehouse multiretailer problem deterministic inventory theory provides streamlined optimization models that attempt to capture trade offs in managing the flow of goods through a supply chain we will consider two well studied deterministic inventory models called the one warehouse multiretailer owmr problem and its special case the joint replenishment problem jrp and give approximation algorithms with worst case performance guarantees that is for each instance of the problem our algorithm produces a solution with cost that is guaranteed to be at most 1 8 times the optimal cost this is called a 1 8 approximation algorithm our results are based on an lp rounding approach we provide the first constant approximation algorithm for the owmr problem and improve the previous results for the jrp

a non autonomous hamiltonian system on time scales abstract in this paper we are concerned with a second order non autonomous hamiltonian system on time scales t of the form u 2 t f t u t a e t 0 t t u 0 u t 0 u 0 u t 0 some results on the existence and multiplicity of periodic solutions are obtained for the hamiltonian system by means of the saddle point theorem the least action principle as well as the three critical point theorem

l2 spaces and boundary value problems on time scales abstract in this paper we consider a second order sturm liouville type boundary value operator of the form l u p u q u on an arbitrary bounded time scale t for suitable functions p q together with suitable boundary conditions operators of this type on time scales have normally been considered in a setting involving banach spaces of continuous functions on t in this paper we introduce a space l 2 t of square integrable functions on t and sobolev type spaces h n t n 1 consisting of l 2 t functions with nth order generalised l 2 t type derivatives we prove some basic functional analytic results for these spaces and then formulate the operator l in this setting in particular we allow p h 1 t while q l 2 t this generalises the usual conditions that p c rd 1 t q c rd 0 t 2 we give some immediate applications of the functional analytic results to l such as positivity injectivity invertibility and compactness of the inverse we also construct a green s function for l the analogues of these results on real intervals are well known and are fundamental to the usual sturm liouville theory on such intervals

a simple forward algorithm to solve general dynamic lot sizing models with n periods in 0n log n or 0n time this paper is concerned with the general dynamic lot size model or generalized wagner whitin model let n denote the number of periods into which the planning horizon is divided we describe a simple forward algorithm which solves the general model in 0n log n time and 0n space as opposed to the well known shortest path algorithm advocated over the last 30 years with 0n2 time a linear i e 0n time and space algorithm is obtained for two important special cases a models without speculative motives for carrying stock i e where in each interval of time the per unit order cost increases by less than the cost of carrying a unit in stock b models with nondecreasing setup costs we also derive conditions for the existence of monotone optimal policies and relate these to known planning horizon and other results from the literature

existence of weak solutions of two point boundary value problems for second order dynamic equations on time scales abstract in this paper by using critical point theory we establish the existence of weak solutions of the two point boundary value problem for a second order dynamic equation on an arbitrary time scale t so that the well known case of differential dynamic systems t r and the recently developed case of discrete dynamic systems t z are unified to the best of our knowledge this is the first time that boundary value problems of dynamic equations on time scales have been dealt with by using critical point theory

renormalization of the asymptotically expanded yang mills spectral action we study renormalizability aspects of the spectral action for the yang mills system on a flat 4 dimensional background manifold focusing on its asymptotic expansion interpreting the latter as a higher derivative gauge theory a power counting argument shows that it is superrenormalizable we determine the counterterms at one loop using zeta function regularization in a background field gauge and establish their gauge invariance consequently the corresponding field theory can be renormalized by a simple shift of the spectral function appearing in the spectral action

representations and ostrowski type inequalities on time scales here we give univariate and multivariate representations of the montgomery type for hybrid functions on time scales based on these we establish univariate and multivariate ostrowski type inequalities on time scales domains these compare the average of a function to its values the estimates involve the higher order delta and nabla derivatives and partial derivatives we finish with applications on the time scales r and z

on green s functions and positive solutions for boundary value problems on time scales in this paper we offer a form of self adjoint differential equations on time scales so that the associated green s function is found symmetric in the usual sense for this purpose together with the delta derivative we employ the nabla derivative as well we introduce the concepts of lebesgue delta and nabla integrals on time scales next sign properties of the green s function are investigated and existence results for positive solutions of nonlinear boundary value problems are established upper and lower bounds for these positive solutions also are given

nonlinear integro differential equations and optimal control problems on time scales this paper is mainly concerned with a class of optimal control problems of a system governed by the nonlinear integro differential equation on time scales using the contraction mapping principle the existence and uniqueness of a weak solution of the nonlinear integro differential equation on time scales are proved discussing l t 1 strong weak lower semicontinuity of integral functional on time scales we give sufficient conditions for the existence of optimal control by the virtue of the integration by parts formula on time scales the necessary conditions of optimality are derived

integration on time scales in this paper we study the process of riemann and lebesgue integration on time scales the relationship of the riemann and lebesgue integrals is considered and a criterion for riemann integrability is established

quantum field theory on curved backgrounds i the euclidean functional integral we give a mathematical construction of euclidean quantum field theory on certain curved backgrounds we focus on generalizing osterwalder schrader quantization as these methods have proved useful to establish estimates for interacting fields on flat space times in this picture a static killing vector generates translations in euclidean time and the role of physical positivity is played by positivity under reflection of euclidean time we discuss the quantization of flows which correspond to classical space time symmetries and give a general set of conditions which imply that broad classes of operators in the classical picture give rise to well defined operators on the quantum field hilbert space in particular killing fields on spatial sections give rise to unitary groups on the quantum field hilbert space and corresponding densely defined self adjoint generators we construct the schrodinger representation using a method which involves localizing certain integrals over the full manifold to integrals over a codimension one submanifold this method is called sharp time localization and implies reflection positivity

periodicity of scalar dynamic equations and applications to population models easily verifiable sufficient criteria are established for the existence of periodic solutions of a class of nonautonomous scalar dynamic equations on time scales which incorporate as special cases many single species models governed by ordinary differential and difference equations when the time scale is the set of all real and all integer numbers respectively

constant approximation algorithms for the one warehouse multiple retailers problem with backlog or lost sales we consider the one warehouse multi retailer owmr problem with deterministic time varying demand in the case where shortages are allowed demand may be either backlogged or lost we present a simple combinatorial algorithm to build an approximate solution from a decomposition of the system into single echelon subproblems we establish that the algorithm has a performance guarantee of 3 for the owmr with backlog under mild assumptions on the cost structure in addition we improve this guarantee to 2 in the special case of the joint replenishment problem jrp with backlog as a by product of our approach we show that our decomposition provides a new lower bound of the optimal cost a similar technique also leads to a 2 approximation for the owmr problem with lost sales in all cases the complexity of the algorithm is linear in the number of retailers and quadratic in the number of time periods which makes it a valuable tool for practical applications to the best of our knowledge these are the fisrt constant approximations for the owmr with shortages

necessary conditions of optimality for a class of optimal control problems on time scales a class of optimal control problems of a system governed by linear dynamic equations on time scales with quadratic cost functional is considered by the lebesgue d integral theory and the sobolev type space h 1 on time scales the weak solution of linear dynamic equations on time scales for both initial value problem and backward problem are introduced therefore the necessary conditions of optimality are presented some typical examples are given for demonstration

basic qualitative and quantitative results for solutions to nonlinear dynamic equations on time scales with an application to economic modelling abstract this article investigates both basic qualitative and basic quantitative properties of solutions to first and higher order dynamic equations on time scales and thus provides a foundation and framework for future advanced nonlinear studies in the field particular focus lies in the existence uniqueness dependency approximation and explicit representation of solutions to nonlinear initial value problems the main tools used are from modern areas of nonlinear analysis including the fixed point theorems of banach and schafer the method of successive approximations a novel definition of measuring distance in metric spaces and normed spaces and a separation of variables technique is introduced to the general time scale setting the new results compliment and extend those of stefan hilger s seminal paper of 1990 as an application of the new results we present and analyse a simple model from economics known as the keynesian cross model with lagged income in the general time scale environment ideas suggesting further applications and possible new directions for the novel results are also presented

the submodular joint replenishment problem the joint replenishment problem is a fundamental model in supply chain management theory that has applications in inventory management logistics and maintenance scheduling in this problem there are multiple item types each having a given time dependent sequence of demands that need to be satisfied in order to satisfy demand orders of the item types must be placed in advance of the due dates for each demand every time an order of item types is placed there is an associated joint setup cost depending on the subset of item types ordered this ordering cost can be due to machine transportation or labor costs for example in addition there is a cost to holding inventory for demand that has yet to be served the overall goal is to minimize the total ordering costs plus inventory holding costs in this paper the cost of an order also known as a joint setup cost is a monotonically increasing submodular function over the item types for this general problem we show that a greedy approach provides an approximation guarantee that is logarithmic in the number of demands then we consider three special cases of submodular functions which we call the laminar tree and cardinality cases each of which can model real world scenarios that previously have not been captured for each of these cases we provide a constant factor approximation algorithm specifically we show that the laminar case can be solved optimally in polynomial time via a dynamic programming approach for the tree and cardinality cases we provide two different linear programming based approximation algorithms that provide guarantees of three and five respectively

on a disconjugacy criterion for second order dynamic equations on time scales in this paper a sufficient condition for disconjugacy of second order differential equations on time scales is given for this purpose a mean value result on time scales is employed

an approximate dynamic programming approach to the joint replenishment problem the main contribution of this paper is to propose a new dynamic programming approach that e approximates the joint replenishment problem with stationary demands and holding costs in its discrete time finite horizon setting our first and foremost objective is to show that the computation time of classical dynamic programming algorithms can be improved on by orders of magnitude when one is willing to lose an e factor in optimality based on synthesizing ideas such as commodity aggregation approximate dynamic programming and a few guessing tricks we show that one can attain any required degree of accuracy in near polynomial time

reflection positivity in higher derivative scalar theories reflection positivity constitutes an integral prerequisite in the osterwalder schrader reconstruction theorem which relates quantum field theories defined on euclidean space to their lorentzian signature counterparts in this work we rigorously prove the violation of reflection positivity in a large class of free scalar fields with a rational propagator this covers in particular higher derivative theories where the propagator admits a partial fraction decomposition as well as degenerate cases including e g p4 type propagators

weak solutions for the dynamic cauchy problem in banach spaces abstract this paper is devoted to unify and extend the results of the existence of the weak solutions of continuous and discrete cauchy problem in banach spaces we offer the existence of the weak solution of dynamic cauchy problem on an infinite time scale the measure of weak noncompactness and the fixed point theorem of kubiaczyk are used to prove the main result

solvability of sturm liouville problems on time scales at resonance under the appropriate growth conditions on the inhomogeneous term we present the existence of the nontrivial solution for sturm liouville boundary value problems on time scales at resonance by making use of coincidence degree theory of mawhin the interesting point is the nonlinearity f which is involved with first order d derivative explicitly

on existence of optimal control governed by a class of the first order linear dynamic systems on time scales in this paper by using the approximation of classical solution we introduce the definition of solution and prove the existence and uniqueness of solutions of the first order linear dynamic systems on time scales existence of lagrange optimal control problem governed by the first order linear dynamic systems on time scales is also presented for illustration some examples of optimal control problems on time scales are also discussed

stability by linear approximation for time scale dynamical systems abstract we study systems on time scales that are generalizations of classical differential or difference equations and appear in numerical methods in this paper we consider linear systems and their small nonlinear perturbations in terms of time scales and of eigenvalues of matrices we formulate conditions sufficient for stability by linear approximation for non periodic time scales we use techniques of central upper lyapunov exponents a common tool of the theory of linear odes to study stability of solutions also time scale versions of the famous chetaev s theorem on conditional instability are proved in a nutshell we have developed a completely new technique in order to demonstrate that methods of non autonomous linear ode theory may work for time scale dynamics

on the connection between the hilger and radon nikodym derivatives abstract we show that the hilger derivative on time scales is a special case of the radon nikodym derivative with respect to the natural measure associated with every time scale moreover we show that the concept of delta absolute continuity agrees with the one from measure theory in this context

multiple positive solutions of singular dirichlet problems on time scales via variational methods abstract this paper is devoted to using variational techniques and critical point theory to derive some sufficient conditions for the existence of multiple positive solutions to a singular second order dynamic equation with homogeneous dirichlet boundary conditions