dimensionally dependent tensor identities by double antisymmetrization some years ago lovelock showed that a number of apparently unrelated familiar tensor identities had a common structure and could all be considered consequences in n dimensional space of a pair of fundamental identities involving trace free p p forms where 2p n we generalize lovelock s results and by using the fact that associated with any tensor in n dimensional space there is associated a fundamental tensor identity obtained by antisymmetrizing over n 1 indices we establish a very general master identity for all trace free k l forms we then show how various other special identities are direct and simple consequences of this master identity in particular we give direct application to maxwell lanczos ricci bel and bel robinson tensors and also demonstrate how relationships between scalar invariants of the riemann tensor can be investigated in a systematic manner

an interval algorithm for constrained global optimization abstract an interval algorithm for bounding the solutions of a constrained global optimization problem is described the problem functions are assumed only to be continuous it is shown how the computational cost of bounding a set which satisfies equality constraints can often be reduced if the equality constraint functions are assumed to be continuously differentiable numerical results are presented

the determination of all syzygies for the dependent polynomial invariants of the riemann tensor i pure ricci and pure weyl invariants in this paper we shall consider all pure ricci and pure weyl scalar invariants of any degree in a four dimensional lorentzian space we present a general graph theoretic based reduction algorithm which decomposes using syzygies any pure invariant in terms of the independent base invariants r1 r2 r3 or w1 w2 e zakhary and j carminati j math phys 42 1474 2001

the convergence speed of interval methods for global optimization abstract three particular algorithms from a class of interval subdivision methods for global optimization are studied the theoretical upper bound on the convergence speed of hansen s method is given the three methods by hansen moore skelboe and a new one with a random actual box selection rule are compared numerically

conditions for the alignment of the principal null directions of two weyl like tensors a possible means to classify the interaction between the weyl and ricci tensors is to look at the number of principal null directions that the weyl and plebanski tensors have in common this paper presents algebraic conditions that can be used to determine this number without explicitly calculating the principal null directions themselves

positivity properties of the bel robinson tensor the bel robinson tensor is a rank 4 tensor constructed from the weyl curvature tensor it is well known that when it is contracted with four future pointing vectors a non negative value is always obtained in this paper we investigate various cases in which the value zero is attained and show that it occurs in some situations where space time is of an algebraically special type this is of importance for expressions of gravitational energy related to the bel robinson tensor some new invariant ways of classifying the petrov type of space times using the bel robinson tensor are also presented

a lanczos potential in kerr geometry a lanczos potential is a tensor of rank 3 for which a certain combination of derivatives always equals the weyl tensor general existence theorems have been proved before and some explicit expressions for lanczos potentials in certain spacetimes have been found in this paper we explicitly study a lanczos potential in kerr spacetime obtained from a previously studied flat connection as an application it is shown that the mass can be expressed as an integral of the lanczos potential

towards a generalized singular value decomposition we suggest a form for and give a constructive derivation of the generalized singular value decomposition of any two matrices having the same number of columns we outline its desirable characteristics and compare it to an earlier suggestion by van loan siam j numer anal 13 1976 pp 76 83 the present form largely follows from the work of van loan but is slightly more general and computationally more amenable than that in the paper cited we also prove a useful extension of a theorem of stewart siam rev 19 1977 pp 634 662 on unitary decompositions of submatrices of a unitary matrix

on interval estimates of perturbations of generalized eigenvalues for diagonalizable pairs abstract on generalized eigenvalue perturbation bounds for diagonalizable matrix pairs upper bounds are discussed all the time in this paper we mainly consider not only upper bounds but also lower bounds of perturbation of generalized eigenvalues for diagonalizable matrix pairs sharper upper bounds and sharp lower bounds on perturbation of generalized eigenvalues for diagonalizable matrix pairs are obtained in terms of the distances of two points in grassmann manifold the main results improve the existing results of chen 2007 1 and numerical examples illustrate the efficiency of the theoretical results

the weyl lanczos equations and the lanczos wave equation in four dimensions as systems in involution the weyl lanczos equations in four dimensions form a system in involution we compute its cartan characters explicitly and use janet riquier theory to confirm the results in the case of all space times with a diagonal metric tensor and for the plane wave limit of space times we write the lanczos wave equation as an exterior differential system and with assistance from janet riquier theory we compute its cartan characters and find that it forms a system in involution we compare these cartan characters with those of the weyl lanczos equations all results hold for the real analytic case

necessary and sufficient conditions for n dimensional conformal einstein spaces via dimensionally dependent identities listing has recently extended results of kozameh newman and tod for four dimensional space times and presented a set of necessary and sufficient conditions for a metric to be locally conformally equivalent to an einstein metric in all semi riemannian spaces of dimension n 4 subject to a nondegeneracy restriction on the weyl tensor by exploiting dimensionally dependent identities we demonstrate how to construct two alternative versions of these necessary and sufficient conditions which we believe will be useful in applications the four dimensional case is discussed in detail and examples are also given in five and six dimensions

empirical evaluation of innovations in interval branch and bound algorithms for nonlinear systems interval branch and bound algorithms for finding all roots use a combination of a computational existence uniqueness procedure and a tessellation process generalized bisection such algorithms identify with mathematical rigor a set of boxes that contains unique roots and a second set within which all remaining roots must lie though each root is contained in a box in one of the sets the second set may have several boxes in clusters near a single root thus the output is of higher quality if there are relatively more boxes in the first set in contrast to previously implemented similar techniques a box expansion technique in this paper based on using an approximate root finder epsilon inflation and exact set complementation decreases the size of the second set increases the size of the first set and never loses roots in addition to the expansion technique use of second order extensions to eliminate small boxes that do not contain roots and interval slopes versus interval derivative matrices are studied these items are evaluated empirically on a significant test problem set within a fortran 90 environment designed for such purposes the results are compared with previous results and show that careful incorporation of the techniques yields both quantitatively and qualitatively superior computer codes

solving a huff like competitive location and design model for profit maximization in the plane abstract a chain wants to set up a single new facility in a planar market where similar facilities of competitors and possibly of its own chain are already present fixed demand points split their demand probabilistically over all facilities in the market proportionally with their attraction to each facility determined by the different perceived qualities of the facilities and the distances to them through a gravitational or logit type model both the location and the quality design of the new facility are to be found so as to maximise the profit obtained for the chain several types of constraints and costs are considered two solution methods are developed and tested the first is a repeated local optimisation heuristic extending earlier proposals to the supplementary design question and the presence of locational constraints the second is an exact global optimisation technique based on reliable computing using interval analysis incorporating several novel features an example and comparative computational results demonstrate that this difficult and very multi modal problem can be solved by such techniques the local optimisation method turns out not to be very robust in its results even after numerous repetitions whereas the global optimisation method yields very useful and complete information on guaranteed near to optimal solutions after an important but still quite acceptable computational effort

providing a basin of attraction to a target region of polynomial systems by computation of lyapunov like functions in this paper we present a method for computing a basin of attraction to a target region for polynomial ordinary differential equations this basin of attraction is ensured by a lyapunov like polynomial function that we compute using an interval based branch and relax algorithm this algorithm relaxes the necessary conditions on the coefficients of the lyapunov like function to a system of linear interval inequalities that can then be solved exactly it iteratively refines these relaxations in order to ensure that whenever a nondegenerate solution exists it will eventually be found by the algorithm application of an implementation to a range of benchmark problems shows the usefulness of the approach

perturbation analysis for the generalized singular value problem this paper discusses perturbation bounds for generalized singular values and for subspaces associated with the generalized singular value decomposition of two matrices having the same number of columns by a limiting procedure from our results we can derive those for the ordinary singular value problem which were proved by l mirsky quart j math oxford 11 1960 pp 50 59 and p a wedin bit 12 1972 pp 99 111

subdivision direction selection in interval methods for global optimization the role of the interval subdivision selection rule is investigated in branch and bound algorithms for global optimization the class of rules that allows convergence for the model algorithm is characterized and it is shown that the four rules investigated satisfy the conditions of convergence a numerical study with a wide spectrum of test problems indicates that there are substantial differences between the rules in terms of the required cpu time the number of function and derivative evaluations and space complexity and two rules can provide substantial improvements in efficiency

on the problem of algebraic completeness for the invariants of the riemann tensor iii we study the set cz of invariants zakhary and carminati j math phys 42 1474 2001 for the class of space times whose ricci tensors possess a null eigenvector we show that all cases are maximally backsolvable in terms of sets of invariants from cz but that some cases are not completely backsolvable and these all possess an alignment between an eigenvector of the ricci tensor with a repeated principal null vector of the weyl tensor we provide algebraically complete sets for each canonically different space time and hence conclude with these results and those of a previous article carminati zakhary and mclenaghan j math phys 43 492 2002 that the cz set is determining or maximal

algebraic invariants of the riemann tensor in a four dimensional lorentzian space a set of 16 scalar invariants is given of the riemann tensor which is shown to contain complete minimal sets in the einstein maxwell and perfect fluid cases all previously known sets fail to be complete in the perfect fluid case

efficiency of a global optimization algorithm an algorithm forr determining the global minimum of an unconstrained function f is considered the algorithm is due to skelboe moore asaithambe and shen the method is applicable to functions f having so called inclusion functions such inclusion functions are easily found using some simple features of interval arithmetic the convergence conditions of the algorithm were studied by ratschek 16 where it was shown that assumptions like continuity convexity bounded number of global or local minimizers etc can be dropped present paper continues the investigations of the algorithm and presents sharp estimates for the convergence order the order depends polynomially on the size of the domain of f and exponentially on the order of the inclusion function chosen and the dimension of the problem

the analytic solutions of a class of constrained matrix minimization and maximization problems with applications in this paper we present the analytic solutions of the following constrained matrix determinant and trace minimization and maximization problems min u 1 ldots u m in mathbb u n vert d

epsilon inflation in verification algorithms epsilon inflation is often used in verification numerics to find an interval vector x e such that some interval function f maps x e into itself we recall algorithms which use epsilon inflation to verify this subset property f x e x e and we derive criteria which guarantee that already finitely many inflation steps are sufficient to prove it these criteria require f to be a p contraction we recall results on p contractions and we derive rules to verify them we also introduce local p contractions and we use this concept to guarantee again the subset property above when using epsilon inflation

an interval step control for continuation methods the authors present a step control for continuation methods that is deterministic in the sense that i it computationally but rigorously verifies that the corrector iteration will converge to a point on the same curve as the previous point i e the predictor corrector iteration will never jump across paths and ii each predictor step is as large as possible subject to verification that the curve is unique with the given interval extension the authors present performance data and comparisons with an approximate step control method pitcon version 6 1 a comparison of plots obtained from both step controls reveals that an approximate step control will behave erratically in situations where the interval step control leads to orderly progression along the curve this is true even if the maximum allowable stepsize for the approximate method is set to be smaller than many of the steps actually taken by the interval algorithm limitations of interval step controls are also discussed

a new verified optimization technique for the packing circles in a unit square problems this paper presents a new verified optimization method for the problem of finding the densest packings of nonoverlapping equal circles in a square in order to provide reliable numerical results the developed algorithm is based on interval analysis as one of the most efficient parts of the algorithm an interval based version of a previous elimination procedure is introduced this method represents the remaining areas still of interest as polygons fully calculated in a reliable way currently the most promising strategy of finding optimal circle packing configurations is to partition the original problem into subproblems still as a result of the highly increasing number of subproblems earlier computer aided methods were not able to solve problem instances where the number of circles was greater than 27 the present paper provides a carefully developed technique resolving this difficulty by eliminating large groups of subproblems together as a demonstration of the capabilities of the new algorithm the problems of packing 28 29 and 30 circles were solved within very tight tolerance values our verified procedure decreased the uncertainty in the location of the optimal packings by more than 700 orders of magnitude in all cases

preconditioners for the interval gauss seidel method interval newton methods in conjunction with generalized bisection can form the basis of algorithms that find all real roots within a specified box bf x subset bf r n of a system of nonlinear equations f x 0 with mathematical certainty even in finite precision arithmetic in such methods the system f x 0 is transformed into a linear interval system 0 f m bf f bf x bf bar x m if interval arithmetic is then used to bound the solutions of this system the resulting box bf bar x contains all roots of the nonlinear system the interval gauss seidel method is a reasonable way of finding such solution bounds for the overall interval newton bisection algorithm to be efficient the image box bf bar x should be as small as possible to do this the linear interval system is multiplied by a preconditioner matrix y before the interval gauss seidel method is applied in this paper a technique for computing such preconditioner matrices y is described this te

perturbation analysis of generalized singular subspaces this paper as a continuation of the paper 20 in numerische mathematik studies the subspaces associated with the generalized singular value decomposition second order perturbation expansions frechet derivatives and condition numbers and perturbation bounds for the subspaces are derived

an extension of analytic solutions of a class of constrained matrix minimization problems abstract in this paper we consider an extension of analytic solutions of a class of constrained matrix minimization problems and propose analytic solutions of the following matrix minimization problem min s k s k h i n l k l k h i m det c i m k 1 s a k s k b k l k min s k s k h i n l k l k h i m t r c i m k 1 s a k s k b k l k where a k c m n b k c n m c is a complex number i m denotes m order identity matrix and det t r denote matrix determinant and trace function the proposed results improve the corresponding existing ones in xu et al 2019 numerical examples are provided to illustrate the efficiency of the proposed theoretical results

brief paper box particle filtering for nonlinear state estimation using interval analysis in recent years particle filters have been applied to a variety of state estimation problems a particle filter is a sequential monte carlo bayesian estimator of the posterior density of the state using weighted particles the efficiency and accuracy of the filter depend mostly on the number of particles used in the estimation and on the propagation function used to re allocate weights to these particles at each iteration if the imprecision i e bias and noise in the available information is high the number of particles needs to be very large in order to obtain good performances this may give rise to complexity problems for a real time implementation this kind of imprecision can easily be represented by interval data if the maximum error is known handling interval data is a new approach successfully applied to different real applications in this paper we propose an extension of the particle filter algorithm able to handle interval data and using interval analysis and constraint satisfaction techniques in standard particle filtering particles are punctual states associated with weights whose likelihoods are defined by a statistical model of the observation error in the box particle filter particles are boxes associated with weights whose likelihood is defined by a bounded model of the observation error experiments using actual data for global localization of a vehicle show the usefulness and the efficiency of the proposed approach

on proving existence of feasible points in equality constrained optimization problems various algorithms can compute approximate feasible points or approximate solutions to equality and bound constrained optimization problems in exhaustive search algorithms for global optimizers and other contexts it is of interest to construct bounds around such approximate feasible points then to verify computationally but rigorously that an actual feasible point exists within these bounds hansen and others have proposed techniques for proving the existence of feasible points within given bounds but practical implementations have not to our knowledge previously been described various alternatives are possible in such an implementation and details must be carefully considered also in addition to hansen s technique for handling the underdetermined case it is important to handle the overdetermined case when the approximate feasible point corresponds to a point with many active bound constraints the basic ideas along with experimental results from an actual implementation are summarized here

numerical solution for bounding feasible point sets finding feasible points is important in optimization there are currently two major classes of algorithms to deal with the problem of feasible points the first class of algorithms of local nature is to find an approximate feasible point given a neighbourhood of an approximate feasible point the second class of algorithms is to prove whether a feasible point exists inside this neighbourhood to the best of our knowledge no methods have been practically implemented to efficiently find the smallest boxes for bounding the feasible points defined by a system of nonlinear and nonconvex inequalities unless the feasible set is convex in this paper we will present a numerical method to find the smallest boxes for bounding the feasible point sets defined by a nonlinear and nonconvex inequality and or a system of nonlinear and nonconvex inequalities two examples have been synthetically constructed and used to show that the proposed numerical method can indeed correctly find all the smallest bounding boxes at any given accuracy efficiently a brief comparison with relevant techniques will be discussed our method may also be thought of as the first solid theoretical basis for multisection and multisplitting in global optimization when compared with those empirical ones in the literature

inclusion functions and global optimization 11 inclusion functions combined with special subdivision strategies are an effective means of solving the global unconstrained optimization problem although these techniques were determined and numerically tested about ten years ago they are nearly unknown and scarcely used in order to make the role of inclusion functions and subdivision strategies more widespread and transparent we will discuss a related simplified basic algorithm it computes approximations of the global minimum and at the same time bounds the absolute approximation error we will show that the algorithm works and converges under more general assumptions than it has been known hitherto that is only appropriate inclusion functions are expected to exist the number of minimal points finite or infinite is not of importance lipschitz conditions or continuity are not assumed as shown in the appendix the required inclusion functions can be constructed and programmed without difficulty in a natural way using interval analysis

a new pruning test for finding all global minimizers of nonsmooth functions a new pruning test is presented via interval slopes for finding all global minimizers of nonsmooth functions of several variables based on an expansion scheme which can be used to achieve better interval slopes and enclosure of function ranges the presented pruning test extends ratz s a nonsmooth global optimization technique using slopes one dimensional case j global optimiz 4 1999 365 393 pruning technique which can be extensively applied in several variable nonsmooth global optimization as a accelerating device this new pruning test is similar to the monotonicity test frequently used in interval methods for smooth problems but it improves the monotonicity test numerical results show that the proposed global optimization algorithm with the new pruning test is better than that with the monotonicity test for smooth functions and superior to other interval algorithm for nonsmooth functions

the lanczos potential for the weyl curvature tensor existence wave equation and algorithms in the last few years renewed interest in the 3 tensor potential l abc proposed by lanczos for the weyl curvature tensor has not only clarified and corrected lanczos9s original work but generalized the concept in a number of ways in this paper we first of all carefully summarize and extend some aspects of these results make some minor corrections and clarify some misunderstandings in the literature the following new results are also presented the computer checked complicated second order partial differential equation for the 3 potential in arbitrary gauge for weyl candidates satisfying bianchi type equations is given in those n dimensional spaces with arbitrary signature for which the potential exists this is easily specialized to lanczos potentials for the weyl curvature tensor it is found that it is only in four dimensional spaces with arbitrary signature that the nonlinear terms disappear and that certain awkward second order derivative terms cancel for four dimensional spacetimes with lorentz signature this remarkably simple form was originally found by illge using spinor methods it is also shown that for most four dimensional vacuum spacetimes any 3 potential in the lanczos gauges which satisfies a simple homogeneous wave equation must be a lanczos potential for the non zero weyl curvature tensor of the background vacuum spacetime this result is used to prove that the form of a possible lanczos potential recently proposed by dolan and kim for a class of vacuum spacetimes is in fact a genuine lanczos potential for these spacetimes

positivity of general superenergy tensors several of the most important results in general relativity require or assume positivity properties of certain tensors the positive energy theorem and the singularity theorems make assumptions about the energy momentum tensor and ricci tensor respectively positivity of the bel robinson tensor is needed in the proof of the global stability of minkowski spacetime senovilla has recently presented a procedure of how to construct a superenergy tensor from any tensor for a maxwell field or a scalar field the procedure yields the usual energy momentum tensor for the weyl tensor and the riemann tensor one obtains the bel robinson tensor and bel tensor respectively in general by considering any tensor as an r fold n 1 n r form one constructs a rank 2r superenergy tensor from it by using spinor methods we prove that the contraction of any such superenergy tensor with 2r future pointing vectors is non negative we refer to this as the dominant superenergy property and it generalizes several previous positivity results obtained for certain tensors as well as it provides a unified way of treating them some more examples are given and applications discussed

a filled function method applied to nonsmooth constrained global optimization the filled function method is an effective approach to find a global minimizer in this paper based on a new definition of the filled function for nonsmooth constrained programming problems a one parameter filled function is constructed to improve the efficiency of numerical computation then a corresponding algorithm is presented it is a global optimization method which modify the objective function as a filled function and which find a better local minimizer gradually by optimizing the filled function constructed on the minimizer previously found illustrative examples are provided to demonstrate the efficiency and reliability of the proposed filled function method

rigorous error bounds for the optimal value in semidefinite programming a wide variety of problems in global optimization combinatorial optimization as well as systems and control theory can be solved by using linear and semidefinite programming sometimes due to the use of floating point arithmetic in combination with ill conditioning and degeneracy erroneous results may be produced the purpose of this article is to show how rigorous error bounds for the optimal value can be computed by carefully postprocessing the output of a linear or semidefinite programming solver it turns out that in many cases the computational costs for postprocessing are small compared to the effort required by the solver numerical results are presented including problems from the sdplib and the netlib lp library these libraries contain many ill conditioned and real life problems

the algebra of two symmetric matrices proving completeness and deriving syzygies for a set of invariants of the riemann tensor a large number of riemann tensor invariants can be written as traces of products of two 3 3 matrices representing the weyl tensor and the weyl like square of the ricci tensor it is pointed out that finding a complete set i for all of these invariants is a simple consequence of the more general problem of finding a complete set of symmetric matrices m in terms of which all symmetric matrix polynomials in these two matrices can be expressed such a set is constructed and a formal proof of its completeness is given several matrix identities and a scalar syzygy obtained recently by sneddon are rederived and their interrelationships clarified they are shown to be ultimately consequences of the cayley hamilton theorem a minimal set of invariants that must be contained in the complete set of invariants of the general problem is identified and it is concluded that no set proposed so far is complete

exterior differential systems janet riquier theory and the riemann lanczos problems in two three and four dimensions we discuss the riemann lanczos problems in two three and four dimensions using the theory of exterior differential systems and janet riquier theory we show that the riemann lanczos problem in two dimensions is always a system in involution for each of the two possible signatures we give the general solution in both instances and show that the occurrence of characteristic coordinates need not affect the result in three dimensions the riemann lanczos problem is not in involution as an identity occurs this does not prevent the existence of singular solutions and we give an example for the reduced godel space time a prolongation of this problem whereby an integrability condition is added leads to a prolonged system in involution the riemann lanczos problem in four dimensions is not in involution and needs to be prolongated as bampi and caviglia suggested but singular solutions of it can be found and we give examples for the godel kasner and debever hubaut space times

the impact of accelerating tools on the interval subdivision algorithm for global optimization abstract the interval subdivision method of moore and skelboe is considered for global optimization the new implementation allows the use of a long list in the procedure which is limited only by the available computer memory standard global optimization test problems are used to measure the efficiency of different versions with or without the monotonicity and the cut off tests of the algorithm the new implementation and the inclusion of the monotonicity test made it possible for the first time for all standard test problems to be solved completely by an interval method moreover in some problems the efficiency of the algorithm was better than that of the best traditional techniques which do not give guaranteed reliability results

a stochastic method for global optimization a stochastic method for global optimization is described and evaluated the method involves a combination of sampling clustering and local search and terminates with a range of confidence intervals on the value of the global optimum computational results on standard test functions are included as well

integral pinched shrinking ricci solitons abstract we prove that a n dimensional 4 n 6 compact gradient shrinking ricci soliton satisfying a l n 2 pinching condition is isometric to a quotient of the round s n the proof relies mainly on sharp algebraic curvature estimates the yamabe sobolev inequality and an improved rigidity result for integral pinched einstein metrics

convergent upper bounds in global minimization with nonlinear equality constraints we address the problem of determining convergent upper bounds in continuous non convex global minimization of box constrained problems with equality constraints these upper bounds are important for the termination of spatial branch and bound algorithms our method is based on the theorem of miranda which helps to ensure the existence of feasible points in certain boxes then the computation of upper bounds at the objective function over those boxes yields an upper bound for the globally minimal value a proof of convergence is given under mild assumptions an extension of our approach to problems including inequality constraints is possible

determination of all syzygies for the dependent polynomial invariants of the riemann tensor iii mixed invariants of arbitrary degree in the ricci spinor in this paper we rigorously prove that the complete set of riemann tensor invariants given by sneddon j math phys 40 5905 1999 is both minimal and complete furthermore we provide a two stage algorithm for the explicit construction of polynomial syzygies relating any dependent riemann tensor invariant to members of the complete set

on closed manifolds with harmonic weyl curvature abstract we derive point wise and integral rigidity gap results for a closed manifold with harmonic weyl curvature in any dimension in particular there is a generalization of tachibana s theorem for non negative curvature operator the key ingredients are new bochner weitzenbock lichnerowicz type formulas for the weyl tensor which are generalizations of identities in dimension four

prolongation methods and cartan characters for the three dimensional riemann lanczos problem the riemann lanczos problem in two dimensions is in involution and its general solution is known the three dimensional riemann lanczos problem is not in involution and needs to be prolongated we write the three dimensional riemann lanczos problem as an exterior differential system and prolong it to second order using algebraic computing we find that we have to add an integrability condition to make it a system in involution we also suggest a prolongation of the three dimensional riemann lanczos problem in the same way as bampi and caviglia did for four dimensions after supplementing this three dimensional system with an integrability condition it becomes involutive with cartan characters 17 8 2 or 20 10 3 if no cyclic conditions are imposed we present the relevant sections of reduce computer codes by means of which these results were obtained

the identities of the algebraic invariants of the four dimensional riemann tensor this paper extends the investigation of the invariants of the riemann tensor to include the invariants that are of odd degree in the trace free ricci tensor it is shown that these invariants can be expressed in terms of 15 such invariants that are irreducible as a consequence it is possible to write down a complete set of invariants of the riemann tensor several syzygies for these invariants have been found and examples of these are given these syzygies suggest there may be several new syzygies of invariants with even degree in the trace free ricci tensor a large number of these have also been found and are discussed in the paper

some np complete problems in quadratic and nonlinear programming in continuous variable smooth nonconvex nonlinear programming we analyze the complexity of checking whether a a given feasible solution is not a local minimum and b the objective function is not bounded below on the set of feasible solutions we construct a special class of indefinite quadratic programs with simple constraints and integer data and show that checking a or b on this class is np complete as a corollary we show that checking whether a given integer square matrix is not copositive is np complete

rigorous lower and upper bounds in linear programming we consider the computation of rigorous lower and upper error bounds for the optimal value of linear programming problems the input data of the lp problem may be exactly given or may vary between given lower and upper bounds the results are then verified for the family of lp problems with input data inside these bounds in many cases only a small computational effort is required for problems with finite simple bounds the rigorous lower bound of the optimal value can be computed with o n2 operations the error bounds can be used as well to perform a sensitivity analysis provided the width of the uncertainties is not too large some numerical examples are presented

linear independence of renormalization counterterms in curved space times of arbitrary dimensionality the counterterms in the lagrangian of a renormalizable quantum field theory that involve the riemann curvature tensor are considered it is proved that for six or fewer dimensions the counterterms not containing derivatives of the riemann tensor are linearly independent it is shown that there appears to be a maximum space time dimension for which identities can exist among invariants involving the product of n riemann tensors without derivatives acting on them space times that would require these products as counterterms for a renormalizable theory have a dimensionality which is one higher than this maximum dimension this makes it plausible that the required set of counterterms not involving derivatives of the riemann tensor is linearly independent in arbitrary dimensions in the appendix a relation cubic in the weyl tensor which relates invariants cubic in the riemann tensor in five dimensions is proved

the wave equation for the lanczos potential i the non local part of the gravitational field in general relativity is described by the 10 component conformal curvature tensor c abcd of weyl for this field lanczos found a tensor potential l abc with 16 independent components we can make l abc have only 10 effective degrees of freedom by imposing the 6 gauge conditions l smallmatrix ab s s endsmallmatrix 0 both fields c abcd l abc satisfy wave equations the wave equation satisfied by l abc is nonlinear even in vacuo however a linear spinor wave equation for the lanczos potential has been found by illge but no correct tensor wave equation for l abc has yet been published here we derive a correct tensor wave equation for l abc and when it is simplified with the aid of some four dimensional identities it is equivalent to illge s wave equation we also show that the nonlinear spinor wave equation of penrose for the weyl field can be derived from illge s spinor wave equation a set of analogues of well known results of classical electromagentic radiation theory can now be given we indicate how a green s function approach to gravitational radiation could be based on our tensor wave equation when a global study of space time is attempted

interval methods and contractor based branch and bound procedures for verified parameter identification of quasi linear cooperative system models abstract a large number of dynamic system models in engineering as well as computational physics chemistry or biology are described after first principle modeling approaches by sets of either discrete time difference equations or by sets of ordinary differential equations in both cases the structure of those system models results from fundamental physical system properties such as conservation laws of mass momentum and energy newton s or kirchhoff s laws as well as a structural subdivision into subsystems or individual compartments regardless of the area of application all of these system models involve parameters which are not perfectly known due to uncertainty resulting from tolerances in the construction of the respective real life systems or which are described by an inherent uncertainty coming from model simplifications or even from a lack of knowledge about detailed microscopic dynamic effects this is especially true for applications in engineering where a trade off between the model accuracy and the resulting computational complexity has to be made in order to obtain mathematical representations which can be evaluated numerically in real time this real time applicability is inevitable if tasks such as feedback control or model based state estimation on the basis of incomplete state measurements are of interest hence all aforementioned tasks depend on the reliable estimation of those parameters of the structurally predefined sets of state equations which are feasible with respect to the knowledge about selected measurable state variables under consideration of the associated measurement tolerances this paper presents computational techniques for a model based identification of system parameters under the a priori knowledge of stability of the underlying open loop dynamics and the structural constraint of cooperativity

constraint aggregation for rigorous global optimization in rigorous constrained global optimization upper bounds on the objective function help to reduce the search space obtaining a rigorous upper bound on the objective requires finding a narrow box around an approximately feasible solution which then must be verified to contain a feasible point approximations are easily found by local optimization but the verification often fails in this paper we show that even when the verification of an approximate feasible point fails the information extracted from the results of the local optimization can still be used in many cases to reduce the search space this is done by a rigorous filtering technique called constraint aggregation it forms an aggregated redundant constraint based on approximate lagrange multipliers or on a vector valued measure of constraint violation using the optimality conditions two sided linear relaxations the gauss jordan algorithm and a directed modified cholesky factorization the information in the redundant constraint is turned into powerful bounds on the feasible set constraint aggregation is especially useful since it also works in a tiny neighborhood of the global optimizer thereby reducing the cluster effect a simple introductory example demonstrates how our new method works extensive tests show the performance on a large benchmark

convergence qualification of adaptive partition algorithms in global optimization following the presentation of a general partition algorithm scheme for seeking the globally best solution in multiextremal optimization problems necessary and sufficient convergence conditions are formulated in terms of respectively implied or postulated properties of the partition operator the convergence results obtained are pertinent to a number of deterministic algorithms in global optimization permitting their diverse modifications and generalizations

space time algebra approach to curvature introducing the notions of vector and bivector differentiation into the dirac algebra considered as a clifford algebra makes possible an extremely concise and geometrically transparent treatment of the curvature tensor and its properties and of related topics such as lorentz invariants characteristic equations petrov types and principal null directions by explicit construction

a proof of jones conjecture abstract in this paper we prove that wright s equation y t y t 1 1 y t has a unique slowly oscillating periodic solution for parameter values 2 1 9 up to time translation this result proves jones conjecture formulated in 1962 that there is a unique slowly oscillating periodic orbit for all 2 furthermore there are no isolas of periodic solutions to wright s equation all periodic orbits arise from hopf bifurcations

the weyl lanczos relations and the four dimensional lanczos tensor wave equation and some symmetry generators we examine symmetry generators for exterior differential systems and for systems of partial differential equations and apply the cartan theory of exterior differential systems to the weyl lanczos equations and to the lanczos wave equation in four dimensions we look at a number of examples of symmetries for the weyl lanczos equations in four dimensions and give examples of isovectors when the solution manifold is the schwarzschild kasner or godel space time solutions of the weyl lanczos system are automatically solutions of the lanczos wave equation we give examples of symmetry generators for the lanczos wave equation and find that they are not automatically symmetry generators for the weyl lanczos equations

the lanczos potential for vacuum space times with an ernst potential by a particular choice of tetrad and coordinate systems we find the lanczos potential for general vacuum stationary axisymmetric space times and for general vacuum cylindrically symmetric solutions this result is particularly elegant and unifies the treatment of the lanczos potential and the ernst potential for vacuum stationary axially symmetric and cylindrically symmetric space times the lanczos potential for such large classes of vacuum space times is expressed very simply in terms of spin coefficients for a null tetrad

the riemann lanczos equations in general relativity and their integrability the aim of this paper is to examine the riemann lanczos equations and how they can be made integrable they consist of a system of linear first order partial differential equations that arise in general relativity whereby the riemann curvature tensor is generated by an unknown third order tensor potential field called the lanczos tensor our approach is based on the theory of jet bundles where all field variables and all their partial derivatives of all relevant orders are treated as independent variables alongside the local manifold coordinates xa on the given space time manifold m this approach is adopted in a cartan s method of exterior differential systems b vessiot s dual method using vector field systems and c the janet riquier theory of systems of partial differential equations all three methods allow for the most general situations under which integrability conditions can be found they give equivalent results namely that involutivity is always achieved at all generic points of the

a new exclusion test for finding the global minimum exclusion algorithms have been used recently to find all solutions of a system of nonlinear equations or to find the global minimum of a function over a compact domain these algorithms are based on a minimization condition that can be applied to each cell in the domain in this paper we consider lipschitz functions of order a and give a new minimization condition for the exclusion algorithm furthermore convergence and complexity results are presented for such algorithm

new interval methods for constrained global optimization interval analysis is a powerful tool which allows to design branch and bound algorithms able to solve many global optimization problems in this paper we present new adaptive multisection rules which enable the algorithm to choose the proper multisection type depending on simple heuristic decision rules moreover for the selection of the next box to be subdivided we investigate new criteria both the adaptive multisection and the subinterval selection rules seem to be specially suitable for being used in inequality constrained global optimization problems the usefulness of these new techniques is shown by computational studies

on effective constraints for the riemann lanczos system of equations there have been conflicting points of view concerning the riemann lanczos problem in three and four dimensions using direct differentiation on the defining partial differential equations massa and pagani in four dimensions and edgar in dimensions n 3 have argued that there are effective constraints so that not all riemann tensors can have lanczos potentials using cartan s criteria of integrability of ideals of differential forms bampi and caviglia have argued that there are no such constraints in dimensions n 4 and that in these dimensions all riemann tensors can have lanczos potentials in this article we give a simple direct derivation of a constraint equation confirm explicitly that known exact solutions of the riemann lanczos problem satisfy it and argue that the bampi and caviglia conclusion must therefore be flawed in support of this we refer to the recent work of dolan and gerber on the three dimensional problem by a method closely related to that of bampi and caviglia they have fou

solving constrained optimization problems using a novel genetic algorithm a novel genetic algorithm is described in this paper for the problem of constrained optimization the algorithm incorporates modified genetic operators that preserve the feasibility of the trial solutions encoded in the chromosomes the stochastic application of a local search procedure and a stopping rule which is based on asymptotic considerations the algorithm is tested on a series of well known test problems and a comparison is made against the algorithms c somga and donlp2

balanced random interval arithmetic in market model estimation the possibility of estimating bounds for the econometric likelihood function using balanced random interval arithmetic is experimentally investigated the experiments on the likelihood function with data from housing starts have proved the assumption that distributions of centres and radii of evaluated balanced random intervals are normal balanced random interval arithmetic can therefore be used to estimate bounds for this function and global optimization algorithms based on this arithmetic are applicable to optimize it the interval branch and bound algorithms with bounds calculated using standard and balanced random interval arithmetic were used to optimize the likelihood function results of the experiments show that when reliability is essential the algorithm with standard interval arithmetic should be used but when speed of optimization is more important the algorithm with balanced random interval arithmetic should be used which in this case finishes faster and provides good although not always optimal values

note on analytic solutions of a class of extended constrained matrix maximization problems abstract this note is devoted to dealing with flaws that existed in a recent paper xu et al 2020 we give corrections of the original 3 9 and 3 12 in theorem 3 2 of xu et al 2020

lipschitz continuity and the termination of interval methods for global optimization abstract in this paper it is shown how the lipschitz continuity of a global optimization problem s objective function influences the termination of a hansen type method the results concern applying different termination criteria to a hansen method and retrieving more information than explicitly supplied by a certain criterion a similar result for moore skelboe methods arises as a consequence

identities of the scalars of the four dimensional riemannian manifold a general algorithm is presented for the calculation of identities relating scalars constructed from appropriate contractions of the riemann tensor ricci tensor and curvature scalar the algorithm is applied to determining the identities among the quartic scalars

old and new results for superenergy tensors from dimensionally dependent tensor identities it is known that some results for spinors and in particular for superenergy spinors are much less transparent and require a lot more effort to establish when considered from the tensor viewpoint in this paper we demonstrate how the use of dimensionally dependent tensor identities enables us to derive a number of 4 dimensional identities by straightforward tensor methods in a signature independent manner in particular we consider the quadratic identity for the bel robinson tensor tabcxtabcy xy tabcdtabcd 4 and also the new conservation law for the chevreton tensor both of which have been obtained by spinor means both of these results are rederived by tensor means for 4 dimensional spaces of any signature using dimensionally dependent identities and moreover we are able to conclude that there are no direct higher dimensional analogs in addition we demonstrate a simple way to show the nonexistense of such identities via counter examples in particular we show that there is no nontrivial bel ten

generalized kronecker and permanent deltas their spinor and tensor equivalents and applications the aim of this paper is fourfold i to introduce a generalized permanent delta on an equal footing with the generalized kronecker delta to use for the symmetries of any tensor or spinor ii to cite an ancillary reference source of comprehensive tensorial and spinorial combinatorial formulas for both iii to table spinor equivalents of these individual tensors and give examples of their usage and iv to tabulate the tensor equivalents of various useful combinations of their spinor forms

on the arbitrariness of the general solution of an involutive partial differential equation the relationship between the strength of a differential equation as introduced by einstein its cartan characters and its hilbert polynomial is studied using the framework of formal theory previous results are extended to nonlinear equations of arbitrary order and to overdetermined systems the problem of computing the number of arbitrary functions in the general solution is treated finally the effect of gauge symmetries is considered

some solutions of the lanczos vacuum wave equation in general relativity the non local part of the gravitational field is described by the 10 degrees of freedom of the weyl conformal curvature tensor c abcd in every space time the weyl field c abcd is derivable from a potential l abc which has at most 16 algebraically independent components reducing to 10 degrees of freedom when the six gauge conditions l smallmatrix ab s s endsmallmatrix 0 are imposed the potential l abc discovered by lanczos was shown by illge to have an extremely simple vacuum wave equation namely square l abc equiv g sm l abc s m 0 using tensor spinor and spin coefficient methods we give some solutions of this new vacuum wave equation in some spacetimes containing one or more preferred vector fields