on the continual rubik s cube abstract let f and g be two continuous functions on the unit sphere s n 1 in r n n 3 and let their restrictions to any one dimensional great circle e coincide after some rotation e of this circle f e g e we prove that in this case f g or f g for all s n 1 this answers the question posed by richard gardner and vladimir golubyatnikov

swarm intelligence and evolutionary algorithms the popularity of metaheuristics especially swarm intelligence and evolutionary algorithms has increased rapidly over the last two decades numerous algorithms are proposed each year and progressively more novel applications are being found however different metaheuristics are often compared by their performance on problems with an arbitrarily fixed number of allowed function calls there are surprisingly few papers that explore the relationship between the relative performance of numerous metaheuristics on versatile numerical real world problems and the number of allowed function calls in this study the performance of 33 various metaheuristics proposed between 1960 and 2016 have been tested on 22 numerical real world problems from different fields of science with the maximum number of function calls varying between 5000 and 500 000 it is confirmed that the algorithms that succeed in comparisons when the computational budget is low are among the poorest performers when the computational budget is high and vice versa among the tested variants particle swarm optimization algorithms and some new types of metaheuristics perform relatively better when the number of allowed function calls is low whereas differential evolution and genetic algorithms perform better relative to other algorithms when the computational budget is large it is difficult to find any metaheuristic that would perform adequately over all of the numbers of function calls tested it was also found that some algorithms may become completely unreliable on specific real world problems even though they perform reasonably on others

splitting and composition methods for explicit time dependence in separable dynamical systems we consider splitting methods for the numerical integration of separable non autonomous differential equations in recent years splitting methods have been extensively used as geometric numerical integrators showing excellent performances both qualitatively and quantitatively when applied on many problems they are designed for autonomous separable systems and a substantial number of methods tailored for different structures of the equations have recently appeared splitting methods have also been used for separable non autonomous problems either by solving each non autonomous part separately or after each vector field is frozen properly we show that both procedures correspond to introducing the time as two new coordinates we generalize these results by considering the time as one or more further coordinates which can be integrated following either of the previous two techniques we show that the performance as well as the order of the final method can strongly depend on the particular choice we present a simple analysis which in many relevant cases allows one to choose the most appropriate split to retain the high performance the methods show on the autonomous problems this technique is applied to different problems and its performance is illustrated for several numerical examples

if you can hide behind it can you hide inside it let k and l be compact convex sets in rn suppose that for a given dimension 1 d n 1 every d dimensional orthogonal projection of l contains a translate of the corresponding projection of k does it follow that the original set l contains a translate of k in other words if k can be translated to hide behind l from any perspective does it follow that k can hide inside l a compact convex set l is defined to be d decomposable if l is a direct minkowski sum affine cartesian product of two or more convex bodies each of dimension at most d a compact convex set l is called d reliable if whenever each d dimensional orthogonal projection of l contains a translate of the corresponding d dimensional projection of k it must follow that l contains a translate of k it is shown that for 1 d n 1 1 d decomposability implies d reliability 2 a compact convex set l in rn is d reliable if and only if for all m d 2 no m unit normals to regular boundary points of l form the outer unit normals of an m 1 dimensional simplex 3 smooth convex bodies are not d reliable 4 a compact convex set l in rn is 1 reliable if and only if l is 1 decomposable i e a parallelotope 5 a centrally symmetric compact convex set l in rn is 2 reliable if and only if l is 2 decomposable however there are non centered 2 reliable convex bodies that are not 2 decomposable as a result of 5 above the only reliable centrally symmetric covers in r3 from the perspective of 2 dimensional shadows are the affine convex cylinders prisms however in dimensions greater than 3 it is shown that 3decomposability is only sufficient and not necessary for l to cover reliably with respect to 3 shadows even when l is assumed to be centrally symmetric consider two compact convex subsetsk and l of n dimensional euclidean space suppose that for a given dimension 1 d 0 is sufficiently small but no longer fits inside d the same construction works for any set k inscribed in d and having strictly less than unit diameter another received by the editors june 4 2009 2000 mathematics subject classification primary 52a20 c 2011 daniel a klain

linearized biogeography based optimization with re initialization and local search biogeography based optimization bbo is an evolutionary optimization algorithm that uses migration to share information among candidate solutions one limitation of bbo is that it changes only one independent variable at a time in each candidate solution in this paper a linearized version of bbo called lbbo is proposed to reduce rotational variance the proposed method is combined with periodic re initialization and local search operators to obtain an algorithm for global optimization in a continuous search space experiments have been conducted on 45 benchmarks from the 2005 and 2011 congress on evolutionary computation and lbbo performance is compared with the results published in those conferences the results show that lbbo provides competitive performance with state of the art evolutionary algorithms in particular lbbo performs particularly well for certain types of multimodal problems including high dimensional real world problems also lbbo is insensitive to whether or not the solution lies on the search domain boundary in a wide or narrow basin and within or outside the initialization domain

some metaheuristics should be simplified abstract users have a large and constantly increasing number of optimization metaheuristics at their disposal in pursuit of ever better approaches popular and successful methods are often improved a number of times in a row or various methods are hybridized one should however pay attention to whether such deliberately improved or hybridized methods do not become unnecessarily complicated or if some of their elements do not introduce a structural bias in the first case the algorithm should be simplified by eliminating the unneeded elements this will make it more clear to the users and as shown in this study may even improve the results in the second case operators that are responsible for over frequent sampling of some part of the search space have to be removed in this study we address the problem of over complexity of metaheuristics focusing on two joint winners of the cec2016 competition on single objective numerical optimization l shade epsin and umoea ii algorithms it is shown that each of them includes a procedure which introduces a structural bias by attracting population towards the origin o as discussed in the text in the case of seven out of 30 benchmark problems considered in the cec2016 competition the objective function values in the origin o are better than those found by the vast majority of algorithms hence structural bias affects the results obtained by l shade epsin and umoea ii in this work we simplify both algorithms by removing operators that are the main cause of structural bias further we show that in the l shade epsin algorithm the ensemble sinusoidal adaptation mechanism of control parameter f that is used during the first half of the search is not needed slightly better results on both artificial benchmarks used in the cec2016 competition and on a wide set of real world problems may be obtained if during that period of the search the value of f is simply set to 0 5 such a simplified algorithm is competitive against a large number of metaheuristics and may be much easier for the users to understand

practical symplectic partitioned runge kutta and runge kutta nystrom methods we present new symmetric fourth and sixth order symplectic partitioned runge kutta and runge kutta nystrom methods we studied compositions using several extra stages optimising the efficiency an effective error ef is defined and an extensive search is carried out using the extra parameters the new methods have smaller values of ef than other methods found in the literature when applied to several examples they perform up to two orders of magnitude better than previously known method which is in very good agreement with the values of ef

regarding the rankings of optimization heuristics based on artificially constructed benchmark functions novel evolutionary algorithms are usually tested on sets of artificially constructed benchmark problems such problems are often created to make the search of one global extremum usually minimum tricky in this paper it is shown that benchmarking heuristics on either minimization or maximization of the same set of artificially created functions with equal bounds and number of allowed function calls may lead to very different ranking of tested algorithms as evolutionary algorithms and other heuristic optimizers are developed in order to be applicable to real world problems such result may raise doubts on the practical meaning of benchmarking them on artificial functions as there is little reason that searching for the minimum of such functions should be more important than searching for their maximum thirty optimization heuristics including a number of variants of differential evolution as well as other kinds of evolutionary algorithms particle swarm optimization direct search methods and following the idea borrowed from no free lunch pure random search are tested in the paper some discussion regarding the choice of the mean or the median performance for comparison is addressed and a short debate on the overall performance of particular methods is given

a positive answer to the busemann petty problem in three dimensions we prove that in e3 the busemann petty problem concerning cen tral sections of centrally symmetric convex bodies has a positive answer together with other results this settles the problem in each dimension

volume bounds for shadow covering for n 2 a construction is given for a large family of compact convex sets k and l in rn such that the orthogonal projection lu onto the subspace u contains a translate of the corresponding projection ku for every direction u while the volumes of k and l satisfy vn k vn l it is subsequently shown that if the orthogonal projection lu onto the subspace u contains a translate of ku for every direction u then the set n n 1 l contains a translate of k it follows that vn k n n 1 n vn l in particular we derive a universal constant bound vn k 2 942 vn l independent of the dimension n of the ambient space related results are obtained for projections onto subspaces of some fixed intermediate co dimension open questions and conjectures are also posed

covering shadows with a smaller volume for n 2 a construction is given for convex bodies k and l in rn such that the orthogonal projection lu onto the subspace u contains a translate of ku for every direction u while the volumes of k and l satisfy vn k vn l a more general construction is then given for n dimensional convex bodies k and l such that each orthogonal projection l onto a k dimensional subspace contains a translate of k while the mth intrinsic volumes of k and l satisfy vm k vm l for all m k for each k 1 n we then define the collection cn k to be the closure under the hausdorff topology of all blaschke combinations of suitably defined cylinder sets prisms it is subsequently shown that if l cn k and if the orthogonal projection l contains a translate of k for every k dimensional subspace of rn then vn k vn l the families cn k called k cylinder bodies of rn form a strictly increasing chain cn 1 cn 2 cn n 1 cn n where cn 1 is precisely the collection of centrally symmetric compact convex sets in rn while cn n is the collection of all compact convex sets in rn members of each family cn k are seen to play a fundamental role in relating covering conditions for projections to the theory of mixed volumes and members of cn k are shown to satisfy certain geometric inequalities related open questions are also posed

invariants for time dependent hamiltonian systems an exact invariant is derived for n degree of freedom hamiltonian systems with general time dependent potentials the invariant is worked out in two equivalent ways in the first approach we define a special ansatz for the invariant and determine its time dependent coefficients in the second approach we perform a two step canonical transformation of the initially time dependent hamiltonian to a time independent one the invariant is found to contain a function of time f 2 t defined as a solution of a linear third order differential equation whose coefficients depend in general on the explicitly known configuration space trajectory that follows from the system s time evolution it is shown that the invariant can be interpreted as the time integral of an energy balance equation our result is applied to a one dimensional time dependent damped non linear oscillator and to a three dimensional system of coulomb interacting particles that are confined in a time dependent quadratic external potential we finally show that our results can be used to assess the accuracy of numerical simulations of time dependent hamiltonian systems

an analytic solution to the busemann petty problem on sections of convex bodies we derive a formula connecting the derivatives of parallel section functions of an origin symmetric star body in r n with the fourier transform of powers of the radial function of the body a parallel section function or ni 1 dimensional x ray gives the ni 1 dimensional volumes of all hyperplane sections of the body orthogonal to a given direction this formula provides a new characterization of intersection bodies in r n and leads to a unifled analytic solution to the busemann petty problem suppose that k and l are two origin symmetric convex bodies inr n such that the ni 1 dimensional volume of each central hyperplane section of k is smaller than the volume of the corresponding section of l is the n dimensional volume of k smaller than the volume of l in conjunction with earlier established connections between the busemann petty problem intersection bodies and positive deflnite distributions our formula shows that the answer to the problem depends on the behavior of the ni 2 nd derivative of the parallel section functions the a rmative answer to the busemann petty problem for n 4 and the negative answer for n 5 now follow from the fact that convexity controls the second derivatives but does not control the derivatives of higher orders

on polytopes with congruent projections or sections abstract let 2 k d 1 and let p and q be two convex polytopes in e d assume that their projections p h q h onto every k dimensional subspace h are congruent in this paper we show that p and q or p and q are translates of each other we also prove an analogous result for sections by showing that p q or p q provided the polytopes contain the origin in their interior and their sections p h q h by every k dimensional subspace h are congruent

intersection bodies positive definite distributions and the busemann petty problem the 1956 busemann petty problem asks whether symmetric convex bodies with larger central hyperplane sections also have greater volume in 1988 lutwak introduced the concept of an intersection body which is closely related to the busemann petty problem we prove that an origin symmetric star body k in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i is an intersection body if and only if x 1 k is a positive definite distribution on inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i where x k min a 0 x ak we use this result to show that for every dimension n there exist polytopes in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i which are intersection bodies for example the cross polytope the unit ball of every subspace of l p 0 p 2 is an intersection body the unit ball of the space l n q 2 q n 5 using lutwak s connection with the busemann petty problem we present new counterexamples to the problem for n 5 and confirm the conjecture of meyer that the answer to the problem is affirmative if the smaller body is a polar projection body

splitting methods for non autonomous separable dynamical systems a large number of splitting methods for autonomous separable systems exist in the literature which have been designed for many different structures of the vector field however the performance of most of these methods is diminished and their orders of accuracy are frequently reduced when applied to non autonomous problems based on the formal solution obtained from the magnus series expansion we show how to modify a standard splitting method for autonomous problems to treat non autonomous systems with similar or better efficiency we illustrate this technique to build new fourth and sixth order schemes whose performance is then illustrated on several numerical examples

intersection bodies and dual mixed volumes the setting for this article is euclidean n space ky n 2 let x denote the set of convex bodies compact convex subsets with nonempty interiors in r and let x denote the subset of x that contains the centered centrally symmetric with respect to the origin bodies for ue s let e denote the hyperplane through the origin that is orthogonal to u and let ki e denote the image of the orthogonal projection of the body ke x onto e we shall use v to denote i dimensional lebesgue measure for v and v we shall usually write v and u shephard 38 poses the question if k l e xe and

on the numerical integration of ordinary differential equations by symmetric composition methods differential equations of the form dot x x a b are considered where the vector fields a and b can be integrated exactly enabling numerical integration of x by composition of the flows of a and b various symmetric compositions are investigated for order complexity and reversibility free lie algebra theory gives simple formulae for the number of determining equations for a method to have a particular order a new more accurate way of applying the methods thus obtained to compositions of an arbitrary first order integrator is described and tested the determining equations are explored and new methods up to 100 times more accurate at constant work than those previously known are given

on a functional equation related to a pair of hedgehogs with congruent projections abstract hedgehogs are geometrical objects that describe the minkowski differences of arbitrary convex bodies in the euclidean space e n we prove that two hedgehogs in e n n 3 coincide up to a translation and a reflection in the origin provided that their projections onto any two dimensional plane are directly congruent and have no direct rigid motion symmetries our result is a consequence of a more general analytic statement about the solutions of a functional equation in which the support functions of hedgehogs are replaced with two arbitrary twice continuously differentiable functions on the unit sphere

sb3a splitting for approximation of invariants in time dependent hamiltonian systems abstract most physical phenomena are described by time dependent hamiltonian systems with qualitative features that should be preserved by numerical integrators used for approximating their dynamics the initial energy of the system together with the energy added or subtracted by the outside forces represent a conserved quantity of the motion for a class of time dependent hamiltonian systems 8 this invariant can be defined by means of an auxiliary function whose dynamics has to be integrated simultaneously with the system s equations we propose splitting procedures featured by a sb 3 a property that allows to construct composition methods with a reduced number of determining order equations and to provide the same high accuracy for both the dynamics and the preservation of the invariant quantity