parity effect of the initial capital based on parrondo s games and the quantum interpretation abstract in our previous study zhu et al quantum game interpretation for a special case of parrondo s paradox physica a 390 2011 579 the capital dependent parrondo s game where one game depends on the capital modulus m 4 was shown not to have a definite stationary probability distribution and that payoffs of the game depended on the parity of the initial capital this paper presents a generalization of these results to even m greater than 4 an intuitive explanation for producing this phenomenon is that the discrete time markov chain of the game is divided into two completely unrelated inner and outer rings the process taking the inner ring or outer ring of the game is determined by the initial capital of parity and then a win or loss of the game is determined quantum game theory is used to further analyze the phenomenon the results show that the explanation of the game corresponding to a stationary probability distribution is that the probability of the initial capital has reached parity

lagrangians of physics and the game of fisher information transfer the lagrangians of physics arise out of a mathematical game between a smart measurer and nature personified by a demon each contestant wants to maximize his level of fisher information i the game is zero sum by conservation of information in the closed system the payoff of the game introduces a variational principle extreme physical information epi which fixes both the lagrangian and the physical constant of each scenario the epi approach provides an understanding of the relationship between measurement and physical law epi also defines a prescription for constructing lagrangians the prior knowledge required for this purpose is a rule of symmetry or conservation that implies a unitary transformation for which i remains invariant as an example when applied to the smart measurement of the space time coordinate of a particle the symmetry used is that between position time space and momentum energy space then the unitary transformation is the fourier one and epi derives the following the equivalence of energy momentum and mass the constancy of planck s parameter h and the lagrangian that implies both the klein gordon equation and the dirac equation of quantum mechanics

frequency domain integrals for stability preservation in galerkin type projection based model order reduction we investigate linear dynamical systems consisting of ordinary differential equations with high dimensionality model order reduction yields alternative systems of much lower dimensions however a

is gauss quadrature better than clenshaw curtis we compare the convergence behavior of gauss quadrature with that of its younger brother clenshaw curtis seven line matlab codes are presented that implement both methods and experiments show that the supposed factor of 2 advantage of gauss quadrature is rarely realized theorems are given to explain this effect first following o hara and smith in the 1960s the phenomenon is explained as a consequence of aliasing of coefficients in chebyshev expansions then another explanation is offered based on the interpretation of a quadrature formula as a rational approximation of log z 1 z 1 in the complex plane gauss quadrature corresponds to pade approximation at z infty clenshaw curtis quadrature corresponds to an approximation whose order of accuracy at z infty is only half as high but which is nevertheless equally accurate near 1 1

adaptively weighted numerical integration in the finite cell method abstract with adaptively weighted aw numerical integration for a given set of quadrature nodes order and domain of integration the quadrature weights are obtained by solving a system of suitable moment fitting equations in least square sense the moments in the moment equations are approximated over a simplified domain that is homeomorphic to the original domain and then are corrected for the deviation from the original domain using shape sensitivity analysis in this paper we demonstrate the application of aw integration scheme in the context of the finite cell method which must perform numerical integration over arbitrary domains without meshing the standard integration technique employed in fcm is the characteristic function method that converts the continuous integrand over a complex domain into a discontinuous integrand over a simple box domain then well known integrand adaptivity techniques are employed to integrate the resulting discontinuous integrand over the box domain although this method is simple to implement it becomes computationally very expensive for realistic complex 3d domains such as sculptures bones and engines in contrast in aw scheme the quadrature weights directly adapt to the complex geometric domain without the need to making the integrand discontinuous leading to superior computational properties in this paper we demonstrate the computational efficiency of aw over the characteristic function method as it requires fewer subdivisions and less time to achieve a given accuracy in both two and three dimensions in addition aw offers a number of advantages including flexibility in the choice of quadrature points and basis functions

numerical integration of functions with a sharp peak at or near one boundary using mo bius transformations abstract a class of coordinate transformations depending on a single parameter is studied as quadrature tool working rules for the choice of the parameter are proposed numerical tests for the method are presented they show that these coordinate transformations when combined with gauss legendre quadrature rules are well suited for the numerical integration of functions possessing a sharp peak at or near one boundary of the interval of integration a method to combine the transformations with automatic quadrature routines is also proposed it seems to be useful for the numerical evaluation of integrals with the same kind of integrand behavior

higher entropic uncertainty relations for anti commuting observables uncertainty relations provide one of the most powerful formulations of the quantum mechanical principle of complementarity yet very little is known about such uncertainty relations for more than two measurements here we show that sufficient unbiasedness for a set of binary observables in the sense of mutual anticommutation is good enough to obtain maximally strong uncertainty relations in terms of the shannon entropy we also prove nearly optimal relations for the collision entropy this is the first systematic and explicit approach to finding an arbitrary number of measurements for which we obtain maximally strong uncertainty relations our results have immediate applications to quantum cryptography

a semiclassical statistical model for quantum dissipation abstract the dynamical problem posed by the coupling of quantal to classical degrees of freedom is tackled within the framework of the quantum friction phenomenon by recourse to statistical methods embedded within jaynes maximum entropy principle

on the computation of fourier transforms of singular functions abstract in this paper we consider a method for the computation of finite fourier transforms of functions having endpoint singularities of algebraic or logarithmic type this method may be considered as a modification of clenshaw curtis quadrature the nonoscillatory and nonsingular part of the integrand is replaced by a truncated chebyshev series approximation and the integral is then evaluated using recurrence relations the numerical stability of these recurrence relations is investigated

approximate inference for disease mapping with sparse gaussian processes gaussian process gp models are widely used in disease mapping as they provide a natural framework for modeling spatial correlations their challenges however lie in computational burden and memory requirements in disease mapping models the other difficulty is inference which is analytically intractable due to the non gaussian observation model in this paper we address both these challenges we show how to efficiently build fully and partially independent conditional fic pic sparse approximations for the gp in two dimensional surface and how to conduct approximate inference using expectation propagation ep algorithm and laplace approximation la we also propose to combine fic with a compactly supported covariance function to construct a computationally efficient additive model that can model long and short length scale spatial correlations simultaneously the benefit of these approximations is computational the sparse gps speed up the computations and reduce the memory requirements the posterior inference via ep and laplace approximation is much faster and is practically as accurate as via markov chain monte carlo

all analytical evaluation of the singular integrals involved in the method of moments surface integral equation methods routinely require the integration of the singular green s function or its gradient over basis functions bf and testing functions tf many techniques have been described in the literature for the fast and accurate computation of these integrals for tf that is located close to the bf in this paper we propose an all analytical formula for the singular part of the integral for both the electric and magnetic field integral equations the method works for any flat polygonal bf and tf of any order and proves to be competitive with existing techniques

adaptive cubature over a collection of triangles using the d transformation abstract we describe an automatic routine to integrate a function over a collection of triangles where on each triangle the function is well behaved or has singularities of certain types at one or more vertices or edges the underlying algorithm is globally adaptive and incorporates the d transformation for extrapolation results from performance profile testing indicate that the routine is superior to other published routines when the singularities are located along edges of the triangles

a transform of complementary aspects with applications to entropic uncertainty relations even though mutually unbiased bases and entropic uncertainty relations play an important role in quantum cryptographic protocols they remain ill understood here we construct special sets of up to 2n 1 mutually unbiased bases mubs in dimension d 2 n which have particularly beautiful symmetry properties derived from the clifford algebra more precisely we show that there exists a unitary transformation that cyclically permutes such bases this unitary can be understood as a generalization of the fourier transform which exchanges two mubs to multiple complementary aspects we proceed to prove a lower bound for min entropic entropic uncertainty relations for any set of mubs and show that symmetry plays a central role in obtaining tight bounds for example we obtain for the first time a tight bound for four mubs in dimension d 4 which is attained by an eigenstate of our complementarity transform finally we discuss the relation to other symmetries obtained by transformations in discrete phase space and note that the extrema of discrete wigner functions are directly related to min entropic uncertainty relations for mubs

a unified approach to newton cotes quadrature formulae in this paper the author introduces a new approach to obtain the cotes numbers for both the open and closed newton cotes quadrature formulae he also introduces a new definition for the degree of exactness for a quadrature formula some numerical results are presented

a new quadrature routine for improper and oscillatory integrals in matlab environment a new quadrature routine based on gaussian quadrature rule has been developed its performance is evaluated for improper integrals rapidly oscillating functions and other types of functions requiring a large number of evaluations this performance is compared against the other quadrature routines written for matlab in terms of capability accuracy and computation time it is found that our routine rates quite favourably

generalized bloch spheres for m qubit states m qubit states are embedded in clifford algebras their probability spectrum then depends on o 2m or o 2m 1 invariants respectively parameter domains for o 2m 1 vector and tensor configurations generalizing the notion of a bloch sphere are derived

practical error estimation in numerical integration theoretical error estimates for quadrature rules involve quantities that are usually difficult if not impossible to obtain in practice various heuristics to obtain computable error estimates are compared by calculating their performance profiles on the lyness family of integrands two sets of tests are used corresponding to adaptive and single rule quadrature in the single rule case gaussian quadrature with error estimate provided by dropping one point from the formula performs best in the adaptive case the best heuristic is non linear extrapolation based on gaussian quadrature

a hierarchical computational model for moving thermal loads and phase changes with applications to selective laser melting computational heat transfer analysis often involves moving fluxes which induce traveling fronts of phase change coupled to one or more field variables examples are the transient simulation of melting welding or of additive manufacturing processes where material changes its state and the controlling fields are temperature and structural deformation one of the challenges for a numerical computation of these processes is their multi scale nature with a highly localized zone of phase transition which may travel over a large domain of a body here a transient local adaptation of the approximation with not only a refinement at the phase front but also a de refinement in regions where the front has past is of advantage because the de refinement can assure a bounded number of degrees of freedom which is independent from the traveling length of the front we present a computational model of this process which involves three novelties a a very low number of degrees of freedom which yet yields a comparatively high accuracy the number of degrees of freedom is additionally kept practically constant throughout the duration of the simulation this is achieved by means of the multi level hp finite element method its exponential convergence is verified for the first time against a semi analytic three dimensional transient linear thermal benchmark with a traveling source term which models a laser beam b a hierarchical treatment of the state variables to this end the state of the material is managed on a separate octree like grid this material grid may refine or coarsen independently of the discretization used for the temperature field this methodology is verified against an analytic benchmark of a melting bar computed in three dimensions in which phase changes of the material occur on a rapidly advancing front c the combination of these technologies to demonstrate its potential for the computational modeling of selective laser melting processes to this end the computational methodology is extended by the finite cell method which allows for accurate simulations in an embedded domain setting this opens the new modeling possibility that neither a scan vectors no a layer of material needs to conform to the discretization of the finite element mesh but can form only a fraction within the discretization of the field and state variables

cooperative quantum parrondo s games abstract coordination and cooperation are among the most important issues of game theory recently the attention turned to game theory on graphs and social networks encouraged by interesting results obtained in quantum evolutionary game analysis we study cooperative parrondo s games in a quantum setup the game is modeled using multidimensional quantum random walks with biased coins we use the ghz and w entangled states as the initial state of the coins our analysis shows that an apparent paradox in cooperative quantum games and some interesting phenomena can be observed

adaptively weighted numerical integration over arbitrary domains abstract in adaptively weighted numerical integration for a given set of quadrature nodes order and domain of integration the quadrature weights are obtained by solving a system of suitable moment fitting equations in least square sense the moments in the moment equations are approximated over a simplified domain that is homeomorphic to the original domain and then are corrected for the deviation from the original domain using shape sensitivity analysis using divergence theorem the moments reduce to integrals over the boundary of the simplified domain the proposed approach supports accurate and efficient computation of quadrature weights for integration of a priori unknown functions over arbitrary 2d and 3d solid domains experimental results 2d indicate that adaptively weighted integration compares favorably with more traditional approaches because the adaptively weighted integration avoids excessive domain subdivision it is useful in many applications and meshfree analysis in particular

quadrature formulas obtained by variable transformation and the de rule abstract this paper gives a survey of the results known to date about quadrature formulas obtained by variable transformation followed by an application of the trapezoidal rule with an equal mesh size it has been shown that a formula obtained by an appropriate transformation is in general efficient and also robust against the end point singularity various kinds of useful transformations together with the asymptotic error behaviors of the resulting quadrature formulas are summarized in particular special emphasis is placed on an asymptotically optimal formula called the double exponential formula abbreviated as the de rule which is characterized by the double exponential decrease of its weights in the neighborhood of the end points of the transformed interval of integration

when not to use an automatic quadrature routine numerical quadrature is required as a component in many large scale and small scale scientific programs frequently the scientist faces a choice either he may use one of the comparatively recent automatic quadrature routines aqr or he may use a standard rule evaluation quadrature routine rear based directly on traditional formulae the purpose of this article is to help the scientist to make an informed choice in the first half of this article we introduce some generic properties of aqrs the more straightforward of these are those concerned with reliability and economy less straightforward is the vital performance profile property the second half of this article is devoted principally to demonstrating how in some calculations the jagged performance profile associated with the aqr can cause other parts of the overall program to be far more expensive than necessary thus while in many programs the use of an aqr can save significant human time in many other programs its use can be a disaster

order preserving mesh spacing for compound quadrature formulas and functions with endpoint singularities for compound quadrature formulas for the numerical evaluation of int a b w x f x dx where the function f as well as the weight function w are allowed to have singularities at the endpoints a and b we show how to preserve the order of convergence which is obtained for smooth w and f by an appropriate choice of the meshpoints the main emphasis is on singularities of algebraic and logarithmic type

rapid analysis of variable stiffness beams and plates legendre polynomial triple product formulation copyright 2017 john wiley sons ltd numerical integration techniques are commonly employed to formulate the system matrices encountered in the analysis of variable stiffness beams and plates using a ritz based approach computing these integrals accurately is often computationally costly herein a novel alternative is presented the recursive analytical polynomial integral definition rapid formulation the rapid formulation offers a significant improvement in the speed of analysis achieved by reducing the number of numerical integrations that are performed by an order of magnitude a common legendre polynomial basis is employed for both trial functions and stiffness load variations leading to a common form for the integrals encountered the legendre polynomial basis possesses algebraic recursion relations that allow these integrals to be reformulated as triple products with known analytical solutions defined compactly using the wigner 3j coefficient the satisfaction of boundary conditions calculation of derivatives and transformation to other bases is achieved through combinations of matrix multiplication with each matrix representing a unique boundary condition or physical effect therefore permitting application of the rapid approach to a variety of problems indicative performance studies demonstrate the advantage of the rapid formulation when compared to direct analysis using matlab s integral and integral2 copyright 2017 john wiley sons ltd

an algebraic cubature formula on curvilinear polygons abstract we implement in matlab a gauss like cubature formula on bivariate domains whose boundary is a piecewise smooth jordan curve curvilinear polygons the key tools are green s integral formula together with the recent software package chebfun to approximate the boundary curve close to machine precision by piecewise chebyshev interpolation several tests are presented including some comparisons of this new routine chebfungauss with the recent splinegauss that approximates the boundary by splines

adaptive change of basis in entropy based moment closures for linear kinetic equations entropy based m n moment closures for kinetic equations are defined by a constrained optimization problem that must be solved at every point in a space time mesh making it important to solve these optimization problems accurately and efficiently we present a complete and practical numerical algorithm for solving the dual problem in one dimensional slab geometries the closure is only well defined on the set of moments that are realizable from a positive underlying distribution and as the boundary of the realizable set is approached the dual problem becomes increasingly difficult to solve due to ill conditioning of the hessian matrix to improve the condition number of the hessian we advocate the use of a change of polynomial basis defined using a cholesky factorization of the hessian that permits solution of problems nearer to the boundary of the realizable set we also advocate a fixed quadrature scheme rather than adaptive quadrature since the latter introduces unnecessary expense and changes the computationally realizable set as the quadrature changes for very ill conditioned problems we use regularization to make the optimization algorithm robust we design a manufactured solution and demonstrate that the adaptive basis optimization algorithm reduces the need for regularization this is important since we also show that regularization slows and even stalls convergence of the numerical simulation when refining the space time mesh we also simulate two well known benchmark problems there we find that our adaptive basis fixed quadrature algorithm uses less regularization than alternatives although differences in the resulting numerical simulations are more sensitive to the regularization strategy than to the choice of basis

bayesian inference and model comparison for metallic fatigue data abstract in this work we present a statistical treatment of stress life s n data drawn from a collection of records of fatigue experiments that were performed on 75s t6 aluminum alloys our main objective is to predict the fatigue life of materials by providing a systematic approach to model calibration model selection and model ranking with reference to s n data to this purpose we consider fatigue limit models and random fatigue limit models that are specially designed to allow the treatment of the run outs right censored data we first fit the models to the data by maximum likelihood methods and estimate the quantiles of the life distribution of the alloy specimen to assess the robustness of the estimation of the quantile functions we obtain bootstrap confidence bands by stratified resampling with respect to the cycle ratio we then compare and rank the models by classical measures of fit based on information criteria we also consider a bayesian approach that provides under the prior distribution of the model parameters selected by the user their simulation based posterior distributions we implement and apply bayesian model comparison methods such as bayes factor ranking and predictive information criteria based on cross validation techniques under various a priori scenarios

highly accurate lagrangian flux calculation via algebraic quadratures on spline approximated donating regions abstract lagrangian flux through a fixed curve segment within a time interval t n t n k can be formulated as an integral at the initial time t n over a compact point set called donating region in which each particle will pass through the curve during the time interval and contribute to the flux based an explicit constructive and analytical solution of the donating region the author proposes algorithms of lagrangian flux calculation lfc as algebraic quadratures over the spline approximated donating regions this generic formulation yields high accuracies up to the 8th order both in time and space as another benefit lfc leads to a conservative semi lagrangian method whose time step size is free of the eulerian stability constraint c r 1 the high accuracy and robustness of the proposed lfc algorithms are demonstrated by various numerical tests with courant numbers ranging from 1 to 10 000 and with accuracies from the 2nd order to the 8th order lfc might also be useful in designing hybrid eulerian methods for complex geometries

social optimality in quantum bayesian games a b s t r a c t a significant aspect of the study of quantum strategies is the exploration of the gametheoretic solution concept of the nash equilibrium in relation to the quantization of a game pareto optimality is a refinement on the set of nash equilibria a refinement on the set of pareto optimal outcomes is known as social optimality in which the sum of players payoffs is maximized this paper analyzes social optimality in a bayesian game that uses the setting of generalized einstein podolsky rosen experiments for its physical implementation we show that for the quantum bayesian game a direct connection appears between the violation of bell s inequality and the social optimal outcome of the game and that it attains a superior socially optimal outcome

twodqd an adaptive routine for two dimensional integration abstract this is an adaptive subroutine that computes an approximation to the integral of a function f x y over a two dimensional region made up of triangles lyness jespersen rules form the basis for a local quadrature module that is used to estimate the integral and the error over each triangle the triangle with the largest error is subdivided and the local quadrature module is applied to each sub triangle to obtain new estimates of the integral and the error this process is repeated until either 1 an error tolerance is satisfied 2 the number of triangles exceeds an input parameter maxtri 3 the number of integrand evaluations exceeds an input parameter mevals or 4 the subroutine senses that round off error is beginning to contaminate the result

quantum strategies win in a defector dominated population a b s t r a c t quantum strategies are introduced into evolutionary games the agents using quantum strategies are regarded as invaders whose fraction generally is 1 of a population in contrast to the 50 of the population that are defectors in this paper the evolution of strategies on networks is investigated in a defector dominated population when three networks square lattice newman watts small world network and scale free network are constructed and three games prisoners dilemma snowdrift and stag hunt are employed as far as these three games are concerned the results show that quantum strategies can always invade the population successfully comparing the three networks we find that the square lattice is most easily invaded by agents that adopt quantum strategies however a scale free network can be invaded by agents adopting quantum strategies only if a hub is occupied by an agent with a quantum strategy or if the fraction of agents with quantum strategies in the population is significant

roundoff errors in the problem of computing cauchy principal value integrals we investigate the possibility of fast accurate and reliable computation of the cauchy principal value integrals mathrm p int a b f x x tau 1 dx a tau b using standard adaptive quadratures in order to properly control the error tolerance for the adaptive quadrature and to obtain a reliable estimation of the approximation error we research the possible influence of round off errors on the computed result as the numerical experiments confirm the proposed method can successfully compete with other algorithms for computing such type integrals moreover the presented method is very easy to implement on any system equipped with a reliable adaptive integration subroutine

automatic integration using asymptotically optimal adaptive simpson quadrature we present a novel theoretical approach to the analysis of adaptive quadratures and adaptive simpson quadratures in particular which leads to the construction of a new algorithm for automatic integration for a given function f in c 4 f c4 with f 4 ge 0 f 4 0 and possible endpoint singularities the algorithm produces an approximation to int a bf x mathrm d x abf x dx within a given varepsilon asymptotically as varepsilon rightarrow 0 0 moreover it is optimal among all adaptive simpson quadratures i e needs the minimal number n f varepsilon n f of function evaluations to obtain an varepsilon approximation and runs in time proportional to n f varepsilon n f

an efficient quadrature for 2 5d boundary element calculations abstract in recent years the boundary element method has become a widely used tool for calculating the mitigation effects of noise barriers however since for large structures calculations in 3d become very inefficient most of the standard implementations are only in 2d this means that the noise source is implicitly assumed to be given by a coherent line source which is not realistic in most cases by using a fourier transform with respect to a spatial coordinate along the length of the structure it is possible to reduce the 3d problem to several 2d problems with distinct wavenumbers which allows the simulation of more realistic noise sources and which is typically referred to as 2 5d bem to that end it is necessary to numerically calculate a fourier like integral over all the 2d solutions in this work an efficient way to calculate this integral is given building on existing approaches using clenshaw curtis filon quadrature and demodulation combined with an adaptive order selection scheme as bem calculations are costly the main focus of the method introduced lies on avoiding too many of these calculations the efficiency of the method is illustrated using two different examples a reflecting cylinder and an l shaped noise barrier

vectorized adaptive quadrature in matlab adaptive quadrature codes process a collection of subintervals one at a time we show how to process them all simultaneously and so exploit vectorization and the use of fast built in functions and array operations that are so important to efficient computation in matlab using algebraic transformations we have made it just as easy for users to solve problems on infinite intervals and with moderate end point singularities as problems with finite intervals and smooth integrands piecewise smooth integrands are handled effectively with breakpoints

matlab program for quadrature in 2d abstract we discuss here the algorithms of twod a m atlab program for approximating integrals over generalized rectangles and sectors capabilities of the language are exploited to make twod very easy to use vectorization is exploited by a novel algorithm to make the program efficient and reliable twod provides for moderate singularities on boundaries of the region

betting on dynamics a physical system in game theoretic terms is represented in this work we first consider an abstract game between a classical and a quantum player whose results represent the two different kinds of non linear dynamics that the system exhibits the existence of winning strategies shows that the dynamics can be fully determined through a game theory analysis moreover by considering a second game we show that different games can be defined to focus on different aspects of the physics of a system different winning strategies and nash equilibrium points emerge in each game these games theory tools are related to physical properties so distinct physical information is obtained for each game in this work special emphasis is made on nash points which add a new conceptual perspective to analyze a physical system

comparison of newton cotes and gaussian methods of quadrature we compare newton cotes and gauss methods of various orders we give two matlab programs that evaluates integrals numerically for given order with given number of points we make extensive tests with various functions and intervals using same number of points for each method and compare errors

connecting complexity with spectral entropy using the laplace transformed solution to the fractional diffusion equation analytical solutions to the fractional diffusion equation are often obtained by using laplace and fourier transforms which conveniently encode the order of the time and the space derivatives and as non integer powers of the conjugate transform variables s and k for the spectral and the spatial frequencies respectively this study presents a new solution to the fractional diffusion equation obtained using the laplace transform and expressed as a fox s h function this result clearly illustrates the kinetics of the underlying stochastic process in terms of the laplace spectral frequency and entropy the spectral entropy is numerically calculated by using the direct integration method and the adaptive gauss kronrod quadrature algorithm here the properties of spectral entropy are investigated for the cases of sub diffusion and super diffusion we find that the overall spectral entropy decreases with the increasing and and that the normal or gaussian case with 1 and 2 has the lowest spectral entropy i e less information is needed to describe the state of a gaussian process in addition as the neighborhood over which the entropy is calculated increases the spectral entropy decreases which implies a spatial averaging or coarse graining of the material properties consequently the spectral entropy is shown to provide a new way to characterize the temporal correlation of anomalous diffusion future studies should be designed to examine changes of spectral entropy in physical chemical and biological systems undergoing phase changes chemical reactions and tissue regeneration