numerical approximation of a one dimensional space fractional advection dispersion equation with boundary layer finite element computations for singularly perturbed convection diffusion equations have long been an attractive theme for numerical analysis in this article we consider the singularly perturbed fractional advection dispersion equation fade with boundary layer behavior we derive a theoretical estimate which shows that the under resolved case corresponds to e

a second order achievable rate region for gaussian multi access channels via a central limit theorem for functions motivated by the growing application of multi access networks with stringent delay constraints we investigate the gaussian multiple access channel mac in the finite blocklength regime by applying central limit theorem clt approximations to non asymptotic information spectrum inner bounds we obtain second order achievable rate regions for the gaussian mac with a positive average error probability and per codeword power constraints our achievability results use spherical inputs uniformly distributed on the power shells which lead to summations of dependent information random variables however we conduct the analysis through a convenient yet powerful form of the clt called the clt for functions

retarding subdiffusion and accelerating superdiffusion governed by distributed order fractional diffusion equations we propose diffusionlike equations with time and space fractional derivatives of the distributed order for the kinetic description of anomalous diffusion and relaxation phenomena whose diffusion exponent varies with time and which correspondingly cannot be viewed as self affine random processes possessing a unique hurst exponent we prove the positivity of the solutions of the proposed equations and establish their relation to the continuous time random walk theory we show that the distributed order time fractional diffusion equation describes the subdiffusion random process that is subordinated to the wiener process and whose diffusion exponent decreases in time retarding subdiffusion this process may lead to superslow diffusion with the mean square displacement growing logarithmically in time we also demonstrate that the distributed order space fractional diffusion equation describes superdiffusion phenomena with the diffusion exponent increasing in time accelerating superdiffusion

on korobov lattice rules in weighted spaces this paper studies the error bounds of multivariate integration in weighted function spaces using lattice rules of the korobov form in which the generating vector for an n point rule with n prime has the form 1 a ad 1 mod n with the parameter a chosen optimally we establish new error bounds for korobov lattice rules in weighted korobov spaces in particular we prove that if the weights decay sufficiently fast the optimal korobov lattice rule has an error bound of order o n alpha 2 delta for arbitrary delta 0 with the implied constant depending at worst polynomially on the dimension here alpha 1 is the smoothness parameter of the weighted korobov spaces we generalize the construction to the case where n is a product of arbitrary distinct prime numbers with the purpose of reducing the construction cost without sacrificing much of the quality of the lattice rules a corresponding result is deduced for weighted sobolev spaces of nonperiodic functions using randomly shifted optimal korobov lattice rules a comparison of the worst case errors for korobov lattice rules and the recent component by component constructions is presented the investigation establishes the usefulness of shifted optimal korobov lattice rules for integration even in high dimensions if the weights which characterize the weighted spaces are suitably chosen

short memory principle and a predictor corrector approach for fractional differential equations fractional differential equations are increasingly used to model problems in acoustics and thermal systems rheology and modelling of materials and mechanical systems signal processing and systems identification control and robotics and other areas of application this paper further analyses the underlying structure of fractional differential equations from a new point of view we apprehend the short memory principle of fractional calculus and farther apply a adams type predictor corrector approach for the numerical solution of fractional differential equation and the detailed error analysis is presented combining the short memory principle and the predictor corrector approach we gain a good numerical approximation of the true solution of fractional differential equation at reasonable computational cost a numerical example is provided and compared with the exact analytical solution for illustrating the effectiveness of the short memory principle

capacity of the trapdoor channel with feedback we establish that the feedback capacity of the trapdoor channel is the logarithm of the golden ratio and provide a simple communication scheme that achieves capacity as part of the analysis we formulate a class of dynamic programs that characterize capacities of unifilar finite state channels the trapdoor channel is an instance that admits a simple closed form solution

matrix approach to discrete fractional calculus ii partial fractional differential equations a new method that enables easy and convenient discretization of partial differential equations with derivatives of arbitrary real order so called fractional derivatives and delays is presented and illustrated on numerical solution of various types of fractional diffusion equation the suggested method is the development of podlubny s matrix approach i podlubny matrix approach to discrete fractional calculus fractional calculus and applied analysis 3 4 2000 359 386 four examples of numerical solution of fractional diffusion equation with various combinations of time space fractional derivatives integer integer fractional integer integer fractional and fractional fractional with respect to time and to the spatial variable are provided in order to illustrate how simple and general is the suggested approach the fifth example illustrates that the method can be equally simply used for fractional differential equations with delays a set of matlab routines for the implementation of the method as well as sample code used to solve the examples have been developed

wave field simulation for heterogeneous porous media with singular memory drag force the objective of this paper is to use biot s theory and the johnson koplik dashen dynamic permeability model in wave field simulation of a heterogeneous porous medium the johnson koplik dashen dynamic permeability model was originally formulated in the frequency domain in this paper the time domain drag force expression of the model is expressed in terms of the shifted fractional derivative of the relative fluid velocity in contrast to the exponential type viscous relaxation models the convolution operator in the johnson koplik dashen dynamic permeability model cannot be replaced by memory variables satisfying first order relaxation differential equations a new method for calculating the shifted fractional derivative without storing and integrating the entire velocity histories is developed using the new method to calculate the fractional derivative the governing equations for the two dimensional porous medium are reduced to a system of first order differential equations for velocities stresses pore pressure and the quadrature variables associated with the drag forces spatial derivatives involved in the first order differential equations system are calculated by fourier pseudospectral method while the time derivative of the system is discretized by a predictor corrector method for the demonstration of our method some numerical results are presented

finite difference approximations for fractional advection dispersion flow equations fractional advection dispersion equations are used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in a porous medium in this paper we develop practical numerical methods to solve one dimensional fractional advection dispersion equations with variable coefficients on a finite domain the practical application of these results is illustrated by modeling a radial flow problem use of the fractional derivative allows the model equations to capture the early arrival of tracer observed at a field site

least squares finite element solution of a fractional order two point boundary value problem in this paper a theoretical framework for the least squares finite element approximation of a fractional order differential equation is presented mapping properties for fractional dimensional operators on suitable fractional dimensional spaces are established using these properties existence and uniqueness of the least squares approximation is proven optimal error estimates are proven for piecewise linear trial elements numerical results are included which confirm the theoretical results

optimal feedback communication via posterior matching in this paper we introduce a fundamental principle for optimal communication over general memoryless channels in the presence of noiseless feedback termed posterior matching using this principle we devise a simple sequential generic feedback transmission scheme suitable for a large class of memoryless channels and input distributions achieving any rate below the corresponding mutual information this provides a unified framework for optimal feedback communication in which the horstein scheme bsc and the schalkwijk kailath scheme awgn channel are special cases thus as a corollary we prove that the horstein scheme indeed attains the bsc capacity settling a longstanding conjecture we further provide closed form expressions for the error probability of the scheme over a range of rates and derive the achievable rates in a mismatch setting where the scheme is designed according to the wrong channel model several illustrative examples of the posterior matching scheme for specific channels are given and the corresponding error probability expressions are evaluated the proof techniques employed utilize novel relations between information rates and contraction properties of iterated function systems

on gaussian channels with feedback under expected power constraints and with non vanishing error probabilities in this paper we consider single and multi user gaussian channels with feedback under expected power constraints and with non vanishing error probabilities in the first of two contributions we study asymptotic expansions for the additive white gaussian noise awgn channel with feedback under the average error probability formalism by drawing ideas from gallager and nakiboglu s work for the direct part and the meta converse for the converse part we establish the varepsilon capacity and show that it depends on varepsilon in general and so the strong converse fails to hold furthermore we provide bounds on the second order term in the asymptotic expansion we show that for any positive integer l the second order term is bounded between a term proportional to ln l n where ln l cdot is the l fold nested logarithm function and a term proportional to n ln n 1 2 where n is the blocklength the lower bound on the second order term shows that feedback does provide an improvement in the maximal achievable rate over the case where no feedback is available in our second contribution we establish the varepsilon capacity region for the awgn multiple access channel with feedback under the expected power constraint by combining ideas from hypothesis testing information spectrum analysis ozarow s coding scheme and power control

least squares spectral method for the solution of a fractional advection dispersion equation fractional derivatives provide a general approach for modeling transport phenomena occurring in diverse fields this article describes a least squares spectral method for solving advection dispersion equations using caputo or riemann liouville fractional derivatives a gauss lobatto jacobi quadrature is implemented to approximate the singularities in the integrands arising from the fractional derivative definition exponential convergence rate of the operator is verified when increasing the order of the approximation solutions are calculated for fractional time and fractional space differential equations comparisons with finite difference schemes are included a significant reduction in storage space is achieved by lowering the resolution requirements in the time coordinate

multidimensional advection and fractional dispersion extension of the fractional diffusion equation to two or three dimensions is not as simple as extension of the second order equation this is revealed by the solutions of the equations unlike the gaussian the most general stable vector cannot be generated with an atomistic measure on the coordinate axes a random combination of maximally skewed stable variables on the unit sphere generates a stable vector that is a general model of a diffusing particle subsets are symmetric stable vectors that have previously appeared in the literature and the well known multidimensional brownian motion a multidimensional fractional differential operator is defined in the process

a space time spectral method for the time fractional diffusion equation in this paper we consider the numerical solution of the time fractional diffusion equation essentially the time fractional diffusion equation differs from the standard diffusion equation in the time derivative term in the former case the first order time derivative is replaced by a fractional derivative making the problem global in time we propose a spectral method in both temporal and spatial discretizations for this equation the convergence of the method is proven by providing a priori error estimate numerical tests are carried out to confirm the theoretical results thanks to the spectral accuracy in both space and time of the proposed method the storage requirement due to the global time dependence can be considerably relaxed and therefore calculation of the long time solution becomes possible

pitfalls in fast numerical solvers for fractional differential equations we consider the problem of implementing fast algorithms for the numerical solution of initial value problems of the form x t f t x t x 0 x0 where x is the derivative of x of order in the sense of caputo and 0 1 we review some of the existing methods and explain their respective strengths and weaknesses we identify and discuss potential problems in the development of generally applicable schemes

a chebyshev pseudospectral method to solve the space time tempered fractional diffusion equation a chebyshev pseudospectral method to solve the space time tempered fractional diffusion equation

finite difference spectral approximations for the time fractional diffusion equation in this paper we consider the numerical resolution of a time fractional diffusion equation which is obtained from the standard diffusion equation by replacing the first order time derivative with a fractional derivative of order a with 0 a 1 the main purpose of this work is to construct and analyze stable and high order scheme to efficiently solve the time fractional diffusion equation the proposed method is based on a finite difference scheme in time and legendre spectral methods in space stability and convergence of the method are rigourously established we prove that the full discretization is unconditionally stable and the numerical solution converges to the exact one with order o dt 2 a n m where dt n and m are the time step size polynomial degree and regularity of the exact solution respectively numerical experiments are carried out to support the theoretical claims

fast algorithms for component by component construction of rank 1 lattice rules in shift invariant reproducing kernel hilbert spaces we reformulate the original component by component algorithm for rank 1 lattices in a matrix vector notation so as to highlight its structural properties for function spaces similar to a weighted korobov space we derive a technique which has construction cost o snlog n in contrast with the original algorithm which has construction cost o sn 2 herein s is the number of dimensions and n the number of points taken prime in contrast to other approaches to speed up construction our fast algorithm computes exactly the same quantity as the original algorithm the presented algorithm can also be used to construct randomly shifted lattice rules in weighted sobolev spaces

on the dispersions of three network information theory problems we analyze the dispersions of distributed lossless source coding the slepian wolf problem the multiple access channel and the asymmetric broadcast channel for the two encoder slepian wolf problem we introduce a quantity known as the entropy dispersion matrix which is analogous to the scalar dispersions that have gained interest recently we prove a global dispersion result that can be expressed in terms of this entropy dispersion matrix and provides intuition on the approximate rate losses at a given blocklength and error probability to gain better intuition about the rate at which the nonasymptotic rate region converges to the slepian wolf boundary we define and characterize two operational dispersions 1 the local dispersion and 2 the weighted sum rate dispersion the former represents the rate of convergence to a point on the slepian wolf boundary whereas the latter represents the fastest rate for which a weighted sum of the two rates converges to its asymptotic fundamental limit interestingly when we approach either of the two corner points the local dispersion is characterized not by a univariate gaussian but a bivariate one as well as a subset of off diagonal elements of the aforementioned entropy dispersion matrix finally we demonstrate the versatility of our achievability proof technique by providing inner bounds for the multiple access channel and the asymmetric broadcast channel in terms of dispersion matrices all our proofs are unified by a so called vector rate redundancy theorem which is proved using the multidimensional berry esseen theorem

error exponents for variable length block codes with feedback and cost constraints variable length block coding schemes are investigated for discrete memoryless channels with ideal feedback under cost constraints upper and lower bounds are found for the minimum achievable probability of decoding error pe min as a function of constraints r p and tau on the transmission rate average cost and average block length respectively for given r and p the lower and upper bounds to the exponent ln pe min tau are asymptotically equal as tau rarr infin the resulting reliability function limtaurarrinfin in pe min tau as a function of r and v is concave in the pair r p and generalizes the linear reliability function of burnashev to include cost constraints the results are generalized to a class of discrete time memoryless channels with arbitrary alphabets including additive gaussian noise channels with amplitude and power constraints

the generalized cattaneo equation for the description of anomalous transport processes the cattaneo equation which describes a diffusion process with a finite velocity of propagation is generalized to describe anomalous transport three possible generalizations are proposed each one supported by a different scheme continuous time random walks non local transport theory and delayed flux force relation the properties of these generalizations are studied in both the long time and the short time regimes in the long time limit we recover the mean square displacement which is characteristic for these anomalous processes as expected the short time behaviour is modified in comparison to generalized diffusion equations

computational aspects of fem approximation of fractional advection dispersion equations on bounded domains in r 2 in this paper we investigate the numerical approximation of the variational solution to the fractional advection dispersion equation fade on bounded domains in r2 more specifically we investigate the computational aspects of the galerkin approximation using continuous piecewise polynomial basis functions on a regular triangulation of the domain the computational challenges of approximating the solution to fractional differential equations using the finite element method stem from the fact that a fractional differential operator is a non local operator several numerical examples are given which demonstrate approximations to fades

fourth order compact scheme for space fractional advection diffusion reaction equations with variable coefficients abstract in this work a new fourth order compact approximation is derived for riemann liouville space fractional derivatives modified wave numbers are obtained for various approximations of fractional derivatives considering these modified wave numbers fourier analysis of differencing errors is presented to quantify the resolution characteristics of various approximations a novel fourth order compact scheme is developed for space fractional advection diffusion reaction equations with variable coefficients in one and two dimensions in the proposed compact scheme the second derivative approximation of unknowns is approximated using the value of these unknowns and their first derivative approximations this splitting of second derivative approximations allows us to obtain a similar system of linear equations as in zhao and tian 2017 which presents only second order accurate finite difference method for advection diffusion reaction equations in one dimension the stability of the proposed compact scheme is demonstrated numerically furthermore the proposed compact scheme is employed to space fractional black scholes equation for pricing european options as an application of fractional derivatives in mathematical finance numerical illustrations are presented to validate the theoretical claims

on gaussian macs with variable length feedback and non vanishing error probabilities we characterize the fundamental limits of transmission of information over a gaussian multiple access channel mac with the use of variable length feedback codes and under a non vanishing error probability formalism we develop new achievability and converse techniques to handle the continuous nature of the channel and the presence of expected power constraints we establish the varepsilon capacity regions and bounds on the second order asymptotics of the gaussian mac with variable length feedback with termination codes and stop feedback codes we show that the former outperforms the latter significantly due to the multi terminal nature of the channel model we leverage tools from renewal theory developed by lai and siegmund to bound the asymptotic behavior of the maximum of a finite number of stopping times

an explicit high order method for fractional advection diffusion equations we propose a high order explicit finite difference method for fractional advection diffusion equations these equations can be obtained from the standard advection diffusion equations by replacing the second order spatial derivative by a fractional operator of order with 1 2 this operator is defined by a combination of the left and right riemann liouville fractional derivatives we study the convergence of the numerical method through consistency and stability the order of convergence varies between two and three and for advection dominated flows is close to three although the method is conditionally stable the restrictions allow wide stability regions the analysis is confirmed by numerical examples

new explicit and accelerated techniques for solving fractional order differential equations abstract this paper proposes new direct and acceleration numerical methods for solving fractional order differential equations fdes for the caputo differential operator with fractional order 0 o h 2 and o h 3 respectively in order to overcome a strong singular issue as 0 we propose the explicit predictor corrector scheme with perturbation technique in case of 1 the truncation errors are reduced by o h and o h 2 in order to accelerate the convergence rate we decompose 2 2 convert the fde into the system of fdes and apply the predictor corrector scheme numerical tests for linear nonlinear variable order and time fractional sub diffusion problems demonstrate that the proposed methods give a prominent performance we also compare the numerical results with other high order explicit and implicit schema

fractional master equations and fractal time random walks fractional master equations containing fractional time derivatives of order 0 ensuremath le 1 are introduced on the basis of a recent classification of time generators in ergodic theory it is shown that fractional master equations are contained as a special case within the traditional theory of continuous time random walks the corresponding waiting time density ensuremath psi t is obtained exactly as ensuremath psi t mathit t mathrm ensuremath omega mathrm ensuremath 1 c mathit e mathrm ensuremath omega mathrm ensuremath omega mathit t mathrm ensuremath omega c where mathit e mathrm ensuremath omega mathrm ensuremath omega x is the generalized mittag leffler function this waiting time distribution is singular both in the long time as well as in the short time limit

a fourth order approximation of fractional derivatives with its applications a new fourth order difference approximation is derived for the space fractional derivatives by using the weighted average of the shifted grunwald formulae combining the compact technique the properties of proposed fractional difference quotient operator are presented and proved then the new approximation formula is applied to solve the space fractional diffusion equations by the energy method the proposed quasi compact difference scheme is proved to be unconditionally stable and convergent in l2 norm for both 1d and 2d cases several numerical examples are given to confirm the theoretical results

the analysis of the impact response of a thin plate via fractional derivative standard linear solid model the impact of a rigid body upon an infinite isotropic plate is investigated for the case when the viscoelastic features of the plate represent themselves only in the place of contact and are governed by the standard linear solid model with fractional derivatives thus the problem concerns the shock interaction of the dropping mass and the target wherein instead of the hertz contact law the generalized fractional derivative standard linear solid law is employed as a law of interaction the part of the plate beyond the contact domain is assumed to be elastic and its behaviour is described by the equations of motion which take rotary inertia and shear deformations into account it is assumed that transient waves generate in the plate at the moment of impact the influence of which on the contact domain is considered using the theory of discontinuities to determine the desired values behind the transverse shear wave front one term ray expansions are used as well as the equations of motion of the falling mass and the contact region as a result we are led to a set of two linear differential equations the solution of which is found analytically by the laplace transform and by the euler substitution method this allows the contact force to be determined as a function of time

the use of finite difference element approaches for solving the time fractional subdiffusion equation in this paper two finite difference element approaches for the time fractional subdiffusion equation with dirichlet boundary conditions are developed in which the time direction is approximated by the fractional linear multistep method and the space direction is approximated by the finite element method the two methods are unconditionally stable and convergent of order o tau q h r 1 in the l 2 norm where q 2 beta or 2 when the analytical solution to the subdiffusion equation is sufficiently smooth beta 0 beta 1 is the order of the fractional derivative tau and h are the step sizes in time and space respectively and r is the degree of the polynomial space the corresponding schemes for the subdiffusion equation with neumann boundary conditions are presented as well where the stability and convergence are shown numerical examples are provided to verify the theoretical analysis comparisons between the algorithms derived in this paper and the existing algorithms are given

component by component construction of good lattice rules this paper provides a novel approach to the construction of good lattice rules for the integration of korobov classes of periodic functions over the unit s dimensional cube theorems are proved which justify the construction of good lattice rules one component at a time that is the lattice rule for dimension s 1 is obtained from the rule for dimension s by searching over all possible choices of the s 1 th component while keeping all the existing components unchanged the construction which goes against accepted wisdom is illustrated by numerical examples the construction is particularly useful if the components of the integrand are ordered in the sense that the first component is more important than the second and so on

constructing randomly shifted lattice rules in weighted sobolev spaces shifted rank 1 lattice rules a special class of quasi monte carlo methods have recently been proposed by the present authors for the integration of functions belonging to certain weighted sobolev spaces the shifts in these rules were generated in a deterministic manner in contrast in this paper we generate these shifts randomly this allows probabilistic estimates for the error in a given integral it also reduces the number of operations required to find the generating vectors for the underlying lattice rules component by component the rules thus constructed achieve a worst case strong tractability error bound in an average or probabilistic sense

design and monitoring of multi arm multi stage clinical trials two arm group sequential designs have been widely used for over 40 years especially for studies with mortality endpoints the natural generalization of such designs to trials with multiple treatment arms and a common control mams designs has however been implemented rarely while the statistical methodology for this extension is clear the main limitation has been an efficient way to perform the computations past efforts were hampered by algorithms that were computationally explosive with the increasing interest in adaptive designs platform designs and other innovative designs that involve multiple comparisons over multiple stages the importance of mams designs is growing rapidly this article provides break through algorithms that can compute mams boundaries rapidly thereby making such designs practical for designs with efficacy only boundaries the computational effort increases linearly with number of arms and number of stages for designs with both efficacy and futility boundaries the computational effort doubles with successive increases in number of stages

analysis of a new finite difference local discontinuous galerkin method for the fractional diffusion wave equation in this paper a finite difference local discontinuous galerkin method for the fractional diffusion wave equation is presented and analyzed we first propose a new finite difference method to approximate the time fractional derivatives and give a semidiscrete scheme in time further we develop a fully discrete scheme for the fractional diffusion wave equation and prove that the method is unconditionally stable and convergent with order o hk 1 t 3 where k is the degree of piecewise polynomial extensive numerical examples are carried out to confirm the theoretical convergence rates

maximum principle for the generalized time fractional diffusion equation abstract in the paper a maximum principle for the generalized time fractional diffusion equation over an open bounded domain g 0 t g r n is formulated and proved the proof of the maximum principle is based on an extremum principle for the caputo dzherbashyan fractional derivative that is given in the paper too the maximum principle is then applied to show that the initial boundary value problem for the generalized time fractional diffusion equation possesses at most one classical solution and this solution continuously depends on the initial and boundary conditions

high order numerical methods for the riesz space fractional advection dispersion equations abstract in this paper we propose high order numerical methods for the riesz space fractional advection dispersion equations rsfade on a finite domain the rsfade is obtained from the standard advection dispersion equation by replacing the first order and second order space derivative with the riesz fractional derivatives of order 0 1 and 1 2 respectively firstly we utilize the weighted and shifted grunwald difference operators to approximate the riesz fractional derivative and present the finite difference method for the rsfade specifically we discuss the crank nicolson scheme and solve it in matrix form secondly we prove that the scheme is unconditionally stable and convergent with the accuracy of o 2 h 2 thirdly we use the richardson extrapolation method rem to improve the convergence order which can be o 4 h 4 finally some numerical examples are given to show the effectiveness of the numerical method and the results are in excellent with theoretical analysis

a lattice boltzmann scheme for cattaneo s diffusion equation abstract a lattice boltzmann lb formulation is developed for the simulation of cattaneo s diffusion equation to do this the collision term is computed from a 1 step back relaxation dynamics which in turn induces a hyperbolic type diffusion effect the fickian lb formulation is recovered in the limit as a relaxation time coefficient goes to zero

finite volume element method and its stability analysis for analyzing the behavior of sub diffusion problems in this paper we analyze the spatially semi discrete piecewise linear finite volume element method for the time fractional sub diffusion problem in two dimensions and give an approximate solution of this problem at first we introduce bilinear finite volume element method with interpolated coefficients and derive some error estimates between exact solution and numerical solution in both finite element and finite volume element methods furthermore we use the standard finite element ritz projection and also the elliptic projection defined by the bilinear form associated with the variational formulation of the finite volume element method finally some numerical examples are included to illustrate the effectiveness of the new technique

a note on the finite element method for the space fractional advection diffusion equation in this paper a note on the finite element method for the space fractional advection diffusion equation with non homogeneous initial boundary condition is given where the fractional derivative is in the sense of caputo the error estimate is derived and the numerical results presented support the theoretical results

on the step by step construction of quasi monte carlo integration rules that achieve strong tractability error bounds in weighted sobolev spaces we develop and justify an algorithm for the construction of quasi monte carlo qmc rules for integration in weighted sobolev spaces the rules so constructed are shifted rank 1 lattice rules the parameters characterising the shifted lattice rule are found component by component the d 1 th component of the generator vector and the shift are obtained by successive 1 dimensional searches with the previous d components kept unchanged the rules constructed in this way are shown to achieve a strong tractability error bound in weighted sobolev spaces a search for n point rules with n prime and all dimensions 1 to d requires a total cost of o n3d2 operations this may be reduced to o n3d operations at the expense of o n2 storage numerical values of parameters and worst ease errors are given for dimensions up to 40 and n up to a few thousand the worst case errors for these rules are found to be much smaller than the theoretical bounds

generalized natural boundary conditions for fractional variational problems in terms of the caputo derivative this paper presents the necessary and sufficient optimality conditions for problems of the fractional calculus of variations with a lagrangian depending on the free end points the fractional derivatives are defined in the sense of caputo

a proof of the strong converse theorem for gaussian multiple access channels we prove the strong converse for the n source gaussian multiple access channel in particular we show that any rate tuple that can be supported by a sequence of codes with asymptotic average error probability 1 must lie in the cover wyner capacity region our proof consists of the following first we perform an expurgation step to convert any given sequence of codes with asymptotic average error probability 1 to codes with asymptotic maximal error probability 1 second we quantize the input alphabets with an appropriately chosen resolution upon quantization we apply the wringing technique by ahlswede on the quantized inputs to obtain further subcodes from the subcodes obtained in the expurgation step so that the resultant correlations among the symbols transmitted by the different sources vanish as the blocklength grows finally we derive upper bounds on achievable sum rates of the subcodes in terms of the type ii error of a binary hypothesis test these upper bounds are then simplified through judicious choices of auxiliary output distributions our strong converse result carries over to the gaussian interference channel under strong interference as long as the sum of the two asymptotic average error probabilities 1