two classes of quadratic apn binomials inequivalent to power functions this paper introduces the first found infinite classes of almost perfect nonlinear apn polynomials which are not carlet charpin zinoviev ccz equivalent to power functions at least for some values of the number of variables these are two classes of apn binomials from f2n to f2n for n divisible by 3 resp 4 we prove that these functions are extended affine ea inequivalent to any power function and that they are ccz inequivalent to the gold kasami inverse and dobbertin functions when n ges 12 this means that for n even they are ccz inequivalent to any known apn function in particular for n 12 20 24 they are therefore ccz inequivalent to any power function

differential properties of x mapsto x 2 t 1 we provide an extensive study of the differential properties of the functions x x2t 1 over bbf 2n for 1 t n we notably show that the differential spectra of these functions are determined by the number of roots of the linear polynomials x2t bx2 b 1 x where b varies in bbf 2n we prove a strong relationship between the differential spectra of x x2t 1 and x x2s 1 for s n t 1 as a direct consequence this result enlightens a connection between the differential properties of the cube function and of the inverse function we also determine the complete differential spectra of x x7 by means of the value of some kloosterman sums and of x x2t 1 for t n 2 n 2 1 n 2

linear codes from some 2 designs a classical method of constructing a linear code over mathrm gf q with a t design is to use the incidence matrix of the t design as a generator matrix over mathrm gf q of the code this approach has been extensively investigated in the literature in this paper a different method of constructing linear codes using specific classes of 2 designs is studied and linear codes with a few weights are obtained from almost difference sets difference sets and a type of 2 designs associated to semibent functions two families of the codes obtained in this paper are optimal the linear codes presented in this paper have applications in secret sharing and authentication schemes in addition to their applications in consumer electronics communication and data storage systems a coding theory approach to the characterization of highly nonlinear boolean functions is presented

new results about the boomerang uniformity of permutation polynomials in eurocrypt 2018 cid et al introduced a new concept on the cryptographic property of s boxes boomerang connectivity table bct for short for evaluating the subtleties of boomerang style attacks very recently bct and the boomerang uniformity the maximum value in bct were further studied by boura and canteaut in this paper aiming at providing new insights we show some new results about bct and the boomerang uniformity of permutations in terms of theory and experiment first we present an equivalent technique to compute bct and the boomerang uniformity which seems to be much simpler than the original definition by cid et al second thanks to carlet s idea we give a characterization of functions f from mathbb f 2 n to itself with boomerang uniformity delta f by means of the walsh transform third by our method we consider boomerang uniformities of some specific permutations mainly the ones with low differential uniformity finally we obtain another class of 4 uniform bct permutation polynomials over mathbb f 2 n

boolean and vectorial plateaued functions and apn functions boolean plateaued functions and vectorial functions with plateaued components play a significant role in cryptography sequences for communications and the related combinatorics and designs our knowledge on them is not at a level corresponding to their importance we introduce new characterizations of plateaued boolean functions we give the characterizations of vectorial functions whose components are all plateaued with possibly different amplitudes that we simply call plateaued by means of the value distributions of their derivatives we characterize similarly those functions whose components are partially bent and autocorrelation functions and of the power moments of their walsh transform this allows us to derive several characterizations of almost perfect nonlinear apn functions in this framework we prove that all the main results known for quadratic apn functions extend to plateaued functions allowing the study of their apn ness to be simplified we show that if additionally the component functions are all unbalanced this study is still simpler the apn ness of such functions depends only on their value distribution this allows proving for instance that any plateaued n n function n even having similar value distribution as the apn power functions is apn and has the same extended walsh spectrum as the apn gold functions as by products we obtain a few other new results for instance any plateaued function in even dimension which is carlet charpin zinoviev ccz equivalent to a gold or kasami apn function is necessarily extended affine ea equivalent to it

pseudospectral solution of the two dimensional navier stokes equations in a disk an efficient and accurate algorithm for solving the two dimensional 2d incompressible navier stokes equations on a disk with no slip boundary conditions is described the vorticity stream function formulation of these equations is used and spatially the vorticity and stream functions are expressed as fourier chebyshev expansions the poisson and helmholtz equations which arise from the implicit explicit time marching scheme are solved as banded systems using a postconditioned spectral tau method the polar coordinate singularity is handled by expanding fields radially over the entire diameter using a parity modified chebyshev series and building partial regularity into the vorticity the no slip boundary condition is enforced by transferring one of the two boundary conditions imposed on the stream function onto the vorticity via a solvability constraint significant gains in run times were realized by parallelizing the code in message passage interface mpi

statistical inference procedures for bivariate archimedean copulas abstract a bivariate distribution function h x y with marginals f x and g y is said to be generated by an archimedean copula if it can be expressed in the form h x y 1 f x g y for some convex decreasing function defined on 0 1 in such a way that 1 0 many well known systems of bivariate distributions belong to this class including those of gumbel ali mikhail haq thelot clayton frank and hougaard frailty models also fall under that general prescription this article examines the problem of selecting an archimedean copula providing a suitable representation of the dependence structure between two variates x and y in the light of a random sample x 1 y 1 x n y n the key to the estimation procedure is a one dimensional empirical distribution function that can be constructed whether the uniform representation of x and y is archimedean or not and independently of their marginals this semiparametric estimator based on a decomposition of kendall s tau statistic

on nonparametric measures of dependence for random variables in 1959 a renyi proposed a set of axioms for a measure of dependence for pairs of random variables in the same year a sklar introduced the general notion of a copula this is a function which links an n dimensional distribution function to its one dimensional margins and is itself a continuous distribution function on the unit n cube with uniform margins we show that the copula of a pair of random variables x y is invariant under a s strictly increasing transformations of x and y and that any property of the joint distribution function of x and y which is invariant under such transformations is solely a function of their copula exploiting these facts we use copulas to define several natural nonparametric measures of dependence for pairs of random variables we show that these measures satisfy reasonable modifications of renyi s conditions and compare them to various known measures of dependence e g the correlation coefficient and spearman s rho

almost perfect nonlinear power functions on gf 2 n the niho case almost perfect nonlinear apn mappings are of interest for applications in cryptography we prove for odd n and the exponent d 22r 2r 1 where 4r 1 0modn that the power functions xd on gf 2n is apn the given proof is based on a new class of permutation polynomials which might be of independent interest our result supports a conjecture of niho stating that the power function xd is even maximally nonlinear or in other terms that the crosscorrelation function between a binary maximum length linear shift register sequences of degree n and a decimation of that sequence by d takes on precisely the three values 1 1 2 n 1 2

two characterizations of crooked functions in this correspondence we give two characterizations of crooked functions one based on the minimum distance of a preparata like code and the other based on the distance regularity of a crooked graph

eigenfrequencies of fractal drums a method for the computation of eigenfrequencies and eigenmodes of fractal drums is presented the approach involves first conformally mapping the unit disk to a polygon approximating the fractal and then solving a weighted eigenvalue problem on the unit disk by a spectral collocation method the numerical computation of the complicated conformal mapping was made feasible by the use of the fast multipole method as described in l banjai l n trefethen a multipole method for schwarz christoffel mapping of polygons with thousands of sides siam j sci comput 25 3 2003 1042 1065 the linear system arising from the spectral discretization is large and dense to circumvent this problem we devise a fast method for the inversion of such a system consequently the eigenvalue problem is solved iteratively we obtain eight digits for the first eigenvalue of the koch snowflake and at least five digits for eigenvalues up to the 20th numerical results for two more fractals are shown

brinkman equation for a corrugated pipe using a spectral galerkin method this study numerically investigates the fully developed flow of a newtonian fluid in a porous saturated corrugated pipe on the basis of a brinkman model the variable coefficient helmholtz equation which is obtained by means of an epitrochoid transformation is solved using a spectral galerkin method the effects of both the darcy number and corrugation on the velocity field are discussed and presented graphically the nature of these effects is documented for the first time

classes of quadratic apn trinomials and hexanomials and related structures a method for constructing differentially 4 uniform quadratic hexanomials has been recently introduced by j dillon we give various generalizations of this method and we deduce the constructions of new infinite classes of almost perfect nonlinear quadratic trinomials and hexanomials from f22m to f22m we check for m 3 that some of these functions are ccz inequivalent to power functions

convergence analysis of spectral collocation methods for a singular differential equation solutions of partial differential equations with coordinate singularities often have special behavior near the singularities which forces them to be smooth special treatment for these coordinate singularities is necessary in spectral approximations in order to avoid degradation of accuracy and efficiency it has been observed numerically in the past that for a scheme to attain high accuracy it is unnecessary to impose all the pole conditions the constraints representing the special solution behavior near singularities in this paper we provide a theoretical justification for this observation specifically we consider an existing approach which uses a pole condition as the boundary condition at a singularity and solves the reformulated boundary value problem with a commonly used gauss lobatto collocation scheme spectral convergence of the legendre and chebyshev collocation methods is obtained for a singular differential equation arising from polar and cylindrical geometries

new families of almost perfect nonlinear power mappings a power mapping f x x sup d over gf p sup n is said to be differentially k uniform if k is the maximum number of solutions x spl isin gf p sup n of f x a f x b where a b spl isin gf p sup n and a spl ne 0 a 2 uniform mapping is called almost perfect nonlinear apn we construct several new infinite families of nonbinary apn power mappings

a note on differential properties of x mapsto x 2 t 1 blondeau and colleagues recently conjectured a characterization of when x2t 1 is an almost perfect nonlinear function on f2n in this paper we will prove the conjecture when n is even the short proof uses a theorem of hou and colleagues concerning reversed dickson polynomials and a theorem of payne

almost perfect nonlinear power functions in odd characteristic in this paper two new families of functions in odd characteristic are constructed and they are proved to be almost perfect nonlinear apn functions some of the open cases which were introduced by helleseth are answered by these new apn functions

a fourier spectral element algorithm for thermal convection in rotating axisymmetric containers we present a fourier spectral element approach for modeling thermal convection in a rotating axisymmetric container following the theory detailed in bernardi et al c bernardi m dauge y maday spectral methods for axisymmetric domains gauthier villars paris 1999 a fourier expansion of the field variables is performed in the periodic direction and the resulting collection of meridional problems is discretized by means of a parallel spectral element method a gauss lobatto jacobi 0 1 quadrature which incorporates the cylindrical radius in its weight is introduced to avoid a potential degeneracy of the discrete set of equations due to those nodes located on the axis of symmetry a second order timestepping scheme is presented which treats the coriolis and viscous forces implicitly numerical comparisons with analytical and published numerical solutions in spherical and cylindrical geometries are presented which highlight the accuracy of the model and demonstrate its spectral convergence properties

a new apn function which is not equivalent to a power mapping a new almost perfect nonlinear function apn on f 2 sup 10 which is not equivalent to any of the previously known apn mappings is constructed this is the first example of an apn mapping which is not equivalent to a power mapping

a new family of ternary almost perfect nonlinear mappings a mapping f x from gf pn to gf pn is differentially k uniform if k is the maximum number of solutions x isin gf pn of f x a f x b where a b isin gf pn and a ne 0 a 2 uniform mapping is called almost perfect nonlinear apn this correspondence describes new families of ternary apn mappings over gf 3n n 3 odd of the form f x uxd xd 2 where d1 3n 1 2 1 and d2 3n 2

handling vectorial functions by means of their graph indicators we characterize the anf and the univariate representation of any vectorial function as parts of the anf and bivariate representation of the boolean function equal to its graph indicator we show how this provides when f is bijective the expression of f 1 and or allows deriving properties of f 1 we illustrate this with examples and with a tight upper bound on the algebraic degree of f 1 by means of that of f we characterize by the fourier hadamard transform by the anf and by the bivariate representation that a given boolean function is the graph indicator of a vectorial function we also give characterizations of those boolean functions that are affine equivalent to graph indicators we express the graph indicators of the sum product composition and concatenation of vectorial functions by means of the graph indicators of the functions we deduce from these results a characterization of the bijectivity of a generic n n function by the fact that some boolean function which appears as a part of the anf resp the bivariate representation of its graph indicator is equal to constant function 1 we also address the injectivity of n m functions finally we study the characterization of the almost perfect nonlinearity of vectorial functions by means of their graph indicators

cartesian authentication codes from functions with optimal nonlinearity in this paper we present several classes of authentication codes using functions with perfect nonlinearity and optimum nonlinearity some of the authentication codes are optimal on the other hand these authentication codes are easy to implement due to their simple algebraic structures

computation and application of copula based weighted average quantile regression copula theories have obtained rapid development with respect to the dependent analysis of financial markets the widely adopted technique for estimating copula is maximum likelihood in distinction this paper proposes weighted average quantile regression to estimate copula which shares robustness from quantile regression and achieves nearly the same efficiency as the maximum likelihood consistency and asymptotic normality of the suggested estimators are established under regularity conditions through solving a quadratic programming we obtain the expression of optimal weights monte carlo simulations are conducted to compare the performance of different estimators the proposed approach is used to investigate the dependent structure and risk measurement of chinese financial markets we propose weighted average quantile method to estimate copulas asymptotic properties of the estimators are provided simulations show satisfactory performance of the proposed estimates the weighted average quantile copula technique is applied to investigate the dependent structure and risk measurement of chinese financial markets

a new spectral boundary integral collocation method for three dimensional potential problems in this work we propose and analyze a new global collocation method for classical second kind boundary integral equations of potential theory on smooth simple closed surfaces gamma subset bbb r 3 under the assumption that gamma is diffeomorphic to the unit sphere partial b the original equation is transferred to an equivalent one on partial b which is solved using collocation onto a nonstandard set of basis functions the collocation points are situated on lines of constant latitude and longitude the interpolation operator used in the collocation method is equivalent to a certain discrete orthogonal pseudospectral projection and this equivalence allows us to establish the fundamental properties of the interpolation process and subsequently to prove that our collocation method is stable and super algebraically convergent in addition we describe a fast method for computing the weakly singular collocation integrals and present some numerical experiments illustrating the use of the method these show that at least for model problems the method attains an exponential rate of convergence and exhibits a good accuracy for very small numbers of degrees of freedom

the weights of the orthogonals of the extended quadratic binary goppa codes starting from results on elliptic curves and kloosterman sums over the finite field ge 2 sup t the authors determine the weights of the orthogonals of some binary linear codes the melas code of length the irreducible cyclic binary code of length 2 sup t 1 and the extended binary goppa codes defined by polynomials of degree two

on almost perfect nonlinear functions over mmb f_2 n we investigate some open problems on almost perfect nonlinear apn functions over a finite field of characteristic 2 we provide new characterizations of apn functions and of apn permutations by means of their component functions we generalize some results of nyberg 1994 and strengthen a conjecture on the upper bound of nonlinearity of apn functions we also focus on the case of quadratic functions we contribute to the current works on apn quadratic functions by proving that a large class of quadratic functions cannot be apn

almost perfect nonlinear power functions on gf 2 sup n the welch case we summarize the state of the classification of almost perfect nonlinear apn power functions x sup d on gf 2 sup n and contribute two new cases to prove these cases we derive new permutation polynomials the first case supports a well known conjecture of welch stating that for odd n 2m 1 the power function x sup 2m 3 is even maximally nonlinear or in other terms that the crosscorrelation function between a binary maximum length linear shift register sequence of degree n and a decimation of that sequence by 2 sup m 3 takes on precisely the three values 1 1 spl plusmn 2 sup m 1

new families of perfect nonlinear polynomial functions abstract a new class of almost perfect nonlinear apn polynomial functions has been recently introduced we give some generalizations of these functions and deduce new families of perfect nonlinear pn functions we show that these pn functions are ccz inequivalent to the known perfect nonlinear functions

radial collocation methods for the onset of convection in rotating spheres the viability of using collocation methods in radius and spherical harmonics in the angular variables to calculate convective flows in full spherical geometry is examined as a test problem the stability of the conductive state of a self gravitating fluid sphere subject to rotation and internal heating is considered a study of the behavior of different radial meshes previously used by several authors in polar coordinates including or not the origin is first performed the presence of spurious modes due to the treatment of the singularity at the origin to the spherical harmonics truncation and to the initialization of the eigenvalue solver is shown and ways to eliminate them are presented finally to show the usefulness of the method the neutral stability curves at very high taylor and moderate and small prandtl numbers are calculated and shown several radial collocation meshes have been tested to solve convective flows in full spherical geometry the inclusion of the origin in the mesh together with a single regularity condition is enough to obtain good results for eigenvalue problems several sources of spurious modes have been detected and methods to eliminate them are proposed the onset of convection in an internally heated sphere has been studied and asymptotic laws for high taylor and moderate prandtl numbers have been found

niho type cross correlation functions via dickson polynomials and kloosterman sums suppose that n 2k is even we study the cross correlation function between two m sequences for niho type decimations d 2 sup k 1 s 1 we develop a new technique to study the value distribution of these cross correlation functions which makes use of dickson polynomials as a first application we derive here the distribution of the six valued cross correlation function for s 3 and odd k up to a term which depends on kloosterman sums in addition applying simpler methods we prove a theorem providing niho type decimations with four valued cross correlation functions and their distribution we conjecture that the latter result actually covers all such decimations

a new family of semifields with 2 parameters abstract a new family of commutative semifields with two parameters is presented its left and middle nucleus are both determined furthermore we prove that for different pairs of parameters these semifields are not isotopic it is also shown that for some special parameters one semifield in this family can lead to two inequivalent planar functions finally using a similar construction new apn functions are given

efficient spectral galerkin methods iii polar and cylindrical geometries we present in this paper several extremely efficient and accurate spectral galerkin methods for second and fourth order equations in polar and cylindrical geometries these methods are based on appropriate variational formulations which incorporate naturally the pole condition s in particular the computational complexities of the chebyshev galerkin method in a disk and the chebyshev legendre galerkin method in a disk or a cylinder are quasi optimal optimal up to a logarithmic term as an indication of efficiency the cpu time for the poisson solver on a disk by our chebyshev galerkin method is only about 70 of the corresponding finite difference code in fishpack

ternary m sequences with three valued cross correlation function new decimations of welch and niho type we show that the cross correlation between two ternary m sequences of period 3 sup n 1 that differ by the decimation d 2 spl middot 3 sup m 1 where n 2m 1 takes on three different values we conjecture the same result for the decimation d 2 spl middot 3 sup r 1 where n is odd and r is defined by the condition 4r 1 spl equiv 0 mod n these two new cases form in a sense ternary counterparts of two previously confirmed binary cases the conjectures of welch and niho 1972

optimal codebooks from binary codes meeting the levenshtein bound in this paper a generic construction of codebooks based on binary codes is introduced with this generic construction a few previous constructions of optimal codebooks are extended and a new class of codebooks almost meeting the levenshtein bound is presented exponentially many codebooks meeting or almost meeting the levenshtein bound from binary codes are obtained in this paper the codebooks constructed in this paper have alphabet size 4 as a byproduct three bounds on the parameters of binary codes are derived

new classes of almost bent and almost perfect nonlinear polynomials new infinite classes of almost bent and almost perfect nonlinear polynomials are constructed it is shown that they are affine inequivalent to any sum of a power function and an affine function

the casl algorithm for quasi geostrophic flow in a cylinder the contour advective semi lagrangian casl algorithm is a hybrid lagrangian eulerian numerical algorithm combining contour dynamics for the advection of a materially conserved dynamical tracer with spectral techniques for the computation of the associated flow field it benefits from the accuracy of contour dynamics and the efficiency of spectral methods and in applications to quasi geostrophic rotating stratified flows has no time step restriction for numerical stability it abandons the expensive aspect of contour dynamics the computation of contour integrals to obtain the flow field and the inaccurate and costly aspect of spectral methods the advection of materially conserved tracers requiring a time step restriction for numerical stability and numerical diffusion the two approaches are tied together by a fast method of converting tracer contours to gridded field values lagrangian eulerian and the bilinear interpolation of the flow field from the grid to points on the contours eulerian lagrangian here we extend this method originally developed for horizontally periodic flows to flows within a cylindrical domain among other applications this permits a comparison with experiments which are often conducted in a cylindrical domain we demonstrate the accuracy of the algorithm in a few test cases then examine the differences between cylindrical and periodic domains for both simple vortex interactions and complex quasi geostrophic turbulence

some new three valued crosscorrelation functions for binary m sequences in 1972 niho gave various conjectures about the crosscorrelation between a binary maximum length linear shift register sequence and a decimation of that sequence by an integer d we prove that the crosscorrelation function for two new values of d takes on precisely three values and thereby confirm two of niho s conjectures

the nonexistence of permutations ea equivalent to certain ab functions carlet and colleagues conjectured that for any almost bent ab function f there exists a linear function l such that f l is a permutation budaghyan and colleagues found a new class of ab functions which is extended affine ea inequivalent to any power functions and can also serve as a counterexample for the conjecture they checked with the help of a computer that there are no linear functions l on f25 such that x2i 1 x2i x tr x2i 1 x l x is a permutation in this paper we prove that there are no permutations ea equivalent to the ab function x2i 1 x2i x tr x2i 1 x on f22m 1 for any m 2 and there are no permutations ea equivalent to the apn function x2i 1 x2i x 1 tr x2i 1 on bbf22m for m 2 either furthermore we present some results about characterizations of permutation polynomials of the type l x2i 1 l x on bbf22m which is essential in the construction of functions carlet charpin zinoviev equivalent to the gold functions we obtain all the linear functions l x such that x l x2i 1 is a permutation on bbf22m when ker l 22m 2

a fast solver of the shallow water equations on a sphere using a combined compact difference scheme the spectral like three point combined compact difference sp ccd scheme is developed for the shallow water models on a sphere which is originally proposed by chu and fan j comput phys 140 1998 370 a standard test set proposed by williamson et al j comput phys 102 1992 211 is used to evaluate the accuracy of the new method it is shown that the accuracy of the method is comparable to that of the spherical harmonics model as a whole the relative efficiency of the method with respect to the spherical harmonics model appears from the resolution of 512 256 and it becomes significant for the higher resolutions the test results suggest that the method could be extended to numerical atmospheric models

a pseudospectral matrix method for time dependent tensor fields on a spherical shell we construct a pseudospectral method for the solution of time dependent non linear partial differential equations on a three dimensional spherical shell the problem we address is the treatment of tensor fields on the sphere as a test case we consider the evolution of a single black hole in numerical general relativity a natural strategy would be the expansion in tensor spherical harmonics in spherical coordinates instead we consider the simpler and potentially more efficient possibility of a double fourier expansion on the sphere for tensors in cartesian coordinates as usual for the double fourier method we employ a filter to address time step limitations and certain stability issues we find that a tensor filter based on spin weighted spherical harmonics is successful while two simplified non spin weighted filters do not lead to stable evolutions the derivatives and the filter are implemented by matrix multiplication for efficiency a key technical point is the construction of a matrix multiplication method for the spin weighted spherical harmonic filter as example for the efficient parallelization of the double fourier spin weighted filter method we discuss an implementation on a gpu which achieves a speed up of up to a factor of 20 compared to a single core cpu implementation

modelling dependence structure with archimedean copulas and applications to the itraxx cds index in this paper we model the dependence structure between credit default swap cds and jump risk using archimedean copulas the paper models and estimates the different relationships that can exist in different ranges of behaviour it studies the bivariate distributions of cds index spreads and the kurtosis of equity return distribution to take into account nonlinear relationships and different structures of dependency we employ three archimedean copula functions gumbel clayton and frank we adopt nonparametric estimation of copula parameters and we find an extreme co movement of cds and stock market conditions in addition tail dependence indicates the extreme co movements and the potential for a simultaneous large loss in stock markets and a significant default risk ignoring the tail dependence would lead to underestimation of the default risk premium

fast shallow water equation solvers in latitude longitude coordinates here we redirect attention to a fast pseudospectral method on the sphere developed by merilees in 1973 recently revived by fornberg in these works the required spatial derivatives are computed by the formal differentiation of one dimensional fourier series approximations to both scalar and vector functions on the surface of the sphere filters must be used to alleviate prohibitive time stepping restrictions and maintain stability on the non isotropic latitude longitude grids merilees original filter was eventually found to be unusable as it was unstable for longer runs in this paper we examine alternatives to merilees filter in particular we first use a harmonic filter that consists of a harmonic analysis followed directly by a synthesis the resulting stability and accuracy are identical to the traditional spectral transform method fewer legendre transforms are required since they are limited to the filter and not used to compute spatial derivatives in theory this approach can also be viewed as a fast spectral method since fast harmonic filters exist in the literature next we examine alternative fast fourier filters with intent to reproduce the accuracy and stability provided by the harmonic filter computational examples are provided with both high order difference and fourier derivative calculations in addition results are presented for both harmonic and fourier filters

finite differences for coarse azimuthal discretization and for reduction of effective resolution near origin of cylindrical flow equations in this paper the errors generated by the computation of derivatives in the azimuthal direction when flow equations are solved in cylindrical coordinates using finite differences are investigated they might be large for coarse discretizations even using high order schemes which led us to design explicit finite differences specially for 8 16 32 and 64 points per circle these schemes are shown to improve accuracy with respect to standard finite differences and to provide solutions for a two dimensional propagation problem similar to those obtained using fourier spectral methods in the direction a method is then presented to alleviate the time step limitation resulting from explicit time integration near cylindrical origin when finite differences are used it consists in calculating azimuthal derivatives at coarser resolutions than permitted by the grid in the same way as usually done using spectral methods in practice a series of doublings of the effective discretization in is implemented thus simulations can for instance be performed on a grid containing n 256 points with a time step 32 times larger with an accuracy comparable to that achieved in corresponding simulations involving fourier spectral methods in the direction

algebraic immunity of s boxes based on power mappings analysis and construction the algebraic immunity of an s box depends on the number and type of linearly independent multivariate equations it satisfies in this paper techniques are developed to find the number of linearly independent multivariate bi affine and quadratic equations for s boxes based on power mappings these techniques can be used to prove the exact number of equations for any class of power mappings two algorithms to calculate the number of bi affine and quadratic equations for any n n s box based on power mapping are also presented the time complexity of both algorithms is only o n 2 to design algebraically immune s boxes four new classes of s boxes that guarantee zero bi affine equations and one class of s boxes that guarantees zero quadratic equations are presented the algebraic immunity of power mappings based on kasami niho dobbertin gold welch and inverse exponents are discussed along with other cryptographic properties and several cryptographically strong s boxes are identified it is conjectured that a known kasami like highly nonlinear power mapping is differentially 4 uniform finally an open problem to find an n n bijective nonlinear s box with more than 5n quadratic equations is solved

computing with functions in spherical and polar geometries i the sphere a collection of algorithms is described for numerically computing with smooth functions defined on the unit sphere functions are approximated to essentially machine precision by using a structure preserving iterative variant of gaussian elimination together with the double fourier sphere method we show that this procedure allows for stable differentiation reduces the oversampling of functions near the poles and converges for certain analytic functions operations such as function evaluation differentiation and integration are particularly efficient and can be computed by essentially one dimensional algorithms a highlight is an optimal complexity direct solver for poisson s equation on the sphere using a spectral method without parallelization we solve poisson s equation with 100 million degrees of freedom in 1 minute on a standard laptop numerical results are presented throughout in a companion paper part ii we extend the ideas presented here to computing with functions on the disk

a finite difference scheme for three dimensional incompressible flows in cylindrical coordinates a finite difference scheme for the direct simulation of the incompressible time dependent three dimensional navier stokes equations in cylindrical coordinates is presented the equations in primitive variables vr v vzandp are solved by a fractional step method together with an approximate factorization technique cylindrical coordinates are singular at the axis the introduction of the radial fluxqr r vron a staggered grid simplifies the treatment of the region atr 0 the method is tested by comparing the evolution of a free vortex ring and its collision with a wall with the theory experiments and other numerical results the formation of a tripolar vortex where the highest vorticity is atr 0 is also considered finally to emphasize the accurate treatment near the axis the motion of a lamb dipole crossing the origin is simulated

efficient spectral galerkin methods iv spherical geometries fast spectral galerkin algorithms are developed for elliptic equations on the sphere the algorithms are based on a double fourier expansion and have quasi optimal optimal up to a logarithmic term computational complexity numerical experiments indicate that they are significantly more efficient and or accurate when compared with the algorithms based on spherical harmonics and on finite difference extensions to problems in spherical layers and to vector equations are also discussed

optimal ternary cyclic codes from monomials cyclic codes are a subclass of linear codes and have applications in consumer electronics data storage systems and communication systems as they have efficient encoding and decoding algorithms perfect nonlinear monomials were employed to construct optimal ternary cyclic codes with parameters 3m 1 3m 1 2m 4 by carlet ding and yuan in 2005 in this paper almost perfect nonlinear monomials and a number of other monomials over gf 3m are used to construct optimal ternary cyclic codes with the same parameters nine open problems on such codes are also presented

binary sequences with three valued cross correlations of different lengths in general it is hard to find the cross correlation distribution between two sequences in particular there are only few cases that the cross correlation is three valued in this paper new pairs of binary sequences with three cross correlation values are presented these two sequences have different lengths 2 2k 1 and 3 2 k 1 respectively we determine the cross correlation distribution between these two sequences we also give the possible correlation values for the shift decimated sequences the magnitude of cross correlation values is shown to be low the techniques presented in this paper may be useful to handle other similar cases

the eps method a new method for constructing pseudospectral derivative operators we develop a new type of derivative matrix for pseudospectral methods the norm of these matrices grows at the optimal rate o n 2 for n by n matrices in contrast to standard pseudospectral constructions that result in o n 4 growth of the norm the smaller norm has a big advantage when using the derivative matrix for solving time dependent problems such as wave propagation the construction is based on representing the derivative operator as an integral kernel and does not rely on the interpolating polynomials in particular we construct second derivative matrices that incorporate dirichlet or neumann boundary conditions on an interval and on the disk but the method can be used to construct a wide variety of commonly used operators for solving pdes and integral equations the construction can be used with any quadrature including traditional gauss legendre quadratures but we have found that by using quadratures based on prolate spheroidal wave functions we can achieve a near optimal sampling rate close to two points per wavelength even for non periodic problems we provide numerical results for the new construction and demonstrate that the construction achieves similar or better accuracy than traditional pseudospectral derivative matrices while resulting in a norm that is orders of magnitude smaller than the standard construction to demonstrate the advantage of the new construction we apply the method for solving the wave equation in constant and discontinuous media and for solving pdes on the unit disk we also present two compression algorithms for applying the derivative matrices in o nlogn operations

binary m sequences with three valued crosscorrelation a proof of welch s conjecture we prove the long standing conjecture of welch stating that for odd n 2m 1 the power function x sup d with d 2 sup m 3 is maximally nonlinear on gf 2 sup n or in other terms that the crosscorrelation function between a binary maximum length linear shift register sequence of degree n and a decimation of that sequence by 2 sup m 3 takes on precisely the three values 1 1 spl plusmn 2 sup m 1

a pseudospectral approach for polar and spherical geometries the singularities that arise in polar and spherical coordinates have in the past caused significant difficulties in obtaining accurate solutions to convection diffusion type problems in many fields astrophysics geophysics meteorology etc viewing pseudospectral methods as a limiting case of finite difference methods rather than as based on expansions in terms of orthogonal functions leads naturally to very simple yet highly effective fast fourier transform fft based pseudospectral methods in such geometries

spectral collocation schemes on the unit disc two spectral collocation schemes on the unit disc are presented the first one is based on the mapping of gordon and hall here the unit square is directly mapped onto the unit disc by means of an interpolation technique unlike other poisson solvers on the unit disc no polar coordinates are involved hence the usual problems with the singularity of polar coordinates are avoided this is also shown for more complex geometries the second method is based on a diameter approach where the collocation nodes are no more clustering in the center numerical results are presented which demonstrate the high accuracy of our new spectral collocation schemes

characterizations of the differential uniformity of vectorial functions by the walsh transform for every positive integers n and m and every even positive integer delta we derive inequalities satisfied by the walsh transforms of all vectorial n m functions and prove that the case of equality characterizes differential delta uniformity this provides a generalization to all differentially delta uniform functions of the characterization of almost perfect nonlinear apn functions due to chabaud and vaudenay by means of the fourth moment of the walsh transform such generalization has been missing since the introduction of the notion of differential uniformity by nyberg in 1994 and since chabaud vaudenay s result in the same year moreover for each even delta geq 2 we find several in fact an infinity of such characterizations in particular when delta 2 and delta 4 we have that for any n n function resp any n n 1 function the arithmetic mean of w f 2 u 1 v 1 w f 2 u 2 v 2 w f 2 u 1 u 2 v 1 v 2 when u 1 u 2 range independently over mathbb f 2 n and v 1 v 2 are nonzero and distinct and range independently over mathbb f 2 m is at least 2 3n and that f is apn resp is differentially 4 uniform if and only if this arithmetic mean equals 2 3n which is the value we would get with a bent function if such function could exist these inequalities give more knowledge on the walsh spectrum of n m functions we deduce in particular a property of the walsh support of highly nonlinear functions we also consider the completely open question of knowing if the nonlinearity of apn functions is necessarily non weak as it is the case for known apn functions we prove new lower bounds which cover all power apn functions and hence a large part of known apn functions which explain why their nonlinearities are not bad and we discuss the question of the nonlinearity of apn quadratic functions since almost all other known apn functions are quadratic

low degree planar polynomials over finite fields of characteristic two abstract planar functions are mappings from a finite field f q to itself with an extremal differential property such functions give rise to finite projective planes and other combinatorial objects there is a subtle difference between the definitions of these functions depending on the parity of q and we consider the case that q is even we classify polynomials of degree at most q 1 4 that induce planar functions on f q by showing that such polynomials are precisely those in which the degree of every monomial is a power of two as a corollary we obtain a complete classification of exceptional planar polynomials namely polynomials over f q that induce planar functions on infinitely many extensions of f q the proof strategy is to study the number of f q rational points of an algebraic curve attached to a putative planar function our methods also give a simple proof of a new partial result for the classification of almost perfect nonlinear functions