monomial bent functions in this correspondence we focus on bent functions of the form f 2 n rarr f 2 where x rarr tr alphaxd the main contribution of this correspondence is that we prove that for n 4r r odd the exponent d 2r 1 2 allows the construction of bent functions this open question has been posed by canteaut based on computer experiments as a consequence for each of the well understood families of bent functions we now know an exponent d that yields to bent functions of the given type

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

nonlocal discrete regularization on weighted graphs a framework for image and manifold processing we introduce a nonlocal discrete regularization framework on weighted graphs of the arbitrary topologies for image and manifold processing the approach considers the problem as a variational one which consists of minimizing a weighted sum of two energy terms a regularization one that uses a discrete weighted dirichlet energy and an approximation one this is the discrete analogue of recent continuous euclidean nonlocal regularization functionals the proposed formulation leads to a family of simple and fast nonlinear processing methods based on the weighted laplace operator parameterized by the degree of regularity the graph structure and the graph weight function these discrete processing methods provide a graph based version of recently proposed semi local or nonlocal processing methods used in image and mesh processing such as the bilateral filter the tv digital filter or the nonlocal means filter it works with equal ease on regular 2 d and 3 d images manifolds or any data we illustrate the abilities of the approach by applying it to various types of images meshes manifolds and data represented as graphs

universal scaling behaviour for iterated maps in the complex plane according to the theory of schroder and siegel certain complex analytic maps possess a family of closed invariant curves in the complex plane we have made a numerical study of these curves by iterating the map and have found that the largest curve is a fractal when the winding number of the map is the golden mean the fractal curve has universal scaling properties and the scaling parameter differs from those found for other types of maps also for this winding number there are universal scaling functions which describe the behaviour asn of theq n th iterates of the map whereq n is thenth fibonacci number

critical behavior of a kam surface i empirical results kolmogorov arnol d moser kam surfaces are studied in the context of a perturbed two dimensional twist map in particular we ask how a kam surface can disappear as the perturbation parameter is increased following greene we use cycles to numerically construct the kam curve and discover that at the critical coupling it shows structure at all length scales aspects of this structure are fitted by a scaling analysis critical indices and scaling functions are determined numerically some evidence is presented which suggests that the results are universal

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

on the global avalanche characteristics between two boolean functions and the higher order nonlinearity the criterion for the global avalanche characteristics gac of cryptographic functions is an important property to measure the correlation between two arbitrary boolean functions we propose two new criteria called the sum of squares indicator and the absolute indicator of the cross correlation between two boolean functions the two indicators generalize the gac criterion based on the properties of the cross correlation function we deduce the rough lower and the rough upper bounds on the two indicators by hamming weights of two boolean functions and generalize some properties between the walsh spectrum and the cross correlation function furthermore we give the tight upper and the tight lower bounds on the two indicators finally we show some relationships between the upper bounds on the two indicators and the higher order nonlinearity

a complete characterization of the discrete p laplacian parabolic equations with q nonlocal reaction with respect to the blow up property abstract in this paper we consider the following discrete p laplacian nonlinear parabolic equations with q nonlocal reaction u t x t p u x t y s u y t q 1 u y t x t s 0 u x t 0 x t s 0 u x 0 u 0 0 x s here s is a network with boundary s the goal of this paper is to characterize completely the parameters p 1 and q 0 to see when the solution blows up vanishes or exists globally indeed the blow up rates for the blow up solutions are derived

secondary constructions of highly nonlinear boolean functions and disjoint spectra plateaued functions in this paper we modify a generalized indirect sum construction to construct functions with high nonlinearity by utilizing the modified construction highly nonlinear functions in n m variables can be obtained from known bent functions in n variables and highly nonlinear functions in m variables it is possible to obtain new n 15 variable functions with nonlinearity 2 n 1 5 1 2 n 1 5 1 2 20x2 n 2 and new 12 variable 2 resilient functions with nonlinearity 2000 and algebraic degree 8 which achieve optimal algebraic immunity moreover the modified construction can also be used as an iterative construction of a quadruple of disjoint spectra plateaued functions in addition we present sufficient conditions for a quadruple of disjoint spectra plateaued functions to have no nonzero linear structure

total blow up versus single point blow up traditional thermal explosion theory is used to describe reaction initiation in condensed explosives and is limited formally to nondeformable materials kassoy and poland 4 significantly extended this theory to develop an ignition model for a reactive gas in a bounded container in order to describe the induction period during this induction period there is a spatially homogeneous pressure rise in the system which causes a com pressive heating effect in the constant volume container mathematically this compressibility of the gas is expressed by means of an integral term in the induction model for the temperature perturbation 0 x t this model is given by e ak 1 j e x t dx y x 2 0 x 0 d x 2 0

generating and counting binary bent sequences two general classes of binary bent sequences bent based and linear based are introduced algorithms that allow easy generation of bent sequences from either class are given based on some simple computation and a computer search the authors conjecture a lower bound on the total number of binary bent sequences of a given order this lower bound is exact for bent sequences of order 16 a list is included from which all such sequences can be derived

the lower bounds on the second order nonlinearity of three classes of boolean functions with high nonlinearity the rth order nonlinearity of boolean functions is an important cryptographic criterion associated with some attacks on stream and block ciphers it is also very useful in coding theory since it is related to the covering radii of reed muller codes this paper tightens the lower bounds of the second order nonlinearity of three classes of boolean functions in the form f x tr x d in n variables where 1 d 2 m 1 3 and n 2m or 2 d 2 m 2 m 1 2 1 n 2m and m is odd or 3 d 2 2 r 2 r 1 1 and n 4r

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

renormalization group operators for maps and universal scaling of universal scaling exponents renormalization group rg methods provide a unifying framework for understanding critical behaviour such as transition to chaos in area preserving maps and other dynamical systems which have associated with them universal scaling exponents recently de la llave et al 2007 10 have formulated the principle of approximate combination of scaling exponents pacse for short which relates exponents for different criticalities via their combinatorial properties the main objective of this paper is to show that certain integrable fixed points of rg operators for area preserving maps do indeed follow the pacse

improved lower bound on the number of balanced symmetric functions over gf p the lower bound on the number of n variable balanced symmetric functions over finite fields gf p presented by cusick et al in t w cusick y li p sta nica balanced symmetric functions over gf p ieee trans inform theory 54 3 2008 1304 1307 is improved in this paper an equivalent characterization is also presented for the general case

a class of exceptional polynomials we present a class of indecomposable polynomials of non primepower degree over the finite field of two elements which are permutation polynomials on infinitely many finite extensions of the field the associated geometric monodromy groups are the simple groups where k 3 and odd the first member of this class was previously found by p muller 17 this realises one of only two possibilities for such a class which remain following deep work of fried guralnick and saxl 7 the other is associated with psl 3 k 3 and odd in fields of characteristic 3

the covering radius of the 2 15 16 reed muller code is at least 16276 two interesting cosets of the first order reed muller code of block length 2 15 are described they provide counterexamples to a conjecture on the covering radius

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

combination laws for scaling exponents and relation to the geometry of renormalization operators the principle of approximate combination of scaling exponents renormalization group has become a standard tool for describing universal properties of different routes to chaos period doubling in unimodal maps quasiperiodic transitions in circle maps dynamics on the boundaries of siegel disks destruction of invariant circles of area preserving twist maps and others the universal scaling exponents for each route are related to the properties of the corresponding renormalization operators

multiple positive solutions of singular and nonsingular discrete problems via variational methods we employ the critical point theory to establish the existence of multiple solutions of some regular as well as singular discrete boundary value problems 2004 elsevier ltd all rights reserved msc 34b16 39a10

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

recursive lower bounds on the nonlinearity profile of boolean functions and their applications the nonlinearity profile of a boolean function i e the sequence of its minimum hamming distances nlr f to all functions of degrees at most r for r ges 1 is a cryptographic criterion whose role against attacks on stream and block ciphers has been illustrated by many papers it plays also a role in coding theory since it is related to the covering radii of reed muller codes we introduce a method for lower bounding its values and we deduce bounds on the second order nonlinearity for several classes of cryptographic boolean functions including the welch and the multiplicative inverse functions used in the s boxes of the advanced encryption standard aes in the case of this last infinite class of functions we are able to bound the whole profile and we do it in an efficient way when the number of variables is not too small this allows showing the good behavior of this function with respect to this criterion as well

omega harmonic functions and inverse conductivity problems on networks in this paper we discuss the inverse problem of identifying the connectivity and the conductivity of the links between adjacent pair of nodes in a network in terms of an input output map to do t

renormalization group analysis of quasi periodicity in analytic maps analytic maps of the form f z e 2 pi i omega z mathcal o z 2 display quasiperiodicity when satisfies a diophantine condition quasiperiodic motion is confined to a neighborhood of the origin known as a siegel domain the boundary of this domain obeys universal scaling relations in this paper we investigate these scaling relations through a renormalization group analysis and we discuss singularities and asymptotic form of the scaling function

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

self similarity of siegel disks and hausdorff dimension of julia sets let f z ez z where is an irrational number of bounded type according to siegel f is linearizable on a disk containing the origin in this paper we show the hausdorff dimension of the julia set j f is strictly less than two and if is a quadratic irrational such as the golden mean then the siegel disk for f is self similar about the critical point in the latter case we also show the rescaled first return maps converge exponentially fast to a system of commuting branched coverings of the complex plane

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

global and blow up solutions to a p laplacian equation with nonlocal source this paper deals with a p laplacian equation ut div u p 2 u uq x t dx with null dirichlet boundary conditions in a bounded domain rn where p 2 q 1 under appropriate hypotheses we establish local theory of the solution and obtain that the solution either exists globally or blows up in finite time

positive solutions for discrete boundary value problems involving the p laplacian with potential terms in this paper we discuss the existence and the uniqueness of discrete boundary value problems involving the p laplacian with potential terms to this end we deal with the strong comparison principle and the dirichlet principle for the p laplacian with potential terms on weighted graphs with boundary moreover we provide a lower bound of the potential terms which guarantees the existence of solutions of the boundary value problem for the operator finally we also discuss inverse conductivity and potential problems for the operator

best affine and quadratic approximations of particular classes of boolean functions in this paper we consider the problem of computing best low order approximations of boolean functions we focus on the best quadratic approximations of a subclass of cubic functions with arbitrary number of variables and we provide formulas for their efficient calculation our methodology is developed upon properties of the best affine approximations of quadratic functions for which formulas for their direct computation not by means of the walsh hadamard transform are given we determine the cubic functions in the above subclass that achieve the maximum second order nonlinearity thus yielding a lower bound for the covering radius of the second order reed muller code ssr rm 2 n in ssr rm 3 n simple extensions of these results to some special cases of higher degree functions are seen to hold furthermore a preliminary analysis of well known constructions for bent functions in terms of their second order nonlinearity is performed that indicates potential weaknesses if construction parameters are not properly chosen

quasiperiodicity in dissipative systems a renormalization group analysis abstract dynamical systems with quasiperiodic behavior i e two incommensurate frequencies may be studied via discrete maps which show smooth continuous invariant curves with irrational winding number in this paper these curves are followed using renormalization group techniques which are applied to a one dimensional system circle and also to an area contracting map of an annulus two fixed points are found representing different types of universal behavior a trivial fixed point for smooth motion and a nontrivial fixed point the latter representsthe incipient breakup of a quasiperiodic motion with frequency ratio the golden mean into a more chaotic flow fixed point functions are determined numerically and via an e expansion and eigenvalues are calculated

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

fixed points of feigenbaum s type for the equationf p λx λf x existence and hyperbolicity of fixed points for the mapn p f x 1f p x withf p p fold iteration and f p 0 are given forp large these fixed points come close to being quadratic functions and our proof consists in controlling perturbation theory about quadratic functions

towards an existence proof of mackay s fixed point it is proved using a computer that there exist symmetric analytic twist maps u and t which satisfy the fixed point equations eqalign u btutb 1 crt btututb 1 cr here b is a diagonal 2 2 matrix if u and t commute then u t is either a fixed point or period three point for mackay s renormalization factor

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

uniform blow up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source in this paper we introduce a new method for investigating the rate and profile of blow up of solutions of diffusion equations with nonlocal nonlinear reaction terms for large classes of equations we prove that the solutions have global blow up and that the rate of blow up is uniform in all compact subsets of the domain this results in a flat blow up profile except for a boundary layer whose thickness vanishes astapproaches the blow up timet in each case the blow up rate of u t is precisely determined furthermore in many cases we derive sharp estimates on the size of the boundary layer and on the asymptotic behavior of the solution in the boundary layer the size of the boundary layer then decays liket t and the solutionu t x behaves like u t d x t tin the boundary layer wheredis the distance to the boundary some fujita type critical exponents results are also given for the cauchy problem

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

universal scalings of universal scaling exponents in the last decades renormalization group rg ideas have been applied to describe universal properties of different routes to chaos quasi periodic period doubling or tripling siegel disc boundaries etc each of the rg theories leads to universal scaling exponents which are related to the action of certain rg operators the goal of this announcement is to show that there is a principle that organizes many of these scaling exponents we give numerical evidence that the exponents of different routes to chaos satisfy approximately some arithmetic relations these relations are determined by combinatorial properties of the route and become exact in an appropriate limit

renormalization for golden circles we prove that twist maps of the cylinder that are attracted by any fixed point of mackay s renormalization operator have a transitive invariant golden circle provided the fixed point satisfies a few simple purely topological conditions these conditions can be verified by finite precision arithmetics they are fulfilled for the simple fixed point and seem to be fulfilled for the critical fixed point taking existence and hyperbolicity of the critical fixed point for granted we conclude that the standard map has a critical invariant golden circle the induced map on the circle is topologically conjugate to a rigid rotation we can show that the conjugator is holder continuous moreover it is not differentiable on a dense set of points

on the lower bounds of the second order nonlinearities of some boolean functions the rth order nonlinearity of a boolean function is an important cryptographic criterion in analyzing the security of stream as well as block ciphers it is also important in coding theory as it is related to the covering radius of the reed muller code r r n in this paper we deduce the lower bounds of the second order nonlinearities of the following two types of boolean functions 1 f l x tr 1 n lx d with d 2 2 r 2 r 1 and l f 2 n where n 6r 2 f x y tr 1 t xy 2 i 1 where x y f 2 t n 2t n 6 and i is an integer such that 1

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

on cryptographic properties of the cosets of r 1 m we introduce a new approach for the study of weight distributions of cosets of the reed muller code of order 1 our approach is based on the method introduced by kasami 1968 using pless 1963 identities by interpreting some equations we obtain a necessary condition for a coset to have a high minimum weight most notably we are able to distinguish such cosets which have three weights only we then apply our results to the problem of the nonlinearity of boolean functions we particularly study the links between this criterion and the propagation characteristics of a function

a renormalization approach to invariant circles in area preserving maps abstract kadanoff and shenker introduced a renormalisation approach to invariant circles in area preserving maps this paper makes more precise the connection between invariant circles and the renormalisation operator restricting attention to noble rotation numbers the stability of a simple fixed point of the renormalisation is analysed corresponding to a linear twist map it is found to be essentially attracting so that noble circles persist under perturbation giving a new view on kam theory shenker and kadanoff found evidence for another fixed point corresponding to a map with a non smooth noble circle further evidence is given in this paper it has essentially only one unstable direction and its stable manifold is believed to give the boundary of the set of twist maps with a noble circle finally noble circles are shown to be locally most robust in an important sense

the blowup property of solutions to some diffusion equations with localized nonlinear reactions abstract in this paper we investigate the blowup property of solutions to the equation u t u u x 0 t where x0 is a fixed point in the domain we show that under certain conditions the solution blows up in finite time moreover we prove that the set of all blowup points is the whole region furthermore the growth rate of solutions near the blowup time is also derived finally the results are generalized to the following nonlocal reaction diffusion equation u t u u dx

on the existence of positive solutions of p laplacian difference equations in this paper by means of fixed point theorem in a cone the existence of positive solutions of p laplacian difference equations is considered

on the second order nonlinearities of some bent functions the rth order nonlinearity of boolean functions plays a central role against several known attacks on stream and block ciphers it plays also an important role in coding theory since its maximum equals the covering radius of the rth order reed muller code but it is difficult to calculate and even to bound in this paper we show lower bounds on the second order nonlinearity of two subclasses of well known bent functions we first improve a known lower bound on the second order nonlinearity of the simplest partial spread bent functions whose nonlinearity profile has been bounded by the second author this improvement allows obtaining a better bound for the whole profile we subsequently give a lower bound on the second order nonlinearity of some infinite class of maiorana mcfarland m m bent functions which generalizes a result by gangopadhyay et al

improving the lower bound on the maximum nonlinearity of 1 resilient boolean functions and designing functions satisfying all cryptographic criteria in this article we improve the lower bound on the maximum nonlinearity of 1 resilient boolean functions for n even by proposing a method of constructing this class of functions attaining the best nonlinearity currently known thus for the first time at least for small values of n the upper bound on nonlinearity can be reached in a deterministic manner in difference to some heuristic search methods proposed previously the nonlinearity of these functions is extremely close to the maximum nonlinearity attained by bent functions and it might be the case that this is the highest possible nonlinearity of 1 resilient functions apart from this theoretical contribution it turns out that the cryptographic properties of these functions are overall good apart from their moderate resistance to fast algebraic attacks faa this weakness is repaired by a suitable modification of the original functions giving a class of balanced functions with almost optimal resistance to faa whose nonlinearity is better than the nonlinearity of other methods

improving the lower bound on the higher order nonlinearity of boolean functions with prescribed algebraic immunity the recent algebraic attacks have received a lot of attention in cryptographic literature the algebraic immunity of a boolean function quantifies its resistance to the standard algebraic attacks of the pseudorandom generators using it as a nonlinear filtering or combining function very few results have been found concerning its relation with the other cryptographic parameters or with the rth order nonlinearity as recalled by carlet at crypto 06 many papers have illustrated the importance of the r th order nonlinearity profile which includes the first order nonlinearity the role of this parameter relatively to the currently known attacks has been also shown for block ciphers recently two lower bounds involving the algebraic immunity on the rth order nonlinearity have been shown by carlet none of them improves upon the other one in all situations in this paper we prove a new lower bound on the rth order nonlinearity profile of boolean functions given their algebraic immunity that improves significantly upon one of these lower bounds for all orders and upon the other one for low orders

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

improving the upper bounds on the covering radii of binary reed muller codes by deriving bounds on character sums of boolean functions and by using the characterizations due to kasami of those elements of the reed muller codes whose hamming weights are smaller than twice and a half the minimum distance we derive an improved upper bound on the covering radius of the reed muller code of order 2 and we deduce improved upper bounds on the covering radii of the reed muller codes of higher orders

algebraic immunity for cryptographically significant boolean functions analysis and construction recently algebraic attacks have received a lot of attention in the cryptographic literature it has been observed that a boolean function f used as a cryptographic primitive and interpreted as a multivariate polynomial over f sub 2 should not have low degree multiples obtained by multiplication with low degree nonzero functions in this paper we show that a boolean function having low nonlinearity is also weak against algebraic attacks and we extend this result to higher order nonlinearities next we present enumeration results on linearly independent annihilators we also study certain classes of highly nonlinear resilient boolean functions for their algebraic immunity we identify that functions having low degree subfunctions are weak in terms of algebraic immunity and we analyze some existing constructions from this viewpoint further we present a construction method to generate boolean functions on n variables with highest possible algebraic immunity spl lceil n 2 spl rceil this construction first presented at the 2005 workshop on fast software encryption fse 2005 has been the first one producing such functions these functions are obtained through a doubly indexed recursive relation we calculate their hamming weights and deduce their nonlinearities we show that they have very high algebraic degrees we express them as the sums of two functions which can be obtained from simple symmetric functions by a transformation which can be implemented with an algorithm whose complexity is linear in the number of variables we deduce a very fast way of computing the output to these functions given their input

universal small scale structure near the boundary of siegel disks of arbitrary rotation number abstract we present numerical evidence that for analytic maps of the complex plane with a siegel disk of any irrational rotation number and a quadratic critical point on its boundary the small scale structure near the critical point can be described asymptotically by a universal two parameter family we propose an explanation in terms of an ergodic attractor for a renormalisation operator