constructing differentially 4 uniform permutations over bbf _ 2 2k via the switching method many block ciphers use permutations defined on f 22k with low differential uniformity high nonlinearity and high algebraic degree as their s boxes to provide confusion it is well known that for a function on f 2n the lowest differential uniformity is 2 and the functions achieving this lower bound are called almost perfect nonlinear apn functions however due to the lack of knowledge on apn permutations on f 22k differentially 4 uniform permutations are usually chosen as s boxes for example the currently endorsed advanced encryption standard chooses one such function the multiplicative inverse function as its s box by a recent survey on differentially 4 uniform permutations over f 22k there are only five known infinite families of such functions and most of them have small algebraic degrees in this paper we apply the powerful switching method to discover many ccz inequivalent infinite families of such functions on f 22k with optimal algebraic degree where k is an arbitrary positive integer this greatly expands the list of differentially 4 uniform permutations and hence provide more choices for the s boxes furthermore lower bounds for the nonlinearity of the functions obtained in this paper are presented and they imply that some infinite families have high nonlinearity

on the connection between sets of operator synthesis and sets of spectral synthesis for locally compact groups we extend the results by froelich and spronk and turowska on the connection between operator synthesis and spectral synthesis for a g to second countable locally compact groups g this gives us another proof that one point subset of g is a set of spectral synthesis and that any closed subgroup is a set of local spectral synthesis furthermore we show that non triangular sets are strong operator ditkin sets and we establish a connection between operator ditkin sets and ditkin sets these results are applied to prove that any closed subgroup of g is a local ditkin set

reflexivity properties of t 0 a construction is given of a reflexive operator t acting on a separable hilbert space h with the property that the direct sum t 0 fails to be reflexive this construction is then used to provide solutions to several other problems which have been studied concerning the direct sum splitting of operator algebras scott brown s technique the theory of bitriangular operators and parareflexivity

compact operators invariant subspaces and spectral synthesis let l be a subspace lattice which contains a sequence pn of commuting projections such that for any subsequence pni vpni i and pni 0 then alg l k h 0 suppose g is a compact abelian group and e g is a closed set there is a family le of commutative subspace lattices for which alg le k h 0 precisely when e is a set of multiplicity in the sense of harmonic analysis by showing that the graph of is a set of uniqueness in 2 2 we obtain a thin set proof that alg 2 m12 contains no nonzero compact operators for an important class of sets k g gamin k amax k iff k is a set of spectral synthesis in the group g x g it follows that if e1and e2 are sets of spectral synthesis in g then alg pe1 bo alg pe2 alg pe1 pe2iff e1 e2 is a set of spectral synthesis in g x g and if semigroups 1 and 2 are sets of spectral synthesis in g then alg 1 bo alg 2 alg 1 2iff 1 2 is a set of spectral synthesis in g g the operator algebra alg p1 p2 is synthetic iff for all n alg p1 pn is synthetic this implies that the operator algebras alg 2 mp are synthetic

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

rank one subspaces of bimodules over maximal abelian selfadjoint algebras spaces of operators that are left and right modules over maximal abelian selfadjoint algebras masa bimodules for short are natural generalizations of algebras with commutative subspace lattices this paper is concerned with density properties of finite rank operators and of various classes of compact operators in such modules it is shown that every finite rank operator of a norm closed masa bimodule m is in the trace norm closure of the rank one subspace of m an important consequence is that the rank one subspace of a strongly reflexive masa bimodule that is one which is the reflexive hull of its rank one operators is dense in the module in the weak operator topology however in contrast to the situation for algebras it is shown that such density need not hold in the ultraweak topology a new method of representing masa bimodules is introduced this uses a novel concept of an topology with the appropriate notion of support a correspondence is established between reflexive masa bimodules and their supports it is shown that if a c2 closed masa bimodule contains a trace class operator then it must contain rank one operators indeed every such operator is in the c2 norm closure of the rank one subspace of the module consequently the weak closure of any masa bimodule of trace class operators is strongly reflexive however the trace norm closure of the rank one subspace need not contain all trace class operators of the module also it is shown that there exists a csl algebra which contains no trace class operators yet contains an operator belonging to cpfor allp 1 from this it follows that a transitive bimodule spanned by the rank one operators it contains need not be dense in cpfor 1 p as an application it is shown that there exists a commutative subspace lattice l such that l is non synthetic but every weakly closed algebra which contains a masa and has invariant lattice l coincides with alg l

computing affine equivalence classes of boolean functions by group isomorphism affine equivalence classification of boolean functions has significant applications in logic synthesis and cryptography previous studies for classification have been limited by the large set of boolean functions and the complex operations on the affine group although there are many research on affine equivalence classification for parts of boolean functions in recent years there are very few results for the entire set of boolean functions the best existing result has been achieved by harrison with 15768919 affine equivalence classes for 6 variable boolean functions this paper presents a concise formula for affine equivalence classification of the entire set of boolean functions as well as a formula for affine classification of boolean functions with distinct on set size respectively the method outlined in this paper greatly simplifies the affine group s action by constructing an isomorphism mapping from the affine group to a permutation group by this method we can compute the affine equivalence classes for up to 10 variables experiment results indicate that our scheme for calculating the affine equivalence classes for more than 6 variables is a significant advancement over previous published methods

operator synthesis and tensor products we show that kraus property s is preserved under taking weak closed sums with masa bimodules of finite width and establish an intersection formula for weak closed spans of tensor products one of whose terms is a masabimodule of finite width we initiate the study of the question of when operator synthesis is preserved under the formation of products and prove that the union of finitely many sets of the form where is a set of finite width while is operator synthetic is under a necessary restriction on the sets again operator synthetic we show that property s is preserved under spatial morita subordinance

the monotone smoothing of scatterplots we consider the problem of summarizing a scatterplot with a smooth monotone curve a solution that combines local averaging and isotonic regression is proposed and we demonstrate its use with two examples in the second example the procedure is applied in a regression setting to some data from box and cox 1964 and it is shown how this new procedure generalizes box and cox s well known family of transformations in the same example the bootstrap is applied to obtain a measure of the variability of the procedure

non parametric estimation of the odds ratios for continuous exposures using generalized additive models with an unknown link function the generalized additive model gam is a powerful and widely used tool that allows researchers to fit non parametrically the effect of continuous predictors on a transformation of the mean response variable such a transformation is given by a so called link function and in gams this link function is assumed to be known nevertheless if an incorrect choice is made for the link the resulting gam is misspecified and the results obtained may be misleading in this paper we propose a modified version of the local scoring algorithm that allows for the non parametric estimation of the link function by using local linear kernel smoothers to better understand the effect that each covariate produces on the outcome results are expressed in terms of the non parametric odds ratio or curves bootstrap techniques were used to correct the bias in the or estimation and to construct point wise confidence intervals a simulation study was carried out to assess the behaviour of the resulting estimates the proposed methodology was illustrated using data from the aids register of galicia nw spain with a view to assessing the effect of the cd4 lymphocyte count on the probability of being aids diagnosed via tuberculosis tb this application shows how the link s flexibility makes it possible to obtain or curve estimates that are less sensitive to the presence of outliers and unusual values that are often present in the extremes of the covariate distributions

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

reduced spectral synthesis and compact operator synthesis we introduce and study the notion of reduced spectral synthesis which unifies the concepts of spectral synthesis and uniqueness in locally compact groups we exhibit a number of examples and prove that every non discrete locally compact group with an open abelian subgroup has a subset that fails reduced spectral synthesis we introduce compact operator synthesis as an operator algebraic counterpart of this notion and link it to other exceptional sets in operator algebra theory studied previously we show that a closed subset e of a second countable locally compact group g satisfies reduced local spectral synthesis if and only if the subset e s t ts 1 is an element of e of g x g satisfies compact operator synthesis we apply our results to questions about the equivalence of linear operator equations with normal commuting coefficients on schatten p classes c 2020 elsevier inc all rights reserved

ranges of bimodule projections and reflexivity abstract we develop a general framework for reflexivity in dual banach spaces motivated by the question of when the weak closed linear span of two reflexive masa bimodules is automatically reflexive we establish an affirmative answer to this question in a number of cases by examining two new classes of masa bimodules defined in terms of ranges of masa bimodule projections we give a number of corollaries of our results concerning operator and spectral synthesis and show that the classes of masa bimodules we study are operator synthetic if and only if they are strong operator ditkin

a generalisation of dillon s apn permutation with the best known differential and nonlinear properties for all fields of size 2 4k 2 the existence of almost perfect nonlinear apn permutations operating on an even number of variables was a long standing open problem until an example with six variables was exhibited by dillon et al in 2009 however it is still unknown whether this example can be generalized to any even number of inputs in a recent work perrin et al described an infinite family of permutations named butterflies operating on 4k 2 variables and with differential uniformity at most 4 which contains the dillon apn permutation in this paper we generalize this family and we completely solve the two open problems raised by perrin et al indeed we prove that all functions in this larger family have the best known nonlinearity we also show that this family does not contain any apn permutation besides the dillon permutation implying that all other functions have differential uniformity exactly four

nonparametric estimation of a generalized additive model with an unknown link function mean of a random scalar y conditional on a random vector x e rd let h x e yix x denote the conditional mean function this paper is concerned with estimating h under weak assumptions about its functional form when x is a multidimensional continuous random variable the paper describes a model that nests single index additive and multiplicative models and that can be estimated with a faster rate of convergence in probability than can a fully nonparametric model nonparametric estimation of h is usually unattractive when x is multidimensional the rate of convergence of nonparametric estimators decreases rapidly as the dimension of x increases and nonparametric estimates can be very imprecise with samples of practical size when x is multidimensional the curse of dimensionality this problem can be overcome through the use of dimension reduction methods they reduce the effective dimension of the estimation problem by making assumptions about the functional form of h that are stronger than those of a fully nonparametric estimator but weaker than those of a finite dimensional parametric model one method of dimension reduction is semiparametric single index modeling the best known and most easily estimated semiparametric single index model assumes that 1 1 h x g 3 x where 3 is an unknown d x 1 vector and g is an unknown function the vector 3 can be estimated with an n 1 2 rate of convergence in probability which is the rate that would be obtained if g were known han 1987 hardle and stoker 1989 horowitz and hardle 1996 ichimura 1993 klein and spady 1993 newey and stoker 1993 powell stock and stoker 1989 sherman 1993 stoker 1991 g can be estimated with the rate of convergence n r 2r 1 where r is an integer that depends on the smoothness of g but not on d dimension reduction can also be achieved by using a nonparametric generalized additive model in such a model