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

embedding problem of fuzzy number space part ii abstract using a concrete structure into which we embed the fuzzy number space e1 several necessary and sufficient conditions of fuzzy set valued functions are given by means of abstract function theory

existence of large solutions for a quasilinear elliptic problem via explosive sub supersolutions abstract we consider the boundary blow up quasilinear elliptic problems div u m 2 u u q m 1 k x g u in a c 2 bounded domain with boundary condition u where m 1 q 0 m m 1 and 0 under suitable growth assumptions on k near the boundary and on g both at zero and at infinity we show the existence of at least one solution in c 1 our proof is based on the method of explosive sub supersolutions which permits positive weights k x which are unbounded and or oscillatory near the boundary

the weight distribution of a class of linear codes from perfect nonlinear functions in this correspondence the weight distribution of a class of linear codes based on perfect nonlinear functions also called planar functions is determined the class of linear codes under study are either optimal or among the best codes known and have nice applications in cryptography

boundary blow up solutions to elliptic systems of competitive type abstract we consider the elliptic system u u p v q v u r v s in where p s 1 q r 0 and r n is a smooth bounded domain subject to different types of dirichlet boundary conditions f u v i u v and sf u v on where 0 under several hypotheses on the parameters p q r s we show existence and nonexistence of positive solutions uniqueness and nonuniqueness we further provide the exact asymptotic behaviour of the solutions and their normal derivatives near some more general related problems are also studied

a new fuzzy compactness defined by fuzzy nets compactness is one of the most important notions in topology and is thoroughly investigated in general topology but it seems that the research about this notion in the area of fuzzy topology is not so effective as in general topology in fact it would seem that there was not even a good definition of compactness in this area until recently indeed soon after the famous article written by zadeh 8 was published chang 1 introduced the notion of fuzzy topological spaces and at the same time a certain fuzzy compactness was given from then on a series of different notions of fuzzy compactness have been discussed 2 3 71 and in 5 they were compared in detail but it seems unsatisfactory that all these fuzzy compactness ideas are defined for the whole fuzzy topological space but not for an arbitrary fuzzy subset therefore they are far from the crisp compactness for example it is impossible to discuss the hereditary property with respect to closed subsets of fuzzy compactness in 1977 pu and liu introduced the notion of fuzzy nets and a new concept of so called q neighborhood was given which could reflect the features of neighborhood structure in fuzzy topological spaces and by this new neighborhood structure the moore smith convergence theory was established spendidly 6 it furnishes us a basic tool to investigate the fuzzy compactness in a more general way to investigate the compactness of arbitrary fuzzy subsets in this paper a new fuzzy compactness defined by fuzzy nets we shall call it n compactness is given but we introduce the notion of r neighborhood see section 2 instead of qneighborhood to establish the moore smith convergence theory for it is perhaps more convennient to use n compactness possesses the following properties 1 n compactness is defined for arbitrary fuzzy subsets 2 for n compactness the hereditary property with respect to closed subsets holds 3 n compactness is a good extension 5 4 the continuous images of ncompact sets are n compact 5 for n compactness the tychonoff product theorem holds 6 if an n compact fuzzy topological space is t2 6 then it will satisfy a stronger separation axiom which will be called t4 in this paper 1 0022 247x 83 3 00

the expressional theorem of a fuzzy unit interval and its application the expressional theorems of a fuzzy unit interval by open intervals and by functions are obtained and by using them the fuzzy valued probability measure is introduced

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

boundary blow up for differential equations with indefinite weight we obtain results of existence and multiplicity of solutions for the second order equation x q t g x 0 with x t defined for all tis an element of 0 1 and such that x t infinity as t 0 and t 1 we assume g having superlinear growth at infinity and q t possibly changing sign on 0 1 c 2002 elsevier science usa all rights reserved

boundary asymptotic and uniqueness of solutions to the p laplacian with infinite boundary values we prove boundary asymptotics to solutions of weighted p laplacian equations that take infinite value on the boundary of a bounded domain uniqueness of such solutions would then follow as a consequence our results extend previously known results by allowing weights that are unbounded in the domain

five families of three weight ternary cyclic codes and their duals as a subclass of linear codes cyclic codes have applications in consumer electronics data storage systems and communication systems as they have efficient encoding and decoding algorithms in this paper five families of three weight ternary cyclic codes whose duals have two zeros are presented the weight distributions of the five families of cyclic codes are settled the duals of two families of the cyclic codes are optimal

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

boundary blow up elliptic problems with nonlinear gradient terms by karamata regular variation theory and perturbation method we show the exact asymptotical behaviour of solutions near the boundary to nonlinear elliptic problems u u q b x g u u 0 in u where is a bounded domain with smooth boundary in rn q 0 g c1 0 g 0 0 g is regularly varying at infinity with index with 0 and b is nonnegative nontrivial in which may be vanishing on the boundary

new families of codebooks achieving the levenstein bound in this paper a construction of codebooks based on a set of bent functions satisfying certain conditions is introduced it includes some earlier constructions of codebooks meeting the levenstein bound as special cases with this construction two new families of codebooks achieving the levenstein bound are obtained the codebooks constructed in this paper could have a very small alphabet size

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

the asymptotic behaviour of solutions with boundary blow up for semilinear elliptic equations with nonlinear gradient terms abstract in this paper we show existence uniqueness and exact asymptotic behaviour of solutions near the boundary to nonlinear elliptic problems u u q k x g u x u where is a bounded domain with smooth boundary in r n 0 q 0 2 lim d x 0 k x d x c 0 0 2 c 0 0 g c 1 0 g 0 0 g is increasing on 0 and is karamata regular variation function at infinity with index 1 and 0

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

optimal uniqueness theorems and exact blow up rates of large solutions in this paper we prove some optimal uniqueness results for large solutions of a canonical class of semilinear equations under minimal regularity conditions on the weight function in front of the non linearity and combine these results with the localization method introduced in lopez gomez the boundary blow up rate of large solutions j differential equations 195 2003 25 45 to prove that any large solution l of u a x up p 1 a 0 must satisfy limx x0l x fx0 dist x i0 pp 1p 1p 1p 1p 1foreachx0 where fx0 t t 0sfx01p 1 p 1p 1ds i0 limt 0fx0 t fx0 t fx0 t 2 and fx0 is any smooth extension of the boundary normal section of a at x0 i e fx0 t a x0 tnx0 t 0 t 0 subsequently nx0 stands for the outward unit normal at x0 therefore the theory can be extended to cover the general case when fx0 l1p 1

general uniqueness results and variation speed for blow up solutions of elliptic equations let omega be a smooth bounded domain in r n we prove general uniqueness results for equations of the form delta u au b x f u in omega subject to u infty on partial omega our uniqueness theorem is established in a setting involving karamata s theory on regularly varying functions which is used to relate the blow up behavior of u x with f u and b x where b equiv 0 on partial omega and a certain ratio involving b is bounded near partial omega a key step in our proof of uniqueness uses a modification of an iteration technique due to safonov

large solutions for an elliptic equation with a nonhomogeneous term abstract in this paper we study the elliptic boundary blow up problem u u p 1 u h x in where p 1 is a smooth bounded domain of r n and h c we are concerned with existence and uniqueness of solutions for quite general functions h which may change sign and are not necessarily bounded near

boundary blow up and degenerate equations let r2 be a bounded domain of class c2 0 1 we show that if u is the solution of u 4exp 2u which tends to as x y then the hyperbolic radius v exp u is also of class c2 up to the boundary the proof relies on new schauder estimates for degenerate elliptic equations of fuchsian type

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

the boundary trace and generalized boundary value problem for semilinear elliptic equations with coercive absorption in the first part of the paper we establish the existence of a boundary trace for positive solutions of the equation u g x u 0 in a smooth domain n for a general class of positive nonlinearities this class includes every space independent monotone increasing g which satisfies the keller osserman condition as well as degenerate nonlinearities g q of the form g q x u d x u q 1u with 2 and q 1 the boundary trace is given by a positive regular borel measure which may blow up on compact sets in the second part we concentrate on the family of nonlinearities g q determine the critical value of the exponent q for fixed 2 and discuss a positive solutions with an isolated singularity for subcritical nonlinearities and b the boundary value problem for u g q x u 0 with boundary data given by a positive regular borel measure possibly unbounded we show that in the subcritical case the problem possesses a unique solution for every such measure 2003 wiley periodicals inc

a necessary and sufficient condition for existence of large solutions to semilinear elliptic equations we consider the semilinear equation u p x f u on a domain rn n 3 where f is a nonnegative nondecreasing continuous function which vanishes at the origin and p is a nonnegative continuous function with the property that any zero of p is contained in a bounded domain in such that p is positive on its boundary for bounded we show that a nonnegative solution u satisfying u x as x exists if and only if the function s s0 f t dt satisfies 1 s 1 2 ds for unbounded including rn we show that a similar result holds where u x as x within and u x as x if p x decays to zero rapidly as x

nondegeneracy and uniqueness for boundary blow up elliptic problems abstract in this paper we use for the first time linearization techniques to deal with boundary blow up elliptic problems after introducing a convenient functional setting we show that the problem u a x u p g x u in with u on has a unique positive solution for large enough and determine its asymptotic behavior as here p 1 a x is a continuous function which can be singular near and g x u is a perturbation term with potential growth near zero and infinity we also consider more general problems obtained by replacing u p by e u or a logistic type function f u

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

a boundary blow up elliptic problem with an inhomogeneous term abstract by a perturbation method and constructing comparison functions we reveal how the inhomogeneous term h affects the exact asymptotic behaviour of solutions near the boundary to the problem u b x g u h x u 0 in u where is a bounded domain with smooth boundary in r n 0 g c 1 0 is increasing on 0 g 0 0 g is regularly varying at infinity with positive index the weight b which is non trivial and non negative in may be vanishing on the boundary and the inhomogeneous term h is non negative in and may be singular on the boundary

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

boundary blow up elliptic problems of bieberbach and rademacher type with nonlinear gradient terms abstract by a perturbation method and constructing comparison functions we show the exact asymptotic behaviour of solutions near the boundary to nonlinear elliptic problems u u q b x e u x u where is a bounded domain with smooth boundary in r n q 0 b is non negative and non trivial in which may be vanishing on the boundary

linear codes from perfect nonlinear mappings and their secret sharing schemes in this paper error correcting codes from perfect nonlinear mappings are constructed and then employed to construct secret sharing schemes the error correcting codes obtained in this paper are very good in general and many of them are optimal or almost optimal the secret sharing schemes obtained in this paper have two types of access structures the first type is democratic in the sense that every participant is involved in the same number of minimal access sets in the second type of access structures there are a few dictators who are in every minimal access set while each of the remaining participants is in the same number of minimal access sets

the asymptotic behaviour of solutions with blow up at the boundary for semilinear elliptic problems abstract by constructing the comparison functions and the perturbed method it is showed that any solution u c 2 to the semilinear elliptic problems u k x g u x u satisfies lim d x 0 u x z d x 2 2 2 c 0 2 1 where is a bounded domain with smooth boundary in r n lim d x 0 k x d x c 0 2 c 0 0 2 2 g c 1 0 g 0 and g s s is increasing on 0 there exists 0 such that lim s g s g s 0 z s d t 2 g t s g t 0 t g s d s

second order expansion for blowup solutions of semilinear elliptic problems abstract in this paper we mainly study the second order expansion of solutions to the elliptic problems u b x f u u 0 x u where is a bounded domain with a smooth boundary in r n n 2 b c which is positive in and may be vanishing on the boundary and f is normalized regularly varying at infinity with index p 1 our analysis is based on the sub supersolution method and karamata regular variation theory

on fuzzy number valued fuzzy measures defined by fuzzy number valued fuzzy integrals ii abstract in this paper we discuss some properties of a fuzzy number valued fuzzy set function defined by a fuzzy number valued fuzzy integral on the fuzzy set and give sufficient conditions that it is a fuzzy number valued fuzzy measure