w2 p estimates for fully nonlinear elliptic equations with oblique boundary conditions abstract we study fully nonlinear elliptic equations with oblique boundary conditions we obtain a global w 2 p estimate n 0 p for viscosity solutions of such problems when the boundary of the domain is in c 2 for every 0 1

affine equivalence of quartic homogeneous rotation symmetric boolean functions homogeneous rotation symmetric boolean functions have been extensively studied in recent years because of their applications in cryptography little is known about the basic question of when two such functions in n variables are affine equivalent the simplest case of quadratic rotation symmetric functions which are generated by cyclic permutations of the variables in a single monomial was only settled in 2009 and the first substantial progress on the much more complicated cubic case came in 2010 in this paper we show that much of the work on the cubic case can be extended to the quartic case we also prove an exact formula for the number and sizes of the affine equivalence classes when n is a prime

large sets of disjoint spectra plateaued functions inequivalent to partially linear functions in this paper we give an efficient method for constructing a large set of disjoint spectra functions without linear structures which are not equivalent to partially linear functions this positively answers the open problem how to construct a large set of disjoint spectra functions which are not linearly equivalent to partially linear functions raised by zhang and xiao at the same time this significantly extends a recent result of zhang where a method of specifying four disjoint spectra functions was given it is demonstrated that such sets can be utilized in the design of highly nonlinear resilient functions in the second part based on a generalization of the indirect sum method we give an alternative approach for designing sets of disjoint spectra functions of even larger cardinality than already given ones but these functions then admit linear structures in addition it is shown that using suitable initial functions in the generalized indirect sum method we can specify highly nonlinear resilient boolean functions in odd number of input variables n whose nonlinearity in many cases exceeds the current best known values moreover we design some balanced functions for odd n that also achieve the highest nonlinearity known

whitham equation with landau damping on a half line abstract we consider the initial boundary value problem on a half line for the evolution equation t r 1 2 x 2 k u x t u x u where r 1 2 sin 2 0 y x y 1 d y is the modified riesz potential and k u 1 2 i x i i e p x p tanh p p u p t u 0 t p d p where x is the heaviside step function we study traditionally important problems of the theory of nonlinear partial differential equations such as global in time existence of solutions to the initial boundary value problem and the asymptotic behavior of solutions for large time

on solutions of the initial value problem for the nonlinear schrodinger equations on considere le probleme aux valeurs initiales pour l equation de schrodinger non lineaire de la forme iu t u f u 2 u t x r r n u 0 x x x r n on etudie la regularite des solutions quand les fonctions initiales ne sont pas regulieres

nonlinear oblique boundary value problems for nonlinear elliptic equations on considere le probleme aux valeurs limites a derivee oblique non lineaire pour des equations uniformement elliptiques quasi lineaires et totalement non lineaires d ordre 2 les operateurs elliptiques satisfont des conditions de structure naturelle on donne des theoremes d existence pour les solutions

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

a neumann problem for the kdv equation with landau damping on a half line abstract we consider the initial boundary value problem on a half line for the kdv equation with landau damping we study traditionally important problems of the theory of nonlinear partial differential equations such as global in time existence of solutions to the initial boundary value problem and the asymptotic behavior of solutions for large time

pseudodifferential operator with a nonanalytic symbol on a half line we study the initial boundary value problem for the nonlinear pseudodifferential equations with nonanalytic symbol we study traditionally important problems of a theory of nonlinear partial differential equations such as global in time existence of solutions to the initial boundary value problem and the asymptotic behavior of solutions for large time

patterson wiedemann type functions on 21 variables with nonlinearity greater than bent concatenation bound nonlinearity is one of the most challenging combinatorial property in the domain of boolean function research obtaining nonlinearity greater than the bent concatenation bound for odd number of variables continues to be one of the most sought after combinatorial research problems the pioneering result in this direction has been discovered by patterson and wiedemann in 1983 ieee it which considered boolean functions on 5 times 3 15 variables that are invariant under the actions of the cyclic group gf 2 5 ast cdot gf 2 3 ast as well as the group of frobenius automorphisms some of these boolean functions possess nonlinearity greater than the bent concatenation bound the next possible option for exploring such functions is on 7 times 3 21 variables however obtaining such functions remained elusive for more than three decades even after substantial efforts as evident in the literature in this paper we exploit combinatorial arguments together with heuristic search to demonstrate such functions for the first time

on the covering radius of first order generalized reed muller codes we generalize to any finite fq fields a theorem about covering radius of codes of strength 2 proved by helleseth and coworkers then using this result and partial covering radius we give bounds for the covering radius of first order generalized reed muller codes finally using magma we get some improvements for f3

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

neumann initial boundary value problem for benjamin ono equation abstract we consider the neumann initial boundary value problem for benjamin ono equation on a half line we study traditionally important problems of the theory of nonlinear partial differential equations such as global in time existence of solutions to the initial boundary value problem and the asymptotic behavior of solutions for large time

hessian estimates in weighted lebesgue spaces for fully nonlinear elliptic equations abstract we prove global regularity in weighted lebesgue spaces for the viscosity solutions to the dirichlet problem for fully nonlinear elliptic equations as a consequence regularity in morrey spaces of the hessian is derived as well

weighted regularity estimates in orlicz spaces for fully nonlinear elliptic equations abstract in this paper we develop a global w 2 p estimate for the viscosity solution of the dirichlet problem of fully nonlinear elliptic equations f d 2 u d u u x f x in u 0 on to a more general function space given an n function and a muckenhoupt weight w we prove that if f belongs to the associated weighted orlicz space l w then d 2 u l w and u satisfies a global w w 2 estimate under a minimal regularity requirement on f in the variable x and a basic geometric assumption on the correct condition on the couple and w is also addressed this result generalizes the w 2 p estimate caffarelli 1989 escauriaza 1993 winter 2009 of calderon and zygmund as well as an analogous one byun et al 2016 in the weighted l p setting

generalized lexicographic multiobjective combinatorial optimization application to cryptography this paper formalizes a family of prioritized multicriteria optimization problems and assesses the corresponding up to date known suboptimal solutions the resulting framework is then employed to characterize and search for boolean functions which are valuable for a robust symmetric mainly block cipher design the proposed optimality definitions generalize the lexicographic method by establishing an ordered sequence of multiobjective combinatorial optimization problems which in turn gathers the relative relevance of the criteria so that the optimal solutions can be obtained from a sequential application of the pareto efficiency the relationship among the different formulable problems is characterized in terms of both their respective solutions sets and computing costs since in practice only a limited set of functions can be evaluated i e are known the best known pareto efficient functions are also defined finally this framework is employed to obtain new functions having known pareto max

systematic methods of constructing bent functions and 2 rotation symmetric bent functions in this paper we first present two systematic constructions of bent functions by modifying the truth tables of rothaus s bent function and maiorana mcfarland s bent function respectively the number of the newly constructed bent functions by modifying the truth table of rothaus s bent function is also determined the methods of constructing self dual bent functions are then given after the dual functions of these bent functions being determined finally as an application two constructions of 2 rotation symmetric bent functions are presented in this paper

regularity for fully nonlinear elliptic equations with oblique boundary conditions in this paper we obtain a series of regularity results for viscosity solutions of fully nonlinear elliptic equations with oblique derivative boundary conditions in particular we derive the pointwise c c 1 and c 2 regularity as byproducts we also prove the a b p maximum principle harnack inequality uniqueness and solvability of the equations

time asymptotics for solutions of the burgers equation with a periodic force we consider the burgers equation with a periodic force frac partial partial t u u cdot nabla u frac12 delta u nabla v x which presents a simplified model for turbulence we are interested in the asymptotic behaviour of solutions for t to infty this problem has been studied by sinai who uses a probabilistic and very technical approach using methods from spectral theory we get similar results this functional analytic approach gives an easier proof for certain initial data periodic or some random perturbations of those we show time convergence towards a deterministic periodic limit solution related to the ground state of a certain schrodinger operator

search for boolean functions with excellent profiles in the rotation symmetric class for the first time boolean functions on 9 variables having nonlinearity 241 are discovered that remained as an open question in literature for almost three decades such functions are found by heuristic search in the space of rotation symmetric boolean functions rsbfs this shows that there exist boolean functions on n odd variables having nonlinearity 2n 1 2n 1 2 if and only if n 7 using similar search technique balanced boolean functions on 9 10 and 11 variables are attained having autocorrelation spectra with maximum absolute value 2lceiln 2rceil on odd number of variables earlier such functions were known for 15 21 variables there was no evidence of such functions at all on even number of variables in certain cases our functions can be affinely transformed to obtain first order resiliency or first order propagation characteristics moreover 10 variable functions having first order resiliency and nonlinearity 492 are presented that had been posed as an open question at crypto 2000 the functions reported in this paper are discovered using a suitably modified steepest descent based iterative heuristic search in the rsbf class along with proper affine transformations it seems elusive to get a construction technique to match such functions

nonlinear evolution equations with a fractional derivative on a half line abstract we study the initial boundary value problem on a half line for the nonlinear evolution equations with a fractional derivative we study traditionally important problems of a theory of nonlinear partial differential equations such as global in time existence of solutions to the initial boundary value problem and the asymptotic behavior of solutions for large time

systematic constructions of rotation symmetric bent functions 2 rotation symmetric bent functions and bent idempotent functions rotation symmetric bent functions and their generation two rotation symmetric bent functions are two classes of cryptographically significant boolean functions however few constructions have been presented in the literature which either have restriction on integer n or have algebraic degree no more than 4 in this paper for any even integer n ge 4 three classes of bent functions are presented respectively most notably the proposed n variable rotation symmetric bent functions and two rotation symmetric bent functions can have any possible algebraic degree ranging from 2 to n 2 besides we obtain bent idempotent functions with the maximal algebraic degree n 2

9 variable boolean functions with nonlinearity 242 in the generalized rotation symmetric class we give a new lower bound to the covering radius of the first order reed muller code rm 1 n where n 9 11 13 equivalently we present the n variable boolean functions for n 9 11 13 with maximum nonlinearity found till now in 2006 9 variable boolean functions having nonlinearity 241 which is strictly greater than the bent concatenation bound of 240 have been discovered in the class of rotation symmetric boolean functions rsbfs by kavut maitra and yucel to improve this nonlinearity result we have firstly defined some subsets of the n variable boolean functions as the generalized classes of k rsbfs and k dsbfs k dihedral symmetric boolean functions where k is a positive integer dividing n secondly utilizing a steepest descent like iterative heuristic search algorithm we have found 9 variable boolean functions with nonlinearity 242 within the classes of both 3 rsbfs and 3 dsbfs thirdly motivated by the fact that rsbfs are invariant under a special permutation of the input vector we have classified all possible permutations up to the linear equivalence of boolean functions that are invariant under those permutations

equivalence of 2 rotation symmetric quartic boolean functions abstract a boolean function in n variables is 2 rotation symmetric if it is invariant under even powers of x 1 x n x 2 x n x 1 but not under the first power ordinary rotation symmetry we call such a function a 2 function a 2 function is called monomial rotation symmetric mrs if it is generated by applying powers of 2 to a single monomial if the quartic mrs 2 function in 2n variables has a monomial x1xqxrxs then we use the notation 2 1 q r s 2n for the function this paper gives a detailed theory of equivalence of quartic mrs 2 functions in 2n variables such a theory was provided for the cubic mrs 2 functions in two 2015 papers of cusick and johns as in the earlier papers the two main topics in the theory are describing the affine equivalence classes of the functions under certain groups of permutations and giving details of the linear recursions that the hamming weights of any sequence of functions 2 1 q r s 2n with q n s s 1 can be shown to satisfy the discussion for both of these topics uses new ideas because the quartic theory naturally divides into two cases

solvability of quasilinear elliptic equations with nonlinear boundary conditions ii abstract it is shown that solvability of the second order quasilinear elliptic equation qu 0 in with first order nonlinear boundary condition nu 0 on can be inferred from appropriate estimates on solutions of the problem qu in nu 0 on as varies over a suitable function class this result improves previous work of the author where estimates were required on solutions of qu in nu on as varies over some function space the value of this improvement is demonstrated by some examples

on viscosity solutions of fully nonlinear equations with measurable ingredients we study fully nonlinear uniformly elliptic equations with measurable ingredients caffarelli s recent work on w2 p estimates for viscosity solutions has led to significant progress in this area here we present a unified treatment of this theory based on an appropriate notion of viscosity solution for instance it is shown that strong solutions are viscosity solutions that viscosity solutions are twice differentiable a e and that the pointwise derivatives satisfy the equation a e an important consequence of our approach is the possibility of passage to various kinds of limits in fully nonlinear equations this extends results of this type due to evans and krylov our work is to some extent expository the main purpose being to provide an easily accessible set of tools and techniques to study equations with measurable ingredients 1996 john wiley sons inc

asymptotic stability of traveling waves for scalar viscous conservation laws with non convex nonlinearity the asymptotic stability of traveling wave solutions with shock profile is considered for scalar viscous conservation lawsu t f u x u xx with the initial datau0 which tend to the constant statesu asx stability theorems are obtained in the absence of the convexity off and in the allowance ofs shock speed f u moreover the rate of asymptotics in time is investigated for the casef u s f u if the integral of the initial disturbance over x is small and decays at the algebraic rate as x then the solution approaches the traveling wave at the corresponding rate ast this rate seems to be almost optimal compared with the rate in the casef u2 2 for which an explicit form of the solution exists the rate is also obtained in the casef u s under some additional conditions proofs are given by applying an elementary weighted energy method to the integrated equation of the original one the selection of the weight plays a crucial role in those procedures

ott sudan ostrovskiy equation on a half line abstract we consider the initial boundary value problem for the ott sudan ostrovskiy equation on a half line we study traditionally important problems of the theory of nonlinear partial differential equations such as global in time existence of solutions to the initial boundary value problem and the asymptotic behavior of solutions for large time

critical exponents for the decay rate of solutions in a semilinear parabolic equation this paper is concerned with the cauchy problem a solution u is said to decay fast if t 1 p 1 u rightarrow 0 as t rightarrow infty uniformly in r and is said to decay slowly otherwise for each nonnegative integer k let lambda k be the set of uniformly bounded functions on r which change sign k times and let p k 1 be defined by p k 1 2 k 1 it is shown that any nontrivial bounded solution with u 0 in lambda k decays slowly if 1 p p k whereas there exists a nontrivial fast decaying solution with u 0 in lambda k if p gep k

on the norm and covering radius of the first order reed muller codes let spl rho 1 m and n 1 m be the covering radius and norm of the first order reed muller code r 1 m respectively it is known that spl rho 1 2k 1 spl les lower bound 2 sup 2k 2 sup 2k 1 2 and n 1 2k 1 spl les 2 lower bound 2 sup 2k 2 sup 2k 1 2 k 0 we prove that spl rho 1 2k 1 spl les 2 lower bound 2 sup 2k 1 2 sup 2k 3 2 and n 1 2k 1 spl les 4 lower bound 2 sup 2k 1 2 sup 2k 3 2 k 0 we also discuss the connections of the two new bounds with other coding theoretic problems

on some cosets of the first order reed muller code with high minimum weight we study a family of particular cosets of the first order reed muller code r 1 m those generated by special codewords the idempotents thus we obtain new maximal weight distributions of cosets of r 1 7 and 84 distinct almost maximal weight distributions of cosets of r 1 9 that is with minimum weight 240 this leads to crypotographic applications in the context of stream ciphers

interior a priori estimates for solutions of fully non linear equations on etend une theorie de perturbations aux solutions d equations uniformement elliptiques d ordre 2 totalement non lineaires

results on rotation symmetric s boxes we give an efficient exhaustive search strategy to enumerate 6x6 bijective rotation symmetric s boxes rssbs having nonlinearity 24 which is found to be the maximum nonlinearity within the class of 6x6 bijective rssbs it is shown that there are 3072 rssbs achieving the cryptographic properties of the inverse function over gf 2 6 i e nonlinearity 24 differential uniformity 4 and algebraic degree 5 such that among them there are only four which are not affine equivalent among these four rssbs we find a non affine transformation under which the cryptographic properties of the inverse function are invariant then we define the generalized classes of k rssbs as the polynomials of gf 2 n with coefficients in gf 2 k where k divides n moreover motivated by the fact that rssbs are symmetric under a special permutation we classify all possible permutations up to the linear equivalence of s boxes that are symmetric under them

2 solvability of the dirichlet problem for nondivergence elliptic equations with vmo coefficients we prove a well posedness result in the class w 2 p w 0 1 p for the dirichlet problem lu f a e in u 0 on we assume the coefficients of the elliptic nondivergence form equation that we study are in vmo l

global w2 δ estimates for singular fully nonlinear elliptic equations with ln right hand side terms abstract we establish in this paper a priori global w 2 estimates for singular fully nonlinear elliptic equations with l n right hand side terms the method is to slide paraboloids and barrier functions vertically to touch the solution of the equation and then to estimate the measure of the contact set in terms of the measure of the vertex point set to derive global estimates from l n data the hardy littlewood maximal functions appropriate localizations and a new type of covering argument are adopted these methods also provide us a more direct proof of the w 2 estimates for nonsingular fully nonlinear elliptic equations established by l a caffarelli and x cabre

generalized maiorana mcfarland construction of resilient boolean functions with high nonlinearity and good algebraic properties a new framework concerning the construction of small order resilient boolean functions whose nonlinearity is strictly greater than 2 n 1 2 n 2 is given first a generalized maiorana mcfarland construction technique is described which extends the current approaches by combining the usage of affine and nonlinear functions in a controllable manner it is shown that for any given m this technique can be used to construct a large class of n variable n both even and odd m resilient degree optimized boolean functions with currently best known nonlinearity this class may also provide functions with excellent algebraic properties measured through the resistance to fast algebraic attacks if the number of n 2 variable affine subfunctions used in the construction is relatively low due to a potentially low hardware implementation cost along with overall good cryptographic properties this class of functions is an attractive candidate for the use in certain stream cipher schemes

a bilinear estimate with applications to the kdv equation u x 0 u0 x where u0 h r our principal aim here is to lower the best index s for which one has local well posedness in h r i e existence uniqueness persistence and continuous dependence on the data for a finite time interval whose size depends on u0 hs equation in 1 1 was derived by korteweg and de vries 21 as a model for long wave propagating in a channel a large amount of work has been devoted to the existence problem for the ivp 1 1 for instance see 9 10 the inverse scattering method applies to this problem and under appropriate decay assumptions on the data several existence results have been established see 5 6 14 28 33 another approach inherited from hyperbolic problems relies on the energy estimates and in particular shows that 1 1 is locally well posed in h r for s 3 2 see 2 3 12 29 30 31 using these results and conservation laws global in time well posedness in h r s 2 was established see 3 12 30 also global in time weak solutions in the energy space h r were constructed in 34 in 13 and 22 a local smoothing effect for solutions of 1 1 was discovered this combined with the conservation laws was used in 13 and 22 to construct global in time weak solutions with data in h r and even in l r in 16 we introduced oscillatory integral techniques to establish local well posedness of 1 1 in h r s 3 4 and hence global in time well posedness in h r s 1 in 16 we showed how to obtain the above mentioned result by picard iteration in an appropriate function space in 4 j bourgain introduced new function spaces adapted to the linear operator t 3 x for which there are good bilinear estimates for the nonlinear term x u 2 using these spaces bourgain was able to establish local well posedness of 1 1 in h r l r and hence by a conservation