properties of the error linear complexity spectrum this paper studies the error linear complexity spectrum of binary sequences with period 2n a precise categorization of those sequences having two distinct critical points in their spectra as well as an enumeration of these sequences is given an upper bound on the maximum number of distinct critical points that the spectrum of a sequence can have is proved and a construction which yields a lower bound on this number is given in the process simpler proofs of some known results on the linear complexity and k error linear complexity of sequences with period 2n are provided

the painleve property for partial differential equations in this paper we define the painleve property for partial differential equations and show how it determines in a remarkably simple manner the integrability the backlund transforms the linearizing transforms and the lax pairs of three well known partial differential equations burgers equation kdv equation and the modified kdv equation this indicates that the painleve property may provide a unified description of integrable behavior in dynamical systems ordinary and partial differential equations while at the same time providing an efficient method for determining the integrability of particular systems

wave propagation on riemannian symmetric spaces abstract let x g k be a riemannian symmetric space of the noncompact type and let l x be the laplace beltrami operator on x we consider on x r a differential equation of the type l x u p t u where p t is a second order differential operator in t r using the radon transform on x we relate huygens principle for the solutions u to this equation to the corresponding question for the solutions v to the equation l a v p t r v on a maximal flat subspace a where r is a certain explicit constant in particular we conclude that huygens principle holds for solutions to the modified wave equation on x obtained by letting p t t 2 r when x is odd dimensional and g has one conjugacy class of cartan subgroups

computing the error linear complexity spectrum of a binary sequence of period 2 sup n binary sequences with high linear complexity are of interest in cryptography the linear complexity should remain high even when a small number of changes are made to the sequence the error linear complexity spectrum of a sequence reveals how the linear complexity of the sequence varies as an increasing number of the bits of the sequence are changed we present an algorithm which computes the error linear complexity for binary sequences of period spl lscr 2 sup n using o spl lscr log spl lscr sup 2 bit operations the algorithm generalizes both the games chan 1983 and stamp martin 1993 algorithms which compute the linear complexity and the k error linear complexity of a binary sequence of period spl lscr 2 sup n respectively we also discuss an application of an extension of our algorithm to decoding a class of linear subcodes of reed muller codes

an algorithm for the k error linear complexity of binary sequences with period 2 sup n certain applications require pseudo random sequences which are unpredictable in the sense that recovering more of the sequence from a short segment must be computationally infeasible it is shown that linear complexity is useful in determining such sequences a generalized linear complexity that has application to the security of stream ciphers is proposed and an efficient algorithm is given for the case in which the sequence is binary with period 2 sup n this algorithm generalizes an algorithm presented by r a games and a h chan 1983

fast rational interpolation reed solomon decoding and the linear complexity profiles of sequences an asymptotically fast algorithm for solving the generalized rational interpolation problem is presented this problem has been studied as part of system theory and is related to the solution of the classical and welch berlekamp 1983 key equations which arise in reed solomon decoding the algorithm can also be used to compute the linear complexity profile of a binary sequence of length m using only o m log m sup 2 log log m bit operations

analysis and efficient implementations of a class of composited de bruijn sequences a binary de bruijn sequence is a sequence of period 2 n 2 n in which every binary n n tuple occurs exactly once in each period a de bruijn sequence has good randomness properties such as long period ideal tuple distribution and high linear complexity and can be generated by a nonlinear feedback shift register nlfsr finding an efficient nlfsr that can generate a de bruijn sequence with a long period is a significant challenge composited construction is a technique for constructing a de bruijn sequence of period 2 n k 2 n k by an nlfsr from a de bruijn sequence of period 2 n 2 n through a composition operation repeatedly applying k k times the goal of this article is to further investigate the composited construction of de bruijn sequences with efficient hardware implementations and determine randomness properties such as linear complexity our contributions in this article are as follows first we present a generalized construction of composited de bruijn sequences that is constructed by adding a combination of conjugate pairs of different lengths in the feedback function of the composited construction which results in generating a class of de bruijn sequences of size 2 k 2 k whereas the original composited construction can generate only two sequences second we investigate the linear complexity and the correlation property of the new class of de bruijn sequences we prove theoretically that the linear complexity of this class of de bruijn sequences is optimal or close to optimal interestingly we also prove that the linear complexities of all the sequences of this class are equal which strengthens etzion s conjecture jcta 1985 ieee it 1999 about the number of de bruijn sequences with equal linear complexity this is the first known construction of de bruijn sequences of an arbitrarily long period whose linear complexities are determined theoretically finally we implement our construction in hardware to demonstrate its practicality we synthesize our implementations for a 65 nm asic and a xilinx spartan fpga and present hardware areas and performances of de bruijn sequences of periods in the range of 2 160 2 160 to 2 1056 2 1056 for instance a class of de bruijn sequences of period 2 160 2 160 resp 2 288 2 288 can be implemented with an area of 3 43 resp 6 71 kges in 65 nm asic and 83 resp 229 slices in spartan6 fpga

equipartition of energy for waves in symmetric space abstract let x g k be an odd dimensional semisimple riemannian symmetric space of the noncompact type and suppose that all cartan subgroups of g are conjugate let u be a real valued classical solution of the modified wave equation utt k u on r x the cauchy data of which are supported in a closed metric ball of radius a at time t 0 here t is the coordinate on r is the nonpositive definite laplace beltrami operator on x and k is a positive constant depending on the root structure of the lie algebra of g we show that the t independent energy functional of u is eventually for t a partitioned into equal potential and kinetic parts specifically half the integrals over x of ut2 and du 2 ku 2 respectively where d is the exterior derivative in x the proof uses helgason s paley wiener theorem for x the classical paley wiener theorem and properties of harish chandra s c function

a fast algorithm for determining the linear complexity of a sequence with period p sup n over gf q a fast algorithm is presented for determining the linear complexity of a sequence with period p sup n over gf q where p is an odd prime and where q is a prime and a primitive root mod p sup 2

comments on a new method to compute the 2 adic complexity of binary sequences we show that there is a very simple approach to determine the 2 adic complexity of periodic binary sequences with ideal two level autocorrelation this is the first main result by h xiong l qu and c li ieee transactions on information theory vol 60 no 4 pp 2399 2406 apr 2014 and the main result by t tian and w qi ieee transactions on information theory vol 56 no 1 pp 450 454 jan 2010

linear complexity of a sequence obtained from a periodic sequence by either substituting inserting or deleting k symbols within one period a unified derivation of the bounds of the linear complexity is given for a sequence obtained from a periodic sequence over gf q by either substituting inserting or deleting k symbols within one period the lower bounds are useful in case of n n k where n and n are the period and the linear complexity of the sequence respectively it is shown that all three different cases can be treated very simply in a unified manner the bounds are useful enough to show how wide the distribution of the linear complexity becomes as k increases although they are not always tight because their derivations do not use the information about the change values

integrability and huygens principle on symmetric spaces the explicit formulas for fundamental solutions of the modified wave equations on certain symmetric spaces are found these symmetric spaces have the following characteristic property all multiplicities of their restricted roots are even as a corollary in the odd dimensional case one has that the huygens principle in hadamard s sense for these equations is fulfilled we consider also the heat and laplace equations on such a symmetric space and give explicitly the corresponding fundamental solutions heat kernel and green s function this continues our previous investigations 16 of the spherical functions on the same symmetric spaces based on the fact that the radial part of the laplace beltrami operator on such a space is related to the algebraically integrable case of the generalised calogero sutherland moser quantum system in the last section of this paper we apply the methods of heckman and opdam to extend our results to some other symmetric spaces in particular to complex and quaternian grassmannians

exact three wave solutions for the kp equation a new type of three wave solution periodic two solitary wave solutions for 1 2 d kadomtsev petviashvili kp equation is obtained using the extended three soliton method and with the help of maple

on the 2 adic complexity and the k error 2 adic complexity of periodic binary sequences a significant difference between the linear complexity and the 2 adic complexity of periodic binary sequences is pointed out in this correspondence based on this observation we present the concept of the symmetric 2 adic complexity of periodic binary sequences the expected value of the 2 adic complexity is determined and a lower bound on the expected value of the symmetric 2 adic complexity of periodic binary sequences is derived we study the variance of the 2 adic complexity of periodic binary sequences and the exact value for it is given because the k adic complexity of periodic binary sequences is unstable we present the concepts of the kappa error 2 adic complexity and the k error symmetric 2 adic complexity and lower bounds on them are also derived in particular we give tighter upper and lower bounds for the minimum k adic complexity of l sequences by substituting two symbols within one period

conformally invariant differential equations and projective geometry abstract we combine harmonic analysis on certain pseudo riemannian symmetric spaces with results on conformally invariant linear and non linear differential equations this gives in many cases part of the decomposition of certain representations of the conformal group of a manifold when restricted to the isometry group

huygens principle for wave equations on symmetric spaces let x gk be a symmetric space lx the laplace betrami operator on x and 2 the sum of the positive restricted roots of x with multiplicity using the fourier transform on x we write down the solution of the modified wave equation 2u t2 lx 2 u on x it is proved using the paley wiener theorem for this fourier transform s helgason ann of math 98 1973 451 480 that the equation satisfies huygens principle if dim x is odd and all cartan subgroups of g are conjugate

an algorithm for the k error linear complexity of sequences over gf p m with period p n p a prime an algorithm is given for the k error linear complexity of sequences over gf pm with period pn p a prime the algorithm is derived by the generalized games chan algorithm for the linear complexity of sequences over gf pm with period pn and by using the modified cost different from that used in the stamp martin algorithm for sequences over gf 2 with period 2n a method is also given for computing an error vector which gives the k error linear complexity

fast algorithms for determining the linear complexity of sequences over gf p sup m with period 2 sup t n we prove a result which reduces the computation of the linear complexity of a sequence over gf pm p is an odd prime with period 2n n is a positive integer such that there exists an element bisingf pm bn 1 to the computation of the linear complexities of two sequences with period n by combining with some known algorithms such as the berlekamp massey algorithm and the games chan algorithm we can determine the linear complexity of any sequence over gf pm with period 2tn such that 2 t pm 1 and gcd n pm 1 1 more efficiently

extended games chan algorithm for the 2 adic complexity of fcsr sequences binary sequences generated by feedback shift registers with carry operation fcsr share many of the important properties enjoyed by sequences generated by linear feedback shift registers we present an fcsr analog of the extended games chan algorithm which efficiently determines the linear complexity of a periodic binary sequence with period length t 2n or pn where p is an odd prime and 2 is a primitive element modulo p2 the algorithm to be presented yields an upper bound for the 2 adic complexity an fcsr analog of the linear complexity of a pn periodic binary sequence

huygens principle and integrable systems abstract the theory of integrable systems in particular of the generalised calogero moser problem is applied to constructing and investigating examples of the hyperbolic equations satisfying huygens principle in hadamard s sense

on the depth distribution of linear codes the depth distribution of a linear code was recently introduced by t etzion see ibid vol 43 pp 1361 3 july 1997 in this correspondence a number of basic and interesting properties for the depth of finite words and the depth distribution of linear codes are obtained in addition we study the enumeration problem of counting the number of linear subcodes with the prescribed depth constraints and derive some explicit and interesting enumeration formulas furthermore we determine the depth distribution of reed muller code rm m r finally we show that there are exactly nine depth equivalence classes for the ternary 11 6 5 golay codes

diversity of interaction solutions to the 2 1 dimensional ito equation abstract we aim to show the diversity of interaction solutions to the 2 1 dimensional ito equation based on its hirota bilinear form the proof is given through maple symbolic computations an interesting characteristic in the resulting interaction solutions is the involvement of an arbitrary function special cases lead to lump solutions lump soliton solutions and lump kink solutions two illustrative examples of the resulting solutions are displayed by three dimensional plots and contour plots

exact three wave solutions for the 3 1 dimensional jimbo miwa equation exact three wave solutions including periodic cross kink wave solutions doubly periodic solitary wave solutions and breather type of two solitary wave solutions for the 3 1 dimensional jimbo miwa equation are obtained using the generalized three wave method it is shown that the generalized three wave method with the help of symbolic computation provides an effective and powerful mathematical tool for solving high dimensional nonlinear evolution equations in mathematical physics

a construction of long period sequences based on lightweight generation and high probability in practice sequences with long period but lightweight generation are always welcome in the applications of communication systems however the generation is not easy from a certain angle of very high probability this paper presents a solution which is performed using cyclic difference sequences as an application the generated sequences can be interleaved to obtain good correlation properties furthermore synchronization with blind detection is often required but difficult to be achieved even harder periodic sequences may be affected to be ultimately periodic sequences with overhead because of device switching or noise the determinations of the ultimate periods of above sequences and corresponding distributions are also investigated in this paper

a generalized recursive construction for de bruijn sequences this paper is concerned with the construction of de bruijn sequences of span n binary sequences of period 2 n in which every binary n tuple appears as some n consecutive terms in one period of the sequence constructions in the literature are based on maximum length linear sequences algorithms which start from scratch and recursive methods which start with a single de bruijn sequence of span n and produce one of span n 1 we give a more general recursive construction which takes two de brnijn sequences of span n and produces a de bruijn sequence of span n 1 in addition for a special case of the construction the complexity shortest linear recursion that generates the sequence of the resulting sequence is determined in terms of the complexities of the ingredient sequences in particular de bruijn sequences of span n 1 with maximum complexity 2 n 1 1 are obtained from maximum complexity sequences of span n reverse complementary de bruijn sequences are also considered

periodic sequences with large k error linear complexity we establish the existence of periodic sequences over a finite field which simultaneously achieve the maximum value for the given period length of the linear complexity and of the k error linear complexity for small values of k this disproves a conjecture of ding xiao and shan 1991 the result is of relevance for the theory of stream ciphers

eventual partition of conserved quantities in wave motion abstract let u be a classical solution to the wave equation in an odd number n of space dimensions with compact spatial support at each fixed time duffin j math anal appl 32 1970 386 391 uses the paley wiener theorem of fourier analysis to show that after a finite time the conserved energy of u is partitioned into equal kinetic and potential parts the wave equation actually has n 2 n 3 2 independent conserved quantities one for each of the standard generators of the conformal group of n 1 dimensional minkowski space of concern in this paper is the zeroth inversional quantity i0 which is commonly used to improve decay estimates which are obtained using conservation of energy we use duffin s method to partition i0 into seven terms each of which after a finite time is explicitly given as a constant coefficient quadratic function of the time zachmanoglou has shown that under the above assumptions if n 3 the spatial l2 norm of u is eventually constant a consequence of the analysis here is a bound on this constant in terms of the energy and the radius of the support of the cauchy data of u at a fixed time

eventual partition of conserved quantities for maxwell s equations in 4 dassios uses a method introduced by duffin in 5 to show that if a classical solution to maxwell s equations has compact spatial support at some time its energy is partitioned after a finite time into exactly equal electric and magnetic parts here we apply the same method to conserved quantities other than the energy these quantities are shown to be eventually partitioned into a number of linear or quadratic functions of the time with constant coefficients

construction of de bruijn sequences of minimal complexity it is well known that the linear complexity of a de bruijn sequence s of length 2 n is bounded below by 2 n 1 n for n geq 3 it is shown that this lower bound is attainable for all n

analyzing the combined multi waves polynomial solutions in a two layer liquid medium abstract in this work we investigate the behavior of mixed wave solutions in two layer liquid with dispersive waveguide these combined multi wave solutions were obtained in polynomial types for the 3 1 dimensional yu toda sasa fukuyama ytsf equation with variable coefficients by virtue of the generalized unified method and symbolic computations we formally derive polynomial solutions for this equation the obtained solutions include multi soliton solutions multi periodic solutions and multi elliptic solutions furthermore the physical insight and the movement role of the waves in the two layers are discussed and analyzed graphically for different selections of the arbitrary functions of the obtained solutions

breather wave solutions for the generalized shallow water wave equation with variable coefficients in the atmosphere rivers lakes and oceans abstract herein a 3 1 dimensional generalized shallow water wave equation with variable coefficients is studied which describes the flow below a pressure surface in a fluid abundant breather wave solutions are constructed via the hirota bilinear form and a novel test function velocity amplitude and shape changes of the breather wave solutions are shown in some 3d graphs and contour plots by selecting different values of the variable coefficients the novel test function is very effective for obtaining the breather wave solutions of the variable coefficient nonlinear partial differential equations

linear complexity of de bruijn sequences old and new results the linear complexity of a de bruijn sequence is the degree of the shortest linear recursion which generates the sequence it is well known that the complexity of a binary de bruijn sequence of length 2 sup n is bounded below by 2 sup n 1 n and above by 2 sup n 1 for n spl ges 3 we briefly survey the known knowledge in this area some new results are also presented in particular it is shown that for each interval of length 2 sup log n 1 in the above range there exist binary de bruijn sequences of length 2 sup n with linear complexity in the interval

support theorem for the fundamental solution to the schrodinger equation on certain compact symmetric spaces abstract in this paper we construct the fundamental solution to the schodinger equation on a compact symmetric space with even root multiplicities using shift operators of heckman and opdam next we prove that the support of the fundamental solution becomes a lower dimensional subset at a rational time whereas its support and its singular support coincide with the whole symmetric space at an irrational time moreover we also show that generalized gauss sums appear in the expression of the fundamental solution

exact three wave solution for higher dimensional kdv type equation a new periodic type of three wave solutions including periodic two solitary solution doubly periodic solitary solution and breather type of two solitary solution for the 1 2 dimensional and 1 3 dimensional kdv type equations are obtained using hirota s bilinear form and generalized three wave type of ansatz approach

an algorithm for computing minimal bidirectional linear recurrence relations we consider the problem of computing a linear recurrence relation or equivalently a linear feedback shift register of minimum order for a finite sequence over a field with the additional requirement that not only the highest but also the lowest coefficient of the recurrence is nonzero such a recurrence relation can then be used to generate the sequence in both directions increasing or decreasing order of indices so we call it bidirectional if the field is finite a sequence is periodic if and only if it admits a bidirectional linear recurrence relation for solving the above problem we propose an algorithm similar to the berlekamp massey algorithm and prove its correctness we describe the set of all solutions to this problem and show that if a sequence admits more than one linear recurrence relation then it admits a bidirectional one we also prove some properties regarding the bidirectionality of the recurrences of the prefixes of the sequence

on the computation of the linear complexity and the k error linear complexity of binary sequences with period a power of two the linear games chan algorithm for computing the linear complexity c s of a binary sequence s of period lscr 2n requires the knowledge of the full sequence while the quadratic berlekamp massey algorithm requires knowledge of only 2c s terms we show that we can modify the games chan algorithm so that it computes the complexity in linear time knowing only 2c s terms the algorithms of stamp martin and lauder paterson can also be modified without loss of efficiency to compute analogs of the k error linear complexity for finite binary sequences viewed as initial segments of infinite sequences with period a power of two we also develop an algorithm which given a constant c and an infinite binary sequence s with period lscr 2n computes the minimum number k of errors and an associated error sequence needed over a period of s for bringing the linear complexity of s below c the algorithm has a time and space bit complexity of o lscr we apply our algorithm to decoding and encoding binary repeated root cyclic codes of length lscr in linear o lscr time and space a previous decoding algorithm proposed by lauder and paterson has o lscr loglscr 2 complexity

rotating table games and derivatives of words we consider two versions of a game for two players a and b the game consists of manipulations of words of length n over an alphabet of size for arbitrary n and for 2 the game is described as follows initially player a puts n drinking glasses on a round table some of which are upside down player b attempts to force player a to set all the glasses in the upright position for this he instructs player a to invert some of the glasses before following the instruction player a has the freedom to rotate the table and then to invert the glasses that are in the locations originally pointed by player b in one version of the game player b is blindfolded and in the other he is not we show that player b has winning strategies for both games iff n and are powers of the same prime in both games we provide optimal winning strategies for b the analysis of the games is closely related to the concept of the derivative of a ary word of length n in particular it is related to the depth of such word which is the smallest k such that the kth derivative of the word is the all zero word we give tight upper bounds on the depth of ary words of length n where and n are powers of the same prime

on the expected value of the linear complexity and the k error linear complexity of periodic sequences rueppel 1986 conjectured that periodic binary sequences have expected linear complexity close to the period length n in this paper we determine the expected value of the linear complexity of n periodic sequences explicitly and confirm rueppel s conjecture for arbitrary finite fields cryptographically strong sequences should not only have a large linear complexity but also the change of a few terms should not cause a significant decrease of the linear complexity this requirement leads to the concept of the k error linear complexity of n periodic sequences we present a method to establish a lower bound on the expected k error linear complexity of n periodic sequences based on the knowledge of the counting function spl nscr sub n sub 0 c i e the number of n periodic sequences with given linear complexity c for some cases we give explicit formulas for that lower bound and we also determine spl nscr sub n 0 c

minimum linear span approximation of binary sequences the determination of the minimum linear span sequence that differs from a given binary sequence of period n 2 sup n 1 by at most one digit is discussed and three methods are presented the sequential divisions method the congruential equations method and the phase synchronization method high level algorithm organizations are provided finally guidelines on sequence characterization and design via the notion of robustness are given

a new method to compute the 2 adic complexity of binary sequences in this paper a new method is presented to compute the 2 adic complexity of pseudo random sequences with this method the 2 adic complexities of all the known sequences with ideal 2 level autocorrelation are determined in a unified way results show that their 2 adic complexities equal their periods in other words their 2 adic complexities attain the maximum in addition 2 adic complexities of two classes of optimal autocorrelation sequences with period n 1mod4 namely legendre sequences and ding helleseth lam sequences are investigated this method also can be used to compute the linear complexity of binary sequences regarded as sequences over other finite fields

a relationship between linear complexity and k error linear complexity linear complexity is an important cryptographic criterion of stream ciphers the k error linear complexity of a periodic sequence of period n is defined as the smallest linear complexity that can be obtained by changing k or fewer bits of the sequence per period this article shows a relationship between the linear complexity and the minimum value k for which the k error linear complexity is strictly less than the linear complexity

group representations arising from lorentz conformal geometry abstract it is shown that there exist conformally covariant differential operators d 2 l k of all even orders 2 l on differential forms of all orders k in the double cover n of the n dimensional compactified minkowski space n these act as intertwining differential operators for natural representations of o 2 n the conformal group of n for even n the resulting decompositions of differential form representations of o 2 n the orthochronous conformal group produce infinite families of unitary representations the most interesting of which are carried by positive mass squared positive frequency quotients for 2l n 2k physically these generalize unitary representations of the conformal group associated with the modified wave operator d 2 0 n 2 2 2 and the maxwell operator on vector potentials d 2 n 2 2 d all the representation spaces produced unitary and nonunitary may be viewed as infinite systems of harmonic oscillators as a by product of the spectral resolution of the d 2 l k one gets some striking wave propagative properties for all of the equations d 2 l k 0 including huygens principle in the curved spacetime n the operators d 2 l k have not been seen before except in the special cases k 0 or n and k n 2 2 l 1 the maxwell operator thus much new information is obtained even in the physical case n 4

on the distribution of de bruijn cr sequences corresp it is shown that the number of de bruijn sequences of order n and linear complexity c is not a multiple of four for every n and c

a generalisation of the discrete fourier transform determining the minimal polynomial of a periodic sequence let s be a periodic sequence whose elements lie in a finite field the authors present an algorithm that calculates the minimal polynomial of s assuming that a period of s is known the algorithm generalises both the discrete fourier transform and the games chan algorithm

a fast algorithm for determining the complexity of a binary sequence with period 2 n corresp the complexity of a periodic sequence s is defined to be the least number of stages in a linear feedback shift register that generates s

reducing the computation of linear complexities of periodic sequences over hbox gf p m the linear complexity of a periodic sequence over gf pm plays an important role in cryptography and communication see menezes van oorschort and vanstone handbook of applied cryptography boca raton fl crc 1997 in this correspondence we prove a result which reduces the computation of the linear complexity and minimal connection polynomial of a period un sequence over gf pm to the computation of the linear complexities and minimal connection polynomials of u period n sequences the conditions u pm 1 and gcd n pm 1 1 are required for the result to hold some applications of this reduction in fast algorithms to determine the linear complexities and minimal connection polynomials of sequences over gf pm are presented

differential difference operators associated to reflection groups there is a theory of spherical harmonics for measures invariant under a finite reflection group the measures are products of powers of linear functions whose zero sets are the mirrors of the reflections in the group times the rotation invariant measure on the unit sphere in rn a commutative set of differential difference operators each homogeneous of degree 1 is the analogue of the set of first order partial derivatives in the ordinary theory of spherical harmonics in the case of r2 and dihedral groups there are analogues of the cauchy riemann equations which apply to gegenbauer and jacobi polynomial expansions the analysis of orthogonality structures for polynomials in several variables is a problem of vast dimensions this paper is part of an ongoing program to establish a workable theory for one particular class the underlying structure is based on finite coxeter groups these are finite groups acting on euclidean space generated by reflections in the zero sets of a collection of linear functions the roots the weight functions for the orthogonality are products of powers of these linear functions restricted to the surface of the unit sphere in addition the weight function is required to be invariant under the action of the group the resulting theory has strong similarities to the theory of spherical harmonics this was established in previous papers of the author 3 4 5 most notably a homogeneous polynomial is orthogonal to all polynomials of lower degree if and only if it is annihilated by a certain second order differential difference operator ordinary partial differentiation acts as an endomorphism on ordinary harmonic functions the use of such operators and their adjoints leads to recurrence formulas and orthogonal decompositions for harmonic polynomials in this paper we construct a commutative set of first order differentialdifference operators associated to the second order operator previously mentioned received by the editors june 1 1987 1980 mathematics subject classification 1985 revision primary 33a45 33a65 20h15 secondary 20c30 42c10 51f15

partial period distribution of fcsr sequences klapper and goresky 1995 introduced feedback with carry shift register fcsr and presented a significant kind of fcsr sequences that is l sequences they showed that the number of 0s and 1s occurring in one of their periods are equal we discuss the partial period distribution of l sequences and show that when the periods become large the proportion of 1s resp 0s occurring in any of their partial periods approximates 50

on the stability of 2 sup n periodic binary sequences the k error linear complexity of a periodic binary sequence is defined to be the smallest linear complexity that can be obtained by changing k or fewer bits per period this contribution focuses on the case of 2n periodic binary sequences for k 1 2 the exact formula for the expected k error linear complexity of a sequence having maximal possible linear complexity 2n and the exact formula of the expected 1 error linear complexity of a random 2n periodic binary sequence are provided for k ges 2 lower and upper bounds on the expected value of the k error linear complexity of a random 2n periodic binary sequence are established

determining the complexity of fh ss sequence by approximate entropy high complexity of frequency hopping fh spreadspectrum ss sequence is of great importance to high security multiple access communication systems for it makes fh ss sequence difficult to be analyzed with the growing development in the design of fh ss sequence in much wider fields the wellknown complexity measuresiquestthe linear complexity lc the linear complexity profile lcp and the k error linear complexity k error lc iquestare widely used but not sufficient to evaluate the complexities of the sequences available such as the cryptographical sequence and the chaotic sequence families in this paper a new complexity metric to evaluate the unpredictability of fh ss sequence based on the approximate entropy apen is proposed in the view of the maximal randomness of the sequences with arbitrary length and the theoretical bounds of the apen are derived from a probabilistic point of view simulations and analysis results show that the proposed apen works effectively to discern the changing complexities of the fh ss sequences with small number of samples which provide superior performance over its candidates

perfect maps given positive integers r s u and spl upsi an r s u spl upsi perfect map pm is defined to be a periodic r spl times s binary array in which every u spl times spl upsi binary array appears exactly once as a periodic subarray perfect maps are the natural extension of the de bruijn sequences to two dimensions in the paper the existence question for perfect maps is settled by giving constructions for perfect maps for all parameter sets subject to certain simple necessary conditions extensive use is made of previously known constructions by finding new conditions which guarantee their repeated application these conditions are expressed as bounds on the linear complexities of the periodic sequences formed from the rows and columns of perfect maps

a fast algorithm for determining the minimal polynomial where of a sequence with period 2p sup n over gf q a fast algorithm is presented for determining the linear complexity and the minimal polynomial of a sequence with period 2p sup n over gf q where p and q are odd prime and q is a primitive root mod p sup 2 the algorithm uses the fact that in this case the factorization of x sup 2p n 1 is especially simple

helmholtz operators and symmetric space duality we consider the property of vanishing logarithmic term vlt for the fundamental solution of the shifted laplace d alembert operators b b a constant on pseudo riemannian reductive symmetric spaces m our main result is that such an operator on the c dual or flensted jensen dual of m has the vlt property if and only if a corresponding operator on m does for lorentzian spaces where the b are hyperbolic vlt is known to be equivalent to the strong huygens principle we use our results to construct a large supply of new space operator pairs satisfying huygens principle

on the k error linear complexity of p m periodic binary sequences in this correspondence we study the statistical stability properties of pm periodic binary sequences in terms of their linear complexity and k error linear complexity where p is n prime number and 2 is a primitive root modulo p2 we show that their linear complexity and k error linear complexity take a value only from some specific ranges we then present the minimum value k for which the k error linear complexity is strictly less than the linear complexity in a new viewpoint different from the approach by meidl we also derive the distribution of pm periodic binary sequences with specific k error linear complexity finally we get an explicit formula for the expectation value of the k error linear complexity and give its lower and upper bounds when k les p 2

the error linear complexity spectrum as a cryptographic criterion of boolean functions the error linear complexity spectrum constitutes a well known cryptographic criterion for sequences indicating how the linear complexity of the sequence decreases as the number of bits allowed to be modified per period increases in this paper via defining an association between 2 n periodic binary sequences and boolean functions on n variables it is shown that the error linear complexity spectrum also provides useful cryptographic information for the corresponding boolean function f namely it yields an upper bound on the minimum hamming distance between f and the set of functions depending on fewer number of variables therefore the prominent lauder paterson algorithm for computing the error linear complexity spectrum of a sequence may also be used for efficiently determining approximations of a boolean function that depend on fewer number of variables moreover it is also shown that through this approach low degree approximations of a boolean function can be also obtained in an efficient way

on the distribution of de bruijn sequences of given complexity the distribution gamma c n of de bruijn sequences of order n and linear complexity c is investigated it is shown that for n geq 4 gamma 2 n 1 n equiv 0 pmod 8 and for k geq 3 gamma 2 2k 1 2k equiv 0 pmod l6 it is also shown that gamma c n equiv 0 pmod 4 for all c and n geq 3 such that cn is even

a lower bound on the linear span of an fcsr we have derived a lower bound on the linear span of a binary sequence generated by a feedback with carry shift register fcsr under the following condition q is a power of a prime such that q r sup e e spl ges 2 and r 2p 1 where both r and p are 2 prime this allows us to design fcsr stream ciphers similar to previously proposed linear feedback shift register lfsr stream ciphers

dynamic behaviors of interaction solutions of 3 1 dimensional shallow water wave equation abstract rational solutions and interaction solutions of the 3 1 dimensional shallow water wave equation are obtained depending on n soliton solution by the hirota bilinear method and long wave limit method and breather lump solutions and line rogue waves are obtained from two soliton solution as well as their interaction solutions are derived from four soliton solution after the collision breathers and lumps remain unchanged but the line rogue wave has a difference furthermore three soliton solution is discussed the interactions between some hybrid solutions such as hybrid solution between the stripe soliton and breather hybrid solution between stripe soliton and lump solution and so on are demonstrated in detail more interestingly stripe soliton remain unchanged but the amplitudes of the periodic line waves lump solutions and rogue waves have a great change after the collision