on the crosscorrelation of sequences over gf p with short periods we investigate the crosscorrelation function cd t spl sigma sub i 1 p sup n 1 spl zeta a sup i t a sup di where spl zeta is a complex primitive pth root of unity a sub i i spl isin n sub 0 is a maximal linear shift register sequence of length p sup n 1 and p is an odd prime for p 3 n odd and d p sup n 1 4 p sup n 1 2 we show that 2 spl middot spl radic p sup n is an upper bound for the absolute value of 1 c sub d t for any odd prime p and p sup k 1 where n gcd n k is not divisible by 4 we determine the maximum absolute value of c sub d t and the number of values of c sub d t

bent function sequences in this paper we construct a new family of nonlinear binary signal sets which achieve welch s lower bound on simultaneous cross correlation and autocorrelation magnitudes given a parameter n with n 0 pmod 4 the period of the sequences is 2 n 1 the number of sequences in the set is 2 n 2 and the cross auto correlation function has three values with magnitudes leq 2 n 2 1 the equivalent linear span of the codes is bound above by sum i 1 n 4 left stackrel n i right these new signal sets have the same size and correlation properties as the small set of kasami codes but they have important advantages for use in spread spectrum multiple access communications systems first the sequences are balances which represents only a slight advantage second the sequence generators are easy to randomly initialize into any assigned code and hence can be rapidly hopped from sequence to sequence for code division multiple access operation most importantly the codes are nonlinear in that the order of the linear difference equation satisfied by the sequence can be orders of magnitude larger than the number of memory elements in the generator that produced it this high equivalent linear span assures that the code sequence cannot be readily analyzed by a sophisticated enemy and then used to neutralize the advantages of the spread spectrum processing

on the correlation distribution of the coulter matthews decimation the distribution of the cross correlation between the ternary m sequence s sub t of period n 3 sup m 1 and the decimated sequences s sub dt and s sub dt 1 of period 3 sup m 1 2 where d 3 sup k 1 2 with k odd and gcd k m 1 is determined the method to find this distribution is related to the result by coulter and matthews that f x x sup d is a planar function over gf 3 sup m

niho type cross correlation functions via dickson polynomials and kloosterman sums suppose that n 2k is even we study the cross correlation function between two m sequences for niho type decimations d 2 sup k 1 s 1 we develop a new technique to study the value distribution of these cross correlation functions which makes use of dickson polynomials as a first application we derive here the distribution of the six valued cross correlation function for s 3 and odd k up to a term which depends on kloosterman sums in addition applying simpler methods we prove a theorem providing niho type decimations with four valued cross correlation functions and their distribution we conjecture that the latter result actually covers all such decimations

pricing general insurance in a reactive and competitive market a simple parameterisation is introduced which represents the insurance market s response to an insurer adopting a pricing strategy determined via optimal control theory claims are modelled using a lognormally distributed mean claim size rate and the market average premium is determined via the expected value principle if the insurer maximises its expected wealth then the resulting bellman equation has a moving boundary in state space that determines when it is optimal to stop selling insurance this stochastic optimisation problem is simplified by the introduction of a stopping time that prevents an insurer leaving and then re entering the insurance market three finite difference schemes are used to verify the existence of a solution to the resulting bellman equation when there is market reaction all of the schemes use a front fixing transformation if the market reacts then it is found that the optimal strategy is altered in that premiums are raised if the strategy is of loss leading type and lowered if it is optimal for the insurer to set a relatively high premium and sell little insurance

ternary m sequences with three valued cross correlation function new decimations of welch and niho type we show that the cross correlation between two ternary m sequences of period 3 sup n 1 that differ by the decimation d 2 spl middot 3 sup m 1 where n 2m 1 takes on three different values we conjecture the same result for the decimation d 2 spl middot 3 sup r 1 where n is odd and r is defined by the condition 4r 1 spl equiv 0 mod n these two new cases form in a sense ternary counterparts of two previously confirmed binary cases the conjectures of welch and niho 1972

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

prime phase sequences with periodic correlation properties better than binary sequences for the case where p is an odd prime n or 2 is an integer and omega is a complex primitive pth root of unity a construction is presented for a family of p sup n p phase sequences symbols of the form omega sup i where each sequence has length p sup n 1 and where the maximum nontrivial correlation value c sub max does not exceed 1 square root p sup n a complete distribution of correlation values is provided as a special case of this construction a previous construction due to sidelnikov 1971 is obtained the family of sequences is asymptotically optimum with respect to its correlation properties and in comparison with many previous nonbinary designs the present design has the additional advantage of not requiring an alphabet of size larger than three the new sequences are suitable for achieving code division multiple access and are easily implemented using shift registers they wee discovered through an application of deligne s bound 1974 on exponential sums of the weil type in several variables the sequences are also shown to have strong identification with certain bent functions

about fractional integrals in the space of locally integrable functions abstract in this paper we study the classical fractional integrals of riemann liouville and weyl considering them as operators defined in the largest functional space where these operators can be defined we describe the fractional derivatives and integrals of riemann liouville as the fractional powers of suitable non negative operators and consequently we get important properties directly this is not the case for the fractional integral of weyl however we can obtain some interesting properties for these operators

on the covering structures of two classes of linear codes from perfect nonlinear functions in this paper the weight distributions of two classes of linear codes based on all known explicit perfect nonlinear functions from fqm to itself are determined using a unified approach all the minimal codewords of these codes are characterized according to their weights which suggests that their covering structures are determined finally all the minimal access sets of the secret sharing schemes based on their dual codes are obtained

exponential sums from half quadratic forms and their applications in this paper we consider a class of exponential sums from some half quadratic binomials the exponential sums are proven to be eleven valued with maximal magnitude 1 2 q sqrt q except for the trivial value q as applications first we investigate the autocorrelation and cross correlation distribution among the sequences in a sequence family second we determine the weight distributions of several classes of linear codes some of the dual codes have minimum distance four which are optimal with respect to the hamming bound our results extend the result by choi et al and show that some correlation values in it do not occur

on the cross correlation of a p ary m sequence of period p 2m 1 and its decimated sequences by p m 1 2 2 p 1 in this paper for an odd prime we investigate into the cross correlation of a p ary m sequence m t of period p n 1 and its d decimated sequences m dt l 0 l m 1 2 where d pm 1 2 2 p l n 2m and m is an odd integer there are pm 1 2 distinct decimated sequences m dt l since gcd d pn 1 pm 1 2 it is shown that the magnitude of the cross correlation values is upper bounded by p 1 2 pn 2 1 we also construct the sequence family f from these sequences where the family size is pm and the correlation magnitude is upper bounded by pm 1 2 pn 2 1

cross correlation distribution of p ary m sequence of period p 4k 1 and its decimated sequences by left p 2k 1 over 2 right 2 for an odd prime p n 4k and d p2k 1 2 2 there are p2k 1 2 distinct decimated sequences s dt l 0lesl p2k 2 of a p ary m sequence s t of period pn 1 because gcd d pn 1 p2k 1 2 in this paper it is shown that the cross correlation function between s t and s dt l 0lesl p2k 1 2 takes the values in 1 1 radic pn 1 radic pn 1 2radic pn and their cross correlation distribution is also derived

a distributional theory of fractional calculus in this paper a theory of fractional calculus is developed for certain spaces d p of generalised functions the theory is based on the construction of fractionalpowers of certain simple differential and integral operators with the parameter suitably restricted these fractional powers are shown to coincide with the riemann liouville and weyl operators of fractional integration and differentiation standard properties associated with fractional integrals and derivatives follow immediately from results obtained previously by the author on fractional powers of operators see 6 7 some spectral properties are also obtained

a comparison of numerical models for one dimensional stefan problems in this paper we present a critical comparison of the suitability of several numerical methods level set moving grid and phase field model to address two well known stefan problems in phase transformation studies melting of a pure phase and diffusional solid state phase transformations in a binary system similarity solutions are applied to verify the numerical results the comparison shows that the type of phase transformation considered determines the convenience of the numerical techniques finally it is shown both numerically and analytically that the solid solid phase transformation is a limiting case of the solid liquid transformation

a new family of p ary sequences with low correlation constructed from decimated sequences in this paper for an odd prime p and an integer n geq 3 a new family of p ary sequences of period p n 1 over 2 with low correlation is constructed the new family is constructed by shifts and additions of two decimated sequences of a p ary m sequence and its family size is 2 p n 1 the complete correlation distribution of this new family is derived it is also shown that the family is optimal with respect to the parameter theta rm rms which denotes the root mean square of all nontrivial correlations compared with the known sequence families our sequence family is new and has a larger family size which is four times of its period

a generalization of the family of p ary decimated sequences with low correlation let p be a prime and n a positive integer let e pn 1 and n pn 1 e in this paper we construct a family s of e2n p ary sequences each member of s has period n and the magnitudes of correlations of members of s are upper bounded by 2 pn 2 en 1

some results on cross correlation distribution between a p ary m sequence and its decimated sequences for an odd prime p and two positive integers m k such that m gcd k m 3 is odd let d be a positive integer satisfying d p k 1 2 mod p m 1 i n this paper the cross correlation between a p ary m sequence and its d decimated sequences is investigated and the cross correlation distribution is completely determined the relationship between the decimations d considered in this paper and some known ones is also studied this paper generalizes some previous results and also gives new decimations which lead to low cross correlation