new families of binary sequences with optimal three level autocorrelation in this correspondence we give several new families of binary sequences of period n with optimal three level autocorrelation where n spl equiv 2 mod 4 these sequences are either balanced or almost balanced our construction is based on cyclotomy

construction of invariant whiskered tori by a parameterization method part ii quasi periodic and almost periodic breathers in coupled map lattices we construct quasi periodic and almost periodic solutions for coupled hamiltonian systems on an infinite lattice which is translation invariant the couplings can be long range provided that they decay moderately fast with respect to the distance for the solutions we construct most of the sites are moving in a neighborhood of a hyperbolic fixed point but there are oscillating sites clustered around a sequence of nodes the amplitude of these oscillations does not need to tend to zero in particular the almost periodic solutions do not decay at infinity we formulate an invariance equation solutions of this equation are embeddings of an invariant torus on which the motion is conjugate to a rotation we show that if there is an approximate solution of the invariance equation that satisfies some non degeneracy conditions there is a true solution close by the proof of this emph a posteriori theorem is based on a nash moser iteration which does not use transformation theory simpler versions of the scheme were developed in e fontich r de la llave y sire emph j differential equations bf 246 3136 2009 one technical tool important for our purposes is the use of weighted spaces that capture the idea that the maps under consideration are local interactions using these weighted spaces the estimates of iterative steps are similar to those in finite dimensional spaces in particular the estimates are independent of the number of nodes that get excited using these techniques given two breathers we can place them apart and obtain an approximate solution which leads to a true solution nearby by repeating the process infinitely often we can get solutions with infinitely many frequencies which do not tend to zero at infinity

on the sidel nikov sequences as frequency hopping sequences a v l lambda fhs denotes a frequency hopping sequence of length v over a frequency set of size l with maximum out of phase hamming autocorrelation lambda recently ding and yin constructed two fhs families for a prime power q satisfying q ef 1 with positive integers e and f theorems 4 and 5 in their paper claim that these two fhs families include optimal q 1 e f fhss and q 1 e 1 f 1 fhss with respect to the lempel greenberger bound respectively in this paper we give counterexamples and make corrections to them furthermore we observe that these fhss are closely related to sidel nikov sequences based on our results on the spectrum of their hamming autocorrelation values we also correct the theorem on the spectrum of hamming distances of nearly equidistant codes derived by sidel nikov

periodic motions of a class of forced infinite lattices with nearest neighbor interaction in this paper by using a priori bounds topological degree and limiting arguments we study the existence of periodic solutions of a class of one dimensional chain of particles periodically perturbed and with nearest neighbor interaction between particles we study the necessary and sufficient conditions for the existence of periodic solutions in two cases when the number of particles is finite and infinite and obtain different results

a new class of sequences with zero or low correlation zone based on interleaving technique by interleaving one length n perfect sequence or ideal sequence according to elaborate phases a new method of construction of zero correlation zone zcz and low correlation zone lcz sequence sets is presented the resultant sequence sets are optimal or almost optimal with respect to tang fan and matsufuji bound furthermore the new method provides flexible choice for the zcz and lcz lengths

two novel methods and multi mode periodic solutions for the fermi pasta ulam model we introduce two novel methods for studying periodic solutions of the fpu model both numerically and rigorously one is a variational approach based on the dual formulation of the problem and the other involves computer assisted proofs these methods are used e g to construct a new type of solutions whose energy is spread among several modes associated with closely spaced resonances

new families of optimal frequency hopping sequence sets frequency hopping sequences fhss are employed to mitigate the interferences caused by the hits of frequencies in frequency hopping spread spectrum systems in this paper we present combinatorial constructions for fhs sets including direct constructions by using cyclotomic classes recursive constructions based on cyclic difference matrices merging blocks and discrete logarithm by these constructions a number of series of new fhs sets are then produced these fhs sets are optimal with respect to the peng fan bounds

theory and applications of q ary interleaved sequences a new class of q ary sequences called interleaved sequence is introduced their periods shift equivalence linear spans and autocorrelation functions are derived the interleaved sequences include a large number of popular sequences such as multiplexed sequences clock controlled sequences kasami 1966 sequences gmw sequences geometric sequences and no 1989 sequences a special class of the interleaved sequences is constructed by mapping gf q sup m sequences into gf q sequences in terms of different bases of gf q sup m over gf q as an application of the theory of interleaved sequences some new families of binary pseudo random sequences are constructed which have large linear spans optimal periodic cross autocorrelation functions balance and the rapidly hopped properties a complete comparison of the new family of sequences with the gold sequence family the kasami small and large set sequence families the bent function sequence family and the no sequence family is discussed this shows that the new sequence families have important advantages for use in spread spectrum multiple access communication systems

frequency hopping code sequence designs having large linear span in frequency hopping spread spectrum multiple access communication systems it is desirable to use sets of hopping patterns that in addition to having good hamming correlation properties and large period are also derived from sequences having large linear span here two such frequency hopping code sequence designs that are based on generalized bent functions and generalized bent sequences are presented the hamming correlation properties of the designs are optimal in the first case and close to optimal in the second in terms of the alphabet size p required to be prime in both cases the period and family size of the two designs are given by p sup 2 p and p sup n p sup n 2 1 n an even integer respectively the finite field sequences underlying the patterns in the first design have linear span exceeding p whereas still larger linear spans when compared to the sequence period can be obtained using the second design method

new classes of optimal frequency hopping sequences by interleaving techniques in this paper we construct new classes of optimal frequency hopping sequences fhss with respect to the lempel greenberger bound and the peng fan bound by interleaving techniques which are used to construct a sequence of length kn from k sequences of length n we first give two generic constructions for optimal fhs sets from some known optimal fhs sets by interleaving techniques and present some examples of new optimal fhs sets we then design new optimal fhss whose parameters include those of the known optimal constructions with length kn and frequency set size n for some positive integers k and n we also construct optimal fhs sets of length kp from power residue sequences for any odd prime p and 2 les k p in particular our constructions give several new parameters not covered in the literature which are summarized in i and ii

on the existence of homoclinic solutions of a class of discrete nonlinear periodic systems abstract in this paper we obtain a new sufficient condition on the existence of homoclinic solutions of a class of discrete nonlinear periodic systems by using critical point theory in combination with periodic approximations we prove that it is also necessary in some special cases

a new construction of frequency hopping sequences with optimal partial hamming correlation frequency hopping sequences fhss with favorable partial hamming correlation properties have important applications in many synchronization and multiple access systems in this paper lower bounds on the partial hamming correlation of fhss and fhs sets are proposed they slightly improve the known bounds by eun et al and zhou et al a construction of fhss and fhs sets having optimal partial hamming correlation with respect to the improved bounds is also presented based on the theory of generalized cyclotomy our construction yields optimal fhss and fhs sets with new and flexible parameters not covered in this paper

iterative deblending of simultaneous source seismic data using seislet domain shaping regularization abstractwe used a novel iterative estimation scheme for separation of blended seismic data from simultaneous sources the scheme is based on an augmented estimation problem that can be solved by iteratively constraining the deblended data using shaping regularization in the seislet domain we formulated the forward modeling operator in the common receiver domain in which two sources were assumed to be blended using a random time shift dithering approach the nonlinear shaping regularization framework offered some freedom in designing a shaping operator to constrain the model in an underdetermined inverse problem we designed the backward operator and the shaping operator for the shaping regularization framework the backward operator can be optimally chosen as half of the identity operator in the two source case and the shaping operator can be chosen as coherency promoting operator the high performance deblending effect of the iterative framework was tested on three numerically blended synthetic data s

new designs for signal sets with low cross correlation balance property and large linear span gf p case new designs for families of sequences over gf p with low cross correlation balance property and large linear span are presented the key idea of the new designs is to use short p ary sequences of period spl upsi with the two level autocorrelation function together with the interleaved structure to construct a set of long sequences with the desired properties the resulting sequences are interleaved sequences of period spl upsi sup 2 there are spl upsi cyclically shift distinct sequences in each family the maximal correlation value is 2 spl upsi 3 which is optimal with respect to the welch bound each sequence in the family is balanced and has large linear span in particular for binary case cross out of phase autocorrelation values belong to the set 1 spl upsi spl upsi 2 2 spl upsi 3 2 spl upsi 1 any sequence where the short sequences are quadratic residue sequences achieves the maximal linear span it is shown that some families of these sequences can be implemented efficiently in both hardware and software

algebraic constructions of optimal frequency hopping sequences frequency hopping fh spread spectrum and direct sequence spread spectrum are two main spread coding technologies frequency hopping sequences are needed in fh code division multiple access cdma systems in this correspondence three classes of optimal frequency hopping sequences are constructed with algebraic methods the three classes are based on perfect nonlinear functions power functions and norm functions respectively both individual optimal frequency hopping sequences and optimal families of frequency hopping sequences are presented

optimal biphase sequences with large linear complexity derived from sequences over z sub 4 new families of biphase sequences of size 2 sup r 1 1 r being a positive integer are derived from families of interleaved maximal length sequences over z sub 4 of period 2 2 sup r 1 these sequences have applications in code division spread spectrum multiuser communication systems the families satisfy the sidelnikov bound with equality on spl theta sub max which denotes the maximum magnitude of the periodic cross correlation and out of phase autocorrelation values one of the families satisfies the welch bound on spl theta sub max with equality the linear complexity and the period of all sequences are equal to r r 3 2 and 2 2 sup r 1 respectively with an exception of the single m sequence which has linear complexity r and period 2 sup r 1 sequence imbalance and correlation distributions are also computed

homoclinic orbits for second order hamiltonian systems possessing superquadratic potentials received by the editors november 27 1990 1991 mathematics subject classification primary 34c99 58e99 58f99 the first author was supported by ministero p i gruppo 40 calcolo delle variazioni the second author s research was sponsored in part by the u s army research office under contract daal03 87 k 0043 the national science foundation under grant mcs 8110556 and the office of naval research under grant n00014 88 ko134 any reproduction for the purpose of the united states govemment is permitted

optimal large linear complexity frequency hopping patterns derived from polynomial residue class rings we construct new sequences over finite rings having optimal hamming correlation properties these sequences are useful in frequency hopping multiple access fhma spread spectrum communication systems our constructions can be classified into linear and nonlinear categories both giving optimal hamming correlations according to lempel greenberger 1974 bound the nonlinear sequences have large linear complexity and can be seen as a generalized version of gmw sequences over fields

optimal frequency hopping sequences via cyclotomy using cyclotomic numbers a simple construction is presented for frequency hopping fh sequences having optimal autocorrelation with respect to the well known lempel greenberger bound some optimal families of fh sequences are constructed the simplicity of this technique makes it attractive for practical use

homoclinic solutions of a class of periodic difference equations with asymptotically linear nonlinearities abstract by using critical point theory and periodic approximations new sufficient conditions are obtained on the existence and nonexistence of homoclinic solutions for a class of discrete nonlinear periodic equations with asymptotically linear nonlinearities these results partially answer an open problem proposed by pankov 2006 2 under rather weaker conditions and greatly improve the related results before

multiple periodic solutions for lattice dynamical systems with superquadratic potentials abstract in this paper we consider one dimensional lattices consisting of infinitely many particles with nearest neighbor interaction the autonomous dynamical system is described by the following infinite system of second order differential equations q i i 1 q i 1 q i i q i q i 1 i z where i denotes the interaction potential between two neighboring particles and q i t is the state of the i th particle supposing i is superquadratic at infinity for all t 0 we obtain a nonzero t periodic solution of finite energy which may be nonconstant in some range of period if in addition i x is even in x we also obtain infinitely many geometrically distinct solutions for any period t 0 in particular a prescribed number of geometrically distinct nonconstant periodic solutions is obtained for some range of period since the functional associated to the above system is invariant under the actions of the non compact group z and the continuous compact group s 1 under our assumptions in order to prove our results we need to extend the abstract critical point theorem about strongly indefinite functional developed by bartsch and ding math nachr 279 2006 1267 1288 to a more general class of symmetry

polyphase power residue sequences polyphase versions of the well known binary legendre or quadratic residue sequences were first described and analysed by sidelnikov over thirty years ago but have since received very little attention here it is shown that these q phase sequences of prime length l can also be constructed from the index sequence of length l or equivalently from the cosets of q th power residues and non residues these sequences are also shown to fall into two classes each of which has well defined periodic autocorrelation functions and merit factors class i sequences exist for prime l q 1 mod 2 q and q even whereas class ii sequences correspond to primes of the form l 1 mod 2 q and are available for all values of q class i sequences have complex out of phase correlation values with magnitudes less than or equal to 5 whereas for class ii sequences these are purely real with magnitude less than or equal to 3 some other properties are also investigated

nontrivial periodic motions for resonant type asymptotically linear lattice dynamical systems abstract in this paper we consider the following one dimensional lattices consisting of infinitely many particles with nearest neighbor interaction q i t i 1 t q i 1 t q i t i t q i t q i 1 t i z where i t x i 2 x 2 v i t x is t periodic in t for t 0 and satisfies i n i for some n n q i t is the state of the i th particle assume that i 0 for some i z and v i t x denoting the derivative of v i respect to x is asymptotically linear with x both at origin and at infinity we would like to point out that this system is resonant both at origin and at infinity and not studied up to now based on some new results concerning the precise computations of the critical groups for a given m z we obtain the existence of nontrivial periodic solutions satisfying q i m n t t q i t for all t r and i z under some additional conditions

discrete breathers localization and transfer of energy in discrete hamiltonian nonlinear systems we present a partial review of results concerning the theory of discrete breathers in nonlinear hamiltonian and discrete systems these special time periodic solutions gained much interest during the last decade because they may be involved in numerous and various phenomena in physics and biophysics where they could produce nontrivial effects of energy focusing and transfer we first review the principles which govern their existence and which are used in the existence proofs available up to now next we discuss their linear stability and the interaction of discrete breathers with small amplitude waves showing also how they could grow or decay we also briefly discuss the existence of intraband dbs in systems with a linear discrete spectrum which are not spatially periodic random or otherwise and where linear waves cannot propagate we also show that nonlinearity may restore the existence of solutions that propagate energy we review some results concerning energy transportation by dbs and especially the main features associated with their mobility we briefly discuss new perspectives opened up by the theory for applications that in particular look especially interesting for biophysics

gmw sequences corresp the difference set design of gordon mills and welch gmw is adapted for use as a pseudorandom number generator statistical properties of the generated binary sequences including periodic correlation linear span and k tuple statistics are derived one mechanization of a gmw sequence generator is suggested and the number of sequences that can be generated with a fixed number of shift register stages and read only memory rom size is evaluated

multiple periodic solutions for two dimensional lattice dynamic systems some existence results for infinitely many periodic motions are obtained for the two dimensional lattice of m n particles with nearest neighbor interaction by means of the critical point theory especially a theorem proposed by p h rabinowitz

discrete nonlinear schrodinger equations with superlinear nonlinearities abstract in this paper we study the discrete nonlinear equation lu n u n n g n u n n z where belongs to a finite spectral gap of the operator l or it is a lower edge of a finite spectral gap by using a variant generalized weak linking theorem for strongly indefinite problem developed by schechter and zou we will study the existence of nontrivial solitons for this equation with gn is super linear

the existence of periodic solutions for a class of nonautonomous lattices with bounded coupling abstract in this paper by using a definition of topological degree and limiting arguments we study the existence of periodic solutions of a class of one dimensional nonautonomous chains with bounded coupling an approach is developed to deal with the case of lack of the a priori bounds for an infinite lattices

lower bounds on the hamming auto and cross correlations of frequency hopping sequences frequency hopping fh sequences have found wide applications in various modern fh spread spectrum communications and radar systems in fh spread spectrum communications the interference occurs when two distinct transmitters use the same frequency simultaneously in order to evaluate the goodness of fh sequence design the hamming correlation function is used as an important measure in this correspondence by considering separately the maximum hamming autocorrelation sidelobe h sub a and the maximum hamming cross correlation h sub c several new lower bounds on the size p of the frequency slot set f the sequence length l the family size m and correlation properties are established the new periodic bounds include the known lempel greenberger bounds as special case when m 2 and are tighter than the seay bounds under certain conditions when m 2 furthermore the new bounds disclose more information on the relationship between the maximum autocorrelation sidelobe and the maximum cross correlation compared with the lempel greenberger bounds and seay bounds besides the aperiodic fh bounds which have not yet been previously reported are also presented and discussed in this correspondence

breathers on diatomic fermi pasta ulam lattices we prove the existence of breathers spatially localized and time periodic oscillations in diatomic fermi pasta ulam fpu chains with arbitrary mass ratio this completes an existence result by livi spicci and mackay valid for large mass ratio the problem is formulated as a mapping in a loop space and analyzed via a discrete spatial centre manifold reduction plane wave solutions of the linearized system have frequencies in a higher optic band or a lower acoustic band for frequencies close to band edges all small amplitude solutions of the nonlinear system lie on a finite dimensional centre manifold which reduces the problem locally to the study of a finite dimensional mapping for good parameter values the map admits homoclinic orbits to 0 corresponding to discrete breathers when the fpu interaction potential satisfies a hardening condition we find breathers with frequencies slightly above the optic band or in the gap slightly above the acoustic band for a potential satisfying the opposite softening condition we obtain breathers with frequencies in the gap slightly below the optic band 2004 elsevier b v all rights reserved

optimal frequency hopping sequences a combinatorial approach frequency hopping multiple access fhma spread spectrum communication systems employing multiple frequency shift keying mfsk as data modulation technique are investigated from a combinatorial approach a correspondence between optimal frequency hopping fh sequences and partition type difference packings is first established by virtue of this correspondence fhma systems with a single optimal fh sequence each are constructed from various combinatorial structures such as affine geometries cyclic designs and difference families combinatorial recursive constructions are also presented many new infinite series of optimal fh sequences are thus obtained these new fh sequences are also useful in ultra wideband uwb communication systems

standing waves for discrete schrodinger equations in infinite lattices with saturable nonlinearities abstract in this paper we consider the periodic discrete nonlinear equation l u n u n g n u n n z lim n u n 0 where l is a jacobi operator and the nonlinearities g n s are asymptotically linear as s in the two different cases is a spectral endpoint of l or it belongs to a finite spectral gap of l we obtain the existence of nontrivial solitons of this equation by using variational methods in particular a necessary and sufficient condition is obtained for the existence of gap solitons of the nonlinear equation here solitons appear when we look for standing waves of some discrete nonlinear schrodinger equations

optimal frequency hopping sequences auto and cross correlation properties frequency hopping fh sequences play a key role in frequency hopping spread spectrum communication systems in order to evaluate the performance of fh sequences lempel and greenberger 1974 and peng and fan 2004 derived lower bounds on their hamming auto and cross correlations in this paper we construct families of fh sequences with hamming correlations meeting those bounds by combinatorial and algebraic techniques we first construct optimal families consisting of a single fh sequence with maximum hamming correlation equal to 2 from a combinatorial approach then we investigate families consisting of multiple fh sequences we provide a combinatorial characterization for such families and present a recursive method to construct them by means of this characterization we also describe two algebraic constructions for such families of fh sequences generalizing those of ding moisio and yuan 2007 as a consequence many new optimal families of fh sequences are obtained

solitary waves with prescribed speed on infinite lattices using a variant of the mountain pass theorem we prove the existence of solitary waves with prescribed speed on infinite lattices of particles with nearest neighbor interaction the problem is to solve a second order forward backward differential difference equation c 1997 academic press

new constructions for optimal sets of frequency hopping sequences in this paper two generic constructions of optimal frequency hopping sequence fhs sets employing d form functions with difference balanced property are presented they generalize the previous constructions of optimal fhs sets using m sequences and produce new optimal fhs sets that cannot be produced by the earlier constructions by choosing appropriate d form functions with difference balanced property both constructions lead to fhss with large linear complexity in addition one of the proposed constructions gives new optimal parameters of fhs sets