on the quadratic spans of debruijn sequences the quadratic span of a periodic binary sequence is defined to be the length of the shortest quadratic feedback shift register fsr that generates it an algorithm for computing the quadratic span of a binary sequence is described the required increase in quadratic span is determined for the special case of when a discrepancy occurs in a linear fsr that generates an initial portion of a sequence the quadratic spans of binary debruijn sequences are investigated an upper bound for the quadratic span of a debruijn sequence of span n is given this bound is attained by the class of debruijn sequences obtained from m sequences it is easy to see that a lower bound is n 1 but a lower bound of n 2 is conjectured the distributions of quadratic spans of debruijn sequences of span 3 4 5 and 6 are presented

on the density of irreducible nfsrs let n be a positive integer an nfsr of n stages is called irreducible if the family of output sequences of any nfsr of stages less than n is not included in that of the nfsr in this paper we prove that the density of the irreducible nfsrs of n stages is larger than 0 39 this implies that it is expected to find an irreducible nfsr of n stages among three randomly chosen nfsrs of n stages

de bruijn sequences adjacency graphs and cyclotomy we study the problem of constructing de bruijn sequences by joining cycles of linear feedback shift registers lfsrs with reducible characteristic polynomials the main difficulty for joining cycles is to find the location of conjugate pairs between cycles and the distribution of conjugate pairs in cycles is defined to be adjacency graphs let l x be a characteristic polynomial and l x l 1 x l 2 x cdots l r x be a decomposition of l x into pairwise co prime factors first we show a connection between the adjacency graph of mathrm fsr l x and the association graphs of mathrm fsr l i x 1 leq i leq r by this connection the problem of determining the adjacency graph of mathrm fsr l x is decomposed to the problem of determining the association graphs of mathrm fsr l i x 1 leq i leq r which is much easier to handle then we study the association graphs of lfsrs with irreducible characteristic polynomials and give a relationship between these association graphs and the cyclotomic numbers over finite fields at last as an application of these results we explicitly determine the adjacency graphs of some lfsrs and show that our results cover the previous ones

finite size scaling analysis of the dilute baxter wu model the finite size scaling method is used to study the critical properties of the spin 1 baxter wu model as function of the fugacity z of the vacant spin zero sites for z 0 the thermal exponent converges very quickly to the exact value yt 3 2 for z 0 yt monotonically increases beyond the value 2 this increase is interpreted as indicating a first order transition out of several possible renormalisation group flows the results seem to favour the one in which the critical baxter wu hamiltonian flows to the fixed point of the 4 state potts model with the amplitude of the marginal operator equal to zero

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

complete monotonicity and logarithmically complete monotonicity properties for the gamma and psi functions we present some complete monotonicity and logarithmically complete monotonicity properties for the gamma and psi functions this extends some known results due to s l qiu and m vuorinen

simple and robust binary self location patterns a simple method to generate a 2 d binary grid pattern which allows for absolute and accurate self location in a finite planar region is proposed the pattern encodes position information in a local way so that reading a small number of its black or white pixels at any place provides sufficient data from which the location can be decoded both efficiently and robustly

an efficient algorithm for the generation of debruijn cycles an algorithm is presented for the generation of binary debruijn sequences using feedback shift registers fsrs the algorithm is based on the principle of cycle joining through which all the cycles in the cycle structure of a fsr can be joined together to one complete cycle thereby producing a debruijn sequence of period 2 sup n where n is the length of the fsr by a proper choice of the feedback function of the fsr o 2 sup 2n sup log2n debruijn sequences of period 2 sup n can be generated requiring only 3n bits of storage and at most 4n fsr shifts for the generation of the next bit in the sequence

driven stability of nonlinear feedback shift registers with inputs driven stable nonlinear feedback shift registers nfsrs with inputs are not only able to limit error propagations in convolutional decoders but also helpful to analyze the period properties of sequences generated by a cascade connection of nfsrs in stream ciphers an nfsr is driven stable if and only if the reachable set is a subset of the basin due to lack of efficient algebraic tools the driven stability of nfsrs with inputs has been much less studied this paper continues to address this research using a boolean control network approach viewing an nfsr with input as a boolean control network we first give its boolean control network representation which is characterized with a state transition matrix some properties of the state transition matrix are then provided based on these explicit forms are given for the reachable set and the set of basin two algorithms for obtaining both the sets are provided as well compared with the exhaustive search and the existing state operator method the boolean control network approach requires lower computational complexity for those nfsrs with their stages greater than 1

a survey of full length nonlinear shift register cycle algorithms shift registers have been used to generate sequences of 0 s and 1 s for over thirty years a wide variety of applications has been made of these sequences principally communications have made use of the sequences generated one particular class of shift register sequences for which applications exist is the full length nonlinear shift register sequences these sequences are periodic and of length 2 n and all 2 n different binary n tuples appear exactly one time in a periodic portion of the sequence in this paper we discuss various algorithms which have been suggested for generating these sequences

algorithms for the generation of full length shift register sequences two algorithms are presented for the generation of full length shift register cycles also referred to as de bruijn sequences the first algorithm generates 2 k cdot g n k full cycles of length 2 n using 3n k cdot g n k bits of storage where k is a free parameter in the range 1 leq k leq 2 n 4 2 and g n k is of the order of n 2 log k the second algorithm generates about 2 n 2 4 full cycles of length 2 n using about n 2 2 bits of storage in both algorithms the time required to produce the next bit from the last n bits is close to n a possible application to the construction of stream ciphers is indicated

constructing de bruijn sequences by concatenating smaller universal cycles abstract we present sufficient conditions for when an ordering of universal cycles 1 2 m for disjoint sets s 1 s 2 s m can be concatenated together to obtain a universal cycle for s s 1 s 2 s m when s is the set of all k ary strings of length n the result of such a successful construction is a de bruijn sequence our conditions are applied to generalize two previously known de bruijn sequence constructions and then they are applied to develop three new de bruijn sequence constructions

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

a scalable method for constructing galois nlfsrs with period 2 n 1 using cross join pairs a method for constructing n stage galois nlfsrs with period 2n 1 from n stage maximum length lfsrs is presented nonlinearity is introduced into state cycles by adding a nonlinear boolean function to the feedback polynomial of the lfsr each assignment of variables for which this function evaluates to 1 acts as a crossing point for the lfsr state cycle the effect of nonlinearity is cancelled and state cycles are joined back by adding a copy of the same function to a later stage of the register the presented method requires no extra time steps and it has a smaller area overhead compared to the previous approaches based on cross join pairs it is feasible for large n

a recipe for randomness despite many diverse theories that address closely related themes e g probability theory algorithmic complexity cryptoanalysis and pseudorandom number generation a near void remains in constructive methods certified to yield the desired random output herein we provide explicit techniques to produce broad sets of both highly irregular finite and normal infinite sequences based on constructions and properties derived from approximate entropy apen a computable formulation of sequential irregularity furthermore for infinite sequences we considerably refine normality by providing methods for constructing diverse classes of normal numbers classified by the extent to which initial segments deviate from maximal irregularity

complete monotonicity of some functions involving polygamma functions in the present paper we establish necessary and sufficient conditions for the functions x a j i x b and a j i x b x j i 1 x b respectively to be monotonic and completely monotonic on 0 where i n a 0 and b 0 are scalars and j i x are polygamma functions

a transformation from the fibonacci to the galois nlfsrs conventional nonlinear feedback shift registers nlfsrs use the fibonacci configuration in which the feedback is applied to the last bit only in this paper we show how to transform a fibonacci nlfsr into an equivalent nlfsr in the galois configuration in which the feedback can be applied to every bit such a transformation can potentially reduce the depth of the circuits implementing feedback functions thus decreasing the propagation time and increasing the throughput

fault tolerant ring embedding in de bruijn networks a method of embedding a ring in a d ary de bruijn multiprocessor network in the event of multiple node processor failures is presented in particular the algorithm guarantees that a 2 sup n n 1 node ring will be found in a binary de bruijn network with a single faulty node where 2 sup n is the total number of nodes in the network it is also shown that a d sup n 1 node ring can always be found in the presence of d 1 link failures when d is a prime power and the network contains d sup n nodes the latter is accomplished by constructing d 1 edge disjoint cycles each of length d sup n 1 a modification of the graph that allows it to admit a hamiltonian cycle in the event of d 1 edge failures is also discussed

monotonicity of quotients of theta functions related to an extremal problem on harmonic measure abstract we prove that for fixed u and v such that u v 0 1 2 the quotients j u i t j v i t j 1 2 3 4 of the theta functions are monotone on 0 t the case v 0 has been used by the second author to study a generalization of gonchar s problem on harmonic measure of radial slits

measuring the clustering effect of bwt via rle abstract the burrows wheeler transform bwt is a reversible transformation on which are based several text compressors and many other tools used in bioinformatics and computational biology the bwt is not actually a compressor but a transformation that performs a context dependent permutation of the letters of the input text that often create runs of equal letters clusters longer than the ones in the original text usually referred to as the clustering effect of bwt in particular from a combinatorial point of view great attention has been given to the case in which the bwt produces the fewest number of clusters cf 5 16 21 23 in this paper we are concerned about the cases when the clustering effect of the bwt is not achieved for this purpose we introduce a complexity measure that counts the number of equal letter runs of a word this measure highlights that there exist many words for which bwt gives an un clustering effect that is bwt produce a great number of short clusters more in general we show that the application of bwt to any word at worst doubles the number of equal letter runs moreover we prove that this bound is tight by exhibiting some families of words where such upper bound is always reached we also prove that for binary words the case in which the bwt produces the maximal number of clusters is related to the very well known artin s conjecture on primitive roots the study of some combinatorial properties underlying this transformation could be useful for improving indexing and compression strategies

convexity of quotients of theta functions abstract for fixed u and v such that 0 u v 1 2 the monotonicity of the quotients of jacobi theta functions namely j u i t j v i t j 1 2 3 4 on 0 t has been established in the previous works of a yu solynin k schiefermayr and solynin and the first author in the present paper we show that the quotients 2 u i t 2 v i t and 3 u i t 3 v i t are convex on 0 t

monotonicity results for dirichlet l functions abstract we present some monotonicity results for dirichlet l functions associated to real primitive characters we show in particular that these dirichlet l functions are far from being logarithmically completely monotonic also we show that unlike in the case of the riemann zeta function the problem of comparing the signs of d k d s k log l s at any two points s 1 s 2 1 is subtler

constructions of snake in the box codes for rank modulation snake in the box code is a gray code which is capable of detecting a single error gray codes are important in the context of the rank modulation scheme which was suggested recently for representing information in flash memories for a gray code in this scheme the codewords are permutations two consecutive codewords are obtained using the push to the top operation and distance measure is defined on permutations in this paper the kendall s metric is used as the distance measure we present a general method for constructing such gray codes we apply the method recursively to obtain a snake of length m2n 1 2n 1 2n 1 m2n 1 for permutations of s2n 1 from a snake of length m2n 1 for permutations of s2n 1 thus we have lim n m2n 1 s2n 1 0 4338 improving on the previous known ratio of lim n 1 n using the general method we also present a direct construction this direct construction is based on necklaces and it might yield snakes of length 2n 1 2 2n 1 for permutations of s2n 1 the direct construction was applied successfully for s7 and s9 and hence lim n m2n 1 s2n 1 0 4743