Specifying periodic words by restrictions

This paper proposes two variants of improved Compression by Substring Enumeration (CSE) with a finite alphabet. In previous studies on CSE, an encoder utilizes inequalities which evaluate the number of occurrences of a substring and a minimal forbidden word (MFW) to be encoded. Also, its codeword length is proportional to the difference between the upper and lower bounds deduced from the inequalities, but the lower bound is not tight. Therefore, in this paper, we derive a new tight lower bound and consequently propose a new CSE algorithm using the new inequality. We also propose a new encoding order of substrings and MFWs which are sorted by row and column marginal totals of the proper substrings and MFWs, instead of lexicographical order used in previous studies. We then propose a new CSE algorithm which is the first proposed CSE algorithm using the new encoding order. Experimental results show that compression ratios of all files of the Calgary corpus in the proposed algorithms are better than those of a previous study on CSE with a finite alphabet. Moreover, compression ratios under the second proposed CSE get better than or equal to that under a well-known compressor for 11 files amongst 14 files in the corpus.

In this paper, we extend the construction by Vu and Gong for families of M-ary sequences of period q - 1 from the array structure of an M-ary Sidelnikov sequence of period q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> - 1, where q is a prime power and M|q - 1. The construction now applies to the cases of using any period q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sup> -1 for 3 ≤ d <; (1/2)(√q - (2/√q) + 1) and q > 27. The proposed construction results in a family of M-ary sequences of period q-1 with: 1) the correlation magnitudes, which are upper bounded by (2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d -1</sup> )√q +1 and 2) the asymptotic size of (M -1)q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d-1</sup> /d as q increases. We also characterize some subsets of the above of size ~(r - 1)q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d-1</sup> /d but with a tighter upper bound (2d - 2)√q + 2 on its correlation magnitude. We discuss reducing both time and memory complexities for the practical implementation of such constructions in some special cases. We further give some approximate size of the newly constructed families in general and an exact count when d is a prime power or a product of two distinct primes. The main results of this paper now give more freedom of tradeoff in the design of M-ary sequence family between the family size and the correlation magnitude of the family.

We consider a single-hop wireless sensor network in which each sensor node has an individual elastic data stream to transmit to each other node. We refer to this traffic pattern as unicast in this paper. The network has multiple slotted channels available for the data transmissions. To guarantee successful unicast within a bounded delay, we consider deterministic schemes that pre-assign each node a periodic schedule sequence to schedule transmitting and receiving at each time slot. The sequence period should be minimized since it upper bounds the unicast delay. We have investigated both synchronous TDMA sequences and asynchronous sequences. Since accurate time synchronization is difficult to achieve in sensor networks, we mainly present analysis and design for asynchronous sequences. In this paper, for a group-based channel assignment, we present a lower bound on the common period and propose a sequence construction method by which the period can achieve the same order as the lower bound. We also analyze optimal transmitting and receiving probabilities for two random schemes and compare their frequency utilization efficiency. Finally, unicast delay and energy consumption performance are compared by simulations.

Given an infinite word with values in a finite alphabet that admit frequencies for all its factors, the smallest frequency function associates with a given positive integer the smallest frequency for factors of this length, and the recurrence function measures the gaps between successive occurrences of factors. We develop in this paper the relations between the growth orders of the smallest frequency and of the recurrence functions. As proved by M. Boshernitzan, linearly recurrent words are characterized in terms of the smallest frequency function. In this paper, we see how to extend this result beyond the case of linearly recurrent words. We first establish a general relation between the smallest frequency and the recurrence functions. Given a lower bound for the smallest frequency function, this relation provides an upper bound for the recurrence function which involves the primitive of this lower bound. We then see how to improve this relation in the case of Sturmian words where the product of the smallest frequency and the recurrence function is bounded for most of the lengths of factors. We also indicate how to construct Sturmian words having a prescribed behaviour for the smallest frequency function, and for the product of the smallest frequency and the recurrence function.

Consider finite sequences $X_{[1,n]}=X_{1},\ldots,X_{n}$ and $Y_{[1,n]}=Y_{1},\ldots,Y_{n}$ of length $n$, consisting of i.i.d. samples of random letters from a finite alphabet, and let $S$ and $T$ be chosen i.i.d. randomly from the unit ball in the space of symmetric scoring functions over this alphabet augmented by a gap symbol. We prove a probabilistic upper bound of linear order in $(\ln(n))^{1/4}n^{3/4}$ for the deviation of the score relative to $T$ of optimal alignments with gaps of $X_{[1,n]}$ and $Y_{[1,n]}$ relative to $S$. It remains an open problem to prove a lower bound. Our result contributes to the understanding of the microstructure of optimal alignments relative to one given scoring function, extending a theory begun in (J. Stat. Phys. 153 (2013) 512–529).

A powerful abelian square-free substitution over 4 letters

Bell–CHSH is an inequality about unconditional expectations: under measurement independence, Bell locality, and bounded outcomes, the CHSH value satisfies S≤2. Experimental correlators, however, are often computed on an accepted subset of trials defined by detection logic, coincidence matching, quality cuts, and analysis windows. We model this by an acceptance probability γ(a,b,λ)∈[0,1] and the resulting accepted hidden-variable law νab obtained by weighting the measurement-independent prior ρ by γ and renormalizing. If νab depends on the setting pair then the four correlators entering CHSH are expectations under four different measures, and a Bell-local measurement-independent model can yield Sobs&gt;2 by selection alone. We quantify the required setting dependence in total variation (TV) distance. For any reference law μ we prove the sharp bound Sobs≤2+2∑q∈QTV(νq,μ) for a CHSH quartet Q. Optimizing over μ yields the intrinsic dispersion bound Sobs≤2+2ΔQ, and, in particular, Sobs≤min{4,2+6DQ}, where DQ is the quartet TV diameter. The constants are optimal. Consequently, reproducing Tsirelson’s value 22 within Bell-local measurement-independent models via setting-dependent acceptance requires ΔQ≥2−1 (hence, DQ≥(2−1)/3). We then propose a two-lane experimental audit protocol: (i) prior-relative fair-sampling diagnostics using tags recorded on all trials, and (ii) prior-free dispersion diagnostics using accepted-tag distributions across settings, with ΔQ,X computable by linear programming on finite tag alphabets.

We investigate two types of altered Fibonacci numbers obtained by adding or subtracting a specific value $\{a\}$ from the square of the $n^{th}$ Fibonacci numbers $G^{(2)}_{F(n)}(a)$ and $H^{(2)}_{F(n)}(a)$. These numbers are significant as they are related to the consecutive products of the Fibonacci numbers. As a result, we establish consecutive sum-subtraction relations of altered Fibonacci numbers and their Binet-like formulas. Moreover, we explore greatest common divisor (GCD) sequences of r-successive terms of altered Fibonacci numbers represented by $\left\{G^{(2)}_{F(n), r}(a)\right\}$ and $\left\{H^{(2)}_{F(n), r}(a)\right\}$ such that $r\in\{1,2,3\}$ and $a\in\{1,4\}$. The sequences are based on the GCD properties of consecutive terms of the Fibonacci numbers and structured as periodic or Fibonacci sequences.

A promising direction for constructing cryptographically stable pseudo-random sequence generators is an application of transformations in a group of points of elliptic and hypereliptic curves. This will allow building evidence-stable crypto algorithms, the problem of finding the private key in which is associated with solving a theoretically complex elliptic curve discrete logarithm problem. This paper proposes a method for generating pseudo-random sequences of the maximal period using transformations on elliptic curves. This method consists in the application of recurrent transformations with sequential formation of elements of points group of elliptic curves. This allows providing the maximum period of pseudo-random sequences with the reduction of the problem of finding the private key to the solution of the theoretically complex elliptic curve discrete logarithm problem. The block diagram of the device for generating pseudo-random sequences and the scheme for generating the internal states of the generator are given. We also present the results of statistical testing of some generators, which show that the generated sequences are indistinguishable in a statistical sense from truly random ones.KeywordsPseudo-random sequencesElliptic curvesMaximum period

PAM sequence design for dimmable visible light communication

The fundamental limits of channels with mismatched decoding are addressed. A general formula is established for the mismatch capacity of a general channel, defined as a sequence of conditional distributions with a general decoding metrics sequence. We deduce an identity between the Verdú-Han general channel capacity formula, and the mismatch capacity formula applied to Maximum Likelihood decoding metric. Further, several upper bounds on the capacity are provided, and a simpler expression for a lower bound is derived for the case of a non-negative decoding metric. The closely related problem of threshold mismatched decoding is also studied, and a general expression for the threshold mismatch capacity is obtained. As an example of threshold mismatch capacity, we state a general expression for the erasures-only capacity of the finite input and output alphabet channel. We observe that for every channel there exists a (matched) threshold decoder which is capacity achieving. Additionally, necessary and sufficient conditions are stated for a channel to have a strong converse.

A detailed study of charge transport properties of synthetic and genomic DNA sequences is reported. Genomic sequences of the Chromosome 22, λ-bacteriophage, and D1s80 genes of Human and Pygmy chimpanzee are considered in this work, and compared with both periodic and quasiperiodic (Fibonacci) sequences of nucleotides. Charge transfer efficiency is compared for all these different sequences, and large variations in charge transfer efficiency, stemming from sequence-dependent effects, are reported. In addition, basic characteristics of tunneling currents, including contact effects, are described. Finally, the thermoelectric power of nucleobases connected in between metallic contacts at different temperatures is presented.

In this paper we study a class of dynamical systems defined by Pfaffian maps. It is a sub-class of o-minimal dynamical systems which capture rich continuous dynamics and yet can be studied using finite bisimulations. The existence of finite bisimulations for o-minimal dynamical and hybrid systems has been shown by several authors; see e.g. Brihaye et al (2004), Davoren (1999), Lafferriere et al (2000). The next natural question to investigate is how the sizes of such bisimulations can be bounded. The first step in this direction was done by Korovina et al (2004) where a double exponential upper bound was shown for Pfaffian dynamical and hybrid systems. In the present paper we improve this bound to a single exponential upper bound. Moreover we show that this bound is tight in general, by exhibiting a parameterized class of systems on which the exponential bound is attained. The bounds provide a basis for designing efficient algorithms for computing bisimulations, solving reachability and motion planning problems.

The paper deals with the problem of the carrier frequency-offset (CFO) estimation for wireless communications. Since the well-known maximum-likelihood (ML) CFO estimator is rather complex, many complexity-reduced CFO estimators have so far been proposed. However, most of them are based on periodic training sequences and have limited estimation range. We propose a general framework called composite frequency- offset estimator (CFE), which can extend the estimation range of any range-limited correlation-based CFO estimator up to the full transmission spectrum. We first formulate the CFO estimation as a problem of composite hypothesis testing, and then solve the problem by the composite hypothesis testing approaches. We show that the composite frequency-offset estimate is the ML estimate as long as the original range-limited estimator is unbiased and attains the Cramer-Rao Lower Bound. The CFE also allows a flexible tradeoff between the estimation range and computational complexity. Finally, numerical results show the efficiency of the proposed method.

The generalized Marcum Q-function, Qm (a,b), can be interpreted geometrically as the probability of a 2m- dimensional, real, Gaussian random vector z2m, whose mean vector has a Frobenius norm of a, lying outside of a hyperball B2m O,b of 2m dimensions, with radius b, and centered at the origin O. Based on this geometric view, we propose some new generic exponential bounds on Qm (a, b) for the case where m is an integer. These generic exponential bounds are obtained by computing the probability of z2m lying outside of some bounding geometrical shapes whose surfaces tightly enclose, or are tightly enclosed by the surface of B2m O,b ldr The bounding geometrical shapes used in the derivation consist of an arbitrarily large number of parts. As their closeness of fit with B2m O,b improves, the generic exponential bounds obtained approach the exact value of Qm (a,b). These generic exponential bounds only involve the exponential function, and thus, are easy to handle in analytical computations. Our numerical results show that when evaluated with a few terms, these generic exponential bounds are much tighter than the existing exponential bounds in the literature for a wide range of arguments. For the case of a > b, our generic upper exponential bound is the first upper exponential bound on Qm (a,b).