Strings Of Ones Research Articles

Numerical representations of AB-type copolymer complexes: analysis of 1H NMR chemical shift patterns in terms of a Smith–Cantor set

When considering the possibility of storing information in the sequence of monomer residues within an AB-type copolymer chain, it is constructive to model that sequence as a string of ones and zeros. The intramolecular environment around any given digit (say a “1”) can then be represented by another string of integers—a code—obtained by summing pairs of digits at equivalent positions, in both directions, from that digit. The code can include only integers 0, 1 and 2, and can represent a number in any base b higher than 2. In base b = 3 the resulting set of codes includes all numbers (because only digits 0, 1 and 2 occur in ternary expansions), but in any base b > 3 the codes define a limited set of numbers comprising a fractal we term a Smith–Cantor set. The 1H NMR spectrum of a random, AB-type co(polyester-imide) shows, on complexation with pyrene, a pattern of complexation shifts approximating very closely to the Smith–Cantor set for which b = 4. Other co(polyimide) complexes show a 1H NMR pattern corresponding to a specific sub-set of this fractal. The sub-set arises from a “stop-at-zero” limitation, whereby digits in the initial string are set to zero for code-generating purposes if they occur beyond a zero, as viewed from the central “1”. The limitation arises in copolymers where pyrene binds by intercalation between pairs of adjacent diimide residues. This numerical approach provides a complete, unifying theory to account for the emergence of fractal character in the 1H NMR spectra of AB-type copolymer complexes.

Integer codes correcting sparse byte errors

In public optical networks, the data are scrambled with a xu + 1 self-synchronous scramblers (SSSs). The reason for this is to avoid long strings of ones or zeros, which might affect the receiver synchronization. Unfortunately, the use of SSSs is always related to the problem of duplication of channel errors. More precisely, each error occurring during the transmission will be duplicated u bits later. In this paper, we present a low-cost solution to this problem based on integer codes capable of correcting sparse byte errors.

All of life in pi? [Reflections

MANY YEARS AGO, WHILE HAVING DINNER WITH AN eminent mathematician, I mentioned a scene regarding the number pi in Carl Sagan's novel, Contact. A scientist from Earth (Jodie Foster in the movie) visits a distant planet and learns that the aliens there have calculated pi out to many places. Very far out in the number's expansion the decimal digits turn to a very long string of ones and zeros–a message embedded in pi.

Estimating survival of thrushes: Statistical model of capture-recapture probabilities

Astochastic modeling technique provides for the correct partition of the “return rate” of marked animals into survival probability (φ) and capture probability (p). Application of this method is enhanced by the program MARK, available free of charge through the Internet (Cooch and White, 2009). The input data for the program are the capture histories of marked animals: strings of ones and zeros indicating the presence or absence of an individual among captures (or sightings) during subsequent recapture seasons. The probability of any history is a product of binomial probabilities φ, p, or their complements (1 − φ) and (1 − p) for each year of observing the individual. Assigning certain values to the parameters φ and p, it is possible to predict the combination of histories of all marked individuals and to assess the prediction probability. The survival parameters for different years and cohorts of individuals can be set either identical or different, and the recapture parameters can be set in different ways. It is possible to constrain the parameters according to a tested hypothesis in the form of a specific model. Within the specified constraints, the program searches for parameter values that describe the observed composition of histories with the maximum likelihood. The program computes the estimates for each parameter, their confidence limits, and the overall model probability. A set of tools is provided for testing model goodness-of-fit under the assumption of independent fates of individuals and their equiprobable survival. Other tools aid in the proper selection of the model that provides for the best parity between detail and precision in describing reality. The method was applied to determine survival probabilities using 20-year capture, marking, and sighting data series for populations of four thrush species (genera Turdus and Zoothera) breeding in the middle taiga subzone of the Enisei River floodplain. The capture probabilities were almost independent of capture efforts but significantly differed among the species and sexes. The estimates for survival probabilities of Siberian migratory thrush populations were lower than those for the resident species from both the tropics and midlatitudes with a marine climate (data by Ricklefs, 1997). Two factors—the average annual temperature, which influences birds during their migrations, and climatic seasonality (temperature difference between summer and winter) in the breeding area—best fit the latitudinal pattern of survival (R 2 = 0.90). The final survival of migrants reflects an adaptive life cycle tradeoff of using superabundant summer resources in the breeding area at the cost of avoidance of severe winter conditions.

Sequential Structuring Element for CFG Induction Using Genetic Algorithm

paper investigates the induction of Context free Grammar with genetic algorithm. The genetic algorithm is not very effective at this (1). To overcome this problem we investigate combined effect of two methods for structuring the chromosomes. The first is to bias the distribution of Non-terminals in the chromosome at the time of chromosome generation as well as updating. The latter approach is sequential structural mapping from chromosome to grammar. It is shown that biasing the chromosome in this fashion produces extremely fast convergence as shown in the result section. Grammars are an extremely general and useful tool with many applications. These include the higher levels of signal processing, such as pattern recognition. However, the application of grammars is limited by the algorithms we can apply to infer them from samples of data. The main contribution of this paper is the effective decoding of the context free grammar from chromosomes with the distinct approaches mentioned. Genetic Algorithms (GAs) were invented by John Holland in the 1960s. In contrast with Evolution Strategies and Evolutionary Programming, Holland's original goal was not to design algorithms to solve specific problems, but rather to formally study the phenomenon of adaptation as it occurs in nature and to develop ways in which the mechanisms of adaptation might be utilized into computer systems. Holland's 1975 book Adaptation in Natural and Artificial Systems presented the Genetic Algorithm as an abstraction of biological evolution and gave a theoretical framework for adaptation under the GA. Holland's GA is a method for moving from one population of (e.g., strings of ones and zeros, or bits) to a new population by using a kind of natural selection together with the genetics-inspired operators of Crossover, Mutation, and Inversion. Each Chromosome consists of Genes (e.g., bits), each Gene being an instance of a particular (e.g., 0 or 1). The Selection Operator chooses those Chromosomes in the Population that will be allowed to reproduce, and on average the fitter Chromosomes produce more offspring than the less fit ones. Crossover exchanges subparts of two Chromosomes, roughly mimicking biological recombination between two single-Chromosome (haploid) organisms; Mutation randomly changes the allele values of some locations in the Chromosome; and Inversion reverses the order of a contiguous section of the Chromosome, thus rearranging the order in which genes are arrayed.

A Difference Equation for Strings of Ones

A Difference Equation for Strings of Ones

Fermat's Last Theorem in Combinatorial Form

(1988). A Difference Equation for Strings of Ones. The American Mathematical Monthly: Vol. 95, No. 7, pp. 636-638.

A Difference Equation for Strings of Ones

A Difference Equation for Strings of Ones

The geometry of markoff numbers

It turns out that v(0) ~ 1/V~ with equality only if 0 is a "noble number ' ' t whose continued fraction expansion ends in a string of ones. In 1879 Markoff improved this result by showing that there is a discrete set of values v i decreasing to 1/3 so that if v(0) > 1/3 then v(0) = v i for some i [8]. The numbers v i are called the M a r k o f f s p e c t r u m and the corresponding O's, M a r k o f f i rra t ional i t ies . Markoff irrationalities have cont inued fraction expansions whose tails satisfy a very special set of rules, often called the Dickson rules [4]. The tail 1,1,1 . . . . is the simplest example. What these rules are will become clear as we proceed. Markoff gave a prescription for determining all of these irrationalities starting from the solutions of a certain diophantine equation and linked his results to the minima of associated binary quadratic forms. Recently there has been a revival of interest in this topic, starting from the realisation that each v i together with its corresponding class of Markoff irrationalities is associated to a simple (non-self-intersecting) loop on the punctured torus, as shown in Figure 1. The details have been worked out most fully by A. Haas [6], based on earlier work of Cohn [2, 3], and Schmidt [10]. Lehner, Scheingorn and Beardon [7] tackle the same problem but base their analysis on a sphere with four punctures. It turns out that almost all the results follow from some rather simple observations about the way in which straight lines cut certain tessellations of the Euclidean and hyperbolic planes and it is these ideas which we want to explain here. Before understanding approximations we shall need to make a fairly lengthy digression to investigate such cutting patterns, for which I offer no apology, for the approach via the patterns is quite as fascinating as Markoff's theory itself.

Evaluation of cellular mobile telephone data receivers

Some measurement techniques and results employed to evaluate advanced mobile phone system (AMPS) data receivers being driven by a Rayleigh fading channel are described. These performance measurements were used as a mechanism for comparing the design effectiveness of various bit clock recovery systems. Of five data receiver types evaluated, two models employed a full-wave rectifier in the bit clock recovery system. In this system nominal diode unbalance may cause the derived bit clock to lock 180° from the phase required to properly decode the incoming bit stream. This condition may even occur at high carrier-to-noise ratios when message structures containing long strings of ones or zeros are received. The other data receiver types employed digital signal processing for clock recovery to circumvent this phase ambiguity problem. The performance data of five data receivers are compared to noncoherent frequency-shift keying (FSK) as a model. In most cases, the test results are in good agreement with this model. The performance measurements presented include derived clock jitter and single and average bit error rates as a function of average carrier-to-noise ratios. The implementation of test instrumentation and the interpretation of test results are discussed. The objective is to stress the capability of simulation measurements to evaluate mobile receiver designs in a laboratory environment.

Program Behavior at the Symbolic Level

No application programmer writes machine-language programs–i.e., strings of ones and zeroes. That primitive pursuit has long been reserved for those few who create the very first modules of a software system for new hardware. Instead, programmers make use of a wide spectrum of symbolic programming languages, ranging from assembly code to high-level languages such as Fortran, Cobol, and the Algol family. Every programming language has semantics which define some abstract machine. For the assembly-language programmer this machine bears a great resemblance to the actual hardware on which the program will be interpreted, but even here the programmer will frequently use system-defined subroutines or macros which represent extensions of the base hardware facilities. The high-level language programmer's abstract machine reflects the control mechanisms and data structures characteristic of the language. The Fortran programmer, for example, can think in terms of multidimensional array structures, DO loops, subprogram facilities, and so on. In principle he need never be concerned with the manner in which his abstract Fortran machine is to be realized by a particular hardware and software system. The user of a modern electronic hand calculator needs no knowledge of the works inside the box, and a modern high-level language system should present to its users an equally consistent environment, completely defined in terms of the syntax and semantics of the source language.