Digit systems over commutative rings

Let epsilon be a commutative ring with identity and P is an element of epsilon[x] be a polynomial. In the present paper we consider digit representations in the residue class ring epsilon[x]/(P). In particular, we are interested in the question whether each A is an element of epsilon[x]/(P) can be represented modulo P in the form e(0)+ e(1)x + ... + e(h)x(h), where the e(i) is an element of epsilon[x]/(P) are taken from a fixed finite set of digits. This general concept generalizes both canonical number systems and digit systems over finite fields. Due to the fact that we do not assume that 0 is an element of the digit set and that P need not be monic, several new phenomena occur in this context.

Constructions of codes through the semigroup ring B[X; 1/2(2)Z(0)] and encoding

For any finite commutative ring B with an identity there is a strict inclusion B[X; Z(0)] subset of B[X; Z(0)] subset of B[X; 1/2(2)Z(0)] of commutative semigroup rings. This work is a continuation of Shah et al. (2011) [8], in which we extend the study of Andrade and Palazzo (2005) [7] for cyclic codes through the semigroup ring B[X; 1/2; Z(0)] In this study we developed a construction technique of cyclic codes through a semigroup ring B[X; 1/2(2)Z(0)] instead of a polynomial ring. However in the second phase we independently considered BCH, alternant, Goppa, Srivastava codes through a semigroup ring B[X; 1/2(2)Z(0)]. Hence we improved several results of Shah et al. (2011) [8] and Andrade and Palazzo (2005) [7] in a broader sense. Published by Elsevier Ltd

A Note on Linear Codes over Semigroup Rings

In this paper, we introduced new construction techniques of BCH, alternant, Goppa, Srivastava codes through the semigroup ring B[X; 1 3Z0] instead of the polynomial ring B[X; Z0], where B is a finite commutative ring with identity, and for these constructions we improve the several results of [1]. After this, we present a decoding principle for BCH, alternant and Goppa codes which is based on modified Berlekamp-Massey algorithm. This algorithm corrects all errors up to the Hamming weight t ≤ r/2, i.e., whose minimum Hamming distance is r + 1.

Cyclic codes through B[X;(a/b)Z_0, with (a/b) in Q^{+} and b=a+1, and Encoding

Let B[X; S] be a monoid ring with any fixed finite unitary commutative ring B and is the monoid S such that b = a + 1, where a is any positive integer. In this paper we constructed cyclic codes, BCH codes, alternant codes, Goppa codes, Srivastava codes through monoid ring . For a = 1, almost all the results contained in [16] stands as a very particular case of this study.

Unconstrained variables of non-commutative open strings

The boundary conditions of the bosonic string theory in non-zero B-field background are equivalent to the second class constraints of a discretized version of the theory. By projecting the original canonical coordinates onto the constraint surface we derive a set of coordinates of string that are unconstrained. These coordinates represent a natural framework for the quantization of the theory.

Duality between non-commutative Yang-Mills-Chem-Simons and non-Abelian self-dual models

By introducing an appropriate parent action and considering a perturbative approach, we establish, up to fourth order terms in the field and for the full range of the coupling constant, the equivalence between the non-commutative Yang-Mills-ChernSimons theory and the non-commutative, non-Abelian self-dual model. In doing this, we consider two different approaches by using both the Moyal star-product and the Seiberg-Witten map. (C) 2003 Elsevier B.V. B.V. All rights reserved.

Duality in noncommutative topologically massive gauge field theory revisited

We introduce a master action in non-commutative space, out of which we obtain the action of the non-commutative Maxwell-Chern-Simons theory. Then, we look for the corresponding dual theory at both first and second order in the non-commutative parameter. At the first order, the dual theory happens to be, precisely, the action obtained from the usual commutative self-dual model by generalizing the Chern-Simons term to its non-commutative version, including a cubic term. Since this resulting theory is also equivalent to the non-commutative massive Thirring model in the large fermion mass limit, we remove, as a byproduct, the obstacles arising in the generalization to non-commutative space, and to the first non-trivial order in the non-commutative parameter, of the bosonization in three dimensions. Then, performing calculations at the second order in the non-commutative parameter, we explicitly compute a new dual theory which differs from the non-commutative self-dual model and, further, differs also from other previous results and involves a very simple expression in terms of ordinary fields. In addition, a remarkable feature of our results is that the dual theory is local, unlike what happens in the non-Abelian, but commutative case. We also conclude that the generalization to non-commutative space of bosonization in three dimensions is possible only when considering the first non-trivial corrections over ordinary space.

Fluctuating commutative geometry

We use the framework of noncommutative geometry to define a discrete model for fluctuating geometry. Instead of considering ordinary geometry and its metric fluctuations, we consider generalized geometries where topology and dimension can also fluctuate. The model describes the geometry of spaces with a countable number n of points. The spectral principle of Connes and Chamseddine is used to define dynamics. We show that this simple model has two phases. The expectation value , the average number of points in the universe, is finite in one phase and diverges in the other. Moreover, the dimension delta is a dynamical observable in our model, and plays the role of an order parameter. The computation of is discussed and an upper bound is found, < 2. We also address another discrete model defined on a fixed d = 1 dimension, where topology fluctuates. We comment on a possible spontaneous localization of topology.

Some comments on the F ring-Prometheus-Pandora environment

The system formed by the F ring and two close satellites, Prometheus and Pandora, has been analysed since the time that Voyager visited the planet Saturn. During the ring plane crossing in 1995 the satellites were found in different positions as predicted by the Voyager data. Besides the mutual effects of Prometheus and Pandora, they are also disturbed by a massive F ring. Showalter et al. [Icarus 100 (1992) 394] proposed that, the core of the ring has a mass which corresponds to a moonlet varying in size from 15 to 70 kin in radius which can prevent the ring from spreading due to dissipative forces, such as Poynting-Robertson drag and collisions. We have divided this work into two parts. Firstly we analysed the secular interactions between Prometheus-Pandora and a massive F ring using the secular theory. Our results show the variation in eccentricity and inclination of the satellites and the F ring taking into account a massive ring corresponding to a moonlet of different sizes. There is also a population of dust particles in the ring in the company of moonlets at different sizes [Icarus 109 (1997) 304]. We also analysed the behaviour of these particles under the effects of the Poynting-Robertson drag and radiation pressure. Our results show that the time scale proposed for a dust particle to leave the ring is much shorter than predicted before even in the presence of a coorbital moonlet. This result does not agree with the confinement model proposed by Dermott et al. [Nature 284 (1980) 309]. In 2004, Cassini mission will perform repeated observations of the whole system, including observations of the satellites and the F ring environment. These data will help us to better understand this system. (C) 2003 COSPAR. Published by Elsevier Ltd. All rights reserved.

Construction and decoding of BCH codes over finite commutative rings

BCH codes over arbitrary finite commutative rings with identity are derived in terms of their locator vector. The derivation is based on the factorization of xs -1 over the unit ring of an appropriate extension of the finite ring. We present an efficient decoding procedure, based on the modified Berlekamp-Massey algorithm, for these codes. The code construction and the decoding procedures are very similar to the BCH codes over finite integer rings. © 1999 Elsevier B.V. All rights reserved.

Strong selection against hybrids at a hybrid zone in the Ensatina ring species complex and its evolutionary implications

The analysis of interactions between lineages at varying levels of genetic divergence can provide insights into the process of speciation through the accumulation of incompatible mutations. Ring species, and especially the Ensatina eschscholtzii system exemplify this approach. The plethodontid salamanders E. eschscholtzii xanthoptica and E. eschscholtzii platensis hybridize in the central Sierran foothills of California. We compared the genetic structure across two transects (southern and northern Calaveras Co.), one of which was resampled over 20 years, and examined diagnostic molecular markers (eight allozyme loci and mitochondrial DNA) and a diagnostic quantitative trait (color pattern). Key results across all studies were: (1) cline centers for all markers were coincident and the zones were narrow, with width estimates of 730 m to 2000 m; (2) cline centers at the northern Calaveras transect were coincident between 1981 and 2001, demonstrating repeatability over five generations; (3) there were very few if any putative F1s, but a relatively high number of backcrossed individuals in the central portion of transects: and (4) we found substantial linkage disequilibrium in all three studies and strong heterozygote deficit both in northern Calaveras, in 2001, and southern Calaveras. Both linkage disequilibrium and heterozygote deficit showed maximum values near the center of the zones. Using estimates of cline width and dispersal, we infer strong selection against hybrids. This is sufficient to promote accumulation of differences at loci that are neutral or under divergent selection, but would still allow for introgression of adaptive alleles. The evidence for strong but incomplete isolation across this centrally located contact is consistent with theory suggesting a gradual increase in postzygotic incompatibility between allopatric populations subject to divergent selection and reinforces the value of Ensatina as a system for the study of divergence and speciation at multiple stages. © 2005 The Society for the Study of Evolution. All rights reserved.

Cytotoxic non-aromatic B-ring flavanones from Piper carniconnectivum C. DC.

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Construction and decoding of BCH codes over chain of commutative rings

In this paper, we present a new construction and decoding of BCH codes over certain rings. Thus, for a nonnegative integer t, let A0 ⊂ A1 ⊂···⊂ At−1 ⊂ At be a chain of unitary commutative rings, where each Ai is constructed by the direct product of appropriate Galois rings, and its projection to the fields is K0 ⊂ K1 ⊂···⊂ Kt−1 ⊂ Kt (another chain of unitary commutative rings), where each Ki is made by the direct product of corresponding residue fields of given Galois rings. Also, A∗ i and K∗ i are the groups of units of Ai and Ki, respectively. This correspondence presents a construction technique of generator polynomials of the sequence of Bose, Chaudhuri, and Hocquenghem (BCH) codes possessing entries from A∗ i and K∗ i for each i, where 0 ≤ i ≤ t. By the construction of BCH codes, we are confined to get the best code rate and error correction capability; however, the proposed contribution offers a choice to opt a worthy BCH code concerning code rate and error correction capability. In the second phase, we extend the modified Berlekamp-Massey algorithm for the above chains of unitary commutative local rings in such a way that the error will be corrected of the sequences of codewords from the sequences of BCH codes at once. This process is not much different than the original one, but it deals a sequence of codewords from the sequence of codes over the chain of Galois rings.