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The family of rings of the form \frac{\mathbb{Z}_{4}\left \langle x,y \right \rangle}{\left \langle x^2-a,y^2-b,yx-xy-2(c+dx+ey+fxy) \right \rangle} is investigated which contains the generalized Hamilton quaternions over $\Z_4$. These rings are local rings of order 256. This family has 256 rings contained in 88 distinct isomorphism classes. Of the 88 non-isomorphic rings, 10 are minimal reversible nonsymmetric rings and 21 are minimal abelian reflexive nonsemicommutative rings. Few such examples have been identified in the literature thus far. The computational methods used to identify the isomorphism classes are also highlighted. Finally, some generalized Hamilton quaternion rings over $\Z_{p^s}$ are characterized.

Background Chronic pain conditions are common among both male and female Iraq/Afghanistan-era veterans and can have substantial negative impacts on quality of life and function. Although in general women tend to report higher levels of pain intensity than men, findings remain mixed on whether gender differences in pain exist in Iraq/Afghanistan-era veterans. Additionally, the relationships between functional impairment, pain intensity, and gender remain unknown. Methods This project examined gender differences in pain intensity and pain interference in 875 male and female Iraq/Afghanistan-era veterans. Nonparametric Wilcoxon rank-tests examined gender differences in pain scores. Multivariable generalized linear regression modeling was used to evaluate the magnitude of pain intensity and interference across levels of chronicity and gender, and to evaluate the role of chronicity in gender effects in measures of pain and function. Results Pain intensity and interference scores were significantly greater among both male and female veterans reporting chronic pain relative to acute pain. Women veterans endorsed higher levels of pain intensity and pain interference compared with men. Results derived from multivariable analyses implicated pain intensity as a factor underlying gender differences in functional impairment among chronic pain sufferers, indicating that gender differences in functional measures were eliminated after controlling statistically for pain intensity. Conclusions Results demonstrate that the effects of functional impairment are impacted by pain intensity, and not by gender.

In a paper on the taxonomy of 2-primal rings, examples of various types of rings that are related to commutativity such as reduced, symmetric, duo, reversible and PS~I were given in order to show that the ring class inclusions were strict. Many of the rings given in the examples were infinite. In this paper, where possible, examples of minimal finite rings of the various types are given. Along with the rings in the previous taxonomy, NI, abelian and reflexive rings are also included.

It has recently been shown that a minimal reversible nonsymmetric ring has order 256 answering a question originally posed in a paper on a taxonomy of 2-primal rings. In a different work, questions on minimal rings having to do with this taxonomy were also answered. In this work it is shown that a minimal abelian reflexive nonsemicommutative ring also has order 256, a related question left open in these previous works. It is also shown here that [Formula: see text] is such a minimal ring. This is a consequence of the main result of the paper which is that a finite abelian reflexive ring of order [Formula: see text] for some prime [Formula: see text] and [Formula: see text] is reversible.

Codes over commutative Frobenius rings are studied with a focus on local Frobenius rings of order 16 for illustration. The main purpose of this work is to present a method for constructing a generating character for any commutative Frobenius ring. Given such a character, the MacWilliams identities for the complete and symmetrized weight enumerators can be easily found. As examples, generating characters for all commutative local Frobenius rings of order 16 are given. In addition, a canonical generator matrix for codes over local non-chain rings is discussed. The purpose is to show that when working over local non-chain rings, a canonical generator matrix exists but is less than useful which emphases the difficulties in working over such rings.

Let [math] be a sequence of [math] pairwise coprime positive integers, [math] , and [math] be a sequence of [math] different colors. Let [math] be an [math] matrix of colors in which row [math] consists of blocks of [math] consecutive entries of the same color with colors 0 through [math] repeated cyclically. The monochromatic column problem is to determine the number of columns of [math] in which every entry is the same color. The solution for a prime number of colors is provided.

In this paper we study polycyclic codes of length $p^{s_1} \times \cdots \times p^{s_n}$\ over $\F_{p^a}$\ generated by a single monomial. These codes form a special class of abelian codes. We show that these codes arise from the product of certain single variable codes and we determine their minimum Hamming distance. Finally we extend the results of Massey et. al. in \cite{MASSEY_1973} on the weight retaining property of monomials in one variable to the weight retaining property of monomials in several variables.

Convolutional codes have appeared in the literature endowed with sufficient additional algebraic structure to be considered as (left) ideals of a (code-ambient) automorphism-twisted polynomial ring with coefficients in a (word-ambient) semisimple finite group ring. In this paper we extend the present scope of the theory by considering a code-ambient twisted polynomial ring having, in addition to an automorphism σ , the action of a σ -derivation δ . In addition, we develop the basic theory without any specific restrictions for the semisimple finite word-ambient ring. This second element therefore extends even the original notions of both cyclic and group convolutional codes considered thus far in the literature. Among other results, in this paper we develop a matrix-based approach to the study of our extended notion of group convolutional codes (and therefore of cyclic convolutional codes as well), inspired by the use of circulant matrices by Gluesing-Luerssen and Schmale, and then use it to extend to this level the results on the existence of dual codes that were originally established by those authors for cyclic codes (in the narrower sense without a σ -derivation). Various examples illustrate the potential value of extending the search for good convolutional codes in this direction.

Dual codes are defined with respect to non-degenerate sesquilinear or bilinear forms over a finite Frobenius ring. These dual codes have the properties one expects from a dual code: they satisfy a double-dual property, they have cardinality complementary to that of the primal code, and they satisfy the MacWilliams identities for the Hamming weight.

Marks showed that F 2 Q 8 , the F 2 group algebra over the quaternion group, is a reversible nonsymmetric ring, then questioned whether or not this ring is minimal with respect to cardinality. In this work, it is shown that the cardinality of a minimal reversible nonsymmetric ring is indeed 256. Furthermore, it is shown that although F 2 Q 8 is a duo ring, there are also examples of minimal reversible nonsymmetric rings which are nonduo.

article number 1750161; International audience; Given a finite ring $A$ which is a free left module over a subring $R$ of $A$, two types of $R$-bases are defined which in turn are used to define duality preserving maps from codes over $A$ to codes over $R$. The first type, pseudo-self-dual bases, are a generalization of trace orthogonal bases for fields. The second are called symmetric bases. Both types are illustrated with skew cyclic codes which are codes that are $A$-submodules of the skew polynomial ring $A[X;\theta]/\langle X^n-1\rangle$ (the classical cyclic codes are the case when $\theta=id$). When $A$ is commutative, there exists criteria for a skew cyclic code over $A$ to be self-dual. With this criteria and a duality preserving map, many self-dual codes over the subring $R$ can easily be found. In this fashion, numerous examples are given, some of which are not chain or serial rings.

Cyclic, negacyclic and constacyclic codes are part of a larger class of codes called polycyclic codes; namely, those codes which can be viewed as ideals of a factor ring of a polynomial ring. The structure of the ambient ring of polycyclic codes over GR(p^a,m) and generating sets for its ideals are considered. Along with some structure details of the ambient ring, the existance of a certain type of generating set for an ideal is proven., arXiv admin note: text overlap with arXiv:0906.4008

In this paper we focus on two generalizations of the notion of cyclicity of codes: polycyclic codes and sequential codes. We establish a duality between these two generalizations and also show connections between them and other well-known generalizations of cyclicity such as the notions of negacyclicity and constacyclicity. In particular, it is shown that a code $C$ is sequential and polycyclic if and only if $C$ and its dual C⊥ are both sequential if and only if $C$ and its dual C⊥ are both polycyclic. Furthermore, any one of these equivalent statements characterizes the family of constacyclic codes.

Repeated root Cyclic and Negacyclic codes over Galois rings have been studied much less than their simple root counterparts. This situation is beginning to change. For example, repeated root codes of length ps, where $p$ is the characteristic of the alphabet ring, have been studied under some additional hypotheses. In each one of those cases, the ambient space for the codes has turned out to be a chain ring. In this paper, all remaining cases of cyclic and negacyclic codes of length ps over a Galois ring alphabet are considered. In these cases the ambient space is a local ring with simple socle but not a chain ring. Nonetheless, by reducing the problem to one dealing with uniserial subambients, a method for computing the Hamming distance of these codes is provided.

Let p"1, p"2,..., p"n be a sequence of n pairwise coprime positive integers, and let P=p"1p"2...p"n. Let 0,1,...,m-1 be a sequence of m different colors. Let A be an nxmP matrix of colors in which row i consists of blocks of p"i consecutive entries of the same color, with colors 0 through m-1 repeated cyclically. The Monochromatic Column problem is to determine the number of columns of A in which every entry is the same color. The solution for m=3 colors is presented and proved.

Human cancer is caused by the accumulation of mutations in oncogenes and tumor suppressor genes. To catalog the genetic changes that occur during tumorigenesis, we isolated DNA from 11 breast and 11 colorectal tumors and determined the sequences of the genes in the Reference Sequence database in these samples. Based on analysis of exons representing 20,857 transcripts from 18,191 genes, we conclude that the genomic landscapes of breast and colorectal cancers are composed of a handful of commonly mutated gene “mountains” and a much larger number of gene “hills” that are mutated at low frequency. We describe statistical and bioinformatic tools that may help identify mutations with a role in tumorigenesis. These results have implications for understanding the nature and heterogeneity of human cancers and for using personal genomics for tumor diagnosis and therapy.

The elucidation of the human genome sequence has made it possible to identify genetic alterations in cancers in unprecedented detail. To begin a systematic analysis of such alterations, we determined the sequence of well-annotated human protein-coding genes in two common tumor types. Analysis of 13,023 genes in 11 breast and 11 colorectal cancers revealed that individual tumors accumulate an average of ∼90 mutant genes but that only a subset of these contribute to the neoplastic process. Using stringent criteria to delineate this subset, we identified 189 genes (average of 11 per tumor) that were mutated at significant frequency. The vast majority of these genes were not known to be genetically altered in tumors and are predicted to affect a wide range of cellular functions, including transcription, adhesion, and invasion. These data define the genetic landscape of two human cancer types, provide new targets for diagnostic and therapeutic intervention, and open fertile avenues for basic research in tumor biology.

Tyrosine phosphorylation, regulated by protein tyrosine phosphatases (PTPs) and kinases (PTKs), is important in signaling pathways underlying tumorigenesis. A mutational analysis of the tyrosine phosphatase gene superfamily in human cancers identified 83 somatic mutations in six PTPs ( PTPRF, PTPRG, PTPRT, PTPN3, PTPN13, PTPN14 ), affecting 26% of colorectal cancers and a smaller fraction of lung, breast, and gastric cancers. Fifteen mutations were nonsense, frameshift, or splice-site alterations predicted to result in truncated proteins lacking phosphatase activity. Five missense mutations in the most commonly altered PTP ( PTPRT ) were biochemically examined and found to reduce phosphatase activity. Expression of wild-type but not a mutant PTPRT in human cancer cells inhibited cell growth. These observations suggest that the mutated tyrosine phosphatases are tumor suppressor genes, regulating cellular pathways that may be amenable to therapeutic intervention.

The early detection of cancers through analysis of circulating DNA could have a substantial impact on morbidity and mortality. To achieve this goal, it is essential to determine the number of mutant molecules present in the circulation of cancer patients and to develop methods that are sufficiently sensitive to detect these mutations. Using a modified version of a recently developed assay for this purpose, we found that patients with advanced colorectal cancers consistently contained mutant adenomatous polyposis coli ( APC ) DNA molecules in their plasma. The median number of APC DNA fragments in such patients was 47,800 per ml of plasma, of which 8% were mutant. Mutant APC molecules were also detected in >60% of patients with early, presumably curable colorectal cancers, at levels ranging from 0.01% to 1.7% of the total APC molecules. These results have implications for the mechanisms through which tumor DNA is released into the circulation and for diagnostic tests based on this phenomenon.

Although most colorectal cancers are chromosomally unstable, the basis for this instability has not been defined. To determine whether genes shown to cause chromosomal instability in model systems were mutated in colorectal cancers, we identified their human homologues and determined their sequence in a panel of colorectal cancers. We found 19 somatic mutations in five genes representing three distinct instability pathways. Seven mutations were found in MRE11, whose product is involved in double-strand break repair. Four mutations were found among hZw10, hZwilch/FLJ10036, and hRod/KNTC, whose products bind to one another in a complex that localizes to kinetochores and controls chromosome segregation. Eight mutations were found in Ding, a previously uncharacterized gene with sequence similarity to the Saccharomyces cerevisiae Pds1, whose product is essential for proper chromosome disjunction. This analysis buttresses the evidence that chromosomal instability has a genetic basis and provides clues to the mechanistic basis of instability in cancers.

Tyrosine kinases (TKs) are central regulators of signaling pathways that control differentiation, transcription, cell cycle progression, apoptosis, motility, and invasion ( [1][1] ). Although a few TK genes have been shown to be mutationally altered in specific human cancers ( [1][1] ), it is not

Protein kinases are enzymes that are important for controlling cellular growth and invasion, and their malfunction is implicated in the development of some tumours. We analysed human colorectal cancers for genetic mutations in 340 serine/threonine kinases and found mutations in eight genes, including in three members of the phosphatidylinositol-3-OH kinase (PI(3)K) pathway. The discovery of this mutational activation of a key cell-signalling pathway may provide new targets for therapeutic intervention.