Abstract:
Let b be a positive integer greater than 1, N a positive integer relatively prime to b, |b|N the order of b in the multiplicative group UN of positive integers less than N and relatively primes to N, and x E UN. It is well known that when we write the fraction x/ N in base b, it is periodic. Let d, k be positive integers with d > 2 and such that |b|N = dk and x /N = 0.a1a2 a|b|N with the bar indicating the period and ai are digits in base b. We separate the period a1a2 a|b|N in d blocks of length k and let Aj = [a(j 1)k+1a(j 1)k+2 ajk]b be the number represented in base b by the j - th block and Sd(x) = d Zigma j=1 Aj . If for all x E UN, the sum Sd(x) is a multiple of bk - 1 we say that N has Midy s property for b and d. In this work we present some interesting properties of the set of positive integers d such that N has Midy's property to for b and d.

Abstract:
Sea A = {a1, a2, . . . , ak} un conjunto de enteros positivos primos relativos entre sí. Dado un entero positivo N, se dice que N es representable por A si existen enteros no negativos x1, x2, . . . , xk tales que N = Pk i=1 aixi. El Problema de Frobenius consiste en encontrar el mayor entero, denotado con g(A), que no es representable por A. En este artículo se presenta un algoritmo para resolver el problema de Frobenius utilizando bases de Gr bner. Al final, en el Apéndice, se presentan los algoritmos desarrollados en este trabajo implementados en el sistema de álgebra computacional MuPAD.

Abstract:
Let $b$ be a positive integer greater than 1, $N$ a positive integer relatively prime to $b$, $ |b|_{N}$ the order of $b$ in the multiplicative group $% \mathbb{U}_{N}$ of positive integers less than $N$ and relatively primes to $% N,$ and $x\in\mathbb{U}_{N}$. It is well known that when we write the fraction $\frac{x}{N}$ in base $b$, it is periodic. Let $d,\,k$ be positive integers with $% d\geq2$ and such that $|b|_{N}=kd$ and $\frac{x}{N}=0.% bar{a_{1}a_{2}...a_{|b|_{N}}}$ with the bar indicating the period and $a_{i}$ are digits in base $b$. We separate the period ${a_{1}a_{2}... a_{|b|_{N}}}$ in $d$ blocks of length $k$ and let $ A_{j}=[a_{(j-1)k+1}a_{(j-1)k+2}...a_{jk}]_{b} $ be the number represented in base $b$ by the $j-th$ block and $% S_{d}(x)=\sum\limits_{j=1}^{d}A_{j}$. If for all $x\in\mathbb{U}_{N}$, the sum $S_{d}(x)$ is a multiple of $b^{k}-1$ we say that $N$ has the Midy's property for $b$ and $d$. In this work we present some interesting properties of the set of positive integers $d$ such that $N$ has the Midy's property for $b$ and $d$.

Abstract:
We introduce a new class of pseudoprimes. In this work we characterize Midy pseudoprimes, give some of their properties and established interesting connections with other known pseudoprimes, in particular we show that every divisor of a Midy pseudoprime is either a prime or a Midy pseudoprime and in the last case it is a strong pseudoprime.

Abstract:
We give new characterizations of the Midy's property and using these results we obtain a new proof of a special case of the Dirichlet's theorem about primes in arithmetic progression.

Abstract:
In this work we present a natural generalization of strong pseudoprime to base $b$, which we have called $q$-pseudoprime to base $b$. It allows us to present another way to define a Midy's number to base $b$ (overpseudoprime to base $b$). Besides, we count the bases $b$ such that $N$ is a $q$-probable prime base $b$ and those ones such that $N$ is a Midy's number to base $b$. Furthemore, we prove that there is not a concept analogous to Carmichael numbers to $q$-probable prime to base $b$ as with the concept of strong pseudoprimes to base $b$.

Abstract:
We introduce a new class of pseudoprimes-so called "overpseudoprimes to base $b$", which is a subclass of strong pseudoprimes to base $b$. Denoting via $|b|_n$ the multiplicative order of $b$ modulo $n$, we show that a composite $n$ is overpseudoprime if and only if $|b|_d$ is invariant for all divisors $d>1$ of $n$. In particular, we prove that all composite Mersenne numbers $2^{p}-1$, where $p$ is prime, are overpseudoprime to base 2 and squares of Wieferich primes are overpseudoprimes to base 2. Finally, we show that some kinds of well known numbers are overpseudoprime to a base $b$.

Abstract:
the objective of this study was to update the current land use map and to classify the areas accordingly to its potential for maize production under rainfed conditions in the state of guanajuato, mexico. the study was carried out from 2004 to 2005. 242 ortophotos obtained during 1993 and 1994 in a 1:20 000 scale and updated at the county level, were utilized. the arcexplorer 2.0 program was used to display and work with the ortophotos. the printed ortophotos were updated with information collected in the field and digitalized and posisionated with the universal transversa of mercator with the parameters datum and spheroid wgs84. the land use then was classified in: rainfed agricultural land, irrigated agricultural land, urban areas, water bodies and of other uses. in comparison with the previous map the area with rainfed agriculture decreased while the irrigated land increased. the urban areas increased from 57 000 ha in 1997 to 104 000 ha in 2005. the intersection of the updated land use maps with the potential production maps allowed for a precise quantification of the high, medium and low levels of land aptitude for maize production under rainfed conditions. with the use of digital ortophotos, geographic information systems and direct field verification, it was possible to update and classify the land use in guanajuato, mexico.

Abstract:
the article explores the political, economic and social globalization process. the literature review of selected documents, books and newspaper/magazine articles serves to identify issues involved in this global phenomenon. as a result, it appears that as long as the current pattern of global capitalist reproduction persists, globalization will tend to favor a small group of highly developed countries to the detriment of the majority of the world？s population, making it difficult to meet the challenges of human development. conclusions are that a change in paradigm and conduct are needed on the part of society to achieve equitable economic and social development.

Abstract:
we present a case of this entity in a caucasian, patient with purpuric macular lesion area with her right posterior tibial, without affecting systemic, chronic and persistent course now, after eighteen months of follow-up with partial improvement after treatment with topical steroids and immunosuppressors. oral contraceptives and mild hypercholesterolemia may be associated data in our case to the entity, althought further studies are needed to confirm this hypothesis.