φ-Prime and φ-Primary Elements in Lattice Modules

In this paper, we introduce φ-prime and φ-primary elements in an L-module M. Many of its characterizations and properties are obtained. By counter examples, it is shown that a φ-prime element of M need not be prime, a φ-primary element of M need not be φ-prime, a φ-primary element of M need not be prime and a φ-primary element of M need not be primary. Finally, some results for almost prime and almost primary elements of an L-module M with their characterizations are obtained. Also, we introduce the notions of n-potent prime (respectively n-potent primary) elements in L and M to obtain interrelations among them where n≥2.

Lessons in Scholarship

This article pays tribute to Professor Robert Lusch and his fine career as a scholar of Marketing. One prime element of Bob’s career is that he took on some very fundamental issues in the field of Marketing. On several occasions I got involved in these issues as well, leading to wonderful intellectual explorations. I’ve organized this piece to share details of three such explorations: I. Bob as Editor II. Bob as Theorist III. Bob as Overseer of the Field IV. Bob as a Fine Individual I reflect on some lessons learned in scholarly pursuit, and close with some personal ruminations about him.

Valency Theory

Valency theory is a grammatical theory which focuses on the verb or the predicate as its center. Modern valency theory was founded in 1959 by Lucien Tesnière and is based on the idea that verbs structure sentences by binding specific elements (complements, actants) as atoms do. Other, freely addable elements are not determined by the verb; these are called supplements, adjuncts, or circonstants. The basic items of valency theory are valency carriers, complements, and supplements. Take for example sentence (1), “He gives the book to Sandra in the library.” While the NPs He and the book and the PP to Sandra in sentence (1) are valency governed complements, the PP in the library is not governed. It is a supplement. Tesnière compares sentences to a stage play, with actors and requisites. The verb is considered the central valency carrier and the complements depend on the valency carrier. In contrast to other projective theories of grammar, such as generative grammar, the binary division of the sentence into subject and predicate is abolished: the prime element of a sentence is the verb, the subject is governed by the verb, and so are the other objects. In valency theory the number of complements that depend on the verb constitutes its valency. There are monovalent (run), bivalent (build), and trivalent verbs (give). The verb run requires a subject to form a minimal sentence and to communicate a scenario, build requires a subject and direct object for this purpose, give a subject, direct, and indirect object. But it is not necessary that every complement be realized. For instance, sentence (2): “He sold the car (to his neighbor)”. A trivalent verb like to sell can easily be realized with only two complements, as shown in example (2). Complements like the directive complement in (2) (called facultative complements) and supplements differ by the fact that complements are determined in their form (syntactic valency) and their meaning (semantic valency) by the valency carrier, while supplements such as temporal or local adjuncts are not. The ability of a valency carrier to determine formal aspects like case marking of its complement(s) is subsumed under syntactic valency and the ability to determine semantic aspects like its thematic role is called semantic valency/specificity. Acknowledgements: For discussion of the material in this article and notes, the author is grateful to Vilmos Ágel, Klaus Fischer, and the reviewers.

Números primos; método gráfico de la conjetura de GOLDBACH

Mediante el uso de Microsoft Excel el siguiente trabajo examina las tablas de multiplicar desde una perspectiva distinta, con un método sencillo para encontrar la secuencia de los números primos en la línea continua de los números naturales , y así luego se identifican gráficamente los números que cumplen con la Conjetura de Goldbach, al realizar una triangulación con líneas que unen la series de los y ; siendo la notación: el cuadrado de los números naturales. A continuación, se trazan diagonales paralelas a las sucesiones y únicamente en cada elemento primo de la línea de los y así se obtienen intersecciones que cumplen con la conjetura fuerte de Goldbach. Se aplican fórmulas para calcular el número mínimo de intersecciones que se generan en un conjunto de los consecutivos. Así mismo, para obtener la conjetura débil de Goldbach, se puede usar el gráfico ya antes mencionado, y se emplean fórmulas combinatorias. Este método permite identificar el intervalo de afectación que tiene un elemento primo en la secuencia de los naturales y modelar una línea continua, que revela un gráfico similar al que se conoce como cometa de Goldbach. Palabras clave: Gráfico, números primos, conjetura de Goldbach. ABSTRACT By the use of Microsoft Excel the following work examines the multiplication tables from a different perspective, with a simple method to find the sequence of the prime numbers in the continuous line of the natural numbers ( ), and then we can graphically identify the numbers that comply with the Goldbach Conjecture, when making a triangulation with lines that join the series of the and , in this article the notation: is the square of the natural numbers. Next, diagonals are drawn parallel to the sequence and only in each prime element of the line of the and thus intersections are obtained that meet the strong conjecture of Goldbach. Formulas are applied to calculate the minimum number of intersections that are generated in a set of consecutive . Likewise, to obtain the weak Goldbach conjecture, the aforementioned graph can be used, and combinatorial formulas are used. This method serves to identify the range of affectation that a prime element has in the sequence of the natural numbers, and to model a continuous line, which reveals a graph similar to what is known as Goldbach's comet. Key words: Graph, prime numbers, Goldbach conjecture.

Números primos; método gráfico de la conjetura de GOLDBACH

Mediante el uso de Microsoft Excel el siguiente trabajo examina las tablas de multiplicar desde una perspectiva distinta, con un método sencillo para encontrar la secuencia de los números primos en la línea continua de los números naturales , y así luego se identifican gráficamente los números que cumplen con la Conjetura de Goldbach, al realizar una triangulación con líneas que unen la series de los y ; siendo la notación: el cuadrado de los números naturales. A continuación, se trazan diagonales paralelas a las sucesiones y únicamente en cada elemento primo de la línea de los y así se obtienen intersecciones que cumplen con la conjetura fuerte de Goldbach. Se aplican fórmulas para calcular el número mínimo de intersecciones que se generan en un conjunto de los consecutivos. Así mismo, para obtener la conjetura débil de Goldbach, se puede usar el gráfico ya antes mencionado, y se emplean fórmulas combinatorias. Este método permite identificar el intervalo de afectación que tiene un elemento primo en la secuencia de los naturales y modelar una línea continua, que revela un gráfico similar al que se conoce como cometa de Goldbach. Palabras clave: Gráfico, números primos, conjetura de Goldbach. ABSTRACT By the use of Microsoft Excel the following work examines the multiplication tables from a different perspective, with a simple method to find the sequence of the prime numbers in the continuous line of the natural numbers ( ), and then we can graphically identify the numbers that comply with the Goldbach Conjecture, when making a triangulation with lines that join the series of the and , in this article the notation: is the square of the natural numbers. Next, diagonals are drawn parallel to the sequence and only in each prime element of the line of the and thus intersections are obtained that meet the strong conjecture of Goldbach. Formulas are applied to calculate the minimum number of intersections that are generated in a set of consecutive . Likewise, to obtain the weak Goldbach conjecture, the aforementioned graph can be used, and combinatorial formulas are used. This method serves to identify the range of affectation that a prime element has in the sequence of the natural numbers, and to model a continuous line, which reveals a graph similar to what is known as Goldbach's comet. Key words: Graph, prime numbers, Goldbach conjecture.

Recent results on weakly factorial domains

In this paper, we will survey recent results on weakly factorial domains base on the results of [11, 13, 14]. LetD be an integral domain, X be an indeterminate over D, d ∈ D, R = D[X,d/X] be a subring of the Laurent polynomial ring D[X,1/X], Γ be a nonzero torsionless commutative cancellative monoid with quotient group G, and D[Γ] be the semigroup ring of Γ over D. Among other things, we show that R is a weakly factorial domain if and only if D is a weakly factorial GCD‐domain and d = 0, d is a unit of D or d is a prime element of D. We also show that if char(D) = 0 (resp., char(D) = p > 0), then D[Γ] is a weakly factorial domain if and only if D is a weakly factorial GCD domain, Γ is a weakly factorial GCD semigroup, and G is of type (0,0,0,…) (resp., (0,0,0,…) except p).

Impact of Microfinance on Consumption of Tribal Groups

Microfinance has emerged as a prime element aimed at reduction of poverty. Microfinance in a broader aspect provides such access to the socially deprived population (i.e., tribes) so that they can uplift from their destitute position. That is why microfinance may be termed as the development finance. This chapter is an attempt to examine the impact of microfinance on the consumption of scheduled tribes in the backward district of Bankura, in West Bengal, India. For the purpose of the present study, the authors selected 50 SHG tribal women households formed under SGSY (now re-structured as NRLM) and 50 non-SHG tribal women households. All the SHG member households and non-SHG member households belong to the category of below poverty line households. The study concludes that the non-SHG households suffer worst and reductions of poverty of SHG households are still in process. Obviously, there is a perceptible positive impact of microfinance for uplifting their living standard.

Prime, weakly prime and almost prime elements in multiplication lattice modules

In this paper, we study multiplication lattice modules. We establish a new multiplication over elements of a multiplication lattice module.With this multiplication, we characterize idempotent element, prime element, weakly prime element and almost prime element in multiplication lattice modules.