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In the development of the theory of autometrized algebra, various types of research have been conducted. However, there are some properties like convex subalgebra, prime convex subalgebra, meet closed sets, regular convex subalgebra, and convex spectral topology on autometrized algebras that have not been studied yet. In this paper, we define the notions of convex subalgebras and congruence relations on an autometrized algebra. We demonstrate that the collection of all convex subalgebras of an autometrized algebra forms a lattice and distributive. In particular, we will show that there exists a one-to-one correspondence between the set of all convex subalgebras and the set of all congruences on an np-autometrized algebra. Furthermore, we explore prime convex subalgebras, meet closed subsets, and regular convex subalgebras and obtain some related results. For instance, we show that in a semiregular np-autometrized l-algebra, the intersection of a chain of prime convex subalgebra is a prime convex subalgebra. We also prove that any convex subalgebra in an autometrized algebra is the intersection of regular convex subalgebras. Lastly, we introduce the convex spectral topology of proper prime convex subalgebras in an autometrized l-algebra and discuss some fundamental facts. We also prove that a convex spectrum is compact in an np-autometrized l-algebra A if and only if A is generated by some element. Specifically, we demonstrate that the convex spectrum is a T1 - space and Hausdorff space. [ABSTRACT FROM AUTHOR]

In this paper, we investigated some related properties and applied the concept of fuzzy congruence relation to a pseudo-UP algebra X. Additionally, we used their level set to define the fuzzy congruence relation on X. The Cartesian product of two fuzzy congruences and the intersection of two fuzzy congruences are discussed. We also provided the connection between the collection of fuzzy pseudo-UP ideals and the collection of fuzzy congruence relations in pseudo-UP algebras. We then demonstrated that there exists an associated algebra X that is a pseudo-UP algebra for each fuzzy ideal ζ and fuzzy congruence ϕζ generated by ζ. Additionally, we created a congruence relation on a pseudo-UP algebra using the fuzzy pseudo-UP ideal and fuzzy congruence relation ϕζ produced by ζ. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the properties of some elements in autometrized algebra. We also introduce some properties of normal autometrized algebra. Furthermore, we obtain normal autometrized l-algebras among the normal autometrized semialgebras satisfying several conditions and discussing some of its properties. We also prove that a normal autometrized semialgebra satisfying certain conditions is a lattice and distributive. Finally, we establish an equivalent condition between normal autometrized l-algebra and representable autometrized algebra with some conditions. We also prove that in a normal autometrized l-algebra satisfying some conditions, there exists a subset which is a convex normal subalgebra. [ABSTRACT FROM AUTHOR]

This paper introduces the concept of subalgebra and discusses some properties of subalgebra. We also examine different types of ideals and how they behave. We also describe some properties of a homomorphism. Finally, we present quotient algebra and discuss isomorphism theorems. [ABSTRACT FROM AUTHOR]

In this paper, we study an excision theorem of the dihedral and reflexive (co)homology theory of associative algebras. That is, for such an extension, we obtain a six-term exact sequence in the dihedral cohomology. Also, we present and prove the relation between cyclic and dihedral cohomology of algebras and some examples. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of sBCI/sBCK/eBCI/eBCK-algebras as a generalization of the notion of BCI/BCK-algebras. This structure is studied in detail. Also we introduce a way to make an eBCK-algebra from a BCK-algebra and vice versa. [ABSTRACT FROM AUTHOR]

In the paper, the authors recover, correct, and extend two representations for derangement numbers in terms of a tridiagonal determinant. [ABSTRACT FROM AUTHOR]

The article focuses on the a theorem about an infinite series which was communicated by Christian Goldbach to Leonhard Euler and is one of the theorems mentioned in Euler's paper "Variae observationes circa series infinitas." Goldbach and Euler's proof is a common example of what some historians consider a misuse of divergent series because it begins by assigning a value to the harmonic series Σ1/n and is then manipulated by subtractions and replacement of other series until the desired result is attained. The proof of the theorem is hereby reviewed in order to vindicate Goldbach and Euler's work. It is shown how the deficiency in the proof can be amended. However, other results obtained in said Euler's paper are not as easily amended.

Over the past 40 years or so, human activities and movements in space-time have attracted considerable research interest in geography. One of the earliest analytical perspectives for the analysis of human activity patterns and movements in space-time is time geography. Despite the usefulness of time geography in many areas of geographical research, there are very few studies that actually implemented its constructs as analytical methods up to the mid-1990s. With increasing availability of geo-referenced individual-level data and improvement in the geo-computational capabilities of Geographical Information Systems (GIS), it is now more feasible than ever before to operationalize and implement time-geographic constructs. This paper discusses recent applications of GIS-based geo-computation and three-dimensional (3-D) geo-visualization methods in time-geographic research. The usefulness of these methods is illustrated through examples drawn from the author's recent studies. The paper attempts to show that GIS provides an effective environment for implementing time-geographic constructs and for the future development of operational methods in time-geographic research. [ABSTRACT FROM AUTHOR]

By the mid-1900s Hugh MacColl had published enough in the way of papers and notes on logic that a more connected presentation could be made. Accordingly, he reworked several items into a book published in 1906 and entitled Symbolic Logic and its Applications. In this paper I comment on some main features of the book and note reactions to it at the time, especially by Russell. Consequences of his ignorance of contemporary literature are also considered. The account begins with his little-known book on algebra. [ABSTRACT FROM AUTHOR]

This paper reports the mathematical generalizations of two groups of algebra students, one which focused primarily on quantitative relationships, and one which focused primarily on number patterns disconnected from quantities. Results indicate that instruction encouraging a focus on number patterns supported generalizations about patterns, procedures, and rules, while instruction encouraging a focus on quantities supported generalizations about relationships, connections between situations, and dynamic phenomena, such as the nature of constant speed. An examination of the similarities and differences in students' generalizations revealed that the type of quantitative reasoning in which students engaged ultimately proved more important in influencing their generalizing than a mere focus on quantities versus numbers. In order to develop powerful, global generalizations about relationships, students had to construct ratios as emergent quantities relating two initial quantities. The role of emergent-ratio quantities is discussed as it relates to pedagogical practices that can support students' abilities to correctly generalize. [ABSTRACT FROM AUTHOR]

The article provides two short and direct proofs of J. S. Pym's result in his 1969 paper "The linking of sets in graphs," regarding the Cantor-Bernstein theorem for paths in graphs. Both proofs are elementary, and they can be examined independently. The first proof employs transfinite induction. This proof starts from the first set of paths which is then transformed step by step into the wanted bijective set by integrating path segments from the second set. The choices made during this inductive process have been made arbitrary. For the second proof, these choices are made explicitly.

The article explores the so-called λ-permutations shown by Steven Krantz and Jeffery McNeal in their paper "Creating more convergent series." They specified that a permutation σ of the positive integers is a λ-permutation if it has two characteristics. The first property is that for every convergent series, the summation of ak from k =1 to infinity, the summation of aσ(k) from k =1 to infinity also converges. The second property is that there is at least one divergent series which is the summation of ak from k =1 to infinity such that the summation of aσ(k) from k =1 to infinity converges.

The article presents information on the properties of Graeco-Latin squares enumerated by mathematician Leonhard Euler. Euler suggested that a Graeco-Latin square of size n could never exist for any n of the form 4k +2, although he was not able to prove it. A Latin square is an n-by-n array of n distinct symbols in which each symbol appears exactly once in each row and column. On the other hand a Graeco-Latin square is an n-by-n array of ordered pairs from a set of n symbols such that in each row and each column of the array, each symbol appears exactly once in each coordinate. In one of his papers on Graeco-Latin squares, Euler used magic squares, which are closely related to Graeco-Latin squares. Magic squares were constructed by using Graeco-Latin squares of orders 3, 4 and 5. He showed that a Graeco-Latin square of order n can be converted into a magic square by the use of an algorithm. One can construct Graeco-Latin squares of every order n except those values for which the prime factorization of n contains only a single factor of 2.

The article presents a discussion on the properties of the zeroes of the Nth partial sum of exponential series. In particular, it discusses their location in the complex plane, the spacing between adjacent zeroes, and the moment sums of the zeroes and their reciprocals. The graphs and calculations presented within this paper were computed with Mathematica 4.2. The fundamental theorem of algebra states that any polynomial of degree N(>0) has exactly N zeroes, provided that zeroes are counted according to multiplicity. Thus, each polynomial PN (z) has exactly N zeroes. G. Szegö's asymptotic theorem and proofs of Gauss-Lucas theorem have been presented. Gauss-Lucas theorem has been applied to the study of the polynomials PN (z). Figures suggest that the distance between adjacent zeroes increases their plots are followed from left to right. For small values of N it can be computationally demonstrated that this is indeed the case. Expressions for the moment generating functions have been derived knowing that these moment generating functions allow computation of the moments of the zeroes without having to compute the zeroes beforehand.

Provides insights on the computation of natural logarithms via long division. Solutions to De l'hospital's pulley problem; Measurement of the curl of paper.

White and Hultquist (1965) developed a method of combining finite fields mapped into a finite commutative ring to provide confounding plans for mixed factorial experiments where the numbers of the levels of factors have to be prime or the power of a prime. This paper extends their procedure to cover mixed factorials where the numbers of levels of factors need to be relatively prime, and not necessarily all prime, thus covering a wider range of mixed factorial experiments amenable to the traditional way of analysis of variance suggested by White and Hultquist. [ABSTRACT FROM AUTHOR]

Least squares estimation and quadratic openloop optimal control share both algebra and intuition; this paper explores the duality to develop the parallels in each. Estimation results suggest control analogs to correlation measures, significance tests, distributed lags and problems of multicollinearity, tests of compatability between restrictions and data, and a version of the Kalman filter that is sometimes computationally expedient. Conversely, the control literature provides methods of incorporating certain types of inequality restrictions and a development of recursive estimation based on the Kalman filter. A by-product is a method of solving control problems using least squares computer programs. [ABSTRACT FROM AUTHOR]

In regression analysis with heteroscedastic and/or correlated errors, the usual assumption is that the covariance matrix Sigma of the errors is completely specified, except perhaps for a scalar multiplier. This condition is relaxed in this paper by assuming only that Sigma has a certain pattern; for example, that Sigma is diagonal or partitionable into a diagonal matrix of sub-matrices. The method used for estimating Sigma is the standard procedure of equating certain quadratic forms of the observations (in this case, squares and products of residuals from regression) to their expectations, and solving for the unknown variances and covariances. A numerical example illustrates the method. [ABSTRACT FROM AUTHOR]

The article offers an alternative elementary proof to the one published by J. P. May in the paper "Munshi's proof of the Nullstellensatz," in which some of the algebraic technicalities are avoided. The alternative proof requires a simple version of the Noether normalization lemma. The proof needs the resultant of two polynomials but in such a simple manner that only one property of it is required. The analysis is confined to the weak form of Nullstellensatz because the strong form is easily derived from the weak form using the Rabinowitsch trick.