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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]

A modified Degasperis-Procesi equation is investigated. The local existence and uniqueness of the strong solution for the equation are established in the Sobolev space Hs(R) with s > 3/2. The L¹(R) local stability for the strong solution is obtained under certain assumptions on the initial data. [ABSTRACT FROM AUTHOR]

In clinical studies, the period before an event occurs is frequently considered as an outcome. The general term used for this analytical method in clinical studies is "survival time analysis". The method employs Cox regression to compare hazards between groups. In this paper, we propose an index defined in the Bayesian framework to compare hazards. This index compares the sizes of the hazards directly. Further, we propose two methods to calculate the sizes of the hazards. One is an approximate method, by which we can easily calculate the size, and the other is an exact method. In addition, we apply the proposed index to two actual clinical studies to assess its usability. [ABSTRACT FROM AUTHOR]

In the paper, the authors find the logarithmically complete monotonicity of the Catalan-Qi function related to the Catalan numbers. [ABSTRACT FROM AUTHOR]

In this article, we introduce the Zweier Paranorm I-convergent double sequence spaces 2ZI(q), 2Z0I(q) and 2Z∞I(q) for q = (qij), a sequence of positive real numbers. We study some algebraic and topological properties on these spaces. [ABSTRACT FROM AUTHOR]

We study capacitated vehicle routing problems (CVRPs) in which the client demands occur over time and the collector or distributor performing the service can decide when to visit the clients. It has been reported in the literature that postponement of collection (or delivery) services may decrease the overall routing cost or distance. We investigate this issue in greater detail and report on experiments in different routing settings to uncover the impact of postponement on routing efficiency. We compare the results by a CVRP local search procedure with those obtained from an analytical, Continuous Approximation model. While the two approaches tend to generate similar results, the differences grow larger when the CVRPs' bin packing characteristics become more intricate. [ABSTRACT FROM AUTHOR]

The article focuses on the analysis of ultrasmall or infinitesimal and ultralarge real numbers. It says that the use of infinitesimals in mathematics started with Leibniz in the 17 century and was used for calculus for formulating rules. It states that infinitesimals are real numbers that are much smaller than to those visible at the macroscopic scale. It adds that it is a relative concept, in which the diameter of a bacterium is considered to be ultrasmall qualified to macroscopic measures.

Benford's Law deals, among other things, with the proportion of numbers whose first significant digit is a 1 (e.g., 0.00131 and 19668 both have first significant digit 1) in a variety of datasets. In these datasets, which arise in various compendiums or as mixtures of various sets of numbers, the proportion of numbers with first significant digit one is 0.3010 which is much higher than the commonsense value of 1 / 9. The reasons for this occurrence have been elusive. Mathematical attempts to explain this phenomenon have been relatively fruitless. Methods involving probability have been somewhat more successful. In this article we give some simple reasons for this occurrence and also give an example of a general mixture of distributions which exactly satisfies this Law. Various other examples and counterexamples are also given. [ABSTRACT FROM AUTHOR]

The article presents a mathematical analysis of approval voting, a stance wherein there is guarantee that some fraction of the population will mutually agree on some candidate despite their distinct preferences. It relates that approval rating is being used by many scientific and mathematical societies, such as the Mathematical Association of America, on their elections. It discusses classical theorems which evaluate the concept of approval rating.

The article presents several mathematical problems and solutions. These include works on positive integers as proposed by Erwin Just from Bronx Community College of the City University of New York, the usual gamma function as proposed by M. L. Glasser from Clarkson University, and partitioning values of a polynomial into sets of equal sum as proposed by Marian Tevita from the National College in Romania. Also explored are oddly triangulated polygons and the characterization of permutation matrices

The article explores the validity of some versions of the Mean Value Theorem for complex-valued functions in the form of an inequality. All of these versions will be proved using full covers of [a, b]. Full covers offer an important tool which can be used to prove many of the classical theorems of calculus and elementary real analysis. According to the author, the meaner the theorem, the stronger it is, does his meanest Mean Value Theorem undertakes issues usually associated with the Lebesgue integral.

The article focuses on the provision of a theorem in which the central moments are replaced by power sums and the data sets are assumed to be of the same size. The result concerning moments can give an analogous result about polynomials when Newton's identities are considered. To derive the main theorems of the authors, a generalized version of Descartes' rule of signs will be used. Under this rule, when a polynomial has real coefficients, the number of positive roots cannot exceed the number of sign changes in the coefficients. The rule of signs is often quoted with the difference in the number of roots and the sign changes should be an even number.

The article explores the mechanics of the television game show "The Weakest Link." Contestants of the game answer general knowledge questions, with each correct answer raising the amount held in a temporary pot. The game involves several mathematical issues such as strategy optimization, existence of Nash equilibria, and voting considerations. The payoff vector as being the amount in the pot after a chain of correct answers was introduced to describe various versions of the game around the world.

The article offers information on a solution to a multiple player gambler's ruin problem. The author presented the given of the ruin problem that aimed to arrive at three results: the probability that player i is the victor; the expected number of stages until one of the players has all the money; and, the expected number of games played between two specified players. The author also outlined the solution that was used to arrive at the desired results. Afterwhich, a proof to solidify the solution was also provided.

The article reports on the sharpening of Wielandt's characterization of the gamma function. It presents the two conditions that is satisfied with Euler's gamma function. According to the author, the two conditions by themselves do not characterize the gamma function and presents characterization of the gamma function through theorems that Wielandt discovered. He adds that a characterization in the real domain, found in 1922 by other scholars, is well known, while Wielandt's theorem is function-theoretic.

The article reports on the proof of some Borwein algorithms for π. It states that Jonathan and Peter Borwein derived many efficient algorithms for computing π in 1987. However, their proofs are difficult for the nonspecialist in the theory of elliptic modular functions. This can be seen by using a formula of Gauss's and elementary algebra which is able to prove the correctness of the two of them. There are two algorithms that was considered: quadratic algorithm and quartic algorithm.

The article reports on the nature of projectile motion in a uniform, downward gravitational field and resisted by a drag force due to the air through which it moves. It states that the nature of the projectile motion lies near the heart of the scientific revolution. The drag force is assumed to act in a direction opposite to that of the projectile's velocity and to depend only on its magnitude, therefore, Coriolis forces and Earth's curvature are ignored. The author notes that the projectile motion under quadratic drag was at the center of an episode in the calculus priority dispute.

The article reports on the complex analysis as catalyst. It states that topology has become part of the bedrock of modern mathematics and now permeates all branches of the discipline. The author also presents several examples from real analysis, group theory, abstract algebra, geometry, partial differential equations, and other subjects. He notes that these examples are given to show how a problem that began as a complex analysis problem could be resolved by techniques from another part of mathematics.

The article presents an elementary proof of the random walk hitting time theorem. This theorem states that a left-continuous random walk on (Z) that start at a nonnegative integer (k) wherein the probability that hits the origin (n) is equal to k/n. It discusses the hitting time theorem which is equivalent to the ballot theorem that dates back to Whitford and Bertrand. It presents equations and solutions to prove these theorems. It also presents minor modification or assumptions in the hitting time theorem.

The article offers information on various mathematical problems and their corresponding solutions. These are edited by Gerald A. Edgar, Doug Hensley, Paul T. Bateman, Richard Stong, Charles Vanden Eynden, Frank B. Miles, Ezra A. Brown, and Ira M. Gessel. Among those who proposed these problems include Bogdan M. Baishanski of the Ohio State University in Columbus, Ohio, one by Daniel Troy of Purdue University in Calumet, Hammond, Indiana, another by Hugh Montgomery of the University of Michigan in Ann Harbor, Michigan and Harold S. Shapiro of the Royal Institute of Technology in Stockholm, Sweden. Accompanying mathematical propositions and their corresponding analysis and computations are also presented.

The article offers information on a strong version of Liouville's theorem. This theorem states that every bounded holomorphic function on C is constant. The authors, through various mathematical problems and their corresponding solutions, present an elementary proof of this theorem. They were also able to obtain a corresponding statement for complex-value functions by applying Theorem 1.1. Moreover, different mathematical propositions and their corresponding numerical analysis and solutions are presented.

Competition has a huge influence on customer buying behaviour and will impact on the optimal price that companies should charge for goods or services. To date, many dynamic pricing models have not modelled competition explicitly. In this paper, we introduce pricing strategies that maximize revenue when selling an inventory of identical items by a fixed time and where there is a competing seller. The model used incorporates a probabilistic formulation of customer demand, which is influenced by the prices offered by the company and the competitor, and the time remaining until the end of the selling period. Calculus of variations is used to solve the problem and simple conditions are given that ensure the uniqueness of a solution. Illustrative examples are included. A practical implementation that uses dynamic updating is proposed and tested using simulated data, showing the effectiveness of the method. [ABSTRACT FROM AUTHOR]

The article presents a numerical analysis of cycles in graphs and groups. It states that N. L. Biggs described a theorem which pertains to this topic, with a brief sketch of J. H. Conway's proof. It also forwards that this theorem seems remarkable for its generality and for the exact number of orbit it contains. Various propositions which illustrate the numerical composition of cycles in graphs and groups are also presented. Moreover, accompanying numerical calculations and derivations relative to this topic are also mentioned.

The article presents a freshman's approach to John Conway's napkin problem. The issue involved in this problem is what is the expected proportion of diners not getting a napkin at a conference dinner. It is also forwarded that this question is posed by Conway at a lunch in Bell Labs in 2001, and it is also described in Peter Winkler's book "Mathematical Puzzles: A Connoisseur's Collection." It is presumed that all diners arrive at one time and find seats at random. Relative to this, two researchers used a systems of equations containing generating functions to assess the generation function of napkinless diners.

The article focuses on the mathematical study of a recursively palindromic (RP) collection of words. It relates that an empty word or an unpaired words is considered as a RP or palindrome. The authors presented theorems for the mathematical analysis of RP words using the proof of recurrent sequence. Analysis of the RP composition is also discussed, in which the authors' theorem is that its number of n is equal to the number of partitions of n into powers of 2, or binary partitions. The parity of the Catalan numbers is also described.

The article describes a mathematical solution for a closed curve problem for plane curves when the period of the curvature and the period of the corresponding curve do not coincide. The authors demonstrated that if the angular variation of the tangent to unit planar curve in one period of the curvature it is a noninteger rational multiple of 2π. They obtained that the associate curve of any periodic function can be forced to close up by scaling or by addition of a constant. The authors' criterion of a closed planar curve is a smooth, regular, periodic map with a minimum period.

The article examines how to transfer some exact amount of liquid from a large container filled with liquid to another container using a universal measuring box whose bases are not rectangular. The authors analyze the maximum capacity of measuring boxes with triangular bases and plane quadrilateral sides. They began with an examination of orthogonal measuring boxes, which have plane trapezoidal sides orthogonal to their bases, followed by an examination of boxes with sides not orthogonal to the base, or boxes with triangular vases and plane quadrilateral sides

The article focuses on the Raabe's test for divergence and convergence of a positive series which determines the behaviour of an associated sequence. It notes that the test is used in most of the textbooks and is often proved through the use of Kummer's test. It demonstrates how the fundamental monograph by K. Knopp explains the test with a distinct sequence and provides as series of equations which proves that the two version of the test are not equivalent. It also includes a parametrized associated sequence which concludes the divergence of the series by the standard of the test.

The article discusses the Fubini principle. It provides a sample equation which demonstrates the use of the principle. It notes that the term came from the analogy with integration and establishes conditions when the order of integration is not a necessity. It also demonstrates the use of permutation as an adjunct to the solutions for the principle. Among the sample problems which cites the application of the principle is the evaluation of the judge to the candidates as either passing or failing and proves the total number of agreements between judges.

The article explores juggling sequences associated with an arbitrary state as well as how to enumerate the number of this kind of sequences, also called siteswap by jugglers. The authors introduce the concepts of primitive juggling sequences and prime juggling sequences, then briefly discuss their difference. They explain why arbitrary products of prime juggling sequences cannot be taken to get new juggling sequences and demonstrates how to enumerate juggling sequences in the ground state and in general state vectors.

The article introduces the ultrafilters which are standard tools in mathematical logic and set theory but are also used in combinatorics and topology. Relative to this, three problems concerning limited solutions with the same conclusion is analyzed. The analogue of the first problem for infinite sets trails to ultrafilters, and shows that this is the right concept needed to formulate the other problems in an infinite settings. Moreover, the original problems in the finite case contemplates the fact that every ultrafilter on a finite set is principal.

The article presents the theory of integration based on two axioms rather than on construction using Riemann sums. It is stated that with axiomatic approach, even the proof of the existence of the definite integral, which uses Reimann sums, becomes a bit more elegant than the conventional approach. An interesting connection between axiomatic approach and the history of calculus is discussed. The theory of integration is strongly recommended by the authors for teaching physics and engineering courses based on calculus.

The article discusses a random walker ranking for National Collegiate Athletic Association (NCAA) I-A college football teams. It focuses on the potential of an algorithm derived by imitating the behavior of voters in offering reasonable rankings. The authors emphasize that the algorithm provided can generate reasonable results in terms of single-voter behavior covering individual walkers who have diverse opinion regarding the best team in the network of teams linked by their head-to-head games. Apart from this, they also highlight that random walker rankings offer various ways of ranking teams with optimum methods and outcome.

The article poses several mathematical problems and solutions. It reports that one problem, proposed by R. V. Heath of New York, New York, searches for the smallest value of n for which n2 triangular numbers 0, 1, 3, 6, 10, . . . . , 1―2n2(n2-1) can be arranged to form a magic square, and a solution is provided by Christian Boyer of Engeun les Bains, France, with its answer. It mentions another problem, proposed by Ovidiu Furdio of Western Michigan University in Kalamazoo, Michigan, which mandates finding all nonnegative integers m and n and real numbers alpha such that the derivativex=0infinity sin2mx―1+xalpha sin2nxmultiply by dx converges.

The article discusses the third mathematical problem of David Hilbert. It mentions that the problem analyzes the question on the possibility of dividing two polyhedrons with the same area and dividing one to the produce the other. The article argues that the problem can be solved and provides equations. The article mentions Max Dehn, who provided evidence that a regular tetrahedron is not equidecomposable with another of the same volume. It also discusses the problems using theorems by mathematicians including Raoul Bricard and Hugo Hadwiger.

Repeat customers are five times more profitable than new customers. Internet vendors have to retain customers to reap the benefits of repeat sales, but more than 50 percent of repeat customers seldom complete a third purchase. One reason is the inability of on-line vendors to manage customers' changing expectations. Vendors need to understand the decision calculus of repeat customers at every stage of their transaction experience with the vendor. This study uses a customer repurchase decision-making model to examine the effect of transaction experience on customers' decision calculus in on-line repurchases. The model shows that the effects of perceived convenience and perceived price change over the transaction experience, whereas the effects of perceived value and pleasure do not. These findings have significant theoretical and practical implications. [ABSTRACT FROM AUTHOR]

The article focuses on the theory of Lie algebra proposed by W. Plesken. Many authors ask as to what is the structure of the Lie algebra and which group the construction produce a simple Lie algebra. A sample of the Plesken Lie algebra of a group is given which involves complex numbers and an equation. In the small examples, the dimension of a smallest nontrivial simple Lie algebra is three, which should serve as a guide to possible examples and which suggests that in such equation, the Schur-Frobenius theory might be involved. In order to understand the group algebra, some materials on representations and bilinear forms were introduced.

The article discusses various developments of a convex polyhedron using tilings. By cutting the surface of the polyhedron, a plane figure will result and that these cuts need not be bounded to edges but can also be made through faces. A plane figure is said to tile the plane if copies of the figure cover the plane with no gaps nor overlaps when placed end to end. A convex polyhedron P is a tile-maker if every development of P tiles the plane while P is said to be a semi-tile-maker if every edge-development of P tiles the plane. Several convex polyhedra that are considered tile-makers and a hypothesis about the convex polyhedra that could possibly be semi-tile-makers are discussed. In addition, the definition of dihedron and almost regular tetrahedron are also included.

The article examines the relation among critical points, external points, and saddle points of a complex analytic function. It also presents aspect of function theory in mathematics. It is reported that the familiar relation between local extremal points and critical points for realvalued functions does not hold analytic functions. It is stated that a nonconstant analytic function has no local extremal points other than its zeros. The contour integral can be evaluated using the residue theorem in many mathematical cases.

The article presents information on the transcendence of Thue-Morse sequence. The Thue-Morse sequence was first introduced by researcher A. Thue in 1912. It was considered nine years later by another researcher M. Morse in a different context. Researcher M. Queffé in 1998 showed that the Thue-Morse continued fractions are transcendental. It is stated that Thue-Morse word t begins with the palindrome abba, and many ways can define the Thue-Morse sequence and each of them gives rise to specific interests, problems, and most of time solutions.

The article presents information on a simplified mathematical approach which is used to measure differentiation in mathematics. This approach uses a local maximal function developed and employed by researchers to deal with limit theorems in various mathematics setting. The principal device of this approach is to differentiate a local maximal function defined in terms of a finite Borel measure and Lebesgue measure. This approach has cut almost half the time spent by researchers in covering mathematical topics.

The article reports on proofs on equations involving constants which are equal to one. It aims to document a case where the constant is evaluated and turned out that it was not equal to one. It presents the theorem which involves the sum in a given equation and denotes the Riemann zeta-function. It also aims to demonstrate a piece of software which can find some sort of identity. It features the Klee-Minty cube which is considered an example of great theoretical importance to analyses of simplex algorithm rules.

The article reports on semi-osculating conics of planar curves focusing on Ghys's theorem. It presents the theorem which involves Schwarzian derivative of every diffeomorphism which is a combination of judicious use of Ghys's theorm and inspiration from Singer's technique which leads to some identified results. It also presents the theorem involving a strictly convex curve whose image is closed in the projective plane. It views a real projective line with a topological and differential structures via stereographic projection.

The article offers a combinatorial method to prove Vandermonde's determinant. It presents card counting questions and their answers as a method of proof for the determinant. It notes that it can be said that a Vandermonde table is good if it has no small cards and if it has at least one small card, it is considered bad. It aims to show that every bad Vandermonde table can be paired up with another bad table with a permutation of opposite parity in order to complete the proof of Vandermonde's expansion.

The article reports on the concept of gamma function which was introduced into analysis by Leonard Euler in his attempts to develop a differential calculus of fractional order. It elaborates on the different approaches in the study of gamma functions including the Eulerian approach, the additive approach and the Gaussian approach. It examines different theorems laid down by mathematicians for the discussion of gamma functions. Corollaries and the remarks regarding the different formulas and theorems are also presented.

The article presents a discussion about the Koch Curve. It states that the mathematically popular Koch curve is usually used as an example to illustrate a continuous but not differentiable function and as an example of a non-rectifiable curve. However, it emphasizes that only the fact that it is not rectifiable is proved, and the proof that it is a curve and that no point does this curve have a tangent line is disregarded. It offers a mathematical and simple geometric proof that the Koch curve is indeed an arc and that it at no point has a tangent line.

The article presents a discussion about an analysis to a solution regarding a broad class of Dirichlet series and show the results in action in a variety of ways. It tells of an acquired integral evaluation that transpires to be a case of a rather pretty class of evaluations and worth creating explicits. It tells of a formal Dirichlet series, with certain real coefficients for other largely standard notations. It offers a solution for alternating zeta-functions, and explains that such functions can be solved by contour integration and consulting a computer algebra system.

The article presents a discussion about the mathematical characterization of ellipses. It states that the best known plane curves are straight lines and circles, which are characterized by constant curvature, and the next most familiar closed plane curves are arguably the ellipses. It expresses how ellipses should be characterized in a simple manner with the use of the curvature function. However, the parameterization of ellipses is not simple, because it involves elliptic functions and curvatures.

The article presents a discussion about a summation formula that is used in certain transformations on generating algebraic and differential functions and the extraction and creation of coefficients. The goal of the article is to offer the method of coefficients. Mathematical functions are manipulated in order to arrive at a single and simple mathematical expression. It mentions that generating functions have emerged as one of the most popular approaches to combanitorial problems, which includes the analysis of algorithms.

The article presents a discussion about the use of pencils to help solve quadratic equations. It offers a diagram relative to curves plotted from various quartic formulas which shows a resemblance to a pencil. It mentions that there is a mathematical notion that a pencil is being used in geometry and topology in previous years. It emphasizes that one of the amazing things about mathematics is that ideas that at first seem unrelated are often related in surprising ways such as the use of pencils as mathematical models for the quadratic equation.