Search

In this paper, we derive a Mittag-Leffler function for real index and establish solutions of special type of fractional order differential equations (FDEs). The same concept is extended to discrete case by replacing polynomials into factorial polynomials and differentiation into l-difference operator. Moreover, numerical examples of our results are stated to validate our findings. The acquired results here have the ability to generate a wide range of formulas in relation to newer results. [ABSTRACT FROM AUTHOR]

Recent results of Kahle and Miller give a method of constructing primary decompositions of binomial ideals by first constructing “mesoprimary decompositions” determined by their underlying monoid congruences. Monoid congruences (and therefore, binomial ideals) can present many subtle behaviors that must be carefully accounted for in order to produce general results, and this makes the theory complicated. In this paper, we examine their results in the presence of a positive A -grading, where certain pathologies are avoided and the theory becomes more accessible. Our approach is algebraic: while key notions for mesoprimary decomposition are developed first from a combinatorial point of view, here we state definitions and results in algebraic terms, which are moreover significantly simplified due to our (slightly) restricted setting. In the case of toral components (which are well-behaved with respect to the A -grading), we are able to obtain further simplifications under additional assumptions. We also provide counterexamples to two open questions, identifying (i) a binomial ideal whose hull is not binomial, answering a question of Eisenbud and Sturmfels, and (ii) a binomial ideal I for which I toral is not binomial, answering a question of Dickenstein, Miller and the first author. [ABSTRACT FROM AUTHOR]

In this work, we develop our idea on the construction of a system of combinatorial generators in a T-ideal of a free associative algebra, which is a full analogy of a Gröbner-Shirshov basis in a polynomial ideal. We prove a theorem on multilinear monomials that enables us to establish the existence of a finite set of combinatorial generators in a metabelian T-ideal. [ABSTRACT FROM AUTHOR]

The unital quantum operation acting on infinite dimensional quantum states fixing a convex cone of density operators is completely characterized. Based on this result, we classify the commutativity of two quantum operations and determine what kind of measurement statistics are preserved by a unital quantum operation. Our work extends the results of Lee et�al (2013 J. Phys. A: Math. Theor.46 205305) to the infinite dimensional case and also corrects part (a) of theorem 6 in Lee et�al (2013 J. Phys. A: Math. Theor.46 205305). [ABSTRACT FROM AUTHOR]

• Square-rooting can be sensitive to nuanced differences in prompting questions. • Students can exhibit apparent conflicts within their mathematical discourses. • The conflicts can be explained with incommensurable precedent pockets. • The commognitive framework can be applied to analyze written data. This study explores the grasp of square roots among eleven students in a remedial mathematics course, with a special focus on where the same student generated apparently conflicting responses. Building on the commognitive framework, the analysis distinguished between routines that individual students consistently implemented in situations where roots "stood alone" and where they were incorporated in more compound exercises, where roots were extracted from square numbers and from squared radicands, where roots were applied to monomials and binomials, and where parameters named with different letters were involved. Differences were found in routines' degree of objectification, procedures, and tasks. These differences are explained with a theoretical account, suggesting that what may seem as a conflict within a student's discourse could be a sensible difference of actions taken in situations that this student construed as different. The contribution of this study to the body of knowledge on teaching and learning of roots and to commognitive research is discussed. [ABSTRACT FROM AUTHOR]

For a Markov chain, both the detailed balance condition and the cycle Kolmogorov condition are algebraic binomials. This remark suggests to study reversible Markov chains with the tool of Algebraic Statistics, such as toric statistical models. One of the results of this study is an algebraic parameterization of reversible Markov transitions and their invariant probability. [ABSTRACT FROM AUTHOR]

Situations often arise in which the matrix of independent variables is not of full column rank. That is, there are one or more linear dependencies among the independent variables. This paper covers in detail the situation in which the rank is one less than full column rank and extends this coverage to include cases of even greater rank deficiency. The emphasis is on the row geometry of the solutions based on the normal equations. The author shows geometrically how constrainedregression/ generalized-inverses work in this situation to provide a solution in the face of rank deficiency. [ABSTRACT FROM AUTHOR]

Abstract: Using Gröbner bases for the construction of public key cryptosystems has been often attempted, but has always failed. We review the reason for these failures, and show that only ideals generated by binomials may give a successful cryptosystem. As a consequence, we concentrate on binomial ideals that correspond to Euclidean lattices. We show how to build a cryptosystem based on lattice ideals and their Gröbner bases, and, after breaking a simple variant, we construct a more elaborate one. In this variant the trapdoor information consists in a “small” change of coordinates that allows one to recover a “fat” Gröbner basis. While finding a change of coordinates giving a fat Gröbner basis is a relatively easy problem, finding a small one seems to be a hard optimization problem. This paper develops the details and proofs related to computer algebra, the cryptographic details related to security, the comparison with other lattice cryptosystems and discusses the implementation. [Copyright &y& Elsevier]

Let s and t be variables. Define polynomials { n} in s, t by {0} = 0, {1} = 1, and { n} = s { n - 1} + t { n - 2} for n ≥ 2. If s, t are integers then the corresponding sequence of integers is called a Lucas sequence. Define an analogue of the binomial coefficients by where { n}! = {1} {2} ⋯ { n}. It is easy to see that is a polynomial in s and t. The purpose of this note is to give two combinatorial interpretations for this polynomial in terms of statistics on integer partitions inside a k × ( n - k) rectangle. When s = t = 1 we obtain combinatorial interpretations of the fibonomial coefficients which are simpler than any that have previously appeared in the literature. [ABSTRACT FROM AUTHOR]

In this short paper we establish identities involving sums of products of binomial coefficients and coefficients that satisfy the general second-order linear recurrence. We obtain generalizations of identities of Carlitz, Prodinger and Haukkanen. [ABSTRACT FROM AUTHOR]

The algebraic area probability distribution of closed planar random walks of length N on a square lattice is considered. The generating function for the distribution satisfies a recurrence relation in which the combinatorics is encoded. A particular case generalizes the q-binomial theorem to the case of three addends. The distribution fits the Lévy probability distribution for Brownian curves with its first-order 1/ N correction quite well, even for N rather small. [ABSTRACT FROM AUTHOR]

In 1960, Sierpiński proved that there exist infinitely many odd positive integers k such that k · 2n + 1 is composite for all integers n ≥ 0. Variations of this problem, using polynomials with integer coefficients, and considering reducibility over the rationals, have been investigated by several authors. In particular, if S is the set of all positive integers d for which there exists a polynomial f(x) ∈ ℤ[x], with f(1) ≠ -d, such that f(x)xn + d is reducible over the rationals for all integers n ≥ 0, then what are the elements of S? Interest in this problem stems partially from the fact that if S contains an odd integer, then a question of Erdös and Selfridge concerning the existence of an odd covering of the integers would be resolved. Filaseta has shown that S contains all positive integers d ≡ 0 (mod 4), and until now, nothing else was known about the elements of S. In this paper, we show that S contains infinitely many positive integers d ≡ 6 (mod 12). We also consider the corresponding problem over 픽p, and in that situation, we show 1 ∈ S for all primes p. [ABSTRACT FROM AUTHOR]

Abstract: We show that there can be no finite list of conditional independence relations which can be used to deduce all conditional independence implications among Gaussian random variables. To do this, we construct, for each a family of conditional independence statements on random variables which together imply that , and such that no subset have this same implication. The proof relies on binomial primary decomposition. [Copyright &y& Elsevier]

Abstract: We study del Pezzo surfaces of degree 1 of the form in the weighted projective space , where k is a perfect field of characteristic not 2 or 3 and . Over a number field, we exhibit an infinite family of (minimal) counterexamples to weak approximation amongst these surfaces, via a Brauer–Manin obstruction. [Copyright &y& Elsevier]

In this article, an attempt has been made to settle the question of existence of unbiased estimator of the key parameter p of the quasi-binomial distributions of Type I (QBD I) and of Type II (QBD II), with/without any knowledge of the other parameter φ appearing in the expressions for probability functions of the QBD's. This is studied with reference to a single observation, a random sample of finite size m as also with samples drawn by suitably defined sequential sampling rules. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider the existence and multiplicity of nodal solutions of semilinear elliptic equations. We prove that a semilinear elliptic equation in large domains does not admit any least energy nodal (sign-changing) solution and in an upper half strip with -holes has at least 2-nodal solutions. [Copyright &y& Elsevier]

Abstract. The paper considers a new algebraic object, the completely automaton binomial algebras, which generalize certain existing classes of algebras. The author presents a classification of semigroup algebras taking into account completely automaton algebras and gives the corresponding examples. A number of standard algorithmic problems are solved for completely automaton binomial algebras: the recognition of a strict and nonstrict polynomial property, the recognition of the right and/or left finite processing, and the construction of the determining regular language for an algebra with finite processing and for monomial subalgebras of a free associative algebra and certain completely automaton algebras. For an automaton monomial algebra, the author constructs the left syzygy module of a finite system of elements and the Gröbner basis of a finitely generated left ideal; also, some algorithmic problems are solved. [ABSTRACT FROM AUTHOR]

We describe exercises in which students use PascGaloisJE to formulate conjectures about certain binomial identities which hold when the binomial coefficients are interpreted as elements in the cyclic group [image omitted] of integers modulo a prime integer p. In addition to having an appealing visual component, these exercises are open-ended and expose students to the investigative nature of mathematical discovery. [ABSTRACT FROM AUTHOR]

A set of new matrix Lie algebra and its corresponding loop algebra are constructed. By making use of Tu scheme, a Liouville integrable multi-component hierarchy of soliton equation is generated. As its reduction cases, the multi-component Tu hierarchy is given. Finally, the multi-component integrable coupling system of Tu hierarchy is presented through enlarging matrix spectral problem. [ABSTRACT FROM AUTHOR]

A new loop algebra containing four arbitrary constants is presented, and the corresponding computing formula of constant γ in the quadratic-form identity is obtained in this paper, which can be reduced to computing formula of constant γ in the trace identity. As application, a new Liouville integrable hierarchy which possess bi-Hamiltonian structure and generalized Burgers hierarchy are derived. [ABSTRACT FROM AUTHOR]

Abstract: Making use a linear operator, we introduce two novel subclasses and of meromorphically -valent functions with positive and negative coefficients, respectively. The main object of this paper is to investigate the various important properties and characteristics of these subclasses of meromorphically -valent functions. We also derive many interesting results for the Hadamard products of functions belonging to the classes and . [Copyright &y& Elsevier]

Abstract: We study the local fluctuations of Brownian motions which are represented by infinite binary strings which are random in the sense of Kolmogorov–Chaitin. We show how the dynamical properties of such a Brownian motion at a point depend on its recursive properties. [Copyright &y& Elsevier]

Abstract: For sums-of-squares formulas of the form where the are bilinear functions of the and . Let denote the smallest possible value of t allowing such a formula to hold. We have two well-known lower bounds on the size of . One was obtained independently by Hopf and Stiefel, and another by Atiyah. These bounds are given by requiring certain binomial coefficients be divisible by certain powers of 2. Although the behavior of the Hopf–Stiefel bound is fairly well understood, the Atiyah bound is not. In this paper we provide an efficient algorithm for computing the Atiyah bound and some results on which of the lower bounds is larger. [Copyright &y& Elsevier]

Abstract: We develop a rank variety for finite-dimensional modules over a certain class of finite-dimensional local -algebras, . Included in this class are the truncated polynomial algebras , with an algebraically closed field and arbitrary. We prove that these varieties characterise projectivity of modules (Dade’s lemma) and examine the implications for the tree class of the stable Auslander–Reiten quiver. We also extend our rank varieties to infinitely generated modules and verify Dade’s lemma in this context. [Copyright &y& Elsevier]

We introduce the concept of Hochschild cohomologies for associative conformal algebras. It is shown that the second cohomology group of a conformal Weyl algebra with values in any bimodule is trivial. As a consequence, we derive that the conformal Weyl algebra is segregated in any extension with nilpotent kernel. [ABSTRACT FROM AUTHOR]

The article presents a mathematical activity which relates the coefficients in the formula to finding a parking spot. First, the students discover Pascal's triangle using a parking spot simulation. Then the activity relates Pascal's triangle to the coefficients in the binomial theorem. Graphing calculators and combinations are used to verify the binomial expansion. This activity supports a variety of learning styles in an enjoyable application. The relationship between the parking spot simulation and Pascal's triangle is explained.

The article discusses the use of the algebraic expression (a ±b)² in determining the pattern for weaving a plaid design. This application of mathematics to weaving was developed by Ada K. Dietz, a mathematics teacher and novice weaver, sometime in the 1940s and 1950s. Dietz discovered that the expression gave the most pleasing patterns and demonstrated this to middle-grade students in either their mathematics or art classes.

Abstract: In this paper we establish a q-analogue of a congruence of Sun concerning the products of binomial coefficients modulo the square of a prime. [Copyright &y& Elsevier]

Abstract: In this paper, we first give an interesting operator identity. Furthermore, using the q-exponential operator technique to the multiple q-binomial theorem and q-Gauss summation theorem, we obtain some transformation formulae and summation theorems of multiple basic hypergeometric series. [Copyright &y& Elsevier]

An algebra A is said to be congruence permutable if any two congruences on it are permutable. This property has been investigated in several varieties of algebras, for example, de Morgan algebras, p-algebras, Kn,o-algebras. In this paper, we study the class of symmetric extended de Morgan algebras that are congruence permutable. In particular we consider the case where A is finite, and show that A is congruence permutable if and only if it is isomorphic to a direct product of finitely many simple algebras. [ABSTRACT FROM AUTHOR]

We give necessary and sufficient conditions that a polynomial f ( x, y) gives an ovoid of PG(3, q). Furthermore by considering Rédei polynomials associated to ovoids of PG(3, q) we obtain sets of equations relating the coefficients of f ( x, y). One such set of equations implies that if xiyj occurs as a term in f ( x, y) + xy then i and j must have disjoint binary expansions. [ABSTRACT FROM AUTHOR]

In this article, we explicitly calculated the values of the representations of the Hecke algebra $\mathcal {H}$, associated with a Gelfand–Graev character of GL4(q), at some of the standard basis elements. [ABSTRACT FROM AUTHOR]

We calculate the Hochschild cohomology of the algebra of smooth functions on a finite-dimensional real torus with coefficients in the adjoint representation, generalizing the previously developed technique to the discrete case for this. [ABSTRACT FROM AUTHOR]

We study a new class of infinite dimensional Lie algebras, which has important applications to the theory of integrable equations. The construction of these algebras is very similar to the one for automorphic functions and this motivates the name automorphic Lie algebras. For automorphic Lie algebras we present bases in which they are quasigraded and all structure constants can be written out explicitly. These algebras have useful factorisations on two subalgebras similar to the factorisation of the current algebra on the positive and negative parts. [ABSTRACT FROM AUTHOR]

A binomial f belonging to a toric ideal I is indispensable if, for any system $\mathcal{F}$ of binomial generators of I, either f or -f belongs to $\mathcal{F}$. In the present paper, we study indispensable binomials of the toric ideals IG arising from a finite graph G. First, we show that the toric ideal IG arising from a finite graph G whose complementary graph is weakly chordal is generated by the indispensable binomials if and only if no complete graph of order ≥4 is a subgraph of G. Second, we completely classify indispensable binomials of the toric ideal IG arising from a finite graph G satisfying the odd cycle condition. Finally, the existence of indispensable binomials of IG will be discussed. [ABSTRACT FROM AUTHOR]

A systematic use of binary codes derived from the Hägg symbol are used to study close-packed polytypes. Seitz operators acting over the corresponding binary codes are defined and used. The number of non-equivalent polytypes of a given length are calculated through the use of the Seitz operators. The same procedure is applied to the problem of counting the number of polytypes complying with a given symmetry group. All counting problems are reduced to an eigenvector problem in the binary code space. The symmetry of the binary codes leads to the different space groups to which polytypes can belong. [ABSTRACT FROM AUTHOR]

I consider the expansion of transcendental functions in a small parameter around rational numbers. This includes in particular the expansion around half-integer values. I present algorithms which are suitable for an implementation within a symbolic computer algebra system. The method is an extension of the technique of nested sums. The algorithms allow in addition the evaluation of binomial sums, inverse binomial sums and generalizations thereof. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

The two main classes of finite dimensional composition algebras, Hurwitz and symmetric, are characterized in terms of the flexible identity, and even of the third power associative identity, over fields of arbitrary characteristic. [ABSTRACT FROM AUTHOR]

Let K be a field. A split basic finite-dimensional K-algebra with left quiver F is monomial if it can be represented as the path algebra KΓ modulo an ideal generated by paths in Γ, and binomial if it can be represented as KΓ modulo relations of the form p = λq, where λ ∈ c K and p and q are paths in Γ. Burgess et al. [Burgess, W. D., Fuller, K. R., Green, E. L., Zacharia, D. (1993). Left monomial rings-A generalization of monomial algebras. Osaka J. Math. 30(3):543-558] define monomial rings, generalizing monomial algebras; we here take this generalization one step further, introducing and studying the class of binomial rings, which includes the classes of binomial algebras, monomial rings, and square-free rings. We further show that every binomial ring has canonically associated with it a binomial algebra, with which the ring shares many properties. [ABSTRACT FROM AUTHOR]

An algebraic formalism for quantum decoherence in systems with continuous evolution spectrum is introduced. A certain subalgebra, dense in the characteristic algebra of the system, is defined in such a way that Riemann-Lebesgue theorem can be used to explain decoherence in a well defined final pointer basis. [ABSTRACT FROM AUTHOR]

Using the WZ-method we find some of the easiest Ramanujan''s formulae and also some new interesting Ramanujan-like sums. [Copyright &y& Elsevier]

Let G be a simple graph on n vertices and J G denote the binomial edge ideal of G in the polynomial ring S = K [ x 1 , ... , x n , y 1 , ... , y n ]. In this article, we compute the second graded Betti numbers of J G , and we obtain a minimal presentation of it when G is a tree or a unicyclic graph. We classify all graphs whose binomial edge ideals are almost complete intersection, prove that they are generated by a d -sequence and that the Rees algebra of their binomial edge ideal is Cohen-Macaulay. We also obtain an explicit description of the defining ideal of the Rees algebra of those binomial edge ideals. [ABSTRACT FROM AUTHOR]

Let G be a simple graph on n vertices. Let L G and I G denote the Lovász–Saks–Schrijver(LSS) ideal and parity binomial edge ideal of G in the polynomial ring S = K [ x 1 , ... , x n , y 1 , ... , y n ] respectively. We classify graphs whose LSS ideals and parity binomial edge ideals are complete intersections. We also classify graphs whose LSS ideals and parity binomial edge ideals are almost complete intersections, and we prove that their Rees algebra is Cohen–Macaulay. We compute the second graded Betti number and obtain a minimal presentation of LSS ideals of trees and odd unicyclic graphs. We also obtain an explicit description of the defining ideal of the symmetric algebra of LSS ideals of trees and odd unicyclic graphs. [ABSTRACT FROM AUTHOR]

A computatiorial algorithm is developed for generating the cumulative distribution of successes in a sequence of dependent trials, characterized by a Markov model, for which tables of the distribution may be compiled. Such tables may be used to assess the impact of such dependence on the calculation of confidence intervals and the testing of hypotheses. Examples are presented to show how such confidence intervals and hypothesis tests deteriorate with increasing dependence. [ABSTRACT FROM AUTHOR]

Abstract: We give a simple proof of Jackson''s terminating q-analogue of Dixon''s identity. [Copyright &y& Elsevier]

Shows that Green's theorem is sufficient in proving the fundamental theorem of algebra. Linearity of derivatives; Binomial theorem.

Provides a simple proof of Descartes' rule of signs in high school algebra. Proof of theorems; Variations in the signs of the sequence of coefficients; Number of positive zeros of p counting multiplicities by z; Nonnegative integers; Induction hypothesis; Arbitrary real numbers.