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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]

• 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]

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: 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: 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: 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]

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]

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]

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