Multiplicative Identities Research Articles

Strolling through common meadows

This paper establishes a connection between rings, lattices and common meadows. Meadows are commutative and associative algebraic structures with two operations (addition and multiplication) with additive and multiplicative identities and for which inverses are total. Common meadows are meadows that introduce, as the inverse of zero, an error term a which is absorbent for addition. We show that common meadows are unions of rings which are ordered by a partial order that defines a lattice. These results allow us to extend some classical algebraic constructions to the setting of common meadows. We also briefly consider common meadows from a categorical perspective.

Prospective teachers’ unified understandings of the structure of identities

United States curriculum standards advise mathematics teachers to teach students to attend to structure and understand how mathematical concepts are related. This requires teachers to have a structural perspective and a coherent, unified understanding of mathematical structures that span curricula. This study explores Prospective Secondary Mathematics Teachers’ (PSMTs) unified understandings of identities and characterizes the structural features of identities that PSMTs attend to. I contribute a theoretical framework of three ways in which PSMTs reason about identities: a do-nothing element, a result of undoing something, and a coordination with inverse, binary operation, and set. I classify the level of coherence of their identity schemas demonstrated as they reasoned about the structural connections among additive, multiplicative, and compositional identities. I illustrate how having unified, coherent understandings of identities can lead PSMTs to reason productively about inverse and identity functions, while having incoherent understandings of identities can lead to inaccurate reasoning about inverse and identity functions. I conclude with teaching implications for fostering PSMTs’ unified understandings of algebraic concepts.

On Simpson’s and Newton’s type inequalities in multiplicative fractional calculus

In this paper, we prove two multiplicative fractional integral identities involving multiplicative differentiable functions. Then, with the help of newly established identities, we establish multiplicative fractional versions of Simpson?s and Newton?s formulas type inequalities for differentiable multiplicative convex functions. It is also shown that the newly proved inequalities are extensions of some existing inequalities within the literature.

Some new midpoint and trapezoidal type inequalities in multiplicative calculus with applications

In this paper, we use multiplicative twice differentiable functions and establish two new multiplicative integral identities. Then, we use convexity for multiplicative twice differentiable functions and establish some new midpoint and trapezoidal type inequalities in the framework of multiplicative calculus. Finally, we give some applications to special means of real numbers to make these inequalities more interesting for the readers.

EFFECTS OF DEMOGRAPHIC STRUCTURE IN GROWTH ACCOUNTING AND LABOUR MARKET DECOMPOSITIONS

Impact of demographic structure on labor market and macroeconomic aggregates might be pronounced in some countries. Despite this fact, only a handful of approaches dealing with quantifications such effects have been derived so far. The aim of this paper is therefore to fill this methodological gap and to introduce methodological approaches for capturing changes in demographic structure, with many applications in growth accounting and labor market decompositions. Firstly, a novel additive decomposition will be presented, as an alternative to traditional models using fixed population weights. This will be followed by the presentation of a multiplicative decomposition, which can be applied to all kinds of growth accounting exercises based on multiplicative identities.

Certain eta-quotients and $$\ell $$-regular overpartitions

Let $${\overline{A}}_{\ell }(n)$$ be the number of overpartitions of n into parts not divisible by $$\ell $$ . In this paper, we prove that $${\overline{A}}_{\ell }(n)$$ is almost always divisible by $$p_i^j$$ if $$p_i^{2a_i}\ge \ell $$ , where j is a fixed positive integer and $$\ell =p_1^{a_1}p_2^{a_2} \dots p_m^{a_m}$$ with primes $$p_i>3$$ . We obtain a Ramanujan-type congruence for $${\overline{A}}_{7}$$ modulo 7. We also exhibit infinite families of congruences and multiplicative identities for $${\overline{A}}_{5}(n)$$ .

Rigid HYM Connections on Tautological Bundles over ALE Crepant Resolutions in Dimension Three

For $G$ a finite subgroup of ${\rm SL}(3,{\mathbb C})$ acting freely on ${\mathbb C}^3{\setminus} \{0\}$ a crepant resolution of the Calabi-Yau orbifold ${\mathbb C}^3\!/G$ always exists and has the geometry of an ALE non-compact manifold. We show that the tautological bundles on these crepant resolutions admit rigid Hermitian-Yang-Mills connections. For this we use analytical information extracted from the derived category McKay correspondence of Bridgeland, King, and Reid [J. Amer. Math. Soc. 14 (2001), 535-554]. As a consequence we rederive multiplicative cohomological identities on the crepant resolution using the Atiyah-Patodi-Singer index theorem. These results are dimension three analogues of Kronheimer and Nakajima's results [Math. Ann. 288 (1990), 263-307] in dimension two.

Lattice points in orthotopes and a huge polynomial Tutte invariant of weighted gain graphs

A gain graph is a graph whose edges are orientably labeled from a group. A weighted gain graph is a gain graph with vertex weights from an abelian semigroup, where the gain group is lattice ordered and acts on the weight semigroup. For weighted gain graphs we establish basic properties and we present general dichromatic and forest-expansion polynomials that are Tutte invariants (they satisfy Tutte's deletion-contraction and multiplicative identities). In order to do that we develop a relative of the Tutte polynomial of a semimatroid. Our dichromatic polynomial includes the classical graph one by Tutte, Zaslavsky's two for gain graphs, Noble and Welsh's for graphs with positive integer weights, and that of rooted integral gain graphs by Forge and Zaslavsky. It is not a universal Tutte invariant of weighted gain graphs; that remains to be found.An evaluation of one example of our polynomial counts proper list colorations of the gain graph from a color set with a gain-group action. When the gain group is Zd, the lists are order ideals in the integer lattice Zd, and there are specified upper bounds on the colors, then there is a formula for the number of bounded proper colorations that is a piecewise polynomial function of the upper bounds, of degree nd where n is the order of the graph.This example leads to graph-theoretical formulas for the number of integer lattice points in an orthotope but outside a finite number of affinographic hyperplanes, and for the number of n×d integral matrices that lie between two specified matrices but not in any of certain subspaces defined by simple row equations.

ε-Complemented algebras

For a nonassociative algebra A, by considering A as a left module over its multiplication algebra M ( A ) , a closure operation (termed the ε-closure) appears on the lattice I A of all ideals of A. For an ideal U of A, the ε-closure of U is the largest ideal of A which satisfies the same “multiplicative identities” as U. An algebra A is said to be ε-complemented if for every ε-closed ideal U of A there exists an ε-closed ideal V of A such that A = U ⊕ V . What is termed the ε ′ -closure appears in a dual fashion in I M ( A ) and the ε ′ -complementarity can be considered in M ( A ) . This paper provides different characterizations of both complementarities. Moreover, we determine the relation between these concepts, the classical complementarity, and the complementarity for the π-closure. We also develop a structure theory for ε-complemented algebras.

Multiplicative Identities for Binomial Coefficients

Multiplicative Identities for Binomial Coefficients

RINGS WITH LEFT MULTIPLICATIVE IDENTITIES

In this paper we introduce the notion of left multiplicative identities for rings and prove some properties of such rings.

The Gender Gap in College: Maximizing the Developmental Potential of Women and Men (review)

Reviewed by: The Gender Gap in College: Maximizing the Developmental Potential of Women and Men Rebecca Ropers-Huilman and Kaaren Williamsen Linda J. Sax. The Gender Gap in College: Maximizing the Developmental Potential of Women and Men. San Francisco: Jossey-Bass, 2008. 352 pp. Hardcover: $40.00. ISBN-13: 978-0787965754. In The Gender Gap in College: Maximizing the Developmental Potential of Women and Men, Linda Sax offers an in-depth analysis of the ways in which men and women experience and are affected by higher education. Emphasizing the need to focus on which types of students benefit from which college experiences, Sax presents her investigation of how women and men: (a) differ when they enter college currently and over time; (b) experience a contraction or expansion of gender differences in their college years; and (c) are affected (whether similarly or differently) by their collegiate experiences. Sax conducts this investigation through the use of two Cooperative Institutional Research Program (CIRP) databases, the Freshman Survey and the College Student Survey, both well-regarded instruments in higher education research communities. Sax argues that our shared wisdom about college student development and engagement looks at students “in the aggregate” and that we have much less understanding about how students experience college based upon important differences such as gender, race, class, and sexual orientation. This, she contends, limits our knowledge base and decreases the likelihood that specific interventions will be successful. In three of her chapters, “Personality and Identity,” “Political and Social Values,” and “Academic Outcomes” she analyzes data for each area and finds factors that are significant for men only, women only, and for both men and women. For example, she finds that feeling supported by faculty members is the most important college factor for both men and women in their quest to develop a meaningful philosophy of life. She also found that, for women, the time spent commuting to and from class was also an important factor. She speculates that this travel time gave women the opportunity for reflection. Men were more likely to develop a philosophy of life if they were paying for college with money that they earned while in college, as opposed to paying by student loans. Sax wonders if the men who incur a great deal of debt worry more about getting a well-paying job after college than creating a meaningful philosophy of life. In the last chapter, Sax highlights areas where she found significant differences for men and women and offers practical advice for creating campus programs and initiatives to take these differences into account. For example, Sax considered financial concerns and student-faculty interaction. Since women from lower socioeconomic backgrounds are entering college at higher rates than men, it should be of no surprise that Sax found that women students in aggregate are more concerned about their financial circumstances. There are other, more surprising findings, including outcomes of student-faculty interactions. [End Page 554] For male students, spending more time with faculty correlated with having less traditional beliefs about gender roles, whereas for women, the opposite was true. Sax calls on the entire campus community to recognize the importance of understanding how different students experience college differently. While we found this book useful in a variety of ways, we also were aware of its limitations. First, there was little attempt to consider the ways in which gender is shaped by the multiplicative (rather than additive) identities that each of us possess. Scholarship from both the field of higher education and other fields (T. Elon Dancy & M. Christopher Brown, 2008; Michael Cuyjet, 2006; Ana Martínez Aléman, 2000; Patrick Dilley, 2002; Laura Rendón, 2008, Patricia Williams, 1992; Adrien Wing, 2003, for example) has critiqued a unilateral approach to studying groups, suggesting that this approach tends to reify the majority (White middle-class heterosexual) experience in ways that may or may not reflect the lived realities of those who do not neatly fit that category. We wonder, even with the cautions offered in the book, how this text will be read and used. Our second critique addresses how the concept of gender was constructed in this text. Gender theory has long suggested that, especially since...

USING COMBINATORIAL APPROACH TO IMPROVE STUDENTS’ LEARNING OF THE DISTRIBUTIVE LAW AND MULTIPLICATIVE IDENTITIES

This article reports an alternative approach, called the combinatorial model, to learning multiplicative identities, and investigates the effects of implementing results for this alternative approach. Based on realistic mathematics education theory, the new instructional materials or modules of the new approach were developed by the authors. From the combinatorial activities based on the things around daily life, the teaching modules assisted students to establish their concept of the distributive law, and to generalize it via the process of progressive mathematizing. The subjects were two classes of 8th graders. The experimental group (n = 32) received a combinatorial approach to teaching by the first author using a problem-centered with double-cycles instructional model, while the control group (n = 30) received a geometric approach to teaching, from the textbook by another teacher who uses lecturing. Data analyses were both qualitative and quantitative. The findings indicated that the experimental group had a better performance than the control group in cognition, such as for the inner-school achievement test, mid-term examination, symbol manipulation, and unfamiliar problem-solving: also in affection, such as the tendency to engage in the mathematics activities and enjoy mathematical thinking.

Extending the reach of computation

The only well-defined mathematical model of the real number system based on the field axioms is the system of terminating decimals. This is reconstructed as the new real number system and built on the basic integers 0 and 1, the additive and multiplicative identities, respectively, and the addition and multiplication tables of elementary arithmetic. Then standard Cauchy sequences are introduced whose Cauchy limits well-define the nonterminating decimals. Cauchy convergence induces the Cauchy norm and the closure of the terminating decimals in the Cauchy norm is the space of nonterminating decimals and the continuum d ∗ that glues together the decimals into the continuum R ∗ which is non-Archimedean and non-Hausdorff. The decimals form a countably infinite, hence, discrete subspace of R ∗ that is both Archimedean and Hausdorff.

Configurations in abelian categories. IV. Invariants and changing stability conditions

This is the last in a series on configurations in an abelian category A . Given a finite poset ( I , ≼ ) , an ( I , ≼ ) -configuration ( σ , ι , π ) is a finite collection of objects σ ( J ) and morphisms ι ( J , K ) or π ( J , K ) : σ ( J ) → σ ( K ) in A satisfying some axioms, where J , K are subsets of I. Configurations describe how an object X in A decomposes into subobjects. The first paper defined configurations and studied moduli spaces of configurations in A , using Artin stacks. It showed well-behaved moduli stacks Obj A , M ( I , ≼ ) A of objects and configurations in A exist when A is the abelian category coh ( P ) of coherent sheaves on a projective scheme P, or mod- K Q of representations of a quiver Q. The second studied algebras of constructible functions and stack functions on Obj A . The third introduced stability conditions ( τ , T , ⩽ ) on A , and showed the moduli space Obj ss α ( τ ) of τ-semistable objects in class α is a constructible subset in Obj A , so its characteristic function δ ss α ( τ ) is a constructible function. It formed algebras H τ pa , H τ to , H ¯ τ pa , H ¯ τ to of constructible and stack functions on Obj A , and proved many identities in them. In this paper, if ( τ , T , ⩽ ) and ( τ ˜ , T ˜ , ⩽ ) are stability conditions on A we write δ ss α ( τ ˜ ) in terms of the δ ss β ( τ ) , and deduce the algebras H τ pa , … , H ¯ τ to are independent of ( τ , T , ⩽ ) . We study invariants I ss α ( τ ) or I ss ( I , ≼ , κ , τ ) ‘counting’ τ-semistable objects or configurations in A , which satisfy additive and multiplicative identities. We compute them completely when A = mod- K Q or A = coh ( P ) for P a smooth curve. We also find invariants with special properties when A = coh ( P ) for P a smooth surface with K P −1 nef, or a Calabi–Yau 3-fold.

NEW EXPLICIT MULTIPLICATIVE RELATIONS BETWEEN GAUSS SUMS

We study multiplicative identities between Gauss sums. If such an identity does not follow from the Davenport–Hasse relation and the norm relation, it is called a sign ambiguity. Until recently only a finite number of explicit sign ambiguities were known. We generalize the first infinite family of sign ambiguities as found by Murray.

Lattice-ordered Fields Determined by d-elements

Most results on the structure of lattice-ordered fields require that the field have a positive multiplicative identity. We construct a functor from the category of lattice-ordered fields with a vector space basis of d-elements to the full subcategory of such fields with positive multiplicative identities. This functor is a left adjoint to the forgetful functor and, in many cases, allows us to write all compatible lattice orders in terms of orders with positive multiplicative identities. We also use these results to characterize algebraically those extensions of totally ordered fields that have vl-bases of d-elements.

Using Clock Arithmetic to Teach Algebra Concepts

Learning algebra concepts can be difficult for middle school students. One reason may be because we work in only one number system, the set of real numbers. Students have only one frame of reference to provide examples of abstract concepts, such as the additive and multiplicative identities, additive and multiplicative inverses, and connections among the operations. These concepts are essential in solving equations. For example, we can think of an equation like 3x + 4 = 7 in the following way: Begin with a number, multiply it by 3, and add 4. If the answer is 7, what number did we start with? To solve this type of equation algebraically, we just undo what was done to the original number, that is, we add -4, the additive inverse of 4, and multiply the result by 1/3, the multiplicative inverse of 3. We can also think of this as subtracting 4 and dividing by 3, because addition and subtraction are inverse operations as are multiplication and division. The ideas of inverse operations and inverse elements are, therefore, central to algebra. Many students understand these ideas well enough to do simple problems like 3x + 4 = 7 but get confused with more difficult problems, such as 3(x + 6) + 2 = 5x + 5, where it is not as easy to see the order in which to undo these operations. We use finite systems to help students understand these key concepts in algebra, including additive and multiplicative identities, additive and multiplicative inverses, closure, and the relationships between addition and subtraction and multiplication and division.

World Population Projections for Greenhouse Gas Emissions Scenarios

A survey is made of the latest world population projections issued by the United Nations, World Bank, U.S. Census Bureau, and International Institute for Applied Systems Analysis. Medium variants from all the organizations show excellent agreement with respect to many features of future world population growth. It appears that little would be gained by obtaining additional regional projections made by governments or organizations other than those listed above. In general, the new range of population projections that are candidates for forthcoming IPCC emissions scenarios are narrower and lower than the previous IPCC IS92 population range: a reflection of updated information on the decline of fertility rates in developing countries and the incorporation of a plausible correlation between mortality rates and fertility rates within the IIASA ‘rapid’ and ‘slow’ demographic transition variants. Comments are made on the schematic approach of forecasting CO2 emissions using multiplicative identities such as ‘IPAT’ (impact/emissions = population × affluence × technology). Although the unqualified IPAT model suggests that emissions should scale linearly with population, a number of caveats to this exist, the most important of which may be factor interactions. A brief review is made of conventional thinking about interactions between population growth and economic development. Correlation studies and theory suggest that population growth has a neutral or, at most, weak negative effect on economic growth. Conversely, it is well established that higher per capita incomes are well correlated with lower fertility and mortality rates in developing countries. Therefore, a plausible first-order relationship worth exploring in the next generation of IPCC scenarios is that scenarios with higher average economic growth rates in the developing world should be associated with lower fertility and mortality rates there. Calculations are presented that illustrate the effect this negative correlation could have had on the range of the older IS92 emission scenarios, assuming that all other factors are unchanged. Finally, some policy issues concerning population and global warming are reviewed in connection with the IPCC’s omission of population policy discussion in its 1995 Second Assessment Report.

Grassmannians, multiplicative ward identities and theta-function identities

The symmetry group Div 0(Σ) on a Riemann surface is used to derive multiplicative Ward identities. These are used to determine the propagator for a spin- J system and to derive Fay's trisecant identity.