Multiplicative Relationships Research Articles

Mathematical modeling as a component of understanding ratio-as-measure: A study of prospective elementary teachers

The authors employed a whole-class teaching experiment in a mathematics class for prospective elementary teachers to study the development of the ability to identify a ratio as the appropriate measure of a given attribute (“ratio-as-measure”). Prior studies as well as pretest data for this study demonstrated that students often invoke additive comparisons when multiplicative comparisons are appropriate. Analysis of the difficulties which students encountered in considering a ratio as an appropriate measure of an attribute of a physical situation led to the hypothesis that an understanding of ratio-as-measure requires, in addition to an understanding of aspects of multiplicative relationships, an understanding of mathematical modeling.

The impedance of scaled transmission lines

It is shown that resistance/capacitance transmission lines with multiplicative scaling relationships in the resistive and capacitive elements exhibit a limited range of spectral responses. At high and low frequencies the characteristic impedance of the first, and in a line of finite length the last, elements of the line dominate. In the intermediate frequency range either constant phase angle response or fractional power response may be obtained. The scaling conditions necessary for the observation of these intermediate-frequency responses are established. It is shown that the scaled transmission lines can be considered as fractal when the scaling derives from fractional dimensional properties in continuous media.

Vector Autoregression and the Study of Politics

In many respects political scientists agree about how best to model political processes. But we disagree about how to translate our theories into structural equations; each of us seems to have our own structural equation model of the same theory. This disagreement is a serious impediment to theory building. Vector autoregression (VAR) is a means of circumventing this problem. We explain the logic of this alternative modeling strategy and examine its relative virtues. In particular, VAR and the more familiar structural equation (SEQ) approaches are compared in terms of their epistemological underpinnings, empirical power, and usefulness in policy analysis. This comparison shows that the two modeling strategies are based on different conceptions of theory and of theory building and that, for the four-six variable systems we usually study, the choice between VAR and SEQ models presents a trade-off between accuracy of causal inference and quantitative precision, respectively. In addition, VAR models have the disadvantage of being unable to incorporate multiplicative and nonlinear relationships as easily as SEQ models. But VAR models have the advantage of providing a more complete treatment of policy endogeneity than SEQ models. These and other contrasts in the two modeling strategies are illustrated in a reanalysis of Alt and Chrystal's (1983) permanent income model of government expenditure.

Numerical Results on Relations Between Fundamental Constants Using a New Algorithm

Let $x = ({x_1},{x_2}, \ldots ,{x_n})$ be a vector of real numbers, x is said to possess an integer relation if there exist integers ${a_i}$ not all zero such that ${a_1}{x_1} + {a_2}{x_2} + \cdots + {a_n}{x_n} = 0$. Beginning ten years ago, algorithms were discovered by one of us which, for any n, are guaranteed to either find a relation if one exists, or else establish bounds within which no relation can exist. One of those algorithms has been employed to study whether or not certain fundamental mathematical constants satisfy simple algebraic polynomials. Recently, one of us discovered a new relation-finding algorithm that is much more efficient, both in terms of run time and numerical precision. This algorithm has now been implemented on high-speed computers, using multiprecision arithmetic. With the help of these programs, several of the previous numerical results on mathematical constants have been extended, and other possible relationships between certain constants have been studied. This paper describes this new algorithm, summarizes the numerical results, and discusses other possible applications. In particular, it is established that none of the following constants satisfies a simple, low-degree polynomial: $\gamma$ (Euler’s constant), $\log \gamma$, $\log \pi$, ${\rho _1}$ (the imaginary part of the first zero of Riemann’s zeta function), $\log {\rho _1}$, $\zeta (3)$ (Riemann’s zeta function evaluated at 3), and $\log \zeta (3)$. Several classes of possible additive and multiplicative relationships between these and related constants are ruled out. Results are also cited for Feigenbaum’s constant, derived from the theory of chaos, and two constants of fundamental physics, derived from experiment.

Numerical results on relations between fundamental constants using a new algorithm

Let x = ( x 1 , x 2 , … , x n ) x = ({x_1},{x_2}, \ldots ,{x_n}) be a vector of real numbers, x is said to possess an integer relation if there exist integers a i {a_i} not all zero such that a 1 x 1 + a 2 x 2 + ⋯ + a n x n = 0 {a_1}{x_1} + {a_2}{x_2} + \cdots + {a_n}{x_n} = 0 . Beginning ten years ago, algorithms were discovered by one of us which, for any n, are guaranteed to either find a relation if one exists, or else establish bounds within which no relation can exist. One of those algorithms has been employed to study whether or not certain fundamental mathematical constants satisfy simple algebraic polynomials. Recently, one of us discovered a new relation-finding algorithm that is much more efficient, both in terms of run time and numerical precision. This algorithm has now been implemented on high-speed computers, using multiprecision arithmetic. With the help of these programs, several of the previous numerical results on mathematical constants have been extended, and other possible relationships between certain constants have been studied. This paper describes this new algorithm, summarizes the numerical results, and discusses other possible applications. In particular, it is established that none of the following constants satisfies a simple, low-degree polynomial: γ \gamma (Euler’s constant), log ⁡ γ \log \gamma , log ⁡ π \log \pi , ρ 1 {\rho _1} (the imaginary part of the first zero of Riemann’s zeta function), log ⁡ ρ 1 \log {\rho _1} , ζ ( 3 ) \zeta (3) (Riemann’s zeta function evaluated at 3), and log ⁡ ζ ( 3 ) \log \zeta (3) . Several classes of possible additive and multiplicative relationships between these and related constants are ruled out. Results are also cited for Feigenbaum’s constant, derived from the theory of chaos, and two constants of fundamental physics, derived from experiment.

Substantive nonadditivity in social science research A note on induced collinearity and measurement and testing of effects

Theories employed to explain regularities in social behavior often contain reference (explicit or implicit) to the presence of nonlinear and/or nonadditive (i.e., multiplicative) relationships among germane variables. While such nonadditive features are theoretically important, the inclusion of quadratic or multiplicative terms in structural equations to model such features can cause significant methodological problems. This paper estimates a set of equations and formally examines how the inclusion of quadratic terms and multiplicative interaction terms contribute to the level of collinearity or ill-conditioning of the input data matrix and the precision of the parameter estimates. Subsequently we examine how effects of explanatory variables in nonadditive models can be measured and tested for statistical significance. The results indicate that collinearity may not be as big a problem for linear structural social science models as is often believed. Further, although collinearity is increased by adding quadratic and/or multiplicative terms, the effects of the collinearity tend to be localized and entail only variables with a common base. The findings suggest the substantive insight gained from including theoretically appropriate nonlinear and nonadditive terms outweigh the methodological problems they create.

Optimization of rated capacities of coal mines in Kuzbass in the framework of a dynamic multiplicative model

The authors analyze the optimization of the rated design capacity of a single coal mine without constraints. The rate of return is used as the target function of the problem. The authors solve the problem in a dynamic statement considering the trends of wholesale coal prices, coal excavation costs, and capital investment in mine construction. The variables optimized in the model are the rated design capacity and the pay reserve of coal in the mine field. The regression relationships of the technical and economic indicators of Kuzbass mines are shown, as are the multiplicative regression relationships of coal mining costs for the Kuzbass production associations. Optimal parameters of the statistical average coal mines of production associations of Kuzbass for the existing times of construction are presented. The authors' estimates shown that the optimal design capacity of a mine is substantially affected by wholesale coal prices.

Joint Effects of Group Composition, Group Norm, Type of Problem and Group vs. Individual Responding

Abstract: Egerbladh, T. & Sjodin, S. 1986: Joint Effects of Group Composition, Group Norm, Type of Problem and Group vs. Individual Responding. Scandinavian Journal of Educational Research 30, 17‐23. The general purpose was to study potential interactions between four factors of educational interest within small group research, that is, type of problem, group composition, group norm and group productivity, the last one being defined as group vs. individual responding. 88 boys and 92 girls from grades 4 and 5 were divided into randomly composed dyads that were homogeneous regarding sex. Three different educational norms were induced by using instructions for cooperation, competition and no instruction. Reliable interactions were obtained for a) Sex x Group Norm, b) Sex x Group Norm x Productivity and c) Type of Problem x Productivity. It is concluded that assumptions of multiplicative relationships between these factors will form important parts of a future model of group problem solving. The educational implications are considered in terms of diversified skills for group activities and subsequent individual retention of knowledge.

Types of life events in relation to symptoms at the climacterium

An increase in life events among the general population of women at the climacterium was shown to be almost exclusively due to an increase in the departure of other persons from the subjects' social field (exits). Most of this increase was found to be due to deaths, especially of significant persons. Using a multivariate analysis technique different classes of life events were shown to correlate differentially with psychological and somatic symptoms. While psychological symptoms were found to be directly and linearly related to stress arising from miscellaneous sources, somatic symptoms were elevated only if such stress was accompanied by stress arising from exits. These results suggest that both additive and multiplicative relationships may hold between life events and distress in this age group.

Formulas and multiplicative relationships for the parameters of subgroups of the modular group

Formulas and multiplicative relationships for the parameters of subgroups of the modular group

Formulas and multiplicative relationships for the parameters of subgroups of the modular group

I. Introduction Let F = PSL(2, Z) be the classical modular group. Let G be a subgroup off of finite index. The most important parameters associated with G are/~ =/~(G), the index of G; t = t(G), the number of parabolic classes of G; e2 = e2(G), the number of elliptic classes of order 2 of G; e3 = e3 (G), the number of elliptic classes of order 3 of G; and g=g(G), the genus of G. These are not independent but are related by the formula bt t e 2 e 3 g= 1 + 12 2 4 3 ' (1) which is just the hyperbolic area formula. The point of view we take here is the following: we assume that a left coset decomposition for F modulo G is known. From this decomposition, we determine the parameters described above. We then apply these results to the case of congruence groups, and prove that/~, t, e2, e3 are multiplicative arithmetic functions (in a well-defined sense) of the level. Finally, we mention briefly some applications to the determination of congruence groups with various restrictions imposed on the parameters; for example, with a fixed number of parabolic classes. If f2 = SL(2, Z) and ~0 is the natural homomorphism of t2 modulo its center C = {I, - I}, then

Perceived Risk and Its Components: A Model and Empirical Test

A theoretical model and measurement system for perceived risk and its components is developed and empirically tested for nine product types. Regression analysis is used to consider both additive and multiplicative relationships. The results generally support the hypothesized model.

Some Theoretical and Methodological Considerations for Measuring Intrasocietal Conflict

Among previous attempts to measure lntrasocietal conflict, the specification of the measurement model has remained implicit. This paper attempts to explicate the principal design considerations which enter into the construction of such a model (i.e., causal versus effect indicators, uni- versus multidimensions, single versus multiple indicators, correlated versus noncorrelated error terms, and additive versus multiplicative relationships). The statistical and theoretical implications of these alternative features are discussed in terms of the current literature dealing with the national measurement of conflict.