I have shown that tropical polynomials with rationalcoefficients can be factored into a product of linear polynomials. My elementary proof explains some of the nuances of functional equivalence and introduces least-coefficient polynomials as representatives of functional equivalences classes. I have given an algorithm to convert a polynomial into least-coefficient form and show that least-coefficient polynomials can be factored by inspection. Overall, my research provides a basis by which others can better understand one-variable polynomials.
Recall that a line of the form ax + by + c = 0 is a rational line if a, b and c are all rational, and a point (x, y) is a rational point if both its coordinates are rational numbers. If a line is drawn through two rational points, then the line will be a rational line. We can see this is true by using the point-slope formula to derive the equation of the line that passes through the given points. Since Q is field, all of the operations used to simplify the point-slope equation will result in rationalcoefficients. Also, if two rational lines intersect, their intersection is a rational point. All of the operations used to solve the system would again be closed since Q is a field.
In this paper, for the first time we study the fairness and correctness in millionaires’ problem with rational players. Rational players are neither ‘good’ nor ‘malicious’, they are utility maximizing. Each rational party wishes to learn the output while allowing as few others as possible to learn the output. Thus, each rational party chooses abort to maximize its utility. We show that the solution of Gordon et al. for millionaires’ problem in non-rational setting no longer remains fair in rational setting. We also propose a modification in the protocol with the help of a third player (explained later) so that fairness, correctness and strict Nash equilibrium can be established.
In [LT] it is proved that, for a local ring essentially of finite type over a field of characteristic 0, the notions of pseudo-rational and rational singularity agree. In [Sm] it is shown that, in positive characteristic, F-rational implies pseudo-rational. Smith uses this to prove that rings of finite type over a field of characteristic 0 that are F-rational type have rational singularities. Here “F-rational type” essentially means that characteristic- p models of the variety are F-rational. More precisely, we next introduce the idea of a model.
The topic of study in this chapter are the eigenvectors of similarity matrices corresponding to coefficients for binary data. Various results on the eigenvector elements of coefficient matrices are presented. It is shown that ordinal information can be obtained from eigenvectors corresponding to the largest eigenvalue of various similarity matrices. Using eigenvectors it is therefore possible to uncover correct orderings of various latent variable models. The point to be made here is that the eigendecomposition of some similarity matrices, especially matrices corresponding to asymmetric coefficients, are more interesting compared to the eigendecomposition of other matrices. Many of the results are perhaps of theoretical interest only, since no new insights are developed compared to existing methodology already available for various nonparametric item response theory models.
One of the fundamental critiques of rational decision-making is that rationality, even if desirable, remains a normative ideal which is impossible to achieve in practice. Our analysis suggests this is narrow view of organizational reality that neglects its materiality, as well as the role of theory. The performative praxis framework suggests that the normative theory of rational choice is a central ingredient in making rational decisions. It maps the processes whereby rational choice theory may influence actual behavior. It highlights how formal and normative knowledge on decision-making sustains rational decision-making practices. It reduces the opposition between normative and descriptive research because it accounts for the role of rationality prescriptors (e.g., consultants prescribing rational decision-making methods). In addition, by incorporating prescriptions, our framework explains how
There is a CW complex T X , which gives a rational homotopical classification of almost free toral actions on spaces in the rational homotopy type of X associated with rational toral ranks and also presents certain relations in them. We call it the rational toral rank complex of X. It represents a variety of toral actions. In this note, we will give eﬀective 2-dimensional examples of it when X is a finite product of odd spheres. This is a combinatorial approach in rational homotopy theory.
Our emphasis in this paper will be on one class of approximation algorithms: Monte Carlo algorithms, which approximate a probability distribution with a set of samples from that distribution. Sophisticated Monte Carlo schemes provide methods for sampling from complex probability distributions (Gilks, Richardson, & Spiegelhalter, 1996), and for recursively updating a set of samples from a distribution as more data are obtained (Doucet, Freitas, & Gordon, 2001). These algorithms provide an answer to the question of how learners with finite memory resources might be able to maintain a distribution over a large hypothesis space. We introduce these algorithms in the general case, and then provide a detailed illustration of how these algorithms can be applied to one of the first rational models of cognition: Anderson’s (1990, 1991) rational model of categorization.
Orthogonal rational functions (ORFs) on a subset S of the real line (see e.g. [2, 5, 6] and [1, Chapt. 11]) are a generalisation of OPs on S in such a way that they are of increasing degree with a given sequence of complex poles, and the OPs result if all the poles are at infinity. Let ϕ j denote the rational func-
(4) The wavelet coefficient will have only few significant coefficients at smooth regions and wavelet coefficients is large only if edges are present within support of wavelet. (5) The wavelet coefficient tends to be approximately decorrelated. The above primary properties give wavelet coefficients of natural image significant statistical structure and they are listed as the following secondary properties (1) The wavelet coefficients have peaky heavy tailed marginal distribution ,or distribution is non Gaussian. (2) Magnitudes of the wavelet coefficient of real image decay exponentially across scales. The persistence of large/small wavelet coefficient magnitude becomes exponentially stronger at finer scales.
The assumptions, on which the method is based, need also to be discussed. For the derivation we assume that the dependent and the independent variables are jointly multivariate normally distributed. This is one of the two main approaches for formulating a regression problem (the other is the approach that assumes that the inde- pendent variables are fixed by design). Even though the two approaches are conceptually very different, it is well known that concerning the estimation of the regression parameters (the coefficients and their variance), they yield exactly the same results. The assumption of multivariate normality is more stringent, but it yields an optimal predictor among all choices, rather than merely among linear predictors. Practically, since the estimators are identical, this means that we can use the expressions derived here, even in the case of binary independent variables and in any case the results are identical with the ones produced by any standard linear regression soft- ware. We need to mention at this point that the method is developed in , which as the authors claimed does not make the assumption of normality, yields estimates for the regression coefficients that differ from the ones produced by standard regression packages.
the ﬁeld of small functions with respect to w. A meromorphic solution w of a diﬀerence equation is called admissible if all coeﬃcients of the equation are in S (w). For example, if a diﬀerence equation has only rational coeﬃcients, then all non-rational meromorphic solutions are admissible; if an admissible solution is rational, then all the coeﬃcients must be constants.