Search results for: monotonicity
Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 6

Search results for: monotonicity

6 Monotonicity of the Jensen Functional for f-Divergences via the Zipf-Mandelbrot Law

Authors: Neda Lovričević, Đilda Pečarić, Josip Pečarić

Abstract:

The Jensen functional in its discrete form is brought in relation to the Csiszar divergence functional, this time via its monotonicity property. This approach presents a generalization of the previously obtained results that made use of interpolating Jensen-type inequalities. Thus the monotonicity property is integrated with the Zipf-Mandelbrot law and applied to f-divergences for probability distributions that originate from the Csiszar divergence functional: Kullback-Leibler divergence, Hellinger distance, Bhattacharyya distance, chi-square divergence, total variation distance. The Zipf-Mandelbrot and the Zipf law are widely used in various scientific fields and interdisciplinary and here the focus is on the aspect of the mathematical inequalities.

Keywords: Jensen functional, monotonicity, Csiszar divergence functional, f-divergences, Zipf-Mandelbrot law

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5 Monotone Rational Trigonometric Interpolation

Authors: Uzma Bashir, Jamaludin Md. Ali

Abstract:

This study is concerned with the visualization of monotone data using a piece-wise C1 rational trigonometric interpolating scheme. Four positive shape parameters are incorporated in the structure of rational trigonometric spline. Conditions on two of these parameters are derived to attain the monotonicity of monotone data and other two are left-free. Figures are used widely to exhibit that the proposed scheme produces graphically smooth monotone curves.

Keywords: trigonometric splines, monotone data, shape preserving, C1 monotone interpolant

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4 Behaviour of Non-local Correlations and Quantum Information Theoretic Measures in Frustrated Molecular Wheels

Authors: Amit Tribedi

Abstract:

Genuine Quantumness present in Quantum Systems is the resource for implementing Quantum Information and Computation Protocols which can outperform the classical counterparts. These Quantumness measures encompass non-local ones known as quantum entanglement (QE) and quantum information theoretic (QIT) ones, e.g. Quantum Discord (QD). In this paper, some well-known measures of QE and QD in some wheel-like frustrated molecular magnetic systems have been studied. One of the systems has already been synthesized using coordination chemistry, and the other is hypothetical, where the dominant interaction is the spin-spin exchange interaction. Exact analytical methods and exact numerical diagonalization methods have been used. Some counter-intuitive non-trivial features, like non-monotonicity of quantum correlations with temperature, persistence of multipartite entanglement over bipartite ones etc. indicated by the behaviour of the correlations and the QIT measures have been found. The measures, being operational ones, can be used to realize the resource of Quantumness in experiments.

Keywords: 0D Magnets, discord, entanglement, frustration

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3 Geometric and Algebraic Properties of the Eigenvalues of Monotone Matrices

Authors: Brando Vagenende, Marie-Anne Guerry

Abstract:

For stochastic matrices of any order, the geometric description of the convex set of eigenvalues is completely known. The purpose of this study is to investigate the subset of the monotone matrices. This type of matrix appears in contexts such as intergenerational occupational mobility, equal-input modeling, and credit ratings-based systems. Monotone matrices are stochastic matrices in which each row stochastically dominates the previous row. The monotonicity property of a stochastic matrix can be expressed by a nonnegative lower-order matrix with the same eigenvalues as the original monotone matrix (except for the eigenvalue 1). Specifically, the aim of this research is to focus on the properties of eigenvalues of monotone matrices. For those matrices up to order 3, there already exists a complete description of the convex set of eigenvalues. For monotone matrices of order at least 4, this study gives, through simulations, more insight into the geometric description of their eigenvalues. Furthermore, this research treats in a geometric and algebraic way the properties of eigenvalues of monotone matrices of order at least 4.

Keywords: eigenvalues of matrices, finite Markov chains, monotone matrices, nonnegative matrices, stochastic matrices

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2 Extensions of Schwarz Lemma in the Half-Plane

Authors: Nicolae Pascu

Abstract:

Aside from being a fundamental tool in Complex analysis, Schwarz Lemma-which was finalized in its most complete form at the beginning of the last century-generated an important area of research in various fields of mathematics, which continues to advance even today. We present some properties of analytic functions in the half-plane which satisfy the conditions of the classical Schwarz Lemma (Carathéodory functions) and obtain a generalization of the well-known Aleksandrov-Sobolev Lemma for analytic functions in the half-plane (the correspondent of Schwarz-Pick Lemma from the unit disk). Using this Schwarz-type lemma, we obtain a characterization for the entire class of Carathéodory functions, which might be of independent interest. We prove two monotonicity properties for Carathéodory functions that do not depend upon their normalization at infinity (the hydrodynamic normalization). The method is based on conformal mapping arguments for analytic functions in the half-plane satisfying appropriate conditions, in the spirit of Schwarz lemma. According to the research findings in this paper, our main results give estimates for the modulus and the argument for the entire class of Carathéodory functions. As applications, we give several extensions of Julia-Wolf-Carathéodory Lemma in a half-strip and show that our results are sharp.

Keywords: schwarz lemma, Julia-wolf-caratéodory lemma, analytic function, normalization condition, caratéodory function

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1 Evaluation of the Effect of Learning Disabilities and Accommodations on the Prediction of the Exam Performance: Ordinal Decision-Tree Algorithm

Authors: G. Singer, M. Golan

Abstract:

Providing students with learning disabilities (LD) with extra time to grant them equal access to the exam is a necessary but insufficient condition to compensate for their LD; there should also be a clear indication that the additional time was actually used. For example, if students with LD use more time than students without LD and yet receive lower grades, this may indicate that a different accommodation is required. If they achieve higher grades but use the same amount of time, then the effectiveness of the accommodation has not been demonstrated. The main goal of this study is to evaluate the effect of including parameters related to LD and extended exam time, along with other commonly-used characteristics (e.g., student background and ability measures such as high-school grades), on the ability of ordinal decision-tree algorithms to predict exam performance. We use naturally-occurring data collected from hundreds of undergraduate engineering students. The sub-goals are i) to examine the improvement in prediction accuracy when the indicator of exam performance includes 'actual time used' in addition to the conventional indicator (exam grade) employed in most research; ii) to explore the effectiveness of extended exam time on exam performance for different courses and for LD students with different profiles (i.e., sets of characteristics). This is achieved by using the patterns (i.e., subgroups) generated by the algorithms to identify pairs of subgroups that differ in just one characteristic (e.g., course or type of LD) but have different outcomes in terms of exam performance (grade and time used). Since grade and time used to exhibit an ordering form, we propose a method based on ordinal decision-trees, which applies a weighted information-gain ratio (WIGR) measure for selecting the classifying attributes. Unlike other known ordinal algorithms, our method does not assume monotonicity in the data. The proposed WIGR is an extension of an information-theoretic measure, in the sense that it adjusts to the case of an ordinal target and takes into account the error severity between two different target classes. Specifically, we use ordinal C4.5, random-forest, and AdaBoost algorithms, as well as an ensemble technique composed of ordinal and non-ordinal classifiers. Firstly, we find that the inclusion of LD and extended exam-time parameters improves prediction of exam performance (compared to specifications of the algorithms that do not include these variables). Secondly, when the indicator of exam performance includes 'actual time used' together with grade (as opposed to grade only), the prediction accuracy improves. Thirdly, our subgroup analyses show clear differences in the effect of extended exam time on exam performance among different courses and different student profiles. From a methodological perspective, we find that the ordinal decision-tree based algorithms outperform their conventional, non-ordinal counterparts. Further, we demonstrate that the ensemble-based approach leverages the strengths of each type of classifier (ordinal and non-ordinal) and yields better performance than each classifier individually.

Keywords: actual exam time usage, ensemble learning, learning disabilities, ordinal classification, time extension

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