Baskakov-Kantorovich operators reproducing affine functions

We present a new Kantorovich modi cation of Baskakov operators which reproduce a ne functions. We present an upper estimate for the rate of convergence of the new operators in polynomial weighted spaces and characterize all functions for which there is convergence in the weighted norm.

A Note on Confusion between Linear and Affine Functions and the Generalized Forms of Gradient

Background: Arguably the most frequently used term in science, particularly in mathematics and statistics, is linear. However, confusion arises from the various meanings of linearity instructed in different levels of mathematical courses. The definition of linearity taught in high school is less correct than the one learned in a linear algebra class. The correlation coefficient of two quantitative variables is a numerical measure of the affinity, not only linearity, of two variables. However, every statistics book loosely says it is a measure of linear relationship. This clearly show that there is some confusion between use of the terms the linear function and affine function. Objective: This article aims at clarifying the confusion between use of the terms linear function and affine function. It also provides more generalized forms of the gradient in different branches of mathematics and show their equivalency. Materials and Methods: We have used the pure analytical deductive methods to proof the statements. Results: We have clearly presented that gradient is the measure of affinity, not just linearity. It becomes a special case of the derivative in calculus, of the least-squares estimate of the regression coefficient in statistics and matrix theory. The gradient can ­­­­be seen in terms of the inverse of the informative matrix in the most general setting of the linear model estimation. Conclusion: The article has been clearly written to show the distinction between the linear and affine functions in a concise and unambiguous manner. We hope that readers will clearly see various generalizations of the gradient and article itself would be a simple exposition, enlightening, and fun to read.

Confidence polytopes for quantum process tomography

In the present work, we propose a generalization of the confidence polytopes approach for quantum state tomography (QST) to the case of quantum process tomography (QPT). Our approach allows obtaining a confidence region in the polytope form for a Choi matrix of an unknown quantum channel based on the measurement results of the corresponding QPT experiment. The method uses the improved version of the expression for confidence levels for the case of several positive operator-valued measures (POVMs). We then show how confidence polytopes can be employed for calculating confidence intervals for affine functions of quantum states (Choi matrices), such as fidelities and observables mean values, which are used both in QST and QPT settings. As we discuss this problem can be efficiently solved using linear programming tools. We also demonstrate the performance and scalability of the developed approach on the basis of simulation and experimental data collected using IBM cloud quantum processor.

Probabilistic Timed Automata with One Clock and Initialised Clock-Dependent Probabilities

Clock-dependent probabilistic timed automata extend classical timed automata with discrete probabilistic choice, where the probabilities are allowed to depend on the exact values of the clocks. Previous work has shown that the quantitative reachability problem for clock-dependent probabilistic timed automata with at least three clocks is undecidable. In this paper, we consider the subclass of clock-dependent probabilistic timed automata that have one clock, that have clock dependencies described by affine functions, and that satisfy an initialisation condition requiring that, at some point between taking edges with non-trivial clock dependencies, the clock must have an integer value. We present an approach for solving in polynomial time quantitative and qualitative reachability problems of such one-clock initialised clock-dependent probabilistic timed automata. Our results are obtained by a transformation to interval Markov decision processes.

Convexity-preserving properties of set-valued ratios of affine functions

"The aim of this paper is to introduce some classes of set-valued functions that preserve the convexity of sets by direct and inverse images. In particular, we show that the so-called set-valued ratios of a ne functions represent such a class. To this aim, we characterize them in terms of vector-valued selections that are ratios of a ne functions in the classical sense of Rothblum."

Assessment of Scratch Programming Language as a Didactic Tool to Teach Functions

The objective of this research is to study the Scratch programming language as a didactic tool to teach functions. The introduction of didactic tools allowing comprehension in simple and attractive ways is required. Given the traditional teaching/learning system, it is necessary to organize participatory and collaborative dynamic classrooms, which allow the interaction of students in activities where the educator modifies his or her traditional role as an advisor and the students take a more active role in learning through their own effort. In this sense, three activities using the Scratch programming language are proposed: the first one refers to the linear and affine functions, while the second one deals with the quadratic function and the third one is related to the exponential function. The participants in this study were 30 future teachers. The study considers the combination of magisterial lessons and active didactic methodologies as demonstration method, cooperative learning and gamification, also including the applied assessment. The activities, methodologies and assessment were evaluated by the participants with results higher than 4 in 5-point Likert scale for all cases, preferring the active methodologies than magisterial lessons.

Modified Operators Interpolating at Endpoints

Some classical operators (e.g., Bernstein) preserve the affine functions and consequently interpolate at the endpoints. Other classical operators (e.g., Bernstein–Durrmeyer) have been modified in order to preserve the affine functions. We propose a simpler modification with the effect that the new operators interpolate at endpoints although they do not preserve the affine functions. We investigate the properties of these modified operators and obtain results concerning iterates and their limits, Voronovskaja-type results and estimates of several differences.