One-variable Functions Research Articles

Clark Measures on Polydiscs Associated to Product Functions and Multiplicative Embeddings

We study Clark measures on the unit polydisc, giving an overview of recent research and investigating the Clark measures of some new examples of multivariate inner functions. In particular, we study the relationship between Clark measures and multiplication; first by introducing compositions of inner functions and multiplicative embeddings, and then by studying products of one-variable inner functions.

Tribodynamic analysis of spur gear drives with uncertain time-variant loads: An interval process approach

Gear drives are inherently subjected to time-variant loads, which always create an uncertainty and may degrade the vibration performance and system reliability. However, due to their complicated statistical properties, it is impractical to employ probability-based dynamic analysis methods. Additionally, most studies neglect the elastohydrodynamic lubrication behaviors between tooth surfaces under time-variant load conditions. A novel tribodynamic analysis method is proposed to evaluate system responses efficiently, in which the uncertain time-variant loads are described using interval processes. Firstly, a deterministic tribodynamic model coupling the system dynamics and elastohydrodynamic lubrication is established. To reduce the computational cost, the interval Karhunen-Loeve expansion is employed to simplify the interval processes using a small number of uncorrelated interval variables, while the Chebyshev subinterval decomposition is carried out to approximate actual responses using one-variable functions. The proposed method shares a comparable accuracy with the time-consuming Monte Carlo simulation and experiments. Parametric studies show that the uncertain time-variant loads significantly affect the uncertainty of vibration and induce a more hazardous lubricating condition. Additionally, the variation trend of uncertain tribodynamic responses is highly correlated with the rotating speed.

Integral of Continuous Functions of Two Variables

Summary We extend the formalization of the integral theory of one-variable functions for Riemann and Lebesgue integrals, showing that the Lebesgue integral of a continuous function of two variables coincides with the Riemann iterated integral of a projective function.

Differentiation on Interval

Summary This article generalizes the differential method on intervals, using the Mizar system [2], [3], [12]. Differentiation of real one-variable functions is introduced in Mizar [13], along standard lines (for interesting survey of formalizations of real analysis in various proof-assistants like ACL2 [11], Isabelle/HOL [10], Coq [4], see [5]), but the differentiable interval is restricted to open intervals. However, when considering the relationship with integration [9], since integration is an operation on a closed interval, it would be convenient for differentiation to be able to handle derivates on a closed interval as well. Regarding differentiability on a closed interval, the right and left differentiability have already been formalized [6], but they are the derivatives at the endpoints of an interval and not demonstrated as a differentiation over intervals. Therefore, in this paper, based on these results, although it is limited to real one-variable functions, we formalize the differentiation on arbitrary intervals and summarize them as various basic propositions. In particular, the chain rule [1] is an important formula in relation to differentiation and integration, extending recent formalized results [7], [8] in the latter field of research.

Ethnomathematics Exploration of Fish Trader Activities in The Traditional Market of Punggulan Village, Asahan Regency

This study aims to explore the activities of fish traders in traditional markets by connecting mathematics and culture, known as ethnomathematics. The type of research used is qualitative research with an ethnographic approach. Data collection techniques used are library data, observation, and interviews. The data analysis technique refers to the Miles and Huberman model: data reduction, data presentation, and conclusion. This study found that the activities of fish traders in traditional markets were grouped into three processes: measurement, packaging, and buying and selling. The concepts found in these three activities are measurement, sets, one-variable linear equations, weight comparisons, social arithmetic, relations and functions, two-variable linear equations, flat-sided spaces, and probability. Thus, the activities of fish traders contain ethnomathematics elements which are simultaneously an application of mathematical theory learned at school. Further researchers are advised to examine the quality of ethnomathematics-based mathematics learning.

Initial Coefficients Upper Bounds for Certain Subclasses of Bi-Prestarlike Functions

In this article, we introduce and study the behavior of the modules of the first two coefficients for the classes NΣ(γ,λ,δ,μ;α) and NΣ*(γ,λ,δ,μ;β) of normalized holomorphic and bi-univalent functions that are connected with the prestarlike functions. We determine the upper bounds for the initial Taylor–Maclaurin coefficients |a2| and |a3| for the functions of each of these families, and we also point out some special cases and consequences of our main results. The study of these classes is closely connected with those of Ruscheweyh who in 1977 introduced the classes of prestarlike functions of order μ using a convolution operator and the proofs of our results are based on the well-known Carathédory’s inequality for the functions with real positive part in the open unit disk. Our results generalize a few of the earlier ones obtained by Li and Wang, Murugusundaramoorthy et al., Brannan and Taha, and could be useful for those that work with the geometric function theory of one-variable functions.

Local visualization of functions in work on optimization

Abstract Optimization is introduced to teaching in secondary school, focussing on functions of one variable with the first derivative criterion for critical points and monotonicity of functions. The second derivative criterion is also used to determine the nature of the critical points. Similarly, in university-level teaching, algebraic work using criteria is the predominant approach adopted for optimization of two-variable functions, but with much less meaning than for one-variable functions. To recover meaning, we designed a learning situation based on local approximations with the first- and second-order Taylor polynomial, both for one- and two-variable functions. The results show that the local visualization for first-order Taylor approximation was quite easy for students to understand (curve and tangent), but local visualization for second-order Taylor approximation (curve and osculatory parabola) was much less so. We also identify a competition between local and global visualization processes, which constitutes a newly identified visualization phenomenon.

Combining the entropy-scaling concept and cubic- or SAFT equations of state for modelling thermal conductivities of pure fluids

The transport properties of a fluid show a complex dependence with temperature and pressure due to the combination of different phenomena occurring at the microscopic scale. The entropy scaling concept aims at describing this complex behavior by expressing reduced transport properties as one-variable functions of the Tv-residual entropy, a thermodynamic quantity that can be straightforwardly estimated with an equation of state (EoS). In this work, a reformulated version of Rosenfeld's original entropy scaling approach is proposed in order to calculate the thermal conductivities of pure fluids. A specifically developed reduced thermal conductivity expression was correlated to a function of the Tv-residual entropy that was recently proposed by our group to correlate viscosities and self-diffusion coefficients. The thermodynamic properties involved in the definition of the entropy-scaling variables (that are the residual entropies, densities, heat capacities) were estimated with either the I-PC-SAFT or the tc-PR equations of state (EoSs) thus leading to the definition of two different models. Each of them was validated against a large database of around 90,000 experimental thermal conductivities encompassing liquid, gas and supercritical states for 119 chemical species belonging to 11 chemical families such as n-alkanes, alkenes, alcohols, HFC-CFC etc. For each model, component-specific, chemical-family specific and universal parameters were proposed. Working with the I-PC-SAFT and tc-PR EoSs, the obtained MAPEs (Mean Absolute Percent Errors) are respectively 3.3% and 3.4% when the model parameters are considered as component-specific, 9.7% and 5.6% when they are selected as chemical-family specific meanwhile they are 11.2% and 9.2% when they are assumed to be universal.

Cellular automata that generate symmetrical patterns give singular functions

In this paper, we mainly study linear one-dimensional and two-dimensional elementary cellular automata that generate symmetrical spatio-temporal patterns. For spatio-temporal patterns of cellular automata from the single site seed, we normalize the number of nonzero states of the patterns, take the limits, and give one-variable functions for the limit sets. We can obtain a one-variable function for each limit set and show that the resulting functions are singular functions, which are non-constant, are continuous everywhere, and have a zero derivative almost everywhere. We show that for Rule 90, a one-dimensional elementary cellular automaton (CA), and a two-dimensional elementary CA, the resulting functions are Salem’s singular functions. We also discuss two nonlinear elementary CAs, Rule 22 and Rule 126. Although their spatio-temporal patterns are different from that of Rule 90, their resulting functions from the number of nonzero states equal the function of Rule 90.

T-adic exponential sums over affinoids

We introduce and develop (π1/c,p)-adic Dwork theory for L-functions of exponential sums associated to one-variable rational functions, interpolating pk-order exponential sums over affinoids. Namely, we prove a generalization of the Dwork-Monsky-Reich trace formula and apply it to establish an analytic continuation of the C-function Cf(s,π). We compute the lower (π1/c,p)-adic bound, the Hodge polygon, for this C-function. Along the way, we also show why a strictly π-adic theory will not work in this case.

On studying equivalent (or not) definitions; the case of limits in R and R 2

This paper initiates a teaching sequence that focuses on building up equivalent definitions to the standard ones for the limit concept in Real Analysis. It comprises two parts: The first provides a classroom assignment where students, guided by Analysis lecturers, are led to develop an alternative definition to the ϵ − δ one for limits of one-variable real functions that are based on the realisation of the interval of δ ’s. In the second one, students are directed by their instructor to restore equivalence between two distinct definitions of limit for a function mapping a subset of R 2 into R . This is done by adding a condition to one of the definitions, resulting in a third definition that is equivalent to the other one.

Subclasses of Multivalent Analytic Functions Associated with a q-Difference Operator

In this article we introduced and studied some inclusion properties for new subclasses of multivalent analytic functions defined by using the q-derivative operator. With the aid of the Jackson q-derivative we defined two new operators that generalize many other previously studied operators, and help us to define two new subclasses of functions with several interesting properties studied in this paper. The methods used for the proof of our results are special tools of the differential subordination theory of one-variable functions.

The reducibility of compressed shifts on a class of quotient modules over the bidisk

In this paper, we show that, for the rational inner function θ(z,w)=zw+b¯w+c¯z+d¯1+bz+cw+dzw, Sz is reducible on the quotient module Kθ=H2⊖θH2 over the bidisk if and only if θ is the product of two one-variable inner functions.

Direct design to stress mapping for cellular structures

This paper aims to instantly predict within any accuracy the stress distribution of cellular structures under parametric design, including the shapes or distributions of the cell geometries, or the magnitudes of external loadings. A classical model reduction technique has to balance the simulation accuracy and interaction speed, and has difficulty achieving this goal. We achieve this by computing offline a design-to-stress mapping that ultimately expresses the stress distribution as an explicit function in terms of its design parameters. The mapping is determined as a solution to an extended finite element analysis problem in a high-dimension space, including both the spatial coordinates and the design parameters. The well-known curse of dimensionality intrinsic to the high-dimension problem is (partly) resolved through a spatial separation using two main techniques. First, the target mapping takes a reduced form as a sum of the products of separated one-variable functions, extending the proper generalized decomposition technique. Second, the simulation problem in a varied computation domain is reformulated as that in a fixed-domain, taking an integration function as the sum of the products of separated one-variable functions, in combination with high-order singular value decomposition. Extensive 2D and 3D examples are shown to demonstrate the approach’s performance.

Minimax Optimization Method in the Two-Dimensional Boundary-Value Inverse Heat Conduction Problem

A method is proposed for the two-dimensional inverse heat conduction problem via reconstruction of the spatial and temporal density of a boundary heat flux. It is based on the optimal control theory for objects with distributed parameters. The method limits the set of desired solutions to the class of physically realized functions, which makes it possible to represent the desired-effect structure as a product of two one-variable functions. The problem of semi-infinite optimization, which minimizes temperature residuals in the uniform estimation metric, is formulated based on the parameterization of the desired characteristic (considered a control action). Analytical solution of the problem with the alternance properties of the desired optimal temperature deviations makes it possible to obtain the optimal values of the parameter vector.

Optimality conditions of the maximum principle type in bilinear control problems

The optimization of a bilinear functional related to a linear state system with a modular control constraint is considered. Exact formulas for the functional increment are used to obtain sufficient conditions for the optimality of extremal controls that supplement the maximum principle. These conditions are represented in the form of inequalities and equalities for one-variable functions on a time interval. The optimization of a quadratic functional with the help of a matrix conjugate function is reduced to the bilinear case.

ON CONNECTING WEYL-ORBIT FUNCTIONS TO JACOBI POLYNOMIALS AND MULTIVARIATE (ANTI)SYMMETRIC TRIGONOMETRIC FUNCTIONS

<p>The aim of this paper is to make an explicit link between the Weyl-orbit functions and the corresponding polynomials, on the one hand, and to several other families of special functions and orthogonal polynomials on the other. The cornerstone is the connection that is made between the one-variable orbit functions of <em>A<sub>1</sub></em> and the four kinds of Chebyshev polynomials. It is shown that there exists a similar connection for the two-variable orbit functions of <em>A<sub>2</sub></em> and a specific version of two variable Jacobi polynomials. The connection with recently studied <em>G<sub>2</sub></em>-polynomials is established. Formulas for connection between the four types of orbit functions of <em>B<sub>n</sub></em> or <em>C<sub>n</sub></em> and the (anti)symmetric multivariate cosine and sine functions are explicitly derived.</p>

ON CONNECTING WEYL-ORBIT FUNCTIONS TO JACOBI POLYNOMIALS AND MULTIVARIATE (ANTI)SYMMETRIC TRIGONOMETRIC FUNCTIONS

<p>The aim of this paper is to make an explicit link between the Weyl-orbit functions and the corresponding polynomials, on the one hand, and to several other families of special functions and orthogonal polynomials on the other. The cornerstone is the connection that is made between the one-variable orbit functions of <em>A<sub>1</sub></em> and the four kinds of Chebyshev polynomials. It is shown that there exists a similar connection for the two-variable orbit functions of <em>A<sub>2</sub></em> and a specific version of two variable Jacobi polynomials. The connection with recently studied <em>G<sub>2</sub></em>-polynomials is established. Formulas for connection between the four types of orbit functions of <em>B<sub>n</sub></em> or <em>C<sub>n</sub></em> and the (anti)symmetric multivariate cosine and sine functions are explicitly derived.</p>

Generating Implications from One-Variable Functions

In this paper, we firstly introduce two new classes of fuzzy implications generated from one-variable functions, called [Formula: see text]- and [Formula: see text]-implications, respectively. Then we give a series of necessary and sufficient conditions that these implications satisfy: left neutrality property, identity principle, ordering principle, law of contraposition, modus ponens and modus tollens, respectively. We also discuss the relations between [Formula: see text]- implication ([Formula: see text]-implications, respectively) and other known classes of fuzzy implications.

Properties of exact solutions to the problem of regularization of the total variation of one-variable functions

In this study, an exact solution to the variational problem of minimization of a functional consisting of an accuracy term and a regularization term is considered. The regularization term is the total variation of a one-variable function. The variational problem under consideration is widely used in the theory of image and signal restoration. As the weighting factor of total variation increases, the exact solution to the variational problem becomes a more smoothed function. A method for calculating exact solutions and the properties of these solutions are described.