On Properties of a Regular Simplex Inscribed into a Ball

Let $B$ be a Euclidean ball in ${\mathbb R}^n$ and let $C(B)$ be a space of continuos functions $f:B\to{\mathbb R}$ with the uniform norm $\|f\|_{C(B)}:=\max_{x\in B}|f(x)|.$ By $\Pi_1\left({\mathbb R}^n\right)$ we mean a set of polynomials of degree $\leq 1$, i.e., a set of linear functions upon ${\mathbb R}^n$. The interpolation projector $P:C(B)\to \Pi_1({\mathbb R}^n)$ with the nodes $x^{(j)}\in B$ is defined by the equalities $Pf\left(x^{(j)}\right)=f\left(x^{(j)}\right)$, $j=1,\ldots, n+1$.The norm of $P$ as an operator from $C(B)$ to $C(B)$ can be calculated by the formula $\|P\|_B=\max_{x\in B}\sum |\lambda_j(x)|.$ Here $\lambda_j$ are the basic Lagrange polynomials corresponding to the $n$-dimensional nondegenerate simplex $S$ with the vertices $x^{(j)}$. Let $P^\prime$ be a projector having the nodes in the vertices of a regular simplex inscribed into the ball. We describe the points $y\in B$ with the property $\|P^\prime\|_B=\sum |\lambda_j(y)|$. Also we formulate some geometric conjecture which implies that $\|P^\prime\|_B$ is equal to the minimal norm of an interpolation projector with nodes in $B$. We prove that this conjecture holds true at least for $n=1,2,3,4$.

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On Some Problems for a Simplex and a Ball in Rn

Modeling and Analysis of Information Systems ◽

10.18255/1818-1015-2018-6-680-691 ◽

2018 ◽

Vol 25 (6) ◽

pp. 680-691

Author(s):

Mikhail V. Nevskii

Keyword(s):

Convex Body ◽

Geometric Interpretation ◽

Unit Cube ◽

Linear Functions ◽

Barycentric Coordinates ◽

Regular Simplex ◽

Euclidean Ball ◽

Lagrange Polynomials ◽

Dimensional Unit ◽

Unit Euclidean Ball

Let $C$ be a convex body and let $S$ be a nondegenerate simplex in ${\mathbb R}^n$. Denote by $\tau S$ the image of $S$ under homothety with a center of homothety in the center of gravity of $S$ and the ratio $\tau$. We mean by $\xi(C;S)$ the minimal $\tau>0$ such that $C$ is a subset of the simplex $\tau S$. Define $\alpha(C;S)$ as the minimal $\tau>0$ such that $C$ is contained in a translate of $\tau S$. Earlier the author has proved the equalities $\xi(C;S)=(n+1)\max\limits_{1\leq j\leq n+1}\max\limits_{x\in C}(-\lambda_j(x))+1$ (if $C\not\subset S$), $\alpha(C;S)=\sum\limits_{j=1}^{n+1} \max\limits_{x\in C} (-\lambda_j(x))+1.$Here $\lambda_j$ are the linear functions that are called the basic Lagrange polynomials corresponding to $S$. The numbers $\lambda_j(x),\ldots, \lambda_{n+1}(x)$ are the barycentric coordinates of a point $x\in{\mathbb R}^n$. In his previous papers, the author investigated these formulae in the case when $C$ is the $n$-dimensional unit cube $Q_n=[0,1]^n$. The present paper is related to the case when $C$ coincides with the unit Euclidean ball $B_n=\{x: \|x\|\leq 1\},$ where $\|x\|=\left(\sum\limits_{i=1}^n x_i^2 \right)^{1/2}.$ We establish various relations for $\xi(B_n;S)$ and $\alpha(B_n;S)$, as well as we give their geometric interpretation. For example, if $\lambda_j(x)=l_{1j}x_1+\ldots+l_{nj}x_n+l_{n+1,j},$ then $\alpha(B_n;S)=\sum\limits_{j=1}^{n+1}\left(\sum\limits_{i=1}^n l_{ij}^2\right)^{1/2}$. The minimal possible value of each characteristics $\xi(B_n;S)$ and $\alpha(B_n;S)$ for $S\subset B_n$ is equal to $n$. This value corresponds to a regular simplex inscribed into $B_n$. Also we compare our results with those obtained in the case $C=Q_n$.

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Linear Interpolation on a Euclidean Ball in Rⁿ

Modeling and Analysis of Information Systems ◽

10.18255/1818-1015-2019-2-279-296 ◽

2019 ◽

Vol 26 (2) ◽

pp. 279-296

Author(s):

Mikhail V. Nevskii ◽

Alexey Yu. Ukhalov

Keyword(s):

Linear Interpolation ◽

Continuous Functions ◽

Linear Functions ◽

Regular Simplex ◽

Euclidean Ball ◽

Integer Number ◽

Space Of Continuous Functions ◽

Lagrange Polynomials ◽

Computational Data ◽

Theoretical Results

For $x^{(0)}\in{\mathbb R}^n, R>0$, by $B=B(x^{(0)};R)$ we denote a Euclidean ball in ${\mathbb R}^n$ given by~the inequality $\|x-x^{(0)}\|\leq R$, $\|x\|:=\left(\sum_{i=1}^n x_i^2\right)^{1/2}$. Put $B_n:=B(0,1)$. We mean by $C(B)$ the space of~continuous functions $f:B\to{\mathbb R}$ with the norm $\|f\|_{C(B)}:=\max_{x\in B}|f(x)|$ and by $\Pi_1\left({\mathbb R}^n\right)$ the set of polynomials in $n$ variables of degree $\leq 1$, i.e. linear functions on ${\mathbb R}^n$. Let $x^{(1)}, \ldots, x^{(n+1)}$ be the~vertices of $n$-dimensional nondegenerate simplex $S\subset B$. The interpolation projector $P:C(B)\to \Pi_1({\mathbb R}^n)$ corresponding to $S$ is defined by the equalities $Pf\left(x^{(j)}\right)=%f_j:=f\left(x^{(j)}\right).$ Denote by $\|P\|_B$ the norm of $P$ as an operator from $C(B)$ into $C(B)$. Let us define $\theta_n(B)$ as minimal value of $\|P\|$ under the condition $x^{(j)}\in B$. In the paper, we obtain the formula to compute $\|P\|_B$ making use of $x^{(0)}$, $R$, and coefficients of basic Lagrange polynomials of $S$. In more details we study the case when $S$ is a regular simplex inscribed into $B_n$. In this situation, we prove that $\|P\|_{B_n}=\max\{\psi(a),\psi(a+1)\},$ where $\psi(t)=\frac{2\sqrt{n}}{n+1}\bigl(t(n+1-t)\bigr)^{1/2}+\bigl|1-\frac{2t}{n+1}\bigr|$ $(0\leq t\leq n+1)$ and integer $a$ has the form $a=\bigl\lfloor\frac{n+1}{2}-\frac{\sqrt{n+1}}{2}\bigr\rfloor.$ For this projector, $\sqrt{n}\leq\|P\|_{B_n}\leq\sqrt{n+1}$. The equality $\|P\|_{B_n}=\sqrt{n+1}$ takes place if and only if $\sqrt{n+1}$ is an integer number. We give the precise values of $\theta_n(B_n)$ for $1\leq n\leq 4$. To supplement theoretical results we present computational data. We also discuss some other questions concerning interpolation on a Euclidean ball.

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Interpolation by linear functions on a Euclidean ball

10.33184/mnkuomsh1t-2021-10-06.59 ◽

2021 ◽

Author(s):

Михаил Невский

Keyword(s):

Linear Functions ◽

Euclidean Ball

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On Optimal Interpolation by Linear Functions on an n-Dimensional Cube

Modeling and Analysis of Information Systems ◽

10.18255/1818-1015-2018-3-291-311 ◽

2018 ◽

Vol 25 (3) ◽

pp. 291-311

Author(s):

Mikhail V. Nevskii ◽

Alexey Yu. Ukhalov

Keyword(s):

Hadamard Matrix ◽

Continuous Functions ◽

Unit Cube ◽

Linear Functions ◽

Optimal Interpolation ◽

Maximum Volume ◽

Space Of Continuous Functions ◽

Lagrange Polynomials ◽

Geometric Character ◽

Dimensional Cube

Let $n\in{\mathbb N}$, and let $Q_n$ be the unit cube $[0,1]^n$. By $C(Q_n)$ we denote the space of continuous functions $f:Q_n\to{\mathbb R}$ with the norm $\|f\|_{C(Q_n)}:=\max\limits_{x\in Q_n}|f(x)|,$ by $\Pi_1\left({\mathbb R}^n\right)$ --- the set of polynomials of $n$ variables of degree $\leq 1$ (or linear functions). Let $x^{(j)},$ $1\leq j\leq n+1,$ be the vertices of $n$-dimnsional nondegenerate simplex $S\subset Q_n$. An interpolation projector $P:C(Q_n)\to \Pi_1({\mathbb R}^n)$ corresponding to the simplex $S$ is defined by equalities $Pf\left(x^{(j)}\right)= f\left(x^{(j)}\right).$ The norm of $P$ as an operator from $C(Q_n)$ to $C(Q_n)$ may be calculated by the formula $\|P\|=\max\limits_{x\in ver(Q_n)} \sum\limits_{j=1}^{n+1} |\lambda_j(x)|.$ Here $\lambda_j$ are the basic Lagrange polynomials with respect to $S,$ $ver(Q_n)$ is the set of vertices of $Q_n$. Let us denote by $\theta_n$ the minimal possible value of $\|P\|.$ Earlier, the first author proved various relations and estimates for values $\|P\|$ and $\theta_n$, in particular, having geometric character. The equivalence $\theta_n\asymp \sqrt{n}$ takes place. For example, the appropriate, according to dimension $n$, inequalities may be written in the form \linebreak $\frac{1}{4}\sqrt{n}$ $<\theta_n$ $<3\sqrt{n}.$ If the nodes of the projector $P^*$ coincide with vertices of an arbitrary simplex with maximum possible volume, we have $\|P^*\|\asymp\theta_n.$When an Hadamard matrix of order $n+1$ exists, holds $\theta_n\leq\sqrt{n+1}.$ In the paper, we give more precise upper bounds of numbers $\theta_n$ for $21\leq n \leq 26$. These estimates were obtained with the application of maximum volume simplices in the cube. For constructing such simplices, we utilize maximum determinants containing the elements $\pm 1.$ Also, we systematize and comment the best nowaday upper and low estimates of numbers $\theta_n$ for a concrete $n.$

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PENGARUH NILAI PENGANTAR AKUNTANSI TERHADAP KEMAMPUAN MAHASISWA MEMAHAMI KOMPUTER AKUNTANSI PADA PROGRAM STUDI PERBANKAN DAN KEUANGAN POLITEKNIK NEGERI MEDAN

KOMIK (Konferensi Nasional Teknologi Informasi dan Komputer) ◽

10.30865/komik.v2i1.925 ◽

2018 ◽

Vol 2 (1) ◽

Author(s):

Jantianus Jantianus ◽

Khairul Khairul

Keyword(s):

Linear Regression ◽

Regression Equation ◽

Linear Functions ◽

First Semester ◽

Normality Test ◽

Introductory Accounting ◽

Linearity Test

Ease of understanding Accounting Computers in principle is influenced by mastery of Introduction to Accounting in a systematic manner, assuming that it is capable of operating computers properly. To find out the magnitude of the influence in this study taken a sample of introductory Accounting values from a number of first semester 2017 students and the same data sample for students of Computer Accounting (Accurate) courses when they are in the fourth semester 2018. Feasibility until the data is tested by the normality test to find out the distribution of data and by linearity test to obtain linear functions. The data that has been obtained and tested for its feasibility is processed by Linear Regression using SPSS 24. From the results of the research that has been done obtained a regression equation: Y = 67,953 0.35X, which describes each increase in the value of introductory Accounting one unit will affect 0.35 to Computer Accounting value, but in testing the hypothesis that the value of Introduction to Accounting obtained by students does not affect their ability to obtain Computer Accounting values, one of the causes of this is due to the lack of skills of students to operate computers.Keywords: influence, value, ability

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Variability of radiosonde-observed precipitable water in the Baltic region*

Hydrology Research ◽

10.2166/nh.2005.0032 ◽

2005 ◽

Vol 36 (4-5) ◽

pp. 423-433 ◽

Cited By ~ 2

Author(s):

E. Jakobson ◽

H. Ohvril ◽

O. Okulov ◽

N. Laulainen

Keyword(s):

Water Vapour ◽

Precipitable Water ◽

Linear Functions ◽

Water Vapour Pressure ◽

Baltic Region ◽

Geographical Latitude ◽

Linear Formula ◽

Annual Means ◽

The Baltic ◽

Two Parameters

The total mass of columnar water vapour (precipitable water, W) is an important parameter of atmospheric thermodynamic and radiative models. In this work more than 60 000 radiosonde observations from 17 aerological stations in the Baltic region over 14 years, 1989–2002, were used to examine the variability of precipitable water. A table of monthly and annual means of W for the stations is given. Seasonal means of W are expressed as linear functions of the geographical latitude degree. A linear formula is also derived for parametrisation of precipitable water as a function of two parameters – geographical latitude and surface water vapour pressure.

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Students’ agency, creative reasoning, and collaboration in mathematical problem solving

Mathematics Education Research Journal ◽

10.1007/s13394-021-00365-y ◽

2021 ◽

Author(s):

Ellen Kristine Solbrekke Hansen

Keyword(s):

Problem Solving ◽

Driving Forces ◽

Mathematical Reasoning ◽

Mathematical Problem ◽

Linear Functions ◽

Shared Understanding ◽

Interaction Patterns ◽

Collaborative Processes ◽

Mathematical Properties ◽

Shared Agency

AbstractThis paper aims to give detailed insights of interactional aspects of students’ agency, reasoning, and collaboration, in their attempt to solve a linear function problem together. Four student pairs from a Norwegian upper secondary school suggested and explained ideas, tested it out, and evaluated their solution methods. The student–student interactions were studied by characterizing students’ individual mathematical reasoning, collaborative processes, and exercised agency. In the analysis, two interaction patterns emerged from the roles in how a student engaged or refrained from engaging in the collaborative work. Students’ engagement reveals aspects of how collaborative processes and mathematical reasoning co-exist with their agencies, through two ways of interacting: bi-directional interaction and one-directional interaction. Four student pairs illuminate how different roles in their collaboration are connected to shared agency or individual agency for merging knowledge together in shared understanding. In one-directional interactions, students engaged with different agencies as a primary agent, leading the conversation, making suggestions and explanations sometimes anchored in mathematical properties, or, as a secondary agent, listening and attempting to understand ideas are expressed by a peer. A secondary agent rarely reasoned mathematically. Both students attempted to collaborate, but rarely or never disagreed. The interactional pattern in bi-directional interactions highlights a mutual attempt to collaborate where both students were the driving forces of the problem-solving process. Students acted with similar roles where both were exercising a shared agency, building the final argument together by suggesting, accepting, listening, and negotiating mathematical properties. A critical variable for such a successful interaction was the collaborative process of repairing their shared understanding and reasoning anchored in mathematical properties of linear functions.

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Breast cancer diagnosis using multiple activation deep neural network

Concurrent Engineering ◽

10.1177/1063293x211025105 ◽

2021 ◽

pp. 1063293X2110251

Author(s):

K Vijayakumar ◽

Vinod J Kadam ◽

Sudhir Kumar Sharma

Keyword(s):

Breast Cancer ◽

Neural Network ◽

Deep Neural Network ◽

Breast Cancer Diagnosis ◽

Activation Function ◽

Linear Functions ◽

Cancer Data ◽

Final Layer ◽

Improved Performance ◽

Hidden Layer

Deep Neural Network (DNN) stands for multilayered Neural Network (NN) that is capable of progressively learn the more abstract and composite representations of the raw features of the input data received, with no need for any feature engineering. They are advanced NNs having repetitious hidden layers between the initial input and the final layer. The working principle of such a standard deep classifier is based on a hierarchy formed by the composition of linear functions and a defined nonlinear Activation Function (AF). It remains uncertain (not clear) how the DNN classifier can function so well. But it is clear from many studies that within DNN, the AF choice has a notable impact on the kinetics of training and the success of tasks. In the past few years, different AFs have been formulated. The choice of AF is still an area of active study. Hence, in this study, a novel deep Feed forward NN model with four AFs has been proposed for breast cancer classification: hidden layer 1: Swish, hidden layer, 2:-LeakyReLU, hidden layer 3: ReLU, and final output layer: naturally Sigmoidal. The purpose of the study is twofold. Firstly, this study is a step toward a more profound understanding of DNN with layer-wise different AFs. Secondly, research is also aimed to explore better DNN-based systems to build predictive models for breast cancer data with improved accuracy. Therefore, the benchmark UCI dataset WDBC was used for the validation of the framework and evaluated using a ten-fold CV method and various performance indicators. Multiple simulations and outcomes of the experimentations have shown that the proposed solution performs in a better way than the Sigmoid, ReLU, and LeakyReLU and Swish activation DNN in terms of different parameters. This analysis contributes to producing an expert and precise clinical dataset classification method for breast cancer. Furthermore, the model also achieved improved performance compared to many established state-of-the-art algorithms/models.

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Drift Analysis and Evolutionary Algorithms Revisited

Combinatorics Probability Computing ◽

10.1017/s0963548318000275 ◽

2018 ◽

Vol 27 (4) ◽

pp. 643-666 ◽

Cited By ~ 13

Author(s):

J. LENGLER ◽

A. STEGER

Keyword(s):

Evolutionary Algorithms ◽

Evolutionary Algorithm ◽

Random Search ◽

Optimization Algorithms ◽

Simple Algorithm ◽

Linear Functions ◽

Search Point ◽

Drift Analysis ◽

Fixed Constant ◽

Greedy Optimization

One of the easiest randomized greedy optimization algorithms is the following evolutionary algorithm which aims at maximizing a function f: {0,1}n → ℝ. The algorithm starts with a random search point ξ ∈ {0,1}n, and in each round it flips each bit of ξ with probability c/n independently at random, where c > 0 is a fixed constant. The thus created offspring ξ' replaces ξ if and only if f(ξ') ≥ f(ξ). The analysis of the runtime of this simple algorithm for monotone and for linear functions turned out to be highly non-trivial. In this paper we review known results and provide new and self-contained proofs of partly stronger results.

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Surrogate-Based Optimization of Horizontal Axis Hydrokinetic Turbine Rotor Blades

Energies ◽

10.3390/en14134045 ◽

2021 ◽

Vol 14 (13) ◽

pp. 4045

Author(s):

David Menéndez Arán ◽

Ángel Menéndez

Keyword(s):

Turbine Blade ◽

Design Method ◽

Optimization Method ◽

Turbine Rotor ◽

Computational Effort ◽

Rotor Blades ◽

Linear Functions ◽

Cavitation Inception ◽

Hydrokinetic Turbine ◽

Blade Geometry

A design method was developed for automated, systematic design of hydrokinetic turbine rotor blades. The method coupled a Computational Fluid Dynamics (CFD) solver to estimate the power output of a given turbine with a surrogate-based constrained optimization method. This allowed the characterization of the design space while minimizing the number of analyzed blade geometries and the associated computational effort. An initial blade geometry developed using a lifting line optimization method was selected as the base geometry to generate a turbine blade family by multiplying a series of geometric parameters with corresponding linear functions. A performance database was constructed for the turbine blade family with the CFD solver and used to build the surrogate function. The linear functions were then incorporated into a constrained nonlinear optimization algorithm to solve for the blade geometry with the highest efficiency. A constraint on the minimum pressure on the blade could be set to prevent cavitation inception.

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