Describing a Set of Points with Elliptical Areas: Mathematical Description and Verification on Operational Tests of Technical Devices

The purpose of this article was to present an algorithm for creating an ellipse for any data set represented on a two-dimensional reference frame. The study objective was to verify the developed method on real results of experimental tests with different subject matter. This article contains a mathematical algorithm to describe a set of points with elliptical areas. In addition, four results of tests with different subject matter are cited, based on which the developed method was verified. The verification of the method included checking the deviation of the geometric dimensions of the ellipse, the number of points contained within the ellipse, and the area of the ellipse. The implemented research methodology allowed to demonstrate the possibility of using the method of describing a set of points with elliptical areas, in order to determine quantitative parameters evaluating the results of the test. The presented results show the method’s applicability for the results obtained in four different operational tests: measurement of the human body’s gravity center position for a person propelling a wheelchair, measurement of marker position using motion capture methods, measurement of particulate emissions when using equipment powered by an internal combustion engine, and measurement of the muscle activity of the upper limb when propelling a hybrid manual-electric wheelchair. The performed experiments demonstrated that the method allows to describe about 85% of all measurement points with an ellipse.

New multiobjective optimization algorithm using NBI-SASP approaches for mechanical structural problems

Various engineering design problems are formulated as constrained multi-objective optimization problems. One of the relevant and popular methods that deals with these problems is the weighted method. However, the major inconvenience with its application is that it does not yield a well distributed set. In this study, the use of the Normal Boundary Intersection approach (NBI) is proposed, which is effective in obtaining an evenly distributed set of points in the Pareto set. Given an evenly distributed set of weights, it can be strictly shown that this approach is absolutely independent of the relative scales of the functions. Moreover, in order to ensure the convergence to the Global Pareto frontier, NBI approach has to be aligned with a global optimization method. Thus, the following paper suggests NBI-Simulated Annealing Simultaneous Perturbation method (NBI-SASP) as a new method for multiobjective optimization problems. The study shall test also the applicability of the NBI-SASP approach using different engineering multi-objective optimization problems and the findings shall be compared to a method of reference (NSGA). Results clearly demonstrate that the suggested method is more efficient when it comes to search ability and it provides a well distributed global Pareto Front.

The Buffon needle problem for Lévy distributed spacings and renewal theory

What is the probability that a needle dropped at random on a set of points scattered on a line segment does not fall on any of them? We compute the exact scaling expression of this hole probability when the spacings between the points are independent identically distributed random variables with a power-law distribution of index less than unity, implying that the average spacing diverges. The theoretical framework for such a setting is renewal theory, to which the present study brings a new contribution. The question posed here is also related to the study of some correlation functions of simple models of statistical physics.

Visual Tracking with Object Center Displacement and CenterNet

Modern artificial intelligence systems have revolutionized approaches to scientific and technological challenges in a variety of fields, thus remarkable improvements in the quality of state-of-the-art computer vision and other techniques are observed; object tracking in video frames is a vital field of research that provides information about objects and their trajectories. This paper presents an object tracking method basing on optical flow generated between frames and a ConvNet method. Initially, optical center displacement is employed to detect possible the bounding box center of the tracked object. Then, CenterNet is used for object position correction. Given the initial set of points (i.e., bounding box) in first frame, the tracker tries to follow the motion of center of these points by looking at its direction of change in calculated optical flow with next frame, a correction mechanism takes place and waits for motions that surpass a correction threshold to launch position corrections.

Numerical solution of the gas compression problem at a specified law of action

Выполнено численное моделирование одномерных течений политропного газа, описывающее сжатие покоящегося газа с плотностью 1 в покоящийся газ, сжатый до значения 10. Описываемое сжатие происходит без ударных волн эффективным с точки зрения энерговложения способом, так как энергия тратится только на сжатие газа, но не на его разгон Controlled thermonuclear fusion (CTF) is an almost unlimited source of energy and scientists have been studying it for several decades. This requires an efficient and stable compression of diyterium-tritium fuel to a very high density. This work addresses shockless one-dimensional (plane, cylindrical and spherical symmetry cases) “compression from rest to rest”, when gas from the initial resting state under the influence of an impenetrable piston is shocklessly transferred to a resting homogeneous state, but compressed by 10000 times. This compression is energetically most advantageous, because work is spent only on the compression, but not on the gas acceleration. Earlier [10] this problem was solved in the opposite direction of time change. In this case, a density jump occurs on the piston which was taken into account in calculations [3] at the final moment of compression. The numerical solution of this problem in the opposite direction of time variation allows calculating the trajectory of the compressing piston in the form of a set of points ( t,r ) at which the gas velocity and density are determined. In this paper, the problem of shockless “compression from rest to rest” is numerically solved in the forward direction of time change if the compressing piston trajectory is known. The compression piston moves along a monotonous trajectory away from the axis or center of symmetry. It is important, when calculating in forward direction of time change, no internal characteristics are initially entered. They, like all gas flow in the calculation area, are determined in the process of direct calculation. This indicates that the trajectory of compressing piston is the recommendation for appropriate physical experiments

Algebraic Points of Degree at Most 5 on the Affine Curve y\(^{2}\) = x\(^{5}\) - 243

In this work, we determine the set of algebraic points of degree at most 5 on the ane curve y2 = x5 - 243. This result extends a result of J.TH Mulholland who described in [4] the set of \(\mathbb{Q}\)- rational points i.e the set of points of degree one over \(\mathbb{Q}\) on this curve.

An extension of Lagrange interpolation to approximate derivative, integral and bivariate function

Lagrange interpolation is the effective method to approximate an arbitrary function by a polynomial. But there is a need to estimate derivative and integral given a set of points. Although it is possible to make Lagrange interpolation first, which produces Lagrange polynomial; after that we take derivative or integral on such polynomial. However this approach does not result out the best estimation of derivative and integral. This research proposes a different approach that makes approximation of derivative and integral based on point data first, which in turn applies Lagrange interpolation into the approximation. Moreover, the research also proposes an extension of Lagrange interpolation to bivariate function, in which interpolation polynomial is converted as two-variable polynomial.

METHOD FOR ELIMINATING ANOMALOUS MEASUREMENTS IN ANALYSIS OF THE MULTI-DIMENSIONAL DATABASE IN SOLVING THE DECISION-MAKING PROBLEM

The paper proposes a method for eliminating abnormal measurements (outliers) to improve the quality of multivariate data in statistical studies. Such a problem arises, for example, in the theory of managerial decision-making, since when calculating estimates of the parameters of probability distributions, the presence of anomalous (that is, those that significantly increase the confidence interval) measurements in the sample can distort the results of a statistical study, and, consequently, the main problem. The peculiarity of the proposed method is a combination of statistical and geometric methods, namely: the Gestwirt estimation method, the Tukey procedure, and a modification of the method for constructing the convex hull of a finite set of points in a multidimensional space. A set of multidimensional data is associated with a set of points of a multidimensional space. To find and eliminate outliers, a sequence of nested convex hulls, polytopes, is constructed, each of which is described by the intersection of half-spaces (support facets). A detailed algorithm for finding anomalous measurements is given. Their elimination corresponds to the successive elimination of the boundary points of nested convex hulls. The Gestwirt estimate gives the condition for stopping the operation of the algorithm. The proposed method does not require large computational costs and can be widely used in solving both theoretical and practical problems related to the processing of multidimensional data. The numerical results of the method with the number of data components 4 and 5 are presented.

Superdifferential Analysis of the Takagi-Van Der Waerden Functions

In this work we completely describe the superdifferential of the Takagi-Van der Waerden functions and, as a consequence, the local maxima of these functions are characterized. Regarding the set of points where the superdifferential is not empty, we calculate its Hausdorff dimension as well as its corresponding Hausdorff measure. To do so, for any even integer greater than or equal to two we determine the 1/2-dimensional Hausdorff measure of the set of points where Takagi-Van der Waerden functions attain their global maximum.