Set Of Planes Research Articles

Solving an EPQ model with doubt fuzzy set: A robust intelligent decision-making approach

This article deals with the novel application of complex neutrosophic set in a production inventory model. First of all, we develop a new set named doubt fuzzy set the real and imaginary parts of which are the membership functions of fuzzy variables. The real part of the set is coined as “true or sure” and that for complex part it is defined as “hesitation or suspect” membership functions or it may be defined as “optimism” and “pessimism” respectively in the psychological view point. Then, we give some definitions of doubt fuzzy set in a rectangular complex plane. Subsequently, we give novel defuzzification method by introducing the concept of power of difficulty/ opportunity power which are complementary to each other. Secondly, we develop a backlogging economic production quantity (EPQ) model the demand function of which is disrupted due to the presence of shortages. Assuming the demand function as complex in nature, we develop four types of doubt fuzzy sets namely proper doubt, harmful doubt, depressive doubt and confident doubt respectively and split the model into four sub-models accordingly. Based on new defuzzification method, we have introduced a new solution algorithm named dynamical doubt fuzzy optimization algorithm (DDFOA). By this new approach we have shown that with the application of learning vector by means of opportunity power/ fitness of various test functions the decision maker can achieve and avail the financial benefit as (s)he wishes to adopt. However, the concept of robust intelligent decision making has been discussed extensively through the numerical illustrations. Finally, sensitivity analysis, graphical illustrations are made to justify the proposed approach.

Hausdorff dimension of planar self-affine sets and measures with overlaps

We prove that if \mu is a self-affine measure in the plane whose defining IFS acts totally irreducibly on \mathbb{RP}^1 and satisfies an exponential separation condition, then its dimension is equal to its Lyapunov dimension.We also treat a class of reducible systems. This extends our previous work on the subject with Bárány to the overlapping case.

Анализ влияния наклона и децентрации ИОЛ на индуцированный астигматизм при исследовании методом трассировки лучей на модели глаза

Purpose. Determination of the influence of the intraocular lens (IOL) inclination and decentration on the magnitude of induced astigmatism in a theoretical study using software on an eye model. Material and methods. An eye model with the specified parameters was performed to carry out the calculation. Modulation of the change in the wavefront was carried out using the Zemax program (LLC, USA). In the model eye, the probability of induction of astigmatism was analyzed depending on the change in the position of the aspherical IOL Acrysof IQ 16.0, 22.0 and 24.0 diopters and the toric model Acrysof IQ Toric SN6AT3, SN6AT4 and SN6AT5. The angle of inclination of the IOL relative to the horizontal plane was examined from 0° to 10° with a step of 1°, decentration from 0.1 to 1.5 mm with a step of 0.1 mm. At each IOL position, aberration data of the 2nd order Z (2, 2) were recorded. The obtained results were used to calculate the critical tilt angle and decentration. Results. The magnitude of induced astigmatism increased with the rise of the degree of inclination and decentration, corresponding to an increase in the optical power of the IOL. Accordingly, critically significant were the values of 3.0° inclination for the aspherical IOL with an optical power of 22.0 and 22.0 diopters and 3.5° for 16.0 diopters, decentration – 0.7 mm and 1.0 mm. The inclination of the TIOL relative to the horizontal plane revealed the induction of the magnitude of inverse astigmatism, according to an increase in the inclination angle of 0.3 diopters at a slope of 3–4°. Conclusions. The tilt and decentration of the aspherical and TIOL patterns induce astigmatism depending on the increase in IOL power. The inclination of the TIOL in relation to the horizontal plane set at 90 ° promotes the induction of astigmatism. Key words: astigmatism, toric intraocular lenses, tilt and decentration of the intraocular lens, ray tracing

Polarization splitting with cubic crystals evaluated with synchrotron radiation.

X-ray polarization-splitting crystals separate incident x rays into two components with perpendicular polarization by Bragg reflections at 45° from paired sets of internal planes. Here, the polarization-splitting properties of a germanium crystal are verified using incompletely polarized synchrotron radiation. Cleaner data would have come from a beam with a higher degree of polarization, which is achievable with small changes in the experimental geometry.

Iterative multilinear optimization for planar model fitting under geometric constraints.

Planes are the core geometric models present everywhere in the three-dimensional real world. There are many examples of manual constructions based on planar patches: facades, corridors, packages, boxes, etc. In these constructions, planar patches must satisfy orthogonal constraints by design (e.g. walls with a ceiling and floor). The hypothesis is that by exploiting orthogonality constraints when possible in the scene, we can perform a reconstruction from a set of points captured by 3D cameras with high accuracy and a low response time. We introduce a method that can iteratively fit a planar model in the presence of noise according to three main steps: a clustering-based unsupervised step that builds pre-clusters from the set of (noisy) points; a linear regression-based supervised step that optimizes a set of planes from the clusters; a reassignment step that challenges the members of the current clusters in a way that minimizes the residuals of the linear predictors. The main contribution is that the method can simultaneously fit different planes in a point cloud providing a good accuracy/speed trade-off even in the presence of noise and outliers, with a smaller processing time compared with previous methods. An extensive experimental study on synthetic data is conducted to compare our method with the most current and representative methods. The quantitative results provide indisputable evidence that our method can generate very accurate models faster than baseline methods. Moreover, two case studies for reconstructing planar-based objects using a Kinect sensor are presented to provide qualitative evidence of the efficiency of our method in real applications.

An Enhanced Ubiquitous-Joint Model for a Rock Mass With Conjugate Joints and Its Application on Excavation Simulation of Large Underground Caverns

A conjugate jointed rock mass (CJRM) is a rock mass with two sets of intersecting joints formed from intact rock under shear. Its mechanical properties and excavation-induced hazards of large underground caverns are different from those of common rock masses because of the unique geological origin thereof. To demonstrate numerically the excavation responses of CJRM, the ubiquitous-joint model is enhanced by consideration of the specific mechanical behaviors of the rock mass. In the enhanced model, CJRM is considered as the composite of columns of rock and two sets of weak planes of joints. The local coordinates, failure modes, and failure sequences of the rock columns and joints are redefined based on the composite characteristics of CJRM, and the failure criteria and plastic potential functions are accordingly modified. The enhanced model is verified numerically by triaxial compression tests and then employed to simulate the excavation of large underground caverns of a pumped storage power station in China. Results show that the modification of the local coordinate system, failure modes, and failure sequences made in the enhanced model is suited to the simulation of the mechanical behaviors of CJRM. Compared with the original ubiquitous-joint model, the enhanced model allows better predictions of the distribution of plastic zones and magnitudes of deformations in simulating underground excavations in CJRM and helps to assess the excavation-triggered hazards more accurately.

Inscribed rectangles in a smooth Jordan curve attain at least one third of all aspect ratios

We prove that for every smooth Jordan curve $\gamma$, if $X$ is the set of all $r\in [0,1]$ so that there is an inscribed rectangle in $\gamma$ of aspect ratio $\mathrm{tan}(r\cdot \pi/4)$, then the Lebesgue measure of $X$ is at least $1/3$. To do this, we study sets of disjoint homologically nontrivial projective planes smoothly embedded in $\mathbb{R}\times \mathbb{R}P^3$. We prove that any such set of projective planes can be equipped with a natural total ordering. We then combine this total ordering with Kemperman's theorem in $S^1$ to prove that $1/3$ is a sharp lower bound on the probability that a Möbius strip filling the $(2,1)$-torus knot in the solid torus times an interval will intersect its rotation by a uniformly random angle.

Continuity of logarithmic capacity

We prove the continuity of logarithmic capacity under Hausdorff convergence of uniformly perfect planar sets. The continuity holds when the Hausdorff distance to the limit set tends to zero at sufficiently rapid rate, compared to the decay of the parameters involved in the uniformly perfect condition. The continuity may fail otherwise.

Subsets and freezing sets in the digital plane

We continue the study of freezing sets for digital images introduced in [L. Boxer and P.C. Staecker, Fixed point sets in digital topology, 1, Applied General Topology 2020; L. Boxer, Fixed point sets in digital topology, 2, Applied General Topology 2020; L. Boxer, Convexity and Freezing Sets in Digital Topology, Applied General Topology, 2021]. We prove methods for obtaining freezing sets for digital images $(X,c_i)$ for $X \subset \mathbb{Z}^2$ and $i \in \{1,2\}$. We give examples to show how these methods can lead to the determination of minimal freezing sets.

Efficiency and localisation for the first Dirichlet eigenfunction

Bounds are obtained for the efficiency or mean to peak ratio $E(\Omega)$ for the first Dirichlet eigenfunction (positive) for open, connected sets $\Omega$ with finite measure in Euclidean space $\R^m$. It is shown that (i) localisation implies vanishing efficiency, (ii) a vanishing upper bound for the efficiency implies localisation, (iii) localisation occurs for the first Dirichlet eigenfunctions for a wide class of elongating bounded, open, convex and planar sets, (iv) if $\Omega_n$ is any quadrilateral with perpendicular diagonals of lengths $1$ and $n$ respectively, then the sequence of first Dirichlet eigenfunctions localises, and $E(\Omega_n)=O\big(n^{-2/3}\log n\big)$. This disproves some claims in the literature. A key technical tool is the Feynman-Kac formula.

Covering Planar Sets

Methods for improving upper and lower bounds for various coverings of planar sets are proposed. New bounds for various numbers of partition constituents are presented, and suggestions for the generalization of the presented methods are offered.

On the Cheeger inequality for convex sets

In this paper, we prove new sharp bounds for the Cheeger constant of planar convex sets that we use to study the relations between the Cheeger constant and the first eigenvalue of the Laplace operator with Dirichlet boundary conditions. This problem is closely related to the study of the so-called Cheeger inequality for which we provide an improvement in the class of planar convex sets. We then provide an existence theorem that highlights the tight relation between improving the Cheeger inequality and proving the existence of a minimizer of the functional J:=λ1/h2 in any dimension n. We finally, provide some new sharp bounds for the first Dirichlet eigenvalue of planar convex sets and a new sharp upper bound for triangles which is better than the conjecture stated in [32] in the case of thin triangles.

On the Intersection Distribution of Degree Three Polynomials and Related Topics

The intersection distribution of a polynomial $f$ over finite field $\mathbb{F}_q$ was recently proposed by Li and Pott [\emph{Finite Fields and Their Applications, 66 (2020)}], which concerns the collective behaviour of a collection of polynomials $\{f(x)+cx \mid c \in\mathbb{F}_q\}$. The intersection distribution has an underlying geometric interpretation, which indicates the intersection pattern between the graph of $f$ and the lines in the affine plane $AG(2,q)$. When $q$ is even, the long-standing open problem of classifying o-polynomials can be rephrased in a simple way, namely, classifying all polynomials which have the same intersection distribution as $x^2$. Inspired by this connection, we proceed to consider the next simplest case and derive the intersection distribution for all degree three polynomials over $\mathbb{F}_q$ with $q$ both odd and even. Moreover, we initiate to classify all monomials having the same intersection distribution as $x^3$, where some characterizations of such monomials are obtained and a conjecture is proposed. In addition, two applications of the intersection distributions of degree three polynomials are presented. The first one is the construction of nonisomorphic Steiner triple systems and the second one produces infinite families of Kakeya sets in affine planes with previously unknown sizes.

Planar point sets determine many pairwise crossing segments

We show that any set of n points in general position in the plane determines n1−o(1) pairwise crossing segments. The best previously known lower bound, Ω(n), was proved more than 25 years ago by Aronov, Erdős, Goddard, Kleitman, Klugerman, Pach, and Schulman. Our proof is fully constructive, and extends to dense geometric graphs.

Integrals with a Meromorphic Function or the Difference of Subharmonic Functions over Discs and Planar Small Sets

Integrals with a Meromorphic Function or the Difference of Subharmonic Functions over Discs and Planar Small Sets

Properties of quasi-Assouad dimension

The connections between quasi-Assouad dimension and tangents are studied. We apply these results to the calculation of the quasi-Assouad dimension for a class of planar self-affine sets. We also show that sets with decreasing gaps have quasi-Assouad dimension 0 or 1 and exhibit an example of a set in the plane whose quasi-Assouad dimension is smaller than that of its projection onto the \(x\)-axis, showing that quasi-Assouad dimension may increase under Lipschitz mappings. Moreover, for closed sets, we show that the Hausdorff dimension is an upper bound for the quasi-lower Assouad dimension.

A Shape Optimization Problem on Planar Sets with Prescribed Topology

We consider shape optimization problems involving functionals depending on perimeter, torsional rigidity and Lebesgue measure. The scaling free cost functionals are of the form P(Omega )T^q(Omega )|Omega |^{-2q-1/2}, and the class of admissible domains consists of two-dimensional open sets Omega satisfying the topological constraints of having a prescribed number k of bounded connected components of the complementary set. A relaxed procedure is needed to have a well-posed problem, and we show that when q<1/2 an optimal relaxed domain exists. When q>1/2, the problem is ill-posed, and for q=1/2, the explicit value of the infimum is provided in the cases k=0 and k=1.

DEVELOPMENT OF A SCIENTIFIC SCHOOL ON APPLIED GEOMETRY FOUNDED BY PROFESSOR OBUKHOVA VIOLETTA SERGEEVNA

The creation of a school on applied geometry at the National University of Bioresources and Nature Management of Ukraine is associated with the names of the so-called prof. Obukhova Violetta and prof. Rozov Seraphim. Thanks to these outstanding personalities, the Department of Drawing Geometry and Mechanical Engineering of the Ukrainian Agricultural Academy (the old name of the department and university) became widely known in the former Soviet Union. Members of the department annually honor the memory of prof. Obukhova V., hold a scientific and practical conference «Obukhov Readings». Now the conference has acquired international status and is held by order of the rector of the university. The conference is attended by teachers from Kiev universities, from universities in other cities, scientists who had to work with V.S. Obukhova or listen to her lectures, being students, doctoral students, graduate students. The development of the school is evidenced by the fact that over the past 15 years, 15 PhD theses have been defended by former graduate students of the department. Some of them are working on doctoral dissertations. Today, the scientific school was headed by Doctor of Technical Sciences, Professor of Pilipakа Sergey. Under his leadership, 17 master's theses and 3 doctoral dissertations were defended. It should be noted that the range of scientific research of the school representatives is quite wide. Many publications focus on bending surfaces based on the invariability of the expression of a linear surface element. The main area of research is geometric modeling of technical forms and automation of their design. In the field of view of scientists - the design of unfolding surfaces, as a bypass single-parameter set of planes, the location on the surfaces of geodetic lines and their design according to a given curve, which should be a geodetic line for the surface and interpolation of a point series in plane and space.

Universal Taylor series on products of planar domains

Using a recent Mergelyan type theorem for products of planar compact sets, we establish generic existence of universal Taylor series on products of planar simply connected domains \({\varOmega }_i\), \(i=1,\ldots ,d\). The universal approximation is realized by partial sums of the Taylor development of the universal function on products of planar compact sets \(K_i\), \(i=1,\ldots ,d\) such that \({\mathbb {C}}-K_i\) is connected and for at least one \(i_0\) the set \(K_{i_0}\) is disjoint from \({\varOmega }_{i_0}\).

Universal Taylor Series in Several Variables Depending on Parameters

We establish generic existence of Universal Taylor Series on products varOmega = prod varOmega _i of planar simply connected domains varOmega _i where the universal approximation holds on products K of planar compact sets with connected complements provided K cap varOmega = emptyset . These classes are with respect to one or several centers of expansion and the universal approximation is at the level of functions or at the level of all derivatives. Also, the universal functions can be smooth up to the boundary, provided that K cap overline{varOmega } = emptyset and {infty } cup [{mathbb {C}} {setminus } overline{varOmega }_i] is connected for all i. All previous kinds of universal series may depend on some parameters; then the approximable functions may depend on the same parameters, as it is shown in the present paper.