33,419 results on '"Regular polygon"'
Search Results
2. New Method for Generating a Regular Polygon
- Author
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Lebamovski, Penio Dimitrov, Kacprzyk, Janusz, Series Editor, Gomide, Fernando, Advisory Editor, Kaynak, Okyay, Advisory Editor, Liu, Derong, Advisory Editor, Pedrycz, Witold, Advisory Editor, Polycarpou, Marios M., Advisory Editor, Rudas, Imre J., Advisory Editor, Wang, Jun, Advisory Editor, Yang, Xin-She, editor, Sherratt, R. Simon, editor, Dey, Nilanjan, editor, and Joshi, Amit, editor
- Published
- 2023
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3. Regular Polygon Central Configuration of the Restricted 1 + N -Body Problem.
- Author
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Chen, Jian, Bi, Peng, and Yang, Mingfang
- Subjects
- *
POLYGONS , *CELESTIAL mechanics - Abstract
We consider the central configurations of the 1 + N -body problem, where N bodies are infinitesimal and the remaining one body is dominant. For regular polygon central configurations, we prove that the masses of all the infinitesimal bodies are equal when N is odd and the masses of the alternate infinitesimal bodies must be equal when N is even. Moreover, in the case of N being even, we present the relationship of the mass parameters between two consecutive infinitesimal bodies. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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4. Solving geometry problems by alternative methods in mathematics education.
- Author
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Aliyev, Samed J., Heydarova, Maftun N., and Aghazade, Shahin M.
- Subjects
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MATHEMATICS education , *PROBLEM solving , *CIRCLE , *GEOMETRY - Abstract
Solving geometry problems is both difficult and interesting. Difficult because there is no general algorithm to solve more or less non-trivial problems as every single problem requires individual and creative approach. At the same time, this is a very interesting activity, because for almost every problem there are plenty of ways to solve it. In this work, we present the method of auxiliary circle divided into equal parts. This method allows finding solution algorithm for some geometry problems which are hard to solve by the method of additional constructions. [ABSTRACT FROM AUTHOR]
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- 2023
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5. TWO NON-CONGRUENT REGULAR POLYGONS HAVING VERTICES AT THE SAME DISTANCES FROM THE POINT.
- Author
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Meskhishvili, Mamuka
- Subjects
- *
GEOMETRICAL constructions - Abstract
For the given regular plane polygon and an arbitrary point in the plane of the polygon, the distances from the point to the vertices of the polygon are defined. We proved that there is one more non-congruent regular polygon having the vertices at the same distances from the point. The sizes of both regular polygons are uniquely determined by these distances. In general case, geometrical construction of the second regular polygon is given. It is proved that there are two points in the plane, which separately have the same set of the distances to the vertices of two non-congruent regular polygons with a shared vertex. [ABSTRACT FROM AUTHOR]
- Published
- 2023
6. Mathematics and Art: Connecting Mathematicians and Artists
- Author
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Malkevitch, Joseph and Sriraman, Bharath, editor
- Published
- 2021
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7. Regular Polygon Central Configuration of the Restricted 1 + N-Body Problem
- Author
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Jian Chen, Peng Bi, and Mingfang Yang
- Subjects
celestial mechanics ,central configuration ,restricted 1 + N-body problem ,regular polygon ,Elementary particle physics ,QC793-793.5 - Abstract
We consider the central configurations of the 1+N-body problem, where N bodies are infinitesimal and the remaining one body is dominant. For regular polygon central configurations, we prove that the masses of all the infinitesimal bodies are equal when N is odd and the masses of the alternate infinitesimal bodies must be equal when N is even. Moreover, in the case of N being even, we present the relationship of the mass parameters between two consecutive infinitesimal bodies.
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- 2023
- Full Text
- View/download PDF
8. Free vibration and buckling of heavy column with regular polygon cross-section.
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Lee, Joon Kyu and Lee, Byoung Koo
- Abstract
This paper deals with the free vibration and buckling of heavy column, considering its own self-weight. The column has a regular polygonal cross-section with a constant area. The column is applied to an external axial load as well as the self-weight. The five end conditions of the column are considered. Based on equilibrium equations of the column element, differential equations governing the vibrational and buckled mode shapes of column are derived. In solution methods, differential equations are numerically integrated by the direct integration method and eigenvalues of the natural frequency, buckling load and self-weight buckling length are calculated by the determinant search method. The numerical results of this study were in good agreement with those of the reference. Parametric study of the end condition, side number and self-weight on the natural frequency and buckling load was carried out. [ABSTRACT FROM AUTHOR]
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- 2022
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9. Buckling optimization of axially functionally graded columns having constant volume.
- Author
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Lee, Joon Kyu and Lee, Byoung Koo
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EULER-Bernoulli beam theory , *YOUNG'S modulus , *FINITE element method , *MODE shapes , *SEARCH algorithms - Abstract
In this article, the buckling optimization of axially functionally graded (AFG) columns to maximize the buckling capacity is studied. Consideration is given to an AFG column having a tapered regular polygon cross section and variable material properties. The governing differential equation is derived based on Euler–Bernoulli beam theory with the relevant boundary conditions and is solved using the direct integration method combined with a determinant search algorithm. The computed buckling loads are compared with those presented in the literature and obtained from finite element analysis. Numerical examples for buckling load and buckled mode shape are given to highlight the effect of parameters related to the Young's modulus, cross-sectional shape, tapering and column volume. In particular, the geometry and material parameters that provide buckling optimization at constant volume of the column are analysed. [ABSTRACT FROM AUTHOR]
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- 2022
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10. Constructive geometric generating of concave pyramids of fourth sort
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Mišić Slobodan Ž. and Backović Marija V.
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concave pyramids ,polyhedral ,equilateral triangle ,regular polygon ,Engineering (General). Civil engineering (General) ,TA1-2040 ,Mechanics of engineering. Applied mechanics ,TA349-359 - Abstract
The paper presents the results of the study of the new set of polihedra, the Concave pyramids of the fourth sort, the construction procedures for generating them and their possible application. Correspondingly to the method of generating the Concave cupolae of fourth sort, the Concave pyramids of fourth sort have the similar logic of origination, and their counterpart in regular faced convex pyramids. They are characterised by the polygonal base, deltahedral surface net, obtained by folding the planar net of unilateral triangles, the polar distribution of the unit space cells with common apex - the top of the Concave pyramid. Polihedral surface of the planar net of Concave pyramids is produced by polar distribution of unit cells, consisting of a spatial sexagon and spatial pentagon - six, or five, unilateral triangles grouped around the common vertex. In the deltahedral surface, the two neighbouring unit cells are joined by means of a unilateral triangle in the zone of the polygonal base and a spatial quadrangle with which they share common sides. The criterion of face regularity is respected, as well as the criterion of multiple axial symmetry. The sort of the Concave pyramids is determined by the number of equilateral triangle rows in thus obtained polyhedron's net. The parameters of the solids were determined constructively by geometric methods.
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- 2021
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11. On inverse construction of isoptics and isochordal-viewed curves.
- Author
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Rochera, David and Bartoň, Michael
- Subjects
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INVERSE problems , *POLYGONS , *LOCUS (Mathematics) - Abstract
Given a regular closed curve α in the plane, a ϕ -isoptic of α is a locus of points from which pairs of tangent lines to α span a fixed angle ϕ. If, in addition, the chord that connects the two points delimiting the visibility angle is of constant length ℓ , then α is said to be (ϕ , ℓ) -isochordal viewed. Some properties of these curves have been studied, yet their full classification is not known. We approach the problem in an inverse manner, namely that we consider a ϕ -isoptic curve c as an input and construct a curve whose ϕ -isoptic is c. We provide thus a sufficient condition that constitutes a partial solution to the inverse isoptic problem. In the process, we also study a relation of isoptics to multihedgehogs. Moreover, we formulate conditions on the behavior of the visibility lines so as their envelope is a (ϕ , ℓ) -isochordal-viewed curve with a prescribed ϕ -isoptic c. Our results are constructive and offer a tool to easily generate this type of curves. In particular, we show examples of (ϕ , ℓ) -isochordal-viewed curves whose ϕ -isoptic is not circular. Finally, we prove that these curves allow the motion of a regular polygon whose vertices lie along the (ϕ , ℓ) -isochordal-viewed curve. [ABSTRACT FROM AUTHOR]
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- 2024
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12. Elliptic Localization of a Moving Object by Transmitter at Unknown Position and Velocity: A Semidefinite Relaxation Approach
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Dominic K. C. Ho, Gang Wang, and Ruichao Zheng
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Mean squared error ,Computer Networks and Communications ,Computer science ,Transmitter ,Regular polygon ,Upper and lower bounds ,Local convergence ,symbols.namesake ,Gaussian noise ,Position (vector) ,symbols ,Relaxation (approximation) ,Electrical and Electronic Engineering ,Algorithm ,Software ,Computer Science::Information Theory - Abstract
This paper investigates the elliptic localization for moving object problem from time delay (TD) and Doppler frequency shift (DFS) measurements, where the transmitter position and velocity are unknown. The transmitter is not perfectly time syncronized such that unknown offsets exist in the TD and DFS measurements. We propose to jointly estimate the object and transmitter positions and velocities and the offsets. Using the TD and DFS measurements from both the indirect and direct paths between the transmitter and the receivers, we formulate a non-convex weighted least squares (WLS) problem. Local convergence may occur when solving the non-convex WLS problem, implying that good estimate is not guaranteed. Thus, we relax the non-convex WLS problem into a convex semidefinite program by applying semidefinite relaxation (SDR). Moreover, we theoretically show that the performance can be improved by using multiple transmitters as compared to that using single transmitter, although more unknown parameters are introduced. We then extend the proposed SDR method to handle the multiple transmitters case. Finally, the mean square error analysis is provided to show that the proposed WLS method reaches the Cramer-Rao lower bound accuracy under small Gaussian noise condition. Simulation results validate the theoretical analysis and show the superior performance over the existing methods.
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- 2023
13. Y-equivalence and rhombic realization of projective-planar quadrangulations.
- Author
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Nakamoto, Atsuhiro and Omizo, Yuta
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POLYGONS , *QUADRILATERALS , *EDGES (Geometry) - Abstract
Let G be a quadrangulation on the projective plane P , i.e., a map of a simple graph on P such that each face is quadrilateral. For a vertex v ∈ V (G) of degree 3 with neighbors v 1 , v 3 , v 5 , a Y-rotation is to delete three edges v v 1 , v v 3 , v v 5 and add v v 2 , v v 4 , v v 6 , where the union of three faces incident to v is surrounded by a closed walk v 1 v 2 v 3 v 4 v 5 v 6 . We say that G is k -minimal if its shortest noncontractible cycle is of length k and if any face contraction yields a noncontractible cycle of length less than k. It was proved that for any k ≥ 3 , any two k -minimal quadrangulations on P are Y -equivalent , i.e., can be transformed into each other by Y-rotations (Nakamoto and Suzuki, 2012). In this paper, we find wider Y-equivalence classes of quadrangulations on P , extending a result on a geometric realization of quadrangulations on P as a rhombus tiling in an even-sided regular polygon (Hamanaka et al., 2020). [ABSTRACT FROM AUTHOR]
- Published
- 2021
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14. Mean value theorems for polynomial solutions of linear elliptic equations with constant coefficients in the complex plane.
- Author
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Trofymenko, Olga D.
- Subjects
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ELLIPTIC equations , *LINEAR equations , *POLYNOMIALS , *MEAN value theorems , *POLYGONS - Abstract
We characterize solutions of the mean value linear elliptic equation with constant coefficients in the complex plane in the case of regular polygon. [ABSTRACT FROM AUTHOR]
- Published
- 2021
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15. Modified Dai-Yuan iterative scheme for nonlinear systems and its application
- Author
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Mohammed Yusuf Waziri, Aliyu Mohammed Awwal, Abubakar Sani Halilu, and Kabiru Ahmed
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Nonlinear system ,Signal processing ,Control and Optimization ,Algebra and Number Theory ,Monotone polygon ,Iterative method ,Computer science ,Applied Mathematics ,Convergence (routing) ,Projection method ,Regular polygon ,Applied mathematics ,Image (mathematics) - Abstract
By exploiting the idea employed in the spectral Dai-Yuan method by Xue et al. [IEICE Trans. Inf. Syst. 101 (12)2984-2990 (2018)] and the approach applied in the modified Hager-Zhang scheme for nonsmooth optimization [PLos ONE 11(10): e0164289 (2016)], we develop a Dai-Yuan type iterative scheme for convex constrained nonlinear monotone system. The scheme's algorithm is obtained by combining its search direction with the projection method [Kluwer Academic Publishers, pp. 355-369(1998)]. One of the new scheme's attribute is that it is derivative-free, which makes it ideal for solving non-smooth problems. Furthermore, we demonstrate the method's application in image de-blurring problems by comparing its performance with a recent effective method. By employing mild assumptions, global convergence of the scheme is determined and results of some numerical experiments show the method to be favorable compared to some recent iterative methods.
- Published
- 2023
16. Incremental subgradient algorithms with dynamic step sizes for separable convex optimizations
- Author
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Xiangmei Wang and Dan Yang
- Subjects
Computer science ,Component (UML) ,General Mathematics ,Convergence (routing) ,Regular polygon ,General Engineering ,Convex function ,Assignment problem ,Subgradient method ,Algorithm ,Separable space - Abstract
We consider the incremental subgradient algorithm employing dynamic step sizes for minimizing the sum of a large number of component convex functions. The dynamic step size rule was firstly introduced by Goffin and Kiwiel [Math. Program., 1999, 85(1): 207-211] for the subgradient algorithm, soon later, for the incremental subgradient algorithm by Nedic and Bertsekas in [SIAM J. Optim., 2001, 12(1): 109-138]. It was observed experimentally that the incremental approach has been very successful in solving large separable optimizations, and that the dynamic step sizes generally have better computational performance than others in the literature. In the present paper, we propose two modified dynamic step size rules for the incremental subgradient algorithm and analyse the convergence properties of them. At last, the assignment problem is considered and the incremental subgradient algorithms employing different kinds of dynamic step sizes are applied to solve the problem. The computational experiments show that the two modified ones converges dramatically faster and stabler than the corresponding one in [SIAM J. Optim., 2001, 12(1): 109-138]. Particularly, for solving large separable convex optimizations, we strongly recommend the second one (see Algorithm 3.3 in the paper) since it has interesting computational performance and is the simplest one.
- Published
- 2022
17. Penetration Depth Between Two Convex Polyhedra: An Efficient Stochastic Global Optimization Approach
- Author
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Mark A. Abramson, Gavin W. Smith, and Griffin D. Kent
- Subjects
Mathematical optimization ,Optimization problem ,Discretization ,Computer science ,Regular polygon ,Polytope ,Computer Graphics and Computer-Aided Design ,Computer graphics ,Polyhedron ,Signal Processing ,Computer Vision and Pattern Recognition ,Focus (optics) ,Global optimization ,Software - Abstract
During the detailed design phase of an aerospace program, one of the most important consistency checks is to ensure that no two distinct objects occupy the same physical space. Since exact geometrical modeling is usually intractable, geometry models are discretized, which often introduces small interferences not present in the fully detailed model. In this paper, we focus on computing the depth of the interference, so that these false positive interferences can be removed, and attention can be properly focused on the actual design. Specifically, we focus on efficiently computing the penetration depth between two polyhedra, which is a well-studied problem in the computer graphics community. We formulate the problem as a constrained five-variable global optimization problem, and then derive an equivalent unconstrained, two-variable nonsmooth problem. To solve the optimization problem, we apply a popular stochastic multistart optimization algorithm in a novel way, which exploits the advantages of each problem formulation simultaneously. Numerical results for the algorithm, applied to 14 randomly generated pairs of penetrating polytopes, illustrate both the effectiveness and efficiency of the method.
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- 2022
18. 드론 무게균형을 위한 원형의 부하 적재 방식.
- Author
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은성배 and 한상숙
- Subjects
POLYGONS ,PROBLEM solving ,SHIPS ,WEIGHTS & measures ,CONTAINER ships - Abstract
Problems that satisfy various constraints while maintaining weight balance in ships or aircraft were studied. In addition, a study was conducted to solve the problem with a mathematical method under the condition that the shape and weight of the load are the same and the m×n (m and n are all odd) mesh structures. The problem is that the existing mathematical weight balancing method is not suitable for circular structures. In this paper, we studied the load stowing problem in a circular space where objects are loaded at the vertices of N equilateral polygons. Assuming that all N conformal polygons have an even number of angles, it was proved that a loading method that always maintains weight balance regardless of the variety of number of loads. By providing the structure and loading method of the drone loading ship, we showed that our method was appropriate. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
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19. CYCLIC AVERAGES OF REGULAR POLYGONAL DISTANCES.
- Author
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Meskhishvili, Mamuka
- Subjects
- *
POWER (Social sciences) , *DISTANCES , *POLYGONS - Abstract
We consider a regular plane polygon with n vertices and an arbitrary point in the plane. Let R be the circumscribed radius of the polygon and L a distance from the point to the centroid of the polygon. Then the averages of the (2m)-th powers of distances from the point to the polygon vertices satisfy the relations Sn(2) = R² + L²; Sn(2m) = (R²> + L²)m + bm 2 ∑ k=1 (m 2k) (2k k) (R² + L²)m-2k (RL)2k, where m = 2,..., n - 1. [ABSTRACT FROM AUTHOR]
- Published
- 2021
20. Top-k Partial Label Machine
- Author
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Dong Yuan, Xiuwen Gong, and Wei Bao
- Subjects
Optimization algorithm ,Linear programming ,Computer Networks and Communications ,Computer science ,business.industry ,Regular polygon ,Pattern recognition ,Computer Science Applications ,Dual (category theory) ,Set (abstract data type) ,Artificial Intelligence ,Partial loss ,Hinge loss ,Artificial intelligence ,Noise (video) ,business ,Software - Abstract
To deal with ambiguities in partial label learning (PLL), the existing PLL methods implement disambiguations, by either identifying the ground-truth label or averaging the candidate labels. However, these methods can be easily misled by the false-positive labels in the candidate label set. We find that these ambiguities often originate from the noise caused by highly correlated or overlapping candidate labels, which leads to the difficulty in identifying the ground-truth label on the first attempt. To give the trained models more tolerance, we first propose the top-k partial loss and convex top-k partial hinge loss. Based on the losses, we present a novel top-k partial label machine (TPLM) for partial label classification. An efficient optimization algorithm is proposed based on accelerated proximal stochastic dual coordinate ascent (Prox-SDCA) and linear programming (LP). Moreover, we present a theoretical analysis of the generalization error for TPLM. Comprehensive experiments on both controlled UCI datasets and real-world partial label datasets demonstrate that the proposed method is superior to the state-of-the-art approaches.
- Published
- 2022
21. On the Δ-interval and the Δ-convexity numbers of graphs and graph products
- Author
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Bijo S. Anand, Mitre Costa Dourado, Prasanth G. Narasimha-Shenoi, and Sabeer Sain Ramla
- Subjects
Combinatorics ,Cardinality ,Chordal graph ,Applied Mathematics ,Regular polygon ,Block (permutation group theory) ,Discrete Mathematics and Combinatorics ,Interval (graph theory) ,Function (mathematics) ,Lexicographical order ,Convexity ,Mathematics - Abstract
Given a graph G and a set S ⊆ V ( G ) , the Δ -interval of S , [ S ] Δ , is the set formed by the vertices of S and every w ∈ V ( G ) forming a triangle with two vertices of S . If [ S ] Δ = S , then S is Δ -convex of G ; if [ S ] Δ = V ( G ) , then S is a Δ -interval set of G . The Δ -interval number of G is the minimum cardinality of a Δ -interval set and the Δ -convexity number of G is the maximum cardinality of a proper Δ -convex subset of V ( G ) . In this work, we show that the problem of computing the Δ -convexity number is W[1]-hard and NP-hard to approximate within a factor O ( n 1 − ɛ ) for any constant ɛ > 0 even for graphs with diameter 2 and that the problem of computing the Δ -interval number is NP-complete for general graphs. For the positive side, we present characterizations that lead to polynomial-time algorithms for computing the Δ -convexity number of chordal graphs and for computing the Δ -interval number of block graphs. We also present results on the Δ -hull, Δ -interval and Δ -convexity numbers concerning the three standard graph products, namely, the Cartesian, the strong and the lexicographic products, in function of these and well-studied parameters of the operands.
- Published
- 2022
22. Convex preferences: An abstract approach
- Author
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Marta Cardin
- Subjects
TheoryofComputation_MISCELLANEOUS ,Settore SECS-S/06 - Metodi mat. dell'economia e Scienze Attuariali e Finanziarie ,Logic ,MathematicsofComputing_NUMERICALANALYSIS ,MathematicsofComputing_GENERAL ,Regular polygon ,Convex preferences ,Convexity ,Convex structure ,Combinatorics ,Areas of mathematics ,Artificial Intelligence ,TheoryofComputation_ANALYSISOFALGORITHMSANDPROBLEMCOMPLEXITY ,Convex space Convexity algebra Convex preference Compatible preference Aggregation of orderings ,Mathematics - Abstract
The notion of abstract convex structure generalizes the standard notion of convexity in linear spaces. We consider abstract convex structures that are combinatorial objects studied in various areas of mathematics and convex algebras as introduced in [8] and we study a general definition of convex preferences. Relations defined by aggregation of orderings are considered.
- Published
- 2022
23. Accelerated Log-Regularized Convolutional Transform Learning and Its Convergence Guarantee
- Author
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Haoli Zhao, Shengli Xie, Zuyuan Yang, Yongcheng Guo, and Zhenni Li
- Subjects
Computer science ,Open problem ,Regular polygon ,Extrapolation ,02 engineering and technology ,Function (mathematics) ,Convolutional neural network ,Computer Science Applications ,Human-Computer Interaction ,CTL ,Control and Systems Engineering ,020204 information systems ,Convergence (routing) ,0202 electrical engineering, electronic engineering, information engineering ,Unsupervised learning ,020201 artificial intelligence & image processing ,Electrical and Electronic Engineering ,Algorithm ,Software ,Information Systems - Abstract
Convolutional transform learning (CTL), learning filters by minimizing the data fidelity loss function in an unsupervised way, is becoming very pervasive, resulting from keeping the best of both worlds: the benefit of unsupervised learning and the success of the convolutional neural network. There have been growing interests in developing efficient CTL algorithms. However, developing a convergent and accelerated CTL algorithm with accurate representations simultaneously with proper sparsity is an open problem. This article presents a new CTL framework with a log regularizer that can not only obtain accurate representations but also yield strong sparsity. To efficiently address our nonconvex composite optimization, we propose to employ the proximal difference of the convex algorithm (PDCA) which relies on decomposing the nonconvex regularizer into the difference of two convex parts and then optimizes the convex subproblems. Furthermore, we introduce the extrapolation technology to accelerate the algorithm, leading to a fast and efficient CTL algorithm. In particular, we provide a rigorous convergence analysis for the proposed algorithm under the accelerated PDCA. The experimental results demonstrate that the proposed algorithm can converge more stably to desirable solutions with lower approximation error and simultaneously with stronger sparsity and, thus, learn filters efficiently. Meanwhile, the convergence speed is faster than the existing CTL algorithms.
- Published
- 2022
24. A note on the convexity number of the complementary prisms of trees
- Author
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P K Neethu and S V Ullas Chandran
- Subjects
Combinatorics ,Disjoint union ,Cardinality ,Applied Mathematics ,Shortest path problem ,Regular polygon ,Convex set ,Discrete Mathematics and Combinatorics ,Tree (graph theory) ,Convexity ,Complement (set theory) ,Mathematics - Abstract
A set of vertices S of a graph G is a (geodesically) convex set, if S contains all the vertices belonging to any shortest path connecting two vertices of S . The cardinality of a maximum proper convex set of G is called the convexity number, con ( G ) , of G . The complementary prism G G ¯ of G is obtained from the disjoint union of G and its complement G ¯ by adding the edges of a perfect matching between them. In this work, we examine the convex sets of the complementary prism of a tree and derive formulas for the convexity numbers of the complementary prisms of all trees.
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- 2022
25. Support Vector Machine Classifier via Soft-Margin Loss
- Author
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Yuan-Hai Shao, Naihua Xiu, Shenglong Zhou, Ce Zhang, and Huajun Wang
- Subjects
Ideal (set theory) ,business.industry ,Computer science ,Applied Mathematics ,Working set ,Regular polygon ,Optimality theory ,Soft margin ,Support vector machine ,Computational Theory and Mathematics ,Artificial Intelligence ,Robustness (computer science) ,Limit point ,Computer Vision and Pattern Recognition ,Artificial intelligence ,business ,Algorithm ,Software - Abstract
Support vector machines (SVM) have drawn wide attention for the last two decades due to its extensive applications, so a vast body of work has developed optimization algorithms to solve SVM with various soft-margin losses. To distinguish all, in this paper, we aim at solving an ideal soft-margin loss SVM: L0/1 soft-margin loss SVM (dubbed as L0/1-SVM). Many of the existing (non)convex soft-margin losses can be viewed as one of the surrogates of the L0/1 soft-margin loss. Despite its discrete nature, we manage to establish the optimality theory for the L0/1-SVM including the existence of the optimal solutions, the relationship between them and P-stationary points. These not only enable us to deliver a rigorous definition of L0/1 support vectors but also allow us to define a working set. Integrating such a working set, a fast alternating direction method of multipliers is then proposed with its limit point being a locally optimal solution to the L0/1-SVM. Finally, numerical experiments demonstrate that our proposed method outperforms some leading classification solvers from SVM communities, in terms of faster computational speed and a fewer number of support vectors. The bigger the data size is, the more evident its advantage appears.
- Published
- 2022
26. Squaring the circle
- Author
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Meskens, Ad, Tytgat, Paul, Meskens, Ad, and Tytgat, Paul
- Published
- 2017
- Full Text
- View/download PDF
27. Pearl
- Author
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Nowlan, Robert A. and Nowlan, Robert A.
- Published
- 2017
- Full Text
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28. Gielis Curves, Surfaces and Transformations
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Gielis, Johan and Gielis, Johan
- Published
- 2017
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29. Pi Formulas
- Author
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Mureşan, Marian and Mureşan, Marian
- Published
- 2017
- Full Text
- View/download PDF
30. Waves and Sensory Perception
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Espinoza, Fernando and Espinoza, Fernando
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- 2017
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31. REGULAR POLYGONS IN 2D OBJECTS SHAPE DESCRIPTION.
- Author
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Kwinta, Andrzej and Bac-Bronowicz, Joanna
- Subjects
- *
GEOMETRIC shapes , *CENTROID , *THREE-dimensional imaging , *POLYGONS , *GEOMETRY , *DATA analysis - Abstract
Many real 3D objects have complex geometric shapes in various types of analyses. Image of these objects is recorded in the form of a 2D map. In the analysis, a simplification of this image to basic 2D figures with defined geometry is often needed. The paper presents an analysis of the geometry of a flat image (an image of a 3D object) using regular polygons. Geometry properties (F form, C centroid, S size) were determined to describe the object. Various criteria of selection of the 'best ' regular n-sided polygon for a given 2D object (solved theoretically) were put forward. In the paper, criteria for describing a 2D object by regular n-sided polygons were defined on the basis of determining the measure of object shape elongation (e). In the 'blind ' theoretical example, it was tested whether the individual shape measures listed in the paper allow for correct identification of the shapes of given 2D objects. The practical application of measures is illustrated by two actual examples. While in the first example the shape of the Canary Islands is analysed, the second example describes the shape of Poland 's borders. Actual examples deliver different results for different measures. In effect, there is no clear objective criterion for selecting a polygon shape. The simplifications of the shape of an object presented in the paper should not be equated with the object 's generalization. Such simplifications are used in GIS to visualize geographic analyses based on the data available in the primary database, because the object will retain the character of the shape in the simplest possible geometry and neighborhood, and does not lose any of the scope and accuracy of the attributes assigned to a given object in the database. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
32. Universal Prediction Band via Semi-Definite Programming
- Author
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Tengyuan Liang
- Subjects
FOS: Computer and information sciences ,Semidefinite programming ,Statistics and Probability ,Heteroscedasticity ,Computer Science - Machine Learning ,Econometrics (econ.EM) ,Nonparametric statistics ,Explained sum of squares ,Regular polygon ,Machine Learning (stat.ML) ,Mathematics - Statistics Theory ,Statistics Theory (math.ST) ,Variance (accounting) ,Machine Learning (cs.LG) ,FOS: Economics and business ,Optimization and Control (math.OC) ,Statistics - Machine Learning ,FOS: Mathematics ,Applied mathematics ,Uncertainty quantification ,Statistics, Probability and Uncertainty ,Mathematics - Optimization and Control ,Mathematics ,Interpolation ,Economics - Econometrics - Abstract
We propose a computationally efficient method to construct nonparametric, heteroscedastic prediction bands for uncertainty quantification, with or without any user-specified predictive model. Our approach provides an alternative to the now-standard conformal prediction for uncertainty quantification, with novel theoretical insights and computational advantages. The data-adaptive prediction band is universally applicable with minimal distributional assumptions, has strong non-asymptotic coverage properties, and is easy to implement using standard convex programs. Our approach can be viewed as a novel variance interpolation with confidence and further leverages techniques from semi-definite programming and sum-of-squares optimization. Theoretical and numerical performances for the proposed approach for uncertainty quantification are analyzed., 21 pages, 4 figures
- Published
- 2022
33. Generalized Nonconvex Approach for Low-Tubal-Rank Tensor Recovery
- Author
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Tingwen Huang, Xinling Liu, Feng Zhang, Jianwen Huang, Hailin Wang, and Jianjun Wang
- Subjects
Rank (linear algebra) ,Computer Networks and Communications ,Rank minimization ,Matrix norm ,Regular polygon ,Mathematical proof ,Computer Science Applications ,Critical point (set theory) ,Artificial Intelligence ,Tensor (intrinsic definition) ,Applied mathematics ,Minification ,Software ,Mathematics - Abstract
The tensor-tensor product-induced tensor nuclear norm (t-TNN) (Lu et al., 2020) minimization for low-tubal-rank tensor recovery attracts broad attention recently. However, minimizing the t-TNN faces some drawbacks. For example, the obtained solution could be suboptimal to the original problem due to its loose approximation. In this article, we extract a unified nonconvex surrogate of the tensor tubal rank as a tighter regularizer, which involves many popular nonconvex penalty functions. An iterative reweighted t-TNN algorithm is proposed to solve the resulting generalized nonconvex tubal rank minimization for tensor recovery. It converges to a critical point globally with rigorous proofs based on the Kurdyka-Łojasiwicz property. Furthermore, we provide the theoretical guarantees for exact and robust recovery by developing the tensor null space property. Extensive experiments demonstrate that our approach markedly enhances recovery performance compared with several state-of-the-art convex and nonconvex methods.
- Published
- 2022
34. Toward a Convex Design Framework for Online Active Fault Diagnosis of LPV Systems
- Author
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Feng Xu, Xueqian Wang, Sorin Olaru, Junbo Tan, Laboratoire des signaux et systèmes (L2S), and Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Design framework ,020301 aerospace & aeronautics ,0209 industrial biotechnology ,Sequence ,Computer science ,Regular polygon ,Characteristic equation ,02 engineering and technology ,Active fault ,[SPI.AUTO]Engineering Sciences [physics]/Automatic ,Computer Science Applications ,Constraint (information theory) ,020901 industrial engineering & automation ,Fractional programming ,0203 mechanical engineering ,Control and Systems Engineering ,Control theory ,Convex optimization ,Electrical and Electronic Engineering ,ComputingMilieux_MISCELLANEOUS - Abstract
This paper focuses on the design of on-line optimal input sequence for robust active fault diagnosis (AFD) of discrete-time linear parameter varying (LPV) systems using set-theoretic methods. Instead of the traditional set-separation constraint conditions leading to the design of off-line input sequence, the proposed approach focuses on on-line (re)shaping of the input sequence based on the real-time information of the output to discriminate system modes at each time instant such that the conservatism of robust AFD has the potential to be further reduced. The criterion on the design of optimal input is characterized based on a non-convex fractional programming problem at each time instant, which is shown to be efficiently solved within a convex optimization framework. Aside this main contribution, by exploiting Lagrange duality, the optimal input is explicitly obtained by solving a characteristic equation. At the end, a physical circuit model is provided to illustrate the effectiveness of the proposed method.
- Published
- 2022
35. On the Analysis of Inexact Augmented Lagrangian Schemes for Misspecified Conic Convex Programs
- Author
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Uday V. Shanbhag, Hesam Ahmadi, and Necdet Serhat Aybat
- Subjects
0209 industrial biotechnology ,020901 industrial engineering & automation ,Control and Systems Engineering ,Augmented Lagrangian method ,Conic section ,Regular polygon ,Applied mathematics ,02 engineering and technology ,Electrical and Electronic Engineering ,Computer Science Applications ,Mathematics - Published
- 2022
36. Private Empirical Risk Minimization With Analytic Gaussian Mechanism for Healthcare System
- Author
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Jiahao Ding, Haixia Zhang, Miao Pan, Sai Mounika Errapotu, Yuanxiong Guo, and Dongfeng Yuan
- Subjects
Information privacy ,Mathematical optimization ,Information Systems and Management ,Computer science ,Gaussian ,Regular polygon ,020207 software engineering ,02 engineering and technology ,symbols.namesake ,0202 electrical engineering, electronic engineering, information engineering ,symbols ,Differential privacy ,Empirical risk minimization ,Convex function ,Classifier (UML) ,Information Systems ,Healthcare system - Abstract
With the wide range application of machine learning in healthcare for helping humans drive crucial decisions, data privacy becomes an inevitable concern due to the utilization of sensitive data such as patients records and registers of a company. Thus, constructing a privacy preserving machine learning model while still maintaining high accuracy becomes a challenging problem. In this paper, we propose two differentially private algorithms, i.e., Output Perturbation with aGM (OPERA) and Gradient Perturbation with aGM (GRPUA) for empirical risk minimization, a useful method to obtain a globally optimal classifier, by leveraging the analytic Gaussian mechanism (aGM) to achieve privacy preservation of sensitive medical data in a healthcare system. We theoretically analyze and prove utility upper bounds of proposed algorithms and compare them with prior algorithms in the literature. The analyses show that in the high privacy regime, our proposed algorithms can achieve a tighter utility bound for both settings: strongly convex and non-strongly convex loss functions. Besides, we evaluate the proposed private algorithms on three benchmark datasets, i.e., Adult, BANK and IPUMS-BR. The simulation results demonstrate that our approaches can achieve higher accuracy and lower objective values compared with existing ones in all three datasets while providing differential privacy guarantees.
- Published
- 2022
37. The Robust Minkowski–Lyapunov Equation
- Author
-
Saša V. Raković
- Subjects
Mathematical analysis ,Regular polygon ,Characterization (mathematics) ,Space (mathematics) ,Computer Science Applications ,symbols.namesake ,Control and Systems Engineering ,Minkowski space ,symbols ,Fundamental solution ,Lyapunov equation ,Uniqueness ,Electrical and Electronic Engineering ,Interior point method ,Mathematics - Abstract
The Lyapunov equation for polytopic linear inclusions over the space of Minkowski functions of nonempty compact and convex sets that contain the origin as an interior point is studied. In particular, necessary and sufficient conditions for the characterization, existence and uniqueness of its fundamental solution are derived.
- Published
- 2022
38. On Boundedness of Maximal Operators Associated with Hypersurfaces
- Author
-
S E Usmanov and I A Ikromov
- Subjects
Statistics and Probability ,Pure mathematics ,Mathematics::Algebraic Geometry ,Hypersurface ,Mathematics::Complex Variables ,Applied Mathematics ,General Mathematics ,Mathematics::Classical Analysis and ODEs ,Regular polygon ,Mathematics::Differential Geometry ,General Medicine ,Value (mathematics) ,Mathematics - Abstract
In this paper, we obtain the criterion of boundedness of maximal operators associated with smooth hypersurfaces. Also we compute the exact value of the boundedness index of such operators associated with arbitrary convex analytic hypersurfaces in the case where the height of a hypersurface in the sense of A. N. Varchenko is greater than 2. Moreover, we obtain the exact value of the boundedness index for degenerated smooth hypersurfaces, i.e., for hypersurfaces satisfying conditions of the classical Hartman-Nirenberg theorem. The obtained results justify the Stein-Iosevich-Sawyer hypothesis for arbitrary convex analytic hypersurfaces as well as for smooth degenerated hypersurfaces. Also we discuss some related problems of the theory of oscillatory integrals.
- Published
- 2022
39. Distributed Adaptive Subgradient Algorithms for Online Learning Over Time-Varying Networks
- Author
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Quanbo Ge, Junlong Zhu, Mingchuan Zhang, Bowei Hao, Qingtao Wu, and Ruijuan Zheng
- Subjects
Computer science ,Generalization ,Online learning ,Regular polygon ,Regret ,Computer Science Applications ,Human-Computer Interaction ,Control and Systems Engineering ,Bounded function ,Electrical and Electronic Engineering ,Performance improvement ,Clipping (computer graphics) ,Subgradient method ,Algorithm ,Software - Abstract
Adaptive gradient algorithms have recently become extremely popular because they have been applied successfully in training deep neural networks, such as Adam, AMSGrad, and AdaBound. Despite their success, however, the distributed variant of the adaptive method, which is expected to possess a rapid training speed at the beginning and a good generalization capacity at the end, is rarely studied. To fill the gap, a distributed adaptive subgradient algorithm is presented, called D-AdaBound, where the learning rates are dynamically bounded by clipping the learning rates. Moreover, we obtain the regret bound of D-AdaBound, in which the objective functions are convex. Finally, we confirm the effectiveness of D-AdaBound by simulation experiments on different datasets. The results show the performance improvement of D-AdaBound relative to existing distributed online learning algorithms.
- Published
- 2022
40. Smart meter data classification using optimized random forest algorithm
- Author
-
Alireza Zakariazadeh
- Subjects
Smart meter ,business.industry ,Computer science ,Applied Mathematics ,Data classification ,Regular polygon ,computer.software_genre ,Computer Science Applications ,Random forest ,Electricity ,Control and Systems Engineering ,Classifier (linguistics) ,Cluster Analysis ,Classification methods ,Data mining ,Electrical and Electronic Engineering ,Cluster analysis ,business ,Instrumentation ,computer ,Algorithms - Abstract
Implementing a proper clustering algorithm and a high accuracy classifier for applying on electricity smart meter data is the first stage in analyzing and managing electricity consumption. In this paper, Random Forest (RF) classifier optimized by Artificial Bee Colony (ABC) which is called Artificial Bee Colony-based Random Forest (ABC-RF) is proposed. Also, in order to determine the representative load curves, the Convex Clustering (CC) is used. The solution paths generated by convex clustering show relationships among clusters that were hidden by static methods such as k-means clustering. To validate the proposed method, a case study that includes a real dataset of residential smart meters is implemented. The results evidence that the proposed ABC-RF method provides a higher accuracy if compared to other classification methods.
- Published
- 2022
41. Yet Another Computation-Oriented Necessary and Sufficient Condition for Stabilizability of Switched Linear Systems
- Author
-
Mirko Fiacchini, GIPSA - Modelling and Optimal Decision for Uncertain Systems (GIPSA-MODUS), GIPSA Pôle Automatique et Diagnostic (GIPSA-PAD), Grenoble Images Parole Signal Automatique (GIPSA-lab), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Grenoble Images Parole Signal Automatique (GIPSA-lab), Université Grenoble Alpes (UGA), and ANR-18-CE40-0010,HANDY,Systèmes Dynamiques Hybrides et en Réseau(2018)
- Subjects
Switched linear systems ,020301 aerospace & aeronautics ,0209 industrial biotechnology ,convex analysis ,Computation ,[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS] ,Linear system ,Mathematics::Optimization and Control ,Regular polygon ,stabilizability ,02 engineering and technology ,Space (mathematics) ,[SPI.AUTO]Engineering Sciences [physics]/Automatic ,Computer Science Applications ,Exponential convergence rate ,020901 industrial engineering & automation ,0203 mechanical engineering ,Cover (topology) ,Computer Science::Systems and Control ,Control and Systems Engineering ,Conic section ,Applied mathematics ,Partition (number theory) ,Electrical and Electronic Engineering ,Mathematics - Abstract
International audience; This paper presents a computational method to test the stabilizability of discrete-time switched linear systems. The existence of a conic cover of the space on whose elements a convex condition holds is proved to be necessary and sufficient for stabilizability. An algorithm for computing a conic partition that satisfies the new necessary and sufficient condition is given. The algorithm, that allows also to determine bounds on the exponential convergence rate, is proved to overcome the conservatism of conditions equivalent to periodic stabilizability and is applied to a four dimensional system.
- Published
- 2022
42. Distributed Nonconvex Event-Triggered Optimization Over Time-Varying Directed Networks
- Author
-
Wei Du, Yang Tang, Chen Liang, Zi wei Dong, Shuai Mao, and Yu-Chu Tian
- Subjects
Mathematical optimization ,Optimization problem ,Computer science ,Regular polygon ,Process (computing) ,Mode (statistics) ,Computer Science Applications ,Transmission (telecommunications) ,Rate of convergence ,Control and Systems Engineering ,Convergence (routing) ,Electrical and Electronic Engineering ,Information Systems ,Data transmission - Abstract
Many problems in industrial smart manufacturing, such as process operational optimization and decision-making, can be regarded as distributed non-convex optimization problems, whose goal is to utilize distributed nodes to cooperatively search for the minimal value of the global objective function. Considering data transmission mode, transmission condition and communication waste in industrial applications, it is meaningful to study the distributed non-convex optimization problem with an event-triggered strategy over time-varying directed networks. To solve such a problem, a distributed non-convex event-triggered algorithm is proposed in this paper. Under some assumptions on local objective functions, gradients and step-sizes, the convergence of the proposed event-triggered algorithm to the local minimum is proved. Moreover, it is obtained that the proposed distributed event-triggered algorithm has a convergence rate of O(1/ln(t)). Finally, two examples in industrial systems are provided to validate the effectiveness of the proposed algorithm.
- Published
- 2022
43. An exact algorithm for two-dimensional vector packing problem with volumetric weight and general costs
- Author
-
Qian Hu, Ting Wang, and Andrew Lim
- Subjects
Work (thermodynamics) ,Mathematical optimization ,Information Systems and Management ,General Computer Science ,Computer science ,Regular polygon ,Function (mathematics) ,Management Science and Operations Research ,Industrial and Manufacturing Engineering ,Piecewise linear function ,Set (abstract data type) ,Exact algorithm ,Bounding overwatch ,Modeling and Simulation ,Column generation - Abstract
The volumetric weight of a package has become an essential factor in calculating the delivery cost of shipments in the international logistics market. In this work, we extend the two-dimensional vector packing problem by considering a more realistic cost structure, which is a general function of volumetric weight. The problem is to pack a set of different items into some identical bins without violating weight limits and volume capacities so that the total delivery cost is minimized. We develop an exact approach based on a branch-and-price algorithm and subset-row inequalities for the problem. To efficiently solve the pricing problem in column generation, a label-setting algorithm with an effective label dominance rule and a bounding procedure is presented. A stronger label dominance rule is derived for the case where the cost function is convex. The computational results show that the exact method is effective in solving the various test instances of the problem. If the volumetric weight is not considered, the exact method can be adapted to solve the two-dimensional vector packing problem with piecewise linear cost function and outperformed the existing exact algorithm by computing 27 optimal solutions for previously open instances.
- Published
- 2022
44. A Convex Neural Network Solver for DCOPF With Generalization Guarantees
- Author
-
Baosen Zhang, Yize Chen, and Ling Zhang
- Subjects
Mathematical optimization ,Control and Optimization ,Karush–Kuhn–Tucker conditions ,Artificial neural network ,Computer Networks and Communications ,Generalization ,Computer science ,Regular polygon ,Systems and Control (eess.SY) ,Solver ,Electrical Engineering and Systems Science - Systems and Control ,Convexity ,Reduction (complexity) ,Electric power system ,Control and Systems Engineering ,Signal Processing ,FOS: Electrical engineering, electronic engineering, information engineering - Abstract
The DC optimal power flow (DCOPF) problem is a fundamental problem in power systems operations and planning. With high penetration of uncertain renewable resources in power systems, DCOPF needs to be solved repeatedly for a large amount of scenarios, which can be computationally challenging. As an alternative to iterative solvers, neural networks are often trained and used to solve DCOPF. These approaches can offer orders of magnitude reduction in computational time, but they cannot guarantee generalization, and small training error does not imply small testing errors. In this work, we propose a novel algorithm for solving DCOPF that guarantees the generalization performance. First, by utilizing the convexity of DCOPF problem, we train an input convex neural network. Second, we construct the training loss based on KKT optimality conditions. By combining these two techniques, the trained model has provable generalization properties, where small training error implies small testing errors. In experiments, our algorithm improves the optimality ratio of the solutions by a factor of five in comparison to end-to-end models.
- Published
- 2022
45. A categorical isomorphism between injective balanced L-S0-convex spaces and fuzzy frames
- Author
-
Changchun Xia
- Subjects
Combinatorics ,Artificial Intelligence ,Logic ,Frame (networking) ,Convex set ,Regular polygon ,Point (geometry) ,Isomorphism ,Categorical variable ,Fuzzy logic ,Injective function ,Mathematics - Abstract
The main purpose of this paper is to show that injective balanced L- S 0 -convex spaces and fuzzy frames are isomorphic from the categorical point of view. Meanwhile, we get that every fuzzy frame equipped with the strong L-filter convex structure is an injective balanced L- S 0 -convex space and conversely, the specialization L-ordered set of an injective balanced L- S 0 -convex space is a fuzzy frame.
- Published
- 2022
46. Rhombus Tilings of an Even-Sided Polygon and Quadrangulations on the Projective Plane.
- Author
-
Hamanaka, Hiroaki, Nakamoto, Atsuhiro, and Suzuki, Yusuke
- Subjects
- *
POLYGONS , *PROJECTIVE planes , *MINIMAL surfaces , *TILES - Abstract
A quadrangulation on a closed surface is k-minimal if its shortest noncontractible cycle is of length k and if any face contraction yields a noncontractible cycle of length less than k. We prove that the rhombus tilings of a regular 2k-gon bijectively correspond to the pairs of a k-minimal quadrangulations on the projective plane and its specified noncontractible k-cycle. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
47. مطالعۀ پدیده شناسی پرشهاي هیدرولیکی چندضلعی
- Author
-
علی اسدي, مجید ملک جعفریان, and علیرضا تیمورتاش
- Subjects
HYDRAULIC jump ,REYNOLDS number ,SURFACE tension ,POLYGONS ,SCIENTISTS - Abstract
Polygonal hydraulic jump is among the subjects widely studied by scientists in recent years. Although this phenomenon was discovered nearly two decades ago, many of the probable reasons behind it remain unknown. Accordingly, the main goal of the present study is to conduct a laboratory-scale phenomenological investigation on polygonal hydraulic jumps. The results indicated that the main cause of polygonal hydraulic jumps is the presence of disturbances and instabilities in flows, systems, or environments. Given the Plateau–Rayleigh instability, in the presence of surface tension and viscosity effects, the disturbances and instabilities create stable circular jumps unstable and then turn them into polygonal jumps. A stable circular jump was created with the elimination of instabilities, and the jump, unlike what observed in previous studies, was stable at a low to high range of Reynolds number. In addition, the behavior of polygonal hydraulic jumps formed in previous studies in the presence of instabilities was investigated. In a constant flow rate, the area inside the jump was equal for all the possible jumps at an error level of less than 10%. Among the polygons with an equal number of sides, which are possibly observed in a particular flow rate, the jump changes into a regular polygon, as the surface tension naturally tends to create the minimum possible surface area for the jump. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
48. EQUIVALENCE CONDITIONS OF TWO SYSTEMS OF VECTORS IN THE TAXICAB PLANE AND ITS APPLICATIONS TO TAXICAB POLYGONS.
- Author
-
ÖREN, IDRIS and ÇOBAN, HÜSNÜ ANIL
- Subjects
- *
MATHEMATICAL equivalence , *POLYGONS , *INVARIANTS (Mathematics) - Abstract
This study presents the conditions of MT (2)-equivalence for two systems of vectors fx1; x2; x3g and {x1, x2, x3} in R2T, where MT (2) is the group of all isometries of the 2-dimensional taxicab space RT². Firstly a minimal complete system of MT (2)-invariants of {x1, x2, x3} is obtained. Then, using the conditions of MT (2)-equivalence, an answer is given to the open prob- lem posed in [10, p.428]. Furthermore, an algorithm is given for constructing taxicab regular polygons in terms of MT (2)-invariants. This algorithm is gen- eral and useful to construct the taxicab regular 2n-gons and gives a tool to solve special cases of the open problem posed in [2, p.32]. Besides, both the conditions of the taxicab regularity of Euclidean regular polygons and Euclidean regularity of taxicab regular polygons are given in terms of MT (2)-invariants. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
49. ALTERNATING COLOURINGS OF THE VERTICES OF A REGULAR POLYGON.
- Author
-
SINGH, SHIVANI and ZELENYUK, YULIYA
- Subjects
- *
COLOR , *MATHEMATICAL equivalence , *ROTATIONAL motion - Abstract
Let $n,r,k\in \mathbb{N}$. An $r$ -colouring of the vertices of a regular $n$ -gon is any mapping $\unicode[STIX]{x1D712}:\mathbb{Z}_{n}\rightarrow \{1,2,\ldots ,r\}$. Two colourings are equivalent if one of them can be obtained from another by a rotation of the polygon. An $r$ -ary necklace of length $n$ is an equivalence class of $r$ -colourings of $\mathbb{Z}_{n}$. We say that a colouring is $k$ -alternating if all $k$ consecutive vertices have pairwise distinct colours. We compute the smallest number $r$ for which there exists a $k$ -alternating $r$ -colouring of $\mathbb{Z}_{n}$ and we count, for any $r$ , 2-alternating $r$ -colourings of $\mathbb{Z}_{n}$ and 2-alternating $r$ -ary necklaces of length $n$. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
50. An improved upper bound on dilation of regular polygons.
- Author
-
Sattari, Sattar and Izadi, Mohammad
- Subjects
- *
POLYGONS , *POINT set theory , *MATHEMATICAL proofs , *TRIANGULATION - Abstract
Dilation of a set of points on the plane is the lowest possible dilation of a plane spanner on the point set. We show that dilation of vertices of any regular polygon is less than 1.4482. We introduce a method for constructing a triangulation of a regular polygon and prove this bound on its dilation. The upper bound on dilation is shown using mathematical proofs and experimental results. The new upper bound improves the previously known bound of 1.48454. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
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