Vertices Of Regular Polygon Research Articles

Задача о беспорядках

The problem of characterizing symmetric necklaces as disorders is considered. Such an idea is an alternative approach to the one using Poya's enumeration theorem, including Burnside's lemma. The relationship between partitions of numbers and permutation types is shown. Disorder types are associated with necklace types considered as permutations in which no element remains in place, as well as with disorders with fixed elements. The distributions of disorders by types for necklaces up to the seventh degree are studied. The number and enumeration of disorders of each type is carried out, and their evenness is established. Particular attention is paid to the study of the symmetric necklaces properties. Classes of symmetry and asymmetry are enumerated for each necklace type considered. At the same time, the concepts of chiral and achiral symmetry are introduced as varieties of axial symmetry. Relationships between the power of symmetry classes and the order of symmetry of necklaces are revealed. The concept of symmetry diagrams for necklaces is introduced. Diagrams could be applied for determining the properties of invariance of symmetric and asymmetric necklaces of a given length. The concept of congruence is used as an equivalence relation between necklaces. It allows implementing a geometric approach to the study of necklaces and the visualization of the results obtained. For this purpose, necklaces are associated with multiconnected non-oriented graphs, the vertices of which are the vertices of regular polygons. In this case, the number of vertices in the graph corresponds to the necklace’s length, and the number of connected components coincides with the number of bead colors. As an application of the results obtained, the possibility of studying the accent dynamics of poetic stanzas vertically using necklaces is considered. Empirical data on the use of various types of seven-line stanzas in poetic practice are presented. It is noted that approximately a quarter of the existing symmetric types of necklaces for various reasons do not find practical application.

Equilibrium, Regular Polygons, and Coulomb-Type Dynamics in Different Dimensions

The equation of motion in ℝ d of n generalized point charges interacting via the s -dimensional Coulomb potential, which contains for d = 2 a constant magnetic field, is considered. Planar exact solutions of the equation are found if either negative n − 1 > 2 charges and their masses are equal or n = 3 and the charges are different. They describe a motion of negative charges along identical orbits around the positive immobile charge at the origin in such a way that their coordinates coincide with vertices of regular polygons centered at the origin. Bounded solutions converging to an equilibrium in the infinite time for the considered equation without a magnetic field are also obtained. A condition permitting the existence of such solutions is proposed.

Complex Outliers of Hermitian Random Matrices

In this paper, we study the asymptotic behavior of the outliers of the sum a Hermitian random matrix and a finite rank matrix which is not necessarily Hermitian. We observe several possible convergence rates and outliers locating around their limits at the vertices of regular polygons as in Benaych-Georges and Rochet (Probab Theory Relat Fields, 2015), as well as possible correlations between outliers at macroscopic distance as in Knowles and Yin (Ann Probab 42(5):1980–2031, 2014) and Benaych-Georges and Rochet (2015). We also observe that a single spike can generate several outliers in the spectrum of the deformed model, as already noticed in Benaych-Georges and Nadakuditi (Adv Math 227(1):494–521, 2011) and Belinschi et al. (Outliers in the spectrum of large deformed unitarily invariant models 2012, arXiv:1207.5443v1 ). In the particular case where the perturbation matrix is Hermitian, our results complete the work of Benaych-Georges et al. (Electron J Probab 16(60):1621–1662, 2011), as we consider fluctuations of outliers lying in “holes” of the limit support, which happen to exhibit surprising correlations.

Intrinsic means on the circle: uniqueness, locus and asymptotics

This paper gives a comprehensive treatment of local uniqueness, asymptotics and numerics for intrinsic sample means on the circle. It turns out that local uniqueness as well as rates of convergence are governed by the distribution near the antipode. If the distribution is locally less than uniform there, we have local uniqueness and asymptotic normality with a square-root rate. With increased proximity to the uniform distribution the rate can be arbitrarily slow, and in the limit, local uniqueness is lost. Further, we give general distributional conditions, e.g., unimodality, that ensure global uniqueness. Along the way, we discover that sample means can occur only at the vertices of a regular polygon which allows to compute intrinsic sample means in linear time from sorted data. This algorithm is finally applied in a simulation study demonstrating the dependence of the convergence rates on the behavior of the density at the antipode.

Regular Tiling Based Flooding Scheme for Energy Efficiency in Wireless Ad Hoc Networks

Flooding is a fundamental operation in wireless ad hoc networks to propagate control messages in the entire network. The traditional flooding scheme generates excessive packet retransmissions, resource contention, and collisions since every node forwards the packet at least once. Several flooding schemes have been proposed to avoid these problems. However, these unnecessary forwarding nodes remain in these flooding schemes. In this paper, we propose a novel flooding scheme termed Regular Tiling (RT) Based Flooding to efficiently minimize the number of forwarding nodes. It employs a regular tiling method. RT Based Flooding, selects nodes that are located closest to vertices of regular polygon, using location information of 1-hop neighbor nodes, as forwarding nodes. We also provide ideal forwarding models to minimize the number of forwarding nodes and prove that our scheme approaches the ideal model through detailed analysis. Simulation shows that the performance of our scheme outperforms recent noteworthy flooding schemes in terms of the number of forwarding nodes, number of collisions, and delivery ratio.

Modes of symmetric composite defects in two-dimensional photonic crystals

We consider the modal fields and resonance frequencies of composite defects in two-dimensional photonic crystals (PCs). Using an asymptotic method based on Green's functions, we show that the coupling matrices for the composite defect can be represented as circulant or block-circulant matrices. Using the properties of these matrices, specifically that their eigenvectors are independent of the values of the matrix elements, we obtain modal properties such as dispersion relations, modal cutoff, degeneracy, and symmetry of the mode fields. Using our formulation, we investigate defects arranged on the vertices of regular polygons as well as PC ring resonators with defects arranged on the edges of polygons. Finally, we discuss the impact of band-edge degeneracies on composite-defect modes.

Point spectrum of the schrödinger operator with point interactions at the vertices of regular N-gons

We present a complete description of the point spectrum of the Laplace operator perturbed by point potentials concentrated at the vertices of regular polygons. We prove a criterion for the absence of points of the point spectrum of a singular perturbed positive self-adjoint operator with the property of cyclicity of defect vectors.

Two layer nested regular polygon central configurations in R3

In this Letter we study the necessary conditions for the masses of the two layer nested regular polygon central configurations in R 3 . If the masses at the corresponding vertices of the upper nested regular polygons have the same ratio, and the masses at the corresponding vertices of the lower nested regular polygons the same ratio, then masses at the vertices of each regular polygon are equal.

On the Stability of Homographic N-body Configurations

AbstractStability of expanding homographic configurations formed by an even number of bodies of equal mass and placed at the vertices of regular polygons and polyhedrons is studied. It is shown that for every configuration considered, there is always an unstable perturbation that divides the system into pairs of bodies.

Contrast Polarity Affects Performance after Figure-Ground Coding

Gilchrist et al (1997 Journal of Experimental Psychology: Human Perception and Performance23 464 – 480) proposed that some aspects of grouping are relatively insensitive to variations in contrast polarity between the elements to be grouped. We assessed the contrast-polarity sensitivity of grouping in a visual search experiment. Display elements were corner-brackets arranged at the vertices of regular polygons (see Donnelly et al, 1991 Journal of Experimental Psychology: Human Perception and Performance17 561 – 570), either aligned with polygon sides (strong-grouping condition), rotated through 20° (weak-grouping condition), or rotated through 180° (open condition). The background was grey; on same-contrast-polarity trials, elements were either all white or all black; on opposite-polarity trials, each element was white and black. The task was to detect a target element rotated 180° with respect to the others. With weak grouping present, opposite contrast polarity slowed reaction times dramatically: they were as slow as those to open displays. A second experiment in which display elements were pacmen showed that the contrast-polarity effect on performance is modulated by figure - ground relations: the dramatic effect of contrast polarity in the weak-grouping condition disappeared, presumably because search focused on the uniform grey illusory surface. These results suggest that grouping operates automatically to produce figure - ground coding of displays, but that contrast polarity differences within a figural surface affect the output of these codes to systems concerned with perceptual discriminations.

Minimal binary trees with a regular boundary: The case of skeletons with five endpoints

Locally minimal binary trees that span the vertices of regular polygons are studied. Their description is given in the dual language, that of diagonal triangulations of polygons. Diagonal triangulations of a special form, called skeletons, are considered. It is shown that planar binary trees dual to skeletons with five endpoints do not occur among locally minimal binary trees that span the vertices of regular polygons.

Invariant tests for uniformity on the vertices of a regular polygon

Beran (1968) and Watson (1974) indicated how to construct optimal invariant tests for local and distant alternatives for distributions on the vertices of regular polygons. Here we give the details of tests which are locally optimal against a linear trend, a single and a double peak in the alternative probabilities. These tests are useful for testing the equality of multinomial probabilities when the labels on the cells can be cyclically permuted without changing the problem e.g. the number of births in the 24 hours of the day since the time origin is biologically arbitrary.

New Method for Calculating Molecular Orbitals with Application to Cyclic Systems

We consider the molecular orbital problem, i.e., the quantum-mechanical problem of a particle (electron) bound to a configuration of $N$ potentials, when the potentials may overlap. We show why, when the individual potentials are spherically symmetric, it is advantageous to write the wave function $\ensuremath{\Psi}$ in the many-center (LCAO) form, $\ensuremath{\Psi}={\ensuremath{\psi}}^{(1)}({\mathrm{r}}_{1})+\ensuremath{\cdots}{\ensuremath{\psi}}^{(N)}({\mathrm{r}}_{N})$, where ${\mathrm{r}}_{i}$ refers to a coordinate system associated with the $i\mathrm{th}$ potential. The main point of this paper is to show that it is not only advantageous to use this form, but that it is also practical to directly determine the "atomic orbitals" ${\ensuremath{\psi}}^{(1)}\ensuremath{\cdots}{\ensuremath{\psi}}^{(N)}$. To show this we define the ${\ensuremath{\psi}}^{(i)}({\mathrm{r}}_{i})$ in a natural way and then use the Schr\"odinger equation to get a set of coupled integral equations for these functions and to get an analogous set for ${\ensuremath{\varphi}}^{(i)}(\ensuremath{\lambda})$, their Fourier transforms. We expand the ${\ensuremath{\varphi}}^{(i)}(\ensuremath{\lambda})$ in partial waves and make it plausible that the resultant sets of equations can be truncated and that frequently only a small number of partial waves need be retained, so that the equations are practical to solve.To show the spirit of the method we discuss, as a simple application, the ground state of cyclic systems, i.e., of potentials arranged at the vertices of regular polygons. The truncation of the equations then gives an approximate equation for the "$S$-wave part" of the orbitals and an expression which enables us to estimate the higher order partial waves from this $S$-wave part. We present methods for approximately solving the $S$-wave equation, which equation has the advantage of strongly resembling the corresponding one for a single potential, except for a "structure factor" which depends on the configuration. These methods include a variational method which does not involve multicenter integrals, a means of reducing the problem to that of solving the Schr\"odinger equation for an effective potential, and a method involving screened nuclear charge. For Coulomb potentials and for $N=2, 3, 4$ ($N=2$ corresponding to the hydrogen molecule ion) we calculate the $S$-wave part of the orbitals using the latter method; in these calculations arbitrary $N$ is not essentially more difficult than $N=2$.As a further application we use the second of the equations mentioned above to calculate, for $N=2$, the $P$-wave or "polarization" terms in the orbitals. We get good agreement for the form of this part of the wave function with the best variational calculation of this kind, that of Dickinson.Finally, we point out that these two applications are the first steps in an iterative procedure which seems to be promising more generally. Namely, starting from an approximation to an orbital which involves, say, one or more dominant partial waves, it would now appear that one can both calculate the other partial-wave components and correct the initial approximation, and so by iteration arrive at the correct form of the orbital.

Symmetrical Antenna Arrays

A relatively simple method is presented for calculating the impedance properties of antenna arrays consisting of n identical radiators oriented at the vertices of regular polygons. All antennas are required to carry currents of equal magnitude, but not necessarily of the same phase. However, the choice of phase angle must be such that the required electrical symmetry of the array is unimpaired. The case of a circular array with a central radiator is discussed. Immediate practical applications of the theory include: (a) Determination of the driving-point impedance of the primary radiator in a corner-reflector antenna. (b) Calculation of the power lost through radiation, or conversely the signal pickup, of certain multiple-wire transmission lines, when operated in a resonant condition. (c) Analyses of problems involving symmetrically disposed fixed antennas for direction-finding purposes. (d) Ascertaining the effectiveness of circular arrays for transmission. Several numerical illustrations of the theory are given.