scholarly journals Determinants of rational knots

2009 ◽  
Vol Vol. 11 no. 2 (Combinatorics) ◽  
Author(s):  
Louis H. Kauffman ◽  
Pedro Lopes
Keyword(s):  
Odd Order ◽  
International Audience ◽  
Even Order ◽  
Rational Knots ◽  
Increasing Sequences

Combinatorics International audience We study the Fox coloring invariants of rational knots. We express the propagation of the colors down the twists of these knots and ultimately the determinant of them with the help of finite increasing sequences whose terms of even order are even and whose terms of odd order are odd.

scholarly journals Mutual information for low-rank even-order symmetric tensor estimation

2020 ◽  
Author(s):  
Clément Luneau ◽  
Jean Barbier ◽  
Nicolas Macris
Keyword(s):  
Mutual Information ◽  
Finite Rank ◽  
Interpolation Method ◽  
Symmetric Tensor ◽  
Low Rank ◽  
Tensor Factorization ◽  
Adaptive Interpolation ◽  
Odd Order ◽  
Rank One ◽  
Even Order

Abstract We consider a statistical model for finite-rank symmetric tensor factorization and prove a single-letter variational expression for its asymptotic mutual information when the tensor is of even order. The proof applies the adaptive interpolation method originally invented for rank-one factorization. Here we show how to extend the adaptive interpolation to finite-rank and even-order tensors. This requires new non-trivial ideas with respect to the current analysis in the literature. We also underline where the proof falls short when dealing with odd-order tensors.

scholarly journals The characterization of elliptic curves over finite fields

1988 ◽  
Vol 45 (2) ◽  
pp. 275-286 ◽  
Author(s):  
J. W. P. Hirschfeld ◽  
J. F. Voloch
Keyword(s):  
Elliptic Curves ◽  
Finite Fields ◽  
Sufficient Conditions ◽  
Maximum Size ◽  
Desarguesian Plane ◽  
Cubic Curve ◽  
Odd Order ◽  
Even Order ◽  
Curves Over Finite Fields

AbstractIn a finite Desarguesian plane of odd order, it was shown by Segre thirty years ago that a set of maximum size with at most two points on a line is a conic. Here, in a plane of odd or even order, sufficient conditions are given for a set with at most three points on a line to be a cubic curve. The case of an elliptic curve is of particular interest.

Chlorite interstratified with a 7 Å mineral: an example from offshore Norway and possible implications for the interpretation of the composition of diagenetic chlorites

Clay Minerals ◽  
1992 ◽  
Vol 27 (4) ◽  
pp. 475-486 ◽  
Author(s):  
S. Hillier ◽  
B. Velde
Keyword(s):  
Alternative Interpretation ◽  
High Porosity ◽  
X Ray Diffraction ◽  
Odd Order ◽  
Even Order ◽  
Diagenetic Minerals ◽  
Structural Formulae ◽  
Xrd Patterns ◽  
Diagenetic History ◽  
Pore Lining

AbstractX-ray diffraction (XRD) patterns of a pore-lining diagenetic chlorite (14 Å) from a reservoir sandstone, offshore Norway, show broad odd-order and sharp even-order basal reflections indicating that it contains 7 Å layers. Using NEWMOD, simulated XRD patterns with 15% 7 Å serpentine layers and a maximum crystallite thickness of 30 layers match the natural mineral well. Microprobe analyses of the 7 Å-14 Å mineral indicate that it is Fe-rich and aluminous suggesting that it is interstratified berthierine-chamosite. Apparent octahedral vacancies, however, suggest a significant dioctahedral component, and an alternative interpretation is interstratified kaolinite-chlorite. Indeed, chemical analyses of the mineral suggest a mixture of chlorite with 15% kaolinite, precisely the proportion of 7 Å layers indicated by XRD. Two other examples from the literature, previously identified as diagenetic chlorite, are probably also 7 Å-14 Å interstratified minerals, and the proportion of 7 Å layers indicated by XRD is also correlated with their structural formulae, if the 7 Å layers are, in fact, kaolinitic. This type of interstratification could explain why Fe-rich diagenetic chlorites appear to be compositionally distinct from metamorphic chlorites. The structure and chemistry of the Norwegian chlorite tend to support the idea that pore-lining chlorites form early in the diagenetic history, inhibiting the precipitation of later diagenetic minerals, and hence preserving abnormally high porosity at greater depths.

Some measurements of multi-point time correlations in grid turbulence

1970 ◽  
Vol 41 (1) ◽  
pp. 169-178 ◽  
Author(s):  
C. W. Van Atta ◽  
T. T. Yeh
Keyword(s):  
Fourth Order ◽  
High Speed ◽  
Longitudinal Velocity ◽  
Joint Probability ◽  
Grid Turbulence ◽  
Odd Order ◽  
Probability Densities ◽  
Even Order ◽  
Point Correlation ◽  
Non Gaussian

Three-point odd-order correlations and four-point even-order correlations of the longitudinal velocity fluctuations in grid-generated turbulence have been measured using linearized hot-wire anemometry, digital sampling, and a high-speed digital computer. The measured correlations are compared with relations between higher-order correlations corresponding to non-Gaussian Gram-Charlier joint probability densities for three and four variables. The fourth-order, three-point Gram-Charlier distribution accurately describes the relation between measured odd-order three-point correlations. The measured fourth-order even-order correlations may be accurately predicted from the two-point correlation using Millionshtchikov's joint-Gaussian hypothesis, except for small values of the separations. The disagreement at small separations cannot be reduced through use of the Gram-Charlier approximation.

scholarly journals Maximal increasing sequences in fillings of almost-moon polyominoes

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Svetlana Poznanović ◽  
Catherine H. Yan
Keyword(s):  
Pigeonhole Principle ◽  
Fixed Size ◽  
Jeu De Taquin ◽  
International Audience ◽  
Increasing Sequences

International audience It was proved by Rubey that the number of fillings with zeros and ones of a given moon polyomino thatdo not contain a northeast chain of a fixed size depends only on the set of column lengths of the polyomino. Rubey’sproof uses an adaption of jeu de taquin and promotion for arbitrary fillings of moon polyominoes and deduces theresult for 01-fillings via a variation of the pigeonhole principle. In this paper we present the first completely bijectiveproof of this result by considering fillings of almost-moon polyominoes, which are moon polyominoes after removingone of the rows. More precisely, we construct a simple bijection which preserves the size of the largest northeast chainof the fillings when two adjacent rows of the polyomino are exchanged. This bijection also preserves the column sumof the fillings. In addition, we also present a simple bijection that preserves the size of the largest northeast chains, therow sum and the column sum if every row of the filling has at most one 1. Thereby, we not only provide a bijectiveproof of Rubey’s result but also two refinements of it. Rubey a montré que le nombre de remplissages d’un polyomino lunaire donné par des zéros et des uns quine contiennent pas de chaîne nord-est d’une taille fixée ne dépend que de l’ensemble des longueurs des colonnesdu polyomino. La preuve de Rubey utilise une adaptation du jeu de taquin et de la promotion sur des remplissagesarbitraires de polyominos lunaires et déduit le résultat pour les remplissages 0/1 par inclusion-exclusion. Dans cetarticle, nous présentons la première preuve bijective de ce résultat en considérant des remplissages de polyominospresque lunaires, qui sont des polyominos lunaires dont on a supprimé une ligne. Plus précisément, nous construisonsune bijection simple qui préserve la taille de la plus longue chaîne nord-est des remplissages lorsque deux lignesadjacentes du polyomino sont échangées. Cette bijection préserve aussi la somme des colonnes des remplissages. Enoutre, nous présentons aussi une bijection simple qui préserve la taille de la plus longue chaîne nord-est, la sommedes lignes et la somme des colonnes si chaque ligne du remplissage contient au plus un 1. Nous fournissons donc nonseulement une preuve bijective du résultat de Rubey, mais aussi deux raffinements de celui-ci.

scholarly journals Influence of Geometric Error of Rollers on Rotational Accuracy of Cylindrical Roller Bearings

2019 ◽  
Vol 37 (4) ◽  
pp. 774-784
Author(s):  
Yongjian Yu ◽  
Guoding Chen ◽  
Jishun Li ◽  
Yujun Xue
Keyword(s):  
Prediction Method ◽  
Geometric Error ◽  
Geometric Errors ◽  
Motion Error ◽  
Roundness Error ◽  
Roller Bearings ◽  
Odd Order ◽  
Even Order ◽  
The Relationship ◽  
Cylindrical Roller

As the rotation of roller bearings is carried out under geometrical constraint of the inner ring, outer ring and multiple rollers, the motion error of the bearing should also be resulted from geometric errors of bearing parts. Therefore, it is crucial to establish the relationship between geometric errors of bearing components and motion error of assembled bearing, which contributes to improve rotational accuracy of assembled bearing in the design and machining of the bearing. For this purpose, considering roundness error and dimension error of the inner raceway, the outer raceway and rollers, a prediction method for rotational accuracy of cylindrical roller bearings is proposed, and the correctness of the proposed prediction method is verified by experimental results. The influences of roller's geometric error distribution, roller's roundness error and the number of rollers on the runout value of inner ring are investigated. The results show that, the roller arrangement with different geometric errors has a significant impact on rotational accuracy of cylindrical roller bearings. The rotational accuracy could be improved remarkably when multiple rollers with different dimension error are distributed alternately according to the size error. Even-order roundness error of rollers has a significant effect on the rotational accuracy, and the decrease level depends on the orders of roundness errors of bearing parts and the number of rollers. But odd-order roundness error of rollers has almost no effect on the rotational accuracy. The rotational accuracy of assembled bearing would be significantly improved or decreased when even order harmonic of rollers and the number of rollers satisfy specific relationships. The greater the order of roundness error of the rollers, the more severe the influence of the roller number on rotational accuracy of assembled bearing. The rotational accuracy can not be always improved with the increase of the number of rollers.

scholarly journals Heron’s cubic root iteration method with a new approach

2021 ◽  
Author(s):  
S. Gadtia ◽  
S. K. Padhan
Keyword(s):  
Iterative Method ◽  
Iteration Method ◽  
Interpolation Formula ◽  
Alternative Proof ◽  
New Approach ◽  
Iteration Formula ◽  
Odd Order ◽  
Even Order ◽  
Lagrange’S Method

Abstract Heron’s cubic root iteration formula conjectured by Wertheim is proved and extended for any odd order roots. Some possible proofs are suggested for the roots of even order. An alternative proof of Heron’s general cubic root iterative method is explained. Further, Lagrange’s interpolation formula for nth root of a number is studied and found that Al-Samawal’s and Lagrange’s method are equivalent. Again, counterexamples are discussed to justify the effectiveness of the present investigations.

scholarly journals Toy model of harmonic and sum frequency generation in 2D dielectric nanostructures

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Jie Xu ◽  
Vassili Savinov ◽  
Eric Plum
Keyword(s):  
Nonlinear Response ◽  
Dipole Moments ◽  
Optical Nonlinearities ◽  
Coulomb Interactions ◽  
Sum Frequency ◽  
Odd Order ◽  
Area And Perimeter ◽  
Even Order ◽  
Dielectric Nanoparticle ◽  
The Individual

AbstractOptical nonlinearities of matter are often associated with the response of individual atoms. Here, using a toy oscillator model, we show that in the confined geometry of a two-dimensional dielectric nanoparticle a collective nonlinear response of the atomic array can arise from the Coulomb interactions of the bound optical electrons, even if the individual atoms exhibit no nonlinearity. We determine the multipole contributions to the nonlinear response of nanoparticles and demonstrate that the odd order and even order nonlinear electric dipole moments scale with the area and perimeter of the nanoparticle, respectively.

Structure functions of temperature fluctuations in turbulent shear flows

1978 ◽  
Vol 84 (3) ◽  
pp. 561-580 ◽  
Author(s):  
R. A. Antonia ◽  
C. W. Van Atta
Keyword(s):  
Temperature Fluctuations ◽  
Structure Functions ◽  
Inertial Subrange ◽  
Turbulent Shear ◽  
Temperature Structure ◽  
Odd Order ◽  
Even Order ◽  
Inner Region ◽  
Heated Jet ◽  
Reynolds Number Dependence

Structure functions of turbulent temperature and velocity fluctuations are measured both for the atmosphere, in the surface layer over land, and for the laboratory, in the inner region of a thermal boundary layer and on the axis of a heated jet. Even-order temperature structure functions, up to order eight, generally compare favourably with the analysis of Antonia & Van Atta over the inertial subrange. The Reynolds number dependence of these structure functions, as predicted by the analysis, is in qualitative agreement with the measured data. Odd-order temperature structure functions depart significantly from the isotropic value of zero, particularly at large time delays. This departure is reasonably well predicted, over the inertial subrange, by postulating a simple ramp model for the temperature fluctuations. Assumptions involved in this model are directly tested by measurements in the heated jet. The ramp structure does not seriously affect either the even-order temperature structure functions or the mixed velocity-temperature functions, which include even-order moments of the temperature difference.

INVESTIGATION OF THE GEOMETRY OF THE D-PARTITION OF ONE-DIMENSIONAL PLANE OF PARAMETER OF THE CHARACTERISTIC EQUATION OF A CONTINUOUS SYSTEM

2021 ◽  
Vol 4 ◽  
pp. 125-136
Author(s):  
Leonid Movchan ◽  
◽  
Sergey Movchan ◽  
Keyword(s):  
Imaginary Part ◽  
Characteristic Equation ◽  
Stability Region ◽  
Practical Interest ◽  
Stability Regions ◽  
Odd Order ◽  
Stability Intervals ◽  
Even Order ◽  
The Stability ◽  
Two Parameters

The paper considers two types of boundaries of the D-partition in the plane of one parameter of linear continuous systems given by the characteristic equation with real coefficients. The number of segments and intervals of stability of the X-partition curve is estimated. The maximum number of stability intervals is determined for different orders of polynomials of the equation of the boundary of the D-partition of the first kind (even order, odd order, one of even order, and the other of odd order). It is proved that the maximum number of stability intervals of a one-parameter family is different for all cases and depends on the ratio of the degrees of the polynomials of the equation of the D-partition curve. The derivative of the imaginary part of the expression of the investigated parameter at the initial point of the D-partition curve is obtained in an analytical form, the sign of which depends on the ratio of the coefficients of the characteristic equation and establishes the stability of the first interval of the real axis of the parameter plane. It is shown that for another type of the boundary of the D-partition in the plane of one parameter, there is only one interval of stability, the location of which, as for the previous type of the boundary of the stability region (BSR), is determined by the sign of the first derivative of the imaginary part of the expression of the parameter under study. Consider an example that illustrates the effectiveness of the proposed approach for constructing a BSR in a space of two parameters without using «Neimark hatching» and constructing special lines. In this case, a machine implementation of the construction of the stability region is provided. Considering that the problem of constructing the boundary of the stability region in the plane of two parameters is reduced to the problem of determining the BSR in the plane of one parameter, then the given estimates of the maximum number of stability regions in the plane of one parameter allow us to conclude about the number of maximum stability regions in the plane of two parameters, which are of practical interest. In this case, one of the parameters can enter nonlinearly into the coefficients of the characteristic equation.

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