scholarly journals On the finiteness of the number of elliptic fields with given degrees of S-units and periodic expansion of √f

2019 ◽  
Vol 488 (3) ◽  
pp. 237-242 ◽  
Author(s):  
V. P. Platonov ◽  
M. M. Petrunin ◽  
Yu. N. Shteinikov

For a field k of characteristic 0, up to a natural equivalence relation, it is proved that the number of nontrivial elliptic fields k(x)(f) with a periodic expansion of f ∈ k((x)), for which the corresponding elliptic curve contains a k-point of even order less or equal than 18 or k-point of odd order less or equal than 11, is finite. In case k is a quadratic extension of Q, all such fields are found.

Author(s):  
Clément Luneau ◽  
Jean Barbier ◽  
Nicolas Macris

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.


1999 ◽  
Vol 09 (09) ◽  
pp. 1803-1813 ◽  
Author(s):  
JÉRÔME E. LOS ◽  
ZBIGNIEW H. NITECKI

An automorphism of a graph is edge-transitive if it acts transitively on the set of geometric edges (components of the complement of the vertices), or equivalently, if there is no nontrivial invariant subgraph. Every such automorphism can be embedded as the restriction to an invariant spine of some orientation-preserving periodic homeomorphism of a punctured surface. We find all the edge-transitive graph automorphisms and for each, find a complete list (up to a natural equivalence relation) of the possible ways that it can be embedded in a periodic homeomorphism.


1989 ◽  
Vol 41 (5) ◽  
pp. 830-854 ◽  
Author(s):  
B. Banaschewski ◽  
A. Pultr

A natural approach to topology which emphasizes its geometric essence independent of the notion of points is given by the concept of frame (for instance [4], [8]). We consider this a good formalization of the intuitive perception of a space as given by the “places” of non-trivial extent with appropriate geometric relations between them. Viewed from this position, points are artefacts determined by collections of places which may in some sense by considered as collapsing or contracting; the precise meaning of the latter as well as possible notions of equivalence being largely arbitrary, one may indeed have different notions of point on the same “space”. Of course, the well-known notion of a point as a homomorphism into 2 evidently fits into this pattern by the familiar correspondence between these and the completely prime filters. For frames equipped with a diameter as considered in this paper, we introduce a natural alternative, the Cauchy points. These are the obvious counterparts, for metric locales, of equivalence classes of Cauchy sequences familiar from the classical description of completion of metric spaces: indeed they are decreasing sequences for which the diameters tend to zero, identified by a natural equivalence relation.


Author(s):  
J. W. P. Hirschfeld ◽  
J. F. Voloch

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.


Clay Minerals ◽  
1992 ◽  
Vol 27 (4) ◽  
pp. 475-486 ◽  
Author(s):  
S. Hillier ◽  
B. Velde

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.


2016 ◽  
Vol 81 (4) ◽  
pp. 1225-1254 ◽  
Author(s):  
RUSSELL MILLER ◽  
KENG MENG NG

AbstractWe introduce the notion of finitary computable reducibility on equivalence relations on the domainω. This is a weakening of the usual notion of computable reducibility, and we show it to be distinct in several ways. In particular, whereas no equivalence relation can be${\rm{\Pi }}_{n + 2}^0$-complete under computable reducibility, we show that, for everyn, there does exist a natural equivalence relation which is${\rm{\Pi }}_{n + 2}^0$-complete under finitary reducibility. We also show that our hierarchy of finitary reducibilities does not collapse, and illustrate how it sharpens certain known results. Along the way, we present several new results which use computable reducibility to establish the complexity of various naturally defined equivalence relations in the arithmetical hierarchy.


1970 ◽  
Vol 41 (1) ◽  
pp. 169-178 ◽  
Author(s):  
C. W. Van Atta ◽  
T. T. Yeh

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.


Author(s):  
Yongjian Yu ◽  
Guoding Chen ◽  
Jishun Li ◽  
Yujun Xue

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.


2021 ◽  
Author(s):  
S. Gadtia ◽  
S. K. Padhan

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.


2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Jie Xu ◽  
Vassili Savinov ◽  
Eric Plum

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.


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