Properties of Dupin Cyclide and Their Application. Part 3: The Problem of Apollonius

2015 ◽  
Vol 3 (4) ◽  
pp. 3-14 ◽  
Author(s):  
Сальков ◽  
Nikolay Sal'kov
Keyword(s):  
Fourth Order ◽  
Second Order ◽  
Practical Application ◽  
Practical Utility ◽  
Engineering Graphics ◽  
The Third ◽  
Focal Surface ◽  
Dupin Cyclide ◽  
Straight Lines

In the first part of work was addressed mainly the issue of properties under Dupin cyclide, and given some examples of their applications: three ways of solving the problem of Apollonius using only compass and ruler, using the identified properties of Dupin cyclid. The second part of work continued with consideration of the use of property under a lie of Dupin. It is determined that the focal surface of cyclid of Dupin is degenerated in the lines and represent curves of the second order. Here under a lie can be defined conic curve and a sphere whose center lies on the focal curve. Polyconic conformity these focal curves is revealed. The article show the formation of the surface of the fourth order on the basis of defocusing curves of the second order. In this issue of the journal the reader is invited to consider the practical application of properties under a lie of Dupin for example of well known problems with on-voltage. If the first part of the work was cited only three ways of solving the problem of Apollonius, in the third part the author considers other possible mates: as at zero the size of the radius of the circle and the demon is of course great. All decisions – both known and not really based on properties of Dupin cyclide. In the course of engineering graphics, introductory tests, as they say now, drawing on architectural faculties there are tasks, dedicated to the mating arcs of circles with straight lines, and circles passing through the points in various combinations. Therefore, the proposed practical application cannot be considered far-fetched – it is based on the practical utility of method.

STANDARD NOMOGRAPHIC FORMS FOR EQUATIONS IN THREE VARIABLES

1935 ◽  
Vol 12 (1) ◽  
pp. 14-40 ◽  
Author(s):  
F. M. Wood
Keyword(s):  
Unit Circle ◽  
Fourth Order ◽  
Third Order ◽  
Rectangular Hyperbola ◽  
Practical Utility ◽  
The Third ◽  
Cubic Curve ◽  
Final Adjustment ◽  
Straight Lines ◽  
Suitable Location

Equations of the third and fourth nomographic order in three variables have been dealt with and classified. Equations of the third order may be reduced to one of two standard forms, α + β + γ = 0 and α + βγ = 0, which give alignment charts composed of three straight lines. Equations of the fourth order may also be reduced to one of two standard forms, resulting in charts composed of (a) two straight lines and a curve, or (b) two scales on a conic, and the third on another curve. Transformations of these four standard forms are given which permit of rapid and easy adjustment of the position and length of the scales for any given example, resulting in a chart of practical utility. Although the underlying theory has been studied by other writers, notably Soreau and Clark, it has possibly never appeared before in such a neat form. On this account, and also because of the standard transformations, it is felt that this article is of particular value.Standard forms have also been developed for third order equations leading to charts composed of two scales on a conic and a third straight scale, and in conclusion a third type of chart, in which all three scales appear on a single cubic curve, has been standardized. The practical value of the last type is questionable, but the conic charts are of use since we may arbitrarily choose the unit circle, or the rectangular hyperbola, for our conic scales. Final adjustment forms which permit suitable location of the scales in particular examples have been obtained in every case.

Properties of Dupin Cyclide and Their Application. Part 4: Applications

2016 ◽  
Vol 4 (1) ◽  
pp. 21-33 ◽  
Author(s):  
Сальков ◽  
Nikolay Sal'kov
Keyword(s):  
Fourth Order ◽  
Second Order ◽  
Dense Packing ◽  
Descriptive Geometry ◽  
Practical Application ◽  
Surfaces Of Revolution ◽  
Special Cases ◽  
Dupin Cyclide ◽  
Different Diameters

In the first and second parts of the work there were considered mainly properties of Dupin cyclide, and given some examples of their application: three ways of solving the problem of Apollonius using only compass and ruler, using the identified properties of cyclide; it is determined that the focal surfaces of Dupin cyclid are degenerated in the lines and represent curves of the second order – herefrom Dupin cyclide can be defined by conic curve and a sphere whose center lies on the focal curve. Polyconic compliance of these focal curves is identified. The formation of the surface of the fourth order on the basis of defocusing curves of the second order is shown. In this issue of the journal the reader is invited to consider the practical application of Dupin cyclide’s properties. The proposed solution of Fermat’s classical task about the touch of the four spheres by the fifth with a ruler and compass, i.e., in the classical way. This task is the basis for the problem of dense packing. In the following there is an application of Dupin cyclide as a transition pipe element, providing smooth coupling of pipes of different diameters in places of their connections. Then the author provides the examples of Dupin cyclide’s application in the architecture as a shell coating. It is shown how to produce membranes from the same cyclide’s modules, from different modules of the same cyclide, from the modules of different cyclides, from cyclides with the inclusion of other surfaces, special cases of cyclides in the educational process. The practical application of the last problem found the place in descriptive geometry at the final geometrical education of architects in the "Construction of surfaces". Here such special cased of cyclides as conical and cylindrical surfaces of revolution.

Conservative Transport Schemes for Spherical Geodesic Grids: High-Order Flux Operators for ODE-Based Time Integration

2011 ◽  
Vol 139 (9) ◽  
pp. 2962-2975 ◽  
Author(s):  
William C. Skamarock ◽  
Almut Gassmann
Keyword(s):  
Fourth Order ◽  
Time Integration ◽  
Higher Order ◽  
Second Order ◽  
Time Step ◽  
Diffusion Term ◽  
The Third ◽  
Integration Schemes ◽  
Order Scheme ◽  
Runge Kutta

Higher-order finite-volume flux operators for transport algorithms used within Runge–Kutta time integration schemes on irregular Voronoi (hexagonal) meshes are proposed and tested. These operators are generalizations of third- and fourth-order operators currently used in atmospheric models employing regular, orthogonal rectangular meshes. Two-dimensional least squares fit polynomials are used to evaluate the higher-order spatial derivatives needed to cancel the leading-order truncation error terms of the standard second-order centered formulation. Positive definite or monotonic behavior is achieved by applying an appropriate limiter during the final Runge–Kutta stage within a given time step. The third- and fourth-order formulations are evaluated using standard transport tests on the sphere. The new schemes are more accurate and significantly more efficient than the standard second-order scheme and other schemes in the literature examined by the authors. The third-order formulation is equivalent to the fourth-order formulation plus an additional diffusion term that is proportional to the Courant number. An optimal value for the coefficient scaling this diffusion term is chosen based on qualitative evaluation of the test results. Improvements using the higher-order scheme in place of the traditional second-order centered approach are illustrated within 3D unstable baroclinic wave simulations produced using two global nonhydrostatic models employing spherical Voronoi meshes.

Study on Tractor-Semitrailer Shimmy of High Speed

2012 ◽  
Vol 433-440 ◽  
pp. 7420-7424
Author(s):  
Mei Zhi Xie ◽  
Bei Li ◽  
Chao Yi Wei ◽  
Feng Yan Yi
Keyword(s):  
Transfer Function ◽  
Dynamic Model ◽  
Fourth Order ◽  
High Speed ◽  
Damping Ratio ◽  
Second Order ◽  
Original Model ◽  
The Third

Through the establishment of dynamic model of tractor-semitrailer, calculate its transfer function. In the case of the third and fourth state of balanced coefficient is very small in the original model, the model of the tractor-semitrailer of fourth-order drop for second-order using MATLAB-modred () function and balreal () function, seek of relationship between damping ratio and the speed of tractor-semitrailer, The results show that: the tractor-semitrailer shimmy of high-speed is speed inversely proportional to the damping ratio, the higher the speed, the smaller the damping ratio, and thus more likely to shock and shimmy.

Applications of the MBPT in the localized representation the behaviour of the localization terms

1988 ◽  
Vol 53 (9) ◽  
pp. 2073-2081 ◽  
Author(s):  
Ede Kapuy ◽  
Zoltán Csépes ◽  
Ferencz Bartha ◽  
Ferencz Bogár ◽  
Cornelia Kozmutza
Keyword(s):  
Fourth Order ◽  
Correlation Energy ◽  
Computer Time ◽  
Second Order ◽  
Saturated Hydrocarbons ◽  
Local Effects ◽  
The Third ◽  
Non Local ◽  
Cyclic Polyenes

The behaviour of the localization corrections of the MBPT is investigated. It is shown that calculating the third and fourth order localization corrections we obtain sufficiently accurate results to the second order correlation energy both for cyclic polyenes and for saturated hydrocarbons. The evaluation of the localization diagrams does not require significant extra computer time. The extra computer time can be recovered if small non-local effects will be neglected.

scholarly journals The Light Curves of Double-Mode Cepheids: The CO Aur Case

1998 ◽  
Vol 185 ◽  
pp. 389-390
Author(s):  
E. Poretti ◽  
I. Pardo
Keyword(s):  
Frequency Analysis ◽  
Fourth Order ◽  
A Priori ◽  
Cross Coupling ◽  
Second Order ◽  
Light Curves ◽  
Third Order ◽  
The Third ◽  
Coupling Terms ◽  
Phase Differences

In two recent papers (Pardo & Poretti 1997; Poretti & Pardo 1997) we analyzed all the available photometry of galactic double-mode Cepheids (DMCs) with the aim of detecting in each case the importance of the harmonics and of the cross coupling terms. We found that no a priori fit can be reliably applied to the measurements of a DMC, but a careful frequency analysis must be done to evaluate the importance of each term. As a further application of this technique, we obtained very precise indications about the properties of the Fourier parameters. When discussing the generalized phase differences Gi,j we demonstrated that plotting them as a function of the order |i|+|j|, there are well-defined regions where they are confined: the second order terms have π < Gi,j < 3π/2; the third order terms have π/2 < Gi,j < π; the fourth order terms cluster around 2π.

scholarly journals Huanglongbing-induced Anatomical Changes in Citrus Fibrous Root Orders

HortScience ◽  
2018 ◽  
Vol 53 (6) ◽  
pp. 829-837 ◽  
Author(s):  
Naveen Kumar ◽  
Fnu Kiran ◽  
Ed Etxeberria
Keyword(s):  
Fourth Order ◽  
Second Order ◽  
Vascular Bundles ◽  
Root Mass ◽  
Starch Granules ◽  
Fibrous Root ◽  
Third Order ◽  
Cortical Cells ◽  
The Third ◽  
Order Root

Citrus fibrous roots are vital for absorption and transport of water, nutrients, and other endogenous plant growth regulators. Efficient functioning of these roots in Huanglongbing (HLB)-affected citrus trees is important for their survival. One-year-old ‘Valencia’ sweet orange (Citrus sinensis L. Osbeck) trees on Swingle citrumelo were budded with HLB-infected budwood to determine the HLB-induced pathological responses at the ultrastructural level of different fibrous root orders. The fibrous root mass was dissected into four root orders: fourth-order (attached to a thick rudimentary taproot), third-order (attached to the fourth-order root), and second-order roots (attached to the third-order root). We were not able to study the ultrastructure of the first-order (attached to the second-order root) roots in this study. Severe loss in fibrous root mass was observed within 1 year following HLB infection. All root orders displayed various degrees of HLB symptoms. The fourth-order roots comprised normal phloem and disintegrated phloem. Some vascular bundles had completely disintegrated phloem tissue, whereas others showed normal ultrastructure. The fourth-order roots were also deficient in starch granules compared with controls. The pattern of phloem disintegration was similar in the third- and second-order roots. A thick layer of necrotic phloem developed near cortical cells, while the rest of the phloem structure remained normal in the third- and second-order roots. Cortical cells of both third and second orders were enriched with starch granules; therefore, soluble carbohydrates are most likely not the limiting factor for root decline in these root orders. The xylem anatomy displayed heptarch to pentarch morphology in the various root orders. These observations confirmed that various root orders in the fibrous root system are distinct and exhibit varied pathological responses during HLB pathogenesis. We propose that photosynthates deprived fourth-order roots in conjunction with necrotic phloem promoted decline in all root orders and impaired the translocation process to aboveground plant parts.

scholarly journals Coefficient functionals for a class of bounded turning functions related to modified sigmoid function

2021 ◽  
Vol 7 (2) ◽  
pp. 3133-3149
Author(s):  
Muhammad Ghaffar Khan ◽  
◽  
Nak Eun Cho ◽  
Timilehin Gideon Shaba ◽  
Bakhtiar Ahmad ◽  
...  
Keyword(s):  
Present Article ◽  
Fourth Order ◽  
Symmetric Functions ◽  
Second Order ◽  
Sigmoid Function ◽  
Hankel Determinant ◽  
Hankel Determinants ◽  
The Third ◽  
Bounded Turning

<abstract><p>The main objective of the present article is to define the class of bounded turning functions associated with modified sigmoid function. Also we investigate and determine sharp results for the estimates of four initial coefficients, Fekete-Szegö functional, the second-order Hankel determinant, Zalcman conjucture and Krushkal inequality. Furthermore, we evaluate bounds of the third and fourth-order Hankel determinants for the class and for the 2-fold and 3-fold symmetric functions.</p></abstract>

The infra-red spectra of crystals

1960 ◽  
Vol 258 (1294) ◽  
pp. 377-401 ◽  
Keyword(s):  
Fourth Order ◽  
Second Order ◽  
Order Moment ◽  
Order Effects ◽  
Ionic Crystals ◽  
Third Order ◽  
Main Band ◽  
The Third ◽  
Cross Terms ◽  
Infra Red

The higher-order effects in the intrinsic infra-red absorption of crystals are investigated in a systematic way. In agreement with a previous paper which dealt with the static dielectric constant, it is found that in the case of ionic crystals the third- and fourth-order potential, the second- and the third-order dipole moment, and the cross-terms between the second-order moment and the third-order potential, all contribute terms of the same order to the infra-red spectrum. In the lowest approximation, the third-order moment and the fourth-order potential only affect the absorption in the immediate neighbourhood of the maximum and hence have little effect on the shape of the spectrum. The broadening of the main band is due mainly to the third-order potential, while the side bands may be caused by the second-order moment as well as by the third-order potential and by cross-terms between the two. But due to an internal field effect, in strongly ionic crystals a large second-order moment automatically leads to a large third-order potential; thus a large second-order moment may increase the width of the main band as well as the intensity of the side bands. Although the intrinsic infra-red absorption of valency crystals, such as diamond or germaniam, is due to the second-order moment only, nevertheless, there is a strong similarity between the expressions for the infra-red absorption of valency crystals and for the side-band absorption of ionic crystals. This similarity suggests that the spectra of all ionic crystals should exhibit a number of secondary maxima. The available experimental evidence does not seem sufficient to decide whether this suggestion is correct.

Children’s Recall of Approximations to English

1973 ◽  
Vol 16 (2) ◽  
pp. 201-212 ◽  
Author(s):  
Elizabeth Carrow ◽  
Michael Mauldin
Keyword(s):  
Language Development ◽  
Fourth Order ◽  
Third Order ◽  
Memory Processes ◽  
The Third ◽  
General Index ◽  
General Linguistic

As a general index of language development, the recall of first through fourth order approximations to English was examined in four, five, six, and seven year olds and adults. Data suggested that recall improved with age, and increases in approximation to English were accompanied by increases in recall for six and seven year olds and adults. Recall improved for four and five year olds through the third order but declined at the fourth. The latter finding was attributed to deficits in semantic structures and memory processes in four and five year olds. The former finding was interpreted as an index of the development of general linguistic processes.

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