Polytopic Prismahedrons

2018 ◽  
pp. 103-144
Keyword(s):  
Chemical Interaction ◽  
Independent Interest ◽  
Geometric Properties ◽  
Simplicial Polytopes ◽  
Geometric Elements

The structure of polytopes - polytopic prismahedrons, which are products of polytopes of lower dimensionality, is investigated. The products of polytopes do not belong to the well-studied class of simplicial polytopes, and therefore their investigations are of independent interest. Analytical dependencies characterizing the structure of the product of polytopes are obtained as a function of the structures of polytope factors. Images of a number of specific polytopic prismahedrons are obtained, tables of structures of polytopic prismahedrons are compiled, depending on the types of polytopes of the factors. The geometric properties of the boundary complexes of polytopic prismehedrons are investigated. Polytopic prismahedrons can be considered as a result of the chemical interaction of molecules, which, from among which there is a polytope of a certain dimension. The possibility of the multivaluedness of the incidence coefficients of geometric elements of polytopic prisms is revealed. It is shown that polytopic prismahedra, due to their nature, as products of polytopes, ensure the filling of n-dimensional spaces during their translation, creating a structure similar to the structure of quasicrystals.

Polytopic Prismahedrons

2018 ◽  
pp. 105-140
Keyword(s):  
Chemical Interaction ◽  
Independent Interest ◽  
Simplicial Polytopes

The structure of polytopes—polytopic prismahedrons, which are products of polytopes of lower dimensionality—is investigated. The products of polytopes do not belong to the well-studied class of simplicial polytopes, and therefore their investigations are of independent interest. Analytical dependencies characterizing the structure of the product of polytopes are obtained as a function of the structures of polytope factors. Images of a number of specific polytopic prismahedrons are obtained, tables of structures of polytopic prismahedrons are compiled, depending on the types of polytopes of the factors. Polytopic prismahedrons can be considered as a result of the chemical interaction of molecules, which, from among which there is a polytope of a certain dimension.

The Geometric Structure of the Product of Polytopes

2021 ◽  
pp. 111-143
Keyword(s):  
Geometric Structure ◽  
Independent Interest ◽  
Higher Dimension ◽  
Geometric Properties ◽  
Simplicial Polytopes

The structure of polytopes of higher dimension (polytopic prismahedrons), which are products of polytopes of lower dimensionality, is investigated. The products of polytopes do not belong to the well-studied class of simplicial polytopes, and therefore, their investigations are of independent interest. Analytical dependencies characterizing the structure of the product of polytopes are obtained as a function of the structures of polytope factors. Images of a number of specific polytopic prismahedrons are obtained, and tables of structures of polytopic prismahedrons are compiled depending on the types of polytopes of the factors. The geometric properties of the boundary complexes of polytopic prismahedrons are investigated.

Dynamic Geometry

1947 ◽  
Vol 40 (4) ◽  
pp. 151-157
Author(s):  
John F. Schacht ◽  
John J. Kinsella
Keyword(s):  
Dynamic Geometry ◽  
Geometric Properties ◽  
Tenth Grade ◽  
Geometric Elements ◽  
Flexible Models

In 1938 Lloyd wrote: “Traditionally, geometry has been studied in the tenth grade as a system of rigid, non-flexible elements. The author advances the thesis that all geometric diagrams should be assumed as flexible linkages. …” The writer felt that the use of motion made the geometric materials more lifelike and, hence, more interesting, He also believed that many geometric properties were elucidated more clearly by the flexible models than by similar static ones. Finally, he pointed out that the variance and invariance of certain geometric elements were more easily demonstrated by jointed rather than fixed devices.

scholarly journals Almost Simplicial Polytopes: The Lower and Upper Bound Theorems

2019 ◽  
Vol 72 (2) ◽  
pp. 537-556
Author(s):  
Eran Nevo ◽  
Guillermo Pineda-Villavicencio ◽  
Julien Ugon ◽  
David Yost
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Independent Interest ◽  
Upper Bound Theorem ◽  
Moment Curve ◽  
Simplicial Polytopes ◽  
Cyclic Polytopes ◽  
Bound Theorem ◽  
The Moment

AbstractWe study $n$-vertex $d$-dimensional polytopes with at most one nonsimplex facet with, say, $d+s$ vertices, called almost simplicial polytopes. We provide tight lower and upper bound theorems for these polytopes as functions of $d,n$, and $s$, thus generalizing the classical Lower Bound Theorem by Barnette and the Upper Bound Theorem by McMullen, which treat the case where $s=0$. We characterize the minimizers and provide examples of maximizers for any $d$. Our construction of maximizers is a generalization of cyclic polytopes, based on a suitable variation of the moment curve, and is of independent interest.

Use of the Magnetostatic Analogue In Electromagnetic Lens Engineering

1970 ◽  
Vol 28 ◽  
pp. 360-361
Author(s):  
John W. Coleman
Keyword(s):  
Boundary Conditions ◽  
High Performance ◽  
Force Fields ◽  
Optical Design ◽  
Direct Conversion ◽  
Design Engineering ◽  
Design Data ◽  
Scalar Potentials ◽  
Geometric Elements ◽  
At Will

In the design engineering of high performance electromagnetic lenses, the direct conversion of electron optical design data into drawings for reliable hardware is oftentimes difficult, especially in terms of how to mount parts to each other, how to tolerance dimensions, and how to specify finishes. An answer to this is in the use of magnetostatic analytics, corresponding to boundary conditions for the optical design. With such models, the magnetostatic force on a test pole along the axis may be examined, and in this way one may obtain priority listings for holding dimensions, relieving stresses, etc..The development of magnetostatic models most easily proceeds from the derivation of scalar potentials of separate geometric elements. These potentials can then be conbined at will because of the superposition characteristic of conservative force fields.

Criteria for evaluation of the coordinate measurements quality of parts geometric parameters at pilot-experimental production

2019 ◽  
pp. 3-8
Author(s):  
N.Yu. Bobrovskaya ◽  
M.F. Danilov
Keyword(s):  
Quality Indicators ◽  
Quality Management System ◽  
Technical Specification ◽  
Experimental Production ◽  
Technological Heredity ◽  
Coordinate Measurements ◽  
Reasonable Choice ◽  
The Stability ◽  
Geometric Elements

The criteria of the coordinate measurements quality at pilot-experimental production based on contemporary methods of quality management system and traditional methods of the measurements quality in Metrology are considered. As an additional criterion for quality of measurements, their duration is proposed. Analyzing the problem of assessing the quality of measurements, the authors pay particular attention to the role of technological heredity in the analysis of the sources of uncertainty of coordinate measurements, including not only the process of manufacturing the part, but all stages of the development of design and technological documentation. Along with such criteria as the degree of confidence in the results of measurements; the accuracy, convergence, reproducibility and speed of the results must take into account the correctness of technical specification, and such characteristics of the shape of the geometric elements to be controlled, such as flatness, roundness, cylindrical. It is noted that one of the main methods to reduce the uncertainty of coordinate measurements is to reduce the uncertainty in the initial data and measurement conditions, as well as to increase the stability of the tasks due to the reasonable choice of the basic geometric elements (measuring bases) of the part. A prerequisite for obtaining reliable quality indicators is a quantitative assessment of the conditions and organization of the measurement process. To plan and normalize the time of measurements, the authors propose to use analytical formulas, on the basis of which it is possible to perform quantitative analysis and optimization of quality indicators, including the speed of measurements.

A system of functions biorthogonal with the derivatives of Chebyshev second-kind polynomials of a complex variable

2020 ◽  
Vol 17 (2) ◽  
pp. 256-277
Author(s):  
Ol'ga Veselovska ◽  
Veronika Dostoina
Keyword(s):  
Complex Plane ◽  
Complex Variable ◽  
Analytic Functions ◽  
Combinatorial Identities ◽  
Independent Interest ◽  
Closed Curves ◽  
In Series ◽  
Derivatives Of

For the derivatives of Chebyshev second-kind polynomials of a complex vafiable, a system of functions biorthogonal with them on closed curves of the complex plane is constructed. Properties of these functions and the conditions of expansion of analytic functions in series in polynomials under consideration are established. The examples of such expansions are given. In addition, we obtain some combinatorial identities of independent interest.

Solubility Enhancement of Poorly Water-Soluble Antifungal Drug Acyclovir by Nanocrystal Formulation Approach

2018 ◽  
Vol 11 (2) ◽  
pp. 4036-4042
Author(s):  
Prakash Goudanavar ◽  
Ankit Acharya ◽  
Vinay C.H
Keyword(s):  
Chemical Interaction ◽  
Research Work ◽  
Antiviral Drug ◽  
In Vitro Release ◽  
Oral Route ◽  
Water Soluble ◽  
Precipitation Method ◽  
Dsc Studies ◽  
Ft Ir

Administration of an antiviral drug, acyclovir via the oral route leads to low and variable bioavailability (15-30%). Therefore, this research work was aimed to enhance bioavailability of acyclovir by nanocrystallization technique. The drug nanocrystals were prepared by anti-solvent precipitation method in which different stabilizers were used. The formed nanocrystals are subjected to biopharmaceutical characterization including solubility, particle size and in-vitro release. SEM studies showed nano-crystals were crystalline nature with sharp peaks. The formulated drug nanocrystals were found to be in the range of 600-900nm and formulations NC7 and NC8 showed marked improvement in dissolution velocity when compared to pure drug, thus providing greater bioavailability. FT-IR and DSC studies revealed the absence of any chemical interaction between drug and polymers used. 

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