The Geometric Theory | ScienceGate

GEOMETRIC THEORY OF MULTIDIMENSIONAL INTERPOLATION

Automation and modeling in design and management of ◽

10.30987/2658-6436-2020-1-9-16 ◽

2020 ◽

Vol 2020 (1) ◽

pp. 9-16

Author(s):

Evgeniy Konopatskiy

Keyword(s):

Response Function ◽

Algebraic Curves ◽

Geometric Theory ◽

Analytical Description ◽

Geometric Interpretation ◽

Affine Geometry ◽

Mathematical Apparatus ◽

Multidimensional Interpolation ◽

Pass Through

The paper presents a geometric theory of multidimensional interpolation based on invariants of affine geometry. The analytical description of geometric interpolants is performed within the framework of the mathematical apparatus BN-calculation using algebraic curves that pass through preset points. A geometric interpretation of the interaction of parameters, factors, and the response function is presented, which makes it possible to generalize the geometric theory of multidimensional interpolation in the direction of increasing the dimension of space. The conceptual principles of forming the tree of the geometric interpolant model as a geometric basis for modeling multi-factor processes and phenomena are described.

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Examples of theories of presheaf type

10.1093/oso/9780198758914.003.0011 ◽

2018 ◽

Author(s):

Olivia Caramello

Keyword(s):

Finite Groups ◽

Linear Independence ◽

Geometric Theory ◽

Vector Spaces ◽

Linear Orders ◽

Locally Finite ◽

Strong Unit ◽

Locally Finite Groups ◽

Simplicial Sets ◽

Separable Extensions

This chapter discusses several classical as well as new examples of theories of presheaf type from the perspective of the theory developed in the previous chapters. The known examples of theories of presheaf type that are revisited in the course of the chapter include the theory of intervals (classified by the topos of simplicial sets), the theory of linear orders, the theory of Diers fields, the theory of abstract circles (classified by the topos of cyclic sets) and the geometric theory of finite sets. The new examples include the theory of algebraic (or separable) extensions of a given field, the theory of locally finite groups, the theory of vector spaces with linear independence predicates and the theory of lattice-ordered abelian groups with strong unit.

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Flat functors and classifying toposes

10.1093/oso/9780198758914.003.0007 ◽

2018 ◽

Author(s):

Olivia Caramello

Keyword(s):

General Theory ◽

Geometric Theory ◽

Small Category ◽

Independent Interest ◽

Yoneda Lemma ◽

Geometric Morphisms ◽

The Way

This chapter develops a general theory of extensions of flat functors along geometric morphisms of toposes; the attention is focused in particular on geometric morphisms between presheaf toposes induced by embeddings of categories and on geometric morphisms to the classifying topos of a geometric theory induced by a small category of set-based models of the latter. A number of general results of independent interest are established on the way, including developments on colimits of internal diagrams in toposes and a way of representing flat functors by using a suitable internalized version of the Yoneda lemma. These general results will be instrumental for establishing in Chapter 6 the main theorem characterizing the class of geometric theories classified by a presheaf topos and for applying it.

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Joseph Fourier 250thBirthday: Modern Fourier Analysis and Fourier Heat Equation in Information Sciences for the XXIst Century

Entropy ◽

10.3390/e21030250 ◽

2019 ◽

Vol 21 (3) ◽

pp. 250

Author(s):

Frédéric Barbaresco ◽

Jean-Pierre Gazeau

Keyword(s):

Fourier Analysis ◽

Heat Equation ◽

Harmonic Analysis ◽

Lie Groups ◽

Coherent States ◽

Geometric Theory ◽

Plancherel Formula ◽

Group Representations ◽

Modern Development ◽

Xxist Century

For the 250th birthday of Joseph Fourier, born in 1768 at Auxerre in France, this MDPI special issue will explore modern topics related to Fourier analysis and Fourier Heat Equation. Fourier analysis, named after Joseph Fourier, addresses classically commutative harmonic analysis. The modern development of Fourier analysis during XXth century has explored the generalization of Fourier and Fourier-Plancherel formula for non-commutative harmonic analysis, applied to locally compact non-Abelian groups. In parallel, the theory of coherent states and wavelets has been generalized over Lie groups (by associating coherent states to group representations that are square integrable over a homogeneous space). The name of Joseph Fourier is also inseparable from the study of mathematics of heat. Modern research on Heat equation explores geometric extension of classical diffusion equation on Riemannian, sub-Riemannian manifolds, and Lie groups. The heat equation for a general volume form that not necessarily coincides with the Riemannian one is useful in sub-Riemannian geometry, where a canonical volume only exists in certain cases. A new geometric theory of heat is emerging by applying geometric mechanics tools extended for statistical mechanics, for example, the Lie groups thermodynamics.

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Numerical simulation and experimental performance research of cylindrical vane pump

Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science ◽

10.1177/09544062211020037 ◽

2021 ◽

pp. 095440622110200

Author(s):

Yiqi Cheng ◽

Xinhua Wang ◽

Waheed Ur Rehman ◽

Tao Sun ◽

Hasan Shahzad ◽

...

Keyword(s):

Numerical Analysis ◽

Rotational Velocity ◽

Geometric Theory ◽

Mechanical Efficiency ◽

Operating Conditions ◽

Experimental Results ◽

Field Analysis ◽

Total Efficiency ◽

Three Dimensional Model ◽

Vane Pump

This study presents a novel cylindrical vane pump based on the traditional working principle. The efficiency of the cylindrical vane pump was verified by experimental validation and numerical analysis. Numerical analysis, such as kinematics analysis, was performed in Pro/Mechanism and unsteady flow-field analysis was performed using ANSYS FLUENT. The stator surface equations were derived using the geometric theory of the applied spatial triangulation function. A three-dimensional model of the cylindrical vane pump was established with the help of MATLAB and Pro/E. The kinematic analysis helped in developing kinematic equations for cylindrical vane pumps and proved the effectiveness of the structural design. The maximum inaccuracy error of the computational fluid dynamics (CFD) model was 5.7% compared with the experimental results, and the CFD results show that the structure of the pump was reasonable. An experimental test bench was developed, and the results were in excellent agreement with the numerical results of CFD. The experimental results show that the cylindrical vane pump satisfied the three-element design of a positive-displacement pump and the trend of changes in efficiency was the same for all types of efficiency under different operating conditions. Furthermore, the volumetric efficiency presented a nonlinear positive correlation with increased rotational velocity, the mechanical efficiency showed a nonlinear negative correlation, and the total efficiency first increased and then decreased. When the rotational velocity was 1.33[Formula: see text] and the discharge pressure was 0.68[Formula: see text], the total efficiency reached its maximum value.

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