scholarly journals On symmetry reduction and some classes of invariant solutions of the (1+3)-dimensional homogeneous Monge-Ampère equation

2021 ◽  
Vol 14 (3) ◽  
pp. 206-218
Author(s):  
Vasyl Fedorchuk ◽  
Volodymyr Fedorchuk
Keyword(s):  
Lie Algebra ◽  
Structural Properties ◽  
Symmetry Reduction ◽  
Two Dimensional ◽  
Invariant Solutions ◽  
Smooth Functions ◽  
3 Dimensional ◽  
Reduced Equations ◽  
Monge Ampere ◽  
The Relationship

We study the relationship between structural properties of the two-dimensional nonconjugate subalgebras of the same rank of the Lie algebra of the Poincaré group P(1,4) and the properties of reduced equations for the (1+3)-dimensional homogeneous Monge-Ampère equation. In this paper, we present some of the results obtained concerning symmetry reduction of the equation under investigation to identities. Some classes of the invariant solutions (with arbitrary smooth functions) are presented.

scholarly journals Building Rubiks Cube: A Function Of Production

2008 ◽  
Vol 12 (1) ◽  
pp. 25-32
Author(s):  
Jose Villacis Gonzalez
Keyword(s):  
The Other ◽  
Two Dimensional ◽  
3 Dimensional ◽  
Production Processes ◽  
Key Factor ◽  
Final Design ◽  
Dimensional Cube ◽  
The One ◽  
The Cost ◽  
The Relationship

The Rubiks cube is a special game and a very particular puzzle. The 3-dimensional cube is made up of six faces, or boundary sections, of the same size. Each face, or section, consists of several two dimensional square parts, or cubelets. Every cubelet has the same surface area, and each of the six faces has the same number of cubelets. Therefore, the cubes surface is entirely covered with isocubelets. The cubelets are painted in six different colours, and it is possible to create a design where each face shows only one colour. Such is the object of the game: to turn the cubelets and sections of the cube so that only one (different) colour shows on each one of the six faces. If one manages to master the puzzle, the cube will show six faces of the same size, each coloured differently. The cubelets and sections of the cube can be turned both horizontally and vertically in order to change colours while trying to determine the appropriate combination to complete the puzzle. This approach is linked to a particular function in microeconomics that deals with the relationship between two magnitudes: on the one hand, the moves needed to achieve the desired final design; and on the other hand, the cost linked to the required production processes. This analytical model must use combinatorial mathematics equipment because, after all, the key factor in solving the Rubiks cube is the way in which the cubelets and sections are arranged.

scholarly journals On the Classification of Symmetry Reductions and Invariant Solutions for the Euler–Lagrange–Born–Infeld Equation

2019 ◽  
Vol 64 (12) ◽  
pp. 1103
Author(s):  
V. M. Fedorchuk ◽  
V. I. Fedorchuk
Keyword(s):  
Lie Algebra ◽  
Structural Properties ◽  
Three Dimensional ◽  
Invariant Solutions ◽  
Symmetry Reductions ◽  
Low Dimension

We study a connection between the structural properties of the low-dimension (dimL ≤ 3) nonconjugate subalgebras of the Lie argebra of the generalized Poincar´e group P(1,4) and the results of symmetry reductions for the Euler–Lagrange–Born–Infeld equation. We have performed the classification of nonsingular manifolds in the space M(1 , 3 ) × R(u) invariant with respect to three-dimensional nonconjugate subalgebras of the Lie algebra of the group P(1,4). The results are used for the classification of symmetry reductions and invariant solutions of the Euler–Lagrange–Born–Infeld equation.

scholarly journals The Fit Matters: Influence of Accelerometer Fitting and Training Drill Demands on Load Measures in Rugby League Players

2018 ◽  
Vol 13 (8) ◽  
pp. 1083-1089 ◽  
Author(s):  
Blake D. McLean ◽  
Cloe Cummins ◽  
Greta Conlan ◽  
Grant Duthie ◽  
Aaron J. Coutts
Keyword(s):  
Team Sport ◽  
Rugby League ◽  
Two Dimensional ◽  
Vertical Load ◽  
3 Dimensional ◽  
Sport Activities ◽  
Load Unit ◽  
Load Vector ◽  
The Relationship ◽  
And Training

Purpose: To determine the relationship between drill type and accelerometer-derived loads during various team-sport activities and examine the influence of unit fitting on these loads. Methods: Sixteen rugby league players were fitted with microtechnology devices in either manufacturer vests or playing jerseys before completing standardized running, agility, and tackling drills. Two-dimensional (2D) and 3-dimensional (3D) accelerometer loads (BodyLoad™) per kilometer were compared across drills and fittings (ie, vest and jersey). Results: When fitted in a vest, 2D BodyLoad was higher during tackling (21.5 [14.8] AU/km) than during running (9.5 [2.5] AU/km) and agility (10.3 [2.7] AU/km). Jersey fitting resulted in more than 2-fold higher BodyLoad during running (2D = 9.5 [2.7] vs 29.3 [14.8] AU/km, 3D = 48.5 [14.8] vs 111.5 [45.4] AU/km) and agility (2D = 10.3 [2.7] vs 21.0 [8.1] AU/km, 3D = 40.4 [13.6] vs 77.7 [26.8] AU/km) compared with a vest fitting. Jersey fitting also produced higher BodyLoad during tackling drills (2D = 21.5 [14.8] vs 27.8 [18.6] AU/km, 3D = 42.0 [21.4] vs 63.2 [33.1] AU/km). Conclusions: This study provides evidence supporting the construct validity of 2D BodyLoad for assessing collision/tackling load in rugby league training drills. Conversely, the large values obtained from 3D BodyLoad (which includes the vertical load vector) appear to mask small increases in load during tackling drills, rendering 3D BodyLoad insensitive to changes in contact load. Unit fitting has a large influence on accumulated accelerometer loads during all drills, which is likely related to greater incidental unit movement when units are fitted in jerseys. Therefore, it is recommended that athletes wear microtechnology units in manufacturer-provided vests to provide valid and reliable information.

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

Reduction of partially invariant submodels of rank 3 defect 1 to invariant submodels

2018 ◽  
Vol 13 (3) ◽  
pp. 59-63 ◽  
Author(s):  
D.T. Siraeva
Keyword(s):  
Equation Of State ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Gas Dynamics ◽  
Hydrodynamic Type ◽  
System Of Equations ◽  
Two Dimensional ◽  
Equations Of Gas Dynamics ◽  
Entropy Functions

Equations of hydrodynamic type with the equation of state in the form of pressure separated into a sum of density and entropy functions are considered. Such a system of equations admits a twelve-dimensional Lie algebra. In the case of the equation of state of the general form, the equations of gas dynamics admit an eleven-dimensional Lie algebra. For both Lie algebras the optimal systems of non-similar subalgebras are constructed. In this paper two partially invariant submodels of rank 3 defect 1 are constructed for two-dimensional subalgebras of the twelve-dimensional Lie algebra. The reduction of the constructed submodels to invariant submodels of eleven-dimensional and twelve-dimensional Lie algebras is proved.

scholarly journals Template-Free Self-Assembly of Two-Dimensional Polymers into Nano/Microstructured Materials

Molecules ◽  
2021 ◽  
Vol 26 (11) ◽  
pp. 3310
Author(s):  
Shengda Liu ◽  
Jiayun Xu ◽  
Xiumei Li ◽  
Tengfei Yan ◽  
Shuangjiang Yu ◽  
...  
Keyword(s):  
Self Assembly ◽  
Recent Progress ◽  
Two Dimensional ◽  
Subsequent Removal ◽  
The Past ◽  
2D Polymers ◽  
Template Free ◽  
Self Assembled ◽  
The Relationship ◽  
Polymer Tubes

In the past few decades, enormous efforts have been made to synthesize covalent polymer nano/microstructured materials with specific morphologies, due to the relationship between their structures and functions. Up to now, the formation of most of these structures often requires either templates or preorganization in order to construct a specific structure before, and then the subsequent removal of previous templates to form a desired structure, on account of the lack of “self-error-correcting” properties of reversible interactions in polymers. The above processes are time-consuming and tedious. A template-free, self-assembled strategy as a “bottom-up” route to fabricate well-defined nano/microstructures remains a challenge. Herein, we introduce the recent progress in template-free, self-assembled nano/microstructures formed by covalent two-dimensional (2D) polymers, such as polymer capsules, polymer films, polymer tubes and polymer rings.

Partially Invariant Solutions for Two-dimensional Ideal MHD Equations

1990 ◽  
Vol 45 (11-12) ◽  
pp. 1219-1229 ◽  
Author(s):  
D.-A. Becker ◽  
E. W. Richter
Keyword(s):  
Differential Equations ◽  
Linear Equations ◽  
Mhd Equations ◽  
Two Dimensional ◽  
Invariant Solutions ◽  
First Order ◽  
Partially Invariant Solutions ◽  
Quasilinear Systems ◽  
Ideal Mhd ◽  
Non Linear Equations

AbstractA generalization of the usual method of similarity analysis of differential equations, the method of partially invariant solutions, was introduced by Ovsiannikov. The degree of non-invariance of these solutions is characterized by the defect of invariance d. We develop an algorithm leading to partially invariant solutions of quasilinear systems of first-order partial differential equations. We apply the algorithm to the non-linear equations of the two-dimensional non-stationary ideal MHD with a magnetic field perpendicular to the plane of motion.

scholarly journals Structure of n-Lie Algebras with Involutive Derivations

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Ruipu Bai ◽  
Shuai Hou ◽  
Yansha Gao
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Two Dimensional

We study the structure of n-Lie algebras with involutive derivations for n≥2. We obtain that a 3-Lie algebra A is a two-dimensional extension of Lie algebras if and only if there is an involutive derivation D on A=A1  ∔  A-1 such that dim A1=2 or dim A-1=2, where A1 and A-1 are subspaces of A with eigenvalues 1 and -1, respectively. We show that there does not exist involutive derivations on nonabelian n-Lie algebras with n=2s for s≥1. We also prove that if A is a (2s+2)-dimensional (2s+1)-Lie algebra with dim A1=r, then there are involutive derivations on A if and only if r is even, or r satisfies 1≤r≤s+2. We discuss also the existence of involutive derivations on (2s+3)-dimensional (2s+1)-Lie algebras.

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