Flat Structures and Algebraic Solutions to Painlevé VI Equation

2017 ◽  
pp. 383-398 ◽  
Author(s):  
Mitsuo Kato ◽  
Toshiyuki Mano ◽  
Jiro Sekiguchi
Keyword(s):  
Flat Structures ◽  
Painlevé Vi Equation ◽  
Algebraic Solutions

scholarly journals Singularity analysis and analytic solutions for the Benney–Gjevik equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Andronikos Paliathanasis ◽  
Genly Leon ◽  
P. G. L. Leach
Keyword(s):  
Evolution Equations ◽  
Singularity Analysis ◽  
Analytic Solutions ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Painlevé Test ◽  
Wave Solutions ◽  
Algebraic Solutions

Abstract We apply the Painlevé test for the Benney and the Benney–Gjevik equations, which describe waves in falling liquids. We prove that these two nonlinear 1 + 1 evolution equations pass the singularity test for the travelling-wave solutions. The algebraic solutions in terms of Laurent expansions are presented.

Exceptional solutions to the Painlevé VI equation associated with the generalized Jacobi weight

2017 ◽  
Vol 06 (01) ◽  
pp. 1750003
Author(s):  
Shulin Lyu ◽  
Yang Chen
Keyword(s):  
Asymptotic Expansions ◽  
Hankel Determinant ◽  
Generalized Jacobi Weight ◽  
Jacobi Weight ◽  
Special Cases ◽  
Painlevé Vi Equation

We consider the generalized Jacobi weight [Formula: see text], [Formula: see text]. As is shown in [D. Dai and L. Zhang, Painlevé VI and Henkel determinants for the generalized Jocobi weight, J. Phys. A: Math. Theor. 43 (2010), Article ID:055207, 14pp.], the corresponding Hankel determinant is the [Formula: see text]-function of a particular Painlevé VI. We present all the possible asymptotic expansions of the solution of the Painlevé VI equation near [Formula: see text] and [Formula: see text] for generic [Formula: see text]. For four special cases of [Formula: see text] which are related to the dimension of the Hankel determinant, we can find the exceptional solutions of the Painlevé VI equation according to the results of [A. Eremenko, A. Gabrielov and A. Hinkkanen, Exceptional solutions to the Painlevé VI equation, preprint (2016), arXiv:1602.04694 ], and thus give another characterization of the Hankel determinant.

Algebraic solutions in determining the coordinates of the radiator by the results of multi-position measurements of field strength

2021 ◽  
Author(s):  
V.A. Ufaev
Keyword(s):  
Maximum Likelihood ◽  
Field Strength ◽  
Reference Point ◽  
Linear Equations ◽  
Type System ◽  
Energy Parameter ◽  
Likelihood Estimation ◽  
Likelihood Method ◽  
Algebraic Solution ◽  
Algebraic Solutions

On the basis of the hypothesis of equality of the measured and true values of the amplitude of the field strength, an algebraic solution for estimating the unknown coordinates and the energy parameter of the radiator is obtained. Initially, by compiling and solving a redefined system of linear equations by pseudo-rotation of matrices, the coordinates of the emitter are determined under the assumption of independence of the distance to the reference point from the coordinates of the emitter. Then make and solve the square equation concerning distance to a reference point with the subsequent estimation of coordinates and an energy parameter. The ambiguity of the algebraic solution is resolved by comparing the maximum likelihood functional and choosing the parameters at which its maximum is reached. According to the simulation of a cellular-type system in multiplicative noise, the results of algebraic solutions by the maximum likelihood method and the calculated ones are close, except for a special zone where anomalous changes occur due to the limitations of the coordinate determination method. Algebraic solutions for maximum likelihood estimation provide an increase in the calculation speed of about 500 times. The proposed principle can be used in solving the ambiguity of algebraic solutions in systems of difference-rangefinder type and in the inverse problem of self-positioning of the receiving point by the amplitude of the electromagnetic field of beacons with a known location. The article contains 4 figures, a list of references from 9 sources.

Algebraic solutions of flow and heat for some nanofluids over deformable and permeable surfaces

2017 ◽  
Vol 27 (10) ◽  
pp. 2259-2267 ◽  
Author(s):  
Mustafa Turkyilmazoglu
Keyword(s):  
Heat Transfer ◽  
Design Methodology ◽  
Temperature Jump ◽  
Volume Fraction ◽  
Velocity Slip ◽  
Jump Conditions ◽  
Content Type ◽  
Layer Flow ◽  
Working Out ◽  
Algebraic Solutions

Purpose This paper aims to working out exact solutions for the boundary layer flow of some nanofluids over porous stretching/shrinking surfaces with different configurations. To serve to this aim, five types of nanoparticles together with the water as base fluid are under consideration, namely, Ag, Cu, CuO, Al2O3 and TiO2. Design/methodology/approach The physical flow is affected by the presence of velocity slip as well as temperature jump conditions. Findings The knowledge on the influences of nanoparticle volume fraction on the practically significant parameters, such as the skin friction and the rate of heat transfer, for the above considered nanofluids, is easy to gain from the extracted explicit formulas. Originality/value Particularly, formulas clearly point that the heat transfer rate is not only dependent on the thermal conductivity of the material but it also highly relies on the heat capacitance as well as the density of the nanofluid under consideration.

scholarly journals Determination of the modal parameters on the thin flat structures

2018 ◽  
Vol 18 ◽  
pp. 91-95 ◽  
Author(s):  
Roman Zajac ◽  
Aleš Prokop ◽  
Kamil Řehák
Keyword(s):  
Modal Parameters ◽  
Flat Structures

scholarly journals Detecting and Visualizing the Communities of Innovation in Beijing-Tianjin-Hebei Urban Agglomeration Based on the Patent Cooperation Network

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Fang Zhou ◽  
Bo Zhang
Keyword(s):  
Deep Understanding ◽  
Urban Agglomeration ◽  
Geographical Information ◽  
Single Center ◽  
Power Construction ◽  
Flat Structures ◽  
Collaborative Innovation ◽  
The Hierarchical Structure ◽  
Cooperation Network ◽  
Communities Of Innovation

For a deep understanding of Beijing-Tianjin-Hebei (BTH) collaborative innovation, we detected and visualized the communities of innovation in BTH Urban Agglomeration based on the patent cooperation network. China Patent Database was connected with Business Registration Database and the Tianyan Check to achieve the geographical information of organizational innovators. Spinglass algorithm was applied and ultimately 12 communities of innovation were detected. Based on the different structure characteristics, we further clustered the 12 communities into four typical structures that are hierarchical, single-center, polycentric, and flat structures. The hierarchical structure is usually large in scale and the cooperative intensity is relatively high. Single-center structure has a center with a high proportion of centrality and the cooperative intensity is relatively low. Polycentric structure has multiple centers with similar proportions of centrality. Flat structure is usually small in scale and has no obvious network center. In the patent cooperative network of BTH Urban Agglomeration, universities and state-owned enterprises occupied the centers and acted important roles to connect other organizations. Some communities of innovation showed significant industry characteristics, mainly involving six industry fields that are electric power, construction, petroleum, metallurgy and materials, municipal transportation, and railway. From the geographical perspective, some communities manifested local attributes and some demonstrated cooperation between regions. Beijing was the center of the Beijing-Tianjin-Hebei patent cooperation network. Compared with the pair of Beijing-Tianjin and the pair of Beijing-Hebei, Tianjin and Hebei were not closely connected. In the future, Beijing-Tianjin-Hebei collaborative innovation should strengthen cooperation between Tianjin and Hebei.

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