scholarly journals A transformation between stationary point vortex equilibria

2020 ◽  
Vol 476 (2240) ◽  
pp. 20200310
Author(s):  
Vikas S. Krishnamurthy ◽  
Miles H. Wheeler ◽  
Darren G. Crowdy ◽  
Adrian Constantin
Keyword(s):  
Finite Number ◽  
Stationary Point ◽  
Point Vortex ◽  
Wide Range ◽  
Finite Sequences ◽  
Free Parameters ◽  
Point Vortex Equilibria ◽  
Isolated Point ◽  
Number Of Iterations ◽  
Finite Number Of Iterations

A new transformation between stationary point vortex equilibria in the unbounded plane is presented. Given a point vortex equilibrium involving only vortices with negative circulation normalized to −1 and vortices with positive circulations that are either integers or half-integers, the transformation produces a new equilibrium with a free complex parameter that appears as an integration constant. When iterated the transformation can produce infinite hierarchies of equilibria, or finite sequences that terminate after a finite number of iterations, each iteration generating equilibria with increasing numbers of point vortices and free parameters. In particular, starting from an isolated point vortex as a seed equilibrium, we recover two known infinite hierarchies of equilibria corresponding to the Adler–Moser polynomials and a class of polynomials found, using very different methods, by Loutsenko (Loutsenko 2004 J. Phys. A: Math. Gen. 37 , 1309–1321 (doi:10.1088/0305-4470/37/4/017)). For the latter polynomials, the existence of such a transformation appears to be new. The new transformation, therefore, unifies a wide range of disparate results in the literature on point vortex equilibria.

scholarly journals Force-enhancing vortex equilibria for two parallel plates in uniform flow

2012 ◽  
Vol 468 (2140) ◽  
pp. 1175-1195 ◽  
Author(s):  
Takashi Sakajo
Keyword(s):  
Stationary Point ◽  
Design Problem ◽  
Flow Problem ◽  
Point Vortex ◽  
Uniform Flow ◽  
Point Vortices ◽  
Multiply Connected Domains ◽  
Parallel Plates ◽  
Single Plate ◽  
Point Vortex Equilibria

A two-dimensional potential flow in an unbounded domain with two parallel plates is considered. We examine whether two free point vortices can be trapped near the two plates in the presence of a uniform flow and observe whether these stationary point vortices enhance the force on the plates. The present study is an extension of previously published work in which a free point vortex over a single plate is investigated. The flow problem is motivated by an airfoil design problem for the double wings. Moreover, it also contributes to a design problem for an efficient wind turbine with vertical blades. In order to obtain the point-vortex equilibria numerically, we make use of a linear algebraic algorithm combined with a stochastic process, called the Brownian ratchet scheme. The ratchet scheme allows us to capture a family of stationary point vortices in multiply connected domains with ease. As a result, we find that stationary point vortices exist around the two plates and they enhance the downward force and the counter-clockwise rotational force acting on the two plates.

scholarly journals Biconjugate residual algorithm for solving general Sylvester-transpose matrix equations

Filomat ◽  
2018 ◽  
Vol 32 (15) ◽  
pp. 5307-5318 ◽  
Author(s):  
Masoud Hajarian
Keyword(s):  
Finite Number ◽  
Efficient Algorithm ◽  
Frobenius Norm ◽  
Matrix Equations ◽  
Numerical Examples ◽  
Least Frobenius Norm Solution ◽  
Residual Algorithm ◽  
Sylvester Matrix Equations ◽  
Number Of Iterations ◽  
Finite Number Of Iterations

The present paper is concerned with the solution of the coupled generalized Sylvester-transpose matrix equations {A1XB1 + C1XD1 + E1XTF1 = M1, A2XB2 + C2XD2 + E2XTF2 = M2, including the well-known Lyapunov and Sylvester matrix equations. Based on a variant of biconjugate residual (BCR) algorithm, we construct and analyze an efficient algorithm to find the (least Frobenius norm) solution of the general Sylvester-transpose matrix equations within a finite number of iterations in the absence of round-off errors. Two numerical examples are given to examine the performance of the constructed algorithm.

scholarly journals On singularity confinement for the pentagram map

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Max Glick
Keyword(s):  
Finite Number ◽  
Configuration Space ◽  
Singular Locus ◽  
Birational Map ◽  
Singularity Confinement ◽  
Nous Montrons ◽  
International Audience ◽  
Pentagram Map ◽  
Number Of Iterations ◽  
Finite Number Of Iterations

International audience The pentagram map, introduced by R. Schwartz, is a birational map on the configuration space of polygons in the projective plane. We study the singularities of the iterates of the pentagram map. We show that a ``typical'' singularity disappears after a finite number of iterations, a confinement phenomenon first discovered by Schwartz. We provide a method to bypass such a singular patch by directly constructing the first subsequent iterate that is well-defined on the singular locus under consideration. The key ingredient of this construction is the notion of a decorated (twisted) polygon, and the extension of the pentagram map to the corresponding decorated configuration space. L'application pentagramme de R. Schwartz est une application birationnelle sur l'espace des polygones dans le plan projectif. Nous ètudions les singularitès des itèrations de l'application pentagramme. Nous montrons qu'une singularitè ``typique'' disparaî t après un nombre fini d'itèrations, un phènomène dècouvert par Schwartz. Nous fournissons une mèthode pour contourner une telle singularitè en construisant la première itèration qui est bien dèfinie. L'ingrèdient principal de cette construction est la notion d'un polygone dècorè et l'extension de l'application pentagramme á l'espace de configuration dècorè.

Solving the coupled Sylvester-like matrix equations via a new finite iterative algorithm

2017 ◽  
Vol 34 (5) ◽  
pp. 1446-1467 ◽  
Author(s):  
Masoud Hajarian
Keyword(s):  
Finite Number ◽  
Iterative Algorithm ◽  
Design Methodology ◽  
Matrix Form ◽  
Matrix Equations ◽  
Content Type ◽  
Numerical Examples ◽  
The Matrix ◽  
Number Of Iterations ◽  
Finite Number Of Iterations

Purpose The purpose of this paper is to obtain an iterative algorithm to find the solution of the coupled Sylvester-like matrix equations. Design/methodology/approach In this work, the matrix form of the conjugate direction (CD) algorithm to find the solution X of the coupled Sylvester-like matrix equations: {A1XB1+M1f1(X)N1=F1,A2XB2+M2f2(X)N2=F2,with fi(X) = X, fi(X) = X¯, fi(X) = XT and fi(X) = XH for i = 1; 2 has been established. Findings It is proven that the algorithm converges to the solution within a finite number of iterations in the absence of round-off errors. Finally, four numerical examples were used to test the proficiency and convergence of the established algorithm. Originality/value The numerical examples have led the author to believe that the generalized CD (GCD) algorithm is efficient and it converges more rapidly in comparison with the CGNR and CGNE algorithms.

A Method for Computing Leontief Multipliers from Rectangular Input—Output Accounts

1978 ◽  
Vol 10 (2) ◽  
pp. 137-143 ◽  
Author(s):  
A P Schinnar
Keyword(s):  
Iterative Method ◽  
Finite Number ◽  
National Accounts ◽  
Technological Changes ◽  
Input Output ◽  
Column Structure ◽  
Multiplier Effect ◽  
Number Of Iterations ◽  
Finite Number Of Iterations

This paper develops an iterative method for computing Leontief multipliers for use in analyses of systems of national accounts. The algorithm entails a finite number of iterations using the row and column structure of the production accounts. It is particularly suitable for evaluating the multiplier effect of technological changes in the input–output coefficients of various industries.

Finding solutions for periodic discrete-time generalized coupled Sylvester matrix equations via the generalized BCR method

2016 ◽  
Vol 40 (2) ◽  
pp. 647-656 ◽  
Author(s):  
Masoud Hajarian
Keyword(s):  
Iterative Method ◽  
Finite Number ◽  
Discrete Time ◽  
Matrix Equations ◽  
Sylvester Matrix ◽  
Bcr Method ◽  
Residual Algorithm ◽  
Sylvester Matrix Equations ◽  
Number Of Iterations ◽  
Finite Number Of Iterations

The periodic discrete-time matrix equations have wide applications in stability theory, control theory and perturbation analysis. In this work, the biconjugate residual algorithm is generalized to construct a matrix iterative method to solve the periodic discrete-time generalized coupled Sylvester matrix equations [Formula: see text] The constructed method is shown to be convergent in a finite number of iterations in the absence of round-off errors. By comparing with other similar methods in practical computation, we give numerical results to demonstrate the accuracy and the numerical superiority of the constructed method.

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