scholarly journals Relative equilibria of point vortices and the fundamental theorem of algebra

2011 ◽  
Vol 467 (2132) ◽  
pp. 2168-2184 ◽  
Author(s):  
Hassan Aref
Keyword(s):  
Fundamental Theorem ◽  
Relative Equilibrium ◽  
Polynomial Equation ◽  
Relative Equilibria ◽  
Point Vortices ◽  
Generating Polynomial ◽  
Second Derivatives ◽  
Fundamental Theorem Of Algebra ◽  
Vortex Systems ◽  
Derivatives Of

Relative equilibria of identical point vortices may be associated with a generating polynomial that has the vortex positions as its roots. A formula is derived that relates the first and second derivatives of this polynomial evaluated at a vortex position. Using this formula, along with the fundamental theorem of algebra, one can sometimes write a general polynomial equation. In this way, results about relative equilibria of point vortices may be proved in a compact and elegant way. For example, the classical result of Stieltjes, that if the vortices are on a line they must be situated at the zeros of the N th Hermite polynomial, follows easily. It is also shown that if in a relative equilibrium the vortices are all situated on a circle, they must form a regular N -gon. Several other results are proved using this approach. An ordinary differential equation for the generating polynomial when the vortices are situated on two perpendicular lines is derived. The method is extended to vortex systems where all the vortices have the same magnitude but may be of either sign. Derivations of the equation of Tkachenko for completely stationary configurations and its extension to translating relative equilibria are given.

Activities for Students: A Graphical Approach to Understanding the Fundamental Theorem of Algebra

2001 ◽  
Vol 94 (9) ◽  
pp. 749-756
Author(s):  
Sudhir Kumar Goel ◽  
Denise T. Reid
Keyword(s):  
Fundamental Theorem ◽  
Polynomial Equation ◽  
Complex Numbers ◽  
Multiple Roots ◽  
Complex Root ◽  
Graphical Approach ◽  
Fundamental Theorem Of Algebra ◽  
Complex Coefficients

The fundamental theorem of algebra states, Every polynomial equation of degree n ≥ 1 with complex coefficients has at least one complex root. This fact implies that these equations have exactly n roots, counting multiple roots, in the set of complex numbers.

Polynomiography: From the Fundamental Theorem of Algebra to Art

Leonardo ◽  
2005 ◽  
Vol 38 (3) ◽  
pp. 233-238 ◽  
Author(s):  
Bahman Kalantari
Keyword(s):  
Fundamental Theorem ◽  
Polynomial Equation ◽  
Computer Visualization ◽  
Iteration Functions ◽  
Fundamental Theorem Of Algebra

The author introduces polynomiography, a bridge between the Fundamental Theorem of Algebra and art. Polynomiography provides a tool for artists to create a 2D image—a polynomiograph—based on the computer visualization of a polynomial equation. The image is dependent upon the solutions of a polynomial equation, various interactive coloring schemes driven by iteration functions and several other parameters under the control of the polynomiographer's choice and creativity. Polynomiography software can mask all of the underlying mathematics, offering a tool that, although easy to use, affords the polynomiographer infinite artistic capabilities.

scholarly journals Collective behaviour of large number of vortices in the plane

2013 ◽  
Vol 469 (2156) ◽  
pp. 20130085 ◽  
Author(s):  
Yuxin Chen ◽  
Theodore Kolokolnikov ◽  
Daniel Zhirov
Keyword(s):  
Vortex Dynamics ◽  
Initial Conditions ◽  
Relative Equilibrium ◽  
Relative Equilibria ◽  
Point Vortices ◽  
Collective Behaviour ◽  
Aggregation Model ◽  
Equal Strength ◽  
Artificial Damping ◽  
Analytical Formulae

We investigate the dynamics of N point vortices in the plane, in the limit of large N . We consider relative equilibria , which are rigidly rotating lattice-like configurations of vortices. These configurations were observed in several recent experiments. We show that these solutions and their stability are fully characterized via a related aggregation model which was recently investigated in the context of biological swarms. By using this connection, we give explicit analytical formulae for many of the configurations that have been observed experimentally. These include configurations of vortices of equal strength; the N +1 configurations of N vortices of equal strength and one vortex of much higher strength; and more generally, N + K configurations. We also give examples of configurations that have not been studied experimentally, including N +2 configurations, where N vortices aggregate inside an ellipse. Finally, we introduce an artificial ‘damping’ to the vortex dynamics, in an attempt to explain the phenomenon of crystallization that is often observed in real experiments. The diffusion breaks the conservative structure of vortex dynamics, so that any initial conditions converge to the lattice-like relative equilibrium.

A Multivariable form of the Fundamental Theorem of Algebra

1983 ◽  
Vol 26 (3) ◽  
pp. 271-272
Author(s):  
Pablo M. Salzberg
Keyword(s):  
Fundamental Theorem ◽  
Homogeneous Polynomial ◽  
Inner Product ◽  
Closed Field ◽  
Sufficient Condition ◽  
Algebraically Closed Field ◽  
Fundamental Theorem Of Algebra ◽  
Image Position ◽  
Derivatives Of

AbstractLet H(x) be a homogeneous polynomial in n indeterminates over an algebraically closed field K. A necesssary and sufficient condition is given for H(x) to admit a factorization of the forma, b∈ Kn, and “∘” is the usual inner product. This condition involves the linear derivatives of H(x).

Earth's gravity field parameters determination by the space geodesy dynamical approach

2017 ◽  
Vol 919 (1) ◽  
pp. 7-12
Author(s):  
N.A Sorokin
Keyword(s):  
Coordinate System ◽  
Gravitational Potential ◽  
Coefficient Matrix ◽  
Measured Quantity ◽  
Second Derivative ◽  
Measured Variable ◽  
Second Derivatives ◽  
Rectangular Coordinates ◽  
Cartesian Coordinate ◽  
Derivatives Of

The method of the geopotential parameters determination with the use of the gradiometry data is considered. The second derivative of the gravitational potential in the correction equation on the rectangular coordinates x, y, z is used as a measured variable. For the calculated value of the measured quantity required for the formation of a free member of the correction equation, the the Cunningham polynomials were used. We give algorithms for computing the second derivatives of the Cunningham polynomials on rectangular coordinates x, y, z, which allow to calculate the second derivatives of the geopotential at the rectangular coordinates x, y, z.Then we convert derivatives obtained from the Cartesian coordinate system in the coordinate system of the gradiometer, which allow to calculate the free term of the correction equation. Afterwards the correction equation coefficients are calculated by differentiating the formula for calculating the second derivative of the gravitational potential on the rectangular coordinates x, y, z. The result is a coefficient matrix of the correction equations and corrections vector of the free members of equations for each component of the tensor of the geopotential. As the number of conditional equations is much more than the number of the specified parameters, we go to the drawing up of the system of normal equations, from which solutions we determine the required corrections to the harmonic coefficients.

Some new generalizations for GA-convex functions

Filomat ◽  
2017 ◽  
Vol 31 (4) ◽  
pp. 1009-1016 ◽  
Author(s):  
Ahmet Akdemir ◽  
Özdemir Emin ◽  
Ardıç Avcı ◽  
Abdullatif Yalçın
Keyword(s):  
Convex Functions ◽  
Integral Identity ◽  
Integral Inequalities ◽  
General Integral ◽  
Second Derivatives ◽  
Derivatives Of

In this paper, firstly we prove an integral identity that one can derive several new equalities for special selections of n from this identity: Secondly, we established more general integral inequalities for functions whose second derivatives of absolute values are GA-convex functions based on this equality.

System of thermodynamic quantities in the McMillan-Mayer solution theory

1985 ◽  
Vol 50 (4) ◽  
pp. 791-798 ◽  
Author(s):  
Vilém Kodýtek
Keyword(s):  
Free Energy ◽  
Generating Function ◽  
Simple Form ◽  
Unit Volume ◽  
Thermodynamic Quantities ◽  
Solution Theory ◽  
Pure Solvent ◽  
Second Derivatives ◽  
Definition Of ◽  
Derivatives Of

The McMillan-Mayer (MM) free energy per unit volume of solution AMM, is employed as a generating function of the MM system of thermodynamic quantities for solutions in the state of osmotic equilibrium with pure solvent. This system can be defined by replacing the quantities G, T, P, and m in the definition of the Lewis-Randall (LR) system by AMM, T, P0, and c (P0 being the pure solvent pressure). Following this way the LR to MM conversion relations for the first derivatives of the free energy are obtained in a simple form. New relations are derived for its second derivatives.

A State-Space Approach to the Synthesis of Random Vertical and Crosslevel Rail Irregularities

1990 ◽  
Vol 112 (1) ◽  
pp. 83-87 ◽  
Author(s):  
R. H. Fries ◽  
B. M. Coffey
Keyword(s):  
Numerical Simulation ◽  
White Noise ◽  
State Space ◽  
Spectral Characteristics ◽  
The State ◽  
Second Derivatives ◽  
Time Histories ◽  
Sample Interval ◽  
Rail Irregularities ◽  
Derivatives Of

Solution of rail vehicle dynamics models by means of numerical simulation has become more prevalent and more sophisticated in recent years. At the same time, analysts and designers are increasingly interested in the response of vehicles to random rail irregularities. The work described in this paper provides a convenient method to generate random vertical and crosslevel irregularities when their time histories are required as inputs to a numerical simulation. The solution begins with mathematical models of vertical and crosslevel power spectral densities (PSDs) representing PSDs of track classes 4, 5, and 6. The method implements state-space models of shape filters whose frequency response magnitude squared matches the desired PSDs. The shape filters give time histories possessing the proper spectral content when driven by white noise inputs. The state equations are solved directly under the assumption that the white noise inputs are constant between time steps. Thus, the state transition matrix and the forcing matrix are obtained in closed form. Some simulations require not only vertical and crosslevel alignments, but also the first and occasionally the second derivatives of these signals. To accommodate these requirements, the first and second derivatives of the signals are also generated. The responses of the random vertical and crosslevel generators depend upon vehicle speed, sample interval, and track class. They possess the desired PSDs over wide ranges of speed and sample interval. The paper includes a comparison between synthetic and measured spectral characteristics of class 4 track. The agreement is very good.

Sign in / Sign up

Export Citation Format

Share Document