On the Non-Existence of Regular Stationary Solutions of Relativistic Field Equations

1943 ◽  
Vol 44 (2) ◽  
pp. 131 ◽  
Author(s):  
A. Einstein ◽  
W. Pauli
Keyword(s):  
Stationary Solutions ◽  
Field Equations ◽  
Relativistic Field

On ring solutions of neural field equations

2021 ◽  
pp. 363-371
Author(s):  
Rachid Atmania ◽  
Evgenii O. Burlakov ◽  
Ivan N. Malkov
Keyword(s):  
Activation Function ◽  
Stationary Solutions ◽  
Neural Field ◽  
Neuronal Activation ◽  
Field Equations ◽  
Broad Ring ◽  
Activation Threshold ◽  
Electrical Potentials ◽  
Radially Symmetric Solutions ◽  
Neural Field Equations

The article is devoted to investigation of integro-differential equation with the Hammerstein integral operator of the following form: ∂_t u(t,x)=-τu(t,x,x_f )+∫_(R^2)▒〖ω(x-y)f(u(t,y) )dy, t≥0, x∈R^2 〗. The equation describes the dynamics of electrical potentials u(t,x) in a planar neural medium and has the name of neural field equation.We study ring solutions that are represented by stationary radially symmetric solutions corresponding to the active state of the neural medium in between two concentric circles and the rest state elsewhere in the neural field. We suggest conditions of existence of ring solutions as well as a method of their numerical approximation. The approach used relies on the replacement of the probabilistic neuronal activation function f that has sigmoidal shape by a Heaviside-type function. The theory is accompanied by an example illustrating the procedure of investigation of ring solutions of a neural field equation containing a typically used in the neuroscience community neuronal connectivity function that allows taking into account both excitatory and inhibitory interneuronal interactions. Similar to the case of bump solutions (i. e. stationary solutions of neural field equations, which correspond to the activated area in the neural field represented by the interior of some circle) at a high values of the neuronal activation threshold there coexist a broad ring and a narrow ring solutions that merge together at the critical value of the activation threshold, above which there are no ring solutions.

The Bianchi Identities in the Generalized Theory of Gravitation

1950 ◽  
Vol 2 ◽  
pp. 120-128 ◽  
Author(s):  
A. Einstein
Keyword(s):  
General Principle ◽  
Field Structure ◽  
Field Equations ◽  
Relativistic Field ◽  
Bianchi Identities ◽  
Principle Of Relativity ◽  
Dependent Variables ◽  
Physical Validity ◽  
Theory Of Gravitation

1. General remarks. The heuristic strength of the general principle of relativity lies in the fact that it considerably reduces the number of imaginable sets of field equations; the field equations must be covariant with respect to all continuous transformations of the four coordinates. But the problem becomes mathematically well-defined only if we have postulated the dependent variables which are to occur in the equations, and their transformation properties (field-structure). But even if we have chosen the field-structure (in such a way that there exist sufficiently strong relativistic field-equations), the principle of relativity does not determine the field-equations uniquely. The principle of “logical simplicity” must be added (which, however, cannot be formulated in a non-arbitrary way). Only then do we have a definite theory whose physical validity can be tested a posteriori.

scholarly journals Component Minimization of the Bargmann?Wigner Wavefunction

1978 ◽  
Vol 31 (2) ◽  
pp. 137 ◽  
Author(s):  
EA Jeffery
Keyword(s):  
Arbitrary Spin ◽  
Spin Operator ◽  
The Other ◽  
Operator Matrices ◽  
Field Equations ◽  
Relativistic Field

The Bargmann-Wigner equations are used to derive relativistic field equations with only 2(2j+ 1) components of the original wavefunction. The other components of the Bargmann-Wigner wavefunction are superfluous and can be defined in terms of the 2(2j+ 1) components. The results are compared with various 2(2j+ 1) theories in the literature. Sylvester's theorem and some properties of induced matrices give simple relationships between the operator matrices of the field equations and the arbitrary spin operator matrices.

scholarly journals The unitary representations of the Poincar\'e group in any spacetime dimension

2021 ◽  
Author(s):  
Xavier Bekaert ◽  
Nicolas Boulanger
Keyword(s):  
Arbitrary Dimension ◽  
Minkowski Spacetime ◽  
Irreducible Representations ◽  
Theoretical Treatment ◽  
Spacetime Dimension ◽  
Field Equations ◽  
Negative Mass ◽  
Relativistic Field ◽  
Fundamental Particles ◽  
Covariant Field

An extensive group-theoretical treatment of linear relativistic field equations on Minkowski spacetime of arbitrary dimension D\geqslant 3D≥3 is presented. An exhaustive treatment is performed of the two most important classes of unitary irreducible representations of the Poincar'e group, corresponding to massive and massless fundamental particles. Covariant field equations are given for each unitary irreducible representation of the Poincar'e group with non-negative mass-squared.

Some solutions of Einstein's equations in general relativity

1968 ◽  
Vol 64 (1) ◽  
pp. 167-170 ◽  
Author(s):  
J. J. J. Marek
Keyword(s):  
General Relativity ◽  
Existence Of Solutions ◽  
Empty Space ◽  
Stationary Solutions ◽  
Einstein’S Equations ◽  
Field Equations ◽  
Einstein's Equations ◽  
New Class

AbstractA new class of axi-symmetric stationary solutions of Einstein's empty space field equations is obtained. Non-existence of solutions of certain other classes is proved.

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