WEAK SOLUTIONS FOR WEAK SINGULARITIES

We revisit the problem of the development of singularities in the gravitational collapse of an inhomogeneous dust sphere. As shown by Yodzis et al1, naked singularities may occur at finite radius where shells of dust cross one another. These singularities are gravitationally weak 2, and it has been claimed that at these singularities, the metric may be written in continuous form 2, with locally L∞ connection coefficients 3. We correct these claims, and show how the field equations may be reformulated as a first order, quasi-linear, non-conservative, non-strictly hyperbolic system. We discuss existence and uniqueness of generalized solutions of this system using bounded functions of bounded variation (BV) 4, where the product of a BV function and the derivative of another BV function may be interpreted as a locally finite measure. The solutions obtained provide a dynamical extension to the future of the singularity.

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On Conformally Covariant Spinor Field Equations

Canadian Journal of Physics ◽

10.1139/p72-280 ◽

1972 ◽

Vol 50 (18) ◽

pp. 2100-2104 ◽

Cited By ~ 2

Author(s):

Mark S. Drew

Keyword(s):

Minkowski Space ◽

Invariant Mass ◽

Spinor Field ◽

Dimensional Space ◽

Flat Space ◽

Field Equations ◽

First Order ◽

Conformally Invariant ◽

Spinor Fields

Conformally covariant equations for free spinor fields are determined uniquely by carrying out a descent to Minkowski space from the most general first-order rotationally covariant spinor equations in a six-dimensional flat space. It is found that the introduction of the concept of the "conformally invariant mass" is not possible for spinor fields even if the fields are defined not only on the null hyperquadric but over the entire manifold of coordinates in six-dimensional space.

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Existence and uniqueness of solutions for the first-order nonlinear differential equations with multipoint boundary conditions

Proceeding of the International Conference "Optimal Control and Differential Games" dedicated to the 110th anniversary of L. S. Pontryagin ◽

10.4213/proc23008 ◽

2018 ◽

Author(s):

Misir Jumail ogly Mardanov ◽

Yagub Amiyar ogly Sharifov

Keyword(s):

Differential Equations ◽

Boundary Conditions ◽

Existence And Uniqueness ◽

Nonlinear Differential Equations ◽

Uniqueness Of Solutions ◽

Multipoint Boundary ◽

First Order ◽

Multipoint Boundary Conditions

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Existence and Uniqueness of Solutions of First-Order Differential Equations

Introduction to Ordinary Differential Equations with Mathematica® ◽

10.1007/978-1-4612-2242-2_5 ◽

1997 ◽

pp. 119-154

Author(s):

Alfred Gray ◽

Michael Mezzino ◽

Mark A. Pinsky

Keyword(s):

Differential Equations ◽

Existence And Uniqueness ◽

Uniqueness Of Solutions ◽

First Order ◽

First Order Differential Equations

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Bounding the Distinguishing Number of Infinite Graphs and Permutation Groups

The Electronic Journal of Combinatorics ◽

10.37236/3046 ◽

2014 ◽

Vol 21 (3) ◽

Cited By ~ 1

Author(s):

Simon M. Smith ◽

Mark E. Watkins

Keyword(s):

Connected Graph ◽

Automorphism Groups ◽

Cardinal Number ◽

Infinite Graph ◽

Infinite Graphs ◽

Finite Radius ◽

Locally Finite ◽

Distinguishing Number ◽

Identity Permutation ◽

Imprimitive Group

A group of permutations $G$ of a set $V$ is $k$-distinguishable if there exists a partition of $V$ into $k$ cells such that only the identity permutation in $G$ fixes setwise all of the cells of the partition. The least cardinal number $k$ such that $(G,V)$ is $k$-distinguishable is its distinguishing number $D(G,V)$. In particular, a graph $\Gamma$ is $k$-distinguishable if its automorphism group $\rm{Aut}(\Gamma)$ satisfies $D(\rm{Aut}(\Gamma),V\Gamma)\leq k$.Various results in the literature demonstrate that when an infinite graph fails to have some property, then often some finite subgraph is similarly deficient. In this paper we show that whenever an infinite connected graph $\Gamma$ is not $k$-distinguishable (for a given cardinal $k$), then it contains a ball of finite radius whose distinguishing number is at least $k$. Moreover, this lower bound cannot be sharpened, since for any integer $k \geq 3$ there exists an infinite, locally finite, connected graph $\Gamma$ that is not $k$-distinguishable but in which every ball of finite radius is $k$-distinguishable.In the second half of this paper we show that a large distinguishing number for an imprimitive group $G$ is traceable to a high distinguishing number either of a block of imprimitivity or of the induced action by $G$ on the corresponding system of imprimitivity. An immediate application is to automorphism groups of infinite imprimitive graphs. These results are companion to the study of the distinguishing number of infinite primitive groups and graphs in a previous paper by the authors together with T. W. Tucker.

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The Problem of Determining of the Source Function and of the Leading Coefficient in the Many-dimensional Semilinear Parabolic Equation

Journal of Siberian Federal University Mathematics & Physics ◽

10.17516/1997-1397-2021-14-4-497-506 ◽

2001 ◽

Vol 14 (4) ◽

pp. 497-506

Author(s):

Svetlana V. Polyntseva ◽

◽

Kira I. Spirina

Keyword(s):

Inverse Problem ◽

Parabolic Equation ◽

Existence And Uniqueness ◽

Source Function ◽

Time Interval ◽

Semilinear Parabolic Equation ◽

Existence And Uniqueness Theorem ◽

Bounded Functions ◽

The Many ◽

Semilinear Parabolic

We consider the problem of determining the source function and the leading coefficient in a multidimensional semilinear parabolic equation with overdetermination conditions given on two different hypersurfaces. The existence and uniqueness theorem for the classical solution of the inverse problem in the class of smooth bounded functions is proved. A condition is found for the dependence of the upper bound of the time interval, in which there is a unique solution to the inverse problem, on the input data

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Weak solvability of the unconditionally stable difference scheme for the coupled sine-Gordon system

Nonlinear Analysis Modelling and Control ◽

10.15388/namc.2020.25.20558 ◽

2020 ◽

Vol 25 (6) ◽

pp. 997-1014

Author(s):

Ozgur Yildirim ◽

Meltem Uzun

Keyword(s):

Difference Scheme ◽

Numerical Method ◽

Existence And Uniqueness ◽

Numerical Experiments ◽

Finite Difference Schemes ◽

Order Of Accuracy ◽

Unconditionally Stable ◽

First Order ◽

The Difference ◽

Weak Solvability

In this paper, we study the existence and uniqueness of weak solution for the system of finite difference schemes for coupled sine-Gordon equations. A novel first order of accuracy unconditionally stable difference scheme is considered. The variational method also known as the energy method is applied to prove unique weak solvability.We also present a new unified numerical method for the approximate solution of this problem by combining the difference scheme and the fixed point iteration. A test problem is considered, and results of numerical experiments are presented with error analysis to verify the accuracy of the proposed numerical method.

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