Search results for "Riemann hypothesis"
showing 10 items of 31 documents
Relativistic simulations of rotational core collapse : I. Methods, initial models, and code tests
2002
We describe an axisymmetric general relativistic code for rotational core collapse. The code evolves the coupled system of metric and fluid equations using the ADM 3+1 formalism and a conformally flat metric approximation of the Einstein equations. The relativistic hydrodynamics equations are formulated as a first-order flux-conservative hyperbolic system and are integrated using high-resolution shock-capturing schemes based on Riemann solvers. We assess the quality of the conformally flat metric approximation for relativistic core collapse and present a comprehensive set of tests which the code successfully passed. The tests include relativistic shock tubes, the preservation of the rotatio…
Approximate Osher–Solomon schemes for hyperbolic systems
2016
This paper is concerned with a new kind of Riemann solvers for hyperbolic systems, which can be applied both in the conservative and nonconservative cases. In particular, the proposed schemes constitute a simple version of the classical Osher-Solomon Riemann solver, and extend in some sense the schemes proposed in Dumbser and Toro (2011) 19,20. The viscosity matrix of the numerical flux is constructed as a linear combination of functional evaluations of the Jacobian of the flux at several quadrature points. Some families of functions have been proposed to this end: Chebyshev polynomials and rational-type functions. Our schemes have been tested with different initial value Riemann problems f…
Models of Computation, Riemann Hypothesis, and Classical Mathematics
1998
Classical mathematics is a source of ideas used by Computer Science since the very first days. Surprisingly, there is still much to be found. Computer scientists, especially, those in Theoretical Computer Science find inspiring ideas both in old notions and results, and in the 20th century mathematics. The latest decades have brought us evidence that computer people will soon study quantum physics and modern biology just to understand what computers are doing.
Riemann solvers in relativistic astrophysics
1999
AbstractOur contribution reviews High Resolution Shock Capturing methods (HRSC) in the field of relativistic hydrodynamics with special emphasis on Riemann solvers. HRSC techniques achieve highly accurate numerical approximations (formally second order or better) in smooth regions of the flow, and capture the motion of unresolved steep gradients without creating spurious oscillations. One objective of our contribution is to show how these techniques have been extended to relativistic hydrodynamics, making it possible to explore some challenging astrophysical scenarios. We will review recent literature concerning the main properties of different special relativistic Riemann solvers, and disc…
Riemann’s Result and Consequences for Physics and Philosophy
2020
Riemann commented on his main result as follows: “The common character of those manifolds whose curvature is constant may also be expressed thus: that figures may be viewed in them without stretching. For clearly figures could not be arbitrarily shifted and turned around in them if the curvature at each point were not the same in all directions at one point as at another, and consequently the same constructions can be made from it; whence it follows that in aggregates with constant curvature, figures may have any arbitrary position given them. The measure-relations of these manifolds depend only on the value of the curvature, and in relation to the analytic expression it may be remarked tha…
Incomplete Riemann Solvers Based on Functional Approximations to the Absolute Value Function
2021
We give an overview on the work developed in recent years about certain classes of incomplete Riemann solvers for hyperbolic systems. These solvers are based on polynomial or rational approximations to |x|, and they do not require the knowledge of the complete eigenstructure of the system, but only a bound on the maximum wave speed. Our solvers can be readily applied to nonconservative hyperbolic systems, by following the theory of path-conservative schemes. In particular, this allows for an automatic treatment of source or coupling terms in systems of balance laws. The properties of our schemes have been tested with some challenging numerical experiments involving systems such as the Euler…
Analysis of eta production using a generalized Lee model
1998
We have investigated the processes N($\pi$, $\pi$)N and N($\pi$, $\eta$)N close to eta threshold using a simple, nonrelativistic Lee model which has the advantage of being analytically solvable. It is then possible to study the Riemann sheets of the S-matrix and the behavior of its resonance poles especially close to threshold. A theoretical simulation of the experimental cusp effect at eta threshold leads to a characteristic distribution of poles on the Riemann sheets. We find a pole located in the $4^{th}$ Riemann sheet that up to now has not been discussed. It belongs to the cusp peak at eta threshold. In addition we obtain the surprising result using the Lee model that the resonance $S_…
On the integration of Riemann-measurable vector-valued functions
2016
We confine our attention to convergence theorems and descriptive relationships within some subclasses of Riemann-measurable vector-valued functions that are based on the various generalizations of the Riemann definition of an integral.
Assessment of a high-resolution central scheme for the solution of the relativistic hydrodynamics equations
2004
We assess the suitability of a recent high-resolution central scheme developed by Kurganov & Tadmor (2000) for the solution of the relativistic hydrodynamics equations. The novelty of this approach relies on the absence of Riemann solvers in the solution procedure. The computations we present are performed in one and two spatial dimensions in Minkowski spacetime. Standard numerical experiments such as shock tubes and the relativistic flat-faced step test are performed. As an astrophysical application the article includes two-dimensional simulations of the propagation of relativistic jets using both Cartesian and cylindrical coordinates. The simulations reported clearly show the capabili…
PRIME NUMBERS, QUANTUM FIELD THEORY AND THE GOLDBACH CONJECTURE
2012
Motivated by the Goldbach conjecture in Number Theory and the abelian bosonization mechanism on a cylindrical two-dimensional spacetime we study the reconstruction of a real scalar field as a product of two real fermion (so-called \textit{prime}) fields whose Fourier expansion exclusively contains prime modes. We undertake the canonical quantization of such prime fields and construct the corresponding Fock space by introducing creation operators $b_{p}^{\dag}$ --labeled by prime numbers $p$-- acting on the vacuum. The analysis of our model, based on the standard rules of quantum field theory and the assumption of the Riemann hypothesis, allow us to prove that the theory is not renormalizabl…