2,432 research outputs found
The Morse theory of \v{C}ech and Delaunay complexes
Given a finite set of points in and a radius parameter, we
study the \v{C}ech, Delaunay-\v{C}ech, Delaunay (or Alpha), and Wrap complexes
in the light of generalized discrete Morse theory. Establishing the \v{C}ech
and Delaunay complexes as sublevel sets of generalized discrete Morse
functions, we prove that the four complexes are simple-homotopy equivalent by a
sequence of simplicial collapses, which are explicitly described by a single
discrete gradient field.Comment: 21 pages, 2 figures, improved expositio
[4 + 2] Cycloadditions of diphenylketene with a highly substituted 1,3-diene
Diphenylketene (2) undergoes [4 + 2] cycloadditions with the s-cis fixed diene (1) to give the dihydropyran (3) and the cyclohexenone (4)
\v{C}ech-Delaunay gradient flow and homology inference for self-maps
We call a continuous self-map that reveals itself through a discrete set of
point-value pairs a sampled dynamical system. Capturing the available
information with chain maps on Delaunay complexes, we use persistent homology
to quantify the evidence of recurrent behavior. We establish a sampling theorem
to recover the eigenspace of the endomorphism on homology induced by the
self-map. Using a combinatorial gradient flow arising from the discrete Morse
theory for \v{C}ech and Delaunay complexes, we construct a chain map to
transform the problem from the natural but expensive \v{C}ech complexes to the
computationally efficient Delaunay triangulations. The fast chain map algorithm
has applications beyond dynamical systems.Comment: 22 pages, 8 figure
The light baryon spectrum in a relativistic quark model with instanton-induced quark forces I. The non-strange baryon spectrum and ground-states
This is the second of a series of three papers treating light baryon
resonances up to 3 GeV within a relativistically covariant quark model based on
the three-fermion Bethe-Salpeter equation with instantaneous two- and
three-body forces. In this paper we apply the covariant Salpeter framework
(which we developed in the first paper) to specific quark model calculations.
Quark confinement is realized by a linearly rising three-body string potential
with appropriate spinorial structures in Dirac-space. To describe the hyperfine
structure of the baryon spectrum we adopt 't Hooft's residual interaction based
on QCD-instanton effects and demonstrate that the alternative
one-gluon-exchange is disfavored phenomenological grounds. Our fully
relativistic framework allows to investigate the effects of the full Dirac
structures of residual and confinement forces on the structure of the mass
spectrum. In the present paper we present a detailed analysis of the complete
non-strange baryon spectrum and show that several prominent features of the
nucleon spectrum such as e.g. the Roper resonance and approximate ''parity
doublets'' can be uniformly explained due to a specific interplay of
relativistic effects, the confinement potential and 't Hooft's force. The
results for the spectrum of strange baryons will be discussed in a subsequent
paper.Comment: 59 p. postscript, including 24 figures and 25 tables, submitted to
Eur.Phys.J.
Coupled Simulation of Transient Heat Flow and Electric Currents in Thin Wires: Application to Bond Wires in Microelectronic Chip Packaging
This work addresses the simulation of heat flow and electric currents in thin
wires. An important application is the use of bond wires in microelectronic
chip packaging. The heat distribution is modeled by an electrothermal coupled
problem, which poses numerical challenges due to the presence of different
geometric scales. The necessity of very fine grids is relaxed by solving and
embedding a 1D sub-problem along the wire into the surrounding 3D geometry. The
arising singularities are described using de Rham currents. It is shown that
the problem is related to fluid flow in porous 3D media with 1D fractures [C.
D'Angelo, SIAM Journal on Numerical Analysis 50.1, pp. 194-215, 2012]. A
careful formulation of the 1D-3D coupling condition is essential to obtain a
stable scheme that yields a physical solution. Elliptic model problems are used
to investigate the numerical errors and the corresponding convergence rates.
Additionally, the transient electrothermal simulation of a simplified
microelectronic chip package as used in industrial applications is presented.Comment: all numerical results can be reproduced by the Matlab code openly
available at https://github.com/tc88/ETwireSi
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