86,477 research outputs found
Implementing Brouwer's database of strongly regular graphs
Andries Brouwer maintains a public database of existence results for strongly
regular graphs on vertices. We implemented most of the infinite
families of graphs listed there in the open-source software Sagemath, as well
as provided constructions of the "sporadic" cases, to obtain a graph for each
set of parameters with known examples. Besides providing a convenient way to
verify these existence results from the actual graphs, it also extends the
database to higher values of .Comment: 18 pages, LaTe
Self-adjoint symmetry operators connected with the magnetic Heisenberg ring
We consider symmetry operators a from the group ring C[S_N] which act on the
Hilbert space H of the 1D spin-1/2 Heisenberg magnetic ring with N sites. We
investigate such symmetry operators a which are self-adjoint (in a sence
defined in the paper) and which yield consequently observables of the
Heisenberg model. We prove the following results: (i) One can construct a
self-adjoint idempotent symmetry operator from every irreducible character of
every subgroup of S_N. This leads to a big manifold of observables. In
particular every commutation symmetry yields such an idempotent. (ii) The set
of all generating idempotents of a minimal right ideal R of C[S_N] contains one
and only one idempotent which ist self-adjoint. (iii) Every self-adjoint
idempotent e can be decomposed into primitive idempotents e = f_1 + ... + f_k
which are also self-adjoint and pairwise orthogonal. We give a computer
algorithm for the calculation of such decompositions. Furthermore we present 3
additional algorithms which are helpful for the calculation of self-adjoint
operators by means of discrete Fourier transforms of S_N. In our investigations
we use computer calculations by means of our Mathematica packages PERMS and
HRing.Comment: 13 page
Note on Ward-Horadam H(x) - binomials' recurrences and related interpretations, II
We deliver here second new recurrence formula,
were array is appointed by sequence of
functions which in predominantly considered cases where chosen to be
polynomials . Secondly, we supply a review of selected related combinatorial
interpretations of generalized binomial coefficients. We then propose also a
kind of transfer of interpretation of coefficients onto
coefficients interpretations thus bringing us back to
and Donald Ervin Knuth relevant investigation decades
ago.Comment: 57 pages, 8 figure
Dynamics on geometrically finite hyperbolic manifolds with applications to Apollonian circle packings and beyond
We present recent results on counting and distribution of circles in a given
circle packing invariant under a geometrically finite Kleinian group and
discuss how the dynamics of flows on geometrically finite hyperbolic
manifolds are related. Our results apply to Apollonian circle packings,
Sierpinski curves, Schottky dances, etc.Comment: To appear in the Proceedings of ICM, 201
A Model Reduction Framework for Efficient Simulation of Li-Ion Batteries
In order to achieve a better understanding of degradation processes in
lithium-ion batteries, the modelling of cell dynamics at the mircometer scale
is an important focus of current mathematical research. These models lead to
large-dimensional, highly nonlinear finite volume discretizations which, due to
their complexity, cannot be solved at cell scale on current hardware. Model
order reduction strategies are therefore necessary to reduce the computational
complexity while retaining the features of the model. The application of such
strategies to specialized high performance solvers asks for new software
designs allowing flexible control of the solvers by the reduction algorithms.
In this contribution we discuss the reduction of microscale battery models with
the reduced basis method and report on our new software approach on integrating
the model order reduction software pyMOR with third-party solvers. Finally, we
present numerical results for the reduction of a 3D microscale battery model
with porous electrode geometry.Comment: 7 pages, 2 figures, 2 table
Mixed Ehrhart polynomials
For lattice polytopes , Bihan (2014)
introduced the discrete mixed volume in analogy
to the classical mixed volume. In this note we initiate the study of the
associated mixed Ehrhart polynomial . We study properties of this polynomial and we
give interpretations for some of its coefficients in terms of (discrete) mixed
volumes. Bihan (2014) showed that the discrete mixed volume is always
non-negative. Our investigations yield simpler proofs for certain special
cases. We also introduce and study the associated mixed -vector. We show
that for large enough dilates the corresponding mixed
-polynomial has only real roots and as a consequence the mixed
-vector becomes non-negative.Comment: 12 page
Apollonian circle packings: Dynamics and Number theory
We give an overview of various counting problems for Apollonian circle
packings, which turn out to be related to problems in dynamics and number
theory for thin groups. This survey article is an expanded version of my
lecture notes prepared for the 13th Takagi lectures given at RIMS, Kyoto in the
fall of 2013.Comment: To appear in Japanese Journal of Mat
A phase-field model for fractures in incompressible solids
Within this work, we develop a phase-field description for simulating
fractures in incompressible materials. Standard formulations are subject to
volume-locking when the solid is (nearly) incompressible. We propose an
approach that builds on a mixed form of the displacement equation with two
unknowns: a displacement field and a hydro-static pressure variable.
Corresponding function spaces have to be chosen properly. On the discrete
level, stable Taylor-Hood elements are employed for the displacement-pressure
system. Two additional variables describe the phase-field solution and the
crack irreversibility constraint. Therefore, the final system contains four
variables: displacements, pressure, phase-field, and a Lagrange multiplier. The
resulting discrete system is nonlinear and solved monolithically with a
Newton-type method. Our proposed model is demonstrated by means of several
numerical studies based on two numerical tests. First, different finite element
choices are compared in order to investigate the influence of higher-order
elements in the proposed settings. Further, numerical results including spatial
mesh refinement studies and variations in Poisson's ratio approaching the
incompressible limit, are presented
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