3,216 research outputs found
Geometrical Ambiguity of Pair Statistics. I. Point Configurations
Point configurations have been widely used as model systems in condensed
matter physics, materials science and biology. Statistical descriptors such as
the -body distribution function is usually employed to characterize
the point configurations, among which the most extensively used is the pair
distribution function . An intriguing inverse problem of practical
importance that has been receiving considerable attention is the degree to
which a point configuration can be reconstructed from the pair distribution
function of a target configuration. Although it is known that the pair-distance
information contained in is in general insufficient to uniquely determine
a point configuration, this concept does not seem to be widely appreciated and
general claims of uniqueness of the reconstructions using pair information have
been made based on numerical studies. In this paper, we introduce the idea of
the distance space, called the space. The pair distances of a
specific point configuration are then represented by a single point in the
space. We derive the conditions on the pair distances that can be
associated with a point configuration, which are equivalent to the
realizability conditions of the pair distribution function . Moreover, we
derive the conditions on the pair distances that can be assembled into distinct
configurations. These conditions define a bounded region in the
space. By explicitly constructing a variety of degenerate point configurations
using the space, we show that pair information is indeed
insufficient to uniquely determine the configuration in general. We also
discuss several important problems in statistical physics based on the
space.Comment: 28 pages, 8 figure
Symmetric Relative Equilibria in the Four-Vortex Problem with Three Equal Vorticities
We examine in detail the relative equilibria of the 4-vortex problem when
three vortices have equal strength, that is, , and is a real parameter. We give the exact number
of relative equilibria and bifurcation values. We also study the relative
equilibria in the vortex rhombus problem.Comment: 21 pages, 4 figures. arXiv admin note: text overlap with
arXiv:1301.6194 by other author
Linear Complexity Hexahedral Mesh Generation
We show that any polyhedron forming a topological ball with an even number of
quadrilateral sides can be partitioned into O(n) topological cubes, meeting
face to face. The result generalizes to non-simply-connected polyhedra
satisfying an additional bipartiteness condition. The same techniques can also
be used to reduce the geometric version of the hexahedral mesh generation
problem to a finite case analysis amenable to machine solution.Comment: 12 pages, 17 figures. A preliminary version of this paper appeared at
the 12th ACM Symp. on Computational Geometry. This is the final version, and
will appear in a special issue of Computational Geometry: Theory and
Applications for papers from SCG '9
Resonance clustering in wave turbulent regimes: Integrable dynamics
Two fundamental facts of the modern wave turbulence theory are 1) existence
of power energy spectra in -space, and 2) existence of "gaps" in this
spectra corresponding to the resonance clustering. Accordingly, three wave
turbulent regimes are singled out: \emph{kinetic}, described by wave kinetic
equations and power energy spectra; \emph{discrete}, characterized by resonance
clustering; and \emph{mesoscopic}, where both types of wave field time
evolution coexist. In this paper we study integrable dynamics of resonance
clusters appearing in discrete and mesoscopic wave turbulent regimes. Using a
novel method based on the notion of dynamical invariant we establish that some
of the frequently met clusters are integrable in quadratures for arbitrary
initial conditions and some others -- only for particular initial conditions.
We also identify chaotic behaviour in some cases. Physical implications of the
results obtained are discussed.Comment: This is Work In Progress carried out in years 2008-2008 and partly
supported by Austrian FWF-project P20164-N18 and 6 EU Programme under the
project SCIEnce, Contract No. 02613
Linear and nonlinear theories of discrete analytic functions. Integrable structure and isomonodromic Green's function
Two discretizations, linear and nonlinear, of basic notions of the complex
analysis are considered. The underlying lattice is an arbitrary quasicrystallic
rhombic tiling of a plane. The linear theory is based on the discrete
Cauchy-Riemann equations, the nonlinear one is based on the notion of circle
patterns. We clarify the role of the rhombic condition in both theories: under
this condition the corresponding equations are integrable (in the sense of 3D
consistency, which yields also the existense of zero curvature representations,
B"acklund transformations etc.). We demonstrate that in some precise sense the
linear theory is a linearization of the nonlinear one: the tangent space to a
set of integrable circle patterns at an isoradial point consists of discrete
holomorphic functions which take real (imaginary) values on two sublattices. We
extend solutions of the basic equations of both theories to Z^d, where d is the
number of different edge slopes of the quasicrystallic tiling. In the linear
theory, we give an integral representation of an arbitrary discrete holomorphic
function, thus proving the density of discrete exponential functions. We
introduce the d-dimensional discrete logarithmic function which is a
generalization of Kenyon's discrete Green's function, and uncover several new
properties of this function. We prove that it is an isomonodromic solution of
the discrete Cauchy-Riemann equations, and that it is a tangent vector to the
space of integrable circle patterns along the family of isomonodromic discrete
power functions.Comment: 40 page
Systolically extremal nonpositively curved surfaces are flat with finitely many singularities
The regularity of systolically extremal surfaces is a notoriously difficult
problem already discussed by M. Gromov in 1983, who proposed an argument toward
the existence of -extremizers exploiting the theory of -regularity
developed by P. A. White and others by the 1950s. We propose to study the
problem of systolically extremal metrics in the context of generalized metrics
of nonpositive curvature. A natural approach would be to work in the class of
Alexandrov surfaces of finite total curvature, where one can exploit the tools
of the completion provided in the context of Radon measures as studied by
Reshetnyak and others. However the generalized metrics in this sense still
don't have enough regularity. Instead, we develop a more hands-on approach and
show that, for each genus, every systolically extremal nonpositively curved
surface is piecewise flat with finitely many conical singularities. This result
exploits a decomposition of the surface into flat systolic bands and
nonsystolic polygonal regions, as well as the combinatorial/topological
estimates of Malestein-Rivin-Theran, Przytycki, Aougab-Biringer-Gaster and
Greene on the number of curves meeting at most once, combined with a kite
excision move. The move merges pairs of conical singularities on a surface of
genus and leads to an asymptotic upper bound on the number
of singularities.Comment: 31 pages, 5 figures, to appear in Journal of Topology and Analysi
Miquel dynamics for circle patterns
We study a new discrete-time dynamical system on circle patterns with the
combinatorics of the square grid. This dynamics, called Miquel dynamics, relies
on Miquel's six circles theorem. We provide a coordinatization of the
appropriate space of circle patterns on which the dynamics acts and use it to
derive local recurrence formulas. Isoradial circle patterns arise as periodic
points of Miquel dynamics. Furthermore, we prove that certain signed sums of
intersection angles are preserved by the dynamics. Finally, when the initial
circle pattern is spatially biperiodic with a fundamental domain of size two by
two, we show that the appropriately normalized motion of intersection points of
circles takes place along an explicit quartic curve.Comment: 34 pages, 24 figures. Final version to appear in Int. Math. Res.
Notice
Variational principles for circle patterns
A Delaunay cell decomposition of a surface with constant curvature gives rise
to a circle pattern, consisting of the circles which are circumscribed to the
facets. We treat the problem whether there exists a Delaunay cell decomposition
for a given (topological) cell decomposition and given intersection angles of
the circles, whether it is unique and how it may be constructed. Somewhat more
generally, we allow cone-like singularities in the centers and intersection
points of the circles. We prove existence and uniqueness theorems for the
solution of the circle pattern problem using a variational principle. The
functionals (one for the euclidean, one for the hyperbolic case) are convex
functions of the radii of the circles. The analogous functional for the
spherical case is not convex, hence this case is treated by stereographic
projection to the plane. From the existence and uniqueness of circle patterns
in the sphere, we derive a strengthened version of Steinitz' theorem on the
geometric realizability of abstract polyhedra.
We derive the variational principles of Colin de Verdi\`ere, Br\"agger, and
Rivin for circle packings and circle patterns from our variational principles.
In the case of Br\"agger's and Rivin's functionals. Leibon's functional for
hyperbolic circle patterns cannot be derived directly from our functionals. But
we construct yet another functional from which both Leibon's and our
functionals can be derived.
We present Java software to compute and visualize circle patterns.Comment: PhD thesis, iv+94 pages, many figures (mostly vector graphics
Some Remarks on Kite Pseudo Effect Algebras
Recently a new family of pseudo effect algebras, called kite pseudo effect
algebras, was introduced. Such an algebra starts with a po-group , a set
and with two bijections Using a clever construction on
the ordinal sum of and we can define a pseudo effect
algebra which can be non-commutative even if is an Abelian po-group. In the
paper we give a characterization of subdirect product of subdirectly
irreducible kite pseudo effect algebras, and we show that every kite pseudo
effect algebra is an interval in a unital po-loop.Comment: arXiv admin note: text overlap with arXiv:1306.030
Constraint capture and maintenance in engineering design
The Designers' Workbench is a system, developed by the Advanced Knowledge Technologies (AKT) consortium to support designers in large organizations, such as Rolls-Royce, to ensure that the design is consistent with the specification for the particular design as well as with the company's design rule book(s). In the principal application discussed here, the evolving design is described against a jet engine ontology. Design rules are expressed as constraints over the domain ontology. Currently, to capture the constraint information, a domain expert (design engineer) has to work with a knowledge engineer to identify the constraints, and it is then the task of the knowledge engineer to encode these into the Workbench's knowledge base (KB). This is an error prone and time consuming task. It is highly desirable to relieve the knowledge engineer of this task, and so we have developed a system, ConEditor+ that enables domain experts themselves to capture and maintain these constraints. Further we hypothesize that in order to appropriately apply, maintain and reuse constraints, it is necessary to understand the underlying assumptions and context in which each constraint is applicable. We refer to them as “application conditions” and these form a part of the rationale associated with the constraint. We propose a methodology to capture the application conditions associated with a constraint and demonstrate that an explicit representation (machine interpretable format) of application conditions (rationales) together with the corresponding constraints and the domain ontology can be used by a machine to support maintenance of constraints. Support for the maintenance of constraints includes detecting inconsistencies, subsumption, redundancy, fusion between constraints and suggesting appropriate refinements. The proposed methodology provides immediate benefits to the designers and hence should encourage them to input the application conditions (rationales)
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