54 research outputs found
Macroscopic network circulation for planar graphs
The analysis of networks, aimed at suitably defined functionality, often
focuses on partitions into subnetworks that capture desired features. Chief
among the relevant concepts is a 2-partition, that underlies the classical
Cheeger inequality, and highlights a constriction (bottleneck) that limits
accessibility between the respective parts of the network. In a similar spirit,
the purpose of the present work is to introduce a new concept of maximal global
circulation and to explore 3-partitions that expose this type of macroscopic
feature of networks. Herein, graph circulation is motivated by transportation
networks and probabilistic flows (Markov chains) on graphs. Our goal is to
quantify the large-scale imbalance of network flows and delineate key parts
that mediate such global features. While we introduce and propose these notions
in a general setting, in this paper, we only work out the case of planar
graphs. We explain that a scalar potential can be identified to encapsulate the
concept of circulation, quite similarly as in the case of the curl of planar
vector fields. Beyond planar graphs, in the general case, the problem to
determine global circulation remains at present a combinatorial problem
On incorrectness of application of the Helmholtz decomposition to microscopic electrodynamics
The integral expressions served to decompose vector field into irrotational
and divergence-free components represent modern version of the Helmholtz
decomposition theorem. These expressions are also widely used to decompose the
electromagnetic fields. However, an appropriate analysis of application of
these expressions to electrodynamics shows that the improper integral arising
in the procedure for calculating these components makes such a decomposition
impossible.Comment: AMS-LaTeX, 6 page
Observation of magnetic fragmentation in spin ice
Fractionalised excitations that emerge from a many body system have revealed
rich physics and concepts, from composite fermions in two-dimensional electron
systems, revealed through the fractional quantum Hall effect, to spinons in
antiferromagnetic chains and, more recently, fractionalisation of Dirac
electrons in graphene and magnetic monopoles in spin ice. Even more surprising
is the fragmentation of the degrees of freedom themselves, leading to
coexisting and a priori independent ground states. This puzzling phenomenon was
recently put forward in the context of spin ice, in which the magnetic moment
field can fragment, resulting in a dual ground state consisting of a
fluctuating spin liquid, a so-called Coulomb phase, on top of a magnetic
monopole crystal. Here we show, by means of neutron scattering measurements,
that such fragmentation occurs in the spin ice candidate NdZrO. We
observe the spectacular coexistence of an antiferromagnetic order induced by
the monopole crystallisation and a fluctuating state with ferromagnetic
correlations. Experimentally, this fragmentation manifests itself via the
superposition of magnetic Bragg peaks, characteristic of the ordered phase, and
a pinch point pattern, characteristic of the Coulomb phase. These results
highlight the relevance of the fragmentation concept to describe the physics of
systems that are simultaneously ordered and fluctuating.Comment: accepted in Nature Physic
FLUID FLOW MODELLING WITH FREE SURFACE
Fluid is a substance that can flow in the form of a liquid or a gas. Based on the movement of the fluid is divided into static and dynamic fluids. This study discusses fluid dynamics, namely modelling fluid flow accompanied by a free surface and an obstacle in the fluid flow. Fluid modelling generally makes some basic assumptions into mathematical equations. The assumptions are incompressible, steady-state and irrotational. The steps to obtain a fluid flow model are using Newton’s second law, the law of conservation of mass, and the law of conservation of momentum to obtain the general Navier-Stokes equation, the designing the Euler free surface equation, the Bernoulli equation, then making a free surface representation and linearizing the wave equation so that it is obtained fluid flow model. The resulting mathematical model is a Laplace equation with boundary conditions in the fluid
On Meshfree GFDM Solvers for the Incompressible Navier-Stokes Equations
Meshfree solution schemes for the incompressible Navier--Stokes equations are
usually based on algorithms commonly used in finite volume methods, such as
projection methods, SIMPLE and PISO algorithms. However, drawbacks of these
algorithms that are specific to meshfree methods have often been overlooked. In
this paper, we study the drawbacks of conventionally used meshfree Generalized
Finite Difference Method~(GFDM) schemes for Lagrangian incompressible
Navier-Stokes equations, both operator splitting schemes and monolithic
schemes. The major drawback of most of these schemes is inaccurate local
approximations to the mass conservation condition. Further, we propose a new
modification of a commonly used monolithic scheme that overcomes these problems
and shows a better approximation for the velocity divergence condition. We then
perform a numerical comparison which shows the new monolithic scheme to be more
accurate than existing schemes
Reference-less complex wavefields characterization with a high-resolution wavefront sensor
Wavefront sensing is a widely-used non-interferometric, single-shot, and
quantitative technique providing the spatial-phase of a beam. The phase is
obtained by integrating the measured wavefront gradient. Complex and random
wavefields intrinsically contain a high density of singular phase structures
(optical vortices) associated with non-conservative gradients making this
integration step especially delicate. Here, using a high-resolution wavefront
sensor, we demonstrate experimentally a systematic approach for achieving the
complete and quantitative reconstruction of complex wavefronts. Based on the
Stokes' theorem, we propose an image segmentation algorithm to provide an
accurate determination of the charge and location of optical vortices. This
technique is expected to benefit to several fields requiring complex media
characterization.Comment: 7 page
Approximation of tensor fields on surfaces of arbitrary topology based on local Monge parametrizations
We introduce a new method, the Local Monge Parametrizations (LMP) method, to
approximate tensor fields on general surfaces given by a collection of local
parametrizations, e.g.~as in finite element or NURBS surface representations.
Our goal is to use this method to solve numerically tensor-valued partial
differential equations (PDE) on surfaces. Previous methods use scalar
potentials to numerically describe vector fields on surfaces, at the expense of
requiring higher-order derivatives of the approximated fields and limited to
simply connected surfaces, or represent tangential tensor fields as tensor
fields in 3D subjected to constraints, thus increasing the essential number of
degrees of freedom. In contrast, the LMP method uses an optimal number of
degrees of freedom to represent a tensor, is general with regards to the
topology of the surface, and does not increase the order of the PDEs governing
the tensor fields. The main idea is to construct maps between the element
parametrizations and a local Monge parametrization around each node. We test
the LMP method by approximating in a least-squares sense different vector and
tensor fields on simply connected and genus-1 surfaces. Furthermore, we apply
the LMP method to two physical models on surfaces, involving a tension-driven
flow (vector-valued PDE) and nematic ordering (tensor-valued PDE). The LMP
method thus solves the long-standing problem of the interpolation of tensors on
general surfaces with an optimal number of degrees of freedom.Comment: 16 pages, 6 figure
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