47,806 research outputs found
Formation control of robots in nonlinear two-dimensional potential
The formation control of multi-agent systems has garnered significant
research attention in both theoretical and practical aspects over the past two
decades. Despite this, the examination of how external environments impact
swarm formation dynamics and the design of formation control algorithms for
multi-agent systems in nonlinear external potentials have not been thoroughly
explored. In this paper, we apply our theoretical formulation of the formation
control algorithm to mobile robots operating in nonlinear external potentials.
To validate the algorithm's effectiveness, we conducted experiments using real
mobile robots. Furthermore, the results demonstrate the effectiveness of
Dynamic Mode Decomposition in predicting the velocity of robots in unknown
environments
Can you hear your location on a manifold?
We introduce a variation on Kac's question, "Can one hear the shape of a
drum?" Instead of trying to identify a compact manifold and its metric via its
Laplace--Beltrami spectrum, we ask if it is possible to uniquely identify a
point on the manifold, up to symmetry, from its pointwise counting function
where here and form an orthonormal
basis for . This problem has a physical interpretation. You are placed
at an arbitrary location in a familiar room with your eyes closed. Can you
identify your location in the room by clapping your hands once and listening to
the resulting echos and reverberations?
The main result of this paper provides an affirmative answer to this question
for a generic class of metrics. We also probe the problem with a variety of
simple examples, highlighting along the way helpful geometric invariants that
can be pulled out of the pointwise counting function .Comment: 26 pages, 1 figur
Optimal Control of the Landau-de Gennes Model of Nematic Liquid Crystals
We present an analysis and numerical study of an optimal control problem for
the Landau-de Gennes (LdG) model of nematic liquid crystals (LCs), which is a
crucial component in modern technology. They exhibit long range orientational
order in their nematic phase, which is represented by a tensor-valued (spatial)
order parameter . Equilibrium LC states correspond to functions
that (locally) minimize an LdG energy functional. Thus, we consider an
-gradient flow of the LdG energy that allows for finding local minimizers
and leads to a semi-linear parabolic PDE, for which we develop an optimal
control framework. We then derive several a priori estimates for the forward
problem, including continuity in space-time, that allow us to prove existence
of optimal boundary and external ``force'' controls and to derive optimality
conditions through the use of an adjoint equation. Next, we present a simple
finite element scheme for the LdG model and a straightforward optimization
algorithm. We illustrate optimization of LC states through numerical
experiments in two and three dimensions that seek to place LC defects (where
) in desired locations, which is desirable in applications.Comment: 26 pages, 9 figure
Band width estimates with lower scalar curvature bounds
A band is a connected compact manifold X together with a decomposition ∂X = ∂−X t ∂+X where ∂±X are non-empty unions of boundary components. If X is equipped with a Riemannian metric, the pair (X, g) is called a Riemannian band and the width of (X, g) is defined to be the distance between ∂−X and ∂+X with respect to g.
Following Gromov’s seminal work on metric inequalities with scalar curvature, the study of Riemannian bands with lower curvature bounds has been an active field of research in recent years, which led to several breakthroughs on longstanding open problems in positive scalar curvature geometry and to a better understanding of the positive mass theorem in general relativity
In the first part of this thesis we combine ideas of Gromov and Cecchini-Zeidler and use the variational calculus surrounding so called µ-bubbles to establish a scalar and mean curvature comparison principle for Riemannian bands with the property that no closed embedded hypersurface which separates the two ends of the band
admits a metric of positive scalar curvature. The model spaces we use for this comparison are warped product over scalar flat manifolds with log-concave warping functions.
We employ ideas from surgery and bordism theory to deduce that, if Y is a closed orientable manifold which does not admit a metric of positive scalar curvature, dim(Y ) 6= 4 and Xn≤7 = Y ×[−1, 1], the width of X with respect to any Riemannian metric with scalar curvature ≥ n(n − 1) is bounded from above by 2π n. This solves, up to dimension 7, a conjecture due to Gromov in the orientable case.
Furthermore, we adapt and extend our methods to show that, if Y is as before and Mn≤7 = Y × R, then M does not admit a metric of positive scalar curvature. This solves, up to dimension 7 a conjecture due to Rosenberg and Stolz in the orientable case.
In the second part of this thesis we explore how these results transfer to the setting where the lower scalar curvature bound is replaced by a lower bound on the macroscopic scalar curvature of a Riemannian band. This curvature condition amounts to an upper bound on the volumes of all unit balls in the universal cover of the band.
We introduce a new class of orientable manifolds we call filling enlargeable and prove: If Y is filling enlargeable, Xn = Y × [−1, 1] and g is a Riemannian metric on X with the property that the volumes of all unit balls in the universal cover of (X, g) are bounded from above by a small dimensional constant εn, then width(X, g) ≤ 1.
Finally, we establish that whether or not a closed orientable manifold is filling enlargeable or not depends on the image of the fundamental class under the classifying map of the universal cover
Richness of dynamics and global bifurcations in systems with a homoclinic figure-eight
We consider 2D flows with a homoclinic figure-eight to a dissipative saddle. We study the rich dynamics that such a system exhibits under a periodic forcing. First, we derive the bifurcation diagram using topological techniques. In particular, there is a homoclinic zone in the parameter space with a non-smooth boundary. We provide a complete explanation of this phenomenon relating it to primary quadratic homoclinic tangency curves which end up at some cubic tangency (cusp) points. We also describe the possible attractors that exist (and may coexist) in the system. A main goal of this work is to show how the previous qualitative description can be complemented with quantitative global information. To this end, we introduce a return map model which can be seen as the simplest one which is 'universal' in some sense. We carry out several numerical experiments on the model, to check that all the objects predicted to exist by the theory are found in the model, and also to investigate new properties of the system
The integral cohomology ring of four-dimensional toric orbifolds
Although toric orbifolds are fundamental objects in toric topology, their
cohomology rings are largely unknown except for very few special cases. The
goal of this paper is to investigate the cohomology rings of 4-dimensional
toric orbifolds. Let is a 4-dimensional toric orbifold
associated to a polygon and a characteristic function . Assuming
is locally smooth over a vertex of , we construct an additive
basis of and express the cup products of the
basis elements in terms of and . Further we derive a formula for
computing cup products in , where is any
general 4-dimensional toric orbifold and is a principal ideal domain
satisfying a mild condition.Comment: 51 pages, 6 figure
Exact -dimensional cosmological-type solutions in gravitational model with Yang-Mills field, Gauss-Bonnet term and -term
We consider -dimensional gravitational model with Yang-Mills
field, Gauss-Bonnet term and -term. We study so-called cosmological
type solutions defined on product manifold , where
is Calabi-Yau manifold. By putting the gauge field 1-form to be
coinciding with 1-form spin connection on , we obtain exact cosmological
solutions with exponential dependence of scale factors (upon -variable),
governed by two non-coinciding Hubble-like parameters: , , obeying . We also present static analogs of these cosmological solutions
(for , and ). The islands of stability
for both classes of solutions are outlined.Comment: 17 pages, 3 figures, LaTex, Revised version: 3 paragraphs are added
into Introduction, new references are included and few references
(self-citations) are omitte
Radial solutions of the Lane-Emden system on Cartan-Hadamard manifolds: asymptotics and rigidity
We investigate existence and qualitative properties of globally defined and
positive radial solutions of the Lane-Emden system, posed on a Cartan-Hadamard
model manifold . We prove that, for critical or supercritical
exponents, there exists at least a one-parameter family of such solutions.
Depending on the stochastic completeness or incompleteness of ,
we show that the existence region stays one dimensional in the former case,
whereas it becomes two dimensional in the latter. Then, we study the
asymptotics at infinity of solutions, which again exhibit a dichotomous
behavior between the stochastically complete (where both components are forced
to vanish) and incomplete cases. Finally, we prove a rigidity result for
finite-energy solutions, showing that they exist if and only if is isometric to
Transport Densities and Congested Optimal Transport in the Heisenberg Group
We adapt the problem of continuous congested optimal transport to the
Heisenberg group with a sub-riemannian metric: we restrict the set of
admissible paths to the absolutely continuous curves which are also horizontal.
We get the existence of equilibrium configurations, known as Wardrop
Equilibria, through the minimization of a convex functional over a suitable set
of measures. To prove existence of such minima, that turn out to be equilibria,
we prove the existence of summable transport densities. Moreover, such
equilibria induces transport plans that solve a Monge-Kantorovic problem
associated with a cost function, depending on the congestion itself, which we
rigorously define
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