47,806 research outputs found

    Process intensification of oxidative coupling of methane

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    Formation control of robots in nonlinear two-dimensional potential

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    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?

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    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 xx on the manifold, up to symmetry, from its pointwise counting function Nx(λ)=∑λj≤λ∣ej(x)∣2, N_x(\lambda) = \sum_{\lambda_j \leq \lambda} |e_j(x)|^2, where here Δgej=−λj2ej\Delta_g e_j = -\lambda_j^2 e_j and eje_j form an orthonormal basis for L2(M)L^2(M). 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 NxN_x.Comment: 26 pages, 1 figur

    Optimal Control of the Landau-de Gennes Model of Nematic Liquid Crystals

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    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 Q=Q(x)Q = Q(x). Equilibrium LC states correspond to QQ functions that (locally) minimize an LdG energy functional. Thus, we consider an L2L^2-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 Q(x)=0Q(x) = 0) in desired locations, which is desirable in applications.Comment: 26 pages, 9 figure

    Band width estimates with lower scalar curvature bounds

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    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

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    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

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    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 X(P,λ)X(P,\lambda) is a 4-dimensional toric orbifold associated to a polygon PP and a characteristic function λ\lambda. Assuming X(P,λ)X(P,\lambda) is locally smooth over a vertex of PP, we construct an additive basis of H∗(X(P,λ);Z)H^*(X(P,\lambda);\mathbb{Z}) and express the cup products of the basis elements in terms of PP and λ\lambda. Further we derive a formula for computing cup products in H∗(X(P,λ);R)H^*(X(P,\lambda);R), where X(P,λ)X(P,\lambda) is any general 4-dimensional toric orbifold and RR is a principal ideal domain satisfying a mild condition.Comment: 51 pages, 6 figure

    Exact (1+3+6)(1 + 3 + 6)-dimensional cosmological-type solutions in gravitational model with Yang-Mills field, Gauss-Bonnet term and Λ\Lambda-term

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    We consider 1010-dimensional gravitational model with SO(6)SO(6) Yang-Mills field, Gauss-Bonnet term and Λ\Lambda-term. We study so-called cosmological type solutions defined on product manifold M=R×R3×KM = R \times R^3 \times K, where KK is 6d6d Calabi-Yau manifold. By putting the gauge field 1-form to be coinciding with 1-form spin connection on KK, we obtain exact cosmological solutions with exponential dependence of scale factors (upon tt-variable), governed by two non-coinciding Hubble-like parameters: H>0H >0, hh, obeying H+2h≠0 H + 2 h \neq 0. We also present static analogs of these cosmological solutions (for H≠0H \neq 0, h≠Hh \neq H and H+2h≠0 H + 2 h \neq 0). 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

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    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 Mn \mathbb{M}^n . 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 Mn \mathbb{M}^n , 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 Mn \mathbb{M}^n is isometric to Rn \mathbb{R}^n

    Transport Densities and Congested Optimal Transport in the Heisenberg Group

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    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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