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"Infinity"
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Hero challenge!
A latest entry in a Disney infinity chapter book series based on the popular video game and starring a diverse cast of favorites characters, from Jack Sparrow and Dash to Anna and Wreck-It Ralph.
Book
Some properties of monoids with infinity
We introduce the notion of PC cancellative additive monoids with infinity and use it to characterize cancellative additive principal ideal domains with infinity. Our characterization improves various known characterizations from the literature, both, in the context of the commutative cancellative monoids, as well as in the context of the analogues of the statements from the commutative ring theory.
Journal Article
The Infinity Laplacian Eigenvalue Problem: Reformulation and a Numerical Scheme
In this work, we present an alternative formulation of the higher eigenvalue problem associated to the infinity Laplacian, which opens the door for numerical approximation of eigenfunctions. A rigorous analysis is performed to show the equivalence of the new formulation to the traditional one. Subsequently, we present consistent monotone schemes to approximate infinity ground states and higher eigenfunctions on grids. We prove that our method converges (up to a subsequence) to a viscosity solution of the eigenvalue problem, and perform numerical experiments which investigate theoretical conjectures and compute eigenfunctions on a variety of different domains.
Journal Article
Dichotomy
Among Zeno paradoxes the most known are Dichotomy, Achilles, Arrow, and Stadium. These argumentations state that movement is impossible since it is not thinkable. According to the ascendant form of dichotomy, a mobile cannot touch its destination since it always must to reach the half of the distance. The solutions of Diogenes, Aristotle and mathematical analysis are not satisfactory. Finally, the difference between rest and movement can be only conventionally established.
Journal Article
Robust adaptive H-infinity based controller for islanded microgrid supplying non-linear and unbalanced loads
This study introduces a proposed control method for microgrids (MGs) in islanded (off-grid) mode. The proposed control method is developed by modifying the droop control method using H-infinity controller. In this control method, the droop control loop, current and voltage control loops are adjusted to respond to system load variation. The proposed method is an adaptive control one as it regulates the system voltage and frequency to their nominal values after system load variations. Also, it is a repetitive control method as it depends on the internal model principle that provides good performance for voltage and current error tracking. To prove the applicability and effectiveness of the proposed method, it is applied to a test system using MATLAB/Simulink under three different loading conditions. The results are compared with those of droop control and they prove the effectiveness of the proposed method in adjusting MGs under the off-grid mode of operation. Also, a system stability analysis is performed based on root locus and system step response. Robustness analysis is performed to prove the ability of the proposed controller to restore the system performance after the fault clearance.
Journal Article
ON THE RATE OF CONVERGENCE OF FULLY CONNECTED DEEP NEURAL NETWORK REGRESSION ESTIMATES
Recent results in nonparametric regression show that deep learning, that is, neural network estimates with many hidden layers, are able to circumvent the so-called curse of dimensionality in case that suitable restrictions on the structure of the regression function hold. One key feature of the neural networks used in these results is that their network architecture has a further constraint, namely the network sparsity. In this paper, we show that we can get similar results also for least squares estimates based on simple fully connected neural networks with ReLU activation functions. Here, either the number of neurons per hidden layer is fixed and the number of hidden layers tends to infinity suitably fast for sample size tending to infinity, or the number of hidden layers is bounded by some logarithmic factor in the sample size and the number of neurons per hidden layer tends to infinity suitably fast for sample size tending to infinity. The proof is based on new approximation results concerning deep neural networks.
Journal Article
Nef divisors of surfaces given by pencils at infinity
We give generators for the nef cone and the cone of curves of rational surfaces obtained by blowing-up the complex projective plane at a set of points \\(\\mathcal{B} \\cup \\mathcal{D}\\), where \\(\\mathcal{B}\\) is the set of (proper and infinitely near) base points of a pencil associated with a curve having one place at infinity, and \\(\\mathcal{D}\\) is a set of finitely many infinitely near free points on the strict transforms of curves of the pencil. We also prove that, when the pencil is given by an AMS-type curve and \\(\\mathcal{D}\\) contains at most two free points on any curve considered, the Cox ring of the obtained surface is finitely generated.
Paper
Multi-model strategies estimation and active intercept control of spacecraft via a zonotopic estimation approach
This paper further explores the challenge of spacecraft interception games under incomplete information. The pursued spacecraft can change multiple evasive strategies, posing a pursuit problem for the interceptor that involves adapting to these changing strategies. A combination of interacting multiple model approach and zonotopic H ∞ observer is proposed to estimate the target’s unknown information. The interception strategies are then updated based on the estimated information. A simulation is given for comparison and to show their effectiveness.
Journal Article
Hyperkähler ambient metrics associated with twistor CR manifolds
Twistor CR manifolds, introduced by LeBrun, are Lorentzian (neutral) CR 5-manifolds defined as \\(\\mathbb{P}^1\\)-bundles over 3-dimensional conformal manifolds. In this paper, we embed a real analytic twistor CR manifold into the twistor space of the anti self-dual Poincaré-Einstein metric whose conformal infinity is the base conformal 3-manifold, and construct the associated Fefferman ambient metric as a neutral hyperk\"ahler metric on the spinor bundle with the zero section removed. We also describe the structure of the Cheng--Yau type K\"ahler-Einstein metric which has the twistor CR manifold as the boundary at infinity.
Paper