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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
Optimality Conditions at Infinity for Nonsmooth Minimax Programming Problems with Some Applications
This paper is devoted to the study of optimality conditions at infinity in nonsmooth minimax programming problems and their applications. By means of the limiting subdifferential and the normal cone at infinity, we derive necessary and sufficient optimality conditions of the Karush–Kuhn–Tucker type for nonsmooth minimax programming problems with constraints. The obtained results are applied to nonsmooth vector optimization problems and robust minimax optimization ones.
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
Generating Functions in \\(R^2n\\) and the Hatcher-Waldhausen map
In this paper, we construct a generating function quadratic at infinity for any exact Lagrangian in \\( R^2n\\) that equals \\( R^n\\) outside a compact set. Such a Lagrangian may be viewed as a Lagrangian filling of the standard Legendrian unknot \\(S^n-1\\) in \\(D^2n\\). Generating functions of the type we construct are related to the space \\( M_ınfty\\) considered by Eliashberg and Gromov. We also show that \\( M_ınfty\\) is the homotopy fiber of the so-called Hatcher--Waldhausen map. This further relates the study of exact Lagrangians (and Legendrians) to algebraic K-theory of spaces. Using this and Bökstedt's result that the Hatcher--Waldhausen map is a rational homotopy equivalence, we prove that the stable Lagrangian Gauss map (relative to the boundary) of the Lagrangian is null-homotopic.
Paper
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
Solutions of Difference Equations Almost Periodic at Infinity
We consider a new class of functions almost periodic at infinity defined by using the subspace of functions that integrally decrease at infinity. We propose four definitions of functions almost periodic at infinity and prove their equivalence. Also, we obtain spectral criteria of almost periodicity at infinity of bounded solutions of systems of linear difference equations and their asymptotic representation.
Journal Article