Search results for "iPSC"
showing 10 items of 125 documents
Extensions and corona decompositions of low-dimensional intrinsic Lipschitz graphs in Heisenberg groups
2020
This note concerns low-dimensional intrinsic Lipschitz graphs, in the sense of Franchi, Serapioni, and Serra Cassano, in the Heisenberg group $\mathbb{H}^n$, $n\in \mathbb{N}$. For $1\leq k\leq n$, we show that every intrinsic $L$-Lipschitz graph over a subset of a $k$-dimensional horizontal subgroup $\mathbb{V}$ of $\mathbb{H}^n$ can be extended to an intrinsic $L'$-Lipschitz graph over the entire subgroup $\mathbb{V}$, where $L'$ depends only on $L$, $k$, and $n$. We further prove that $1$-dimensional intrinsic $1$-Lipschitz graphs in $\mathbb{H}^n$, $n\in \mathbb{N}$, admit corona decompositions by intrinsic Lipschitz graphs with smaller Lipschitz constants. This complements results that…
Vertical versus horizontal Sobolev spaces
2020
Let $\alpha \geq 0$, $1 < p < \infty$, and let $\mathbb{H}^{n}$ be the Heisenberg group. Folland in 1975 showed that if $f \colon \mathbb{H}^{n} \to \mathbb{R}$ is a function in the horizontal Sobolev space $S^{p}_{2\alpha}(\mathbb{H}^{n})$, then $\varphi f$ belongs to the Euclidean Sobolev space $S^{p}_{\alpha}(\mathbb{R}^{2n + 1})$ for any test function $\varphi$. In short, $S^{p}_{2\alpha}(\mathbb{H}^{n}) \subset S^{p}_{\alpha,\mathrm{loc}}(\mathbb{R}^{2n + 1})$. We show that the localisation can be omitted if one only cares for Sobolev regularity in the vertical direction: the horizontal Sobolev space $S_{2\alpha}^{p}(\mathbb{H}^{n})$ is continuously contained in the vertical Sobolev sp…
Adaptive control of uncertain nonlinear systems with quantized input signal
2018
Abstract This paper proposes new adaptive controllers for uncertain nonlinear systems in the presence of input quantization. The control signal is quantized by a class of sector-bounded quantizers including the uniform quantizer, the logarithmic quantizer and the hysteresis quantizer. To clearly illustrate our approaches, we will start with a class of single-loop nonlinear systems and then extend the results to multi-loop interconnected nonlinear systems. By using backstepping technique, a new adaptive control algorithm is developed by constructing a new compensation method for the effects of the input quantization. A hyperbolic tangent function is introduced in the controller with a new tr…
Adaptive Control of Quantized Uncertain Nonlinear Systems
2017
Abstract This paper proposes a new adaptive controller for uncertain nonlinear systems in presence of quantized input signal and unknown external disturbance. A hysteresis quantizer is incorporated to reduce chattering phenomenon. By proposing a new transformation of the final control signal, using the sector-bound property of the quantizer and introducing a hyperbolic tangent function, the effects from input quantization and external disturbance are effectively compensated and the Lipschitz condition required for the nonlinear functions in the systems is removed. Besides showing global stability, tracking error performance is also established and can be adjusted by tuning certain design pa…
Adaptive consensus of uncertain nonlinear systems with event triggered communication and intermittent actuator faults
2019
This paper investigates distributed consensus tracking problem for uncertain nonlinear systems with event-triggered communication. The common desired trajectory information and each subsystem's state will be broadcast to their linked subsystems only when predefined triggering conditions are satisfied. Compared with the existing related literature, the main features of the results presented in this paper include four folds. (i) A totally distributed adaptive control scheme is developed for multiple nonlinear systems without Lipschitz condition, while with parametric uncertainties. (ii) The derivative of desired trajectory function is allowed unknown by all subsystems and directed communicati…
Nuclear inclusions of pathogenic ataxin-1 induce oxidative stress and perturb the protein synthesis machinery
2020
Spinocerebellar ataxia type-1 (SCA1) is caused by an abnormally expanded polyglutamine (polyQ) tract in ataxin-1. These expansions are responsible for protein misfolding and self-assembly into intranuclear inclusion bodies (IIBs) that are somehow linked to neuronal death. However, owing to lack of a suitable cellular model, the downstream consequences of IIB formation are yet to be resolved. Here, we describe a nuclear protein aggregation model of pathogenic human ataxin-1 and characterize IIB effects. Using an inducible Sleeping Beauty transposon system, we overexpressed the ATXN1(Q82) gene in human mesenchymal stem cells that are resistant to the early cytotoxic effects caused by the expr…
Ahlfors-regular distances on the Heisenberg group without biLipschitz pieces
2015
We show that the Heisenberg group is not minimal in looking down. This answers Problem 11.15 in `Fractured fractals and broken dreams' by David and Semmes, or equivalently, Question 22 and hence also Question 24 in `Thirty-three yes or no questions about mappings, measures, and metrics' by Heinonen and Semmes. The non-minimality of the Heisenberg group is shown by giving an example of an Ahlfors $4$-regular metric space $X$ having big pieces of itself such that no Lipschitz map from a subset of $X$ to the Heisenberg group has image with positive measure, and by providing a Lipschitz map from the Heisenberg group to the space $X$ having as image the whole $X$. As part of proving the above re…
From Feynman–Kac formulae to numerical stochastic homogenization in electrical impedance tomography
2016
In this paper, we use the theory of symmetric Dirichlet forms to derive Feynman–Kac formulae for the forward problem of electrical impedance tomography with possibly anisotropic, merely measurable conductivities corresponding to different electrode models on bounded Lipschitz domains. Subsequently, we employ these Feynman–Kac formulae to rigorously justify stochastic homogenization in the case of a stochastic boundary value problem arising from an inverse anomaly detection problem. Motivated by this theoretical result, we prove an estimate for the speed of convergence of the projected mean-square displacement of the underlying process which may serve as the theoretical foundation for the de…
Products of snowflaked Euclidean lines are not minimal for looking down
2017
We show that products of snowflaked Euclidean lines are not minimal for looking down. This question was raised in Fractured fractals and broken dreams, Problem 11.17, by David and Semmes. The proof uses arguments developed by Le Donne, Li and Rajala to prove that the Heisenberg group is not minimal for looking down. By a method of shortcuts, we define a new distance $d$ such that the product of snowflaked Euclidean lines looks down on $(\mathbb R^N,d)$, but not vice versa.
Nonsmooth Optimization Methods
1999
From the previous chapters we know that after the discretization, elliptic and parabolic hemivariational inequalities can be transformed into substationary point type problems for locally Lipschitz superpotentials and as such will be solved. There is a class of mathematical programming methods especially developed for this type of problems. The aim of this chapter is to give an overview of nonsmooth optimization techniques with special emphasis on the first and the second order bundle methods. We present their basic ideas in the convex case and necessary modifications for nonconvex optimization. We shall use them in the next chapter for the numerical realization of several model examples. L…