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Register a new userA nonlinear Lazarev-Lieb theorem: $L^2$-orthogonality via motion planning
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Lazarev and Lieb showed that finitely many integrable functions from the unit interval to $mathbb{C}$ can be simultaneously annihilated in the $L^2$ inner product by a smooth function to the unit circle. Here we answer a question of Lazarev and Lieb proving a generalization of their result by lower bounding the equivariant topology of the space of smooth circle-valued functions with a certain $W^{1,1}$-norm bound. Our proof uses a relaxed notion of motion planning algorithm that instead of contractibility yields a lower bound for the $mathbb{Z}/2$-coindex of a space.
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The $(k,a)$-generalised Fourier transform is the unitary operator defined using the $a$-deformed Dunkl harmonic oscillator. The main aim of this paper is to prove $L^p$-$L^q$ boundedness of $(k, a)$-generalised Fourier multipliers. To show the boundedness we first establish Paley inequality and Hausdorff-Young-Paley inequality for $(k, a)$-generalised Fourier transform. We also demonstrate applications of obtained results to study the well-posedness of nonlinear partial differential equations.
Wavelet set wavelets were the first examples of wavelets that may not have associated multiresolution analyses. Furthermore, they provided examples of complete orthonormal wavelet systems in $L^2(mathbb{R}^d)$ which only require a single generating wavelet. Although work had been done to smooth these wavelets, which are by definition discontinuous on the frequency domain, nothing had been explicitly done over $mathbb{R}^d$, $d >1$. This paper, along with another one cowritten by the author, finally addresses this issue. Smoothing does not work as expected in higher dimensions. For example, Bin Hans proof of existence of Schwartz class functions which are Parseval frame wavelets and approximate Parseval frame wavelet set wavelets does not easily generalize to higher dimensions. However, a construction of wavelet sets in $hat{mathbb{R}}^d$ which may be smoothed is presented. Finally, it is shown that a commonly used class of functions cannot be the result of convolutional smoothing of a wavelet set wavelet.
Let $(Omega,{mathcal F},P)$ be a probability space and $L^0({mathcal F})$ the algebra of equivalence classes of real-valued random variables defined on $(Omega,{mathcal F},P)$. A left module $M$ over the algebra $L^0({mathcal F})$(briefly, an $L^0({mathcal F})$-module) is said to be regular if $x=y$ for any given two elements $x$ and $y$ in $M$ such that there exists a countable partition ${A_n,nin mathbb N}$ of $Omega$ to $mathcal F$ such that ${tilde I}_{A_n}cdot x={tilde I}_{A_n}cdot y$ for each $nin mathbb N$, where $I_{A_n}$ is the characteristic function of $A_n$ and ${tilde I}_{A_n}$ its equivalence class. The purpose of this paper is to establish the fundamental theorem of affine geometry in regular $L^0({mathcal F})$-modules: let $V$ and $V^prime$ be two regular $L^0({mathcal F})$-modules such that $V$ contains a free $L^0({mathcal F})$-submodule of rank $2$, if $T:Vto V^prime$ is stable and invertible and maps each $L^0$-line segment onto an $L^0$-line segment, then $T$ must be $L^0$-affine.
Nonlinear programming targets nonlinear optimization with constraints, which is a generic yet complex methodology involving humans for problem modeling and algorithms for problem solving. We address the particularly hard challenge of supporting domain experts in handling, understanding, and trouble-shooting high-dimensional optimization with a large number of constraints. Leveraging visual analytics, users are supported in exploring the computation process of nonlinear constraint optimization. Our system was designed for robot motion planning problems and developed in tight collaboration with domain experts in nonlinear programming and robotics. We report on the experiences from this design study, illustrate the usefulness for relevant example cases, and discuss the extension to visual analytics for nonlinear programming in general.
We investigate dynamical analogues of the $L^2$-Betti numbers for modules over integral group ring of a discrete sofic group. In particular, we show that the $L^2$-Betti numbers exactly measure the failure of addition formula for dynamical invariants.
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