Discrete Approximate Solution of Differential Inclusion with Normal Cone and Prox-Regular Set.

Our aim is to calculate the discrete approximate solution of di⁄erential inclusion with normal cone and prox-regular set, the question is how to calculate this solution? We use the discrete approximation property of a new variant of nonconvex sweeping processes involving normal cone and a nite element method. Knowing that The majority of mathematicians have proved only the existence and uniqueness of the solution for this type of inclusions, like: Mordukhovich, Thibault, Aubin, Messaoud, ...etc.

Computability-theoretic complexity of effective Banach spaces

<p><b>We investigate the geometry of effective Banach spaces, namely a sequenceof approximation properties that lies in between a Banach space having a basis and the approximation property.</b></p> <p>We establish some upper bounds on suchproperties, as well as proving some arithmetical lower bounds. Unfortunately,the upper bounds obtained in some cases are far away from the lower bound.</p> <p>However, we will show that much tighter bounds will require genuinely newconstructions, and resolve long-standing open problems in Banach space theory.</p> <p>We also investigate the effectivisations of certain classical theorems in Banachspaces.</p>

Computability-theoretic complexity of effective Banach spaces

<p><b>We investigate the geometry of effective Banach spaces, namely a sequenceof approximation properties that lies in between a Banach space having a basis and the approximation property.</b></p> <p>We establish some upper bounds on suchproperties, as well as proving some arithmetical lower bounds. Unfortunately,the upper bounds obtained in some cases are far away from the lower bound.</p> <p>However, we will show that much tighter bounds will require genuinely newconstructions, and resolve long-standing open problems in Banach space theory.</p> <p>We also investigate the effectivisations of certain classical theorems in Banachspaces.</p>

Ordered Weighted Averaging (OWA), Decision Making Under Uncertainty, and Deep Learning: How Is This All Related?

Among many research areas to which Ron Yager contributed are decision making under uncertainty (in particular, under interval and fuzzy uncertainty) and aggregation – where he proposed, analyzed, and utilized the use of Ordered Weighted Averaging (OWA). The OWA algorithm itself provides only a specific type of data aggregation. However, it turns out that if we allows several OWA stages one after another, we get a scheme with a universal approximation property – moreover, a scheme which is perfectly equivalent to deep neural networks. In this sense, Ron Yager can be viewed as a (grand)father of deep learning. We also show that the existing schemes for decision making under uncertainty are also naturally interpretable in OWA terms.

Compact reduction in Lipschitz-free spaces

We prove a general principle satisfied by weakly precompact sets of Lipschitz-free spaces. By this principle, certain infinite dimensional phenomena in Lipschitz-free spaces over general metric spaces may be reduced to the same phenomena in free spaces over their compact subsets. As easy consequences we derive several new and some known results. The main new results are: $\mathcal {F}(X)$ is weakly sequentially complete for every superreflexive Banach space $X$, and $\mathcal {F}(M)$ has the Schur property and the approximation property for every scattered complete metric space $M$.

A Local-in-Time Theory for Singular SDEs with Applications to Fluid Models with Transport Noise

We establish a local theory, i.e., existence, uniqueness and blow-up criterion, for a general family of singular SDEs in Hilbert spaces. The key requirement relies on an approximation property that allows us to embed the singular drift and diffusion mappings into a hierarchy of regular mappings that are invariant with respect to the Hilbert space and enjoy a cancellation property. Various nonlinear models in fluid dynamics with transport noise belong to this type of singular SDEs. By establishing a cancellation estimate for certain differential operators of order one with suitable coefficients, we give the detailed constructions of such regular approximations for certain examples. In particular, we show novel local-in-time results for the stochastic two-component Camassa–Holm system and for the stochastic Córdoba–Córdoba–Fontelos model.

FAT-based robust adaptive control of cooperative multiple manipulators without velocity measurement

This article addresses the problem of pose and force control in a cooperative system comprised of multiple n-degree-of-freedom (n-DOF) electrically driven robotic arms that move a payload. The proposed controller should be capable of maintaining the position and orientation of the payload in the desired path. In addition, the force exerted by robot end effectors on the object must remain limited. The system has unmodeled dynamics, and measuring the robot joint velocities is impossible. Therefore, a FAT-based observer–controller is designed to estimate the uncertainty and velocities based on universal approximation property of Fourier series expansion. The stability of the system is confirmed based on Lyapunov’s stability theorem. Finally, the proposed adaptive controller–observer is applied on two 3-DOF cooperative robotic arms carrying a payload, and the results are precisely analyzed. The results of the proposed approach are also compared with two state-of-art powerful approximation method.