In this paper, we propose sufficient conditions to guarantee that a linear temporal logic formula of the form p Until q , denoted by pUq , is satisfied for a hybrid system. Roughly speaking, the ...formula pUq is satisfied means that the solutions, initially satisfying proposition p , keep satisfying this proposition until proposition q is satisfied. To certify such a formula, connections to invariance notions – specifically, conditional invariance and eventual conditional invariance – as well as finite-time convergence properties are established. As a result, sufficient conditions involving the data of the hybrid system and an appropriate choice of Lyapunov-like functions, such as barrier functions, are derived. Examples illustrate the results throughout the paper.
Surprisingly, general estimators for nonlinear continuous time models based on stochastic differential equations are yet lacking. Most applications still use the Euler-Maruyama discretization, ...despite many proofs of its bias. More sophisticated methods, such as Kessler's Gaussian approximation, Ozak's Local Linearization, Aït-Sahalia's Hermite expansions, or MCMC methods, lack a straightforward implementation, do not scale well with increasing model dimension or can be numerically unstable. We propose two efficient and easy-to-implement likelihood-based estimators based on the Lie-Trotter (LT) and the Strang (S) splitting schemes. We prove that S has L p convergence rate of order 1, a property already known for LT. We show that the estimators are consistent and asymptotically efficient under the less restrictive one-sided Lipschitz assumption. A numerical study on the 3-dimensional stochastic Lorenz system complements our theoretical findings. The simulation shows that the S estimator performs the best when measured on precision and computational speed compared to the state-of-the-art.
This book introduces sparse and redundant representations with a focus on applications in signal and image processing. It details mathematical modeling for signal sources along with how to use the ...model for tasks such as denoising, restoration and separation.
Public key cryptography is a major interdisciplinary subject with many real-world applications, such as digital signatures. A strong background in the mathematics underlying public key cryptography ...is essential for a deep understanding of the subject, and this book provides exactly that for students and researchers in mathematics, computer science and electrical engineering. Carefully written to communicate the major ideas and techniques of public key cryptography to a wide readership, this text is enlivened throughout with historical remarks and insightful perspectives on the development of the subject. Numerous examples, proofs and exercises make it suitable as a textbook for an advanced course, as well as for self-study. For more experienced researchers it serves as a convenient reference for many important topics: the Pollard algorithms, Maurer reduction, isogenies, algebraic tori, hyperelliptic curves and many more.
We present finite-element numerical algorithms for the identification of vortices in quantum fluids described by a macroscopic complex wave function. Their implementation using the free software ...FreeFem++ is distributed withthis paper as a post-processing toolbox that can be used to analyse numerical or experimental data. Applications for Bose-Einstein condensates (BEC) and superfluid helium flows are presented. Programs are tested and validated using either numerical data obtained by solving the Gross-Pitaevskii equation or experimental images of rotating BEC. Vortex positions are computed as topological defects (zeros) of the wave function when numerical data are used. For experimental images, we compute vortex positions as local minima of the atomic density, extracted after a simple image processing. Once vortex centers are identified, we use a fit with a Gaussian to precisely estimate vortex radius. For vortex lattices, the lattice parameter (inter-vortex distance) is also computed. The post-processing toolbox offers a complete description of vortex configurations in superfluids. Tests for two-dimensional (giant vortex in rotating BEC, Abrikosov vortex lattice in experimental BEC) and three-dimensional (vortex rings, Kelvinwaves and quantum turbulence fields in superfluid helium) configurations show the robustness of the software. The communication with programs providing the numerical or experimental wave function field is simple and intuitive.The post-processing toolbox can be also applied for the identification of vortices in superconductors.
This paper proposes an optimal control problem for a parabolic equation with a nonlocal nonlinearity.The system is described by a parabolic equation involving a nonlinear term that depends on the ...solutionand its integral over the domain. We prove the existence and uniqueness of the solution to the system andthe boundedness of the solution. Regularity results for the control-to-state operator, the cost functionaland the adjoint state are also proved. We show the existence of optimal solutions and derive first-ordernecessary optimality conditions. In addition, second-order necessary and sufficient conditions for optimalityare established
