Functions preserving nonnegativity of matrices

The main goal of this work is to determine which entire functions preserve nonnegativity of matrices of a fixed order $n$ -- i.e., to characterize entire functions $f$ with the property that $f(A)$ is entrywise nonnegative for every entrywise nonnegative matrix $A$ of size $n\times n$. Towards this goal, we present a complete characterization of functions preserving nonnegativity of (block) upper-triangular matrices and those preserving nonnegativity of circulant matrices. We also derive necessary conditions and sufficient conditions for entire functions that preserve nonnegativity of symmetric matrices. We also show that some of these latter conditions characterize the even or odd functions that preserve nonnegativity of symmetric matrices.Comment: 20 pages; expanded and corrected to reflect referees' remarks; to appear in SIAM J. Matrix Anal. App

Perturbing non-real eigenvalues of non-negative real matrices

AbstractLet σ=(ρ,b+ic,b-ic,λ4,…,λn) be the spectrum of an entry non-negative matrix and t⩾0. Laffey [T. J. Laffey, Perturbing non-real eigenvalues of nonnegative real matrices, Electron. J. Linear Algebra 12 (2005) 73–76] has shown that σ=(ρ+2t,b-t+ic,b-t-ic,λ4,…,λn) is also the spectrum of some nonnegative matrix. Laffey (2005) has used a rank one perturbation for small t and then used a compactness argument to extend the result to all nonnegative t. In this paper, a rank two perturbation is used to deduce an explicit and constructive proof for all t⩾0

Digital waveguide modeling for wind instruments: building a state-space representation based on the Webster-Lokshin model

This paper deals with digital waveguide modeling of wind instruments. It presents the application of state-space representations for the refined acoustic model of Webster-Lokshin. This acoustic model describes the propagation of longitudinal waves in axisymmetric acoustic pipes with a varying cross-section, visco-thermal losses at the walls, and without assuming planar or spherical waves. Moreover, three types of discontinuities of the shape can be taken into account (radius, slope and curvature). The purpose of this work is to build low-cost digital simulations in the time domain based on the Webster-Lokshin model. First, decomposing a resonator into independent elementary parts and isolating delay operators lead to a Kelly-Lochbaum network of input/output systems and delays. Second, for a systematic assembling of elements, their state-space representations are derived in discrete time. Then, standard tools of automatic control are used to reduce the complexity of digital simulations in the time domain. The method is applied to a real trombone, and results of simulations are presented and compared with measurements. This method seems to be a promising approach in term of modularity, complexity of calculation and accuracy, for any acoustic resonators based on tubes

Heuristics for the refinement of assumptions in generalized reactivity formulae

Reactive synthesis is concerned with automatically generating implementations from formal specifications. These specifications are typically written in the language of generalized reactivity (GR(1)), a subset of linear temporal logic capable of expressing the most common industrial specification patterns, and describe the requirements about the behavior of a system under assumptions about the environment where the system is to be deployed. Oftentimes no implementation exists which guarantees the required behavior under all possible environments, typically due to missing assumptions (this is usually referred to as unrealizability). To address this issue, new assumptions need to be added to complete the specification, a problem known as assumptions refinement. Since the space of candidate assumptions is intractably large, searching for the best solutions is inherently hard. In particular, new methods are needed to (i) increase the effectiveness of the search procedures, measured as the ratio between the number of solutions found and of refinements explored; and (ii) improve the results' quality, defined as the weakness of the solutions. In this thesis we propose a set of heuristics to meet these goals, and a methodology to assess and compare assumptions refinement methods based on quantitative metrics. The heuristics are in the form of algorithms to generate candidate refinements during the search, and quantitative measures to assess the quality of the candidates. We first discuss a heuristic method to generate assumptions that target the cause of unrealizability. This is done by selecting candidate refinement formulas based on Craig's interpolation. We provide a formal underpinning of the technique and evaluate it in terms of our new metric of effectiveness, as defined above, whose value is improved with respect to the state of the art. We demonstrate this on a set of popular benchmarks of embedded software. We then provide a formal, quantitative characterization of the permissiveness of environment assumptions in the form of a weakness measure. We prove that the partial order induced by this measure is consistent with the one induced by implication. The key advantage of this measure is that it allows for prioritizing candidate solutions, as we show experimentally. Lastly, we propose a notion of minimal refinements with respect to the observed counterstrategies. We demonstrate that exploring minimal refinements produces weaker solutions, and reduces the amount of computations needed to explore each refinement. However, this may come at the cost of reducing the effectiveness of the search. To counteract this effect, we propose a hybrid search approach in which both minimal and non-minimal refinements are explored.Open Acces

Isotropic 3D topological phases with broken time reversal symmetry

Axial vectors, such as current or magnetization, are commonly used order parameters in time-reversal symmetry breaking systems. These vectors also break isotropy in three dimensional systems, lowering the spatial symmetry. We demonstrate that it is possible to construct a fully isotropic and inversion-symmetric three-dimensional medium where time-reversal symmetry is systematically broken. We devise a cubic crystal with scalar time-reversal symmetry breaking, implemented by hopping through chiral magnetic clusters along the crystal bonds. The presence of only the spatial symmetries of the crystal -- finite rotation and inversion symmetry -- is sufficient to protect a topological phase. The realization of this phase in amorphous systems with average continuous rotation symmetry yields a statistical topological insulator phase. We demonstrate the topological nature of our model by constructing a bulk integer topological invariant, which guarantees gapless surface spectrum on any surface with several overlapping Dirac nodes, analogous to crystalline mirror Chern insulators. We also show the expected transport properties of a three-dimensional statistical topological insulator, which remains critical on the surface for odd values of the invariant.Comment: 18 pages, 4 figure

Core-Collapse Supernova Simulations With Spectral Two-Moment Neutrino Transport

The primary focus of this dissertation is to develop a next-generation, state-of-the-art neutrino kinetics capability that will be used in the context of the next-generation, state-of-the-art core-collapse supernova (CCSN) simulation frameworks \thornado\ and \FLASH.\index{CCSN} \thornado\ is a \textbf{t}oolkit for \textbf{h}igh-\textbf{or}der \textbf{n}eutrino-r\textbf{ad}iation hydr\textbf{o}dynamics, which is a collection of modules that can be incorporated into a simulation code/framework, such as \FLASH, together with a nuclear equation of state (EOS)\index{EOS} library, such as the \WeakLib\ EOS tables. The first part of this work extends the \WeakLib\ code to compute neutrino interaction rates from~\cite{Bruenn_1985} and produce corresponding opacity tables.\index{Bruenn 1985} The processes of emission, absorption, scattering of neutrinos from nucleons and nuclei, neutrino--electron scattering, and neutrino pair production and annihilation are included. The second part of this dissertation builds the special-relativity-corrected (\Ov) neutrino radiation module in \thornado, based on the spectral two-moment method.\index{\Ov} This part of the work involved studying the accuracy, efficiency, and robustness of the numerical solver. We propose a special kind of implicit-explicit scheme, PDARSs, based on efficiency, diffusion accuracy, and physics-preserving (positivity-preserving and realizability-preserving) requirements. \index{PD-ARS} Emission, absorption, scattering of neutrinos from nucleons and nuclei, neutrino--electron scattering, and neutrino pair production and annihilation are included as neutrino--matter couplings. The third part of this work builds interfaces between \FLASH\ and \thornado, \FLASH\ and \WeakLib, and \thornado\ and \WeakLib\ for simulations with the \FLASH\ hydrodynamics module, \WeakLib\ EOS module, and \thornado\ neutrino kinetics module. This part of the work includes data mapping between finite-volume grids and finite-element grids, time-step balancing between hydrodynamics time steps and radiation transport time steps, and GPU enhancement. The fourth part of this work makes a detailed comparison of the results of a spherically symmetric simulation performed by \FLASH+\thornado\ with the result of the \chimera\ code, which is a sophisticated, mature, and evolving code with spectral flux-limited diffusion (one-moment) neutrino kinetics and improved input physics~\citep{bruenn_etal_2020}. This part of the work demonstrates the ability of \FLASH+\thornado\ to perform CCSN simulations and quantifies the potential differences between the two codes caused by the different neutrino kinetics treatments, as well as other differences. Supported by all of the above work, spherically symmetric CCSN simulations with spectral two-moment neutrino kinetics were performed for three low-mass progenitors of 9-, 10-, and 11-Solar-mass (\solarmass) from~\cite{sukhbold_etal_2016}

Aircraft adaptive learning control

The optimal control theory of stochastic linear systems is discussed in terms of the advantages of distributed-control systems, and the control of randomly-sampled systems. An optimal solution to longitudinal control is derived and applied to the F-8 DFBW aircraft. A randomly-sampled linear process model with additive process and noise is developed