Use Of Hilbert Spaces Research Articles

The Sound of Swarm. Auditory Description of Swarm Robotic Movements

Movements of robots in a swarm can be mapped to sounds, highlighting the group behavior through the coordinated and simultaneous variations of musical parameters across time. The vice versa is also possible: Sound parameters can be mapped to robotic motion parameters, giving instructions through sound. In this article, we first develop a theoretical framework to relate musical parameters such as pitch, timbre, loudness, and articulation (for each time) with robotic parameters such as position, identity, motor status, and sensor status. We propose a definition of musical spaces as Hilbert spaces and musical paths between parameters as elements of bigroupoids, generalizing existing conceptions of musical spaces. The use of Hilbert spaces allows us to build up quantum representations of musical states, inheriting quantum computing resources, already used for robotic swarms. We present the theoretical framework and then some case studies as toy examples. In particular, we discuss a 2D video and matrix simulation with two robo-caterpillars; a 2D simulation of 10 robo-ants with Webots; a 3D simulation of three robo-fish in an underwater search-and-rescue mission.

Orthogonality and Dimensionality

In this article, we present what we believe to be a simple way to motivate the use of Hilbert spaces in quantum mechanics. To achieve this, we study the way the notion of dimension can, at a very primitive level, be defined as the cardinality of a maximal collection of mutually orthogonal elements (which, for instance, can be seen as spatial directions). Following this idea, we develop a formalism based on two basic ingredients, namely an orthogonality relation and matroids which are a very generic algebraic structure permitting to define a notion of dimension. Having obtained what we call orthomatroids, we then show that, in high enough dimension, the basic ingredients of orthomatroids (more precisely the simple and irreducible ones) are isomorphic to generalized Hilbert lattices, so that the latter are a direct consequence of an orthogonality-based characterization of dimension.

Hysteresis loops predicted by isoenergy density theory for polycrystals. Part II: cyclic heating and cooling effects predicted from non-equilibrium theory for 6061-T6 aluminum, SAE 4340 steel and Ti–8Al–1Mo–1V titanium cylindrical bars

Mathematical tools and physical models should fit like hand and glove. Traditionally, such an understanding has worked surprisingly well in mechanics and physics. Differential calculus enabled the determination of motion of celestial bodies. Heisenberg's use of Hilbert space applied to calculations in quantum mechanics. These foot steps, however, failed to continue in the field of material science when attempting to address the evolution of material damage. On one hand, the mathematical formulation of the Riemann–Hilbert problem achieved great success for solving macrocrack problems but whether the same tool could be used for lower scale defects seems to be of less concern. This is a surprise because it is not obvious by any means whether microcracks can be distinguished from macrocrack by size difference alone, particularly when they are both treated in a single formulation. There are several issues of the fatigue problem that must be addressed. To begin with, local failure initiating near a surface should be distinguished from global failure that correspond to separation of a specimen into two or more pieces. The interim stages of initiation and termination would depend on the ductility and brittleness of the material in addition to the history of cyclic loading. Three different materials 6061-T6 aluminum, SAE 4340 steel and Ti–8Al–1Mo–1V titanium will therefore be analyzed to show how their uniaxial properties would compare with the dissipation energy density functions and oscillation in temperature. Hysteresis loops for the local material points are determined. Their averages for the ASTM standard hour-glass specimen are then found. They give the global hysteresis loops corresponding to measurements from the uniaxial tests. The local dissipative energy density loops represent the irreversible nature of the materials are also taken into consideration. They are calculated for the soft aluminum and the hard titanium with steel being in between. Obtained also are the cyclic cooling and heating characteristics of the material under fatigue; they fluctuate about the ambient temperature. The H -function designating the order and disorder of the fatigue process is also found to be oscillatory in character. Unlike entropy, it is not non-increasing or non-decreasing. It can change sign in a given process. This implies that the order of a system can be enhanced and then impeded or vice versa. Such a behavior is shown to prevail during the initial stage of the fatigue.