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Ubiquitous computing systems and the digital economy
Published in James Juniper, The Economic Philosophy of the Internet of Things, 2018
For his part, Jean-Yves Girard (2007) appeals to quantum-theoretic notions of an operator algebra: My personal bias, the one followed in this paper, is that the real hypostases are very different from our familiar (mis)conceptions: I shall thence propose a disturbing approach to foundations. This viewpoint is by no means “non standard”, it is on the contrary most standard; but it relies on ideas developed in the last century and prompted by quantum physics, the claim being that operator algebra is more primitive than set theory . . . in the non commutative geometry of Connes, a paradigm violently anti-set-theoretic, based upon the familiar result: A commutative operator algebra is a function space.
The Shuffle Product and Symmetric Groups
Published in K. D. Elworthy, W. Norrie Evenitt, E. Bruce Lee, Differential equations, dynamical systems, and control science, 2017
A.A. Agrachev, R.V. Gamkrelidze
where τi- are reals and φn(τ1,…, τn) belong to the appropriate associative algebra — to an operator algebra, to a tensor algebra or, in the simplest case, to a (commutative) field of scalars. An expression of the form (1) is of a very general nature and describes an arbitrary evolutionary process on the interval [t, T], when the state of the process at every time instant does not depend on the future. Taking into account this connection, we call the series of the form (1) a chronological series. It is natural to presuppose that the product of two chronological series is again a chronological series, though an explicit formula representing a product of two chronological series in the form (1) is not at all trivial and is connected with interesting combinatorial and algebraic structures.
Random Feedback Control, Group Invariants and Pattern Classification, Quantum Mechanics in Levy Noise, State Observer and Trajectory Tracking
Published in Harish Parthasarathy, Electromagnetics, Control and Robotics, 2023
[ 5] The next topic of research concerns developing algorithms for quantum filtering using the Hudson-Parthasarathy Belavkin approach. Noise, in quantam system is modelled using the Hudson-Parthasarathy quantum stochastic calculus based on the differentials of the creation annihilation and conservation processes which are simply families of operators in the infinite dimensional Boson Fock-space. Using this calculus the Hudson-Parthasarathy noisy Schrodinger equation can be set up. The Heisenberg evolution of observables based on this equation can then also be written down. This evolution appears as a continuous family of homomorphism on an initial Von-Neumann operator algebra. When measurements are carried out, the evolution of the state may get disturbed due to the Heisenberg uncertainty principle. In order to avoid this, Belavkin in a series of remarkable papers suggested how non-demolition measurement can be made on the system plus bath. These measurements consist of unitarily transforming input noise processes with the unitary transformation coming from the noisy Schrodinger equation. The Belavkin filter is essentially a real time implementation of the conditional expectation of the state of the system at time t (which is the Heisenberg evolved observable) given the measurement Abeliean, non-demolition Von-Neumann algebra upto time t. It is a generalized non-commutative version of the classical Kushner filter. The quantum information processing group at the NSIT shall be focussing on implementing the Hudon-Parthasarathy noisy Schrodinger equation as well as the Belavkin filter using finite dimensional linear algebra and MATLAB. The Belavkin filter would enable us to get refined estimates of observables like the spin of the electron or its position, momentum, kinetic energy, potential energy and angular momentum form noisy meaurement of a subset of these variables.
Edge counts for the auxiliary pair graph within the graphical unitary group approach
Published in Molecular Physics, 2021
Table 4 lists the permutations among the 16 segment shapes produced by the symmetry operators. The set of all symmetry operators together with operator algebra is the Klein four-group. Equivalence relations on the set of all segment shapes can be defined using subgroups of V: if there exists an operator in the subgroup that relates one shape to another then the shapes are equivalent. Each equivalence relation partitions the segment shapes into equivalence classes; the equivalence class of shape x is . If then each shape is in a distinct equivalence class. If then there are 10 equivalence classes: 4 singletons, where , and 6 doubletons. If then there are also 10 equivalence classes: 4 singletons, where , and 6 doubletons. If then there are seven equivalence classes: two singletons, where , three doubletons, and two classes of cardinality four. Table 4 is ordered by ascending equivalence class size for the equivalence relation with empty lines separating the classes.
A biaxial nematic liquid crystal composed of matchbox-symmetric molecules
Published in Molecular Physics, 2020
Robert A. Skutnik, Immanuel S. Geier, Martin Schoen
We emphasise that the expansion in Wigner rotation matrices of the interaction potential and of the orientation-dependent pair correlation function can also be cast in terms of the contraction of Cartesian tensors. These tensors describe the coupling between uniaxial degrees of freedom on the one hand and biaxial degrees of freedom on the other hand. The latter formulation is identical with the one proposed by Sonnet et al. [5] and by Sonnet and Virga [36]. We have verified here that both formulations are indeed the same. The proof of this equivalence rests upon a ladder-operator algebra [34] borrowed from the quantum-mechanical theory of angular momentum [35].
Recent research and development activities on space robotics and AI
Published in Advanced Robotics, 2021
Richard Doyle, Takashi Kubota, Martin Picard, Bernd Sommer, Hiroshi Ueno, Gianfranco Visentin, Richard Volpe
In addition to hardware and software systems that have been broadly discussed in this paper, JPL robotics also develops modeling, simulation and visualization for a variety of applications. One body of work is based on the Spatial Operator Algebra formulation [68], implemented in the Dynamics and Real-Time Simulation (DARTS) software system [69]. This system is used for everything from robotics research, to medical research in protein folding, to flight project for Mars, Europa and the International Space Station.