Use Of Commutators Research Articles

Explicit Commutativity for Lamé Linear Time‐Varying Differential Systems

ABSTRACTThis article studies the commutativity and sensitivity of the Lamé linear time‐varying systems (LTVSs), and investigates the effects of disturbances on such systems. The commutative pair for the Lamé LTVS of order is found, that is, a new Lamé LTVS of order is derived using the explicit commutative theories for zero initial conditions (ICs). For the case of nonzero ICs, the commutativity between the connected input–output of Lamé systems and is studied. New and simple explicit commutative theories and conditions for second‐order LTVSs are derived, simplifying the use of commutativity for practical and industrial scenarios. These findings enable us to analyse the commutativity, sensitivity, robustness and stability of Lamé systems, and to determine the effects of disturbances. The explicit results presented in this article are supported by simulations and verified by examples and constitute a significant contribution to science and engineering applications.

Decomposition of Fourth-Order Euler-Type Linear Time-Varying Differential System into Cascaded Two Second-Order Euler Commutative Pairs

This paper presents decomposition of the fourth-order Euler-type linear time-varying system (LTVS) as a commutative pair of two second-order Euler-type systems. All necessary and sufficient conditions for the decomposition are deployed to investigate the commutativity, sensitivity, and the effect of disturbance on the fourth-order LTVS. Some systems are commutative, and some are not commutative, while some are commutative under certain conditions. Based on this fact, the commutativity of fourth-order Euler-type LTVS is investigated by introducing the commutative requirements, theories, and conditions. The fourth-order Euler-type LTVSs are investigated into commutative pairs of twice Euler-type second-order linear time-varying systems (LTVSs). The decomposition theories and conditions are derived, proved, and solved to simplify the use of commutativity for practical and industrial uses. Some fourth-order systems are sensitive toward change in initial conditions or parameters while others are not, and the effect due to disturbance also varies within systems. Furthermore, the stability and robustness of systems have so many issues. But we consider fourth-order Euler-type LTVS to observe, investigate, and tackle these issues. Lastly, the realization of fourth-order LTVS from cascaded two second-order systems can be laboratory experimented which is an open problem for future engineers to investigate. However, the theoretical results show a good agreement with the simulation results is considered in this work. Perhaps it might have unlimited physical applications in science and engineering as well as theoretical contribution. But beyond any reasonable doubt, the novelty is guaranteed because this study is the first of its kind that introduces the decomposition of the fourth-order Euler-type linear time-varying system (LTVS) as a commutative pair of two second-order Euler-type systems. Illustrative examples are presented to support the results.

New methods in spectral theory of N-body Schrödinger operators

We develop a new scheme of proofs for spectral theory of the [Formula: see text]-body Schrödinger operators, reproducing and extending a series of sharp results under minimum conditions. Our main results include Rellich’s theorem, limiting absorption principle bounds, microlocal resolvent bounds, Hölder continuity of the resolvent and a microlocal Sommerfeld uniqueness result. We present a new proof of Rellich’s theorem which is unified with exponential decay estimates studied previously only for [Formula: see text]-eigenfunctions. Each pair-potential is a sum of a long-range term with first-order derivatives, a short-range term without derivatives and a singular term of operator- or form-bounded type, and the setup includes hard-core interaction. Our proofs consist of a systematic use of commutators with ‘zeroth order’ operators. In particular, they do not rely on Mourre’s differential inequality technique.

0446 A Novel System for Enabling High-Density EEG Recordings in a Mouse

Abstract Introduction Recent advances in micro-electromechanical system (MEMS) technology have promoted the development of microelectrode arrays (MEA) that allow high resolution recordings in neuroscience research. However, applying MEA in studies in freely moving mice remains very challenging due to the large number of electrical connections required in this type of studies. The use of commutators for a large number of connections is not practical, and headmounts/loggers placed on the animal head are too heavy for small animals such as mice. Therefore, there is a need for a better compact system for using MEA in mice. Herein, we designed such a system and successfully recorded high-density-EEG in freely moving mice. Methods We designed a system in which forty flexible ultrathin wires are connected to the headstage enclosed in a container held close to the mouse. The container also houses a logger and battery connected to the headstage. This recording system allows minimizing weighted pressure on the animal using a counterbalance, so that the animal can freely move in the cage. Results We tested the system using a signal generator and mouse EEG arrays (NeuroNexus). When potentials produced by the signal generator were recorded via the wires, recorded traces were indistinguishable from the traces that were recorded when the signal generator was connected directly to the logger. We then implanted mice with EEG electrode arrays under surgical anesthesia. The high-density EEG recordings were performed one and four weeks after the surgery. High-quality EEG signals were observed in all the channels of the 32-channel logger (SpikeGadgets) in freely moving mice. Conclusion We successfully developed and tested a novel system for enabling high-density EEG recordings in freely moving mice. We expect that this system will be useful for recording biopotentials from different types of MEA in freely moving mice. Support NIH 1R43OD023231 (LG), NIH 1RF1AG061774 (DG), and NIH 5R21NS106406 (DG)

Non-existence of generalized splitting methods with positive coefficients of order higher than four

We prove that generalized exponential splitting methods making explicit use of commutators of the vector fields are limited to order four when only real coefficients are admitted. This generalizes the restriction to order two for classical splitting methods with only positive coefficients.

Можливості модернізації комп’ютерної мережі інформаційно-телекомунікаційної системи Придніпровської залізниці

Modernization variants of the computer network of Dnieper railway, namely – the complete replacement of concentrators by commutators; the use of commutators and partial introduction of routers; the organization of VLANs on commutators; the integration of virtual networks by means of routers have been proposed. Appropriate simulation models, with the help of which the investigation of the average reaction time was carried out, have been developed in Cisco Packet Tracer. The best value gives the variant of the integration of virtual networks by routers: the utilization of switching within VLAN provides high-speed connection between workstations and servers, and routing between VLAN makes it possible to allocate resources and to reduce broadcast traffic. The evaluation of the dependency of commutator productive capacity on the quantity of its ports as well as the work technique of a port (Ethernet, Fast Ethernet) and a matrix of an average values of traffic intensity among the ports has been fulfilled with thw help og the developed «SPA» model. The investigation of maximum and average time of waiting in a queue as well as the reaction time by means of different methods of frame flow control in a two-way alternative operation has been conducted with the help of the developed simulation model “Fast Ethernet”. It has been found out that the method of aggressive medium capture is more effective than the method of back pressure to the network station

Transformations of the unitary group on a Hilbert space

Let H be an infinite dimensional separable complex Hilbert space and U be the group of all unitary operators on H. Motivated by the algebraic properties of surjective isometries of U that have recently been revealed, and also by some classical results related to automorphisms of the unitary groups of operator algebras, we determine the structures of bijective transformations of U that respect certain algebraic operations. These are, among others, the usual product of operators, the Jordan triple product, the inverted Jordan triple product, and the multiplicative commutator. Our basic approach to obtain these results is the use of commutativity preserving transformations on the unitary group.

Isothermal Quantum Hydrodynamics: Derivation, Asymptotic Analysis, and Simulation

This article is devoted to the reformulation of an isothermal version of the quantum hydrodynamic model derived by Degond and Ringhofer in [J. Statist. Phys., 112 (2003), pp. 587–628] (which will be referred to as the quantum Euler system). We write the model under a simpler (differential) form. The derivation is based on an appropriate use of commutators. Starting from the quantum Liouville equation, the system of moments is closed by a density operator which minimizes the quantum free energy. Some properties of the model are then exhibited, and most of them rely on a gauge invariance property of the system. Several simplifications of the model are also written for the special case of irrotational flows. The second part of the paper is devoted to a formal analysis of the asymptotic behavior of the quantum Euler system in three situations: at the semiclassical limit, at the zero‐temperature limit, and at a diffusive limit. The remarkable fact is that in each case we recover a known model: respectively, the isothermal Euler system, the Madelung equations, and the entropic quantum drift‐diffusion model. Finally, we give in the third part some preliminary numerical simulations.

First derivative of the period function with applications

Given a centre of a planar differential system, we extend the use of the Lie bracket to the determination of the monotonicity character of the period function. As far as we know, there are no general methods to study this function, and the use of commutators and Lie bracket was restricted to prove isochronicity. We give several examples and a special method which simplifies the computations when a first integral is known.

First derivative of the period function with applications

Given a centre of a planar differential system, we extend the use of the Lie bracket to the determination of the monotonicity character of the period function. As far as we know, there are no general methods to study this function, and the use of commutators and Lie bracket was restricted to prove isochronicity. We give several examples and a special method which simplifies the computations when a first integral is known.

Use of Multiplicative Commutativity by School Children and Street Sellers

We analyzed use of the commutative property for solving multiplication problems by children who learn about multiplication in schools and by street vendors who solve multiplication problems through repeated addition. Subjects were Brazilian street vendors with irregular school attendance, who had received no, or very little, school instruction on multiplication, and 1st- to 3rd-grade Brazilian school children. Results from 2 studies show that use of commutativity to solve multiplication problems is closely related to use of multiplication. Street sellers who rely exclusively on repeated addition to solve such problems may, however, use multiplicative commutativity if it represents a clear reduction in the number of computational steps needed to reach a solution. But their justifications for its use are often based on knowledge about multiplication.

A design for a visual motion transducer

Autonomous vehicles could benefit greatly from visual-motion sensors of sufficiently low cost that 10 or 20 of them could be distributed around the vehicle's periphery. A design is outlined here that builds on well-known schemes using gratings. The grating principle is illustrated by the fact that a man with a torch, walking at night behind a railing, seems to flash. The frequency of flashing is proportional to his velocity. A major drawback is that backward and forward motion are not distinguished. The key development here is the use of commutation as a means of modulating the grating output signal. This is equivalent, in the illustration above, to simulating stroboscopic motion of the railing. Thus, when the man is stationary, there is flashing at a resting frequency. When he moves one way the frequency increases, and for motion the opposite way, frequency decreases. Results from an analogue implementation of the visual motion transducer are presented. The current transducer measures translational motion across the grating. The design is also shown to be capable of extension to direct measurement of the divergence and curl of the flow field.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Commutation in flow injection analysis

Recent advances in flow injection analysis, including the introduction of zone-sampling and zone-trapping processes and the incorporation of ion exchangers, have involved electronically operated injector/commutators. Other techniques based on manual commutation, such as manifold modifications and intermittent and alternating streams, have been reported. A comprehensive review of these techniques is given. The use of commutation for difference injection procedures (loop-based, time-based, hydrodynamic, sequential and nested injection) is emphasized. An alternative stopped-flow approach without stopping the pump is suggested.

Commutation of Prison Sentences: Practice, Promise, and Limitation

Executive clemency procedures offer an ancient but still modern mechanism for righting injustice and coping with problems arising when the judicial system fails to adjust to social needs. As the indeterminate sentence gives way to determinate sanctions and prison crowding grows acute, clemency, particularly in the form of commutation of sentences of prisoners, offers one means of adaptation to correctional population pressures. This article examines the legal and administrative structures, policies, and actual uses of sentence commutation in the United States. A survey of state commutation practices finds that there is wide variation in structure and policy among the states, that in general commutation is granted quite sparingly, that in response to sentencing changes states have both increased and decreased its use, and that political and practical considerations limit the use of commutation as a prison populaton safety valve.

On Minimizing the Number of Multiplications Necessary for Matrix Multiplication

This paper develops an algorithm to multiply a $p \times 2$ matrix by a $2 \times n$ matrix in $\lceil (3pn + \max (n,p)) 2\rceil $ multiplications without use of commutativity of matrix elements. The algorithm minimizes the number of multiplications for matrix multiplication without commutativity for the special cases $p = 1$ or $2,n = 1,2, \cdots $ and $p = 3,n = 3$. It is shown that commutativity actually reduces the number of multiplications required.

The classification of electromagnetic machinery

The author proposes to classify all dynamo electric machinery according to five sets of characteristics, as follows: 1. Type of field, as constant intensity, variable intensity, multiple polarity and homopolar. 2. Method of disposal of secondary power, which may be zero wattless, utilized in separate apparatus or transformed and returned to the line. 3. Use of commutators, which may be on either the primary or the secondary, or may be absent. 4. Method of magnetization, namely, high frequency magnetization as in the ordinary induction motor and low frequency magnetization as in synchronous and compensated motors. 5. Method of connection. Many types may be connected either in series or shunt. A tabulation of these characteristics is given, by reference to which any dynamo electric machine can be classified.

The Classification of Electromagnetic Machinery

The author proposes to classify all dynamo-electric machinery according to five sets of characteristics, as follows: 1. Type of field, as constant intensity, variable intensity, multiple polarity and homopolar. 2. Method of disposal of secondary power, which may be zero, wattless, utilized in separate apparatus or transformed and returned to the line. 3. Use of commutators, which may be on either the primary or the secondary, or may be absent. 4. Method of magnetization, namely, high-frequency magnetization as in the ordinary induction motor and low-frequency magnetization as in synchronous and compensated motors. 5. Method of connection. Many types may be connected either in series or shunt. A tabulation of these characteristics is given, by reference to which any dynamo-electric machine can be classified.

Experimental researches in electricity. Twenty-fourth series. On the possible relation of gravity to electricity

Under the full persuasion that all the forces of nature are mutually dependent, and often, if not always, convertible more or less into each other, the author endeavoured to connect gravity and magnetic or electric action together by experimental results, and though the conclusions were, when cleared from all error, of a negative nature, he still thinks that the principle followed and the experiments themselves deserve to be recorded. Considering that some condition of the results produced by gravity ought to present itself, having a relation to the dual or antithetical character of the magnetic or electric forces, it seemed to the author that the approximation of two gravitating bodies towards each other, and their separation, were the only points which offered this kind of coincidence; and therefore, using the earth as one gravitating body, he employed a cylinder of metal, glass, resins, or other substances, as the other, and endeavoured to ascertain when the latter was allowed to fall, being surrounded by a helix of wire, whether any electric current was generated. Sometimes the cylinder was allowed to fall through the helix; at other times with the helix; and occasionally the helix was made the falling body. But when the various sources of error which sprung up were gradually removed, no traces of electric action remained which could be referred to the power of gravity. In order to obtain a greater effect, an aparatus was employed (being nearly that, used in the 23rd Series of these Researches) by which the effect of raising a body from the earth could be combined with that of a falling body by the fit use of commutators (if any action at all were produced). The apparatus was very good, and gave exceedingly delicate results, as was shown by other consequences of its action; but in respect of gravity it produced no effect whatever. Notwithstanding his failure in obtaining any experimental relation between gravity and magnetic or electric force, the author still expresses his conviction that there is a relation, and his hopes that it may be hereafter practically demonstrated.