Commutativity Condition Research Articles

Generalized derivation on semiprime and prime Banach algebras

Abstract In this study, the commutativity conditions with generalized derivations, which have not been examined before, are discussed. Under these conditions, the subject of generalized derivations in (semi)prime Banach algebras is studied. New fundamental results will be provided for researchers in this field and generalize some of the results found in the literature.

High Order Splitting Methods for SDEs Satisfying a Commutativity Condition

High Order Splitting Methods for SDEs Satisfying a Commutativity Condition

Higher-order adaptive methods for exit times of Itô diffusions

Abstract We construct a higher-order adaptive method for strong approximations of exit times of Itô stochastic differential equations (SDEs). The method employs a strong Itô–Taylor scheme for simulating SDE paths, and adaptively decreases the step size in the numerical integration as the solution approaches the boundary of the domain. These techniques complement each other nicely: adaptive timestepping improves the accuracy of the exit time by reducing the magnitude of the overshoot of the numerical solution when it exits the domain, and higher-order schemes improve the approximation of the state of the diffusion process. We present two versions of the higher-order adaptive method. The first one uses the Milstein scheme as the numerical integrator and two step sizes for adaptive timestepping: $h$ when far away from the boundary and $h^2$ when close to the boundary. The second method is an extension of the first one using the strong Itô–Taylor scheme of order 1.5 as the numerical integrator and three step sizes for adaptive timestepping. Under some regularity assumptions, we show that for any $\xi>0$, the strong error is ${\mathcal{O}}(h^{1-\xi })$ and ${\mathcal{O}}(h^{3/2-\xi })$ for the first and second method, respectively. Provided quite restrictive commutativity conditions hold for the diffusion coefficient, we further show that the expected computational cost for both methods is ${\mathcal{O}}(h^{-1} \log (h^{-1}))$. This results in a near doubling/trebling of the strong error rate compared to the standard Euler–Maruyama-based approach, while the computational cost rate is kept close to order one. Numerical examples that support the theoretical results are provided, and we discuss the potential for extensions that would further improve the strong convergence rate of the method.

Some Commutativity Conditions for Rings

During the last 55 years there have been many results concerning conditions that force a ring to be commutative. These results were stimulated by Jacobson’s famous result and were extensively devel- oped by Herstein. This paper will survey the area by organizing the results according to whether they come from variations on Herstein’s conditions, depend on general polynomial conditions, depend on the presence of a derivation, or whether a ring has special properties that make commutativity more easily accessible. Finally, the most recent conditions concern product sets and lead to results in a new area of inquiry.

Frobenius-Perron theory of the bound quiver algebras containing loops

The Frobenius-Perron dimension of a matrix, also known as the spectral radius, is a useful tool for studying linear algebras and plays an important role in the classification of the representation categories of algebras. In this paper, we study the Frobenius-Perron theory of the representation categories of bound quiver algebras containing loops, and find a way to calculate the Frobenius-Perron dimensions of these algebras satisfying the commutativity condition of loops. As an application, we prove that the Frobenius-Perron dimension of the representation category of a modified ADE bounded quiver algebra is equal to the maximal number of loops at each vertex. Finally, we point out that there also exist infinite dimensional algebras whose Frobenius-Perron dimensions is equal to the maximal number of loops by giving an example.

Periodic two-dimensional descriptor systems

In this note, we analyze the compatibility conditions of 2D descriptor systems with periodic coefficients and we derive a special coordinate system in which these conditions reduce to simple matrix commutativity conditions. We also show that the compatibility of the different trajectories in such a periodic 2D descriptor system can elegantly be formulated in terms of so-called matrix relations of regular pencils, which were introduced in [Benner and Byers. An arithmetic for matrix pencils: Theory and new algorithms. Numer. Math., 103(4):539-573, 2006]. We then show that these ideas can be extended to multidimensional periodic descriptor systems and briefly discuss the difference between the case of complex and real coefficient matrices.

Certain Fixed-Point Results via DS-Weak Commutativity Condition in Neutrosophic Metric Spaces With Application to Non-linear Fractional Differential Equations

This study demonstrates that, for the non-linear contractive conditions in Neutrosophic metric spaces, a common fixed-point theorem may be proved without requiring the continuity of any mappings. A novel commutativity requirement for mappings weaker than the compatibility of mappings is used to demonstrate the conclusion. We provide several examples to illustrate our major idea. Also, we provide an application to the non-linear fractional differential equation to show the validity of our main result.

Explicit commutativity conditions for second-order linear time-varying systems with non-zero initial conditions

Explicit commutativity conditions for second-order linear time-varying systems with non-zero initial conditions

Strong convergence of an adaptive time-stepping Milstein method for SDEs with monotone coefficients

We introduce an explicit adaptive Milstein method for stochastic differential equations with no commutativity condition. The drift and diffusion are separately locally Lipschitz and together satisfy a monotone condition. This method relies on a class of path-bounded time-stepping strategies which work by reducing the stepsize as solutions approach the boundary of a sphere, invoking a backstop method in the event that the timestep becomes too small. We prove that such schemes are strongly L_2 convergent of order one. This order is inherited by an explicit adaptive Euler–Maruyama scheme in the additive noise case. Moreover we show that the probability of using the backstop method at any step can be made arbitrarily small. We compare our method to other fixed-step Milstein variants on a range of test problems.

Iterative Design Algorithm for Robust Disturbance-Rejection Control

An iterative design algorithm is developed for robust disturbance–rejection control of uncertain systems with time-varying parameter perturbations in this paper. For more design degrees of freedom, a generalized equivalent-input-disturbance estimator is adopted to approximate the effect of both disturbances and uncertainties. By the bound real lemma, the H∞ norm is used to evaluate the robust disturbance–rejection performance of the closed-loop uncertain system. To avoid the constraints introduced by the widely used commutative condition, the control gains are divided into two groups and calculated by steps. Further, two robust quadratic stability conditions are derived, and an iterative design algorithm is developed to optimize the robust H∞ disturbance–rejection performance. Finally, the effectiveness and advantages of the developed method are demonstrated by a case study of a suspension system of modern vehicles.

Controllability of Continuous-Time Bilinear Systems Without Using the Lie-Algebraic Conditions: A New Perspective

In this paper, we propose a new perspective to study controllability of continuous-time bilinear systems without using the Lie-algebraic conditions. Specifically, we first consider controllability of the bilinear systems in the single-input case under a commutativity condition. We show that, although the Lie algebra rank condition, which is necessary for a classical controllability result to work, does not fit such bilinear systems, they can still be controllable. Our approach to prove controllability is using controllability of the discrete-time counterparts of the continuous-time systems, and we derive a necessary and sufficient controllability criterion without the Lie algebra rank condition, which is algebraically verifiable for any finite dimension. More importantly, through this controllability study, we propose a new perspective to deal with the controllability problems of continuous-time bilinear systems by changing the verification of the Lie algebra rank condition to solving two linear algebra problems. Examples are given to illustrate the obtained controllability results of this paper.

Purifying teleportation

Coupling to the environment typically suppresses quantum properties of physical systems via decoherence mechanisms. This is one of the main obstacles in practical implementations of quantum protocols. In this work we show how decoherence effects can be reversed/suppressed during quantum teleportation in a network scenario. Treating the environment quantumly, we show that under a general pure dephasing coupling, performing a second teleportation step can probabilistically reverse the decoherence effects if certain commutativity conditions hold. This effect is purely quantum and most pronounced for qubit systems, where in 25 % of instances the decoherence can be reversed completely. As an example, we show the effect in a physical model of a qubit register coupled to a bosonic bath. We also analyze general d-dimensional systems, identifying all instances of decoherence suppression. Our results are proof-of-concept but we believe will be relevant for the emerging field of quantum networks as teleportation is the key building block of network protocols.

Improved Properties of Positive Solutions of Higher Order Differential Equations and Their Applications in Oscillation Theory

In this article, we present new criteria for testing the oscillation of solutions of higher-order neutral delay differential equation. By deriving new monotonic properties of a class of the positive solutions of the studied equation, we establish better criteria for oscillation. Furthermore, we improve these properties by giving them an iterative character, allowing us to apply the criteria more than once. The results obtained in this paper are characterized by the fact that they do not require the existence of unknown functions and do not need the commutation condition to composition of the delay functions, which are necessary conditions for the previous related results.

Model Hamiltonians of open quantum optical systems: Evolvement from hermiticity to adjoint commutativity

In the conventional quantum mechanics of conserved systems, Hamiltonian is assumed to be a Hermitian operator. However, when it comes to quantum systems in presence of dissipation and/or noise, including open quantum optical systems, the strict hermiticity requirement is nor longer necessary. In fact, it can be substantially relaxed: the non-Hermitian part of a Hamiltonian is allowed, in order to account for effects of dissipative environment, whereas its Hermitian part would be describing subsystem’s energy. Within the framework of the standard approach to dissipative phenomena based on a master equation for the reduced density operator, we propose a replacement of the hermiticity condition by a more general condition of commutativity between Hermitian and anti-Hermitian parts of a Hamiltonian. As an example, we consider a dissipative two-mode quantum system coupled to a single-mode electromagnetic wave, where we demonstrate that the adjoint-commutativity condition does simplify the parametric space of the model.

Untwining multiple parameters at the exclusive zero-coincidence points with quantum control

In this paper we address a special case of ‘sloppy’ quantum estimation procedures which happens in the presence of intertwined parameters. A collection of parameters are said to be intertwined when their imprinting on the quantum probe that mediates the estimation procedure, is performed by a set of linearly dependent generators. Under this circumstance the individual values of the parameters can not be recovered unless one tampers with the encoding process itself. An example is presented by studying the estimation of the relative time-delays that accumulate along two parallel optical transmission lines. In this case we show that the parameters can be effectively untwined by inserting a sequence of balanced beam splitters (and eventually adding an extra phase shift on one of the lines) that couples the two lines at regular intervals in a setup that remind us a generalized Hong-Ou-Mandel interferometer. For the case of two time delays we prove that, when the employed probe is the frequency-correlated biphoton state, the untwining occurs in correspondence of exclusive zero-coincidence (EZC) point. Furthermore we show the statistical independence of two time delays and the optimality of the quantum Fisher information at the EZC point. Finally we prove the compatibility of this scheme by checking the weak commutativity condition associated with the symmetric logarithmic derivative operators.

Common Fixed Point Theorems For Weakly Compatible Mappings On Complete Metric Spaces

Metrical common fixed point theorems invariably require a commutativity condition and a condition on the continuity of involved mappings besides a contractive condition and researches in this domain are aimed at weakening one or more of these conditions. In this paper, we prove a common fixed point theorem that enhance the scope of proving common fixed point theorems from the class of compatible continuous mappings to a wider class of mappings which includes pairwise coincidentally commuting (weak compatible) mappings without any continuity requirement.

Veracity: declarative multicore programming with commutativity

There is an ongoing effort to provide programming abstractions that ease the burden of exploiting multicore hardware. Many programming abstractions ( e.g. , concurrent objects, transactional memory, etc.) simplify matters, but still involve intricate engineering. We argue that some difficulty of multicore programming can be meliorated through a declarative programming style in which programmers directly express the independence of fragments of sequential programs. In our proposed paradigm, programmers write programs in a familiar, sequential manner, with the added ability to explicitly express the conditions under which code fragments sequentially commute. Putting such commutativity conditions into source code offers a new entry point for a compiler to exploit the known connection between commutativity and parallelism. We give a semantics for the programmer’s sequential perspective and, under a correctness condition, find that a compiler-transformed parallel execution is equivalent to the sequential semantics. Serializability/linearizability are not the right fit for this condition, so we introduce scoped serializability and show how it can be enforced with lock synthesis techniques. We next describe a technique for automatically verifying and synthesizing commute conditions via a new reduction from our commute blocks to logical specifications, upon which symbolic commutativity reasoning can be performed. We implemented our work in a new language called Veracity, implemented in Multicore OCaml. We show that commutativity conditions can be automatically generated across a variety of new benchmark programs, confirm the expectation that concurrency speedups can be seen as the computation increases, and apply our work to a small in-memory filesystem and an adaptation of a crowdfund blockchain smart contract.

Entanglement Dynamics Governed by Time-Dependent Quantum Generators

In the article, we investigate entanglement dynamics defined by time-dependent linear generators. We consider multilevel quantum systems coupled to an environment that induces decoherence and dissipation, such that the relaxation rates depend on time. By applying the condition of partial commutativity, one can precisely describe the dynamics of selected subsystems. More specifically, we investigate the dynamics of entangled states. The concurrence is used to quantify the amount of two-qubit entanglement in the time domain. The framework appears to be an efficient tool for investigating quantum evolution of entangled states driven by time-local generators. In particular, non-Markovian effects can be included to observe the restoration of entanglement in time.

A derivative-free Milstein type approximation method for SPDEs covering the non-commutative noise case

We propose a derivative-free Milstein type scheme to approximate the mild solution of stochastic partial differential equations (SPDEs) that do not need to fulfill a commutativity condition for the noise term. The newly developed derivative-free Milstein type scheme differs significantly from schemes that are appropriate for the case of commutative noise. As a key result, the new derivative-free Milstein type scheme needs only two stages that are specifically tailored based on a technique that, compared to the original Milstein scheme, allows for a reduction of the computational complexity by one order of magnitude. Moreover, the proposed derivative-free Milstein scheme can flexibly be combined with some approximation method for the involved iterated stochastic integrals. As the main result, we prove the strong L^2-convergence of the introduced derivative-free Milstein type scheme, especially if it is combined with any suitable approximation algorithm for the necessary iterated stochastic integrals. We carry out a rigorous analysis of the error versus computational cost and derive the effective order of convergence for the derivative-free Milstein type scheme in the case that the truncated Fourier series algorithm for the approximation of the iterated stochastic integrals is applied. As a further novelty, we show that the use of approximations of iterated stochastic integrals based on truncated Fourier series together with the proposed derivative-free Milstein type scheme improves the effective order of convergence compared to that of the Euler scheme and the original Milstein scheme. This result is contrary to well known results in the finite dimensional SDE case where the use of merely truncated Fourier series does not improve the effective order of convergence in the L^2-sense compared to that of the Euler scheme.

Furstenberg systems of Hardy field sequences and applications

We study measure preserving systems, called Furstenberg systems, that model the statistical behavior of sequences defined by smooth functions with at most polynomial growth. Typical examples are the sequences (\({n^{{3 \over 2}}}\)), (n log n), and (\([{n^{{3 \over 2}}}]\)), α ∈ ℝ ℚ, where the entries are taken mod 1. We show that their Furstenberg systems arise from unipotent transformations on finite-dimensional tori with some invariant measure that is absolutely continuous with respect to the Haar measure and deduce that they are disjoint from every ergodic system. We also study similar problems for sequences of the form (\(g({S^{[{n^{{3 \over 2}}}]}}y)\)), where S is a measure preserving transformation on the probability space (Y, ν), g ∈ L∞(ν), and y is a typical point in Y. We prove that the corresponding Furstenberg systems are strongly stationary and deduce from this a multiple ergodic theorem and a multiple recurrence result for measure preserving transformations of zero entropy that do not satisfy any commutativity conditions.