Commutators Of Operators Research Articles

A fast method for simultaneous reconstruction and segmentation in X-ray CT application

In this paper, we propose a new method to solve the minimization problem in a simultaneous reconstruction and segmentation (SRS) model for X-ray computed tomography (CT). The SRS model uses Bayes' rule and the maximum a posteriori (MAP) estimate on the hidden Markov measure field model (HMMFM). The original method [Romanov M, Dahl AB, Dong Y, Hansen PC. Simultaneous tomographic reconstruction and segmentation with class priors. Inverse Problems Sci Eng. 2016;24(8):1432–1453] includes a subproblem with logarithmic-summation (log-sum) term, which is non-separable to the classification index. This subproblem was solved by Frank–Wolfe algorithm, which is very time consuming especially when dealing with large-scale CT problems. The starting point of this paper is the commutativity of log-sum operations, where the log-sum problem could be transformed into a sum-log problem by introducing an auxiliary variable. The corresponding sum-log problem for the SRS model is separable. By applying the primal-dual algorithm, the sum-log problem turns into several easy-to-solve convex subproblems. In addition, we introduce an improved model by adding Tikhonov regularization on the SRS model, and give some convergence results for the proposed methods. Experimental results demonstrate that the proposed methods produce comparable results compared with the original SRS method with much less CPU time.

ECROs: building global scale systems from sequential code

To ease the development of geo-distributed applications, replicated data types (RDTs) offer a familiar programming interface while ensuring state convergence, low latency, and high availability. However, RDTs are still designed exclusively by experts using ad-hoc solutions that are error-prone and result in brittle systems. Recent works statically detect conflicting operations on existing data types and coordinate those at runtime to guarantee convergence and preserve application invariants. However, these approaches are too conservative, imposing coordination on a large number of operations. In this work, we propose a principled approach to design and implement efficient RDTs taking into account application invariants. Developers extend sequential data types with a distributed specification, which together form an RDT. We statically analyze the specification to detect conflicts and unravel their cause. This information is then used at runtime to serialize concurrent operations safely and efficiently. Our approach derives a correct RDT from any sequential data type without changes to the data type's implementation and with minimal coordination. We implement our approach in Scala and develop an extensive portfolio of RDTs. The evaluation shows that our approach provides performance similar to conflict-free replicated data types for commutative operations, and considerably improves the performance of non-commutative operations, compared to existing solutions.

Commutators of operators in Musielak–Orlicz–Morrey spaces on differential forms

Commutators of operators in Musielak–Orlicz–Morrey spaces on differential forms

SIMULATION OF COMMUTATION PROCESSES OF MULTI-LEVEL VOLTAGE INVERTER SWITCHES

An algorithm for controlling of commutator’s transistors of a single-phase multi-level voltage inverter is presented. The structure of the power circuit of the considered converter, in contrast to most existing circuits, does not depend on the levels number of the output curve due to the using of an universal source of levels, which are formed by output capacitors of two pulsed DC-converters. The commutator control algorithm ensures the formation of the required output curve of the inverter and excludes the occurrence of emergency situations during level commutating. Features of the converter and commutator operation in modes of active power transmission from pulse converters to the load and perception of the reactive power of the inverter load by them are considered.
 The commutator operation algorithm determines the sequence of control pulses, applied to the base of transistors, the sequence being synchronized with the process of the output voltage curve forming, as well as the direction of the inverter input current. At the same time, features of the level commutating are considered. They differ in the transition direction in value of level voltages: «up» from a lower value to a higher one and «down» from a higher value to a lower one. The sequence of supply and removal of pulses from control electrodes of switches and commutations caused by them, when the inverter input current is positive, are given. The similar commutating of levels is realized, when the inverter input current is negative. Wherein indexes of switches are rearranged in accordance with the direction replacement of their switching of the commutator circuit.
 A commutator model is realized using Micro-Cap 12 to demonstrate the operation of the algorithm. The transistor MJ15003 model is used as a commutator’s switch. In the model, output capacitors of pulse converters are represented by voltage sources, an autonomous inverter – by an active inductive load. The commutating from a higher voltage level to a lower one with a positive input current of the inverter is considered as an example. Simulation results confirm the performance of the algorithm.

Regularity of Commutators of Maximal Operators with Lipschitz Symbols

This paper is devoted to studying Sobolev regularity properties of commutators of Hardy–Littlewood maximal operator and its fractional case with Lipschitz symbols, both in the global and local case. Some new pointwise estimates for the weak gradients of the above commutators will be established. As applications, some bounds for the above commutators on the Sobolev spaces will be obtained.

Erratum to: The boundedness of commutators of sublinear operators on Herz Triebel-Lizorkin spaces

Erratum to: The boundedness of commutators of sublinear operators on Herz Triebel-Lizorkin spaces

The commutators of Bochner-Riesz operators for elliptic operators

We study $L^p$-boundedness of commutators of Bochner-Riesz operators for elliptic self-adjoint operators which satisfy the finite speed of propagation property for the corresponding wave equation. Our results can be applied to Schrödinger operators with inverse square potentials on $\mathbb{R}^n$, elliptic operators on compact manifolds, and Schrödinger operators on asymptotically conic manifolds Our proof is new even for the commutator of the classical Bochner-Riesz operator when $L$ is the Laplace operator $\Delta=\sum_{i=1}^n\partial_{x_i}^2$ on the Euclidean space $\mathbb{R}^n$.

Extrapolation to product Herz spaces and some applications

Abstract This paper extends the extrapolation theory to product Herz spaces. To prove the main result, we first investigate the dual space of the product Herz space, and then show the boundedness of the strong maximal operator on product Herz spaces. By using this extrapolation theory, we establish the John–Nirenberg inequality, the characterization of little bmo, the Fefferman–Stein vector-valued inequality, the boundedness of the bi-parameter singular integral operator, the strong fractional maximal operator, and the bi-parameter fractional integral operator on product Herz spaces. We also give a new characterization of little bmo via the boundedness of the commutators of some bi-parameter operators on product Herz spaces. Even in the one-parameter setting, some of our results are new.

Development of inequalities and characterization of equality conditions for the numerical radius

Let A be a bounded linear operator on a complex Hilbert space and ℜ(A) (ℑ(A)) denote the real part (imaginary part) of A. Among other refinements of the lower bounds for the numerical radius of A, we prove thatw(A)≥12‖A‖+12|‖ℜ(A)‖−‖ℑ(A)‖|andw2(A)≥14‖A⁎A+AA⁎‖+12|‖ℜ(A)‖2−‖ℑ(A)‖2|, where w(A) and ‖A‖ are the numerical radius and operator norm of A, respectively. We study the equality conditions for w(A)=12‖A⁎A+AA⁎‖ and prove that w(A)=12‖A⁎A+AA⁎‖ if and only if the numerical range of A is a circular disk with center at the origin and radius 12‖A⁎A+AA⁎‖. We also obtain upper bounds for the numerical radius of commutators of operators which improve on the existing ones.

Commutation principles for optimization problems on spectral sets in Euclidean Jordan algebras

The commutation principle of Ramírez et al. (SIAM J Optim 23:687–694, 2013) proved in the setting of Euclidean Jordan algebras says that when the sum of a real valued function h and a spectral function $$\Phi $$ is minimized/maximized over a spectral set E, any local optimizer a at which h is Fréchet differentiable operator commutes with the derivative $$h^{\prime }(a)$$ . In this note, we describe some analogs of the above result by assuming the existence of a subgradient in place of the derivative (of h) and obtaining strong operator commutativity relations. We show, for example: if a solves the problem $$\underset{E}{\max }\,(h+\Phi )$$ , then a strongly operator commutes with every element in the subdifferential of h at a; If E and h are convex and a solves the problem $$\underset{E}{\min }\,h$$ , then a strongly operator commutes with the negative of some element in the subdifferential of h at a. These results improve known operator commutativity relations for linear h and for solutions of variational inequality problems. We establish these results via a geometric commutation principle that is valid not only in Euclidean Jordan algebras, but also in a broader setting.

Simultaneous spectral decomposition in Euclidean Jordan algebras and related systems

This article deals with necessary and sufficient conditions for a family of elements in a Euclidean Jordan algebra to have simultaneous (order) spectral decomposition. Motivated by a well-known matrix theory result that any family of pairwise commuting complex Hermitian matrices is simultaneously (unitarily) diagonalizable, we show that in the setting of a general Euclidean Jordan algebra, any family of pairwise operator commuting elements has a simultaneous spectral decomposition, i.e. there exists a common Jordan frame relative to which every element in the given family has the eigenvalue decomposition of the form . The simultaneous order spectral decomposition further demands the ordering of eigenvalues . We characterize this by a pairwise strong operator commutativity condition or, equivalently, , where denotes the vector of eigenvalues of x written in the decreasing order. Going beyond Euclidean Jordan algebras, we formulate commutativity conditions in the setting of the so-called Fan–Theobald–von Neumann system that includes normal decomposition systems (Eaton triples) and certain systems induced by hyperbolic polynomials.

Variational inequalities for the commutators of rough operators with BMO functions

Starting with the relatively simple observation that the variational estimates of the commutators of the standard Calderón-Zygmund operators with the bounded mean oscillation (BMO) functions can be obtained from their weighted variational estimates, we establish similar variational estimates for the commutators of the BMO functions with rough singular integrals, which do not admit any weighted variational estimates. The proof involves several Littlewood-Paley-type inequalities with the commutators as well as Bony decomposition and related para-product estimates.

The boundedness of commutators of rough p-adic fractional Hardy type operators on Herz-type spaces

In this paper, we obtain some inequalities about commutators of a rough p-adic fractional Hardy-type operator on Herz-type spaces when the symbol functions belong to two different function spaces.

Commutative matching Rota-Baxter operators, shuffle products with decorations and matching Zinbiel algebras

The Rota-Baxter algebra and the shuffle product are both algebraic structures arising from integral operators and integral equations. Free commutative Rota-Baxter algebras provide an algebraic framework for integral equations with the simple Riemann integral operator. The Zinbiel algebras form a category in which the shuffle product algebra is the free object. Motivated by algebraic structures underlying integral equations involving multiple integral operators and kernels, we study commutative matching Rota-Baxter algebras and construct the free objects making use of the shuffle product with multiple decorations. We also construct free commutative matching Rota-Baxter algebras in a relative context, to emulate the action of the integral operators on the coefficient functions in an integral equation. We finally show that free commutative matching Rota-Baxter algebras give the free matching Zinbiel algebra, generalizing the characterization of the shuffle product algebra as the free Zinbiel algebra obtained by Loday.

On local and integrated stress-tensor commutators

We discuss some general aspects of commutators of local operators in Lorentzian CFTs, which can be obtained from a suitable analytic continuation of the Euclidean operator product expansion (OPE). Commutators only make sense as distributions, and care has to be taken to extract the right distribution from the OPE. We provide explicit computations in two and four-dimensional CFTs, focusing mainly on commutators of components of the stress-tensor. We rederive several familiar results, such as the canonical commutation relations of free field theory, the local form of the Poincaré algebra, and the Virasoro algebra of two-dimensional CFT. We then consider commutators of light-ray operators built from the stress-tensor. Using simplifying features of the light sheet limit in four-dimensional CFT we provide a direct computation of the BMS algebra formed by a specific set of light-ray operators in theories with no light scalar conformal primaries. In four-dimensional CFT we define a new infinite set of light-ray operators constructed from the stress-tensor, which all have well-defined matrix elements. These are a direct generalization of the two-dimensional Virasoro light-ray operators that are obtained from a conformal embedding of Minkowski space in the Lorentzian cylinder. They obey Hermiticity conditions similar to their two-dimensional analogues, and also share the property that a semi-infinite subset annihilates the vacuum.

SetSketch

MinHash and HyperLogLog are sketching algorithms that have become indispensable for set summaries in big data applications. While HyperLogLog allows counting different elements with very little space, MinHash is suitable for the fast comparison of sets as it allows estimating the Jaccard similarity and other joint quantities. This work presents a new data structure called SetSketch that is able to continuously fill the gap between both use cases. Its commutative and idempotent insert operation and its mergeable state make it suitable for distributed environments. Fast, robust, and easy-to-implement estimators for cardinality and joint quantities, as well as the ability to use SetSketch for similarity search, enable versatile applications. The presented joint estimator can also be applied to other data structures such as MinHash, HyperLogLog, or Hyper-MinHash, where it even performs better than the corresponding state-of-the-art estimators in many cases.

Weighted Estimates for Vector-Valued Intrinsic Square Functions and Commutators in the Morrey-Type Spaces

In this paper, the boundedness properties of vector-valued intrinsic square functions and their vector-valued commutators with \(BMO(\mathbb {R}^{n})\) functions are discussed. We first show the weighted strong-type and weak-type estimates of vector-valued intrinsic square functions in the Morrey-type spaces. Then, we obtain weighted strong-type estimates of vector-valued analogues of commutators in Morrey-type spaces. In the endpoint case, we establish the weighted weak \(L\log L\)-type estimates for these vector-valued commutators in the setting of weighted Lebesgue spaces. Furthermore, we prove weighted endpoint estimates of these commutator operators in Morrey-type spaces. In particular, we can obtain strong-type and endpoint estimates of vector-valued intrinsic square functions and their commutators in the weighted Morrey spaces and the generalized Morrey spaces.

Thomson-Coil Actuator System for Enhanced Active Resonant DC Circuit Breakers

Enhanced active resonant (EAR) dc circuit breakers (DCCBs) are a promising set of recently proposed DCCB concepts. As other DCCBs, EAR DCCBs still require a fast mechanical switch. The requirements on the actuator of the mechanical switch depend on the DCCB concept and the dc grid and are derived here for an EAR DCCB. Thomson-coil actuators (TCAs) can open and close mechanical switches sufficiently fast to satisfy the requirements. This work studies experimentally a TCA system with active damping for an off-the-shelf industrial vacuum interrupter used as mechanical switch in an EAR DCCB. The prototype is explained in detail, and extensive measurement results are presented, showing that active damping must be perfectly timed to be effective. A novel Thomson-coil (TC) driver is proposed and studied experimentally, which operates the TCA more efficiently by recycling energy during the actuation. Moreover, the novel TC driver reduces the capacitive storage by 50% and allows for faster recharging with lower charging current. Finally, the autoreclosing and proactive commutation operation of the TCA system and the interruption capability of the prototype EAR DCCB are demonstrated experimentally.

A Description of Commutant of Multiplication Operators on Bergman Spaces of the Polydisk

A Description of Commutant of Multiplication Operators on Bergman Spaces of the Polydisk

Musielak-Orlicz-Bumps and Bloom Type Estimates for Commutators of Calderón-Zygmund and Fractional Integral Operators on Generalized Zygmund Spaces Via Sparse Operators

We study continuity properties for commutators of Calderon-Zygmund and fractional integral operators between generalized Zygmund spaces of L log L type, in the variable exponent setting with different weights. In order to reach this goal we use two different approaches: the first one is related to generalized bump conditions on a pair of weights, allowing us to handle with a wide class of symbol involved with the commutator. The other approaches give Bloom type estimates restricting the class of symbols. The techniques involved in both type of results are related with the theory of sparse domination.