Dissipative Operators Research Articles

Pick’s theorems for dissipative operators

Let H be a complex Hilbert space and let A be a bounded linear transformation on H. For a complex-valued function f, which is analytic in a domain D of the complex plane containing the spectrum of A, let f (A) denote the operator on H defined by means of the Riesz-Dunford integral. In the present paper, several (presumably new) versions of Pick?s theorems are proved for f (A), where A is a dissipative operator (or a proper contraction) and f is a suitable analytic function in the domain D.

Hypocoercivity for a Linearized Multispecies Boltzmann System

A new coercivity estimate on the spectral gap of the linearized Boltzmann collision operator for multiple species is proved. The assumptions on the collision kernels include hard and Maxwellian potentials under Grad's angular cutoff condition. Two proofs are given: a non-constructive one, based on the decomposition of the collision operator into a compact and a coercive part, and a constructive one, which exploits the “cross effects” coming from collisions between different species and which yields explicit constants. Furthermore, the essential spectra of the linearized collision operator and the linearized Boltzmann operator are calculated. Based on the spectral-gap estimate, the exponential convergence towards global equilibrium with explicit rate is shown for solutions to the linearized multispecies Boltzmann system on the torus. The convergence is achieved by the interplay between the dissipative collision operator and the conservative transport operator and is proved by using the hypocoercivity method of Mouhot and Neumann.

Dissipative and nonunitary solutions of operator commutation relations

We study the (generalized) semi-Weyl commutation relations $$ U_gAU_g^*=g(A) \quad \text{ on }\quad \Dom(A), $$ where $A$ is a densely defined operator and $G\ni g\mapsto U_g$ is a unitary representation of the subgroup $G$ of the affine group $\cG$, the group of affine transformations of the real axis preserving the orientation. If $A$ is a symmetric operator, the group $G$ induces an action/flow on the operator unit ball of contractive transformations from $\Ker (A^*-iI)$ to $\Ker (A^*+iI)$. We establish several fixed point theorems for this flow. In the case of one-parameter continuous subgroups of linear transformations, self-adjoint (maximal dissipative) operators associated with the fixed points of the flow give rise to solutions of the (restricted) generalized Weyl commutation relations. We show that in the dissipative setting, the restricted Weyl relations admit a variety of non-unitarily equivalent representations. In the case of deficiency indices $(1,1)$, our general results can be strengthened to the level of an alternative.

Recent Trends, Challenges and Scope in Single-Phase Liquid Flow Heat Transfer in Microchannels-A Review

Miniaturization of electronic gadgets has increased the power density and reduced the sizes of these gadgets drastically. Heat dissipation and stable operation of these electronic devices is challenge for researchers. Microchannel passages providing high surface area to volume ratio gives high heat transfer rates from small areas has emerged as potential heat dissipating and thermal control devices for Micro-Electro Mechanical Systems (MEMS). High performance microchannel heat exchangers are also attractive in applications where space and/or weight constraints dictate the size of a heat exchanger or where performance enhancement is desired. In this review paper a small attempt is made to look into the recent developments and methods to enhance the heat transfer performance of microchannels by using channel geometry, coolant and structural materials. It has been also observed that Nanofluids as coolants in microchannels have excellent potential to enhance the heat transfer performance and are quickly establishing as future coolant to be reckoned with, further challenges faced by researchers and future research scope in the field of single-phase liquid flow heat transfer in microchannels are identified.

Observation of Dissipative Bright Soliton and Dark Soliton in an All-Normal Dispersion Fiber Laser

This paper proposes a novel way for controlling the generation of the dissipative bright soliton and dark soliton operation of lasers. We observe the generation of dissipative bright and dark soliton in an all-normal dispersion fiber laser by employing the nonlinear polarization rotation (NPR) technique. Through adjusting the angle of the polarizer and analyzer, the mode-locked and non-mode-locked regions can be obtained in different polarization directions. Numerical simulation shows that, in an appropriate pump power range, the dissipative bright soliton and dark soliton can be generated simultaneously in the mode-locked and non-mode-locked regions, respectively. If the pump power exceeds the top limit of this range, only dissipative soliton will exist, whereas if it is below the lower bound of this range, only dark soliton will exist.

Non-self-adjoint fourth-order dissipative operators and the completeness of their eigenfunctions

Non-self-adjoint fourth-order dissipative operators and the completeness of their eigenfunctions

Dissipative soliton operation of a diode pumped Yb:NaY(WO₄)₂ laser.

We report on the dissipative soliton operation of a diode pumped Yb:NaY(WO₄)₂ (Yb:NYW) solid-state laser. The dissipative solitons and their features as the net cavity group velocity dispersion is changed from the normal to the anomalous dispersion regime are experimentally investigated. Taking advantage of the dissipative soliton shaping of the mode-locked pulses we have generated stable near transform-limited pulses as short as 54 fs. To our knowledge, this is so far the shortest pulse directly obtained from the mode-locked Yb:NYW oscillator.

Small-scale behavior of Hall magnetohydrodynamic turbulence.

Decaying Hall magnetohydrodynamic (HMHD) turbulence is studied using three-dimensional (3D) direct numerical simulations with grids up to 768(3) points and two different types of initial conditions. Results are compared to analogous magnetohydrodynamic (MHD) runs and both Laplacian and Laplacian-squared dissipative operators are examined. At scales below the ion inertial length, the ratio of magnetic to kinetic energy as a function of wave number transitions to a magnetically dominated state. The transition in behavior is associated with the advection term in the momentum equation becoming subdominant to dissipation. Examination of autocorrelation functions reveals that, while current and vorticity structures are similarly sized in MHD, HMHD current structures are narrower and vorticity structures are wider. The electric field autocorrelation function is significantly narrower in HMHD than in MHD and is similar to the HMHD current autocorrelation function at small separations. HMHD current structures are found to be significantly more intense than in MHD and appear to have an enhanced association with strong alignment between the current and magnetic field, which may be important in collisionless plasmas where field-aligned currents can be unstable. When hyperdiffusivity is used, a longer region consistent with a k(-7/3) scaling is present for right-polarized fluctuations when compared to Laplacian dissipation runs.

Dissipative operators and additive perturbations in locally convex spaces

Let be a densely defined operator on a Banach space X. Characterizations of when generates a C0-semigroup on X are known. The famous result of Lumer and Phillips states that it is so if and only if is dissipative and is dense in X for some . There exists also a rich amount of Banach space results concerning perturbations of dissipative operators. In a recent paper Tyran–Kaminska provides perturbation criteria of dissipative operators in terms of ergodic properties. These results, and others, are shown to remain valid in the setting of general non–normable locally convex spaces. Applications of the results to concrete examples of operators on function spaces are also presented.

Characterization of a Serializer ASIC Chip for the Upgrade of the ATLAS Muon Detector

We report on the design of a serializer ASIC to be used in the ATLAS forward muon detector for trigger data transmission. We discuss the performance of a prototype chip covering power dissipation, latency and stable operating line rate. Tests show that the serializer is capable of running at least at 5.76 Gbps with a bit error ratio below 1x10^{-15}, and a power consumption of 200 mW running at 4.8 Gbps. The latency between the start of loading 30 bits into the serializer to the transmission of the first bit from the serializer is measured to be about 6 ns.

The Optimal Convergence Rates for the Multi-dimensional Chemotaxis Model in Critical Besov Spaces

In this paper, we are concerned with the Cauchy problem to the multi-dimensional (N?4$N\geq4$) chemotaxis model. We prove the optimal convergence rates of the strong solutions to the system for initial data close to a stable equilibrium state in critical Besov spaces. Our main ideas are based on the low-high frequency decomposition and the smooth effect of dissipative operator.

The decay of wall-bounded MHD turbulence at low

We present direct numerical simulations of decaying magnetohydrodynamic (MHD) turbulence at low magnetic Reynolds number. The domain considered is bounded by periodic boundary conditions in the two directions perpendicular to the magnetic field and by two plane Hartmann walls in the third direction. Regimes of high magnetic fields (Hartmann number of up to 896) are reached thanks to a new spectral method using the eigenvectors of the dissipation operator. The decay is found to proceed through two phases: first, energy and integral length scales vary rapidly during a two-dimensionalisation phase extending over approximately a Hartmann friction time. During this phase, the evolution of the former appears significantly more impeded by the presence of walls than that of the latter. Once the large scales are nearly quasi-two-dimensional, the decay results from the competition of a two-dimensional dynamics driven by dissipation in the Hartmann boundary layers and the three-dimensional dynamics of smaller scales. In the later stages of the decay, three-dimensionality subsists under the form of barrel-shaped structures. A purely quasi-two dimensional decay entirely dominated by friction in the Hartmann layers is not reached because of residual dissipation in the bulk. However, this dissipation is not generated by the three-dimensionality that subsists, but by residual viscous friction due to horizontal velocity gradients. In the process, the energy in the velocity component aligned with the magnetic field is found to be strongly suppressed, as is skewness. This result reproduces the experimental findings of Kolesnikov & Tsinober (Fluid Dyn., vol. 9, 1974, pp. 621–624), where, as in the present simulations, Hartmann walls were present.

Unidirectional dissipative soliton operation in an all-normal-dispersion Yb-doped fiber laser without an isolator.

We demonstrate self-started unidirectional dissipative soliton operation and noise-like pulse operation in an all-normal-dispersion bidirectional Yb-doped fiber laser mode-locked by nonlinear polarization rotation. The laser works unidirectionally once mode-locking is achieved due to the cavity directional nonlinearity asymmetry along with the nonlinear polarization rotation mode-locking mechanism.

Information Fusion Identification for Shearer Cutting Status through Improved Support Vector Data Description

In order to research the identification methods for shearer cutting status, some relevant patents and scientific references are studied and analyzed. A new method based on the integration of particle swarm optimization (PSO) and support vector data description (SVDD) is proposed to identify the shearer cutting status. The PSO is provided to solve the constraint optimization problem of SVDD. Some key technologies are proposed to balance the search ability and enhance the population diversity of PSO, and the flowchart of proposed method is designed. Furthermore, some simulation examples are carried out and the compared results indicate that the proposed method is feasible and efficient. Finally, an industrial application example of coal mining face is demonstrated to specify the effect of proposed system. Keywords: Dissipative operation, identification method, inertia weight, particle swarm optimization algorithm, shearer cutting status, support vector data description.

Global well-posedness and blow-up of solutions for the Camassa–Holm equations with fractional dissipation

Consideration in this paper is the effect of varying fractional dissipation with the dissipative operator power \( \gamma \ge 0\) on the well-posedness of the Camassa–Holm equations with fractional dissipation. It is shown that the zero-filter limit (\(\alpha \rightarrow 0\)) of the Camassa–Holm equation with fractional dissipation is the fractal Burgers equation. It is known that in the supercritical case \(\gamma \in [0, 1)\), the fractal Burgers equation blows up in finite time in \(H^s({\mathbb {R}})\) with \(s>\frac{3}{2}-\gamma \). It is established here that the dissipative Camassa–Holm equation is globally well-posed in the critical Sobolev space \(H^{\frac{3}{2}-\gamma }({\mathbb {R}})\) with the fractional parameter \(\gamma \in [\frac{1}{2}, 1)\). Moreover, it is also demonstrated that the solution of the dissipative Camassa–Holm equation blows up in finite time in the particular supercritical case when \( \gamma = 0\).

Investigation of the spectral properties of a fourth order differential operator with finite transmission conditions

In this paper, the spectral analysis of a singular dissipative fourth order differential operator in lim-4 case with finite transmission conditions is investigated. For this purpose, the inverse operator with explicit form is used. Finally, with the help of Livsic's theorem, it is proved that all root vectors of the fourth order dissipative operator in lim-4 case with finite transmission conditions are complete in the Hilbert space. Copyright © 2015 John Wiley & Sons, Ltd.

Regularity and capacity for the fractional dissipative operator

This paper is devoted to exploring some analytic–geometric properties of the regularity and capacity associated with the so-called fractional dissipative operator ∂t+(−Δ)α, naturally establishing a diagonally sharp Hausdorff dimension estimate for the blow-up set of a weak solution to the fractional dissipative equation (∂t+(−Δ)α)u(t,x)=F(t,x) subject to u(0,x)=0. The methods used in this paper rely on effectively controlling the time-dependent non-local kernels and potentials with fractional order α∈(0,1), dual representation of the capacity and Frostman type theorem from geometric measure theory.

A method for spectral DNS of low Rm channel flows based on the least dissipative modes

We put forward a new type of spectral method for the direct numerical simulation of flows where anisotropy or very fine boundary layers are present. The main idea is to take advantage of the fact that such structures are dissipative and that their presence should reduce the number of degrees of freedom of the flow, when paradoxically, their fine resolution incurs extra computational cost in most current methods. The principle of this method is to use a functional basis with elements that already include these fine structures so as to avoid these extra costs. This leads us to develop an algorithm to implement a spectral method for arbitrary functional bases, and in particular, non-orthogonal ones. We construct a basic implementation of this algorithm to simulate magnetohydrodynamic (MHD) channel flows with an externally imposed, transverse magnetic field, where very thin boundary layers are known to develop along the channel walls. In this case, the sought functional basis can be built out of the eigenfunctions of the dissipation operator, which incorporate these boundary layers, and it turns out to be non-orthogonal. We validate this new scheme against numerical simulations of freely decaying MHD turbulence based on a finite volume code and it is found to provide accurate results. Its ability to fully resolve wall-bounded turbulence with a number of modes close to that required by the dynamics is demonstrated on a simple example. This opens the way to full-blown simulations of MHD turbulence under very high magnetic fields. Until now such simulations were too computationally expensive. In contrast to traditional methods the computational cost of the proposed method, does not depend on the intensity of the magnetic field.

Brushless Servomotor’s Thermal Behavior Computer Assisted Evaluation for a Linear Motion NC Axis Experimental Stand

The paper presents works carried out by the authors in the field of NC axes’ structural and functional optimization. This paper includes the results obtained by using a MathCAD application (developed in the doctoral thesis of the second author) for the servomotor's thermal behavior computer assisted evaluation. The analyzed servomotor is included in the driving system of a linear motion NC axis experimental testing stand, (existing in MMS department from EMTS Faculty). The NC axis of the experimental stand integrates a FAGOR brushless servomotor, a 1:1 ratio belt drive intermediary transmission, a ball screw - bearings assembly (lead screw - ball nut - bearings) and a driven element guided through a ball rail system. The MathCAD application was developed in order to perform specific calculation for servomotor's thermal energy dissipation and maximum operating temperature evaluation, and allows to perform the assisted final check and optimum selection of the electrical driving servomotors based on these results. The paper presents, through some screenshots from running the MathCAD application, the computer assisted evaluation procedure and results of the brushless servomotor's thermal behavior analysis corresponding to analyzed NC axis (included in the experimental stand).

Quantum fluctuations of one- and two-dimensional spatial dissipative solitons in a nonlinear interferometer: I. One-dimensional dark solitons

Quantum fluctuations of one-dimensional dark dissipative solitons sustained by an external radiation in an interferometer with a Kerr nonlinearity are analyzed theoretically. The stability region of classical solitons in this interferometer is studied. The boundaries of this region are determined, and types of excited solitons are classified. Quantum fluctuations of solitons are analyzed in an approximation linear in fluctuations. This problem was solved by linearizing the quantum Langevin equation in a neighborhood of a classical solution for the main type of a soliton from the obtained stability region. The main attention has been paid to studying quantum fluctuations of collective variables of dissipative solitons, namely, the coordinate of the center and momentum of the soliton. Based on the expansion of solutions of the linearized equation in eigenfunctions of the discrete spectrum of this equation, a solution describing quantum fluctuations of these variables is constructed. Using this expansion scheme made it possible to give a rigorous definition of the dissipative soliton position fluctuation operator. The study performed based on this scheme has made it also possible to construct a solution for a one-dimensional dark relaxing dissipative soliton. This soliton generalizes the stationary soliton with allowance for the shift of its center and deformation of its profile followed by the recovery of its initial shape. Average squares of quantum fluctuations of collective variables are calculated. A domain of parameters in which there exist quantum states of solitons with an initially high degree of squeezing with respect to the momentum is found. It is shown that such states are in correspondence with significantly higher velocities of soliton center drift. An experiment that could detect the relative squeezing with respect to the momentum due to the soliton center drift is discussed.