Behavior Of Function Research Articles

Boundary Behavior of Conformal Maps on Domains with Corners

This paper investigates the behavior of conformal maps near corner points and other non-differentiable boundary points of planar domains. Conformal maps that preserve angles locally will exhibit singular behavior when approaching corners or other less smooth parts of the boundary. We classify types of isolated corner singularities and characterize the magnitude, derivatives, and integral properties of analytic functions near such points. Explicit mappings are constructed between model domains with cusps, wedges, slits, and logarithmic-type corner points. The behavior of the mapping functions is analyzed as the boundary coordinates approach the singular points. We establish several theorems describing the boundary limits, convergence, boundary correspondences, and boundary integrals of these conformal maps on domains with corners. The mapping properties provide insight into the effect of geometric singularities on analytic functions in application areas such as physics, fluid flow, and engineering problems involving complex mappings. The boundary behavior classifications developed here expand the mathematical understanding of conformal maps on domains with sharp corners or discontinuities.

Majorana edge states in Kitaev chains of the BDI symmetry class

Majorana edge states in Kitaev chains possessing an effective time reversal symmetry with one fermionic site per unit cell are studied. For a semi-infinite chain the equations for the wave functions of Majorana zero modes can be reduced to a single Wiener-Hopf equation, which has an exact analytical solution. We use this solution to determine the asymptotic behaviors of the Majorana wave functions at large distances from the edge of the chain for several infinite-range models described in the literature with focus on a model with slow power-law falloff of pairing and hopping amplitudes. For these models we also determine the asymptotic behavior of the energy of the fermionic state composed of two Majorana modes in the limit of long (finite) Kitaev chains.

Impact of tidal environment on galaxy clustering in GAMA

ABSTRACT We constrain models of the galaxy distribution in the cosmic web using data from the Galaxy and Mass Assembly (GAMA) survey. We model the redshift-space behaviour of the 2-point correlation function (2pcf) and the recently proposed Voronoi volume function (VVF) – which includes information beyond two-point statistics. We extend the standard halo model using extra satellite degrees of freedom and two assembly bias parameters: αcen and αsat, which correlate the occupation numbers of central and satellite galaxies with their host halo’s tidal environment, respectively. We measure $\alpha _{\rm sat}=1.44^{+0.25}_{-0.43}$ and $\alpha _{\rm cen}=-0.79^{+0.29}_{-0.11}$ using a combination of 2pcf and VVF measurements, representing a detection of assembly bias at the 3.3σ (2.4σ) significance level for satellite (central) galaxies. This result remains robust to possible anisotropies in the halocentric distribution of satellites as well as technicalities of estimating the data covariance. We show that the growth rate (fσ8) deduced using models with assembly bias is about 7 per cent (i.e. 1.5σ) lower than if assembly bias is ignored. When projected on to the Ωm–σ8 plane, the model constraints without assembly bias overlap with Planck expectations, while allowing assembly bias introduces significant tension with Planck, preferring either a lower Ωm or a lower σ8. Finally, we find that the all-galaxy weak-lensing signal is unaffected by assembly bias, but the central and satellite sub-populations individually show significantly different signals in the presence of assembly bias. Our results illustrate the importance of accurately modelling galaxy formation for cosmological inference from future surveys.

Electrical resistivity and the Hall effect in the doped Mott-Hubbard material with strong spin-charge coupling

The kinetic characteristics of the doped Mott-Hubbard material are considered within the realistic spin-fermion model which takes into account the strong spin-charge coupling. The kinetic equation constructed on the basis of the mechanism of carrier scattering on the spin fluctuations is solved using the multi-moment method, which allows one to analyze the temperature behavior of nonequilibrium distribution function in the problems of electrical resistivity ρ and the Hall coefficient R H . The calculated dependences ρ(T) and R H (T) for the underdoped and optimally doped regimes demonstrate good qualitative agreement with the experimental data. In particular, the Hall coefficient calculated for the underdoped regime reproduces the experimentally observed sharp drop and even a change in sign at low temperatures.

Different glassy characteristics are related to either caging or dynamical heterogeneity.

Despite the enormous theoretical and application interests, a fundamental understanding of the glassy dynamics remains elusive. The static properties of glassy and ordinary liquids are similar, but their dynamics are dramatically different. What leads to this difference is the central puzzle of the field. Even the primary defining glassy characteristics, their implications, and if they are related to a single mechanism remain unclear. This lack of clarity is a severe hindrance to theoretical progress. Here, we combine analytical arguments and simulations of various systems in different dimensions and address these questions. Our results suggest that the myriad of glassy features are manifestations of two distinct mechanisms. Particle caging controls the mean, and coexisting slow- and fast-moving regions govern the distribution of particle displacements. All the other glassy characteristics are manifestations of these two mechanisms; thus, the Fickian yet non-Gaussian nature of glassy liquids is not surprising. We discover a crossover, from stretched exponential to a power law, in the behavior of the overlap function. This crossover is prominent in simulation data and forms the basis of our analyses. Our results have crucial implications on how the glassy dynamics data are analyzed, challenge some recent suggestions on the mechanisms governing glassy dynamics, and impose strict constraints that a correct theory of glasses must have.

Adaptive Neural Coordinated Control for Multiple Euler-Lagrange Systems With Periodic Event-Triggered Sampling.

This article addresses the event-triggered coordinated control problem for multiple Euler-Lagrange systems subject to parameter uncertainties and external disturbances. Based on the event-triggered technique, a distributed coordinated control scheme is first proposed, where the neural network-based estimation method is incorporated to compensate for parameter uncertainties. Then, an input-based continuous event-triggered (CET) mechanism is developed to schedule the triggering instants, which ensures that the control command is activated only when some specific events occur. After that, by analyzing the possible finite-time escape behavior of the triggering function, the real-time data sampling and event monitoring requirement in the CET strategy is tactfully ruled out, and the CET policy is further transformed into a periodic event-triggered (PET) one. In doing so, each agent only needs to monitor the triggering function at the preset periodic sampling instants, and accordingly, frequent control updating is further relieved. Besides, a parameter selection criterion is provided to specify the relationship between the control performance and the sampling period. Finally, a numerical example of attitude synchronization for multiple satellites is performed to show the effectiveness and superiority of the proposed coordinated control scheme.

Holographic Einstein rings of an AdS black hole in massive gravity

In the context of holography, the Einstein ring of an AdS black hole (BH) in massive gravity (MG) is depicted. An oscillating Gaussian source on one side of the AdS boundary propagates in bulk, and we impose a response function to explain it. Using a wave optics imaging system, we obtain the optical appearance of the Einstein ring. Our research reveals that the ring can change into a luminosity-deformed ring or light spots depending on the variation of parameters and observational positions. When observers are positioned at the north pole, the holographic profiles always appear as a ring with concentric stripe surroundings, and a bright ring appears at the location of the photon sphere of the BH. To investigate the distinct features of the MG AdS BH from previous studies, we discussed the relation between the temperature T and the event horizon ue , which shows the decaying behavior with the increasing values of the event horizon ue , when the graviton parameter m is fixed. These in turn influence the behavior of the response function and the Einstein ring. For example, the amplitude of the lensed response function |〈O〉| increases with the increasing values of the graviton parameter m, for the fixed value of the horizon ue . On the other hand, the amplitude of the lensed response function |〈O〉| decreases with the increasing values of the horizon ue , for the fixed value of graviton parameter m. These differences are also reflected in the Einstein ring, where the intensities and the locations of the Einstein ring significantly vary according to the numerical values of the involved parameters. These findings are also observed in the brightness profiles and the best fit comparison between the results obtained by wave optics and geometric optics for different values of graviton parameter m. In this perspective, our study contributes to a better understanding of the analytical studies of holographic theory, which can be used to evaluate different types of BHs for a fixed wave source and optical system.

Classical wave methods and modern gauge transforms: spectral asymptotics in the one dimensional case

In this article, we consider the asymptotic behaviour of the spectral function of Schrödinger operators on the real line. Let H: L^{2}(mathbb{R})to L^{2}(mathbb{R}) have the form \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ H:=-\\frac{d^{2}}{dx^{2}}+Q, $$\\end{document} where Q is a formally self-adjoint first order differential operator with smooth coefficients, bounded with all derivatives. We show that the kernel of the spectral projector, {1}_{(-infty ,rho ^{2}]}(H), has a complete asymptotic expansion in powers of ρ. This settles the 1-dimensional case of a conjecture made by the last two authors.

On the Two-Point Function of the Ising Model with Infinite-Range Interactions

In this article, we prove some results concerning the truncated two-point function of the infinite-range Ising model above and below the critical temperature. More precisely, if the coupling constants are of the form Jx=ψ(x)e-ρ(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$J_{x}=\\psi (x)\ extsf{e}^{-\\rho (x)}$$\\end{document} with ρ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho $$\\end{document} some norm and ψ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi $$\\end{document} an subexponential correction, we show under appropriate assumptions that given s∈Sd-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s\\in \\mathbb {S}^{d-1}$$\\end{document}, the Laplace transform of the two-point function in the direction s is infinite for β=βsat(s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta =\\beta _\ extrm{sat}(s)$$\\end{document} (where βsat(s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta _\ extrm{sat}(s)$$\\end{document} is a the biggest value such that the inverse correlation length νβ(s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u _{\\beta }(s)$$\\end{document} associated to the truncated two-point function is equal to ρ(s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho (s)$$\\end{document} on [0,βsat(s)))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$[0,\\beta _\ extrm{sat}(s)))$$\\end{document}. Moreover, we prove that the two-point function satisfies up-to-constants Ornstein-Zernike asymptotics for β=βsat(s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta =\\beta _\ extrm{sat}(s)$$\\end{document} on Z\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Z}$$\\end{document}. As far as we know, this constitutes the first result on the behaviour of the two-point function at βsat(s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta _\ extrm{sat}(s)$$\\end{document}. Finally, we show that there exists β0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta _{0}$$\\end{document} such that for every β>β0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta >\\beta _{0}$$\\end{document}, νβ(s)=ρ(s)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u _{\\beta }(s)=\\rho (s)$$\\end{document}. All the results are new and their proofs are built on different results and ideas developed in Duminil-Copin and Tassion (Commun Math Phys 359(2):821–822, 2018) and Aoun et al. in (Commun Math Phys 386:433–467, 2021).

Multiple Terms Identification of Time Fractional Diffusion Equation with Symmetric Potential from Nonlocal Observation

This paper considers a simultaneous identification problem of a time-fractional diffusion equation with a symmetric potential, which aims to identify the fractional order, the potential function, and the Robin coefficient from a nonlocal observation. Firstly, the existence and uniqueness of the weak solution are established for the forward problem. Then, by the asymptotic behavior of the Mittag-Leffler function, the Laplace transform, and the analytic continuation theory, the uniqueness of the simultaneous identification problem is proved under some appropriate assumptions. Finally, the Levenberg–Marquardt method is employed to solve the simultaneous identification problem for finding stably approximate solutions of the fractional order, the potential function, and the Robin coefficient. Numerical experiments for three test cases are given to demonstrate the effectiveness of the presented inversion method.

Matter growth perturbations and cosmography in modified torsion cosmology

We examine the cosmic evolution of the growth of perturbations with respect to matter content at early and recent past era of the Universe under the framework of non-zero torsion cosmology. Some cosmographic parameters are also discussed. To analyze this cosmic scenario, we formulate the cosmic models with two dark matter considerations and apply spherical collapse formalism. We explore that for model-1 (pressureless matter), the density contrast starts growing in early era of the Universe and with the Universe’s overall expansion, density contrast grows faster for different choices of torsion parameter. In model-2 (matter with non-zero pressure), the density contrast shows enormous growth as compared to overall expansion of the Universe for the perturbed region, which indicates the collapse of the perturbed region and formulation of new large scale matter or galaxy. Further, we analyze the behavior of the growth function for both models with different values of torsion parameter. For model-1, the growth function increases smoothly for different choices of torsion parameter but for the model-2 the behavior of the growth function with nonzero DM and torsion significantly deviates from pressureless DM with zero torsion (Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}CDM). We also investigate different cosmographic parameters for both models and analyze their behavior with different choices of torsion parameter and Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}CDM. The results show the power-law expansion rate of the accelerating Universe with respect to redshift function.

Multiple testing of composite null hypotheses for discrete data using randomized p-values.

P-values that are derived from continuously distributed test statistics are typically uniformly distributed on (0,1) under least favorable parameter configurations (LFCs) in the null hypothesis. Conservativeness of a p-value P (meaning that P is under the null hypothesis stochastically larger than uniform on (0,1)) can occur if the test statistic from which P is derived is discrete, or if the true parameter value under the null is not an LFC. To deal with both of these sources of conservativeness, we present two approaches utilizing randomized p-values. We illustrate their effectiveness for testing a composite null hypothesis under a binomial model. We also give an example of how the proposed p-values can be used to test a composite null in group testing designs. We find that the proposed randomized p-values are less conservative compared to nonrandomized p-values under the null hypothesis, but that they are stochastically not smaller under the alternative. The problem of establishing the validity of randomized p-values has received attention in previous literature. We show that our proposed randomized p-values are valid under various discrete statistical models, which are such that the distribution of the corresponding test statistic belongs to an exponential family. The behavior of the power function for the tests based on the proposed randomized p-values as a function of the sample size is also investigated. Simulations and a real data example are used to compare the different considered p-values.

Mirror symmetry for five-parameter Hulek-Verrill manifolds

We study the mirrors of five-parameter Calabi-Yau threefolds first studied by Hulek and Verrill in the context of observed modular behaviour of the zeta functions for Calabi-Yau manifolds. Toric geometry allows for a simple explicit construction of these mirrors, which turn out to be familiar manifolds. These are elliptically fibred in multiple ways. By studying the singular fibres, we are able to identify the rational curves of low degree on the mirror manifolds. This verifies the mirror symmetry prediction obtained by studying the mirror map near large complex structure points. We undertake also an extensive study of the periods of the Hulek-Verrill manifolds and their monodromies. On the mirror, we compute the genus-zero and -one instanton numbers, which are labelled by 5 indices, as h^{1,1}\!=5h1,1=5. There is an obvious permutation symmetry on these indices, but in addition there is a surprising repetition of values. We trace this back to an S_{6}S6 symmetry made manifest by certain constructions of the complex structure moduli space of the Hulek-Verrill manifold. Among other consequences, we see in this way that the moduli space has six large complex structure limits. It is the freedom to expand the prepotential about any one of these points that leads to this symmetry in the instanton numbers. An intriguing fact is that the group that acts on the instanton numbers is larger than S_6S6 and is in fact an infinite hyperbolic Coxeter group, that we study. The group orbits have a “web” structure, and with certain qualifications the instanton numbers are only nonzero if they belong to what we term “positive webs”. This structure has consequences for instanton numbers at all genera.

Binding Energy Stark-Shift, Polarizability and Dipole Moment Response of Shallow Donor Impurity in GaAs Quantum Dots

In this study, we have examined, under the influence of an electric field applied along the z-direction, the binding energy Stark-shift, the dipole moment and the polarizability of a confined shallow donor impurity in GaAs conical-shaped quantum dots (CSQD). With square infinite confinement system, the calculations are based on the approximation of the effective mass by using the finite difference method. Our results show that increasing the radius of the CSQD structure and the electric field intensity increases the Stark shift binding energy and it has a mixed behavior as a function of the impurity position. Furthermore, the polarizability and the dipole moment vary in a quasi-linear way as a function of the dot radius and they follow a decreasing function as a function of the electric field intensity. These two physical parameters have a double behavior, they decrease with the position of the impurity in the strong confinement regime and they increase in the top regions of the quantum dot. These results provide a lot of information about the behavior of the electronic wave function which give more interesting ideas for the fabrication of optoelectronic devices.

Atomic screening and e+e− pair photoproduction at low energies

The effect of screening by atomic electrons on the behavior of electron and positron wave functions in the continuous spectrum at small distances is studied. It is shown that these asymptotics are affected not only by the static potential of atomic electrons, but also by the polarization potential, as well as by the exchange interaction, which is essential for nonrelativistic electrons. A simple analytical expression is obtained for the photoproduction cross section of e+e− pair in an atomic field near the threshold. The spectrum and angular distribution of the produced particles are considered. It is shown that screening significantly affects the cross section in the near-threshold region.

Transport theory and the founding of condensed matter physics

In 1966, Woon Siong Gan coined and invented the name transport theory in condensed matter physics. Today, transport theory is the backbone theory of condensed matter physics and the whole condensed matter physics can be represented by transport theory. His PhD thesis pioneered the application of statistical mechanics to ultrasound propagation in semiconductor in the presence of high magnetic fields and low temperatures with the phase transition from the spherical energy surface of metal to the warped energy surface of semiconductor. The usual treatment is using the many-body theory of quantum field theory. Phase transition is an important topic in condensed matter physics. Thus, his PhD thesis also played a role in the founding of condensed matter physics. In this paper, transport theory is applied to phase transition, an important topic in condensed matter physics. The advantages of using transport theory is its broad coverage of both particles interaction and description of the singularity characteristics of phase transition. An example for illustration is that of magnetization which has Ising model describing spins interaction and the Lee Yang theory describing the singularity behavior of the partition function at the critical point of phase transition.

Two-dimensional Cattaneo-Hristov heat diffusion in the half-plane

In this paper, Cattaneo-Hristov heat diffusion is discussed in the half plane for the first time, and solved under two different boundary conditions. For the solution purpose, the Laplace, and the sine- and exponential- Fourier transforms with respect to time and space variables are applied, respectively. Since the fractional term in the problem is the Caputo-Fabrizio derivative with the exponential kernel, the solutions are in terms of time-dependent exponential and spatial-dependent Bessel functions. Behaviors of the temperature functions due to the change of different parameters of the problem are interpreted by giving 2D and 3D graphics.

Statistics of pressure fluctuations in turbulent kinetic plasmas

ABSTRACTIn this study, we explore the statistics of pressure fluctuations in kinetic collisionless turbulence. A 2.5D kinetic particle-in-cell simulation of decaying turbulence is used to investigate pressure balance via the evolution of thermal and magnetic pressure in a plasma with β of order unity. We also discuss the behaviour of thermal, magnetic, and total pressure structure functions and their corresponding wavenumber spectra. The total pressure spectrum exhibits a slope of −7/3 extending for about a decade in the ion-inertial range. In contrast, shallower −5/3 spectra are characteristic of the magnetic pressure and thermal pressure. The steeper total pressure spectrum is a consequence of cancellation caused by density-magnetic field magnitude anti-correlation. Further, we evaluate higher order total pressure structure functions in an effort to discuss intermittency and compare the power exponents with higher order structure functions of velocity and magnetic fluctuations. Finally, applications to astrophysical systems are also discussed.

Position control based on Gaussian function applied on magneto-rheological fluid disc brake of a hybrid mechanical device (MR brake - DC motor)

This article presents an angular position control, based on the Gaussian function, of a Magneto-Rheological fluid disc brake (MR brake) driven by a DC motor. Our proposed control strategy is to apply a continuous magnetic flux density to the MR brake, which will be maximum when the proportional controller of the DC motor reaches the desired position to brake the hybrid device. The MR brake controller activates a braking torque that adopts the behavior of the Gaussian function instead of a pulsed braking torque as provided by other commonly used controllers (On-Off controllers). The response of the MR brake controller, which is presented in a closed-loop feedback system, depends on the angular position error of the shaft and a tuning parameter representing the critical angular position at which the magnetic flux density, which is applied to the MR brake, reaches 60.65% of its maximum value. The advantage is to avoid knowing the dynamic parameters, such as the inertia of the mechanical device or its speed, and to reject these perturbations by a simple tuning parameter of the MR brake. To show the effectiveness of the proposed controller, the dynamic model of a slider-crank mechanism is considered. The results showed similar behavior as conventional controllers, where overshoot and oscillations were minimized. This behavior has been obtained in other research articles using controllers that require a greater amount of data processing.

A stronger form of Yamamoto's theorem on singular values

For a matrix T∈Mm(C), let |T|:=T⁎T. For A∈Mm(C), we show that the matrix sequence {|An|1n}n∈N converges to a positive-semidefinite matrix H whose jth-largest eigenvalue is equal to the jth-largest eigenvalue-modulus of A (for 1≤j≤m). In fact, we give an explicit description of the spectral projections of H in terms of the eigenspaces of the diagonalizable part of A in its Jordan-Chevalley decomposition. This gives us a stronger form of Yamamoto's theorem which asserts that limn→∞⁡sj(An)1n is equal to the jth-largest eigenvalue-modulus of A, where sj(An) denotes the jth-largest singular value of An. Moreover, we also discuss applications to the asymptotic behaviour of the matrix exponential function, t↦etA.