Positive Imaginary Part Research Articles

Cosmological observatories

Abstract We study the static patch of de Sitter space in the presence of a timelike boundary. We impose that the conformal class of the induced metric and the trace of the extrinsic curvature, K, are fixed at the boundary. We present the thermodynamic structure of de Sitter space subject to these boundary conditions, for static and spherically symmetric configurations to leading order in the semiclassical approximation. In three spacetime dimensions, and taking K constant on a toroidal Euclidean boundary, we find that the spacetime is thermally stable for all K. In four spacetime dimensions, the thermal stability depends on the value of K. It is established that for sufficiently large K, the de Sitter static patch subject to conformal boundary conditions is thermally stable. This contrasts the Dirichlet problem for which the region encompassing the cosmological horizon has negative specific heat. We present an analysis of the linearised Einstein equations subject to conformal boundary conditions. In the worldline limit of the timelike boundary, the underlying modes are linked to the quasinormal modes of the static patch. In the limit where the timelike boundary approaches the cosmological event horizon, the linearised modes are interpreted in terms of the shear and sound modes of a fluid dynamical system. Additionally, we find modes with a frequency of positive imaginary part. Measured in a local inertial reference frame, and taking the stretched cosmological horizon limit, these modes grow at most polynomially.

Norm and numerical radius inequalities for operator matrices

Operator matrices have played a significant role in studying Hilbert space operators. In this paper, we discuss further properties of operator matrices and present new estimates for the operator norms and numerical radii of such operators. Moreover, operator matrices with positive real and imaginary parts will be discussed, and sharper bounds will be shown for such classes.

Long‐time asymptotics and the radiation condition with time‐periodic boundary conditions for linear evolution equations on the half‐line and experiment

AbstractThe asymptotic Dirichlet‐to‐Neumann (D‐N) map is constructed for a class of scalar, constant coefficient, linear, third‐order, dispersive equations with asymptotically time/periodic Dirichlet boundary data and zero initial data on the half‐line, modeling a wavemaker acting upon an initially quiescent medium. The large time asymptotics for the special cases of the linear Korteweg‐de Vries and linear Benjamin–Bona–Mahony (BBM) equations are obtained. The D‐N map is proven to be unique if and only if the radiation condition that selects the unique wave number branch of the dispersion relation for a sinusoidal, time‐dependent boundary condition holds: (i) for frequencies in a finite interval, the wave number is real and corresponds to positive group velocity, and (ii) for frequencies outside the interval, the wave number is complex with positive imaginary part. For fixed spatial location , the corresponding asymptotic solution is (i) a traveling wave or (ii) a spatially decaying, time‐periodic wave. The linearized BBM asymptotics are found to quantitatively agree with viscous core‐annular fluid experiments.

Complex modal analysis of structural vibration and acoustic radiation of plates

Inhomogeneous damping distribution leads to the occurrence of complex modes of structures. Complex modes' vibration and acoustic radiation characteristics are different from real modes. This paper studies the complex modal characteristics of inhomogeneously damped plates for forced vibration. We used the state space method to obtain the complex eigenvalues and eigenvectors of the vibration system and calculated the relationship between modal coordinates and frequency under unit force. The results show that the absolute values of the modal coordinates of two conjugate complex eigenvectors are different, and the structure's vibration is dominated by the eigenvectors corresponding to eigenvalues with positive imaginary parts near natural frequencies. When investigating the structural sound radiation in the vibration mode space, each mode's acoustic radiation is not independent but coupled with each other. The influence of the coupling between complex modes on the calculation results of the acoustic radiation power of the plate is studied in this paper. The results show that the coupling effect between modes cannot be ignored when calculating the acoustic radiation power based on complex modes. Finally, the error of calculating acoustic power using the decoupling approximation instead of the complex mode method is analyzed.

A Cross-Power Spectral Density Method for Locating Oscillation Sources Using Synchrophasor Measurements

This paper proposes a new measurement-based approach for locating the sources of forced or poorly damped oscillations in a power system. The approach is based on the concept of cross-correlation in the frequency domain known as cross-power spectral density (CPSD). CPSDs of synchronized voltage magnitude and voltage angle versus active power and reactive power signals obtained by phasor measurement units (PMUs) are computed using fast Fourier transform. The largest positive imaginary part of a CPSD is used as an indicator of the oscillation source. The type of an oscillation source is determined by comparing the spectral densities of active and reactive power. In addition, preprocessing of the signals is performed via variational mode decomposition for extracting the dynamic component of the signals. The proposed approach was able to successfully identify all submitted test cases in the IEEE-NASPI Oscillation Source Location Contest. Several case studies presented in this paper highlight the advantages of the proposed approach compared to the state-of-the-art dissipating energy flow method.

Asymmetric phase modulation of light with parity-symmetry broken metasurfaces

The design of wavefront-shaping devices is conventionally approached using real-frequency modeling. However, since these devices interact with light through radiative channels, they are by default non-Hermitian objects having complex eigenvalues (poles and zeros) that are marked by phase singularities in a complex frequency plane. Here, by using temporal coupled mode theory, we derive analytical expressions allowing to predict the location of these phase singularities in a complex plane and as a result, allowing to control the induced phase modulation of light. In particular, we show that spatial inversion symmetry breaking—implemented herein by controlling the coupling efficiency between input and output radiative channels of two-port components called metasurfaces—lifts the degeneracy of reflection zeros in forward and backward directions, and introduces a complex singularity with a positive imaginary part necessary for a full 2π-phase gradient. Our work establishes a general framework to predict and study the response of resonant systems in photonics and metaoptics.

Charge instability of JMaRT geometries

We perform a detailed study of linear perturbations of the JMaRT family of non-BPS smooth horizonless solutions of type IIB supergravity beyond the near-decoupling limit. In addition to the unstable quasi normal modes (QNMs) responsible for the ergo-region instability, already studied in the literature, we find a new class of ‘charged’ unstable modes with positive imaginary part, that can be interpreted in terms of the emission of charged (scalar) quanta with non zero KK momentum. We use both matched asymptotic expansions and numerical integration methods. Moreover, we exploit the recently discovered correspondence between JMaRT perturbation theory, governed by a Reduced Confluent Heun Equation, and the quantum Seiberg-Witten (SW) curve of N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 SYM theory with gauge group SU(2) and Nf = (0, 2) flavours.

The combinatorics of supertorus sheaf cohomology

Affine superspace C1|n has a single bosonic coordinate z and n fermionic coordinates θ1,…,θn. Let M be the supertorus obtained by quotienting C1|n by the abelian group generated by the maps S:(z,θ1,…,θn)↦(z+1,θ1,…,θn) and T:(z,θ1,…,θn)↦(z+t,θ1+α1,…,θn+αn) where t∈C has positive imaginary part and α1,…,αn are independent fermionic parameters. We compute the zeroth and first cohomology groups of the structure sheaf O of M as doubly graded Sn-modules, exhibiting an instance of Serre duality between these groups. We use skein relations and noncrossing matchings to give a combinatorial presentation of H0(M,O) in terms of generators and relations.

ON THE INTEGRAL FORM OF RANK 1 KAC–MOODY ALGEBRAS

In this paper we shall prove that the ℤ-subalgebra generated by the divided powers of the Drinfeld generators xr±\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {x}_r^{\\pm } $$\\end{document} (r ∈ ℤ) of the Kac–Moody algebra of type A22\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\ extrm{A}}_2^{(2)} $$\\end{document} is an integral form (strictly smaller than Mitzman’s; see [Mi]) of the enveloping algebra, we shall exhibit a basis generalizing the one provided in [G] for the untwisted affine Kac–Moody algebras and we shall determine explicitly the commutation relations. Moreover, we prove that both in the untwisted and in the twisted case the positive (respectively negative) imaginary part of the integral form is an algebra of polynomials over ℤ.

Investigation of eigenfunctions of perturbation of the radial component of flow velocity

The flow of a thermoviscous model fluid in an annular channel with a given temperature field is considered. The problem of the stability of the flow of a thermoviscous fluid is solved on the basis of a generalized equation by the spectral method of expansion in Chebyshev polynomials of the first kind. The influence of taking into account the exponential dependence of fluid viscosity on temperature and channel geometry on the spectral characteristics of the equation of hydrodynamic stability of incompressible fluid flow in an annular channel is investigated. The eigenvalue spectra were obtained numerically. The spectral characteristics determine the structure of the eigenfunctions and the critical parameters of the flow of a thermoviscous fluid. In this case, the eigenfunctions demonstrate the behavior of transverse velocity disturbances, their possible growth or decay over time. Graphs of the eigenfunctions of the generalized flow stability equation in an annular channel have been constructed. It is shown that the structure of the spectra largely depends on both the properties of the liquid, determined by the functional dependence of viscosity, and on the geometry of the channel. The stability of the flow of a thermoviscous fluid depends on the presence of an eigenvalue with a positive imaginary part among the entire set of found eigenvalues for fixed parameters of the Reynolds number and wave number. It has been established that at small values of the thermoviscosity parameter the spectrum is comparable to the spectrum for isothermal fluid flow in a flat channel, however, as it increases, the number of eigenvalues and their density increase, that is, there are a greater number of points at which the problem has non-zero amplitudes of transverse velocity disturbances. It is worth noting that the diversity of eigenvalue spectra corresponds to the diversity of eigenfunctions, which have a nontrivial distribution of the oscillation amplitude over the cross section in each case. Smooth curves are obtained for a narrow channel, as for the case of a flat channel. However, as the ratio of the channel radii increases, “jumps” appear. Also, note that the eigenfunctions do not have the property of symmetry; this follows from the fact that the velocity profile in the unperturbed state also does not have symmetry. The maximum values of the eigenfunctions are shifted to the right from the center of the channel, which corresponds to the fact that disturbances arise and grow intensively near the hot wall.

Ultracold spin-balanced fermionic quantum liquids with renormalized P -wave interactions

We consider a spin-balanced degenerate gas of spin-1/2 fermions whose dynamics is governed by low-energy $P$-wave interactions, characterized by the scattering volume $a_1$ and effective momentum $r_1$. The energy per particle $\bar{\cal{E}}$ in the many-body system is calculated by resumming the ladder diagrams comprising both particle-particle and hole-hole intermediate states, following the novel advances recently developed by us in Ann.Phys. 437,168741(2022). This allows to obtain a renormalized result for $\bar{\cal{E}}$ within generic cutoff regularization schemes, with $\bar{\cal{E}}$ directly expressed in terms of the scattering parameters $a_1$ and $r_1$, once the cut off is sent to infinity. The whole set of possible values of $a_1$ and $r_1$ is explored for the first time in the literature looking for minima in the energy per particle with $\bar{\cal{E}}$ given as described. They are actually found, but a further inspection reveals that the associated scattering parameters give rise to resonance poles in the complex momentum-plane with positive imaginary part, which is at odds with the Hermiticity of the Hamiltonian. We also determine that these conflictive poles, with a pole-position momentum that is smaller in absolute value than the Fermi momentum of the system, clearly impact the calculation of $\bar{\cal{E}}$. As a result, we conclude that unpolarized spin-1/2 fermionic normal matter interacting in $P$-wave is not stable. We also study three universal parameters around the unitary limit. Finally, the whole set of values for the parameters $a_1$, $r_1$ is characterized according to whether they give rise to unallowed poles and, if so, by attending to their pole positions relative to the Fermi momentum of the system explored.

The Loschmidt spectral form factor

The Spectral Form Factor (SFF) measures the fluctuations in the density of states of a Hamiltonian. We consider a generalization of the SFF called the Loschmidt Spectral Form Factor, textrm{tr}left[{e}^{i{H}_1T}right]textrm{tr}left[{e}^{-i{H}_2T}right] , for H1 − H2 small. If the ensemble average of the SFF is the variance of the density fluctuations for a single Hamiltonian drawn from the ensemble, the averaged Loschmidt SFF is the covariance for two Hamiltonians drawn from a correlated ensemble. This object is a time-domain version of the parametric correlations studied in the quantum chaos and random matrix literatures. We show analytically that the averaged Loschmidt SFF is proportional to eiλTT for a complex rate λ with a positive imaginary part, showing in a quantitative way that the long-time details of the spectrum are exponentially more sensitive to perturbations than the short-time properties. We calculate λ in a number of cases, including random matrix theory, theories with a single localized defect, and hydrodynamic theories.

Loewner’s theorem for maps on operator domains

AbstractThe classical Loewner’s theorem states that operator monotone functions on real intervals are described by holomorphic functions on the upper half-plane. We characterize local order isomorphisms on operator domains by biholomorphic automorphisms of the generalized upper half-plane, which is the collection of all operators with positive invertible imaginary part. We describe such maps in an explicit manner, and examine properties of maximal local order isomorphisms. Moreover, in the finite-dimensional case, we prove that every order embedding of a matrix domain is a homeomorphic order isomorphism onto another matrix domain.

Negative Series Resistance (Rs) and Real Part of Impedance (Z′), and Positive and Negative Imaginary Part of Impedance (Z″) at a High Frequency of Au/CNTS/n-Si/Al Structure

Here in this article, presented negative values of series resistance (Rs), the negative real part of impedance (Z″), the positive and negative values of imaginary part of impedance (Z″) seemed at high frequency for all voltages, temperatures and frequencies of Au/CNTS/n-Si/Al. At all frequencies the Rs has positive values, increase with decreasing frequencies except at frequency 2 × 107 Hz (Rs) has negative values reached to about −5200 Ω. At f = 2 × 107 Hz Z″ has negative values reached to −0.5 Ω and at other frequencies, the Z″ has positive values, growth with decreasing frequency. The Z″ has positive values at frequencies (2 × 107, 1 × 107, 1 × 103, 1 × 102, 10) Hz, whilst at frequencies (1 × 106, 1 × 105, 1 × 104) Hz the Z″ has negative values. At high frequencies, this assembly Au/CNTS/n-Si/Al behaves as a tunneling diode has negative resistance. At other frequencies this structure Au/CNTS/n-Si/Al behaves as a normal diode, and negative resistance is disappeared. So this structure Au/CNTS/n-Si/Al is applied as tunneling diode at high frequency and normal diode by changing of frequencies. The electrical properties and conduction mechanism of this structure were investigated.

Monotonicity of the principal pivot transform

We prove that the principal pivot transform (also known as the partial inverse, sweep operator, or exchange operator in various contexts) maps matrices with positive imaginary part to matrices with positive imaginary part. We show that the principal pivot transform is matrix monotone by establishing Hermitian square representations for the imaginary part and the derivative.

Investigation of the spectral characteristics of a thermoviscous fluid flow in an annular channel

The flow of a thermoviscous model fluid in an annular channel with a given temperature field is considered. The problem of the stability of the flow of a thermoviscous fluid is solved on the basis of the generalized equation by the spectral method of expansion in Chebyshev polynomials of the first kind. The effect of taking into account the exponential dependence of the fluid viscosity on temperature and channel geometry on the spectral characteristics of the equation of hydrodynamic stability of an incompressible fluid flow in a flat channel for various wall temperatures is studied. Spectral patterns of eigenvalues of the generalized equation are constructed. The spectral characteristics determine the structure of the eigenfunctions and the critical parameters of the flow of a thermoviscous fluid. In this case, the eigenfunctions demonstrate the behavior of transverse velocity perturbations, their possible growth or decay with time. It is shown that the structure of the spectra largely depends both on the properties of the liquid, determined by the index of the functional dependence of viscosity, and on the geometry of the channel. It has been established that for small values of the thermoviscosity parameter, the spectrum is comparable to the spectrum for an isothermal fluid flow in a flat channel, however, as it increases, the number of eigenvalues and their density increase, that is, there are more points at which the problem has nonzero amplitudes of transverse velocity perturbations. The stability of a thermoviscous fluid flow depends on the presence of an eigenvalue with a positive imaginary part among the entire set of found eigenvalues for fixed parameters of the Reynolds number and wave number. It is shown that, at fixed values of the Reynolds number and wave number, the flow can become unstable with an increase in the thermoviscosity parameter.

Fate of the conformal order

We use holographic correspondence to study transport of the conformal plasma in ${\mathbb{R}}^{2,1}$ in a phase with a spontaneously broken global ${\mathbb{Z}}_{2}$ symmetry. The dual black branes in a Poincare patch of asymptotically ${\mathrm{AdS}}_{4}$ have ``hair''---a condensate of the order parameter for the broken symmetry. This hair affects both the hydrodynamic and the nonhydrodynamic quasinormal modes of the black branes. Nonetheless, the shear viscosity of the conformal order is universal, the bulk viscosity vanishes and the speed of the sound waves is ${c}_{s}^{2}=\frac{1}{2}$. We compute the low-lying spectrum of the nonhydrodynamic modes. We identify a quasinormal mode associated with the fluctuations of the ${\mathbb{Z}}_{2}$ order parameter with the positive imaginary part. The presence of this mode in the spectrum renders the holographic conformal order perturbatively unstable. Correspondingly, the dual black branes violate the correlated stability conjecture.

Photoelasticity of VO2 nanolayers in insulating and metallic phases studied by picosecond ultrasonics

We use a picosecond ultrasonic technique to evaluate the photoelastic parameters at the wavelength of 1028 nm in epitaxial vanadium dioxide $({\mathrm{VO}}_{2})$ nanolayers grown on c-cut sapphire substrates. In the experiments, we monitor the picosecond evolution of the reflectivity of ${\mathrm{VO}}_{2}$ in insulating and metallic phases under the impact of a picosecond longitudinal strain pulse injected into the nanolayer from the side of the substrate. We show that in a 145-nm-thick granular nanolayer, the temporal features of the reflectivity are clearly dependent on the phase state of ${\mathrm{VO}}_{2}$, showing the change of the photoelastic parameters upon the insulator-metal transition. Analytical consideration and numerical simulations of the optical response to the picosecond strain pulse show that the temporal evolution of the reflectivity strongly depends on the complex photoelastic parameter. The analysis enables us to obtain the values of the photoelastic parameters in the studied nanolayer in both insulating and metallic phases. We find that for a 145-nm film of ${\mathrm{VO}}_{2}$ in an insulating state the imaginary part of the photoelastic constant is negligible. This means that in the insulating phase the strain does not affect the optical absorption of ${\mathrm{VO}}_{2}$. In the metallic phase, the photoelastic parameter of ${\mathrm{VO}}_{2}$ is found to be similar to that typical for metals with positive real and negative imaginary parts. We further show that the optical response to the strain pulse in the layer consisting of disconnected ${\mathrm{VO}}_{2}$ nanohillocks with an average height of 70 nm is governed by their morphology and is different from what is predicted in the plane ${\mathrm{VO}}_{2}$ films.

Upper bounds of some special zeros for the Rankin-Selberg L-function

Abstract In this paper, we prove some conditional results about the order of zero at central point s = 1/2 of the Rankin-Selberg L-function L(s, πf × π͠′ f ). Then, we give an upper bound for the height of the first zero with positive imaginary part of L(s, πf × π͠′ f ). We apply our results to automorphic L-functions.

Geometric Properties of Measures Related to Holomorphic Functions Having Positive Imaginary or Real Part

In this paper, we study the properties of a certain class of Borel measures on {mathbb {R}}^n that arise in the integral representation of Herglotz–Nevanlinna functions. In particular, we find that restrictions to certain hyperplanes are of a surprisingly simple form and show that the supports of such measures cannot lie within particular geometric regions, e.g., strips with positive slope. Corresponding results are derived for measures on the unit poly-torus with vanishing mixed Fourier coefficients. These measures are closely related to functions mapping the unit polydisk analytically into the right half-plane.