Positive Real Part Research Articles

Herglotz's representation and Carathéodory's approximation

AbstractHerglotz's representation of holomorphic functions with positive real part and Carathéodory's theorem on approximation by inner functions are two well‐known classical results in the theory of holomorphic functions on the unit disc. We show that they are equivalent. On a multi‐connected domain , a version of Heglotz's representation is known. Carathéodory's approximation was not known. We formulate and prove it and then show that it is equivalent to the known form of Herglotz's representation. Additionally, it also enables us to prove a new Heglotz's representation in the style of Korányi and Pukánszky. Of particular interest is the fact that the scaling technique of the disc is replaced by Carathéodory's approximation theorem while proving this new form of Herglotz's representation. Carathéodory's approximation theorem is also proved for operator‐valued functions on a multi‐connected domain.

FUNCTIONS WHOSE DERIVATIVE HAS POSITIVE REAL PART

It is well-known that a normalized analytic function for which the quantity $1+zf''(z)/f'(z)$ lies in the half-plane $\real w>-1/2$ is close-to-convex and hence univalent. In this paper, we show that the derivative of the function $f$ has positive real part if the quantity $1+\alpha zf''(z)/f'(z)$ with $\alpha>0$ lies in the sector $|\arg w| <\arctan( \alpha )$.

Radius problem associated with certain ratios and linear combinations of analytic functions

Abstract For normalized starlike functions f : 𝔻 → ℂ, we consider the analytic functions g : 𝔻 → ℂ defined by g(z) = (1 + z(f″(z))/f′(z))/(zf′(z)/f(z)) and g(z) = (1 − α)(zf′(z))/f(z) + α(1 + (zf″(z))/f′(z)), 0 ≤ α ≤ 1. We determine the largest radius ρ with 0 < ρ ≤ 1 such that g(ρ z) is subordinate to various functions with positive real part.

Arithmetic Bohr radius for the Minkowski space

Abstract The main aim of this paper is to study the arithmetic Bohr radius for holomorphic functions defined on a Reinhardt domain in ℂ n {\mathbb{C}^{n}} with positive real part. The present investigation is motivated by the work of Lev Aizenberg [Proc. Amer. Math. Soc. 128 (2000), 2611–2619]. A part of our study in the present paper includes a connection between the classical Bohr radius and the arithmetic Bohr radius of unit ball in the Minkowski space ℓ q n {\ell^{n}_{q}} , 1 ≤ q ≤ ∞ {1\leq q\leq\infty} . Further, we determine the exact value of a Bohr radius in terms of arithmetic Bohr radius.

New Well-Posed boundary conditions for semi-classical Euclidean gravity

We consider four-dimensional Euclidean gravity in a finite cavity. Dirichlet conditions do not yield a well-posed elliptic system, and Anderson has suggested boundary conditions that do. Here we point out that there exists a one-parameter family of boundary conditions, parameterized by a constant p, where a suitably Weyl rescaled boundary metric is fixed, and all give a well-posed elliptic system. Anderson and Dirichlet boundary conditions can be seen as the limits p → 0 and ∞ of these. Focussing on static Euclidean solutions, we derive a thermodynamic first law. Restricting to a spherical spatial boundary, the infillings are flat space or the Schwarzschild solution, and have similar thermodynamics to the Dirichlet case. We consider smooth Euclidean fluctuations about the flat space saddle; for p > 1/6 the spectrum of the Lichnerowicz operator is stable — its eigenvalues have positive real part. Thus we may regard large p as a regularization of the ill-posed Dirichlet boundary conditions. However for p < 1/6 there are unstable modes, even in the spherically symmetric and static sector. We then turn to Lorentzian signature. For p < 1/6 we may understand this spherical Euclidean instability as being paired with a Lorentzian instability associated with the dynamics of the boundary itself. However, a mystery emerges when we consider perturbations that break spherical symmetry. Here we find a plethora of dynamically unstable modes even for p > 1/6, contrasting starkly with the Euclidean stability we found. Thus we seemingly obtain a system with stable thermodynamics, but unstable dynamics, calling into question the standard assumption of smoothness that we have implemented when discussing the Euclidean theory.

Instability of periodic waves for the Korteweg–de Vries–Burgers equation with monostable source

In this paper, it is proved that the KdV–Burgers equation with a monostable source term of Fisher–KPP type has small-amplitude periodic traveling wave solutions with finite fundamental period. These solutions emerge from a subcritical local Hopf bifurcation around a critical value of the wave speed. Moreover, it is shown that these periodic waves are spectrally unstable as solutions to the PDE, that is, the Floquet (continuous) spectrum of the linearization around each periodic wave intersects the unstable half plane of complex values with positive real part. To that end, classical perturbation theory for linear operators is applied in order to prove that the spectrum of the linearized operator around the wave can be approximated by that of a constant coefficient operator around the zero solution, which intersects the unstable complex half plane.

Generalized Bounded Turning Functions Connected with Gregory Coefficients

In this research article, we introduce new family RG of holomorphic functions, which is related to the generalized bounded turning and generating functions of Gregory coefficients. Leveraging the concept of functions with positive real parts, we acquire the first five coefficients for the functions belonging to this newly defined family, demonstrating their sharpness. Furthermore, we find the third Hankel determinant for functions in the class RG. Moreover, the sharp bounds for logarithmic and inverse coefficients of functions belonging to the under-considered class RG are estimated.

Modeling Excitable Cells with Memristors

This paper presents an in-depth analysis of an excitable membrane of a biological system by proposing a novel approach that the cells of the excitable membrane can be modeled as the networks of memristors. We provide compelling evidence from the Chay neuron model that the state-independent mixed ion channel is a nonlinear resistor, while the state-dependent voltage-sensitive potassium ion channel and calcium-sensitive potassium ion channel function as generic memristors from the perspective of electrical circuit theory. The mechanisms that give rise to periodic oscillation, aperiodic (chaotic) oscillation, spikes, and bursting in an excitable cell are also analyzed via a small-signal model, a pole-zero diagram of admittance functions, local activity, the edge of chaos, and the Hopf bifurcation theorem. It is also proved that the zeros of the admittance functions are equivalent to the eigen values of the Jacobian matrix, and the presence of the positive real parts of the eigen values between the two bifurcation points lead to the generation of complicated electrical signals in an excitable membrane. The innovative concepts outlined in this paper pave the way for a deeper understanding of the dynamic behavior of excitable cells, offering potent tools for simulating and exploring the fundamental characteristics of biological neurons.

Stability analysis of electro-osmotic flow in a rotating microchannel

We investigate the linear stability analysis of rotating electro-osmotic flow in confined and unconfined configurations by appealing to the Debye–Hückel approximation. Pertaining to flow in confined and unconfined domains, the stability equations are solved using the Galerkin method to obtain the stability picture. Both qualitative and quantitative aspects of Ekman spirals are examined in stable and unstable scenarios within the unconfined domain. Within the confined domain, the variation of the real growth rate and the transition to instability are analysed using the modified Routh–Hurwitz criteria, employed for the first time in this context. The stability of the underlying flow, characterized by the number of roots with a positive real part, is determined by establishing a Routhian table. The inferences of this analysis show that the velocity plane produces intriguing closed Ekman spirals, which diminish in size with an increase in the rotation speed $\omega$ . The Ekman spirals in the stable region exhibit a distinct discontinuity, indicating the dissipation of disturbances over time. In the confined domain, the flow appears consistently stable for a set of involved parameters pertinent to this analysis, such as electrokinetic parameter $K=1.5$ and rotational parameter $\omega$ approximately up to $6$ . However, the flow instabilities become evident for $K=1.5$ and $\omega \geq 6$ .

Toeplitz Determinant for the Inverse of a Function Whose Derivative Has a Positive Real Part

Abstract Let R denote the family of analytic and univalent functions f(z) in the unit disk Ω = {z ∈ C: |z | &lt; 1} such that Re f′(z) &gt; 0. We investigated the Toeplitz determinant for the inverse of the function to f ∈ R and gave the upper bound estimates of the determinants T 2(2), T 2(3), T 3(1) and T 3(2) .

Electrodynamic Formulas for Media with Negative Permittivity

Electrodynamic formulas widely used in the description of objects containing dielectric media give results contrary to physics in the case when the permittivity of the medium takes a negative value. The problem is solved by refined formulas that allow calculating the electric potentials of the point charge and the point dipole moment, the capacitance of the capacitor, as well as the energy density of the electromagnetic field and the quality factor of the material. The formulas are valid for any media, both with positive and negative real part of the complex permittivity.

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.

Detection and Elimination of the Causes of Unstable System Operation

Small-signal stability, which is also called static stability or modal analysis, deals with system stability in case of small disturbances such as consumption and production variations on hourly and daily basis. Advantage of this type of analysis is its global character because it provides all system state matrix eigenvalues, i.e. system poles, within single system calculation. Existence of single system pole with positive real part indicates unstable system. Measure of relative participation of a particular state variable, which is correlated to particular generator, and particular system pole is provided by calculating participation factors. Sorting of participation factors for all system poles in descending order and establishing a correlation to certain state variable of particular generator provides feedback to generators that are dominant causes of existence of system poles with positive real part. The paper presents calculation of system eigenvalues and establishment of user feedback to generator causing appearance of unstable pole. The establishment of this feedback and elimination of unstable system pole by changing generator parameters is demonstrated on the example of the real distribution system of Leskovac branch with over 2500 nodes and more than 40 synchronous generators connected.

Оптические свойства сверхтонких пленок Pd и Pt на кварцевой подложке и на пленках триоксида вольфрама

Based on ellipsometry and transmission spectra, optical parameters of palladium and platinum films were determined. The Pd and Pt films had a thickness of 5 – 7 nm and were studied both on a SiO2 and WO3 substrates. Despite the extremely small thicknesses, the parameters of most films were well described by isotropic dielectric constant. An interesting feature was that films deposited directly on a SiO2 substrate had a positive (rather than negative, characteristic of a metal) real part of the effective dielectric constant, while the films deposited on WO3 exhibited metallic properties for unannealed films and properties characteristic of metal-dielectric composites for films that have been annealed (some of the films annealed at 300°C in argon retained their metallic properties).

The sharp bound of the third Hankel determinant of the kth-root transformation for bounded turning functions

The objective of this paper is to estimate the sharp bound of the third Hankel determinant for the kth-root transformation to the class of functions whose derivative has a positive real part satisfying the normalized conditions f(0) = 0 and f′(0) = 1 in the open unit disk D := {z ∈ C : |z| &lt; 1}.

Integral Operators Applied to Classes of Convex and Close-to-Convex Meromorphic p-Valent Functions

We consider a newly introduced integral operator that depends on an analytic normalized function and generalizes many other previously studied operators. We find the necessary conditions that this operator has to meet in order to preserve convex meromorphic functions. We know that convexity has great impact in the industry, linear and non-linear programming problems, and optimization. Some lemmas and remarks helping us to obtain complex functions with positive real parts are also given.

Certain Class of Close-to-Convex Univalent Functions

The purpose of this paper was to define a new class of close-to-convex function, denoted by CV(δ,α), which is a subclass of all functions that are univalent in D and have positive coefficients normalized by the conditions f(0)=0, f′(0)=1, if it satisfies such a condition that is dependent on positive real part. Furthermore, we proved how the power series distribution is essential for determining the sufficient and necessary condition on any function f in class CV(δ,α).

The fastest stabilization of second-order switched systems with all modes unstable via an optimal state-dependent switching rule revisited

In this paper, an optimal state-dependent switching rule design method is proposed for the fastest asymptotic stabilization of second-order switched systems, wherein all eigenvalues of each subsystem are positive real parts. First, the definition of an optimal invertible transformation is proposed based on the physical meaning of a vibrational system with one degree-of-freedom. Then, the formulas of both the optimal invertible transformations and the optimal switching lines are calculated. In this way, the designed optimal state-dependent switching rule can minimize the energy increment of unstable subsystem operation and maximize the energy loss of system switching simultaneously, achieving the fastest asymptotic stability. Moreover, the critical stability condition for a general switched system is investigated, demonstrating that this state-dependent switching rule is optimal. Finally, three examples are provided to verify the effectiveness and superiority of the results.

Inequalities for the Coefficients of Schwarz Functions

The relation between a considered family of analytic functions and the class {mathcal {P}} of functions with a positive real part is one of the main tools used in solving various extremal problems, among others coefficient problems. Another approach can be useful in solving such tasks. This approach is to exploit the correspondence between a considered family and the family {mathcal {B}}_0 of bounded analytic functions omega such that omega (0)=0. Such functions appear in the well-known Schwarz lemma, so they are called Schwarz functions. In the literature, there are numerous coefficient functionals discussed for functions in {mathcal {P}}. On the other hand, relative functionals for functions in {mathcal {B}}_0 are not so commonly studied. Consequently, we do not know so much about coefficient inequalities for Schwarz functions. We shall fill the gap to some extent considering two types of functionals. The first one is a Zalcman-type functional c_{n}-c_{k}c_{n-k}; the other one is the Hankel determinant c_{n-1}c_{n+1}-c_{n}{}^2. For these functionals, bounds with respect to a fixed first coefficient c_1 (or a few initial coefficients) are obtained. Some generalizations of these functionals are also given. All results are sharp.

State‐dependent switching stabilization of third‐order switched linear systems consisting of two subsystems with fully positive real part characteristic roots

In the present work, we investigate the stability problem for a third‐order switched linear system in which the eigenvalues of each subsystem are all positive real parts. Firstly, we introduce a degenerated linear mechanical model with two state variables, in which one of the mechanical equations degenerates to a nonholonomic constraint, thereby such a mode being treated as a mechanical system with 1.5‐degrees of freedom (DOFs). Secondly, we establish the corresponding relationship between the mechanical system with 1.5‐DOFs and a standard third‐order linear system so as to define an energy function with actual physical meaning. Thirdly, an invertible transformation is introduced to convert a general third‐order linear system to a standard one so as to calculate energy function easily. On this basis, we reasonably introduce an intermediate subsystem for a general third‐order switched linear system with two unstable subsystems, such that each subsystem and the intermediate subsystem can be converted to standard third‐order linear systems simultaneously via an identical invertible transformation. Accordingly, we construct an energy ratio function with the help of the energy functions of the intermediate subsystem and two subsystems. Fourthly, based on the obtained energy ratio function, a proper state‐dependent switching law is designed to guarantee asymptotic stability of switched systems. That is, the energy loss from switching is large enough to offset the energy increased from unstable subsystems operation in a switching loop. Finally, simulation results demonstrate the effectiveness of the proposed state‐dependent switching approach.