Positive Real Part Research Articles

Ruin problems in the generalized Erlang(n) risk model

For actuarial applications we consider the Sparre–Andersen risk model when the interclaim times are Generalized Erlang(n) distributed. Unlike the standard Erlang(n) case, the roots of the generalized Lundberg’s equation with positive real parts can be multiple. This has a significant impact in the formulae for ruin probabilities that have to be found. We start by addressing the problem of solving an integro–differential equation that is satisfied by the survival probability, as well as other probabilities related, and present a method to solve such equation. This is done by considering the roots with positive real parts of the generalized Lundberg’s equation, and then establishing a one-one relation between them and the solutions of the integro–differential equation mentioned above. We first study the cases when all the roots are single and when there are roots with higher multiplicity. Secondly, we show that it is possible to have double roots but no higher multiplicity. Also, we show that the number of double roots depend on the choice of the parameters of the generalized Erlang(n) distribution, with a maximum number depending on n being even or odd. Afterwards, we extend our findings above for the computation of the distribution of the maximum severity of ruin as well as, considering an interest force, to the study the expected discounted future dividends, prior to ruin. Our findings show an alternative and more general method to the one provided by Albrecher et al. (Insur Math Econ 37:324–334, 2005), by considering a general claim amount distribution.

A Herglotz-type representation for vector-valued holomorphic mappings on the unit ball of [formula omitted

The generalization of the Carathéodory class, those analytic functions on the open unit disk having positive real part and taking the value 1 at the origin, to the open unit ball B of Cn is the family M of all holomorphic mappings f:B→Cn such that f(0)=0, Df(0)=I, and Re〈f(z),z〉>0 for all z∈B∖{0}, where Df is the Fréchet derivative of f, I is the identity operator on Cn, and 〈⋅,⋅〉 is the Hermitian inner product in Cn. We present an integral representation for functions in the class M in terms of probability measures on the unit sphere S=∂B similar to the well-known Herglotz representation of the Carathéodory class. This representation follows, in part, from a new integral formula of Cauchy type that reproduces a continuous f:B‾→Cn whose restriction to B is holomorphic by using a fixed vector-valued kernel and the scalar values 〈f(u),u〉, u∈S. Not every probability measure on S corresponds to a mapping in M, and we conclude by examining several additional properties the representing measures must satisfy.

Unstable transient response of gyroscopic systems with stable eigenvalues

Gyroscopic conservative dynamical systems may exhibit flutter instability that leads to a pair of complex conjugate eigenvalues, one of which has a positive real part and thus leads to a divergent free response of the system. When dealing with non-conservative systems, the pitch fork bifurcation shifts toward the negative real part of the root locus, presenting a pair of eigenvalues with equal imaginary parts, while the real parts may or may not be negative. Several works study the stability of these systems for relevant engineering applications such as the flutter in airplane wings or suspended bridges, brake squeal, etc. and a common approach to detect the stability is the complex eigenvalue analysis that considers systems with all negative real part eigenvalues as stable systems. This paper studies analytically and numerically the cases where the free response of these systems exhibits a transient divergent time history even if all the eigenvalues have negative real part thus usually considered as stable, and relates such a behaviour to the non orthogonality of the eigenvectors. Finally, a numerical method to evaluate the presence of such instability is proposed.

Absolute Stability of Nonlinear Systems with Piecewise Linear Sector Condition

The circle theorem is one of the most important methods for study of absolute stability of nonlinear systems. In circle theorem, it is assumed that the nonlinearity term is located between linear bounds of a typical sector. In this paper the absolute stability of nonlinear systems with generalized sector condition is studied in which the bounds of the sector are piecewise linear in general. Therefore this method could be applied to the sector with nonlinear bounds. In this paper using modified Nyquist criterion, it is proved that the study of absolute stability of a nonlinear system using circle theorem could be reduced to the study of stability of equalized linear system. The aim of this paper is to define a pseudo circle region that the Nyquist plot of the linear system with nonlinearity in sector condition with piecewise bounds doesn’t have any intersection with that region and encircle it m times in the counterclockwise direction where m is the number of poles of linear part of the system with positive real parts.

Upper Bound of the Second Order Hankel Determinant for a Class of Functions with Positive Real Part

Upper Bound of the Second Order Hankel Determinant for a Class of Functions with Positive Real Part

Some characteristic properties of analytic functions

Abstract In this paper, we consider a class L(λ, μ; ϕ) of analytic functions f defined in the open unit disk U satisfying the subordination condition that , where is the Sălăgean operator and ϕ(z) is a convex function with positive real part in U. We obtain some characteristic properties giving the coefficient inequality, radius and subordination results, and an inclusion result for the above class when the function ϕ(z) is a bilinear mapping in the open unit disk. For these functions f (z) ; sharp bounds for the initial coefficient and for the Fekete-Szegö functional are determined, and also some integral representations are given.

Research on Geometric Mappings in Complex Systems Analysis

We mainly discuss the properties of a new subclass of starlike functions, namely, almost starlike functions of complex order λ, in one and several complex variables. We get the growth and distortion results for almost starlike functions of complex order λ. By the properties of functions with positive real parts and considering the zero of order k, we obtain the coefficient estimates for almost starlike functions of complex order λ on D. We also discuss the invariance of almost starlike mappings of complex order λ on Reinhardt domains and on the unit ball B in complex Banach spaces. The conclusions contain and generalize some known results.

ANALYSIS AND DESIGN OF ANTI-CONTROLLED HIGHER-DIMENSIONAL HYPERCHAOTIC SYSTEMS VIA LYAPUNOV-EXPONENT GENERATING ALGORITHMS

Based on Lyapunov-exponent generation and the Gram-Schimdt orthogonalization, analysis and design of some anti-controlled higher-dimensional hyperchaotic systems are investigated in this paper. First, some theoretical results for Lyapunov-exponent generating algorithms are proposed. Then, the relationship between the number of Lyapunov exponents and the number of positive real parts of the eigenvalues of the Jacobi matrix is qualitatively described and analyzed. By configuring as many as possible positive real parts of the Jacobian eigenvalues, a simple anti-controller of the form <i>b</i> sin(<i>&#963;x</i>) for higher-dimensional linear systems is designed, so that the controlled systems can be hyperchaotic with multiple positive Lyapunov exponents. Utilizing the above property, one can resolve the positive Lyapunov exponents allocation problem by purposefully designing the number of positive real parts of the corresponding eigenvalues. Two examples of such anti-controlled higherdimensional hyperchaotic systems are given for demonstration.

Вычислительные алгоритмы исследования устойчивости динамических систем в задачах схемотехнического анализа

The article presents efficient computational algorithms for analysis of the stability of the dynamical systems. The algorithms are based on the principle of modal approximation. A numerical algorithm for computing of the poles of the transfer function is proposed, which ensures determination of the poles located in the right half-plane. In order to detect the pole with the positive real part the suggested algorithm exploits the property of the damping factor ς i =- αi αi 2 + βi 2 for the obtained poles λi= αi+i βi . This factor is negative for the poles with positive real parts. The following sequence of the form ς1 α1 , β1 - ς0 < ς2 α2 , β2 - ς0 <...< ςk αk , βk - ς0 is constructed for the computed poles. The sequence is converged to the pole with the positive real part under ς0 <1 . The paper presents an algorithm for computation of the poles of the transfer function with the maximal sensitivity with respect to the circuit parameters. The suggested algorithm allows us to detect the parameters with the maximal impact on the shift of the poles and to determine the critical parameter values corresponding to the boundary of instability.

Local stability analysis of differential equations with state-dependent delay

In the present article, we discuss some aspects of the local stability analysis for a class of abstract functional differential equations. This is done under smoothness assumptions which are often satisfied in the presence of a state-dependent delay. Apart from recapitulating the two classical principles of linearized stability and instability, we deduce the analogon of the Pliss reduction principle for the class of differential equations under consideration. This reduction principle enables to determine the local stability properties of a solution in the situation where the linearization does not have any eigenvalues with positive real part but at least one eigenvalue on the imaginary axis.

Sign patterns that require [formula omitted] exist for each n ≥ 4

The refined inertia of a square real matrix A is the ordered 4-tuple (n+,n−,nz,2np), where n+ (resp., n−) is the number of eigenvalues of A with positive (resp., negative) real part, nz is the number of zero eigenvalues of A, and 2np is the number of nonzero pure imaginary eigenvalues of A. For n≥3, the set of refined inertias Hn={(0,n,0,0),(0,n−2,0,2),(2,n−2,0,0)} is important for the onset of Hopf bifurcation in dynamical systems. We say that an n×n sign pattern A requires Hn if Hn={ri(B)|B∈Q(A)}. Bodine et al. conjectured that no n×n irreducible sign pattern that requires Hn exists for n sufficiently large, possibly n≥8. However, for each n≥4, we identify three n×n irreducible sign patterns that require Hn, which resolves this conjecture.

A generalization of Livingston's coefficient inequalities for functions with positive real part

For functions p(z)=1+∑n=1∞pnzn holomorphic in the unit disk, satisfying Rep(z)>0, we generalize two inequalities proved by Livingston [10,11] and simplify their proofs. One of our results states that |pn−wpkpn−k|≤2max⁡{1,|1−2w|}, w∈C. Another result involves certain determinants whose entries are the coefficients pn. Both results are sharp. As applications we provide a simple proof of a theorem of Brown [2] and various inequalities for the coefficients of holomorphic self-maps of the unit disk.

Motion stability analysis of non-sinusoidal oscillation of mold driven by servomotor

The investments of the electro-hydraulic servo system of the mold non-sinusoidal oscillator are great, the modification ratio of the mechanical type is unable to be adjusted online, and some continuous casters suffer from server resonance during the casting. A mold non-sinusoidal oscillation mechanism driven by servomotor is proposed and the prototype is produced in the lab, the investment is low and the modification ratio is can be adjusted online, and the stability problem is studied. At first the dynamics model of the servomotor non-sinusoidal oscillation is established, and the kinematics differential function is deduced. Furthermore, based on the harmonic balance method, the eigenvalues of the system are solved; the criterion of the stability of the system is put forward. In addition, the eigenvalues and harmonic with different oscillating parameters are analyzed. Analytical results show that the real parts of the eigenvalues are positive, the system will be unstable, and the resonance will occur when the positive real parts of the eigenvalues are extremum. A foundation is established for solving the running smooth problem and next application of this mechanism.

Hyperchaotic encryption based on multi-scroll piecewise linear systems

A hyperchaotic multi-scroll piecewise linear system in R4 is binarized to generate a pseudo-random sequence which encrypt a grayscale image via symmetric-key algorithm. The sequence is analyzed throughout statistical tests according to the National Institute of Standards and Technology (NIST) specifications. The scrolls of the system are the result of a switching law that changes between the saddle hyperbolic equilibria of piecewise linear systems with eigenvalues as follows: two negative real and one pair of complex conjugate eigenvalues with positive real part. Thus, the encryption quality is evaluated depending on the variation of the number of scrolls.

Coefficient estimate of p-valent Bazilevic functions with a bounded positive real part

By considering a $p-$valent Bazilevi\v{c} function in the open unit disk$\triangle$ which maps $\triangle$ onto the strip domain $w$ with$p\alpha &lt; \Re\, w &lt; p \beta,$ we estimate bounds of coefficients and solve Fekete-Szeg\"{o} problem forfunctions in this class.\\

Exponentially weighted resolvent estimates for complex Ornstein–Uhlenbeck systems

In this paper we study differential operators of the form \begin{align*} \left[\mathcal{L}_\infty v \right](x) = A\triangle v(x) + \left\langle Sx,\nabla v(x) \right\rangle - Bv(x), \,x \in \mathbb{R}^d, \,d \geqslant 2, \end{align*} for matrices $A,B\in\mathbb{C}^{N,N}$, where the eigenvalues of $A$ have positive real parts. The sum $A\triangle v(x) + \left\langle Sx, \nabla v(x) \right\rangle$ is known as the Ornstein-Uhlenbeck operator with an unbounded drift term defined by a skew-symmetric matrix $S\in\mathbb{R}^{d,d}$. Differential operators such as $\mathcal{L}_{\infty}$ arise as linearizations at rotating waves in time-dependent reaction diffusion systems. The results of this paper serve as foundation for proving exponential decay of such waves. Under the assumption that $A$ and $B$ can be diagonalized simultaneously we construct a heat kernel matrix $H(x,\xi,t)$ of $\mathcal{L}_{\infty}$ that solves the evolution equation $v_t=\mathcal{L}_{\infty}v$. In the following we study the Ornstein-Uhlenbeck semigroup \begin{align*} \left[ T(t)v\right](x) = \int_{\mathbb{R}^d} H(x,\xi,t) v(\xi) d\xi,\,x \in \mathbb{R}^d,\, t>0, \end{align*} in exponentially weighted function spaces. This is used to derive resolvent estimates for $\mathcal{L}_{\infty}$ in exponentially weighted $L^p$-spaces $L^p_{\theta} (\mathbb{R}^d,\mathbb{C}^N)$, $1\leqslant p<\infty$, as well as in exponentially weighted $C_{\mathrm{b}}$-spaces $C_{\mathrm{b},\theta}(\mathbb{R}^d,\mathbb{C}^N)$.

Bound for the fifth coefficient of certain starlike functions

For two different choices of φ, the sharp bound for the fifth coefficient of a normalized analytic function f satisfying zf′(z)/f(z)≺φ(z) is obtained by using a bound for a polynomial in the coefficients of functions with positive real part. Our proof uses a characterization of functions with positive real part in terms of certain positive semi-definite Hermitian form and certain well-known coefficient inequalities for functions with positive real part are shown to follow easily from this characterization.

Quaternionic Hardy spaces in the open unit ball and half space and Blaschke products

The Hardy spaces H2(B) and H2(H+), where B and H+ denote, respectively, the open unit ball of the quaternions and the half space of quaternions with positive real part, as well as Blaschke products, have been intensively studied in a series of papers where they are used as a tool to prove other results in Schur analysis. This paper gives an overview on the topic, collecting the various results available.

Constructing hyperchaotic systems at will

SummaryOver the last two decades, chaotification has received increasing attention from various communities of science and engineering. This paper aims at developing a unified chaotification framework for generating desired higher‐dimensional dissipative hyperchaotic systems by using a single‐parameter controller. The main idea is to use a block diagonal matrix in the design of the nominal system, so as to construct a desired hyperchaotic system. Specifically, the proposed scheme is to assign desired closed‐loop poles to the controlled system, allocating the corresponding numbers of eigenvalues with positive real parts at two types of saddle‐focus equilibria. For the non‐degenerate case, the number of positive Lyapunov exponents of the controlled system is given by the largest number of positive Lyapunov exponents of the desired hyperchaotic system. Two representative examples are given to illustrate and verify the proposed design method. Finally, the digital signal processor is employed to implement the above 10‐dimensional hyperchaotic systems, and the experimental observation is also given. Copyright © 2015 John Wiley &amp; Sons, Ltd.

Transient Analysis of Fluid Flow Models via Matrix Decomposition

□ We present a new approach to carry out transient analysis for Markov modulated fluid flows. The system of partial differential equations for the time-dependent distributions is transformed into a system of ordinary differential equations by Laplace transform. This system of ordinary differential equations is solved by means of decomposing its coefficient matrix into two parts, one with eigenvalues with negative real parts and the other with positive real parts. The results we obtained are equivalent, but in simpler form, to those of Ahn and Ramaswami, which are derived by probabilistic arguments.