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Polynomial control on stability, inversion and powers of matrices on simple graphs
Shin, Chang Eon,Sun, Qiyu Elsevier 2019 Journal of functional analysis Vol.276 No.1
<P><B>Abstract</B></P> <P>Spatially distributed networks of large size arise in a variety of science and engineering problems, such as wireless sensor networks and smart power grids. Most of their features can be described by properties of their state-space matrices whose entries have indices in the vertex set of a graph. In this paper, we introduce novel algebras of Beurling type that contain matrices on a connected simple graph having polynomial off-diagonal decay, and we show that they are Banach subalgebras of B ( <SUP> ℓ p </SUP> ) , 1 ≤ p ≤ ∞ , the space of all bounded operators on the space <SUP> ℓ p </SUP> of all <I>p</I>-summable sequences. The <SUP> ℓ p </SUP> -stability of state-space matrices is an essential hypothesis for the robustness of spatially distributed networks. In this paper, we establish the equivalence among <SUP> ℓ p </SUP> -stabilities of matrices in Beurling algebras for different exponents 1 ≤ p ≤ ∞ , with quantitative analysis for the lower stability bounds. Admission of norm-control inversion plays a crucial role in some engineering practice. In this paper, we prove that matrices in Beurling subalgebras of B ( <SUP> ℓ 2 </SUP> ) have norm-controlled inversion and we find a norm-controlled polynomial with close to optimal degree. Polynomial estimate to powers of matrices is important for numerical implementation of spatially distributed networks. In this paper, we apply our results on norm-controlled inversion to obtain a polynomial estimate to powers of matrices in Beurling algebras. The polynomial estimate is a noncommutative extension about convolution powers of a complex function and is applicable to estimate the probability of hopping from one agent to another agent in a stationary Markov chain on a spatially distributed network.</P>
ON SOME UPPER BOUNDS OF THE EXPONENTIAL FUNCTION
Bae, Jae-Gug The Honam Mathematical Society 2008 호남수학학술지 Vol.30 No.2
In this paper, we introduce some algebraic, rational and polynomial upper bounds of the exponential function on the interval [0,1). And then we compare the acuteness of these bounds.
Sheza M. El-Deeb,Gangadharan Murugusundaramoorthy,Alhanouf Alburaikan 경남대학교 수학교육과 2022 Nonlinear Functional Analysis and Applications Vol.27 No.2
In this paper, we introduce new subclasses of analytic and bi-univalent functions associated with the Mittag-Leffler-type Borel distribution by using the Legendre polynomi- als. Furthermore, we find estimates on the first two Taylor-Maclaurin coefficients |a2| and |a3| for functions in these subclasses and obtain Fekete-Szegő problem for these subclasses. We also state certain new subclasses of Σ and initial coefficient estimates and Fekete-Szegő inequalities.
On σ-nil ideals of bounded index of σ-nilpotence
Hong, C.Y.,Kim, N.K.,Lee, Y.,Nielsen, P.P. Academic Press 2012 Journal of algebra Vol.371 No.-
We investigate properties of σ-nil subsets with bounded index of σ-nilpotence, beginning with a classification of many of the possible types of nilpotence available in the σ-skewed case. In the process we introduce a σ-analog of the bounded nilradical of a ring. In many situations we completely describe the bounded nilradical for skew polynomial rings in terms of the σ-bounded nilradical in the coefficient ring. We also construct an example demonstrating that the bounded nilradical for skew polynomial rings is more complicated than one might initially guess. Further extensions are explored when one assumes additional restrictions on σ.
Spectral p-dilations and polynomially bounded operators
Lee, Mi-Young,Lee, Sang-Hun Korean Mathematical Society 1995 대한수학회지 Vol.32 No.4
Let $H$ be a separable, infinite dimensional, complex Hilbert space and let $L(H)$ denote the algebra of all bounded linear operators on $H$.
Iterative LMI Approach to Robust State-feedback Control of Polynomial Systems with Bounded Actuators
Tanagorn Jennawasin,Michihiro Kawanishi,Tatsuo Narikiyo,David Banjerdpongchai 제어·로봇·시스템학회 2019 International Journal of Control, Automation, and Vol.17 No.4
This paper presents a novel approach to state-feedback stabilization of polynomial systems with bounded actuators. To overcome limitation of the existing approaches, we introduce additional variables that separate the system matrices and the Lyapunov matrices. Therefore, parameterization of the state-feedback controllers is independent of the Lyapunov matrices. The proposed design condition is bilinear in the decision variables, and hence we provide an iterative algorithm to solve the design problem. At each iteration, the design condition is cast as convex optimization using the sum-of-squares technique and can be efficiently solved. In addition, the novel parameter-dependent Lyapunov functions are readily applied to robust state-feedback stabilization of polynomial systems subject to parametric uncertainty. Effectiveness of the proposed approach is demonstrated by numerical examples.
DILATIONS FOR POLYNOMIALLY BOUNDED OPERATORS
EXNER, GEORGE R.,JO, YOUNG SOO,JUNG, IL BONG Korean Mathematical Society 2005 대한수학회지 Vol.42 No.5
We discuss a certain geometric property $X_{{\theta},{\gamma}}$ of dual algebras generated by a polynomially bounded operator and property ($\mathbb{A}_{N_0,N_0}$; these are central to the study of $N_{0}\timesN_{0}$-systems of simultaneous equations of weak$^{*}$-continuous linear functionals on a dual algebra. In particular, we prove that if T $\in$ $\mathbb{A}$$^{M}$ satisfies a certain sequential property, then T $\in$ $\mathbb{A}^{M}_{N_0}(H) if and only if the algebra $A_{T}$ has property $X_{0, 1/M}$, which is an improvement of Li-Pearcy theorem in [8].
APPROXIMATION BY INTERPOLATING POLYNOMIALS IN SMIRNOV-ORLICZ CLASS
Akgun Ramazan,Israfilov Daniyal M. Korean Mathematical Society 2006 대한수학회지 Vol.43 No.2
Let $\Gamma$ be a bounded rotation (BR) curve without cusps in the complex plane $\mathbb{C}$ and let G := int $\Gamma$. We prove that the rate of convergence of the interpolating polynomials based on the zeros of the Faber polynomials $F_n\;for\;\bar G$ to the function of the reflexive Smirnov-Orlicz class $E_M (G)$ is equivalent to the best approximating polynomial rate in $E_M (G)$.
Improved subdivision scheme for the root computation of univariate polynomial equations
Ko, K.H.,Kim, K. Elsevier [etc.] 2013 Applied mathematics and computation Vol.219 No.14
In this paper, a subdivision method of computing the roots of a univariate polynomial equation is proposed using novel bounding methods. The equation is converted into a Bezier curve, and the root computation problem is transformed into the geometric problem of intersection computation between the curve and an axis, which can be solved using a bound of the curve and subdivision. Four different bounding schemes are compared, and new hybrid bounding schemes are proposed for use in the root computation. In particular, the convex hull and the quasi-interpolating bound are combined to produce a smaller polygonal bound, which is then used for the root computation of the input equation. The new bounding scheme provides improved robustness and performance compared to the existing convex hull-based method, e.g., the Projected Polyhedron algorithm. Examples are provided to demonstrate the performance of the proposed method, which shows that the proposed approach is superior to the existing one.
NILRADICALS OF POWER SERIES RINGS AND NIL POWER SERIES RINGS
HUH, CHAN,KIM, CHOL ON,KIM, EUN JEONG,KIM, HONG KEE,LEE, YANG Korean Mathematical Society 2005 대한수학회지 Vol.42 No.5
Klein proved that polynomial rings over nil rings of bounded index are also nil of bounded index; while Puczylowski and Smoktunowicz described the nilradical of a power series ring with an indeterminate. We extend these results to those with any set of commuting indeterminates. We also study prime radicals of power series rings over some class of rings containing the case of bounded index, finding some examples which elaborate our arguments; and we prove that R is a PI ring of bounded index then the power series ring R[[X]], with X any set of indeterminates over R, is also a PI ring of bounded index, obtaining the Klein's result for polynomial rings as a corollary.