A numerical radius inequality and an estimate for the numerical radius of the Frobenius companion matrix

Fuad Kittaneh

Displaying similar documents to “A numerical radius inequality and an estimate for the numerical radius of the Frobenius companion matrix”

The joint essential numerical range, compact perturbations, and the Olsen problem

Vladimír Müller (2010)

Studia Mathematica

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Let T₁,...,Tₙ be bounded linear operators on a complex Hilbert space H. Then there are compact operators K₁,...,Kₙ ∈ B(H) such that the closure of the joint numerical range of the n-tuple (T₁-K₁,...,Tₙ-Kₙ) equals the joint essential numerical range of (T₁,...,Tₙ). This generalizes the corresponding result for n = 1. We also show that if S ∈ B(H) and n ∈ ℕ then there exists a compact operator K ∈ B(H) such that $| | (S - K) ⁿ | | = {| | S ⁿ | |}_{e}$ . This generalizes results of C. L. Olsen.

Numerical solution of Black-Scholes option pricing with variable yield discrete dividend payment

Rafael Company, Lucas Jódar, Enrique Ponsoda (2008)

Banach Center Publications

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This paper deals with the construction of numerical solution of the Black-Scholes (B-S) type equation modeling option pricing with variable yield discrete dividend payment at time $t_{d}$ . Firstly the shifted delta generalized function $δ (t - t_{d})$ appearing in the B-S equation is approximated by an appropriate sequence of nice ordinary functions. Then a semidiscretization technique applied on the underlying asset is used to construct a numerical solution. The limit of this numerical solution is independent...

The $☆$ -product approach for linear ODEs: A numerical study of the scalar case

Pozza, Stefano, Van Buggenhout, Niel

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Solving systems of non-autonomous ordinary differential equations (ODE) is a crucial and often challenging problem. Recently a new approach was introduced based on a generalization of the Volterra composition. In this work, we explain the main ideas at the core of this approach in the simpler setting of a scalar ODE. Understanding the scalar case is fundamental since the method can be straightforwardly extended to the more challenging problem of systems of ODEs. Numerical examples illustrate...

A note on the number of zeros of polynomials in an annulus

Xiangdong Yang, Caifeng Yi, Jin Tu (2011)

Annales Polonici Mathematici

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Let p(z) be a polynomial of the form $p (z) = \sum_{j = 0}^{n} a_{j} z^{j}$ , $a_{j} \in - 1, 1$ . We discuss a sufficient condition for the existence of zeros of p(z) in an annulus z ∈ ℂ: 1 - c < |z| < 1 + c, where c > 0 is an absolute constant. This condition is a combination of Carleman’s formula and Jensen’s formula, which is a new approach in the study of zeros of polynomials.

A lower bound sequence for the minimum eigenvalue of Hadamard product of an $M$ -matrix and its inverse

Wenlong Zeng, Jianzhou Liu (2022)

Czechoslovak Mathematical Journal

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We propose a lower bound sequence for the minimum eigenvalue of Hadamard product of an $M$ -matrix and its inverse, in terms of an $S$ -type eigenvalues inclusion set and inequality scaling techniques. In addition, it is proved that the lower bound sequence converges. Several numerical experiments are given to demonstrate that the lower bound sequence is sharper than some existing ones in most cases.

Guaranteed and fully computable two-sided bounds of Friedrichs’ constant

Vejchodský, Tomáš

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This contribution presents a general numerical method for computing lower and upper bound of the optimal constant in Friedrichs’ inequality. The standard Rayleigh-Ritz method is used for the lower bound and the method of $𝑎 𝑝𝑟𝑖𝑜𝑟𝑖 - 𝑎 𝑝𝑜𝑠𝑡𝑒𝑟𝑖𝑜𝑟𝑖 𝑖𝑛𝑒𝑞𝑢𝑎𝑙𝑖𝑡𝑖𝑒𝑠$ is employed for the upper bound. Several numerical experiments show applicability and accuracy of this approach.

Numerical radius inequalities for Hilbert $C^{*}$ -modules

Sadaf Fakri Moghaddam, Alireza Kamel Mirmostafaee (2022)

Mathematica Bohemica

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We present a new method for studying the numerical radius of bounded operators on Hilbert $C^{*}$ -modules. Our method enables us to obtain some new results and generalize some known theorems for bounded operators on Hilbert spaces to bounded adjointable operators on Hilbert $C^{*}$ -module spaces.

A survey of some recent results on Clifford algebras in $ℝ^{4}$

Drahoslava Janovská, Gerhard Opfer (2023)

Applications of Mathematics

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We will study applications of numerical methods in Clifford algebras in $ℝ^{4}$ , in particular in the skew field of quaternions, in the algebra of coquaternions and in the other nondivision algebras in $ℝ^{4}$ . In order to gain insight into the multidimensional case, we first consider linear equations in quaternions and coquaternions. Then we will search for zeros of one-sided (simple) quaternion polynomials. Three different classes of zeros can be distinguished. In general, the quaternionic coefficients...

Explicit estimates for the summatory function of Λ(n)/n from the one of Λ(n)

Olivier Ramaré (2013)

Acta Arithmetica

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We prove that the error term $\sum_{n \leq x} Λ (n) / n - l o g x + γ$ differs from (ψ(x)-x)/x by a well controlled function. We deduce very precise numerical results from the formula obtained.

Inequalities concerning polar derivative of polynomials

Arty Ahuja, K. K. Dewan, Sunil Hans (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In this paper we obtain certain results for the polar derivative of a polynomial $p (z) = c_{n} z^{n} + \sum_{j = μ}^{n} c_{n - j} z^{n - j}$ , $1 \leq μ \leq n$ , having all its zeros on $| z | = k$ , $k \leq 1$ , which generalizes the results due to Dewan and Mir, Dewan and Hans. We also obtain certain new inequalities concerning the maximum modulus of a polynomial with restricted zeros. [Editor’s note: There are flaws in the paper, see M. A. Qazi, Remarks on some recent results about polynomials with restricted zeros, Ann. Univ. Mariae Curie-Skłodowska Sect. A 67 (2), (2013),...

On distance between zeros of solutions of third order differential equations

N. Parhi, S. Panigrahi (2001)

Annales Polonici Mathematici

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The lower bounds of the spacings b-a or a’-a of two consecutive zeros or three consecutive zeros of solutions of third order differential equations of the form y”’ + q(t)y’ + p(t)y = 0 (*) are derived under very general assumptions on p and q. These results are then used to show that $t_{n + 1} - t ₙ \to \infty$ or $t_{n + 2} - t ₙ \to \infty$ as n → ∞ under suitable assumptions on p and q, where ⟨tₙ⟩ is a sequence of zeros of an oscillatory solution of (*). The Opial-type inequalities are used to derive lower bounds of the spacings d-a...

On the Bishop-Phelps-Bollobás theorem for operators and numerical radius

Sun Kwang Kim, Han Ju Lee, Miguel Martín (2016)

Studia Mathematica

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We study the Bishop-Phelps-Bollobás property for numerical radius (for short, BPBp-nu) of operators on ℓ₁-sums and $ℓ_{\infty}$ -sums of Banach spaces. More precisely, we introduce a property of Banach spaces, which we call strongly lush. We find that if X is strongly lush and X ⊕₁ Y has the weak BPBp-nu, then (X,Y) has the Bishop-Phelps-Bollobás property (BPBp). On the other hand, if Y is strongly lush and $X \oplus_{\infty} Y$ has the weak BPBp-nu, then (X,Y) has the BPBp. Examples of strongly lush spaces are C(K)...

The image of multilinear polynomials evaluated on $3 \times 3$ upper triangular matrices

Thiago Castilho de Mello (2021)

Communications in Mathematics

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We describe the images of multilinear polynomials of arbitrary degree evaluated on the $3 \times 3$ upper triangular matrix algebra over an infinite field.

The joint essential numerical range of operators: convexity and related results

Chi-Kwong Li, Yiu-Tung Poon (2009)

Studia Mathematica

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Let W(A) and $W_{e} (A)$ be the joint numerical range and the joint essential numerical range of an m-tuple of self-adjoint operators A = (A₁, ..., Aₘ) acting on an infinite-dimensional Hilbert space. It is shown that $W_{e} (A)$ is always convex and admits many equivalent formulations. In particular, for any fixed i ∈ 1, ..., m, $W_{e} (A)$ can be obtained as the intersection of all sets of the form $c l (W (A ₁, . . ., A_{i + 1}, A_{i} + F, A_{i + 1}, . . ., A ₘ))$ , where F = F* has finite rank. Moreover, the closure cl(W(A)) of W(A) is always star-shaped with the elements in...

Zeros of a certain class of Gauss hypergeometric polynomials

Addisalem Abathun, Rikard Bøgvad (2018)

Czechoslovak Mathematical Journal

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We prove that as $n \to \infty$ , the zeros of the polynomial $_{2} F_{1} [\begin{matrix} - n, α n + 1 \\ α n + 2 \end{matrix}; z]$ cluster on (a part of) a level curve of an explicit harmonic function. This generalizes previous results of Boggs, Driver, Duren et al. (1999–2001) to the case of a complex parameter $α$ and partially proves a conjecture made by the authors in an earlier work.

A matrix constructive method for the analytic-numerical solution of coupled partial differential systems

Lucas Jódar, Enrique A. Navarro, M. V. Ferrer (1995)

Applications of Mathematics

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In this paper we construct analytic-numerical solutions for initial-boundary value systems related to the equation $u_{t} - A u_{x x} - B u = 0$ , where $B$ is an arbitrary square complex matrix and $A$ ia s matrix such that the real part of the eigenvalues of the matrix $\frac{1}{2} (A + A^{H})$ is positive. Given an admissible error $ε$ and a finite domain $G$ , and analytic-numerical solution whose error is uniformly upper bounded by $ε$ in $G$ , is constructed.

The new iteration methods for solving absolute value equations

Rashid Ali, Kejia Pan (2023)

Applications of Mathematics

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Many problems in operations research, management science, and engineering fields lead to the solution of absolute value equations. In this study, we propose two new iteration methods for solving absolute value equations $A x - | x | = b$ , where $A \in ℝ^{n \times n}$ is an $M$ -matrix or strictly diagonally dominant matrix, $b \in ℝ^{n}$ and $x \in ℝ^{n}$ is an unknown solution vector. Furthermore, we discuss the convergence of the proposed two methods under suitable assumptions. Numerical experiments are given to verify the feasibility, robustness...