Notes on the Theory of Equations

1939 ◽  
Vol 23 (256) ◽  
pp. 376-379
Author(s):  
E. P. Lewis
Keyword(s):  
Real Root ◽  
Real Roots ◽  
The Third ◽  
Complex Roots ◽  
Image Position

Multiply throughout by a 2 and write y for ax+ b ; the equation becomes where H ≡ ac − b 2, G ≡ a2d − 3abc + 2b3. Since in an equation with real coefficients complex roots occur in conjugate pairs, (i) must have at least one real root; so if α is this root, (i) may be written Accordingly the two remaining roots are also real if But since α satisfies (i), and so Hence if (i) has three real roots, G2 +4H 3 ≤ 0; and clearly, when G2 +4H 3 = 0, two roots are numerically equal to and the third to .

scholarly journals Application of a Generalized Secant Method to Nonlinear Equations with Complex Roots

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 169
Author(s):  
Avram Sidi
Keyword(s):  
Nonlinear Equations ◽  
Local Convergence ◽  
Real Root ◽  
Numerical Procedure ◽  
Secant Method ◽  
Convergence Theory ◽  
Simple Complex ◽  
Real Roots ◽  
Complex Roots ◽  
Appl Math

The secant method is a very effective numerical procedure used for solving nonlinear equations of the form f(x)=0. In a recent work (A. Sidi, Generalization of the secant method for nonlinear equations. Appl. Math. E-Notes, 8:115–123, 2008), we presented a generalization of the secant method that uses only one evaluation of f(x) per iteration, and we provided a local convergence theory for it that concerns real roots. For each integer k, this method generates a sequence {xn} of approximations to a real root of f(x), where, for n≥k, xn+1=xn−f(xn)/pn,k′(xn), pn,k(x) being the polynomial of degree k that interpolates f(x) at xn,xn−1,…,xn−k, the order sk of this method satisfying 1<sk<2. Clearly, when k=1, this method reduces to the secant method with s1=(1+5)/2. In addition, s1<s2<s3<⋯, such that limk→∞sk=2. In this note, we study the application of this method to simple complex roots of a function f(z). We show that the local convergence theory developed for real roots can be extended almost as is to complex roots, provided suitable assumptions and justifications are made. We illustrate the theory with two numerical examples.

scholarly journals V. A New Method of resolving Cubic Equations

1805 ◽  
Vol 5 (1) ◽  
pp. 99-116 ◽  
Author(s):  
James Ivory
Keyword(s):  
Real Root ◽  
The Other ◽  
New Method ◽  
Real Roots ◽  
The One ◽  
Image Position

1. I Divide cubic equations into two varieties or species: the one, comprehending all cubic equations with three real roots; the other, all those with only one real root.2. Let φ denote any angle whatever, and let τ = tan φ, the radius being unity: let also : then from the doctrine of angular sections we havewhich being reduced to the form of an equation, isZ3-3τZ2-3Z+τ=0.

scholarly journals Dislocation Arrangement in a Thick LEO GaN Film on Sapphire

2000 ◽  
Vol 5 (S1) ◽  
pp. 97-103
Author(s):  
Kathleen A. Dunn ◽  
Susan E. Babcock ◽  
Donald S. Stone ◽  
Richard J. Matyi ◽  
Ling Zhang ◽  
...  
Keyword(s):  
High Resolution ◽  
Electron Diffraction ◽  
Threading Dislocations ◽  
X Ray Diffraction ◽  
Burgers Vector ◽  
X Ray ◽  
The Third ◽  
Dislocation Arrangement ◽  
Image Position ◽  
Gan Film

Diffraction-contrast TEM, focused probe electron diffraction, and high-resolution X-ray diffraction were used to characterize the dislocation arrangements in a 16µm thick coalesced GaN film grown by MOVPE LEO. As is commonly observed, the threading dislocations that are duplicated from the template above the window bend toward (0001). At the coalescence plane they bend back to lie along [0001] and thread to the surface. In addition, three other sets of dislocations were observed. The first set consists of a wall of parallel dislocations lying in the coalescence plane and nearly parallel to the substrate, with Burgers vector (b) in the (0001) plane. The second set is comprised of rectangular loops with b = 1/3 [110] (perpendicular to the coalescence boundary) which originate in the coalescence boundary and extend laterally into the film on the (100). The third set of dislocations threads laterally through the film along the [100] bar axis with 1/3<110>-type Burgers vectors These sets result in a dislocation density of ∼109 cm−2. High resolution X-ray reciprocal space maps indicate wing tilt of ∼0.5º.

On Graeffe's Method for Complex Roots of Algebraic Equations

1924 ◽  
Vol 22 (2) ◽  
pp. 83-87 ◽  
Author(s):  
S. Brodetsky ◽  
G. Smeal
Keyword(s):  
Nineteenth Century ◽  
Real Axis ◽  
Practical Method ◽  
Algebraic Equations ◽  
Argand Diagram ◽  
Considerable Difficulty ◽  
The Real ◽  
Real Roots ◽  
Bipolar Coordinates ◽  
Complex Roots

The only really useful practical method for solving numerical algebraic equations of higher orders, possessing complex roots, is that devised by C. H. Graeffe early in the nineteenth century. When an equation with real coefficients has only one or two pairs of complex roots, the Graeffe process leads to the evaluation of these roots without great labour. If, however, the equation has a number of pairs of complex roots there is considerable difficulty in completing the solution: the moduli of the roots are found easily, but the evaluation of the arguments often leads to long and wearisome calculations. The best method that has yet been suggested for overcoming this difficulty is that by C. Runge (Praxis der Gleichungen, Sammlung Schubert). It consists in making a change in the origin of the Argand diagram by shifting it to some other point on the real axis of the original Argand plane. The new moduli and the old moduli of the complex roots can then be used as bipolar coordinates for deducing the complex roots completely: this also checks the real roots.

THE SHARP BOUND FOR THE HANKEL DETERMINANT OF THE THIRD KIND FOR CONVEX FUNCTIONS

2018 ◽  
Vol 97 (3) ◽  
pp. 435-445 ◽  
Author(s):  
BOGUMIŁA KOWALCZYK ◽  
ADAM LECKO ◽  
YOUNG JAE SIM
Keyword(s):  
Convex Functions ◽  
Analytic Functions ◽  
Sharp Inequality ◽  
Sharp Bound ◽  
Hankel Determinant ◽  
The Third ◽  
Image Position

We prove the sharp inequality $|H_{3,1}(f)|\leq 4/135$ for convex functions, that is, for analytic functions $f$ with $a_{n}:=f^{(n)}(0)/n!,~n\in \mathbb{N}$, such that $$\begin{eqnarray}Re\bigg\{1+\frac{zf^{\prime \prime }(z)}{f^{\prime }(z)}\bigg\}>0\quad \text{for}~z\in \mathbb{D}:=\{z\in \mathbb{C}:|z|<1\},\end{eqnarray}$$ where $H_{3,1}(f)$ is the third Hankel determinant $$\begin{eqnarray}H_{3,1}(f):=\left|\begin{array}{@{}ccc@{}}a_{1} & a_{2} & a_{3}\\ a_{2} & a_{3} & a_{4}\\ a_{3} & a_{4} & a_{5}\end{array}\right|.\end{eqnarray}$$

On the “Third Definition” of the Topology on the Spectrum of a C*-Algebra

1971 ◽  
Vol 23 (3) ◽  
pp. 445-450 ◽  
Author(s):  
L. Terrell Gardner
Keyword(s):  
Hilbert Space ◽  
Quotient Space ◽  
Principal Result ◽  
Irreducible Representations ◽  
C Algebra ◽  
Fell Topology ◽  
The Third ◽  
Definition Of ◽  
Image Position

0. In [3], Fell introduced a topology on Rep (A,H), the collection of all non-null but possibly degenerate *-representations of the C*-algebra A on the Hilbert space H. This topology, which we will call the Fell topology, can be described by giving, as basic open neighbourhoods of π0 ∈ Rep(A, H), sets of the formwhere the ai ∈ A, and the ξj ∈ H(π0), the essential space of π0 [4].A principal result of [3, Theorem 3.1] is that if the Hilbert dimension of H is large enough to admit all irreducible representations of A, then the quotient space Irr(A, H)/∼ can be identified with the spectrum (or “dual“) Â of A, in its hull-kernel topology.

On Stability of Solutions of Certain Differential Equations of the Third Order

1967 ◽  
Vol 10 (5) ◽  
pp. 681-688 ◽  
Author(s):  
B.S. Lalli
Keyword(s):  
Differential Equation ◽  
Asymptotic Stability ◽  
Lyapunov Function ◽  
Differential Equations ◽  
Global Asymptotic Stability ◽  
Sufficient Conditions ◽  
Stability Of Solutions ◽  
Third Order ◽  
The Third ◽  
Image Position

The purpose of this paper is to obtain a set of sufficient conditions for “global asymptotic stability” of the trivial solution x = 0 of the differential equation1.1using a Lyapunov function which is substantially different from similar functions used in [2], [3] and [4], for similar differential equations. The functions f1, f2 and f3 are real - valued and are smooth enough to ensure the existence of the solutions of (1.1) on [0, ∞). The dot indicates differentiation with respect to t. We are taking a and b to be some positive parameters.

scholarly journals On Appell's Function P (θ, φ)

1932 ◽  
Vol 3 (1) ◽  
pp. 53-55
Author(s):  
Pierre Humbert
Keyword(s):  
Third Order ◽  
The Third ◽  
Direct Generalization ◽  
Circular Functions ◽  
Image Position

§1. Appell's functions, P (θ, φ), Q (θ, φ) and R (θ, φ) are defined by the expansionwhere j3 = 1, affording, both for the third order and the field of two variables, a very direct generalization of the circular functions, as

Polynomials With {0, +1, -1} Coefficients and a Root Close to a Given Point

1997 ◽  
Vol 49 (5) ◽  
pp. 887-915 ◽  
Author(s):  
Peter Borwein ◽  
Christopher Pinner
Keyword(s):  
Algebraic Number ◽  
Real Root ◽  
Small Degree ◽  
Pisot Number ◽  
Roots Of Unity ◽  
Best Approximations ◽  
Rate Of Approximation ◽  
Root Of Unity ◽  
Extremal Polynomials ◽  
Image Position

AbstractFor a fixed algebraic number α we discuss how closely α can be approximated by a root of a {0, +1, -1} polynomial of given degree. We show that the worst rate of approximation tends to occur for roots of unity, particularly those of small degree. For roots of unity these bounds depend on the order of vanishing, k, of the polynomial at α.In particular we obtain the following. Let BN denote the set of roots of all {0, +1, -1} polynomials of degree at most N and BN(α k) the roots of those polynomials that have a root of order at most k at α. For a Pisot number α in (1, 2] we show thatand for a root of unity α thatWe study in detail the case of α = 1, where, by far, the best approximations are real. We give fairly precise bounds on the closest real root to 1. When k = 0 or 1 we can describe the extremal polynomials explicitly.

scholarly journals Asymptotic formulae for linear equations

1972 ◽  
Vol 13 (2) ◽  
pp. 147-152 ◽  
Author(s):  
Don B. Hinton
Keyword(s):  
Asymptotic Behaviour ◽  
Fourth Order ◽  
Linear Equations ◽  
Order Equation ◽  
Third Order ◽  
Fourth Order Equation ◽  
The Third ◽  
Asymptotic Formulae ◽  
Image Position

Numerous formulae have been given which exhibit the asymptotic behaviour as t → ∞solutions ofwhere F(t) is essentially positive and Several of these results have been unified by a theorem of F. V. Atkinson [1]. It is the purpose of this paper to establish results, analogous to the theorem of Atkinson, for the third order equationand for the fourth order equation

Sign in / Sign up

Export Citation Format

Share Document