Which is bigger? An intriguing ‘double alternation’

2014 ◽  
Vol 98 (541) ◽  
pp. 67-72
Author(s):  
Takeshi Hokuto ◽  
Mitsuhiro Kumano
Keyword(s):  
The Third ◽  
Double Alternation ◽  
Image Position

The following three inequalities hold:The first inequality is trivial. The second one was proved without calculating aids in note [1], and the third along similar lines in note [2]. The author of note [2] also suggested an extension to the relation betweenHow best to continue the sequence of inequalities is not obvious and we return to that point shortly. Before doing so, we note that an interesting geralisation is to replace π by a variable x, and to determine the precise interval of x in which the regularity of the ‘alternation of inequality signs’ is maintained. We need no longer consider particular properties of π.

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.

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

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

scholarly journals Conjugacy classes in sylow p-subgroups of GL(n, q), IV

1994 ◽  
Vol 36 (1) ◽  
pp. 91-96 ◽  
Author(s):  
A. Vera-López ◽  
J. M. Arregi
Keyword(s):  
Conjugacy Classes ◽  
The Third ◽  
New Information ◽  
Image Position

In this paper we give new information about the conjugacy vector of the group , the Sylow p-subgroup of GL(n, q) consisting of the upper unitriangular matrices. The first two components of this vector are given in [4]. Here, we obtain the third component, that is, the number of conjugacy classes whose centralizer has qn+l elements. Besides, we give the whole set of numbers which compose this vector:

scholarly journals I.—Clay Disks from Tarentum

1883 ◽  
Vol 4 ◽  
pp. 156-157
Author(s):  
P. G.
Keyword(s):  
Fourth Century ◽  
The Other ◽  
Third Century ◽  
Artistic Value ◽  
The Third ◽  
Image Position

Among the objects brought from Tarentum by the Rev. G. J. Chester are certain disks of clay of some interest, though not of artistic value. They are circular and flat or cheese-like in form, with a diameter of 3½ to 3¾ inches, and a thickness of about ¾ of an inch. The inscriptions are impressed in the clay by means of a stamp, and run thus:The order in date is that followed in the list. No. 1 is oldest, and the shape of the м seems to indicate that it may date from the fourth century B.C.; the other three are probably not earlier than the third century. Later they can scarcely be, for after that time the obol gave way to the Roman denarius and sestertius as a measure of value at Tarentum.

scholarly journals Integrals involving Bessel and Legendre functions

1965 ◽  
Vol 14 (4) ◽  
pp. 269-272 ◽  
Author(s):  
J. S. Lowndes
Keyword(s):  
Plane Wave ◽  
Bessel Function ◽  
Scattered Wave ◽  
Approximate Expression ◽  
Apex Angle ◽  
Legendre Functions ◽  
The Third ◽  
Image Position

Felsen (1) has shown that when a plane wave is incident along the axis of a rigid cone of narrow apex angle an approximate expression for the scattered wave involves an integral of the formwhere is a Bessel function of the third kind, k a constant and , is the Legendre (conical) function of the first kind.

scholarly journals On the Number of Ways of Colouring a Map

1930 ◽  
Vol 2 (2) ◽  
pp. 83-91 ◽  
Author(s):  
George D. Birkhoff
Keyword(s):  
Present Note ◽  
The Third ◽  
Image Position

It is well known that any map of n regions on a sphere may be coloured in five or fewer colours. The purpose of the present note is to prove the followingTheorem. If Pn(λ)denotes the number of ways of colouring any ma: of n regions on the sphere in λ (or fewer) colours, then(1)This inequality obviously holds for λ = 1, 2, 3 so that we may confine attention to the case λ > 4. Furthermore it holds for n = 3, 4 since the first region may be coloured in λ ways, the second in at least λ — 1 ways, the third in at least λ — 2 ways, and the fourth, if there be one, in at least λ — 3 ways.

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