scholarly journals The convolution x-r*xs

1976 ◽  
Vol 17 (1) ◽  
pp. 53-56 ◽  
Author(s):  
B. Fisher
Keyword(s):  
Differentiable Function ◽  
Convolution Of Distributions ◽  
Definition Of ◽  
Image Position ◽  
Infinitely Differentiable Function

In a recent paper [1], Jones extended the definition of the convolution of distributions so that further convolutions could be defined. The convolution w1*w2 of two distributions w1 and w2 was defined as the limit ofithe sequence {wln*w2n}, provided the limit w exists in the sense thatfor all fine functions ф in the terminology of Jones [2], wherew1n(x) = wl(x)τ(x/n), W2n(x) = w2(x)τ(x/n)and τ is an infinitely differentiable function satisfying the following conditions:(i) τ(x) = τ(—x),(ii)0 ≤ τ (x) ≤ l,(iii)τ (x) = l for |x| ≤ ½,(iv) τ (x) = 0 for |x| ≥ 1.

The product of the distributions x−r and δ(r−1)(x)

1972 ◽  
Vol 72 (2) ◽  
pp. 201-204 ◽  
Author(s):  
B. Fisher
Keyword(s):  
Differentiable Function ◽  
Open Interval ◽  
Image Position ◽  
Infinitely Differentiable Function

The product of two distributions f and g on the open interval (a, b), where −∞ ≤ a < b ∞, was defined in (1) as the limit of the sequence {fn·gn} provided this sequence is regular in (a, b), wherefor n = 1, 2, … and ρ is a fixed infinitely differentiable function having the following properties:

scholarly journals A result on the convolution of distributions

1975 ◽  
Vol 19 (4) ◽  
pp. 393-395 ◽  
Author(s):  
B. Fisher
Keyword(s):  
Previous Definition ◽  
Convolution Of Distributions ◽  
Definition Of ◽  
Image Position

In a recent paper (1), Jones extended the definition of the convolution of two distributions to cover certain pairs of distributions which could not be convolved in the sense of the previous definition. The convolution ω1 * ω2 of two distributions ω1 and ω2 was defined as the limit of the sequence ω1n * ω2n, provided the limit ω exists in the sense thatfor all fine functions φ in the terminology of Jones (2) whereand τ is an infinitely differentiate function satisfying the following conditions:

Some results on the product of distributions

1973 ◽  
Vol 73 (2) ◽  
pp. 317-325 ◽  
Author(s):  
B. Fisher
Keyword(s):  
Differentiable Function ◽  
Open Interval ◽  
Image Position ◽  
Infinitely Differentiable Function

The product of two distributions f and g on the open interval (a, b), where −∞ ≤ a < b ≤ ∞, was defined(1) as the limit of the sequence {fngn}, provided this sequence is regular on (a, b), wherefor n = 1, 2, … and ρ is a fixed infinitely differentiable function having the following properties:(1) ρ(x) = 0 for |x| ≥ 1,(2) ρ(x) ≥ 0,(3) ρ(x) = ρ(−x),

Pascal's Prism

2012 ◽  
Vol 96 (536) ◽  
pp. 213-220
Author(s):  
Harlan J. Brothers
Keyword(s):  
Probability Theory ◽  
Fractal Geometry ◽  
Euclidean Geometry ◽  
Fibonacci Series ◽  
Pascal’S Triangle ◽  
Number Sequences ◽  
Definition Of ◽  
Image Position ◽  
Pascal's Triangle

Pascal's triangle is well known for its numerous connections to probability theory [1], combinatorics, Euclidean geometry, fractal geometry, and many number sequences including the Fibonacci series [2,3,4]. It also has a deep connection to the base of natural logarithms, e [5]. This link to e can be used as a springboard for generating a family of related triangles that together create a rich combinatoric object.2. From Pascal to LeibnizIn Brothers [5], the author shows that the growth of Pascal's triangle is related to the limit definition of e.Specifically, we define the sequence sn; as follows [6]:

The Static Aeroelasticity of Slender Configurations

1963 ◽  
Vol 14 (1) ◽  
pp. 75-104 ◽  
Author(s):  
G. J. Hancock
Keyword(s):  
Static Stability ◽  
Aerodynamic Forces ◽  
Flexible Aircraft ◽  
The Past ◽  
Plate Models ◽  
Non Linear ◽  
The Future ◽  
Static Aeroelasticity ◽  
Definition Of ◽  
Image Position

SummaryThe validity and applicability of the static margin (stick fixed) Kn,where as defined by Gates and Lyon is shown to be restricted to the conventional flexible aircraft. Alternative suggestions for the definition of static margin are put forward which can be equally applied to the conventional flexible aircraft of the past and the integrated flexible aircraft of the future. Calculations have been carried out on simple slender plate models with both linear and non-linear aerodynamic forces to assess their static stability characteristics.

The Multiplicative Groups of Quasifields

1987 ◽  
Vol 39 (4) ◽  
pp. 784-793 ◽  
Author(s):  
Michael J. Kallaher
Keyword(s):  
Zero Element ◽  
Binary Operations ◽  
Definition Of ◽  
Image Position ◽  
Multiplicative Groups ◽  
Multiplicative Mapping

Let (Q, +, ·) be a finite quasifield of dimension d over its kernel K = GF(q), where q = pk with p a prime and k ≧ 1. (See p. 18-22 and p. 74 of [7] or Section 5 of [9] for the definition of quasifield.) For the remainder of this article we will follow standard conventions and omit, whenever possible, the binary operations + and · in discussing a quasifield. For example, the notation Q will be used in place of the triple (Q, +, ·) and Q* will be used to represent the multiplicative loop (Q − {0}, ·).Let m be a non-zero element of the quasifield Q; the right multiplicative mapping ρm:Q → Q is defined by1

scholarly journals On the Inversion of the Gauss Transformation

1957 ◽  
Vol 9 ◽  
pp. 459-464 ◽  
Author(s):  
P. G. Rooney
Keyword(s):  
Differential Operator ◽  
Recent Work ◽  
Suitable Definition ◽  
Inversion Theory ◽  
The Subject ◽  
Definition Of ◽  
Image Position ◽  
Gauss Transformation ◽  
Operational Methods

The inversion theory of the Gauss transformation has been the subject of recent work by several authors. If the transformation is defined by1.1,then operational methods indicate that,under a suitable definition of the differential operator.

On the Definition of C*-Algebras II

1985 ◽  
Vol 37 (4) ◽  
pp. 664-681 ◽  
Author(s):  
Zoltán Magyar ◽  
Zoltán Sebestyén
Keyword(s):  
Hilbert Space ◽  
Banach Algebra ◽  
Representation Theorem ◽  
Banach Algebras ◽  
Bounded Operators ◽  
Fundamental Representation ◽  
C Algebras ◽  
Definition Of ◽  
Image Position

The theory of noncommutative involutive Banach algebras (briefly Banach *-algebras) owes its origin to Gelfand and Naimark, who proved in 1943 the fundamental representation theorem that a Banach *-algebra with C*-condition(C*)is *-isomorphic and isometric to a norm-closed self-adjoint subalgebra of all bounded operators on a suitable Hilbert space.At the same time they conjectured that the C*-condition can be replaced by the B*-condition.(B*)In other words any B*-algebra is actually a C*-algebra. This was shown by Glimm and Kadison [5] in 1960.

Sign in / Sign up

Export Citation Format

Share Document