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:

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.

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]:

Neomisticius platypi n. sp. and N. variabilis n. sp. (Tylenchomorpha: Anguinidae) from dead oak trees in Japan

Nematology ◽  
2021 ◽  
pp. 1-21
Author(s):  
Natsumi Kanzaki ◽  
Hayato Masuya ◽  
Keiko Hamaguchi
Keyword(s):  
New Species ◽  
Vas Deferens ◽  
Morphological Characteristics ◽  
Platypus Quercivorus ◽  
Oak Trees ◽  
Two New Species ◽  
Previous Definition ◽  
Rectal Glands ◽  
Stylet Length ◽  
Definition Of

Summary Two new Neomisticius species, typologically and phylogenetically close to each other, are described and illustrated from dead Quercus trees and an ambrosia beetle, Platypus quercivorus. The two species share some stylet morphological characteristics, namely, they both possess a long conus occupying more than half of the total stylet length, a long crustaformeria composed of more than 160 cells (eight rows of more than 20 cells each), and a short and broad female tail with a digitate tip. They are distinguished from each other by N. variabilis n. sp. having a wide, spindle-shaped male bursa with a blunt terminus and N. platypi n. sp. having an oval bursa with a rounded terminus. In addition, the males and females of both species have three large rectal glands and the posterior end of the male testis (distal end of the vas deferens) bears three cells that seemingly function as a valve between the vas deferens and the cloacal tube. These characteristics have not been reported in other tylenchids. Currently, the genus contains only three species: the two new species and N. rhizomorphoides, which has a normal stylet with a short conus, a short crustaformeria, and lacks rectal glands and valve cells in the vas deferens. Therefore, the two new species are readily distinguished from N. rhizomorphoides and, based on the previous definition, may even represent a new genus. However, considering their phylogenetic closeness and biological similarities (e.g., association with ambrosia beetles), the generic definition of Neomisticius was emended to include these new species.

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.

Abstract P533: Incidence and Outcomes of Thrombectomy Rescue Therapies in Acute Basilar Artery Occlusions

Stroke ◽  
2021 ◽  
Vol 52 (Suppl_1) ◽  
Author(s):  
Ronda Lun ◽  
Greg B Walker ◽  
David Weisenburger-Lile ◽  
Bertrand Lapergue ◽  
Adrien Guenego ◽  
...  
Keyword(s):  
Primary Outcome ◽  
Secondary Analysis ◽  
Hematoma Expansion ◽  
Primary Analysis ◽  
Predictive Values ◽  
Previous Definition ◽  
Secondary Analyses ◽  
Revised Definitions ◽  
Definition Of ◽  
Sensitivity Specificity

Background: Hematoma expansion (HE) is an important therapeutic target in intracerebral hemorrhage. Recently proposed HE definitions have not been validated, and no previous definition has accounted for withdrawal of care (WOC). Objective: To compare conventional and revised definitions of hematoma expansion (HE), while accounting for WOC. Methods: We analyzed data from the ATACH-2 trial, comparing revised definitions of HE incorporating intraventricular hemorrhage (IVH) expansion to the conventional definition of “≥6 mL or ≥33%”. The primary outcome was modified Rankin Scale of 4-6 at 90-days. We calculated the incidence, sensitivity, specificity, positive and negative predictive values, and c- statistic for all definitions of HE. Definitions were compared using non-parametric methods. Secondary analyses were performed after removing patients who experienced WOC. Results: Primary analysis included 948 patients. Using the conventional definition, the sensitivity was 37.1% and specificity was 83.2% for the primary outcome. Sensitivity improved with all three revised definitions (53.3%, 48.7%, and 45.3%, respectively), with minimal change to specificity (78.4%, 80.5%, and 81.0%, respectively). The greatest improvement was seen with the definition “≥6 mL or ≥33% or any IVH”, with increased c -statistic from 60.2% to 65.9% (p < 0.001). Secondary analysis excluded 46 participants who experienced WOC. The revised definitions outperformed the conventional definition in this population as well, with the greatest improvement in c -statistic using “≥6 mL or ≥33% or any IVH” (58.1% vs 64.1%, p < 0.001). Conclusions: HE definitions incorporating intraventricular expansion outperformed conventional definitions for predicting poor outcome, even after accounting for care limitations.

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.

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.

The þ-function in λ-K-conversion

1937 ◽  
Vol 2 (4) ◽  
pp. 164-164 ◽  
Author(s):  
A. M. Turing
Keyword(s):  
Positive Integer ◽  
Natural Number ◽  
Definition Of ◽  
Image Position

In the theory of conversion it is important to have a formally defined function which assigns to any positive integer n the least integer not less than n which has a given property. The definition of such a formula is somewhat involved: I propose to give the corresponding formula in λ-K-conversion, which will (naturally) be much simpler. I shall in fact find a formula þ such that if T be a formula for which T(n) is convertible to a formula representing a natural number, whenever n represents a natural number, then þ(T, r) is convertible to the formula q representing the least natural number q, not less than r, for which T(q) conv 0.2 The method depends on finding a formula Θ with the property that Θ conv λu·u(Θ(u)), and consequently if M→Θ(V) then M conv V(M). A formula with this property is,The formula þ will have the required property if þ(T, r) conv r when T(r) conv 0, and þ(T, r) conv þ(T, S(r)) otherwise. These conditions will be satisfied if þ(T, r) conv T(r, λx·þ(T, S(r)), r), i.e. if þ conv {λptr·t(r, λx·p(t, S(r)), r)}(þ). We therefore put,This enables us to define also a formula,such that (T, n) is convertible to the formula representing the nth positive integer q for which T(q) conv 0.

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