Carathéodory's Theorem

1972 ◽  
Vol 15 (2) ◽  
pp. 293-293 ◽  
Author(s):  
W. D. Cook ◽  
R. J. Webster
Keyword(s):  
Carathéodory’S Theorem ◽  
Simple Generalization

Despite the abundance of generalizations of Carathéodory's theorem occurring in the literature (see [1]), the following simple generalization involving infinite convex combinations seems to have passed unnoticed. Boldface letters denote points of Rn and Greek letters denote scalars.

On automorphisms of monotone transformation posemigroups

2021 ◽  
pp. 2250032
Author(s):  
Dilawar Juneed Mir ◽  
Aftab Hussain Shah ◽  
Shabir Ahmad Ahanger
Keyword(s):  
Partial Transformation ◽  
Monotone Transformation ◽  
Full Transformation ◽  
Simple Generalization

In this paper, we provide a simple generalization of results of Sullivan for [Formula: see text] the full transformation monotone pomonoid and for [Formula: see text] the partial transformation monotone pomonoid by showing that every automorphism of [Formula: see text] and [Formula: see text] is inner induced by the elements of [Formula: see text] the pogroup of all ordered bijections on [Formula: see text]. We also show that [Formula: see text] is isomorphic to [Formula: see text]. Finally, we apply these results to get some more results in this direction.

The distribution of the sequence {nξ}(n = 0, 1, 2, …)

1965 ◽  
Vol 61 (3) ◽  
pp. 665-670 ◽  
Author(s):  
John H. Halton
Keyword(s):  
Unit Interval ◽  
Simple Generalization ◽  
Unit Hypercube

Introduction and statement of results. We shall describe how, for successive integers N, the points {nξ}, with n = 0, 1, …,N – 1, are distributed in the closed unit interval U = [0, 1]; by showing how successive points {Nξ,} modify the partition of U produced by the previous points. The simple generalization to the k-dimensional sequence {nξ} = ({nξ(1)},{nξ(2)}, …,{nξ(k)}), in the unit hypercube Uk, is also made.

A simple generalization of Luenberger method for pole assignment

1974 ◽  
Vol 62 (3) ◽  
pp. 412-414 ◽  
Author(s):  
S. Beghelli ◽  
G. Bertoni ◽  
G. Capitani
Keyword(s):  
Pole Assignment ◽  
Simple Generalization

The shift lattice: an interpretation of some infinitely adaptive structures

1993 ◽  
Vol 440 (1908) ◽  
pp. 23-36 ◽  
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
Periodic Systems ◽  
Adaptive Structures ◽  
One Dimensional ◽  
Modulated Structures ◽  
Simple Generalization ◽  
The One

The structures of various ordered, but non-periodic, systems have been investigated and exhibit features which can be directly described by means of a construction which the authors call the shift lattice , which is a simple generalization of the concept of the lattice. This paper is devoted to a description of the properties of the one-dimensional shift lattice and its Fourier transform. Its applications to the phases related to L–Ta 2 O 5 and some Bi 2 TeO 5 -related systems are outlined and its relation to the theory of modulated structures and their Fourier transforms is briefly discussed.

Gauge-invariant formulation of axion-electrodynamics

2021 ◽  
Author(s):  
Stanley A. Bruce
Keyword(s):  
Dirac Fermion ◽  
Dirac Field ◽  
Gauge Invariant ◽  
Fermion Fields ◽  
Simple Generalization ◽  
Invariant Formulation ◽  
Em Field

In this paper, we propose a simple generalization of axion-electrodynamics (AED) for the general case in which Dirac fermion fields and scalar/pseudoscalar axion-like fields are present in the local [Formula: see text]([Formula: see text])[Formula: see text] gauge-invariant Lagrangian of the system. Our primary goal (which is not explored here) is to understand and predict novel phenomena that have no counterpart in standard (pseudoscalar) AED. With this end in view, we discuss on very general grounds, possible processes in which a Dirac field is coupled to axionic fields via the electromagnetic (EM) field.

scholarly journals Highest energy proton–nucleus cross-sections

2018 ◽  
Vol 33 (40) ◽  
pp. 1850242 ◽  
Author(s):  
L. Stodolsky
Keyword(s):  
Cosmic Rays ◽  
Cross Sections ◽  
High Energy ◽  
High Energy Proton ◽  
Proton Proton ◽  
Energy Proton ◽  
Black Disc ◽  
Simple Generalization ◽  
Very High Energy ◽  
Very High

The description of very high energy proton–proton cross-sections in terms of a “black disc” with an “edge” allows a simple generalization to highest energy proton–nucleus cross-sections. This results in a leading ln2W term and a ln W term whose coefficient depends linearly on the radius of the nucleus (W the c.m. energy). The necessary parameters are determined from the fits to p–p data. Since the coefficient of the ln W term is rather large, it is doubtful that the regime of ln2W dominance can be reached with available energies in accelerators or cosmic rays. However, the ln W term can be relevant for highest energy cosmic rays in the atmosphere, where a large increase for the cross-section on nitrogen is expected. Tests of the theory should be possible by studying the coefficient of ln W at p-nucleus colliders.

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