scholarly journals Comparing Sunspot Equilibrium And Lottery Equilibrium Allocations: The Finite Case*

2004 ◽  
Vol 45 (2) ◽  
pp. 351-386 ◽  
Author(s):  
Rod Garratt ◽  
Todd Keister ◽  
Karl Shell
Keyword(s):  
Finite Case ◽  
Sunspot Equilibrium

LINEAR AND NONLINEAR INTERSUBBAND OPTICAL ABSORPTION OF FINITE AND INFINITE SEMI-PARABOLIC QUANTUM WELLS

2011 ◽  
Vol 25 (07) ◽  
pp. 497-507 ◽  
Author(s):  
M. J. KARIMI ◽  
A. KESHAVARZ ◽  
A. POOSTFORUSH
Keyword(s):  
Refractive Index ◽  
Optical Properties ◽  
Optical Absorption ◽  
Quantum Well ◽  
Quantum Wells ◽  
Resonant Peak ◽  
Energy Eigenvalues ◽  
Finite Case ◽  
Infinite Case ◽  
Index Changes

In this work, the optical absorption coefficients and the refractive index changes for the infinite and finite semi-parabolic quantum well are calculated. Numerical calculations are performed for typical GaAs / Al x Ga 1-x As semi-parabolic quantum well. The energy eigenvalues and eigenfunctions of these systems are calculated numerically. Optical properties are obtained using the compact density matrix approach. Results show that the energy eigenvalues and the matrix elements of the infinite and finite cases are different. The calculations reveal that the resonant peaks of the optical properties of the finite case occur at lower values of the incident photon energy with respect to the infinite case. Results indicate that the maximum value of the refractive index changes for the finite case are greater than that of the infinite case. Our calculations also show that in contrast to the infinite case, the resonant peak value of the total absorption coefficient in the case of the finite well is a non-monotonic function of the semi-parabolic confinement frequency.

SYMMETRY GEOMETRY BY PAIRINGS

2018 ◽  
Vol 106 (03) ◽  
pp. 342-360 ◽  
Author(s):  
G. CHIASELOTTI ◽  
T. GENTILE ◽  
F. INFUSINO
Keyword(s):  
Closure Operator ◽  
Local Symmetry ◽  
Global Symmetry ◽  
Symmetry Relation ◽  
Basic Tool ◽  
Mathematical Structures ◽  
Finite Case ◽  
Set Systems ◽  
Power Set ◽  
Abstract Simplicial Complex

In this paper, we introduce asymmetry geometryfor all those mathematical structures which can be characterized by means of a generalization (which we call pairing) of a finite rectangular table. In more detail, let$\unicode[STIX]{x1D6FA}$be a given set. Apairing$\mathfrak{P}$on$\unicode[STIX]{x1D6FA}$is a triple$\mathfrak{P}:=(U,F,\unicode[STIX]{x1D6EC})$, where$U$and$\unicode[STIX]{x1D6EC}$are nonempty sets and$F:U\times \unicode[STIX]{x1D6FA}\rightarrow \unicode[STIX]{x1D6EC}$is a map having domain$U\times \unicode[STIX]{x1D6FA}$and codomain$\unicode[STIX]{x1D6EC}$. Through this notion, we introduce a local symmetry relation on$U$and a global symmetry relation on the power set${\mathcal{P}}(\unicode[STIX]{x1D6FA})$. Based on these two relations, we establish the basic properties of our symmetry geometry induced by$\mathfrak{P}$. The basic tool of our study is a closure operator$M_{\mathfrak{P}}$, by means of which (in the finite case) we can represent any closure operator. We relate the study of such a closure operator to several types of others set operators and set systems which refine the notion of an abstract simplicial complex.

Generalized Frobenius Algebras and Hopf Algebras

2014 ◽  
Vol 66 (1) ◽  
pp. 205-240 ◽  
Author(s):  
Miodrag Cristian Iovanov
Keyword(s):  
Hopf Algebras ◽  
Topological Algebra ◽  
Homological Algebra ◽  
General Concept ◽  
Frobenius Algebra ◽  
Theoretic Approach ◽  
Topological Algebras ◽  
Infinite Dimensional ◽  
Finite Case ◽  
Frobenius Algebras

Abstract“Co-Frobenius” coalgebras were introduced as dualizations of Frobenius algebras. We previously showed that they admit left-right symmetric characterizations analogous to those of Frobenius algebras. We consider the more general quasi-co-Frobenius (QcF) coalgebras. The first main result in this paper is that these also admit symmetric characterizations: a coalgebra is QcF if it is weakly isomorphic to its (left, or right) rational dual Rat(C*) in the sense that certain coproduct or product powers of these objects are isomorphic. Fundamental results of Hopf algebras, such as the equivalent characterizations of Hopf algebras with nonzero integrals as left (or right) co-Frobenius, QcF, semiperfect or with nonzero rational dual, as well as the uniqueness of integrals and a short proof of the bijectivity of the antipode for such Hopf algebras all follow as a consequence of these results. This gives a purely representation theoretic approach to many of the basic fundamental results in the theory of Hopf algebras. Furthermore, we introduce a general concept of Frobenius algebra, which makes sense for infinite dimensional and for topological algebras, and specializes to the classical notion in the finite case. This will be a topological algebra A that is isomorphic to its complete topological dual Aν. We show that A is a (quasi)Frobenius algebra if and only if A is the dual C* of a (quasi)co-Frobenius coalgebra C. We give many examples of co-Frobenius coalgebras and Hopf algebras connected to category theory, homological algebra and the newer q-homological algebra, topology or graph theory, showing the importance of the concept.

scholarly journals On Ramsey-Minimal Infinite Graphs

2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Jordan Barrett ◽  
Valentino Vito
Keyword(s):  
Ramsey Theory ◽  
Infinite Graphs ◽  
Minimal Graph ◽  
Finite Graphs ◽  
Minimal Graphs ◽  
Finite Case ◽  
Ramsey Minimal Graph ◽  
General Conditions

For fixed finite graphs $G$, $H$, a common problem in Ramsey theory is to study graphs $F$ such that $F \to (G,H)$, i.e. every red-blue coloring of the edges of $F$ produces either a red $G$ or a blue $H$. We generalize this study to infinite graphs $G$, $H$; in particular, we want to determine if there is a minimal such $F$. This problem has strong connections to the study of self-embeddable graphs: infinite graphs which properly contain a copy of themselves. We prove some compactness results relating this problem to the finite case, then give some general conditions for a pair $(G,H)$ to have a Ramsey-minimal graph. We use these to prove, for example, that if $G=S_\infty$ is an infinite star and $H=nK_2$, $n \geqslant 1$ is a matching, then the pair $(S_\infty,nK_2)$ admits no Ramsey-minimal graphs.

Decomposition of Witt Rings

1982 ◽  
Vol 34 (6) ◽  
pp. 1276-1302 ◽  
Author(s):  
Andrew B. Carson ◽  
Murray A. Marshall
Keyword(s):  
Quadratic Forms ◽  
Classification Problem ◽  
Bilinear Forms ◽  
Interesting Problem ◽  
Witt Ring ◽  
Witt Rings ◽  
Finite Case ◽  
Characteristic 2 ◽  
Definition Of

We take the definition of a Witt ring to be that given in [13], i.e., it is what is called a strongly representational Witt ring in [8]. The classical example is obtained by considering quadratic forms over a field of characteristic ≠ 2 [17], but Witt rings also arise in studying quadratic forms or symmetric bilinear forms over more general types of rings [5,7, 8, 9]. An interesting problem in the theory is that of classifying Witt rings in case the associated group G is finite. The reduced case, i.e., the case where the nilradical is trivial, is better understood. In particular, the above classification problem is completely solved in this case [4, 12, or 13, Corollary 6.25]. Thus, the emphasis here is on the non-reduced case. Although some of the results given here do not require |G| < ∞, they do require some finiteness assumption. Certainly, the main goal here is to understand the finite case, and in this sense this paper is a continuation of work started by the second author in [13, Chapter 5].

A New Characterization of Finite Prime Fields

1968 ◽  
Vol 11 (3) ◽  
pp. 381-382 ◽  
Author(s):  
Carlton J. Maxson
Keyword(s):  
Multiplicative Group ◽  
Near Field ◽  
Finite Case ◽  
Prime Fields

Let N ≡ <N, +,.> be a (right) near-ring with 1 (we say N is a unitary near-ring)[1] and recall that a near-field is a unitary near-ring in which <N - {0}, . > is a multiplicative group. In [2], Beidelman characterizes near-fields as those unitary near-rings without non-trivial N-subgroups. We show that in the finite case this absence of non-trivial N-subgroups is equivalent to the absence of non-trivial left ideals.

scholarly journals 12. Solar electrodynamics

1958 ◽  
Vol 6 ◽  
pp. 105-113 ◽  
Author(s):  
T. G. Cowling
Keyword(s):  
Magnetic Fields ◽  
Solar Magnetic Field ◽  
Charged Particles ◽  
Coronal Heating ◽  
Magnetic Control ◽  
Solar Magnetic Fields ◽  
Historical Account ◽  
Turbulent Field ◽  
Sunspot Equilibrium ◽  
General Field

A historical account of the subject's development is attempted. Prior to 1940, the most significant papers were those by Larmor (1919) and Cowling (1934) on dynamo theories of solar fields: by Kiepenheuer (1935) on the corona; and by Ferraro (1937) on isorotation. These indicated the importance of electromagnetic forces and were groping towards the idea of frozen-in fields. The latter idea was, however, not clearly stated before Alfvén's 1941–2 papers.Theory since then is divided into sections concerned with mechanical effects of magnetic fields, theories of sunspots, and the nature and origin of solar magnetic fields. The first includes theories of magnetic control of support of coronal filaments and prominences (van de Hulst, Alfvén, Dungey) and theories of magnetic influence on sunspot equilibrium. The second includes Alfvén's and Walén's theories of the solar cycle, and Biermann's explanation of sunspot coolness in terms of magnetic inhibition of convection. Sunspot theories, being discussed more fully by Biermann, are considered only briefly.Electromagnetic heating covers theories of coronal heating and flares, discharge phenomena, particle acceleration and radio emission. Many of the older theories (Alfvén's on coronal heating, Giovanelli's on flares, that of Bagge and Biermann on cosmic rays) are set aside because of their neglect of self-induction effects and inadequacy of the mechanism of conversion. The relative motion of charged particles and neutral atoms (Piddington, Cowling) is described as supplying a powerful heating effect.As regards the magnitude of the general solar magnetic field, it is suggested that the observed value can be discarded only if decisive reasons are given. Other theories having so far proved inadequate, dynamo theories of the origin of solar fields are regarded as the most promising. These can be partial, as when a toroidal field capable of explaining spot fields is supposed to be generated from the general field (Walén and others), or when a turbulent field is supposed to be generated from a smaller regular field (Alfvén and others): or total, when a simultaneous explanation of all fields is attempted (e.g. Parker). A general appraisal is made of the different theories.

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