scholarly journals The many faces of far-from-equilibrium thermodynamics: Deterministic chaos, randomness, or emergent order?

MRS Bulletin ◽  
2019 ◽  
Vol 44 (2) ◽  
pp. 130-133 ◽  
Author(s):  
Atanu Chatterjee ◽  
Germano Iannacchione
Keyword(s):  
Deterministic Chaos ◽  
Equilibrium Thermodynamics ◽  
Emergent Order ◽  
Far From Equilibrium ◽  
The Many ◽  
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Abstract

Unitarily Invariant Operator Norms

1983 ◽  
Vol 35 (2) ◽  
pp. 274-299 ◽  
Author(s):  
C.-K. Fong ◽  
J. A. R. Holbrook
Keyword(s):  
Hilbert Space ◽  
Convex Functions ◽  
Invariant Operator ◽  
Space Operator ◽  
Numerical Radius ◽  
The Past ◽  
Operator Norms ◽  
Power Inequality ◽  
The Many ◽  
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1.1. Over the past 15 years there has grown up quite an extensive theory of operator norms related to the numerical radius1of a Hilbert space operator T. Among the many interesting developments, we may mention:(a) C. Berger's proof of the “power inequality”2(b) R. Bouldin's result that3for any isometry V commuting with T;(c) the unification by B. Sz.-Nagy and C. Foias, in their theory of ρ-dilations, of the Berger dilation for T with w(T) ≤ 1 and the earlier theory of strong unitary dilations (Nagy-dilations) for norm contractions;(d) the result by T. Ando and K. Nishio that the operator radii wρ(T) corresponding to the ρ-dilations of (c) are log-convex functions of ρ.

On a sequence of algebraic formulae of Ramanujan

2012 ◽  
Vol 96 (536) ◽  
pp. 201-206
Author(s):  
Juan Pla
Keyword(s):  
Quadratic Form ◽  
Complex Numbers ◽  
The Many ◽  
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Among the many astonishing formulae stated by Ramanujan we find in his Notebooks the following sequence (see [1], p 96, Entry 43):which hold for any triple of complex numbers (a, b, c) such that a + b + c = 0.Ramanujan concluded this list by ‘And so on’, which suggests that he had some kind of method or algorithm allowing him to extend this list to any even power of the quadratic form p(a, b, c) = ab + bc + ca, when a + b + c = 0. To explain this, Bruce Brendt details a theorem by S. Bhargava [2], which can be used to produce identities of the kind above, and infers that Ramanujan is likely to have used the same proof to establish his identities (see [1], pp. 97-100)

scholarly journals Multiscale Thermodynamics

Entropy ◽  
2021 ◽  
Vol 23 (2) ◽  
pp. 165
Author(s):  
Miroslav Grmela
Keyword(s):  
Complex Systems ◽  
Equilibrium Thermodynamics ◽  
Classical Equilibrium ◽  
Far From Equilibrium ◽  
Special Case

Multiscale thermodynamics is a theory of the relations among the levels of investigation of complex systems. It includes the classical equilibrium thermodynamics as a special case, but it is applicable to both static and time evolving processes in externally and internally driven macroscopic systems that are far from equilibrium and are investigated at the microscopic, mesoscopic, and macroscopic levels. In this paper we formulate multiscale thermodynamics, explain its origin, and illustrate it in mesoscopic dynamics that combines levels.

God's Use of theIdem per Idemto Terminate Debate

1978 ◽  
Vol 71 (3-4) ◽  
pp. 193-201 ◽  
Author(s):  
Jack R. Lundbom
Keyword(s):  
Divine Name ◽  
The Many ◽  
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Twice in the book of Exodus where tradition preserves the revelation of the divine name to Moses, God employs a peculiar idiom which S. R. Driver has called theidem per idem. In Exod 3:14 God says:I will be what I will beand again in 33:19 he tells his servant:But I will be gracious to whom I will be graciousand I will show mercy on whom I will show mercyTheidem per idemis a tautology of sorts which Driver says is employed “where the means or the desire to be more explicit does not exist.” Driver calls the idiom Semitic, and indeed it is, as one can see by perusing the many examples from Hebrew and Arabic cited earlier by Paul de Lagarde in hisPsalterium Iuxta Hebraeos Hieronymi. But it is also found, as we shall see in a moment, in other languages both ancient and modern.

scholarly journals Systems of derivations on topological algebras of power series

1975 ◽  
Vol 19 (3) ◽  
pp. 358-370
Author(s):  
Henry J. Schultz
Keyword(s):  
Power Series ◽  
Regularity Condition ◽  
Banach Algebras ◽  
Binomial Coefficient ◽  
Continuous Functions ◽  
Analytic Structure ◽  
Topological Algebras ◽  
Linear Maps ◽  
The Many ◽  
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If Do, D1, … are linear maps from an algebra A to an algebra B, both over the complexes, then {Do, D1, …} is a system of derivations if for all a, b in A and for all nonnegative integers k, we have Where C(k, i) is the binomial coefficient k!/i! (k—i)!. By (1.1) we see that Do must be a homomorphism and in case Do = I, where I is the identity map, D1 is a derivation and, for k ≧ 2, the Dk are higher derivations in the sense of Jacobson (1964), page 191. Gulick (1970), Theorem 4.2, proved that if A is a commutative regular semi-simple F-algebra with identity and {DO, D1, …} is a system of derivations from A to B = C(S(A)), the algebra of all continuous functions on the spectrum of A, where Dox = x, then the Dk are all continuous. Carpenter (1971), Theorem 5, shows that the regularity condition is unnecessary and Loy (1973) generalizes this a bit further. One of the many interesting features of systems of derivations is that they help determine analytic structure in Banach algebras (see for example, Miller (to appear)).

scholarly journals On a generalisation of a result of Ramanujan connected with the exponential series

1981 ◽  
Vol 24 (3) ◽  
pp. 179-195
Author(s):  
R. B. Paris
Keyword(s):  
Asymptotic Expansion ◽  
Positive Integer ◽  
Exponential Series ◽  
Large Positive Integer ◽  
The Many ◽  
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One of the many interesting problems discussed by Ramanujan is an approximation related to the exponential series for en, when n assumes large positive integer values. If the number θn is defined byRamanujan (9) showed that when n is large, θn possesses the asymptotic expansionThe first demonstrations that θn lies between ½ and and that θn decreases monotoni-cally to the value as n increases, were given by Szegö (12) and Watson (13). Analogous results were shown to exist for the function e−n, for positive integer values of n, by Copson (4). If φn is defined bythen πn lies between 1 and ½ and tends monotonically to the value ½ as n increases, with the asymptotic expansionA generalisation of these results was considered by Buckholtz (2) who defined, in a slightly different notation, for complex z and positive integer n, the function φn(z) by

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