Legendre transforms

2018 ◽  
pp. 317-327
Author(s):  
Baidyanath Patra
Keyword(s):  
Legendre Transforms

scholarly journals Legendre transforms and Apéry's sequences

1995 ◽  
Vol 58 (3) ◽  
pp. 358-375 ◽  
Author(s):  
Asmus L. Schmidt
Keyword(s):  
Legendre Transform ◽  
Image Position ◽  
Legendre Transforms

AbstractThis article studies particular sequences satisfying polynomial recurrences, among those Apéry's sequence which is shown to be the Legendre transform of the sequence. This results in the construction of simultaneous approximations of π 2/8 and ζ(3).

scholarly journals Model reduction by mean-field homogenization in viscoelastic composites. I. Primal theory

2020 ◽  
Vol 476 (2242) ◽  
pp. 20200407 ◽  
Author(s):  
Martín I. Idiart ◽  
Noel Lahellec ◽  
Pierre Suquet
Keyword(s):  
Strain Field ◽  
Mean Field ◽  
Inelastic Strain ◽  
Mathematical Structure ◽  
Schwarz Inequality ◽  
Internal Variables ◽  
Legendre Transform ◽  
Time Step ◽  
Legendre Transforms ◽  
Viscoelastic Composites

A homogenization scheme for viscoelastic composites proposed by Lahellec & Suquet (2007 Int. J. Solids Struct. 44 , 507–529 ( doi:10.1016/j.ijsolstr.2006.04.038 )) is revisited. The scheme relies upon an incremental variational formulation providing the inelastic strain field at a given time step in terms of the inelastic strain field from the previous time step, along with a judicious use of Legendre transforms to approximate the relevant functional by an alternative functional depending on the inelastic strain fields only through their first and second moments over each constituent phase. As a result, the approximation generates a reduced description of the microscopic state of the composite in terms of a finite set of internal variables that incorporates information on the intraphase fluctuations of the inelastic strain and that can be evaluated by mean-field homogenization techniques. In this work we provide an alternative derivation of the scheme, relying on the Cauchy–Schwarz inequality rather than the Legendre transform, and in so doing we expose the mathematical structure of the resulting approximation and generalize the exposition to fully anisotropic material systems.

Conjugates and Legendre Transforms of Convex Functions

1967 ◽  
Vol 19 ◽  
pp. 200-205 ◽  
Author(s):  
R. T. Rockafellar
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Strict Convexity ◽  
Legendre Transformation ◽  
Legendre Transform ◽  
Implicit Functions ◽  
The One ◽  
Legendre Transforms ◽  
Effective Domain ◽  
The Relationship

Fenchel's conjugate correspondence for convex functions may be viewed as a generalization of the classical Legendre correspondence, as indicated briefly in (6). Here the relationship between the two correspondences will be described in detail. Essentially, the conjugate reduces to the Legendre transform if and only if the subdifferential of the convex function is a one-to-one mapping. The one-to-oneness is equivalent to differentiability and strict convexity, plus a condition that the function become infinitely steep near boundary points of its effective domain. These conditions are shown to be the very ones under which the Legendre correspondence is well-defined and symmetric among convex functions. Facts about Legendre transforms may thus be deduced using the elegant, geometrically motivated methods of Fenchel. This has definite advantages over the more restrictive classical treatment of the Legendre transformation in terms of implicit functions, determinants, and the like.

Legendre transforms in the Ising model

1974 ◽  
Vol 21 (1) ◽  
pp. 963-970 ◽  
Author(s):  
A. N. Vasil'ev ◽  
R. A. Radzhabov
Keyword(s):  
Ising Model ◽  
Legendre Transforms
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