scholarly journals On a property of the Fenchel transform

2021 ◽  
Vol 22 (3) ◽  
pp. 474-478
Author(s):  
Aisylu Azatovna Farvazova
Keyword(s):  
Fenchel Transform

scholarly journals Prox-regular sets and Legendre-Fenchel transform related to separation properties

Optimization ◽  
2020 ◽  
pp. 1-33
Author(s):  
Samir Adly ◽  
Florent Nacry ◽  
Lionel Thibault
Keyword(s):  
Fenchel Transform ◽  
Regular Sets

scholarly journals Work fluctuations in the active Ornstein–Uhlenbeck particle model

2021 ◽  
Vol 2021 (12) ◽  
pp. 123202
Author(s):  
Massimiliano Semeraro ◽  
Antonio Suma ◽  
Isabella Petrelli ◽  
Francesco Cagnetta ◽  
Giuseppe Gonnella
Keyword(s):  
Numerical Solutions ◽  
Free Particle ◽  
Bath Temperature ◽  
Rate Function ◽  
Potential Interaction ◽  
Two Dimensions ◽  
Particle Model ◽  
Confining Potential ◽  
Fenchel Transform ◽  
Work Distribution

Abstract We study the large deviations of the power injected by the active force for an active Ornstein–Uhlenbeck particle (AOUP), free or in a confining potential. For the free-particle case, we compute the rate function analytically in d-dimensions from a saddle-point expansion, and numerically in two dimensions by (a) direct sampling of the active work in numerical solutions of the AOUP equations and (b) Legendre–Fenchel transform of the scaled cumulant generating function obtained via a cloning algorithm. The rate function presents asymptotically linear branches on both sides and it is independent of the system’s dimensionality, apart from a multiplicative factor. For the confining potential case, we focus on two-dimensional systems and obtain the rate function numerically using both methods (a) and (b). We find a different scenario for harmonic and anharmonic potentials: in the former case, the phenomenology of fluctuations is analogous to that of a free particle, but the rate function might be non-analytic; in the latter case the rate functions are analytic, but fluctuations are realised by entirely different means, which rely strongly on the particle-potential interaction. Finally, we check the validity of a fluctuation relation for the active work distribution. In the free-particle case, the relation is satisfied with a slope proportional to the bath temperature. The same slope is found for the harmonic potential, regardless of activity, and for an anharmonic potential with low activity. In the anharmonic case with high activity, instead, we find a different slope which is equal to an effective temperature obtained from the fluctuation–dissipation theorem.

scholarly journals The Joly topology and the Mosco-Beer topology revisited

1993 ◽  
Vol 48 (3) ◽  
pp. 353-363 ◽  
Author(s):  
Dominique Azé ◽  
Jean-Paul Penot
Keyword(s):  
Vector Space ◽  
Convex Functions ◽  
Normed Vector Space ◽  
Fenchel Transform ◽  
Proper Convex ◽  
Normed Vector

Some extensions to the non reflexive case of continuity results for the Legendre-Fenchel transform are presented following an approach due to J.-L. Joly. We compare the topology introduced by J.-L. Joly and the Mosco-Beer topology introduced by G. Beer. In particular, in the case of the space of closed proper convex functions defined on the dual of a normed vector space they coincide.

scholarly journals On a Differential Inclusion Involving Dirichlet–Laplace Operators of Fractional Orders

2020 ◽  
Vol 43 (6) ◽  
pp. 4089-4106
Author(s):  
Rafał Kamocki
Keyword(s):  
Boundary Conditions ◽  
Differential Inclusion ◽  
Laplace Operator ◽  
Spectral Decomposition ◽  
Convex Analysis ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Fenchel Transform ◽  
The Laplace Operator ◽  
Fractional Orders

Abstract In this paper, we investigate a nonlinear differential inclusion with Dirichlet boundary conditions containing a weak Laplace operator of fractional orders (defined via the spectral decomposition of the Laplace operator $$-{\varDelta }$$ - Δ under Dirichlet boundary conditions). Using variational methods, we characterize solutions of such a problem. Our approach is based on tools from convex analysis (properties of a Legendre–Fenchel transform).

Sign in / Sign up

Export Citation Format

Share Document