A variational problem of one-dimensional nonequilibrium gasdynamics

1968 ◽  
Vol 1 (2) ◽  
pp. 16-22 ◽  
Author(s):  
N. S. Galyun ◽  
A. N. Kraiko
Keyword(s):  
Variational Problem ◽  
One Dimensional

scholarly journals ON A LATTICE GENERALISATION OF THE LOGARITHM AND A DEFORMATION OF THE DEDEKIND ETA FUNCTION

2020 ◽  
Vol 102 (1) ◽  
pp. 118-125
Author(s):  
LAURENT BÉTERMIN
Keyword(s):  
Variational Problem ◽  
Theta Function ◽  
Dedekind Eta Function ◽  
Dimensional Lattice ◽  
One Dimensional ◽  
Natural Logarithm ◽  
Two Parameters

We consider a deformation $E_{L,\unicode[STIX]{x1D6EC}}^{(m)}(it)$ of the Dedekind eta function depending on two $d$-dimensional simple lattices $(L,\unicode[STIX]{x1D6EC})$ and two parameters $(m,t)\in (0,\infty )$, initially proposed by Terry Gannon. We show that the minimisers of the lattice theta function are the maximisers of $E_{L,\unicode[STIX]{x1D6EC}}^{(m)}(it)$ in the space of lattices with fixed density. The proof is based on the study of a lattice generalisation of the logarithm, called the lattice logarithm, also defined by Terry Gannon. We also prove that the natural logarithm is characterised by a variational problem over a class of one-dimensional lattice logarithms.

The Poincaré-Cartan forms of one-dimensional variational integrals

2020 ◽  
Vol 70 (6) ◽  
pp. 1381-1412
Author(s):  
Veronika Chrastinová ◽  
Václav Tryhuk
Keyword(s):  
Differential Equations ◽  
Variational Problem ◽  
Ordinary Differential Equations ◽  
Differential Forms ◽  
Point Of View ◽  
Full Generality ◽  
One Dimensional ◽  
Variational Integrals

AbstractFundamental concepts for variational integrals evaluated on the solutions of a system of ordinary differential equations are revised. The variations, stationarity, extremals and especially the Poincaré-Cartan differential forms are relieved of all additional structures and subject to the equivalences and symmetries in the widest possible sense. Theory of the classical Lagrange variational problem eventually appears in full generality. It is presented from the differential forms point of view and does not require any intricate geometry.

A one-dimensional variational problem with gradient constraint

2004 ◽  
Vol 83 (7) ◽  
pp. 747-756
Author(s):  
G. Cardone ◽  
A. Corbo Esposito † ◽  
V. V. Zhikov ‡
Keyword(s):  
Variational Problem ◽  
One Dimensional ◽  
Gradient Constraint ◽  
Dimensional Variational Problem
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