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Q: Uniqueness of the Legendre-Fenchel Transformation

Zeno TecchiolliWhat is the relation between two different functions, say $g(x)$ and $f(x)$, which have the same Legendre–Fenchel transformation $h(s)$? \begin{equation} h(s) = \sup_{x\in I}\{sx - f(x)\} \quad \quad x \in I \end{equation} \begin{equation} h(s) = \sup_{x\in I'}\{sx - g(x)\} \quad \quad ...

In mathematics and physics, the Legendre transformation, named after Adrien-Marie Legendre, is an involutive transformation on the real-valued convex functions of one real variable. In physical problems, it is used to convert functions of one quantity (such as position, pressure, or temperature) into functions of the conjugate quantity (momentum, volume, and entropy, respectively). In this way, it is commonly used in classical mechanics to derive the Hamiltonian formalism out of the Lagrangian formalism and in thermodynamics to derive the thermodynamic potentials, as well as in the solution of...
 
 
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6:41 PM
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Q: Homomorphism of formal groups induced by nonzero isogeny

bellowLet $K$ be a field of characteristic $p > 0$, let $E_1/ K$ and $ E_2/K$ be elliptic curves, and let $\phi : E_1 \mapsto E_2$ be a nonzero isogeny defined over $K$. Further, let $f: \hat{E_1} \mapsto \hat{E_2} $ be the homomorphism of formal groups induced by $\phi$. My question: How does an isoge...

In mathematics, in particular, in algebraic geometry, an isogeny is a morphism of algebraic groups (a.k.a. group varieties) that is surjective and has a finite kernel. If the groups are abelian varieties, then any morphism f : A → B of the underlying algebraic varieties which is surjective with finite fibres is automatically an isogeny, provided that f(1A) = 1B. Such an isogeny f then provides a group homomorphism between the groups of k-valued points of A and B, for any field k over which f is defined. The terms "isogeny" and "isogenous" come from the Greek word ισογενη-ς, meaning "equal in kind...
 

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