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Register a new userCalculus of variations on locally finite graphs
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Let $G=(V,E)$ be a locally finite graph. Firstly, using calculus of variations, including a direct method of variation and the mountain-pass theory, we get sequences of solutions to several local equations on $G$ (the Schrodinger equation, the mean field equation, and the Yamabe equation). Secondly, we derive uniform estimates for those local solution sequences. Finally, we obtain global solutions by extracting convergent sequence of solutions. Our method can be described as a variational method from local to global.
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We provide a gentle introduction, aimed at non-experts, to Borel combinatorics that studies definable graphs on topological spaces. This is an emerging field on the borderline between combinatorics and descriptive set theory with deep connections to many other areas. After giving some background material, we present in careful detail some basic tools and results on the existence of Borel satisfying assignments: Bore
Divergence-free symmetric tensors seem ubiquitous in Mathematical Physics. We show that this structure occurs in models that are described by the so-called second variational principle, where the argument of the Lagrangian is a closed differential form. Divergence-free tensors are nothing but the second form of the Euler--Lagrange equations. The symmetry is associated with the invariance of the Lagrangian density upon the action of some orthogonal group.
For an infinite penny graph, we study the finite-dimensional property for the space of harmonic functions, or ancient solutions of the heat equation, of polynomial growth. We prove the asymptotically sharp dimensional estimate for the above spaces.
We develop an algorithmic framework for graph colouring that reduces the problem to verifying a local probabilistic property of the independent sets. With this we give, for any fixed $kge 3$ and $varepsilon>0$, a randomised polynomial-time algorithm for colouring graphs of maximum degree $Delta$ in which each vertex is contained in at most $t$ copies of a cycle of length $k$, where $1/2le tle Delta^frac{2varepsilon}{1+2varepsilon}/(logDelta)^2$, with $lfloor(1+varepsilon)Delta/log(Delta/sqrt t)rfloor$ colours. This generalises and improves upon several notable results including those of Kim (1995) and Alon, Krivelevich and Sudakov (1999), and more recent ones of Molloy (2019) and Achlioptas, Iliopoulos and Sinclair (2019). This bound on the chromatic number is tight up to an asymptotic factor $2$ and it coincides with a famous algorithmic barrier to colouring random graphs.
The calculus of variations is a field of mathematical analysis born in 1687 with Newtons problem of minimal resistance, which is concerned with the maxima or minima of integral functionals. Finding the solution of such problems leads to solving the associated Euler-Lagrange equations. The subject has found many applications over the centuries, e.g., in physics, economics, engineering and biology. Up to this moment, however, the theory of the calculus of variations has been confined to Newtons approach to calculus. As in many applications negative values of admissible functions are not physically plausible, we propose here to develop an alternative calculus of variations based on the non-Newtonian approach first introduced by Grossman and Katz in the period between 1967 and 1970, which provides a calculus defined, from the very beginning, for positive real numbers only, and it is based on a (non-Newtonian) derivative that permits one to compare relative changes between a dependent positive variable and an independent variable that is also positive. In this way, the non-Newtonian calculus of variations we introduce here provides a natural framework for problems involving functions with positive images. Our main result is a first-order optimality condition of Euler-Lagrange type. The new calculus of variations complements the standard one in a nontrivial/multiplicative way, guaranteeing that the solution remains in the physically admissible positive range. An illustrative example is given.
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