Related papers: Heat content determines planar triangles

We establish sharp higher-order heat estimates with complete bound on the noncommutative tori \(\mathbb{T}_{\theta}^{n}\) and show the optimality in the small-time order. As an application in polynomial semilinear heat equations on…

We formulate three-dimensional equations for the finite temperature in-matter three-body problem. Our approach takes into account the full infinite series for the effective pair-interaction kernel, so that all possible two-body…

A fixed set of vertices in the plane may have multiple planar straight-line triangulations in which the degree of each vertex is the same. As such, the degree information does not completely determine the triangulation. We show that even if…

In this article we prove some theorems related to the triplets of triangles, homological two by two. These theorems will be used later to build triplets of triangles two by two tri-homological.

The proof uses the property that the vertices of a triangulated planar graph can be four coloured if the triangles can have a +1 or -1 orientation so that the sum of the triangle orientations around each vertex is a multiple of 3. Such…

We formulate conjectures regarding percolation on planar triangulations suggested by assuming (quasi) invariance under coarse conformal uniformization.

We consider the $3$-state Potts generating function $T(\nu,w)$ of planar triangulations; that is, the bivariate series that counts planar triangulations with vertices coloured in $3$ colours, weighted by their size (number of vertices,…

We obtain monotonicity and convexity results for the heat content of domains in Riemannian manifolds and in Euclidean space subject to various initial temperature conditions. We introduce the notion of a strictly decreasing temperature set,…

We find a simple exact model of radiating stellar collapse, with a shear-free and non-accelerating interior matched to a Vaidya exterior. The heat flux is subject to causal thermodynamics, leading to self-consistent determination of the…

We address the enumeration of q-coloured planar maps counted bythe number of edges and the number of monochromatic edges. We prove that the associated generating function is differentially algebraic,that is, satisfies a non-trivial…

We show that it is possible to decide whether a given planar diagram correctly depicts the spatial figure consisting of a planar quadrangle together with its shadow in another plane.

We consider the semilinear heat equation, to which we add a nonlinear gradient term, with a critical power. We construct a solution which blows up in finite time. We also give a sharp description of its blow-up profile. The proof relies on…

This article assesses the current state of understanding of coronal heating, outlines the key elements of a comprehensive strategy for solving the problem, and warns of obstacles that must be overcome along the way.

The heat trace of a planar polygon contains corner terms depending only on the opening angles, while the heat trace of a smooth planar domain contains curvature terms along the boundary. We show that, for curvilinear polygons, these two…

We have found some patterns in some triangles.

We investigate heat kernel estimates of the form $p_{t}(x, x)\geq c_{x}t^{-\alpha},$ for large enough $t,$ where $\alpha$ and $c_{x}$ are positive reals and $c_{x}$ may depend on $x,$ on manifolds having at least one end.

The heat capacity of solids at intermediate-to-high temperatures is of fundamental importance to several fields ranging from geology to material science. It depends on a variety of factors, with anharmonicity and, ultimately, melting…

Two species of particles in a binary granular system typically do not have the same mean kinetic energy, in contrast to the equipartition of energy required in equilibrium. We investigate the role of the heating mechanism in determining the…

We define the notions of outer medians and outer median triangles. We show that outer median triangles enjoy similar properties to that of the median triangle.

We study the heat content asymptotics with either Dirichlet or Robin boundary conditions where the initial temperature exhibits radial blowup near the boundary. We show that there is a complete small-time asymptotic expansion and give…