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A morph is a continuous transformation between two representations of a graph. We consider the problem of morphing between contact representations of a plane graph. In an F-contact representation of a plane graph G, vertices are realized by internally disjoint elements from a family F of connected geometric objects. Two such elements touch if and only if their corresponding vertices are adjacent. These touchings also induce the same embedding as in G. In a morph between two F-contact representations we insist that at each time step (continuously throughout the morph) we have an F-contact representation. |
Heron of Alexandria is a well-known author in the field of mathematics and engineering, but his work is of great interest for understanding ancient construction problems related to architecture. The formulas translated and commented on by Heiberg (1914a, b) are analyzed through geometric diagrams and applied to famous Roman domed architecture. |
We prove limit laws for infinite horizon planar periodic Lorentz gases when, as time n tends to infinity, the scatterer size ρ may also tend to zero simultaneously at a sufficiently slow pace. In particular we obtain a non-standard Central Limit Theorem as well as a Local Limit Theorem for the displacement function. To the best of our knowledge, these are the first results on an intermediate case between the two well-studied regimes with superdiffusive nlogn−−−−−√ scaling (i) for fixed infinite horizon configurations—letting first n→∞ and then ρ→0—studied e.g. by Szász and Varjú (J Stat Phys 129(1):59–80, 2007) and (ii) Boltzmann–Grad type situations—letting first ρ→0
and then n→∞—studied by Marklof and Tóth (Commun Math Phys 347(3):933–981, 2016) . |
For a d-dimensional random vector X, let pn,X(θ) be the probability that the convex hull of n independent copies of X contains a given point θ. We provide several sharp inequalities regarding pn,X(θ) and NX(θ) denoting the smallest n for which pn,X(θ)≥1/2. As a main result, we derive the totally general inequality 1/2≤αX(θ)NX(θ)≤3d+1
, where αX(θ) (a.k.a. the Tukey depth) is the minimum probability that X is in a fixed closed halfspace containing the point θ. We also show several applications of our general results: one is a moment-based bound on NX(E[X])
, which is an important quantity in randomized approaches to cubature construction or measure reduction problem. |
We study the eigenvalues and eigenfunctions of one-dimensional weighted fractal Laplacians. These Laplacians are defined by self-similar measures with overlaps. We first prove the existence of eigenvalues and eigenfunctions. We then set up a framework for one-dimensional measures to discretize the equation defining the eigenvalues and eigenfunctions, and obtain numerical approximations of the eigenvalue and eigenfunction by using the finite element method. Finally, we show that the numerical eigenvalues and eigenfunctions converge to the actual ones and obtain the rate of convergence. |
We introduce an analogue to the amalgamation of metric spaces into the setting of Lorentzian pre-length spaces. This provides a very general process of constructing new spaces out of old ones. The main application in this work is an analogue of the gluing theorem of Reshetnyak for CAT(k) spaces, which roughly states that gluing is compatible with upper curvature bounds. Due to the absence of a notion of spacelike distance in Lorentzian pre-length spaces we can only formulate the theorem in terms of (strongly causal) spacetimes viewed as Lorentzian length spaces. |
We construct optimal flat functions in Carleman–Roumieu ultraholomorphic classes associated to general strongly nonquasianalytic weight sequences, and defined on sectors of suitably restricted opening. A general procedure is presented in order to obtain linear continuous extension operators, right inverses of the Borel map, for the case of regular weight sequences in the sense of Dyn’kin. Finally, we discuss some examples (including the well-known q-Gevrey case) where such optimal flat functions can be obtained in a more explicit way. |
We consider a two-dimensional determinantal point process arising in the random normal matrix model and which is a two-parameter generalization of the complex Ginibre point process. In this paper, we prove that the probability that no points lie on any number of annuli centered at 0 satisfies large n asymptotics of the form... |
Cluster algebras were introduced by Fomin and Zelevinsky as a tool in the study of Lusztig’s dual canonical bases. Since their inception they have found application in a variety of different areas in mathematics, nevertheless a fundamental problem in the theory remains constructing bases with “good” properties. |
The super-Jack polynomials, introduced by Kerov, Okounkov and Olshanski, are polynomials in n+m
variables, which reduce to the Jack polynomials when n=0 or m=0 and provide joint eigenfunctions of the quantum integrals of the deformed trigonometric Calogero–Moser–Sutherland system. We prove that the super-Jack polynomials are orthogonal with respect to a bilinear form of the form (p,q)↦(Lpq)(0), with Lp
quantum integrals of the deformed rational Calogero–Moser–Sutherland system. In addition, we provide a new proof of the Lassalle–Nekrasov correspondence between deformed trigonometric and rational harmonic Calogero–Moser–Sutherland systems and infer orthogonality of super-Hermite polynomials, which provide joint eigenfunctions of the latter system. |
