Related papers: Continuous limits of discrete perimeters
Take a set of balls in $\mathbb R^d$. We find a subset of pairwise disjoint balls whose combined perimeter controls the perimeter of the union of the original balls. This can be seen as a boundary version of the Vitali covering lemma. We…
We define convexity canonically in the setting of monoids. We show that many classical results from convex analysis hold for functions defined on such groups and semigroups, rather than only on vector spaces. Some examples and…
In this paper we study a continuity of the "values" of modular functions at the real quadratic numbers which are defined in terms of their cycle integrals along the associated closed geodesics. Our main theorem reveals a more finer…
We give three proofs of the fact that a smoothly bounded, convex domain in R^n has smooth defining functions whose Hessians are non-negative definite in a neighborhood of the boundary of the domain.
We show how our recent results on compositions of d.c. functions (and mappings) imply positive results on extensions of d.c. functions (and mappings). Examples answering two natural relevant questions are presented. Two further theorems,…
We investigate properties of the set of discrete Morse functions on a simplicial complex as defined by Forman. It is not difficult to see that the pairings of discrete Morse functions of a finite simplicial complex again form a simplicial…
We investigate the variational structure of discrete Laplace-type equations that are motivated by discrete integrable quad-equations. In particular, we explain why the reality conditions we consider should be all that are reasonable, and we…
The class of ordinary linear constant coefficient differential equations is naturally embedded into a wider class by associating differential equations to algebraic curves.
We study the properties of the set where a generalized function of bounded variation has infinite approximate limit, highlighting in this way the main geometric difference with functions of bounded variation. To this aim we prove a new…
We study a one-dimensional ordinary differential equation modelling optical conveyor belts, showing in particular cases of physical interest that periodic solutions exist. Moreover, under rather general assumptions it is proved that the set…
Subaddivity type matrix inequalities for concave funcions and symetric norms are given.
In this paper, we propose the Fourier Discrepancy Function, a new discrepancy to compare discrete probability measures. We show that this discrepancy takes into account the geometry of the underlying space. We prove that the Fourier…
We study the boundary behaviour of a variant of the Fridman's invariant function (defined in terms of the Bergman metric) on Levi corank one domains, strongly pseudoconvex domains, smoothly bounded convex domains in $ \mathbb{C}^n $ and…
In \cite{X-Z DCS1}, we introduced discrete conformal structures on surfaces with boundary via an axiomatic framework, and provided a classification of such discrete conformal structures. The present work focuses on the rigidity and…
We consider ruled surfaces with finite multiplicity. We study behaviors of the striction curves and the singularities of the ruled surfaces. We also give geometric meanings of invariants related to the ruled surfaces.
We solve two continuous extremal problems on the classes of monotone functions: in the first problem we find extremal values for a line integral of a coordinate-wise monotone function of two variables from a rearrange\-ment-invariant class…
D-finite functions and P-recursive sequences are defined in terms of linear differential and recurrence equations with polynomial coefficients. In this paper, we introduce a class of numbers closely related to D-finite functions and…
We pose some open problems related to boundedness of real-valued functions on balleans and coarse spaces. Also we prove that the Bergman property of groups is a coarse invariant. A special attention is payed to balleans on groups.
The work consists of solutions of metric problems for convex and finite subsets of geodesic spaces.
A sufficient condition for $\bar{\partial}$ to have closed range is given for pseudoconvex, possibly unbounded domains in $\mathbb{C}^n$.