Related papers: A note on the Kronecker--Weyl equidistribution the…

We prove an abstract index formula about Sinnott's symbol between two different lattices. We also develop the theory of the universal distribution and predistribution in a double complex point of view. The theory of spectral sequence is…

A Kleinian manifold Y is a quotient of a rank-one symmetric space of non-compact type by a convex-cocompact discrete group of isometries. We describe the spectral decomposition of the space of square integrable sections of locally…

We show that certain statements related to the Fourier-Walsh expansion of functions with respect to a biased measure on the discrete cube can be deduced from the respective results for the uniform measure by a simple reduction. In…

This paper surveys recent developments in the sampling discretization of integral and uniform norms for functions in general finite-dimensional spaces. These results generalize the classical Marcinkiewicz-Zygmund inequalities for…

We establish two surprising types of Weyl's laws for some compact $\mathrm{RCD}(K, N)$/Ricci limit spaces. The first type could have power growth of any order (bigger than one). The other one has an order corrected by logarithm similar to…

In this paper, we finish the basic development of the discrete octonionic analysis by presenting a Weyl calculus-based approach to bounded domains in $\mathbb{R}^{8}$. In particular, we explicitly prove the discrete Stokes formula for a…

The geometry of divisors on algebraic curves has been studied extensively over the years. The foundational results of this Brill-Noether theory imply that on a general curve, the spaces parametrizing linear series (of fixed degree and…

We study the continuous map induced on spectra by a separable extension of tensor-triangulated categories. We determine the image of this map and relate the cardinality of its fibers to the degree of the extension. We then prove a weak form…

We prove asymptotic equidistribution results for pieces of large closed horospheres in cofinite hyperbolic manifolds of arbitrary dimension. This extends earlier results by Hejhal and Str\"ombergsson in dimension 2. Our proofs use spectral…

We extend the Nielsen theory of coincidence sets to equalizer sets, the points where a given set of (more than 2) mappings agree. On manifolds, this theory is interesting only for maps between spaces of different dimension, and our results…

For a complete noncompact Riemannian manifold with nonnegative Ricci curvature, we show that bounded biharmonic functions are constant and the space consists of biharmonic functions with polynomial growth of a fixed rate is finite…

For distributions, we build a theory of higher order pointwise differentiability comprising, for order zero, {\L}ojasiewicz's notion of point value. Results include Borel regularity of differentials, higher order rectifiability of the…

Starting from the hypothesis that both physics, in particular space-time and the physical vacuum, and the corresponding mathematics are discrete on the Planck scale we develop a certain framework in form of a '{\it cellular network}'…

We study the simultaneous reductions at several supersingular primes of elliptic curves with complex multiplication. We show -- under additional congruence assumptions on the CM order -- that the reductions are surjective (and even become…

We examine some kinds of discrete symmetries which are dynamically preserved, using the (generalized) Gowdy models of the first kind.

Copulas have now become ubiquitous statistical tools for describing, analysing and modelling dependence between random variables. Sklar's theorem, "the fundamental theorem of copulas", makes a clear distinction between the continuous case…

We propose the notion of a supercategory as an alternative approach to supermathematics. We show that this setting is rich to carry out many of the basic constructions of supermathematics. We also prove generalizations of a number of…

We derive point-wise and integral rigidity/gap results for a closed manifold with harmonic Weyl curvature in any dimension. In particular, there is a generalization of Tachibana's theorem for non-negative curvature operator. The key…

We study the convergence of solutions of the discrete nonlinear Klein-Gordon equation on an infinite lattice in the continuum limit, using recent tools developed in the context of nonlinear discrete dispersive equations. Our approach relies…

In the conformal class of Euclidean space, we give some volume comparison theorems with help of Q-curvature. Meanwhile, for compact four dimensional manifolds with non-negative scalar curvature, we give a volume rigidity theorem with…