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Helly's theorem is a classical result concerning the intersection patterns of convex sets in $\mathbb{R}^d$. Two important generalizations are the colorful version and the fractional version. Recently, B\'{a}r\'{a}ny et al. combined the…
Let $X$ be a smooth projective variety defined on a finite field $\mathbb{F}_q$. On $X$ there is a special morphism $Fr_X$, which raises coordinates to exponent $q$: $t\mapsto t^q$. The two main results in this paper are: Result 1: If…
We present the foundational theory of condensed sets and basic condensed algebra after having introduced key concepts from category theory and homological algebra. In the later sections, we indicate the relevance of condensed mathematics to…
Reference analysis produces objective Bayesian inference, in the sense that inferential statements depend only on the assumed model and the available data, and the prior distribution used to make an inference is least informative in a…
This article proposes a unified analytical approach leading to a partial resolution of the Erdos-Straus, Sierpinski conjectures, and their generalization. We introduce an equivalent reformulation of these conjectures while constructing two…
The aim of this paper is to establish some metrical coincidence and common fixed point theorems with an arbitrary relation under an implicit contractive condition which is general enough to cover a multitude of well known contraction…
This paper has been withdrawn and replaced by arXiv:1309.5035. In this paper we describe some examples of so called spherical functors between triangulated categories, which generalize the notion of a spherical object. We also give…
We introduce a new version of arithmetic in all finite types which extends the usual versions with primitive notions of extensionality and extensional equality. This new hybrid version allows us to formulate a strong form of extensionality,…
This paper studies fundamental questions concerning category-theoretic models of induction and recursion. We are concerned with the relationship between well-founded and recursive coalgebras for an endofunctor. For monomorphism preserving…
We prove the global triangulation conjecture for families of refined p-adic representations under a mild condition. That is, for a refined family, the associated family of (phi, Gamma)-modules admits a global triangulation on a Zariski open…
A popular scientific contribution should not contradict any established facts and ought to be understandable. I complied with both these requirements and am offering a sufficiently full introduction to probability theory. Furthermore, I…
In this article, we introduce the notion of {\it concentric twin cotorsion pair} on a triangulated category. This notion contains the notions of $t$-structure, cluster tilting subcategory, co-$t$-structure and functorally finite rigid…
We introduce the notion of a prile of one-sided triangulated categories. Roughly speaking, a prile consists of two one-sided triangulated categories having a common full subcategory which inherits a pretriangulated structure from these…
We show that the traditional criterion for a simplex to belong to the Delaunay triangulation of a point set is equivalent to a criterion which is a priori weaker. The argument is quite general; as well as the classical Euclidean case, it…
The Implicit and Inverse Function Theorems are special cases of a general Implicit/Inverse Function Theorem which can be easily derived from either theorem. The theorems can thus be easily deduced from each other via the generalized…
A generalized numerical semigroup is a submonoid of $\mathbb{N}^d$ with finite complement in it. In this work we study some properties of three different classes of generalized numerical semigroups. In particular, we prove that the first…
In this paper we introduce the notion of generalized Lie algebroid and we develop a new formalism necessary to obtain a new solution for the Weistein's Problem. Many applications emphasize the importance and the utility of this new…
We establish a simple generalization for the famous theorem of Morley about trisectors in triangle with a purely synthetic proof using only angle chasing and similar triangles. Furthermore, based on the converse construction, another simple…
This is an exposition of some basic ideas in the realm of Global Inverse Function theorems. We address ourselves mainly to readers who are interested in the applications to Differential Equations. But we do not deal with those applications…
The $F$-triangle is a refined face count for the generalised cluster complex of Fomin and Reading. We compute the $F$-triangle explicitly for all irreducible finite root systems. Furthermore, we use these results to partially prove the…