Related papers: The Interplay between Additive and Multiplicative …
There are different meanings of foundation of mathematics: philosophical, logical, and mathematical. Here foundations are considered as a theory that provides means (concepts, structures, methods etc.) for the development of whole…
Bishop's informal set theory is briefly discussed and compared to Lawvere's Elementary Theory of the Category of Sets (ETCS). We then present a constructive and predicative version of ETCS, whose standard model is based on the constructive…
We develop the theory of arrangements of spheres. Consider a finite collection of codimension-$1$ subspheres in a positive-dimensional sphere. There are two posets associated with this collection: the poset of faces and the poset of…
This is a survey of old and new problems and results in additive number theory.
Using ideas from synthetic topology, a new approach to descriptive set theory is suggested. Synthetic descriptive set theory promises elegant explanations for various phenomena in both classic and effective descriptive set theory.…
The theory of condensed mathematics by Dustin Clausen and Peter Scholze claims that topological spaces should be replaced by the definition of condensed sets. The main purpose of this paper is to investigate in which way the theory of…
To enable the study of open sets in computational approaches to mathematics, lots of extra data and structure on these sets is assumed. For both foundational and mathematical reasons, it is then a natural question, and the subject of this…
We begin with a context more general than set theory. The basic ingredients are essentially the object and functor primitives of category theory, and the logic is weak, requiring neither the Law of Excluded Middle nor quantification. Inside…
The purpose of the article is to provide partial proofs for two conjectures given by Witte and Forrester in "Moments of the Gaussian $\beta$ Ensembles and the large $N$ expansion of the densities" with the use of the topological recursion…
The assignment of classifying spectra to saturated fusion systems was suggested by Linckelmann and Webb and has been carried out by Broto, Levi and Oliver. A more rigid (but equivalent) construction of the classifying spectra is given in…
This paper investigates the behavior of statistical ensembles under iteration map induced by discrete integrable Hamiltonian systems in deterministic case and stochastic case, addressing the problem from two perspectives: the Law of Large…
This essay advocates the view that any problem that has a meaningful empirical content, can be formulated in constructive, more definitely, finite terms. We consider combinatorial models of dynamical systems and approaches to statistical…
Richard Stanley played a crucial role, through his work and his students, in the development of the relatively new area known as combinatorial representation theory. In the early stages, he has the merit to have pointed out to…
A commutative order in a central simple algebra over a number field is said to be selective if it embeds in some, but not all, the maximal orders in the algebra. We completely characterize selective orders in central division algebras, of…
Motivated by a problem in ergodic Ramsey theory, Furstenberg and Katznelson introduced the notion of strong stationarity, showing that certain recurrence properties hold for arbitrary measure preserving systems if they are valid for…
This is the first paper that provides a systematic treatment of the $r$-dimensional PTE problem in additive number theory, abbreviated by PTE$_r$, through its connection with combinatorial design theory, the branch of combinatorial…
We introduce the notion of a hyper-atom and prove a basic property of this object. This new method allows to improve several results in the classical critical pair theory including its cornerstone: the Kemperman Structure Theorem.
With the inflation of the data, clustering analysis, as a branch of unsupervised learning, lacks unified understanding and application of its mathematical law. Based on the view of fixed point, this paper restates the model-based clustering…
A factorisation problem in the symmetric group is central if conjugate permutations always have the same number of factorisations. We give the first fully combinatorial proof of the centrality of transitive star factorisations that is valid…
The paper contains an exposition of part of topology using partitions of unity. The main idea is to create variants of the Tietze Extension Theorem and use them to derive classical theorems. This idea leads to a new result generalizing…