Related papers: Slalom numbers
The paper investigates the problem of performing correlation analysis when the number of observations is very large. In such a case, it is often necessary to combine the random observations to achieve dimensionality reduction of the…
We discuss various forms of the Luxemburg norm in spaces of random vectors with coordinates belonging to the classical Orlicz spaces of exponential type. We prove equivalent relations between some kinds of these forms. We also show when the…
The success of machine learning has resulted from its structured representation of data. Similar data have close internal representations as compressed codes for classification or emerged labels for clustering. We observe that the frequency…
Our goal is to study the pseudo-intersection and tower numbers on uncountable regular cardinals, whether these two cardinal characteristics are necessarily equal, and related problems on the existence of gaps. First, we prove that either…
This paper surveys some combinatorial aspects of Smith normal form, and more generally, diagonal form. The discussion includes general algebraic properties and interpretations of Smith normal form, critical groups of graphs, and Smith…
We discuss isoperimetric inequalities for convex sets. These include the classical isoperimetric inequality and that of Brunn-Minkowski, Blaschke-Santalo, Busemann-Petty and their various extensions. We show that many such inequalities…
We introduce the split principles and show that they bear tight connections to large cardinal properties such as inaccessibility, weak compactness, subtlety, almost ineffability and ineffability, as well as classical combinatorial objects…
We introduce characteristic numbers of a finite commutative unital $\mathbb{C}$-algebra, which are numerical invariants arising from algebraic intersection theory. We characterize Gorenstein and local complete intersection algebras in terms…
The main results of this paper interpret mixed volumes of lattice polytopes as mixed multiplicities of ideals and mixed multiplicities of ideals as Samuel's multiplicities. In particular, we can give a purely algebraic proof of Bernstein's…
This paper is intended to survey the basics of localization and anti-localization cardinals on the reals, and its interplay with notions and cardinal characteristics related to measure and category.
An algorithm for resolution of singularities in characteristic zero is described. It is expressed in terms of multi-ideals, that essentially are defined as a finite sequence of pairs, each one consiting of a sheaf of ideals and a positive…
The present work provides an original framework for random matrix analysis based on revisiting the concentration of measure theory from a probabilistic point of view. By providing various notions of vector concentration ($q$-exponential,…
We investigate some topological properties of random geometric complexes and random geometric graphs on Riemannian manifolds in the thermodynamic limit. In particular, for random geometric complexes we prove that the normalized counting…
For a strongly inacessible cardinal $\kappa$, we investigate the relationships between the following ideals: - the ideal of meager sets in the ${<}\kappa$-box product topology - the ideal of "null" sets in the sense of [Sh:1004]…
We derive modular parametrizations for certain infinite series whose summands involve central binomial coefficients and higher-order harmonic numbers. When the rates of convergence are certain rational numbers, modularity allows us to…
We use bounds of character sums and some combinatorial arguments to show the abundance of very smooth numbers which also have very few non-zero binary digits.
Results on matrix canonical forms are used to give a complete description of the higher rank numerical range of matrices arising from the study of quantum error correction. It is shown that the set can be obtained as the intersection of…
How many odd numbers are there? How many even numbers? From Galileo to Cantor, the suggestion was that there are the same number of odd, even and natural numbers, because all three sets can be mapped in one-one fashion to each other. This…
Using the language of Riordan arrays, we look at two related iterative processes on matrices and determine which matrices are invariant under these processes. In a special case, the invariant sequences that arise are conjectured to have…
For each countable ordinal $\alpha \ge 2$, the ideals $\mathsf{conv}_\alpha$ were introduced in ``Critical ideals for countable compact spaces'' (to appear in Fund. Math., see also arXiv:2503.12571) to characterize compact countable spaces…