Related papers: Ranks on the Baire class $\xi$ functions
The rank of a finite algebraic structure with a single binary operation is the minimum number of elements needed to express every other element under the closure of the operation. In the case of groups, the previous best algorithm for…
In problems involving approximation, completion, denoising, dimension reduction, estimation, interpolation, modeling, order reduction, regression, etc, we argue that the near-universal practice of assuming that a function, matrix, or tensor…
Certain subclasses of $B_1(K)$, the Baire-1 functions on a compact metric space $K$, are defined and characterized. Some applications to Banach spaces are given.
This paper introduces the ring of all real valued Baire one functions, denoted by $B_1(X)$ and also the ring of all real valued bounded Baire one functions, denoted by $B_1^*(X)$. Though the resemblance between $C(X)$ and $B_1(X)$ is the…
We consider the problem of counting the number of possible sets of rankings (called ranking patterns) generated by unfolding models of codimension one. We express the ranking patterns as slices of the braid arrangement and show that all…
Generalizing the continuous rank function of Barja-Pardini-Stoppino, in this paper we consider cohomological rank functions of $\mathbb Q$-twisted (complexes of) coherent sheaves on abelian varieties. They satisfy a natural transformation…
In this paper we study partitions whose successive ranks belong to a given set. We enumerate such partitions while keeping track of the number of parts, the largest part, the side of the Durfee square, and the height of the Durfee…
$\Pi_1^1$-ranks are a natural tool for studying coanalytic sets in descriptive set theory. In the book "Classical descriptive set theory", Kechris provided a technique to build $\Pi_1^1$-ranks using derivatives. In this note we will prove a…
The permanent vs. determinant problem is one of the most important problems in theoretical computer science, and is the main target of geometric complexity theory proposed by Mulmuley and Sohoni. The current best lower bound for the…
Rankings are ubiquitous across many applications, from search engines to hiring committees. In practice, many rankings are derived from the output of predictors. However, when predictors trained for classification tasks have intrinsic…
An action of a compact Lie group is called equivariantly formal, if the Leray--Serre spectral sequence of its Borel fibration degenerates at the E_2-term. This term is as prominent as it is restrictive. In this article, also motivated by…
For any given Cuntz semigroup, we introduce a function associated to it, called the ``Rank Ratio Function". This function ensures global control over the oscillation of the rank of any pair of elements in the Cuntz semigroup and, also,…
Convex rank tests are partitions of the symmetric group which have desirable geometric properties. The statistical tests defined by such partitions involve counting all permutations in the equivalence classes. Each class consists of the…
Some aspects of the development of physics and the mathematics set one think about relation between complex numbers and reality around us. If number to spot as the relation of two quantities, from the fact of existence of complex numbers…
This is a paper that aims to interpret the cardinality of a set in terms of Baire Category, i.e. how many closed nowhere dense sets can be deleted from a set before the set itself becomes negligible. . To do this natural tree-theoretic…
The analytic rank of a tensor, first defined by Gowers and Wolf in the context of higher-order Fourier analysis, is defined to be the logarithm of the bias of the tensor. We prove that it is a subadditive measure of rank: that is, the…
We consider the discrete assignment problem in which agents express ordinal preferences over objects and these objects are allocated to the agents in a fair manner. We use the stochastic dominance relation between fractional or randomized…
In this paper we study the complexity of the problems: given a loop, described by linear constraints over a finite set of variables, is there a linear or lexicographical-linear ranking function for this loop? While existence of such…
A rational function is the ratio of two complex polynomials in one variable without common roots. Its degree is the maximum of the degrees of the numerator and the denominator. Rational functions belong to the same class if one turns into…
The linear continuity of a function defined on a vector space means that its restriction on every affine line is continuous. For functions defined on $\mathbb R^m$ this notion is near to the separate continuity for which it is required only…