Related papers: Generalized Lorenz dominance orders
The goal of this paper is to describe an elementary combinatorial heuristic that predicts Hardy and Littlewood's extended Goldbach's conjecture. We examine common features of other heuristics in additive prime number theory, such as…
Grokking refers to a delayed generalization following overfitting when optimizing artificial neural networks with gradient-based methods. In this work, we demonstrate that grokking can be induced by regularization, either explicit or…
The problem of when the vanishing of a (generalized) Schur complement of a block matrix (corresponding to the leading principal subblock) implies that the other (generalized) Schur complement (corresponding to the trailing principal…
Algorithmic generalization in machine learning refers to the ability to learn the underlying algorithm that generates data in a way that generalizes out-of-distribution. This is generally considered a difficult task for most machine…
One of the most important features observed in real networks is that, as a network's topology evolves so does the network's ability to perform various complex tasks. To explain this, it has also been observed that as a network grows certain…
A sequence of vertices in a graph is called a legal dominating sequence if every vertex in the sequence dominates at least one vertex not dominated by those that precede it, and at the end all vertices of the graph are dominated. The Grundy…
We prove a moment majorization principle for matrix-valued functions with domain $\{-1,1\}^{m}$, $m\in\mathbb{N}$. The principle is an inequality between higher-order moments of a non-commutative multilinear polynomial with different random…
We extend Elitzur's theorem to systems with symmetries intermediate between global and local. In general, our theorem formalizes the idea of {\it dimensional reduction}. We apply the results of this generalization to many systems that are…
Quasi-trees generalize trees in that the unique "path" between two nodes may be infinite and have any countable order type. They are used to define the rank-width of a countable graph in such a way that it is equal to the least upper-bound…
Given any finite subset $A$ of order $n$ of a distributive lattice and $k\in\{1,...,n\}$, there is a natural extension of the median operation to $n$ variables which generalizes the notion of the $k$th smallest element of $A$. By applying…
It is well known that the Lorenz system has $Z_2$-symmetry. Using introducted in math.DS/0105147 topological covering-coloring a new representation for the Lorenz system is obtained. Deleting coloring leads to the factorized Lorenz system…
In this paper we develop a general ergodic approach which reveals the underpinnings of the effect of arithmetic operations involving normal and deterministic numbers. This allows us to recast in new light and amplify the result of Rauzy,…
Given a graph $G$ consider a procedure of building a dominating set $D$ in $G$ by adding vertices to $D$ one at a time in such a way that whenever vertex $x$ is added to $D$ there exists a vertex $y\in N_G[x]$ that becomes dominated only…
Suppose that Y is a cyclic cover of projective space branched over a hyperplane arrangement D, and that U is the complement of the ramification locus in Y. The first theorem implies that the Beilinson-Hodge conjecture holds for U if certain…
The theory of majorizing measures, extensively developed by Fernique, Talagrand and many others, provides one of the most general frameworks for controlling the behavior of stochastic processes. In particular, it can be applied to derive…
Understanding and accurately following instructions is critical for large language models (LLMs) to be effective across diverse tasks. In this work, we rigorously examine the key factors that enable models to generalize to unseen…
In this paper a generalization of Urysohn's metrization theorem is given for higher cardinals. Namely, it is shown that a topological space with a basis of cardinality at most $|\omega_\mu|$ or smaller is $\omega_\mu$-metrizable if and only…
We prove the theorems which are equivalent to the Roland's results such that a new form of them allows to consider some generalizations. In particular, we give generators of primes more than a fixed prime.
We study regularization in the context of small sample-size learning with over-parameterized neural networks. Specifically, we shift focus from architectural properties, such as norms on the network weights, to properties of the internal…
In the long paper "Family Blowup formula, Admissible Graphs and the Enumeration of Singular Curves (I)" (appearing in JDG), the author solved the enumeration problem of nodal (or general singular) curve counting on algebraic surfaces by…