Related papers: A Note on the Size-Sensitive Packing Lemma
We upgrade [1] to a complete proof of the conjecture NP = PSPACE. [1]: L. Gordeev, E. H. Haeusler, Proof Compression and NP Versus PSPACE, Studia Logica (107) (1): 55-83 (2019)
We report a recent developement on the theory of upper conical densities. More precicely, we look at what can be said in this respect for other measures than just the Hausdorff measure. We illustrate the methods involved by proving a result…
This note serves to provide additional details for the proof of Lemma 3.6 in our paper [Liu, Zhang and Zhang, Comm. Math. Sci., 3(2005), pp.201-218]. Moreover, we will also present an alternative, yet simpler, proof based on arguments in…
The Kakeya conjecture is generally formulated as one the following statements: every compact/Borel/arbitrary subset of ${\mathbb R}^n$ that contains a (unit) line segment in every direction has Hausdorff dimension $n$; or, sometimes, that…
Datasets containing both categorical and continuous variables are frequently encountered in many areas, and with the rapid development of modern measurement technologies, the dimensions of these variables can be very high. Despite the…
Measure homology is a variation of singular homology designed by Thurston in his discussion of simplicial volume. Zastrow and Hansen showed independently that singular homology (with real coefficients) and measure homology coincide…
We resolve a problem posed by Mattila, Serapioni and Serra Cassano concerning the role of density assumptions in the characterization of rectifiable sets of low codimension in Heisenberg groups. Specifically, we prove that the positive…
Large language models (LLMs) have rapidly advanced and are increasingly capable of tackling complex scientific problems, including those in physics. Despite this progress, current LLMs often fail to emulate the concise, principle-based…
The basic disentanglement theorem established by the present authors states that estimates on a weighted geometric mean over (convex) families of functions can be disentangled into quantitatively linked estimates on each family separately.…
Human logic has gradually shifted from intuition-driven inference to rigorous formal systems. Motivated by recent advances in large language models (LLMs), we explore whether LLMs exhibit a similar evolution in the underlying logical…
We track the motion of a horizontally vibrated amorphous assembly of bidisperse hard disks, for densities ranging across the jamming transition. We derive on very general grounds a bound on the dynamical susceptibility in terms of the…
A recent result on size functions is extended to higher homology modules: the persistent homology based on a multidimensional measuring function is reduced to a 1-dimensional one. This leads to a stable distance for multidimensional…
Small additive ensembles of symbolic rules offer interpretable prediction models. Traditionally, these ensembles use rule conditions based on conjunctions of simple threshold propositions $x \geq t$ on a single input variable $x$ and…
Large language Models (LLMs) are highly sensitive to variations in prompt formulation, which can significantly impact their ability to generate accurate responses. In this paper, we introduce a new task, Prompt Sensitivity Prediction, and a…
We give a revised version of Schmidt's treatment of forms in many variables, which allows us to prove a Hasse principle under more lenient conditions on the number of variables than what had previously been thought possible with these…
We give a short and self-contained proof of Levi's Extension Lemma for pseudoline arrangements.
Tsetlin machine (TM) is a logic-based machine learning approach with the crucial advantages of being transparent and hardware-friendly. While TMs match or surpass deep learning accuracy for an increasing number of applications, large clause…
The central result of this paper is the small-is-very-small principle for restricted sequential theories. The principle says roughly that whenever the given theory shows that a property has a small witness, i.e. a witness in every definable…
In this paper I explore a nonstandard formulation of Hausdorff dimension. By considering an adapted form of the counting measure formulation of Lebesgue measure, I prove a nonstandard version of Frostman's lemma and show that Hausdorff…
In the constraint satisfaction problem (CSP) corresponding to a constraint language (i.e., a set of relations) $\Gamma$, the goal is to find an assignment of values to variables so that a given set of constraints specified by relations from…