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We deal with interval parametric systems of linear equations and the goal is to solve such systems, which basically comes down to finding an enclosure for a parametric solution set. Obviously we want this enclosure to be as tight as…
In this paper we consider ill-posed inverse problems, both linear and nonlinear, by a heavy ball method in which a strongly convex regularization function is incorporated to detect the feature of the sought solution. We develop ideas on how…
We introduce a unified framework for studying persistence phenomena in commutative algebra via filtrations of ideals. For a filtration $\mathcal{F} = \{I_i\}_{i \in \mathbb{N}}$, we define $\mathcal{F}$-persistence and $\mathcal{F}$-strong…
We study ideal cotorsion pairs associated to weak proper classes of triangles in extension closed subcategories of triangulated categories. This approach allows us to extend the recent ideal approximations theory developed by Fu, Herzog et…
The classification of isoparametric hypersurfaces with four principal curvatures in the sphere interplays in a deep fashion with commutative algebra, whose abstract and comprehensive nature might obscure a differential geometer's insight…
With infinitely many high-quality data points, infinite computational power, an infinitely large foundation model with a perfect training algorithm and guaranteed zero generalization error on the pretext task, can the model be used for…
The paper proposes and studies new classical, type-free theories of truth and determinateness with unprecedented features. The theories are fully compositional, strongly classical (namely, their internal and external logics are both…
Inspired by recent breakthroughs in predictive modeling, practitioners in both industry and government have turned to machine learning with hopes of operationalizing predictions to drive automated decisions. Unfortunately, many social…
This paper investigates how global decision problems over arithmetically represented domains acquire reflective structure through class-quantification. Arithmetization forces diagonal fixed points whose verification requires reflection…
We develop a general theory for class-sized symmetric systems as a natural extension of symmetric systems with respect to class forcing. In particular, adapting the usual notions of pretameness and tameness for class forcing, we present…
Weak proregularity of an ideal in a commutative ring is a subtle generalization of the noetherian property of the ring. Weak proregularity is of special importance for the study of derived completion, and it occurs quite often in…
Let $\mathcal{N}$ be the $\sigma$-ideal of the null sets of reals. We introduce a new property of forcing notions that enable control of the additivity of $\mathcal{N}$ after finite support iterations. This is applied to answer some open…
An ideal of polynomials is symmetric if it is closed under permutations of variables. We relate general symmetric ideals to the so called Specht ideals generated by all Specht polynomials of a given shape. We show a connection between the…
We find the general behaviour of homogeneous and isotropic cosmological models in some fourth-order theories of gravity. Explicit, exact, general solutions are given for both empty universes and those filled with a perfect fluid. For the…
Based on the work of Shelah, Kellner, and T\u{a}nasie (Fund. Math., 166(1-2):109-136, 2000 and Comment. Math. Univ. Carolin., 60(1):61-95, 2019), and the recent developments in the third author's master's thesis, we develop a general theory…
This is an expository paper about several sophisticated forcing techniques closely related to standard finite support iterations of ccc partial orders. We focus on the four topics of ultrapowers of forcing notions, iterations along…
A new notion of typicality for arbitrary probability measures on standard Borel spaces is proposed, which encompasses the classical notions of weak and strong typicality as special cases. Useful lemmas about strong typical sets, including…
I prove preservation theorems for countable support iteration of proper forcing concerning certain classes of capacities and submeasures. New examples of forcing notions and connections with measure theory are included.
We give some sufficient and necessary conditions on a forcing notion Q for preserving the forcing notion ([omega]^{aleph_0},supseteq^*) is proper. They cover many reasonable forcing notions.
We introduce a new method for building models of CH, together with $\Pi_2$ statements over $H(\omega_2)$, by forcing. Unlike other forcing constructions in the literature, our construction adds new reals, although only $\aleph_1$-many of…