Related papers: The linear refinement number and selection theory
In \cite{liu2022practical}, a general algorithm is developed to efficiently obtain the best accuracy using the regular refinement. The adaptive refinement allows for obtaining an accuracy with a smaller number of DoFs compared with the…
We prove two $\mathrm{ZFC}$ inequalities between cardinal invariants. The first inequality involves cardinal invariants associated with an analytic P-ideal, in particular the ideal of subsets of $\omega$ of asymptotic density $0$. We obtain…
In this work we study the uncountable Borel chromatic numbers, defined by Geschke (2011) as cardinal characteristics of the continuum, of low complexity graphs. We show that a strong form of locally countable graphs with compact totally…
We obtain a small ultrafilter number at $\aleph_{\omega_1}$. Moreover, we develop a version of the overlapping strong extender forcing with collapses which can keep the top cardinal $\kappa$ inaccessible. We apply this forcing to construct…
This paper contributes to the theory of large cardinals beyond the Kunen inconsistency, or choiceless large cardinal axioms, in the context where the Axiom of Choice is not assumed. The first part of the paper investigates a periodicity…
In this paper we demonstrate that it is consistent, relative to the existence of a supercompact cardinal, that there is no linear order which is minimal with respect to being non $\sigma$-scattered. This shows that a theorem of Laver, which…
In this paper we study existence, regularity, and approximation of solution to a fractional semilinear elliptic equation of order $s \in (0,1)$. We identify minimal conditions on the nonlinear term and the source which leads to existence of…
We develop the notion of coherent ultrafilters (extenders without normality or well-foundedness). We then use definable coherent ultraproducts to characterize any extension of a model $M$ in any fragment of $\mathbb{L}_{\infty, \omega}$…
We reformulate slightly Russell's notion of typicality, so as to eliminate its circularity and make it applicable to elements of any first-order structure. We argue that the notion parallels Martin-L\"{o}f (ML) randomness, in the sense that…
We derive sharp upper and lower bounds on the number of intersection points and closed regions that can occur in sets of line segments with certain structure, in terms of the number of segments. We consider sets of segments whose underlying…
We investigate the relationship between variants of the uniformization property for ladder system colorings and fragments of Martin's Axiom. The well-known forcing properties of having precaliber $\aleph_1$ and being $\sigma$-centered…
A partition is finitary if all its blocks are finite. For a cardinal $\mathfrak{a}$ and a natural number $n$, let $\mathrm{fin}(\mathfrak{a})$ and $\mathscr{B}_{n}(\mathfrak{a})$ be the cardinalities of the set of finite subsets and the set…
This article critically reappraises arguments in support of Cantor's theory of transfinite numbers. The following results are reported: i) Cantor's proofs of nondenumerability are refuted by analyzing the logical inconsistencies in…
Motivated by applications in automated verification of higher-order functional programs, we develop a notion of constrained Horn clauses in higher-order logic and a decision problem concerning their satisfiability. We show that, although…
We continue investigating variants of the splitting and reaping numbers introduced in arXiv:1808.02442. In particular, answering a question raised there, we prove the consistency of $\mathrm{cof}(\mathcal{M})<\mathfrak{s}_{\frac{1}{2}}$ and…
We consider some coloring issues related to the famous Erd\H {o}s Discrepancy Problem. A set of the form $A_{s,k}=\{s,2s,\dots,ks\}$, with $s,k\in \mathbb{N}$, is called a \emph{homogeneous arithmetic progression}. We prove that for every…
Calibration of mean estimates for predictions is a crucial property in many applications, particularly in the fields of financial and actuarial decision-making. In this paper, we first review classical approaches for validating…
We mainly investigate model of set theory with restricted choice, e.g., ZF + DC + "the family of countable subsets of lambda is well ordered for every lambda" (really local version for a given lambda). In this frame much of pcf theory can…
We introduce $\mathsf{LEM}$, a type-assignment system for the linear $ \lambda $-calculus that extends second-order $\mathsf{IMLL}_2$, i.e., intuitionistic multiplicative Linear Logic, by means of logical rules that weaken and contract…
The paper establishes several inequalities between cardinal characteristics of the continuum. In particular, it is shown that the partition splitting number is not larger than the uniformity of the meagre ideal; not all sets of reals having…