Challenge Problems for a Theory of Degree Multiplication (with partial answer key)

This paper offers a theory of degree multiplication in natural language semantics. Motivation for the development such a theory comes from proportional readings of quantity words and rate expressions such as miles per hour. After laying out a set of ‘challenge problems’ that any good theory of degree multiplication should be able to handle, I set about solving them, borrowing mathematical tools from quantity calculus. These algebraic foundations are integrated into a compositional Montagovian framework, yielding a system that can solve, or partially solve, some of the problems.

Cardinal estimates involving the weak Lindelöf game

We show that if X is a first-countable Urysohn space where player II has a winning strategy in the game $$G^{\omega _1}_1({\mathcal {O}}, {\mathcal {O}}_D)$$ G 1 ω 1 ( O , O D ) (the weak Lindelöf game of length $$\omega _1$$ ω 1 ) then X has cardinality at most continuum. This may be considered a partial answer to an old question of Bell, Ginsburg and Woods. It is also the best result of this kind since there are Hausdorff first-countable spaces of arbitrarily large cardinality where player II has a winning strategy even in the weak Lindelöf game of countable length. We also tackle the problem of finding a bound on the cardinality of a first-countable space where player II has a winning strategy in the game $$G^{\omega _1}_{fin}({\mathcal {O}}, {\mathcal {O}}_D)$$ G fin ω 1 ( O , O D ) , providing some partial answers to it. We finish by constructing an example of a compact space where player II does not have a winning strategy in the weak Lindelöf game of length $$\omega _1$$ ω 1 .

Groundstates for Choquard type equations with weighted potentials and Hardy–Littlewood–Sobolev lower critical exponent

We are concerned with a class of Choquard type equations with weighted potentials and Hardy–Littlewood–Sobolev lower critical exponent − Δ u + V ( x ) u = I α ∗ [ Q ( x ) | u | N + α N ] Q ( x ) | u | α N − 1 u , x ∈ R N . $$\begin{array}{} \displaystyle -{\it\Delta} u+V(x)u=\left(I_{\alpha}\ast [Q(x)|u|^{\frac{N+\alpha}{N}}]\right)Q(x)|u|^{\frac{\alpha}{N}-1}u, \quad x\in \mathbb R^N. \end{array}$$ By using variational approaches, we investigate the existence of groundstates relying on the asymptotic behaviour of weighted potentials at infinity. Moreover, non-existence of non-trivial solutions is also considered. In particular, we give a partial answer to some open questions raised in [D.~Cassani, J. Van Schaftingen and J. J. Zhang, Groundstates for Choquard type equations with Hardy-Littlewood-Sobolev lower critical exponent, Proceedings of the Royal Society of Edinburgh, Section A Mathematics, 150(2020), 1377–1400].

Some remarks on the metrizability of some metric-like structures

The main purpose of this article is to provide alternative proofs of the metrizability of metric-like spaces like b-metric spaces, \mathcal{F}-metric spaces, and \theta-metric spaces. We improve upon the metrizability result of An et al. [Topology Appl. 185–186 (2015)] for b-metric spaces. Moreover, we provide two shorter proofs of the metrizability of \mathcal{F}-metric spaces, recently introduced by Jleli and Samet. Furthermore, we give a partial answer to an open problem regarding the openness of \mathcal{F}-open balls in \mathcal{F}-metric spaces. Finally, we give an alternative proof of the metrizability of \theta-metric spaces.

Density and capacity of balleans generated by filters

UDC 519.51 We consider a ballean with an infinite support and a free filter on and define for every and The ballean will be called the <em>ballean-filter mix</em> of and and denoted by It was introduced in [O. V. Petrenko, I. V. Protasov, <em>Balleans and filters</em>, Mat. Stud., <strong>38</strong>, No. 1, 3–11 (2012)] and was used to construction of a non-metrizable Frechet group ballean. In this paper some cardinal invariants are compared. In particular, we give a partial answer to the question: if we mix an ordinal unbounded ballean with a free filter of the subsets of its support, will the mix-structure's density be equal to its capacity, as it holds in the original balleans?

Climate Change and Cultural Cognition

How should we form beliefs concerning global climate change? For most of us, directly evaluating the evidence isn’t feasible; we lack expertise. So, any rational beliefs we form will have to be based in part on deference to those who have it. But in this domain, questions about how to identify experts can be fraught. This chapter discusses a partial answer to the question of how we in fact identify experts: Dan Kahan’s cultural cognition thesis, according to which we treat experts on factual questions of political import only insofar as they share our moral and cultural values. The chapter then poses some normative questions about cultural cognition: is it a species of irrationality that must be overcome if we are to communicate scientific results effectively, or is it instead an inescapable part of rational belief management? Ultimately, it is argued that cultural cognition is substantively unreasonable, though not formally irrational.

Triharmonic Riemannian submersions from 3-dimensional space forms

We show that any triharmonic Riemannian submersion from a 3-dimensional space form into a surface is harmonic. This is an affirmative partial answer to the submersion version of the generalized Chen conjecture. Moreover, a non-existence theorem for f -biharmonic Riemannian submersions is also presented.

Equivariant wrapped Floer homology and symmetric periodic Reeb orbits

The aim of this article is to apply a Floer theory to study symmetric periodic Reeb orbits. We define positive equivariant wrapped Floer homology using a (anti-)symplectic involution on a Liouville domain and investigate its algebraic properties. By a careful analysis of index iterations, we obtain a non-trivial lower bound on the minimal number of geometrically distinct symmetric periodic Reeb orbits on a certain class of real contact manifolds. This includes non-degenerate real dynamically convex star-shaped hypersurfaces in ${\mathbb {R}}^{2n}$ which are invariant under complex conjugation. As a result, we give a partial answer to the Seifert conjecture on brake orbits in the contact setting.

Basarab loop and the generators of its total multiplication group

A loop (Q; ·) is called a Basarab loop if the identities: (x · yxρ)(xz) = x · yz and (yx) · (xλz · x) = yz · x hold. It was shown that the left, right and middle nuclei of the Basarab loop coincide, and the nucleus of a Basarab loop is the set of elements x whose middle inner mapping Tx are automorphisms. The generators of the inner mapping group of a Basarab loop were refined in terms of one of the generators of the total inner mapping group of a Basarab loop. Necessary and su_cient condition(s) in terms of the inner mapping group (associators) for a loop to be a Basarab loop were established. It was discovered that in a Basarab loop: the mapping x ↦ Tx is an endomorphism if and only if the left (right) inner mapping is a left (right) regular mapping. It was established that a Basarab loop is a left and right automorphic loop and that the left and right inner mappings belong to its middle inner mapping group. A Basarab loop was shown to be an automorphic loop (A-loop) if and only if it is a middle automorphic loop (middle A-loop). Some interesting relations involving the generators of the total multiplication group and total inner mapping group of a Basarab loop were derived, and based on these, the generators of the total inner mapping group of a Basarab loop were finetuned. A Basarab loop was shown to be a totally automorphic loop (TA-loop) if and only if it is a commutative and exible loop. These aforementioned results were used to give a partial answer to a 2013 question and an ostensible solution to a 2015 problem in the case of Basarab loop.

On G-mappings defined by G-methods and G-topological groups

In this note, the concepts of (G1,G2)-open, (G1,G2)-closed, (G1,G2)-quotient and (G1,G2)-perfect mappings on arbitrary sets are introduced and some theorems on them are established firstly. In particular, some results improve the corresponding results in [17]. Secondly, we give a partial answer to the question posed by L. Liu [14]. Finally, some properties of G-topological groups, G-connectedness and totally G-disconnectedness in G-topological groups are discussed.