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In this paper, we first investigate the regularity of finite energy solutions to the following Wolff type integral system: { u (x) = R 1 (x) W β , γ (v p u r) (x) , u (x) > 0 , x ∈ R N , v (x) = R 2 (x) W β , γ (u q v s) (x) , v (x) > 0 , x ∈ R N , where γ > 1 , β > 0 , β γ < N , W β , γ is the Wolff potential, R 1 , R 2 are double bounded functions in R N and p , q > max ⁡ { 1 , γ − 1 } , r , s ≥ 0 with p − s ≥ q − r > − γ + 1. We exploit the regularity lifting lemma to obtain the optimal integrability, boundedness and decaying property of finite energy solutions to the system. Secondly, we establish sharp pointwise estimates of positive finite energy solutions to the p -Laplacian and k -Hessian systems related to the above integral system, by using the previous regularity results, the interior Hölder estimates of solutions for the corresponding differential systems and a doubling lemma of Poláčik, Quittner and Souplet (Duke Math. J.,2007). These extend Vétois's decay results (Indiana Univ. Math. J.,2019) on the positive solutions of Laplacian systems to the p -Laplacian and k -Hessian cases. We remark that our methods do not need Harnack type inequalities and it can be applied to deal with the solutions without radial structures. As far as we know, this is the first attempt to derive sharp decay estimates for possibly non-radial solutions involving k -Hessian operators. We also obtain some decay estimates of the gradients at infinity. [ABSTRACT FROM AUTHOR]

There are several classical characterisations of the valuative dimension of a commutative ring. Constructive versions of this dimension have been given and proven to be equivalent to the classical notion within classical mathematics, and they can be used for the usual examples of commutative rings. To the contrary of the classical versions, the constructive versions have a clear computational content. This paper investigates the computational relationship between three possible constructive definitions of the valuative dimension of a commutative ring. In doing so, it proves these constructive versions to be equivalent within constructive mathematics. [ABSTRACT FROM AUTHOR]

In this paper, we give an example of a statement concerning two-player zero-sum games which is undecidable, meaning that it can neither be proven or disproven by the standard axioms of mathematics. Earlier work has shown that there exist "paradoxical" two-player zero-sum games with unbounded payoffs, in which a standard calculation of the two players' expected utilities of a mixed strategy profile yield a positive sum. We show that whether or not a modified version of this paradoxical situation, with bounded payoffs and a weaker measurability requirement, exists is an unanswerable question. Our proof relies on a mixture of techniques from set theory and ergodic theory. [ABSTRACT FROM AUTHOR]

The paper is dedicated to analysing the role of "denominate numbers" in textbooks for Polish schools, whose author or co-author was Stefan Banach. Banach used the concepts of "number" and "denominate number" in his textbooks. The ways of introducing these concepts, as well as the associated manner of understanding numeration, are analysed in the paper. The methods of manipulating "denominate numbers" through various procedures are also discussed. The analysis is carried out in the context of the Polish ministerial ordinances related to teaching mathematics and the content of other textbooks, especially those used by Banach as a secondary school student. [ABSTRACT FROM AUTHOR]

Leibniz's manuscripts on perspective geometry remained unpublished and unknown until very recently. Among them, Scientia perspectiva stands out as the most complex and the most original. In this paper, we offer a thorough analysis of this manuscript, showing how Leibniz moves from perspective concepts fairly common at that time to a completely new idea of the practice that could have affected its entire history. This new science represents not only Leibniz's unique contribution to the development of perspective but also casts a new light on his own notion of space and geometry and their philosophical grounding. [ABSTRACT FROM AUTHOR]

It is shown in this paper that suitable weak solutions to the 6D steady incompressible Navier-Stokes are Hölder continuous at 0 provided that ∫ B 1 | u (x) | 3 d x + ∫ B 1 | f (x) | q d x or ∫ B 1 | ∇ u (x) | 2 d x + ∫ B 1 | ∇ u (x) | 2 d x (∫ B 1 | u (x) | d x) 2 + ∫ B 1 | f (x) | q d x with q > 3 is sufficiently small, which implies that the 2D Hausdorff measure of the set of singular points is zero. For the boundary case, we also obtain that 0 is regular provided that ∫ B 1 + | u (x) | 3 d x + ∫ B 1 + | f (x) | 3 d x or ∫ B 1 + | ∇ u (x) | 2 d x + ∫ B 1 + | f (x) | 3 d x is sufficiently small. These results improve previous regularity theorems by Dong-Strain ([8] , Indiana Univ. Math. J., 2012), Dong-Gu ([7] , J. Funct. Anal., 2014), and Liu-Wang ([27] , J. Differential Equations, 2018), where either the smallness of the pressure or the smallness of the scaling invariant quantities on all balls is necessary. [ABSTRACT FROM AUTHOR]

In this note, we provide a corrected proof of Lemma 3.2 in [4]. We also point out some improvements to Remark 5.5 in the same paper. [ABSTRACT FROM AUTHOR]

In this paper, we provide generating functions and counting formulas for spanning trees and spanning forests in hypergraphs in two different ways: (1) We represent spanning trees and spanning forests in hypergraphs through Berezin-Grassmann integrals on Zeon algebra and hyper-Hafnians (orders and signs are not considered); (2) We establish a Hyper-Pfaffian-Cactus Spanning Forest Theorem through Berezin-Grassmann integrals on Grassmann algebra (orders and signs are considered), which generalizes the Hyper-Pfaffian-Cactus Theorem by Abdesselam (2004) [1] and Pfaffian matrix tree theorem by Masbaum and Vaintrob (2002) [15]. • We provide generating functions and counting formulas for spanning trees and spanning forests in hypergraphs through Berezin-Grassmann integrals on Zeon algebra and hyper- Hafnians when orders and signs are not considered. • We establish a Hyper-Pfaffian-Cactus Spanning Forest Theorem through Berezin-Grassmann integrals on Grassmann algebra (orders and signs are considered), which generalizes the Hyper-Pfaffian-Cactus Theorem by Abdesselam [Advances in Applied Mathematics (2004) Vol. 33: 51-70] and Pfaffian matrix tree theorem by Masbaum and Vaintrob [Internat. Math. Res. Notices (2002) Vol. 27: 1397-1426]. [ABSTRACT FROM AUTHOR]

In the paper, we construct compact embedded convex λ -hypersurfaces which are diffeomorphic to a sphere and are not isometric to a standard sphere. As the special case of our result, we solve Sun's problem (Int. Math. Res. Not. (2021) 11818–11844). In this sense, one can not expect to have Alexandrov type theorem for λ -hypersurfaces. [ABSTRACT FROM AUTHOR]

• Variational Laplace (VL) is a scheme for Bayesian modelling. • VL is widely used in neuroimaging, in particular DCM. • This paper provides a tutorial explanation of the math and algorithms. • New standalone code is provided to enable re-implementation. • The supplementary materials provide worked derivations. This article details a scheme for approximate Bayesian inference, which has underpinned thousands of neuroimaging studies since its introduction 15 years ago. Variational Laplace (VL) provides a generic approach to fitting linear or non-linear models, which may be static or dynamic, returning a posterior probability density over the model parameters and an approximation of log model evidence, which enables Bayesian model comparison. VL applies variational Bayesian inference in conjunction with quadratic or Laplace approximations of the evidence lower bound (free energy). Importantly, update equations do not need to be derived for each model under consideration, providing a general method for fitting a broad class of models. This primer is intended for experimenters and modellers who may wish to fit models to data using variational Bayesian methods, without assuming previous experience of variational Bayes or machine learning. Accompanying code demonstrates how to fit different kinds of model using the reference implementation of the VL scheme in the open-source Statistical Parametric Mapping (SPM) software package. In addition, we provide a standalone software function that does not require SPM, in order to ease translation to other fields, together with detailed pseudocode. Finally, the supplementary materials provide worked derivations of the key equations. [ABSTRACT FROM AUTHOR]

In 1804 the chair of Elementary Mathematics at Prague University became vacant and a selection procedure, which consisted of a written and an oral examination, was announced. Only Bernard Bolzano and Ladislav Jandera took part in it. Jandera was appointed to the chair, whereas Bolzano became Professor of "Religious Doctrine." In this paper we examine the context of this Concursprüfung , the performance of both candidates and the outcome, based on a number of related papers preserved at the Czech National Archives. In addition to this, we publish for the first time, albeit only in the online version of this paper, the transcription and English translation of Bolzano's written examination. [ABSTRACT FROM AUTHOR]

The proof of Theorem 0.4 in "Successive minima of line bundles" contains an error. We give a modified statement of Theorem 0.4, which is weaker than the original one. Since Theorem 0.1 in the paper is a special case of Theorem 0.4, we give a new proof of Theorem 0.1. This proof improves a lower bound in Theorem 0.1. [ABSTRACT FROM AUTHOR]

• Coordination of scientific research communities supported mathematical modelling of COVID-19. • Pre-existing research collaborations contributed to the success of coordination initiatives. • National infrastructure is key to support nationwide research networks. • Evidence synthesis important in supporting scientists providing advice to government. In March 2020 mathematics became a key part of the scientific advice to the UK government on the pandemic response to COVID-19. Mathematical and statistical modelling provided critical information on the spread of the virus and the potential impact of different interventions. The unprecedented scale of the challenge led the epidemiological modelling community in the UK to be pushed to its limits. At the same time, mathematical modellers across the country were keen to use their knowledge and skills to support the COVID-19 modelling effort. However, this sudden great interest in epidemiological modelling needed to be coordinated to provide much-needed support, and to limit the burden on epidemiological modellers already very stretched for time. In this paper we describe three initiatives set up in the UK in spring 2020 to coordinate the mathematical sciences research community in supporting mathematical modelling of COVID-19. Each initiative had different primary aims and worked to maximise synergies between the various projects. We reflect on the lessons learnt, highlighting the key roles of pre-existing research collaborations and focal centres of coordination in contributing to the success of these initiatives. We conclude with recommendations about important ways in which the scientific research community could be better prepared for future pandemics. This manuscript was submitted as part of a theme issue on "Modelling COVID-19 and Preparedness for Future Pandemics". [ABSTRACT FROM AUTHOR]

In this paper, we study existence of isometric embedding of S q m into S p n , where 1 ≤ p ≠ q ≤ ∞ and n ≥ m ≥ 2. We show that for all n ≥ m ≥ 2 if there exists a linear isometry from S q m into S p n , where (q , p) ∈ (1 , ∞ ] × (1 , ∞) ∪ (1 , ∞) ∖ { 3 } × { 1 , ∞ } and p ≠ q , then we must have q = 2. This mostly generalizes a classical result of Lyubich and Vaserstein. We also show that whenever S q embeds isometrically into S p for (q , p) ∈ (1 , ∞) × [ 2 , ∞) ∪ [ 4 , ∞) × { 1 } ∪ { ∞ } × (1 , ∞) ∪ [ 2 , ∞) × { ∞ } with p ≠ q , we must have q = 2. Thus, our work complements work of Junge, Parcet, Xu and others on isometric and almost isometric embedding theory on non-commutative L p -spaces. Our methods rely on several new ingredients related to perturbation theory of linear operators, namely Kato-Rellich theorem, theory of multiple operator integrals and Birkhoff-James orthogonality, followed by thorough and careful case by case analysis. The question whether for m ≥ 2 and 1 < q < 2 , S q m embeds isometrically into S ∞ n , was left open in Bull. London Math. Soc. 52 (2020) 437-447. [ABSTRACT FROM AUTHOR]