Maximal ideals of regulous functions are not finitely generated

We prove that every maximal ideal in the ring of k-regulous functions, $k\in \mathbb{N}$, on a smooth real affine algebraic variety of dimension $d\ge 2$ is not finitely generated.     

Almost slender rings and compact rings

Classical results concerning slenderness for commutative integral domains are generalized to commutative rings with zero divisors. This is done by extending the methods from the domain case and bringing them in connection with results on the linear topologies associated to non-discrete Hausdorff filtrations. In many cases a weakened notion “almost slenderness” of slenderness is appropriate for rings with zero divisors. Special results for countable rings are extended to rings said to be of “bounded type” (including countable rings, ‘small’ rings, and, for instance, rings that are countably generated as algebras over an Artinian ring).More precisely, for a ring R of bounded type it is proved that R is slender if R is reduced and has no simple ideals, or if R is Noetherian and has no simple ideals; moreover, R is almost slender if R is not perfect (in the sense of H. Bass). We use our methods to study various special classes of rings, for instance von Neumann regular rings and valuation rings. Among other results we show that the following two rings are slender: the ring of Puiseux series over a field and the von Neumann regular ring $k^{\mathbb{N}}/k^{(\mathbb{N})}$ over a von Neumann regular ring k.For a Noetherian ring R we prove that R is a finite product of local complete rings iff R satisfies one of several (equivalent) conditions of algebraic compactness. A 1-dimensional Noetherian ring is outside this ‘compact’ class precisely when it is almost slender. For the rings of classical algebraic geometry we prove that a localization of an algebra finitely generated over a field is either Artinian or almost slender. Finally, we show that a Noetherian ring R is a finite product of local complete rings with finite residue fields exactly when there exists a map of R-algebras $R^{\mathbb{N}}\to R$ vanishing on $R^{(\mathbb{N})}$.     

Integral representations of martingales for progressive enlargements of filtrations

We work in the setting of the progressive enlargement $\mathbb{G}$ of a reference filtration $\mathbb{F}$ through the observation of a random time $\tau$. We study an integral representation property for some classes of $\mathbb{G}$-martingales stopped at $\tau$. In the first part, we focus on the case where $\mathbb{F}$ is a Poisson filtration and we establish a predictable representation property with respect to three $\mathbb{G}$-martingales. In the second part, we relax the assumption that $\mathbb{F}$ is a Poisson filtration and we assume that $\tau$ is an $\mathbb{F}$-pseudo-stopping time. We establish integral representations with respect to some $\mathbb{G}$-martingales built from $\mathbb{F}$-martingales and, under additional hypotheses, we obtain a predictable representation property with respect to two $\mathbb{G}$-martingales.     

Equivalence of bar induction and bar recursion for continuous functions with continuous moduli

We compare Brouwer's bar theorem and Spector's bar recursion for the lowest type in the context of constructive reverse mathematics. To this end, we reformulate bar recursion as a logical principle stating the existence of a bar recursor for every function which serves as the stopping condition of bar recursion. We then show that the decidable bar induction is equivalent to the existence of a bar recursor for every continuous function from ${\mathbb{N}}^{\mathbb{N}}$ to $\mathbb{N}$ with a continuous modulus. We also introduce fan recursion, the bar recursion for binary trees, and show that the decidable fan theorem is equivalent to the existence of a fan recursor for every continuous function from ${\{0,1\}}^{\mathbb{N}}$ to $\mathbb{N}$ with a continuous modulus. The equivalence for bar induction holds over the extensional version of intuitionistic arithmetic in all finite types augmented with the characteristic principles of Gödel's Dialectica interpretation. On the other hand, we show the equivalence for fan theorem without using such extra principles.     

The extension theorem for the Lee and Euclidean weights over Z/pkZ

The Lee and Euclidean weights have the extension property over the local rings $\mathbb{Z}/p^k\mathbb{Z}$, p prime. The non-vanishing of certain Fourier coefficients is established by expressing the coefficients in terms of generalized Bernoulli numbers and making use of knowledge of the locations of zeros of Dirichlet L-functions.     

On the weak representation property in progressively enlarged filtrations with an application in exponential utility maximization

In this paper we show that the weak representation property of a semimartingale $X$ with respect to a filtration $\mathbb{F}$ is preserved in the progressive enlargement $\mathbb{G}$ by a random time $\tau$ avoiding $\mathbb{F}$-stopping times and such that $\mathbb{F}$ is immersed in $\mathbb{G}$. As an application of this, we can solve an exponential utility maximization problem in the enlarged filtration $\mathbb{G}$ following the dynamical approach, based on suitable BSDEs, both over the fixed-time horizon $[0,T]$, $T>0$, and over the random-time horizon $[0,T\wedge \tau ]$.     

Minimal reversible nonsymmetric rings

Marks showed that ${\mathbb{F}}_2{Q}_8$, the ${\mathbb{F}}_2$ group algebra over the quaternion group, is a reversible nonsymmetric ring, then questioned whether or not this ring is minimal with respect to cardinality. In this work, it is shown that the cardinality of a minimal reversible nonsymmetric ring is indeed 256. Furthermore, it is shown that although ${\mathbb{F}}_2{Q}_8$ is a duo ring, there are also examples of minimal reversible nonsymmetric rings which are nonduo.     

Construction of the circle in UniMath

We show that the type $\mathrm{T}\mathbb{Z}$ of $\mathbb{Z}$-torsors has the dependent universal property of the circle, which characterizes it up to a unique homotopy equivalence. The construction uses Voevodsky's Univalence Axiom and propositional truncation, yielding a stand-alone construction of the circle not using higher inductive types.     

Low-energy points on the sphere and the real projective plane

We present a generalization of a family of points on ${\mathbb{S}}^2$, the Diamond ensemble, containing collections of N points on ${\mathbb{S}}^2$ with very small logarithmic energy for all $N\in \mathbb{N}$. We extend this construction to the real projective plane ${\mathbb{RP}}^2$ and we obtain upper and lower bounds with explicit constants for the Green and logarithmic energy on this last space.     

On a characterization of path connected topological fields

The aim of this paper is to give a characterization of path connected topological fields, inspired by the classical Gelfand correspondence between a compact Hausdorff topological space X and the space of maximal ideals of the ring of real valued continuous functions $C(X,\mathbb{R})$. More explicitly, our motivation is the following question: What is the essential property of the topological field $F=\mathbb{R}$ that makes such a correspondence valid for all compact Hausdorff spaces? It turns out that such a perfect correspondence exists if and only if F is a path connected topological field.     

Hod up to ADR+Θ is measurable

Suppose M is a transitive class size model of $A{D}_{\mathbb{R}}+\text{“Θ is regular”}$. M is a minimal model of $A{D}_{\mathbb{R}}+\text{“Θ is measurable”}$ if (i) $\mathbb{R},Ord?M$ (ii) there is $\mu \in M$ such that $M?\text{“}\mathit{\mu }\text{ is a normal }\mathbb{R}\text{-complete measure on Θ”}$ and (iii) for any transitive class size $N?M$ such that $\mathbb{R}?N$, $N?\text{“there is no }\mathbb{R}\text{-complete measure on Θ”}$. Continuing Trang's work in [8], we compute HOD of a minimal model of $A{D}_{\mathbb{R}}+\text{“Θ is measurable”}$.