We show that, up to automorphisms of $\mathbb{P}^{2}_{\mathbb{C}}$, there are fourteen homogeneous convex foliations of degree $5$ on $\mathbb{P}^{2}_{\mathbb{C}}$. We establish some properties of the Fermat foliation $\mathcal{F}_{0}^{d}$ of degree $d\geq2$ and of the Hilbert modular foliation $\mathcal{F}_{H}^{5}$ of degree $5$. As a consequence, we obtain that every reduced convex foliation of degree $5$ on $\mathbb{P}^{2}_{\mathbb{C}}$ is linearly conjugated to one of the two foliations $\mathcal{F}_{0}^{5}$ or $\mathcal{F}_{H}^{5}$, which is a partial answer to a question posed in 2013 by D. Marín and J. V. Pereira. We end with two conjectures about the Camacho–Sad indices along the line at infinity at the non radial singularities of the homogeneous convex foliations of degree $d\geq2$ on $\mathbb{P}^{2}_{\mathbb{C}}$.

We show that, in the category of groups, every singly-generated class which is closed under isomorphisms, direct limits, and extensions is also singly-generated under isomorphisms anddirect limits, and in particular is co-reflective. In this way, we extend to the group-theoretic frameworkthe topological analogue proved by Chachólski, Parent, and Stanley in 2004. We also establish several new relations between singly-generated closed classes.

Let $F$ be a global function field of characteristic $p$ with ring of integers $A$ and let $\Phi$ be a Hayes module on the Hilbert class field $H_A$ of $F$. We prove an Iwasawa Main Conjecture for the $\mathbb{Z}_p^\infty$-extension $\mathcal{F}/F$ generated by the $\mathfrak{p}$-powertorsion of $\Phi$ ($\mathfrak{p}$ a prime of $A$). The main tool is a Stickelberger series whose specializationprovides a generator for the Fitting ideal of the class group of $\mathcal{F}$. Moreover we prove that the same series, evaluated at complex or $\mathfrak{p}$-adic characters, interpolates the Goss Zeta-function or some $\mathfrak{p}$-adic $L$-function, thus providing the link between the algebraic structure (class groups) and the analytic functions, which is the crucial part of Iwasawa Main Conjecture.

Given a set-theoretic solution $(X,r)$ of the Yang–Baxter equation, we denote by $M=M(X,r)$ the structure monoid and by $A=A(X,r)$, respectively $A'=A'(X,r)$, the left, respectively right, derived structure monoid of $(X,r)$. It is shown that there exist a left action of $M$ on $A$ and a right action of $M$ on $A'$ and $1$-cocycles $\pi$ and $\pi'$ of $M$ with coefficients in $A$ and in $A'$ with respect to these actions, respectively. We investigate when the $1$-cocycles are injective, surjective, or bijective. In case $X$ is finite, it turns out that $\pi$ is bijective if and only if $(X,r)$ is left non-degenerate, and $\pi'$ is bijective if and only if $(X,r)$ is right non-degenerate. In case $(X,r) $ is left non-degenerate, in particular $\pi$ is bijective, we define a semi-truss structure on $M(X,r)$ and then we show that this naturally induces a set-theoretic solution $(\overline M, \overline r)$ on the least cancellative image $\overline M= M(X,r)/\eta$ of $M(X,r)$. In case $X$ is naturally embedded in $M(X,r)/\eta$, for example when $(X,r)$ is irretractable, then $\overline r$ is an extension of $r$. It is also shown that non-degenerate irretractable solutions necessarily are bijective.

This article investigates the monodromy conjecture for a space monomial curve that appears as the special fiber of an equisingular family of curves with a plane branch as generic fiber. Roughly speaking, the monodromy conjecture states that every pole of the motivic, or related, Igusa zeta function induces an eigenvalue of monodromy. As the poles of the motivic zeta function associated with such a space monomial curve have been determined in earlier work, it remains to study the eigenvalues of monodromy. After reducing the problem to the curve seen as a Cartier divisor on a generic embedding surface, we construct an embedded $\mathbb Q$-resolution of this pair and use an A'Campo formula in terms of this resolution to compute the zeta function of monodromy. Combining all results, we prove the monodromy conjecture for this class of monomial curves.

We study the structure of groups of finite (Prüfer) rank in a very wide class of groups and also of central extensions of such groups. As a result we are able to improve, often substantially, on a number of known numerical bounds, in particular on bounds for the rank (resp.\ Hirsch number) of the derived subgroup of a group in terms of the rank (resp.\ Hirsch number) of its central quotient and on bounds for the rank of a group in terms of its Hirsch number.

We construct the first example of a $5$-dimensional simply connected compact manifold that admits a K-contact structure but does not admit any semi-regular Sasakian structure.For this, we need two ingredients: (a) to construct a suitable simply connected symplectic $4$-manifold with disjoint symplectic surfaces spanning the homology, all of them of genus $1$ except for one of genus $g>1$; (b) to prove a bound on the second Betti number $b_2$ of an algebraic surface with $b_1=0$ and having disjoint complex curves spanning the homology, all of them of genus $1$ except for one of genus $g>1$.

In this paper we study the initial value problem for infinite dimensional fractional non-autonomous reaction-diffusion equations. Applying general time-splitting methods, we prove the existence of solutions globally defined in time using convex sets as invariant regions. We expose examples where biological and pattern formation systems, under suitable assumptions, achieve global existence. We also analyze the asymptotic behavior of solutions.

Given a compact Riemannian manifold $(M^n,g)$ with boundary $\partial M$, we give an estimate for the quotient $\frac{\int_{\partial M} f\,d\mu_g}{\int_M f\,d\mu_g}$, where $f$ is a smooth positive function defined on $M$ that satisfies some inequality involving the scalar Laplacian. By the mean value lemma established [39], we provide a differential inequality for $f$ which, under some curvature assumptions, can be interpreted in terms of Bessel functions. As an application of our main result, a new inequality is given for Dirichlet and Robin Laplacian. Also, a new estimate is established for the eigenvalues of the Dirac operator that involves a positive root of Bessel function besides the scalar curvature. Independently, we extend the Robin Laplacian on functions to differential forms. We prove that this natural extension defines a self-adjoint and elliptic operator whose spectrum is discrete and consists of positive real eigenvalues. In particular, we characterize its first eigenvalue and provide a lower bound of it in terms of Bessel functions.

We prove that if $X$ and $Y$ are first countable compact Hausdorff spaces, then the set of all diameter-preserving linear bijections from $C(X)$ to $C(Y)$ is algebraically reflexive.

We study noninvolutive set-theoretic solutions $(X,r)$ of the Yang–Baxter equations in terms of the properties of the canonically associated braided monoid $S(X,r)$, the quadratic Yang–Baxter algebra $A= A(\mathbf{k}, X, r)$ over a field $\mathbf{k}$, and its Koszul dual $A^{!}$. More generally, we continue our systematic study of nondegenerate quadratic sets $(X,r)$ and their associated algebraic objects. Next we investigate the class of (noninvolutive) square-free solutions $(X,r)$. This contains the self distributive solutions (quandles). We make a detailed characterization in terms of various algebraic and combinatorial properties each of which shows the contrast between involutive and noninvolutive square-free solutions. We introduce and study a class of finite square-free braided sets $(X,r)$ of order $n\geq 3$ which satisfy the minimality condition, that is, $\dim_{\mathbf{k}} A_2 =2n-1$. Examples are some simple racks of prime order $p$. Finally, we discuss general extensions of solutions and introduce the notion of a generalized strong twisted union of braided sets. We prove that if $(Z,r)$ is a nondegenerate $2$-cancellative braided set splitting as a generalized strong twisted union of $r$-invariant subsets $Z = X\mathbin{\natural}^{\ast} Y$, then its braided monoid $S_Z$ is a generalized strong twisted union $S_Z= S_X\mathbin{\natural}^{\ast} S_Y$ of the braided monoids $S_X$ and $S_Y$. We propose a construction of a generalized strong twisted union $Z = X\mathbin{\natural}^{\ast} Y$ of braided sets $(X,r_X)$ and $(Y,r_Y)$, where the map $r$ has a high, explicitly prescribed order.

The purpose of this paper is to study the properties of the irrational-slope Thompson's group $F_\tau$ introduced by Cleary [11]. We construct presentations, both finite and infinite, and we describe its combinatorial structure using binary trees. We show that its commutator group is simple. Finally, inspired by the case of Thompson's group $F$, we define a unique normal form for the elements of the group and study the metric properties for the elements based on this normal form. As a corollary, we see that several embeddings of $F$ in $F_\tau$ are undistorted.

We study a two-dimensional discrete directional maximal operator along the set of the prime numbers. We show existence of a set of vectors, which are lattice points in a sufficiently large annulus, for which the $\ell^2$ norm of the associated maximal operator, with supremum taken over all large scales, grows with an epsilon power in the number of vectors. This paper is a follow-up to a prior work on the discrete directional maximal operator along the integers by the first and third author.

Given $\{W^{(m)}(t),\, t \in [0,T]\}_{m \ge 1}$, a sequence of approximations to a standard Brownian motion $W$ in $[0,T]$ such that $W^{(m)}(t)$ converges almost surely to $W(t)$, we show that, under regular conditions on the approximations, the multiple ordinary integrals with respect to $dW^{(m)}$ converge to the multiple Stratonovich integral. We are integrating functions of the type $$ f(t_1,\dotsc,t_n)=f_1(t_1)\dotsm f_n(t_n) I_{\{t_1\le \dotsb \le t_n\}}, $$ where for each $i \in \{1,\dotsc,n\}$, $f_i$ has continuous derivatives in $[0,T]$. We apply this result to approximations obtained from uniform transport processes.