We will give a new proof of the existence of hypercylinder expander of the inverse mean curvature flow which is a radially symmetric homothetic soliton of the inverse mean curvature flow in $\mathbb {R}^{n}\times \mathbb {R}$ , $n\ge 2$ , of the form $(r,y(r))$ or $(r(y),y)$ , where $r=|x|$ , $x\in \mathbb {R}^{n}$ , is the radially symmetric coordinate and $y\in \mathbb {R}$ . More precisely, for any $\lambda>\frac {1}{n-1}$ and $\mu>0$ , we will give a new proof of the existence of a unique even solution $r(y)$ of the equation $\frac {r^{\prime \prime }(y)}{1+r^{\prime }(y)^{2}}=\frac {n-1}{r(y)}-\frac {1+r^{\prime }(y)^{2}}{\lambda (r(y)-yr^{\prime }(y))}$ in $\mathbb {R}$ which satisfies $r(0)=\mu $ , $r^{\prime }(0)=0$ and $r(y)>yr^{\prime }(y)>0$ for any $y\in \mathbb {R}$ . We will prove that $\lim _{y\to \infty }r(y)=\infty $ and $a_{1}:=\lim _{y\to \infty }r^{\prime }(y)$ exists with $0\le a_{1}<\infty $ . We will also give a new proof of the existence of a constant $y_{1}>0$ such that $r^{\prime \prime }(y_{1})=0$ , $r^{\prime \prime }(y)>0$ for any $0<y<y_{1}$ , and $r^{\prime \prime }(y)<0$ for any $y>y_{1}$ .

In response to an open problem raised by S. Rabinowitz, we prove that

$$ \begin{align*} \begin{array} [c]{l} \left( \left( x^{2}+y^{2}\right) {}^{2}+8y\left( y^{2}-3x^{2}\right) \right) {}^{2}+432y\left( y^{2}-3x^{2}\right) \left( 351-10\left( x^{2}+y^{2}\right) \right) \\ =567^{3}+28\left( x^{2}+y^{2}\right) {}^{3}+486\left( x^{2}+y^{2}\right) \left( 67\left( x^{2}+y^{2}\right) -567\times18\right) \end{array} \end{align*} $$

In this paper, we give characterizations of the category of finitely generated projective modules having a right rejective chain. By focusing on the characterizations, we give sufficient conditions for right rejective chains to be total right rejective chains. Moreover, we prove that Nakayama algebras with heredity ideals, locally hereditary algebras and algebras of global dimension at most two satisfy the sufficient conditions. As an application, we show that these algebras are right-strongly quasi-hereditary algebras.

In 1955, Lehto showed that, for every measurable function $\psi $ on the unit circle $\mathbb T,$ there is a function f holomorphic in the unit disc, having $\psi $ as radial limit a.e. on $\mathbb T.$ We consider an analogous problem for solutions f of homogenous elliptic equations $Pf=0$ and, in particular, for holomorphic functions on Riemann surfaces and harmonic functions on Riemannian manifolds.

A $k_{\omega }$ -space X is a Hausdorff quotient of a locally compact, $\sigma $ -compact Hausdorff space. A theorem of Morita’s describes the structure of X when the quotient map is closed, but in 2010 a question of Arkhangel’skii’s highlighted the lack of a corresponding theorem for nonclosed quotient maps (even from subsets of $\mathbb {R}^n$ ). Arkhangel’skii’s specific question had in fact been answered by Siwiec in 1976, but a general structure theorem for $k_{\omega }$ -spaces is still lacking. We introduce pure quotient maps, extend Morita’s theorem to these, and use Fell’s topology to show that every quotient map can be “purified” (and thus every $k_{\omega }$ -space is the image of a pure quotient map). This clarifies the structure of arbitrary $k_{\omega }$ -spaces and gives a fuller answer to Arkhangel’skii’s question.

For $c \in \mathbb {Q}$ , consider the quadratic polynomial map $\varphi _c(z)=z^2-c$ . Flynn, Poonen, and Schaefer conjectured in 1997 that no rational cycle of $\varphi _c$ under iteration has length more than $3$ . Here, we discuss this conjecture using arithmetic and combinatorial means, leading to three main results. First, we show that if $\varphi _c$ admits a rational cycle of length $n \ge 3$ , then the denominator of c must be divisible by $16$ . We then provide an upper bound on the number of periodic rational points of $\varphi _c$ in terms of the number s of distinct prime factors of the denominator of c. Finally, we show that the Flynn–Poonen–Schaefer conjecture holds for $\varphi _c$ if $s \le 2$ , i.e., if the denominator of c has at most two distinct prime factors.

For every pseudovariety $\mathbf {V}$ of finite monoids, let $\mathbf {LV}$ denote the pseudovariety of all finite semigroups all of whose local submonoids belong to $\mathbf {V}$ . In this paper, it is shown that, for every nontrivial semidirectly closed pseudovariety $\mathbf {V}$ of finite monoids, the pseudovariety $\mathbf {LV}$ of finite semigroups is also semidirectly closed if, and only if, the given pseudovariety $\mathbf {V}$ is local in the sense of Tilson. This finding resolves a long-standing open problem posed in the second volume of the classic monograph by Eilenberg.

We obtain a characterization of the unital C*-algebras with the property that every element is a limit of products of positive elements, thereby answering a question of Murphy and Phillips.

Fix a poset Q on $\{x_1,\ldots ,x_n\}$ . A Q-Borel monomial ideal $I \subseteq \mathbb {K}[x_1,\ldots ,x_n]$ is a monomial ideal whose monomials are closed under the Borel-like moves induced by Q. A monomial ideal I is a principal Q-Borel ideal, denoted $I=Q(m)$ , if there is a monomial m such that all the minimal generators of I can be obtained via Q-Borel moves from m. In this paper we study powers of principal Q-Borel ideals. Among our results, we show that all powers of $Q(m)$ agree with their symbolic powers, and that the ideal $Q(m)$ satisfies the persistence property for associated primes. We also compute the analytic spread of $Q(m)$ in terms of the poset Q.

For functions in $C^k(\mathbb {R})$ which commute with a translation, we prove a theorem on approximation by entire functions which commute with the same translation, with a requirement that the values of the entire function and its derivatives on a specified countable set belong to specified dense sets. Using this theorem, we show that if A and B are countable dense subsets of the unit circle $T\subseteq \mathbb {C}$ with $1\notin A$ , $1\notin B$ , then there is an analytic function $h\colon \mathbb {C}\setminus \{0\}\to \mathbb {C}$ that restricts to an order isomorphism of the arc $T\setminus \{1\}$ onto itself and satisfies $h(A)=B$ and $h'(z)\not =0$ when $z\in T$ . This answers a question of P. M. Gauthier.

Motivated by the recent result in Samei and Wiersma (2020, Advances in Mathematics 359, 106897) that quasi-Hermitian groups are amenable, we consider a generalization of this property on discrete groups associated to certain Roe-type algebras; we call it uniformly quasi-Hermitian. We show that the class of uniformly quasi-Hermitian groups is contained in the class of supramenable groups and includes all subexponential groups. We also show that they are invariant under quasi-isometry.

Thurston norms are invariants of 3-manifolds defined on their second homology and understanding the shape of their dual unit balls is a widely open problem. In this article, we provide a large family of polytopes in $\mathbb {R}^{2g}$ that appear like dual unit balls of Thurston norms, generalizing Thurston’s construction for polygons in $\mathbb {R}^2$ .

À mes enseignants, premiers

We show that the Tits index $E_{8,1}^{133}$ cannot be obtained by means of the Tits construction over a field with no odd degree extensions. The proof uses a general method coming from the theory of symmetric spaces. We construct two cohomological invariants, in degrees $6$ and $8$ , of the Tits construction and the more symmetric Allison–Faulkner construction of Lie algebras of type $E_8$ and show that these invariants can be used to detect the isotropy rank.

We compare the Pontryagin duals of fine Selmer groups of two congruent p-adic Galois representations over admissible pro-p, p-adic Lie extensions $K_\infty $ of number fields K. We prove that in several natural settings the $\pi $ -primary submodules of the Pontryagin duals are pseudo-isomorphic over the Iwasawa algebra; if the coranks of the fine Selmer groups are not equal, then we can still prove inequalities between the $\mu $ -invariants. In the special case of a $\mathbb {Z}_p$ -extension $K_\infty /K$ , we also compare the Iwasawa $\lambda $ -invariants of the fine Selmer groups, even in situations where the $\mu $ -invariants are nonzero. Finally, we prove similar results for certain abelian non-p-extensions.

We show that any weakly separated Bessel system of model spaces in the Hardy space on the unit disc is a Riesz system and we highlight some applications to interpolating sequences of matrices. This will be done without using the recent solution of the Feichtinger conjecture, whose natural generalization to multidimensional model subspaces of ${\mathrm {H}}^2$ turns out to be false.

For two $\sigma $ -unital $C^*$ -algebras, we consider two equivalence bimodules over them, respectively. Then, by taking the crossed products by the equivalence bimodules, we get two inclusions of $C^*$ -algebras. Furthermore, we suppose that one of the inclusions of $C^*$ -algebras is irreducible, that is, the relative commutant of one of the $\sigma $ -unital $C^*$ -algebras in the multiplier $C^*$ -algebra of the crossed product is trivial. We will give a sufficient and necessary condition that the two inclusions are strongly Morita equivalent. Applying this result, we will compute the Picard group of a unital inclusion of unital $C^*$ -algebras induced by an equivalence bimodule over the unital $C^*$ -algebra under the assumption that the unital inclusion of unital $C^*$ -algebras is irreducible.

We show that every Lorentz sequence space $d(\textbf {w},p)$ admits a 1-complemented subspace Y distinct from $\ell _p$ and containing no isomorph of $d(\textbf {w},p)$ . In the general case, this is only the second nontrivial complemented subspace in $d(\textbf {w},p)$ yet known. We also give an explicit representation of Y in the special case $\textbf {w}=(n^{-\theta })_{n=1}^\infty $ ( $0<\theta <1$ ) as the $\ell _p$ -sum of finite-dimensional copies of $d(\textbf {w},p)$ . As an application, we find a sixth distinct element in the lattice of closed ideals of $\mathcal {L}(d(\textbf {w},p))$ , of which only five were previously known in the general case.

We introduce shadow structures for singular knot theory. Precisely, we define two invariants of singular knots and links. First, we introduce a notion of action of a singquandle on a set to define a shadow counting invariant of singular links which generalize the classical shadow colorings of knots by quandles. We then define a shadow polynomial invariant for shadow structures. Lastly, we enhance the shadow counting invariant by combining both the shadow counting invariant and the shadow polynomial invariant. Explicit examples of computations are given.

Let q be a prime number and $K = \mathbb Q(\theta )$ be an algebraic number field with $\theta $ a root of an irreducible trinomial $x^{6}+ax+b$ having integer coefficients. In this paper, we provide some explicit conditions on $a, b$ for which K is not monogenic. As an application, in a special case when $a =0$ , K is not monogenic if $b\equiv 7 \mod 8$ or $b\equiv 8 \mod 9$ . As an example, we also give a nonmonogenic class of number fields defined by irreducible sextic trinomials.

We give a family of real quadratic fields such that the 2-class field towers over their cyclotomic $\mathbb Z_2$ -extensions have metabelian Galois groups of abelian invariants $[2,2,2]$ . We also consider the boundedness of the Galois groups in relation to Greenberg’s conjecture, and calculate their class-2 quotients with an explicit example.