In this paper, we consider a conilpotent coalgebra $C$ over a field $k$. Let $\Upsilon :\ C{{-\mathsf{Comod}}}\longrightarrow C^*{{-\mathsf{Mod}}}$ be the natural functor of inclusion of the category of $C$-comodules into the category of $C^*$-modules, and let $\Theta :\ C{{-\mathsf{Contra}}}\longrightarrow C^*{{-\mathsf{Mod}}}$ be the natural forgetful functor. We prove that the functor $\Upsilon$ induces a fully faithful triangulated functor on bounded (below) derived categories if and only if the functor $\Theta$ induces a fully faithful triangulated functor on bounded (above) derived categories, and if and only if the $k$-vector space $\textrm {Ext}_C^n(k,k)$ is finite-dimensional for all $n\ge 0$. We call such coalgebras “weakly finitely Koszul”.

We study the distribution of consecutive sums of two squares in arithmetic progressions. We show that for any odd squarefree modulus q, any two reduced congruence classes $a_1$ and $a_2$ mod q, and any $r_1,r_2 \ge 1$, a positive density of sums of two squares begin a chain of $r_1$ consecutive sums of two squares, all of which are $a_1$ mod q, followed immediately by a chain of $r_2$ consecutive sums of two squares, all of which are $a_2$ mod q. This is an analog of the result of Maynard for the sequence of primes, showing that for any reduced congruence class a mod q and for any $r \ge 1$, a positive density of primes begin a sequence of r consecutive primes, all of which are a mod q.

We present a short and simple proof of the celebrated hypergraph container theorem of Balogh–Morris–Samotij and Saxton–Thomason. On a high level, our argument utilises the idea of iteratively taking vertices of largest degree from an independent set and constructing a hypergraph of lower uniformity which preserves independent sets and inherits edge distribution. The original algorithms for constructing containers also remove in each step vertices of high degree, which are not in the independent set. Our modified algorithm postpones this until the end, which surprisingly results in a significantly simplified analysis.

This article is the second in a series investigating cartesian closed varieties. In first of these, we showed that every non-degenerate finitary cartesian variety is a variety of sets equipped with an action by a Boolean algebra B and a monoid M which interact to form what we call a matched pair ${\left[\smash{{B} \mathbin{\mid}{M} }\right]}$. In this article, we show that such pairs ${\left[\smash{{B} \mathbin{\mid}{M} }\right]}$ are equivalent to Boolean restriction monoids and also to ample source-étale topological categories; these are generalizations of the Boolean inverse monoids and ample étale topological groupoids used to encode self-similar structures such as Cuntz and Cuntz–Krieger $C^\ast$-algebras, Leavitt path algebras, and the $C^\ast$-algebras associated with self-similar group actions. We explain and illustrate these links and begin the programme of understanding how topological and algebraic properties of such groupoids can be understood from the logical perspective of the associated varieties.

We determine the list of automorphism groups for smooth plane septic curves over an algebraically closed field $K$ of characteristic $0$, as well as their signatures. For each group, we also provide a geometrically complete family over $K$, which consists of a generic defining polynomial equation describing each locus up to $K$-projective equivalence. Notably, we present two distinct examples of what we refer to as final strata of smooth plane curves.

In the present work, we investigate the Lie algebra of the Formanek-Procesi group $\textrm {FP}(A_{\Gamma })$ with base group $A_{\Gamma }$ a right-angled Artin group. We show that the Lie algebra $\textrm {gr}(\textrm {FP}(A_{\Gamma }))$ has a presentation that is dictated by the group presentation. Moreover, we show that if the base group $G$ is a finitely generated residually finite $p$-group, then $\textrm { FP}(G)$ is residually nilpotent. We also show that $\textrm {FP}(A_{\Gamma })$ is a residually torsion-free nilpotent group.

We prove that the class of separably algebraically closed valued fields equipped with a distinguished Frobenius endomorphism $x \mapsto x^q$ is decidable, uniformly in q. The result is a simultaneous generalization of the work of Chatzidakis and Hrushovski (in the case of the trivial valuation) and the work of the first author and Hrushovski (in the case where the fields are algebraically closed).

The logical setting for the proof is a model completeness result for valued fields equipped with an endomorphism $\sigma $ which is locally infinitely contracting and fails to be onto. Namely, we prove the existence of a model complete theory $\widetilde {\mathrm {VFE}}$ amalgamating the theories $\mathrm {SCFE}$ and $\widetilde {\mathrm {VFA}}$ introduced in [5] and [11], respectively. In characteristic zero, we also prove that $\widetilde {\mathrm {VFE}}$ is NTP$_2$ and classify the stationary types: they are precisely those orthogonal to the fixed field and the value group.

Étant donnée une suite $A = (a_n)_{n\geqslant 0}$ d’entiers naturels tous au moins égaux à 2, on pose $q_0 = 1$ et, pour tout entier naturel n, $q_{n+1} = a_n q_n$. Tout nombre entier naturel $n\geqslant 1$ admet une unique représentation dans la base A, dite de Cantor, de la forme

$$ \begin{align*} m = \sum_{j\geqslant 0}\varepsilon_j(m)q_j\quad \text{avec} \quad 0\leqslant \varepsilon_j(m) < a_j = \frac{q_{j+1}}{q_j}. \end{align*} $$

$$ \begin{align*} S = \sum_{n \leqslant x}\Lambda(n) f(n) \end{align*} $$

$\Lambda $

$(a_n)_{n\geqslant 0}$

Nous étudions plusieurs exemples dans la base $A = (j+2)_{j\geqslant 0}$, dite factorielle. En particulier, si $s_A$ désigne la fonction somme de chiffres dans cette base et p parcourt la suite des nombres premiers, nous montrons que la suite $(s_A(p))_{p\in \mathcal {P}}$ est bien répartie dans les progressions arithmétiques, et que la suite $(\alpha s_A(p))_{p\in \mathcal {P}}$ est équirépartie modulo $1$ pour tout nombre irrationnel $\alpha $.

In this paper, we investigate hypersurfaces of $\mathbb{S}^2\times \mathbb{S}^2$ and $\mathbb{H}^2\times \mathbb{H}^2$ with recurrent Ricci tensor. As the main result, we prove that a hypersurface in $\mathbb{S}^2\times \mathbb{S}^2$ (resp. $\mathbb{H}^2\times \mathbb{H}^2$) with recurrent Ricci tensor is either an open part of $\Gamma \times \mathbb{S}^2$ (resp. $\Gamma \times \mathbb{H}^2$) for a curve $\Gamma$ in $\mathbb{S}^2$ (resp. $\mathbb{H}^2$), or a hypersurface with constant sectional curvature. The latter has been classified by H. Li, L. Vrancken, X. Wang, and Z. Yao very recently.

An element of a group is called strongly reversible or strongly real if it can be expressed as a product of two involutions. We provide necessary and sufficient conditions for an element of $\mathrm{SL}(n,\mathbb{C})$ to be a product of two involutions. In particular, we classify the strongly reversible conjugacy classes in $\mathrm{SL}(n,\mathbb{C})$.

We study several basic problems about colouring the $p$-random subgraph $G_p$ of an arbitrary graph $G$, focusing primarily on the chromatic number and colouring number of $G_p$. In particular, we show that there exist infinitely many $k$-regular graphs $G$ for which the colouring number (i.e., degeneracy) of $G_{1/2}$ is at most $k/3 + o(k)$ with high probability, thus disproving the natural prediction that such random graphs must have colouring number at least $k/2 - o(k)$.

In this article, we deal with non-existence results, i.e., Liouville type results, for positive radial solutions of quasilinear elliptic equations with weights both in the entire $\mathbb R^N$ and in a ball, in the latter case under Dirichlet boundary conditions. The presence of weights, possibly singular or degenerate, makes the study fairly delicate. The proofs use a Pohozaev type identity combined with an accurate qualitative analysis of solutions. In the last part of the article, a non-existence theorem is proved for a Dirichlet problem with a convection term.

We provide a characterization of equivariant Fock covariant injective representations for product systems. We show that this characterization coincides with Nica covariance for compactly aligned product systems over right least common multiple semigroups of Kwaśniewski and Larsen and with the Toeplitz representations of a discrete monoid of Laca and Sehnem. By combining with the framework established by Katsoulis and Ramsey, we resolve the reduced Hao–Ng isomorphism problem for generalized gauge actions by discrete groups.

Asymptotic homogenisation is considered for problems with integral constraints imposed on a slowly varying microstructure; an insulator with an array of perfectly dielectric inclusions of slowly varying size serves as a paradigm. Although it is well-known how to handle each of these effects (integral constraints, slowly varying microstructure) independently within multiple scales analysis, additional care is needed when they are combined. Using the flux transport theorem, the multiple scales form of an integral constraint on a slowly varying domain is identified. The proposed form is applied to obtain a homogenised model for the electric potential in a dielectric composite, where the microstructure slowly varies and the integral constraint arises due to a statement of charge conservation. A comparison with multiple scales analysis of the problem with established approaches provides validation that the proposed form results in the correct homogenised model.

A p-arithmetic subgroup of $\mathbf {SL}_2(\mathbb {Q})$ like the Ihara group $\Gamma := \mathbf {SL}_2(\mathbb {Z}[1/p])$ acts by Möbius transformations on the Poincaré upper half plane $\mathcal H$ and on Drinfeld’s p-adic upper half plane ${\mathcal H_p := \mathbb {P}_1(\mathbb {C}_p)-\mathbb {P}_1(\mathbb {Q}_p)}$. The diagonal action of $\Gamma $ on the product is discrete, and the quotient $\Gamma \backslash (\mathcal H_p\times \mathcal H)$ can be envisaged as a ‘mock Hilbert modular surface’. According to a striking prediction of Neková$\check {\text {r}}$ and Scholl, the CM points on genuine Hilbert modular surfaces should give rise to ‘plectic Heegner points’ that encode nontrivial regulators attached, notably, to elliptic curves of rank two over real quadratic fields. This article develops the analogy between Hilbert modular surfaces and their mock counterparts, with the aim of transposing the plectic philosophy to the mock Hilbert setting, where the analogous plectic invariants are expected to lie in the alternating square of the Mordell–Weil group of certain elliptic curves of rank two over $\mathbb {Q}$.

We study the long-time asymptotics for the solution of the modified Camassa–Holm (mCH) equation with step-like initial data.

\begin{align*} &m_{t}+\left (m\left (u^{2}-u_{x}^{2}\right )\right )_{x}=0, \quad m=u-u_{xx}, \\[3pt] & {u(x,0)=u_0(x)\to \left \{ \begin{array}{l@{\quad}l} 1/c_+, &\ x\to +\infty, \\[3pt] 1/c_-, &\ x\to -\infty, \end{array}\right .} \end{align*}

$c_+$

$c_-$

$(y,t)$

$(\xi, c)$

$c=c_+/c_-$

$\{ (\xi, c)\,:\, \xi \in \mathbb{R}, \ c\gt 0, \ \xi =y/t\}$

$u(y,t)$

$\mathcal{O}(t^{-2})$

$\mathcal{O}(t^{-1})$

We consider the chemotaxis-Navier–Stokes system with generalised fluid dissipation in $\mathbb{R}^3$:

\begin{eqnarray*} \begin{cases} \partial _t n+u\cdot \nabla n=\Delta n- \nabla \cdot (\chi (c)n \nabla c),\\[5pt] \partial _t c+u \cdot \nabla c=\Delta c-nf(c),\\[5pt] \partial _t u +u \cdot \nabla u+\nabla P=-(\!-\Delta )^\alpha u-n\nabla \phi, \\[5pt] \nabla \cdot u=0, \end{cases} \end{eqnarray*}

$\alpha \gt \frac {3}{4}$

$\tilde {\textrm {a}}$

$\alpha \gt \frac {1}{2}$

$\alpha \geq \frac {5}{4}$

$\frac {3}{4}\lt \alpha \lt \frac {5}{4}$

$L^2$

We describe algebraically, diagrammatically, and in terms of weight vectors, the restriction of tensor powers of the standard representation of quantum $\mathfrak {sl}_2$ to a coideal subalgebra. We realize the category as a module category over the monoidal category of type $\pm 1$ representations in terms of string diagrams and via generators and relations. The idempotents projecting onto the quantized eigenspaces are described as type $B/D$ analogues of Jones–Wenzl projectors. As an application, we introduce and give recursive formulas for analogues of $\Theta$-networks.

Bioreactor scaffolds must be designed to facilitate adequate nutrient delivery to the growing tissue they support. For perfusion bioreactors, the dominant transport process is determined by the scale of fluid velocity relative to diffusion and the geometry of the scaffold. In this paper, models of nutrient transport in a fibrous bioreactor scaffold are developed using homogenisation via multiscale asymptotics. The scaffold is modelled as an ensemble of aligned strings surrounded by viscous, slowly flowing fluid. Multiple scales analysis is carried out for various parameter regimes which give rise to macroscale transport models that incorporate the effects of advection, reaction and diffusion. Multiple scales in both space and time are employed when macroscale advection balances macroscale diffusion. The microscale model is solved to obtain the effective diffusion coefficient and simple solutions to the macroscale problem are presented for each regime.

Let f be a non-constant meromorphic function. We define its linear differential polynomial $ L_k[f] $ by

\begin{equation*}L_k[f]=\displaystyle b_{-1}+\sum_{j=0}^{k}b_jf^{(j)}, \text{where}\; b_j (j=0, 1, 2, \ldots, k) \; \text{are constants with}\; b_k\neq 0.\end{equation*}

$ L_k[f] $

$ f^n+g^n=1 $