As noted by Landau and Lifshitz in their work Course of Theoretical Physics (Vol. 6, Pergamon Press, Oxford, 2nd edition, 1987), “if the relaxation time of these processes is long, a considerable dissipation of energy occurs when the fluid is compressed or expanded,” then the volume viscosity coefficient becomes relatively large, making the shear viscosity coefficient much smaller in comparison. In this article, we take this phenomenon into consideration and investigate isentropic compressible magnetohydrodynamic flow with zero shear viscosity in a periodic domain. We prove the global existence of smooth solutions when the initial data are close to a background magnetic field. In addition, stability and large-time decay rates are also obtained. Mathematically, the zero shear viscosity makes the elliptic operator “$\mu \Delta +\nu \nabla \text{div}\, $” lose its uniform ellipticity, rendering the classical dissipation mechanisms for velocity inapplicable. The stabilizing effect of the background magnetic field plays a crucial role in our analysis.

We show that induction along a Frobenius extension of Hopf algebras is a Frobenius monoidal functor in great generality, in particular, for all finite-dimensional and all pointed Hopf algebras. As an application, we show that induction functors from unimodular Hopf subalgebras to small quantum groups at roots of unity are Frobenius monoidal functors and classify such unimodular Hopf subalgebras. Moreover, we present stronger conditions on Frobenius extensions under which the induction functor extends to a braided Frobenius monoidal functor on categories of Yetter–Drinfeld modules. We show that these stronger conditions hold for any extension of finite-dimensional semisimple and co-semisimple (or, more generally, unimodular and dual unimodular) Hopf algebras.

We give a model-independent definition of limits for diagrams valued in an $(\infty ,n)$-category. We show that this definition is compatible with the existing notion of homotopy $2$-limits for $2$-categories, with the existing notion of $(\infty ,1)$-limits for $(\infty ,1)$-categories, and with itself across different values of n.

In Benedetto et al. (2025, J. Lond. Math. Soc., 112, Article no. e70257), we provided an explicit description of the arboreal Galois group of the postcritically finite polynomial $f(z) = z^2 +c$ in the special case when the critical point $0$ is periodic under the action of $f(z)$. In the current article, we complete the picture for all postcritically finite polynomials by addressing the cases when $0$ is strictly preperiodic for the polynomial $f(z)$.

Let $(F_r M)_{r\in {\mathbb Z}}$ be a finite filtration on an object M of a neutral Tannakian category ${\mathbf {T}}$ in characteristic zero. Let ${\mathfrak {u}}(M)={\mathfrak {u}}^F(M)$ be the Lie algebra of the subgroup of the Tannakian fundamental group of M that acts trivially on the associated graded $\mathrm {Gr}^FM$. The filtration $F_\bullet M$ induces a filtration on the internal Hom ${\underline {\mathrm{Hom}}}(M,M)$, which in turn induces a filtration $F_\bullet {\mathfrak {u}}(M)$ on ${\mathfrak {u}}(M)$. This filtration on ${\mathfrak {u}}(M)$ is concentrated in negative degrees. In this article, we give a description of the graded piece $\mathrm {Gr}^F_{-1}{\mathfrak {u}}( M)$ in terms of the extensions $F_{r+1}M/F_{r-1}M\in \mathrm {Ext}^1(\mathrm {Gr}^F_{r+1}M, \mathrm {Gr}^F_{r}M)$. In particular, these extensions determine $\mathrm {Gr}^F_{-1}{\mathfrak {u}}(M)$. Note that here we neither assume the filtration is functorial, nor we assume that $\mathrm {Gr}^FM$ is semisimple. The problem of studying ${\mathfrak {u}}(M)$ in this generality is motivated by the desire to understand Tannakian groups associated with a mixed motive and its realizations, including realizations for which semisimplicity of realizations of pure motives is not known and realizations that lack an interesting functorial weight filtration. We also give two related applications. The first is an equivalent condition in the generality described above for when ${\mathfrak {u}}(M)$ coincides with its trivial upper bound $F_{-1}{\underline {\mathrm{Hom}}}(M,M)$. This result generalizes earlier criteria for maximality of ${\mathfrak {u}}(M)$ obtained by various authors in special contexts or under limiting conditions. In the second application, we apply the results to the setting of a neutral Tannakian category with a functorial weight filtration $W_\bullet $. Combining the constructions of [7] with our general maximality criterion, we prove a result about the structure of the set of isomorphism classes of objects M for which $\mathrm {Gr}^WM$ is isomorphic to a given graded object A and ${\mathfrak {u}}^W(M)=W_{-1}{\underline {\mathrm{Hom}}}(M,M)$.

We classify translatively exponential and $\mathrm {GL}(2,\mathbb Z)$ covariant valuations on lattice polygons with values in the space of real (complex) measurable functions. A typical example of such valuations is induced by the Laplace transform, but as it turns out there are many more. The argument uses the ergodicity of the linear action of $\mathrm {SL}(2,\mathbb Z)$ on $\mathbb R^2$, and some elementary properties of the Fibonacci numbers.

We study the weighted compactness and boundedness of Toeplitz operators on the Fock spaces. Fix $\alpha>0$. Let $T_{\varphi }$ be the Toeplitz operator on the Fock space $F^2_{\alpha }$ over $\mathbb {C}^n$ with symbol $\varphi \in L^{\infty }$. For $1<p<\infty $ and any finite sum T of finite products of Toeplitz operators $T_{\varphi }$’s, we show that T is compact on the weighted Fock space $F^p_{\alpha ,w}$ if and only if its Berezin transform vanishes at infinity, where w is a restricted $A_p$-weight on $\mathbb {C}^n$. Concerning boundedness, for $1\leq p<\infty $, we characterize the r-doubling weights w such that $T_{\varphi }$ is bounded on the weighted spaces $L^p_{\alpha ,w}$ via a $\varphi $-adapted $A_p$-type condition. Our method also establishes a two-weight inequality for the Fock projections in the case of r-doubling weights. Moreover, we characterize the corresponding weighted compactness of Bergman–Toeplitz operators, which answers a question raised by Stockdale and Wagner (2023, Math. Z. 305, Article no. 10).

Let V be a finite-dimensional complex vector space. Assume that V is a direct sum of subspaces each of which is equipped with a nondegenerate symmetric or skew-symmetric bilinear form. In this article, we introduce a stratification of the Grassmannian $\text {Gr}_k(V)$ related to the action of the appropriate product of orthogonal and symplectic groups, and we study the topology of this stratification. The main results involve sheaves with coefficients in a field of characteristic other than $2$. We prove that there are “enough” parity sheaves, and that the hypercohomology of each parity sheaf also satisfies a parity-vanishing property. This situation arises in the following context: let x be a nilpotent element in the Lie algebra of either $G = \text {Sp}_N(\mathbb {C})$ or $G = \text {SO}_N(\mathbb {C})$, and let $V = \ker x \subset \mathbb {C}^N$. Our stratification of $\text {Gr}_k(V)$ is preserved by the centralizer $G^x$, and we expect our results to have applications in Springer theory for classical groups.

A graph G is H-induced-saturated if G is H-free but deleting any edge or adding any edge creates an induced copy of H. There are nontrivial graphs H, such as $P_4$, for which no finite H-induced-saturated graph G exists. We show that for every finite graph H that is not a clique or an independent set, there always exists a countable H-induced-saturated graph. In fact, we show that a far stronger property can be achieved: there is a countably infinite H-free graph G such that any graph $G'\ne G$ obtained by making a locally finite set of changes to G contains a copy of H.

Graph burning is a discrete process that models the spread of influence through a network using a fire as a proxy for the type of influence being spread. This process was recently extended to apply to hypergraphs in both round-based and lazy settings. We introduce a variant of hypergraph burning that uses an alternative propagation rule for how the fire spreads – if some fixed proportion of vertices are on fire in a hyperedge, then in the next round, the entire hyperedge catches fire.

We obtain bounds on the burning numbers of general hypergraphs, and introduce the concept of the burning distribution, which describes how the burning numbers change as the proportion parameter ranges over $(0,1)$. We also obtain computational results which suggest there is a strong correlation between the automorphism group order and the lazy burning number of a balanced incomplete block design.

In this article, we investigate a high-order quasilinear hyperbolic equation that involves Kirchhoff damping and logarithmic source:

$$ \begin{align*}u_{tt}-{\operatorname*{div}}(|\nabla u|^{p-2}\nabla u)+\sigma(\|\nabla u\|_{2}^{2} )u_t+\Delta^{2}u=|u|^{q-2}u\log|u|,\end{align*} $$

$\Omega \times (0,T_{\mathrm {max}})$

$\Omega \subset \mathbb {R}^{n}$

$p,q>2$

$\sigma (\|\nabla u\|_{2}^{2} )$

$u_t$

$q\geq p$

$q>p$

$q<p$

This article studies the stability of the $L_p$ torsional measure in dimension $n \geq 3$. We show that the ball is the unique domain for the solution to the related overdetermined boundary value problem. Two concepts of relative asymmetry distance and $L^c$-distance have been used to estimate the stability of the $L_p$ torsional measure, based on different stability results of the $L_p$-width functionals. The result has been uniformly given via relative asymmetry distance for all $1<p \neq n+2$. In terms of $L^c$-distance, the result has been split into two cases: $p\in (1,n)$, and $n< p\neq n+2$.

We investigated the symplectic geometry of homogeneous spaces associated with semisimple Lie groups, focusing on cotangent bundles of maximal flag manifolds. Our work provides an explicit description of the canonical symplectic structure on these spaces using connections and curvatures of principal bundles naturally associated with the underlying Lie groups. We extend classical results concerning the exactness of symplectic forms on adjoint orbits, previously known for specific Lie algebras, to arbitrary simple Lie groups. In particular, we identify conditions under which the Kostant–Kirillov–Souriau form on a regular adjoint orbit coincides with the canonical symplectic form of the cotangent bundle, yielding exact symplectic structures. The approach combines differential-geometric techniques with Lie-theoretic constructions, offering a unifying framework that connects the geometry of coadjoint orbits with symplectic structures on homogeneous spaces.

The well-known proof of Beurling’s Theorem in the Hardy space $H^2$, which describes all shift-invariant subspaces, rests on calculating the orthogonal projection of the unit constant function onto the subspace in question. Extensions to other Hardy spaces $H^p$ for $0 < p < \infty $ are usually obtained by reduction to the $H^2$ case via inner–outer factorization of $H^p$ functions. In this article, we instead explicitly calculate the metric projection of the unit constant function onto a shift-invariant subspace of the Hardy space $H^p$ when $1<p<\infty $. This problem is equivalent to finding the best approximation in $H^p$ of the conjugate of an inner function. In $H^2$, this approximation is always a constant, but in $H^p$, when $p\neq 2$, this approximation turns out to be zero or a non-constant outer function. Further, we determine the exact distance between the unit constant and any shift-invariant subspace and propose some open problems. Our results use the notion of Birkhoff–James orthogonality and Pythagorean inequalities, along with an associated dual extremal problem, which leads to some interesting inequalities. Further consequences shed light on the lattice of shift-invariant subspaces of $H^p$, as well as the behavior of the zeros of optimal polynomial approximants in $H^p$.

An automorphism of the free group $F_n$ is called pure symmetric if it sends each generator to a conjugate of itself. The group $\mathrm {PSAut}_n$ of all pure symmetric automorphisms and its quotient $\mathrm {PSOut}_n$ by the group of inner automorphisms are called the McCool groups. In this article, we prove that every BNSR-invariant $\Sigma ^m$ of a McCool group is either dense or empty in the character sphere, and we characterize precisely when each situation occurs. Our techniques involve understanding higher generation properties of abelian subgroups of McCool groups, coming from the McCullough–Miller space. We also investigate further properties of the second invariant $\Sigma ^2$ for McCool groups using a general criterion due to Meinert for a character to lie in $\Sigma ^2$.

The objective of this article is twofold. In the first half of the article, we investigate upper parts of the hyperspace convergences determined by uniform convergence of distance functionals on a bornology under different metrizations of a metrizable space. To do this, a new covering property associated with the underlying bornology is introduced. An independent study of this new covering notion in relation to some well-known notions, such as strong uniform continuity, is also presented. In the second half, we study the infima of hyperspace convergences (induced by distance functionals) determined by a family of (uniformly) equivalent metrics. In particular, we establish the existence of the minimum element for the collection of upper Attouch–Wets convergences corresponding to all equivalent metrics on a metrizable space X. We show that such a minimum element exists if and only if X has a compatible Heine–Borel metric. Our findings give several new insights into the theory of hyperspace convergences.

I provide several natural properties of group actions which translate into fragments of axiom of choice in the associated permutation models of choiceless set theory.

Let $\omega (n)$ denote the number of distinct prime factors of a natural number n. In 1940, Erdős and Kac established that $\omega (n)$ obeys the Gaussian distribution over natural numbers, and in 2004, the third author generalized their theorem to all abelian monoids. In this article, we extend her theorem to any subsets of an abelian monoid satisfying some additional conditions, and apply this result to the subsets of h-free and h-full elements. We study generalizations of several arithmetic functions, such as the prime counting omega functions and the divisor function in a unified framework. Finally, we apply our results to number fields, global function fields, and geometrically irreducible projective varieties, demonstrating the broad relevance of our approach.

In their 2016 paper on exotic Bailey–Slater SPT-functions, Garvan and Jennings-Shaffer introduced many new spt-crank-type functions and proposed a conjecture that the spt-crank-type functions $M_{C1}(m,n)$ and $M_{C5}(m,n)$ are both nonnegative for all $m\in \mathbb {Z}$ and $n\in \mathbb {N}.$ Applying Wright’s circle method, Jang and Kim showed that $M_{C1}(m,n)$ and $M_{C5}(m,n)$ are both positive for a fixed integer m and large enough integers $n.$ Up to now, no complete proof of this conjecture has been given. In this article, we provide a complete proof for this conjecture by using the theory of lattice points. Our proof is quite different from that of Jang and Kim.

The metrics induced on free boundary minimal surfaces in geodesic balls in the upper unit hemisphere and hyperbolic space can be characterized as critical metrics for the functionals $\Theta _{r,i}$ and $\Omega _{r,i}$, introduced recently by Lima, Menezes, and the second author. In this article, we generalize this characterization to free boundary minimal submanifolds of higher dimension in the same spaces. We also introduce some functionals of the form different from $\Theta _{r,i}$ and show that the critical metrics for them are the metrics induced by free boundary minimal immersions into a geodesic ball in the upper unit hemisphere. In the case of surfaces, these functionals are bounded from above and not bounded from below. Moreover, the canonical metric on a geodesic disk in a 3-ball in the upper unit hemisphere is maximal for this functional on the set of all Riemannian metric of the topological disk.