We investigate the qualitative properties of positive solutions to mixed local–nonlocal equations with indefinite nonlinearities, emphasizing the interaction between classical and fractional Laplacians. We first establish maximum principles and prove strict monotonicity along the $x_1$-direction for mixed elliptic operators. By combining a mollified first eigenfunction with a suitable sub-solution, we derive nonexistence results for the mixed operator ${(-\Delta )}^s - \Delta $ via a contradiction argument. These results are further extended to the parabolic setting, incorporating both the Marchaud-type fractional time derivative and the classical first-order derivative, revealing new qualitative features under dual nonlocality. A key aspect of our approach is a careful adaptation of the method of moving planes to the mixed local–nonlocal context. By addressing the distinct scaling behaviors of local and nonlocal terms, the method yields monotonicity and Liouville-type results without standard decay assumptions and provides a framework potentially applicable to a broader class of mixed elliptic and parabolic problems.

Splitting invariants describe how a plane curve “splits” by the pull-back under a Galois cover over the projective plane whose branch locus contains no component of the plane curve. They enable us to distinguish the embedded topology of several plane curves with the same fundamental group of the complements. In this note, we introduce a generalization of splitting invariants, called the G-combinatorial type, for plane curves by using the modified plumbing graph defined by Hironaka. We prove the invariance of the G-combinatorial type under certain homeomorphisms based on the arguments of graph manifolds by Waldhausen and plumbing graphs by Neumann. Furthermore, we distinguish the embedded topology of quasi-triangular curves by the G-combinatorial type, which are generalization of triangular curves studied by Artal Bartolo et al. (2020, Collect. Math. 71, 427–441).

In this article, we first investigate weighted Minkowski-type inequalities for nearly spherical sets in space forms, focusing on the sets that are $C^1$-close to geodesic spheres. Our results generalize the work of Glaudo (2022, Adv. Math. 408, 108595) by incorporating broader geometric settings and convex weight functions. Additionally, we establish quantitative stability estimates for weighted Alexandrov–Fenchel-type inequalities in $\mathbb {R}^{n+1}$ and $\mathbb {H}^{n+1}$, extending the earlier results of VanBlargan and Wang (2024, Commun. Contemp. Math. 26, 2350026) and Zhou and Zhou (2024, J. Geom. Anal. 34, 376). These inequalities hold for nearly spherical sets that are $W^{2,\infty }$-close to geodesic spheres coupled with general convex weights.

We prove that the Weil representation over a non-archimedean local field can be realized with coefficients in a number field. We give an explicit descent argument to describe precisely which number field the Weil representation descends to. Our methods also apply over more general coefficient fields, such as $\ell $-modular coefficient fields, as well as coefficient rings, such as rings of integers, that is, in families. We also prove that the theta correspondence over a perfect field is valid if and only if it is valid over the algebraic closure of this perfect field. These two results together show that the classical local theta correspondence is rational in the sense that it can be defined over a number field and it is compatible to Galois automorphism.

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.

The theta cycle of a modular form modulo a prime $p\geq 5$ is well understood. By contrast, the theta cycle modulo a power of p is still mysterious and experimentally erratic. Here, we completely determine the theta cycle of a weight $k < p$ modular form modulo $p^2$ on the initial segment of length p and we prove exact values or nontrivial bounds for the weight filtrations on $p-2$ further segments of length $p - k + 1$. In particular, asymptotically as $p \rightarrow \infty ,$ we establish 50% of the theta cycle exactly, and we provide nontrivial bounds for 100% of it. We determine the first two low points exactly and $\left \lfloor \frac {p-k+1}{2} \right \rfloor $ further low points at regular positions. Moreover, we detect low points at exceptional positions which solve a quadratic equation modulo p, and which disturb the otherwise regular structure in the segments that we exhibit.

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:

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

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

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

$p,q>2$

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

$u_t$

$q\geq p$

$q>p$

$q<p$

The main result of this article is the construction of a new class of weight shifting operators, similar to the theta operators of de Shalit and Goren (2019, Algebra Number Theory, 13, 1829–1877), Eischen et al. (2021, Algebra Number Theory, 15, 1469–1504), and others, which are defined on the lower Ekedahl–Oort strata of the geometric special fiber of unitary Shimura varieties of signature $(n-1, 1)$ at a good prime p, split in the reflex field E, which we assume to be quadratic imaginary. These operators act on certain graded sheaves which are obtained from the arithmetic structure of the EO strata, in particular the p-rank on each stratum. We expect these operators to have applications to the study of Hecke-eigensystems of $({\mathrm {mod}}\, p)$ modular forms and generalizations of the weight part of Serre’s conjecture.

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 $H^p$.