Given a number field F with ring of integers $\mathcal {O}_{F}$, one can associate to any torsion free subgroup of $\operatorname {SL}(2,\mathcal {O}_{F})$ of finite index a complete Riemannian manifold of finite volume with fibered cusp ends. For natural choices of flat vector bundles on such a manifold, we show that analytic torsion is identified with the Reidemeister torsion of the Borel-Serre compactification. This is used to obtain exponential growth of torsion in the cohomology for sequences of congruence subgroups.

We study the well-posedness of solutions to the general nonlinear parabolic equations with merely integrable data in time-dependent Musielak–Orlicz spaces. With the help of a density argument, we establish the existence and uniqueness of both renormalized and entropy solutions. Moreover, we conclude that the entropy and renormalized solutions for this equation are equivalent. Our results cover a variety of problems, including those with Orlicz growth, variable exponents and double-phase growth.

We provide a first-order homogenization result for quadratic functionals. In particular, we identify the scaling of the energy and the explicit form of the limiting functional in terms of the first-order correctors. The main novelty of the paper is the use of the dual correspondence between quadratic functionals and PDEs, combined with a refinement of the classical Riemann–Lebesgue lemma.

We introduce a notion of stratification for rigidly-compactly generated tensor-triangulated categories relative to the homological spectrum and develop the fundamental features of this theory. In particular, we demonstrate that it exhibits excellent descent properties. In conjunction with Balmer’s Nerves of Steel conjecture, we conclude that classical stratification also admits a general form of descent. This gives a uniform treatment of several recent stratification results and provides a complete answer to the question: When does stratification descend? As a new application, we extend earlier work on the tensor triangular geometry of equivariant module spectra from finite groups to compact Lie groups.

We prove that the average size of a mixed character sum

\begin{equation*}\sum_{1\leqslant n \leqslant x} \chi(n) e(n\theta) w(n/x)\end{equation*}

$w$

$\sqrt{x}$

$\theta$

$\chi$

$r$

$x\leqslant r$

$o(\sqrt{x})$

$\theta$

An avoshift is a subshift where for each set C from a suitable family of subsets of the shift group, the set of all possible valid extensions of a globally valid pattern on C to the identity element is determined by a bounded subpattern. This property is shared (for various families of sets C) by, for example, cellwise quasigroup shifts, totally extremally permutive (TEP) subshifts, and subshifts of finite type (SFTs) with a safe symbol. In this paper, we concentrate on avoshifts on polycyclic groups, when the sets C are what we call ‘inductive intervals’. We show that then, avoshifts are a recursively enumerable subset of subshifts of finite type. Furthermore, we can effectively compute lower-dimensional projective subdynamics and certain factors (avofactors), and we can decide equality and inclusion for subshifts in this class. These results were previously known for group shifts, but our class also covers many non-algebraic examples as well as many SFTs without dense periodic points. The theory also yields new proofs of decidability of inclusion for SFTs on free groups, and SFTness of subshifts with the topological strong spatial mixing property.

We consider $L^q$-spectra of planar graph-directed self-affine measures generated by diagonal or anti-diagonal matrices. Assuming the directed graph is strongly connected and the system satisfies the rectangular open set condition, we obtain a general closed form expression for the $L^q$-spectra. Consequently, we obtain a closed form expression for box dimensions of associated planar graph-directed box-like self-affine sets. We also provide a precise answer to a question posed by Fraser [On the $L^q$-spectrum of planar self-affine measures. Trans. Amer. Math. Soc. 368(8) (2016), 5579–5620] concerning the $L^q$-spectra of planar self-affine measures generated by diagonal matrices. An interesting observation of the closed form expression is that it is possible to calculate the $L^q$-spectrum of a measure without involving the exact $L^q$-spectra of its projections to the axes.

We consider induced representations $\operatorname {\mathrm {Ind}}_{\mathrm {P}(F)}^{\operatorname {\mathrm {G}}(F)} \pi $, where $\mathrm {P}$ is a maximal parabolic subgroup of a reductive group $\operatorname {\mathrm {G}}$ over a p-adic field F, and $(\pi , V)$ is a unitary supercuspidal representation of $\operatorname {\mathrm {M}}(F)$, $\operatorname {\mathrm {M}}$ being some Levi subgroup of $\mathrm {P}$. Imposing a certain ‘Heisenberg parabolic subgroup’ assumption on $\mathrm {P}$, we apply the method of Goldberg, Shahidi and Spallone to obtain an expression for a certain constant $R(\tilde {\pi })$, which captures the residue of a family $s \mapsto A(s, \pi , w_0)$ of intertwining operators associated to this situation, in terms of harmonic analysis on the twisted Levi subgroup $\tilde {\operatorname {\mathrm {M}}}(F) := \operatorname {\mathrm {M}}(F) w_0$. For $\operatorname {\mathrm {G}}$ absolutely almost simple and simply connected of type $G_2$ or $D_4$ (resp., $B_3$), and $\mathrm {P}$ satisfying the ‘Heisenberg’ condition, if the central character of $\pi $ is nontrivial (resp., trivial) on $\operatorname {\mathrm {A}}_{\operatorname {\mathrm {M}}}(F)$, where $\operatorname {\mathrm {A}}_{\operatorname {\mathrm {M}}}$ is the connected centre of $\operatorname {\mathrm {M}}$, our formula for $R(\tilde {\pi })$ can be rewritten in terms of the Langlands parameter of $\pi $, in the spirit of a prediction of Arthur. For the same collection of $\operatorname {\mathrm {G}}$ and $\mathrm {P}$, when these central character conditions are not satisfied, Arthur’s prediction combined with our formula for $R(\tilde {\pi })$ suggests a harmonic analytic formula for a product of one or two $\gamma $-factors associated to the situation.

This paper introduces a parallelizable lossless image compression algorithm designed for three-channel standard images and two-channel pathology images. The proposed algorithm builds on the Quite OK Image Format (QOI) by addressing its limitations in parallelizability and compression efficiency, thereby enhancing both the compression ratio and processing speed. By incorporating image context and optimizing pixel traversal sequences, the algorithm enables effective parallel processing, achieving rapid compression of million-pixel pathology images within milliseconds, and is scalable to larger whole-slide images. It also delivers exceptional performance in terms of both speed and compression ratio for standard images. Additionally, the low complexity lossless compression for images (LOCO-I) context prediction algorithm used in joint photographic experts group lossless standard (JPEG-LS) is parallelized to improve compression efficiency and speed. By implementing full-process parallelization across the entire compression workflow rather than confining parallelization to individual steps, this approach significantly enhances overall time performance.

Given a weakly almost additive sequence of continuous functions with bounded variation ${\mathcal {F}}=\{\log f_n\}_{n=1}^{\infty }$ on a subshift X over finitely many symbols, we study properties of a function f on X such that $\lim _{n\to \infty }({1}/{n})\int \log f_n\,d\mu =\int f\,d\mu $ for every invariant measure $\mu $ on X. Under some conditions, we construct a function f on X explicitly, and study a relation between the property of ${\mathcal {F}}$ and some particular types of f. As applications, we study images of Gibbs measures for continuous functions under one-block factor maps. We investigate a relation between the almost additivity of the sequences associated to relative pressure functions and the fiber-wise sub-positive mixing property of a factor map. For a special type of one-block factor maps between shifts of finite type, we study necessary and sufficient conditions for the image of a one-step Markov measure to be a Gibbs measure for a continuous function.

For upper semi-continuous potentials defined on shifts over countable alphabets, this paper ensures sufficient conditions for the existence of a maximizing measure. We resort to the concept of blur shift, introduced by T. Almeida and M. Sobottka as a compactification method for countable alphabet shifts consisting of adding new symbols given by blurred subsets of the alphabet. Our approach extends beyond the Markovian case to encompass more general countable alphabet shifts. In particular, we guarantee a convex characterization and compactness for the set of blur invariant probabilities with respect to the discontinuous shift map.

In this work, we study a class of elliptic problems involving nonlinear superpositions of fractional operators of the form

\begin{equation*}A_{\mu,p}u := \int_{[0,1]} (-\Delta)_{p}^{s} u \, d\mu(s),\end{equation*}

$\mu$

$[0,1]$

$p$

$p$

$p$

$p$

For a complete discrete valuation field K, we show that one may always glue a separated formal algebraic space $\mathfrak {X}$ over $\mathcal {O}_K$ to a separated algebraic space U over K along an open immersion of rigid spaces $j\colon \mathfrak {X}^{\mathrm {rig}}\to U^{\mathrm {an}}$, producing a separated algebraic space X over $\mathcal {O}_K$. This process gives rise to an equivalence between such ‘gluing triples’ $(U,\mathfrak {X},j)$ and separated algebraic spaces X over $\mathcal {O}_K$, which one might interpret as a version of the Beauville–Laszlo theorem for algebraic spaces rather than coherent sheaves. Moreover, an analogous equivalence exists over any excellent base. Examples due to Matsumoto imply that the result of such a gluing might be a genuine algebraic space (not a scheme) even if U and the special fibre of $\mathfrak {X}$ are projective. The proof is a combination of the Nagata compactification theorem for algebraic spaces and of Artin’s contraction theorem. We give multiple examples and applications of this idea.

In this paper, we study the quantitative recurrence properties in the case of $\mathbb {Z}$-extension of Axiom A flows on a Riemannian manifold. We study the asymptotic behavior of the first return time to a small neighborhood of the starting point. We establish results of almost everywhere convergence, and of convergence in distribution with respect to any probability measure absolutely continuous with respect to the infinite invariant measure. In particular, our results apply to geodesic flows on the $\mathbb {Z}$-cover of compact smooth surfaces of negative curvature.

Inspired by Bhatt–Scholze [BS22], in this article, we introduce prismatic cohomology for rigid analytic spaces with l.c.i. singularities, with coefficients over Fontaine’s de Rham period ring ${\mathrm {B_{dR}^+}}$.

In this paper, we consider asymptotic behaviours of multiscale multivalued stochastic systems with small noises. First of all, for general, fully coupled systems for multivalued stochastic differential equations of slow and fast motions with small noises in the slow components, we prove an averaging principle in the strong convergence sense. Moreover, a convergence rate is given in a special case. Next, for these systems, we establish the large deviation principle by the weak convergence approach. Then, for a special case, the rate function is explicitly characterized. Finally, we explain our results with an example.

We study the sets of points where a Lévy function and a translated Lévy function share a given couple of Hölder exponents, and we investigate how their Hausdorff dimensions depend on the translation parameter.

We study the relationship between the enumerative geometry of rational curves in local geometries and various versions of maximal contact logarithmic curve counts. Our approach is via quasimap theory, and we show versions of the [vGGR19] local/logarithmic correspondence for quasimaps, and in particular for normal crossings settings, where the Gromov-Witten theoretic formulation of the correspondence fails. The results suggest a link between different formulations of relative Gromov-Witten theory for simple normal crossings divisors via the mirror map. The main results follow from a rank reduction strategy, together with a new degeneration formula for quasimaps.