Papers

In this paper, we consider a free boundary problem with fractional diffusion $(-\Delta)^s$ $(0<s<1)$, as a model for species spreading, which can be viewed as a natural extension of the free boundary model in \cite{CDLL2019}, where the nonlocal diffusion term is defined via a continuous integrable kernel function $J(x)$. This fractional diffusion model can also be viewed as a nonlocal version of the free boundary model in \cite{DuLin2010}, whose local diffusion term is given by the classical Laplacian. We first establish the global existence and uniqueness of the solution, which is the main contribution of this paper. This well-posedness question has been open for some time now due to the difficulties caused by the singularity and non-integrability of the associated kernel function of $(-\Delta)^s$, and the lack of general enough regularity results for operators extending the standard fractional Laplacian. A crucial step that enables us to answer this question relies on an approximation approach, which breaks an associated linear fractional parabolic initial boundary value problem with curved boundaries into a sequence of approximating problems, where the functions representing the curved boundaries are replaced by approximating step functions. We then show that, for Fisher-KPP type nonlinear growth terms, the long-time dynamics of the model exhibits a spreading-vanishing dichotomy, similar to the earlier models of \cite{DuLin2010, CDLL2019}. More precise descriptions of the spreading profile of the solution will be considered in a separate work.

Social guidance, as a universal mechanism to promote collective cooperation, effectively maintains and enhances the level of population cooperation by correcting the behavior of defectors. However, existing guidance models have two key limitations: first, they assume that guidance actions are inevitably successful; second, they neglect information sharing among guides, leading to repeated guidance of the same successfully guided defector. Additionally, the psychological characteristics of individuals are often not fully considered in existing models. Here, we construct a stochastic sequence guidance model by introducing random ordering of guides and termination conditions for guidance actions to optimize the efficiency of guidance. Besides, we incorporate individual psychological traits such as persuasion-prone (where the success rate of persuasion increases with attempts) and persuasion-averse (where the success rate of persuasion decreases with attempts) into the modeling framework. We find that the introduction of random sequence guidance can not only effectively promote cooperation but also mitigate the second-order free-rider problem. Importantly, cooperation thrives with stronger persuasion, lower costs, and larger groups.

We introduce a deep kernel hedging framework that combines the flexibility of deep learning with the structural inductive bias of kernel methods. The hedging functional is restricted to a reproducing kernel Hilbert space whose kernel is parameterized through a neural network embedding of the input features. The framework minimizes a regularized empirical risk under convex loss functions and can accommodate path-dependent information through truncated time-augmented signature features. We derive a generalized representer theorem for the joint hedging problem, reducing the empirical optimization to a finite-dimensional problem. To further reduce the computational cost associated with large kernel matrices, we develop a scalable random Fourier feature approximation and establish convergence guarantees. The random Fourier parameters are sampled once and remain fixed throughout training, while the deep kernel adapts to market data through the learned neural representation. We evaluate the performance of the proposed deep kernel approach on both synthetic and real data and compare it with standard kernel methods and classical deep hedging architectures. Numerical results indicate competitive and robust hedging performance, particularly in low-data regimes, which highlights the benefits of combining expressive neural representations with the inductive bias of kernel methods.

Spencer's 1985 “six standard deviations suffice” theorem shows that every $A \in [-1,1]^{n \times n}$ has a sign vector $x \in \{-1,1\}^n$ with $\|Ax\|_\infty \le 6\sqrt{n}$. We show the upper bound $\sqrt{3\operatorname{arsinh}(10)}\sqrt{n}+4 < 2.9992 \sqrt{n} + 4$ by directly rounding the minimizer of a potential function to a vertex of the cube. We also show that, for every power of two $n \ge 2^{50}$, there exists a matrix $A \in \{-1,1\}^{n \times n}$ such that $\|Ax\|_\infty >1.0000002\sqrt{n}$ for every choice of signs $x \in \{-1,1\}^n$. The construction simply replaces a $2^{-22}$ fraction of the columns of a Hadamard matrix with independent random sign vectors. This is the first improvement over the $\sqrt{n}$ lower bound of Olson and Spencer (1978), which uses a Hadamard matrix.

This thesis studies how disorder and fluctuations shape large-scale growth phenomena in three related settings: fluctuating interfaces, random spring chains, and random multiplicative growth. The first part concerns interface growth and a phenomenon denoted anomalous scaling, where local and global exponents differ.To clarify its origin, we study two models in which this mismatch has different causes. The first is a heterogeneous elastic line, where we show that the apparent anomaly is a purely statistical effect. The second is the stochastic porous medium equation, a strongly nonlinear model in which the anomaly is genuine. The second part concerns Anderson localization, which appears here through the spectral properties of the heterogeneous elastic lines. We study chains with both random masses and random spring constants, with particular emphasis on the strong-disorder regime, where standard weak-disorder expansions break down. We develop a new combinatorial method to probe this regime. The last part concerns random multiplicative growth with redistribution. In such models, changes in wealth, population, or mass, for instance, are proportional to the amount already present, so that small differences are amplified over time. This naturally generates broad distributions and provides a simple mechanism for the emergence of inequality and concentration, with applications to population dynamics, wealth distribution, city growth, and ecology. We in particular study models in which each site has its own quenched growth rate, representing a persistent advantage or disadvantage, and is also subject to transient fluctuations. Redistribution competes with these mechanisms by homogenizing the system. This competition leads to rich phase diagrams.

We construct some part of the plectic Galois action on the locally analytic $p$-adically completed cohomology of Hilbert modular varieties, predicted by the plectic conjecture of Nekovář-Scholl. As a byproduct, we prove a vanishing result for the completed cohomology of Hilbert modular varieties below middle degree. To do this, we make use of the partial Sen operators constructed earlier by a geometric method.

We prove an $\Omega(n^2)$ lower bound on the determinantal complexity of the power sum polynomial $\sum_{i=1}^n x_i^n$ over the field of complex numbers. A similar result was claimed in a recent paper of Sheshadri (arXiv:2606.13628), via an AI-assisted and AI-written proof. Assuming its correctness, this was the first super-linear lower bound for this fundamental algebraic problem for any explicit polynomial. However, the authors of this note were unable to follow the details and verify the argument in arXiv:2606.13628, in spite of considerable effort on their part. The proof we provide here is short, (almost) self-contained and seemingly simpler.

Classical statistical theory is insufficient to explain the successes of general-purpose AI models, because it depends on handcrafted inductive biases that it cannot justify. No Free Lunch (NFL) theorems force any learner that beats chance on some environments to underperform on others. We might hope that past experience informs which environments to expect, but NFL applies equally to meta-learning. Thus, any method that makes meaningful predictions necessarily begins with an inductive bias external to the data. Choosing to bias toward short programs yields Solomonoff induction (SI), whose performance is competitive against all computable learners - albeit up to "constants" that become large when comparing against specialized methods that exploit background information. We therefore relativize SI to an information vantage point, biasing toward short programs with access to all preexisting information. This reframes the inductive bias: instead of seeking some absolute notion of simplicity, we favor accessibility with respect to our vantage point. An algorithm can only outpredict the relativized SI to the extent that its code contains additional information about the data, and no algorithm can generate such information. While SI is incomputable and hence not a practical algorithm, it provides a formal optimum for inference in the limit of infinite compute, and there is evidence to suggest that frontier AI systems roughly approximate it. Thus, the only known answer to meta-NFL is rooted in algorithmic information theory, which we should expect to play a fundamental role in explaining the generalization behavior of modern (and future) AI systems.

The $\triangle$-convexity space on a finite, simple graph $G = (V,E)$ is the collection $\mathcal{C}$ of subsets $S \subseteq V(G)$ such that whenever $x \in V(G)$ forms a triangle with two vertices in $S$, we have $x \in S$. The members of $\mathcal{C}$ are called convex sets, and the convex hull of a set $S \subseteq V(G)$, denoted $\operatorname{Hull}(S)$, is the smallest member of $\mathcal{C}$ that contains $S$. The Carathéodory (resp., exchange) number, $c_{\triangle}(G)$ (resp., $e_{\triangle}(G)$), is the size of a largest Carathéodory (resp., exchange) independent subset of $V(G)$. It was shown by Anand et al. (JCMCC 126, 2025, 11–27) that $c_{\triangle}(G) \leq t(G)+1$ and $e_{\triangle}(G) \leq t(G) + 2$, where $t(G)$ is the number of triangles in $G$, and that these bounds are tight. They also computed $c_{\triangle}(G)$ and $e_{\triangle}(G)$ for a block graph $G$ in terms of the number and arrangement of non-$K_2$ blocks in $G$. In this paper, we point out a gap in the proof in Anand et al. of the first inequality, $c_{\triangle}(G) \leq t(G)+1$, which has consequences for the proof of the second inequality, $e_{\triangle}(G) \leq t(G) + 2$, as well. Moreover, the tightness results are inadvertently applied as characterizations of the extremal graphs, leading to incorrect computations of the Carathéodory and exchange numbers of block graphs in certain cases. We fix these gaps by giving a full proof of the first inequality via a different route from that in Anand et al. Together with the argument in Anand et al., this also completes the proof of the second inequality. Our proof also leads to a characterization of the extremal graphs for each bound, which we use to compute the Carathéodory and exchange numbers of block graphs and to identify the extremal block graphs. We also determine $e_{\triangle}(G)$ exactly for $k$-trees for every $k \geq 2$.

In this largely expository article we give an introduction to higher dimensional local class field theory, more precisely class field theory for smooth projective schemes over local fields. We begin with a quick summary of some of the main results of classical local class field theory. Then we explain the main definitions and statements in the higher dimensional setting. We give a new and simple proof of the $\ell$-part, which is originally due to Jannsen–Saito and Forré. Inspired by this proof, we propose a conjecture which would imply the remaining open cases with $p$-coefficients and which improves our understanding of the $p$-part. A particular goal of the exposition is to outline analogies between the classical and the higher dimensional theory and on how the unit filtration and ramification, i.e. the $p$- and the $\ell$-part, can be interpreted in the higher dimensional setting.

We prove a rigidity theorem: a globally symmetric complex Finsler space $(M, J, F)$ is necessarily a Kähler-Berwald space, namely $F$ must be a Kähler-Berwald metric. We also obtain a Schwarz lemma for holomorphic mappings $f$ from an arbitrary bounded symmetric domain $\mathfrak{D}$ into itself whenever $\mathfrak{D}$ is endowed with an $\mbox{Aut}(\mathfrak{D})$-invariant Kähler-Berwald metric $F$ such that its holomorphic sectional curvature is bounded below and above by negative constants $-K_1<0$ and $-K_2<0$, respectively. The novelty of this Schwarz lemma is that the Lu constant of $(\mathfrak{D},F)$ is optimal for each given $F$ whenever rank$(\mathfrak{D})\geq 2$.

In this paper, we introduce and investigate the notion of phi-semi contractibility for Banach algebras. We establish a complete characterization of phi-semi contractibility through the existence of specific elements satisfying particular algebraic conditions. We thoroughly examine the hereditary properties of this concept, including its behavior under closed ideals, continuous homomorphisms with dense range, and projective tensor products. Notably, we prove that the projective tensor product A tensor B is phi tensor psi-semi contractible precisely when both A and B are semi contractible with respect to their respective characters, where phi belongs to the character space of A and psi belongs to the character space of B. Furthermore, we introduce and analyze the approximate version of phi-semi contractibility, providing equivalent characterizations.

We study the wall crossing for the moduli space of unmarked cubic surfaces at the log Calabi-Yau threshold. We prove the existence of a good moduli space for the corresponding moduli stack of boundary polarized Calabi-Yau pairs with Gorenstein underlying surfaces and describe the wall crossing morphisms explicitly. More precisely, the morphism from the K-moduli space to the log Calabi-Yau moduli space is an isomorphism, whereas the morphism from the KSBA moduli space contracts a divisor isomorphic to $\mathbb{P}^3$ to a point and is an isomorphism away from this divisor.

Non-stationary online learning has attracted much attention in recent years, as static regret is insufficient to guide algorithm design in changing environments. To address this limitation, interval regret and dynamic regret have been introduced as two representative performance metrics that strengthen static regret in complementary directions. Interval regret requires an online algorithm to achieve competitive static regret over every local time interval, whereas dynamic regret evaluates performance against an arbitrary sequence of time-varying comparators. Despite their importance, the relation between these metrics has long remained unclear. Prior work has often regarded interval regret as the stronger notion, based on the intuition that local guarantees should naturally induce global guarantees. Consequently, it is widely conjectured that an algorithm with optimal interval regret should automatically attain optimal dynamic regret. In this paper, we first establish a negative result that refutes this intuition of a metric-level implication. Specifically, for both convex and curved functions (including exp-concave and strongly convex functions), we show that there exist instances in which an algorithm with optimal interval regret nevertheless fails to achieve optimal dynamic regret. We then show how to leverage local adaptivity to obtain optimal dynamic regret. In particular, optimal dynamic regret can be attained by invoking an interval regret minimization process over an enlarged Euclidean ball containing the original convex feasible domain and using a suitable domain-converted surrogate loss. This reduction applies to both convex and curved functions. As a byproduct, we obtain the first proper and efficient algorithm with optimal dynamic regret for exp-concave functions, improving prior results while significantly simplifying the analysis.

For $\alpha\in(0,n)$, $p\in(1,\frac{n}{\alpha})$, $q:=\frac{np}{n-\alpha p}$, and $W\in\mathscr A_{p,q}$, we obtain the following matrix-weighted boundedness of the fractional integral $I_\alpha$: for any $\vec{f}\in L^p(W^p)$, \begin{align*} \left\|I_\alpha\vec f\right\|_{L^q(W^q)} \lesssim[W]_{\mathscr A_{p,q}}^{ (1-\frac{\alpha}{n})\max\{1,\frac{p'}{q}\}} \left\|\vec f\right\|_{L^p(W^p)}, \end{align*} where the implicit positive constant is independent of $W$ and $\vec{f}$. For integer $n\geq2$, $p\in[1,n)$, $q:=\frac{np}{n-p}$, and $W\in\mathscr A_{p,q}$, we establish the following matrix-weighted Sobolev inequality: for any smooth $\mathbb{C}^d$-valued function $\vec{f}$ with compact support, \begin{align*} \left\|\vec{f}\right\|_{L^{q}(W^q)} \lesssim[W]_{\mathscr A_{p,q}}^{\frac{n-1}{n}} \left\|WD\vec{f}\right\|_{L^p(\mathbb R^n,\mathbb C^{d\times n})}, \end{align*} where the implicit positive constant is independent of $W$ and $\vec{f}$ and where $D\vec{f}$ is the Jacobian matrix of $\vec{f}$. Surprisingly, in both bounds above the exponents of $[W]_{\mathscr A_{p,q}}$ coincide with the corresponding known optimal scalar exponents, and hence are optimal, which is different from the Calderón–Zygmund operator case. A key idea to obtain the above matrix-weighted boundedness of the fractional integrals is to establish the off-diagonal chain-packing principle of dyadic cubes, which is inspired by a recent work of A. K. Lerner.

We study the Heisenberg dynamics of continuum fermions in $\IR^d$ interacting through a pair potential. For a large class of nonconstant pair potentials and under suitable assumptions on the dispersion relation and external potential, we prove that the dynamics fails to be pointwise-norm continuous even on the gauge-invariant CAR algebra. An automatic-continuity argument then shows that neither the CAR algebra nor its gauge-invariant subalgebra is invariant under the dynamics, with non-invariance occurring for almost all times.

We study embeddings of Cohen–Macaulay local rings into their canonical modules such that the quotient of the cokernel by a regular sequence has the residue field as a direct summand. Motivated by almost Gorenstein rings, we prove an Ext-vanishing criterion for finite projective dimension, which implies G-regularity and the generalized Auslander–Reiten condition. We characterize this summand condition for one-dimensional fiber products and numerical semigroup rings. For Stanley–Reisner rings of graphs, we characterize the corresponding condition for graded embeddings in terms of the graph and show that it is equivalent to a strict multiplicity inequality.

The principal matroid determinant $E_L$ of a linear space $L \subseteq \mathbb{P}^n$ has been introduced in recent work by Matsubara-Heo and Telen. In this paper, we give the complete factorization of this polynomial into its irreducible components, proving a conjecture of the aforementioned authors. Our methods rely on bounding local multiplicities and an étale-local description of the strata of reciprocal linear spaces developed by Elias, Proudfoot and Wakefield. We also discuss analogous questions for other coordinate-wise powers of linear spaces.

Let $T$ be the classical Takagi function, and $L(y):=\{ x \in [0,1] : T(x) = y \}$ be its level set at height $y$. For each positive integer $m$, let $S_m$ be the set of $y$ for which $L(y)$ has exactly $m$ points. We prove that $S_{2n}$ has positive Lebesgue measure for every positive integer $n$, confirming a conjecture of Allaart.

The pseudo-parabolic term $u_{xxt}$ serves as a canonical example of higher-order viscosity that provides effective regularization while capturing the refined physical mechanism. How it alters the dynamics of prototype equations remains a fundamental issue, which is not yet fully understood. The goal of this paper is to show that this term induces a sharp transition in traveling wave structure, via studying the pseudo-parabolic Fisher-KPP equation $$ u_t - \tau u_{xxt} = D u_{xx} + u(1-u). $$ We completely characterize how the parameter ratio $\tau/D$ acts as a critical switch: it preserves the monotonic structure of classical traveling waves when $\tau/D \leq 1$, yet it triggers a qualitative shift to oscillatory traveling waves when $\tau/D > 1$. Numerical simulations support these findings. The threshold $\tau/D=1$ thus captures the dual role of the pseudo-parabolic term: it acts as a structure-preserving viscosity when small, but when large it reflects dominant capillary hysteresis, generating the oscillations that explain the saturation overshoot, a phenomenon that contradicts classical diffusion models yet is widely observed in two-phase flow.

Vehicle-to-Everything (V2X) cooperative perception improves 3-D detection by sharing intermediate features, but dense remote features may repeat context that the ego agent can infer locally. Most communication-efficient designs optimize masks or codes empirically, leaving a more basic question open: which remote evidence is indispensable given the receiver's own observation? We introduce a closure-fidelity perspective on ego conditioned remote perception. Under a finite deductive abstraction and explicit conditions, its rate–distortion function decomposes over an irredundant core, and the exact zero-distortion rate becomes $P_A H(\pi_A)$. This analysis suggests a concrete design principle: transmit compact evidence and recover derivable context with bounded receiver-side inference. Guided by this principle, SemRD-V2X is an operational neural proxy that combines exact-budget BEV support selection, pointwise channel compression, and masked shared-weight reconstruction before standard fusion. Experiments on simulated V2XSet and real-world DAIR-V2X validate the resulting design. In a controlled five-run V2XSet comparison against a locally reproduced V2X-ViT-v1 baseline on one Tesla V100, SemRD-V2X reduces the analytical feature payload by $26.6\times$ while improving AP@0.5/AP@0.7 by 4.13/8.57 points, with 3.81% additional mean compute latency. These results position closure fidelity as both an analytical lens and an actionable design principle for communication-efficient cooperative perception.

We study the temporal exploration problem on temporal digraphs. We prove that a lifetime of $O(n^2)$ suffices to guarantee the existence of a temporal exploration on always-unilateral temporal digraphs. We complement this with a $\Omega(n^2)$ lower bound, even in the case where each snapshot has maximum undirected degree 2; for always-strong temporal digraphs, the lower bound still holds even if the maximum undirected degree is 3. This stands in stark contrast with the undirected setting. For the large minimum degree setting, we show that a lifetime of $4n/3 - 1$ is sufficient and necessary for guaranteeing the existence of a temporal exploration on temporal digraphs where each snapshot is semicomplete. For always-strong temporal digraphs where each snapshot has minimum undirected degree at least $n - c - 1$, we prove that a lifetime of $O(cn)$ guarantees the existence of a temporal exploration, and we also prove that this is asymptotically tight. From a computational perspective, our results for temporal semicomplete digraphs also yield a polynomial-time, factor-$4/3$ algorithm for deciding if a temporal semicomplete digraph admits a temporal exploration within the first $\ell$ snapshots. We complement this showing that no polynomial-time, factor-$(4/3 - \epsilon)$ approximation algorithm exists, even if every snapshot is a tournament, unless P$=$NP.

We study the Riesz transform $\mathcal{R}=\partial(-\Delta)^{-\frac{1}{2}}$ on uniform local trees, metric measure spaces that are locally real trees and whose canonical Dirichlet form is built from weak derivatives along the skeleton. The reference measure $m$ may be singular with respect to the length measure $\nu$, so boundedness of $\mathcal R$ is understood from $L^{p}(m)$ to $L^{p}(\nu)$. In contrast with fractal-like manifolds and cable systems, the diffusion is sub-Gaussian at small scales and Gaussian at large scales. Under uniform volume growth, two-sided heat kernel estimates and a pointwise gradient estimate for the heat kernel, we prove that a local Dini condition on the scale function implies boundedness of $\mathcal{R}$ on $L^{p}$ for every $p\in[2,\infty)$, and hence the reverse Riesz inequality for every $p\in(1,2]$. Conversely, boundedness of $\mathcal{R}$ for some $p<2$, or a reverse Riesz inequality for some $p>2$, forces the space to be one-dimensional at small scales. We show that a reverse Hölder inequality for harmonic functions yields the gradient estimate, and verify all hypotheses for spaces carrying a geometric group action whose generators have bounded displacement. As an application, for the alternating Vicsek fractafold in $\mathbb Z^{d}$ we determine the exact ranges of $p$ for which the Riesz and reverse Riesz inequalities hold.

Given a metric space $Y$ and a compact subspace $X$, we present a method for constructing an inverse sequence of finite topological spaces by sampling points from neighborhoods of $X$ in $Y$. We show that these inverse sequences can be employed to define shape theory and to approximate shape invariants, such as Čech homology groups.

Building on the James–Stein approach to leading eigenvector estimation (Goldberg and Kercheval, Proc. Natl. Acad. Sci. USA 120, e2207046120, 2023), we develop a data-adaptive augmented James–Stein shrinkage framework for estimating leading eigenvectors and eigenspaces under a generalized spiked population model, in the high-dimensional regime where the dimension $p$ and sample size $n$ grow proportionally. For each spiked eigenvector, we construct an augmented target subspace that combines auxiliary information, either from domain knowledge or prior information, with the remaining sample spiked eigenvectors. This augmentation allows information shared across the sample spiked components to be exploited while retaining a fully data-driven shrinkage rule. We show that the resulting eigenvector estimator strictly improves upon standard PCA whenever the target subspace contains nonvanishing information about the population eigenvector, while asymptotically reverting to PCA when the target is uninformative. The individual estimators further yield a nested sequence of estimators for all leading spiked eigenspaces, with analogous dominance properties. The proposed estimator also strictly improves upon the existing HDLSS-motivated shrinkage estimator in the proportional high-dimensional regime. Simulation studies demonstrate substantial finite-sample gains and robustness to misspecification of the number of spikes.

We prove that the pushforwards of Gauss–Manin connections on smooth rigid analytic varieties along Zariski-open immersions are coadmissible and holonomic over Ardakov–Wadsley's sheaf D-cap of infinite order differential operators. This may be viewed as a rigid analytic variant of the classical theorem of Griffiths, Deligne, and Katz that Gauss-Manin systems have regular singularities after compactification. The proof combines p-adic de Rham comparison and Diao–Lan–Liu–Zhu's logarithmic Riemann–Hilbert correspondence with extendability results for log-connections with nilpotent residues. In the algebraic snc case, we also establish weak holonomicity via a p-adic Bernstein-Sato criterion of Bitoun and the first author.

Let $d\geq 2$ and $T^*_{\Omega,b}$ be the commutator of the rough maximal singular integral on $\mathbb{R}^d$ defined by $$T^*_{\Omega,b}f(x)=\sup_{\varepsilon>0}\left|\int_{|x-y|>\varepsilon}\bigl(b(x)-b(y)\bigr)\frac{\Omega(x-y)}{|x-y|^d}f(y)\,dy\right|.$$ Assume that $\Omega\in L(\log L)^2(\mathbb{S}^{d-1})$ has mean zero and that $b\in\operatorname{BMO}(\mathbb{R}^d)$. We prove that, for every $\lambda>0$, \[\bigl|\{x\in\mathbb{R}^d:T^*_{\Omega,b}f(x)>\lambda\}\bigr|\lesssim_{\Omega,b}\int_{\mathbb{R}^d}\frac{|f(x)|}{\lambda}\log\!\left(e+\frac{|f(x)|}{\lambda}\right)\,dx.\] The argument starts from a Calderón-Zygmund decomposition of $f$ and a dyadic linearization of the maximal truncation. A decomposition of $\Omega$ by size, followed by regularization of the spatial cutoffs and a microlocal decomposition, reduces the problem to a family of estimates with quantitative decay. The required decay follows by interpolating localized $L^1$ and $L^3$ bounds and using an analytic-family argument for the commutators.

K-means is a widely adopted clustering approach in signal processing and machine learning. In this paper, we study K-means clustering through a cardinality-constrained formulation on a compact embedded submanifold. We replace the cardinality constraint with a difference-of-convex (DC) penalty and establish a global error bound to prove that the penalized and constrained formulations share the same global minimizers whenever the penalty parameter exceeds a finite threshold. To solve the resulting nonsmooth Riemannian DC problem, we reformulate it as a minimax problem and propose RADA-DC, a Riemannian alternating descent ascent method combining dual regularization with DC linearization. Under standard assumptions and suitable parameter choices, RADA-DC finds an $\epsilon$-Riemannian critical point within $O(\epsilon^{-3})$ iterations. We conduct experiments on synthetic and real-world datasets to demonstrate that the proposed method outperforms the tested baselines, including K-means++, in solution quality at competitive computational cost when the number of clusters is large.

We study $\mathcal F$-transitivity of backward shifts on finite- and infinite-width directed lattice graphs. For the unilateral and bilateral finite-width lattices on weighted $\ell^p$- or $c_0$-spaces, we obtain exact weight characterizations of $\mathcal F$- and $\widetilde{\mathcal F}$-transitivity for an arbitrary Furstenberg family $\mathcal F$. If $\mathcal F$ is finitely invariant, these conditions also characterize topological $\mathcal F$-recurrence. For the infinite quadrant lattice with radial weights on weighted $\ell^2$-spaces, a finite-dimensional reduction and sharp smallest-singular-value estimates for Pascal-type transfer matrices yield equivalent characterizations of $\mathcal F$-transitivity, hypercyclicity and mixing. This provides a partial answer, in the radial $\ell^2$-setting, to the open problem posed by Baranov, Lishanskii and Papathanasiou concerning hypercyclicity on the infinite lattice graph. Examples at the critical exponential growth rate illustrate the finer dynamical information provided by our criteria.

Let $\mathrm{D}$ be a quaternion division algebra over a non-Archimedean local field $\mathrm{F}$ of characteristic zero, and let $\mathrm{G}_n=\mathrm{GL}_n(\mathrm{D})$. Let $\mathrm{H}_{1,n-1}=\{\mathrm{diag}(g_1,g_2)\in\mathrm{G}_1,\ g_2\in\mathrm{G}_{n-1}\}$ and, for $s \in \mathbb{R}$, define $\chi_s(\mathrm{diag}(g_1,g_2))=\nu(g_1)^{2s}\nu(g_2)^{-2s}$. We classify, for $n=3,4$, the irreducible smooth representations of $\mathrm{G}_n$ admitting a generalized linear period with respect to $(\mathrm{H}_{1,n-1},\chi_s)$. Motivated by these results, we conjecture a complete classification for all $n>2$. Assuming this conjecture, we characterize such representations in terms of Langlands parameters: an irreducible smooth representation $\pi$ of $\mathrm{G}_n$ admits such a period if and only if $\mathfrak{L}(\pi)$ contains a Weil-Deligne subrepresentation isomorphic to $\mathfrak{L}(\nu^{-2s})$, the $(2n-4)$-dimensional parameter of $\nu^{-2s}$ on $\mathrm{G}_{n-2}$, and the four-dimensional quotient is the Langlands parameter of either the trivial representation of $\mathrm{G}_2$, with $s=\pm\frac{n-2}{2}$, or an irreducible infinite-dimensional $\mathrm{H}_{1,1}$-distinguished representation of $\mathrm{G}_2$. We also verify the Lapid-Prasad conjecture in this setting: the $L$-packet of an irreducible representation of $\mathrm{G}_n$ admitting a linear period with respect to $\mathrm{H}_{1,n-1}$ is invariant under $\rho\mapsto\widetilde{\rho}^{\theta}$.