Papers

We study homomorphisms, with a focus on isomorphisms, between the tracial metric reduced products of finite dimensional unitary groups and of matrix algebras. A variant of Ulam stability for unitary groups and a classification of the almost surjective continuous homomorphisms between finite dimensional unitary groups are proved and then used to show that all isomorphisms of product form of these tracial reduced products are induced by almost permutations of the coordinates and coordinatewise application of automorphisms. We prove coordinate recognition for these reduced products and obtain under set theoretic assumptions rigidity and classification results for their full automorphism groups. For tracial reduced matrix algebras we obtain such rigidity result in the more general context of center-preserving $*$-homomorphisms.

In this paper, for odd $n$, we consider the graph $T_n$ with vertex set $\{x_1,....,x_n,x_{n+1}\}$ and edge set $\{x_{1}x_{2},x_{2}x_{3},...,x_{n-1}x_{n}\} \cup \{x_{\frac{n+1}{2}}x_{n+1}\}$ and prove that the Ramsey number $R(T_n,T_n)$ is equal to $\frac{3n+1}{2}$ for $n \in [1]_{4}$.

We study online optimization for a broad class of structured non-convex non-smooth problems where each loss is a composition of a difference-of-convex function with a smooth mapping, and the feasible region is defined by constraint functions of the same kind. We propose a time-smoothed proximal linear algorithm and a local-regret measure based on a proximal residual mapping. We show that this residual is a proper stationarity measure for the original problem: its fixed-point condition implies first-order stationarity. Our analysis relies on a tangent-cone characterization for a feasible region described by composite difference-of-convex constraints, which is of independent interest and allows each update to be computed via a convex optimization oracle, despite the non-convexity of the problem. We establish a local-regret bound and a bound on the total number of inner convex subproblems. We also derive an error bound connecting the proximal residual to the distance to stationarity, providing a quantitative certificate of approximate stationarity.

Let $A(n,d,w)$ denote the maximum size of a binary constant-weight code of length $n$, minimum distance $d$, and weight $w$. We construct explicit codes proving $A(23,6,10)\ge 2979$ and $A(24,6,10)\ge 4214$. These improve the best surviving explicit codes of sizes 2969 and 4174 and surpass the corresponding 1990 bounds 2970 and 4200 of Brouwer, Shearer, Sloane and Smith, whose code listings were lost. We also obtain $A(23,6,11)\ge 3539$ and $A(24,6,8)\ge 1855$. All four bounds are now listed in Brouwer's online table. The constructions use a coordinate decomposition in which one half is fixed to a known code and the complementary half is selected from its full cross-compatible pool using CHILS for maximum-weight independent set. For the 2969-word $A(23,6,10)$ incumbent, exact computations with two solver families prove insertion maximality and exclude every improving exchange deleting at most three codewords. We also analyze codes invariant under prime-order permutations: several cycle types are excluded exactly, the $5+1^{18}$ type has upper bound 499, and reproducible heuristic saturation evidence is reported for the remaining types, with $13+1^{10}$ left open. Code files, an independent validator, model descriptions, and computational logs are released.

In this paper, we investigate the generalized friendship paradox for eigenvectors (alternatively called the eigen friendship paradox and abbreviated hereafter as EFP) in the setting of inhomogeneous Erdős–Rényi random graphs whose edge probabilities are generated by a continuous graphon. We consider the adjacency matrix of the graph and take the entries of the eigenvector corresponding to its largest eigenvalue as the vertex attributes. It was shown in \cite{hazra2026generalized} that the generalized friendship paradox holds in this setting. We study the empirical distribution of the resulting bias values across the vertices and derive its limiting distribution explicitly in terms of the principal eigenvalue and the corresponding eigenfunction of the integral operator whose kernel is the underlying graphon.

This paper proposes FullPASS, an in-band full-duplex architecture for pinching-antenna systems (PASSs) based on two parallel waveguides. The FullPASS transceiver simultaneously communicates with a single full-duplex user terminal over the same time-frequency resource: the transmit waveguide delivers the downlink signal, while the receive waveguide collects the uplink signal. Candidate pinching elements are placed along both waveguides, and the FullPASS transceiver jointly selects the active transmit and receive elements. We derive a geometry-based channel model for the downlink, uplink, and transmit-to-receive self-interference paths, including free-space propagation, in-waveguide propagation phase and attenuation, and the attenuation caused by upstream activated elements along each waveguide. The joint activation problem is formulated as a binary optimization that maximizes the bidirectional sum spectral efficiency while keeping the self-interference leakage at the FullPASS receiver below a prescribed threshold. To solve the resulting nonconvex combinatorial problem, we develop a two-stage algorithm. The first stage uses phase-anchored second-order-cone relaxations and deterministic rounding to generate binary trial activation patterns under a simplified propagation model. The second stage applies alternating best-improvement local search with add, remove, and swap operations evaluated under the full propagation model. Simulations show that the proposed method achieves an average sum spectral efficiency within 0.95% of exhaustive search on both the 13-by-13 and 15-by-15 candidate grids. On the 15-by-15 grid, it reduces the average runtime by more than one order of magnitude relative to exhaustive search and remains applicable to substantially larger candidate sets.

We introduce RELTA-SGLD, a taming scheme that stabilizes superlinear stochastic-gradient updates while reducing unnecessary suppression of the original learning drift. A threshold determines where the taming turns on, while a relative-growth principle derived from the one-step Lyapunov stability condition determines the required taming strength. Together, they produce a lighter $\lambda$-scale denominator and preserve a nonvanishing far-tail return. As a consequence, we prove polynomial moment stability and first-order stationary accuracy in both $W_1$ and $W_2$ for nonconvex SGLD with superlinearly growing stochastic-gradient oracles, improving the corresponding half-order and quarter-order bounds for comparable stochastic-gradient tamed schemes. On Fashion-MNIST under active stabilization pressure, RELTA improves the mean learning metrics over both untamed SGLD and TUSLA and remains competitive with a tuned AdamW reference. In an ordinary-training regime, its lighter localized denominator reduces unnecessary perturbation of the original update and maintains nearly untamed learning dynamics.

Let $O_n$ be the odometer semigroup, and let $\mathcal A_\xi=C^*(S_1,\ldots,S_n,W_\xi)\subseteq\mathcal B(\mathcal F_n^2)$ be the $C^*$-algebra generated by the left creation operators and the scalar odometer map associated with a symbol $\xi\in\mathcal F_n^2$. We show that $\mathcal A_\xi$ contains the compact operators for every scalar symbol. For an isometric scalar symbol, we prove that $W_\xi$ is Fredholm if and only if the associated inner function is a finite Blaschke product. We further show that the image of $\mathcal A_\xi$ in the Calkin algebra is canonically isomorphic to the odometer boundary quotient $\mathcal Q(O_n)$. If the associated finite Blaschke product has degree $d$, then $\operatorname{ind}(W_\xi)=-d$. For $d\geq 1$, we obtain $K_0(\mathcal A_\xi)\cong\mathbb Z\oplus\mathbb Z_{d(n-1)}$ and $K_1(\mathcal A_\xi)=0$, whereas for $d=0$, $K_0(\mathcal A_\xi)\cong\mathbb Z^2$ and $K_1(\mathcal A_\xi)\cong\mathbb Z$. Consequently, for fixed $n\geq 2$, finite Blaschke symbols of distinct degrees generate non-isomorphic $C^*$-algebras.

Power grids, markets, and interacting populations, settle into feedback driven equilibria observed through unknown sensors. Our Equilibrium Causal Game (ECG) joins a game to its cyclic causal model, hidden inputs, sensor map, and rules for interventions and equilibrium selection; interventions edit declared objects and recompute equilibrium. Under stated conditions, ECG-separation is sound but incomplete in our examples. Back-door/half-trek routes identify observed queries. Yet for an untouched rotationally symmetric Gaussian block, second moments determine only a source-frame rotation, across which distinct-variable effects generically change. Unknown sensing creates a separate ambiguity. In passive stable linear models without self-effects, unknown wiring and full-rank unknown sensing leave $B$ completely unidentified for $d\ge2$. Under LiNG, non-Gaussianity removes the source rotation; mechanism interventions separate sensing from interactions. With unknown support, invariant sensing, aligned responses, and well-posed single-target interventions identify $(H,B)$ up to declared equivalence. Of $d$ targets, $d-1$ suffice exactly when the sole untargeted node directly parents all others; otherwise $d$ are needed. Acquisition probes are excluded; known wiring gives no universal count. With nonlinear sensing, isotropic Gaussian source blocks admit hidden twists within and across blocks in labelled environments preserving required radial laws. Conversely, under stated positivity, informative one-block changes, rank, and irreducibility conditions, the finest independent source-block representation is identified within the stated alternative class up to block permutation and blockwise coordinate changes, but not downstream mechanisms or the sensor/interaction split. Together, these results show which causal conclusions equilibrium data support and which require targeted experiments.

The study of holomorphic self-mappings of the unit disc commuting under the composition goes back to A.L. Shields (1964), W.A. Pranger (1970), D.F. Behan (1973), and C.C. Cowen (1984). In many situations, the centralizer of a holomorphic self-map ${\varphi:\mathbb D \to \mathbb D}$, i.e. the semigroup $\mathcal Z_\forall(\varphi):=\{\psi\in\mathsf{Hol}(\mathbb D):\psi\circ\varphi=\varphi\circ\psi\}$ turns out to be commutative. However, this does not hold for the case of a parabolic self-map $\varphi$ of positive hyperbolic step, which is analyzed in detail in this paper. We investigate the relationships among commutativity, simultaneous linearization, and holomorphic models. In particular, we obtain existence and uniqueness results for the simultaneous linearization of commuting pairs $\varphi$, $\psi\in \mathcal Z_\forall(\varphi)$. Furthermore, extending this notion to arbitrary families of holomorphic self-mappings, we show that a given (finite or infinite) family in the centralizer ${\Delta\subset\mathcal Z_\forall(\varphi)}$ can be simultaneously linearized together with$~\varphi$ if and only if any two elements of$~\Delta$ commute with each other. This gives a far reaching extension of Cowen's result concerning commuting pairs in$~\mathsf{Hol}(\mathbb D)$.

We study additive and multiplicative densities of subsets of $\N$ along prescribed Følner sequences. We prove that additive upper density one implies multiplicative density one along a suitable multiplicative Følner sequence. We also prove independence in the following sense: given an additive Følner sequence $(G_n)_n$, a multiplicative Følner sequence $(F_n)_n$, and any $(\alpha,\beta)\in[0,1]^2$, we construct a single set $A\subseteq\N$ such that $\dens_{(G_n)_n}(A)=\alpha$ and $\md_{(F_n)_n}(A)=\beta$. Prime-valuation coordinates yield exact density formulas and random models for one local condition and, under summability assumptions, countably many; in the finite-coordinate case they also give exact higher-order correlations. Finally, we realize these multiplicative correlations as correlations in symbolic dynamical systems and obtain criteria for ergodicity and mixing.

Following an idea of Demarche and using computer calculations of the authors and ideas of the LLM Claude Fable 5, we construct an explicit 10-dimensional complex two-step nilpotent Lie algebra that is isomorphic to its complex conjugate but cannot be defined over the field of real numbers R.

We introduce point conditional probability spaces (PCPSs) as primitive building blocks for stochastic choice. This concept goes back to Rényi (1955), who proposed conditional probability spaces (CPSs) as a basis for probability theory. Luce (1959) noted the connection between CPSs and stochastic choice, and Cerreia-Vioglio et al. (2021) have developed the connection further. A PCPS is a CPS each of whose component probability measures concentrates on a singleton selection. We build a four-level PCPS-based hierarchy of families of stochastic choice rules. Level 1 consists of PCPSs, Level 2 is made up of finitely additive "conditionally consistent" mixtures of PCPSs, Level 3 comprises all finitely additive probabilistic mixtures of PCPSs, and Level 4 consists of all finitely additive signed probabilistic mixtures of PCPSs. We construct our hierarchy at a general measure-theoretic level that encompasses infinite choice sets. We also connect each level of our hierarchy to well-known axioms for stochastic choice, namely, the Weak Axiom of Stochastic Revealed Preference, Independence of Irrelevant Alternatives, and no Dutch book. We establish the relationship between total orders and PCPSs and demonstrate a sense in which PCPSs can be a more parsimonious representation of choice.

Bilinear systems with quadratic outputs (BQO) have recently emerged as an important system class, arising naturally in applications where both the dynamics and the quantities of interest depend nonlinearly on the state. Despite the growing interest in this class of systems, a systematic $\mathcal{H}_2$ framework for BQO systems has been lacking. In this paper, we develop such a framework by establishing an $\mathcal{H}_2$ inner product and norm for BQO systems, deriving output bounds in terms of the $\mathcal{H}_2$ norm, and obtaining first-order optimality conditions for $\mathcal{H}_2$ optimal model reduction. Building on these theoretical foundations, we propose an algorithm that computes a reduced BQO system satisfying these optimality conditions, and thus generalizing existing $\mathcal{H}_2$ optimal methods for bilinear and linear quadratic output systems. The effectiveness of the proposed framework is demonstrated on two numerical examples.

Organic electrochemical transistor (OECT)-based molecular communication (MC) receivers connect transport and binding to device current, noise, calibration, and detection. We present MC-BRIDGE (Molecular Communication Bioelectronic Receiver-chain Integrated Design and Guided Evaluation), a modular framework linking release, extracellular diffusion, finite-area observation, stochastic binding, and OECT transduction to charge-domain detection. On an independent electrical time grid, it generates sequence-wide multichannel colored noise, performs control-channel referencing and charge integration, and supports molecule-shift-keying (MoSK), concentration-shift-keying (CSK), and Hybrid decisions. Through common module interfaces, it estimates symbol error rate (SER) and decoded-symbol mutual information and analyzes inter-symbol interference (ISI). At nominal separation, the passive finite-area observer retains 27.5 percent of the center-point decision-charge magnitude, while correlation between the selective and control channels determines whether control referencing helps. After adaptive search, held-out records with seeds disjoint from search and calibration test the selected and next-lower budgets, yielding a tested-grid upper bound on the minimum budget meeting the SER target. This passive-field test resets receptor occupancy, excludes ISI, and calibrates thresholds on separate records at each operating point. Retaining transport and receptor memory instead produces high SER. Thus, geometry, covariance, calibration, and memory can change receiver conclusions.

Discretizing integral formulations of the Helmholtz equation yields dense linear systems. Hence, simulating acoustic models at larger scales or higher frequencies is typically constrained by memory capacity. Fast algorithms, such as hierarchical matrix compression, reduce the memory footprint substantially while controlling the approximation error in matrix-vector multiplications. However, the commonly used Adaptive Cross Approximation suffers from early-convergence problems, where the iterative construction of low-rank decompositions stops before reaching the targeted error tolerance. This failure arises when the error estimator does not capture significant components of the matrix structure under partial pivoting. This manuscript proposes a new diagonal convergence criterion, additional matrix elements for the pivoting strategy, an extended admissibility condition, and a sustained convergence check to improve the robustness of hierarchical matrix compression. These modifications improve compression reliability without increasing memory. We tested our compression strategy on various discretized volume and boundary integral operators. The computational results show that our approach successfully compresses all benchmark matrices within predefined tolerances, thereby resolving the early-convergence issues encountered in standard algorithms. This robust matrix compression was achieved at the same memory footprint as alternative compression strategies. Furthermore, a complexity analysis shows log-linear memory scaling with mesh refinement at constant frequency. Finally, we successfully applied our robust matrix compression algorithm to a coupled system of volume and boundary integral operators that models transcranial ultrasound propagation. This confirms the feasibility of our robust algorithm to accelerate large-scale simulations with high-resolution meshes in a biomedical application.

Gaussian process (GP) modeling is widely used in computational science and engineering. However, fitting a GP to high-dimensional inputs remains challenging due to the curse of dimensionality. While various methods have been proposed to reduce input dimensionality, they typically follow a two-stage approach, performing dimension reduction and GP fitting separately. We introduce a Bayesian framework that seamlessly integrates dimensionality reduction with GP modeling and inference. Our approach, built on a hierarchical Bayesian model with priors on the Stiefel manifold, enforces orthonormality on the projection matrix and enables posterior inference via Hamiltonian Monte Carlo with geodesic flow. Additionally, we extend this framework by incorporating Deep Gaussian Processes (DGP) with built-in dimension reduction, providing a more flexible and powerful tool for complex datasets. Through extensive numerical studies, we demonstrate that while the proposed Bayesian method incurs higher computational costs, it improves predictive performance and uncertainty quantification, providing a principled and robust alternative to existing methods.

This paper presents some results for the slice regular functions and hyperholomorphic functions theories, as consequences of new relationships between the global operator $G$, Fueter operator $\mathcal D$ and Laplacian operator $\Delta_{\mathbb{R}^{4}}$. Although our results are presented in the quaternionic setting, a fine interpretation in terms of vector calculus of the slice regular function theory yields an interesting relationship between this theory and harmonic analysis. Moreover, we introduce and study a submodule of the hyperholomorphic functions associated to the unit ball $\mathbb B^4(0,1)$ given in terms of new series expansions of these functions.

This paper constructs a half-order differentiation operator on the graded monomial space spanned by non-negative integer and half-integer powers of x, with x > 0. The algebraic stage uses neither an integral kernel, a limiting process, nor the Gamma function. The operator is assumed to act in the form $D^{1/2}x^\beta=c(\beta)x^{\beta-1/2}$. Requiring two successive applications of the operator to reproduce the ordinary first derivative yields the fundamental recurrence relation $c(\beta)c(\beta-1/2)=\beta$. Expanding the recurrence produces two linked coefficient families, one for integer powers and one for half-integer powers. Their formulas contain a free normalization constant $c_0=c(0)$, which cancels under composition. Thus, the compositional requirement fixes the relative coefficients but not the scale of each individual half-step. Ratios of even and odd double factorials lead naturally to a partial Wallis product; however, the Wallis product alone does not determine $c_0$. Imposing compatibility with the monomial formula for the left Riemann-Liouville half-derivative with lower limit 0 selects the value $c_0=1/\sqrt{\pi}$. With this normalization, $D^{1/2}x^\beta=\frac{\Gamma(\beta+1)}{\Gamma(\beta+1/2)}x^{\beta-1/2}$ holds for every $\beta\in\{0,1/2,1,3/2,\ldots\}$. The resulting monomial formula agrees with the corresponding Riemann-Liouville formula. For positive integer powers it also agrees with the Caputo formula, whereas for the constant function it agrees only with the Riemann-Liouville rule. The resulting linear compositional operator lowers the exponent by exactly one half and is fully specified on the stated graded monomial space. Extension to broader function spaces, construction of an integral or convolution kernel, and the analysis of non-locality remain open problems.

We construct infinite families of irreducible exotic proper knotted surfaces in $\mathbb{R}^4$, making progress on a question of Gompf. Here irreducible means these surfaces are not end-sums of standard surfaces with exotic planes. To prove the topological equivalence, we give a highly flexible construction of exotic open 2-handles, which generalizes several similar constructions in the literature. We distinguish exotic surfaces through the genus functions and end Floer homology of their double branched covers. By studying these generalized handles further, we construct a new family of topologically slice links.

We study the uniform $L^\infty(\Omega)$ a priori boundedness of positive weak solutions to the fractional semilinear Dirichlet problem $(-\Delta)^s u = f(u)$ in a bounded, convex, $C^{1,1}$ domain $\Omega \subset \mathbb{R}^N$ with homogeneous exterior condition $u\equiv 0$ in $\mathbb{R}^N\setminus\Omega$. We consider slightly superlinear nonlinearities of the form $f(t) = t^q L(t)$, where $1 \le q \le \frac{N+2s}{N-2s}$ and $L$ is a slowly varying function. Although uniform estimates are well-established in the strictly subcritical regime $q < \frac{N+2s}{N-2s}$, the slightly subcritical case, $q = \frac{N+2s}{N-2s}$, is highly challenging due to the potential formation of bubbling profiles. In this work, we isolate a structural condition on the slowly varying perturbation, namely $$ \lim_{t \to \infty} \frac{t \, |L'(t)|}{L^{\frac{N}{2s}}(t)} = \infty, $$ which acts as an asymptotic barrier that prevents mass concentration. Under this assumption, we establish global uniform $L^\infty(\Omega)$ bounds for positive solutions, significantly expanding the class of known nonlinearities for which such estimates hold.

We introduce and solve a new family of two-dimensional string theories obtained by coupling Liouville CFT to the generalized Runkel-Watts CFT. By taking an appropriate limit of the complex Liouville string, we derive an all-genus formula for its string amplitudes, and formulate a duality with a matrix integral. We explain that it can be interpreted as 2d SU(2) Yang-Mills theory coupled to gravity. We show that it also directly relates to other minimal string constructions such as the A-series minimal string.

We complete the classification of mutual arrangements of a smooth real algebraic or real pseudoholomorphic quartic curve and three lines under condition that each oval of the quartic intersects the union of the lines. This classification was started in a recent preprint by Maletto. There is one arrangement which is realizable pseudoholomorphically but not algebraically. It can be constructed in different ways, in particular, by a combinatorial patchworking on an irregular triangulation. This is the first example of a combinatorial patchworking which produces a PL curve in $RP^2$ whose arrangement relative to the coordinate axes is algebraically unrealizable.

Adversarial attacks can degrade the legitimate decision performance in wireless autoencoder communications. However, in complex scenarios with multiple adversaries, the cumulative leakage interference (CLI) caused by the multiple parallel attacks increases the chance of detecting the attacks, while dynamical environments also make the fixed attack strategies difficult to have stable effectiveness. To jointly enhance the undetectability, aggressivity and adaptability of adversarial attacks, we propose a deep learning based intelligent attack framework. Specifically, considering the CLI caused by the multiple parallel attacks, a deep neural network based transmit power control is established to reduce the interference leakage by regulating the transmit power of these adversaries, thereby improving the undetectability. Furthermore, to enhance the attack effectiveness and stability in the dynamic environment, the conditional generative adversarial attack is further developed. The generator takes the attack channel information as the conditional input to produce the perturbating signals to mislead the discriminator by making the attacked received signals resemble the clean received signals, while the discriminator distinguishes between the two under the same condition. Through the adversarial training, the generator can learn to create adaptive perturbating signals with enhanced attack performance. Simulation results demonstrate that the proposed framework outperforms benchmarks in terms of attack undetectability, aggressivity and adaptability.

Fermi coordinates provide a natural local coordinate system near a plane curve. Nevertheless, they are not conformal and are not generated directly by the curvature form. The present work asks whether a real analytic plane curve determines a distinguished conformal coordinate net generated by its curvature. Let \[ \omega_F=-k\,ds \] be the Frenet curvature 1-form of an analytic curve. We show that its holomorphic continuation \[ \Omega=K(q)\,dq \] naturally determines a conformal coordinate net through the reconstruction formula \[ z_q=\exp\left(i\int\Omega\right). \] The holomorphic differential \(\Omega\) is identified with a conformal connection differential associated with the coordinate net and whose restriction to the initial curve coincides with the original curvature form. For a pair of dual conformal nets, the derivative of the conformal transition map admits the geometric representation \[ z_q=-\overline{\left(\frac{\widetilde\omega}{\omega}\right)}, \] where \omega and \widetilde{\omega} denote the complex curvature coefficients of the two nets. Although neither curvature coefficient is holomorphic in general, their normalized conjugate dual ratio is holomorphic. Thus an analytic plane curve determines, through the holomorphic continuation of its curvature form, a distinguished conformal coordinate net in its neighborhood.

Dittert's conjecture asserts that, among nonnegative $n\times n$ matrices whose entries sum to $n$, the functional $\phi(A)=\prod_{i=1}^n r_i+\prod_{j=1}^n c_j-\operatorname{per}(A)$ is uniquely maximized by the uniform matrix $J_n/n$. This paper proves the conjecture for $n=16$. The key observation is that, for a near-maximizer, the deficits of the row-sum and column-sum products satisfy a single joint constraint rather than two independent bounds. Combining this joint-deficit estimate with a Pinsker-type subset-sum bound yields a sharper scalar dilation to a doubly superstochastic matrix. The Knopp-Sinkhorn boundary lower bound for permanents then excludes maximizers with a zero entry, and Hwang's positive-support theorem identifies the unique maximizer. Together with Pang's result for $n\ge 17$ (arXiv:2606.01531), this establishes Dittert's conjecture for every $n\ge 16$.

Wahl's conjecture states that two-dimensional singularities admitting a rational homology disk smoothing are weighted homogeneous. Assuming the conjecture, we classify all normal degenerations of the complex projective plane with only rational singularities. They are precisely the surfaces classified by Manetti and Hacking–Prokhorov, which are controlled by the Markov equation, together with six new degenerations containing four non-log canonical singularities.

We develop a strategy for the classification of singular fibers in smooth fibered surfaces by exploiting the fact that, in the tame case, a fibration with smooth general fiber of genus $g > 1$ corresponds to a representable morphism from a stacky base to $\overline{\mathcal{M}}_g$ thanks to the root stack valuative criterion for properness of tame stacks. This framework serves as another justification of Dokchitser's classification of reduction types independently of Winters' abstract result.

Let $p_n$ denote the $n$-th prime. We prove that for every nonzero polynomial $f(x)\in\mathbb{Z}[x]$, there exist infinitely many positive integers $n$ such that $p_n\nmid f(n)$.

Controlling the Lipschitz constant of a neural network is a standard way to promote robustness and stability. Most existing constraining strategies are designed for Euclidean spaces. In this work, we construct and analyze a class of 1-Lipschitz neural networks on Hadamard manifolds. Our layers are of gradient-descent type, $1$-Lipschitz, and quasi-$\alpha$-firmly nonexpansive. The core building blocks of the proposed architecture are Busemann functions, and we exploit the properties of Busemann gradient flows to design $1$-Lipschitz geometry-preserving layers. We provide explicit constructions and examples for hyperbolic manifolds and the manifold of symmetric positive definite (SPD) matrices. We test the proposed architecture in two numerical experiments: robust classification on the Poincaré disk and masked-Wishart covariance reconstruction. On the Poincaré disk, the proposed networks yield robust classifiers under hyperbolic perturbations. On the SPD manifold, we train SPD-valued denoisers and adopt them as a Plug-and-Play prior for a masked-Wishart covariance reconstruction problem. We show improved results from the nonexpansive denoiser over static, data-only, and Log-Euclidean denoising baselines, and empirically test its convergence properties.