Papers

Let $f_{\lambda}(z)=\lambda e^z$, where $0<\lambda<1/e$. In 1984, Devaney and Krych showed that the Julia set consists of pairwise disjoint hairs. Viana further in 1988 proved that these hairs are $C^{\infty}$. In this paper we show that hairs are not analytic except for trivial ones. This solves a long-standing open question.

In this note, we prove uniform upper bounds for the volume of the level set $$\{x\in\Omega: c\le f(x)<c+\delta\},\quad c\in\mathbb R,\;\delta>0,$$ for strictly convex and hyperbolic functions $f$ defined on a convex domain $\Omega\subset\mathbb{R}^n$ ($n\ge 2$). In particular, under a Hessian lower bound $D^2f\ge I_n$, we obtain the sharp volume bound $$|S^{n-1}|\, r(\Omega)^{n-2}\,\delta,$$ where $r(\Omega)=\frac12\,{\mathrm{diam}\Omega}$. As an application, we derive $L^2$ estimates for oscillatory integrals with convex phases.

We study how the affine geometry of finite ReLU networks is created at random initialization and reorganized by supervised training. We call a sign-changing zero of a hidden preactivation an activation switch and a point where the scalar network output is nondifferentiable a scalar kink. For one-dimensional input, conditioning on the preceding layers makes each preactivation Gaussian and affine on the cells of a random finite partition. This yields an exact finite-width conditional Kac-Rice formula for the expected number of activation switches along an input interval. For fixed depth and proportionally growing widths, the resulting switch intensities converge to explicit deterministic limits. A visibility estimate shows that the expected number of switches that do not produce scalar kinks is negligible, yielding an explicit leading formula for the expected number of scalar kinks and hence affine regions. In higher input dimensions d >= 2, the analogous conditional surface formula yields the leading expected (d-1)-dimensional Hausdorff measure of the scalar kink set. On the Breast Cancer Wisconsin data, the initialization formula accurately predicts switch counts along held-out segments. After training, switch counts decrease along within-class segments and increase along between-class segments. Thus, training redistributes rather than merely contracts affine complexity.

We determine the small-diffusion limit of the principal eigenvalue for a one-dimensional cooperative elliptic system with component-dependent drifts and Neumann boundary conditions. For scalar equations, this limit is determined by local data at the drift equilibria and boundary points. We show that such localization fails for systems: switching between components can make some intervals spectrally relevant. We first prove that the logarithmic transforms of all component eigenfunctions converge to the same function, which solves a scalar Hamilton-Jacobi equation. The associated Aubry set determines the asymptotic behavior of the principal eigenvalue. Under nondegeneracy assumptions, this set decomposes into isolated points, regular intervals, and composite intervals containing common drift zeros. We then identify the effective value of each class through local Ornstein-Uhlenbeck spectra and interval transport problems, and establish that the limit of the principal eigenvalue is the minimum of these class values. A unique minimizing class also determines the common logarithmic eigenfunction profile.

For a graph $G$, we write $\mathrm{mad}(G)$ for its maximum average degree and $\mathrm{diam} G$ for its diameter. Let $R_k(G)$ be the graph whose vertices are the proper colorings of $G$ with $k$ colors, where two colorings are adjacent when they differ at one vertex. Feghali (JCTB, 2021) proved that, for fixed integers $d,k\ge 1$ with $k\ge d+1$ and every $\varepsilon>0$, every $n$-vertex graph $G$ satisfying $\mathrm{mad}(G)\le d-\varepsilon$ has $\mathrm{diam} R_k(G)=O_{d,k,\varepsilon}(n(\log n)^{d-1})$. In this article, we prove that \[ \mathrm{diam} R_k(G)=O_{d,k,\varepsilon}\!\left( n(\log n)^{\left\lfloor (d-1)/(k-d)\right\rfloor} \right), \] which extends the result proved by Feghali directly. The proof uses a partition into independent layers and removes $k-d$ colors at each recursive stage. We also improve the bound on the number of layers and determine the best possible linear coefficient in the forest case. More precisely, for $0<\varepsilon<2$, every $n$-vertex graph $G$ with $\mathrm{mad}(G)\le2-\varepsilon$ satisfies $\mathrm{diam} R_3(G)\le\rho_M n$, where $M=\lfloor2/\varepsilon\rfloor$, $\rho_M=\max_{1\le m\le M}D_m/m$, and $D_m$ is the largest diameter of $R_3(T)$ over all trees $T$ on $m$ vertices. Moreover, the coefficient $\rho_M$ is best possible.

We consider a problem of optimal proportional reinsurance-dividend distribution under a Brownian risk model, where both the drift and volatility coefficients are subject to endogenous regime-switching. Dividend payments are subject to fixed transaction costs. The problem is formulated as a two-dimensional stochastic control problem, and we prove that the value function is the unique viscosity solution of the associated Hamilton-Jacobi-Bellman equation with nonlocal operator. For almost all parameter configurations, we explicitly characterize the optimal strategy that maximizes the expected total discounted dividends net of transaction costs until ruin. The optimal dividend policy is a two-barrier impulsive strategy, while the optimal reinsurance proportion is given in feedback form. Numerical examples are provided to illustrate the optimality results.

We study the construction of positively curved Riemannian metrics by non-isometric circle actions. For a family of homotopy eleven-spheres whose Eells–Kuiper invariants form the even subgroup of $\mathbb{Z}/992$, we construct, on each member, a smooth background metric $q$ and three effective circle actions with generators $W_1,W_2,W_3$ such that the metric determined by $g^{-1}=q^{-1}+\sum_{a=1}^3W_a\otimes W_a$ has positive sectional curvature. Each action is non-isometric for every partial metric, including its incoming metric and the final metric. We first describe the sphere by gauge transformations of the quaternionic Hopf bundle. We then construct compatible metrics on two disks and smooth their inverse metrics while preserving the action formula. A local conjugation makes the circle actions non-isometric. For each fixed member, we obtain an explicit positive lower bound $2^{-54}(1+M_{0,k}+M_{1,k})^{-28}$, where $M_{0,k}$ and $M_{1,k}$ are norms of the curvature and its first covariant derivative for its fixed connection. The bound may depend on the member of the family.

E-processes provide a flexible framework for anytime-valid inference, with the conventional rejection boundary $1/\alpha$ typically justified by Ville's inequality. Such a boundary is universal but can be conservative, as it does not exploit additional information about the null distribution. We propose a reference-null calibration framework that uses independent null samples to construct sharper rejection thresholds while preserving type-I error control. To study the statistical gain from sharper thresholds, we specialize the framework to conformal martingales. Building on histogram-based betting, we incorporate Krichevsky–Trofimov smoothing and develop a restart-mixture construction for distribution shift detection. We establish quantitative results for detection power and detection delay and characterize how the reduction in the rejection boundary translates into power and delay gains relative to the conventional Ville boundary. Numerical experiments corroborate the theoretical findings across a range of distributional changes. Finally, we apply the proposed calibration strategy to an existing e-process for online LLM watermark detection, showing that the method can improve sequential detection performance without modifying the underlying e-process. These results demonstrate that reference-null calibration provides a general and modular way to enhance the efficiency of e-process-based sequential inference.

In this article, we describe a method to obtain lower bounds for the 2kth moment of values of L-functions localised at the central point for all real k > 0. In particular, we handle the families of quadratic twists of self-dual irreducible unitary cuspidal automorphic representations of GL_2(A_Q) and GL_3(A_Q).

In this paper, we investigate the geometry of the Gieseker moduli space of semistable rank $2$ torsion-free sheaves on $\mathbb{P}^3$ with Chern classes $(c_1, c_2, c_3) = (-1, c_2, c_2^2-2)$ for $c_2 \geq 2$. We use the modular Serre correspondence to relate these moduli spaces to spaces of pairs and to suitable Hilbert schemes of one-dimensional subschemes in $\mathbb{P}^3$. For $c_2 \geq 4$, we prove that the moduli space is irreducible of dimension $c_2^2+3c_2+5$. For the case $c_2 = 3$, relying on a known geometric description of a relevant Hilbert scheme, we show that the moduli space $\mathcal{M}(-1,3,7)$ consists of exactly two irreducible components, namely the generic component of reflexive sheaves and a $T$-component, and we prove that their intersection is nonempty.

We establish a dimension-free vector-valued Poincaré-type inequality on the discrete cube for Banach spaces with Rademacher type $p$, $1\le p\le2$. As a consequence, we obtain $$ T_p^R(X)\le T_p^E(X)\le p\,T_p^R(X). $$ This gives an alternative proof that Rademacher type and Enflo type coincide, while improving the constant $\pi/\sqrt{2}$ in the quantitative estimate of Ivanisvili, van Handel, and Volberg \cite{IVV} to $p$.

The compact four-dimensional manifold $\mathbb{CP}_2\# 3\overline{\mathbb{CP}_2}$ is known to admit a toric Kähler-Einstein metric $g_{\text{KE}}$, but the metric is not known in closed form, which makes it difficult to draw conclusions about its geometry. In this article, we use a combination of analytic and computer-assisted techniques to produce an approximate Einstein metric $g$ described explicitly herein, and also prove that the true Einstein metric $g_{\text{KE}}$ is close to $g$, where both the closeness and the topology are described explicitly. As an application, we prove bounds on the first invariant eigenvalue of the Laplace-Beltrami operator, and prove that this Kähler-Einstein metric does not have positive holomorphic sectional curvature everywhere.

Soft state representations assign states nonnegative class weights that sum to one, but agreement with reference labels does not guarantee accurate prediction of the next weight vector. We derive lower confidence bounds on the minimum population root-mean-square prediction error over linear predictors with a prescribed spectral-norm limit. The bounds compare within-class successor variation with retained within-class input variation, with an alternative based on reference-label deviation. They are estimated from independent transition pairs without fitting a prediction matrix to the evaluation data. A bound above a chosen error tolerance excludes that accuracy for every admissible predictor at the stated confidence level; a bound of zero does not establish whether an accurate predictor exists. For coordinates constructed from Kernel Affine Hull Machine reconstruction scores, we relate the reference class's affinity advantage over competitors to label agreement. The class-centered population bound equals the optimum in a four-state example. Using 117,000 previously generated evaluation datasets from fixed four-state designs, we compare the finite-sample certificates with an empirical-risk-minimization baseline. For one Van der Pol representation selected before evaluation, the label-based bound is positive in all 64 independent evaluation datasets at the fitted matrix's spectral norm. Separately, exact matrix propagation under deterministic dynamics yields a Koopman interpretation of the coordinate-function span. The certificates help identify forecasting targets that cannot be reached by refitting a fixed representation within a specified predictor class and evaluation distribution.

In 1975, Kaplansky conjectured that every five-dimensional division algebra over a sufficiently large finite field is a field or a twisted field. We disprove this conjecture. For every prime power $q=p^e\equiv1\pmod3$ and every $n\ge5$ with $\gcd(n,6)=1$, we construct semifields of order $q^n$. For fixed $q,n$, the family represents $\varphi(n)$ isotopy classes if $p\equiv1\pmod3$, and $\varphi(n)/2$ if $p\equiv2\pmod3$, where $\varphi$ is Euler's totient function. Using new isotopy invariants and the structural properties of our construction, we prove that none of these semifields is isotopic to a finite field or an Albert's generalized twisted field. In particular, the five-dimensional specialization gives infinitely many pairwise nonisotopic counterexamples to Kaplansky's conjecture over arbitrarily large finite fields. More generally, for each prime dimension $n\ge5$, the examples occur over arbitrarily large fields in every characteristic other than three, contradicting the classification asserted by Menichetti in 1996, whose proof contains gaps.

Little is known about the dynamics of Toeplitz operators on the Bergman space, and even in the case of harmonic symbols, few results are available. In this paper, we study the dynamical properties of Toeplitz operators with symbols of the form $a \bar{z} + p (z)$ on the Bergman space, where $a\neq 0$ and $p$ is analytic on the closed unit disk. We obtain some necessary conditions and some sufficient conditions for such operators to be hypercyclic, weakly mixing, mixing, chaotic, or frequently hypercyclic. As an application, we obtain a complete characterization of hypercyclicity for Toeplitz operators with harmonic linear polynomial symbols. We also show that Toeplitz operators with the same symbol may exhibit different hypercyclic behavior on the Hardy and Bergman spaces.

We introduce curve band depth (CBD), a band-based data depth for samples of unparameterized planar curves. CBD is motivated by band depth and modified band depth for functional data, but targets trajectory data. Unlike the halfspace-based curve depth of \citet{de2021depth} and the curve stabbing depth of \citet{durocher2023csd}, CBD is defined through a geometric band region generated by two curves, and measures the arc-length proportion of a target curve lying inside such bands. We develop a CBD family consisting of an integral version (int-CBD), an infimal version (inf-CBD), and a fast-walk variant (FW-CBD). The fast-walk band is a narrower band construction contained in the global convex-combination band. We establish boundedness, vanishing at infinity, and similarity invariance for these constructions, together with a Borel-measurability result for the induced depth maps under a mild measurability assumption. A length-penalized variant is proposed for samples with heterogeneous curve lengths. We implement the methods via arc-length sampling and polygonal approximations, and evaluate them through classification of overlapping handwriting data and MNIST-derived digit curves, online-signature screening on MOBISIG, and an exploratory clustering task based on decomposed band contributions.

We study two-dimensional steady incompressible Euler flows with finite total curvature. A distinguished family of such flows arises from finite-Morse-index solutions to semilinear elliptic equations in two dimensions. We establish three main properties of finite-total-curvature flows in the $L^\infty\cap H^1_{\rm loc}$ class. First, in the plane and the half-plane, every such flow without stagnation points is a parallel shear flow. Second, in the plane, the half-plane, and the infinite strip, the pressure converges uniformly to a constant at each end at infinity. Third, in the plane, the half-plane, the infinite strip, and the periodic strip, we establish sharp lower bounds for the total curvature in terms of the pressure oscillation. The rigidity result in fact extends to a broader class of flows: in the plane and the half-plane, any steady Euler flow without stagnation points is a parallel shear flow provided that $\nabla P\in L^q$ for some $1\le q\le 2$. Combined with standard elliptic estimates for the pressure, this result yields a concise new proof of Hamel and Nadirashvili's rigidity theorems, with the regularity assumption sharpened optimally to $H^1_{\rm loc}$. The pressure serves as a unifying quantity throughout the analysis.

We establish Morrey-type conditions ensuring that a finite signed Radon measure belongs to the dual of the energy space of solutions to nonlinear parabolic equations of $p$-Laplacian type. More precisely, for the parabolic cylinders $Q_{r,r^p}(z):=B_r(x)\times(t-r^p,t+r^p)$, we prove that \[ |\mu|\bigl(Q_{r,r^p}(z)\cap(\Omega\times (0,T))\bigr) \leq Mr^{n+p-\vartheta}, \qquad \vartheta<p, \] implies duality, and this threshold is sharp in general. More generally, for cylinders with time length proportional to $r^q$, $q>1$, the sufficient threshold is $\vartheta<\min\{p,q\}$. Finally, we discuss the resulting variational, energy, and renormalized solution theories for nonlinear parabolic equations with measure data, relating our conclusions to the existing literature.

We prove the logarithmic Weyl law predicted by Connes and Moscovici for large positive eigenvalues of the distinguished self-adjoint extension of the operator $P=-\partial_x(x^2-1)\partial_x-4\pi^2x^2$ on the real line. More precisely, we use the WKB method and show a Bohr-Sommerfeld quantization rule characterizing sufficiently large eigenvalues. A particular difficulty is that the associated $h$-dependent Lagrangian submanifolds are non-compact, which requires a renormalization of the action integral. Furthermore, to deal with the degeneracy of the coefficients at $x =\pm 1$, we use a quasimode matching argument based on radial estimates and Lagrangian regularity.

For a class of non-stationary linear processes $(X_{n})_{n}$ with weakly dependent heavy-tailed innovations we prove a convergence result for the space-time point processes $N_{n}=\sum_{i=1}^{n}\delta_{(i/n, X_{i}/a_{n})}$. Using this result and the continuous mapping theorem we then derive functional convergence of the corresponding partial maxima process in the space of càdlàg functions on $[0,1]$.

In this paper we investigate existence and multiplicity of solutions for a critical growth anisotropic quasilinear elliptic system, depending on two real parameters $\lambda,\,\mu$, that is coupled through a subcritical perturbation term and involves a negative critical part. We identify a specific scaling for the system and introduce a parameter $\gamma$ associated with this scaling, which governs the geometry of the corresponding variational functional. We establish three distinct types of multiplicity results, depending on the regimes $\gamma = 1$, $\gamma > 1$, and $\gamma < 1$. The multiple results of multiplicity of solutions are based on a recent abstract critical point theorem for symmetric functionals on product spaces when $\gamma\geq1$ and on a truncation argument for $\gamma<1$.

We establish a quaternionic Hansen–Pedersen inequality for real operator-convex functions on right quaternionic Hilbert spaces. The proof is based on the continuous real functional calculus for selfadjoint quaternionic operators, a direct-sum functional-calculus identity, and a block-unitary construction. As an application, let \(T = U|T|\) be an injective \(p\)-hyponormal operator. We prove that, for \(0 < r \leq 1/2\), the transform \(S_{r}(T) = U|T|^{r}U\) is \(2p\)-hyponormal when \(0 < p \leq 1/2\) and hyponormal when \(1/2 < p \leq 1\). We also show that \(S_{r}(T)\) preserves log-hyponormality for \(r > 0\). These results illustrate the use of quaternionic Jensen-type operator inequalities in the study of transforms associated with the polar decomposition.

In recent work, authors defined $\bar{b}^k_{r,s}(n)$ to be the number of overcolored partition $k$-tuples wherein both even and odd parts are colored with $r$ and $s$ colors, respectively. This function generalizes the overcubic partition $k$-tuples studied by Chacon and Sellers. The present work gives a uniform framework for studying congruences for $\bar{b}^k_{r,s}(n)$ modulo odd primes. For example, we prove that, for $n\ge 1$, \begin{align*} \bar{b}_{4,2}^4(5n)\equiv 0\pmod{5}. \end{align*}

Networks of interacting Hebbian networks have recently been shown to perform a task beyond associative memory, namely pattern disentanglement: when fed with a spurious mixture of stored patterns, the different modules spontaneously specialize on, and retrieve, the different constituents of the mixture. So far, this capability has only been established for pairwise interactions, which limits the number of patterns that can be handled. Here we introduce a dense extension of these modular networks, in which both the auto-associative couplings within each module and the hetero-associative couplings among modules are promoted to higher-order Hebbian interactions. We show that, with a suitable choice of the interaction orders, the network disentangles mixtures while storing a number of patterns that scales linearly with the module size, a regime where its pairwise counterpart fails. Through a statistical-mechanical analysis based on Guerra's interpolation, we derive the self-consistency equations for the order parameters and draw the phase diagrams identifying the region where disentanglement is achieved; these predictions are confirmed by Monte Carlo simulations. Finally, we show that disentanglement provides a natural decoding primitive, and we illustrate it with two applications: the explicit reconstruction of all the hidden patterns from the Hebbian tensors and a stream of unlabeled mixtures, and a proof-of-concept communication protocol in which each message token is transmitted as a masked mixture of hidden patterns and decoded by the network dynamics. Owing to its attractor-based decoding, the protocol degrades gracefully under strong channel corruption, where conventional secure-transmission pipelines fail abruptly.

Let $X$ be a quasi-projective scheme of dimension $d$ over an infinite field $k$, such that $X$ is reduced after removing all components of dimension $<d$, and let $X^*\subset X$ be a closed subset of dimension $<d$ such that $X\setminus X^*$ is regular of dimension $d$. Using a modification $\text{CH}^d(X, X^*)$ of the Chow group of 0-cycles on a singular variety defined by C. Weibel and the author in 1985, we construct a Chern class map $c_d:K_0(X)\to \text{CH}^d(X, X^*)$, a cycle class map from $\text{CH}^d(X, X^*)$ onto a subgroup $F^dK_0(X)$ of $K_0(X)$, and prove a Riemann-Roch theorem computing the two compositions of these two maps.

In this paper, we provide a rigorous study of the direct scattering problem proposed by Kodama, Ablowitz and Satsuma for the complete integrability theory of the Intermediate Long Wave (ILW) equation. We derive the Lax operators for ILW, and study their spectral theory. We show finiteness and simplicity of the discrete spectrum, existence and uniqueness of the Jost solutions, and construct the scattering data for the Lax operator. We also show that the Jost solutions satisfy a nonlocal Riemann-Hilbert problem whose singularity data are given by the scattering data, and that the scattering potential can be reconstructed from the Jost solutions. This work paves the way for a solution to ILW via the inverse scattering transform.

The least regular norm of a factorization of antipodal differences through sign differences on \((\mathbb Z/m\mathbb Z)^n\) is \(m/2\) when \(4\mid m\). On these tori, when \(2\sqrt n\le m\le4\sqrt n\), normalized divergence equations admit selections with pointwise norm bound \(3/2\), while the least linear lifting norm and global Lipschitz selection constant equal \(\sqrt n/2\). Higher-moment lifting bounds follow from a quoted sharp \(L_1\) inequality.

We study the fractional perimeter per period of planar sets with prescribed horizontal period and area per period. We prove that sufficiently wide strips are the unique minimizers among all measurable competitors, up to vertical translation and null sets. For period one, a half-width of at least $14$ suffices for every fractional exponent $s\in(0,1)$. The proof first replaces each vertical section by a centered interval of the same length. We then establish a quantitative lower bound for the perimeter excess in terms of the deviation of the resulting half-width profile from its mean. The main difficulty is that the perimeter of a strip is a concave function of its width. We show that, at large mean width, the interaction cost of unequal sections dominates the corresponding concavity deficit and leaves a positive remainder for every nonconstant profile. Estimates at small and large horizontal separations make the sufficient width bound independent of $s$. The argument applies to arbitrary nonnegative integrable profiles, including unbounded profiles and profiles that vanish on sets of positive measure.

We prove that any block of a profinite group with a topologically finitely generated defect group $D$ is isomorphic to a block of a virtually pro-$p$ group, and if $D$ is finitely presented then so is the basic algebra of the block over $k$. Using these general results we then prove that any block with defect group $D=\mathbb Z_p^n$ is Morita equivalent to $\mathcal O_\alpha[[\mathbb Z_p^n\rtimes E]]$ for a finite $p'$-group $E$ acting faithfully and a $2$-cocycle $\alpha\in H^2(E,k^\times)$. In particular, Donovan's conjecture holds for such blocks of profinite groups.

We prove that, in phenotype-structured populations under uniform competition, the long-term phenotypic distribution does not depend on the proliferation rates: it is the stationary distribution of the switching dynamics alone. The density u(x,t) of cells with phenotype x evolves by advection and diffusion in phenotype space (switching) and by proliferation at a phenotype-dependent rate r(x,t), modulated by a factor g(U) common to all phenotypes, where U is the total population. This extends to continuous phenotypes a recent result for compartmental models. On a bounded interval with no-flux boundary conditions, the total population converges monotonically to the carrying capacity U*, the density converges in L1 to U*psi, where psi is the stationary density of the switching dynamics, and the nonlinear model is asymptotically equivalent to the linear one; by an explicit estimate, the imprint of fitness decays at the spectral gap of the switching dynamics. On the whole real line we prove the same convergence under two natural conditions, confinement and no escape to infinity, covering Ornstein-Uhlenbeck dynamics, multi-well landscapes, and heavy-tailed densities. If g'(U*) < 0 and psi satisfies a Poincare inequality, convergence is exponential in a weighted L2 norm, while no uniform L1 rate holds for Ornstein-Uhlenbeck dynamics. The proof follows the compartmental one: under uniform competition the reaction term has one sign, so its size is the growth rate of the total population, integrable in time, and the switching dynamics contracts the zero-mass perturbations it produces. Since it uses only the Markov semigroup generated by switching, the proof extends to general mechanisms; we apply it to nonlocal switching by jumps and to diffusions with non-gradient drift in bounded domains of R^d, whose stationary density is not of Boltzmann form and carries a circulating flux.