Papers

We study the dynamics of Polya-Szegö inner functions and discuss some of their basic properties such as equivalent conditions for simple and double parabolicity. We show that a simply parabolic Polya-Szegö inner function admits forward and backward quotient half-cylinders, which allows one to enrich its dynamics with a Lavaurs map. To proceed, we restrict our attention to simply parabolic inner functions with finite Lyapunov exponent: $\int_{\mathbb{R}} \log |F'| d\ell < \infty$. We define a geodesic flow on the Riemann surface lamination associated to the Lavaurs semigroup and show that it is ergodic. As an application, we establish the Orbit Counting Theorem up to a Cesàro average for Lavaurs semigroups. If we additionally assume that $F$ is a parabolic one component inner function, then the geodesic flow is mixing and the full Orbit Counting Theorem holds.

We study the well-posedness problem for the extended Schrödinger-Benjamin-Ono system (eSBO) on the real line. This system couples a Schrödinger field $u$ with a Benjamin-Ono type field $v$, including a term of the form $\partial_{x}(v^2)$. This latter term, just as in the case of the Benjamin-Ono equation, causes the system to become quasilinear and unsolvable via Picard iteration. We prove that eSBO is locally well-posed in $H^{s+\frac 12}(\mathbb{R})\times H^{s}(\mathbb{R})$ for any $s\geq 0$. In particular, this result covers the energy space at $s=\frac 12$, yielding global well-posedness in $H^{1}(\mathbb{R})\times H^{\frac 12}(\mathbb{R})$ with a small $L^2$-assumption on the Schrödinger part of the initial data.

Motivated by the introduction of Anosov-like actions by Thomas Barthelmé, Steven Frankel, and Kathryn Mann, in this paper we study group actions on the plane that preserve a pair of transverse singular foliations, as well as a strong Markovian family, namely, a collection of rectangles serving as an analogue of a Markov partition for group actions. We prove that any such action is Anosov-like. In particular, the existence of a strong Markovian family constrains the behavior of the invariant bifoliation and endows the action with characteristic features of hyperbolic dynamics, such as the existence of infinitely many elements with fixed points.

Recently, Ehret and Gilliers introduced the notion of a (trivially) restricted post-Lie algebra, recovering the concepts of a restricted Lie algebra and a restricted pre-Lie algebra. In this paper, we specifically introduce restricted Rota-Baxter Lie algebras of arbitrary weight with an intrinsic graph subalgebra characterization. We show that, via the splitting property, they give rise to restricted post-Lie algebras, and furthermore possess a novel replication property. We then present two natural constructions of such restricted Rota-Baxter structures in prime characteristic: one arising from Rota-Baxter associative algebras of arbitrary weight, and the other from Rota-Baxter Lie algebras of weight $1$. The Rota-Baxter $p$-envelopes of a Rota-Baxter Lie algebra are also examined.

After fixing a pattern $\mathbf{m}$ of zeroes and poles, we introduce a Moduli stack $\Gamma\mathcal{M}_{g,n}^{ex, \mathbf{m}}$ over $\mathbb{F}_p$ that parametrizes smooth marked curves together with a non-zero differential form that is the differential of a meromorphic function. Furthermore, we consider the stack $\mathcal{M}_{g,n}^{ex, \mathbf{m}}$ that parametrizes those divisors on smooth curves that appear as divisors of exact differential forms. By introducing a local-global principle for first order deformations of the objects that we consider, we show smoothness of these stacks and compute their dimension.

We use a family of Rogers–Ramanujan-type combinatorial identities of Dousse–Konan involving coloured partitions to prove classical freeness of the simple vertex operator algebras based on $\widehat{\mathfrak{sl}}_n$ at level $1$. These identities are used to produce Gröbner bases for the relevant arc algebras.

Faster-than-Nyquist (FTN) signaling is gaining attention as a smart way to pack more data into limited spectrum by intentionally breaking the traditional symbol-spacing rules. This article takes a fresh look at FTN's potential to boost capacity, examining how performance varies across different acceleration factors and signal-to-noise ratio (SNR) definitions. Beyond the theory, we explore what it takes to make FTN work in practice, such as dealing with power amplifier constraints, managing high peak-to-average power, and designing practical coding strategies. We also highlight real-world issues like spectrum sharing, short-packet communication, and receiver complexity. With applications ranging from low-latency links to integrated sensing and satellite systems, FTN offers a compelling path forward for future wireless technologies.

We characterize when two sequences of composition operators admit disjointly universal Blaschke products and singular inner functions. The characterizations we provide depend on geometric features of the symbols like their hyperbolic derivatives and pseudo hyperbolic distances. To achieve our results, we build a disjoint universality criterion for sequences of maps that act on a metrizable, complete topological semigroup.

Ensemble Kalman Filter (EnKF) based data assimilation algorithms synthesize predictive numerical forecast models with accumulated data as time evolves and account for model uncertainty and noisy measurements. The computational cost of these algorithms can be expensive, in particular for highly dimensional dynamical systems. Often, EnKF based algorithms have traded accuracy for reduced computational cost. In this paper, we present a novel parallelizable Conjugate Gradient-based Ensemble Kalman Filter (CGD-EnKF) algorithm that maintains comparable computational cost to efficient algorithms while realizing better state estimation accuracy in select cases. Here, we established the new approach by reformulating a matrix inverse calculation with a classical Conjugate Gradient (CGD) method. In addition, we discuss the upper error bound under CGD, error convergence to the classical EnKF result, and the computational complexity of the algorithm. We also showcase the CGD-EnKF-Reduced algorithm that is shown to be further computationally efficient for highly dimensional dynamical systems under small ensemble formulation. Numerical examples demonstrate the performance of our proposed algorithms and analytical properties, highlighting their comparability and advantages with respect to some benchmark EnKF algorithms.

The special orthogonal group $\mathbb{SO}_n$ is a Lie group whose geometry and local structure are encoded by the exponential map in its Lie algebra $\mathbf{Skew}_n$, the set of skew-symmetric matrices. The associated multi-valued inverse problem – the matrix logarithm – in $\mathbb{SO}_n$ exhibits a highly nontrivial local diffeomorphism structure, which differs from the matrix logarithm for invertible matrices. This work characterizes the local diffeomorphism structure of the exponential in the set of skew-symmetric matrices where its derivative is invertible. We show that this set with an invertible derivative can be organized into diffeomorphic regions, using a canonical alignment of Schur decompositions. In particular, the region that contains the principal logarithm has a special multiplicity structure: each matrix in $\mathbb{SO}_n$ admits at most two skew-symmetric preimages in this region. Based on this geometric framework, we introduce the diffeomorphic logarithm of special orthogonal matrices together with an efficient and stable algorithm. Moreover, it is applied to the Karcher mean problem in $\mathbb{SO}_n$, demonstrating continuous behavior of the mean under perturbations of the data, which is not captured by the principal logarithm.

For graphs $F$ and $H$, let $\ex(n,F,H)$ denote the maximum number of copies of $F$ in an $n$-vertex $H$-free graph. Janzer, Longbrake and Yepremyan recently proved that, for fixed $3<a\le b$ and sufficiently large $t$, \[ \ex(n,K_{a,b},K_{3,t})=\Theta(n^3). \] We make their threshold explicit, showing that this conclusion holds for all $t\ge \tau(b):=2\max\{3,\lceil b/2\rceil\}+1.$ In particular, for every even $b\ge 6$, this matches the necessary threshold $t=b+1$. The main new ingredient is an explicit finite-field point set whose plane sections are controlled directly, rather than through a general bounded-complexity algebraic lemma. This direct line-and-conic section analysis gives the required \(K_{3,t}\)-freeness while preserving many coplanar \(b\)-element subsets.

Sparse matrix kernels form the computational backbone of scientific computing, traditionally relying on C/C++ and Fortran implementations that prioritize performance over memory safety. This work evaluates Rust as a systems-level alternative for sparse linear algebra by implementing and benchmarking three core workloads: sparse matrix-vector multiplication (SpMV), Lanczos-based Krylov methods, and matrix-exponential evaluation. We compare native Rust code against established baselines (Intel oneMKL, Eigen, PETSc, and PSBLAS) across a suite of representative matrices. Our results show that Rust's sparse kernels achieve performance comparable to Eigen and PSBLAS, tracking the state-of-the-art for CSC formats, while trailing PETSc's advanced blocked CSR optimizations. By analyzing compile-time monomorphization, SIMD vectorization, and FFI boundaries, we assess the practical impact of Rust's safety model and ecosystem readiness. The study provides concrete, evidence-based guidance for modernizing high-performance numerical software stacks.

In this paper, we study the local Lie group structure associated with the space of admissible bisections of a local Lie groupoid over a compact manifold. We further investigate the relation of this local Lie group to the Lie algebra of sections of the associated Lie algebroid. In addition, we prove that the globalizability of a local Lie groupoid implies the globalizability of its associated local Lie group of bisections

We introduce a family of Gaussian fields with an inhomogeneous cooling variance profile, indexed by $\alpha\in(0,1/2]$, whose covariance is log–log-correlated at $\alpha=1/2$ and approaches the log-correlated regime as $\alpha\downarrow0$. We consider both one dimensional such objects (which we call cooling Branching Brownian Motions) as well as the related Gaussian fields. We identify the centering of the maximum at the terminal time $T$ and prove tightness of the recentered maximum. While the exponent in the first-order growth varies linearly with $\alpha$, giving a leading order of $T^{1-\alpha}$, the second-order correction exhibits a phase transition at $\alpha=1/3$. We also show that any subsequential limit cannot be a randomly shifted Gumbel law, in contrast with the logarithmically correlated case.

We study the product $C$ of two algebraic cones equipped with algebraic structures given by contractions. First we show that any holomorphic tensor on a quotient of $C$ by a group containing a contraction on both factors is invariant under the Zariski closure of this contraction when the factors have dimension $\geq 2$. We then give an explicit embedding of the cone of a Sasaki manifold to a normal variety. Using it and the result on algebraic cones, we prove that any holomorphic tensor on the product of two Sasaki manifolds is invariant under the flows of the Reeb fields.

We establish well-posedness of the stationary Kolmogorov equation on a bounded domain, with either spherical or Gaussian velocities, subject to either inflow boundary conditions or specular reflection. For the sake of completeness, we also include the problem on the torus, which has already been solved by Albritton, Armstrong, Mourrat, and Novack (2024). Although the natural trace estimate with weight $|n_x\cdot v|$ does not hold in general in either setting, we provide a natural functional framework in which the trace is defined via the transport operator. We prove a Poincaré-type inequality with trace, which is an essential step towards the well-posedness of the inflow problem without damping. Moreover, we obtain a one-sided energy estimate for the trace with weight $|n_x\cdot v|$, in which the outflow (resp. inflow) flux is bounded by the energy inside the domain and the inflow (resp. outflow) flux. Finally, for the bounded-velocity model, we complement the existing well-posedness theory for the stationary Kolmogorov equation with inflow conditions on the hypoelliptic boundary and Dirichlet conditions on the velocity boundary.

Measuring a quantum computation in a basis adapted to a symmetry it carries reduces the repeated measurements, commonly referred to as “shots”, needed to read a statistical answer. Detecting the symmetry a quantum state carries has many uses: certifying a claimed symmetry, identifying a conserved-charge sector, flagging symmetry-breaking as an error signature, and selecting a compression or readout basis; shot-count reduction is developed here as one exemplary case. Existing methods assume the symmetry is known in advance; we remove that assumption. When it is unknown, the carried symmetry is discovered from the data by a symmetry test that scores candidate groups, and the largest passing group is exploited as the measurement basis. We state the pipeline precisely, prove the selection rule is unbiased, and charge discovery in full. Two conditions are treated, both detected by the same score with a different projection: a weak condition, commutation with the representation, and a strong condition, confinement to a single charge sector, the distinction drawn in the quantum-reference-frame literature. A single circuit, a controlled twirl followed by a SWAP test, discovers both: discarding the group register tests the weak condition, post-selecting it the strong one. The framework is general over finite groups, with cyclic (Fourier), dihedral, and symmetric-group (Schur-Weyl) examples; strong confinement to the symmetric, or Dicke, subspace is an exponential reduction. Seeded demonstrations show the loop wins net of discovery: weak matching on momentum readout reduces shots by a factor widening from ten to several thousand, and strong matching on a two-system target by a further factor of the subsystem size. Blind symmetry matching is a practical primitive for the common case where the matched basis cannot be written down in advance.

We prove that collapsed metric measure spaces with Ricci curvature bounded below and essential dimension two are topological surfaces, possibly with boundary.

Stochastic oscillations can emerge from a two-population model as triggered by endogenous finite size fluctuations. Here, an extended dynamical scenario is considered in which a third fluctuating species is added to a proto-typical scheme of neuronal interaction. As we shall prove both analytically and numerically, the third added species can enhance or even suppress the emergence of quasi-cycles, namely the coherent oscillations of the two original populations, as instigated by the demographic noise component. In general, investigating the coupled dynamics of noisy oscillators of the type considered could yield an extended framework for synchronization studies, beyond the pioneering setting introduced by Kuramoto.

Stochastic momentum methods such as heavy ball (HB), Nesterov momentum, and variants of Accelerated SGD (ASGD) [Kidambi et al., 2018] are widely used in modern training, but their stochastic benefits depend on two distinct quantities: serial runtime, the number of iterations needed to reach a target accuracy, and compute efficiency (CE), the inverse total gradient-query or FLOP cost. Larger batches reduce serial runtime without hurting CE only when the contraction gap grows linearly with batch size. We study stochastic HB and ASGD for consistent linear regression with Gaussian covariates and prove finite-dimensional, discrete-time lower bounds on their batch-size tradeoffs. Our first result shows that HB does not improve the CE frontier over SGD for arbitrary spectra; rather, it preserves SGD-level CE over a larger batch-size window, allowing larger batches to reduce serial runtime until HB reaches its deterministic accelerated scale. This window can be a factor $\sqrt{\kappa}$ larger than the SGD critical batch size. For ASGD, the picture is more spectrum-dependent: for rapidly decaying power-law spectra, ASGD improves small-batch CE over HB/SGD, but as batch size grows it trades this CE advantage for improved serial runtime. Synthetic linear-regression experiments verify these qualitative regimes, including near-overlap of ASGD and HB for slowly decaying spectra and the predicted CE–serial tradeoff for rapidly decaying spectra.

Let $f(z)=\prod_{j=1}^{n}(z-a_j)$ be monic, with all zeros in the closed unit disk, and put $E_f=\{z\in\mathbb{C}: |z|\leq 1,\ |f(z)|\leq 1\}$. Let $S(n)$ be the largest possible shortest length of a path in $E_f$ joining $0$ to $\partial\mathbb{D}$, where the maximum is taken over all such polynomials of degree $n$. We prove that, for all sufficiently large $n$, $c\sqrt{\log n}\leq S(n)\leq \pi n$ with an absolute constant $c>0$. This proves the qualitative unboundedness predicted by Erdős. The proof combines an explicit geometric maze, Green-function and Faber-polynomial estimates, analytic quantization of circle measures, and a reciprocal-sweeping upper bound.

In this work we consider the pressureless Euler-Poisson system on $\mathbb{R}^3$ describing ion dynamics, which is a common model in the study of cold plasma. We prove global regularity and scattering for small, irrotational velocity fields around a constant density profile. At the heart of the result is a dispersive mechanism leading to amplitude decay at the full rate in $3$D. This fast decay feature and a favourable structure of the nonlinearity allow to control quadratic interactions globally.

We prove a `nonlinear graviton theorem' for higher-spin self-dual gravity. We consider small deformations of the complex structure of the non-projective twistor space that are bounded in a specified region near the origin and investigate the space $M_{HS}$ of holomorphically embedded complex planes $\mathbb{C}^2$ that intersect the origin. We show that this space is an infinite dimensional complex manifold with a canonical projection onto a four-dimensional holomorphic self-dual spacetime $\mathcal{M}$, and discuss the geometry induced on this new higher-spin space. Solutions of higher-spin self-dual gravity are then obtained by choosing an embedding of spacetime $\mathcal{M}$ into higher-spin space $M_{HS}$, with higher-spin symmetries arising from the different choices of embedding. Integrability of the theory is manifested in the form of a Lax pair for the system that we present. We conjecture that chiral higher-spin gravity can similarly be realized by considering deformations that are unconstrained at the origin.

We extend on the work of Bloch, Gillet, Soulé on non-archimedean Arakelov geometry, by proving in this context explicit formulas for Green currents, which are analogs of known formulas in complex geometry. More specifically, we prove an analog of the Poincaré-Lelong formula, as well as an equivalent of a formula by Bost, Gillet, Soulé, which expresses a Green current for the zero-locus of a regular section of a vector bundle. As corollaries, we also obtain non-archimedean counterparts of Levine formula and Martinelli formula.

This is the third and the last of a series of three papers, in which we prove a formula expressing the modular height of a unitary Shimura variety over a CM number field in terms of the logarithmic derivative of the Hecke L-function associated with the CM extension. The main idea of our proof is to compare the holomorphic projection of the derivative of a certain mixed Eisenstein-theta series and the arithmetic degree of a generating series of divisors on unitary Shimura varieties. In this paper, we compute the arithmetic degree of the arithmetic generating series of divisors on unitary Shimura varieties, and then, combining with the results from the first two papers in this series, derive the modular height formula for unitary Shimura varieties.

This manuscript reports on an ongoing project. The finite full-column-rank computations underlying the Key Vanishing Lemmas isolated below are currently undergoing exact computer-assisted verification. Conditional on successful completion of that verification, the arguments presented here would establish the two positivity endpoints of Demailly's invariant two-jet Riemann–Roch calculation: every very general surface in $\mathbb{P}^3$ of degree $15$ would be Kobayashi hyperbolic, and the complement of every general smooth plane curve of degree $11$ would be Kobayashi hyperbolic. At the logarithmic endpoint the same conditional argument would also give a Second Main Theorem with coefficient $1422$; the preceding work treated all degrees $d\geqslant12$. The argument separates an infinite geometric range from a finite algebraic obstruction.An $\mathcal{O}(3)$ slanted-vector-field package combines a quadratic acceleration with a classical matrix lift. Their incompatible actions on the invariant fibre ring, together with the common-twist, prime-ideal, and relative-family interfaces, exclude the dependence alternative for every weight $m\geqslant3$. This improves the differentiation range from $t\geqslant8$ to $t\geqslant4$ and leaves, beyond the twist-one ranges, only five compact and thirteen logarithmic systems. The residual systems are reduced by symmetry. A two-term Fermat perturbation carries an involution, so a hypothetical negatively twisted two-jet differential has an invariant or anti-invariant component. Imposing this eigencondition before elimination adds exact linear constraints without adding unknowns. The required full-column-rank statements are isolated in the Key Vanishing Lemmas as finite computational targets. This paper records their finite reduction and proof-to-certificate interface. Their exact computer-assisted verification is in progress.

How can training data be used to compare local updates to the current model, choose an update, and retain valid bounds for the selected update's population-risk change? We construct lower and upper confidence fields that jointly cover the population-risk change of every update in a possibly continuous local update space. The fields can therefore be used both to compare the updates and to select an update; a negative upper endpoint certifies improvement over the reference model. For linear risk changes in possibly infinite-dimensional feature spaces, cross-calibration uses the discrepancy between two balanced folds to calibrate the full-sample estimation error. Under covariance-aligned sub-Gaussian tails, covariance-estimation stability, and sufficient sample size, the cross-calibrated field has finite-sample simultaneous coverage. The confidence field's directional widths are governed by a population ridge effective dimension rather than the ambient feature dimension. For losses formed locally by continuous selection among finitely many smooth branches, separate uniform bounds for the linear Taylor field, Taylor remainder, and branch-interface discrepancy extend the field to every local update.

The low-altitude economy (LAE) heavily relies on aerial vehicles, yet these platforms remain vulnerable to environmental and security risks, necessitating robust airspace monitoring. Integrated sensing and communication (ISAC) as one of the key technologies of 6G provides potential solutions for safe LAE. However, conventional antenna arrays face limitations in cost, scalability, and coverage, especially directly above the base station, due to hardware complexity and degraded angular resolution. By exploiting the recently proposed ray antenna array (RAA), this paper considers a RAA-enhanced low-altitude ISAC system. RAA architecture employs multiple ray-arranged arrays directly connected without phase shifters, significantly reducing hardware costs while supporting flexible beamforming via dynamic ray selection. Moreover, RAA can provide uniform angular resolution and eliminates coverage holes, making it particularly suitable for low-altitude ISAC. In this paper, we formulate an optimization problem for joint ray selection and beamforming to enhance sensing coverage under communication constraints. An efficient alternating optimization algorithm is proposed to solve this problem. Analytical and simulation results demonstrate that RAA achieves higher sensing signal-to-noise ratio compared to traditional arrays, offering a cost-effective and high-performance solution for achieving low-altitude ISAC.

We study bilevel optimisation problems in which the inner problem is represented as a set-valued, parametric constraint. We develop relevant optimistic and pessimistic calculus rules, derive corresponding optimality conditions, and formulate nonsmooth adjoint inclusions based on both the Fréchet and limiting coderivatives. Founded on these results, we propose a single-loop algorithm that accommodates a wide range of inner and adjoint steps, including those of primal-dual methods. We prove its convergence. Numerical experiments on total variation regularised inverse problems demonstrate the practicality of the approach.

In a previous work, the topology of inversion-symmetric one-dimensional periodic media was characterized through the pole-zero pattern of an impedance-like function associated with Bloch waves. This construction reproduces the Berry–Zak invariant and provides a criterion for topological interface states. In the present work, we give a geometric interpretation of this formalism. We show that poles and zeros arise naturally from the action of inversion symmetry on the projectivized space of Cauchy data. The corresponding Dirichlet and Neumann states are identified with the two fixed points of the induced $\mathbb Z_2$ action on the Riemann sphere. The key observation is that Bloch eigenvectors are naturally constructed on the universal covering of the Brillouin circle. The topology of the associated Real eigenline bundle is encoded in the action of the deck transformation group on lifted eigenvectors. This action is described by a monodromy sign $\rho\in\{\pm1\}$, determined by the inversion representations carried by the band at the fixed points of the Brillouin zone. We show that this monodromy defines a natural rank-one local system over the Brillouin circle. The corresponding Real line bundle is classified by its first Stiefel–Whitney class, which coincides with the associated $\mathbb Z_2$ pole-zero invariant. This establishes a geometric connection between the pole-zero formalism, Berry–Zak phases, Real bundles and local coefficient systems.