The relationship between linear codes in the Hamming metric and projective algebraic varieties has led to deep interactions between coding theory and algebraic geometry, with classical examples such as Reed-Solomon codes and the rational normal curve. On the other hand, the sum-rank metric has recently gained attention due to applications in network coding, distributed storage, and post-quantum cryptography, with linearized Reed-Solomon codes emerging as optimal constructions. Despite recent advances, their structural and geometric properties are still not fully understood, and existing distinguishers remain limited. In this paper, we develop a geometric framework for linearized Reed-Solomon codes by considering a $q$-analogue of the rational normal curve. This yields a geometric characterization for certain parameter choices and reveals that the corresponding sets of points satisfy unexpectedly many $(q+1)$-degree hypersurface conditions. Our approach extends Schur-product-based techniques from the Hamming and rank-metric settings to the sum-rank metric case. Finally, we study the Hilbert function of the associated coordinate ring, providing a detailed description of its behavior and identifying its regularity, which also sheds new light on Gabidulin codes.

We prove the global-in-time existence of weak solutions for the Maxwell--Stefan--Fourier system with physically motivated degenerate thermal conductivity $κ(θ)=θ^α$, for $0<α\leq2$. In contrast to existing nonisothermal compressible theories based on nondegenerate conductivities, the entropy inequality no longer gives control of the quantities $\nablaθ$ and $\nabla \logθ$. The main new ingredient is a renormalized energy estimate, which yields compactness of the temperature and allows the identification of the degenerate heat flux.

Variational methods are widely used to approximate posterior distributions in Bayesian inference when exact computation is infeasible. We study when such approximations inherit posterior consistency. Our first result shows that, on a general metric space, a uniform bound on the Kullback--Leibler divergence from the approximating measures to a tight sequence of target measures forces the approximating sequence to be tight. It follows that if the target posteriors converge weakly to a Dirac mass at the true parameter, then any variational sequence with bounded Kullback--Leibler divergence to the targets is also consistent. We also give simple logarithmic-moment conditions that verify this boundedness condition, and illustrate them for smooth generalised posterior distributions.

We develop an embedded Trefftz-DG method for the Oseen problem and prove a complete stability and quasi-optimality theory in standard DG norms. The key ingredient is a construction of a suitable local complement space to the Trefftz space, on which the Oseen operator is stably invertible. We also derive a reduced formulation of the method, the resulting system is posed in terms of the velocity unknown only, a crucial step in the analysis especially for the nonlinear Navier-Stokes problem in Part II.

This paper introduces Bott-Chern and Aeppli homotopy sets for a fibrant class of bisimplicial sets and establishes their basic properties. In positive bidegrees, Bott-Chern homotopy sets carry natural monoid structures, while Aeppli homotopy sets carry natural group structures. They are related by a loop-space comparison: after a bidegree shift, the Aeppli homotopy groups of X are naturally identified with the Bott-Chern homotopy monoids of the loop space of X. In particular, the Bott-Chern homotopy monoids of loop spaces are groups. To justify our definitions, we show that the Bott-Chern homotopy monoids of a bisimplicial abelian group are naturally isomorphic to the Bott-Chern homology groups of its associated normalized Moore bicomplex. An analogous statement holds for Aeppli homotopy.

We develop and analyze an embedded Trefftz-DG method for the steady incompressible Navier-Stokes equations, based on the reduced Oseen discretization from Part I. The main difficulty is that the reduced Trefftz space depends on the convection field, so successive Picard iterates live in different discrete spaces. We address this by constructing projections between convection-dependent Trefftz spaces and using them to control the reduced Oseen solution map. Under suitable resolution and small-data assumptions, we prove existence of discrete solutions, uniqueness, and convergence of the Picard iteration. We also derive an a priori error analysis by relating the method to the underlying DG discretization, thereby inheriting convergence properties from compatible DG Navier-Stokes analyses. Numerical experiments on standard incompressible-flow benchmarks illustrate the theory.

Although the successive cancellation list (SCL) decoding of polar codes exhibits excellent performance, it retains many decoding paths in the list with negligible contribution to the final output, resulting in high sorting and computational complexity. In this letter, we propose a novel pruning strategy to mitigate the decoding complexity. By leveraging the blockwise soft output extraction process of soft-output SCL and soft-output fast SCL decoding, we provide an accurate approximation of the probability that a decoding path is correct, and thus accordingly prune the paths failing to meet a pre-defined reliability threshold. The complexity reduction achieved by the proposed soft-output-based pruned SCL (SOP-SCL) decoder and its fast version, SOP-FSCL decoder, is significant, without any compromise in error-correction performance. Meanwhile, they also prove to be more efficient than state-of-the-art pruned polar decoders.

In this paper, we develop a general theory of twist deformations for Hopf coquasigroups in a symmetric monoidal category. To this end, we first introduce and study non-coassociative bimonoids endowed with left and right codivisions, and establish their connection with left and right Hopf coquasigroups. Next, motivated by the classical theory of Drinfeld twists for Hopf algebras, we define twists for non-coassociative bimonoids and prove that they induce deformations of Hopf coquasigroup structures through suitable modifications of the coproduct. In particular, we obtain explicit deformation procedures for right and left Hopf coquasigroups and analyze the corresponding antipodes. Finally, we apply the general theory to construct nontrivial examples arising from Hopf coquasigroups associated with the sphere ${\sf S}^7$, obtaining new examples of twisted Hopf coquasigroups that are neither commutative nor cocommutative.

Assume that \(M_n\), the canonical inner model with \(n\) Woodin cardinals, exists. We force a model with continuum \(\aleph_2\) in which every \(\boldsymbolΣ^1_{n+2}\) set of reals is Lebesgue measurable and has the Baire property, the \(Σ^1_{n+2}\)- and \(Π^1_{n+3}\)-uniformization properties hold, and the reals admit a \(Δ^1_{n+3}\)-definable well-order. Thus regularity up to a fixed finite projective level, together with a definable well-order of the reals at the adjacent level, does not force the determinacy strength which would normally explain that regularity, even when this package is strengthened by adjacent \(Σ\)- and \(Π\)-uniformization. In particular, this gives a negative answer to a local form of Woodin's twelfth Delfino problem asked by Friedman-Schindler.

Assuming that $M_n$, the canonical inner model with $n$ Woodin cardinals, exists, we force a model in which every $\boldsymbolΣ^1_{n+2}$ set is Lebesgue measurable and has the Baire property, and in which $Σ^1_{n+2+m}$-uniformization holds for every $m\inω$. Additionally, this universe has a $Δ^1_{n+3}$-definable wellorder of the reals. This answers a question of S. D. Friedman and R. Schindler from 1999. In the case $n=1$, the construction also gives a model with one Woodin cardinal in which all $Σ^1_3$ sets are measurable with respect to the random, Cohen, Sacks and Miller notions of measurability, while a $Δ^1_4$-definable wellorder of the reals exists answering an instance of a question of S. D. Friedman and D. Schrittesser.

In this paper, we study a class of monotone generalized Nash equilibrium problems (GNEPs) with jointly linear constraints. The players' strategy spaces are real Hilbert spaces, while the joint constraint is formulated in a Banach space. To solve such problems, we propose a novel symmetric Gauss-Seidel (sGS) based alternating proximal augmented Lagrangian method (sGS-APALM) which incorporates newly designed quadratic surrogates. In contrast to existing regularization and ALM-type methods, the proposed method avoids solving coupled Nash equilibrium subproblems at each iteration and instead updates the players' strategies alternately by solving a sequence of unconstrained quadratic programs. Moreover, unlike many existing splitting-based methods, our global convergence analysis and convergence rate estimation require only monotonicity and Lipschitz continuity of the pseudo-gradient mapping, without imposing stronger assumptions such as strong monotonicity or co-coercivity. Finally, we apply the method to a class of risk-neutral PDE-constrained GNEPs with joint state constraints, and preliminary numerical results demonstrate its efficiency and effectiveness.

We investigate two problems concerning spaceability in function spaces. First, we study the spaceability of sets of nowhere regular periodic functions, for various notions of regularity, including Hölder regularity and real analyticity. Second, we show that the set of real analytic functions is spaceable in $C((0,1))$. Moreover, we provide a complete characterization of the closed subspaces of $C((0,1))$ consisting of real analytic functions. Our work solves several open questions posed by Bernal-González et al. [4, 5, 6].

We develop and analyze a fully discrete scheme for the weakly damped semilinear wave equation that combines a Symmetric Interior Penalty Discontinuous Galerkin (SIPG) spatial discretization with a hybrid Crank--Nicolson/second-order Backward Differentiation Formula (CN--BDF2) time integrator. A chord-slope linearization of the nonlinear reaction term is employed, which preserves an exact discrete gradient structure and, crucially, requires {no global Lipschitz continuity assumption} on the nonlinearity. Stability of the fully discrete solution is established through a Lyapunov-based analysis-rather than spectral arguments-by constructing a discrete Lyapunov functional that yields existence, uniqueness, and uniform boundedness of the numerical solution. Under standard regularity assumptions, optimal a~priori error estimates of order $\mathcal{O}(h^{k}+τ^{2})$ in the DG energy norm and $\mathcal{O}(h^{k+1}+τ^{2})$ in the $L^{2}$-norm are proved, where $h$ is the mesh size, $τ$ the time step, and $k$ the polynomial degree. Numerical experiments on two-dimensional problems with linear, cubic, and trigonometric nonlinearities confirm the theoretical convergence rates and illustrate the long-time energy-dissipation properties guaranteed by the Lyapunov structure.

We study an extension of Ramanujan's identities for odd zeta values by Lim and introduce Jacobi analogues of classical Eichler integrals of Eisenstein series. In negative weight we construct explicit completions and embed these objects into a modular framework by showing that they are (singular) harmonic Maass--Jacobi forms. We further describe their non-holomorphic parts in terms of Eichler integrals, establish Ramanujan-type inversion formulas, and study their behavior under the Maass raising and lowering operators and at torsion points.

Three important problems in Fourier analysis are the Fourier restriction problem, the $L^p$-improving problem, and the Fourier decay problem. Positive results for any of these problems require a quantitative understanding of various geometric properties of the given measure, including curvature and arithmetic resonance. In this paper we establish a unified framework for providing negative results for all three problems (that is, we provide explicit obstructions to a measure satisfying certain Fourier restriction, $L^p$-improving, or Fourier decay estimates) by quantifying flat parts of the measure in the spirit of the well-known Knapp examples from harmonic analysis. Our main interest is in the application of these abstract results in various concrete settings where we use analytic and fractal geometric concepts to force `flatness'. Our framework applies generally and this allows us to unify and extend various parts of the literature. Some representative applications include: (i) we bound the Fourier dimension of the surface measure on a compact $C^2$ surface of arbitrary dimension above by the smallest ambient rank of a point on the surface; (ii) we prove that the Fourier dimension of a smooth curve in $\mathbb{R}^d$ is at most $4/(d+1)$ and so such curves cannot be Salem for $d \geq 4$ with analogous results for higher dimensional submanifolds; (iii) we obtain explicit upper bounds for the Fourier dimension of the Patterson-Sullivan measure for parabolic Kleinian group actions, as well as ergodic measures on self-affine sets; (iv) we establish novel connections between Fourier restriction/decay and a priori unrelated concepts in fractal geometry including the Assouad spectrum of projections and slices, and a strong form of tube-nullity. We establish several auxiliary results along the way, including a precise characterisation of L^2-flattening in terms of the Fourier spectrum.

We prove two new results about probability preserving actions $T:\mathbb{Z}^d \curvearrowright (X,μ)$. First, for a function $f\in L^2(μ)$, we provide an explicit formula for the $L^2(μ)$-limit of the average \[\frac{1}{|Q_N^\mathcal{P}|}\sum_{v \in Q_N^\mathcal{P}} T_v f\] where $\mathcal{P}\subset \mathbb{Z}^d$ is the set of primitive vectors, i.e. those for which the greatest common divisor of its components is $1$, and $Q_N^\mathcal{P}= [-N,N]^d\cap \mathcal{P}$. Second, for a set $A\subset X$ with $μ(A)>0$, we provide a spectral condition under which the set of $\varepsilon$-expansive directions \[\left\{ v\in \mathbb{Z}^d \, : \, μ\left(\bigcup_{n\in \mathbb{Z}} T_{nv}A\right)>1-\varepsilon\right\}\] has lower density very close to $1$. As an application of our techniques we are also able to prove a quantitative variant of a twisted multiple recurrence theorem of Björklund, Fish and the first author (arXiv:2503.02501).

We study structure within the homotopy groups of the Davis-Januszkiewicz space DJ(K) associated with a simplicial complex K. The inclusion of each vertex in K induces a map from the two-sphere into DJ(K). These maps generate a quasi-Lie subalgebra QL(K) via the Whitehead product and a Pi-subalgebra S(K) via the Whitehead product and composition. We describe the quasi-Lie subalgebra QL(K), and show that the Pi-subalgebra S(K) coincides with the whole of the homotopy groups of DJ(K) if and only if K is a flag complex. Extensions to more general polyhedral products are also considered.

Working over a (left) coherent ring, we consider the class of weakly Ding injective complexes. These are the cycles of the exact complexes of FP-injective complexes that stay exact when applying $\Hom(A,-)$ for any FP-injective complex $A$. We study the cotorsion pair generated by the class of all such complexes, and exhibit it as part of an abelian model structure. As an application we show that when $R$ is a Ding-Chen ring, its stable chain complex category is compactly generated and triangle equivalent to the stable category of four Frobenius categories. They are the categories of all (i) complexes of Gorenstein injective modules, (ii) complexes of Gorenstein projective modules, (iii) complexes of Gorenstein flat-cotorsion modules, and (iv) complexes of Gorenstein FP-pro-injective modules.

Given a square matrix, the permanent is a determinant-like function without signs. In this paper, we study inequalities involving permanents of certain submatrices. We first focus on a family of zero-one totally nonnegative matrices that arise as anti-adjacency matrices of unit interval orders. For these matrices, we prove a collection of inequalities comparing products of permanents of consecutive principal submatrices with products of permanents of parity-selected principal submatrices. We also study a related combinatorial problem involving two families of permutations: a Young subgroup and a set of parity alternating permutations. We construct a bijective map from the first family to the second family, and verify computationally for n at most 13 that each permutation is below its image in Bruhat order. We conjecture that this property holds for all n. If true, this would imply one of the main permanent inequalities for all totally nonnegative matrices in the balanced case. More broadly, we conjecture that the full family of inequalities holds for all totally nonnegative matrices.

We present an algorithm for computing the join of two elements in the weak order of the Coxeter group of type B. This extends Markowsky's algorithm for computing joins of standard permutations to signed permutations, and allows us to confirm a conjecture of Dyer concerning a geometric interpretation of these joins.

In this paper, we study the dynamical Prandtl equation, which plays an important role in the study of the vanishing viscosity limit of the Navier--Stokes equations. Our focus is on the (Sobolev) well-posedness regime, where the given data satisfy a crucial monotonicity condition. In this case, local classical solutions have been constructed in the pioneering works of Oleinik \cite{O68,OS99}. More recently, global weak solutions were obtained in \cite{XZ04} by Xin and Zhang, and in \cite{XZZ24} by Xin, Zhang, and Zhao, where the uniqueness and interior H"older estimates of the solutions were established (in Crocco coordinates). Using a precise description of the fundamental solution to the Kolmogorov equation in the half-space, we first obtain the H"older regularity of local weak solutions up-to-boundary. We also provide a detailed proof of higher-order regularity estimates using this H"older regularity, together with H"ormander's hypoelliptic estimates, which are nontrivial. Up-to-boundary smoothness of solutions (in Crocco coordinates) is important in order to conclude the smoothness of the Prandtl solutions in the physical variables, even in the interior. It is also physically significant for applications to the Boundary Layer Theory where the dynamical Prandtl equation is essential. Using these smoothing estimates, we then prove the global existence and regularity of classical solutions to the dynamical Prandtl equation under monotonicity assumptions, which was listed by Oleinik and Samokhin in \cite{OS99} as one of the open problems. We also develop a self-contained local existence theory using weighted energy estimates and further expand the theory of global weak solutions. The main point is to incorporate all physical types of asymptotic matching of the boundary layer with the outer flow, which is expected to be useful for applications to the Navier--Stokes equations.

Let $\mathcal{H}$ be a family of graphs. A graph is said to be $\mathcal{H}$-free if it contains no subgraph isomorphic to a graph in $\mathcal{H}$. The planar Turán number $ex_{_\mathcal{P}}(n,\mathcal{H})$ is defined as the maximum number of edges in an $\mathcal{H}$-free planar graph on $n$ vertices. In this paper, we determine the exact value of $ex_{_\mathcal{P}}(n,\{K_{4}, Θ_{6}^{1}\})$ and a tight upper bound of $ex_{_\mathcal{P}}(n,\{K_{4}, Θ_{6}^{2}\})$.

We show that if a graded monotone Lagrangian $L_0$ has a non-vanishing disk potential, then for every smooth isotopy $\{L_s\}_{s\in[0,1]}$ of Lagrangians starting from it and for every tame almost complex structure $J$, each $L_s$ bounds a $J$-holomorphic disk of Maslov index two. The main input is a non-archimedean analytic potential function, defined as an invariant up to analytic isomorphisms, generalizing the classical disk potential of a monotone Lagrangian. The techniques are inspired by recent developments in the Strominger-Yau-Zaslow mirror construction via family Floer theory and non-archimedean geometry. We also discuss applications such as recovering a simple case of Audin's conjecture.

Let $Φ$ be a univalent function in $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$% , $Φ(\mathbb{D})$ is symmetric with respect to the real axis, starlike with respect to $Φ(0)=1$, and $Φ^{\prime }(0)>0$. Let $\mathcal{C}% (Φ)$ denote the class of Ma-Minda convex functions. In this article, we present the bounds on $||a_{3}|-|a_{2}||$ for Taylor's coefficients of the function $f$ in the class $\mathcal{C}(Φ)$. We also establish the same bounds for the inverse coefficients. All the bounds we study here are sharp. We also present the conditions such that the bounds on $|a_{3}|-|a_{2}||$ and $|A_{3}|-|A_{2}||$ are invariant, where $A_{2}$ and $A_{3}$ are the first two coefficients of the Taylor series of the inverse functions of $f\in \mathcal{C}(Φ).$ Thus provides examples of invariance and nonvariance among the subclasses of convex functions.

Lagrangian sensors have shown promise to improve operator awareness of conditions inside a chemical reactor but three-dimensional tracking remains a mostly unsolved challenge. We explore a setup where in-silico sensors, based on a recently proposed real-world design, are tracked using data from an accelerometer and magnetometer available from a built-in inertial measurement unit. Filtering algorithms, using a bespoke dynamical model, are used to process these readings into position estimates. We compare tracking performance of an extended Kalman filter, a particle filter and the unscented Kalman filter implemented in the pykalman library. Our numerical experiments track in-silico particles moving in an analytically given three dimensional vortex as well as in the experimentally measured flow-field of a lab-scale stirred tank reactor. Using the Maxey-Riley-Gatignol equations for the movement of inertial particles as ground-truth, we demonstrate that trajectories can be reconstructed from noisy synthetic data with errors below 10%.

This article investigates stochastic processes taking values in metric spaces that lack a topological vector space structure, a regime characterized by intricate interplay between topological, geometric, and temporal dependence structures. It is formally established that spaces admitting an isometric Hilbertian embedding constitute a strict subclass within the much broader class of metric spaces possessing the ball property. While traditional kernel methods are susceptible to geometric distortion when the underlying space cannot be isometrically embedded into a Hilbert space, we bypass such limitations by exploiting a fundamental structural property inherent to this broader class; namely, that Borel probability measures are uniquely determined by their values on balls. These separating classes provide the foundation for the subsequently introduced measure-theoretic inference methodology. We derive uniform convergence of a family of time-dependent random measures, alongside weak invariance principles for the corresponding nonstationary random fields. This framework explicitly exposes how dependence and geometric complexity influence sample path regularity. Furthermore, because the rapid decay of small-ball probabilities can prohibit the existence of limiting distributions for supremum-based discrepancy measures, we develop $L^p$-based alternatives. By directly leveraging the introduced convergence results, this approach circumvents the need for higher-order $U$-process formulations. Finally, for spaces that do admit an isometric Hilbertian embedding, and where $U$-processes naturally arise, we establish limit theory for both degenerate and nondegenerate multi-parameter $U$-processes, and demonstrate that local discrepancy tests maintain asymptotic stability under dynamic parameter regimes.

Rigidity, stability and local minimizing properties of Einstein metrics as critical points of quadratic Riemannian functionals defined by $L^2$-norms of Ricci curvature, scalar curvature, Weyl curvature and Riemannian curvature have been extensively studied. However, there are non-Einstein critical points of these functionals that are not so well understood. In this paper, we study Ricci solitons, a generalization of Einstein metrics, that are critical points of a special quadratic curvature functional and analyze their rigidity.

We consider the uniqueness of the solution of the dissipative surface quasi-geostrophic equation, without assuming time-continuity and smallness of the solutions. We show that the uniqueness holds in the scale-critical Lebesgue spaces and non-homogeneous Besov spaces. The proof is based on the energy method, inspired by the approach introduced by Lions and Masmoudi (2001) in the study of uniqueness for the Navier-Stokes equations. A key ingredient of the argument is the justification of the energy inequality via the smoothing effect of the fractional heat semigroup together with an iteration scheme based on the structure of the integral equation.

Forecasting the evolution of complex dynamical systems remains a fundamentally challenging task, primarily due to pronounced nonlinear interactions, high-dimensional state spaces, and the concomitant requirement for rigorous and reliable uncertainty quantification. Contemporary reduced-order modelling (ROM) frameworks frequently exhibit inherent trade-offs among predictive accuracy, numerical stability, and interpretability, and thus often fail to achieve an optimal balance among these competing objectives. To address these limitations, we propose a framework for forecasting complex dynamical systems via a kernel autonomous ordinary differential equation approach based on Gaussian Processes and Quadratic Order Model Reduction. Our base method, the Gaussian Process Ordinary Differential Equations model, allows accurate short-term forecasting with uncertainty quantification, and it provably converges to the real autonomous equation in the smooth case. We integrate it with quadratic order reduced-order modelling and sphere projection for learning the latent dynamics efficiently while preserving stability. Numerical experiments demonstrate that our full model outperforms ROM forecasting methods such as Extended Dynamic Mode Decomposition, Bagging Optimised Dynamic Mode Decomposition and Linear and Nonlinear Disambiguation Optimisation in terms of accuracy or computational costs. These results demonstrate the potential of the framework as a robust and stable tool for forecasting complex dynamical systems with rigorous uncertainty quantification.

We consider a scheduling problem for star observations using a telescope. Identical nights are available to observe stars that have been identified as of interest by astronomers. Each star yields a profit if it is observed; in that case, the observation must respect a visibility window and a minimum observation duration. The objective is to maximize the total profit of the observations that are actually performed. By interpreting nights as machines and stars as jobs to be processed, this problem is a classic scheduling setting of parallel machines. In the practical star-observation problem, the ability to observe depends on meteorological and atmospheric conditions, which are not known when the schedule is computed. A possible modeling is to consider that each night can be either perfect and everything is observable, either terrible and nothing is observable. The challenge is therefore to propose a ''high-quality'' schedule without knowing the actual number of nights (machines) that will ultimately be available. We focus here on evaluating the worst case within the robust optimization paradigm.