Papers

We study a singularly perturbed harmonic problem, with nonlinear Neumann boundary conditions of signed exponential type, arising in galvanic corrosion applications. We focus on the low-conductivity regime, which leads to a singular limit in which the solution presents a jump in the boundary. We prove convergence of the solution traces in the corresponding fractional Sobolev spaces, identify the interior limit solution, and obtain a sharp logarithmic energy expansion for smooth boundary junctions. The proofs rely mostly on trace Moser–Trudinger estimates for showing well-posedness and regularity, and on mollifier approximation techniques for the singular limit analysis. Numerical experiments test the predicted convergence rate.

The Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime, which was first proposed by Friedmann (1922–1924) and Lemaître (1927) and subsequently developed by Robertson and Walker (1935), is an isotropic and homogeneous cosmological model of the universe. This paper addresses a significant gap in the differential geometry literature by providing a comprehensive examination of the curvature properties of the FLRW spacetime. It is demonstrated that the FLRW spacetime satisfies the curvature condition R \cdot R - Q(S, R)=L_C Q(g, C) alongside several pseudosymmetric-type conditions related to the conformal and conharmonic curvature tensors. Furthermore, the Tachibana tensors Q(g,C) and Q(S, C) are found to exhibit a linear dependence on the tensor $(C \cdot R + R \cdot C)$. Additionally, the spacetime is shown to be a 2-quasi-Einstein manifold, generalized Roter type and Ein(3). The Ricci tensor is shown to be neither cyclic parallel nor of Codazzi type, yet it satisfies several compatibility requirements concerning the R, C, P, K and W curvature tensors. A thorough analysis of Ricci solitons and curvature inheritance properties reveals that the spacetime admits almost Ricci soliton and $\eta$-Ricci Yamabe soliton structures with respect to the non-Killing soliton vector fields $\frac{\partial}{\partial t}$ and $\frac{\partial}{\partial r}$. Moreover, the spacetime admits generalized curvature inheritance symmetry properties for the Riemann curvature tensor, as well as for the Weyl conformal, concircular, and conharmonic curvature tensors with respect to the coordinate vector field $\frac{\partial}{\partial t}$ and the gradient of $t$. Later, a comparison of the FLRW and Lemaître–Tolman–Bondi (LTB) spacetimes is provided in terms of various curvature-related geometric properties and physical characteristics. Finally, a noteworthy conclusion of the entire study is presented.

In this paper, we study the Bohr phenomenon for a generalized subclass of harmonic mappings defined by a second-order differential inequality in the unit disk. Specifically, we consider the class $\mathcal{BH}_0(\gamma, \delta)$, which extends several known subclasses of harmonic and analytic functions. By employing sharp coefficient estimates and growth results, we establish improved versions of Bohr-type inequalities, including refined Bohr radii and Bohr–Rogosinski radii for this class. Furthermore, we derive generalized inequalities involving higher-order coefficient sums and area terms, thereby extending classical Bohr inequalities in a harmonic setting. The sharpness of the obtained results is verified through extremal functions. In addition, we obtain Landau-type theorems for the class $\mathcal{BH}_0(\gamma, \delta)$, providing explicit bounds for the radius of univalence and the size of schlicht disks contained in the image domain. Our results not only unify and extend several earlier works but also provide new insights into the geometric behavior of harmonic mappings under differential constraints.

We describe a multi-neck spacetime wormhole with a simple metric tensor and a simple injective map without coordinate patching. An intra-universe, non-thin-shell, non-spherically-symmetric 3-neck spacetime wormhole is geometrically constructed by spherically inverting a 3-torus. We place the resulting Dupin hypercyclide in a synchronous reference frame. The three necks are arranged around a central point and satisfy topological and geometric spacetime wormhole definitions. Asserting this metric tensor as an exact solution of Einstein's field equations in global coordinates generates diagonal Ricci and stress-energy tensors, and a Riemann curvature tensor with only six nonzero entries. The local inertial frame at every point of the coordinate system is comoving with the triple wormhole. This non-vacuum solution answers affirmatively the question posed by Einstein and Rosen (1935) of whether or not multi-neck solutions exist. The wormhole solution contains negative energy density as is expected to hold the necks open; however, geodesic paths through each neck exist which encounter only positive energy density. The spatial manifold is a trivariate Dupin hypercyclide. The spherically inverted equal-radii 3-torus is unbounded, asymptotically flat and admits a global isothermal coordinate system that further simplifies the curvature tensors.

Decarbonizing electricity generation, heating, and transportation simultaneously requires integrated planning tools that can coordinate multiple energy production sources and demand points while remaining reliable under uncertainty. This paper develops a carbon-aware and uncertainty-resilient optimization framework for a grid-connected multiple source hub that co-optimizes electricity, heating, cooling, and transport energy services with an explicit hydrogen sub-hub. The proposed model is formulated as an MILP over a 25-year planning horizon (2025-2050). The hub integrates renewable electricity (Photovoltaic and wind), dispatchable resources (including natural-gas-based conversion), storage systems, demand response, and a hydrogen subsystem comprising an electrolyzer and hydrogen storage to supply hydrogen-vehicle demand and provide temporal flexibility. Two policy archetypes are examined: a Carbon Tax (price instrument), and a Net-Zero pathway (quantity instrument). To hedge feasibility-critical operational uncertainty, the deterministic model is extended using budgeted robust optimization and a tunable uncertainty budget. The developed scheme is applied to the province of Ontario, Canada; the results indicate substantial long-term hydrogen expansion, with the electrolyzer capacity increasing from 300~MW (2025) to 3,800~MW (2050) and hydrogen storage from 2,000~MWh to 37,000~MWh (2050), accompanied by sharply higher hydrogen production. Compared with deterministic solutions, robust solutions preserve feasibility under adverse realizations but incur a moderate robustness premium of approximately 6.6-9.0% in total cost across the policy cases studied, while slightly reducing hydrogen utilization and renewable share and increasing reliance on dispatchable balancing.

We consider polynomial maps of affine space over an algebraically closed field of characteristic zero. We prove that every irreducible component of the zero locus of the Jacobian determinant corresponds to either a contracted divisor or a branching divisor. We further consider polynomial maps of degree two without contracted divisors and show that the Jacobian determinant is irreducible, anti-invariant under the Galois involution, and coincides with the defining equation of the unique branching divisor.

We present MARUT, a scalable multi-GPU computational fluid dynamics (CFD) framework designed for high-fidelity simulations of compressible flows spanning subsonic to hypersonic regimes, including chemically reacting nonequilibrium flows with finite-rate chemistry and adaptive mesh refinement (AMR). The framework addresses a central challenge in contemporary scientific computing: the development of numerically accurate and computationally scalable algorithms capable of resolving strongly nonlinear, multiscale flow physics on emerging heterogeneous supercomputing architectures. Built around a distributed-memory MPI-parallel infrastructure and implemented natively on NVIDIA GPUs, MARUT combines high-order spectral discontinuous Galerkin discretisations with strong-stability-preserving Runge–Kutta time integration to achieve low-dissipation and high-resolution representation of shocks, vortical structures and reactive interfaces. Dynamic AMR further enables efficient concentration of computational resources in localized regions of physical complexity, thereby substantially reducing computational cost while preserving solution fidelity. MARUT is designed to maintain strong parallel efficiency through GPU-resident computations and scalable MPI communication strategies, achieving near-linear strong scaling across multiple GPUs. The solver is validated against a broad suite of canonical benchmark problems involving inviscid, viscous, and reactive compressible flows, including subsonic, transonic, supersonic, and hypersonic configurations with multi-species nonequilibrium chemistry. The numerical predictions show close agreement with established reference solutions. Beyond its immediate performance characteristics, the framework reflects the broader transition of computational science towards modular, adaptive and AI-compatible simulation ecosystems.

Lezama \cite{LezamaLatorre2017} introduced the notion of semi-graded ring with the aim of generalizing $\mathbb{Z}$-graded rings and several families of noncommutative rings of polynomial type non-$\mathbb{N}$-graded such as the skew Poincaré-Birkhoff-Witt extensions defined by him \cite{GallegoLezama2010}. In a series of papers, \cite{Lezama2020, Lezama2021, LezamaGomez2019, LezamaLatorre2017}, he studied problems of non-commutative projective algebraic geometry generalizing the original ideas of Artin et al. \cite{Artin1992, ArtinSchelter1987, ArtinTateVandenBergh2007, ArtinTateVandenBergh1991, ArtinZhang1994} on $\mathbb{N}$-graded rings, in the categorical context of the category $\mathsf{SGR}-R$ of left semi-graded modules over a semi-graded ring $R$. In this note we prove that $\mathsf{SGR}-R$ possesses a canonical set of free generators via shifted twists, which endows the category with a Grothendieck structure and guarantees the existence of enough injective and projective objects. This categorical robustness allows us to formulate a semi-graded analogue of Baer's criterion for injectivity and to establish a first approach to the dual theory of projective resolutions using shifted twists.

We study adversarial online learning with hidden-convex losses, i.e., nonconvex losses that become convex after a nonlinear reparameterization. Ghai, Lu and Hazan (2022) proved that, under geometric and smoothness assumptions, online gradient descent (OGD) on such nonconvex losses approximately simulates online mirror descent (OMD) on the underlying convex losses with a suitable regularizer, yielding $\mathcal{O}(T^{2/3})$ regret. They left open whether the optimal $\Theta(\sqrt{T})$ regret from online convex optimization can be recovered in this hidden-convex setting. We answer this question affirmatively. More specifically, via a sharper discrete-time algorithmic equivalence argument, we prove that OGD achieves $\mathcal{O}(\sqrt{T})$ regret under the same assumptions, matching the optimal worst-case rate for adversarial online convex optimization. We also address another open question of Ghai, Lu and Hazan (2022) by clarifying the geometry required for this algorithmic equivalence. We replace the diagonal-Jacobian sufficient condition with a necessary-and-sufficient Hessian compatibility condition, thereby expanding the class of admissible reparameterizations. We complement our tight regret bound with a lower bound showing that OGD suffers $\Omega(T)$ regret even with a stepsize of order $1/\sqrt{T}$, for a smooth Hessian incompatible reparameterization between convex domains and an adversarial sequence of hidden-convex losses. Finally, we extend our analysis to one-point bandit feedback and prove a $\mathcal{O}(T^{3/4})$ expected regret bound for bandit OGD with spherical smoothing, matching its classical rate on convex losses.

Motivated by the emergence of local groundwater exchanges, we construct and analyze stochastic models of dynamic groundwater markets. Our primary focus is endogenizing the price formation and groundwater pumping strategies in a closed market with stochastic groundwater allocations and opportunities for intertemporal transfer through rights banking. In our model, several agents, interpreted as farmers or agricultural districts, make competitive decisions on water consumption to produce a basket of goods, as well as on trading allocations among themselves, or banking them for future periods. We define the respective discrete-time non-zero-sum non-cooperative game and construct its sub-game perfect Nash equilibria characterized by the groundwater price process $\{p^\circ(t)\}$. We furthermore construct an algorithm to determine equilibrium strategies and prices through a machine learning approach on top of best-response iterations. Extensive numerical experiments illustrate dynamic phenomena, including the role of groundwater recharge dynamics, agents' risk aversion and groundwater allocations. Our model provides insights into competitive effects in environmental markets with banking features.

Policy learning in modern operations environments faces a fundamental tension between limited operational data and the large, often continuous, state and action spaces over which good decisions must be identified and deployed. We study value-based policy learning in stochastic optimal control: a greedy policy induced by an estimate of the optimal action-value function $Q^*$ is deployed, and its performance is measured by regret. The empirical success of this approach calls for statistical insight into the structures that enable fast regret convergence. We show that, in continuous action spaces, fast policy learning is induced by three geometric structures: a growth exponent $p$, which quantifies how quickly $Q^*$ separates suboptimal actions from its maximizers; a margin-mass exponent $m$, which controls how much deployment mass lies on states with weak growth; and an action-wise regularity exponent $q$, which measures the smoothness of the $Q^*$-estimation error across actions. Given a $n^{-1/2}$-accurate estimator of $Q^*$, we show that the minimax-optimal policy regret convergence rate is \[ \widetilde{\Theta}\left( n^{-\min\left\{\frac{p}{2(p-q)},\frac{m+1}{2m}\right\}} \right), \] up to a logarithmic factor at the boundary between the two regimes. The exponent $q$ is crucial: $q>0$ yields faster-than-$n^{-1/2}$ regret. This regime is natural in operations applications. In particular, we verify $q>0$ under mild regularity conditions in dynamic inventory control and service allocation examples, while the mechanism underlying this fast rate regime extends beyond these settings.

Let $k$ be an $F$-finite and infinite field of characteristic $p>2$. We show, there exist infinitely many $F$-finite local domains $(R,\mathfrak{m})$ which are not $\mathbb{Q}$-Gorenstein and $\tau_{\mathrm{b}}(R;\mathfrak{m}^t)$ has all but finitely many irrational $F$-jumping numbers.

We develop a framework for studying how global spectral structure emerges from interacting local sectors in stratified operadic systems. The central object is the interaction residue, which measures the failure of exact spectral decomposition across interfaces. Under suitable localization assumptions, the global spectrum decomposes into local spectral sectors together with interface-generated residue contributions. The theory introduces a classification of spectral defects based on interface geometry and algebraic structure. Point interfaces produce isolated spectral contributions; line and surface interfaces produce extended spectral regimes. Non-semisimple operator structure generates nilpotent defects associated with Jordan blocks and generalized eigenspaces, yielding a two-dimensional defect taxonomy combining geometric localization with Jordan complexity. Several structural results are established, including interface localization, rigidity and vanishing criteria, refinement functoriality, and deformation stability. Under a local triviality condition, the residue is homotopy invariant, preserving its homology and Betti numbers throughout admissible deformations. The framework is illustrated through explicit operator and block matrix examples demonstrating how localized interactions generate localized spectral defects. The nilpotent sector connects the theory with classical operator theory via generalized eigenvectors, functional calculus, and perturbative Jordan splitting. Overall, the framework provides a unified viewpoint for understanding how interface interactions and non-semisimple structure influence global spectral behavior.

In this paper, we use affine surfaces to describe completely the real-time trajectories of a homogeneous vector field in $\mathbf{C}^{2}$. We prove the existence of a continuous "rotation number" in $\mathbf{C}^{2}$ which is constant along real-time trajectories. A description of the trajectories depending on their rotation number is given.

We show that, given $d \geq 4$ and two closed connected oriented PL $4$-manifolds $M$ and $N$ such that $N$ has a handle decomposition with no $1$- and $3$-handles, there exists a $d$-fold (simple) branched covering $p \colon M \rightarrow N$ if and only if there is an isometric embedding of lattices $d \cdot I_N \hookrightarrow I_M$. Here $I_N$ and $I_M$ respectively denote the intersection lattices of $N$ and $M$. In particular, we characterize the manifolds which are branched covers of the K3 surface.

We determine the $RO(C_2)$-graded Hurewicz images of the $C_2$-equivariant Eilenberg–MacLane spectra $H\underline{\mathbb F_2}$, $H\underline{\mathbb Z}$ and $H\underline{A}$, where $\underline{\mathbb F_2}$ and $\underline{\mathbb Z}$ denote the constant Mackey functors with values in $\mathbb F_2$ and $\mathbb Z$, respectively, and $\underline A$ denotes the Burnside Mackey functor. Surprisingly, the answer is closely tied to the problem of vector fields on spheres: the element $\frac{\theta}{\rho^k\tau^n}$ in the negative cone of the homotopy groups of $H\underline{\mathbb F_2}$ lies in the Hurewicz image if and only if $S^n$ admits $k$ linearly independent vector fields. Moreover, using the Generalized Leibniz Rule and the Generalized Mahowald Trick introduced by arXiv:2412.10879, we show that there are nonzero Adams differentials of arbitrary length supported by filtration-$0$ elements in the genuine $C_2$-equivariant Adams spectral sequence.

Uncertainty relations are usually formulated as trade-offs between two or more observables. Yet the uncertainty of a single observable can already contain an intrinsically quantum contribution arising from its noncommutativity with the state. In the sense of Luo, such a contribution can be quantified by a quantum uncertainty $Q_\rho(A)$, with prominent examples including the Wigner–Yanase and Wigner–Yanase–Dyson skew informations and the quantum Fisher information. Here we ask how strong such single-observable uncertainty relations can be when the quantum state is fixed. We solve this fixed-state optimization problem by determining the largest coefficient $c_{\rm opt}(\rho)$ for which $V_\rho(A)\ge c_{\rm opt}(\rho)Q_\rho(A)$ holds for every observable $A$, and show that it depends only on the smallest and largest eigenvalues of $\rho$. The same optimization principle extends to a new power-commutator family $\frac{1}{2}|[\rho^s,A]|^2$ $(s\ge 1/2)$, providing a direct measure of state–observable noncommutativity beyond the standard quantum-uncertainty framework. More importantly, the optimal noncommutative contribution can be combined with the maximal classical contribution compatible with the state, yielding a sharpened variance decomposition without weakening the optimal bound. For states with exactly two distinct eigenvalues, this relation becomes an identity for every observable; in particular, qubit variances decompose exactly into classical and noncommutative parts. Finally, the corresponding variance-product bounds are no weaker than the Robertson relation for qubit spin observables and can be strictly stronger, showing that a substantial part of the uncertainty usually expressed as a trade-off between distinct observables is already encoded in their individual noncommutativity with the state.

Most generalized fractional operators rely on prescribed memory kernels, restricting hereditary behavior to predefined forms and limiting flexibility in modeling diverse memory effects. Motivated by these limitations, this paper develops a generator-based framework for fractional calculus in which memory laws are systematically generated through a dynamic memory generator in the Laplace domain. The resulting construction produces dynamic-memory kernels via inverse Laplace transforms, leading to generalized dynamic-memory fractional integrals together with Riemann–Liouville and Caputo dynamic-memory fractional derivatives. Fundamental analytical properties are established, including inverse relations, composition formulas, admissibility conditions, semigroup structures, and consistency principles. In addition, a unified convolution-symbol operational calculus and generalized dynamic-memory Mittag–Leffler functions are developed. Unlike fixed-kernel formulations, the proposed framework can generate singular, nonsingular, tempered, logarithmic, oscillatory, and multiscale memory behaviors within a single analytical setting. Numerous classical and modern fractional operators are recovered as special cases, demonstrating the unifying capability and flexibility of the developed theory.

Learned physics simulators are often evaluated by one-step or short-horizon prediction error, but these metrics can miss failures in temporal composition and long-horizon rollout. For autonomous, state-complete systems, exact solution maps satisfy a semigroup law: direct evolution over $s+t$ should agree with evolution over $s$ followed by $t$. We propose normalized semigroup error as a post hoc, model-agnostic diagnostic comparing these direct and composed learned predictions. On one-dimensional heat and Burgers dynamics with time-conditioned ConvNet and FNO baselines, semigroup error is positively associated with rollout degradation, with trajectory-level Spearman correlation $\rho = 0.635$ and $95%$ CI $[0.621, 0.649]$. Semigroup regularization has mixed effects, supporting semigroup consistency primarily as an evaluation diagnostic rather than a universally beneficial training objective.

Generalized inverses play a fundamental role in numerical linear algebra, particularly when matrices are rectangular, singular, or rank deficient. Even when the input matrix is sparse, generalized inverses such as the M-P pseudoinverse are typically dense, leading to high storage requirements, expensive matrix-vector multiplications, and reduced efficiency. We investigate computing sparse symmetric generalized inverses for sparse symmetric matrices, extending previous work on sparse generalized inverses for rectangular rank-deficient matrices. We consider minimizing the vector 1-norm over the set of symmetric generalized inverses, using vector 1-norm minimization as a surrogate for sparsity promotion. We give a new characterization of symmetric generalized inverses of symmetric matrices, which yields a compact affine reformulation of the problem. From this, we develop a Douglas-Rachford splitting (DRS) algorithm equipped with a closed-form projection onto the feasible affine space. Computational experiments compare the proposed DRS approach with exact linear-optimization formulations solved by a commercial optimizer, as well as with local-search heuristics. The results demonstrate that the proposed formulation and algorithm produce generalized inverses that are substantially sparser than the Moore-Penrose pseudoinverse while maintaining significantly smaller 1-norms than competing approaches. Moreover, the DRS algorithm exhibits superior scalability relative to exact linear-optimization formulations, successfully solving instances far beyond the reach of commercial solvers. As an application, we investigate the computation of least-squares solutions for problems involving many right-hand-side vectors and sparse rank-deficient design matrices.

We present a fast Jacobi-like algorithm for computing the eigenvalues, and optionally the eigenvectors, of a real normal matrix. The method gains a computational advantage by using Paardekooper's method for skew-symmetric matrices The method is most efficient for matrices where most eigenvalues are complex, such as random orthogonal matrices arising in the context of statistics on manifolds. In this case, the method is faster than the other Jacobi-like algorithms. In the last section of this paper, we also give explicit formulas for the nearest symmetric skew-Hamiltonian and the nearest ortho-symplectic matrix. These problems arise in the design and the analysis of the algorithm.

The detection of unmanned aerial vehicles (UAVs) is important for the protection of civilian and military infrastructure. In this paper we propose a cost effective UAV detection system using sound signals obtained from microphones. The recorded signals are passed through a signal processing pipeline which employs interpretable adaptive feature extractors using so-called rational Gaussian wavelets. These adaptive wavelet transformations are embedded into and trained together with an underlying small neural network which detects and classifies UAVs based on the obtained features. This leads to a physically interpretable machine learning algorithm that in addition to classifying UAVs is also capable of detecting UAV swarms. We demonstrate our results using data collected in indoor studio and noisy outdoor environments. We conclude that the proposed method outperforms traditional machine learning approaches for detecting and classifying single UAVs as well as drone swarms, while retaining a high degree of interpretability. Our implementation of the proposed methods is made publicly available for reproducibility.

In this article, we provide the first theoretical framework guaranteeing that computers can, in principle, be used to analyze the parameter space of complex Hémaps. More precisely, we obtain computability results for hyperbolic polynomial diffeomorphisms of $\mathbb{C}^2$, for which Hénon maps are prototypical examples. Specifically, we establish computability of the Julia set for hyperbolic maps, semi-decidability of hyperbolicity, and lower computability of the hyperbolicity locus in the parameter space of generalized Hénon mappings of fixed degree at least two. Our approach builds upon techniques developed in our's recent previous works on polynomial maps of $\mathbb{C}$ and polynomial skew products of $\mathbb{C}^2$. In the setting of polynomial diffeomorphisms of $\mathbb{C}^2$, however, establishing hyperbolicity for the Julia set is considerably more difficult, as it requires identifying unstable (and stable) cone fields that are preserved and expanded by $Df$ (respectively $Df^{-1}$), and also due to the lack of algorithmically detectable quantitative shadowing.

This work addresses the problem of learning the dynamics of high-dimensional probability densities over time using unlabeled samples, without assuming access to trajectory information. We introduce two-parameter flows that learn only sampling-time transports from a base distribution to each marginal and then extract a physics-time velocity by regressing on coupled synthetic trajectories. We prove that the resulting physics-time dynamics are unique and inherit regularity from the sampling-time transports. Because we can build on standard, well-developed conditional flow matching techniques for learning the base-to-marginal transports, our approach scales to high dimensions and avoids per-step optimal-transport couplings, while allowing admissible non-gradient dynamics that can naturally explain rotational or circulating physics phenomena.

This paper has a threefold aim. On the one hand, we provide a complete description of Laguerre-Hahn forms of class zero. This fills a gap in the literature: more precisely, up to an affine change of variables, there are ten families, including two new ones analogous to the Bessel classical family. On the other hand, we establish the structure relations for all these families, correcting those that have previously been reported in the literature. At last, as an application, using an algorithm previously obtained, with the aid of symbolic computations, we derive four new structure relations and a new fourth-order differential equation for one of the new families analogous to Bessel.

For every integer \(n\ge 3\), every \(1\le \ell\le n-2\), and every sufficiently large integer \(m\), we construct harmonic functions \(u_{m,\ell}\) on the unit ball \(B_1(0)\subset\mathbb{R}^n\) such that the frequency is bounded independently of \(m\), every point of the nodal set \(\{u_{m,\ell}=0\}\cap B_{1/2}(0)\) is regular, but the Betti numbers satisfy \begin{align*} b_\ell\bigl(\{u_{m,\ell}=0\}\cap B_{1/2}(0)\bigr)\ge 2m. \end{align*} Thus bounded frequency, even together with regularity of the nodal set, does not imply a uniform topological bound. In particular, these examples give counterexamples to the claimed global Betti-number bound of Lin and Liu.

We investigate the stability of quantum positivity cones under nonlinear operator means. Specifically, we examine how Kubo–Ando means interact with the separable, positive partial transpose (PPT), and Schmidt-number cones. By analyzing the curvature of operator monotone functions at the identity, we give a strict rigidity phenomenon: weighted arithmetic means are the only Kubo–Ando means that preserve the separable cone in all dimensions. We show that the strictly positive curvature of any non-arithmetic mean explicitly forces a violation of the PPT condition, even in the foundational two-qubit setting, and can strictly increase the Schmidt number of the resulting operator. Finally, using the Choi–Jamiołkowski correspondence, we translate these geometric obstructions to the map-theoretic setting, concluding that convex mixing is the uniquely permissible Kubo–Ando operation for preserving entanglement-breaking quantum channels.

We establish uncertainty principles on compact Riemannian manifolds without boundary by combining restriction estimates for orthonormal systems with spectral projection bounds for Laplace-Beltrami and Schrödinger operators. Our results relate the size of the support of spectrally localized functions to the cardinality of the underlying spectral cluster and to Fourier-ratio type quantities. We obtain analogues for Schrödinger operators with singular potentials belonging to Kato and scaling-critical classes. As an application, we prove uniqueness results for recovery from incomplete spectral data on compact manifolds. Under curvature assumptions, including nonpositive and negative sectional curvatures, we also prove logarithmically improved uncertainty principles associated with shrinking spectral windows.

Let $\mathbb{F}_q$ denote the finite field of order $q$. For $q$ odd, we investigate the planarity over $\mathbb{F}_{q^3}$ of the family $$ f_{E,A,B,C,D}(X) := EX^2+ AX^{q+1}+ BX^{q^2+1}+CX^{2q} +DX^{2q^2}\in \mathbb{F}_{q}[X]. $$ Using results from the theory of q-polynomials, we establish conditions under which these polynomials are planar functions. In particular, we provide characterizations for the planarity property and present new families of planar trinomials, quadrinomials, and pentanomials.

Composite optimization problems, where a smooth loss is combined with a nonsmooth regularizer, are common in machine learning and inverse problems. In this work, we study a proximal extension of NAG-GS, a semi-implicit accelerated method obtained from a Gauss-Seidel discretization of an inertial dynamics. The proposed method, called Prox-NAG-GS, keeps the coupled structure of NAG-GS for the smooth part and replaces the second update by a proximal step. It therefore applies to objectives of the form $F=f+r$, where $f$ is smooth and $r$ is convex and proximable. We derive deterministic convergence guarantees for this method. The analysis has to account for a specific feature of the scheme. Prox-NAG-GS keeps two coupled sequences: an $x$-sequence, on which the gradient of the smooth term is evaluated, and a $v$-sequence, produced by the proximal update. The gradient is evaluated at $x_{k+1}$, whereas the proximal step returns $v_{k+1}$, which creates a mismatch absent from the standard proximal-gradient analysis. Under the sufficient condition that the proximal quadratic parameter is at least as large as the smoothness constant of $f$, we control this mismatch through an augmented Lyapunov function involving both sequences. This gives a linear convergence result in the strongly convex composite case. In the convex case, the same Lyapunov structure yields an $O(1/k)$ rate for the best iterate and for the averaged iterate. We test the method on deterministic Elastic Net and Group Lasso problems, and on stochastic sparse softmax-regression benchmarks. In the deterministic tests, Prox-NAG-GS reaches the same solutions as the baselines with substantially fewer iterations; for Group Lasso this also gives the best wall-clock time. In the stochastic tests, Prox-NAG-GS compares favorably with Prox-SGD in terms of data-fit reduction and gives similar test accuracies.