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3e8a1aacf2ae5faba8500c165b19e5076e7e23f0ea2973e8e6b356f76df24e6e
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2026-01-21T00:00:00-05:00
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Going beyond density functional theory accuracy: Leveraging experimental data to refine pre-trained machine learning interatomic potentials
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arXiv:2506.10211v2 Announce Type: replace-cross Abstract: Machine learning interatomic potentials (MLIPs) are inherently limited by the accuracy of the training data, usually consisting of energies and forces obtained from quantum mechanical calculations, such as density functional theory (DFT). Since DFT itself is based on several approximations, MLIPs may inherit systematic errors that lead to discrepancies with experimental data. In this paper, we use a trajectory re-weighting technique to refine DFT pre-trained MLIPs to match the target experimental Extended X-ray Absorption Fine Structure (EXAFS) spectra. EXAFS spectra are sensitive to the local structural environment around an absorbing atom. Thus, refining an MLIP to improve agreement with experimental EXAFS spectra also improves the MLIP prediction of other structural properties that are not directly involved in the refinement process. We combine this re-weighting technique with transfer learning and a minimal number of training epochs to avoid overfitting to the limited experimental data. The refinement approach demonstrates significant improvement for two MLIPs reported in previous work, one for an established nuclear fuel: uranium dioxide (UO2) and second one for a nuclear fuel candidate: uranium mononitride (UN). We validate the effectiveness of our approach by comparing the results obtained from the original (unrefined) DFT-based MLIP and the EXAFS-refined MLIP across various properties, such as lattice parameters, bulk modulus, heat capacity, point defect energies, elastic constants, phonon dispersion spectra, and diffusion coefficients. An accurate MLIP for nuclear fuels is extremely beneficial as it enables reliable atomistic simulation, which greatly reduces the need for large number of expensive and inherently dangerous experimental nuclear integral tests, traditionally required for the qualification of efficient and resilient fuel candidates.
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https://arxiv.org/abs/2506.10211
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ea7ad3d30113463c381547aecb40be2784f82c10e2a14806510cbe6f25f358cb
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2026-01-21T00:00:00-05:00
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Toward a Circular Nanotechnology for Biofuels: Integrating Sustainable Synthesis, Recovery, and Performance Optimization
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arXiv:2506.17548v2 Announce Type: replace-cross Abstract: This review exhaustively evaluates the role of nanomaterials across the synthesis, characterization and application stages of biofuel systems. Common types of nanomaterials that are used for biofuel applications include metal oxides, carbon-based structures, and hybrids, which are evaluated for their effectiveness in efficient biofuel production. The properties of such nanomaterials are being utilized as an aid to produce biofuels through improved catalysis, enzyme immobilization and thermal stability. Common synthesis methods, such as sol-gel, coprecipitation, and green synthesis, are compared, alongside characterization tools, such as TEM, SEM, FTIR, and BET. This study focuses on transesterification, biomass pretreatment, and fermentation processes, where nanomaterials significantly improve yield and reusability. There are several challenges, despite the merits of using nanomaterials, and the trade-offs include cost, scalability, and environmental impact, which further expand into evaluating the life cycle of such materials. This review outlines the practical potential of nanomaterials in enabling efficient and sustainable biofuel production.
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https://arxiv.org/abs/2506.17548
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4b1699a262d2a9375b5373162eb5eae79f523430d5704b928061db1334f0f6b0
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2026-01-21T00:00:00-05:00
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Classification of curl forces for all space dimensions
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arXiv:2507.09817v2 Announce Type: replace-cross Abstract: We present a decomposition of classical potentials into a conservative (gradient) component and a non-conservative component. The latter generalizes the curl component of the force in the three-dimensional case. The force is transformed into a differential $1$-form, known as the work form. This work form is decomposed into an exact (gradient) component and an antiexact component, which in turn generalizes the curl part of the force. The antiexact component is subsequently decomposed using the Frobenius theorem. This local decomposition is a useful tool for identifying the specific components of classical potentials.
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https://arxiv.org/abs/2507.09817
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f2abc4cb1deb5673b9b70399990fe5ea112c9d175ac170d808253879bd3bb9c5
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2026-01-21T00:00:00-05:00
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Nonlinear phase synchronization and the role of spacing in shell models
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arXiv:2507.14142v2 Announce Type: replace-cross Abstract: A shell model can be considered as a chain of triads, where each triad can be interpreted as a nonlinear oscillator that can be mapped to a spinning top. Investigating the relation between phase dynamics and intermittency in a such a chain of nonlinear oscillators, it is found that synchronization is linked to increased energy transfer. In particular, the results provide evidence that the observed systematic increase of intermittency, as the shell spacing is decreased, is associated with strong phase alignment among consecutive triadic phases, facilitating the energy cascade. It is shown that while the overall level of synchronization can be quantified using a Kuramoto order parameter for the global phase coherence in the inertial range, a local, weighted Kuramoto parameter can be used for the detection of burst-like events propagating across shells in the inertial range. This novel analysis reveals how partially phase-locked states are associated with the passage of extreme events of energy flux. Applying this method to helical shell models, reveals that for a particular class of helical interactions, a reduction in phase coherence correlates with suppression of intermittency. When inverse cascade scenarios are considered using two different shell models including a non local helical shell model, and a local standard shell model with a modified conservation law, it was shown that a particular phase organization is needed in order to sustain the inverse energy cascade. It was also observed that the PDFs of the triadic phases were peaked in accordance with the basic considerations of the form of the flux, which suggests that a triadic phase of \pi/2 and -\pi/2 maximizes the forward and the inverse energy cascades respectively.
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https://arxiv.org/abs/2507.14142
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62e5f1659a7a4ef9a2df4f3327b7f7e722d7852f03fe60ee245c78b4a22a20c3
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2026-01-21T00:00:00-05:00
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Enhanced sensitivity to trace $^{238}$U impurity of sapphire via coincidence neutron activation analysis
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arXiv:2508.04232v3 Announce Type: replace-cross Abstract: Sapphire has mechanical and electrical properties that are advantageous for the construction of internal components of radiation detectors such as time projection chambers and bolometers. However, it has proved difficult to assess its $\rm ^{232}Th$ and $\rm ^{238}U$ content down to the picogram per gram level. This work reports an experimental verification of a computational study that demonstrates $\gamma\gamma$ coincidence counting, coupled with neutron activation analysis (NAA), can reach ppt sensitivities. Combining results from $\gamma\gamma$ coincidence counting with those of earlier single-$\gamma$ counting based NAA shows that a sample of Saint Gobain sapphire has $\rm ^{232}Th$ and $\rm ^{238}U$ concentrations of $<0.26$ ppt and $<2.3$ ppt, respectively; the best constraints on the radiopurity of sapphire.
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https://arxiv.org/abs/2508.04232
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ab620a68b329c8a4120ddd541fed6b9034d540c4a7e8ec848d90546cba085203
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2026-01-21T00:00:00-05:00
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Robust Control and Entanglement of Qudits in Neutral Atom Arrays
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arXiv:2508.16294v2 Announce Type: replace-cross Abstract: Quantum devices comprised of elementary components with more than two stable levels - so-called qudits - enrich the accessible Hilbert space, enabling applications ranging from fault-tolerant quantum computing to simulating complex many-body models. While several quantum platforms are built from local elements that are equipped with a rich spectrum of stable energy levels, schemes for the efficient control and entanglement of qudits are scarce. Importantly, no experimental demonstration of multi-qudit control has been achieved to date in neutral atom arrays. Here, we propose a general scheme for controlling and entangling qudits and perform a full analysis for the case of qutrits, encoded in ground and metastable states of alkaline earth atoms. We find an efficient implementation of single-qudit gates via the simultaneous driving of multiple transition frequencies. For entangling operations, we provide a concrete and intuitive recipe for the controlled-Z (CZ) gate for any local dimension d, realized through alternating single qudit and entangling pulses that simultaneously drive up to two Rydberg transitions. We further prove that two simultaneous Rydberg tones are, in general, the minimum necessary for implementing the CZ gate with a global drive. The pulses we use are optimally-controlled, smooth, and robust to realistic experimental imperfections, as we demonstrate using extensive noise simulations. This amounts to a minimal, resource-efficient, and practical protocol for realizing a universal set of gates. Our scheme for the native control of qudits in a neutral atom array provides a high-fidelity route toward qudit-based quantum computation, ready for implementation on near-term devices.
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https://arxiv.org/abs/2508.16294
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1146c03293d42fc0c3d263a090d949effbdf6d07cb6b55e839ce79ddf14bd762
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2026-01-21T00:00:00-05:00
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Floquet-engineered moire quasicrystal patterns of ultracold Bose gases in twisted bilayer optical lattices
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arXiv:2508.21093v5 Announce Type: replace-cross Abstract: We investigate the formation of moire quasicrystal patterns in Bose gasses confined in twisted bilayer optical lattices via Floquet-engineered intralayer atomic interactions. Dynamical evolutions of the total density wave amplitude exhibit the stage for the emergence of moire quasicrystal patterns, where the pattern formation is closely associated with the momenta of collective modes excited by the weak periodic drive. Through analyzing the radial and angular density wave amplitude, we find that these new collective modes are only coupled radially and cannot be decoupled eventually. The symmetry of quasicrystal patterns can be easily manipulated by the modulation frequencies and amplitudes. Reducing the frequencies and increasing the amplitudes can both facilitate lattice symmetry breaking and the subsequent emergence of rotational symmetry. Notably, a twelve-fold quasicrystal pattern emerges under specific parameters, closely resembling the moire quasicrystal in twisted bilayer graphene. The momentum-space distributions also exhibit high rotational symmetry, which is consistent with the real-space patterns at specific evolution times. Our findings establish a new quantum platform for exploring quasicrystals and their symmetry properties in ultracold bosonic systems.
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https://arxiv.org/abs/2508.21093
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d61eda08b5f50d8551a5b4f675495e5614faac740edb0a06dddf3cd79e33116a
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2026-01-21T00:00:00-05:00
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Coherent Two-State Oscillations in False Vacuum Decay Regimes
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arXiv:2509.04272v2 Announce Type: replace-cross Abstract: Coherent two-state oscillations are observed in numerical simulations of the one-dimensional transverse-longitudinal-field Ising model (TLFIM) within false vacuum decay regimes. Starting from the false vacuum (a nearly fully polarized ferromagnetic state), we show that in moderate-sized systems, at resonances $h\approx 2J/n$ (with longitudinal field $h$, transverse field $J$, and an integer $n$), the expected decay can give way to coherent oscillations between the false vacuum and a symmetric resonant state. The oscillation frequency, i.e., the tunneling splitting, is observed notably to exhibit a superradiant-like $\sqrt{L}$ enhancement, as confirmed by a Schrieffer-Wolff analysis. In large chains, coherence remains for $n\gtrsim L/2$ due to bubble-size blockade and is robust against stronger transverse fields; for small $n$, long-range interactions can stabilize the oscillations by lifting multi-bubble degeneracies, establishing a robust many-body coherence mechanism beyond perturbative and finite-size limits.
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https://arxiv.org/abs/2509.04272
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972de93ba001c5f21b0ec3cd3a3cc91357d71958aad13720275d224e56a1d840
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2026-01-21T00:00:00-05:00
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Transfer tensor analysis of localization in the Anderson and Aubry-Andr\'e-Harper models
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arXiv:2509.21374v2 Announce Type: replace-cross Abstract: We use the transfer tensor method to analyze localization and transport in simple disordered systems, specifically the Anderson and Aubry-Andr\'e-Harper models. Emphasis is placed on the memory effects that emerge when ensemble-averaging over disorder, even when individual trajectories are strictly Markovian. We find that transfer tensor memory effects arise to remove fictitious terms that would correspond to redrawing static disorder at each time step, which would create a temporally uncorrelated dynamic disorder. Our results show that while eternal memory is a necessary condition for localization, it is not sufficient. We determine that signatures of localization and transport can be found within the transfer tensors themselves by defining a metric called "outgoing-pseudoflux". This work establishes connections between theoretical research on dynamical maps and Markovianity and localization phenomena in physically realizable model systems.
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https://arxiv.org/abs/2509.21374
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f12150ff10f92846190602ec3ae1e57c7535546e24bdbc04168228560b15e78c
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2026-01-21T00:00:00-05:00
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Vector Nematodynamics with Symmetry-driven Energy Exchange
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arXiv:2510.24177v2 Announce Type: replace-cross Abstract: We review inadequacy of existing nematodynamic theories and suggest a novel way of establishing relations between nematic orientation and flow based on the \emph{local} symmetry between simultaneous rotation of nematic alignment and flow, which establishes energy exchange between the the two without reducing the problem to near-equilibrium conditions and invoking Onsager's relations. This approach, applied in the framework of the vector-based theory with a variable modulus, involves antisymmetric interactions between nematic alignment and flow and avoids spurious instabilities in the absence of an active inputs.
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https://arxiv.org/abs/2510.24177
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f4b185e33945e0fb037e6b4b6357bf9a8ecd2685eecf038478b63c09125ea13a
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2026-01-21T00:00:00-05:00
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Beyond MMD: Evaluating Graph Generative Models with Geometric Deep Learning
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arXiv:2512.14241v2 Announce Type: replace-cross Abstract: Graph generation is a crucial task in many fields, including network science and bioinformatics, as it enables the creation of synthetic graphs that mimic the properties of real-world networks for various applications. Graph Generative Models (GGMs) have emerged as a promising solution to this problem, leveraging deep learning techniques to learn the underlying distribution of real-world graphs and generate new samples that closely resemble them. Examples include approaches based on Variational Auto-Encoders, Recurrent Neural Networks, and more recently, diffusion-based models. However, the main limitation often lies in the evaluation process, which typically relies on Maximum Mean Discrepancy (MMD) as a metric to assess the distribution of graph properties in the generated ensemble. This paper introduces a novel methodology for evaluating GGMs that overcomes the limitations of MMD, which we call RGM (Representation-aware Graph-generation Model evaluation). As a practical demonstration of our methodology, we present a comprehensive evaluation of two state-of-the-art Graph Generative Models: Graph Recurrent Attention Networks (GRAN) and Efficient and Degree-guided graph GEnerative model (EDGE). We investigate their performance in generating realistic graphs and compare them using a Geometric Deep Learning model trained on a custom dataset of synthetic and real-world graphs, specifically designed for graph classification tasks. Our findings reveal that while both models can generate graphs with certain topological properties, they exhibit significant limitations in preserving the structural characteristics that distinguish different graph domains. We also highlight the inadequacy of Maximum Mean Discrepancy as an evaluation metric for GGMs and suggest alternative approaches for future research.
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https://arxiv.org/abs/2512.14241
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31c9198c0c65d7948409d56ce7a58bd8e429dd784fce54c6d5e7ce47da01dbae
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2026-01-21T00:00:00-05:00
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Robustness of the Frank-Wolfe Method under Inexact Oracles and the Cost of Linear Minimization
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arXiv:2601.11548v1 Announce Type: new Abstract: We investigate the robustness of the Frank-Wolfe method when gradients are computed inexactly and examine the relative computational cost of the linear minimization oracle (LMO) versus projection. For smooth nonconvex functions, we establish a convergence guarantee of order $\mathcal{O}(1/\sqrt{k}+\delta)$ for Frank-Wolfe with a $\delta$--oracle. Our results strengthen previous analyses for convex objectives and show that the oracle errors do not accumulate asymptotically. We further prove that approximate projections cannot be computationally cheaper than accurate LMOs, thus extending to the case of inexact projections. These findings reinforce the robustness and efficiency of the Frank-Wolfe framework.
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https://arxiv.org/abs/2601.11548
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44b6504270b6ff6408188336e6df342e1565dc4e3e491f71f30b4a26c2d2b3c0
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2026-01-21T00:00:00-05:00
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Minimal Perimeter Triangle in Nonconvex Quadrilateral:Generalized Fagnano Problem
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arXiv:2601.11552v1 Announce Type: new Abstract: In 1775, Fagnano introduced the following geometric optimization problem: inscribe a triangle of minimal perimeter in a given acute-angled triangle. A widely accessible solution is provided by the Hungarian mathematician L. Fejer in 1900. This paper presents a specific generalization of the classical Fagnano problem, which states that given a nonconvex quadrilateral (having one reflex angle and others are acute angles), find a triangle of minimal perimeter with exactly one vertex on each of the sides that do not form reflex angle, and the third vertex lies on either of the sides forming the reflex angle. We provide its geometric solution. Additionally, we establish an upper bound for the classic Fagnano problem, demonstrating that the minimal perimeter of the triangle inscribed in a given acute-angled triangle cannot exceed twice the length of any of its sides.
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https://arxiv.org/abs/2601.11552
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24826f64081f0d638eed7dd0c9ab93ff3841c74aedc1c7511b2277cc179ec331
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2026-01-21T00:00:00-05:00
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A Generalized Waist Problem: Optimality Condition and Algorithm
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arXiv:2601.11554v1 Announce Type: new Abstract: Many years ago John Tyrell a lecturer at King's college London challenged his Ph.D. students with the following puzzle: show that there is a unique triangle of minimal perimeter with exactly one vertex to lie on one of three given lines, pairwise disjoint and not all parallel in the space. The problem in literature is known as the waist problem, and only convexity rescued in this case. Motivated by this we generalize it by replacing lines with a number of convex sets in the Euclidean space and ask to minimize the sum of distances connecting the sets by means of closed polygonal curve. This generalized problem significantly broadens its geometric and practical scope in view of modern convex analysis. We establish the existence of solutions and prove its uniqueness under the condition that at least one of the convex sets is strictly convex and all are in general position: each set can be separated by convex hull of others. A complete set of necessary and sufficient optimality conditions is derived, and their geometric interpretations are explored to link these conditions with classical principles such as the reflection law of light. To address this problem computationally, we develop a projected subgradient descent method and prove its convergence. Our algorithm is supported by detailed numerical experiments, particularly in cases involving discs and spheres. Additionally, we present a real-world analogy of the problem in the form of inter-island connectivity, illustrating its practical relevance. This work not only advances the theory of geometric optimization but also contributes effective methods and insights applicable to facility location, network design, robotics., computational geometry, and spatial planning.
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https://arxiv.org/abs/2601.11554
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618eeb9fddd76bac2838228c7fc125095e798f3af6d03c3bb95814d2a13b5ff7
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2026-01-21T00:00:00-05:00
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A Generalized $(k,m)$ Heron Problem:Optimality Conditions and Algorithm
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arXiv:2601.11555v1 Announce Type: new Abstract: This paper presents a new extension of the classical Heron problem, termed the generalized $(k,m)$-Heron problem, which seeks an optimal configuration among $k$ feasible and $m$ target non-empty closed convex sets in $\mathbb{R}^n$. The problem is formulated as finding a point in each set that minimizes the pairwise distances from the points in the $k$-feasible sets to the points in the $m$-target sets. This formulation leads to a convex optimization framework that generalizes several well-known geometric distance problems. Using tools from convex analysis, we establish fundamental results on existence, uniqueness, and first-order optimality conditions through subdifferential calculus and normal cone theory. Building on these insights, a Projected Subgradient Algorithm (PSA) is proposed for numerical solution, and its convergence is rigorously proved under a diminishing step-size rule. Numerical experiments in $\mathbb{R}^2$ and $\mathbb{R}^3$ illustrate the algorithm's stability, geometric accuracy, and computational efficiency. Overall, this work provides a comprehensive analytical and algorithmic framework for multi-set geometric optimization with promising implications for location science, robotics, and computational geometry.
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https://arxiv.org/abs/2601.11555
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2cad51d6100e1d8bdcd761f50fa1ef8ff073da9bf9c7493cd1b001fd86a48186
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2026-01-21T00:00:00-05:00
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The global well-posedness for master equations of mean field games of controls
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arXiv:2601.11588v1 Announce Type: new Abstract: In this manuscript, we establish the global well-posedness for master equations of mean field games of controls, where the interaction is through the joint law of the state and control. Our results are proved under two different conditions: the Lasry-Lions monotonicity and the displacement $\lambda$-monotonicity, both considered in their integral forms. We provide a detailed analysis of both the differential and integral versions of these monotonicity conditions for the corresponding nonseparable Hamiltonian and examine their relation. The proof of global well-posedness relies on the propagation of these monotonicity conditions in their integral forms and a priori uniform Lipschitz continuity of the solution with respect to the measure variable.
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https://arxiv.org/abs/2601.11588
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f5d208e81ed8971b5ab218e60b32f7b7f8ef0b5aab1c9495b77d5100e75a104d
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2026-01-21T00:00:00-05:00
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Poisson semigroup and the Gruet formula for the heat kernels on spaces of constant curvature
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arXiv:2601.11596v1 Announce Type: new Abstract: This paper is concerned with the Poisson and heat equations on spaces of constant curvature. More explicitly we provide new methods for obtaining old and new explicit formulas for the Poisson and heat semigroups on the Euclidean, spherical and hyperbolic spaces $\R^n$, $\S^n$ and $\H^n$ . We obtain the Gruet formula for the heat kernels in Euclidean and spherical spaces $\R^n$ and $\S^n$, which are new and we provide a new elementary method to derive the classical Gruet formula Gruet\cite{Gruet} for the kernel of the heat semigroup on the hyperbolic space $\H^n$.
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https://arxiv.org/abs/2601.11596
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7abd9c3c84a054bfe40933efc01a814519839e2b14e6c3b821b6c5a153025da4
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2026-01-21T00:00:00-05:00
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Boundary Delocalization and Spectral Packets for Dirichlet Eigenfunctions
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arXiv:2601.11605v1 Announce Type: new Abstract: We establish a boundary delocalization principle for high-frequency Dirichlet eigenfunctions on smooth strictly convex domains. The main result excludes persistent boundary concentration at the level of individual eigenmodes when compared to short spectral packets of sublinear length. Quantitatively, we compare boundary energies of single eigenfunctions to packet sums over frequency windows of size N_k = o(k), without asserting any asymptotic gain in magnitude. The main mode-to-packet estimate relies only on the Rellich identity. For the multi-mode bias exclusion we additionally use the boundary local Weyl law to obtain a packet zero-mean cancellation estimate. This mode-to-packet comparison is independent of eigenvalue monotonicity and is stable under eigenvalue crossings.
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https://arxiv.org/abs/2601.11605
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9372ef587b036fca6bde0fb381b6d0924b6da81a3c6c4d307d9ca0ef55d1375b
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2026-01-21T00:00:00-05:00
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Concatenated Matrix SVD: Compression Bounds, Incremental Approximation, and Error-Constrained Clustering
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arXiv:2601.11626v1 Announce Type: new Abstract: Large collections of matrices arise throughout modern machine learning, signal processing, and scientific computing, where they are commonly compressed by concatenation followed by truncated singular value decomposition (SVD). This strategy enables parameter sharing and efficient reconstruction and has been widely adopted across domains ranging from multi-view learning and signal processing to neural network compression. However, it leaves a fundamental question unanswered: which matrices can be safely concatenated and compressed together under explicit reconstruction error constraints? Existing approaches rely on heuristic or architecture-specific grouping and provide no principled guarantees on the resulting SVD approximation error. In the present work, we introduce a theory-driven framework for compression-aware clustering of matrices under SVD compression constraints. Our analysis establishes new spectral bounds for horizontally concatenated matrices, deriving global upper bounds on the optimal rank-$r$ SVD reconstruction error from lower bounds on singular value growth. The first bound follows from Weyl-type monotonicity under blockwise extensions, while the second leverages singular values of incremental residuals to yield tighter, per-block guarantees. We further develop an efficient approximate estimator based on incremental truncated SVD that tracks dominant singular values without forming the full concatenated matrix. Therefore, we propose three clustering algorithms that merge matrices only when their predicted joint SVD compression error remains below a user-specified threshold. The algorithms span a trade-off between speed, provable accuracy, and scalability, enabling compression-aware clustering with explicit error control. Code is available online.
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https://arxiv.org/abs/2601.11626
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19d7092cf3798ac6af13a0dc50c163bb49fb2d588f38543e4ac3819ffc6a2b5b
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2026-01-21T00:00:00-05:00
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Qualitative analysis and numerical investigations of time-fractional Zika virus model arising in population dynamics
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arXiv:2601.11636v1 Announce Type: new Abstract: Epidemic models play a crucial role in population dynamics, offering valuable insights into disease transmission while aiding in epidemic prediction and control. In this paper, we analyze the mathematical model of the time-fractional Zika virus transmission for human and mosquito populations. The fractional derivative is considered in the Caputo sense of order $\alpha\in(0,1).$ We begin by conducting a qualitative analysis using the stability theory of differential equations. The existence and uniqueness of the solution are established, and the model's stability is examined through Hyers-Ulam stability analysis. Furthermore, an efficient difference scheme utilizing the standard L1 technique is developed to simulate the model and analyze the solution's behavior under key parameters. The resulting nonlinear algebraic system is solved using the Newton-Raphson method. Finally, illustrative examples are presented to validate the theoretical findings. Graphical results indicate that the fractional model provides deeper insights and a better understanding of disease dynamics. These findings aid in controlling the virus through contact precautions and recommended therapies while also helping to predict its future spread.
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https://arxiv.org/abs/2601.11636
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51d0f729a014dd33d6d30efeb8e8566f0066bfa2fed916acd2212e19122ea7fa
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2026-01-21T00:00:00-05:00
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A note on Reeb spaces of some explicit real analytic functions
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arXiv:2601.11648v1 Announce Type: new Abstract: Reeb spaces of smooth functions are fundamental and strong tools in understanding manifolds via smooth functions with mild critical points. They are defined as the natural spaces of all connected components of level sets. They are also important objects in related studies. Realization of graphs as Reeb spaces of smooth functions of certain nice classes is of such studies. In this paper, we present Reeb spaces of explicit real analytic functions which are not finite graphs. Related problems were started by Sharko, in 2006, who has studied smooth functions with critical points represented by certain elementary polynomials, and followed by a study of Masumoto and Saeki, which is on smooth functions on closed surfaces under an extended situation, and a study of Michalak, which is on Morse functions on closed manifolds. The author has contributed to this by respecting topologies of level sets, and real algebraic construction.
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https://arxiv.org/abs/2601.11648
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7936867cf10fb89619e02c0d20e3129f35d0c45e050a6e24170984e2ffa86d70
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2026-01-21T00:00:00-05:00
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Dirichlet Extremals for Discrete Plateau Problems in GT-Bezier Spaces via PSO
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arXiv:2601.11677v1 Announce Type: new Abstract: We study a discrete analogue of the parametric Plateau problem in a non-polynomial tensor-product surface spaces generated by the generalized trigonometric (GT)--B\'ezier basis. Boundary interpolation is imposed by prescribing the boundary rows and columns of the control net, while the interior control points are selected by a Dirichlet principle: for each admissible choice of B\'ezier basis shape parameters, we compute the unique Dirichlet-energy extremal within the corresponding GT--B\'ezier patch space, which yields a parameter-dependent symmetric linear system for the interior control net under standard nondegeneracy assumptions. The remaining design freedom is thereby reduced to a four-parameter optimization problem, which we solve by particle swarm optimization. Numerical experiments show that the resulting two-level procedure consistently decreases the Dirichlet energy and, in our tests, often reduces the realized surface area relative to classical Bernstein--B\'ezier Dirichlet patches and representative quasi-harmonic and bending-energy constructions under identical boundary control data. We further adapt the same Dirichlet-extremal methodology to a hybrid tensor-product/bilinear Coons framework, obtaining minimality-biased TB--Coons patches from sparse boundary specifications.
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https://arxiv.org/abs/2601.11677
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6ce58fe5e6eb7ec1d84719a551c21c617806cda27a445609bc48aaccb409230e
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2026-01-21T00:00:00-05:00
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Five Circles: Real Analysis Theorems equivalent to Completeness
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arXiv:2601.11681v1 Announce Type: new Abstract: This is an exposition of the work of O. Riemenschneider about five ''circles'' of implications relating real analysis theorems each equivalent to the Dedekind completeness of the real field. These circles cover five elements of real function theory: convergence, connectedness, differentiability, compactness and integration.
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https://arxiv.org/abs/2601.11681
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f10de7fafa73b95487259523ab34e0662d61e09ff6387010cbdfd54ba4ee0d15
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2026-01-21T00:00:00-05:00
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Age-Based Scheduling for a Memory-Constrained Quantum Switch
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arXiv:2601.11698v1 Announce Type: new Abstract: In a time-slotted system, we study the problem of scheduling multipartite entanglement requests in a quantum switch with a finite number of quantum memory registers. Specifically, we consider probabilistic link-level entanglement (LLE) generation for each user, probabilistic entanglement swapping, and one-slot decoherence. To evaluate the performance of the proposed scheduling policies, we introduce a novel age-based metric, coined age of entanglement establishment (AoEE). We consider two families of low-complexity policies for which we obtain closed-form expressions for their corresponding AoEE performance. Optimizing over each family, we obtain two policies. Further, we propose one more low-complexity policy and provide its performance guarantee. Finally, we numerically compare the performance of the proposed policies.
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https://arxiv.org/abs/2601.11698
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b29e40cee457c2492d923e75e6544df758eb4635568f5aa417840cd10ec260c3
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2026-01-21T00:00:00-05:00
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Stability and Accuracy Trade-offs in Statistical Estimation
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arXiv:2601.11701v1 Announce Type: new Abstract: Algorithmic stability is a central concept in statistics and learning theory that measures how sensitive an algorithm's output is to small changes in the training data. Stability plays a crucial role in understanding generalization, robustness, and replicability, and a variety of stability notions have been proposed in different learning settings. However, while stability entails desirable properties, it is typically not sufficient on its own for statistical learning -- and indeed, it may be at odds with accuracy, since an algorithm that always outputs a constant function is perfectly stable but statistically meaningless. Thus, it is essential to understand the potential statistical cost of stability. In this work, we address this question by adopting a statistical decision-theoretic perspective, treating stability as a constraint in estimation. Focusing on two representative notions-worst-case stability and average-case stability-we first establish general lower bounds on the achievable estimation accuracy under each type of stability constraint. We then develop optimal stable estimators for four canonical estimation problems, including several mean estimation and regression settings. Together, these results characterize the optimal trade-offs between stability and accuracy across these tasks. Our findings formalize the intuition that average-case stability imposes a qualitatively weaker restriction than worst-case stability, and they further reveal that the gap between these two can vary substantially across different estimation problems.
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https://arxiv.org/abs/2601.11701
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98dfbb9b1239d0c6b2538ad821dd3c61477bd64cbd9317ad8b812e8646b3ded2
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2026-01-21T00:00:00-05:00
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Detecting Mutual Excitations in Non-Stationary Hawkes Processes
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arXiv:2601.11717v1 Announce Type: new Abstract: We consider the problem of learning the network of mutual excitations (i.e., the dependency graph) in a non-stationary, multivariate Hawkes process. We consider a general setting where baseline rates at each node are time-varying and delay kernels are not shift-invariant. Our main results show that if the dependency graph of an $n$-variate Hawkes process is sparse (i.e., it has a maximum degree that is bounded with respect to $n$), our algorithm accurately reconstructs it from data after observing the Hawkes process for $T = \mathrm{polylog}(n)$ time, with high probability. Our algorithm is computationally efficient, and provably succeeds in learning dependencies even if only a subset of time series are observed and event times are not precisely known.
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https://arxiv.org/abs/2601.11717
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8cd694e77d63cae37956c1bedf139d4926328500c4c6a2699cac259c44f518d4
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2026-01-21T00:00:00-05:00
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Asymptotically Optimal Tests for One- and Two-Sample Problems
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arXiv:2601.11727v1 Announce Type: new Abstract: In this work, we revisit the one- and two-sample testing problems: binary hypothesis testing in which one or both distributions are unknown. For the one-sample test, we provide a more streamlined proof of the asymptotic optimality of Hoeffding's likelihood ratio test, which is equivalent to the threshold test of the relative entropy between the empirical distribution and the nominal distribution. The new proof offers an intuitive interpretation and naturally extends to the two-sample test where we show that a similar form of Hoeffding's test, namely a threshold test of the relative entropy between the two empirical distributions is also asymptotically optimal. A strong converse for the two-sample test is also obtained.
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https://arxiv.org/abs/2601.11727
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4a2b5e4d30636c992597412b869e2a49aca8e36b4cfc81b97176cec978f923c8
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2026-01-21T00:00:00-05:00
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Positive energy-momentum theorems for asymptotically AdS spin initial data sets with charge
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arXiv:2601.11728v1 Announce Type: new Abstract: For complete spin initial data sets with an asymptotically anti--de Sitter end, we introduce a charged energy--momentum defined as a linear functional arising from the Einstein--Maxwell constraints. Under a dominant energy condition adapted to the presence of a negative cosmological constant, we establish positive energy--momentum theorems, showing in particular that this functional is non--negative on a natural real cone. We place particular emphasis on the case where the manifold carries a compact inner boundary. In the time--symmetric setting, this yields a mass--charge inequality for asymptotically hyperbolic manifolds with charge.
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https://arxiv.org/abs/2601.11728
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77cbe4cc5d1a29537c5fa36ef6a15192fc138c9c0eb0155d14b51375a0e156ce
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2026-01-21T00:00:00-05:00
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Construction of a Gibbs measure for the zonal Dirac equation
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arXiv:2601.11730v1 Announce Type: new Abstract: We propose a framework to construct Gibbs measures for the Dirac equation. We consider the Dirac equation on the sphere with a "Hartree-type" nonlinearity. We consider a zonal model, that is the analog of a spherically symmetric model but on the sphere. We build a Gibbs measure for this model. With a compactness argument, we prove the existence of a random variable that is a weak solution to the Dirac equation and whose law is the Gibbs measure at all times.
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https://arxiv.org/abs/2601.11730
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ac7ad8d43aae9a64876c4b5cef8e5f4b514c7f0fdca900f4ef35ec03585d67c7
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2026-01-21T00:00:00-05:00
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Multiary gradings
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arXiv:2601.11738v1 Announce Type: new Abstract: This article develops a comprehensive theory of multiary graded polyadic algebras, extending the classical concept of group-graded algebras to higher-arity structures. We introduce the notion of grading by multiary groups and investigate various compatibility conditions between the arity of algebra operations and grading group operations. Key results include quantization rules connecting arities, classification of graded homomorphisms, and concrete examples including ternary superalgebras and polynomial algebras over $n$-ary matrices. The theory reveals fundamentally new phenomena not present in the binary case, such as the existence of higher power gradings and nontrivial constraints on arity compatibility.
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https://arxiv.org/abs/2601.11738
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aff2645393eae7dde5d7842af261cdbd7a8ffcd78438dd7754e61fca832cd140
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2026-01-21T00:00:00-05:00
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Integrated Optimization of Scheduling and Flexible Charging in Mixed Electric-Diesel Urban Transit Bus Systems
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arXiv:2601.11751v1 Announce Type: new Abstract: The transition of transit fleets to alternative powertrains offers a potential pathway to reducing the cost of mobility. However, the limited range and long charging durations of battery electric buses (BEBs) introduce significant operational complexities, necessitating innovative scheduling and charging strategies. This study proposes an integrated mixed-integer linear programming model to optimize vehicle scheduling and charging strategies for mixed fleets of BEBs and diesel buses. Unlike existing models, which often assume a fixed BEB fleet size or restrict charging to a single charger type, our approach simultaneously determines the optimal fleet composition, scheduling, and flexible partial charging strategy incorporating both slow and fast chargers at garages and terminal stations. The model minimizes combined fleet purchase and operational costs. A queuing strategy is introduced, departing from traditional first-come, first-served methods by dynamically allocating waiting and charging times based on operational priorities and resource availability, improving overall scheduling efficiency. To overcome computational complexities arising from numerous variables, a column generation framework is developed, facilitating scalable solutions for large-scale transit networks. Numerical experiments using real-world transit data from the Chicago Transit Authority and the Pace suburban bus systems demonstrate the model's effectiveness. Results indicate that while a full transition to alternative powertrains results in a modest cost increase, optimal mixed-fleet configurations can actually reduce total system costs. Furthermore, sensitivity analyses reveal that restricting charging to garages significantly increases fleet size and operational costs, underscoring the potential of distributed opportunistic charging.
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https://arxiv.org/abs/2601.11751
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5a1c281ceeae288970db1b03b1adf7e4f651443050ce95623cd2854539cadc6a
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2026-01-21T00:00:00-05:00
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Existence of Decreasing Nambu Solutions to the Rainbow Ladder Gap Equation of QCD by Cone Compression
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arXiv:2601.11752v1 Announce Type: new Abstract: Studying Nambu solutions of the rainbow-ladder gap equation in QCD at zero temperature and chemical potential, we prove that the mass function emerges continuously from zero as the interaction strength is increased past the critical point for all positive, asymptotically perturbative kernels almost everywhere continuous in $L^1$ using the Krasnosel'skii-Guo Cone Compression Theorem. We prove that the coupled system of equations must have a positive, continuous Nambu solution with decreasing mass function for all current quark masses for a class of models which includes the physical point of a popular model of QCD by using a hybrid Krasnosel'skii-Schauder Fixed Point Theorem.
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https://arxiv.org/abs/2601.11752
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d647934ff2898e5babe7b03d5dec1965e5ea04cb36d11c3f12c3cfb6ee8a8483
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2026-01-21T00:00:00-05:00
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A volume formula for Reuleaux polyhedra
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arXiv:2601.11756v1 Announce Type: new Abstract: A ball polyhedron is a finite intersection of congruent balls in $\mathbb{R}^3$. These shapes arise in various contexts in discrete and convex geometry. We focus on Reuleaux polyhedra, the subclass of ball polyhedra whose centers and vertices coincide. Building on Bogosel's recent work on the volume of Meissner polyhedra, we derive a formula for the volume of Reuleaux polyhedra in terms of their edges.
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https://arxiv.org/abs/2601.11756
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6554e790f6c200ec1ebee512b5f1bbe034454096170a9f5bc9ee7d352a760b40
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2026-01-21T00:00:00-05:00
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Nonautonomous Linear Systems: Exponential Dichotomy and its Applications
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arXiv:2601.11759v1 Announce Type: new Abstract: The first purpose of this work is to provide a friendly introduction to the theory of nonautonomous linear systems of ordinary differential equations, the property of exponential dichotomy and its corresponding spectral theory. The second purpose of this work is disseminate the linearization results carried out by the authors in a nonautonomous framework. The actual structure of this work is a consequence of several elective courses (2014, 2016, 2019, 2021 and 2023) carried out by the authors for undergraduate and graduated students at the Department of Mathematics of the Universidad de Chile. The monography assumes a good knowledge of multivariate calculus, linear algebra and ordinary differential equations.
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https://arxiv.org/abs/2601.11759
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3d0cf74d5138488e9fbcef8c751c243d915b4c6935b27ffa5a56998d4fd6a13b
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2026-01-21T00:00:00-05:00
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Solving High-Dimensional PDEs Using Linearized Neural Networks
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arXiv:2601.11771v1 Announce Type: new Abstract: Linearized shallow neural networks that are constructed by fixing the hidden-layer parameters have recently shown strong performance in solving partial differential equations (PDEs). Such models, widely used in the random feature method (RFM) and extreme learning machines (ELM), transform network training into a linear least-squares problem. In this paper, we conduct a numerical study of the variational (Galerkin) and collocation formulations for these linearized networks. Our numerical results reveal that, in the variational formulation, the associated linear systems are severely ill-conditioned, forming the primary computational bottleneck in scaling the neural network size, even when direct solvers are employed. In contrast, collocation methods combined with robust least-squares solvers exhibit better numerical stability and achieve higher accuracy as we increase neuron numbers. This behavior is consistently observed for both ReLU$^k$ and $\tanh$ activations, with $\tanh$ networks exhibiting even worse conditioning. Furthermore, we demonstrate that random sampling of the hidden layer parameters, commonly used in RFM and ELM, is not necessary for achieving high accuracy. For ReLU$^k$ activations, this follows from existing theory and is verified numerically in this paper, while for $\tanh$ activations, we introduce two deterministic schemes that achieve comparable accuracy.
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https://arxiv.org/abs/2601.11771
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3bed93fb0a3fe59c759050b611a69bdcb36690b61c2ee5a2bcdc638ce6ebb3c0
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2026-01-21T00:00:00-05:00
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On the Narrow 2-Class Field Tower of Some Real Quadratic Number Fields: Lengths Heuristics Follow-Up
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arXiv:2601.11773v1 Announce Type: new Abstract: In this article we continue the investigation of the length of the narrow $2$-class field tower of real quadratic number fields $\mathrm{k}$ whose discriminants are not a sum of two squares and for which their $2$-class groups are elementary of order $4$. Letting $\mathrm{G}$ equal the Galois group of the second Hilbert narrow $2$-class field over $\mathrm{k}$, and $[\mathrm{G}_i]$ denote the lower central series of $\mathrm{G}$, we give heuristic evidence that the length of the narrow $2$-class field tower of $\mathrm{k}$ is equal to $2$ when $\mathrm{G}/\mathrm{G}_3$ is of type $64.150$ (in the tables of Hall and Senior). We also give the formulation of the relevant unit groups of the narrow Hilbert $2$-class field for these fields.
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https://arxiv.org/abs/2601.11773
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79a30fd1dd26a336568f413b2da2b1ddb217295ef70a5ac53b55333dfe49fb5a
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2026-01-21T00:00:00-05:00
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Mixed-Integer Reaggregated Hull Reformulation of Special Structured Generalized Linear Disjunctive Programs
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arXiv:2601.11782v1 Announce Type: new Abstract: Generalized Disjunctive Programming (GDP) provides a powerful framework for combining algebraic constraints with logical disjunctions. To solve these problems, mixed-integer reformulations are required, but traditional reformulation schemes, such as Big-M and Hull, either yield a weak continuous relaxation or result in a bloated model size. Castro and Grossmann showed that scheduling problems can be formulated as GDP by modeling task orderings as disjunctions with algebraic timing constraints. Moreover, in their work, a particular representation of the single-unit scheduling problem, namely using a time-slot concept, can be reformulated as a tight yet compact mixed-integer linear program with notable computational performance. Based on that observation, and focusing on the case where the constraints in disjunctions are linear and share the same coefficients, we connect the characterization of the convex hull of these disjunctive sets by Jeroslow and Blair with Castro and Grossmann's time-slot reaggregation strategy to derive a unified reformulation methodology. We test this reformulation in two problems, single-unit scheduling and two-dimensional strip-packing. We derive new formulations of the general precedence concept of single-unit scheduling and symmetry-breaking formulations of the strip-packing problem, yielding mixed-integer programs with strong theoretical guarantees, particularly compact formulations in terms of continuous variables, and efficient computational performance when solving them with commercial mixed-integer solvers for these problems.
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https://arxiv.org/abs/2601.11782
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fadfe53b83059430e08a0b32f48c0c6890d52a4ec0b492d451b7540397e2698a
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2026-01-21T00:00:00-05:00
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Projected Stochastic Momentum Methods for Nonlinear Equality-Constrained Optimization for Machine Learning
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arXiv:2601.11795v1 Announce Type: new Abstract: Two algorithms are proposed, analyzed, and tested for solving continuous optimization problems with nonlinear equality constraints. Each is an extension of a stochastic momentum-based method from the unconstrained setting to the setting of a stochastic Newton-SQP-type algorithm for solving equality-constrained problems. One is an extension of the heavy-ball method and the other is an extension of the Adam optimization method. Convergence guarantees for the algorithms for the constrained setting are provided that are on par with state-of-the-art guarantees for their unconstrained counterparts. A critical feature of each extension is that the momentum terms are implemented with projected gradient estimates, rather than with the gradient estimates themselves. The significant practical effect of this choice is seen in an extensive set of numerical experiments on solving informed supervised machine learning problems. These experiments also show benefits of employing a constrained approach to supervised machine learning rather than a typical regularization-based approach.
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https://arxiv.org/abs/2601.11795
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f9c20f57660bdc72209e19873d00f7e5e8fda14cf4d4bfca72831e1d9da8f1d4
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2026-01-21T00:00:00-05:00
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The Noisy Quantitative Group Testing Problem
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arXiv:2601.11797v1 Announce Type: new Abstract: In this paper, we study the problem of quantitative group testing (QGT) and analyze the performance of three models: the noiseless model, the additive Gaussian noise model, and the noisy Z-channel model. For each model, we analyze two algorithmic approaches: a linear estimator based on correlation scores, and a least squares estimator (LSE). We derive upper bounds on the number of tests required for exact recovery with vanishing error probability, and complement these results with information-theoretic lower bounds. In the additive Gaussian noise setting, our lower and upper bounds match in order.
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https://arxiv.org/abs/2601.11797
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13b6c1531bbcc2a993fa5064a0cd1ce72291f7d107f1a6d7b7e69597f36d87e6
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2026-01-21T00:00:00-05:00
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Distance of Quadratic Algebraic Numbers from the Middle-Third Cantor Set
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arXiv:2601.11799v1 Announce Type: new Abstract: We study the distance from quadratic irrational numbers to the middle-third Cantor set $C$. Mahler asked whether $C$ contains any irrational algebraic numbers; this remains open even for quadratic irrationals. Rather than assuming an answer to this problem, we obtain uniform lower bounds for the distance from a quadratic irrational $\alpha$ to $C$ in terms of the height $H$ of the minimal polynomial of $\alpha$. We encode $\alpha$ by its orbit under the map $x \mapsto 3x \bmod 1$ and define the exit time $\operatorname{exit}(\alpha)$ as the first iterate that enters the middle interval $[1/3,2/3]$. Our main unconditional result is a quadratic exit bound $\operatorname{exit}(\alpha) \le A (\log_3 H)^2 + B$ for absolute constants $A,B > 0$, valid for all quadratic irrationals whose orbit stays a fixed small distance away from the coarse Cantor boundaries. As a consequence we obtain a distance lower bound $\operatorname{dist}(\alpha,C) \ge H^{-\kappa \log H}$ for some constant $\kappa > 0$. On the dynamical side we classify orbits by an $L/M/R$ coding and prove that the total number of visits to the right interval $[2/3,1)$ is $O(\log H)$. A finite case analysis on a bounded portion of the orbit is reduced to checking a finite list of explicit affine inequalities on subintervals of $[0,1]$, which we verify with short computer scripts; all Diophantine and dynamical estimates are proved by hand.
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https://arxiv.org/abs/2601.11799
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e20f5a60d1362030bc23945c176db69469987e578dec52a83a08e38f8774e805
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2026-01-21T00:00:00-05:00
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On large periodic traveling surface waves in porous media
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arXiv:2601.11800v1 Announce Type: new Abstract: We study large traveling surface waves within a two-dimensional finite depth, free boundary, homogeneous, incompressible and viscous fluid governed by Darcy's law. The fluid is bound by a gravitational force to a flat rigid bottom and meets an atmosphere of constant pressure at the top with its free surface, where it does not experience any capillarity effects. Additionally, the fluid is subject to a fixed, but arbitrarily selected, forcing data profile with variable amplitude. We use the Riemann mapping to equivalently reformulate the resulting two-dimensional free boundary problem as a single one-dimensional fully nonlinear pseudodifferential equation for a function describing the domain's geometry. By discovering a hidden ellipticity in the reformulated equation, we are able to import a global implicit function theorem to construct a connected set of traveling waves, containing both the quiescent solution and large amplitude members. We find that either solutions continue to exist for arbitrarily large data amplitude or else one of a finite number of meaningful breakdown scenarios must occur. This work stands as the first non perturbative construction of large traveling surface waves in any free boundary viscous fluid without surface tension.
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https://arxiv.org/abs/2601.11800
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bb200d4cc039f8694786e74b06450af59b7875f401627268106590981b999874
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2026-01-21T00:00:00-05:00
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Classification of dynamics for a two person model of planned behavior
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arXiv:2601.11804v1 Announce Type: new Abstract: We study a dynamical system modeling the Theory of Planned Behavior (TPB) in which each individual's behavioral intention evolves continuously under an ODE driven by internal attitudes, perceived social norms, and perceived behavioral control. Actions occur as discrete threshold events: when intention reaches a fixed threshold it is reset to 0 and produces a transient "nudge" that jumps to 1 and then decays exponentially. This yields a hybrid ODE-threshold system with psychologically interpretable parameters. We derive a partial classification in the general case of n individuals. Focusing on the two-individual case (n=2), we obtain explicit formulas for trajectories between action events and derive bounds for first-action times. In the mixed setting where one individual is intrinsically increasing and the other is not, we identify a scalar invariant, M, measuring the net effect of one period of excitation. We prove that non-positive M is equivalent to a partial-action state (only the intrinsically active individual acts countable infinitely often), while positive M is equivalent to full action (both individuals act countably infinitely often). Finally, we demonstrate numerically that these analytic boundaries partition the parameter space with near-perfect agreement, and we provide exploratory simulations suggesting analogous structures for three individuals.
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https://arxiv.org/abs/2601.11804
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d7cba75602109d67a48b30257011a823dcf4b5fec6caa48f94fa7120f4d1c7fd
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2026-01-21T00:00:00-05:00
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Rank of normal functions and Betti strata
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arXiv:2601.11805v1 Announce Type: new Abstract: In a recent work of the authors, we proved the generic positivity of the Beilinson-Bloch heights of the Gross-Schoen and Ceresa cycles. The geometric part of the proof was to prove the maximality of the rank of the associated normal function and the Zariski closedness of the Betti strata. In this paper, we generalize these geometric results to an arbitrary family of homologically trivial cycles. More generally, we prove a formula to compute the Betti rank and prove the Zariski closedness of the Betti strata, for any admissible normal function of a variation of Hodge structures of weight $-1$. We also define and prove results about degeneracy loci. In the end, we go back to the arithmetic setting and ask some questions about the rationality of the Betti strata and the torsion loci.
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https://arxiv.org/abs/2601.11805
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8a6664a4eed469fad75521b98e87688994a8e1165a22438077e78fd5b516766b
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2026-01-21T00:00:00-05:00
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Infinitesimal invariants of mixed Hodge structures II: Log Clemens conjecture and log connectivity
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arXiv:2601.11810v1 Announce Type: new Abstract: Following previous work, we continue the study of infinitesimal methods in mixed Hodge theory. In the first part, inspired by the deformation theory of curves on Calabi-Yau threefolds, we study deformations of smooth $\mathbb{Q}$-log Calabi-Yau pairs $(X,Y)$. We prove unobstructedness results for these pairs under Fano hypotheses. We define families of infinitesimal Abel-Jacobi maps associated with these deformation problems and show that they control the first-order deformations of smooth curves embedded in the pair. Crucially, for the $\frac{1}{2}$-log Calabi-Yau case, we establish an exact duality between deformations and obstructions, recovering the symmetry found in the absolute Calabi-Yau setting. We apply this framework to the cubic threefold, proposing a relative generalization of the Clemens conjecture regarding the injectivity of the infinitesimal Abel-Jacobi map, and establishing a criterion for its non-vanishing. In the second part, we define infinitesimal invariants for normal functions using extension classes and the log-Leray filtration. Relying on the theory of generalized Jacobian rings developed by Asakura and Saito, we prove a logarithmic Nori connectivity theorem for the universal family of open hypersurfaces, we also deduce a sharp algebraic criterion for the properness of the Hodge loci for open hypersurfaces, generalizing the proof of Carlson-Green-Griffiths-Harris.
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https://arxiv.org/abs/2601.11810
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3371208c13d3686116bfbd149181bdbb4104a39a57576f2ad7b2ca056481686c
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2026-01-21T00:00:00-05:00
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The Clemens Conjectures for Cubic Threefolds relative to a Hyperplane
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arXiv:2601.11813v1 Announce Type: new Abstract: We propose an analogue of the Clemens conjectures for $\frac{1}{2}$-log Calabi-Yau threefolds, specifically for the pair $(X, Y)$ where $X$ is a cubic threefold and $Y$ is a hyperplane section. By exploiting a perfect deformation/obstruction duality specific to the $\frac{1}{2}$-log setting, we formulate conjectures regarding the injectivity of the relative infinitesimal Abel-Jacobi map and the finiteness of rational curves with fixed intersection on $Y$.
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https://arxiv.org/abs/2601.11813
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7d407027bf0f3ab7a8d1adc92578b6917211825069708186fe45aee3950c7e3c
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2026-01-21T00:00:00-05:00
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The maximal mean equicontinuous factor via regional mean sensitivity
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arXiv:2601.11814v1 Announce Type: new Abstract: For actions of amenable groups, mean equicontinuity-a natural relaxation of equicontinuity obtained by averaging metrics along orbits-is well known to yield a maximal mean equicontinuous factor. In 2021, Li and Yu introduced the notion of weak sensitivity in the mean for actions of $\mathbb{Z}$ to gain a deeper understanding of this phenomenon, building on earlier work by Qiu and Zhao. We demonstrate that this relation is insufficient for actions of non-Abelian groups. To overcome this limitation, we introduce the regional mean sensitive relation, which more precisely captures the dynamical behaviour underlying the maximal mean equicontinuous factor. We discuss its fundamental properties and highlight its advantages in the non-Abelian setting. In particular, we show that mean equicontinuity is equivalent to the nonexistence of non-diagonal regional mean sensitive pairs. For this, we work in the context of actions of $\sigma$-compact and locally compact amenable groups.
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https://arxiv.org/abs/2601.11814
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c64f5629d121e2d63c8619c39c4d52687cb89b23c26ccd297ab8eb30b89cdc64
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2026-01-21T00:00:00-05:00
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Bayesian ICA for Causal Discovery
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arXiv:2601.11815v1 Announce Type: new Abstract: Causal discovery based on Independent Component Analysis (ICA) has achieved remarkable success through the LiNGAM framework, which exploits non-Gaussianity and independence of noise variables to identify causal order. However, classical LiNGAM methods rely on the strong assumption that there exists an ordering under which the noise terms are exactly independent, an assumption that is often violated in the presence of confounding. In this paper, we propose a general information-theoretic framework for causal order estimation that remains applicable under arbitrary confounding. Rather than imposing independence as a hard constraint, we quantify the degree of confounding by the multivariate mutual information among the noise variables. This quantity is decomposed into a sum of mutual information terms along a causal order and is estimated using Bayesian marginal likelihoods. The resulting criterion can be interpreted as Bayesian ICA for causal discovery, where causal order selection is formulated as a model selection problem over permutations. Under standard regularity conditions, we show that the proposed Bayesian mutual information estimator is consistent, with redundancy of order $O(\log n)$. To avoid non-identifiability caused by Gaussian noise, we employ non-Gaussian predictive models, including multivariate $t$ distributions, whose marginal likelihoods can be evaluated via MCMC. The proposed method recovers classical LiNGAM and DirectLiNGAM as limiting cases in the absence of confounding, while providing a principled ranking of causal orders when confounding is present. This establishes a unified, confounding-aware, and information-theoretically grounded extension of ICA-based causal discovery.
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https://arxiv.org/abs/2601.11815
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c68d81d6216c6d745ef2ddea1c93636cacc7febdae6ccdf0be055a1846453ddc
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2026-01-21T00:00:00-05:00
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Large deviations and the matrix product ansatz
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arXiv:2601.11820v1 Announce Type: new Abstract: We consider probability measures on $A^N$, the set of sequences of symbols on a finite alphabet $A$ of length $N$, that give a weight to each sequence in terms of a collection of matrices with non-negative entries and having rows and columns labeled by a finite or countable set $B$. We prove for such kind of measures large deviations principles for several empirical measures. Our approach is based on a simultaneous combination of an enlargement of the state space to sequences on $A\times B$ and a spectral conjugation that produces a stochastic matrix, as discussed in \cite{GI1}. As a result we describe the measures as hidden Markov measures and can deduce the large deviations results by contraction from the corresponding ones for the enlarged Markov chain. The measure on the enlarged state space is a Markov bridge. The invariant measures of several non equilibrium models of interacting particle systems can be represented by the so called {\it Matrix Product Ansatz} that corresponds to measures of the type that we consider and with matrices labeled by $B$ that is typically countable infinite. The large deviations behavior is different in the cases with $B$ finite or countable. In the finite case we give a variational formula for both the algebraic and the spatial empirical measures, that can be solved in special cases. For the infinite case, we illustrate the method through an example that is the invariant measure of the boundary driven TASEP model in a special regime. We recover in this way the celebrated results in \cite{Der4,Derr7}, and in particular we obtain a variational representation of the rate function similar to that in \cite{Bryc}. Our approach is general and can in principle be applied to any measure represented by the matrix product ansatz with matrices having positive entries.
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https://arxiv.org/abs/2601.11820
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4cb0ca154968a3cc598815d12ef9d7f999f7f7ebd3aecb65f0fde009bc6606c2
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2026-01-21T00:00:00-05:00
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A high-order augmented Lagrangian method with arbitrarily fast convergence
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arXiv:2601.11826v1 Announce Type: new Abstract: We propose a high-order version of the augmented Lagrangian method for solving convex optimization problems with linear constraints, which achieves arbitrarily fast -- and even superlinear -- convergence rates. First, we analyze the convergence rates of the high-order proximal point method under certain uniform convexity assumptions on the energy functional. We then introduce the high-order augmented Lagrangian method and analyze its convergence by leveraging the convergence results of the high-order proximal point method. Finally, we present applications of the high-order augmented Lagrangian method to various problems arising in the sciences, including data fitting, flow in porous media, and scientific machine learning.
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https://arxiv.org/abs/2601.11826
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07c379d04179640fa9f30540f9613143ce8e0fa8ac9bba0351c8f058b0aea578
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2026-01-21T00:00:00-05:00
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Topological and Purely Topological Alignment Dynamics
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arXiv:2601.11828v1 Announce Type: new Abstract: We study the Euler Alignment system of collective behavior, equipped with `topological' interaction protocols, which were introduced to the mathematical literature by Shvydkoy and Tadmor. Interactions subject to these protocols may depend on both the Euclidean distance between agents and on the mass distribution between them -- the `topological' component. When the interaction protocol is regular, we prove sufficient conditions for the existence of global-in-time classical solutions, related to the initial nonnegativity of a conserved quantity of the system. The remainder of our results explore the case where the interactions are `purely' topological and the interactions do not depend on the Euclidean distance. We show that in this case, the system decouples into an autonomous velocity equation in mass coordinates together with a scalar conservation law with time-dependent flux determined by the velocity. We analyze the long-time behavior for the dynamics associated to both regular and singular protocols.
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https://arxiv.org/abs/2601.11828
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90c8192423e65dc5e93cd3259eb6ae7f91ac625c4166674f56f5d263e01dabcb
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2026-01-21T00:00:00-05:00
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Fractional Supershifts and their associated Cauchy Evolution problems
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arXiv:2601.11829v1 Announce Type: new Abstract: In this work, we extend the notion of supershifts and superoscillation sequence to fractional Fock spaces based on Gelfond-Leontiev fractional derivatives. We first introduce the fractional supershifts sequence, and then discuss the associated evolution Cauchy problem with the fractional supershifts as initial condition.
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https://arxiv.org/abs/2601.11829
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892732e5b66ca83ad0cbc21049161ab58bb6e1b01bada34af136464313dea99a
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2026-01-21T00:00:00-05:00
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Subspaces of $L^2(\mathbb{R}^n)$ Invariant Under Shifts by a Crystal Group
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arXiv:2601.11839v1 Announce Type: new Abstract: For a crystal group $\Gamma$ in dimension $n$, a closed subspace $\mathcal{V}$ of $L^2(\mathbb{R}^n)$ is called $\Gamma$--shift invariant if, for every $f\in\mathcal{V}$, the shifts of $f$ by every element of $\Gamma$ also belong to $\mathcal{V}$. The main purpose of this paper is to provide a characterization of the $\Gamma$--shift invariant closed subspaces of $L^2(\mathbb{R}^n)$.
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https://arxiv.org/abs/2601.11839
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972bbfc8b8e57fdc1fa04b22c6f5aee6f6cb0446df6cb6b759f2f0630673b219
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2026-01-21T00:00:00-05:00
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Greedily Constructing Small Quasi-Kernels
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arXiv:2601.11847v1 Announce Type: new Abstract: In a digraph $D$,a quasi-kernel is an independent set $Q$ such that for every vertex $u$, there is a vertex $v \in Q$ satisfying $\text{dist}(v,u)\leq 2$. In 1974 Chv\'atal and Lov\'asz showed every digraph contains a quasi-kernel. In 1976, P. L. Erd\H{o}s and Sz\'ekely conjectured that every sourceless digraph has a quasi-kernel of order at most $\frac{n}{2}$. Despite significant recent attention by the community the problem remains far from solved, with no bound of the form $(1-\epsilon)n$ known. We introduce a polynomial time algorithm which greedily constructs a small quasi-kernel. Using this algorithm we show that if $D$ is a $\vec{K}_{1,d}$-free digraph, then $D$ has a quasi-kernel of order at most $\frac{(d^2 - 2d + 2)n}{d^2-d+1}$. By refining this argument we prove that for any $D$ with maximum out-degree $3$ this algorithm constructs a quasi-kernel of order at most ${4n}/{7}$. Finally, we consider the problem in digraphs forbidding certain orientation of short cycles as subgraphs, concluding that all orientations $D$ of a graph $G$ with girth at least $7$ have a quasi-kernel of order at most $\frac{(d^2+4)n}{(d+2)^2}$, where $d$ is the maximum out-degree of $D$.
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https://arxiv.org/abs/2601.11847
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9233c47f2f68ce300e906773b12d35458cd52afd227d40eee97f3f2a7484b3dc
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2026-01-21T00:00:00-05:00
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Weighted fractional ultrahyperbolic diffusion on geometrically deformed domains
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arXiv:2601.11851v1 Announce Type: new Abstract: Standard fractional models on manifolds often conflate geometric anisotropy with medium heterogeneity. In this Letter, we overcome this rigidity by deriving the fundamental solution for a weighted space-time fractional ultrahyperbolic operator, denoted by $(-\Box_{\phi,\omega})^{\beta}$. Using a novel spectral approach based on the Weighted Fourier Transform, we explicitly \textbf{decouple the medium density from the geometric deformation}. A crucial finding is the emergence of a \textbf{geometry-independent drift mechanism} driven purely by the inhomogeneity of the medium. The Green's function is obtained in closed form via the Fox H-function, providing a unified and computable framework for anomalous transport in complex, structurally deformed media.
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https://arxiv.org/abs/2601.11851
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99e9feb74035c7b57bfb9ee57bc96fac6830bec4ddc9fa532c0ebf812fced54e
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2026-01-21T00:00:00-05:00
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Global weak solutions to the isentropic compressible Navier--Stokes equations with vacuum and unbounded density in a half-plane under Dirichlet boundary conditions
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arXiv:2601.11852v1 Announce Type: new Abstract: We establish the global existence of a class of weak solutions to the isentropic compressible Navier--Stokes equations in a half-plane with Dirichlet boundary conditions, allowing for vacuum both in the interior and at infinity, under a suitably small initial total energy. The solutions constructed here admit unbounded densities and lie in an intermediate regularity regime between the finite-energy weak solutions of Lions--Feireisl and the framework of Hoff. This result generalizes previous works of Hoff (Comm. Pure Appl. Math. 55 (2002), pp. 1365--1407) and Perepelitsa (Arch. Ration. Mech. Anal. 212 (2014), pp. 709--726) concerning discontinuous solutions by allowing vacuum states and unbounded density. Our analysis relies on the Green function method and new estimates involving the specific structure of the equations and the geometry of the half-plane. To the best of our knowledge, this is the first result concerning global weak solutions within Hoff's framework on an unbounded domain that simultaneously accommodates Dirichlet boundary conditions and far-field vacuum. The intermediate-regularity class developed here may be viewed as a natural extension of Hoff's theory, precisely tailored to overcome the two corresponding obstructions: the lack of global space-time control of the effective viscous flux arising from far-field vacuum and the absence of boundary-induced regularity gains in the no-slip setting.
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https://arxiv.org/abs/2601.11852
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d23e05716943992e0332417ddf65e93266708d5ba0a5adf78f5c61604deb4e02
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2026-01-21T00:00:00-05:00
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New examples of twisted Brill-Noether loci II
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arXiv:2601.11855v1 Announce Type: new Abstract: Our purpose in this paper is to construct new examples of twisted Brill Noether loci on curves of genus g greater than 2 with negative expected dimension. We begin by completing the proof of Butler's conjecture for coherent systems of certain type establishing the birationality, smoothness, and irreducibility of the corresponding loci. We also produce new points on the BN map.
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https://arxiv.org/abs/2601.11855
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0a701fd794d210e485d9959c1808973d247b3d54923c3baed3baf0a5ba471ca3
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2026-01-21T00:00:00-05:00
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On the R\'enyi Rate-Distortion-Perception Function and Functional Representations
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arXiv:2601.11862v1 Announce Type: new Abstract: We extend the Rate-Distortion-Perception (RDP) framework to the R\'enyi information-theoretic regime, utilizing Sibson's $\alpha$-mutual information to characterize the fundamental limits under distortion and perception constraints. For scalar Gaussian sources, we derive closed-form expressions for the R\'enyi RDP function, showing that the perception constraint induces a feasible interval for the reproduction variance. Furthermore, we establish a R\'enyi-generalized version of the Strong Functional Representation Lemma. Our analysis reveals a phase transition in the complexity of optimal functional representations: for $0.5 1$, the representation collapses to one with finite support, offering new insights into the compression of shared randomness under generalized notions of mutual information.
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https://arxiv.org/abs/2601.11862
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49ed853490701c30c58b037ae4213bfad06353f295370f2f9246df70567c2b8f
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2026-01-21T00:00:00-05:00
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Open book decompositions with page a four-punctured sphere
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arXiv:2601.11871v1 Announce Type: new Abstract: In this paper, we study contact structures supported by open book decompositions whose pages are four-punctured spheres. The paper is split into two parts. In the first part, we find infinitely many overtwisted, right-veering monodromies on the four-punctured sphere. This is done using the techniques developed by Ito-Kawamuro in the papers arXiv:1112.5874, arXiv:1310.6404. Although most of the monodromies that we show are overtwisted are pseudo-Anosov, we are also able to classify precisely which reducible monodromies on the four-punctured sphere are tight. In the second part of the paper, we reprove part of a result of Lekili arXiv:1008.3529 by classifying which reducible mondromies have non-zero Heegaard Floer invariant. This is done by using the bordered contact invariants of Min-Varvarezos arXiv:2410.05511.
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https://arxiv.org/abs/2601.11871
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8514479bdf38988a522505030476420f81a9c353ac95e344d516226fa113da22
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2026-01-21T00:00:00-05:00
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Simple, subdirectly irreducible weakly dicomplemented lattices
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arXiv:2601.11873v1 Announce Type: new Abstract: In this work, we exhibit several subclasses of weakly dicomplemented lattices (WDLs) based on their skeletons and dual skeletons. We investigate normal filters (resp. ideals) and show that the set of normal filters (resp. ideals) forms a complete lattice, which is not a sublattice of the lattice of all filters (ideals). The normal filter (ideal) generated by a subset and the join of two normal filters (resp. ieals) are characterized. We further prove that the lattice of normal filters is isomorphic to the lattice of normal ideals, and that the only class of filters (or ideals) that generate a congruence in WDLs is the class of normal filters. For distributive WDLs, the congruences generated by filters are characterized. Using normal filters, we characterize simple, subdirectly irreducible, and regular WDLs. Moreover, it is shown that the congruences generated by normal filters are permutable, and that regular distributive WDLs are congruence-permutable and verify the congruence extension property (CEP). Finally, we prove that, under certain conditions, the lattice of normal filters is isomorphic to the lattice of filters of the Boolean center of a distributive WDL. It is also established that the lattice of normal filters of a WDL $L$ embeds into the lattice of normal filters of the power $L^{X}$ of $L$.
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https://arxiv.org/abs/2601.11873
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daa22ab354b96d6a396bf1bb83380a98475971b16b8547201e2fbdabb3b38572
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2026-01-21T00:00:00-05:00
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On the eigenvalues of cyclic covers of Paley graphs
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arXiv:2601.11877v1 Announce Type: new Abstract: We study covering graphs of the Paley graph associated to a finite field of characteristic p in the case where the covering transformation group is cyclic of prime order distinct from p. When the field has q = p elements, we show that the eigenvalues of the adjacency matrix determine the graph isomorphism class among translation invariant covers. When q = p^r > p, we construct examples of cospectral covering graphs that are not isomorphic as graphs.
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https://arxiv.org/abs/2601.11877
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c4bea8695c0f6921807e178f8bb38f11a5ed8d25d11d5cbf36559bc8ae14b3c7
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2026-01-21T00:00:00-05:00
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Lowest eigenvalues and higher order elliptic differential operators
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arXiv:2601.11882v1 Announce Type: new Abstract: Let $(M,g)$ be a closed, smooth Riemannian manifold of dimension $m \geq 1$. It is not difficult to produce an example of an elliptic differential operator on $(M,g)$ that has the property that there exists a sign-changing eigenfunction that is associated with the lowest eigenvalue. Indeed, $\Delta_g^2 + \lambda_2 \Delta_g$ does the job, where $\Delta_g:=div_g \nabla_g$. and where $\lambda_2$ is the second lowest eigenvalue of the operator $-\Delta_g$. The question that remains is how rare are elliptic differential operators whose lowest eigenvalue has this property. In this paper, the author proves that elliptic operators of the form $\Delta_g^2 - div_g(T-\lambda_2 g^{-1}) d$, where $T$ is a negative semi-definite $(2,0)$-tensor field on $M$, and where $g^{-1}$ is the inverse metric tensor, have the property that there exists a sign-changing eigenfunction that is associated with the lowest eigenvalue of the operator. This suggests that there are a lot of fourth-order elliptic operators with the property that there exists a sign-changing eigenfunction that is associated with the lowest eigenvalue of the operator.
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https://arxiv.org/abs/2601.11882
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43264a095408549ea9e2aae11fe40e6bfc620c78ed99ceb9be26a1f37795b9c4
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2026-01-21T00:00:00-05:00
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Analytic Regularization of a Ramanujan Machine Conjecture
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arXiv:2601.11892v1 Announce Type: new Abstract: We provide a formal analytic derivation of a continued fraction identity for $-\pi/4$ recently conjectured by the Ramanujan Machine~\cite{Raayoni2021}. By utilizing the contiguous relations of the Gauss hypergeometric function ${}_2F_1(a, b; c; z)$, we establish that the conjectured polynomial architecture is a regularized representation of the transcendental ratio $\mathcal{R}(1/2, 0, 1/2; -1)$. Through an explicit equivalence transformation $\mathcal{T}$ defined by a linear scaling sequence, we map the Gaussian unit-denominator expansion to the conjectured form, thereby recovering the quadratic partial numerators $(n-1)^2$ and linear partial denominators $-(2n-1)$. Convergence is rigorously established via the limit-periodicity of the transformed coefficients, which reside on the Worpitzky boundary $L=1/4$.
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https://arxiv.org/abs/2601.11892
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b960fecd5e4f1bf5ccfd7647302b237efd4f6c79ab3bb108c64a35f62ce06569
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2026-01-21T00:00:00-05:00
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A Separable and Asymptotic-Preserving Dynamical Low-Rank Method for the Vlasov--Poisson--Fokker--Planck System
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arXiv:2601.11900v1 Announce Type: new Abstract: We present a dynamical low-rank (DLR) method for the Vlasov--Poisson--Fokker--Planck (VPFP) system. Our main contributions are two-fold: (i) a conservative spatial discretization of the Fokker--Planck operator that factors into velocity-only and space-only components, enabling efficient low-rank projection, and (ii) a time discretization within the DLR framework that properly handles stiff collisions. We propose both first-order and second-order low-rank IMEX schemes. For the first-order scheme, we prove an asymptotic-preserving (AP) property when the field fluctuation is small. Numerical experiments demonstrate accuracy, robustness, and AP property at modest ranks.
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https://arxiv.org/abs/2601.11900
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41396daec2d1584111dd7fdb99f12f2ef4074048721bfa99ab9c6dbb5978fcef
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2026-01-21T00:00:00-05:00
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The Inverse Symplectic Eigenvalue Problem of a Graph
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arXiv:2601.11912v1 Announce Type: new Abstract: Symplectic geometry plays an increasingly important role in mathematics, physics and applications, and naturally gives rise to interesting matrix families and properties. One of these is the notion of symplectic eigenvalues, whose existence for positive definite matrices is known as Williamson's theorem or decomposition. This notion of symplectic eigenvalues gives rise to inverse problems. We introduce the inverse symplectic eigenvalue problem for positive definite matrices described by a labeled graph and solve it for several families of labeled graphs and all labeled graphs of order four. To solve these problems we develop various tools such as the Strong Symplectic Spectral Property (SSSP) and its consequences such as the Supergraph Theorem, the Bifurcation Theorem, and the Matrix Liberation Lemma for symplectic eigenvalues, graph couplings to describe collections of labelings of a graph that produce the same symplectic eigenvalues, and coupled graph zero forcing. We establish numerous results for symplectic positive definite matrices, including a sharp lower bound on the number of nonzero entries of such a matrix (or equivalently, the number of edges in its graph). This lower bound is a consequence of a lower bound on the sum of number of nonzero entries in an irreducible positive definite matrix and its inverse.
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https://arxiv.org/abs/2601.11912
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eaafba0f9e26f0184cac47eab3957ae8cf853f1c8c817dce83079b5cd1119bfb
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2026-01-21T00:00:00-05:00
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Rate-Distortion-Classification Representation Theory for Bernoulli Sources
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arXiv:2601.11919v1 Announce Type: new Abstract: We study task-oriented lossy compression through the lens of rate-distortion-classification (RDC) representations. The source is Bernoulli, the distortion measure is Hamming, and the binary classification variable is coupled to the source via a binary symmetric model. Building on the one-shot common-randomness formulation, we first derive closed-form characterizations of the one-shot RDC and the dual distortion-rate-classification (DRC) tradeoffs. We then use a representation-based viewpoint and characterize the achievable distortion-classification (DC) region induced by a fixed representation by deriving its lower boundary via a linear program. Finally, we study universal encoders that must support a family of DC operating points and derive computable lower and upper bounds on the minimum asymptotic rate required for universality, thereby yielding bounds on the corresponding rate penalty. Numerical examples are provided to illustrate the achievable regions and the resulting universal RDC/DRC curves.
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https://arxiv.org/abs/2601.11919
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46a7d2afe47fe2eea14066cd682b9f79a6b64b2e222fdc716fc0d0b1618bf26e
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2026-01-21T00:00:00-05:00
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Phase-IDENT: Identification of Two-phase PDEs with Uncertainty Quantification
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arXiv:2601.11922v1 Announce Type: new Abstract: We propose a novel method, Phase-IDENT, for identifying partial differential equations (PDEs) from noisy observations of dynamical systems that exhibit phase transitions. Such phenomena are prevalent in fluid dynamics and materials science, where they can be modeled mathematically as functions satisfying different PDEs within distinct regions separated by phase boundaries. Our approach simultaneously identifies the underlying PDEs in each regime and accurately reconstructs the phase boundaries. Furthermore, by incorporating change point detection techniques, we provide uncertainty quantification for the detected boundaries, enhancing the interpretability and robustness of our method. We conduct numerical experiments on a variety of two-phase PDE systems under different noise levels, and the results demonstrate the effectiveness of the proposed approach.
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https://arxiv.org/abs/2601.11922
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825f8dfc2ad8ce3b4f16e198f64d71bdecf2c62872c1ba64d99eac1df9f7e548
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2026-01-21T00:00:00-05:00
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Exact Redundancy for Symmetric Rate-Distortion
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arXiv:2601.11927v1 Announce Type: new Abstract: For variable-length coding with an almost-sure distortion constraint, Zhang et al. show that for discrete sources the redundancy is upper bounded by $\log n/n$ and lower bounded (in most cases) by $\log n/(2n)$, ignoring lower order terms. For a uniform source with a distortion measure satisfying certain symmetry conditions, we show that $\log n/(2n)$ is achievable and that this cannot be improved even if one relaxes the distortion constraint to be in expectation rather than with probability one.
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https://arxiv.org/abs/2601.11927
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b8511b9a2236bad4c2509ac793fc05da5b5e642bff82812594ddecb9ac200779
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2026-01-21T00:00:00-05:00
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Classification of connected proper pairs in the affine transformation group
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arXiv:2601.11933v1 Announce Type: new Abstract: Let $(L, H)$ be closed subgroups of a locally compact group $G$. The pair $(L, H)$ is said to be proper if the action of $L$ on the homogeneous space $G/H$ is proper. We give a complete list of connected closed proper pairs in the affine transformation group of $\mathbb{R}^2$. This result extends Kobayashi's classification (1992) of connected closed subgroups of the affine transformation group of $\mathbb{R}^2$acting properly on $\mathbb{R}^2$.
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https://arxiv.org/abs/2601.11933
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7a57092746630b802f61ea6d709ae18084039db813737f655eee825025836b27
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2026-01-21T00:00:00-05:00
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The nonlinear estimates on quantum Besov space
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arXiv:2601.11934v1 Announce Type: new Abstract: The superposition operators have been widely studied in nonlinear analysis, which are essential for the well-posedness theory of nonlinear equations. In this paper, we investigate the boundedness estimates of superposition operators with non-smooth symbols on quantum Besov spaces, which significantly generalize McDonald's results \cite{McNLE} for infinitely differentiable symbols and have rich applications in the well-posedness theory of noncommutative PDEs. As a byproduct, we prove the equivalence of the two descriptions of quantum Besov spaces, resolving the conjecture proposed in \cite[Remark 3.16]{McNLE}. The new ingredients in the proof also involve quantum chain rule and nonlinear interpolation.
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https://arxiv.org/abs/2601.11934
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20d87e6ea020c96b31d80f8e5e9017b16dee0255cf4ff93f0824ac2596ed011f
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2026-01-21T00:00:00-05:00
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Small-Error Cascaded Group Testing
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arXiv:2601.11945v1 Announce Type: new Abstract: Group testing concerns itself with the accurate recovery of a set of "defective" items from a larger population via a series of tests. While most works in this area have considered the classical group testing model, where tests are binary and indicate the presence of at least one defective item in the test, we study the cascaded group testing model. In cascaded group testing, tests admit an ordering, and test outcomes indicate the first defective item in the test under this ordering. Under this model, we establish various achievability bounds for several different recovery criteria using both non-adaptive and adaptive (with "few" stages) test designs.
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https://arxiv.org/abs/2601.11945
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c506b13712e21290a3017517d97dec8ce793468ad19a5461ccf8793b24e5b2c5
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2026-01-21T00:00:00-05:00
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Observer design and boundary output feedback stabilization for semilinear parabolic system over general multidimensional domain
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arXiv:2601.11948v1 Announce Type: new Abstract: This paper investigates the output feedback stabilization of parabolic equation with Lipschitz nonlinearity over general multidimensional domain using spectral geometry theories. First, a novel nonlinear observer is designed, and the error system is shown to achieve any prescribed decay rate by leveraging the Berezin-Li-Yau inequality from spectral geometry, which also provides effective guidance for sensor placement. Subsequently, a finite-dimensional state feedback controller is proposed, which ensures the quantitative rapid stabilization of the linear part. By integrating this control law with the observer, an efficient boundary output feedback control strategy is developed. The feasibility of the proposed control design is rigorously verified for arbitrary Lipschitz constants, thereby resolving a persistent theoretical challenge. Finally, a numerical case study confirms the effectiveness of the approach.
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https://arxiv.org/abs/2601.11948
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8e780fd1e5100d940908c547c12ba3b2be4aaf6b4d7d473e002682049bceac31
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2026-01-21T00:00:00-05:00
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Computations of higher elliptic units
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arXiv:2601.11961v1 Announce Type: new Abstract: In this paper we present a conjecture on the construction of generalised elliptic units above number fields with exactly one complex place. These elliptic units obtained as values of multiple elliptic Gamma functions. These form a collection of multivariate meromorphic functions which were studied in the late 1990s and early 2000s in mathematical physics. Our construction extends the scheme of a recent article by Bergeron, Charollois and Garc\'ia where they constructed conjectural elliptic units above complex cubic fields using the elliptic Gamma function. The elliptic units we construct are expected to generate specific abelian extensions of the base field where they are evaluated, thus giving a conjectural solution to Hilbert's 12th problem for the number fields with exactly one complex place. We provide several examples to support our conjecture in optimal cases for cubic, quartic and quintic fields.
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https://arxiv.org/abs/2601.11961
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bdf14b0fed89a4105a882655b75536d21965ad2ca64659c37c7414f45a4dd242
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2026-01-21T00:00:00-05:00
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A Survey on Spherical Designs: Existence, Numerical Constructions, and Applications
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arXiv:2601.11963v1 Announce Type: new Abstract: This paper provides a survey of spherical designs and their applications, with a particular emphasis on the perspective of ``numerical analysis''. A set \(X_N\) of \(N\) points on the unit sphere \(\mathbb{S}^d\) is called a \textit{spherical \(t\)-design} if the average value of any polynomial of degree at most \(t\) over \(X_N\) equals its average over the entire sphere. Spherical designs represent one of the most significant topics in the study of point distributions on spheres. They are deeply connected to algebraic combinatorics, discrete geometry, differential geometry, approximation theory, optimization, coding theory, quantum physics, and other fields, which have led to the development of profound and elegant mathematical theories. This article reviews fundamental theoretical results, numerical construction methods, and applied outcomes related to spherical designs. Key topics covered include existence proofs, optimization-based construction techniques, fast computational algorithms, and applications in interpolation, numerical integration, hyperinterpolation, signal and image processing, as well as numerical solutions to partial differential and integral equations.
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https://arxiv.org/abs/2601.11963
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dd00d50d1bcc595a70ada60186d6f59d08e6926b4529e9bb7af3920652af8005
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2026-01-21T00:00:00-05:00
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On efficient estimates of the rate of convergence for Markov chains
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arXiv:2601.11973v1 Announce Type: new Abstract: The paper presents efficient approaches for evaluating convergence rate in total variation for finite and general linear Markov chains. The motivation for studying convergence rate in this metric is its usefulness in various limit theorems. For homogeneous Markov chains the goal is to compare several different methods: (1) the second eigenvalue for the transition matrix method (the method no. 1), (2) the method based on Markov -- Dobrushin's ergodic coefficient, and the new spectral method developed in earlier works, as well as modifications of they both by iterations (the ``other methods''). We answer the question whether or not the ``other methods'' may provide the optimal or close to optimal convergence rate in the case of homogeneous Markov chains. The answer turns out to be positive for appropriate modifications of both ``other methods''. The analogues of these ``other methods'' for the non-homogeneous Markov chains are also presented. The work is theoretical. However, the methods of computing efficient bounds of convergence rates may be in demand in various applied areas.
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https://arxiv.org/abs/2601.11973
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51710b2452d9c3c4ceeebc538679704f8bc552f648de7a3fb4517031a5ff3f42
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2026-01-21T00:00:00-05:00
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The linearization approach to the Calder\'on problem revisited: reconstruction via the Born approximation
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arXiv:2601.11975v1 Announce Type: new Abstract: Linearization techniques are widely used in the analysis and numerical solution of the Calder\'on inverse problem, even if their theoretical basis is not fully understood. In this article, we study the effectiveness of linearization for reconstructing a conductivity from its Dirichlet-to-Neumann (DtN) map, combining rigorous analysis with numerical experiments. In particular, we prove that any DtN map arising from a radial conductivity in the unit ball of $\mathbb{R}^d$ admits an exact representation as a linearized DtN map for a uniquely determined integrable function, the Born approximation. We linearize on a family of background conductivities that includes the constant case, giving a rigorous foundation for linearization-based methods in this framework. We also characterize the Born approximation as a solution of a generalized moment problem. Since this moment problem is formally well-defined even for non-radial conductivities, we use it to develop a numerical algorithm to reconstruct the Born approximation of a general conductivity on the unit disk. We provide numerical experiments to test the resolution and robustness of the Born approximation in different situations. Finally, we show how it can be used as the starting point of an algorithm for reconstructing a conductivity from its DtN map.
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https://arxiv.org/abs/2601.11975
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6fd4f6684742b9e200a1bd11c90a9b75607fab8313d49e604e77bc2ca454a396
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2026-01-21T00:00:00-05:00
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The small cancellation flat torus theorem
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arXiv:2601.11991v1 Announce Type: new Abstract: We establish Flat Torus Theorem type results for groups acting on small cancellation complexes satisfying C(6), C(4)-T(4) and C(3)-T(6) conditions. For C(3)-T(6) complexes the result closely parallels the CAT(0) setting. For C(6) complexes we prove an analogous theorem using a refined notion of flat, exploiting the relationship between C(6) complexes and their duals. In the C(4)-T(4) case we demonstrate that genuine flats do not necessarily exist, providing an explicit example of a C(4)-T(4) complex with an action of $\mathbb{Z}^2$ without invariant flat, and hence not admitting any CAT(0) metric. We introduce the notion of quasi-flats and prove a Flat Torus Theorem for quasi-flats by passing to quadric complexes via quadrization and invoking the Quadric Flat Torus Theorem of Hoda-Munro.
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https://arxiv.org/abs/2601.11991
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6757848f785f9bc307da01e721b3d2cf185ea724cebe7f6b43de414c10e0165f
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2026-01-21T00:00:00-05:00
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On examples of duals Saito's basis of some inhomogeneous divisors, and application
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arXiv:2601.11992v1 Announce Type: new Abstract: We investigate a class of non-quasi-homogeneous free divisors in the sense of Saito. These divisors are defined by equations of the form $D:= \{h=0\}$ on $\mathbb{C}^p$, where the polynomial $h$ is specific linear combination of monomials involving the product of coordinates. For this class, we explicitly construct a Saito basis for the module of logarithmic vector fields $Der(logD)$. This construction is then applied to the setting of logarithmic Poisson geometry. Focusing on the example defined by $h=xy+x^{2}y^{2}+x^3y^3$ on the Poisson algebra $(\mathcal{A}=\mathbb{C}[x,y], \{-,-\}_{h})$, where the Poisson bracket is induced by the bivector $\pi = h\partial x\wedge\partial y$. We define the associated Koszul bracket on the module of logarithmic 1-forms. This enables us to prove that $\pi$ endows the sheaf of logarithmic 1-forms $\Omega^{1}(log D )$ with a Lie-Rinehart algebra structure. Furthermore, we introduce and provide explicit descriptions for the resulting cohomology theory, which we term the logarithmic Poisson cohomology $H_{log}^{\bullet} $ of $\{-,-\}_{h}$. As a related and foundational computation, we also calculate the corresponding logarithmic De Rham cohomology $H^{\bullet}_{DR}$ for the divisor $D$ and we make a generalization in dimension 2.
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https://arxiv.org/abs/2601.11992
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3c3b7c11351c109d966d5f07439fc809ccc06842bddf82e42fc923e557dbb1e1
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2026-01-21T00:00:00-05:00
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Conjugacy limits of certain subgroups in $\SL(2,\mathbb{R})\ltimes\mathbb{R}^2$
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arXiv:2601.11994v1 Announce Type: new Abstract: We study conjugacy limits of certain of subgroups inside $\SL(2,\R)\ltimes\R^2$. These subgroups have a common feature that any two in the same category are conjugates of each other.
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https://arxiv.org/abs/2601.11994
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2c2b3849da1195e574c388d4b5e34443e6c69f658c3cbd55b9c7138046584ed4
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2026-01-21T00:00:00-05:00
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Microscopic derivation of a one-dimensional lubrication model with roughness
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arXiv:2601.11999v1 Announce Type: new Abstract: We derive a hydrodynamic model for the motion of inertial particles with a spherical hard core, interacting through lubrication forces and pairwise repulsive forces. The repulsion arises from the assumption that each particle is surrounded by a thin rough layer of reduced permeability. We prove that, as the number of particles tends to infinity (and their size tends to 0), the microscopic dynamics converges to a macroscopic hydrodynamic model in which congestion effects are encoded directly into the macroscopic interaction forces, depending on a local critical density transported by the flow. In particular, we extend the work of Lefebvre-Lepot and Maury where non-inertial particles, submitted to only a lubrication force were considered, and present the convergence proof when inertial effects and roughness are taken into account.
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https://arxiv.org/abs/2601.11999
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931b5fc8c832ad84f3a6ff3149b3f7ee822181fbe31680b1fcc723951949b01d
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2026-01-21T00:00:00-05:00
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A Multi-Level Deep Framework for Deep Solvers of Partial Differential Equations
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arXiv:2601.12000v1 Announce Type: new Abstract: In this paper, inspired by the multigrid method, we propose a multi-level deep framework for deep solvers. Overall, it divides the entire training process into different levels of training. At each level of training, an adaptive sampling method proposed in this paper is first employed to obtain new training points, so that these points become increasingly concentrated in computational regions corresponding to high-frequency components. Then, the generalization ability of deep neural networks are utilized to update the PDEs for the next level of training based on the results from all previous levels. Rigorous mathematical proofs and detailed numerical experiments are employed to demonstrate the effectiveness of the proposed method.
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https://arxiv.org/abs/2601.12000
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577a69d077cd021e07145c13a2639f231dd029c0343cbf5984c8fad15f852432
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2026-01-21T00:00:00-05:00
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Structure of ind-pro completions of Noetherian rings
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arXiv:2601.12016v1 Announce Type: new Abstract: We prove some results on the structure of ind-pro completions of Noetherian rings along flags of prime ideals. In particular, we compute the Krull dimension and deduce the criterion on semilocality in the case of essentially of finite type algebras over a field. We also show that ind-pro completion inherits properties of the base ring such as normality, regularity, local equidimensionality, etc.
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https://arxiv.org/abs/2601.12016
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766b82807d43d04e06984e452d9dc17199cb67ed091e818310d7cb34c07b379f
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2026-01-21T00:00:00-05:00
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Deformation rigidity of some simple affine VOAs
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arXiv:2601.12017v1 Announce Type: new Abstract: In this paper, we prove that simple affine vertex operator algebras with positive integral levels admit only trivial first-order deformations. Therefore, the deformation rigidity conjecture of strongly rational vertex operator algebras holds for these cases. We also show that the same holds simple affine vertex operator algebra of $\mathfrak{sl}_2$ at the non-integral admissible level $-4/3$. Therefore, neither $C_2$-cofiniteness nor rationality is a necessary condition for deformation rigidity of VOAs. We conjecture that the same should hold for every simple affine VOA that does not coincide with the corresponding universal affine VOA.
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https://arxiv.org/abs/2601.12017
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340a1f3b2579c7a1b16d95fa9439c4e3f67ddbac25c60eacb11145984de43fab
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2026-01-21T00:00:00-05:00
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On Multilinear Forms for Mod $p$ Representations of $\mathrm{GL}_2(\mathbb{Q}_p)$
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arXiv:2601.12021v1 Announce Type: new Abstract: Motivated by the study of trilinear forms for complex representations, we investigate the space of $G$-invariant linear forms on tensor products of irreducible admissible representations of $G = \mathrm{GL}_2(\mathbb{Q}_p)$ over $\overline{\mathbb{F}}_p$. Our main result is a complete vanishing theorem: for any $n \ge 1$ and $n$ infinite-dimensional irreducible admissible representations $\pi_1,\dots,\pi_n$ of $G$, \[ \operatorname{Hom}_G(\pi_1 \otimes \cdots \otimes \pi_n, \mathbb{1}) = 0. \] A refined version holds for $B^+ := \begin{pmatrix} p^{\mathbb{Z}} & \mathbb{Q}_p \\ 0 & 1 \end{pmatrix}$-invariant forms when at least one $\pi_i$ is supersingular. The proof proceeds by a detailed analysis of certain subgroups, reducing the problem from $G$ to $B^+$ and ultimately to the representation theory of $\mathbb{Z}_p$. We also deduce partial extensions of the result to $\mathrm{GL}_2(F)$ for finite extensions $F/\mathbb{Q}_p$.
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https://arxiv.org/abs/2601.12021
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71fccbba3afb27001f4d0a006d4c50e589d3e81eff6b03ade246a9ecc9a45120
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2026-01-21T00:00:00-05:00
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Generalizing the Fano inequality further
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arXiv:2601.12027v1 Announce Type: new Abstract: Interactive statistical decision making (ISDM) features algorithm-dependent data generated through interaction. Existing information-theoretic lower bounds in ISDM largely target expected risk, while tail-sensitive objectives are less developed. We generalize the interactive Fano framework of Chen et al. by replacing the hard success event with a randomized one-bit statistic representing an arbitrary bounded transform of the loss. This yields a Bernoulli f-divergence inequality, which we invert to obtain a two-sided interval for the transform, recovering the previous result as a special case. Instantiating the transform with a bounded hinge and using the Rockafellar-Uryasev representation, we derive lower bounds on the prior-predictive (Bayesian) CVaR of bounded losses. For KL divergence with the mixture reference distribution, the bound becomes explicit in terms of mutual information via Pinsker's inequality.
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https://arxiv.org/abs/2601.12027
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5b8e9944a1dd27c11f3fc800fe3af4f06fe055a8914decfe3f2e6329edae262c
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2026-01-21T00:00:00-05:00
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High-Dimensional $p$-Normed Flows
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arXiv:2601.12036v1 Announce Type: new Abstract: We generalize Tutte's integer flows and the $d$-dimensional Euclidean flows of Mattiolo, Mazzuoccolo, Rajn\'{i}k, and Tabarelli to \emph{$d$-dimensional $p$-normed nowhere-zero flows} and define the corresponding flow index $\phi_{d,p}(G)$ to be the infimum over all real numbers $r$ for which $G$ admits a $d$-dimensional $p$-normed nowhere-zero $r$-flow. For any bridgeless graph $G$ and any $p\ge 1$, we establish general upper bounds, including $\phi_{2,p}(G) \le 3$, $\phi_{3,p}(G) \le 1+\sqrt{2}$, and tight bounds for graphs admitting a $4$-NZF. For graphs with oriented $(k+1)$-cycle $2l$-covers, we show that $\phi_{k,p}(G) = 2$, which implies $\phi_{2,p}(G) = 2$ for graphs admitting a nowhere-zero $3$-flow and $\phi_{3,p}(G) = 2$ for those admitting a nowhere-zero $4$-flow. These results extend classical flow theory to arbitrary norms, provide supporting evidences for Tutte's $5$-flow Conjecture and Jain's $S^2$-Flow Conjecture, and connect combinatorial flows with geometric and topological perspectives.
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https://arxiv.org/abs/2601.12036
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fe3fff5f293ad878da9e01a6a8bac8bf96217b74c3691b79844e7fc9099d4018
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2026-01-21T00:00:00-05:00
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Critical partition regular functions for compact spaces
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arXiv:2601.12041v1 Announce Type: new Abstract: We study ideal-based refinements of sequential compactness arising from the class FinBW(I), consisting of topological spaces in which every sequence admits a convergent subsequence indexed by a set outside a given ideal I. A central theme of this work is the existence of critical ideals whose position in the Katetov order determines the relationship between a fixed class of spaces and the corresponding FinBW(I) classes. Building on earlier results characterizing several classical topological classes via such ideals, we extend this theory to a broader framework based on partition regular functions, which unifies ordinary convergence with other non-classical convergence notions such as IP- and Ramsey-type convergence. Furthermore, we investigate the existence of critical ideals associated with function classes motivated by Mazurkiewicz's theorem on uniformly convergent subsequences.
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https://arxiv.org/abs/2601.12041
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ab8c114220ef2b4d5633a9d0fc22f1d8e8a4e49369c7d010457aa851a6bd207c
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2026-01-21T00:00:00-05:00
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Koopman Spectral Computation Beyond The Reflexive Regime: Endpoint Solvability Complexity Index And Type-2 Links
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arXiv:2601.12044v1 Announce Type: new Abstract: We study the Solvability Complexity Index (SCI) of Koopman operator spectral computation in the information-based framework of towers of algorithms. Given a compact metric space $(\mathcal{X},d)$ with a finite Borel measure $\omega$ on $\mathcal{X}$ and a continuous nonsingular map $F:\mathcal{X}\to \mathcal{X}$, our focus is the Koopman operator $\mathcal{K}_F$ acting on $L^p(\mathcal{X},\omega)$ for $p\in\{1,\infty\}$ for the computational problem \[ \Xi_{\sigma_{\mathrm{ap}}}(F) :=\sigma_{\mathrm{ap}}\!\bigl(\mathcal{K}_F\bigr), \] with input access given by point evaluations of $F\mapsto F(x)$ (and fixed quadrature access to $\omega$). We clarify how the $L^1$ case can be brought into the same oracle model as the reflexive regime $1<\infty$ by proving a uniform finite-dimensional quadrature compatibility, while highlighting the fundamentally different role played by non-separability at $p=\infty$. Beyond Koopman operators, we also construct a prototype family of decision problems $(\Xi_m)_{m\in\mathbb N}$ realizing prescribed finite tower heights, providing a reusable reduction source for future SCI lower bounds. Finally, we place these results deeper in the broader computational landscape of Type-2/Weihrauch theory.
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https://arxiv.org/abs/2601.12044
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aae787edc0cc5afc932b04765a5ec2d2d0e8eb3bfa15451b638aa9c29ea15a21
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2026-01-21T00:00:00-05:00
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Geometric realisations of type $\tilde{A}_n$ preprojective algebras in homological mirror symmetry
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arXiv:2601.12045v1 Announce Type: new Abstract: The type $A_n$-singularity $\mathbb{C}^2/\mathbb{Z}_{n+1}$ can be resolved by hyper-K\"ahler manifolds $X_{\zeta}$ with underlying smooth manifolds diffeomorphic to the resolution of singularities $X_{\text{res}}$, whose hyper-K\"ahler structure depends on a parameter $\zeta\in H_2(X_{\text{res}};\mathbb{R})$. The structure as a complex manifold of each such hyper-K\"ahler manifold is equivalent to the resolution of singularities at the poles and the structure of a Milnor fibre with roots determined by $\zeta$ elsewhere; the symplectic structure is exact along the equator and is deformed by areas depending on $\zeta$ on the exceptional $(-2)$-spheres away from the equator. We show that removing suitable divisors $D_u$ from a fixed $X_{\zeta}$ varying with $u$ in the underlying upper hemisphere of the $S^2$-family of K\"ahler-structures yields a log Calabi--Yau hyper-K\"ahler family (in particular a family of log Calabi--Yau submanifolds), and that mirror symmetry is satisfied (partly conjectural in one direction) for this family by hyper-K\"ahler rotation, in particular by interchanging the structures over the equator and the pole. We furthermore show homological mirror symmetry after adding the missing divisors, which is related to attaching stops and computing singularity categories of certain Landau--Ginzburg potentials on the $A$-side and $B$-side, respectively. More concretely: we compute wrapped Fukaya categories and compare them with (previous and new) computations of derived categories of coherent sheaves and derived categories of singularities in algebraic geometry. We show that the relevant categories (with two exceptions) are triangulated equivalent to module categories over the additive and the multiplicative preprojective algebras of type $\tilde{A}_n$, or to deformations of these algebras depending on the parameters $\zeta$.
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https://arxiv.org/abs/2601.12045
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227015bf2659bad67839f0f6d3560003d63fcce7d3a714dc1962bd761f72e775
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2026-01-21T00:00:00-05:00
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Partition identities associated with $A_r$-Surface singularities
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arXiv:2601.12048v1 Announce Type: new Abstract: We prove a family of partition identities involving integer partitions in three colors. The conditions imposed on the types of partitions appearing in these identities involve constraints that arise in the Rogers-Ramanujan and Andrews-Gordon identities, as well as in their recent extensions. The identities established in this paper are associated with the $A_r$ surface singularities via the arc HP-series, which provides a measure of singularities of algebraic varieties defined using arc spaces.
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https://arxiv.org/abs/2601.12048
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f67de4ddffc117d8863f4e0227888c0c30be8eccafcafc2cf9ea4dda8bd2938e
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2026-01-21T00:00:00-05:00
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Function Computation Over Multiple Access Channels via Hierarchical Constellations
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arXiv:2601.12050v1 Announce Type: new Abstract: We study function computation over a Gaussian multiple-access channel (MAC), where multiple transmitters aim at computing a function of their values at a common receiver. To this end, we propose a novel coded-modulation framework for over-the-air computation (OAC) based on hierarchical constellation design, which supports reliable computation of multiple function outputs using a single channel use. Moreover, we characterize the achievable computation rate and show that the proposed hierarchical constellations can compute R output functions with decoding error probability epsilon while the gap to the optimal computation rate scales as O(\log_2(1/\epsilon)/K) for independent source symbols, where K denotes the number of transmitters. Consequently, this gap vanishes as the network size grows, and the optimal rate is asymptotically attained. Furthermore, we introduce a shielding mechanism based on variable-length block coding that mitigates noise-induced error propagation across constellation levels while preserving the superposition structure of the MAC. We show that the shielding technique improves reliability, yielding a gap that scales optimally as O(\log_2\ln{(1/\epsilon)}), regardless of the source distribution. Together, these results identify the regimes in which uncoded or lightly coded OAC is information-theoretically optimal, providing a unified framework for low-latency, channel-agnostic function computation.
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https://arxiv.org/abs/2601.12050
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db8a9b2ab0f42278cb425643aaa8e9826c608ab97debb327866aa4d11974099d
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2026-01-21T00:00:00-05:00
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Ramanujan polar graphs
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arXiv:2601.12057v1 Announce Type: new Abstract: Recently, a construction of minimal codes arising from a family of almost Ramanujan graphs was shown. Ramanujan graphs are examples of expander graphs that minimize the second-largest eigenvalue of their adjacency matrix. We call such graphs Ramanujan, since all known non-trivial constructions imply the Ramanujan conjecture on arithmetical functions. In this paper, we prove that some families of tangent graphs of finite classical polar spaces satisfy Ramanujan's condition. If the polarity is unitary, or it is orthogonal and the quadric is over the binary field, the tangent graphs are strongly regular, and we know their spectrum. By direct computation, it is possible to show which families of tangent graphs are Ramanujan.
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https://arxiv.org/abs/2601.12057
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47e051eb3212cddb9b2eb14d2e85c5009c67d187d5540902dad50139467a684a
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2026-01-21T00:00:00-05:00
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Magnetic spectral inverse problems on compact Anosov manifolds
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arXiv:2601.12058v1 Announce Type: new Abstract: In this paper, we establish positive results for two spectral inverse problems in the presence of a magnetic potential. Exploiting the principal wave trace invariants, we first show that on closed Anosov manifolds with simple length spectrum, one can recover an electric and a magnetic (up to a natural gauge) potential from the spectrum of the associated magnetic Schr\"odinger operator. This extends a particular instance of a recent positive result on the spectral inverse problem for the Bochner Laplacian in negative curvature, obtained by M.Ceki\'c and T.Lefeuvre (2023). Similarly, we prove that the spectrum of the magnetic Dirichlet-to-Neumann map (or Steklov operator) determines at the boundary both a magnetic potential, up to gauge, and an electric potential, provided the boundary is Anosov with simple length spectrum. Under this assumption, one can actually show that the magnetic Steklov spectrum determines the full Taylor series at the boundary of any smooth magnetic field and electric potential. As a simple consequence, in this case, both an analytic magnetic field and an analytic electric potential are uniquely determined by their Steklov spectrum.
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https://arxiv.org/abs/2601.12058
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baf3ab922ce3b492d7def1e795f35b15734b164c1b08f53c37f3ecab17065a9a
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2026-01-21T00:00:00-05:00
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Almost coherent rings
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arXiv:2601.12059v1 Announce Type: new Abstract: Inspired from the work of P. Scholze on the finiteness of \(\mathbf{F}_{p}\)-cohomology groups of proper rigid-analytic varieties over \(p\)-adic fields, Zavyalov recently introduced the notion of almost coherent rings, which plays a key role in the almost ring theory. In this paper, we characterize almost coherent rings in terms of almost flat modules and almost absolutely pure modules, integrating numerous classical results into almost mathematics. Besides, we show that every almost coherent $R$-module is not almost isomorphic to a coherent $R$-module, giving a negative answer to a question proposed in [14,B. Zavyalov, {\it Almost coherent modules and almost coherent sheaves}, Memoirs of the European Mathematical Society 19. Berlin: European Mathematical Society (EMS), 2025].
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https://arxiv.org/abs/2601.12059
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fd644640558719750f046a9efe42c02a6485b73c28b0da414134cef4e81a938a
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2026-01-21T00:00:00-05:00
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Invariant Means on $VN^n(G)$
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arXiv:2601.12063v1 Announce Type: new Abstract: Let $G$ be a locally compact group, and $VN^n(G)$ is the dual of the multidimensional Fourier algebra $A^n(G)$. In this article, we define invariant means on $VN^n(G)$ and prove that the set of all invariant means on $VN^n(G)$ is non-empty. Further, we investigated the invariant means on $VN^n(G)$ for discrete and non-discrete cases of $G$. Also, we show that if $H$ is an open subgroup of $G$, then the number of invariant means on $VN^n(H)$ is the same as that of $VN^n(G)$. Finally, we study invariant means on the dual of the algebra $A_0^n(G)$, the closure of Fourier algebra $A^n(G)$ in the cb-multiplier norm.
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https://arxiv.org/abs/2601.12063
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3bc84425d2959ef4339c95a3de0514c94633048a9e9609ef1bf5f84822d2db1f
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2026-01-21T00:00:00-05:00
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Expansion and Bounds for the Bias of Empirical Tail Value-at-Risk
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arXiv:2601.12064v1 Announce Type: new Abstract: Tail Value-at-Risk (TVaR) is a widely adopted risk measure playing a critically important role in both academic research and industry practice in insurance. In data applications, TVaR is often estimated using the empirical method, owing to its simplicity and nonparametric nature. The empirical TVaR has been explicitly advocated by regulatory authorities as a standard approach for computing TVaR. However, prior literature has pointed out that the empirical TVaR estimator is negatively biased, which can lead to a systemic underestimation of risk in finite-sample applications. This paper aims to deepen the understanding of the bias of the empirical TVaR estimator in two dimensions: its magnitude as well as the key distributional and structural determinants driving the severity of the bias. To this end, we derive a leading-term approximation for the bias based on its asymptotic expansion. The closed-form expression associated with the leading-term approximation enables us to obtain analytical insights into the structural properties governing the bias of the empirical TVaR estimator. To account for the discrepancy between the leading-term approximation and the true bias, we further derive an explicit upper bound for the bias. We validate the proposed bias analysis framework via simulations and demonstrate its practical relevance using real data.
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https://arxiv.org/abs/2601.12064
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19dad45790c7ec16dc102ae8b8a67dfaf5a62aa3649200a11aff39a1c475c0c9
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2026-01-21T00:00:00-05:00
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Boojums in Liquid Crystals Around a Colloid
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arXiv:2601.12065v1 Announce Type: new Abstract: We study the Landau-de Gennes theory in the one constant limit. The bulk domain is the exterior of a spherical colloid. A Rapini-Papoular surface potential is imposed on the colloid surface, supplemented by a homogeneous far-field condition at spatial infinity. Under the axially symmetric ansatz and the Lyuksyutov constraint, we show that energy minimizers exhibit boojum disclinations at the two poles of the colloid. The local structure of these boojum disclinations is also characterized.
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https://arxiv.org/abs/2601.12065
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7d5642f88b17e99ed773d84bec00dd391ee0bf3fbdeaa6038542856d5120427f
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2026-01-21T00:00:00-05:00
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Boundary Perturbations of Steklov Eigenvalues
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arXiv:2601.12077v1 Announce Type: new Abstract: We consider the dependence of non-zero Steklov eigenvalues on smooth perturbations of the domain boundary. We prove that these eigenvalues are generically simple under such boundary perturbations. This result complements our previous work on metric perturbations, thereby establishing generic simplicity Steklov eigenvalues under both fundamental geometric variations.
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https://arxiv.org/abs/2601.12077
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f3f94b45e8bfafb8d07325c6c8149c430ef16b2a11e21a4d804083588bbe187b
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2026-01-21T00:00:00-05:00
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Sharpness of the Osgood Criterion for the Continuity Equation with Divergence-free Vector Fields
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arXiv:2601.12096v1 Announce Type: new Abstract: For any modulus of continuity $\omega$ that fails the Osgood condition, we construct a divergence-free velocity field $v \in C_t C^\omega_x$ for which the associated ODE admits at least two distinct flow maps. In other words, non-uniqueness does not occur merely for a single or even finitely many trajectories, but instead on a set of initial conditions $E$ of positive Lebesgue measure. In fact, the set $E$ has full measure inside a cube where the construction is supported. Moreover, we also construct a divergence-free velocity field $v \in C_{t}C^\omega_x$ for which the associated continuity equation admits two distinct solutions $\mu^1$ and $\mu^2$ which are absolutely continuous with respect to Lebesgue measure for almost every time, and start from the same initial datum $\bar \mu \ll \mathscr{L}^{d}$. Our construction introduces two novel ideas: (i) We introduce the notion of "parallelization", where at each time, the velocity field consists of simultaneous motion across multiple nested spatial scales. This differs from most explicit constructions in the literature on mixing or anomalous dissipation, where the velocity on different scales acts at separate times. This is crucial to cover the whole class of non-Osgood moduli of continuity. (ii) Inspired by a recent work of Bru\`e, Colombo and Kumar, we develop a new fixed-point framework that naturally incorporates the parallelization mechanism. This framework allows us to construct anomalous solutions of the continuity equation that belong to $L^1(\mathbb{R}^d)$ a.e. in time.
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https://arxiv.org/abs/2601.12096
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Academic Papers
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554dd61d1c11d4b45a741ca9420381734954231104e31a1126923b6c352374a0
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2026-01-21T00:00:00-05:00
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Higher integrability of solutions to elliptic equations under additional sign constraints
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arXiv:2601.12100v1 Announce Type: new Abstract: Solutions to elliptic equations often exhibit higher regularity properties such as \emph{higher integrability}. That is, for instance, a solution $u$ to a system that a priori only satisfies $ u \in W^{1,r}$ is more regular and even in the Sobolev space $W^{1,s}$ for some $s>r$. Under additional constraints of the sign of specific terms such as $(\partial_i u)$ this improvement of regularity can be sharpened further. In this work, we consider two examples of such higher integrability results: First, we show a version of M\"uller's result on the higher integrability of the determinant for maps $u \in W^{1,n} $ such that $\mathrm{det}(\nabla u) \geq 0$ (or $ \mathrm{det}_-(\nabla u) \in L \log L$). Second, we consider (very weak) solutions to the $p$-Laplace equation that satisfy sign constraints for their partial derivatives, i.e. that $(\partial_i u)_- $ is of higher integrability than $(\partial_i u)_+$. To prove our results, we use the method of Lipschitz truncation; for the second example we further develop a variation of this technique, the \emph{asymmetric} Lipschitz truncation.
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https://arxiv.org/abs/2601.12100
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Academic Papers
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c68b920f4b93ee5747af6efd7fb91e8c92448bfade1ed073027e796591f8967b
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2026-01-21T00:00:00-05:00
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On the Construction and Correlation Properties of Permutation-Interleaved Zadoff-Chu Sequences
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arXiv:2601.12107v1 Announce Type: new Abstract: Constant amplitude zero auto-correlation (CAZAC) sequences are widely applied in waveforms for radar and communication systems. Motivated by a recent work [Berggren and Popovi\'c, IEEE Trans. Inf. Theory 70(8), 6068-6075 (2024)], this paper further investigates the approach to generating CAZAC sequences by interleaving Zadoff-Chu (ZC) sequences with permutation polynomials (PPs). We propose one class of high-degree PPs over the integer ring Z N , and utilize them and their inverses to interleave ZC sequences for constructing CAZAC sequences. It is known that a CAZAC sequence can be extended to an equivalence class by five basic opertations. We further show that the obtained CAZAC sequences are not covered by the equivalence classes of ZC sequences and interleaved ZC sequences by quadratic PPs and their inverses, and prove the sufficiency of the conjecture by Berggren and Popovi\'c in the aforementioned work. In addition, we also evaluate the aperiodic auto-correlation of certain ZC sequences from quadratic PPs.
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https://arxiv.org/abs/2601.12107
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Academic Papers
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