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5b024f90671ab2ffa93e70a93966eeb2b2dcdf9af08abb5583e1e2ab51f4e38d
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2026-01-13T00:00:00-05:00
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Heat as a gauge connection
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arXiv:2503.08753v2 Announce Type: replace-cross Abstract: We show that heat defines a gauge connection on a line bundle over work configurations. Vanishing curvature is equivalent to the local existence of entropy and temperature functions such that heat can be expressed as $TdS$. A conjecture of Jauch, that entropy and temperature arise from a conservation law, is shown to follow as a special case. Global equilibrium may nevertheless fail in the presence of a thermal analogue of geometric phase.
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https://arxiv.org/abs/2503.08753
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Academic Papers
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530a906f4f97372027b0daf5d40451951f02d8fe3f6396d8eac381d79e6ac18e
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2026-01-13T00:00:00-05:00
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Observation of a generalized Gibbs ensemble in photonics
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arXiv:2503.16526v2 Announce Type: replace-cross Abstract: In generic classical and quantum many-body systems, where typically energy and particle number are the only conserved quantities, stationary states are described by thermal equilibrium. In contrast, integrable systems showcase an infinite hierarchy of conserved quantities that inhibits conventional thermalization, forcing relaxation to a Generalized Gibbs Ensemble (GGE) -- a concept first introduced in quantum many-body physics. In this study, we provide experimental evidence for the emergence of a GGE in a photonic system. By investigating partially coherent waves propagating in a normal dispersion optical fiber, governed by the one-dimensional defocusing nonlinear Schroedinger equation, we directly measure the density of states of the spectral parameter (rapidity) to confirm the time invariance of the full set of conserved charges. We also observe the relaxation of optical power statistics to the GGE's theoretical prediction, obtained using the experimentally measured density of states. These complementary measurements unambiguously establish the formation of a GGE in our photonic platform, highlighting its potential as a powerful tool for probing many-body integrability and bridging classical and quantum integrable systems.
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https://arxiv.org/abs/2503.16526
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Academic Papers
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f14bf8392d8e7b2d5d1de220575a5b23e2e5a70b4e1335b0d098e2a49c73c783
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2026-01-13T00:00:00-05:00
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LOGLO-FNO: Efficient Learning of Local and Global Features in Fourier Neural Operators
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arXiv:2504.04260v2 Announce Type: replace-cross Abstract: Modeling high-frequency information is a critical challenge in scientific machine learning. For instance, fully turbulent flow simulations of the Navier-Stokes equations at Reynolds numbers 3500 and above can generate high-frequency signals due to swirling fluid motions caused by eddies and vortices. Faithfully modeling such signals using neural nets depends on the accurate reconstruction of moderate to high frequencies. However, it has been well known that neural nets exhibit spectral or frequency bias towards learning low-frequency components. Meanwhile, Fourier Neural Operators (FNOs) have emerged as a popular class of data-driven models for surrogate modeling and solving PDEs. Although impressive results were achieved on several PDE benchmark problems, FNOs perform poorly in learning non-dominant frequencies characterized by local features. This limitation stems from spectral bias inherent in neural nets and the explicit exclusion of high-frequency modes in FNOs and their variants. Therefore, to mitigate these issues and improve FNO's spectral learning capabilities to represent a broad range of frequency components, we propose two key architectural enhancements: (i) a parallel branch performing local spectral convolution (ii) a high-frequency propagation module. Moreover, we propose a novel frequency-sensitive loss based on radially binned spectral errors. This introduction of a parallel branch for local convolution reduces the trainable parameters by up to 50% while achieving the accuracy of FNO that relies solely on global convolution. Moreover, our findings demonstrate that the proposed model improves stability over longer rollouts. Experiments on six challenging PDEs in fluid mechanics, wave propagation, and biological pattern formation, and the qualitative and spectral analysis of predictions, show the effectiveness of our method over SOTA neural operator families of baselines.
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https://arxiv.org/abs/2504.04260
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Academic Papers
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db4ef4eb0658159a154ee855879b2e47d406976f22454d30110d9ce3a8cf449f
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2026-01-13T00:00:00-05:00
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Radiation-induced Instability of Organic-Inorganic Halide Perovskite Single Crystals
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arXiv:2504.06222v2 Announce Type: replace-cross Abstract: Organic-inorganic halide perovskites (OIHPs) are promising optoelectronic materials, but their instability under radiation environments restricts their durability and practical applications. Here we employ electron and synchrotron X-ray beams, individually, to investigate the radiation-induced instability of two types of OIHP single crystals (FAPbBr3 and MAPbBr3). Under the electron beam, we observe that 3-point star-style cracks grow on the surface of FAPbBr3, and bricklayer-style cracks are formed on the surface of MAPbBr3. Under the X-ray beam, a new composition without organic components appears in both FAPbBr3 and MAPbBr3. Such cracking and composition changes are attributed to the volatilization of organic components. We propose a volume-strain-based mechanism, in which the energy conversion results from the organic cation loss. Using nanoindentation, we reveal that beam radiations reduce the Youngs modulus and increase the hardness of both OIHPs. This study provides valuable insights into the structural and mechanical stabilities of OIHP single crystals in radiation environments.
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https://arxiv.org/abs/2504.06222
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Academic Papers
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aa201b1ff4a3c3c2ae054ca7f5f081647e38c5daed22dda247592f21c4f0b2f7
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2026-01-13T00:00:00-05:00
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Spin polarized enantio-sensitive multipolar photoelectron currents
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arXiv:2505.23460v3 Announce Type: replace-cross Abstract: Photoelectron circular dichroism (PECD) manifests as a forward-backward asymmetry of electron emission in the direction orthogonal to the light polarization plane via one-photon ionization of chiral molecules with circularly polarized light. Multi-polar `PECD' currents, i.e., currents resolved along multiple directions, have also been predicted using two mutually-orthogonal linearly polarized light with carrier frequencies $\omega$ and $2\omega$. These currents arise from the interference between the one- and two-photon transitions. Here, we will show that photoelectron spin detection already reveals enantio-sensitive multi-polar currents in the one-photon regime since the two axes can be marked by the photoelectron momentum $\unitvec{k}$ and spin-detection axis $\unitvec{s}$. Specifically, we consider one-photon ionization of an isotropic ensemble of randomly oriented chiral molecules and show that the direction of the resulting photoelectron current is enantio-sensitively `locked' to the photoelectron's spin, which is mediated by two mechanisms. First, is the Bloch pseudovector which enables a collinear locking forming either a spin-sink or source for opposite enantiomers. Second, is the spin torque pseudovector that enables orthogonal locking forming a spin vortex in the polarization plane that rotates in opposite directions for opposite enantiomers. The former effect is a spin analog of photoelectron vortex dichroism (\href{https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.129.233201}{Phys. Rev. Lett. \textbf{129}, 233201, 2022}) wherein the detected photoelectron spin encodes molecular chirality while the latter is reminiscent of the Rashba effect in solids
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https://arxiv.org/abs/2505.23460
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Academic Papers
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95c9bed2f41a7df1c3ba8b433b3b1eb9ff7805054d291baf743e104ac534cdc4
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2026-01-13T00:00:00-05:00
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Uncovering the Computational Roles of Nonlinearity in Sequence Modeling Using Almost-Linear RNNs
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arXiv:2506.07919v2 Announce Type: replace-cross Abstract: Sequence modeling tasks across domains such as natural language processing, time series forecasting, and control require learning complex input-output mappings. Nonlinear recurrence is theoretically required for universal approximation of sequence-to-sequence functions, yet linear recurrent models often prove surprisingly effective. This raises the question of when nonlinearity is truly required. We present a framework to systematically dissect the functional role of nonlinearity in recurrent networks, identifying when it is computationally necessary and what mechanisms it enables. We address this using Almost Linear Recurrent Neural Networks (AL-RNNs), which allow recurrence nonlinearity to be gradually attenuated and decompose network dynamics into analyzable linear regimes, making computational mechanisms explicit. We illustrate the framework across diverse synthetic and real-world tasks, including classic sequence modeling benchmarks, a neuroscientific stimulus-selection task, and a multi-task suite. We demonstrate how the AL-RNN's piecewise linear structure enables identification of computational primitives such as gating, rule-based integration, and memory-dependent transients, revealing that these operations emerge within predominantly linear backbones. Across tasks, sparse nonlinearity improves interpretability by reducing and localizing nonlinear computations, promotes shared representations in multi-task settings, and reduces computational cost. Moreover, sparse nonlinearity acts as a useful inductive bias: in low-data regimes or when tasks require discrete switching between linear regimes, sparsely nonlinear models often match or exceed fully nonlinear architectures. Our findings provide a principled approach for identifying where nonlinearity is functionally necessary, guiding the design of recurrent architectures that balance performance, efficiency, and interpretability.
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https://arxiv.org/abs/2506.07919
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363548a1d6edb944d6a470e4e2df70098579b999a5bc7aabd19d8e1fb1529aed
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2026-01-13T00:00:00-05:00
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Phase-Space Topology in a Single-Atom Synthetic Dimension
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arXiv:2506.24020v4 Announce Type: replace-cross Abstract: We investigate topological features in the synthetic Fock-state lattice (FSL) of a single-atom system described by the quantum Rabi model. By diagonalizing the Hamiltonian, we identify a zero-energy defect state localized at a domain wall of the FSL, whose spin polarization is topologically protected. To address the challenge of applying band topology to the FSL, we introduce a physically motivated and directly measurable topological invariant based on phase-space geometry-the phase-space winding number. We show that the Zak phase, computed using a phase-space parameter, is related to the invariant. This quantized geometric phase reflects the spin polarization of the defect state, demonstrating a bulk-boundary correspondence. The resulting phase-space topology reveals the emergence of single-atom dressed states with contrasting properties-topologically protected spin states and driving-tunable bosonic states. Our results establish phase-space topology as a novel framework for exploring topological physics in single-atom synthetic dimensions, uncovering quantum-unique topological protection distinct from classical analogs.
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https://arxiv.org/abs/2506.24020
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Academic Papers
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632d7f3af1054113402d93bd3b18abc71b8df88a3015cc3b93620114d76dbe0f
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2026-01-13T00:00:00-05:00
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Three-boson scattering hypervolume for a nonzero orbital angular momentum
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arXiv:2507.20787v2 Announce Type: replace-cross Abstract: We analyze the zero energy collision of three identical bosons in the same internal state with total orbital angular momentum $L=2$, assuming short range interactions. By solving the Schr\"odinger equation asymptotically, we derive two expansions of the wave function when three bosons are far apart or a pair of bosons and the third boson are far apart. The scattering hypervolume $D$ is defined for this collision. Unlike the scattering hypervolume defined by one of us in 2008, whose dimension is length to the fourth power, the dimension of $D$ studied in the present paper is length to the eighth power. We then derive the expression of $D$ when the interaction potentials are weak, using the Born's expansion. We also calculate the energy shift of such three bosons with three different momenta $\hbar \mathbf{k_{1}}$, $\hbar\mathbf{k_{2}}$ and $\hbar\mathbf{k_{3}}$ in a large periodic box. The obtained energy shift depends on $D^{(0)}/\Omega^{2}$ and $D/\Omega^{2}$, where $D^{(0)}$ is the three-body scattering hypervolume defined for the three-body $L=0$ collision and $\Omega$ is the volume of the periodic box. We also calculate the contribution of $D$ to the three-body T-matrix element for low-energy collisions. We then calculate the shift of the energy and the three-body recombination rate due to $D^{(0)}$ and $D$ in the dilute homogeneous Bose gas. The contribution to the three-body recombination rate constant from $D$ is proportional to $T^2$ if the temperature $T$ is much larger than the quantum degeneracy temperature but still much lower than the temperature scale at which the thermal de Broglie wave length becomes comparable to the physical range of interaction.
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https://arxiv.org/abs/2507.20787
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Academic Papers
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35bdef70bf5d352e39d2975705d5835f272d1576622c341c381d35a383c0b211
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2026-01-13T00:00:00-05:00
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Beyond asymptotic reasoning: the practicalities of a quantum ground state projector based on the wall-Chebyshev expansion
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arXiv:2508.00533v3 Announce Type: replace-cross Abstract: We consider a quantum algorithm for ground-state preparation based on a Chebyshev series approximation to the wall function. In a classical setting, this approach is appealing as it guarantees rapid convergence. We analyze the asymptotic scaling and success probabilities of different quantum implementations and provide numerical benchmarks, comparing the performance of the wall-Chebyshev projectors with current state-of-the-art approaches. We find that this approach requires fewer serial applications of the Hamiltonian oracle to achieve a given ground state fidelity, but is severely limited by exponentially decaying success probability. However, we find that some implementations maintain non-trivial success probability in regimes where wall-Chebyshev projection leads to a fidelity improvement over other approaches. As the wall-Chebyshev projector is highly robust to loose known upper bounds on the true ground state energy, it offers a potential resource trade-off, particulary in the early fault-tolerant regime of quantum computation.
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https://arxiv.org/abs/2508.00533
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aa03c15c8b9b98c2c21b93deb746d1f927a49c05a58805d2dbf935f03199b3a0
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2026-01-13T00:00:00-05:00
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First Light from Beam Neutrinos on an LAPPD in ANNIE
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arXiv:2508.11111v2 Announce Type: replace-cross Abstract: The Accelerator Neutrino Neutron Interaction Experiment (ANNIE) is both a physics experiment and a technology testbed for next-generation light-based neutrino detection. In this paper, we report the first demonstration of a fully integrated Large Area Picosecond Photodetector (LAPPD) operating in a running neutrino beam experiment. Particular focus is given to the design, commissioning, and successful deployment of the Packaged ANNIE LAPPD (PAL), a waterproof, self-triggering module incorporating fast waveform digitization and precision timing synchronized to the ANNIE detector subsystems. We identify beam-correlated LAPPD data frames consistent with charged-current neutrino interactions observed in multiple detector subsystems, establishing the first detection of neutrino-induced Cherenkov light with an LAPPD. These results validate the system-level performance of LAPPDs under realistic experimental conditions-including long-term stability, timing synchronization, and event matching with conventional PMT and muon detector systems-marking a critical step toward their deployment in future large-scale neutrino and particle detectors.
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https://arxiv.org/abs/2508.11111
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10a4b9369f84e7b31a617ac43f157e08a6f168a93f19324fcd2119ff5adc589a
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2026-01-13T00:00:00-05:00
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Thermoelectric power factors of defective scandium nitride nanostructures from first principles
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arXiv:2509.14762v3 Announce Type: replace-cross Abstract: The thermoelectric properties of scandium nitride are strongly influenced by structural and electronic factors arising from defects and impurities. Nevertheless, the mechanisms by which these microscopic features affect transport are not yet fully understood. Experiments show a large variability in the electronic transport properties, with a strong dependence on the experimental conditions, and attempts to improve thermoelectric efficiency often lead to conflicting effects. In this work, we employ the Landauer approach to analyze the effects of different kinds of structural defects and impurities on electronic transport in scandium nitride. This approach allows us to relate the transport mechanisms to the structural and electronic modifications introduced in the lattice, with atomistic resolution. In light of these new insights, we propose a rationale relating part of the experimental variability to its microscopic origin.
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https://arxiv.org/abs/2509.14762
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957ce32eb619cda92b6e800c81e4a2507a951132423d7bcda1c283f028d99ad2
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2026-01-13T00:00:00-05:00
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Dynamically stable optical trapping of thermophoretically active Janus colloids
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arXiv:2509.21088v2 Announce Type: replace-cross Abstract: The ability to optically trap and manipulate artificial microswimmers such as active Janus particles (JPs) provides a breakthrough in active matter research and applications. However, it presents significant challenges because of the asymmetry in the optical properties of JPs and remains incomprehensible. Illustrating the interplay between optical and thermophoretic forces, we demonstrate dynamically stable optical trapping of Pt-silica JPs, where the force-balanced position evolves spontaneously within a localized volume around the focal point and in a vertically shifted annular confinement at low and high laser powers, respectively. Intriguingly, the orientational and orbital dynamics of JP remain strongly coupled in the delocalized confinement. Furthermore, we demonstrate simultaneous optical trapping of multiple JPs. This first report on thermophoresis of Pt-silica JPs and localized-to-delocalized crossover in the position distributions of an optically trapped active JP, verifying theoretical predictions, advances our understanding on confined active matter and their experimental realizations.
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https://arxiv.org/abs/2509.21088
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Academic Papers
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37e0f24b95e24320b3e075df016f0f301744313e88cbfce56ec21bfc8dc08bda
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2026-01-13T00:00:00-05:00
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Tunable optical lattices for the creation of matter-wave lattice solitons
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arXiv:2509.22471v2 Announce Type: replace-cross Abstract: We present experimental techniques that employ an optical accordion lattice with dynamically tunable spacing to create and study bright matter-wave solitons in optical lattices. The system allows precise control of lattice parameters over a wide range of lattice spacings and depths. We detail calibration methods for the lattice parameters that are adjusted to the varying lattice spacing, and we demonstrate site-resolved atom number preparation via microwave addressing. Lattice solitons are generated through rapid quenches of the atomic interaction strength and the external trapping potential. We systematically optimize the quench parameters, such as duration and final scattering length, to maximize soliton stability. Our results provide insight into nonlinear matter-wave dynamics in discretized systems and establish a versatile platform for the controlled study of lattice solitons.
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https://arxiv.org/abs/2509.22471
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394fc1e375799fa97baa42973066649b03c53e64d6058baeb7bbfb51662dd0a1
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2026-01-13T00:00:00-05:00
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SymBoltz.jl: a symbolic-numeric, approximation-free and differentiable linear Einstein-Boltzmann solver
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arXiv:2509.24740v2 Announce Type: replace-cross Abstract: SymBoltz is a new Julia package that solves the linear Einstein-Boltzmann equations. It features a symbolic-numeric interface for specifying equations, is free of approximation switching schemes and is compatible with automatic differentiation. Cosmological models are built from replaceable physical components in a way that scales well in model space, or alternatively written as one compact system of equations. The modeler should simply write down their equations, and SymBoltz solves them and eliminates friction in the modeling process. Symbolic knowledge enables powerful automation of tasks, such as separating computational stages like the background and perturbations, generating the analytical Jacobian matrix and its sparsity pattern, and interpolating arbitrary variables from the solution. Implicit solvers integrate the full stiff equations at all times without approximations, which greatly simplifies the code. Performance remains as good as in existing approximation-based codes due to high-order implicit methods that take long time steps, fast generated code, optimal handling of the Jacobian and efficient sparse matrix methods. Automatic differentiation gives exact derivatives of any output with respect to any input, which is important for gradient-based Markov chain Monte Carlo methods in large parameter spaces, training of emulators, Fisher forecasting and sensitivity analysis. The main features form a synergy that reinforces the design of the code. Results agree with established codes to 0.1% with standard precision. More work is needed to implement additional features and for fast reverse-mode automatic differentiation of scalar loss functions. SymBoltz is available at https://github.com/hersle/SymBoltz.jl with single-command installation and extensive documentation, and welcomes all contributions.
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https://arxiv.org/abs/2509.24740
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2a497ec37aa21b1ed3f3f45726a6df0d00d16d12ab885439ea2a23f4ea509569
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2026-01-13T00:00:00-05:00
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Machine Learning Interatomic Potentials Enable Molecular Dynamics Simulations of Doped MoS2
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arXiv:2510.05339v2 Announce Type: replace-cross Abstract: We present the first computational framework for molecular dynamics simulation of MoS2 doped with 25 elements spanning metals, non-metals, and transition metals using Meta's Universal Model for Atoms machine learning interatomic potential (MLIP). Benchmarking against density functional theory calculations demonstrates the accuracy of the MLIP for simulating doped-MoS2 systems and highlights opportunities for improvement. Using the MLIP, we perform heating-cooling simulations of doped-MoS2 supercells. The simulations capture complex phenomena including dopant clustering, MoS2 layer fracturing, interlayer diffusion, and chemical compound formation at orders-of-magnitude reduced computational cost compared to density functional theory. This work provides an open-source computational workflow for application-oriented design of doped-MoS2, enabling high-throughput screening of dopant candidates and optimization of compositions for targeted tribological, electronic, and optoelectronic performance. The MLIP bridges the accuracy-efficiency gap between first-principles methods and empirical potentials, and the framework offers unprecedented opportunities for large-scale materials discovery in two-dimensional doped material systems.
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https://arxiv.org/abs/2510.05339
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Academic Papers
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9d2f1fde1afe15f193fffa47b351d239020997c0eb924f153b01d83093b0bc38
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2026-01-13T00:00:00-05:00
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Thermodynamics of data
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arXiv:2510.06818v2 Announce Type: replace-cross Abstract: The recently introduced concept of generalized thermodynamics is explored here in the context of 1d, 2d and 3d data analysis, performed on samples drawn from a 3d X-ray soil sample image. Different threshold levels are used to binarize the 3d sample, wherefrom relative frequencies of binary patterns are found and then used to address finite size scaling behavior of the response functions as a function of the disorder parameter (equivalent of temperature in thermodynamics). It is found that for different threshold levels response functions for increasing sample sizes approach the thermodynamic limit from different directions, with a crossover reminiscent of a transition from open to periodic boundaries of the Ising model, implying existence of a characteristic correlation scale. It is argued here that this characteristic scale corresponds to the "natural" properties of the data, where correlations within finite size samples are neither underestimated nor overestimated. In the current context of soil this scale may be related to the so-called Representative elementary volume (REV), while in other situations this characteristic scale should be interpreted in the context of the phenomenon under study.
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https://arxiv.org/abs/2510.06818
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11bd8a22fd25a9d4472ec96a11ea6b2f4f4ea5d198ed2bfc279318aab0a9ffb0
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2026-01-13T00:00:00-05:00
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Quantum Monte Carlo study of low-dimensional Fermi fluids of dipolar atoms
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arXiv:2510.19533v2 Announce Type: replace-cross Abstract: Fermionic cold atoms in optical traps provide viable quantum simulators of correlation effects in electronic systems. For dressed Rydberg atoms in two-dimensional traps with out-of-plane dipole moments, a realistic model of the pairwise interaction is of repulsive dipolar $1/r^3$ form at long range, softened to a constant at short range. This study provides parameterizations of fixed-node diffusion Monte Carlo energy data for ferromagnetic (one-component) and paramagnetic (two-component) two-dimensional homogeneous Fermi fluids of interacting dipolar atoms. We find itinerant ferromagnetism to be unstable within our parameter spaces for dipolar interactions both with and without softening. Our parameterization of the energy as a function of density will enable density functional theory to support experimental studies of inhomogeneous fermionic cold atom systems.
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https://arxiv.org/abs/2510.19533
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24d05da480cd5ab9fecb61500594fa58b878f884f0ed2a0968d10808c56b226d
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2026-01-13T00:00:00-05:00
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Transverse superconducting diode without parity and time-reversal violation
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arXiv:2511.01560v2 Announce Type: replace-cross Abstract: The superconducting diode effect (SDE) is characterized by its nonreciprocal nature in critical supercurrents. However, realizing a longitudinal SDE typically requires simultaneous time-reversal ($\mathcal{T}$) and inversion ($\mathcal{P}$) symmetry breaking in the device, raising challenges in applications. In this Letter, we reveal that an off-axis direct-current bias applied to a planar anisotropic superconductor can convert intrinsic anisotropy into transverse nonreciprocity, generating ultra-tunable SDE without breaking either $\mathcal{P}$ or $\mathcal{T}$ symmetry. Using both Ginzburg-Landau theory and self-consistent mean-field calculations, we show that diode efficiency can be continuously tuned via bias current amplitude. Notably, when the injected bias current exceeds a critical threshold, the system is driven into a ``unidirectional superconductivity" regime, where transverse dissipationless currents are permitted in only one direction. Based on this mechanism, we propose the ``current-gated orthogonal superconducting transistor (CGOST)" and demonstrate its utility in tunable supercurrent range controllers and half-wave rectifiers. Our findings open new avenues for nonreciprocal superconducting electronics.
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https://arxiv.org/abs/2511.01560
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9169c1c0e64cc95f3c49c22765305992f7bfefa05c0dae58a042a6ae2a869832
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2026-01-13T00:00:00-05:00
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Implementation of transformer-based LLMs with large-scale optoelectronic neurons on a CMOS compatible platform
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arXiv:2511.04136v2 Announce Type: replace-cross Abstract: The recent rapid deployment of datacenter infrastructures for performing large language models (LLMs) and related artificial intelligence (AI) applications in the clouds is predicted to incur an exponentially growing energy consumption in the near-term future. In this paper, we propose and analyze the implementation of the transformer model, which is the cornerstone of the modern LLMs, with novel large-scale optoelectronic neurons (OENs) constructed over a complementary metal-oxide-semiconductor (CMOS) compatible platform. With all of the required optoelectronic devices and electronic circuits integrated in a chiplet only about 2 cm by 3 cm in size, 175 billon parameters in the case of GPT-3 are shown to perform inference at an unprecedented speed of 12.6 POPS using only 40 nm CMOS process node, orchestrated by an optoelectronic version of systolic array with no data skew and negligible propagation delay, along with a high power efficiency of 74 TOPS/W and a high area efficiency of 19 TOPS/mm^2. The influence of the quantization formats and the hardware induced errors are numerically investigated, and are shown to have a minimal impact. Our study presents a new yet practical path toward analog neural processing units (NPUs) to complement existing digital processing units.
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https://arxiv.org/abs/2511.04136
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aa6617b8a857dd1b9be4bbcb22f066fdee0bc8a548647b4ea482ce9162b6f934
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2026-01-13T00:00:00-05:00
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Partial Collapse and Ensemble Invariance under Continuous Quantum Measurement
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arXiv:2512.22235v2 Announce Type: replace-cross Abstract: Wavefunction collapse is commonly associated with unavoidable physical disturbance of the measured system. Here we show that in driven-dissipative quantum systems, continuous measurement can induce strong trajectory-level collapse while leaving the ensemble-averaged steady state strictly invariant. We identify measurement-invariant steady states whose unconditional density matrix remains unchanged under continuous monitoring, despite pronounced measurement-induced localization in conditioned quantum trajectories. This separation between trajectory-level collapse and ensemble invariance defines a regime of partial collapse, in which measurement-induced localization is continuously counteracted by dissipative dynamics. We derive a necessary and sufficient condition for steady-state invariance under continuous measurement and identify Liouvillian symmetry as a concrete dynamical mechanism enforcing it. Our results clarify the distinction between conditional collapse and physical disturbance in open quantum systems and provide a framework for non-invasive continuous monitoring in driven-dissipative settings.
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https://arxiv.org/abs/2512.22235
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bdc06cc1678748bf8d9ea5cdca0b704a108e7eae729da395cb7d78e66fecd4fa
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2026-01-13T00:00:00-05:00
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Casimir Arc Plate Geometry: Computational Analysis of Thickness Constraints for Gold and Silver Nanomembranes in MEMS Applications
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arXiv:2512.22352v2 Announce Type: replace-cross Abstract: A theoretical analysis of the Casimir interaction between an arc and plate is conducted, which remains unexplored despite its relevance to Micro-Electro-Mechanical Systems (MEMS) fabrication. The configuration consists of a rigid finite plate and a flexible curved nanomembrane, with radius 100 micrometers, initially concave toward the rigid plate. The maximum thickness is evaluated for which the nanomembrane undergoes a change in curvature: from concave to convex with respect to the plate, due to the Casimir interaction. The Casimir energy for a curved surface is derived using the Proximity Force Approximation (PFA) with next-to-leading-order (NTLO) corrections. Kirchhoff-Love theory for a thin isotropic plate of constant thickness is used to estimate the bending energy. Material-dependent effects on the Casimir interaction are evaluated by comparing Au and Ag plates. The maximum thickness is derived where U_Casimir > U_bending for distances in the range of 0.1-1 micrometers. Results show curvature reversal occurs for nanomembranes with nanoscale thicknesses at the studied distances. Silver nanomembranes tolerate greater thickness than gold nanomembranes due to material-dependent properties. Comparison between NTLO-corrected PFA and perturbative PFA confirms the accuracy of the NTLO approach. The Casimir arc-to-plate geometry in MEMS enables Casimir-based actuation, enhances devices reliability, and prevents stiction. These findings provide thickness constraints for MEMS design and performance, accounting for the Casimir force.
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https://arxiv.org/abs/2512.22352
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41c08144c10024e20021523447c9b4f55c732e8640d27c714023fcc25961ce0e
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2026-01-13T00:00:00-05:00
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Local Scale Invariance in Quantum Theory: A Non-Hermitian Pilot-Wave Formulation
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arXiv:2601.03567v2 Announce Type: replace-cross Abstract: We show that Weyl's abandoned idea of local scale invariance has a natural realization at the quantum level in pilot-wave (deBroglie-Bohm) theory. We obtain the Weyl covariant derivative by complexifying the electromagnetic gauge coupling parameter. The resultant non-hermiticity has a natural interpretation in terms of local scale invariance in pilot-wave theory. The conserved current density is modified from $|\psi|^2$ to the local scale invariant, trajectory-dependent ratio $|\psi|^2/ \mathbf{1}^2[\mathcal{C}]$, where $\mathbf 1[\mathcal C]$ is a scale factor that depends on the pilot-wave trajectory $\mathcal C$ in configuration space. Our approach is general, and we implement it for the Schr\"odinger, Pauli, and Dirac equations coupled to an external electromagnetic field. We also implement it in quantum field theory for the case of a quantized axion field interacting with a quantized electromagnetic field. We discuss the equilibrium probability density and show that the corresponding trajectories are unique.
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https://arxiv.org/abs/2601.03567
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763d2c05fec2330d5ae3a08c3dd5d1d8bd5bfa545c442989e5cc8f076a28819f
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2026-01-13T00:00:00-05:00
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Self-Organized Criticality from Protected Mean-Field Dynamics: Loop Stability and Internal Renormalization in Reflective Neural Systems
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arXiv:2601.04450v2 Announce Type: replace-cross Abstract: The reflective homeostatic dynamics provides a minimal mechanism for self-organized criticality in neural systems. Starting from a reduced stochastic description, we demonstrate within the MSRJD field-theoretic framework that fluctuation effects do not destabilize the critical manifold. Instead, loop corrections are dynamically regularized by homeostatic curvature, yielding a protected mean-field critical surface that remains marginally stable under coarse-graining. Beyond robustness, we show that response-driven structural adaptation generates intrinsic parameter flows that attract the system toward this surface without external fine tuning. Together, these results unify loop renormalization and adaptive response in a single framework and establish a concrete route to autonomous criticality in reentrant neural dynamics.
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https://arxiv.org/abs/2601.04450
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f36b1df459edf9cd7569c74bd9489b2442740271767af66df93d4eb99fa1437a
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2026-01-13T00:00:00-05:00
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Crystal Generation using the Fully Differentiable Pipeline and Latent Space Optimization
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arXiv:2601.04606v2 Announce Type: replace-cross Abstract: We present a materials generation framework that couples a symmetry-conditioned variational autoencoder (CVAE) with a differentiable SO(3) power spectrum objective to steer candidates toward a specified local environment under the crystallographic constraints. In particular, we implement a fully differentiable pipeline to enable batch-wise optimization on both direct and latent crystallographic representations. Using the GPU acceleration, this implementation achieves about fivefold speed compared to our previous CPU workflow, while yielding comparable outcomes. In addition, we introduce the optimization strategy that alternatively performs optimization on the direct and latent crystal representations. This dual-level relaxation approach can effectively escape local minima defined by different objective gradients, thus increasing the success rate of generating complex structures satisfying the target local environments. This framework can be extended to systems consisting of multi-components and multi-environments, providing a scalable route to generate material structures with the target local environment.
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https://arxiv.org/abs/2601.04606
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3410e3c9230d7b11c771dece429961beac27cc8fe5a22775fc118c93905b137e
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2026-01-13T00:00:00-05:00
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Vorticity-Crystalline Order Coupling in Supersolids: Excitations and Re-entrant Phases
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arXiv:2601.05846v2 Announce Type: replace-cross Abstract: Rotation is a natural tool in ultracold gases to break time-reversal symmetry, yet its impact on the collective excitations of supersolids remains largely unexplored. We show theoretically that tuning the rotation frequency, rather than the interparticle interactions, can trigger the superfluid-to-supersolid transition in Bose-Einstein condensates (dBECs). Computing excitation spectra in the presence of vortices and persistent currents, we uncover a vortex-driven de-softening mechanism whereby quantized vorticity elevates the gapless Goldstone mode to a finite-energy roton, restoring superfluidity. This effect results in re-entrant supersolid phases as a function of rotation frequency, revealing a fundamental coupling between topological defects and crystalline order.
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https://arxiv.org/abs/2601.05846
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37dd75f455599b214361af1faeb848a6cc85ee2593957d78a839db31005a0077
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2026-01-13T00:00:00-05:00
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A $3\times3$ linear $q$-difference system with $E_8^{(1)}$-symmetry
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arXiv:2601.06070v1 Announce Type: new Abstract: We present a linear $q$-difference equation of rank $3$, which admits the affine Weyl group symmetry of type $E_8^{(1)}$. We further compare this equation with Moriyama-Yamada's quantum curve which has $W(E_8^{(1)})$-symmetry. The symmetry of our equation is provided by the $q$-middle convolution, defined by Sakai-Yamaguchi and reformulated by Arai-Takemura. In this paper, we provide a reconstruction of the $q$-middle convolution via a $q$-Okubo type equation.
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https://arxiv.org/abs/2601.06070
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6baf66f9f94380a085e05d42e50ad13909a80fee8e5b131acfcd703067fb4425
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2026-01-13T00:00:00-05:00
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Port--Hamiltonian Diffusion Models: A Control-Theoretic Perspective on Generative Modeling
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arXiv:2601.06071v1 Announce Type: new Abstract: Diffusion models have recently achieved remarkable success in generative modeling, yet they are commonly formulated as black-box stochastic systems with limited interpretability and few structural guarantees. In this paper, we establish a control-theoretic foundation for diffusion models by embedding them within the port--Hamiltonian (PH) systems framework. We show that the score function can be interpreted as the gradient of a learnable Hamiltonian energy, allowing both the forward and reverse diffusion processes to be formulated as structured PH dynamics. The reverse-time generative process is further interpreted as a feedback-controlled PH system, where dissipation plays a fundamental role in stabilizing sampling dynamics. This formulation yields intrinsic stability guarantees that are independent of score estimation accuracy. A simple analytical example illustrates the proposed framework.
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https://arxiv.org/abs/2601.06071
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d2706dfc905d76fd2ebc8f21826cfe4993332a31a854f676c63ef50356994ca9
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2026-01-13T00:00:00-05:00
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A Characterization of Quadrics Among Affine Hyperspheres by Section-Centroid Location
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arXiv:2601.06107v1 Announce Type: new Abstract: A theorem of Meyer and Reisner characterizes ellipsoids by the collinearity of centroids of parallel sections: if $\Omega\subset\mathbb{R}^{n+1}$ is a convex body such that for every $n$-dimensional subspace $M\subset\mathbb{R}^{n+1}$ the centroids of the sections $(x+M)\cap \Omega$ are collinear, then $\Omega$ is an ellipsoid. We study natural extensions of this centroid-collinearity condition to unbounded convex sets. In particular, we show that among affine hyperspheres, precisely the ellipsoids, paraboloids and one sheet of a two-sheeted hyperboloid satisfy this property. We also identify additional assumptions under which any convex hypersurface with this property must necessarily be a quadric.
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https://arxiv.org/abs/2601.06107
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595a24818f3809153cd57d65d498cc2d83409cafcfa2852b32ed7c72bde6f4c7
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2026-01-13T00:00:00-05:00
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Quantitative local inversion and a minimax depth barrier for canonical systems from fixed-height Weyl-Schur seam data
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arXiv:2601.06128v1 Announce Type: new Abstract: Fix $\Lambda>0$ and consider trace-normed $2\times 2$ canonical systems $\begin{pmatrix}0&-1\\1&0\end{pmatrix}Y'(s)=zH(s)Y(s)$ on $[0,\Lambda]$, extended by the free tail $H(s)=\frac12 I$ for $s\ge\Lambda$. The Weyl-Titchmarsh coefficient $m_{H,\Lambda}$ and its Cayley transform $v$ encode the spectral measure. We study the finite sampling operator (the seam map) that sends $H$ (or a finite-dimensional parameter $\theta$) to the values $v(z_k)$ at points $z_k=x_k+i\eta$ with $\eta>0$, $k=1,\dots,M$. For piecewise-constant (block) Hamiltonians, with block length $\ell=\Lambda/N$, we analyze stability and optimal sampling. We prove three types of results. (1) The free tail forces the transfer matrix at $\Lambda$ to have entire entries of exponential type at most $\Lambda$; consequently $m_{H,\Lambda}$ and $v$ extend meromorphically to $\mathbb{C}$, showing that the free-tail class is a strict spectral subclass. (2) Fixed-height data is intrinsically smoothed: $\operatorname{Im} m(\cdot+i\eta)$ is the Poisson extension of the spectral measure, yielding an exponential minimax obstruction to recovering oscillatory spectral features from finitely many noisy samples. (3) Linearization at the free Hamiltonian identifies seam sampling with an evaluation operator in a Paley-Wiener/de Branges space, so stability is governed by kernel frame bounds; shifted equispaced designs give optimal tight frames and explicit conditioning estimates. Finally, a perturbation formula expresses $Dm$ as an $L^2([0,\Lambda];\mathbb{R}^2,H)$ pairing with $\Delta H$, linking depth attenuation to decay of Weyl solutions.
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https://arxiv.org/abs/2601.06128
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b43478a07e8a8f059a5372136defb0c42999c3c063a1583fd10fecc0f45c71a1
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2026-01-13T00:00:00-05:00
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Derivative for Functions $f : G \to H$, Where $G$ Is a Metric Divisible Group
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arXiv:2601.06130v1 Announce Type: new Abstract: In this paper, a derivative for functions $f : G \to H$, where $G$ is any metric divisible group and $H$ is a metric Abelian group with a group metric, is defined. Basic differentiation theorems are stated and demonstrated. In particular, we obtain the Chain Rule
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https://arxiv.org/abs/2601.06130
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c4cc61b3a97ae9c5412fcaa1adf0cad225272b52a27141f33252732e572e001c
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2026-01-13T00:00:00-05:00
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Algebraic Classification of All 880 Fourth-Order Magic Squares and the Discovery of Complete Alternating Magic Squares
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arXiv:2601.06131v1 Announce Type: new Abstract: In this paper, we introduce a newly defined algebraic invariant for square matrices termed the \emph{Alternating Power Difference (APD)}. The APD is defined as the signed sum of the powers of diagonal sums along permutations of the symmetric group, distinguishing between even and odd permutations. It serves as a measure of the broken even-odd symmetry inherent in a matrix through higher-order moments. We applied this invariant to all 880 essentially different normal $4\times4$ magic squares (excluding symmetries) and defined the \emph{First Appearance Degree} $m_1$ as the minimum power at which the APD first becomes non-zero. Through an exhaustive computational search, we found that these magic squares are categorized into three clearly separated classes: $m_1=3$ (240 squares), $m_1=4$ (624 squares), and $m_1=\infty$ (16 squares). In particular, the case $m_1=\infty$ identifies exceptionally rare magic squares for which the APD vanishes at all degrees. We refer to these as \emph{Complete Alternating Magic Squares} and demonstrate that they possess a strong algebraic symmetry undetectable by conventional geometric classifications or link-line patterns. Furthermore, we reveal that the APD-based classification refines the classical link-line classification based on complementary sum pairs, showing that each geometric type is clearly distinguished by its first appearance degree. All results in this paper are based on exhaustive computations and are fully reproducible. Our findings suggest that the APD is an effective new invariant for detecting hidden algebraic structures in magic squares and related combinatorial matrices.
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https://arxiv.org/abs/2601.06131
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62d13bf5f3e252bca455f9eb96eb83dd7495607a0c217d68602a6e8fdbfd31f8
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2026-01-13T00:00:00-05:00
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The Geometric Origin of the Cayley-Hamilton Theorem: A Constructive Proof via Dimensional Syzygy
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arXiv:2601.06136v1 Announce Type: new Abstract: We demonstrate that the Cayley-Hamilton theorem is a derived consequence of a more fundamental dimensional constraint: the syzygy formed by the tensor product of two Levi-Civita symbols, which vanishes identically in m-dimensional space. By shifting perspective from the tensor A to the isotropic operators that induce A's invariants through contraction, we reveal that the Cayley-Hamilton identity emerges when this vanishing operator acts on the m-fold tensor product of A. The intrinsic tensorial form of the theorem--invariant coefficients multiplying tensor powers--is inherited from the contraction structure rather than imposed ad hoc. We provide explicit verification for two-dimensional space and a dimension-independent proof using Laplace expansion combined with Newton-Girard identities. This framework clarifies why the theorem's structure depends on ambient dimension and suggests extensions to higher-order tensors where classical characteristic polynomial methods fail.
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https://arxiv.org/abs/2601.06136
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65843a31cebe875c12ce587b797820addb3cce01f03f04050cb2eb6a204ac5ea
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2026-01-13T00:00:00-05:00
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New ideas to the design of algorithms based on derivatives
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arXiv:2601.06146v1 Announce Type: new Abstract: This article proposes new perspectives for developing derivative based numerical algorithms, supported by the introduction of a generalized derivative operators. It demonstrates that these operators have the potential to enhance and extend existing derivativebased numerical methods. To this end, two iterative derivative driven methods are examined and refined: the Newton Raphson method and the Gradient method. For both approaches, generalized derivatives are introduced with the goal of reducing the number of iterations required for convergence. These modifications are presented through geometric interpretations of the proposed constructions, which clearly illustrate their convergenceaccelerating properties. The concluding remarks emphasize the significant opportunity to advance and refine numerical algorithms through the use of generalized derivatives.
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https://arxiv.org/abs/2601.06146
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2b7e4e9df3944886faf5aa122018652bca0e196079f3602eb45d14c6978160f7
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2026-01-13T00:00:00-05:00
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Analyze the Density of Words over Morphism $\{a,b\}$
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arXiv:2601.06150v1 Announce Type: new Abstract: In this paper, we analyze the density of the Fibonacci word and its derived forms by examining the morphisms associated with each. It offers a comparative analysis of the density of Fibonacci numbers alongside other words derived from Fibonacci word. Fibonacci words over the alphabet $\{a,b\}$, we define a novel \emph{power} operation that yields a formal linear combination in the free abelian group generated by all finite words.
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https://arxiv.org/abs/2601.06150
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45e097eadcb464b4ad1abc68d2e87ec1e7a12db0fcdf17e2bbdd0858941db9d8
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2026-01-13T00:00:00-05:00
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A 920-block explicit construction guaranteeing a triple intersection with every 6-subset of [60]
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arXiv:2601.06179v1 Announce Type: new Abstract: We present an explicit family $\mathcal{B}$ of $920$ subsets of size $6$ of $[60]=\{1,\dots,60\}$ with the property that every $6$-subset $S\subset[60]$ intersects at least one block $B\in\mathcal{B}$ in at least three elements, i.e.\ $|S\cap B|\ge 3$. The construction is purely combinatorial, based on a partition of the ground set into pairs and a pigeonhole argument. We also record a simple counting lower bound and discuss how different partitions of the ten base blocks affect the emergence of triple intersections.
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https://arxiv.org/abs/2601.06179
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6578bb095b7aa658df862cb5008ecff5d8949544a1582c5ebdb080015b7159a1
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2026-01-13T00:00:00-05:00
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Consciousness in a Higher Categorical Context
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arXiv:2601.06192v1 Announce Type: new Abstract: We provide two representations of the Segal category $\mathcal{X}$ modeling natural phenomena, the first one being based on the concept of micro-reversibility, producing a long sequence $\Sigma$ of categories as a resolution of $\mathcal{X}$, the second one providing graded categories cofibered in groupoids over the categories of $\Sigma$, using the concept of consciousness as impetus. We show those two representations are dual to each other.
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https://arxiv.org/abs/2601.06192
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ad222b67c8f810c2626082454ca4b8f4cd4c64275ab1278d62a52eab4c0da408
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2026-01-13T00:00:00-05:00
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An Extension of the Collatz Conjecture modulo $2^p+2^q$
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arXiv:2601.06208v1 Announce Type: new Abstract: In this paper, we will introduce an extension to the Collatz's conjecture. This conjecture may be seen as a general conjecture that unifies the Collatz one together with many other similar conjectures. For instance, we propose our new conjecture modulo $10$ which may be stated as follows. Starting from any positive integer, if it is a multiple of $10$ then divide it by 10, otherwise, multiply it by $12$, add $8$ times its last digit and divide the result by $10$. Repeat the process infinitely. Regardless the starting number, the process eventually reaches $4$ after a finite number of iterations. The genaral conjecture studied here will encompasse the classical Collatz conjecture togher with our proposed one modulo $10$.
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https://arxiv.org/abs/2601.06208
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d43b8e274cecf645f98929ded545a7403d5917000ef77a9266016e0356502b6b
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2026-01-13T00:00:00-05:00
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Applications of an identity of Bat{\i}r
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arXiv:2601.06210v1 Announce Type: new Abstract: Based on an interesting identity of Bat{\i}r we derive new identities for double sums involving famous number sequences. We also prove some double sum identities for binomial transform pairs.
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https://arxiv.org/abs/2601.06210
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8af563c7472004e708d7283a6974d3368de183ea2b95f0b41ba168feaabc232e
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2026-01-13T00:00:00-05:00
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Realising all countable groups as quasi-isometry groups
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arXiv:2601.06261v1 Announce Type: new Abstract: Given any countable group $G$, we construct uncountably many quasi-isometry classes of proper geodesic metric spaces with quasi-isometry group isomorphic to $G$. Moreover, if the group $G$ is a hyperbolic group, the spaces we construct are hyperbolic metric spaces. We make use of a rigidity phenomenon for quasi-isometries exhibited by many symmetric spaces, called strong quasi-isometric rigidity. Our method involves the construction of new examples of strongly quasi-isometrically rigid spaces, arising as graphs of strongly quasi-isometrically rigid rank-one symmetric spaces.
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https://arxiv.org/abs/2601.06261
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ae61309c10e39123ca9dcaed66bc4938f6c0b2c8cddf627ada37b5e611620bf6
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2026-01-13T00:00:00-05:00
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A characterisation of probabilistic metrizability for approach spaces
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arXiv:2601.06269v1 Announce Type: new Abstract: Characterisations of metrizable topological spaces or metrizable uniform spaces are well known. A natural counterpart to being metrizable for topological spaces can be expressed in terms of probabilistic metrizability for approach spaces. The notion of a probabilistic metrizable approach space is based on a well known concrete functor $\Gamma$, as introduced in [9], from the category of probabilistic metric spaces with respect to a continuous arbitrary t-norm to the category of approach spaces. A characterization of those probabilistic metrizable approach spaces is still missing and in the first part of this paper we solve this problem. A natural counterpart to being metrizable for uniform spaces can be expressed in terms of probabilistic metrizability for uniform gauge spaces. In the second part of the paper we start from another concrete functor $\Lambda$, as described in [7], on the category of probabilistic metric spaces with respect to a continuous t-norm to the category of uniform gauge spaces. In a similar way as for the functor $\Gamma$ we obtain a characterisation of probabilistic metrizability of uniform gauge spaces. The last section of the paper is devoted to an isomorphic description of the category of probabilistic metric spaces. This problem is not new. Previous attempts in providing isomorphic descriptions of the category of probabilistic metric spaces worked with collections of (pseudo)metrics. These attempts were only formulated in restricted cases. Our isomorphic description is in terms of objects that are sets endowed with a collection of distances, where the distances involved do not satisfy the triangle inequality but fulfil a mixed triangle condition instead.
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https://arxiv.org/abs/2601.06269
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32523a520f12a1ef0e48e40195e8ff76003084b4de1ce616eafacc990a7c33c1
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2026-01-13T00:00:00-05:00
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Eigenvalues of $p$-adic random matrices
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arXiv:2601.06283v1 Announce Type: new Abstract: We develop the basic theory of eigenvalues of $p$-adic random matrices, analogous to the classical theory for random matrices over $\mathbb{R}$ and $\mathbb{C}$. Such eigenvalue statistics were proposed as a model for the zeroes of $p$-adic $L$-functions by Ellenberg-Jain-Venkatesh, who computed the limiting distribution of the number of eigenvalues in a unit disc. We compute the full joint distribution of the $n$ eigenvalues of an $n \times n$ matrix with Haar distribution, obtaining Coulomb gas type formulas as in the archimedean case, with Vandermonde terms leading to eigenvalue repulsion. From these Coulomb gas density functions we derive asymptotics of eigenvalue statistics as $n \to \infty$. These include exact computations, such as a closed form $$\rho(x,y) = 1 - \theta_3(-\sqrt{p};||x-y||^2/p)$$ for the limiting pair correlation of eigenvalues in $\mathbb{Z}_p$, and similar results in quadratic extensions. Such formulas yield concrete numerical predictions on zeroes of $p$-adic $L$-functions. For eigenvalues in arbitrary extensions of $\mathbb{Q}_p$ we also give precise estimates on their pair-repulsion and expected number of eigenvalues in each extension. Finally, we compute the asymptotic probability that all eigenvalues lie in $\mathbb{Z}_p$. Our proofs combine results from several distinct areas: $p$-adic orbital integrals, roots of random $p$-adic polynomials, the Sawin-Wood moment method for random modules, and Markov chains associated with measures on integer partitions.
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https://arxiv.org/abs/2601.06283
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80bccd9e494dea462bf5e83f33286c968eae65868510ed36bba24da57d2e1aa8
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2026-01-13T00:00:00-05:00
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The discrete second moment of mixed derivatives of the Riemann zeta function
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arXiv:2601.06292v1 Announce Type: new Abstract: We establish the full asymptotic for the discrete second moment of the Riemann zeta function of mixed derivatives evaluated at the zeta zeros, providing both unconditional and conditional error terms. This was first studied by Gonek, where only the leading order asymptotic was given, later extended by Conrey--Snaith and Milinovich to include the lower order terms for the first derivative. We extend the case of the first derivative to all derivatives.
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https://arxiv.org/abs/2601.06292
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d497718285c0decc5c153e044b60e7d7e953b1dab119a839fa576183af033b5d
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2026-01-13T00:00:00-05:00
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Some minimum topological spaces, and vector lattices
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arXiv:2601.06310v1 Announce Type: new Abstract: We investigate the existence of compact Hausdorff spaces $X$ that are minimum with respect to $cX=K$ for some fixed covering operator $c$ and compact Hausdorff space $K$ with $cK=K$. Then, using the Yosida representation theorem, we show how that situation relates to the existence of Archimedean vector lattices $A$ with distinguished strong unit that are minimum with respect to $hA=H$ for some fixed hull operator $h$ and vector lattice $H$ with $hH=H$. Among others, we obtain answers for $c=g$ (the Gleason covering operator), $c=qF$ (the quasi-$F$ covering operator), $h = u$ (the uniform completion operator), and $h=e$ (the essential completion operator).
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https://arxiv.org/abs/2601.06310
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b1463532cf1942a5b5c45f9488305bc207ba89c2a634c5a8ae177579e83af570
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2026-01-13T00:00:00-05:00
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Estimation of the intercept parameter in integrated Galton-Watson processes
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arXiv:2601.06317v1 Announce Type: new Abstract: We study estimation of the intercept parameter in an integrated Galton-Watson process, a basic building-block for many count-valued time series models. In this unit root setting, the ordinary least squares estimator is inconsistent, whereas an existing weighted least squares (WLS) estimator is consistent only in the case where the process is transient, a condition that depends on the unknown intercept parameter . We propose an alternative WLS estimator based on the new weight function of $1/t$, and show that it is consistent regardless of whether the process is transient or null recurrent, with a convergence rate of $\sqrt{\ln n}$.
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https://arxiv.org/abs/2601.06317
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0a93fb98e78235204bdef4a69bdaf127acccfd11c73ece822aaa634b034ae8e2
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2026-01-13T00:00:00-05:00
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Multi-fidelity constraints in blackbox optimization
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arXiv:2601.06321v1 Announce Type: new Abstract: This work studies constrained blackbox optimization problems that cannot be solved in reasonable time due to prohibitive computational costs. This challenge is especially prevalent in industrial applications, where blackbox evaluations are costly. However, constraints can be evaluated at various fidelities at a lower computational cost. More specifically, this work targets situations in which the infeasibility of each individual constraint can be detected at lower fidelities, and where a large discrete number of fidelities are available. Moreover, highly discontinuous problems which may fail to evaluate are considered, such that direct search methods are preferred to model-based ones. To this effect, the Interruptible Direct Search (IDS) and the Dynamic Interruptible Direct Search (DIDS) algorithms are proposed to leverage feasibility assessments from various fidelity levels to avoid high cost evaluations. The results show highly increased performances from NOMAD when it is paired with IDS or DIDS.
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https://arxiv.org/abs/2601.06321
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aa19c9c3da54d8a79a22044d94e7d3a71aff64310eba799f4711021dbf29966e
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2026-01-13T00:00:00-05:00
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Uniformly affine actions on Banach spaces: growth of cocycles
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arXiv:2601.06322v1 Announce Type: new Abstract: We investigate growth properties of cocycles with values in uniformly bounded representations on super-reflexive Banach spaces; this includes $L^p$-spaces for $1<\infty$ as well as Hilbert spaces. We then study the generalized Hilbert compression of cocycles arising in this setting for the Property (T) groups $\mathrm{Sp}(n,1)$, $n\ge 2$, and establish the existence of uniformly Lipschitz affine actions with optimal growth.
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https://arxiv.org/abs/2601.06322
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9f231fa9530dbac76685fdd71ca44fab82af28da6258e1904c84dbdfaac8ced0
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2026-01-13T00:00:00-05:00
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Estimating the Evolution of Solution Norms in Vector Delay Nonlinear Systems: Stability and Boundedness
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arXiv:2601.06324v1 Announce Type: new Abstract: Existing methods rarely capture the temporal evolution of solution norms in vector nonlinear DDEs with variable delays and coefficients, often leading to overly conservative boundedness and stability criteria. We develop a framework that constructs scalar counterparts of vector DDEs whose solutions upper-bound the evolution of the original solution norms when the corresponding history functions are matched. This reduction enables boundedness and stability assessment of vector DDEs through the dynamics of their scalar counterparts, using straightforward simulations or simplified analytical reasoning. New boundedness and stability criteria and the estimates of the radii of balls containing history functions that yield bounded or stable solutions for the original vector systems were derived and validated through representative simulations.
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https://arxiv.org/abs/2601.06324
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419d1ce55f52a9250f6c6d39b3de575646d6dbb1e75832754586dc84f6a883a9
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2026-01-13T00:00:00-05:00
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Bilateral Solution Bounds and Successive Estimation of Boundedness and Stability Regions for Vector Delay Nonlinear Time-Varying Systems
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arXiv:2601.06330v1 Announce Type: new Abstract: Stability and boundedness analysis for vector nonlinear systems with variable delays and coefficients remains challenging due to the conservatism of existing methods. Moreover, estimates of the transient behavior of solution norms remain insufficiently developed. This paper presents an approach to estimate the temporal evolution of solution norms and applies it to the analysis of boundedness and stability of vector nonlinear systems with variable delays and coefficients. The method is based on a novel scheme for successive approximations of the original solutions, complemented by the estimates of the corresponding residual norms. This leads to the construction of a scalar nonlinear delay equation whose solutions provide upper bounds for the evolution of residual norms. As a result, bilateral bounds on the original solution norms are obtained, yielding effective boundedness and stability criteria and enabling estimation of the associated regions. Simulations demonstrate that the proposed approximations rapidly approach the reference boundaries of the regions of interest as the iteration count increases. Moreover, the bilateral bounds progressively approach each other and the norm of the reference solution when the initial function remains within the considered regions.
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https://arxiv.org/abs/2601.06330
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1d2e0f1b41449b427afe39cde5f072aef7f8744670a40b0b5d9b5b499cee1890
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2026-01-13T00:00:00-05:00
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On invariant subalgebras when the ISR property fails
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arXiv:2601.06350v1 Announce Type: new Abstract: We classify all $G$-invariant von Neumann subalgebras in $L(G)$ for $G=\mathbb{Z}^2\rtimes SL_2(\mathbb{Z})$. This is the first result on classifying $G$-invariant von Neumann subalgebras in $L(G)$ for i.c.c. groups $G$ without the invariant von Neumann subalgebras rigidity property (ISR property for short) as introduced in Amrutam-Jiang's work. As a corollary, we show that $L(\mathbb{Z}^2\rtimes \{\pm I_2\})$ is the unique maximal Haagerup $G$-invariant von Neumann subalgebra in $L(G)$, where $I_2$ denotes the identity matrix in $SL_2(\mathbb{Z})$.
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https://arxiv.org/abs/2601.06350
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3dbdc1ca041105d8863892ba70c493e74cf6ad9db6add76b7289da64b726d4f0
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2026-01-13T00:00:00-05:00
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Classification of Invariant Subalgebras in a class of factors with property (T)
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arXiv:2601.06353v1 Announce Type: new Abstract: Let $n\geq 2$ and $G_n=\mathbb{Z}^n\rtimes SL_n(\mathbb{Z})$. We classify all $G_n$-invariant von Neumann subalgebras in $L(G_n)$. For $n=2$, this gives an alternative proof of the previous result of Jiang-Liu. For $n\geq 3$, this gives the first class of property (T) groups without the invariant subalgebras rigidity property but invariant subalgebras in the corresponding group factors can still be classified. As a corollary, $L(G_n)$ admits a unique maximal Haagerup $G_n$-invariant von Neumann subalgebra.
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https://arxiv.org/abs/2601.06353
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df06d2a8f028e16b81db41713e105de65663ecede9ed2500c7adcf2a3a8f74c4
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2026-01-13T00:00:00-05:00
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Uniform hypergraphs of girth $6$ and $8$ from generalized polygons
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arXiv:2601.06374v1 Announce Type: new Abstract: Let $ex_r(N,g)$ be the maximum number of edges in an $r$-uni\-form hypergraph on $N$ vertices with girth at least $g$. We are interested in the asymptotic behavior of this value when $N$ is increasing but parameters $g\in\{6,8\}$ and $r\geq3$ are fixed. It is shown that for some positive constants $c$ and $d$, any integer $r\geq3$ and all sufficiently large integers $N$ the inequalities $ex_r(N,6)\geq N^{\frac{11}{8}-\frac{c}{\sqrt{\log N}}}$ and $ex_r(N,8)\geq N^{\frac{11}{9}-\frac{d}{\sqrt{\log N}}}$ hold.
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https://arxiv.org/abs/2601.06374
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0ade046fcfcb642fa6f1ada1b828f269fd69d7f12531488d2cb80859caf8e06b
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2026-01-13T00:00:00-05:00
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Toricness and smoothness criteria for spherical varieties
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arXiv:2601.06376v1 Announce Type: new Abstract: We prove equivalent numerical conditions for a complete spherical variety to admit a toric structure, and for the smoothness of an arbitrary spherical variety along any given G-orbit. The conditions are in terms of spherical skeletons, a coarse ''subset'' of the Luna-Vust data of a spherical variety. Our smoothness criterion improves upon classical criteria by removing the dependency on external reference tables.
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https://arxiv.org/abs/2601.06376
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a28459dc566daa0b467ded060d9e034a9f5764f6f899ce70b840e559f35e735e
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2026-01-13T00:00:00-05:00
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Cr\'onica de un contraejemplo
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arXiv:2601.06379v1 Announce Type: new Abstract: In the 1960s, John Nash proposed a method to resolve singularities. Five decades of encouraging results could not prevent an unexpected ending: the method does not work in general. In this note (written in Spanish), we tell the story of the rise and fall of the Nash blowup.
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https://arxiv.org/abs/2601.06379
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7e53ee722c39d4ffb02889364de7f5535254c7c47d1a0e3df204868f1cf0c90c
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2026-01-13T00:00:00-05:00
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Well-posedness of state-dependent rank-based interacting systems
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arXiv:2601.06383v1 Announce Type: new Abstract: We study the existence and uniqueness of rank-based interacting systems of stochastic differential equations. These systems can be seen as modifications with state-dependent coefficients of the Atlas model in mathematical finance. The coefficients of the underlying SDEs are possibly discontinuous. We first establish strong well-posedness for a planar system with rank-dependent drift coefficients, and non-rank-dependent and non-uniformly elliptic diffusion coefficients. We then state weak well-posedness for two classes of high-dimensional rank-based interacting SDEs with elliptic diffusion coefficients. Finally, we address the positivity of solutions in the case where the diffusion coefficients vanish at zero.
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https://arxiv.org/abs/2601.06383
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2f1735bed0a6a833bb7f588c8506bee0fd859379a5b9e21d7d5c0935e36f167a
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2026-01-13T00:00:00-05:00
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Integration of branched rough paths
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arXiv:2601.06399v1 Announce Type: new Abstract: When the one-form is $Lip\left(\gamma-1\right) $ with $\gamma >p\geq 1$, we construct the integral of a branched $p$-rough path, which defines another branched $p$-rough path. We derive a quantitative bound for this integral and prove that it depends continuously on the driving branched rough path in rough path metric. Moreover, we prove that the first level branched rough integral coincides with a first level integral of the associated $\Pi$-rough path.
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https://arxiv.org/abs/2601.06399
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75218325786470c354a56e93843759d3ffa8a33e5eb98a9d99a0a0dbfcad5adc
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2026-01-13T00:00:00-05:00
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Dynamics for a viscoelastic beam equation with past history and nonlocal boundary dissipation
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arXiv:2601.06414v1 Announce Type: new Abstract: This article aims to study the long-time dynamics of the linear viscoelastic plate equation $\displaystyle{u_{tt}+\Delta^2 u-\int_{\tau}^tg(t-s)\Delta^2u(s)ds=0}$ subject to nonlinear and nonlocal boundary conditions. This model, with $\tau=0$, was first considered by Cavalcanti (Discrete Contin. Dyn. Syst., 8(3), 675-695, 2002), where results of global existence and uniform decay rates of energy have been established. In this work, by taking $\tau=-\infty$, and considering the autonomous equivalent problem we prove that the dynamical system $(\mathcal{H},S_t)$ generated by the weak solutions has a compact global attractor $\mathfrak{A}$ (in the topology of the weak phase space $\mathcal{H}$), which in subcritical case has finite dimension and smoothness. Furthermore, when the force follows the {\it Hook Law}, we prove that $(\mathcal{H},S_t)$ possesses a (generalized) fractal exponential attractor $\mathfrak{A}_{\exp}$ with finite dimension in a space $\widetilde{\mathcal{H}}\supset\mathcal{H}$.
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https://arxiv.org/abs/2601.06414
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7eff80e72592d2dfc5e1dee010653ce203f6551b3abf3d74089cef4bc3cf756d
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2026-01-13T00:00:00-05:00
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Isoperimetric estimates in the product of small and large volume manifolds
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arXiv:2601.06421v1 Announce Type: new Abstract: Let $(M^m,g)$, $(N^n,h)$ be closed Riemannian manifolds, $m,n\geq 2$, with concave isoperimetric profiles and volumes $V_M$, $V_N$ respectively. We consider a one parameter family of product manifolds of the same volume, $(X,G_{\lambda})=(M^m\times N^n,\lambda^{2n}g+ \lambda^{-2m}h)$, $\lambda>0$, and estimate a lower bound for their isoperimetric profile for big $\lambda$. In particular, we show that for $\alpha \in (\frac{3}{4},1)$ and $v_0 \in (0, V_MV_N)$, there is some $\lambda_{0}>0$, such that for $\lambda>\lambda_0$, we can bound the isoperimetric profile of $(X,G_{\lambda})$: $$ \alpha^{4} f_{M,\lambda}(v_0) \leq I_{(X,G_{\lambda})}(v_0)\leq f_{M,\lambda}(v_0)$$ where $f_{M,\lambda}(v)= \lambda^{-n} V_N I_{(M,g)}(\frac{v}{V_N})$ and $ I_{(M,g)}$ is the isoperimetric profile of $(M,g)$. Moreover if $(M,g)=(S^m,g_0)$, the $m-$sphere with the round metric, in this setting, we show that some regions of the type ${ D^{\lambda}(r)\times N_{\lambda} }$, are actual isoperimetric regions in $ (S^m\times N^n,\lambda^{2n}g_0+ \lambda^{-2m}h)$ when $\lambda$ is big enough; being $D^{\lambda}(r)$ a disk on $(S^m,\lambda^{2n}g_0)$ and $N_{\lambda}=(N, \lambda^{-2m}h)$.
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https://arxiv.org/abs/2601.06421
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33a1f53be789583324272aee14df2b3e8c898ce63fa9e3875ba8f5e87ecb087e
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2026-01-13T00:00:00-05:00
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Regions surrounded by cylinders of real algebraic manifolds and natural decompositions
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arXiv:2601.06454v1 Announce Type: new Abstract: The author has been interested in regions surrounded by cylinders of real algebraic hypersurfaces and their shapes and polynomials associated to them. Here, we formulate and investigate natural decompositions into such cylinders of real algebraic hypersurfaces. Especially, intersections of these cylinders of real algebraic hypersurfaces, which give important information on regions, are investigated via singularity theory. This is a kind of natural problems on real geometry. This also comes from construction of explicit real algebraic maps onto explicit regions in real affine spaces on real algebraic manifolds. More generally, we are interested in difficulty in explicit construction of real algebraic objects, where existence and approximation has been well-known, since pioneering studies by Nash and Tognoli, in the latter half of 20th century. This also comes from interest in singularity theory of differentiable, smooth or real algebraic functions and maps, especially, explicit construction.
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https://arxiv.org/abs/2601.06454
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91ac1df61be4a96c4ad14e93a8b9970a6815dd6ef677067e0b38c706a2bbbcd1
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2026-01-13T00:00:00-05:00
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A Note on Pseudofinite W*-Probability Spaces
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arXiv:2601.06455v1 Announce Type: new Abstract: We introduce pseudofinite W*-probability spaces. These are W*-probability spaces that are elementarily equivalent to Ocneanu ultraproducts of finite-dimensional von Neumann algebras equipped with arbitrary faithful normal states. We are particularly interested in the case where these finite-dimensional von Neumann algebras are full matrix algebras: the pseudofinite factors. We show that these are indeed factors. We see as a consequence that pseudofinite factors are never of type $\mathrm{III}_0$. Mimicking the construction of the Powers factors, we give explicit families of examples of matrix algebra ultraproducts that are $\mathrm{III}_\lambda$ factors for $\lambda \in (0,1]$. We show that these examples share their universal theories with the corresponding Powers factor and thus have uncomputable universal theories. Finally, we show that pseudofinite factors are full. This generalizes a theorem of Farah-Hart-Sherman which shows that pseudofinite tracial factors do not have property $\Gamma$. It has the consequence that hyperfinite factors of type $\mathrm{III}$ (the Powers factors) are never pseudofinite. Our proofs combine operator algebraic insights with routine continuous logic syntactic arguments: using \L os' theorem to prove that certain sentences which are true for all matrix algebras are inherited by their ultraproducts.
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https://arxiv.org/abs/2601.06455
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800992c1b16577494d537cfdb97b0bf2179a128e10bf26335d12d16d34022413
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2026-01-13T00:00:00-05:00
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Non-Linear Generalization of the DLR Equations: $q$-Specifications and $q$-Equilibrium Measures
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arXiv:2601.06470v1 Announce Type: new Abstract: We introduce a {\it non-linear} generalization of the classical Dobrushin-Lanford-Ruelle (DLR) framework by developing the concept of a $q$-specification and the associated $q$-equilibrium measures. These objects arise naturally from a family of non-linear $q$-stochastic operators acting on the space of probability measures. A $q$-equilibrium measure is characterized as a fixed point of such operators, providing a non-linear analogue of the Gibbs equilibrium in the sense of DLR. We establish general conditions ensuring the existence and uniqueness of $q$-equilibrium measures and demonstrate how quasilocality plays a decisive role in their construction. Moreover, we exhibit examples of $q$-specifications with an empty set of $q$-equilibrium measures. We characterize the set of $q$-equilibrium measures by studying the dynamical systems generated by a class of $q$-stochastic operators. As a concrete application, we show that for the one-dimensional Ising model at sufficiently low temperatures, multiple $q$-equilibrium measures may exist, even though the classical Gibbs measure remains unique. Our results reveal that the $q$-specification formalism extends the DLR theory from linear to non-linear settings and opens a new direction in the study of Gibbs measures and equilibrium states of physical systems.
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https://arxiv.org/abs/2601.06470
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5b34fe2985166c22b5c4944877a28d88f73f526bd27633dbf8d9f383a6001600
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2026-01-13T00:00:00-05:00
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Herzog ideals and $F$-singularities
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arXiv:2601.06476v1 Announce Type: new Abstract: In this paper we study the connection between Herzog ideals (i.e., ideals with a squarefree Gr\"obner degeneration) and $F$-singularities. More precisely, we show that, in positive characteristic, homogeneous Herzog ideals define $F$-anti-nilpotent rings, and we inquire, in characteristic 0, on a surprising relationship between being Herzog ideals after a change of coordinates and defining rings of dense open $F$-pure type.
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https://arxiv.org/abs/2601.06476
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d94d8a957963885290a5f34474025340430480a8b2264b49642fd7e5694177ef
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2026-01-13T00:00:00-05:00
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Plastic limit of a viscoplastic Burgers equation -- A toy model for sea-ice dynamics
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arXiv:2601.06489v1 Announce Type: new Abstract: We study the plastic Burgers equation in one space dimension, i.e., the Burgers equation featuring an additional term formally given by the p-Laplacian with p=1, or rather, by the multivalued subdifferential of the total variation functional. Our study highlights that the interplay of the advection term with the stresses given by the multivalued 1-Laplacian is a crucial feature of this model. Eventhough it is an interesting model in itsef, it can also be regarded as a one-dimensional version of the momentum balance of Hibler's model for sea-ice dynamics. Therein, the stress tensor is given by a term with similar properties as the 1-Laplacian in order to account for plastic effects of the ice. For our analysis we start out from a viscoplastic Burgers equation, i.e., a suitably regularized version of the plastic Burgers equation with a small regularization parameter $\varepsilon>0$. For the viscoplastic Burgers equation, we construct a global BV-solution. In the singular limit $\varepsilon\to0$ we deduce the existence of a BV-solution for the plastic Burgers equation. In addition we show that the term arising as the limit of the regularized stresses is indeed related to an element of the subdifferential of the total variation functional.
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https://arxiv.org/abs/2601.06489
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356a6388f11e1b75f00b24f019e5cb91bbf718916a716fd6f0d3b9f162f1a578
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2026-01-13T00:00:00-05:00
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Shadowing, chain transitive sets with nonempty interior and attractor boundaries
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arXiv:2601.06523v1 Announce Type: new Abstract: We examine certain phenomena in $C^1$-dynamics from a viewpoint of shadowing and improve a known result on hyperbolic sets. We also review a result on the stability of attractor boundaries from the same viewpoint and derive several additional results.
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https://arxiv.org/abs/2601.06523
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c929b6785575a951fcc5696bbe29068a16d7e0028323e0e26c3da96c09af784c
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2026-01-13T00:00:00-05:00
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Geometric structures arising from the deformation of groups of Heisenberg type
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arXiv:2601.06526v1 Announce Type: new Abstract: Motivated by the desire of finding a geometric interpretation to the Yamabe equation on groups of Heisenberg type, we define a geometric structure on manifolds modelled locally on these groups, which we call contact structure of Heisenberg type. In the case of the Heisenberg group is equivalent to contact Riemannian manifolds. We define a natural connection on these structures, we compute the formula for the conformal change of scalar curvature, and introduce the Yamabe problem for these manifods.
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https://arxiv.org/abs/2601.06526
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0166dabd6a4f01c6eae365e38d34c9682486dc72de493371e2920dddfebaa827
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2026-01-13T00:00:00-05:00
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Hurwitz spaces and Inverse Galois Theory
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arXiv:2601.06532v1 Announce Type: new Abstract: Hurwitz spaces which parametrize branched covers of the line play a prominent role in inverse Galois theory. This paper surveys fifty years of works in this direction with emphasis on recent advances. Based on the Riemann-Hurwitz theory of covers, the geometric and arithmetic setup is first reviewed, followed by the semi-modern developments of the 1990--2010 period: large fields, compactification, descent theory, modular towers. The second half of the paper highlights more recent achievements that have reshaped the arithmetic of Hurwitz spaces, notably via the systematic study of the ring of components. These include the construction of components defined over ${\mathbb Q}$, and the Ellenberg-Venkatesh-Westerland approach to rational points over finite fields, applied to the Cohen-Lenstra heuristics and the Malle conjecture over function fields ${\mathbb F}_q(T)$.
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https://arxiv.org/abs/2601.06532
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46995461ccba623cb3181d04fd6ef105e461f0aab1c1611df4476b1f340a6c1e
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2026-01-13T00:00:00-05:00
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Exponential dichotomy and $(L^p,L^q)$-admissibility
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arXiv:2601.06534v1 Announce Type: new Abstract: We consider the notion of an exponential dichotomy with respect to a family of norms for an evolutionary family in a Banach space, and we characterize it by the admissibility of the pair $(L^p,L^q)$ for $p,q \in [1,\infty]$ with $p\ge q$. We then use this characterization to establish the robustness of an exponentially dichotomic evolutionary family with respect to a family of norms.
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https://arxiv.org/abs/2601.06534
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fd79f252f623a35cedbca3dfb1393a657ac2fa47d66f3b19d97602585cfc8fae
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2026-01-13T00:00:00-05:00
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Splitting of Liftings in Product Spaces II
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arXiv:2601.06538v1 Announce Type: new Abstract: Let $(X, \mfA,P)$ and $(Y, \mfB,Q)$ be two probability spaces, $R$ be their skew product on the product $\sigma$-algebra $\mfA\otimes\mfB$ and $\{(\mfA_y,S_y)\colon y\in{Y}\}$ be a $Q$-disintegration of $R$. Then let $\mfA\dd\mfB$ be the $\sigma$-algebra generated $\mfA\otimes\mfB$ and by the family $\mcM:=\{E\subset{X\times{Y}}\colon \exists\;N\in\mfB_0\;\forall\;y\notin{N}\;\wh{S_y}(E^y)=0\}$ and $\wh{R_{\dd}}$ be the extension of $R$ such that $\mcM$ becomes the family of $\wh{R_*}$-zero sets ($\wh{S_y}$ is the completion of $S_y$ and $\mfB_0=\{B\in\mfB: Q(B)=0\}$). We prove that there exist a lifting $\pi$ on $\mcL^{\infty}(\wh{R_{\dd}})$ and liftings $\sigma_y$ on $\mcL^{\infty}(\wh{S_y})$ , $y\in Y$, such that \[ [\pi(f)]^y= \sigma_y\Bigl([\pi(f)]^y\Bigr) \qquad\mbox{for every} \quad y\in Y\quad\mbox{and every}\quad f\in\mcL^{\infty}(\wh{R_{\dd}}). \] In case of a separable $P$ and in case when $R\ll{P}\times{Q}$ a characterization of stochastic processes possessing an equivalent measurable version is presented. The theorem is a generalization and correction of \cite[Theorem 3.8]{mu25}.
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https://arxiv.org/abs/2601.06538
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24d2782730fcb5fe1cd924b8bdc4f8aaaaa2ef0df40b1153c874e80fd2af6ec9
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2026-01-13T00:00:00-05:00
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A generalization of $q$-deformation of graphic arrangements to simplicial complexes
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arXiv:2601.06546v1 Announce Type: new Abstract: The purpose of this thesis is to introduce two new kinds of hyperplane arrangements, inspired by the graphic arrangements and $q$-deformations of graphic arrangements. In this thesis, the author extends the definition of $q$-deformation to simplicial complexes, with the conjecture by Nian, Tsujie, Uchiumi and Yoshinaga. The author also investigates a special case called graphic monomial arrangement, including the characteristic polynomials and freeness with a further extension to fields with primitive roots.
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https://arxiv.org/abs/2601.06546
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f5a06eedd7bd2bb0577e2ea9e6c197e0c3cc02315b7279c8bcaeafb936ea2935
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2026-01-13T00:00:00-05:00
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Homology of degenerate real projective quadrics
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arXiv:2601.06548v1 Announce Type: new Abstract: Homology of non degenerate real projective quadrics was studied by Steenrod and Tucker. We Compute the rational and the $\mathbb{Z}/2\mathbb{Z}$ homology of degenerate real projective quadrics. This allows to determine the integer homology of these quadrics.
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https://arxiv.org/abs/2601.06548
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3098557abc74a94e7ae8586670e7d3d06eac4005c481bd33fe129c31a913e506
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2026-01-13T00:00:00-05:00
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Cone Conditions for the Curvature Operator of the Second Kind on Einstein Manifolds
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arXiv:2601.06556v1 Announce Type: new Abstract: In this note, we study Einstein manifolds whose curvature operator of the second kind $\mathring{R}$ satisfies the cone condition \[ \alpha^{-1}\big(\sum_{i=1}^{[\alpha]} \lambda_i+ (\alpha - [\alpha] ) \lambda_{[\alpha] + 1} \big) \ge -\theta \bar{\lambda} \] for some real number $\alpha \in [1, (n+2)(n-1)/2)$. Here $[\alpha] :=\max\{ m \in \mathbb{Z}: m \leq \alpha\}$, $\theta>-1$ and $\lambda_1 \le \cdots \le \lambda_{(n+2)(n-1)/2}$ are the eigenvalues of $\mathring{R}$ and $\bar{\lambda}$ is their average. The main result states that any closed Einstein manifold of dimension $n \ge 4$ with $\mathring{R}$ satisfies the cone condition is flat or a round sphere. These results generalize recent works corresponding to $\alpha \in \mathbb Z_+$ of the authors \cite{CW24-1,CW25-2} and Fu-Lu \cite{FL25}.
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https://arxiv.org/abs/2601.06556
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c7b25dfc3e3e1b2c270e5123206c3308aa27191c2a33f6e24353561b36f65ea4
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2026-01-13T00:00:00-05:00
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Path Types in Algebraic Type Theory
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arXiv:2601.06567v1 Announce Type: new Abstract: A new approach to the semantics of identity types in intensional Martin-L\"of type theory is proposed, assuming only a category with finite limits and an interval. The specification of \emph{extensional} identity types in the original presentation of natural models paralleled that of the other type formers $\Sigma$ and $\Pi$, but the treatment of the \emph{intensional} case there was less uniform. It was later reformulated to an account based on polynomials; here a further improvement in the style of the other type formers is achieved by employing an interval, in order to give a single pullback specification of a model with \emph{path types}. The interval is also used to specify a (Hurewicz) fibration structure on the universe of the model. It is shown that the combination of these two conditions suffices to model the intensional identity rules, assuming only finite limits. The addition of an interval also relates the current treatment to that of cubical type theory.
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https://arxiv.org/abs/2601.06567
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f7c21dd59eeaa040510420fe123c81423a7e53b2ebce04748566275ecb5143e6
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2026-01-13T00:00:00-05:00
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Commutativity criteria for prime rings with involution via pairs of endomorphisms
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arXiv:2601.06576v1 Announce Type: new Abstract: The aim of this article is to investigate central-valued identities involving pairs of endomorphisms on prime rings equipped with an involution of the second kind. Extending the recent contributions of Mir et al. (2020) and Boua et al. (2024), we establish several new commutativity criteria for such rings in the presence of two distinct nontrivial endomorphisms. Our approach provides a unified technique that covers multiple classes of $\ast$-identities and yields generalizations of earlier single-endomorphism results. Moreover, explicit counterexamples are constructed to demonstrate the necessity of the hypotheses on primeness and on the nature of the involution.
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https://arxiv.org/abs/2601.06576
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dee9c2e04f301ec3c6010392981d162230a898c74effd8722a1ea3a57fd66187
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2026-01-13T00:00:00-05:00
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Quiver presentations for band algebras are defined over the integers
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arXiv:2601.06587v1 Announce Type: new Abstract: A band is a semigroup in which each element is idempotent. In recent years, there has been a lot of activity on the representation theory of the subclass of left regular bands due to connections to Markov chains associated to hyperplane arrangements, oriented matroids, matroids and CAT(0) cube complexes. We prove here that the integral semigroup algebra of a band is isomorphic to the integral path algebra of a quiver modulo an admissible ideal. This leads to a uniform bound quiver presentation for band algebras over all fields. Also, we answer a question of Margolis, Saliola and Steinberg by proving that the integral semigroup algebra of a CW left regular band is isomorphic to the quotient of the integral path algebra of the Hasse diagram of its support semilattice modulo the ideal generated by the sum of all paths of length two. This includes, for example, hyperplane face semigroup algebras.
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https://arxiv.org/abs/2601.06587
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7de31075be580bd656b271af38a220e048cca2ca0087af0925f8c8dfc1529cab
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2026-01-13T00:00:00-05:00
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FlagAlgebraToolbox: Flag Algebra Computations in SageMath
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arXiv:2601.06590v1 Announce Type: new Abstract: We introduce FlagAlgebraToolbox, an extension of SageMath capable of automating flag algebra calculations and optimizations. FlagAlgebraToolbox has a simple interface, can handle a wide range of combinatorial theories, can numerically optimize extremal combinatorial problems and round the results to produce exact proofs. We present the core concepts used in the toolbox, with example workflows.
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https://arxiv.org/abs/2601.06590
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fc1f1005061d7a954a3a04b373ecaa67bd39d8be7c4d85b8a088c6077d4f6041
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2026-01-13T00:00:00-05:00
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G\"odel-Dummett and $\mathsf{BD_2}$: Linearity and Depth-Two Branching in Kripke Semantics
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arXiv:2601.06593v1 Announce Type: new Abstract: We study the semantic relationship between G\"odel-Dummett logic $\mathsf{GL}$ and bounded-depth-2 logic $\mathsf{BD_2}$, two well-known intermediate logics. While $\mathsf{GL}$ imposes linearity on Kripke frames, $\mathsf{BD_2}$ bounds their depth to two. We prove these logics are incomparable (neither contains the other) through minimal frame conditions. Notably, their combination $\mathsf{GL+BD_2}$ collapses to the logic of one or two world frames, bringing it remarkably close to classical logic. This illustrates how controlling breadth and depth in intuitionistic semantics leads to mutually exclusive structural constraints. Finally, we give a conceptual and philosophical interpretation of the previous results. This is an extended abstract of work in progress. Comments and suggestions welcome at: vicent.navarro@ub.edu
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https://arxiv.org/abs/2601.06593
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8ca87dd1ecd5507b2acc54e80fbdff3eba6f070585a44cd063b26f9a6fab116b
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2026-01-13T00:00:00-05:00
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Radial measures of pseudo-cones
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arXiv:2601.06594v1 Announce Type: new Abstract: We consider $C$-pseudo-cones, that is, closed convex sets $K \subset{\mathbb R}^n$ with $o\notin K\subset C$, for which $C$ is the recession cone. Here $C$ is a given closed convex cone in ${\mathbb R}^n$, pointed and with nonempty interior. We define a class of measures for such pseudo-cones and show how they can be interpreted as derivative measures. For a subclass of these measures, namely for dual curvature measures with negative indices, we solve a Minkowski type existence problem.
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https://arxiv.org/abs/2601.06594
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11f16fe54cd46b0b23d1f452c0b073d1587f91efbc85d2e047c98a47a5f9a8e4
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2026-01-13T00:00:00-05:00
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Minimality of free-boundary axial hyperplanes in high dimensional circular cones via calibration
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arXiv:2601.06601v1 Announce Type: new Abstract: Consider an $(n+1)$-dimensional circular cone. Using a calibration argument, we prove that if $n \geq 4$ and the aperture of the cone is sufficiently large, the intersection of the cone with an axial hyperplane is area-minimizing with respect to free-boundary variations inside the cone.
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https://arxiv.org/abs/2601.06601
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662e7a86dd0ed3e87b1ad0e0cf6fa4ff776c5c5fb0b4d8d70a45abf5cf788984
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2026-01-13T00:00:00-05:00
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Families of Toeplitz operators, family index and deformation quantization
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arXiv:2601.06619v1 Announce Type: new Abstract: Given a contact fibration, we construct smooth families of Szeg\"o projections on the fibers. This allows us to define smooth families of Toeplitz operators. We apply these operators to construct a deformation quantization of prequantizable symplectic fibrations, recovering a result of Kravchenko in an analytic way. We also derive a family index for these families of Toeplitz operators. To this end, we generalize an index formula of Baum and van Erp to families.
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https://arxiv.org/abs/2601.06619
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32309a0b7458a91af1adc37d8157cc4bcd1ffceb3c48648154226adf404132b0
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2026-01-13T00:00:00-05:00
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Inexact DC Algorithms in Hilbert Spaces with Applications to PDE-Constrained Optimization
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arXiv:2601.06622v1 Announce Type: new Abstract: In this paper, we design and apply novel inexact adaptive algorithms to deal with minimizing difference-of-convex (DC) functions in Hilbert spaces. We first introduce I-ADCA, an inexact adaptive counterpart of the well-recognized DCA (difference-of-convex algorithm), that allows inexact subgradient evaluations and inexact solutions to convex subproblems while still guarantees global convergence to stationary points. Under a Polyak-Lojasiewicz type property for DC objectives, we obtain explicit convergence rates for the proposed algorithm. Our main application addresses elliptic optimal control problems with control constraints and nonconvex $L^{1-2}$ sparsity-enhanced regularizers admitting a DC decomposition. Employing I-ADCA and appropriate versions of finite element discretization leads us to an efficient procedure for solving such problems with establishing its well-posedness and error bound estimates confirmed by numerical experiments.
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https://arxiv.org/abs/2601.06622
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5d27b9de2a09259c58ca41bf23dce0c8167d0ceab51ef341665f85a31b2b98d5
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2026-01-13T00:00:00-05:00
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Necessary and sufficient condition for existence at resonance for eigenvalues of multiplicity two
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arXiv:2601.06623v1 Announce Type: new Abstract: We establish necessary and sufficient condition for existence of solutions for a class of semilinear Dirichlet problems with the linear part at resonance at eigenvalues of multiplicity two. The result is applied to give a condition for unboundness of all solutions of the corresponding semilinear heat equation.
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https://arxiv.org/abs/2601.06623
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a007f651bd0d1a6b7215ca4d1528d4dc0056b59f7a88f78a2f2edfc9eb4f4320
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2026-01-13T00:00:00-05:00
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Khintchin conjecture and related topics
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arXiv:2601.06626v1 Announce Type: new Abstract: Motivated by Khintchin's 1923 conjecture, refuted by Marstrand in 1970, we study the Khintchin class of functions associated to a given increasing sequence of integers. When the Khintchin class contains L^p(\mathbb{T}), we call the sequence a L^p-Khintchin sequence. We establish basic properties of Khintchin sequences, provide several constructions, and propose open problems for further research. We also initiate the study of Khintchin sequences of group endomorphisms on compact abelian groups. Under a Fourier-tightness assumption, we show that ergodicity (respectively, weakly mixing or strongly mixing) of a skew product of endomorphisms is equivalent to the corresponding property of the base system, supporting the idea that typical fiber orbits in such skew products should form Khintchin sequences.
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https://arxiv.org/abs/2601.06626
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Academic Papers
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9b08eb02a1918d5a82f02a55fae94fa55e39c2ad4fd81a21b06eaa157e5c2db1
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2026-01-13T00:00:00-05:00
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Sharp Bohr-Rogosinski radii for Schwarz functions and Euler operators in C^n
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arXiv:2601.06630v1 Announce Type: new Abstract: This paper is devoted to the investigation of multidimensional analogues of refined Bohr-type inequalities for bounded holomorphic mappings on the unit polydisc $\mathbb{D}^n$. We establish a sharp extension of the classical Bohr inequality, proving that the Bohr radius remains $R_n = 1/(3n)$ for the family of holomorphic functions bounded by unity in the multivariate setting. Further, we provide a definitive resolution to the Bohr-Rogosinski phenomenon in several complex variables by determining sharp radii for functional power series involving the class of Schwarz functions $\omega_{n,m}\in\mathcal{B}_{n,m}$ and the local modulus $|f(z)|$. By employing the radial (Euler) derivative operator $Df(z) = \sum_{k=1}^{n} z_k \frac{\partial f(z)}{\partial z_k}$, we obtain refined growth estimates for derivatives that generalize well-known univariate results to $\mathbb{C}^n$. Finally, a multidimensional version of the area-based Bohr inequality is established. The optimality of the obtained constants is rigorously verified, demonstrating that all established radii are sharp.
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https://arxiv.org/abs/2601.06630
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eacf77e3694f162de9b9d0c4b6b16de1238b007f7d9520708e44d7f63151e8d1
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2026-01-13T00:00:00-05:00
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Comminution as a Non-Hermitian Quantum Field Theory: Log-Size Jump Generators, Branching Embeddings, and the Airy Solvable Sector
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arXiv:2601.06635v1 Announce Type: new Abstract: Pure-breakage population balance equations (PBEs) give the standard deterministic description of fragmentation and comminution. They predict mean particle size distributions, but they do not determine fluctuations, size-size correlations, or universality under coarse-graining. We develop a field-theoretic framework anchored in the PBE kernel inputs (selection rate and daughter distribution) and compatible with the stochastic Doi-Peliti approach. From homogeneous kernels we derive an exact Markov jump generator in log-size for a mass-weighted (tagged-mass) distribution, with a jump law that is a probability density fixed by the daughter distribution. The generator is generically non-self-adjoint, admits a Lindblad embedding, and has a second-quantized extension. The deterministic PBE appears as the one-body sector, while multi-point correlators encode finite-population fluctuations. We also give a binary-fragmentation embedding whose mean-field limit reproduces the PBE but whose higher correlators capture multiplicative cascade noise. For a linear Airy-type kernel, long-wavelength coarse-graining yields an effective Airy operator as a solvable quadratic sector about a stationary profile, producing explicit mode-sum formulas for equal-time connected two-point correlations. Overall, the framework separates what is fixed by kernel data from what requires additional stochastic modeling and links comminution kernels to universality classes.
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https://arxiv.org/abs/2601.06635
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e29685a50c7f69ce3c27c415b9b649f63972f0ec24298b316947867aec385e20
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2026-01-13T00:00:00-05:00
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On Asymptotic Properties of Certain $B$-Splines in Terms of Theta-like Functions
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arXiv:2601.06643v1 Announce Type: new Abstract: The asymptotic behavior of the Mellin transform of the associated $B$-splines $B_N^*(t) :=t^{-N}B_N(t)$ with special knots in terms of theta-like functions is found. The proof is based on polynomial interpolation of power functions and properties of certain theta-like functions. Pointwise asymptotics of $B_N^*$ and $B_N$ are discussed as well.
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https://arxiv.org/abs/2601.06643
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6066e90d582507c815485a1739dfacc4414002e73b4a0494987f49f99471bd0a
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2026-01-13T00:00:00-05:00
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A finite-termination algorithm for testing copositivity over the positive semidefinite cone
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arXiv:2601.06648v1 Announce Type: new Abstract: This paper proposes an efficient algorithm for testing copositivity of homogeneous polynomials over the positive semidefinite cone. The algorithm is based on a novel matrix optimization reformulation and requires solving a hierarchy of semidefinite programs. Notably, it always terminates in finitely many iterations. If a homogeneous polynomial is copositive over the positive semidefinite cone, the algorithm provides a certificate; otherwise, it returns a vector that refutes copositivity. Building on a similar idea, we further propose an algorithm to test copositivity over the direct product of the positive semidefinite cone and the nonnegative orthant. Preliminary numerical experiments demonstrate the effectiveness of the proposed methods.
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https://arxiv.org/abs/2601.06648
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612fdb62985344cdc8a64080d597528a42b3ec98cd24b42032b9532dd4611121
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2026-01-13T00:00:00-05:00
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Rational surgery exact triangles in Heegaard Floer homology
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arXiv:2601.06654v1 Announce Type: new Abstract: We construct a new family of surgery exact triangles in Heegaard Floer theory over the field with two elements. This family generalizes both Ozsv\'{a}th and Szab\'{o}'s $n$- and $1/n$-surgery exact triangles for positive integers $n$ and the author's recent 2-surgery exact triangle to all positive rational slopes. The construction reduces to a combinatorial problem that involves triangle and quadrilateral counting maps in a genus 1 Heegaard diagram. The main contribution of this paper is solving this combinatorial problem, which is particularly tricky for slopes $r\neq n,1/n$; one key idea is to use an involution that is closely related to the ${\rm Spin}^{c}$ conjugation symmetry.
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https://arxiv.org/abs/2601.06654
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48e1f5cc8ec657e647d04157bf36d20ee2a6f73a30fc901203794b1eecbfdf4b
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2026-01-13T00:00:00-05:00
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Relative Invariants from Moving Frames on an Extended Manifold
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arXiv:2601.06660v1 Announce Type: new Abstract: A constructive modification of the moving frame method is developed in this paper for the construction of relative invariants of regular Lie group actions. Let a relative invariant $I$ of weight $\omega$ transform according to the rule $$ I(g \cdot \boldsymbol x) = \mu(g, \boldsymbol x)^{\omega} I(\boldsymbol x), $$ where $\mu: G \times \mathcal{M} \to \mathbb{R}^\times$ is a scalar multiplier (1-cocycle). It is shown that the cocycle property of $\mu$ is equivalent to the well-definedness of the twisted group action on the extended manifold $\widehat{\mathcal{M}} = \mathcal{M} \times \mathbb{R}^\times$, and that relative invariants on $\mathcal{M}$ are in one-to-one correspondence with absolute invariants of this action on $\widehat{\mathcal{M}}$. The main result is that, given a moving frame, the invariantization of the multiplier is a canonical relative invariant of weight $-1$. This enables the constructive realization of any weight and yields an explicit formula for an arbitrary relative invariant in terms of the fundamental absolute invariants and the invariantized multiplier. Examples are provided to demonstrate the application of the proposed approach for the projective group $PGL(3, \mathbb{R})$.
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https://arxiv.org/abs/2601.06660
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08661244470ef65001e4da5edf010deeacde5d9675fa566b5173d76204c0ad96
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2026-01-13T00:00:00-05:00
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Reduction and classification of higher-order Markov chains for categorical data
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arXiv:2601.06674v1 Announce Type: new Abstract: Categorical time series models are powerful tools for understanding natural phenomena. Most available models can be formulated as special cases of $m$-th order Markov chains, for $m\geq 1$. Despite their broad applicability, theoretical research has largely focused on first-order Markov chains, mainly because many properties of higher-order chains can be analyzed by reducing them to first-order chains on an enlarged alphabet. However, the resulting first-order representation is sparse and possesses a highly structured transition kernel, a feature that has not been fully exploited. In this work, we study finite-alphabet Markov chains with arbitrary memory length and introduce a new reduction framework for their structural classification. We define the skeleton of a transition kernel, an object that captures the intrinsic pattern of transition probability constraints in a higher-order Markov chain. We show that the class structure of a binary matrix associated with the skeleton completely determines the recurrent classes and their periods in the original chain. We also provide an explicit algorithm for efficiently extracting the skeleton, which in many cases yields substantial computational savings. Applications include simple criteria for irreducibility and essential irreducibility of higher-order Markov chains and a concrete illustration based on a 10th-order Markov chain.
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https://arxiv.org/abs/2601.06674
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c96c12bd8faf17781689af58601223cb1eecbae7dfb04216cf7357e03416ebac
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2026-01-13T00:00:00-05:00
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Comparing Two Notions of Coaction Invariance of Ideals in $\mathrm{C}^*$-Algebras
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arXiv:2601.06679v1 Announce Type: new Abstract: Given a coaction $\delta$ of a locally compact group $G$ on a $\mathrm{C}^*$-algebra $A$, we study the relationship between two different forms of coaction invariance of ideals of $A$ and the ideals of the corresponding crossed product $\mathrm{C}^*$-algebra $A \rtimes_{\delta} G$. In particular, we characterize when these two notions of invariance are equivalent.
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https://arxiv.org/abs/2601.06679
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c209963c1735c49a5d99b2fef5711b2a837986b2f1dbad7897a49008da7a35e0
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2026-01-13T00:00:00-05:00
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Amenability constants for unconditional sums of Banach algebras
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arXiv:2601.06680v1 Announce Type: new Abstract: We study Johnson amenability for unconditional direct sums of Banach algebras. Given a family $(A_i)_{i\in I}$ of Banach algebras and a Banach sequence lattice $E$ on~$I$, the $E$-sum $\bigl(\bigoplus_{i\in I} A_i\bigr)_{\!E}$ carries a natural Banach algebra structure via coordinatewise multiplication. Under the hypothesis that $C_E := \sup\{\|\chi_F\|_E : F \subseteq I \text{ finite}\} < \infty$, we prove that this $E$-sum is amenable if and only if the amenability constants of the summands are uniformly bounded, and we establish the two-sided estimate \[ \sup_{i\in I}\operatorname{AM}(A_i) \;\le\; \operatorname{AM}\Bigl(\bigl(\textstyle\bigoplus_{i\in I} A_i\bigr)_{\!E}\Bigr) \;\le\; C_E^2 \sup_{i\in I}\operatorname{AM}(A_i). \] We show that the factor $C_E^2$ is sharp by exhibiting finite-dimensional examples where equality holds. We further prove that finiteness of $C_E$ is necessary whenever infinitely many summands are non-zero and the sum admits a bounded approximate identity. Finally, we investigate weak amenability of $E$-sums. We prove that weak amenability passes to summands, that $E$-sums of commutative weakly amenable algebras are weakly amenable, and contrasting sharply with the Johnson amenability picture that for $1 < p < \infty$, the $\ell_p$-sum of infinitely many copies of a non-commutative weakly amenable algebra fails to be weakly amenable. In the $c_0$-type regime ($C_E < \infty$), we establish a two-sided estimate for weak amenability constants analogous to that for Johnson amenability.
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https://arxiv.org/abs/2601.06680
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52b24038106eafdfd5752a21d298d4841e69def46933a62e536c23d09b76bfda
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2026-01-13T00:00:00-05:00
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Extinction and persistence criteria in non-local Klausmeier model of vegetation dynamics on flat landscapes
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arXiv:2601.06681v1 Announce Type: new Abstract: This paper investigates the dynamics of vegetation patterns in water-limited ecosystems using a generalized Klausmeier model that incorporates non-local plant dispersal within a finite habitat. We establish the well-posedness of the system and provide a rigorous analysis of the conditions required for vegetation survival. Our results identify a critical patch size governed by the trade-off between local growth and boundary losses; habitats smaller than this threshold lead to inevitable extinction. Furthermore, we derive a critical maximal biomass density below which the population collapses to a desert state, regardless of the domain size. We determine stability criteria for stationary solutions and describe the emergence of stable, non-trivial biomass distributions. Numerical experiments comparing sub-Gaussian and super-Gaussian kernels confirm that non-local dispersal mechanisms, particularly those with fat tails, enhance ecosystem resilience by allowing vegetation to persist in smaller, fragmented habitats than predicted by classical local diffusion models.
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https://arxiv.org/abs/2601.06681
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0d6de4887a8112c7b354c9dcb45e3660a909dfc17bd0f2b0c0d01ad1f0b55c3e
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2026-01-13T00:00:00-05:00
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Inverse problem for the divisor of the good Boussinesq equation
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arXiv:2601.06683v1 Announce Type: new Abstract: A third-order operator with periodic coefficients is an L-operator in the Lax pair for the Boussinesq equation on a circle. The projection of the divisor of the Floquet solution poles for this operator coincides with the spectrum of the three-point Dirichlet problem. The sign of the norming constant of the three-point problem determines the sheet of the Riemann surface on which the pole lies. We solve the inverse problem for a third-order operator with three-point Dirichlet conditions when the spectrum and norming constant are known. We construct a mapping from the set of coefficients to the set of spectral data and prove that this mapping is an analytic bijection in the neighborhood of zero.
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https://arxiv.org/abs/2601.06683
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e3dcfc89192fa6954998699439d4427e62b0ab1a871d056c17d50f582a5c0e2b
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2026-01-13T00:00:00-05:00
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On De Concini-Kac forms of quantum groups
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arXiv:2601.06696v1 Announce Type: new Abstract: Quantum groups of semisimple Lie algebras at roots of unity admit several different forms. Among them is the De Concini-Kac form, which is the easiest to define but, perhaps, hardest to study. In this paper, we propose a suitable modification to the De Concini-Kac form, namely the even part algebra, which has some appealing features. Notably, it behaves uniformly with respect to the order of the roots of unity and admits an adjoint action of the Lusztig form. We revisit several results due to De Concini-Kac-Procesi and Tanisaki for the even part algebra. Namely, we give conceptual definitions of the Frobenius and Harish-Chandra centers and describe the entire center in terms of these two subalgebras getting a complete quantum analog of the Veldkamp theorem on the center of the universal enveloping algebras in positive characteristic. We investigate the Azumaya locus of the even part algebra over its center. We also show that the locally finite part of the even part algebra under the adjoint action of the Lusztig form is isomorphic to the reflection equation algebra, which is the quantized coordinate algebra with the product twisted by $R$-matrix. Some results on Lusztig forms at roots of unity are revisited and proved in greater generality including Kempf vanishing theorem and good filtrations on the quantized coordinate algebra.
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https://arxiv.org/abs/2601.06696
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63472845d3e2cb17aa3b80bc87e46cce3a8d51943b0b3f40c0149b1ba78561a1
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2026-01-13T00:00:00-05:00
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On a stochastic Cahn-Hilliard-Brinkman model
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arXiv:2601.06698v1 Announce Type: new Abstract: In this paper, we consider a stochastic version of the Cahn-Hilliard-Brinkman model in a smooth two- or three-dimensional domain with dynamical boundary conditions. The system describes creeping two-phase flows and is basically a coupling of the Brinkman equation for the velocity field that governs the flow through the porous media coupled with convective Cahn-Hilliard equations for the phase field, both with two independent sources of randomness given by general multiplicative-type Wiener noises in the Cahn-Hilliard equations. The existence of a weak solution, both in the probabilistic and PDEs sense, is proved. Our construction of a solution is based on the classical Faedo-Galerkin approximation, the Yosida approximation and uses a compactness method. Our paper is the first attempt to generalize the paper \cite{Colli+Knopf+Schimperna+Signor_2024} to a stochastic setting.
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https://arxiv.org/abs/2601.06698
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4db4506523a4de65cbfdeb69bcfcf08cf09f3a792241c4608f6b274a536e8f8f
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2026-01-13T00:00:00-05:00
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Upper bound for the total mean curvature of spin fill-ins
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arXiv:2601.06713v1 Announce Type: new Abstract: Gromov conjectured that the total mean curvature of the boundary of a compact Riemannian manifold can be estimated from above by a constant depending only on the boundary metric and on a lower bound for the scalar curvature of the fill-in. We prove Gromov's conjecture if the manifolds are spin and the mean curvature is non-negative.
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https://arxiv.org/abs/2601.06713
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50e35a9d3c49f55d1721c2175e03f2a7ac58ffac07a14b7d9e721b565a3d3181
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2026-01-13T00:00:00-05:00
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Reconfiguration of Hamiltonian Cycles in Rectangular Grid Graphs
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arXiv:2601.06731v1 Announce Type: new Abstract: An \textit{\(m \times n\) grid graph} is the induced subgraph of the square lattice whose vertex set consists of all integer grid points \(\{(i,j) : 0 \leq i < m,\ 0 \leq j < n\}\). Let $H$ and $K$ be Hamiltonian cycles in an $m \times n$ grid graph $G$. We study the problem of reconfiguring $H$ into $K$, \textcolor{blue}{\textbullet} where the Hamiltonian cycles are viewed as vertices of a reconfiguration graph \textcolor{blue}{\textbullet}, using a sequence of local transformations called \textit{moves}. A \textit{box} of $G$ is a unit square face. A box with vertices $a, b, c, d$ is \textit{switchable} in $H$ if exactly two of its edges belong to $H$, and these edges are parallel. Given such a box with edges $ab$ and $cd$ in $H$, a \textit{switch move} removes $ab$ and $cd$, and adds $bc$ and $ad$. A \textit{double-switch move} consists of performing two consecutive switch moves. If, after a double-switch move, we obtain a Hamiltonian cycle, we say that the double-switch move is \textit{valid}. We prove that any Hamiltonian cycle $H$ can be transformed into any other Hamiltonian cycle $K$ via a sequence of valid double-switch moves, such that every intermediate graph remains a Hamiltonian cycle. Moreover, assuming $n \geq m$, the number of required moves is bounded by $mn^2$.
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https://arxiv.org/abs/2601.06731
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Academic Papers
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886d4835f562bc44959e0f102ba3078ff033ac90fb3f02552cc68e1bb0fa84a1
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2026-01-13T00:00:00-05:00
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Probabilities of random monomial ideals associated to large graphs
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arXiv:2601.06739v1 Announce Type: new Abstract: Inspired by the Erd\H{o}s R\'enyi model, we propose a new model for freesquare random monomial ideals generated by edges and covers of a graph. This permit us to investigate the conditions of normality for which we obtain asymptotic results. We also elaborate on asymptotic results for other invariants such as the Krull dimension (for which we obtain threshold function), the regularity and the $v$-number.
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https://arxiv.org/abs/2601.06739
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42bc47d61afe9341da149839be8ca7ededa839d63109058afe454cfabbf7145a
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2026-01-13T00:00:00-05:00
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Minimal reduction type in classical cases
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arXiv:2601.06744v1 Announce Type: new Abstract: We prove Yun's minimal reduction conjecture for all classical groups. More precisely, for any topologically nilpotent regular semisimple element $\gamma$, we show that the associated minimal reduction set $\mathrm{RT}_{\mathrm{min}}(\gamma)$ consists of a single nilpotent orbit. This result confirms and extends Yun's earlier work in types A and C, and resolves the remaining cases in types B and D. Moreover, we provide an explicit and effective procedure for determining $\mathrm{RT}_{\mathrm{min}}(\gamma)$.
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https://arxiv.org/abs/2601.06744
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17ff0d0bf4aaa3d168b713d035a10204425d60e6ec04e9b71b4a7d6ea724b407
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2026-01-13T00:00:00-05:00
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Reconfiguration of Hamiltonian Paths and Cycles in Rectangular Grid Graphs
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arXiv:2601.06749v1 Announce Type: new Abstract: \noindent An \textit{\(m \times n\) grid graph} is the induced subgraph of the square lattice whose vertex set consists of all integer grid points \(\{(i,j) : 0 \leq i < m,\ 0 \leq j < n\}\). Let $H$ and $K$ be Hamiltonian cycles in an $m \times n$ grid graph $G$. We study the problem of reconfiguring $H$ into $K$ using a sequence of local transformations called \textit{moves}. A \textit{box} of $G$ is a unit square face. A box with vertices $a, b, c, d$ is \textit{switchable} in $H$ if exactly two of its edges belong to $H$, and these edges are parallel. Given such a box with edges $ab$ and $cd$ in $H$, a \textit{switch move} removes $ab$ and $cd$, and adds $bc$ and $ad$. A \textit{double-switch move} consists of performing two consecutive switch moves. If, after a double-switch move, we obtain a Hamiltonian cycle, we say that the double-switch move is \textit{valid}. We prove that any Hamiltonian cycle $H$ can be transformed into any other Hamiltonian cycle $K$ via a sequence of valid double-switch moves, such that every intermediate graph remains a Hamiltonian cycle. This result extends to Hamiltonian paths. In that case, we also use single-switch moves and a third operation, the \textit{backbite move}, which enables the relocation of the path endpoints.
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https://arxiv.org/abs/2601.06749
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9f465cf309b794408127b307bd3813696dffb823038f6fbd1ee75d5b87bf104e
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2026-01-13T00:00:00-05:00
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New Calabi-Yau Metrics of Taub-NUT Type on C^{N+1}
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arXiv:2601.06756v1 Announce Type: new Abstract: We construct a class of complete non-flat Calabi-Yau metrics on C^{N+1} for every N >= 3, which generalize the Taub-NUT metrics from C^2 and C^3 and whose tangent cone at infinity is R^N. The construction relies on the generalized Gibbons-Hawking ansatz. A key obstacle is that the volume-form defect of the ansatz fails to decay near certain components of the discriminant locus, producing singularities more severe than those encountered in dimension three, we resolve this by a gluing procedure.
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https://arxiv.org/abs/2601.06756
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Academic Papers
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