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1130d6e1bc2fc77bd3fdffd11c0c142fb78fdae706a2390a161f7f203adbdc60
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2026-01-23T00:00:00-05:00
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Can a Quantum Computer Simulate Nuclear Magnetic Resonance Spectra Better than a Classical One?
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arXiv:2508.06448v2 Announce Type: replace-cross Abstract: The simulation of the spectra measured in nuclear magnetic resonance (NMR) spectroscopy experiments is a computationally non-trivial problem which, due to its natural interpretation as a quantum spin problem, maps in a straightforward way to a quantum computer. As such, it represents a problem for which such a device may provide some practical advantage over traditional computing methods. In order to understand the extent to which such problems may indeed provide examples of useful quantum advantage, it is important to understand the limitations of existing classical simulation methods. In this work, we benchmark our own classical solver designed to study such problems. This solver uses a clustering approximation to achieve a resource scaling which is linear in the total number of nuclear spins in a given molecule, for a fixed cluster size. The success of such an approximation would present a stark repudiation to the common claim that such problems require an exponential scaling of resources, the very claim which makes simulating an NMR spectra a candidate for quantum advantage. Our benchmarking results indicate that our approximation performs well throughout, and even somewhat beyond, the more typical experimental regimes. We discuss what implications this may have for future efforts to demonstrate quantum advantage in the context of NMR.
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https://arxiv.org/abs/2508.06448
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72ad602f2985f4745e679108f4de0870aacc775611f6a2d240f590bc164414ba
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2026-01-23T00:00:00-05:00
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TeMFpy: a Python library for converting fermionic mean-field states into tensor networks
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arXiv:2510.05227v2 Announce Type: replace-cross Abstract: We introduce TeMFpy, a Python library for converting fermionic mean-field states to finite or infinite matrix product state (MPS) form. TeMFpy includes new, efficient, and easy-to-understand algorithms for both Slater determinants and Pfaffian states. Together with Gutzwiller projection, these also allow the user to build variational wave functions for various strongly correlated electron systems, such as quantum spin liquids. We present all implemented algorithms in detail and describe how they can be accessed through TeMFpy, including full example workflows. TeMFpy is built on top of TeNPy and, therefore, integrates seamlessly with existing MPS-based algorithms.
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https://arxiv.org/abs/2510.05227
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2c5b19fd59341380e319f64473f83d052c6902f4ed6d01b994bc994acbdef298
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2026-01-23T00:00:00-05:00
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Single-fluid model for rotating annular supersolids and its experimental implications
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arXiv:2510.26753v2 Announce Type: replace-cross Abstract: The famous two-fluid model of finite-temperature superfluids has been recently extended to describe the mixed classical-superfluid dynamics of the newly discovered supersolid phase of matter. We show that for rigidly rotating supersolids one can derive a more appropriate single-fluid model, in which the seemingly classical and superfluid contributions to the motion emerge from a spatially varying phase of the global wavefunction. That allows to design experimental protocols to excite and detect the peculiar rotation dynamics of annular supersolids, including partially quantized supercurrents, in which each atom brings less than $\hbar$ unit of angular momentum. Our results are valid for a more general class of density-modulated superfluids.
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https://arxiv.org/abs/2510.26753
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212a96d7fe1f27ce182fc2dedc0c5e50bfb8e4a7e14d013d61f9f21d7abb22b4
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2026-01-23T00:00:00-05:00
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Emergent clusters in strongly confined systems
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arXiv:2511.00234v2 Announce Type: replace-cross Abstract: Driven suspensions, where energy is input at a particle scale, are a framework for understanding general principles of out-of-equilibrium organization. A large number of simple interacting units can give rise to non-trivial structure and hierarchy. Rotationally driven colloidal particles are a particularly nice model system for exploring this pattern formation, as the dominant interaction between the particles is hydrodynamic. Here, we use experiments and large-scale simulations to explore how strong confinement alters dynamics and emergent structure at the particle scale in these driven suspensions. Surprisingly, we find that large-scale (many times the particle size) density fluctuations emerge as a result of confinement, and that these density fluctuations sensitively depend on the degree of confinement. We extract a characteristic length scale for these fluctuations, demonstrating that the simulations quantitatively reproduce the experimental pattern. Moreover, we show that these density fluctuations are a result of the large-scale recirculating flow generated by the rotating particles inside a sealed chamber. This surprising result shows that even when system boundaries are far away, they can cause qualitative changes to mesoscale structure and ordering.
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https://arxiv.org/abs/2511.00234
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f5eb897f441ca050a88a06207c9c9cf6690eb781ae828763f8710efb3ee86bbd
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2026-01-23T00:00:00-05:00
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Precision Bounds for Characterising Quantum Measurements
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arXiv:2512.20091v2 Announce Type: replace-cross Abstract: Quantum measurements, alongside quantum states and processes, form a cornerstone of quantum information processing. However, unlike states and processes, their efficient characterisation remains relatively unexplored. We resolve this asymmetry by introducing a comprehensive framework for efficient detector estimation that reveals the fundamental limits to extractable parameter information and errors arising in detector analysis - the detector quantum Fisher information. Our development eliminates the need to optimise for the best probe state, while highlighting aspects of detector analysis that fundamentally differ from quantum state estimation. Through proofs, examples and experimental validation, we demonstrate the relevance and robustness of our proposal for current quantum detector technologies. By formalising a dual perspective to state estimation, our framework completes and connects the triad of efficient state, process, and detector tomography, advancing quantum information theory with broader implications for emerging technologies reliant on precisely calibrated measurements.
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https://arxiv.org/abs/2512.20091
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910edae438721d2cd7515fe3819dbd515f6de248352a6febd55d379ac5215f01
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2026-01-23T00:00:00-05:00
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Calibration Method of Spacecraft-Inertial Sensor Center-of-Mass Offset for the Taiji Gravitational Wave Detection Mission under Science Mode
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arXiv:2512.20468v2 Announce Type: replace-cross Abstract: Accurately calibrating the center-of-mass (CoM) offset between the spacecraft (SC) and the inertial sensor test mass (TM) is crucial for space-based gravitational-wave (GW) antennas, such as LISA and Taiji. Current calibration methods require additional spacecraft maneuvers that disrupt science data continuity and inter-satellite links, compromising the coherence of gravitational wave signals. Here, we present a maneuver-free calibration scheme that directly estimates the CoM offset vector using only standard science-mode measurements from inertial sensors, interferometers, and differential wavefront sensors. By embedding the CoM offset induced coupling acceleration as an extended state in a model-based adaptive Kalman filter, we achieve estimation accuracy of 0.01-1.5 mm across all axes with a maximum error below 1%. This approach enables continuous, high-precision calibration during nominal observation runs, ensuring continuous and coherent gravitational wave data collection while maintaining the required precision, and also facilitating advanced DFACS functions such as performance evaluations and fault diagnosis. For LISA-like missions, where data continuity is paramount for detecting faint gravitational wave signals, this method will enhance scientific output and reliability.
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https://arxiv.org/abs/2512.20468
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db8c7b5c5805475cbc5ade84692dadf411e4c70b9a8a5d84af02397688faebd6
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2026-01-23T00:00:00-05:00
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Elasticity without a reference state: continuum mechanics of active tension nets
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arXiv:2601.08968v2 Announce Type: replace-cross Abstract: A constitutive relation between stress and strain relative to a reference state is the basic assumption of elasticity theory. However, in living matter, stress is governed by (motor molecule) activity rather than a constitutive law. What paradigm takes the place of elasticity in this setting? Here, we derive a continuum theory of active mechanics by taking the continuum limit of the Active Tension Network model of 2d epithelia. Instead of a reference state, we start from a prescribed active force configuration, encoded in a Riemannian "tension metric". Intuitively, one expects cells to adjust their positions to achieve force balance by rearranging local sources of active stress. More precisely, the cell positions define an embedding of the tension metric into 2d physical space, which determines the macroscopic physical stress. For free boundaries, tissue adopts a certain intrinsically defined shape, the force-balanced embedding with minimal internal stress. Boundary forces then deform this embedding. The resulting stress transformation yields an effective stress-strain relation. Key elements of elasticity hence emerge from a "stress-only" starting point, explaining how tissue shape can be adiabatically controlled by active stress during morphogenesis. Plastic behavior arises from topological cell rearrangement, which we represent by a continuous reparameterization of the tension metric, providing a principled continuum theory of emergent elasto-plastic flow. To express this physics, we use the mathematics of isothermal coordinates and quasi-conformal maps. An explicit coarse-graining study, described in a companion paper, substantiates the results of our continuum analysis. The present theory elucidates the unconventional mechanics of living tissues and may apply to 2d active and granular materials more generally.
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https://arxiv.org/abs/2601.08968
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aedc1bd5aa83fe48b5ebf73dd3b56a03a3165afc4d0d99cfdf51900f1b461709
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2026-01-23T00:00:00-05:00
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Resolving the band alignment of InAs/InAsSb mid-wave-infrared type-II superlattices
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arXiv:2601.15053v2 Announce Type: replace-cross Abstract: In this work, three InAs/InAs$_{0.65}$Sb$_{0.35}$ superlattices with different periods were investigated using photoluminescence and photoreflectance measurements and their band structure was simulated using a 14 bulk-band kp model. The structures were studied by analyzing the evolution of the spectral features in temperature and excitation power to determine the origin of optical transitions. After identifying which of these are related to the superlattice mini-bands, a rich collection of observed higher-order optical transitions was compared with refractive-index calculations. This procedure was used to adjust the parameters of the theoretical model, namely the bowing parameters of the InAsSb valence band offset and bandgap. It was also shown that the spectroscopy of the higher-order states combined with numerical modeling of the refractive index is a powerful tool for improvement of the material parameters, presenting a new approach to material studies of advanced semiconductor heterostructures.
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https://arxiv.org/abs/2601.15053
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9a84f7e0a37f84626065a01d19173ac3b868436aaa9bf9d9ca865ab35fe90e54
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2026-01-23T00:00:00-05:00
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A new iterative three-point method for solving systems of nonlinear equations
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arXiv:2601.15323v1 Announce Type: new Abstract: A three-point iterative method for solving scalar non-linear equations was selected and then adapted to solve systems of non-linear equations. Subsequently, by applying Taylor's theorem to functions of $\R^{n}$ in $\R^{n}$, it is shown that the new method also has a sixth order of convergence. It is confirmed that the theoretical order of convergence coincides with the computational order of convergence by the numerical solution of two problems. Finally, its computational efficiency is calculated and subsequently compared with that of other three-point methods of fifth and sixth order convergence that also solve systems of non-linear equations.
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https://arxiv.org/abs/2601.15323
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42a37ed08e08f3b5f1273ccc35e85717151abc4da086b53304507d2ec25557e4
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2026-01-23T00:00:00-05:00
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G\"unter Hellwig (1926-2004) -- in memoriam
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arXiv:2601.15329v1 Announce Type: new Abstract: G\"unter Hellwig was the author of influential textbooks on PDEs and differential operators of mathematical physics, an enthusiastic and inspiring teacher to generations of engineers, organiser of PDE conferences at Oberwolfach and a pioneer in index theory.
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https://arxiv.org/abs/2601.15329
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3f722ac8646061e2fcf34147cc20e4cab283b89b856bc8a57661e0f7742ee295
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2026-01-23T00:00:00-05:00
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Conjectures on Sums of Consecutive Primes
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arXiv:2601.15346v1 Announce Type: new Abstract: We study additive properties of consecutive prime numbers and the primality of the sums they generate. For a given prime number $p_n$, we consider the sums \[ S_k(p_n) = p_n + p_{n+1} + \cdots + p_{n+k-1}, \] where $k \ge 3$ is an odd integer. We first formulate an existence conjecture asserting that, for every prime number $p_n$, there exists at least one odd length $k \ge 3$ such that $S_k(p_n)$ is itself a prime number. An exhaustive computational verification covering the first one million prime numbers revealed no counterexamples. We then propose a strengthened conjecture according to which, for every prime number $p_n$, there exist infinitely many odd lengths $k$ such that $S_k(p_n)$ is prime. This strong version is supported by a probabilistic heuristic showing that the series of the corresponding primality probabilities diverges, suggesting that the phenomenon is not exceptional but recurrent. We also analyze the possible modular obstructions, showing that they are local in nature and cannot persist when the length $k$ varies among odd integers. A Diophantine interpretation of the problem is proposed, together with a conceptual comparison with the generalized Goldbach conjecture. Finally, we discuss the role of the Generalized Riemann Hypothesis (GRH) in controlling the distribution of the sums under consideration. These structural, modular, Diophantine, and probabilistic (heuristic) arguments support both conjectures and formalize heuristic theorems of Cram\'er, GRH, and Hardy--Littlewood type explaining the expected absence of counterexamples.
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https://arxiv.org/abs/2601.15346
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d9b4604c92b15bb04f21c90f9336ceba1d61a2f8879f799931530eb2a3764971
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2026-01-23T00:00:00-05:00
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Maximal Green Sequences for Cluster Algebras Associated to Closed Orbifolds
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arXiv:2601.15389v1 Announce Type: new Abstract: It is known that the existence of a maximal green sequence for a quiver associated to surfaces is equivalent to the equality of the cluster algebra and upper cluster algebra generated by the quiver. This paper makes the first steps in investigating this behavior in the generalised case of cluster algebras from orbifolds; determining when such surfaces admit a diagram with a maximal green sequence. Specifically, we will provide a triangulation for the orientable surfaces of genus $n$ with an arbitrary number of orbifold points and arbitrary number of punctures, determine when it has a maximal green sequence, and construct one if it exists.
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https://arxiv.org/abs/2601.15389
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a27ff5d9bec18bccd1e3af0bf0e0d8f63c09403f731c38e5becbb827a61051cf
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2026-01-23T00:00:00-05:00
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Understanding FISTA's weak convergence: A step-by-step introduction to the 2025 milestone
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arXiv:2601.15398v1 Announce Type: new Abstract: Beck and Teboulle's FISTA for finding the minimizer of the sum of two convex functions is one of the most important algorithms of the past decades. While function value convergence of the iterates was known, the actual convergence of the iterates remained elusive until October 2025 when Jang and Ryu, as well as Bo\c{t}, Fadili, and Nguyen proved weak convergence. In this paper, we provide a gentle self-contained introduction to the proof of their remarkable result.
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https://arxiv.org/abs/2601.15398
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f3a27624c7c8c1b9ae4818e7526dbbb0e52fe3bca53a24bc3e96519283a04c0c
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2026-01-23T00:00:00-05:00
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The Geometry of Rough Path Space
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arXiv:2601.15402v1 Announce Type: new Abstract: We describe $H^p(V)$, a subset of $p$-rough path space $\Omega_p(V)$ which is a vector space under an addition operation $\boxplus$ and a scalar multiplication $\odot$. We show that the domain of $\boxplus$ can be extended to $\Omega_p(V)\times H^p(V)$, allowing any $p$-rough path $X$ to be additively perturbed by an $H\in H^p(V)$. We prove associativity $(X\boxplus H)\boxplus \tilde H = X\boxplus (H\boxplus \tilde H)$ and trivial kernel $X\boxplus H = X \Leftrightarrow H = 1$, where $1$ is the additive zero in $(H^p(V),\boxplus,\odot)$. Finally, we show that enlarging $H^p(V)$ to almost rough paths $H^{am,p}(V)$ does not enlarge the set of displacements of a given $X$, i.e. $\{X\boxplus H: H\in H^p(V)\}=\{X\boxplus H: H\in H^{am,p}(V)\}$.
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https://arxiv.org/abs/2601.15402
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b23f170a33cf6157da5e6f6f9f608fd03752c1aa0aaa9ba5a0753338d6290ae0
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2026-01-23T00:00:00-05:00
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F-Purity of Binomial Edge Ideals
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arXiv:2601.15403v1 Announce Type: new Abstract: In 2012, K. Matsuda introduced the class of weakly closed graphs and investigated when binomial edge ideals are F-pure. He proved that weakly closed binomial edge ideals are F-pure whenever the base field has positive characteristic. He conjectured that: (i) when the base field has characteristic two, every F-pure binomial edge ideal comes from a weakly closed graph; and (ii) that every binomial edge ideal is F-pure provided that the characteristic of the residue field is sufficiently large. In this paper, we resolve both of Matsuda's conjectures. We confirm Matsuda's first conjecture, showing that the binomial edge ideal of a graph defines an F-pure quotient in characteristic 2 if and only if the graph is weakly closed. We also show that Matsuda's second conjecture is false in a very strong way by showing that graphs containing asteroidal triples, such as the net, define non-F-pure binomial edge ideals in any positive characteristic. Our results yield a complete classification of F-pure binomial edge ideals of chordal graphs as well as large families of standard graded algebras that are F-injective but neither F-pure nor F-rational in all characteristics.
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https://arxiv.org/abs/2601.15403
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8417f5d11c7748833f40f14443bcd694fb77b30423428281db55545c0bcb0cd8
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2026-01-23T00:00:00-05:00
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Cancellation elements in multiplicative lattices
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arXiv:2601.15405v1 Announce Type: new Abstract: We extend to multiplicative lattices a theorem of Anderson and Roitman characterizing the cancellation ideals of a commutative ring.
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https://arxiv.org/abs/2601.15405
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b904e21f473510965ccb23ebd2d3deedb9ab32c37340a0f9780ceef6868d55f5
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2026-01-23T00:00:00-05:00
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On the diagonal of low bidegree hypersurfaces
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arXiv:2601.15409v1 Announce Type: new Abstract: We study the existence of a decomposition of the diagonal for bidegree hypersurfaces in a product of projective spaces. Using a cycle theoretic degeneration technique due to Lange, Pavic and Schreieder, we develop an inductive procedure that allows one to raise the degree and dimension starting from the quadric surface bundle of Hassett, Pirutka and Tschinkel. Furthermore, we are able to raise the dimension without raising the degree in a special case, showing that a very general $(3,2)$ complete intersection in $\mathbb P^4\times \mathbb P^3$ does not admit a decomposition of the diagonal. As a corollary of these theorems, we show that in a certain range, bidegree hypersurfaces which were previously only known to be stably irrational over fields of characteristic zero by results of Moe, Nicaise and Ottem, are not retract rational over fields of characteristic different from two.
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https://arxiv.org/abs/2601.15409
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b00287ca78f73b54827aaedeb8582759014c0b686645b3aa21ad91f69b99f78b
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2026-01-23T00:00:00-05:00
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What is... hierarchical hyperbolicity?
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arXiv:2601.15410v1 Announce Type: new Abstract: This is a very short introduction to hierarchically hyperbolic spaces and groups. It is aimed at non-experts, including anyone who may encounter a group with some similarities to mapping class groups.
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https://arxiv.org/abs/2601.15410
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b00464cd20c0937fad3f85c11ec359b460b0aa41e0a2f12462735d8ad379d476
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2026-01-23T00:00:00-05:00
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Asymptotic behaviour of coupled random dynamical systems with multiscale aspects
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arXiv:2601.15411v1 Announce Type: new Abstract: We examine a class of stochastic differential inclusions involving multiscale effects designed to solve a class of generalized variational inequalities. This class of problems contains constrained convex non-smooth optimization problems, constrained saddle-point problems and various equilibrium problems in economics and engineering. In order to respect constraints we adopt a penalty approach, introducing an explicit time-dependency into the evolution system. The resulting dynamics are described in terms of a non-autonomous stochastic evolution equation governed by maximally monotone operators in the drift and perturbed by a Brownian motion. We study the asymptotic behavior, as well as finite time convergence rates in terms of gap functions. The condition we use to prove convergence involves a Legendre transform of the function describing the set C, a condition first used by Attouch and Czarnecki (J. Differ. Equations, Vol. 248, Issue 6, 2010) in the context of deterministic evolution equations. We also establish a large deviations principle showing that individual trajectories exhibit exponential concentration around the solution set. Finally we show how our continuous-time approach relates to penalty-regulated algorithms of forward-backward type after performing a suitable Euler-Maruyama discretisation.
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https://arxiv.org/abs/2601.15411
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e414307af54501ef2d2868b47b93288fe4507f588e2d18a4cdfab9abc3e452a1
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2026-01-23T00:00:00-05:00
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Counting point configurations in projective space
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arXiv:2601.15421v1 Announce Type: new Abstract: We investigate the enumerative geometry of point configurations in projective space. We define "projective configuration counts": these enumerate configurations of points in projective space such that certain specified subsets are in fixed relative positions. The $\mathbb{P}^1$ case recovers cross-ratio degrees, which arise naturally in numerous contexts. We establish two main results. The first is a combinatorial upper bound given by the number of weighted transversals of a bipartite graph. The second is a recursion that relates counts associated to projective spaces of different dimensions, by projecting away from a given point. Key inputs include the Gelfand-MacPherson correspondence, the Jacobi-Trudi and Thom-Porteous formulae, and the notion of surplus from matching theory of bipartite graphs.
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https://arxiv.org/abs/2601.15421
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6b69a7d512d10971333ac2c350545271f41d18b90b6757a609624afaf0fed6b1
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2026-01-23T00:00:00-05:00
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Isotropic meta Kazhdan--Lusztig combinatorics I: Ext-quiver presentation for the Hecke category
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arXiv:2601.15426v1 Announce Type: new Abstract: We provide an ${\rm Ext}$-quiver and relations presentation for the basic algebra of the anti-spherical Hecke categories of isotropic Grassmannians, $H_{(D_n, A_{n-1})}$, in terms of cup-cap meta Kazhdan--Lusztig combinatorics and Temperley--Lieb diagrammatics.
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https://arxiv.org/abs/2601.15426
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3966cecb124fb93ff9ece6a4431be1c3c189f560d4375ac9504902a57d9b589b
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2026-01-23T00:00:00-05:00
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A numerical characterization of Dunkl systems
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arXiv:2601.15430v1 Announce Type: new Abstract: We give a numerical characterization of weighted hyperplane arrangements arising from Dunkl systems.
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https://arxiv.org/abs/2601.15430
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62b2ce3e013d691748a7980a2712e91c130098a894744b302335f3e48716c179
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2026-01-23T00:00:00-05:00
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Discrete log-concavity and threshold phenomena for atomic measures
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arXiv:2601.15444v1 Announce Type: new Abstract: We investigate threshold phenomena for random polytopes $K_N=\conv\{X_1,\dots,X_N\}$ generated by i.i.d.\ samples from an atomic law $\mu$. We identify and provide a missing justification in the discrete-hypercube threshold argument of Dyer--F\"uredi--McDiarmid, where the supporting half-space estimate is derived via a smooth (gradient/uniqueness) step that can fail at boundary contact points. We then compare threshold-driving mechanisms in the continuous log-concave setting -- through the Cram\'{e}r transform and Tukey's half-space depth -- with their discrete analogues. Within this framework, we establish a sharp threshold for lattice $p$-balls $\mathbb{Z}^n \cap rB_p^n$. Finally, we present structural counterexamples showing that sharp thresholds need not hold in general discrete log-concave settings.
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https://arxiv.org/abs/2601.15444
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a1c358cb7455d168dc197a5d7dd870a4a2396ea7fd95d8b0071012e037d859cf
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2026-01-23T00:00:00-05:00
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On certain bilinear sums with modular square roots and applications
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arXiv:2601.15448v1 Announce Type: new Abstract: We extend bounds on additive energies of modular square roots by Dunn, Kerr, Shparlinski, Shkredov and Zaharescu and apply these results to obtain bounds on certain bilinear exponential sums with modular square roots. From here, we make partial progress on the large sieve for square moduli.
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https://arxiv.org/abs/2601.15448
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45e601bfa15d1df3458000da6f3bb647989775117ec2d9029052077e10a66d6b
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2026-01-23T00:00:00-05:00
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Variance bounds in product measures without exponential tails
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arXiv:2601.15450v1 Announce Type: new Abstract: We establish analogs of Cheeger's inequality for probability measures with heavy tails. As one of the principal applications, suppose $\lambda > 3$ and define the (Pareto) probability measure $\mu_{\lambda}$ on $[1,\infty)$ by $d\mu_{\lambda}(x) = (\lambda - 1) x^{-\lambda}$. Let $\mu_{\lambda}^n$ denote the product measure of $\mu_{\lambda}$ on $\mathbb{R}^n$. Then, for any $1$-Lipschitz function (with respect to the Euclidean distance) $f : \mathbb{R}^n \to \mathbb{R}$, we obtain the variance bound $\operatorname{Var}_{\mu_{\lambda}^n}(f) \le C(\lambda)\, n^{\frac{2}{\lambda - 1}}$, where $C(\lambda)$ is an explicit constant depending only on $\lambda$. This improves upon the existing bound $\operatorname{Var}_{\mu_{\lambda}^n}(f) = O(n)$ derived from the Efron--Stein inequality. Moreover, this bound is asymptotically tight when considering the $1$-Lipschitz function $f(x) = |x|_{\infty}$ corresponding to the $L^{\infty}$ norm. In probabilistic terms, suppose $X_1, \dots, X_n$ are i.i.d.\ random variables with distribution $\mu_{\lambda}$. Then, for any $1$-Lipschitz function $f$, we have $\operatorname{Var}(f(X_1, \dots, X_n)) \le C'(\lambda)\operatorname{Var}(\max\{X_1, \dots, X_n\}) = \Theta\!\left(n^{\frac{2}{\lambda - 1}}\right)$, where $C'(\lambda)$ is another explicit constant depending only on $\lambda$.
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https://arxiv.org/abs/2601.15450
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1ccbcbd2809ac11e7edc3f57a086e95f0b7240bee02ad669c8f58a1e22fd59bd
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2026-01-23T00:00:00-05:00
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Generalized Ramsey Numbers in the Hypercube
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arXiv:2601.15451v1 Announce Type: new Abstract: We study the generalized Ramsey numbers $f(Q_n, C_{k}, q)$, that is, the minimum number of colors needed to edge-color the hypercube $Q_n$ so that every copy of the cycle $C_{k}$ has at least $q$ colors. Our main result is that for any integers $k,q$ satisfying $k \geq 6$ and $3 \leq q \leq k/2+1$, we have $f(Q_n, C_{k}, q)= o\left( n^{\frac{k/2-1}{k-q+1}} \right).$ We also prove a few other upper and lower bounds in the special cases $k=4$ and $k=6$. This continues the line of research initiated by Faudree, Gy\'arf\'as, Lesniak, and Schelp and Mubayi and Stading who studied the case $k=q$, and by Conder who considered the case $k=6$ and $q=2$.
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https://arxiv.org/abs/2601.15451
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db2200a19aff361e5348098501dcd3a0e8acc4f22f832c25aff04a9538c42b95
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2026-01-23T00:00:00-05:00
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The paper "On the constant in a transference inequality for the vector-valued Fourier transform" revisited
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arXiv:2601.15454v1 Announce Type: new Abstract: The standard proof of the equivalence of Fourier type on \(\mathbb R^d\) and on the torus \(\mathbb T^d\) is usually stated in terms of an implicit constant which can be expressed in terms of the global minimiser of the functions \[f_r(x)=\sum_{m\in\mathbb{Z}}\left|\frac{\sin(\pi(x+m))}{\pi(x+m)}\right|^{2r},\qquad x\in [0,1], \ r\ge 1.\] The aim of this note is to provide a short proof of a result of the authors which states that each \(f_r\) takes a global minimum at the point \(x = \frac12\).
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https://arxiv.org/abs/2601.15454
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01b84ef5e4418d9836cd039e941aae0245cc88e8d6886114673662165a57a0de
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2026-01-23T00:00:00-05:00
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Brauer groups of varieties over local fields of finite characteristic
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arXiv:2601.15461v1 Announce Type: new Abstract: We show that the non-log version of Kato's ramification filtration on the Brauer group of a separated and finite type regular scheme over a positive characteristic local field coincides with the evaluation filtration. This extends a recent result of Bright-Newton to positive characteristics. Among several applications, we extend some results of Ieronymou, Saito-Sato and Kai to positive characteristics.
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https://arxiv.org/abs/2601.15461
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513a80fb2995b47fd796ce905373fabec54b7d29be239ab08bb7399053475ee6
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2026-01-23T00:00:00-05:00
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Determinants of modular Collatz graphs and variants
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arXiv:2601.15463v1 Announce Type: new Abstract: The determinants of modular Collatz graphs and the modular Conway amusical permutation graph are determined, and some interesting number theoretic properties are described.
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https://arxiv.org/abs/2601.15463
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f2912960037a463593d9e630209596698bd50e8bedebc69b592b8330aec7a997
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2026-01-23T00:00:00-05:00
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Gorenstein flat preenvelopes and weakly Ding injective covers
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arXiv:2601.15469v1 Announce Type: new Abstract: We consider a (left) coherent ring R. We prove that if the character module of every Ding injective (left) R-module is Gorenstein flat, then the class of Gorenstein flat (right) R-modules, GF, is preenveloping. We show that this is the case when every injective (left) R-module has finite flat dimension. In particular, GF is preenveloping over any Ding-Chen ring.\\ The proofs use the class of weakly Ding injective (left) R-modules, wDI. We show that, when wDI is closed under extensions, the following statements are equivalent:\\ 1. The character module of every Ding injective left R-module is a Gorenstein flat right R-module.\\ 2. The class of weakly Ding injective left R-modules is closed under direct limits.\\ 3. The class of weakly Ding injective modules is covering.\\ The equivalent statements (1)-(3) imply that GF is preenveloping
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https://arxiv.org/abs/2601.15469
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8e7e4c8f95c7cfae7fa1a1307d6b2f69eb97b801f5a81d7b81d8fe098e40152b
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2026-01-23T00:00:00-05:00
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Folklore in Multi-Objective Optimisation
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arXiv:2601.15499v1 Announce Type: new Abstract: In this paper, we present and prove some results in multi-objective optimisation that are considered folklore. For the most part, proofs for these results exist in special cases, but they are used in more general settings since their proofs can be (largely) transferred. We do this transfer explicitly and try to state the results as generally as possible. In particular, we also aim at providing clean and complete proofs for results where the original papers are not rigorous.
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https://arxiv.org/abs/2601.15499
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1041b6d6675182c8a127c4c5387bf70d825d82b8abb5d0fe601d364f5c951fcd
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2026-01-23T00:00:00-05:00
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On orthogonality graphs of Okubo algebras
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arXiv:2601.15501v1 Announce Type: new Abstract: The orthogonality graph of an Okubo algebra with isotropic norm over an arbitrary field $\mathbb{F}$ is considered. Its connected components are described, and their diameters are computed. It is shown that there exist at most two shortest paths between any pair of vertices, and the conditions under which the shortest path is unique are determined.
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https://arxiv.org/abs/2601.15501
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6a9eab183670f6216624b58c28f12f158f6442d49e53bd36e02e14bec931c04e
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2026-01-23T00:00:00-05:00
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Maps on Surfaces as a Structural Framework for Genus-One Virtual Knot Classification
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arXiv:2601.15512v1 Announce Type: new Abstract: We develop a purely combinatorial framework for the systematic enumeration of knot and link diagrams supported on the thickened torus $T^2\times I$. Using the theory of maps on surfaces, cellular $4$--regular torus projections are encoded by permutation pairs $(\alpha,\sigma)$, and unsensed projection classes are enumerated completely and without duplication via canonical representatives. For a fixed projection, crossing assignments are encoded by bit data, and an immediate Reidemeister~II reduction supported by a bigon face is characterized directly in terms of these bits. The genus-one generalized Kauffman-type bracket is then evaluated as a state sum entirely within the permutation model, without drawing diagrams in a fundamental polygon. The implementation is validated against published genus-one classifications for $N\le 5$ under explicit comparison conventions, with remaining discrepancies explained at the level of global conventions. Beyond the published range, we compute projection and diagram data for crossing numbers up to $N=8$ and provide a public reference implementation together with machine-readable datasets. Via the standard correspondence between virtual knots and knots in thickened surfaces, this yields a canonical and fully reproducible genus-one framework for virtual knot tabulation.
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https://arxiv.org/abs/2601.15512
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d64af6f41772fb4ec72104233fc010f8f072f58c7e496623c0374c6a92ef9a7f
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2026-01-23T00:00:00-05:00
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Colour ratio in Prim's ranking of bipartite graphs
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arXiv:2601.15520v1 Announce Type: new Abstract: We consider a complete bipartite graph of size $n$ endowed with i.i.d. uniform edge weights and run Prim's Algorithm to obtain a ranking of its vertices. Let $\rho^{(n)}_k$ be the proportion of black vertices among the first $k$ vertices in this ranking. We characterise the limit behaviour of $\rho^{(n)}_k$ as both $n$ and $k$ tend to infinity. Our results show that in general the limit of $\rho^{(n)}_k$, when existing, differs from the overall proportion of the black vertices in the graph.
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https://arxiv.org/abs/2601.15520
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048db026c602933e3a9f331aecc7cf0fdfcdcda73a632eb68701fb0c48ab126a
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2026-01-23T00:00:00-05:00
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Length minimization of filling pairs on hyperbolic surfaces
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arXiv:2601.15524v1 Announce Type: new Abstract: A filling pair $(\alpha, \beta)$ of a surface $S_g$ is a pair of simple closed curves in minimal position such that the complement of $\alpha\cup\beta$ in $S_g$ is a disjoint union of topological disks. A filling pair is said to be minimally intersecting if the number of intersections between them, or equivalently, the number of complementary disks, is minimal among all filling pairs of $S_g$. For surfaces of genus $g \geq 3$, minimal filling pairs are well understood, whereas in genus two, such a pair divides the surface into exactly two disks. In this paper, we classify all minimal filling pairs up to the action of the mapping class group in genus two and determine the length of the shortest minimal filling pair.
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https://arxiv.org/abs/2601.15524
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c9fb8bc96f34794c2e90f549a140167d553417f3c3a3f315eb1734d30a044187
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2026-01-23T00:00:00-05:00
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The Frog Model on $\mathbb{Z}$ with Discrete Weibull Lifetimes and Random Parameter $p$
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arXiv:2601.15526v1 Announce Type: new Abstract: We study the frog model on $\mathbb{Z}$ with particle wise discrete Weibull lifetimes. Each particle has an i.i.d. survival parameter $\pi\in(0,1)$; conditionally on $\pi=p$, its lifetime $\Xi$ satisfies \[ P(\Xi\ge k\mid \pi=p)=p^{k^{\gamma}},\qquad k\in\mathbb{N}_0,\gamma>0. \] The law of $\pi$ has right edge density \[ f_\pi(u)\sim(1-u)^{\beta-1},L\big((1-u)^{-1}\big)\qquad (u\uparrow 1), \] with $\beta>0$ and $L$ slowly varying; let $\eta$ denote the common law of the i.i.d. initial occupation numbers $\{\eta_x\}_{x\in\mathbb{Z}}$. The survival parameter distribution strictly extends the Beta family, while the lifetime distribution extends the geometric case. We prove a sharp extinction and survival dichotomy with the $\gamma-$dependent threshold \[ \beta_c:=\frac{1}{2\gamma}. \] If $\beta>\beta_c$ and $E(\eta)<\infty$, the process becomes extinct almost surely; if $\beta<\beta_c$ and $P(\eta=0)<1$, it survives with positive probability. At the boundary $\beta=\beta_c$ we provide explicit criteria in terms of $\limsup/\liminf$ of $L(n^{2\gamma})$. The case $\gamma=1$ (geometric lifetimes) recovers the benchmark $\beta_c=\frac{1}{2}$ and the critical refinements previously obtained for random geometric lifetimes.
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https://arxiv.org/abs/2601.15526
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d2054677ba8720a649bd9298aca723336dd6450f9b4b685a2b4f3af200c80814
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2026-01-23T00:00:00-05:00
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Palindromicity of multivariate Eulerian polynomials
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arXiv:2601.15527v1 Announce Type: new Abstract: We lift to the multivariate Eulerian polynomials the identity implying that univariate Eulerian polynomials are palindromic. As a consequence of this generalization, we obtain nice combinatorial identities that can be directly extracted from this polynomial relation and the bijection between permutations involved in the proof of the identity.
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https://arxiv.org/abs/2601.15527
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9cf4aebdde995c86b4624a31acc993f9841fe83015dae0a27ea2ad1016e2bf41
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2026-01-23T00:00:00-05:00
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Variable Stepsize Distributed Forward-Backward Splitting Methods as Relocated Fixed-Point Iterations
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arXiv:2601.15531v1 Announce Type: new Abstract: We present a family of distributed forward-backward methods with variable stepsizes to find a solution of structured monotone inclusion problems. The framework is constructed by means of relocated fixed-point iterations, extending the approach introduced in arXiv:2507.07428 to conically averaged operators, thus including iteration operators for methods of forward-backward type devised by graphs. The family of methods we construct preserve the per-iteration computational cost and the convergence properties of their constant stepsize counterparts. Specifically, we show that the resulting methods generate a sequence that converges to a fixed-point of the underlying iteration operator, whose shadow sequences converge to a solution of the problem. Numerical experiments illustrate the behaviour of our framework in structured sparse optimisation problems.
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https://arxiv.org/abs/2601.15531
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7eff0bd126a880e869f47ba53b9c90aed97afe8aac4d513ed75395f72f0e27c6
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2026-01-23T00:00:00-05:00
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The formal theory of tangentads PART II
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arXiv:2601.15534v1 Announce Type: new Abstract: Tangent category theory is a well-established categorical framework for differential geometry. A long list of fundamental geometric constructions, such as the tangent bundle functor, vector fields, Euclidean spaces, and vector bundles have been successfully generalized and internalized within tangent categories. Over the past decade, the theory has also been extended in several directions, yielding concepts such as tangent monads, tangent fibrations, tangent restriction categories, and reverse tangent categories. It is natural to wonder how these new flavours of the theory interact with the geometric constructions. How does a tangent monad or a tangent fibration lift to the tangent category of differential bundles of a tangent category? What is the correct notion of connections for a tangent restriction category? In previous work, we introduced tangentads, a unifying framework that generalizes many tangent-like notions, and developed a formal theory of vector fields for tangentads. In this paper, we extend this formal theory to three further fundamental constructions. These are differential objects, which generalize Euclidean spaces, differential bundles, which represent vector bundles in tangent category theory, and connections on differential bundles, which are the analogue of Koszul connections. These notions are introduced in the general theory of tangentads via appropriate universal properties. We then extend some of the main results of tangent category theory, including the equivalence between differential objects and differential bundles over the terminal object, and show that connections admit well-defined notions of covariant derivative, curvature, and torsion. Finally, we construct connections using PIE limits and apply our framework to several concrete instances of tangentads.
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https://arxiv.org/abs/2601.15534
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d2265b70b7c83994bc7f1aa6a5936eb7ee04f83e7d32da4852b14c918fe65820
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2026-01-23T00:00:00-05:00
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Rationality of the trivial lattice rank weighted motivic height zeta function for elliptic surfaces
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arXiv:2601.15543v1 Announce Type: new Abstract: Let $k$ be a perfect field with $\mathrm{char}(k)\neq 2,3$, set $K=k(t)$, and let $\mathcal{W}_n^{\min}$ be the moduli stack of minimal elliptic curves over $K$ of Faltings height $n$ from the height-moduli framework of Bejleri-Park-Satriano applied to $\overline{\mathcal{M}}_{1,1}\simeq \mathcal{P}(4,6)$. For $[E]\in \mathcal{W}_n^{\min}$, let $S \to \mathbb{P}^1_{k}$ be the associated elliptic surface with section. Motivated by the Shioda-Tate formula, we consider the trivariate motivic height zeta function \[ \mathcal{Z}(u,v;t):= \sum_{n\ge0}\Bigl(\sum_{[E]\in \mathcal{W}_n^{\min}} u^{T(S)}v^{\mathrm{rk}(E/K)}\Bigr)t^n \in K_0(\mathrm{Stck}_k)[u,v][[t]] \] which refines the height series by weighting each height stratum with the trivial lattice rank $T(S)$ and the Mordell--Weil rank $\mathrm{rk}(E/K)$. We prove rationality for the trivial lattice specialization $Z_{\mathrm{Triv}}(u;t)=\mathcal{Z}(u,1;t)$ by giving an explicit finite Euler product. We conjecture irrationality for the N\'eron-Severi $Z_{\mathrm{NS}}(w;t)=\mathcal{Z}(w,w;t)$ and the Mordell-Weil $Z_{\mathrm{MW}}(v;t)=\mathcal{Z}(1,v;t)$ specializations.
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https://arxiv.org/abs/2601.15543
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6ab0371539cfd83e70ca206f25dbe5d817a0cca433fe2ed2aaca776b7b6a02e3
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2026-01-23T00:00:00-05:00
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Computability of $\mathcal{G}$-Beroulli Measures and Measures of Maximal Entropy on Coded Shift Spaces
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arXiv:2601.15548v1 Announce Type: new Abstract: In this paper, we investigate the computability of $\mathcal{G}$-Bernoulli measures, with a particular focus on measures of maximal entropy (MMEs) on coded shift spaces. Coded shifts are natural generalizations of sofic shifts and are defined as the closure of all bi-infinite concatenations of words (generators) drawn from a countable generating set $\mathcal{G}$. We begin by establishing a computability criterion for $\mathcal{G}$-Bernoulli measures which are invariant measures given by assigning probability weights to the generators. We then apply this criterion to the setting in which the concatenation entropy exceeds the residual entropy, showing that in this case the unique measure of maximal entropy $\mu_{\rm max}$ on $X$ is computable, provided the Vere--Jones parameter $\kappa$ of $\mathcal{G}$ is computable, based on having oracle access to the generators and the language of $X$. As a consequence, the unique MME is computable for several well-known classes of shift spaces, including $S$-gap shifts, multiple-gap shifts, and $\beta$-shifts. Moreover, the two ergodic MMEs of the Dyck shift are also computable. Finally, we examine the opposite situation, where the residual entropy exceeds the concatenation entropy and the MME is known to be non-unique in general. We show that even when $\mu_{\rm max}$ is unique and the parameter $\kappa$ is computable, the measure $\mu_{\rm max}$ may still fail to be computable.
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https://arxiv.org/abs/2601.15548
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f598799a2149fb4e0eac87b5b32f85ff2c15493e7d3ff58c99056398c97ecd9d
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2026-01-23T00:00:00-05:00
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On the nilpotent residue non-abelian Hodge correspondence for higher-dimensional quasiprojective varieties
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arXiv:2601.15553v1 Announce Type: new Abstract: In arXiv:2408.16441, the authors proved that on a projective log smooth variety $(\bar{X}, D)$ there is a continuous bijection between the moduli space $M^{\mathrm{nilp}}_{\mathrm{Dol}}(\bar{X}, D)$ of logarithmic Higgs bundles with nilpotent residues and the moduli space $M^{\mathrm{nilp}}_{\mathrm{DR}}(\bar{X}, D)$ of logarithmic connections with nilpotent residues. In this note, we argue that the map is a homeomorphism.
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https://arxiv.org/abs/2601.15553
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17bd430e62a8c10540734d32cd5a8d59ce7fe3526495671d4a226485108fd0ab
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2026-01-23T00:00:00-05:00
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Primes and almost primes between cubes
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arXiv:2601.15564v1 Announce Type: new Abstract: In this paper we study the problem of detecting prime numbers between all consecutive cubes. Firstly, we use a large computation to show that there is always a prime between $n^3$ and $(n+1)^3$ for $n^3\leq 1.649\cdot 10^{40}$. In addition, we use this computation and a sieve-theoretic argument to show that there exists a number with at most 2 prime factors (counting multiplicity) between $n^3$ and $(n+1)^3$ for all $n\geq 1$. Our sieving argument uses a logarithmic weighting procedure attributed to Richert, which yields significant numerical improvements over previous approaches.
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https://arxiv.org/abs/2601.15564
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4c9c531fae79436da6b6a036196149cfd4de6a991f50ea41d49606bee2311693
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2026-01-23T00:00:00-05:00
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Non-universality of ternary quadratic forms over fields containing $\sqrt2$
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arXiv:2601.15568v1 Announce Type: new Abstract: We prove Kitaoka's conjecture for all totally real number fields of degree 4 -- namely, there is no positive definite classical quadratic form in three variables which is universal. To achieve this, we study the fields (often without restricting the degree) where 2 is a square, because in this arguably most difficult case, the recent results connecting Kitaoka's conjecture to sums of integral squares do not apply. We also prove some other properties of ternary quadratic forms over fields containing $\sqrt2$, for example in relation to the lifting problems for universal quadratic forms and for criterion sets.
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https://arxiv.org/abs/2601.15568
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448739a7ab16ee41756e3d4fc3dd0e8aebb1e9d157c1c4568c363ca0385bef80
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2026-01-23T00:00:00-05:00
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The second Delannoy category
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arXiv:2601.15574v1 Announce Type: new Abstract: In recent work, Harman and Snowden constructed a symmetric tensor category associated to an oligomorphic group equipped with a measure. The oligomorphic group $\mathbb{G}$ of order preserving automorphisms of the real line admits exactly four measures. The category $\mathcal{C}$ associated to the first measure is called the (first) Delannoy category; it is semi-simple and pre-Tannakian, with numerous special properties. In this paper, we study the (non-abelian) category $\mathcal{A}$ associated to the second measure, which we call the second Delannoy category. We construct a new pre-Tannakian category $\mathcal{D}$ together with a fully faithful tensor functor $\Psi \colon \mathcal{A} \to \mathcal{D}$. The category $\mathcal{D}$ is the correct ``abelian version'' of the second Delannoy category. Like $\mathcal{C}$, it has remarkable properties: for instance, it is non-semi-simple, but behaves uniformly in the coefficient field (e.g., it has the same Grothendieck ring and $\mathrm{Ext}^1$ quiver over any field). Additionally, we completely solve the problem of understanding how $\mathcal{A}$ relates to general pre-Tannakian categories. We show that $\mathcal{A}$ admits exactly two local abelian envelopes: the functor $\Psi$, and a previously constructed functor $\Phi \colon \mathcal{A} \to \mathcal{C}$. This is the first case where the local envelopes of a category have been completely determined, outside of cases where there is at most one envelope. This work opens the door to constructing abelian versions of other oligomorphic tensor categories that do not admit a unique envelope.
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https://arxiv.org/abs/2601.15574
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4dc003bd431ba5d6988b15859e4997c54fe17879c5928314aae91061821755aa
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2026-01-23T00:00:00-05:00
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Open problems in K-stability of Fano varieties
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arXiv:2601.15576v1 Announce Type: new Abstract: In this note, we discuss a number of open problems in K-stability theory.
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https://arxiv.org/abs/2601.15576
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cba2cbd719b15a3cc534c24bff88064f7f24ea348ca7f7513374b602e0ce9487
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2026-01-23T00:00:00-05:00
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Global solution curves for first order periodic problems, with applications
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arXiv:2601.15579v1 Announce Type: new Abstract: Using continuation methods and bifurcation theory, we study the exact multiplicity of periodic solutions, and the global solution structure, for periodic problems of first order. The results are applied to a population model with fishing, and to the existence and stability of limit cycles. We also describe in detail our numerical computations of curves of periodic solutions, and of limit cycles.
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https://arxiv.org/abs/2601.15579
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cf7cd0360878d78d44959cb2f0c10c90ad235da759280c4f6a7eb61ffad5e6c1
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2026-01-23T00:00:00-05:00
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A Wild Steiner-Lehmus Chase
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arXiv:2601.15591v1 Announce Type: new Abstract: We present a proof the Steiner-Lehmus equal bisectors theorem by applying the Law of sines in rapid succession to a side-by-side comparison. For nearly two centuries, the quest for a direct proof has sustained interest in proving and reproving this theorem. We suggest that a second driving force may also be at play.
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https://arxiv.org/abs/2601.15591
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0f0e2599564a683882a6dd19f18517543c14d961ea91b4a68a8c37b730e24928
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2026-01-23T00:00:00-05:00
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Overpartitions with repeated smallest non-overlined part
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arXiv:2601.15601v1 Announce Type: new Abstract: Inspired by Andrews' and Bachraoui's work on partitions with repeated smallest part, we extend the concept to overpartitions.
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https://arxiv.org/abs/2601.15601
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4811fd0e2e5651815207a3ca92d741745445bde86c6af24bb96856c170dd5e67
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2026-01-23T00:00:00-05:00
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Barcode entropy and relative symplectic cohomology
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arXiv:2601.15606v1 Announce Type: new Abstract: In this paper, we study the barcode entropy--the exponential growth rate of the number of not-too-short bars--of the persistence module associated with the relative symplectic cohomology $SH_M(K)$ of a Liouville domain $K$ embedded in a symplectic manifold $M$. Our main result establishes a quantitative link between this Floer-theoretic invariant and the dynamics of the Reeb flow on $\partial K$. More precisely, we show that the barcode entropy of the relative symplectic cohomology $SH_M(K)$ is bounded above by a constant multiple of the topological entropy of the Reeb flow on the boundary of the domain, where the constant depends on the embedding of $K$ into $M$.
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https://arxiv.org/abs/2601.15606
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2843ac54cfc51d3c7a97d5b35f62e7386b7818d6c36fb08ed355072c5305693c
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2026-01-23T00:00:00-05:00
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Lead distance under a pickoff limit in Major League Baseball: A sequential game model
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arXiv:2601.15608v1 Announce Type: new Abstract: Major League Baseball (MLB) recently limited pitchers to three pickoff attempts, creating a cat-and-mouse game between pitcher and runner. Each failed attempt adds pressure on the pitcher to avoid using another, and the runner can intensify this pressure by extending their leadoff toward the next base. We model this dynamic as a two-player zero-sum sequential game in which the runner first chooses a lead distance, and then the pitcher chooses whether to attempt a pickoff. We establish optimality characterizations for the game and present variants of value iteration and policy iteration to solve the game. Using lead distance data, we estimate generalized linear mixed-effects models for pickoff and stolen base outcome probabilities given lead distance, context, and player skill. We compute the game-theoretic equilibria under the two-player model, as well as the optimal runner policy under a simplified one-player Markov decision process (MDP) model. In the one-player setting, our results establish an actionable rule of thumb: the Two-Foot Rule, which recommends that a runner increase their lead by two feet after each pickoff attempt.
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https://arxiv.org/abs/2601.15608
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3967fbd69606c01cc96c63ac1d79ae57687ee4c8cfc58db0bfcbf5a9ca7fc09c
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2026-01-23T00:00:00-05:00
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On the Zeros of the Riemann Zeta Function with Two Ordinate Shifts
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arXiv:2601.15610v1 Announce Type: new Abstract: We prove that for any fixed real numbers y_1, y_2 not equal to 0, and constant C > 0, there exists a threshold T_* = T_*(y_1, y_2, C) > 0 such that for all T >= T_*, the interval [T, T(1 + epsilon)], with epsilon = exp(-C sqrt(log T)), contains at least one gamma satisfying zeta(1/2 + i gamma) = 0, zeta(1/2 + i (gamma + y_1)) != 0, and zeta(1/2 + i (gamma + y_2)) != 0. This extends earlier work by Banks (for a single shift y) to two distinct shifts y_1, y_2. Our argument is based on the behavior of zeta and L functions in zero-free regions via Perron's formula.
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https://arxiv.org/abs/2601.15610
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713870fccde1421477f44a80db1f029f54bd084061162d2af7f3888e0a5beb22
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2026-01-23T00:00:00-05:00
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Degree-choosability of proper conflict-free list coloring of sparse graphs
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arXiv:2601.15611v1 Announce Type: new Abstract: Given a graph $G$ and a mapping $f:V(G) \to \mathbb{N}$, an $f$-list assignment of $G$ is a function that maps each $v \in V(G)$ to a set of at least $f(v)$ colors. For an $f$-list assignment $L$ of a graph $G$, a proper conflict-free $L$-coloring of $G$ is a proper coloring $\phi$ of $G$ such that for every vertex $v \in V(G)$, $\phi(v) \in L(v)$ and some appears precisely once in the neighborhood of $v$. We say that $G$ is proper conflict-free $f$-choosable if for every $f$-list assignment $L$ of $G$, there exists a proper conflict-free $L$-coloring of $G$. If $G$ is proper conflict-free $f$-choosable and there is a constant $k$ such that $f(v)= d_G(v)+k$ for every vertex $v$ of $G$, then we say $G$ is proper conflict-free $({\rm degree}+k)$-choosable. In this paper, we consider graphs with a bounded maximum average degree. We show that every graph with the maximum average degree less than $\frac{10}{3}$ is proper conflict-free $({\rm degree}+3)$-choosable, and that every graph with the maximum average degree less than $\frac{18}{7}$ is proper conflict-free $({\rm degree}+2)$-choosable. As a result, every planar graph with girth at least $5$ is proper conflict-free $({\rm degree}+3)$-choosable, and every planar graph with girth at least $9$ is proper conflict-free $({\rm degree}+2)$-choosable.
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https://arxiv.org/abs/2601.15611
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d3e2809b272dfa6b80cf6eb9442d4ca98a216a7550b1b0054dad14ddfd6f66bd
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2026-01-23T00:00:00-05:00
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Lucas sequences, Pell's equations, and automorphisms of K3 surfaces
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arXiv:2601.15617v1 Announce Type: new Abstract: We have the correspondences between Lucas sequences, Pell's equations, and the automorphisms of K3 surfaces with Picard number 2. Using these correspondences, we determine the intersections of some Lucas sequences.
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https://arxiv.org/abs/2601.15617
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8e4ecb7a2ed781b9d16e17d9299dcdabf7e642b81509e4cb246f46217d62128f
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2026-01-23T00:00:00-05:00
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Existence and uniqueness of $L^1$-solutions to time-fractional nonlinear diffusion equations
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arXiv:2601.15618v1 Announce Type: new Abstract: We establish the global existence and uniqueness of $L^1$-solutions to the Cauchy problem for time-fractional porous medium type nonlinear diffusion equations. Furthermore, we give the mass conservation law for $L^1$-solutions to time-fractional fast diffusion equations, and prove that the finite-time extinction does not occur for any nonnegative $L^1$-solutions.
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https://arxiv.org/abs/2601.15618
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59b8d7d727f7db9c5d92772edb537420fb7c734ce22889eca3a3d85f5630b407
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2026-01-23T00:00:00-05:00
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Limit behavior of linearly edge-reinforced random walks on the half-line
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arXiv:2601.15627v1 Announce Type: new Abstract: Motivated by the article [M. Takei, Electron. J. Probab. 26 (2021), article no. 104], we study the limit behavior of linearly edge-reinforced random walks on the half-line $\mathbb{Z}_+$ with reinforcement parameter $\delta>0$, and each edge $\{x,x+1\}$ has the initial weight $x^{\alpha}\ln^{\beta}x$ for $x > 1$ and $1$ for $x = 0, 1$. The aim of this paper is to study the almost sure limit behavior of the walk in the recurrent regime, and extend the results of Takei mentioned above.
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https://arxiv.org/abs/2601.15627
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44d467acd111a2b276deb7ac3842f5b8188234ec339344fc5699a067d2b942bc
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2026-01-23T00:00:00-05:00
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The $V_1$- and $V_2$-polynomials of a long virtual knot
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arXiv:2601.15634v1 Announce Type: new Abstract: We introduce two polynomial invariants $V_1(K;t)$ and $V_2(K;t)$ of a long virtual knot $K$, which generalize the degree-two finite type invariants $v_{2,1}$ and $v_{2,2}$ of Goussarov, Polyak, and Viro. We establish their fundamental properties and show that any pair of Laurent polynomials can be realized as $(V_1(K;t),V_2(K;t))$ for some long virtual knot $K$. While these polynomials are not finite type invariants of any degree with respect to virtualizations, their first derivatives at $t=1$ define finite type invariants of degree three. As an application, we obtain an explicit Gauss diagram formula for the $\alpha_3$-invariant.
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https://arxiv.org/abs/2601.15634
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d961e0821a50d8d66bf5d7e2e12653d2110dcfc58d161d79bfc72899c24f5ce7
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2026-01-23T00:00:00-05:00
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Collaboration versus Specialization in Service Systems with Impatient Customers
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arXiv:2601.15636v1 Announce Type: new Abstract: We study tandem queueing systems in which servers work more efficiently in teams than on their own and customers are impatient in that they may leave the system while waiting for service. Our goal is to determine the server assignment policy that maximizes the long-run average throughput. We show that when each server is equally skilled at all tasks, the optimal policy has all the servers working together at all times. We also provide a complete characterization of the optimal policy for Markovian systems with two stations and two servers when each server's efficiency may be task dependent. We show that the throughput is maximized under the policy which assigns one server to each station (based on their relative skill at that station) unless station 2 has no work (in which case both servers work at station 1) or the number of customers in the buffer reaches a threshold whose value we characterize (in which case both servers work at station 2). We study how the optimal policy varies with the level of server synergy (including no synergy) and also compare the optimal policy for systems with different customer abandonment rates (including no abandonments). Finally, we investigate the case where the synergy among collaborating servers can be task-dependent and provide numerical results.
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https://arxiv.org/abs/2601.15636
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403990bf744b37d04bd62ab71e227baa1ffcca81a580d7f98bd41e3076d4cdae
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2026-01-23T00:00:00-05:00
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Linear stability of the first bifurcation in a tumor growth free boundary problem via local bifurcation structure
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arXiv:2601.15647v1 Announce Type: new Abstract: In this paper, we consider a 3-dimensional free boundary problem modeling tumor growth with the Robin boundary condition. The system involves a positive parameter $\mu$ which reflects the intensity of tumor aggressiveness. Huang, Zhang and Hu [Nonlinear Anal. Real World Appl. 2017(35), 483-502] have shown that for each $\mu_n$ ($n$ even) in a strictly increasing sequence $\{ \mu_n \}(n\geq 2)$, there exists a stationary bifurcation solution $(\sigma_n(\varepsilon),p_n(\varepsilon),r_n(\varepsilon))$ with $\mu = \mu_n(\varepsilon)$ bifurcating from $\mu_n$. We first derive that the bifurcation curve $(r_2(\varepsilon),\mu_2(\varepsilon))$ exhibits a transcritical bifurcation with $\mu_2'(0)<0$. Moreover, we show that the stationary bifurcation solution $(\sigma_2(\varepsilon),p_2(\varepsilon),r_2(\varepsilon))$ is linearly unstable for small $|\varepsilon|$ under non-radially symmetric perturbations. In contrast to the linear stability of the radially symmetric stationary solution, the lack of explicit expressions for bifurcation solutions adds great difficulty in analyzing their linear stability. The novelty of this paper lies in the use of the bifurcation curve's structure to overcome the above difficulties. Moreover, this linear stability result is not established using the standard method, due to an eight-dimensional generalized kernel at eigenvalue 0 for the linearized operator.
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https://arxiv.org/abs/2601.15647
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8c695b9b3bee1fd43afdfb93faacf6acf2f44be32a0411791a2dbdc851ded021
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2026-01-23T00:00:00-05:00
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Iterative Derivations on Central Simple Algebras
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arXiv:2601.15648v1 Announce Type: new Abstract: We prove that an iterative derivation $\delta_F$ on a field $F$ can be extended to an iterative derivation $\delta_A$ on a central simple $F-$algebra $A$ if the characteristic of $F$ does not divide the exponent of $A$ in the Brauer group of $F.$ For a central simple $F-$algebra with an iterative derivation, we show the existence of a unique (up to isomorphism) Picard-Vessiot splitting field and from the nature its Galois group, we also describe the structure of the central simple algebra in terms of its $\delta_A-$right ideals.
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https://arxiv.org/abs/2601.15648
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f649d854bb10f251b9ecd089bba2a473b235b3826e220c498e411e697fc12054
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2026-01-23T00:00:00-05:00
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An index theory for transverse trajectories
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arXiv:2601.15651v1 Announce Type: new Abstract: In this work, we present an alternative definition of the Le Roux index, which generalizes the Poincar\'e-Hopf index for non-singular planar flows to the broader setting of Brouwer homeomorphisms. This new approach answers a question raised by Le Roux by establishing a connection between the index of a Brouwer homeomorphism and the structure of its transverse foliations, in the sense of Le Calvez.
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https://arxiv.org/abs/2601.15651
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14827942cd28cc37be4cc56135784da0949d33b76a98fc90118abe8ef7b2a1ff
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2026-01-23T00:00:00-05:00
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Construction and Box-counting Dimension of the Edelstein Hidden Variable Fractal Interpolation Function
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arXiv:2601.15658v1 Announce Type: new Abstract: This paper presents the construction of a hidden variable fractal interpolation function using Edelstein contractions in an iterated function system based on a finite collection of data points. The approach incorporates an iterated function system where variable functions act as vertical scaling factors leading to a generalised vector-valued fractal interpolation function. Furthermore, the paper rigorously examines the smoothness of the constructed function and establishes an upper bound for the box-counting dimension of its graph.
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https://arxiv.org/abs/2601.15658
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b0b4ea20a63152ea536a166d0924e539440b979588e1b8cf963f046ff0ccb00c
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2026-01-23T00:00:00-05:00
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Local smoothing estimates for bilinear Fourier integral operators
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arXiv:2601.15667v1 Announce Type: new Abstract: We formulate a local smoothing conjecture for bilinear Fourier integral operators in every dimension $d \ge 2,$ derived from the celebrated linear case due to Sogge, which we refer to as the \emph{bilinear smoothing conjecture}. We show that the linear local smoothing conjecture implies this bilinear version. As a consequence of our approach and due to the recent progress on the subject, we establish local smoothing estimates for Fourier integral operators in dimension $d=2,$ that is, on $\mathbb{R}^2_x \times \mathbb{R}_t$. Also, a partial progress is presented for the high-dimensional case $d\geq 3.$ In particular, our method allows us to deduce that the bilinear local smoothing conjecture holds for all odd dimensions $d$.
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https://arxiv.org/abs/2601.15667
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684cbae8253189415e4c803434b7dd04cfaca9829c097ab2b500e78e4a6b0ff9
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2026-01-23T00:00:00-05:00
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Geometric wavefront sets of genuine Iwahori-spherical representations
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arXiv:2601.15670v1 Announce Type: new Abstract: For Iwahori-spherical genuine representations of central covers with positive real Satake parameters, we prove the upper bound inequality for their geometric wavefront sets, formulated for general genuine representations in an earlier work by Gao--Liu--Lo--Shahidi. Meanwhile, we show the equality is attained for covers of type A groups and for some representations of covers of the exceptional groups. We also verify the equality for certain Iwahori-spherical representations occurring in regular unramified principal series; this uses and generalizes the earlier work of Karasiewicz--Okada--Wang on theta representations. Lastly, we determine the leading coefficients in the Harish-Chandra character expansion of a theta representation when its geometric wavefront set is of a special type.
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https://arxiv.org/abs/2601.15670
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eea71fef8fc06ea9032c2848f1e254c9e3f7a7a5ebfc102864d6c5c4ee8014de
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2026-01-23T00:00:00-05:00
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Arithmetic Properties of Colored Partitions Restricted by Parity of the Parts
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arXiv:2601.15680v1 Announce Type: new Abstract: Let $a_{r,s}(n)$ denote the number of mutlicolored partitions of $n$, wherein both even parts and odd parts may appear in one of $r$-colors and $s$-colors, respectively, for fixed $r,s\ge 1$. The paper aims to study arithmetic properties satisfied by $a_{r,s}(n)$, using elementary generating function manipulations and classical $q$-series techniques.
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https://arxiv.org/abs/2601.15680
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9e8f558f4700bea2aebaac012bcc15bd0ea61863156b7a32945d8bc982fe24de
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2026-01-23T00:00:00-05:00
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Parallelizable Riemannian Alternating Direction Method of Multipliers for Non-convex Pose Graph Optimization
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arXiv:2601.15684v1 Announce Type: new Abstract: Pose graph optimization (PGO) is fundamental to robot perception and navigation systems, serving as the mathematical backbone for solving simultaneous localization and mapping (SLAM). Existing solvers suffer from polynomial growth in computational complexity with graph size, hindering real-time deployment in large-scale scenarios. In this paper, by duplicating variables and introducing equality constraints, we reformulate the problem and propose a Parallelizable Riemannian Alternating Direction Method of Multipliers (PRADMM) to solve it efficiently. Compared with the state-of-the-art methods that usually exhibit polynomial time complexity growth with graph size, PRADMM enables efficient parallel computation across vertices regardless of graph size. Crucially, all subproblems admit closed-form solutions, ensuring PRADMM maintains exceptionally stable performance. Furthermore, by carefully exploiting the structures of the coefficient matrices in the constraints, we establish the global convergence of PRADMM under mild conditions, enabling larger relaxation step sizes within the interval $(0,2)$. Extensive empirical validation on two synthetic datasets and multiple real-world 3D SLAM benchmarks confirms the superior computational performance of PRADMM.
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https://arxiv.org/abs/2601.15684
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62bb9de4530eb81d79b9be1b36efe25dca10d3db6e19ddb3c890e1564df2560a
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2026-01-23T00:00:00-05:00
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Global regularity for the Navier-Stokes equations with application to global solvability for the Euler equations
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arXiv:2601.15685v1 Announce Type: new Abstract: We show that any Leray-Hopf weak solution to the $d$-dimensional Navier-Stokes equations $(d\geq 3)$ with initial values $u_0\in H^{s}(\mathbb R^d)$, $s\geq -1+\frac{d}{2}$, belongs to $L^\infty(0,\infty; H^{s}(\mathbb R^d))$ and thus it is globally regular. For the proof, first, we construct a supercritical space which has very sparse inverse logarithmic weight in the frequency domain, compared to the critical homogeneous Sobolev $\dot{H}^{-1+d/2}$-norm. Then we obtain the energy estimates of high frequency parts of the solution which involve the supercritical norm as a factor of the upper bounds. Finally, we superpose the energy norm of high frequency parts of the solution to get estimates of the critical and subcritical norms independent of the viscosity coefficient for the weak solution via the re-scaling argument.
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https://arxiv.org/abs/2601.15685
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8a7dc2d51180829ee83fb972174c9d8fd57b6562fbd072417a196b4fcfff4779
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2026-01-23T00:00:00-05:00
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Symbolic Rees algebras of space monomial primes of degree 5
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arXiv:2601.15692v1 Announce Type: new Abstract: Let K be a field of characteristic 0. Let P_K(5,103,169) be the defining ideal of the space monomial curve {(t^5,t^{103},t^{169})}. In this paper we shall prove that the symbolic Rees algebra R_s(P_K(5,103,169)) is not Noetherian, that is, is not finitely generated over K.
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https://arxiv.org/abs/2601.15692
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32a41ea4994ef3408857fe9b9ea73bc299e1be59b875f4121473fa81c1bc2712
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2026-01-23T00:00:00-05:00
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Rokhlin dimension for actions of residually compact groups
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arXiv:2601.15694v1 Announce Type: new Abstract: We introduce the concept of Rokhlin dimension for actions of residually compact groups on C*-algebras, which extends and unifies previous notions for actions of compact groups, residually finite groups and the reals. We then demonstrate that finite nuclear dimension (respectively, absorption of a strongly self-absorbing C*-algebra) is preserved under the formation of crossed products by residually compact group actions with finite Rokhlin dimension (respectively, finite Rokhlin dimension with commuting towers). Furthermore, if second countable residually compact group contains a non-open cocompact closed subgroup, then crossed products arising from actions with finite Rokhlin dimension are stable. Finally, we study the relationship between the tube dimension of a topological dynamical system and the Rokhlin dimension of the induced C*-dynamical system.
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https://arxiv.org/abs/2601.15694
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be90f7aef90a780ad4bd6eff7d4b1846f01eacb739c8aa52b22d8e75c5ec419a
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2026-01-23T00:00:00-05:00
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Maximal Fuchsian subgroups of the $d=2$ Bianchi group
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arXiv:2601.15700v1 Announce Type: new Abstract: Let $\Gamma$ denote the $d = 2$ Bianchi group $\operatorname{PSL}(2,\mathbb{Z}[\sqrt{-2}])$. We give an explicit description of all conjugacy classes of maximal nonelementary Fuchsian subgroups of $\Gamma$ as integral orders of certain indefinite quaternion algebras over $\mathbb{Q}$. Using this description, we also provide the covolumes corresponding to each conjugacy class. As an application, we compute the limit $\lim_{x\to\infty} \frac{\Pi(x)}{x}$ where $\Pi(x)$ counts the number of primitive totally geodesic immersed surfaces in the manifold $\Gamma\backslash\mathbb{H}^3$ with area less than $x$.
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https://arxiv.org/abs/2601.15700
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33e0def09deb660104a4933ab2cf97295d88d2d52ee706a3ae9d11e710b2c569
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2026-01-23T00:00:00-05:00
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On mode transition algebras for $\mathbb{Z}$-graded vertex algebras and applications to bosonic ghosts
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arXiv:2601.15701v1 Announce Type: new Abstract: We study the mode transition algebras and Zhu algebras in the setting of $\mathbb{Z}$-graded vertex algebras, with particular focus on the Weyl vertex algebra at central charge 2 (also known as bosonic ghosts or the $\beta\gamma$-system). We show that the mode transition algebras of the Weyl vertex algebra at central charge 2 admit unity elements that form a family of strong unities in the sense of Damiolini-Gibney-Krashen. The existence of unities for the mode transition algebra of the Weyl vertex algebra at central charge 2 allows us to explicitly construct all higher level Zhu algebras of the Weyl vertex algebra at central charge 2. We further analyze weak modules of the Weyl vertex algebra at central charge 2 induced from Zhu algebras, proving that every such module is already induced from the level-zero Zhu algebra. We then prove that all indecomposable reducible weight modules induced from a Zhu algebra are not weakly interlocked, and hence not strongly interlocked in the sense of Barron-Batistelli-Orosz Hunziker-Yamskulna. More generally, we show that the property of being weakly interlocked is preserved under the action of an invertible Li's $\mathbf{\Delta}$ operator. As an application, we prove that all indecomposable reducible weight modules of the Weyl vertex algebra at central charge 2 obtained via spectral flow of Zhu-induced modules are likewise not weakly interlocked. These results clarify the role of being weakly interlocked in the modularity properties of bosonic ghost modules previously studied by Ridout-Wood and Allen-Wood.
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https://arxiv.org/abs/2601.15701
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99e7155fab13dbfdabceb2e39359919d2272200860f34121f10fd72c347fb153
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2026-01-23T00:00:00-05:00
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Four-dimensional Lorentzian algebraic Ricci solitons
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arXiv:2601.15730v1 Announce Type: new Abstract: We describe four-dimensional Lorentzian algebraic Ricci solitons. In sharp contrast with the Riemannian situation, any connected and simply connected four-dimensional Lie group admits a left-invariant Lorentz metric which is a Ricci soliton.
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https://arxiv.org/abs/2601.15730
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5a625c54bba502740ab5221c10dd44891aa7662bdc240219e548c76dd32d2143
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2026-01-23T00:00:00-05:00
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A sequential linear complementarity problem method for generalized Nash equilibrium problems
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arXiv:2601.15742v1 Announce Type: new Abstract: We propose a sequential linear complementarity problem (SLCP) method for solving generalized Nash equilibrium problems (GNEPs). By introducing a novel merit function that utilizes the specific structure of GNEPs, we establish global convergence of the method. The conditions guaranteeing global convergence are analogous to those for the classical sequential quadratic programming method with exact Lagrange Hessians, making this a natural and reasonable generalization. Moreover, we provide a detailed analysis of the solvability of the mixed linear complementarity subproblems, which are formulated as affine GNEPs. Sufficient characterizations for the local superlinear convergence are also derived, highlighting the efficiency of the proposed method. Finally, numerical experiments demonstrate the practical performance and effectiveness of the SLCP method in comparison with existing approaches.
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https://arxiv.org/abs/2601.15742
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31addc7d807cdf0391b401ff9a32dc4ba58fde2e6083f0b9f6080d53e435f918
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2026-01-23T00:00:00-05:00
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Rankin--Cohen brackets in Representation Theory
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arXiv:2601.15750v1 Announce Type: new Abstract: The Rankin--Cohen brackets provide a basic example of ``non-elementary" differential symmetry breaking operators. They can be interpreted as bi-differential operators remarkable for reflecting the structure of fusion rules for holomorphic discrete series representations of the Lie group $SL(2,\mathbb R)$ and are intimately connected to classical special polynomials. In this introductory article, we explore the combinatorial structure of these operators and discuss a general framework for constructing their higher-dimensional analogues from the representation-theoretic perspective on branching problems. The exposition is based on lectures delivered by the authors during the thematic semester ``Representation Theory and Noncommutative Geometry", held in Spring 2025 at the Henri Poincar\'e Institute in Paris.
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https://arxiv.org/abs/2601.15750
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2eb96bb18cecc90f8e313a60e2b825be38d11b4254296f47217c94384855ecb2
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2026-01-23T00:00:00-05:00
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Stochastically forced compressible Navier-Stokes equations with slip boundary conditions of friction type
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arXiv:2601.15768v1 Announce Type: new Abstract: We study a mathematical model of a compressible viscous fluid driven by stochastic forces under slip boundary conditions of friction type. We introduce a notion of a weak solution that is analytically and probabilistically consistent with this model. Our main result establishes the existence of such weak solutions under slip boundary conditions on bounded domains with $C^{2+\nu}$-boundary ($\nu>0$). The proof of this result combines an extended version of the four-layer approximation scheme on the torus by Breit/Feireisl/Hofmanov\'{a} (2018) with the convex approximation method for absolute value functions studied by Ne\v{c}asov\'{a}/Ogorzaly/Scherz (2023).
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https://arxiv.org/abs/2601.15768
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f892695687a92eb2ba51e3c55e73e7e2ddcfaaca8ddec93637f80538869cf7a7
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2026-01-23T00:00:00-05:00
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Representations of the modular group into the isometries of $\mathrm{SL}_3(\mathbb{R})/\mathrm{SO}(3)$
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arXiv:2601.15781v1 Announce Type: new Abstract: We describe a connected component of the space of conjugacy classes of representations of the modular group $\mathrm{PSL}_2(\mathbb{Z})$ into the isometry group of the symmetric space $\mathrm{SL}_3(\mathbb{R})/\mathrm{SO}(3)$. This connected component contains the family of representations constructed by Schwartz via Pappus' theorem, as well as their Anosov deformations studied by Barbot, Lee, and Val\'erio. We show that certain representations in this component (far from the Schwartz representations) are Anosov.
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https://arxiv.org/abs/2601.15781
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040879d3901e236f869738f97d13d17f015977986deedf24351d390c3fda5c39
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2026-01-23T00:00:00-05:00
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Shuriken Graphs Arising from Clean Graphs of Rings and Their Properties Relative to Base Graphs
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arXiv:2601.15783v1 Announce Type: new Abstract: Let $R$ be a finite ring with identity. The idempotent graph $I(R)$ is the graph whose vertex set consists of the non-trivial idempotent elements of $R$, where two distinct vertices $x$ and $y$ are adjacent if and only if $xy = yx = 0$. The clean graph $Cl_2(R)$ is a graph whose vertices are of the form $(e, u)$, where $e$ is a nonzero idempotent element and $u$ is a unit of $R$. Two distinct vertices $(e,u)$ and $(f, v)$ are adjacent if and only if $ef = fe = 0$ or $uv = vu = 1$. The shuriken graph operation is an operation that arises from the structure of the clean graph and depends on the structure of the associated idempotent graph. In this paper, we study the graph obtained from the shuriken operation and examine how its properties depend on those of the base graph. In particular, we investigate several graph invariants, including the clique number, chromatic number, independence number, and domination number. Moreover, we analyze topological indices and characterize Eulerian and Hamiltonian properties of the resulting shuriken graphs in terms of the properties of the base graphs.
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https://arxiv.org/abs/2601.15783
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405d87a8e617543cedea0697003883953531d3c4d0350e1b4ea7ed403fe0de8f
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2026-01-23T00:00:00-05:00
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Remarks about symmetry-type conditions of conditional bases of Banach spaces
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arXiv:2601.15784v1 Announce Type: new Abstract: We investigate the existence of equivalent p-norms, 0< p 1, under which conditional symmetric or spreading bases in quasi-Banach spaces become isometric. For spreading bases (which need not be unconditional or even Schauder bases), we develop new techniques involving the geometry of spreading sequences and their associated spreading models. We prove that any spreading basis is automatically seminormalized, M-bounded, and uniformly spreading, which allows the construction of an isometric renorming via its spreading model. For symmetric bases, we show they are necessarily spreading and uniformly symmetric, enabling a direct application of a renorming lemma for uniformly bounded semigroups of operators. Consequently, any quasi-Banach space with a symmetric basis admits a renorming making all permutations isometries, and any spreading basis admits a renorming making all increasing maps isometries. These results extend and unify classical isometric renorming theorems for unconditional, subsymmetric, and symmetric Schauder bases to the conditional, non-Schauder setting.
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https://arxiv.org/abs/2601.15784
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a02a0d0f9162459468cbdf52add17204e3323cfbe6e0c9456a56d22ed88e6157
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2026-01-23T00:00:00-05:00
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A half-space Liouville theorem for anisotropic minimal graph with free boundary
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arXiv:2601.15788v1 Announce Type: new Abstract: In this paper we prove the following Liouville-type theorem: any anisotropic minimal graph with free boundary in the half-space must be flat, provided that the graph function has at most one-sided linear growth. This extends the classical results of Bombieri-De Giorgi-Miranda and Simon to an appropriate free boundary setting.
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https://arxiv.org/abs/2601.15788
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15944db2af95a08ba835667f81932ad0162320151490875506d02b0cdf25af6d
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2026-01-23T00:00:00-05:00
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Localization of complementarity eigenvalues
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arXiv:2601.15789v1 Announce Type: new Abstract: Let A, B be symmetric n x n real matrices with B positive definite and strictly diagonally dominant. We derive two localization sets for the complementarity eigenvalues of (A, B), the tightest one assuming additionally that A is copositive. This extends He-Liu-Shen sets to the case where B is not the identity. Moreover, we compare the computable bounds obtained from these new sets with the extreme classical generalized eigenvalues.
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https://arxiv.org/abs/2601.15789
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6b65056df27318dbcda015da53cb5e5180a532936be79ff0a632964674940a4f
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2026-01-23T00:00:00-05:00
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Superpositions of CARMA processes
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arXiv:2601.15796v1 Announce Type: new Abstract: We introduce supCARMA processes, defined as superpositions of L\'evy-driven CARMA processes with respect to a L\'evy basis, as a natural extension of the superpositions of Ornstein-Uhlenbeck type processes. We then focus on supCAR$(2)$ processes and show that they can be classified into three distinct types determined by the eigenstructure of the underlying CAR$(2)$ matrix. For each type we provide conditions for existence and derive explicit expressions for the correlation function. The resulting correlation structures may exhibit long-range dependence and can be non-monotone. These features make supCAR$(2)$ processes a flexible class for modeling time series with oscillatory correlations or strong dependence.
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https://arxiv.org/abs/2601.15796
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85f01118deebb221fc08b394fb5e55a2efac301586d849c67d796fa377075430
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2026-01-23T00:00:00-05:00
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Equivariant linear isometries and infinite little discs operads via transfer systems
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arXiv:2601.15800v1 Announce Type: new Abstract: In this article, we apply the recently developed theory of transfer systems to study the relationship between $G$-equivariant linear isometries and infinite little discs operads, for a finite group $G$. This framework allows us to reduce involved topological problems to discrete problems regarding the subgroup structure and representation theory of the group $G$. Our main result is an example of this: we classify the $G$-universes $\mathcal{U}$ for which the linear isometries operad $\mathcal{L}(\mathcal{U})$ and the infinite little discs operad $\mathcal{D}(\mathcal{U})$ are homotopically equivalent. To achieve this, we use ideas that originate from the work of Balchin-Barnes-Roitzheim on the combinatorics of transfer systems on a total order. Additionally, the use of transfer systems gives us insight into the algebraic structures that arise from equivariant homotopy theory. Compatible pairs of transfer systems provide rules for when multiplicative transfer maps can be paired with additive transfer maps. In the case that the group $G$ is abelian, we provide conditions for when the pair $(\mathcal{L}(\mathcal{U}),\mathcal{D}(\mathcal{U}))$ defines a maximally compatible pair of transfer systems. As a consequence, we contribute to a recent conjecture about equivariant operad pairs.
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https://arxiv.org/abs/2601.15800
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Academic Papers
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dd8c736771324354bf7521debd84a147f008d0f8fd9d11f8a9410e09dad23006
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2026-01-23T00:00:00-05:00
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Optimal stochastic impulse control problem with delay with actions decided at the execution time
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arXiv:2601.15803v1 Announce Type: new Abstract: In this paper, we consider a class of stochastic impulse control problem when there is a fixed delay $\Delta$ between the decision and execution times. The dynamics of the controlled system between two impulses is an arbitrary adapted stochastic process. Unlike the most existing literature, we consider the problem when the impulse sizes are decided at the execution time in both risk-neutral and risk-sensitive cases. This model fits more, in the real life, for some problems such as the pricing of swing options. The horizon T of the problem can be finite or infinite. In each case we show the existence of an optimal strategy. The main tools we use are the notions of reflected Backward Stochastic Differential Equations (BSDEs for short) and the Snell envelope of processes.
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https://arxiv.org/abs/2601.15803
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Academic Papers
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fb7faf767b8681277003d4d58fdd820782aa6708ae41581718243c6ade3e00f7
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2026-01-23T00:00:00-05:00
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New results on Fourier multipliers on $L^p$: a perspective through unimodular symbols
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arXiv:2601.15815v1 Announce Type: new Abstract: The paper focuses on the behaviour of unimodular Fourier multipliers with exponential growth in the context of weighted $L^p$-spaces. Our main result shows that much of the general theory of multipliers is approachable through the theory of unimodular multipliers. Indeed, we show that a bounded measurable function $m$ is a multiplier on $L^p$ for $1\leq p0$ and $0<1$. We then apply this principle to obtain new results related to the boundedness of homogeneous rough operators, singular operators along curves and oscillatory integrals. A key ingredient in our study is an extension of the classical Stein's theorem on analytic families of operators that studies the behaviour of the derivative operator when $\theta \to 0$.
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https://arxiv.org/abs/2601.15815
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Academic Papers
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91acc790d7c1c9a16fd28ee899036d194eb3e389f3018df0c8d4cf1d3660f0da
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2026-01-23T00:00:00-05:00
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Weakly pancyclic vertices in dense nonbipartite graphs
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arXiv:2601.15822v1 Announce Type: new Abstract: Let $G$ be a graph of girth $g$ and circumference $c.$ A vertex $v$ of $G$ is called weakly pancyclic if $v$ lies on an $\ell$-cycle for every integer $\ell$ with $g\le \ell\le c.$ We prove that if $G$ is a nonbipartite graph of order $n\ge 5$ and size at least $\left\lfloor(n-1)^2/4\right\rfloor+2,$ then $G$ contains three weakly pancyclic vertices, with one exception. This strengthens a result of Brandt from 1997. We also pose a related problem.
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https://arxiv.org/abs/2601.15822
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Academic Papers
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a4465d784eba7cf49fdf624884906612da369265f3be0f3a99e81df86831e51b
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2026-01-23T00:00:00-05:00
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Reversibility and symmetry of affine toral automorphisms
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arXiv:2601.15827v1 Announce Type: new Abstract: We study reversibility and strong reversibility of affine automorphisms of the two-torus, written as $f_{A,\bar{a}}(\bar{x})=A\bar{x}+\bar{a} \ (\mathrm{mod}\ \mathbb{Z}^2)$. We derive explicit criteria for the reversibility of such maps in terms of the matrix $A$ and the translation $\bar{a}$. If $1$ is not an eigenvalue of $A$, reversibility of the affine map coincides with reversibility of $A$. When $1$ is an eigenvalue, additional arithmetic obstructions appear. We also provide a simple geometric condition, based on Pick's Theorem, that guarantees the existence of fixed points, along with a description of the dynamics of affine toral automorphisms. We also compute the entropy and characterize when conjugacy classes in the affine group are finite or uncountable.
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https://arxiv.org/abs/2601.15827
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Academic Papers
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26dc9fd75f271f07ab26fa01d686c5bbb96e3a57b22e8fb268b2e7ac63f05024
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2026-01-23T00:00:00-05:00
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$C^\ast$-extreme points of unital completely positive maps invariant under group action
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arXiv:2601.15840v1 Announce Type: new Abstract: In this work, we study a sub-collection of unital completely positive maps from a unital $C^\ast$-algebra $\mathcal{A}$ to $\mathcal{B}(\mathcal{H})$, the algebra of bounded linear operators on a Hilbert space $\mathcal{H}$ in the setting of $C^\ast$-convexity. Let $\tau$ be an action of a group $G$ on the $C^\ast$-algebra $\mathcal{A}$ through $C^\ast$-automorphisms. We focus our attention to the set of all unital completely positive maps from $\mathcal{A}$ to $\mathcal{B}(\mathcal{H})$, which remain invariant under $\tau$. We denote this collection by the notation $\text{UCP}^{G_\tau} \big(\mathcal{A}, \mathcal{B} (\mathcal{H} ) \big)$. This collection forms a $C^\ast$-convex set. We characterize the set of $C^\ast$-extreme points of $\text{UCP}^{G_\tau} \big(\mathcal{A}, \mathcal{B} (\mathcal{H} ) \big)$. Further, we conclude the article by proving the Krein--Milman type theorem in the setting of $C^\ast$-convexity for the set $\text{UCP}^{G_\tau} \big(\mathcal{A}, \mathcal{B} (\mathcal{H} ) \big)$.
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https://arxiv.org/abs/2601.15840
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Academic Papers
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411de06acb1987b6b703ff4c226060d962ecec29b6a990f0941571d86bfb73c1
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2026-01-23T00:00:00-05:00
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Riemann-Hilbert approach for the nonlocal modified Korteweg-de Vries equation with a step-like oscillating background
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arXiv:2601.15841v1 Announce Type: new Abstract: This work focuses on the Cauchy problem for the nonlocal modified Korteweg-de Vries equation $$ u_t(x,t)+6u(x,t)u(-x,-t)u_x(x,t)+u_{xxx}(x,t)=0, $$ with the oscillating step-like boundary conditions: $u(x,t)\to 0$ as $x\to-\infty$ and $u(x,t)\backsimeq A\cos(2Bx+8B^3t)$ as $x\to\infty$, where $A,B>0$ are arbitrary constants. The main goal is to develop the Riemann-Hilbert formalism for this problem, paying a particular attention to the case of the ``pure oscillating step'' initial data, that is $u(x,0)=0$ for $x\frac{A}{4}$, and $B=\frac{A}{4}$.
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https://arxiv.org/abs/2601.15841
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Academic Papers
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5c4cb24272a6cd63f8ebbe61e3f6c7650afbeb384185e23ada580038c2843fdb
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2026-01-23T00:00:00-05:00
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Weak Centrality: AF-algebras, C(X)-algebras, and group C*-algebras
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arXiv:2601.15843v1 Announce Type: new Abstract: We first prove that every AF-algebra is weakly central, thereby resolving a question left open by Archbold--Gogi\'c. We then establish a new characterization of weak centrality for unital $C^*$-algebras in terms of $C(X)$-algebras. The paper concludes with an appendix that examines weak centrality in full group $C^*$-algebras and places these examples within the hierarchy of group classes.
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https://arxiv.org/abs/2601.15843
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Academic Papers
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6b78b303c4ecc461d3499910354376b334dadd3fc51c756fc37a2cc87e7ec49f
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2026-01-23T00:00:00-05:00
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Quantitative Borg-Levinson theorem for the magnetic Sch\"odinger operator with unbounded electrical potential
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arXiv:2601.15847v1 Announce Type: new Abstract: The first author established in [8] a quantitative Borg-Levinson theorem for the Schr\"odinger operator with unbounded potential. In the present work, we extend the results in [8] to the magnetic Schr\"odinger operator. We discuss both the isotropic and anisotropic cases. We establish H\"older stability inequalities of determining the electrical potential or magnetic field from the corresponding boundary spectral data.
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https://arxiv.org/abs/2601.15847
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b2acbeb606a810a96d9d09607a702e624156f8097509a9a6728b0f06b401c866
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2026-01-23T00:00:00-05:00
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Quadratic discrepancy estimates for probability measures on the Heisenberg group
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arXiv:2601.15850v1 Announce Type: new Abstract: We initiate the study of quadratic discrepancy for finite point sets on the Heisenberg group $\mathbb H^n$ with respect to upper Ahlfors regular probability measures. For a natural family of test sets given by left translations and dilations of cylindrically defined neighborhoods, we introduce an $L^2$-discrepancy and establish a Roth-type lower bound depending on the homogeneous dimension of $\mathbb H^n$. This result extends classical discrepancy estimates from the Euclidean and compact settings to a non-commutative, step-two nilpotent Lie group. It should be viewed as a first step toward the development of a discrepancy theory on the Heisenberg group.
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https://arxiv.org/abs/2601.15850
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Academic Papers
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1192e09f68d1daa2f7accf27cdae496f7eaa9bf0caa07fb67111ca3aca05d8e6
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2026-01-23T00:00:00-05:00
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Synthetic Differential Jet Bundles are Reduced
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arXiv:2601.15862v1 Announce Type: new Abstract: We have previously observed that the theory of solutions of partial differential equations, regarded as diffieties inside jet bundles, acquires a powerful comonadic formulation after passage from the category of Fr\'echet smooth manifolds to the Cahiers topos of formal smooth sets (a well-adapted model for Synthetic Differential Geometry). However, the tacit assumption that this passage preserves the projective limits that define infinite jet bundles had remained unproven. Here we provide a detailed proof.
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https://arxiv.org/abs/2601.15862
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59007c6fcb80b77c0e1eb5400d2a4cb5f4d98fb8f82d5b9d3d779416324ddea2
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2026-01-23T00:00:00-05:00
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Tangle structure trees
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arXiv:2601.15870v1 Announce Type: new Abstract: We introduce a comprehensive data structure, tangle structure trees, which simultaneously displays all the $\mathcal{F}$-tangles of an abstract separation system for very general obstruction sets $\mathcal{F}$. It simultaneously also displays certificates $\sigma\in\mathcal{F}$ for any non-existence of such tangles, or for the non-extendability of low-order tangles to higher-order ones. Our theorem can be applied to produce the structures of the classical tree-of-tangles and tangle-tree duality theorems, both for graph tangles and for their known generalizations to more general separation systems. It extends those theorems to obstruction sets $\mathcal{F}$ that need not define profiles (as they must in trees of tangles) or consist of stars of separations (as they must in tangle-tree duality). Our existence proof for these structure trees is constructive. The construction has been implemented in open-source software available for tangle detection and further analysis.
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https://arxiv.org/abs/2601.15870
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Academic Papers
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a9e02732245e4e7d89478ddf6bc0b80bba6b2b9bf08921daf12c43f1ff461d65
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2026-01-23T00:00:00-05:00
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Visibility of Lattice Points across Polynomials
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arXiv:2601.15877v1 Announce Type: new Abstract: The visibility of lattice points from the origin along a polynomial family of curves constitutes a significant generalization of visibility along straight lines. Following the classical notion, where the density equals 1/2, and its generalization to monomial curves of the form y = a x^b, where the density equals 1/(b+1), we study a family of polynomial curves defined by y = q(a_n x^n + ... + a_1 x), where q is a positive rational number. We introduce a new criterion based on a polynomial greatest common divisor condition that provides a lower bound on the number of visible lattice points in N^2. Conversely, we derive conditions under which a given lattice point becomes the next visible point along such a polynomial curve. Using the principle of inclusion-exclusion, we also obtain an exact double-sum formula for the number of pairs (a, b) less than or equal to N that are visible with respect to this polynomial family. Finally, we extend the framework to related problems and pose several open questions concerning gap distributions and quantitative bounds for non-visible points. This work provides a broader theoretical foundation for lattice point visibility beyond linear and monomial settings.
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https://arxiv.org/abs/2601.15877
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Academic Papers
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ee21b2e3e0dfa40d7efc708758c03f374577b4889cf846aff9f33a1600f90476
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2026-01-23T00:00:00-05:00
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Recovery of nonlinear material parameters in a quasilinear Lam\'e system
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arXiv:2601.15881v1 Announce Type: new Abstract: We investigate the inverse problem of determining nonlinear elastic material parameters from boundary stress measurements corresponding to prescribed boundary displacements. The material law is described by a nonlinear, space-independent elastic tensor depending on both the displacement and the strain, and gives rise to a general class of quasilinear Lam\'e systems. We prove the unique and stable recovery of a wide class of space-independent nonlinear elastic tensors, including the identification of two nonlinear isotropic Lam\'e moduli as well as certain anisotropic tensors. The boundary measurements are assumed to be available at a finite number of boundary points and, in the isotropic case, at a single point. Moreover, the measurements are generated by boundary displacements belonging to an explicit class of affine functions. The analysis is based on structural properties of nonlinear Lam\'e systems, including asymptotic expansions of the boundary stress and tensorial calculus.
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https://arxiv.org/abs/2601.15881
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Academic Papers
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4475f97929b3a982cf4799362109a2c4d240e05844bcea45f6f0cfcb900c500f
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2026-01-23T00:00:00-05:00
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Directional polynomial frames on spheres
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arXiv:2601.15883v1 Announce Type: new Abstract: We introduce a general framework for the construction of polynomial frames in $L^2(\mathbb{S}^{d-1})$, $d \geq 3$, where the frame functions are obtained as rotated versions of an initial sequence of polynomials $\Psi^j$, $j\in \mathbb{N}_0$. The rotations involved are discretized using suitable quadrature rules. This framework includes classical constructions such as spherical needlets and directional wavelet systems, and at the same time permits the systematic design of new frames with adjustable spatial localization, directional sensitivity, and computational complexity. We show that a number of frame properties can be characterized in terms of simple, easily verifiable conditions on the Fourier coefficients of the functions $\Psi^j$. Extending an earlier result for zonal systems, we establish sufficient conditions under which the frame functions are optimally localized in space with respect to a spherical uncertainty principle, thus making the corresponding systems a viable tool for position-frequency analyses. To conclude this article, we explicitly discuss examples of well-localized and highly directional polynomial frames.
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https://arxiv.org/abs/2601.15883
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5bcd9969bced4c96285302100326f20477cffd6af314e9541987a379ae3db083
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2026-01-23T00:00:00-05:00
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2-Equivariant 2-Vector bundles and 2K-theories
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arXiv:2601.15893v1 Announce Type: new Abstract: We construct a theory of 2-vector bundles over a Lie groupoid, with fibers modeled by the bicategory of super algebras, bimodules and intertwiners. We demonstrate that these 2-vector bundles form a symmetric monoidal 2-stack. From this structure, we define the 2K-theory as the Grothendieck group of the internal equivalence classes of the 2-vector bundle over the given Lie groupoid, and we construct the spectra representing this theory. We then extend this framework to the equivariant setting. For any Lie groupoid equipped with an action by a coherent 2-group, we introduce the bicategory of 2-equivariant 2-vector bundles over it. This leads to the definition of 2-equivariant 2K-theory as the Grothendieck group of the internal equivalence classes in the bicategory. Furthermore, we define a higher analogue of orbifold, which generalizes Lie groupoids with a 2-group action, and construct the bicategory of 2-orbifold 2-vector bundles. Finally, we can define the 2-orbifold 2K-theory.
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https://arxiv.org/abs/2601.15893
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Academic Papers
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e513c7162f742856303b155421dfd0fc3df03bdeef23aca5b4c898020a34eab7
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2026-01-23T00:00:00-05:00
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Convergence to shock profiles for Burgers equation with singular fast-diffusion and boundary effect
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arXiv:2601.15900v1 Announce Type: new Abstract: In this paper, we study the asymptotic stability of viscous shock profile for the Burgers equation $u_t +f(u)_x = (\frac{u_{x}}{u^{1-m}})_x$ on the half-space $(0,+\infty)$, subject to the boundary conditions $u|_{x=0}=u_->0$ and $u|_{x=+\infty}=0$. Here, the parameter $\frac{1}{2}<1$ measures the strength of fast diffusion. A key challenge arises from the pronounced singularity in the diffusivity $\left(\frac{u_x}{u^{1-m}} \right)_x$ at $u=0$ and the boundary layer. We demonstrate that the long-time behavior of $u$ converges to a shifted shock profile $U(x-st-d(t))$, where $d(t)$ is governed by the boundary layer dynamics at $x=0$ and driven by the initial data $u(x,0)$. To overcome the singularity from fast diffusion compounded by the bad effect of boundary layer for wave stability, some new techniques for weighted energy estimates are introduced artfully.
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https://arxiv.org/abs/2601.15900
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Academic Papers
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2c3b9e1ad08f745643f585d7693b963385dd2ba7627cee22b84316b95dc22435
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2026-01-23T00:00:00-05:00
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Metric constructions and fixed point theorems in product spaces
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arXiv:2601.15907v1 Announce Type: new Abstract: The paper studies a general scheme for constructing metrics on a product of metric spaces by means of a family of continuous convex functions. This construction includes the conventional $p$-metrics and generates metrics that are topologically equivalent to the conventional ones. As an application, we study fixed point and approximate fixed point properties for nonexpansive maps on a product space equipped with the constructed metric. We show that existing fixed point results of this type are consequences of our framework. Examples are provided to illustrate the established results. The construction machinery is also used to study products of length and geodesic spaces. The obtained results encompass existing ones and provide a background for potential studies of fixed point properties on these product spaces.
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https://arxiv.org/abs/2601.15907
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Academic Papers
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273166b908ec31e5e29cf0c95137a2e663dee791c109dd7ebfcc2c9fcb93b2ac
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2026-01-23T00:00:00-05:00
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On the escape rate for intermittent maps with holes shrinking around the indifferent fixed point
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arXiv:2601.15908v1 Announce Type: new Abstract: We study non-uniformly expanding maps of the unit interval with a parabolic fixed point at the origin that admit an ergodic absolutely continuous invariant measure, which may be finite or infinite. By introducing a hole defined by an interval containing the parabolic fixed point, we analyze the escape rate of the resulting open system and its asymptotic behavior as the hole shrinks. Our approach relies on the transfer operator associated with the dynamical system and on the relationship between the transfer operators of the original system and its induced version. The results extend to this general framework previous investigations which considered special cases.
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https://arxiv.org/abs/2601.15908
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Academic Papers
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