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b9856d8baea09787b49b9f69244a26492de101c0fae7ce5cbbf57df5a9473931
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2026-01-01T00:00:00-05:00
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Enhanced premelting at the ice-rubber interface using all-atom molecular dynamics simulation
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arXiv:2508.20448v2 Announce Type: replace-cross Abstract: The ice-rubber interface is critical in applications such as tires and shoe outsoles, yet its molecular tribology remains unclear. Using all-atom molecular dynamics simulations, we studied premelting layers at the basal face of ice in contact with styrene-butadiene rubber from 254 to 269 K. Despite its hydrophobicity, rubber enhances structural disorder of interfacial water, promoting premelting. In contrast, water mobility is suppressed by confinement from polymer chains, leading to glassy dynamics distinct from the ice-vapor interface. Near the melting point, rubber chains become more flexible and penetrate the premelting layer, forming a mixed rubber-water region that couples the dynamics of both components. These results suggest that nanoscale roughness and morphology of hydrophobic polymers disrupt ice hydrogen-bond networks, thereby enhancing premelting. Our findings provide molecular-level insight into ice slipperiness and inform the design of polymer materials with controlled ice adhesion and friction.
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https://arxiv.org/abs/2508.20448
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Academic Papers
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a80922ac2fd2e44f4faef128c77cbfe66173cbdd4b1b4728bafdb9897b7dbc9e
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2026-01-01T00:00:00-05:00
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Setting limits on blazar-boosted dark matter with xenon-based detectors
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arXiv:2509.07265v2 Announce Type: replace-cross Abstract: Dual-phase xenon time-projection chambers achieve optimal sensitivity to dark matter in the mass range from about 10 to 1000~GeV/$c^{2}$. However, sub-GeV dark-matter particles do not produce nuclear recoils above detection thresholds in these detectors. Blazar-boosted dark matter provides a way to overcome this limitation: relativistic jets in active galactic nuclei can accelerate light dark matter in their host-galaxy halos to energies capable of producing detectable nuclear-recoil signals in xenon-based detectors on Earth. We present the first blazar-boosted dark-matter search that incorporates full detector-response modeling, using public data from XENON1T and LZ for the blazar TXS 0506+056. We model dark matter-proton scattering in the jet environment, tracing the full process from acceleration in the jet to the detector response on Earth, and we investigate the impact of the host-galaxy dark-matter density profile on the predicted signals. We set model-dependent exclusion regions on the dark matter-nucleon scattering cross section for dark matter with mass $m_\chi \simeq 1~\mathrm{MeV}$. Using XENON1T data, the excluded cross-section range spans approximately $5.8\times10^{-31}$ to $6.3\times10^{-29}~\mathrm{cm}^{2}$, while LZ effective-field-theory searches exclude cross sections between $9.9\times10^{-32}$ and $2.5\times10^{-28}~\mathrm{cm}^{2}$. Our results show that astrophysical uncertainties -- particularly those associated with the dark-matter distribution near the supermassive black hole -- are the dominant limitation of this search, rather than detector-related effects. The resulting limits are therefore model-dependent and should be regarded as exploratory, highlighting both the potential and the present theoretical uncertainties of blazar-boosted dark matter as a probe of light dark matter.
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https://arxiv.org/abs/2509.07265
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cce9bd41d1b27114a017c8c59959624d3c690c8f362af36cc066ceca17cce4f9
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2026-01-01T00:00:00-05:00
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Constraints on inelastic dark matter from the CDEX-1B experiment
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arXiv:2510.07800v2 Announce Type: replace-cross Abstract: We present limits on spin-independent inelastic weakly interacting massive particles (WIMP)-nucleus scattering using the 737.1 kg$\cdot$day dataset from the CDEX-1B experiment. Expected nuclear recoil spectra for various inelastic WIMP masses $m_\chi$ and mass splittings $\delta$ are calculated under the standard halo model. An accurate background model of CDEX-1B is constructed by simulating all major background sources. The model parameters are then determined through maximum likelihood estimation and Markov chain Monte Carlo fitting. The resulting 90\% confidence level upper limits on the WIMP-nucleon cross section $\sigma_{\mathrm{n}}$ exclude certain DAMA/LIBRA allowed regions: the $\chi^2 < 4$ regions for $\delta < 30$ keV at $m_\chi = 250$ GeV and the $\chi^2 < 9$ region for $\delta < 50$ keV at $m_\chi = 500$ GeV. The method is applicable to other inelastic dark matter scenarios, and the upcoming CDEX-50 experiment is expected to improve sensitivity by four orders of magnitude.
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https://arxiv.org/abs/2510.07800
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12cf59885d43ba134e16a14ded9569eece88959dc77436a8275efb670e45fc5d
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2026-01-01T00:00:00-05:00
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GoodRegressor: A General-Purpose Symbolic Regression Framework for Physically Interpretable Materials Modeling
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arXiv:2510.18325v4 Announce Type: replace-cross Abstract: Symbolic regression offers a promising route toward interpretable machine learning, yet existing methods suffer from poor predictability and computational intractability when exploring large expression spaces. I introduce GoodRegressor, a general-purpose C++-based framework that resolves these limitations while preserving full physical interpretability. By combining hierarchical descriptor construction, interaction discovery, nonlinear transformations, statistically rigorous model selection, and stacking ensemble, GoodRegressor efficiently explores symbolic model spaces such as $1.44 \times 10^{457}$, $5.99 \times 10^{124}$, and $4.20 \times 10^{430}$ possible expressions for oxygen-ion conductors, NASICONs, and superconducting oxides, respectively. Across these systems, it produces compact equations that surpass state-of-the-art black-box models and symbolic regressors, improving $R^2$ by $4 \sim 40$ %. The resulting expressions reveal physical insights, for example, into oxygen-ion transport through coordination environment and lattice flexibility. Independent ensemble runs yield nearly identical regressed values and the identical top-ranked candidate, demonstrating high reproducibility. With scalability up to $10^{4392}$ choices without interaction terms, GoodRegressor provides a foundation for general-purpose interpretable machine intelligence.
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https://arxiv.org/abs/2510.18325
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671962be6fd2d0dec8164eb9811890733b808f9a424193ea08359ed74c0d28e4
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2026-01-01T00:00:00-05:00
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A mini-review on combinatorial solutions to the Marcus-Lushnikov irreversible aggregation
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arXiv:2512.11459v3 Announce Type: replace-cross Abstract: Over the past decade, a combinatorial framework for discrete, finite, and irreversibly aggregating systems has emerged. This work reviews its progress, practical applications, and limitations. We outline the approach's assumptions and foundations, based on direct enumeration of system states, contrasting with classical Smoluchowski and Marcus-Lushnikov methods. Using the constant kernel as an example, we derive combinatorial expressions for the average number of clusters of a given size and their standard deviation, and present the complete probability distribution for cluster counts. The method is then extended to several kernels (additive, product, linear-chain, condensation) by explicitly enumerating ways to form clusters of a given size. For general kernels, approximate solutions are obtained via recursive expressions, enabling predictions without explicit solutions. Applications to aerosol growth and planetesimal formation are demonstrated, with comparisons to numerical results. We summarize issues of validity and precision and propose open problems. The appendix includes partial Bell polynomials, generating functions, Lagrange inversion, potential applications, and links between combinatorial and scaling solutions of the Smoluchowski equation.
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https://arxiv.org/abs/2512.11459
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280c619e123e510c3f02c5d8e36d5e35f7aa562ee196599e672d7a0547432fbe
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2026-01-01T00:00:00-05:00
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Measuring the time-scale-dependent information flow between maternal and fetal heartbeats during the third trimester
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arXiv:2512.22270v2 Announce Type: replace-cross Abstract: Prenatal maternal stress alters maternal-fetal heart rate coupling, as demonstrated by the Fetal Stress Index derived from bivariate phase-rectified signal averaging. Here, we extend this framework using information-theoretical measures to elucidate underlying mechanisms. In 120 third-trimester pregnancies (58 stressed, 62 control), we computed transfer entropy (TE), entropy rate (ER), and sample entropy (SE) under multiple conditioning paradigms, employing mixed linear models for repeated measures. We identify dual coupling mechanisms at the short-term (0.5 - 2.5 s), but not long-term (2.5 - 5 s) time scales: (1) stress-invariant state-dependent synchronization, with maternal decelerations exerting approximately 60% coupling strength on fetal heart rate complexity - a fundamental coordination conserved across demographics; and (2) stress-sensitive temporal information transfer (TE), showing exploratory associations with maternal cortisol that require replication. A robust sex-by-stress interaction emerged in TE from mixed models, with exploratory female-specific coupling patterns absent in males. Universal acceleration predominance was observed in both maternal and fetal heart rates, stronger in fetuses and independent of sex or stress. We provide insight into the dependence of these findings on the sampling rate of the underlying data, identifying 4 Hz, commonly used for ultrasound-derived fetal heart rate recordings, as the necessary and sufficient sampling rate regime to capture the information flow. Information-theoretical analysis reveals that maternal-fetal coupling operates through complementary pathways with differential stress sensitivity, extending the Fetal Stress Index by elucidating causal foundations. Future studies should explore additional information-theoretical conditional approaches to resolve stress-specific and time-scale-specific differences in information flow.
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https://arxiv.org/abs/2512.22270
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44ebdab467d44f6d15d10c0aeb464f30f08609d542c639ec00bbb2abe53305c3
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2026-01-01T00:00:00-05:00
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Pair Space in Classical Mechanics III. Some Four-Body Central Configurations
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arXiv:2512.23730v1 Announce Type: new Abstract: We study central configurations in the four body problem, i.e., configurations in which the forces on all the bodies point to a fixed, single point in space. The newly formulated pair-space formalism yields a set of vectorial equations that fully characterize such configurations. We investigate a sub-class of solutions in which at least two pairs of inter-body distances are equal. The only such non-collinear configurations are the tetrahedron (the unique non-planar configuration), kites and the isosceles trapezium. The specific shapes (internal angles) are determined by the ratio of the masses of the bodies. Mathematical expression are given for all these relations.
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https://arxiv.org/abs/2512.23730
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930078b4fafdfb172cc61df0a19b380872abe23822adf07adf8db33588ae523c
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2026-01-01T00:00:00-05:00
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Solvability of the B\'ezout Equation for Banach Algebra-Valued $H^\infty$ Functions on the Polydisk
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arXiv:2512.23733v1 Announce Type: new Abstract: In connection with the still unsolved multidimensional corona problem for algebras of bounded holomorphic functions on convex domains, we study the solvability of the B\'ezout equation for the algebra of bounded holomorphic functions on the polydisk with values in a complex Banach algebra. Assuming local solvability of the B\'ezout equation on a special open cover of the maximal ideal space of the algebra, we combine a dimension-induction scheme with a careful analysis of the topological structure of this space to glue local solutions into a global one. As a corollary, we obtain the solvability of the B\'ezout equation for a broader class of subalgebras containing the slice algebra of bounded holomorphic functions, the case of the latter having been previously proved by the first author
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https://arxiv.org/abs/2512.23733
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ad8a5f418e7c43c8391a149e0fb4b31e35acce428f025342e1c582d55fd5c039
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2026-01-01T00:00:00-05:00
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Linear Preservers of Real Matrix Classes Admitting a Real Logarithm
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arXiv:2512.23735v1 Announce Type: new Abstract: In real Lie theory, matrices that admit a real logarithm reside in the identity component $\mathrm{GL}_n(\mathbb{R})_+$ of the general linear group $\mathrm{GL}_n(\mathbb{R})$, with logarithms in the Lie algebra $\mathfrak{gl}_n(\mathbb{R})$. The exponential map \[ \exp : \mnr \to \mathrm{GL}_n(\mathbb{R}) \] provides a fundamental link between the Lie algebra and the Lie group, with the logarithm as its local inverse. In this paper, we characterize all bijective linear maps $\varphi : \mnr \to \mnr$ that preserve the class of matrices admitting a real logarithm (principal logarithm). We show that such maps are exactly those of the form \[ \varphi(A) = c\, P A P^{-1} \quad \text{or} \quad \varphi(A) = c\, P A^{T} P^{-1}, \] for some $P \in \mathrm{GL}_n(\mathbb{R})$ and $c > 0$. The proof proceeds in two stages. First, we analyze preservers within the class of standard linear transformations. Second, using Zariski denseness, we prove that any bijective linear map preserving matrices with real logarithms (principal logarithm) must preserve $\mathrm{GL}_n(\mathbb{R})$, which then implies the map is of the standard form.
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https://arxiv.org/abs/2512.23735
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9d021e96cef4d363e55a95fe94275acda0d9b0a8ed62e84d865e6d86e78ebeae
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2026-01-01T00:00:00-05:00
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A Discrete Logarithm Construction for Orthogonal Double Covers of the Complete Graph by Hamiltonian Paths
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arXiv:2512.23802v1 Announce Type: new Abstract: During their investigation of power-sequence terraces, Anderson and Preece briefly mention a construction of a terrace for the cyclic group $\mathbb{Z}_n$ when $n$ is odd and $2n+1$ is prime; it is built using the discrete logarithm modulo $2n+1$. In this short note we see that this terrace gives rise to an orthogonal double cover (ODC) for the complete graph $K_n$ by Hamiltonian paths. This gives infinitely many new values for which such an ODC is known.
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https://arxiv.org/abs/2512.23802
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38002830fad0b2160f880ed5eb8a16ff95cc89e31fef173c43e221fa30fd4241
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2026-01-01T00:00:00-05:00
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A new kind of automorphic form and a proof of the essential transformation laws
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arXiv:2512.23823v1 Announce Type: new Abstract: We utilize the structure of quasiautomorphic forms over an arbitrary Hecke triangle group to define a new vector analogue of an automorphic form. We supply a proof of the functional equations that hold for these functions modulo the group generators.
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https://arxiv.org/abs/2512.23823
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70c10e07957a75b331aa655a4b9986946af5d04bd98a2506799f0410fd0a4d9e
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2026-01-01T00:00:00-05:00
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Gradings on the Hecke category, and categorification with unequal parameters
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arXiv:2512.23827v1 Announce Type: new Abstract: We classify gradings on the Hecke category that refine the standard integer grading. We also classify object-preserving autoequivalences of the Hecke category. We obtain a natural bigrading on the Hecke category which is related to the Frobenius automorphism. We also obtain an exotic grading in special characteristic that can be used to categorify many Hecke algebras with unequal parameters, including all Hecke algebras with unequal parameters for all finite and affine Weyl groups. This paper is a replacement for arXiv:2305.08278, which is now obsolete.
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https://arxiv.org/abs/2512.23827
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cc1d76caaf755ff3a3193e4fc356213922fa114c9419ec4d2344c60d4e13f7b9
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2026-01-01T00:00:00-05:00
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Hoffman-London graphs: When paths minimize $H$-colorings among trees
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arXiv:2512.23828v1 Announce Type: new Abstract: Given a graph $G$ and a target graph $H$, an $H$-coloring of $G$ is an adjacency-preserving vertex map from $G$ to $H$. The number of $H$-colorings of $G$, $\hom(G,H)$, has been studied for many classes of $G$ and $H$. In particular, extremal questions of maximizing and minimizing $\hom(G,H)$ have been considered when $H$ is a clique or $G$ is a tree. In this paper, we develop a new technique using automorphisms of $H$ to show that $\hom(T,H)$ is minimized by paths as $T$ varies over trees on a fixed number of vertices. We introduce the term Hoffman-London to refer to graphs that are minimal in this sense. In particular, we define an automorphic similarity matrix which is used to compute $\hom(T,H)$ and give matrix conditions under which $H$ is Hoffman-London. We then apply this technique to identify several families of graphs that are Hoffman-London, including loop threshold graphs and some with applications in statistical physics (e.g. the Widom-Rowlinson model). By combining our approach with a few other observations, we fully characterize the minimizing trees for all graphs $H$ on three or fewer vertices.
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https://arxiv.org/abs/2512.23828
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c108a21b53efc03a30dfa1bd4afa0e3d0e486b2cde2d5a7733aa35baa947aca9
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2026-01-01T00:00:00-05:00
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Fractal Mehler kernels and nonlinear geometric flows
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arXiv:2512.23830v1 Announce Type: new Abstract: In this paper we introduce a two-parameter family of Mehler kernels and connect them to a class of Baouendi-Grushin flows in fractal dimension. We also highlight a link with a geometric fully nonlinear equation and formulate two questions.
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https://arxiv.org/abs/2512.23830
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55e4d6b84b06bdb38bb1d8372a3d1e2b514f8aee0a88f2489a05bbfdaf42322b
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2026-01-01T00:00:00-05:00
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Absolutely partially hyperbolic surface endomorphisms are dynamically coherent
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arXiv:2512.23831v1 Announce Type: new Abstract: We show that if an endomorphism $f:\mathbb{T}^2 \to \mathbb{T}^2$ is absolutely partially hyperbolic, then it has a center foliation. Moreover, the center foliation is leaf conjugate to that of its linearization.
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https://arxiv.org/abs/2512.23831
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7f9413cab2366124a11f6add197252d531ca9c61654b1df721fde26f764dfbca
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2026-01-01T00:00:00-05:00
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Prime ideals in the Boolean polynomial semiring
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arXiv:2512.23839v1 Announce Type: new Abstract: In this article, we disprove a conjecture of F. Alarc\'on and D. Anderson and give a complete classification of the prime ideals in the one variable polynomial semiring with coefficients in Boolean semifield. We group the prime ideals of $\mathbb{B}[x]$ into three classes, indexed by integers.
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https://arxiv.org/abs/2512.23839
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d108f5aa90e340bc5a2954e30e2b03e6be79069c3bb0aeb4de08f314b3d2a3be
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2026-01-01T00:00:00-05:00
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Multigraphs and Time Ordered Isserlis-Wick formulae
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arXiv:2512.23845v1 Announce Type: new Abstract: Given a m-dimensional Gaussian process and polynomial m variables with real coefficients, we calculate the induced path odered exponenial in two different ways: one is purely algebraic in spirit and the other one is diagrammatic in spirit and uses multigraph labelings (and is inspired by the use of Feynman diagrams in quantum field theory).
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https://arxiv.org/abs/2512.23845
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b4d632242d82ad80547918fe7d5f1abf3743cb5ceb1d22d9942ece0f2c3803ab
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2026-01-01T00:00:00-05:00
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A positive eigenvalue result for semilinear differential equations in Banach spaces with functional initial conditions
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arXiv:2512.23876v1 Announce Type: new Abstract: We study the existence of positive eigenvalues with associated nonnegative mild eigenfunctions for a class of abstract initial value problems in Banach spaces with functional, possibly nonlocal, initial conditions. The framework includes periodic, multipoint, and integral average conditions. Our approach relies on nonlinear analysis, topological methods, and the theory of strongly continuous semigroups, yielding results applicable to a wide range of models. As an illustration, we apply the abstract theory to a reaction-diffusion equation with a nonlocal initial condition arising from a heat flow problem.
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https://arxiv.org/abs/2512.23876
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d3570d7c8bf9eac3c809918b0832df6c7d33654ab4890c2cffb1d340be3cb679
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2026-01-01T00:00:00-05:00
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Powers of Hamiltonian cycles in randomly augmented P\'osa-Seymour graphs
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arXiv:2512.23886v1 Announce Type: new Abstract: We study the question of the least number of random edges that need to be added to a P\'osa-Seymour graph, that is, a graph with minimum degree exceeding $\frac k{k+1}n$, to secure the existence of the $m$-th power of a Hamiltonian cycle, $m>k$. It turns out that, depending on $k$ and $m$, this quantity may be captured by two types of thresholds, with one of them, called over-threshold, becoming dominant for large $m$. Indeed, for each $k\ge2$ and $m>m_0(k)$, we establish asymptotically tight lower and upper bounds on the over-thresholds (provided they exist) and show that for infinitely many instances of $m$ the two bounds coincide. In addition, we also determine the thresholds for some small values of $k$ and $m$.
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https://arxiv.org/abs/2512.23886
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a8b1f9825ab6d230351270998648a1dc69c6b75b0210f566f971ed1e6585e1e8
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2026-01-01T00:00:00-05:00
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Tree-independence number VII. Excluding a star
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arXiv:2512.23887v1 Announce Type: new Abstract: We prove that for every fixed integer $s$ and every planar graph $H$, the class of $H$-induced-minor-free and $K_{1,s}$-induced-subgraph-free graphs has polylogarithmic tree-independence number. This is a weakening of a conjecture of Dallard, Krnc, Kwon, Milani\v{c}, Munaro, \v{S}torgel, and Wiederrecht.
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https://arxiv.org/abs/2512.23887
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73e078df47d1d5cf56367019418c1dcee873bd1f18bd0741945e0688872df4e6
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2026-01-01T00:00:00-05:00
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Algorithms for numerical semigroups with fixed maximum primitive
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arXiv:2512.23891v1 Announce Type: new Abstract: We present an algorithm to explore various properties of the numerical semigroups with a given maximum primitive. In particular, we count the number of such numerical semigroups and verify that there is no counterexample to Wilf's conjecture among the numerical semigroups with maximum primitive up to \(60\).
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https://arxiv.org/abs/2512.23891
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66cfb96dc7f63a8ef0cb95f84c2a65e7cd438a26065082ef88d56d3b645c3399
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2026-01-01T00:00:00-05:00
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On $GL(1|1)$ Higgs bundles
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arXiv:2512.23909v1 Announce Type: new Abstract: We investigate the moduli space of holomorphic $GL(1|1)$ Higgs bundles over a compact Riemann surface. The supergroup $GL(1|1)$, the simplest non-trivial example beyond abelian cases, provides an ideal setting for developing supergeometric analogues of classical results in Higgs bundle theory. We derive an explicit description of the moduli space and we study the analogue of the Narasimhan-Seshadri theorem as well as the nonabelian Hodge correspondence. Furthermore, we formulate and solve the corresponding Hitchin equations, demonstrating their compatibility with fermionic contributions. As a highlight, we discuss the related Hitchin system on $\mathbb{P}^1$ and its integrability.
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https://arxiv.org/abs/2512.23909
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1082e21e456b62793118b3ff8a7c30a009110c45a30ad5f8aecdf06b882d7b6f
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2026-01-01T00:00:00-05:00
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From Stable Rank One to Real Rank Zero: A Note on Tracial Approximate Oscillation Zero
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arXiv:2512.23911v1 Announce Type: new Abstract: We present a relation between stable rank one and real rank zero via the method of tracial oscillation. Let $A$ be a simple separable $C^*$-algebra of stable rank one. We show that $A$ has tracial approximate oscillation zero and, as a consequence, the tracial sequence algebra $l^\infty(A)/J_A$ has real rank zero, where $J_A$ is the trace-kernel ideal with respect to 2-quasitraces. We also show that for a $C^*$-algebra $B$ that has non-trivial 2-quasitraces, $B$ has tracial approximate oscillation zero is equivalent to $l^\infty(B)/J_B$ has real rank zero.
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https://arxiv.org/abs/2512.23911
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a829950c5dbeea42a1962d5011190a1d364181c625324a4a2dca2acb751d30f4
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2026-01-01T00:00:00-05:00
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$p$-Adic $\lambda$ Functions for Cyclic Mumford Curve
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arXiv:2512.23913v1 Announce Type: new Abstract: We express the branch points cross ratio of cyclic Mumford curves as quotients of $p$-adic theta functions evaluated at the p-adic period matrix
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https://arxiv.org/abs/2512.23913
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c4b396fba7947c943cf1b008115568b304a7944c0b4375bf79d2bd0c7912b96f
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2026-01-01T00:00:00-05:00
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The probability of isomorphic group structures of isogenous elliptic curves over finite fields
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arXiv:2512.23921v1 Announce Type: new Abstract: Let l be a prime number and let E and E' be l-isogenous elliptic curves defined over Q. In this paper we determine the proportion of primes p for which E(F_p) is isomorphic to E'(F_p). Our techniques are based on those developed in \cite{ck} and \cite{rnt}.
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https://arxiv.org/abs/2512.23921
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268e98d01279af5f6c30099e5c582583ef8773133290b2bed168e5b6ed49db4a
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2026-01-01T00:00:00-05:00
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Stable envelopes for critical loci
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arXiv:2512.23929v1 Announce Type: new Abstract: This is the first in a sequence of papers devoted to stable envelopes in critical cohomology and critical $K$-theory for symmetric GIT quotients with potentials and related geometries, and their applications to geometric representation theory and enumerative geometry. In this paper, we construct critical stable envelopes and establish their general properties, including compatibility with dimensional reductions, specializations, Hall products, and other geometric constructions. In particular, for tripled quivers with canonical cubic potentials, the critical stable envelopes reproduce those on Nakajima quiver varieties. These set up foundations for applications in subsequent papers.
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https://arxiv.org/abs/2512.23929
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6240769cfee07aeb70ad686de8b70190b0ec6257dfa9af3bc386101307659530
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2026-01-01T00:00:00-05:00
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On strongly multiplicative sets
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arXiv:2512.23935v1 Announce Type: new Abstract: A multiplicative subset S of a ring R is called strongly multiplicative if (\cap_{i \in \Delta} s_i R) \cap S is non-empty for each family (s_i)_{i \in \Delta} of S. In this paper, we study how these sets help stabilize localization and ideal operations. We show that localization and arbitrary intersections commute, meaning S^{-1}(\cap I_\alpha) = \cap S^{-1}I_\alpha$ for any family of ideals, if and only if S is strongly multiplicative. Furthermore, we characterize total quotient rings and strongly zero-dimensional rings in terms of strongly multiplicative sets. We also answer an open question by Hamed and Malek about whether this condition is needed for S-minimal primes to exist. In addition, we prove a Strong Krull's Separation Lemma, which guarantees a maximal ideal disjoint from S. Finally, we demonstrate that if S is a strongly multiplicative set and S is not contained in U(R), then S-minimal primes are not classical prime ideals, and we provide an algorithmic approach to constructing such ideals.
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https://arxiv.org/abs/2512.23935
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c190a4758ea01baa63afd23861683307acf433eaa7d0ea28b67272f6963b33f0
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2026-01-01T00:00:00-05:00
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Positive specializations of K-theoretic Schur P- and Q-functions
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arXiv:2512.23944v1 Announce Type: new Abstract: Yeliussizov has classified the positive specializations of symmetric Grothendieck functions, defined in several different ways, providing a K-theoretic lift of the classical Edrei-Thoma theorem. This note studies the analogous classification problem for Ikeda and Naruse's K-theoretic Schur P- and Q-functions, which are the shifted versions of symmetric Grothendieck functions. Our results extend a shifted variant of the Edrei-Thoma theorem due to Nazarov. We also discuss an application to the problem of determining the extreme harmonic functions on a filtered version of the shifted Young lattice.
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https://arxiv.org/abs/2512.23944
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2122f122e5154f7b78482ebf09ee7baf7470707a436ae7a01eb7d0da145c0e90
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2026-01-01T00:00:00-05:00
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A nonlinear instability result to the Navier-Stokes equations with Navier slip boundary conditions
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arXiv:2512.23946v1 Announce Type: new Abstract: In this paper, we investigate the instability of the trivial steady states to the incompressible viscous fluid with Navier-slip boundary conditions. For the linear instability, the existence of infinitely many normal mode solutions to the linearized equations is shown via the operator method of Lafitte and Nguyen (2022). Hence, we prove the nonlinear instability by adapting the framework of Desjardins and Grenier (2003) studying some classes of viscous boundary layers to obtain two separated solutions at escaping time. Our work performs a different approach from that of Ding, Li and Xin (2018).
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https://arxiv.org/abs/2512.23946
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854bc5763193fd46a3d9fcf80f11a69e40ffc9134463374ffce0ea86469b5347
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2026-01-01T00:00:00-05:00
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Comonotone approximation and interpolation by entire functions II
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arXiv:2512.23949v1 Announce Type: new Abstract: A theorem of Hoischen states that given a positive continuous function $\varepsilon:\mathbb{R}\to\mathbb{R}$, an integer $n\geq 0$, and a closed discrete set $E\subseteq\mathbb{R}$, any $C^n$ function $f:\mathbb{R}\to\mathbb{R}$ can be approximated by an entire function $g$ so that for $k=0,\dots,n$, and $x\in\mathbb{R}$, $|D^{k}g(x)-D^{k}f(x)|<\varepsilon(x)$, and if $x\in E$ then $D^{k}g(x)=D^{k}f(x)$. The approximating function $g$ is entire and hence piecewise monotone. Building on earlier work, for $n\leq 3$, we determine conditions under which when $f$ is piecewise monotone we can choose $g$ to be comonotone with $f$ (increasing and decreasing on the same intervals), and under which the derivatives of $g$ can be taken to be comonotone with the corresponding derivatives of $f$ if the latter are piecewise monotone. The proof for $n\leq 3$ establishes the theorem for all $n$, assuming a conjecture (shown in previous work with Haris and Madhavendra to hold for $n\leq 3$) regarding the set of $2(n+1)$-tuples $(f(0),Df(0),\dots,D^nf(0),f(1),Df(1),\dots,D^nf(1))$ of the values at the endpoints of the derivatives of a $C^n$ function $f$ on $[0,1]$ for which $D^nf$ is increasing and not constant.
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https://arxiv.org/abs/2512.23949
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2120bbe557065e0437bc59d1f249bd6adaa59880676f9bc3eec7c9792329e0e6
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2026-01-01T00:00:00-05:00
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Concentration and fluctuations of sine-Gordon measure around topological multi-soliton manifold
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arXiv:2512.23957v1 Announce Type: new Abstract: We study the sine-Gordon measure defined on each homotopy class. The energy space decomposes into infinitely many such classes indexed by the topological degree $Q \in \mathbb{Z}$. Even though the sine-Gordon action admits no minimizer in homotopy classes with $|Q| \ge 2$, we prove that the Gibbs measure on each class nevertheless concentrates and exhibits Ornstein-Uhlenbeck fluctuations near the multi-soliton manifold in the joint low-temperature and infinite-volume limit. Furthermore, we show that the joint distribution of the multi-soliton centers coincides with the ordered statistics of independent uniform random variables, so that each soliton's location follows a Beta distribution.
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https://arxiv.org/abs/2512.23957
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0e8b582992da3801d10a19cc2963d15fd7a6773d09f7549361ca088f1bd193b9
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2026-01-01T00:00:00-05:00
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Adjoint L-Infinity Actions and Conserved Charges in GR
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arXiv:2512.23970v1 Announce Type: new Abstract: In this work we compute the conserved currents and charges associated to the action of an infinitesimal isometry (Killing field) in Einstein--Cartan--Palatini gravity. We offer a new approach to these quantities through the formalism of $L_\infty$-algebras and the work of \'{C}iri\'{c}, Giotopoulos, Radovanovi\'{c}, and Szabo, and Costello and Gwilliam. We demonstrate our approach by computing the entropy of the Schwarzchild black hole. Along the way, we prove a purely algebraic result about the existence and utility of a higher (a full $\infty$) version of the adjoint action of an $L_\infty$-algebra.
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https://arxiv.org/abs/2512.23970
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8885da2551a73cf7879f123099763eca80fe5cae9762ed62a200c43d47b122cc
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2026-01-01T00:00:00-05:00
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Sub-structure in module category of Virasoro vertex operator algebras $L(c_{p,q}, 0)$
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arXiv:2512.23980v1 Announce Type: new Abstract: We determine all premodular subcategories and modular tensor subcategories in the module categories of Virasoro vertex operator algebras $L(c_{p,q},0)$ and the module categories of the simple current extensions of $L(c_{p,p+1},0)$.
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https://arxiv.org/abs/2512.23980
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f643c84fc9038b4f5da94ac3ac1d3a65b3a8bcc6ed44adaa7cb3b6945b3eca5d
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2026-01-01T00:00:00-05:00
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A Short Proof that Every Claw-Free Cubic Graph is (1,1,2,2)-Packing Colorable
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arXiv:2512.24001v1 Announce Type: new Abstract: It was recently proved that every claw-free cubic graph admits a (1, 1, 2, 2)-packing coloring--that is, its vertex set can be partitioned into two 1-packings and two 2-packings. This result was established by Bre\v{s}ar, Kuenzel, and Rall [Discrete Mathematics 348 (8) (2025), 114477]. In this paper, we provide a simpler and shorter proof.
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https://arxiv.org/abs/2512.24001
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3d778db66eaeb6bec402e38dfba78cc583cfaea01d4db8542d10645015470609
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2026-01-01T00:00:00-05:00
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Integrality of a trigonometric determinant arising from a conjecture of Sun
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arXiv:2512.24012v1 Announce Type: new Abstract: In this paper we resolve a conjecture of Zhi-Wei Sun concerning the integrality and arithmetic structure of certain trigonometric determinants. Our approach builds on techniques developed in our previous work, where trigonometric determinants were studied via special values of Dirichlet $L$-functions. The method is refined by establishing a connection between odd characters modulo $4n$ and even characters modulo $n$. The results highlight a close connection between trigonometric determinant matrices, Fourier-analytic structures, and arithmetic invariants.
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https://arxiv.org/abs/2512.24012
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c4fdee2f1c5dacb56e431b42eb0a839d6193bdc5b6e53c2992d097bf83750467
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2026-01-01T00:00:00-05:00
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Moduli of surfaces fibered in (log) Calabi-Yau pairs II: elliptic surfaces
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arXiv:2512.24017v1 Announce Type: new Abstract: This paper continues the study initiated in [ISZ25] on the moduli of surfaces admitting lc-trivial fibrations. Using the techniques developed in [ISZ25], we (1) provide a classification of the surfaces appearing on the boundary of the KSBA-moduli space of elliptic surfaces with a bisection (2) recover the results of a series of papers on the moduli stacks of elliptic surfaces with a section [AB22, Inc20, Bru15]. Notably, our proof of (2) avoids the use of explicit steps of an MMP, such as the "La Nave flip" from [LN02], which plays a central role in [AB22,Inc20].
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https://arxiv.org/abs/2512.24017
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f1aab343a47bd6d8dd36ca40564f184214e105f2b4cbf740dcc6418ee31b9d72
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2026-01-01T00:00:00-05:00
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A regularity theory for second-order parabolic partial differential equations in weighted mixed norm Sobolev-Zygmund spaces
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arXiv:2512.24020v1 Announce Type: new Abstract: We develop an optimal regularity theory for parabolic partial differential equations in weighted mixed norm Sobolev-Zygmund spaces. The results extend the classical Schauder estimates to coefficients that are merely measurable in time and to the critical case of integer-order regularity. In addition, nonzero initial data are treated in the optimal trace space via a sharp trace theorem.
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https://arxiv.org/abs/2512.24020
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630750dd4f40f3ed7fc557c76c7ade468532f9b65352a04dd8e639679cc4f30c
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2026-01-01T00:00:00-05:00
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Filtered cospans and interlevel persistence with boundary conditions
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arXiv:2512.24025v1 Announce Type: new Abstract: We develop the notion of a "filtered cospan" as an algebraic object that stands in the same relation to interlevel persistence modules as filtered chain complexes stand with respect to sublevel persistence modules. This relation is expressed via a functor from a category of filtered cospans to a category of persistence modules that arise in Bauer-Botnan-Fluhr's study of relative interlevel set homology. We associate a filtered cospan to a Morse function $f:X\to [-\Lambda,\Lambda]$ such that $\partial X$ is the union of the regular level sets $f^{-1}(\{\pm\Lambda\})$; this allows us to capture the interlevel persistence of such a function in terms of data associated to Morse chain complexes. Similar filtered cospans are associated to simplicial and singular chain complexes, and isomorphism theorems are proven relating these to each other and to relative interlevel set homology. Filtered cospans can be decomposed, under modest hypotheses, into certain standard elementary summands, giving rise to a notion of persistence diagram for filtered cospans that is amenable to computation. An isometry theorem connects interleavings of filtered cospans to matchings between these persistence diagrams.
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https://arxiv.org/abs/2512.24025
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ce8d882506f2db366f5af7993820403b019ab79625d5da9ae3594583c886e259
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2026-01-01T00:00:00-05:00
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On the finiteness of the group associated with weighted walks in multidimensional orthants
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arXiv:2512.24027v1 Announce Type: new Abstract: In the study of walks with small steps confined to multidimensional orthants, a certain group of transformations plays a central role. In particular, several techniques to potentially compute the generating function, including the orbit sum method, can only be applied when this group is finite. In this note, we present three new results concerning this group. First, in two dimensions, we provide a complete characterization of the weight parameters that yield a finite group. In higher dimensions, we show that whenever the group is finite, it must necessarily be isomorphic to a simpler reflection group. Finally, in dimension three, we give a full classification of the parameters leading to a finite group that also satisfies an additional Weyl property.
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https://arxiv.org/abs/2512.24027
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f0fce38a6b77a4af3c615fcd0759765bb41e0019a48caabe0cd45d0f79c1944c
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2026-01-01T00:00:00-05:00
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Secondary Term for the Mean Value of Maass Special $L$-values
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arXiv:2512.24028v1 Announce Type: new Abstract: In this paper, we discover a secondary term in the asymptotic formula for the mean value of Hecke--Maass special $L$-values $ L (1/2+it_f, f) $ with the average over $f (z)$ in an orthonormal basis of (even or odd) Hecke--Maass cusp forms of Laplace eigenvalue $1/4 + t_f^2$ ($t_f > 0$). To be explicit, we prove $$ \sum_{t_f \leqslant T} \omega_f L (1/2+it_f, f) = \frac {T^2} {\pi^2} + \frac {8T^{3/2}} {3\pi^{3/2} } + O \big(T^{1+\varepsilon}\big), $$ for any $\varepsilon > 0$, where $\omega_f$ are the harmonic weights. This provides a new instance of (large) secondary terms in the moments of $L$-functions -- it was known previously only for the smoothed cubic moment of quadratic Dirichlet $L$-functions. The proof relies on an explicit formula for the smoothed mean value of $L (1/2+it_f, f)$.
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https://arxiv.org/abs/2512.24028
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0189bae7aef5359e6c7e749acc3af71d7dea79f61d6981c63fe8dc7e9b881e87
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2026-01-01T00:00:00-05:00
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Whittaker modules and representations of finite $W$-algebras of queer Lie superalgebras
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arXiv:2512.24030v1 Announce Type: new Abstract: We study various categories of Whittaker modules over the queer Lie superalgebras $\mathfrak q(n)$. We formulate standard Whittaker modules and reduce the problem of composition factors of these standard Whittaker modules to that of Verma modules in the BGG categories $\mathcal O$ of $\mathfrak q(n)$. We also obtain an analogue of Losev-Shu-Xiao decomposition for the finite $W$-superalgebras $U(\mathfrak q(n), E)$ of $\mathfrak q(n)$ associated to an odd nilpotent element $E\in \mathfrak q(n)_{\bar{1}}$. As an application, we establish several equivalences of categories of Whittaker $\mathfrak q(n)$-modules and analogues of BGG category of $U(\mathfrak q(n), E)$-modules. In particular, we reduce the multiplicity problem of Verma modules over $U(\mathfrak q(n), E)$ to that of the Verma modules in the BGG categories $\mathcal O$ of $\mathfrak q(n)$.
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https://arxiv.org/abs/2512.24030
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710eefbaa12c2c7449487a6c635ef6966af58c27a6b81bcd4fdec74c0c94730e
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2026-01-01T00:00:00-05:00
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An anti-classification theorem for minimal homeomorphisms on the torus
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arXiv:2512.24031v1 Announce Type: new Abstract: We show that it is impossible to classify topological conjugacy relation of minimal homeomorphisms on the torus by countable structures.
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https://arxiv.org/abs/2512.24031
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890491b6898abcf31e5ada600a2928eb55c2c306c5c6d93ab05c6d020ccb544b
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2026-01-01T00:00:00-05:00
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Jordan Nilpotent Group Rings of index $4$
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arXiv:2512.24033v1 Announce Type: new Abstract: Let $RG$ be the group ring of an arbitrary group $G$ over an associative non-commutative ring $R$ with identity. In this paper, we have obtained the necessary and sufficient conditions under which $RG$ is Jordan nilpotent of index $4$.
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https://arxiv.org/abs/2512.24033
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33c59c31ddbcc3419903c53dc6e3a021977c1b525d7efa699bae88133f19f8e1
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2026-01-01T00:00:00-05:00
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Push-forward of smooth measures and strong Thom stratifications
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arXiv:2512.24034v1 Announce Type: new Abstract: We study the collection of measures obtained via push-forward along a map between smooth varieties over p-adic fields. We investigate when the stalks of this collection are finite-dimensional. We provide an algebro-geometric criterion ensuring this property. This criterion is formulated in terms of a canonical subvariety of the cotangent bundle of the source of the map.
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https://arxiv.org/abs/2512.24034
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5dcdcb8db678a58b6353371648c14d6069aec347350c4a89ac9adb162a83a2a3
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2026-01-01T00:00:00-05:00
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Local Asymptotic Normality for Mixed Fractional Brownian Motion with $0<H<3/4$
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arXiv:2512.24042v1 Announce Type: new Abstract: This paper establishes the Local Asymptotic Normality (LAN) property for the mixed fractional Brownian motion under high-frequency observations with Hurst index $H \in (0, 3/4)$. The simultaneous estimation of the volatility and the Hurst index encounters a degeneracy problem in the Fisher information matrix.
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https://arxiv.org/abs/2512.24042
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5890cf06b39ef13fe8a84c0d67b52f1f72bc2de27b56c340fb4b7411f037ada6
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2026-01-01T00:00:00-05:00
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On spectral equations for an evolution operator of a $q$-oscillator lattice
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arXiv:2512.24043v1 Announce Type: new Abstract: We propose a set of algebraic equations describing eigenvalues and eigenstates of a relativistic evolution operator for a two-dimensional $q$-oscillator Kagom\'e lattice. Evolution operator is constructed with the help of $q$-oscillator solution of the Tetrahedron Equation. We focus on the unitary regime of the evolution operator, so our results are related to 3d integrable systems of the quantum mechanics. Our conjecture is based on a two-dimensional lattice version of the coordinate Bethe-Ansatz.
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https://arxiv.org/abs/2512.24043
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5eeb7ed451d61f09ab60d0787a8cd8cc794bcffb54aa6e94ac76f390f21876a1
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2026-01-01T00:00:00-05:00
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Quantum two-dimensional superintegrable systems in flat space: exact-solvability, hidden algebra, polynomial algebra of integrals
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arXiv:2512.24045v1 Announce Type: new Abstract: In this short review paper the detailed analysis of six two-dimensional quantum {\it superintegrable} systems in flat space is presented. It includes the Smorodinsky-Winternitz potentials I-II (the Holt potential), the Fokas-Lagerstrom model, the 3-body Calogero and Wolfes (equivalently, $G_2$ rational, or $I_6$) models, and the Tremblay-Turbiner-Winternitz (TTW) system with integer index $k$. It is shown that all of them are exactly-solvable, thus, confirming the Montreal conjecture (2001); they admit algebraic forms for the Hamiltonian and both integrals (all three can be written as differential operators with polynomial coefficients without a constant term), they have polynomial eigenfunctions with the invariants of the discrete symmetry group of invariance taken as variables, they have hidden (Lie) algebraic structure $g^{(k)}$ with various $k$, and they possess a (finite order) polynomial algebras of integrals. Each model is characterized by infinitely-many finite-dimensional invariant subspaces, which form the infinite flag. Each subspace coincides with the finite-dimensional representation space of the algebra $g^{(k)}$ for a certain $k$. In all presented cases the algebra of integrals is a 4-generated $(H, I_1, I_2, I_{12}\equiv[I_1, I_2])$ infinite-dimensional algebra of ordered monomials of degrees 2,3,4,5, which is a subalgebra of the universal enveloping algebra of the hidden algebra.
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https://arxiv.org/abs/2512.24045
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934d149c26640ac758e47785968b075d0f6c83c7d21298c249c75d485ebc059b
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2026-01-01T00:00:00-05:00
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Yaglom theorem for critical branching random walk on $\mathbb{Z}^d$
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arXiv:2512.24047v1 Announce Type: new Abstract: We study the critical branching random walk on $\mathbb{Z}^d$ started from a distant point $x$ and conditioned to hit some compact set $K$ in $\mathbb{Z}^d$. We are interested in the occupation time in $K$ and present its asymptotic behaviors in different dimensions. It is shown in this work that the occupation time is of order $\|x\|^{4-d}$ in dimensions $d\leq 3$, of order $\log\|x\|$ in dimension $d=4$, and of order 1 in dimensions $d\geq 5$. The corresponding weak convergences are also established. These results answer a question raised by Le Gall and Merle (Elect. Comm. in Probab. 11 (2006), 252-265).
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https://arxiv.org/abs/2512.24047
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b5917a60f438bf9dea84173802672999074531b25dc85b30a0e7fd4003bb865a
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2026-01-01T00:00:00-05:00
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Polynomial functors over free nilpotent groups
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arXiv:2512.24048v1 Announce Type: new Abstract: Let $k$ be a unital commutative ring. In this paper, we study polynomial functors from the category of finitely generated free nilpotent groups to the category of $k$-modules, focusing on comparisons across different nilpotency classes and polynomial degrees. As a consequence, we obtain refinements of parts of the results of Baues and Pirashvili on polynomial functors over free nilpotent groups of class at most 2, which also recover several folklore results for free groups and free abelian groups. Furthermore, we investigate a modular analogue, formulated using dimension subgroups over a field of positive characteristic instead of lower central series. To prove the main results, we establish general criteria that guarantee equivalences between the categories of polynomial functors of different degrees or with different base categories. They are described by using a two-sided ideal of a monad associated with the base category, which encodes polynomiality of a specific degree. Inspired by the main results, we also investigate an analogous ideal for analytic functors, and show that, in most cases, no such an ideal exists.
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https://arxiv.org/abs/2512.24048
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3a8ec4dcd341114035ef2ea446e0abd291f26a21473e83968c96421a77a386dd
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2026-01-01T00:00:00-05:00
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Complexity and convergence analysis of a single-loop SDCAM for Lipschitz composite optimization and beyond
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arXiv:2512.24059v1 Announce Type: new Abstract: We develop and analyze a single-loop algorithm for minimizing the sum of a Lipschitz differentiable function $f$, a prox-friendly proper closed function $g$ (with a closed domain on which $g$ is continuous) and the composition of another prox-friendly proper closed function $h$ (whose domain is closed on which $h$ is continuous) with a continuously differentiable mapping $c$ (that is Lipschitz continuous and Lipschitz differentiable on the convex closure of the domain of $g$). Such models arise naturally in many contemporary applications, where $f$ is the loss function for data misfit, and $g$ and $h$ are nonsmooth functions for inducing desirable structures in $x$ and $c(x)$. Existing single-loop algorithms mainly focus either on the case where $h$ is Lipschitz continuous or the case where $h$ is an indicator function of a closed convex set. In this paper, we develop a single-loop algorithm for more general possibly non-Lipschitz $h$. Our algorithm is a single-loop variant of the successive difference-of-convex approximation method (SDCAM) proposed in [22]. We show that when $h$ is Lipschitz, our algorithm exhibits an iteration complexity that matches the best known complexity result for obtaining an $(\epsilon_1,\epsilon_2,0)$-stationary point. Moreover, we show that, by assuming additionally that dom $g$ is compact, our algorithm exhibits an iteration complexity of $\tilde{O}(\epsilon^{-4})$ for obtaining an $(\epsilon,\epsilon,\epsilon)$-stationary point when $h$ is merely continuous and real-valued. Furthermore, we consider a scenario where $h$ does not have full domain and establish vanishing bounds on successive changes of iterates. Finally, in all three cases mentioned above, we show that one can construct a subsequence such that any accumulation point $x^*$ satisfies $c(x^*)\in$ dom $h$, and if a standard constraint qualification holds at $x^*$, then $x^*$ is a stationary point.
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https://arxiv.org/abs/2512.24059
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343b6c12758d007bca0ed337503dfeb6a7abdef4e2504b3f85f8e3e8ca33739b
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2026-01-01T00:00:00-05:00
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Propagation of chaos for the homogeneous Boltzmann equation with moderately soft potentials
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arXiv:2512.24065v1 Announce Type: new Abstract: We show that the Kac particle system converges, as the number of particles tends to infinity, to the solution of the homogeneous Boltzmann equation, in the regime of moderately soft potentials, $\gamma \in (-2,0)$ with the common notation. This proves the propagation of chaos. We adapt the recent work of Imbert, Silvestre and Villani, to show that the Fisher information is nonincreasing in time along solutions to the Kac master equation. This estimate allows us to control the singularity of the interaction.
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https://arxiv.org/abs/2512.24065
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6b749839c259ebaac0143cdeaf79ffe2a0ad5d64a9f3c99a7e9fa29a79c65d7e
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2026-01-01T00:00:00-05:00
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Short sums of trace functions over function fields and their applications
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arXiv:2512.24080v1 Announce Type: new Abstract: For large enough (but fixed) prime powers $q$, and trace functions to squarefree moduli in $\mathbb{F}_q[u]$ with slopes at most $1$ at infinity, and no Artin--Schreier factors in their geometric global monodromy, we come close to square-root cancellation in short sums. A special case is a function field version of Hooley's Hypothesis $R^*$ for short Kloosterman sums. As a result, we are able to make progress on several problems in analytic number theory over $\mathbb{F}_q[u]$ such as Mordell's problem on the least residue class not represented by a polynomial and the variance of short Kloosterman sums.
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https://arxiv.org/abs/2512.24080
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ecf06ad8860c0b6631bcb04521926fc228b3fa51fcf6051c6bdd1a978ad75f1f
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2026-01-01T00:00:00-05:00
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On generalized metric structures
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arXiv:2512.24082v1 Announce Type: new Abstract: Let $M$ be a smooth manifold, let $TM$ be its tangent bundle and $T^{*}M$ its cotangent bundle. This paper investigates integrability conditions for generalized metrics, generalized almost para-complex structures, and generalized Hermitian structures on the generalized tangent bundle of $M$, $E=TM \oplus T^{*}M$. In particular, two notions of integrability are considered: integrability with respect to the Courant bracket and integrability with respect to the bracket induced by an affine connection. We give sufficient criteria that guarantee the integrability for the aforementioned generalized structures, formulated in terms of properties of the associated $2$-form and connection. Extensions to the pseudo-Riemannian setting and consequences for generalized Hermitian and K\"ahler structures are also discussed. We also describe relationship between generalized metrics and weak metric structures.
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https://arxiv.org/abs/2512.24082
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ebe9d2aa2b145a63fdde12cf58412616b31eca4c665ae680c0454e7e113b9355
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2026-01-01T00:00:00-05:00
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Rank three representations of Painlev\'e systems: II. de Rham structure, Fourier--Laplace transformation
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arXiv:2512.24083v1 Announce Type: new Abstract: We use formal microlocalization to describe the Fourier--Laplace transformation between rank 3 and rank 2 D-module representations of Painleve systems. We conclude the existence of biregular morphism between the corresponding de Rham complex structures.
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https://arxiv.org/abs/2512.24083
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b1ac6dfd06042710443d87332b11c1abe18b5272e289c8740c5f94c4f712e47c
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2026-01-01T00:00:00-05:00
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Dirac solitons in one-dimensional nonlinear Schr\"odinger equations
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arXiv:2512.24089v1 Announce Type: new Abstract: In this paper we study a family of one-dimensional stationary cubic nonlinear Schr\"odinger (NLS) equations with periodic potentials and linear part displaying Dirac points in the dispersion relation. By introducing a suitable periodic perturbation, one can open a spectral gap around the Dirac-point energy. This allows to construct standing waves of the NLS equation whose leading-order profile is a modulation of Bloch waves by means of the components of a spinor solving an appropriate cubic nonlinear Dirac (NLD) equation. We refer to these solutions as Dirac solitons. Our analysis thus provides a rigorous justification for the use of the NLD equation as an effective model for the original NLS equation.
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https://arxiv.org/abs/2512.24089
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c8b7d231abf81b18c91d65f8bd92415e4432cb5af3afabc815b2d7334ff587e7
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2026-01-01T00:00:00-05:00
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Herman Rings: Structure, Dynamics, and Open Problems
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arXiv:2512.24118v1 Announce Type: new Abstract: The existence of the Herman ring of a function adds interest and complexity to the dynamics of the function. We present a detailed and understandable summary of the core discoveries and recent developments on the Herman ring of rational and transcendental meromorphic functions. It is demonstrated that the Herman ring is intriguing on its own and valuable in terms of overall dynamics. Finally, the results and potential future research problems are briefly discussed.
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https://arxiv.org/abs/2512.24118
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456f55002d3a6247ea25b25f72090b7b10211649ec52de70df3b4bb853d4d0b1
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2026-01-01T00:00:00-05:00
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A goodness-of-fit test for the Zeta distribution with unknown parameter
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arXiv:2512.24128v1 Announce Type: new Abstract: We introduce a new goodness-of-fit test for count data on $\mathbb{N}$ for the Zeta distribution with unknown parameter. The test is built on a Stein-type characterization that uses, as Stein operator, the infinitesimal generator of a birth-death process whose stationary distribution is Zeta. The resulting $L^2$-type statistic is shown to be omnibus consistent, and we establish the limit null behavior as well as the validity of the associated parametric bootstrap procedure. In a Monte Carlo simulation study, we compare the proposed test with the only existing Zeta-specific procedure of Meintanis (2009), as well as with more general competitors based on empirical distribution functions, kernel Stein discrepancies and other Stein-type characterizations.
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https://arxiv.org/abs/2512.24128
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06824a16a95c580559eecb195c690efc0e255e212d585d4efaaff961c9462e94
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2026-01-01T00:00:00-05:00
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Bicombing the mapping class group and Teichm\"uller space via stable cubical intervals
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arXiv:2512.24136v1 Announce Type: new Abstract: In this mostly expository article, we provide a new account of our proof with Minsky and Sisto that mapping class groups and Teichm\"uller spaces admit bicombings. More generally, we explain how the hierarchical hull of a pair of points in any colorable hierarchically hyperbolic space is quasi-isometric to a finite CAT(0) cube complex of bounded dimension, with the added property that perturbing the pair of points results in a uniformly bounded change to the cubical structure. Our approach is simplified and new in many aspects.
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https://arxiv.org/abs/2512.24136
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caf6f023bd3a5a3d1e0d72ea7d1182904863fd6152704749c031fbd777a28b53
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2026-01-01T00:00:00-05:00
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Large values of quadratic character sums revisited
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arXiv:2512.24147v1 Announce Type: new Abstract: We study large values of quadratic character sums with summation lengths exceeding the square root of the modulus. Assuming the Generalized Riemann Hypothesis, we obtain a new Omega result.
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https://arxiv.org/abs/2512.24147
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e0a4e607be16c479f758560674ff90077c10660ef7a186189e329e66cee05db7
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2026-01-01T00:00:00-05:00
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Some Congruences Involving Fourth Powers of Generalized Central Trinomial Coefficients
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arXiv:2512.24148v1 Announce Type: new Abstract: Let $ p \ge 5 $ be a prime and let $ b, c \in \mathbb{Z} $. Denote by $ T_k(b,c) $ the generalized central trinomial coefficient, i.e., the coefficient of $ x^k $ in $ (x^2 + bx + c)^k $. In this paper, we establish congruences modulo $ p^3 $ and $ p^4 $ for sums of the form $$ \sum_{k=0}^{p-1} (2k+1)^{2a+1}\,\varepsilon^{k}\,\frac{T_k(b,c)^4}{d^{2k}}, $$ where $ a \in \left\lbrace 0,1\right\rbrace $, $ \varepsilon \in \{1,-1\} $, and $ d = b^2 - 4c $ satisfies $ p \nmid d $. In particular, for the special case $ b = c = 1 $, we show that \begin{align*} \sum_{k=0}^{p-1}\left( 2k+1\right) ^{3} \frac{T_{k}^4}{9^k}\equiv -\frac{3p}{4}+\frac{3p^2}{4}\left( \frac{q_p(3)}{4}-1\right) \pmod{p^3}, \end{align*} where $T_k$ is the central trinomial coefficient and $q_p(a)$ is the Fermat quotient.
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https://arxiv.org/abs/2512.24148
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7faf3d21ccac1c572af7fa99ff7cf86392bb32137340e5a71a98e437dc324cf3
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2026-01-01T00:00:00-05:00
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Transitive partially hyperbolic diffeomorphisms in dimension three
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arXiv:2512.24151v1 Announce Type: new Abstract: We prove that any $C^{1+\alpha}$ transitive conservative partially hyperbolic diffeomorphism of a closed 3-manifold with virtually solvable fundamental group is ergodic. Consequently, in light of \cite{FP-classify}, this establishes the equivalence between transitivity and ergodicity for $C^{1+\alpha}$ conservative partially hyperbolic diffeomorphisms in \emph{any} closed 3-manifold. Moreover, we provide a characterization of compact accessibility classes under transitivity, thereby giving a precise classification of all accessibility classes for transitive 3-dimensional partially hyperbolic diffeomorphisms.
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https://arxiv.org/abs/2512.24151
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f5cbc38fbb6cd52c71b1113fd7b6c007632f78a7857ae95c0d4e987bd10e6221
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2026-01-01T00:00:00-05:00
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Minimal Polynomials in Spin Representations of Symmetric and Alternating Groups
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arXiv:2512.24158v1 Announce Type: new Abstract: We determine the minimal polynomial of each element of the double cover $G$ of the symmetric or alternating group in every irreducible spin representation of $G$.
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https://arxiv.org/abs/2512.24158
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372374ad91da06827e594555a15ca11a8dc27eda90a8f0c7fd8ea5dcc9fc2666
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2026-01-01T00:00:00-05:00
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Admissible HYM metrics on klt KE varieties and the MY equality for big anticanonical K-stable varieties
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arXiv:2512.24161v1 Announce Type: new Abstract: This short note includes three results: $(1)$ If a reflexive sheaf $\mathcal{E}$ on a log terminal K\"{a}hler-Einstein variety $(X,\omega)$ is slope stable with respect to a singular K\"{a}hler-Einstein metric $\omega$, then $\mathcal{E}$ admits an $\omega$-admissible Hermitian-Yang-Mills metric. $(2)$ If a K-stable log terminal projective variety with big anti-canonical divisor satisfies the equality of the Miyaoka-Yau inequality in the sense of \cite{IJZ25}, then its anti-canonical model admits a quasi-\'{e}tale cover from $\mathbb{C}P^n$. $(3)$ There exists a holomorphic rank 3 vector bundle on a compact complex surface which is semistable for some nef and big line bundle, but it is not semistable for any ample line bundles.
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https://arxiv.org/abs/2512.24161
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7d138c812b95c1c3a57ac972ef2fe7adb63b29b7d2bb24e2588563e051a7aab8
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2026-01-01T00:00:00-05:00
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Cheeger Bounds for Stable Phase Retrieval in Reproducing Kernel Hilbert Spaces
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arXiv:2512.24169v1 Announce Type: new Abstract: Phase retrieval seeks to reconstruct a signal from phaseless intensity measurements and, in applications where measurements contain errors, demands stable reconstruction. We study local stability of phase retrieval in reproducing kernel Hilbert spaces. Motivated by Grohs-Rathmair's Cheeger-type estimate for Gabor phase retrieval, we introduce a kernel Cheeger constant that quantifies connectedness relative to kernel localization. This notion yields a clean stability certificate: we establish a unified lower bound over both real and complex fields, and in the real case also an upper bound, each in terms of the reciprocal kernel Cheeger constant. Our framework treats finite- and infinite-dimensional settings uniformly and covers discrete, semi-discrete, and continuous measurement domains. For generalized wavelet phase retrieval from (semi-)discrete frames, we bound the kernel Cheeger constant by the Cheeger constant of a data-dependent weighted graph. We further characterize phase retrievability for generalized wavelet transforms and derive a simple sufficient criterion for wavelet sign retrieval in arbitrary dimension for transforms associated with irreducibly admissible matrix groups.
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https://arxiv.org/abs/2512.24169
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d12cff853313789b92ea19fef6c91692c8d687c418232994a37600bf9295a019
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2026-01-01T00:00:00-05:00
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On the MLC Conjecture and the Renormalization Theory in Complex Dynamics
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arXiv:2512.24171v1 Announce Type: new Abstract: In this Note, we present recent developments in the Renormalization Theory of quadratic polynomials and discuss their applications, with an emphasis on the MLC conjecture, the problem of local connectivity of the Mandelbrot set, and on its geometric counterparts.
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https://arxiv.org/abs/2512.24171
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26ff587310f0145d674167cba27dfc369aba6c766207731816925ecc3b274418
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2026-01-01T00:00:00-05:00
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Tensor-Network Analysis of Root Patterns in the XXX Model with Open Boundaries
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arXiv:2512.24182v1 Announce Type: new Abstract: The string hypothesis for Bethe roots represents a cornerstone in the study of quantum integrable systems, providing access to physical quantities such as the ground-state energy and the finite-temperature free energy. While the $t-W$ scheme and the inhomogeneous $T-Q$ relation have enabled significant methodological advances for systems with broken $U(1)$ symmetry, the underlying physics induced by symmetry breaking remains largely unexplored, due to the previously unknown distributions of the transfer-matrix roots. In this paper, we propose a new approach to determining the patterns of zero roots and Bethe roots for the $\Lambda-\theta$ and inhomogeneous Bethe ansatz equations using tensor-network algorithms. As an explicit example, we consider the isotropic Heisenberg spin chain with non-diagonal boundary conditions. The exact structures of both zero roots and Bethe roots are obtained in the ground state for large system sizes, up to ($N\simeq 60$ and $100$). We find that even in the absence of $U(1)$ symmetry, the Bethe and zero roots still exhibit a highly structured pattern. The zero roots organize into bulk strings, boundary strings, and additional roots, forming two dominant lines with boundary-string attachments. Correspondingly, the Bethe roots can be classified into four distinct types: regular roots, line roots, arc roots, and paired-line roots. These structures are associated with a real-axis line, a vertical line, characteristic arcs in the complex plane, and boundary-induced conjugate pairs. Comparative analysis reveals that the $t-W$ scheme generates significantly simpler root topologies than those obtained via off-diagonal Bethe Ansatz.
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https://arxiv.org/abs/2512.24182
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dbc00364c7907090ba914f363aa97033ff280c6c8785170e02bf032e83b23cf6
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2026-01-01T00:00:00-05:00
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On the 1-leg Donaldson-Thomas $\mathbb{Z}_2\times\mathbb{Z}_2$-vertex
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arXiv:2512.24196v1 Announce Type: new Abstract: We introduce a notion of restricted pyramid configurations for computing the 1-leg Donaldson-Thomas $\mathbb{Z}_2\times\mathbb{Z}_2$-vertex. We study a special type of restricted pyramid configurations with the prescribed 1-leg partitions, and find one unique class of them satisfying the symmetric interlacing property. This leads us to obtain an explicit formula for a class of 1-leg Donaldson-Thomas $\mathbb{Z}_2\times\mathbb{Z}_2$-vertex through establishing its connection with 1-leg Donaldson-Thomas $\mathbb{Z}_4$-vertex using the vertex operator methods of Okounkov-Reshetikhin-Vafa and Bryan-Young.
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https://arxiv.org/abs/2512.24196
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dfae03aa3ba7ef4d6fa4ce315fa1cca8a11df12abdd941353a00ec56322738da
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2026-01-01T00:00:00-05:00
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Gaussian free fields on Hamming graphs and lattice spin systems
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arXiv:2512.24199v1 Announce Type: new Abstract: We discuss a class of discrete Gaussian free fields on Hamming graphs, where interactions are determined solely by the Hamming distance between vertices. The purpose of examining this class is that it differs significantly from the commonly discussed spin system on the integer lattice with nearest-neighbour interactions. After introducing general results on the partition function and covariance for the class of Gaussian free fields, we present detailed properties of some specific models. Group-theoretic arguments and the Fourier transform give some explicit results.
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https://arxiv.org/abs/2512.24199
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5bc18005ec82fe890f4061776222954a30b5a9c5e5afa6d0b9a12aab3068e976
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2026-01-01T00:00:00-05:00
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Instanton 2-torsion and fibered knots
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arXiv:2512.24206v1 Announce Type: new Abstract: We prove that the unreduced singular instanton homology $I^\sharp(Y,K;\mathbb{Z})$ has $2$-torsion for any null-homologous fibered knot $K$ of genus $g>0$ in a closed $3$-manifold $Y$ except for $\#^{2g}S^1\times S^2$. The main technical result is a formula of $I^\sharp(Y,K;\mathbb{C})$ via sutured instanton theory, by which we can compare the dimensions of $I^\sharp(Y,K;\mathbb{F}_2)$ and $I^\sharp(Y,K;\mathbb{C})$. As a byproduct, we show that $I^\sharp(S^3,K;\mathbb{C})$ for a knot $K\subset S^3$ admitting lens space surgeries is determined by the Alexander polynomial, while some special cases of torus knots have been previously studied by many people. Another byproduct is that the next-to-top Alexander grading summand of instanton knot homology $KHI(S^3,K,g(K)-1)$ is non-vanishing when $K$ has unknotting number one, which generalizes the Baldwin--Sivek's result in the fibered case. Finally, we discuss the relation to the Heegaard Floer theory.
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https://arxiv.org/abs/2512.24206
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3b43bb1ecff4f29a66d8efd254b204a7a34c368db80f58805282d09d452f574f
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2026-01-01T00:00:00-05:00
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Notes on the LVP and the CVP in $p$-adic Fields
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arXiv:2512.24207v1 Announce Type: new Abstract: This paper explores computational methods for solving the Longest Vector Problem (LVP) and Closest Vector Problem (CVP) in $p$-adic fields. Leveraging the non-Archimedean property of $p$-adic norms, we propose a polynomial time algorithm to compute orthogonal bases for $p$-adic lattices when the $p$-adic field is given by a minimal polynomial. The method utilizes the structure of maximal orders and $p$-radicals in extension fields of $\mathbb{Q}_{p}$ to efficiently construct uniformizers and residue field bases, enabling rapid solutions for LVP and CVP. In addition, we introduce the characterization of norms on vector spaces over $\mathbb{Q}_p$.
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https://arxiv.org/abs/2512.24207
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b443321ea0b8a14e1f73e3d66208df99110518fbec7af7f41b7c9209ed130d0d
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2026-01-01T00:00:00-05:00
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An Equivalence Result on the Order of Differentiability in Frobenius' Theorem
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arXiv:2512.24218v1 Announce Type: new Abstract: This paper examines the simplest case of total differential equations that appears in the theory of foliation structures, without imposing the smoothness assumptions. This leads to a peculiar asymmetry in the differentiability of solutions. To resolve this asymmetry, this paper focuses on the differentiability of the integral manifold. When the system is locally Lipschitz, a solution is ensured to be only locally Lipschitz, but the integral manifolds must be $C^1$. When the system is $C^k$, we can only ensure the existence of a $C^k$ solution, but the integral manifolds must be $C^{k+1}$. In addition, we see a counterexample in which the system is $C^1$, but there is no $C^2$ solution. Moreover, we characterize a minimizer of an optimization problem whose objective function is a quasi-convex solution to a total differential equation. In this connection, we examine two necessary and sufficient conditions for the system in which any solution is quasi-convex.
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https://arxiv.org/abs/2512.24218
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9facd6690d53c31b03092c12744abe8e02fabc38b257cfaff5d1b2309e201994
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2026-01-01T00:00:00-05:00
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Node-Kayles on Trees
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arXiv:2512.24221v1 Announce Type: new Abstract: Node-Kayles is a well-known impartial combinatorial game played on graphs, where players alternately select a vertex and remove it along with its neighbors. By the Sprague-Grundy theorem, every position of an impartial game corresponds to a non-negative integer called its Grundy value. In this paper, we investigate the Grundy value sequences of $n$-regular trees as well as graphs formed by joining two $n$-regular trees with a path of length $k$. We derive explicit formulas and recursive relations for the associated Grundy value sequences. Furthermore, we prove that these sequences are eventually periodic and determine both their preperiod lengths and their periods.
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https://arxiv.org/abs/2512.24221
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c1ce1476ef33e7182e9aa219b02f8230afb31173a9753d2a89a32b40d9ba28b1
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2026-01-01T00:00:00-05:00
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Explicit bounds for the graphicality of the prime gap sequence
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arXiv:2512.24230v1 Announce Type: new Abstract: We establish explicit unconditional results on the graphic properties of the prime gap sequence. Let \( p_n \) denote the \( n \)-th prime number (with $p_0=1$) and \( \mathrm{PD}_n = (p_\ell - p_{\ell-1})_{\ell=1}^n \) be the sequence of the first \( n \) prime gaps. Building upon the recent work by Erd\H{o}s \emph{et al}, which proved the graphic nature of \( \mathrm{PD}_n \) for large \( n \) unconditionally, and for all \( n \) under RH, we provide the first explicit unconditional threshold such that:\\ (1) For all \( n \geq \exp\exp(30.5) \), \( \mathrm{PD}_n \) is graphic.\\ (2) For all \( n \geq \exp\exp(34.5) \), every realization \( G_n \) of \( \mathrm{PD}_n \) satisfies that \( (G_n, p_{n+1}-p_n) \) is DPG-graphic. Our proofs utilize a more refined criterion for when a sequence is graphic, and better estimates for the first moment of large prime gaps proven through an explicit zero-free region and explicit zero-density estimate for the Riemann zeta function.
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https://arxiv.org/abs/2512.24230
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126794e93a3a7346b982fb8de56437dbd5655f21f59bd1b3ca52c55b2aa5df17
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2026-01-01T00:00:00-05:00
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Multi-bump solutions for sublinear elliptic equations with nonsymmetric coefficients
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arXiv:2512.24234v1 Announce Type: new Abstract: We investigate the existence of nonnegative bump solutions to the sublinear elliptic equation \[ \begin{cases} -\Delta v - K(x)v + |v|^{q-2}v = 0 & \text{in } \mathbb{R}^N, \\ v(x) \to 0 & \text{as } |x| \to \infty, \end{cases} \] where $q \in (1,2)$, $ N \geq 2$, and the potential $K \in L^p_{\mathrm{loc}}(\mathbb{R}^N)$ with $p > N/2$ is a function without any symmetry assumptions. Under the condition that $\|K - 1\|_{L^p_{\mathrm{loc}}}$ is sufficiently small, we construct infinitely many solutions with arbitrarily many bumps. The construction is challenged by the sensitive interaction between bumps, whose limiting profiles have compact support. The key to ensuring their effective separation lies in obtaining sharp estimates of the support sets. Our method, based on a truncated functional space, provides precisely such control. We derive qualitative local stability estimates in region-wise maximum norms that govern the size of each bump's essential support, confining its core to a designated region and minimizing overlap. Crucially, these estimates are uniform in the number of bumps, which is the pivotal step in establishing the existence of solutions with infinitely many bumps.
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https://arxiv.org/abs/2512.24234
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9fdfc38621583ffd7b44eee02e86363b30bdc231fdcc0f6bed27bff28b3a51c5
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2026-01-01T00:00:00-05:00
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Semiclassical Limits of Strongly Parabolic Higgs Bundles and Hyperpolygon Spaces
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arXiv:2512.24236v1 Announce Type: new Abstract: We investigate the Hitchin hyperk\"ahler metric on the moduli space of strongly parabolic $\mathfrak{sl}(2,\C)$-Higgs bundles on the $n$-punctured Riemann sphere and its degeneration obtained by scaling the parabolic weights $t\alpha$ as $t\to0$. Using the parabolic Deligne--Hitchin moduli space, we show that twistor lines of hyperpolygon spaces arise as limiting initial data for twistor lines at small weights, and we construct the corresponding real-analytic families of $\lambda$-connections. On suitably shrinking regions of the moduli space, the rescaled Hitchin metric converges, in the semiclassical limit, to the hyperk\"ahler metric on the hyperpolygon space $\mathcal X_\alpha$, which thus serves as the natural finite-dimensional model for the degeneration of the infinite-dimensional hyperk\"ahler reduction. Moreover, higher-order corrections of the Hitchin metric in this semiclassical regime can be expressed explicitly in terms of iterated integrals of logarithmic differentials on the punctured sphere.
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https://arxiv.org/abs/2512.24236
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72e77fd5ee0474ebc2400e386ea52737d899f8dd6266a859daa36fc38635a8d3
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2026-01-01T00:00:00-05:00
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Geometric Eisenstein series in non-abelian Hodge theory and hyperholomorphic branes from supersymmetry
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arXiv:2512.24239v1 Announce Type: new Abstract: Using geometric Eisenstein series, foundational work of Arinkin and Gaitsgory constructs cuspidal-Eisenstein decompositions for ind-coherent nilpotent sheaves on the de Rham moduli of local systems. This article extends these constructions to coherent (not ind-coherent) nilpotent sheaves on the Dolbeault, Hodge and twistor moduli from non-abelian Hodge theory. We thus account for Higgs bundles, Hodge filtrations and hyperk\"ahler rotations of local systems. In particular, our constructions are shown to decompose a hyperholomorphic sheaf theory of so-called BBB-branes into cuspidal and Eisenstein components. Our work is motivated, on the one hand, by the `classical limit' or `Dolbeault geometric Langlands conjecture' of Donagi and Pantev, and on the other, by attempts to interpret Kapustin and Witten's physical duality between BBB-branes and BAA-branes in 4D supersymmetric Yang--Mills theories as a mathematical statement within the geometric Langlands program.
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https://arxiv.org/abs/2512.24239
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3bf1bb57e9ee010da697550d0d2e3a2b0401dcecb374d81ee441eb185ba2b5f7
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2026-01-01T00:00:00-05:00
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Spanning Components and Surfaces Under Minimum Vertex Degree
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arXiv:2512.24242v1 Announce Type: new Abstract: We study minimum vertex-degree conditions in 3-uniform hypergraphs for (tight) spanning components and (combinatorial) surfaces. Our main results show that a 3-uniform hypergraph $G$ on $n$ vertices contains a spanning component if $\delta_1(G) \gtrsim \tfrac{1}{2} \binom{n}{2}$ and a spanning copy of any surface if $\delta_1(G) \gtrsim \tfrac{5}{9} \binom{n}{2}$, which in both cases is asymptotically optimal. This extends the work of Georgakopoulos, Haslegrave, Montgomery, and Narayanan who determined the corresponding minimum codegree conditions in this setting.
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https://arxiv.org/abs/2512.24242
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4c60cc56cd8b942bb841f8ade16f379f3bb5976a6816f65d88f5cc4e233d2515
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2026-01-01T00:00:00-05:00
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On the Schwarz Lemma for Bergman metrics of bounded domains
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arXiv:2512.24244v1 Announce Type: new Abstract: We present a new Schwarz Lemma for bounded domains with Bergman metrics. The key ingredient of our proof is the Cauchy-Schwarz inequality from probability theory.
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https://arxiv.org/abs/2512.24244
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ea95d2f5fc29a1f705f5d1647486250ef9354991c4e161fc2c3e696f45e811d0
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2026-01-01T00:00:00-05:00
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On the Consistency of Combinatorially Symmetric Sign Patterns and the Class of 2-Consistent Sign Patterns
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arXiv:2512.24248v1 Announce Type: new Abstract: A sign pattern is a matrix that has entries from the set $\{+,-,0\}$. An $n\times n$ sign pattern $\mathcal{P}$ is called consistent if every real matrix in its qualitative class has exactly $k$ real eigenvalues and $n-k$ nonreal eigenvalues for some integer $k$, with $1\leq k\leq n$. In the article \cite{1}, the authors established a necessary condition for irreducible, tridiagonal patterns with a $0$-diagonal to be consistent. Subsequently, they proposed that this condition is also sufficient for such patterns to be consistent. In this article, we first demonstrate that this proposition does not hold. We characterize all irreducible, tridiagonal sign patterns with a $0$-diagonal of order at most five that are consistent. Moreover, we establish useful, necessary conditions for irreducible, combinatorially symmetric sign patterns to be consistent. Finally, we introduce the class $\Delta$ of all $2$-consistent sign patterns and provide several necessary conditions for sign patterns to belong to this class.
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https://arxiv.org/abs/2512.24248
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30039cb9463454c87ccb1d4e162fd2f329f9e6b26f238c796c5db654f9019303
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2026-01-01T00:00:00-05:00
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Simple factor graphs associated with split graphs
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arXiv:2512.24252v1 Announce Type: new Abstract: We introduce and study a loopless multigraph associated with a split graph $S$: the factor graph of $S$, denoted by $\Phi(S)$, which encodes the combinatorial information about 2-switch transformations over $S$. This construction provides a cleaner, compact and non-redundant alternative to the graph $A_4(S)$ by Barrus and West, for the particular case of split graphs. If $\Phi(S)$ is simple and connected, we obtain a precise description of the underlying structure of $S$, particularly when $\Phi(S)$ is complete, highlighting the usefulness of the factor graph for understanding 2-switch dynamics in balanced and indecomposable split graphs, as well as its 2-switch-degree classification.
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https://arxiv.org/abs/2512.24252
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17a5802d975997f99d60374e69714763f6ce7503d3d323802d7f33506f0756a2
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2026-01-01T00:00:00-05:00
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Complete lift of control system
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arXiv:2512.24262v1 Announce Type: new Abstract: We study affine control systems on smooth manifolds and their complete lifts to the tangent bundle, providing an explicit geometric description of the solutions of the lifted system. We show that, although controllability of the complete lift implies controllability of the original system, the lifted system is never controllable due to intrinsic geometric constraints. By introducing chain controllability, we prove that controllability of the original system guarantees chain controllability of its complete lift.
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https://arxiv.org/abs/2512.24262
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7097fa20e356e900fe0611599e08e3194f21ff223374bb3100754009580989ce
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2026-01-01T00:00:00-05:00
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Construction of sign k-potent sign patterns and conditions for such sign patterns to allow k-potence
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arXiv:2512.24264v1 Announce Type: new Abstract: A sign pattern is a matrix whose entries are from the set $\{+,-, 0\}$. A square sign pattern $A$ is called sign $k$-potent if $k$ is the smallest positive integer for which $A^{k+1}=A$, and for $k=1$, $A$ is called sign idempotent. In 1993, Eschenbach \cite{01} gave an algorithm to construct sign idempotent sign patterns. However, Huang \cite{02} constructed an example to show that matrices obtained by Eschenbach's algorithm were not necessarily sign idempotent. In \cite{03}, Park and Pyo modified Eschenbach's algorithm to construct all reducible sign idempotent sign patterns. In this paper, we give an example to establish that the modified algorithm by Park and Pyo does not always terminate in a single iteration; the number of iterations, depending on the order of the sign pattern, could be large. In this paper, we give a new algorithm that terminates in a single iteration to construct all possible sign idempotent sign patterns. We also provide an algorithm for constructing sign $k$-potent sign patterns. Further, we give some necessary and sufficient conditions for a sign $k$-potent sign pattern to allow $k$-potence.
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https://arxiv.org/abs/2512.24264
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dff5adf5579f33e5d8d5232915d059aa03997610ba9b3033395132a5e21cc0da
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2026-01-01T00:00:00-05:00
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On the word problem for just infinite groups
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arXiv:2512.24266v1 Announce Type: new Abstract: In this note we establish that the word problem is algorithmically decidable for finitely generated just infinite groups given by a recursively enumerable set of relations. The proof does not use the Wilson--Grigorchuk theorem on the classification of just infinite groups, and the argument proceeds directly from the definition, using ideas from classical results on decidability of the word problem: Kuznetsov's theorem on simple groups and the Dyson--Mostowski theorem on residually finite groups.
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https://arxiv.org/abs/2512.24266
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Academic Papers
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98801ea4e049dfbc9d510cdcd6ceac2aec491b39d748ad67b4e4c402a3156cd3
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2026-01-01T00:00:00-05:00
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Proper moduli spaces of orthosymplectic complexes
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arXiv:2512.24275v1 Announce Type: new Abstract: We apply the formalism of Alper-Halpern-Leistner-Heinloth to construct proper good moduli spaces for moduli stacks of Bridgeland semistable orthosymplectic complexes on a complex smooth projective variety, which we propose as a candidate for compactifying moduli spaces of principal bundles for the orthogonal and symplectic groups. We also prove some results on good moduli spaces of fixed point stacks and mapping stacks from finite groupoids.
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https://arxiv.org/abs/2512.24275
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Academic Papers
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404d5073c0c9f8d200e11c2e1be78a5decd8bd45fe01ed87c1ff261266a93fd0
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2026-01-01T00:00:00-05:00
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Tropical methods for building real space sextics with totally real tritangent planes
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arXiv:2512.24277v1 Announce Type: new Abstract: This paper proposes the use of combinatorial techniques from tropical geometry to build the 120 tritangent planes to a given smooth algebraic space sextic. Although the tropical count is infinite, tropical tritangents come in 15 equivalence classes, each containing the tropicalization of exactly eight classical tritangents. Under mild genericity conditions on the tropical side, we show that liftings of tropical tritangents are defined over quadratic extensions of the ground field over which the input sextic curve is defined. When the input curve is real, we prove that every complex liftable member of a given tropical tritangent class either completely lifts to the reals or none of its liftings are defined over the reals. As our main application we use these methods to build examples of real space sextics with 64 and 120 totally real tritangents, respectively. The paper concludes with a discussion of our results in the arithmetic setting.
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https://arxiv.org/abs/2512.24277
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Academic Papers
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927ee310bc5a275e5ff496d5942bd90247c29746a83acd3b03b21d6dc3953361
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2026-01-01T00:00:00-05:00
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On $\mathrm{Ext}^{\bullet}$ between locally analytic generalized Steinberg with applications
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arXiv:2512.24279v1 Announce Type: new Abstract: Let $n\geq 2$ be an integer, $p$ be a prime number and $K$ be a finite extension of $\mathbb{Q}_p$. Motivated by Schraen's thesis and Gehrmann's definition of automorphic simple $\mathscr{L}$-invariants, we study the first non-vanishing extension groups between a pair of locally $K$-analytic generalized Steinberg representations of $\mathrm{GL}_n(K)$. We study subspaces of these extension groups defined by using either relative conditions with respect to Lie subalgebras of $\mathfrak{s}\mathfrak{l}_{n}$ (isomorphic to $\mathfrak{s}\mathfrak{l}_{m}$ for some $2\leq m<n$) or maps between locally $K$-analytic generalized Steinberg representations of $\mathrm{GL}_n(K)$ with different highest weights. The applications of these computations are two-fold. On one hand, we prove that a certain universal successive extension of filtered $(\varphi,N)$-modules can be realized as the space of homomorphisms from a suitable shift of the dual of locally $K$-analytic Steinberg representation into the de Rham complex of the Drinfeld upper-half space, generalizing one main result of Schraen's thesis from $\mathrm{GL}_{3}(\mathbb{Q}_p)$ to $\mathrm{GL}_{n}(K)$. On the other hand, we give a definition of higher $\mathscr{L}$-invariants for $\mathrm{GL}_n(K)$ (which we call Breuil-Schraen $\mathscr{L}$-invariants) and discuss its possible explicit relation to Fontaine-Mazur $\mathscr{L}$-invariants, using ideas from Breuil-Ding's higher $\mathscr{L}$-invariants for $\mathrm{GL}_{3}(\mathbb{Q}_p)$.
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https://arxiv.org/abs/2512.24279
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Academic Papers
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f32aa1ad31dddc7eba18fa43ddad8b7665f43877d9a1e9e9c97c314cd89ae762
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2026-01-01T00:00:00-05:00
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On the Picard-Lindel\"of Argument and the Banach-Caccioppoli Contraction Mapping Principle
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arXiv:2512.24283v1 Announce Type: new Abstract: The aim of this note is to present the simple observation that a slight refinement of the Contraction Mapping Principle allows one to recover the precise convergence rate in the Picard-Lindel\"of Theorem.
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https://arxiv.org/abs/2512.24283
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Academic Papers
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68879815cba4e01e5d967806c0ca620aacc80d97285fe0ba57a99f62435e687b
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2026-01-01T00:00:00-05:00
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On Characterizations of W-weighted DMP and MPD Inverses
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arXiv:2512.24285v1 Announce Type: new Abstract: Recently, the weak Drazin inverse and its characterization have been crucial studies for matrices of index k. In this article, we have revisited W-weighted DMP and MPD inverses and constructed a general class of unique solutions to certain matrix equations. Moreover, we have generalized the W-weighted Drazin inverse of Meng, 2017 using the minimal rank Wweighted weak Drazin inverse. In addition to that, we have derived several equivalent properties of W-weighted DMP and MPD inverses for minimal rank W-weighted weak Drazin inverse of rectangular matrices. Furthermore, some projection-based results are discussed for the characterization of minimal rank W-weighted Drazin inverse, along with some new expressions that are derived for MPD and DMP inverses. Thereby, we have elaborated certain expressions of the perturbation formula for W-weighted weak MPD and DMP inverses. As an application, we establish the reverse and forward order laws using the W-weighted weak Drazin inverse and the minimal rank W-weighted weak Drazin inverse, and apply these results to solve certain matrix equation.
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https://arxiv.org/abs/2512.24285
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Academic Papers
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a4aaf7b97e6f9d390ce5deebc64d3667df0166516574584007aa2c8acc086518
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2026-01-01T00:00:00-05:00
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Adaptive Algorithms for Nonconvex Bilevel Optimization under P{\L} Conditions
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arXiv:2512.24291v1 Announce Type: new Abstract: Existing methods for nonconvex bilevel optimization (NBO) require prior knowledge of first- and second-order problem-specific parameters (e.g., Lipschitz constants and the Polyak-{\L}ojasiewicz (P{\L}) parameters) to set step sizes, a requirement that poses practical limitations when such parameters are unknown or computationally expensive. We introduce the Adaptive Fully First-order Bilevel Approximation (AF${}^2$BA) algorithm and its accelerated variant, A${}^2$F${}^2$BA, for solving NBO problems under the P{\L} conditions. To our knowledge, these are the first methods to employ fully adaptive step size strategies, eliminating the need for any problem-specific parameters in NBO. We prove that both algorithms achieve $\mathcal{O}(1/\epsilon^2)$ iteration complexity for finding an $\epsilon$-stationary point, matching the iteration complexity of existing well-tuned methods. Furthermore, we show that A${}^2$F${}^2$BA enjoys a near-optimal first-order oracle complexity of $\tilde{\mathcal{O}}(1/\epsilon^2)$, matching the oracle complexity of existing well-tuned methods, and aligning with the complexity of gradient descent for smooth nonconvex single-level optimization when ignoring the logarithmic factors.
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https://arxiv.org/abs/2512.24291
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Academic Papers
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svg
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2a0d1f2b419584a31738ccd9d999267bf8dc88c88087ae3f9f3e938df9dea159
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2026-01-01T00:00:00-05:00
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Quasi Neighborhood Balanced Coloring of Graphs
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arXiv:2512.24293v1 Announce Type: new Abstract: For a simple graph G = (V, E), a coloring of vertices of G using two colors, say red and blue, is called a quasi neighborhood balanced coloring if, for every vertex of the graph, the number of red neighbors and the number of blue neighbors differ by at most one. In addition, there must be at least one vertex in G for which this difference is exactly one. If a graph G admits such a colouring, then G is said to be a quasi-neighbourhood balanced colored graph. We also define variants of such a coloring, like uniform quasi neighborhood balanced coloring, positive quasi neighborhood balanced coloring and negative quasi neighborhood balanced coloring based on the color of the extra neighbor of every vertex of odd degree of the graph G. We present several examples of graph classes that admit the various variants of quasi neighborhood balanced coloring. We also discuss various graph operations involving such graphs. Furthermore, we prove that there is no forbidden subgraph characterization for the class of quasi neighborhood balanced coloring and show that the problem of determining whether a given graph has such a coloring is NP-complete.
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https://arxiv.org/abs/2512.24293
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Academic Papers
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svg
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f2e646238e6a7a3338f9828e941fb488ac746dfe0a506e8876c6b388dac54d79
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2026-01-01T00:00:00-05:00
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Optimization over Trained Neural Networks: Going Large with Gradient-Based Algorithms
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arXiv:2512.24295v1 Announce Type: new Abstract: When optimizing a nonlinear objective, one can employ a neural network as a surrogate for the nonlinear function. However, the resulting optimization model can be time-consuming to solve globally with exact methods. As a result, local search that exploits the neural-network structure has been employed to find good solutions within a reasonable time limit. For such methods, a lower per-iteration cost is advantageous when solving larger models. The contribution of this paper is two-fold. First, we propose a gradient-based algorithm with lower per-iteration cost than existing methods. Second, we further adapt this algorithm to exploit the piecewise-linear structure of neural networks that use Rectified Linear Units (ReLUs). In line with prior research, our methods become competitive with -- and then dominant over -- other local search methods as the optimization models become larger.
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https://arxiv.org/abs/2512.24295
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Academic Papers
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svg
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499e8815b2e8eb8b442932f6c48d8466e194630df1ed85bbd740b5d9d89a11ac
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2026-01-01T00:00:00-05:00
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On Trivial Cyclically Covering Subspaces of $\mathbb{F}_q^n$ in Non-Coprime Characteristic
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arXiv:2512.24301v1 Announce Type: new Abstract: A subspace $U$ of $\mathbb{F}_q^n$ is called \textit{cyclically covering} if the whole space $\mathbb{F}_q^n$ is the union of the cyclic shifts of $U$. The case $\mathbb{F}_q^n$ itself is the only covering subspace, is of particular interest. Recently, Huang solved this problem completely under the condition $\gcd(n, q)=1$ using primitive idempotents and trace functions, and explicitly posed the non-coprime case as an open question. This paper provides a complete answer to Huang's question. We prove that if $n = p^k m$ where $p = \operatorname{char}(\mathbb{F}_q)$ and $\gcd(m, p)=1$, then $h_q(p^k m) = 0$ if and only if $h_q(m) = 0$. This result fully reduces the non-coprime case to the coprime case settled by Huang. Our proof employs the structure theory of cyclic group algebras in modular characteristic.
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https://arxiv.org/abs/2512.24301
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Academic Papers
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6521ca0ad51c55229d986f6a2f8562fee238db9096a6982147cfad362b2e4fc5
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2026-01-01T00:00:00-05:00
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Approximation algorithms for integer programming with resource augmentation
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arXiv:2512.24302v1 Announce Type: new Abstract: The classic algorithm [Papadimitriou, J.ACM '81] for IPs has a running time $n^{O(m)}(m\cdot\max\{\Delta,\|\textbf{b}\|_{\infty}\})^{O(m^2)}$, where $m$ is the number of constraints, $n$ is the number of variables, and $\Delta$ and $\|\textbf{b}\|_{\infty}$ are, respectively, the largest absolute values among the entries in the constraint matrix and the right-hand side vector of the constraint. The running time is exponential in $m$, and becomes pseudo-polynomial if $m$ is a constant. In recent years, there has been extensive research on FPT (fixed parameter tractable) algorithms for the so-called $n$-fold IPs, which may possess a large number of constraints, but the constraint matrix satisfies a specific block structure. It is remarkable that these FPT algorithms take as parameters $\Delta$ and the number of rows and columns of some small submatrices. If $\Delta$ is not treated as a parameter, then the running time becomes pseudo-polynomial even if all the other parameters are taken as constants. This paper explores the trade-off between time and accuracy in solving an IP. We show that, for arbitrary small $\varepsilon>0$, there exists an algorithm for IPs with $m$ constraints that runs in ${f(m,\varepsilon)}\cdot\textnormal{poly}(|I|)$ time, and returns a near-feasible solution that violates the constraints by at most $\varepsilon\Delta$. Furthermore, for $n$-fold IPs, we establish a similar result -- our algorithm runs in time that depends on the number of rows and columns of small submatrices together with $1/\varepsilon$, and returns a solution that slightly violates the constraints. Meanwhile, both solutions guarantee that their objective values are no worse than the corresponding optimal objective values satisfying the constraints. As applications, our results can be used to obtain additive approximation schemes for multidimensional knapsack as well as scheduling.
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https://arxiv.org/abs/2512.24302
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Academic Papers
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a412d2decea8b05ab4a6da16fd599f8673e499b78c3c1a6e0ec570a38319e40c
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2026-01-01T00:00:00-05:00
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Universality of cutoff for independent random walks on the circle conditioned not to intersect
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arXiv:2512.24307v1 Announce Type: new Abstract: In the present paper, we consider a class of Markov processes on the discrete circle which has been introduced by K\"onig, O'Connell and Roch. These processes describe movements of exchangeable interacting particles and are discrete analogues of the unitary Dyson Brownian motion: a random number of particles jump together either to the left or to the right, with trajectories conditioned to never intersect. We provide asymptotic mixing times for stochastic processes in this class as the number of particles goes to infinity, under a sub-Gaussian assumption on the random number of particles moving at each step. As a consequence, we prove that a cutoff phenomenon holds independently of the transition probabilities, subject only to the sub-Gaussian assumption and a minimal aperiodicity hypothesis. Finally, an application to dimer models on the hexagonal lattice is provided.
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https://arxiv.org/abs/2512.24307
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Academic Papers
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svg
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e5226a391aac42b8ccf26335a12efaf0c628cb30759f497a95db4578cea128b4
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2026-01-01T00:00:00-05:00
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1-Lefschetz contact solvmanifolds
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arXiv:2512.24311v1 Announce Type: new Abstract: We study the contact Lefschetz condition on compact contact solvmanifolds, as introduced by B.\ Cappelletti-Montano, A.\ De Nicola and I.\ Yudin. We seek to fill the gap in the literature concerning Benson-Gordon type results, characterizing $1$-Lefschetz contact solvmanifolds. We prove that the $1$-Lefschetz condition on Lie algebras is preserved via $1$-dimensional central extensions by a symplectic cocycle, thereby establishing that a unimodular symplectic Lie algebra $(\mathfrak{h}, \omega)$ is $1$-Lefschetz if and only if its contactization $(\mathfrak{g}, \eta)$ is $1$-Lefschetz. We achieve this by showing an explicit relation for the relevant cohomology degrees of $\mathfrak{h}$ and $\mathfrak{g}$. Using this, we show how the commutators $[\mathfrak{h},\mathfrak{h}]$ and $[\mathfrak{g},\mathfrak{g}]$ are related, especially when the $1$-Lefschetz condition holds. By specializing to the nilpotent setting, we prove that $1$-Lefschetz contact nilmanifolds equipped with an invariant contact form are quotients of a Heisenberg group, and deduce that there are many examples of compact $K$-contact solvmanifolds not admitting compatible Sasakian structures. We also construct examples of completely solvable $1$-Lefschetz solvmanifolds, some having the $2$-Lefschetz property and some failing it.
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https://arxiv.org/abs/2512.24311
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Academic Papers
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svg
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078471bf209041cf006a38e35b02a036e0c41b79bd5bcc9444aa42626863ba07
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2026-01-01T00:00:00-05:00
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Discrete-Time Mean Field Type Games: Probabilistic Setup
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arXiv:2512.24313v1 Announce Type: new Abstract: We introduce a general probabilistic framework for discrete-time, infinite-horizon discounted Mean Field Type Games (MFTGs) with both global common noise and team-specific common noises. In our model, agents are allowed to use randomized actions, both at the individual level and at the team level. We formalize the concept of Mean Field Markov Games (MFMGs) and establish a connection between closed-loop policies in MFTGs and Markov policies in MFMGs through different layers of randomization. By leveraging recent results on infinite-horizon discounted games with infinite compact state-action spaces, we prove the existence of an optimal closed-loop policy for the original MFTG when the state spaces are at most countable and the action spaces are general Polish spaces. We also present an example satisfying our assumptions, called Mean Field Drift of Intentions, where the dynamics are strongly randomized, and we establish the existence of a Nash equilibrium using our theoretical results.
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https://arxiv.org/abs/2512.24313
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Academic Papers
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svg
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cb8303901b7fed4ba6992428f6eb67b9c8a6e33dc10e8f6b3becfe92b02f10ae
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2026-01-01T00:00:00-05:00
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On the $\tau$-tilting finiteness and silting-discreteness of graded (skew-) gentle algebras
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arXiv:2512.24316v1 Announce Type: new Abstract: This paper investigates finiteness conditions for gentle and skew-gentle algebras. First, we prove that a skew-gentle algebra is $\tau$-tilting finite if and only if it is representation-finite, which extends the result for gentle algebras by Plamondon (2019). Second, using surface models, we characterize silting-discreteness for the perfect derived categories of graded gentle and skew-gentle algebras. Specifically, for a graded gentle algebra, silting-discreteness is equivalent to its associated surface being of genus zero with non-zero winding numbers for all simple closed curves. We further extend this geometric characterization to graded skew-gentle algebras via orbifold surface models.
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https://arxiv.org/abs/2512.24316
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Academic Papers
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svg
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ba00fe0e5474ff668d0dab2f24a93207f101d4409157252f1a9b438e14e06edc
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2026-01-01T00:00:00-05:00
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Fractal behavior of tensor powers of tilting modules of $\text{SL}_2$
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arXiv:2512.24317v1 Announce Type: new Abstract: Given a group $G$ and $V$ a representation of $G$, denote the number of indecomposable summands of $V^{\otimes k}$ by $b_k^{G, V}$. Given a tilting representation $T$ of $\text{SL}_2(K)$ where $K=\overline{K}$ and of characteristic $p>2$, we show that $Ck^{-\alpha_p}(\text{dim} T)^k0$ where $\alpha_p=1-(1/2)\log_p(\frac{p+1}{2}).$
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https://arxiv.org/abs/2512.24317
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Academic Papers
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svg
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1318112dc85b5842492ad21545ed2857cc4b9e66942468aa58e309fefbac9602
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2026-01-01T00:00:00-05:00
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Isomorphism types of definable (maximal) cofinitary groups
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arXiv:2512.24318v1 Announce Type: new Abstract: Kastermans proved that consistently $\bigoplus_{\aleph_1} \mathbb{Z}_2$ has a cofinitary representation. We present a short proof that $\bigoplus_{\mathfrak{c}} \mathbb{Z}_2$ always has an arithmetic cofinitary representation. Further, for every finite group $F$ we construct an arithmetic maximal cofinitary group of isomorphism type $(\ast_{\mathfrak{c}} \mathbb{Z}) \times F$. This answers an implicit question by Schrittesser and Mejak whether one may construct definable maximal cofinitary groups not decomposing into free products.
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https://arxiv.org/abs/2512.24318
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Academic Papers
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d1cd36e547fa1b9cee1d0af714e46e7400684c82df698cf3618e8041beae9aa0
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2026-01-01T00:00:00-05:00
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Arithmetic in the Boij S\"oderberg Cone
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arXiv:2512.24320v1 Announce Type: new Abstract: We study two long-standing conjectures concerning lower bounds for the Betti numbers of a graded module over a polynomial ring. We prove new cases of these conjectures in codimensions five and six by reframing the conjectures as arithmetic problems in the Boij-S\"oderberg cone. In this setting, potential counterexamples correspond to explicit Diophantine obstructions arising from the numerics of pure resolutions. Using number-theoretic methods, we completely classify these obstructions in the codimension three case revealing some delicate connections between Betti tables, commutative algebra and classical Diophantine equations. The new results in codimensions five and six concern Gorenstein algebras where a study of the variety determined by these Diophantine equations is sufficient to resolve the conjecture in this case.
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https://arxiv.org/abs/2512.24320
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Academic Papers
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