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b15c7082beed48db63304eb32977aab0fed358f3e6c23a431f3b7a899b55ab43
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2026-02-02T00:00:00-05:00
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Geometric Quantization by Paths, Part III: The Metaplectic Anomaly
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arXiv:2601.23259v1 Announce Type: new Abstract: In the previous parts of this work, we established the Prequantum Groupoid $\mathbf{T}_\omega$ as the universal geometric container for quantum mechanics. This approach, which we call the "Geometric Quantization by Paths" (GQbP) framework, replaces the traditional construction of principal bundles with the distillation of the space of histories. In this third part, we cross the "Threshold of Analysis" by constructing the intrinsic observable algebra of the system. The harmonic oscillator is treated here as a validation case, demonstrating that the standard resolution via complex polarization and half-forms is naturally integrated into the GQbP framework. Starting from the complexified groupoid, we define the algebra using symplectic half-densities to ensure a canonical convolution product. We then show that the transition to a polarized representation forces a factorization of these densities. The action of the symmetry group on the polarized half-forms generates a divergence term, which we identify as the source of the zero-point energy of the harmonic oscillator, $E_0 = n\hbar/2$. This derivation resolves the "Metaplectic Anomaly" as a necessary geometric consequence of the intrinsic quantization process.
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https://arxiv.org/abs/2601.23259
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4edd0702b5ab316dba7350ab94e86981769d99460e40632658be2b2090e6d33b
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2026-02-02T00:00:00-05:00
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On graphs with girth at least five achieving Steffen's edge coloring bound
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arXiv:2601.23274v1 Announce Type: new Abstract: Vizing and Gupta showed that the chromatic index $\chi'(G)$ of a graph $G$ is bounded above by $\Delta(G) + \mu(G)$, where $\Delta(G)$ and $\mu(G)$ denote the maximum degree and the maximum multiplicity of $G$, respectively. Steffen refined this bound, proving that $\chi'(G) \leq \Delta(G) + \left\lceil \mu(G)/\left\lfloor g(G)/2 \right\rfloor \right\rceil$, where $g(G)$ is the girth of the graph $G$. A {\it ring graph} is a graph obtained from a cycle by duplicating some edges. The equality in Steffen's bound is achieved by ring graphs of the form $\mu C_g$, obtained from an odd cycle $C_g$ by duplicating each edge $\mu$ times. We answer two questions posed by Stiebitz et al. regarding the characterization of graphs which achieve Steffen's bound. In particular, we show that if $G$ is a critical graph which achieves Steffen's bound with $g(G)\geq 5$ and $\chi'(G)\geq \Delta+2$, then $G$ must be a ring graph of odd girth.
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https://arxiv.org/abs/2601.23274
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23ff4c96f7fa2358a0e6e71f1fad75aff9ab16a2274476e19f6218a9c1912d6e
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2026-02-02T00:00:00-05:00
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Variational Tail Bounds for Norms of Random Vectors and Matrices
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arXiv:2503.17300v4 Announce Type: cross Abstract: We propose a variational tail bound for norms of random vectors under moment assumptions on their one-dimensional marginals. A simplified version of the bound that parametrizes the ``aggregating distribution'' using a certain pushforward of the Gaussian distribution is also provided. We apply the proposed method to reproduce some of the well-known bounds on norms of Gaussian random vectors, and also obtain dimension-free tail bounds for the Euclidean norm of random vectors with arbitrary moment profiles. Furthermore, we reproduce a dimension-free concentration inequality for sum of independent and identically distributed positive semidefinite matrices with sub-exponential marginals, and obtain a concentration inequality for the sample covariance matrix of sub-exponential random vectors. We also obtain a tail bound for the operator norm of a random matrix series whose random coefficients may have arbitrary moment profiles. Furthermore, we use coupling to formulate an abstraction of the proposed approach that applies more broadly.
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https://arxiv.org/abs/2503.17300
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47640ab38559e41b9fc49da7ef6b6bdcd55ffbf53cd197c441175771079663b7
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2026-02-02T00:00:00-05:00
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Optimizing Shanghai's Household Waste Recycling Collection Program by Decision-Making based on Mathematical Modeling
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arXiv:2507.03844v1 Announce Type: cross Abstract: In this article, we will discuss the optimization of Shanghai's recycling collection program, with the core of the task as making a decision among the choice of the alternatives. We will be showing a vivid and comprehensive application of the classical mathematical multi-criteria decision model: Analytical Hierarchy Process (AHP), using the eigenvector method. We will also seek the key criteria for the sustainability development of human society, by assessing the important elements of waste recycling.First, we considered the evaluation for a quantified score of the benefits and costs of recycling household glass wastes in Shanghai, respectively. In the evaluation of each score, we both adopted the AHP method to build a hierarchical structure of the problem we are facing. We first identified the key assessment criteria of the evaluation, on various perspectives including direct money costs and benefits, and further environmental and indirect considerations. Then, we distributed questionnaires to our school science teachers, taking the geometric mean, to build the pairwise comparison matrix of the criterion. After the theoretical modeling works are done, we began collecting the essential datasets for the evaluation of each score, by doing research on the official statistics, Internet information, market information and news reports. Sometimes, we proceed a logical pre-procession of the data from other data, if the data wanted isn't directly accessible. Then, we crucially considered the generalization of our mathematical model. We considered from several perspectives, including the extension of assessment criteria, and the consideration of the dynamic interdependency between the wastes, inside a limited transportation container.
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https://arxiv.org/abs/2507.03844
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c48869df8c27f6b7b88bbffad329c656a3ac77ead176f9843efd2d2596510d9b
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2026-02-02T00:00:00-05:00
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Emergent spatial organization of competing species under environmental stress and cooperation
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arXiv:2601.22177v1 Announce Type: cross Abstract: Understanding how species persist under interacting stressors is a central challenge in ecology. We develop a spatially explicit reaction-diffusion framework to investigate competing species in landscapes shaped by climate variability, pollution, resource heterogeneity, and cooperation. Here, temperature follows low-frequency oscillations, while pollution and resources diffuse from localized sources. Growth is governed by a dynamic carrying capacity integrating abiotic stress with an endogenous, pollution-sensitive cooperation field. Numerical simulations reveal the spontaneous emergence of persistent spatial organization, including dominance segregation and stable competitive boundaries. Quantitative analyses-using boundary geometry, fractal dimension, and spatial entropy-demonstrate a transition from intermixed initial states to low-complexity, quasi-stationary configurations. Coexistence occurs through distinct strategies: one species occupies more area, while the other maintains higher local densities. Cooperation enhances resilience but collapses in polluted zones, creating heterogeneous "social buffering." We further introduce a hybrid inverse modeling framework using a Swin Transformer to infer high-dimensional parameters from only two temporal snapshots. Trained on synthetic data, the model accurately recovers demographic, diffusive, and environmental-sensitivity parameters. While it achieves reliable short-term spatial predictions, long-term forecasts diverge due to the intrinsic sensitivity of nonlinear systems. This unified framework links sparse observations to mechanistic dynamics, advancing biodiversity forecasting under accelerating global change.
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https://arxiv.org/abs/2601.22177
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3e8d476d8f5313968386061c539b56f6cc88cdf1b05c3c2197a3bea536ff3b82
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2026-02-02T00:00:00-05:00
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Efficient learning of logical noise from syndrome data
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arXiv:2601.22286v1 Announce Type: cross Abstract: Characterizing errors in quantum circuits is essential for device calibration, yet detecting rare error events requires a large number of samples. This challenge is particularly severe in calibrating fault-tolerant, error-corrected circuits, where logical error probabilities are suppressed to higher order relative to physical noise and are therefore difficult to calibrate through direct logical measurements. Recently, Wagner et al. [PRL 130, 200601 (2023)] showed that, for phenomenological Pauli noise models, the logical channel can instead be inferred from syndrome measurement data generated during error correction. Here, we extend this framework to realistic circuit-level noise models. From a unified code-theoretic perspective and spacetime code formalism, we derive necessary and sufficient conditions for learning the logical channel from syndrome data alone and explicitly characterize the learnable degrees of freedom of circuit-level Pauli faults. Using Fourier analysis and compressed sensing, we develop efficient estimators with provable guarantees on sample complexity and computational cost. We further present an end-to-end protocol and demonstrate its performance on several syndrome-extraction circuits, achieving orders-of-magnitude sample-complexity savings over direct logical benchmarking. Our results establish syndrome-based learning as a practical approach to characterizing the logical channel in fault-tolerant quantum devices.
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https://arxiv.org/abs/2601.22286
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87c0e3fe2ae76b5119649569eaa00201235cff065f9ab758e1ec686a05b433b2
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2026-02-02T00:00:00-05:00
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Quantum bootstrap product codes
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arXiv:2601.22363v1 Announce Type: cross Abstract: Product constructions constitute a powerful method for generating quantum CSS codes, yielding celebrated examples such as toric codes and asymptotically good low-density parity check (LDPC) codes. Since a CSS code is fully described by a chain complex, existing product formalisms are predominantly homological, defined via the tensor product of the underlying chain complexes of input codes, thereby establishing a natural connection between quantum codes and topology. In this Letter, we introduce the \textit{quantum bootstrap product} (QBP), an approach that extends beyond this standard homological paradigm. Specifically, a QBP code is determined by solving a consistency condition termed the ``bootstrap equation''. We find that the QBP paradigm unifies a wide range of important codes, including general hypergraph product (HGP) codes of arbitrary dimensions and fracton codes typically represented by the X-cube code. Crucially, the solutions to the bootstrap equation yield chain complexes where the chain groups and associated boundary maps consist of multiple components. We term such structures \textit{fork complexes}. This structure elucidates the underlying topological structures of fracton codes, akin to foliated fracton order theories. Beyond conceptual insights, we demonstrate that the QBP paradigm can generate self-correcting quantum codes from input codes with constant energy barriers and surpass the code-rate upper bounds inherent to HGP codes. Our work thus substantially extends the scope of quantum product codes and provides a versatile framework for designing fault-tolerant quantum memories.
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https://arxiv.org/abs/2601.22363
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fd46ba5e64febb245aa946363d044ee81d25404d011004379de70bd2ee91bef8
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2026-02-02T00:00:00-05:00
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Spin quantum Hall transition on random networks: exact critical exponents via quantum gravity
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arXiv:2601.22639v1 Announce Type: cross Abstract: We solve the problem of the spin quantum Hall transition on random networks using a mapping to classical percolation that focuses on the boundary of percolating clusters. Using tools of two-dimensional quantum gravity, we compute critical exponents that characterize this transition and confirm that these are related to the exponents for the regular (square) network through the KPZ relation. Our results demonstrate the relevance of the geometric randomness of the networks and support conclusions of numerical simulations of random networks for the integer quantum Hall transition.
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https://arxiv.org/abs/2601.22639
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140702b46423f2c28830949314ecc503575fb104db5e058edaf85c31ff87071f
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2026-02-02T00:00:00-05:00
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Geometric Selection Rules for Singularity Formation in Modified Gravity
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arXiv:2601.22739v1 Announce Type: cross Abstract: We argue that the polynomial degeneracies of curvature invariants can act as geometric selection rules for spacetime singularities in modified theories of gravity. The degeneracies arise purely from the algebraic structure of Riemannian geometry and impose non-trivial constraints on the effective energy-momentum tensor. We derive these constraints for metric $f(R)$ gravity and a wide class of scalar-tensor theories to show that a singularity formation is generally occluded by curvature and/or scalar-induced anisotropies. Therefore, formation of a singularity in modified theories of gravity is not always a generic outcome but can occur only along algebraically admissible branches selected by Riemannian curvature invariants.
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https://arxiv.org/abs/2601.22739
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7e2bc0db8f563fb0561c5b222285e1a291788a8251ec6d4d268a348d81f039cb
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2026-02-02T00:00:00-05:00
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A Framework for the Bayesian Calibration of Complex and Data-Scarce Models in Applied Sciences
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arXiv:2601.22890v1 Announce Type: cross Abstract: In this work, we review the theory involved in the Bayesian calibration of complex computer models, with particular emphasis on their use for applications involving computationally expensive simulations and scarce experimental data. In the article, we present a unified framework that incorporates various Bayesian calibration methods, including well-established approaches. Furthermore, we describe their implementation and use with a new, open-source Python library, ACBICI (A Configurable BayesIan Calibration and Inference Package). All algorithms are implemented with an object-oriented structure designed to be both easy to use and readily extensible. In particular, single-output and multiple-output calibration are addressed in a consistent manner. The article completes the theory and its implementation with practical recommendations for calibrating the problems of interest. These guidelines -- currently unavailable in a unified form elsewhere -- together with the open-source Python library, are intended to support the reliable calibration of computational codes and models commonly used in engineering and related fields. Overall, this work aims to serve both as a comprehensive review of the statistical foundations and (computational) tools required to perform such calculations, and as a practical guide to Bayesian calibration with modern software tools.
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https://arxiv.org/abs/2601.22890
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23c04294104498e587d04eae8ad5223ec246a5343cc63d5353d4ff3c8dac2b5e
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2026-02-02T00:00:00-05:00
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A categorical account of the Metropolis-Hastings algorithm
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arXiv:2601.22911v1 Announce Type: cross Abstract: Metropolis-Hastings (MH) is a foundational Markov chain Monte Carlo (MCMC) algorithm. In this paper, we ask whether it is possible to formulate and analyse MH in terms of categorical probability, using a recent involutive framework for MH-type procedures as a concrete case study. We show how basic MCMC concepts such as invariance and reversibility can be formulated in Markov categories, and how one part of the MH kernel can be analysed using standard CD categories. To go further, we then study enrichments of CD categories over commutative monoids. This gives an expressive setting for reasoning abstractly about a range of important probabilistic concepts, including substochastic kernels, finite and $\sigma$-finite measures, absolute continuity, singular measures, and Lebesgue decompositions. Using these tools, we give synthetic necessary and sufficient conditions for a general MH-type sampler to be reversible with respect to a given target distribution.
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https://arxiv.org/abs/2601.22911
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4fbfb786e9a555c91323c0eeeb0733ca7fdd0180e335da9e7c0b06584aef4c2e
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2026-02-02T00:00:00-05:00
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Open strings on knot complements
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arXiv:2601.22922v1 Announce Type: cross Abstract: Using skein valued holomorphic curve counting techniques, we give a flow loop formula for the skein valued partition function of the Lagrangian knot complement of a fibered knot (of the $A$-model open topological strings with Lagrangian $A$-branes wrapping the complement) in the cotangent bundle of the three-sphere and in the resolved conifold. For torus knots we show that the partition function in the cotangent bundle localizes on two or three holomorphic annuli and give a corresponding generalized quiver structure for the partition function in the resolved conifold. We connect the formula to the augmentation curve, the representation variety of the knot contact homology algebra of the knot, generated by Reeb chords of its Legendrian conormal and with differential given by holomorphic disks interpolating between words of Reeb chords. The curve admits a quantization as a $q$-difference equation for the generating function of symmetrically colored HOMFLYPT-polynomials of the knot or, geometrically, for the $U(1)$-partition function of the knot conormal. For $(2,2p+1)$-torus knots we show that, after a change of variables, the partition function of the knot complement also satisfies this $q$-difference equation. This gives another geometrically defined coordinate chart for the $D$-module defined by the quantized augmentation polynomial.
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https://arxiv.org/abs/2601.22922
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12fa46db082abb0b14362d4923af1b889246deda78ddbd9b28d61a11e64e8266
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2026-02-02T00:00:00-05:00
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Causal spinfoam vertex for 4d Lorentzian quantum gravity
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arXiv:2601.23162v1 Announce Type: cross Abstract: We introduce a new causal spinfoam vertex for $4$d Lorentzian quantum gravity. The causal data are encoded in Toller $T$-matrices, which add to Wigner $D$-matrices $T^{(+)}+T^{(-)}=D$, and for which we provide a Feynman $\mathrm{i}\varepsilon$ representation. We discuss how the Toller poles cancel in the EPRL vertex, how the Livine-Oriti model is obtained in the Barrett-Crane limit, and how spinfoam causal data are distinct from Regge causal data. In the large-spin limit, we show that only Lorentzian Regge geometries with causal data compatible with the spinfoam data are selected, resulting in a single exponential $\exp(+\mathrm{i}\, S_{\mathrm{Regge}}/\hbar)$ and a new form of causal rigidity.
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https://arxiv.org/abs/2601.23162
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a0d1363b2bc1365dd695445b74e531fd86e7c195c1708fe75f99f09c06e0cdd8
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2026-02-02T00:00:00-05:00
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A unified theory of order flow, market impact, and volatility
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arXiv:2601.23172v1 Announce Type: cross Abstract: We propose a microstructural model for the order flow in financial markets that distinguishes between {\it core orders} and {\it reaction flow}, both modeled as Hawkes processes. This model has a natural scaling limit that reconciles a number of salient empirical properties: persistent signed order flow, rough trading volume and volatility, and power-law market impact. In our framework, all these quantities are pinned down by a single statistic $H_0$, which measures the persistence of the core flow. Specifically, the signed flow converges to the sum of a fractional process with Hurst index $H_0$ and a martingale, while the limiting traded volume is a rough process with Hurst index $H_0-1/2$. No-arbitrage constraints imply that volatility is rough, with Hurst parameter $2H_0-3/2$, and that the price impact of trades follows a power law with exponent $2-2H_0$. The analysis of signed order flow data yields an estimate $H_0 \approx 3/4$. This is not only consistent with the square-root law of market impact, but also turns out to match estimates for the roughness of traded volumes and volatilities remarkably well.
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https://arxiv.org/abs/2601.23172
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386d3e8105652fc1fcd22a65f5d276d440f0246a4f2ce8ddd9b720796d557976
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2026-02-02T00:00:00-05:00
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Cubical approximation for directed topology I
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arXiv:1012.0509v3 Announce Type: replace Abstract: Topological spaces - such as classifying spaces, configuration spaces and spacetimes - often admit extra temporal structure. Qualitative invariants on such directed spaces often are more informative yet more difficult to calculate than classical homotopy invariants on underlying spaces because directed spaces rarely decompose as homotopy colimits of simpler directed spaces. Directed spaces often arise as geometric realizations of simplicial sets and cubical sets equipped with temporal structure encoding the orientations of simplices and 1-cubes. In an attempt to develop calculational tools for directed homotopy theory, we prove appropriate simplicial and cubical approximation theorems. We consequently show that geometric realization induces an equivalence between weak homotopy diagram categories of cubical sets and directed spaces and that its right adjoint satisfies an excision theorem. Along the way, we give criteria for two different homotopy relations on directed maps in the literature to coincide.
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https://arxiv.org/abs/1012.0509
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2e42457864e9b80fb6a6d149d2058ef540f5a27f2093743f064c205f96a8c881
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2026-02-02T00:00:00-05:00
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The Partition-Frequency Enumeration Matrix
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arXiv:2102.04191v3 Announce Type: replace Abstract: We develop a calculus that gives an elementary approach to enumerate partition-like objects using an infinite upper-triangular number-theoretic matrix. We call this matrix the Partition-Frequency Enumeration (PFE) matrix. This matrix unifies a large number of results connecting number-theoretic functions to partition-type functions. The calculus is extended to arbitrary generating functions, and functions with Weierstrass products. As a by-product, we recover (and extend) some well-known recurrence relations for many number-theoretic functions, including the sum of divisors function, Ramanujan's $\tau$ function, sums of squares and triangular numbers, and for $\zeta(2n)$, where $n$ is a positive integer. These include classical results due to Euler, Ewell, Ramanujan, Lehmer and others. As one application, we embed Ramanujan's famous congruences $p(5n+4)\equiv 0$ (mod $5)$ and $\tau(5n+5)\equiv 0$ (mod $5)$ into an infinite family of such congruences.
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https://arxiv.org/abs/2102.04191
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ced94bc4851cca75042a62f214c89c9bc5b78f9d8345e2083829fdcdb9e47b3f
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2026-02-02T00:00:00-05:00
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On quasi-isospectrality of potentials and Riemannian manifolds
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arXiv:2202.06110v5 Announce Type: replace Abstract: In this article, we study quasi-isospectral operators as a generalization of isospectral operators. The paper contains both expository material and original results. We begin by reviewing known results on isospectral potentials on compact manifolds and finite intervals, and then introduce the notion of quasi-isospectrality. We next investigate the BMT method as a systematic approach to constructing quasi-isospectral Sturm-Liouville operators on a finite interval, and apply it to several boundary value problems. Our main result shows that any two quasi-isospectral closed manifolds of odd dimension are, in fact, isospectral. In addition, we extend classical compactness results for isospectral potentials on low-dimensional manifolds to the quasi-isospectral setting via heat trace asymptotics.
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https://arxiv.org/abs/2202.06110
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03df4170f5ff83d3fb2f2889a0001751e42de90c64ff298f806b1bebfebbcd41
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2026-02-02T00:00:00-05:00
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On classification of continuous first order theories
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arXiv:2205.12051v3 Announce Type: replace Abstract: We give several new characterizations of $IP$ (the independence property) and $SOP$ (the strict order property) for continuous first order logic and study their relations to the function theory and the Banach space theory. We suggest new dividing lines of unstable theories by the study of subclasses of Baire-1 functions and argue why one should not expect a perfect analog of Shelah's theorem, namely a theory is unstable iff it has $IP$ or $SOP$, for real-valued logics, especially for continuous logic.
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https://arxiv.org/abs/2205.12051
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646a251945e468aaef41fac0da9e1aca8a373db402be293951cb77bbc96e2c37
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2026-02-02T00:00:00-05:00
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Ramification theory from homotopical point of view, I
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arXiv:2206.02401v2 Announce Type: replace Abstract: We prove the compatibility of pushforward along a proper morphism of an \'{e}tale constructible sheaf and the pushforward of its characteristic cycle up to $p$-torsion. This was conjectured by Takeshi Saito. For this, we revisit the construction of the characteristic cycle, due to Saito and Beilinson, from more homotopical point of view. In particular, the language of $\infty$-categories is indispensable to carry this out.
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https://arxiv.org/abs/2206.02401
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5a6884b5174181555a87e9d8e0361a7e98f89218135b90dcd86104cc67edf6c8
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2026-02-02T00:00:00-05:00
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Cardinal Optimizer (COPT) User Guide
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arXiv:2208.14314v4 Announce Type: replace Abstract: Cardinal Optimizer is a high-performance mathematical programming solver for efficiently solving largescale optimization problem. This documentation provides basic introduction to the Cardinal Optimizer.
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https://arxiv.org/abs/2208.14314
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a0d3bc38e0edf1369143a5f9c58043f5b9ce30ae2b4cb5172efaf322ff77afe1
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2026-02-02T00:00:00-05:00
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Conjectures on the reduced Kronecker coefficients
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arXiv:2210.14668v4 Announce Type: replace Abstract: We formulate a series of conjectures on the stable tensor product of irreducible representations of symmetric groups, which are closely related to the reduced Kronecker coefficients. These conjectures are certain generalizations of Okounkov's conjecture on the log-concavity of the Littlewood--Richardson coefficients and the Schur log-concavity theorem of Lam--Postnikov--Pylyavskyy. We prove our conjectures in some special cases and discuss some implications of these conjectures.
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https://arxiv.org/abs/2210.14668
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bbe2c2ac37bd1888f847a4777faecf07d95926d7ff325e6a41a455025b5b3a2c
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2026-02-02T00:00:00-05:00
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Optimal Stabilization of Periodic Orbits: A Symplectic Geometry Approach
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arXiv:2211.11955v3 Announce Type: replace Abstract: In this contribution, the optimal stabilization problem of periodic orbits is studied via invariant manifold theory and symplectic geometry. The stable manifold theory for the optimal point stabilization case is generalized to the case of periodic orbit stabilization, where a normally hyperbolic invariant manifold plays the role of a hyperbolic equilibrium point. A sufficient condition for the existence of an NHIM of an extended Hamiltonian system is derived in terms of a periodic Riccati differential equation. It is shown that the problem of optimal orbit stabilization has a solution if a linearized periodic system is stabilizable and detectable. A moving orthogonal coordinate system is employed along the periodic orbit, which is a natural framework for orbital stabilization and linearization along the orbit. Two illustrative examples are presented: the first involves stabilizing a spring-mass oscillator at a target energy level, and the second addresses an orbit transfer problem for a satellite-a classic scenario in orbital mechanics. In both cases, we show that the proposed nonlinear feedback controller outperforms traditional linear control.
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https://arxiv.org/abs/2211.11955
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0ab5285cc2e7c4dffbf77d8ec8a8f734e502e013606e70054bb2195791f9d555
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2026-02-02T00:00:00-05:00
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A universal sheaf of algebras governing representations of vector fields on quasi-projective varieties
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arXiv:2302.07918v2 Announce Type: replace Abstract: We construct a quasi-coherent sheaf of associative algebras which controls a category of $AV$-modules over a smooth quasi-projective variety. We establish a local structure theorem, proving that in \'etale charts these associative algebras decompose into a tensor product of the algebra of differential operators and the universal enveloping algebra of the Lie algebra of power series vector fields vanishing at the origin.
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https://arxiv.org/abs/2302.07918
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a5959470dd4837d8a9ef1de202e09cffe4e4496680a5fbf796e0407b3775445c
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2026-02-02T00:00:00-05:00
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On the finite time blow-ups for solutions of nonlinear differential equations
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arXiv:2303.10153v2 Announce Type: replace Abstract: We study systems of nonlinear ordinary differential equations where the dominant term, with respect to large spatial variables, causes blow-ups and is positively homogeneous of a degree $1+\alpha$ for some $\alpha>0$. We prove that the asymptotic behavior of a solution $y(t)$ near a finite blow-up time $T_*$ is $(T_*-t)^{-1/\alpha}\xi_*$ for some nonzero vector $\xi_*$. Specific error estimates for $|(T_*-t)^{1/\alpha}y(t)-\xi_*|$ are provided. In some typical cases, they can be a positive power of $(T_*-t)$ or $1/|\ln(T_*-t)|$. This depends on whether the decaying rate of the lower order term, relative to the size of the dominant term, is of a power or logarithmic form. Similar results are obtained for a class of nonlinear differential inequalities with finite time blow-up solutions. Our results cover larger classes of nonlinear equations, differential inequalities and error estimates than those in the previous work.
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https://arxiv.org/abs/2303.10153
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64590f84b0ccdb84d6cecc54c36029a510983a7676e26ceead28790c19b604eb
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2026-02-02T00:00:00-05:00
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Higher topological complexity of Seifert fibered manifolds
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arXiv:2304.01274v3 Announce Type: replace Abstract: In this article, we investigate the higher topological complexity of oriented Seifert fibered manifolds that are Eilenberg--MacLane spaces $K(G,1)$ with infinite fundamental group $G$. We first refine the cohomological lower bounds for higher topological complexity by introducing the notion of higher topological complexity weights. As an application, we show that the $r^{\text{th}}$ topological complexity of these manifolds lies in $\{3r-1, 3r, 3r+1\}$, and characterize large families where the value is $3r$ or $3r+1$. Additionally, we establish a sufficient condition for higher topological complexity to be exactly $3r$ when the base surface is orientable and aspherical. Finally, we show that the higher topological complexity of the wedge of finitely many closed, orientable, aspherical $3$-manifolds is exactly $3r+1$.
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https://arxiv.org/abs/2304.01274
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ab7d71686996085a0beac0818f4aadaf31f6d6c6634a3263209f505af918bacf
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2026-02-02T00:00:00-05:00
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Amending the Lonely Runner Spectrum Conjecture
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arXiv:2306.10417v2 Announce Type: replace Abstract: Let $||x||$ be the absolute distance from $x$ to the nearest integer. For a set of distinct positive integral speeds $v_1, \ldots, v_n$, we define its maximum loneliness, also known as the gap $\delta$, to be $$ML(v_1,\ldots,v_n) = \max_{t \in \mathbb{R}}\min_{1 \leq i \leq n} || tv_i||.$$ The Loneliness Spectrum Conjecture, recently proposed by Kravitz (2021), asserts that $$\exists s \in \mathbb{N}, \text{ML}(v_1,\ldots,v_n) = \frac{s} {sn + 1} \text{ or } \text{ML}(v_1,\ldots,v_n) \geq \frac{1}{n}. $$ We disprove the Loneliness Spectrum Conjecture for $n = 4$ with an infinite family of counterexamples and propose an alternative conjecture. We confirm the amended conjecture for $n = 4$ whenever there exists a pair of speeds with a common factor of at least $3$ and also prove some related results.
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https://arxiv.org/abs/2306.10417
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a4bea096c14814fd13ccbcdd51509a7089ddb2fae14c5773a86e2d6d9fc9c2ef
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2026-02-02T00:00:00-05:00
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Tilings in quasi-random $k$-partite hypergraphs
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arXiv:2306.10539v2 Announce Type: replace Abstract: Given $k\ge 2$ and two $k$-graphs ($k$-uniform hypergraphs) $F$ and $H$, an $F$-factor in $H$ is a set of vertex disjoint copies of $F$ that together cover the vertex set of $H$. Lenz and Mubayi were first to study the $F$-factor problems in quasi-random $k$-graphs with a minimum degree condition. Recently, Ding, Han, Sun, Wang and Zhou gave the density threshold for having all $3$-partite $3$-graphs factors in quasi-random $3$-graphs with vanishing minimum codegree condition $\Omega(n)$. In this paper, we consider embedding factors when the host $k$-graph is $k$-partite and quasi-random with partite minimum codegree condition. We prove that if $p>1/2$ and $F$ is a $k$-partite $k$-graph with each part having $m$ vertices, then for $n$ large enough and $m\mid n$, any $p$-dense $k$-partite $k$-graph with each part having $n$ vertices and partite minimum codegree condition $\Omega(n)$ contains an $F$-factor. We also present a construction showing that $1/2$ is best possible. Furthermore, for $1\leq \ell \leq k-2$, by constructing a sequence of $p$-dense $k$-partite $k$-graphs with partite minimum $\ell$-degree $\Omega(n^{k-\ell})$ having no $K_k(m)$-factor, we show that the partite minimum codegree constraint can not be replaced by other partite minimum degree conditions. On the other hand, we prove that $n/2$ is the asymptotic partite minimum codegree threshold for having all fixed $k$-partite $k$-graph factors in sufficiently large host $k$-partite $k$-graphs even without quasi-randomness.
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https://arxiv.org/abs/2306.10539
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d72286bcc2947772de21b7ff28f42cded04cfcc6e4d4ba93bb91c92e31cbf9c1
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2026-02-02T00:00:00-05:00
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Quartic Gauss sums over primes and metaplectic theta functions
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arXiv:2306.11875v5 Announce Type: replace Abstract: We improve 1987 estimates of Patterson for sums of quartic Gauss sums over primes. Our Type-I and Type-II estimates feature new ideas, including use of the quadratic large sieve over $\mathbb{Q}(i)$, and Suzuki's evaluation of the Fourier-Whittaker coefficients of quartic theta functions at squares. We also conjecture asymptotics for certain moments of quartic Gauss sums over primes.
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https://arxiv.org/abs/2306.11875
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454eef2a36a0f23a878b28a1b8ed7144b18629631e825ff151798f9206aef15b
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2026-02-02T00:00:00-05:00
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The center of the asymptotic Hecke category and unipotent character sheaves
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arXiv:2307.07276v3 Announce Type: replace Abstract: In 2015, Lusztig [Bull. Inst. Math. Acad. Sin. (N.S.)10(2015), no.1, 1-72] showed that for a connected reductive group over an algebraic closure of a finite field the associated (geometric) Hecke category admits a truncation in a two-sided Kazhdan--Lusztig cell, making it a categorification of the asymptotic algebra (J-ring), and that the categorical center of this "asymptotic Hecke category" is equivalent to the category of unipotent character sheaves supported in the cell. Subsequently, Lusztig noted that an asymptotic Hecke category can be constructed for any finite Coxeter group using Soergel bimodules. Lusztig conjectured that the centers of these categories are modular tensor categories (which was then proven by Elias and Williamson) and that for non-crystallographic finite Coxeter groups the S-matrices coincide with the Fourier matrices that were constructed in the 1990s by Lusztig, Malle, and Brou\'e--Malle. If the conjecture is true, the centers may be considered as categories of "unipotent character sheaves" for non-crystallographic finite Coxeter groups. In this paper, we show that the conjecture is true for dihedral groups and for some (we cannot resolve all) cells of H3 and H4. The key ingredient is the method of H-reduction and the identification of the (reduced) asymptotic Hecke category with known categories whose center is already known as well. We conclude by studying the asymptotic Hecke category and its center for some infinite Coxeter groups with a finite cell.
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https://arxiv.org/abs/2307.07276
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1c1f37ca95de9616034c7b9cbc1c1d6cbcefcc0f84ca0bc1dd2cea458b4b8982
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2026-02-02T00:00:00-05:00
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Logarithmic Asymptotic Relations Between $p$-Values and Mutual Information
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arXiv:2308.14735v2 Announce Type: replace Abstract: We establish a precise connection between statistical significance in dependence testing and information-theoretic dependence as quantified by Shannon mutual information (MI). In the absence of prior distributional information, we consider a maximum-entropy model and show that the probability associated with the realization of a given magnitude of MI takes an exponential form, yielding a corresponding tail-probability interpretation of a $p$-value. In contingency tables with fixed marginal frequencies, we analyze Fisher's exact test and prove that its $p$-value $P_F$ satisfies a logarithmic asymptotic relation of the form $MI=-(1/N)\log P_F + O(\log(N+1)/N)$ as the sample size $N\to\infty$. These results clarify the role of MI as the exponential rate governing the asymptotic behavior of $p$-values in the settings studied here, and they enable principled comparisons of dependence across datasets with different sample sizes. We further discuss implications for combining evidence across studies via meta-analysis, allowing mutual information and its statistical significance to be integrated in a unified framework.
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https://arxiv.org/abs/2308.14735
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565f8326d1f18b5ce26a3c6a74fc279fe9908a710cab4d3de7a774eb8f5a1c03
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2026-02-02T00:00:00-05:00
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Frank--Wolfe algorithms for piecewise star-convex functions with a nonsmooth difference-of-convex structure
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arXiv:2308.16444v4 Announce Type: replace Abstract: In the present paper, we formulate two versions of Frank--Wolfe algorithm or conditional gradient method to solve the DC optimization problem with an adaptive step size. The DC objective function consists of two components; the first is thought to be differentiable with a continuous Lipschitz gradient, while the second is only thought to be convex. The second version is based on the first and employs finite differences to approximate the gradient of the first component of the objective function. In contrast to past formulations that used the curvature/Lipschitz-type constant of the objective function, the step size computed does not require any constant associated with the components. For the first version, we established that the algorithm is well-defined of the algorithm and that every limit point of the generated sequence is a stationary point of the problem. We also introduce the class of weak-star-convex functions and show that, despite the fact that these functions are non-convex in general, the rate of convergence of the first version of the algorithm to minimize these functions is ${\cal O}(1/k)$. The finite difference used to approximate the gradient in the second version of the Frank-Wolfe algorithm is computed with the step-size adaptively updated using two previous iterations. Unlike previous applications of finite difference in the Frank-Wolfe algorithm, which provided approximate gradients with absolute error, the one used here provides us with a relative error, simplifying the algorithm analysis. In this case, we show that all limit points of the generated sequence for the second version of the Frank-Wolfe algorithm are stationary points for the problem under consideration, and we establish that the rate of convergence for the duality gap is ${\cal O}(1/\sqrt{k})$.
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https://arxiv.org/abs/2308.16444
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0fa8046c2b7cbfc44d5ffa091317b3bd9c457509d2fdfb7ea4a0b445cbcb981d
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2026-02-02T00:00:00-05:00
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High-Dimensional Bernstein Von-Mises Theorems for Covariance and Precision Matrices
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arXiv:2309.08556v3 Announce Type: replace Abstract: This paper aims to examine the characteristics of the posterior distribution of covariance/precision matrices in a "large $p$, large $n$" scenario, where $p$ represents the number of variables and $n$ is the sample size. Our analysis focuses on establishing asymptotic normality of the posterior distribution of the entire covariance/precision matrices under specific growth restrictions on $p_n$ and other mild assumptions. In particular, the limiting distribution turns out to be a symmetric matrix variate normal distribution whose parameters depend on the maximum likelihood estimate. Our results hold for a wide class of prior distributions which includes standard choices used by practitioners. Next, we consider Gaussian graphical models which induce sparsity in the precision matrix. Asymptotic normality of the corresponding posterior distribution is established under mild assumptions on the prior and true data-generating mechanism.
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https://arxiv.org/abs/2309.08556
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e834b3bb80c9d231cd34cea0ef6bb41e443204918096ae6e7c0877b0a007975e
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2026-02-02T00:00:00-05:00
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Imaginaries, products and the adele ring
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arXiv:2309.11678v3 Announce Type: replace Abstract: We describe the imaginary sorts of infinite products in terms of imaginary sorts of the factors. We extend the result to certain reduced powers and then to infinite products $\prod_{i\in I} M_i$ enriched with a predicate for the ideal of finite subsets of $I$. As a special case, using the Hils-Rideau-Kikuchi uniform $p$-adic elimination of imaginaries, we find the imaginary sorts of the ring of rational adeles. Our methods include the use of the Harrington-Kechris-Louveau Glimm-Efros dichotomy both for transitioning from monadic second order imaginaries to first-order reducts, and for proving a certain ``one-way'' model-theoretic orthogonality within the adelic imaginaries.
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https://arxiv.org/abs/2309.11678
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fa718c1f6de10ccc484a62bff1151b99b239da3996fbc3aaf15271597711be20
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2026-02-02T00:00:00-05:00
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Cubical Approximation for Directed Topology II
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arXiv:2309.16619v4 Announce Type: replace Abstract: The paper establishes an equivalence between directed homotopy categories of (diagrams of) cubical sets and (diagrams of) directed topological spaces. This equivalence both lifts and extends an equivalence between classical homotopy categories of cubical sets and topological spaces. Some simple applications include combinatorial descriptions and subsequent calculations of directed homotopy monoids and directed singular 1-cohomology monoids. Another application is a characterization of isomorphisms between small categories up to zig-zags of natural transformations as directed homotopy equivalences between directed classifying spaces. Cubical sets throughout the paper are taken to mean presheaves over the minimal symmetric monoidal variant of the cube category. Along the way, the paper characterizes morphisms in this variant as the interval-preserving lattice homomorphisms between finite Boolean lattices.
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https://arxiv.org/abs/2309.16619
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65cfe4d86791b48c47bb68070eb4181e966549d6b931a89a5d2113dc7959311a
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2026-02-02T00:00:00-05:00
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Balanced metrics, Zoll deformations and isosystolic inequalities in $\mathbb{C}P^n$
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arXiv:2310.10877v2 Announce Type: replace Abstract: The k-systole of a Riemannian manifold is the infimum of the volume over all homologically non-trivial k-cycles. In this paper we discuss the behavior of the dimension two and co-dimension two systole of the complex projective space for distinguished classes of metrics, namely the homogeneous metrics and the balanced metrics. In particular, we argue that every homogeneous metric maximizes the systole in its volume-normalized conformal class, as well as that each K\"ahler metric locally minimizes the systole on the set of volume-normalized balanced metrics. The proof demands the implementation of integral geometric techniques, and a careful analysis of the second variation of the systole functional. As an application, we characterize the systolic behavior of almost-Hermitian 1-parameter Zoll-like deformations of the Fubini-Study metric.
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https://arxiv.org/abs/2310.10877
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54a3df4dea48c7dbe3125f3f6c729328106fb985cd3f885b6472c9c5bd82d1e4
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2026-02-02T00:00:00-05:00
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Optimal sampling for stochastic and natural gradient descent
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arXiv:2402.03113v2 Announce Type: replace Abstract: We consider the problem of optimising the expected value of a loss functional over a nonlinear model class of functions, assuming that we have only access to realisations of the gradient of the loss. This is a classical task in statistics, machine learning and physics-informed machine learning. A straightforward solution is to replace the exact objective with a Monte Carlo estimate before employing standard first-order methods like gradient descent, which yields the classical stochastic gradient descent method. But replacing the true objective with an estimate ensues a generalisation error. Rigorous bounds for this error typically require strong compactness and Lipschitz continuity assumptions while providing a very slow decay with sample size. To alleviate these issues, we propose a version of natural gradient descent that is based on optimal sampling methods. Under classical assumptions on the loss and the nonlinear model class, we prove that this scheme converges almost surely monotonically to a stationary point of the true objective. Under Polyak-Lojasiewicz-type conditions, this provides bounds for the generalisation error. As a remarkable result, we show that our stochastic optimisation scheme achieves the linear or exponential convergence rates of deterministic first order descent methods under suitable conditions.
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https://arxiv.org/abs/2402.03113
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fb94f9e49cd934642d139c2847d0b64d32b75bc5d9ed7e581801bfefa92d41b6
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2026-02-02T00:00:00-05:00
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Cyclotomic Factors and LRS-Degeneracy
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arXiv:2403.08751v4 Announce Type: replace Abstract: We present three new, practical algorithms for polynomials in $\mathbb{Z}[x]$: one to test if a polynomial is cyclotomic, one to determine which cyclotomic polynomials are factors, and one to determine whether the given polynomial is LRS-degenerate. A polynomial is "LRS-degenerate" iff it has two distinct roots $\alpha, \beta$ such that $\beta = \zeta \alpha$ for some root of unity $\zeta$. All three algorithms are based on "intelligent brute force". The first two produce the indexes of the cyclotomic polynomials; the third produces a list of degeneracy orders. The algorithms are implemented in CoCoALib.
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https://arxiv.org/abs/2403.08751
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ca50342dfe4f0093065ceba8d714d253bcc1e0b4419323b2a36ebe1c9b3c2545
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2026-02-02T00:00:00-05:00
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Quasi-invariant lifts of completely positive maps for groupoid actions
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arXiv:2405.07859v2 Announce Type: replace Abstract: Let $G$ be a locally compact, Hausdorff, second countable groupoid and $A$ be a separable, $C_0(G^{(0)})$-nuclear, $G$-$C^*$-algebra. We prove the existence of quasi-invariant, completely positive and contractive lifts for equivariant, completely positive and contractive maps from $A$ into a separable, quotient $C^*$-algebra. Along the way, we construct the Busby invariant for $G$-actions.
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https://arxiv.org/abs/2405.07859
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88e10ef0178caf1e7616c7cb0c7e1f2fe33d53cb6231a6fe581d26b95a69ee75
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2026-02-02T00:00:00-05:00
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Asymptotic vanishing of cohomology in triangulated categories
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arXiv:2405.12763v2 Announce Type: replace Abstract: Given a graded-commutative ring acting centrally on a triangulated category, our main result shows that if cohomology of a pair of objects of the triangulated category is finitely generated over the ring acting centrally, then the asymptotic vanishing of the cohomology is well-behaved. In particular, enough consecutive asymptotic vanishing of cohomology implies all eventual vanishing. Several key applications are also given.
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https://arxiv.org/abs/2405.12763
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64fa69c606749080135ceccda86357218f509de08fe826c552f5e48074a9db77
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2026-02-02T00:00:00-05:00
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SLE and its partition function in multiply connected domains via the Gaussian Free Field and restriction measures
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arXiv:2405.20148v2 Announce Type: replace Abstract: One way to uniquely define Schramm-Loewner Evolution (SLE) in multiply connected domains is to use the restriction property. This gives an implicit definition of a $\sigma$-finite measure on curves; yet it is in general not clear how to construct such measures nor whether the mass of these measures, called the partition function, is finite. We provide an explicit construction of the such conformal restriction SLEs in multiply connected domains when $\kappa = 4$ using the Gaussian Free Field (GFF). In particular, both when the target points of the curve are on the same or on distinct boundary components, we show that there is a mixture of laws of level lines of GFFs that satisfies the restriction property. This allows us to give an expression for the partition function of $\mathrm{SLE}_4$ on multiply connected domains and shows that the partition function is finite, answering the question raised in [Lawler, J. Stat. Phys. 2009]. In a second part, we provide a second construction of $\mathrm{SLE}_\kappa$ in multiply-connected domains for the whole range $\kappa \in (8/3,4]$; specific, however, to the case of the two target points belonging to the same boundary components. This is inspired by [Werner, Wu, Electron. J. Probab. 2013] and consists of a mixture of laws on curves obtained by following $\mathrm{CLE}_\kappa$ loops and restriction hulls attached to parts of the boundary of the domain. In this case as well, we obtain as a corollary the finiteness of the partition function for this type of $\mathrm{SLE}_\kappa$.
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https://arxiv.org/abs/2405.20148
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d1be80a5d754322c8ecebe6da55d3bc6545c7dd99d46a053021c065cae2374b2
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2026-02-02T00:00:00-05:00
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Enumeration of minimal transversals of hypergraphs of bounded VC-dimension
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arXiv:2407.00694v4 Announce Type: replace Abstract: We consider the problem of enumerating all minimal transversals (also called minimal hitting sets) of a hypergraph $\mathcal{H}$. An equivalent formulation of this problem known as the \emph{transversal hypergraph} problem (or \emph{hypergraph dualization} problem) is to decide, given two hypergraphs, whether one corresponds to the set of minimal transversals of the other. The existence of a polynomial time algorithm to solve this problem is a long standing open question. In \cite{fredman_complexity_1996}, the authors present the first sub-exponential algorithm to solve the transversal hypergraph problem which runs in quasi-polynomial time, making it unlikely that the problem is (co)NP-complete. In this paper, we show that when one of the two hypergraphs is of bounded VC-dimension, the transversal hypergraph problem can be solved in polynomial time, or equivalently that if $\mathcal{H}$ is a hypergraph of bounded VC-dimension, then there exists an incremental polynomial time algorithm to enumerate its minimal transversals. This result generalizes most of the previously known polynomial cases in the literature since they almost all consider classes of hypergraphs of bounded VC-dimension. As a consequence, the hypergraph transversal problem is solvable in polynomial time for any class of hypergraphs closed under partial subhypergraphs. We also show that the proposed algorithm runs in quasi-polynomial time in general hypergraphs and runs in polynomial time if the conformality of the hypergraph is bounded, which is one of the few known polynomial cases where the VC-dimension is unbounded.
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https://arxiv.org/abs/2407.00694
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1af3e5ebb14413d4cc815044104a8d752d8a50b6c3032c5ff86417025805ef6d
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2026-02-02T00:00:00-05:00
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A unified theory of regular functions of a hypercomplex variable
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arXiv:2408.01523v2 Announce Type: replace Abstract: This work proposes a unified theory of regularity in one hypercomplex variable: the theory of $T$-regular functions. In the special case of quaternion-valued functions of one quaternionic variable, this unified theory comprises Fueter-regular functions, slice-regular functions and a recently-discovered function class. In the special case of Clifford-valued functions of one paravector variable, it encompasses monogenic functions, slice-monogenic functions, generalized partial-slice monogenic functions, and a variety of function classes not yet considered in literature. For $T$-regular functions over an associative $*$-algebra, this work provides integral formulas, series expansions, an Identity Principle, a Maximum Modulus Principle and a Representation Formula. It also proves some foundational results about $T$-regular functions over an alternative but nonassociative $*$-algebra, such as the real algebra of octonions.
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https://arxiv.org/abs/2408.01523
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ad335e9d14ee5ec2e42d39d2093910f7880b8f3a45f779ee92931c9328ff797a
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2026-02-02T00:00:00-05:00
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Homogenization of Poisson-Nernst-Planck equations for multiple species in a porous medium
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arXiv:2408.08831v3 Announce Type: replace Abstract: We rigorously derive a homogenized model for the Poisson--Nernst--Planck (PNP) equations for the case of multiple species defined on a periodic porous medium in spatial dimensions two and three. This extends the previous homogenization results for the PNP equations concerning two species. Here, the main difficulty is that the microscopic concentrations remain uniformly bounded in a space with relatively weak regularity. Therefore, the standard Aubin-Lions-Simon type compactness results for porous media, which give strong convergence of the microscopic solutions, become inapplicable in our weak setting. We overcome this problem by constructing suitable cut-off functions. The cut-off function, together with the application of a previously known energy functional, yields strong convergence of the microscopic concentrations in $L^1_t L^r_x$, for some $r>2$, enabling us to pass to the limit in the nonlinear drift term. Finally, we derive the homogenized equations by means of two-scale convergence in $L^p_t L^q_x$ setting.
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https://arxiv.org/abs/2408.08831
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4c81044926293a33ae700b9b87a34a26fc66a25d68c94017a295df0ff3182030
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2026-02-02T00:00:00-05:00
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Lannes' $T$-functor and mod-$p$ cohomology of profinite groups
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arXiv:2408.12488v2 Announce Type: replace Abstract: The Lannes-Quillen theorem relates the mod-$p$ cohomology of a finite group $G$ with the mod-$p$ cohomology of centralizers of abelian elementary $p$-subgroups of $G$, for $p>0$ a prime number. This theorem was extended to profinite groups whose mod-$p$ cohomology algebra is finitely generated by Henn. In a weaker form, the Lannes-Quillen theorem was then extended by Symonds to arbitrary profinite groups. Building on Symonds' result, we formulate and prove a full version of this theorem for all profinite groups. For this purpose, we develop a theory of products for families of discrete torsion modules, parameterized by a profinite space, which is dual, in a very precise sense, to the theory of coproducts for families of profinite modules, parameterized by a profinite space, developed by Haran, Melnikov and Ribes. In the last section, we give applications to the problem of conjugacy separability of $p$-torsion elements and finite $p$-subgroups.
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https://arxiv.org/abs/2408.12488
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628af47f8e6a9596708a6853d2030fb4a5f4fc571d25dd1f5b24bc85581bf7ea
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2026-02-02T00:00:00-05:00
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Multivariate functorial difference
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arXiv:2409.09494v2 Announce Type: replace Abstract: Partial difference operators for a large class of functors between presheaf categories are introduced, extending our difference operator from \cite{Par24} to the multivariable case. These combine into the Jacobian profunctor which provides the setting for a lax chain rule. We introduce a functorial version of multivariable Newton series whose aim is to recover a functor from its iterated differences. Not all functors are recovered but we get a best approximation in the form of a left adjoint, and the induced comonad is idempotent. Its fixed points are what we call soft analytic functors, a generalization of the multivariable analytic functors of Fiore et al.~\cite{FioGamHylWin08}.
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https://arxiv.org/abs/2409.09494
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5ca7b65bdd6f85010fb8142bfc6a34324ac5cffa585677022d3dd6c8c19c0a17
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2026-02-02T00:00:00-05:00
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Positively closed $Sh(B)$-valued models
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arXiv:2409.11231v3 Announce Type: replace Abstract: We study positively closed and strongly positively closed topos-valued models of coherent theories. Positively closed is a global notion (it is defined in terms of all possible outgoing homomorphisms), while strongly positively closed is a local notion (it only concerns the definable sets inside the model). For $\mathbf{Set}$-valued models of coherent theories they coincide. We prove that if $\mathcal{E}=Sh(B,\tau _{coh})$ for a complete Boolean algebra, then positively closed but not strongly positively closed $\mathcal{E}$-valued models of coherent theories exist, yet, there is an alternative local property which characterizes positively closed $\mathcal{E}$-valued models. A large part of our discussion is given in the context of infinite quantifier geometric logic, dealing with the fragment $L^g_{\kappa \kappa }$ where $\kappa $ is weakly compact.
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https://arxiv.org/abs/2409.11231
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a6ab91406717f5cb75b1c6b91d1d6e0d7eb44bbe82b9dc264fff8c5b760a6886
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2026-02-02T00:00:00-05:00
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On the tails of log-concave density estimators
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arXiv:2409.17910v3 Announce Type: replace Abstract: It is shown that the nonparametric maximum likelihood estimator of a univariate log-concave probability density satisfies desirable consistency properties in the tail regions. Specifically, let $P$ and $f$ denote the true underlying distribution and density, respectively. If $\hat{f}_n$ is the estimated log-concave density, and $\hat{\varphi}_n = \log \hat{f}_n$, then we specify sequences $(b_n)_{n\in \mathbb{N}}$ such that $P([b_n,\infty)) \to 0$ at a specific speed, ensuring that the absolute errors or absolute relative errors of $\hat{f}_n, \ \hat{\varphi}_n$ and $\hat{\varphi}_n'$ converge to zero uniformly on sets $[a, b_n]$. The main tools, besides characterizations of $\hat{f}_n$, are exponential and maximal inequalities for truncated moments of log-concave distributions, which are of independent interest.
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https://arxiv.org/abs/2409.17910
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4342f52debdfd1138b15a18b3f31d49bea3830586852854eccc610e1129d7eea
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2026-02-02T00:00:00-05:00
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On the Uniqueness of the Norton-Sullivan Quasiconformal extension
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arXiv:2409.19805v2 Announce Type: replace Abstract: We show that the extension map \[ \mathcal{E}_{NS}(f)(z)=\frac{f(x+y)+f(x-y)}{2}+i\frac{f(x+y)-f(x-y)}{2}\mbox{ for all }z=x+iy\in\mathbb{H}\,, \] defined by Norton and Sullivan in '96, is the only locally linear extension map taking bi-Lipschitz functions on $\mathbb{R}$ to quasiconformal functions on $\mathbb{H}$, modulo the action of a group isomorphic to the linear group. In fact, we discovered many other extension like this one (lying in the orbit of such group action), such as: $f(x)\mapsto f(x)+i(f(x)-f(x-y))$.
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https://arxiv.org/abs/2409.19805
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f17e3af352f01e999a27b5627bdbda35123f1388e14157eb4b8f3157b3d73970
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2026-02-02T00:00:00-05:00
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Large Deviations of Mean-Field Jump-Markov Processes on Structured Sparse Disordered Graphs
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arXiv:2410.13682v2 Announce Type: replace Abstract: We prove a Large Deviation Principle for {\color{blue} jump-Markov } Processes on sparse large disordered network with disordered connectivity. The network is embedded in a geometric space, with the probability of a connection a (scaled) function of the spatial positions of the nodes. This type of model has numerous applications, including neuroscience, epidemiology and social networks. We prove that the rate function (that indicates the asymptotic likelihood of state transitions) is the same as for a network with all-to-all connectivity. We apply our results to a stochastic $SIS$ epidemiological model on a disordered networks, and determine Euler-Lagrange equations that dictate the most likely transition path between different states of the network.
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https://arxiv.org/abs/2410.13682
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7bd26e8715944660910e8135cc315466fbd7d5843544cc9eb7daeeb022aab389
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2026-02-02T00:00:00-05:00
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Zarankiewicz bounds from distal regularity lemma
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arXiv:2410.13695v3 Announce Type: replace Abstract: Since K\H{o}v\'ari, S\'os, and Tur\'an proved upper bounds for the Zarankiewicz problem in 1954, much work has been undertaken to improve these bounds, and some have done so by restricting to particular classes of graphs. In 2017, Fox, Pach, Sheffer, Suk, and Zahl proved better bounds for semialgebraic binary relations, and this work was extended by Do in the following year to arbitrary semialgebraic relations. In this paper, we show that Zarankiewicz bounds in the shape of Do's are enjoyed by all relations satisfying the distal regularity lemma, an improved version of the Szemer\'edi regularity lemma satisfied by relations definable in distal structures (a vast generalisation of o-minimal structures).
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https://arxiv.org/abs/2410.13695
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6fa1fa729fc2de2ca97b3c645197c3ee7e01f08b4cb435b135ed1b4bb2a6fa02
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2026-02-02T00:00:00-05:00
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State Estimation Using Sparse DEIM and Recurrent Neural Networks
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arXiv:2410.15982v3 Announce Type: replace Abstract: Sparse Discrete Empirical Interpolation Method (S-DEIM) was recently proposed for state estimation in dynamical systems when only a sparse subset of the state variables can be observed. The S-DEIM estimate involves a kernel vector whose optimal value is inferred through a data assimilation algorithm. This data assimilation step suffers from two drawbacks: (i) It requires the knowledge of the governing equations of the dynamical system, and (ii) It is not generally guaranteed to converge to the optimal kernel vector. To address these issues, here we introduce an equation-free S-DEIM framework that estimates the optimal kernel vector from sparse observational time series using recurrent neural networks (RNNs). We show that the recurrent architecture is necessary since the kernel vector cannot be estimated from instantaneous observations. But RNNs, which incorporate the past history of the observations in the learning process, lead to nearly optimal estimations. We demonstrate the efficacy of our method on three numerical examples with increasing degree of spatiotemporal complexity: a conceptual model of atmospheric flow known as the Lorenz-96 system, the Kuramoto-Sivashinsky equation, and the Rayleigh-Benard convection. In each case, the resulting S-DEIM estimates are satisfactory even when a relatively simple RNN architecture, namely the reservoir computing network, is used. More specifically, our RNN-based S-DEIM state estimations reduce the relative error between 42% and 58% when compared to Q-DEIM which ignores the kernel vector by setting it equal to zero.
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https://arxiv.org/abs/2410.15982
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9019772725da3a3efd6920f39049806d4e391c14f682bc87864b4898f36d430f
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2026-02-02T00:00:00-05:00
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Gelfand-Fuks cohomology of vector fields on algebraic varieties
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arXiv:2410.20316v2 Announce Type: replace Abstract: For an affine algebraic variety, we introduce algebraic Gelfand-Fuks cohomology of polynomial vector fields with coefficients in differentiable $AV$-modules. Its complex is given by cochains that are differential operators in the sense of Grothendieck. Using the jets of vector fields, we compute this cohomology for varieties with uniformizing parameters. We prove that in this case, Gelfand-Fuks cohomology with coefficients in a tensor module decomposes as a tensor product of the de Rham cohomology of the variety and the cohomology of the Lie algebra of vector fields on affine space, vanishing at the origin. We explicitly compute this cohomology for affine space, the torus, and Krichever-Novikov algebras.
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https://arxiv.org/abs/2410.20316
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06f487da2ce5ccf4b8168a6641a19a61289adae25a10f37bbd486f5ad4450718
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2026-02-02T00:00:00-05:00
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On uniqueness in structured model learning
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arXiv:2410.22009v3 Announce Type: replace Abstract: This paper addresses the problem of uniqueness in learning physical laws for systems of partial differential equations (PDEs). Contrary to most existing approaches, it considers a framework of structured model learning, where existing, approximately correct physical models are augmented with components that are learned from data. The main results of the paper are a uniqueness and a convergence result that cover a large class of PDEs and a suitable class of neural networks used for approximating the unknown model components. The uniqueness result shows that, in the limit of full, noiseless measurements, a unique identification of the unknown model components as functions is possible as classical regularization-minimizing solutions of the PDE system. This result is complemented by a convergence result showing that model components learned as parameterized neural networks from incomplete, noisy measurements approximate the regularization-minimizing solutions of the PDE system in the limit. These results are possible under specific properties of the approximating neural networks and due to a dedicated choice of regularization. With this, a practical contribution of this analytic paper is to provide a class of model learning frameworks different to standard settings where uniqueness can be expected in the limit of full measurements.
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https://arxiv.org/abs/2410.22009
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34d0f6fcb5b8c25c0a20f91c22850236366cd1dad8c324db3e375e87d6773f5c
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2026-02-02T00:00:00-05:00
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Generalized random processes related to Hadamard operators and Le Roy measures
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arXiv:2410.22880v3 Announce Type: replace Abstract: The definition of generalized random processes in Gel'fand sense allows to extend well-known stochastic models, such as the fractional Brownian motion, and study the related fractional pde's, as well as stochastic differential equations in distributional sense. By analogy with the construction (in the infinite-dimensional white-noise space) of the latter, we introduce two processes defined by means of Hadamard-type fractional operators. When used to replace the time derivative in the governing p.d.e.'s, the Hadamard-type derivatives are usually associated with ultra-slow diffusions. On the other hand, in our construction, they directly determine the memory properties of the so-called Hadamard fractional Brownian motion (H-fBm) and its long-time behaviour. Still, for any finite time horizon, the H-fBm displays a standard diffusing feature. We then extend the definition of the H-fBm from the white noise space to an infinite dimensional grey-noise space built on the Le Roy measure, so that our model represents an alternative to the generalized grey Brownian motion. In this case, we prove that the one-dimensional distribution of the process satisfies a heat equation with non-constant coefficients and fractional Hadamard time-derivative. Finally, once proved the existence of the distributional derivative of the above defined processes and derived an integral formula for it, we construct an Ornstein-Uhlenbeck type process and evaluate its distribution.
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https://arxiv.org/abs/2410.22880
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3001454712d321b529efd2a58648dad3ca0d8d37085bfba9843493d62f98d91d
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2026-02-02T00:00:00-05:00
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Index estimates for constant mean curvature surfaces in three-manifolds by energy comparison
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arXiv:2411.02932v2 Announce Type: replace Abstract: We prove a linear upper bound on the Morse index of closed constant mean curvature (CMC) surfaces in orientable three-manifolds in terms of genus, number of branch points and a Willmore-type energy.
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https://arxiv.org/abs/2411.02932
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052c157d8c30154248835bfa67dae554b1b3b8ae95fc15843d808daaa8042478
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2026-02-02T00:00:00-05:00
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Well-Posedness of the Linear Regularized 13-Moment Equations Using Tensor-Valued Korn Inequalities
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arXiv:2501.14108v2 Announce Type: replace Abstract: In this paper, we finally prove the well-posedness of the linearized R13 moment model, which describes, e.g., rarefied gas flows. As an extension of the classical fluid equations, moment models are robust and have been frequently used, yet they are challenging to analyze due to their additional equations. By effectively grouping variables, we identify a 2-by-2 block structure, allowing us to analyze well-posedness within the abstract LBB framework for saddle point problems. Due to the unique tensorial structure of the equations, in addition to an interesting combination of tools from Stokes' and linear elasticity theory, we also need new coercivity estimates for tensor fields. These Korn-type inequalities are established by analyzing the symbol map of the symmetric and trace-free part of tensor derivative fields. Together with the corresponding right inverse of the tensorial divergence, we obtain the existence and uniqueness of weak solutions. This result also serves as the basis for future numerical analysis of corresponding discretization schemes.
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https://arxiv.org/abs/2501.14108
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57154cccf5b9803c05dfbea046926ef05c903c6ce0ec17424be8e9a6449a70ed
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2026-02-02T00:00:00-05:00
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Families of singular algebraic varieties that are rationally elliptic spaces
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arXiv:2501.17970v3 Announce Type: replace Abstract: We discuss families of hypersurfaces with isolated singularities in projective space with the property that the sum of the ranks of the rational homotopy and the homology groups is finite. They represent infinitely many distinct homotopy types with all hypersurfaces having a nef canonical or anti-canonical class. In the appendix we show that such an infinite family of smooth rationally elliptic 3-folds does not exist.
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https://arxiv.org/abs/2501.17970
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b1a66250bfc905b9073d87c7bddbbf32b07ea929daa72ac471fb0af2a5e8a384
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2026-02-02T00:00:00-05:00
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Inexact Moreau Envelope Lagrangian Method for Non-Convex Constrained Optimization under Local Error Bound Conditions on Constraint Functions
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arXiv:2502.19764v2 Announce Type: replace Abstract: In this paper, we investigate how structural properties of the constraint system impact the oracle complexity of smooth non-convex optimization problems with convex inequality constraints over a simple polytope. In particular, we show that, under a local error bound condition with exponent $d\in[1,2]$ on constraint functions, an inexact Moreau envelope Lagrangian method can attain an $\epsilon$-Karush--Kuhn--Tucker point with $\tilde O(\epsilon^{-2d})$ gradient oracle complexity. When $d=1$, this result matches the best-known complexity in literature up to logarithmic factors. Importantly, the assumed error bound condition with any $d\in[1,2]$ is strictly weaker than the local linear independence constraint qualification that is required to achieve the best-known complexity. Our results clarify the interplay between error bound conditions of constraints and algorithmic complexity, and extend complexity guarantees to a broader class of constrained non-convex problems.
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https://arxiv.org/abs/2502.19764
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d43f2a3b3d408032cb164fa6132b1ea7ae5f5966166093041fe3db193194136f
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2026-02-02T00:00:00-05:00
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Dynamic Programming in Ordered Vector Space
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arXiv:2503.06055v2 Announce Type: replace Abstract: New approaches to the theory of dynamic programming view dynamic programs as families of policy operators acting on partially ordered sets. In this paper, we extend these ideas by shifting from arbitrary partially ordered sets to ordered vector spaces. The integrated algebraic and order structure in such spaces leads to sharper fixed point results. These fixed point results can then be exploited to obtain optimality properties. We illustrate our results through applications ranging from firm management to data valuation. These applications include features from the recent literature on dynamic programming, including risk-sensitive preferences, nonlinear discounting, and state-dependent discounting. In all cases we establish existence of optimal policies, characterize them in terms of Bellman optimality relationships, and prove convergence of major algorithms.
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https://arxiv.org/abs/2503.06055
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fbdcb7b42a5790e9b5cf5f2ca060c219ed9ab909abeb0dcecca694dd8dd4df7a
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2026-02-02T00:00:00-05:00
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Supersimplicity and arithmetic progressions
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arXiv:2503.08258v3 Announce Type: replace Abstract: The main motivation for this article is to explore the connections between the existence of certain combinatorial patterns (as in van der Corputs's theorem on arithmetic progressions of length $3$) with well-known tools and theorems for definable groups in simple theories. In the last sections of this article, we apply our model-theoretic results to bound the number of initial points starting few arithmetic progression of length $3$ in the structure of the additive group of integers with a predicate for the prime integers, assuming Dickson's conjecture, or with a predicate for the square-free integers, as well as for asymptotic limits of finite fields. Our techniques yield similar results for the elements appearing as distances in skew-corners and for S\'ark\"ozy's theorem on the distance of distinct elements being perfect squares.
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https://arxiv.org/abs/2503.08258
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abf898b1ec75cc1d9423edea3efeefcfe19a605c225e8578010f04e3640c14c2
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2026-02-02T00:00:00-05:00
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Some Geometric Aspects Related to Lim's Condition
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arXiv:2504.09464v2 Announce Type: replace Abstract: In their seminal work, Lau and Mah (1986) study $w^*$-normal structure in the space of operators $\mathcal{L}(H)$, on a Hilbert space $H$, using a geometric property of the dual unit ball called Lim's condition. In this paper, we study a weaker form of Lim's condition, which we call property ($\ddagger$), for $C^\ast$-algebras, uniform algebras, and $L^1$-predual spaces. In the case of a $C^\ast$-algebra, we prove that property $(\ddagger)$ is equivalent to Lim's condition and consequently, we obtain a geometric characterization of $C^*$-algebras which are $c_0$-direct sum of finite-dimensional operator spaces. For a uniform algebra, we extend a result of Lau and Mah to show that property $(\ddagger)$ implies that the space is finite-dimensional. In the case of an $L^1$-predual space, we show that this condition implies $k$-smoothness of the norm in the sense considered in Lin and Rao (2007).
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https://arxiv.org/abs/2504.09464
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add656c85a6a014aa999524475b8866204998b5361e8f2a18ca9f8139580ebf3
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2026-02-02T00:00:00-05:00
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On anti-coproximinal and strongly anti-coproximinal subspaces of function spaces
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arXiv:2504.13464v2 Announce Type: replace Abstract: The purpose of this article is to study the anti-coproximinal and strongly anti-coproximinal subspaces of the Banach space of all bounded (continuous) functions. We obtain a tractable necessary condition for a subspace to be stronsgly anti-coproximinal. We prove that for a subspace $\mathbb{Y}$ of a Banach space $\mathbb{X}$ to be strongly anti-coproximinal, $\mathbb Y$ must contain all w-ALUR points of $\mathbb{X}$ and intersect every maximal face of $B_{\mathbb{X}}.$ We also observe that the subspace $\mathbb{K}(\mathbb{X}, \mathbb{Y})$ of all compact operators between the Banach spaces $ \mathbb X $ and $ \mathbb Y$ is strongly anti-coproximinal in the space $\mathbb{L}(\mathbb{X}, \mathbb{Y})$ of all bounded linear operators between $ \mathbb X $ and $ \mathbb Y$, whenever $\mathbb{K}(\mathbb{X}, \mathbb{Y})$ is a proper subset of $\mathbb{L}(\mathbb{X}, \mathbb{Y}),$ and the unit ball $B_{\mathbb{X}}$ is the closed convex hull of its strongly exposed points.
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https://arxiv.org/abs/2504.13464
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fdf3415be737f6e1117758bea7c498aa39da821a59e929623021604051378f4b
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2026-02-02T00:00:00-05:00
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Geometry of regular semisimple Lusztig varieties
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arXiv:2504.15868v2 Announce Type: replace Abstract: Lusztig varieties are subvarieties in flag manifolds $G/B$ associated to an element $w$ in the Weyl group $W$ and an element $x$ in $G$, introduced in Lusztig's papers on character sheaves. We study the geometry of these varieties when $x$ is regular semisimple. In the first part, we establish that they are normal, Cohen-Macaulay, of pure expected dimension and have rational singularities. We then show that the cohomology of ample line bundles vanishes in positive degrees, in arbitrary characteristic. This extends to nef line bundles when the base field has characteristic zero or sufficiently large characteristic. Along the way, we prove that Lusztig varieties are Frobenius split in positive characteristic and that their open cells are affine. We also prove that the open cells in Deligne-Lusztig varieties are affine, settling a question that has been open since the foundational paper of Deligne and Lusztig. In the second part, we explore their relationship with regular semisimple Hessenberg varieties. Both varieties admit Tymoczko's dot action of $W$ on their (intersection) cohomology. We associate to each element $w$ in $W$ a Hessenberg space using the tangent cone of the Schubert variety associated with $w$, and show that the cohomology of the associated regular semisimple Lusztig varieties and Hessenberg varieties is isomorphic as graded $W$-representations when they are smooth. This relationship extends to the level of varieties: we construct a flat degeneration of regular semisimple Lusztig varieties to regular semisimple Hessenberg varieties. In particular, this proves a conjecture of Abreu and Nigro on the homeomorphism types of regular semisimple Lusztig varieties in type $A$, and generalizes it to arbitrary Lie types.
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https://arxiv.org/abs/2504.15868
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ad98f0e66e2791ffbc0e0830781a0836615b03947c48f7caa7f1eee145e59ed4
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2026-02-02T00:00:00-05:00
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Discrete analogues of second-order Riesz transforms
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arXiv:2504.18739v2 Announce Type: replace Abstract: Discrete analogues of classical operators in harmonic analysis have been widely studied, revealing deep connections with areas such as ergodic theory and analytic number theory. This line of research is commonly known as \emph{Discrete Analogues in Harmonic Analysis (DAHA)}. In this paper, we study the $\ell^p$ norms of discrete analogues of second-order Riesz transforms. Using probabilistic methods, we construct a new class of second-order discrete Riesz transforms $\mathcal{R}^{(jk)}$ on the lattice $\mathbb{Z}^d$, $d \ge 2$. We show that for $1<\infty$, their $\ell^p(\mathbb{Z}^d)$ norms coincide with those of the classical second-order Riesz transforms $R^{(jk)}$ on $L^p(\mathbb{R}^d)$ when $j \neq k$, and are comparable up to dimensional constants when $j = k$. The operators $\mathcal{R}^{(jk)}$ differ from the discrete analogue $R^{(jk)}_{\mathrm{dis}}$ by convolution with an $\ell^1(\mathbb{Z}^d)$ function. Applications are given to the DAHA of the Beurling--Ahlfors operator. We also show that $\mathcal{R}^{(jk)}$ arise as discrete analogues of certain Calder\'on--Zygmund operators $\mathbf{R}^{(jk)}$, which differ from $R^{(jk)}$ by convolution with an $L^1(\mathbb{R}^d)$ function. Finally, we conjecture that the $L^p$ norms of $\mathcal{R}^{(jk)}$, $R^{(jk)}_{\mathrm{dis}}$, and $\mathbf{R}^{(jk)}$ agree with those of the classical Riesz transforms, known to equal the corresponding martingale transform norms.
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https://arxiv.org/abs/2504.18739
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2b31a92ee2c9684bc9d9e453733ff288b786cc162b0d4c8cefd5e6450fe20de3
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2026-02-02T00:00:00-05:00
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Weakly Einstein curvature tensors
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arXiv:2504.18752v2 Announce Type: replace Abstract: We classify weakly Einstein algebraic curvature tensors in an oriented Euclidean 4-space, defined by requiring that the three-index contraction of the curvature tensor against itself be a multiple of the inner product. This algebraic formulation parallels the geometric notion of weakly Einstein Riemannian four-manifolds, which include conformally flat scalar-flat, and Einstein manifolds. Our main result provides a complete classification of non-Einstein weakly Einstein curvature tensors in dimension four, naturally dividing them into three disjoint five-dimensional families of algebraic types. These types are explicitly constructed using bases that simultaneously diagonalize both the Einstein tensor and the (anti)self-dual Weyl tensors, which consequently proves that such simultaneous diagonalizability follows from the weakly Einstein property. We also point out that our classification has immediate applications, and describe how some known geometric examples that are neither Einstein, nor conformally flat scalar-flat (namely, the EPS space and certain K\"ahler surfaces) fit within our classification framework.
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https://arxiv.org/abs/2504.18752
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f9cc38c952e7785b30279a0250d441dfa90bfe3d79358b9158665612cc964823
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2026-02-02T00:00:00-05:00
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Sharp asymptotics for $N$-point correlation functions of coalescing heavy-tailed random walk
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arXiv:2505.05000v2 Announce Type: replace Abstract: We study a system of coalescing continuous-time random walks starting from every site on $\mathbb{Z}$, where the jump increments lie in the domain of attraction of an $\alpha$-stable distribution with $\alpha\in(0,1]$. We establish sharp asymptotics for the $N$-point correlation function of the system. Our analysis relies on two precise tail estimates for the system density, as well as the non-collision probability of $N$ independent random walks with arbitrary fixed initial configurations. In addition, we derive refined estimates for heavy-tailed random walks, which may be of independent interest.
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https://arxiv.org/abs/2505.05000
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81774848c78b48231475e070fc6ff5f87f443d1cd580dd8d982dc5a481f25ce3
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2026-02-02T00:00:00-05:00
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A Dantzig-Wolfe Decomposition Method for Quasi-Variational Inequalities
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arXiv:2505.08108v2 Announce Type: replace Abstract: We propose an algorithm to solve quasi-variational inequality problems, based on the Dantzig-Wolfe decomposition paradigm. Our approach solves in the subproblems variational inequalities, which is a simpler problem, while restricting quasi-variational inequalities in the master subproblems, making them generally (much) smaller in size when the original problem is large-scale. We prove global convergence of our algorithm, assuming that the mapping of the quasi-variational inequality is either single-valued and continuous or it is set-valued maximally monotone. Quasi-variational inequalities serve as a framework for several equilibrium problems, and we apply our algorithm to an important example in the field of economics, namely the Walrasian equilibrium problem formulated as a generalized Nash equilibrium problem. Our numerical assessment demonstrates good performance and usefullness of the approach for the large-scale cases.
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https://arxiv.org/abs/2505.08108
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a88da1bd6b0da8b0db6e3f87671ca6d7f000fcd3346b7cbea75da8a60400dbbf
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2026-02-02T00:00:00-05:00
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Bias-Optimal Bounds for SGD: A Computer-Aided Lyapunov Analysis
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arXiv:2505.17965v2 Announce Type: replace Abstract: The non-asymptotic analysis of Stochastic Gradient Descent (SGD) typically yields bounds that decompose into a bias term and a variance term. In this work, we focus on the bias component and study the extent to which SGD can match the optimal convergence behavior of deterministic gradient descent. Assuming only (strong) convexity and smoothness of the objective, we derive new bounds that are bias-optimal, in the sense that the bias term coincides with the worst-case rate of gradient descent. Our results hold for the full range of constant step-sizes $\gamma L \in (0,2)$, including critical and large step-size regimes that were previously unexplored without additional variance assumptions. The bounds are obtained through the construction of a simple Lyapunov energy whose monotonicity yields sharp convergence guarantees. To design the parameters of this energy, we employ the Performance Estimation Problem framework, which we also use to provide numerical evidence for the optimality of the associated variance terms.
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https://arxiv.org/abs/2505.17965
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a94eefd88b5806b5a5efc5bd4d21727bd85c0b1c1ffe2b0ff9ffa400ab64defe
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2026-02-02T00:00:00-05:00
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Calder\'{o}n-Zygmund estimates for double phase problems with matrix weights
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arXiv:2505.20856v3 Announce Type: replace Abstract: We establish an optimal Calder\'{o}n-Zygmund theory for nonuniformly elliptic double phase problems with matrix weights. For $11$, $$ (|\M F|^p+a(x)|\M F|^q)\in L^\gamma_{\mathrm{loc}} \;\Longrightarrow\; (|\M Du|^p+a(x)|\M Du|^q)\in L^\gamma_{\mathrm{loc}}. $$ Our argument combines a freezing of the logarithm of the matrix field, $\log \M$, with a fractional maximal-operator method governed by the Muckenhoupt-Wheeden $\mathcal{A}_{p,s}$ classes (where $1/s=1/p-\alpha/(nq)$). This yields scale-invariant comparison and level-set estimates and precludes Lavrentiev gaps at the sharp threshold $q/p\le 1+\alpha/n$. Our result recovers the identity case $\,\M\equiv {\rm I}_n\,$, i.e., the classical (unweighted) Calder\'{o}n-Zygmund theory for double-phase problems.
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https://arxiv.org/abs/2505.20856
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26f86005980561b45bba2c26c3d66a4d1c2910096640c9fa8ea1bf36e26afeb0
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2026-02-02T00:00:00-05:00
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Classification of exact structures using the Ziegler spectrum
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arXiv:2506.02304v2 Announce Type: replace Abstract: Given an idempotent complete additive category, we show the there is an explicitly constructed topological space such that the lattice of exact substructures is anti-isomorphic to the lattice of closed subsets. In the special case that the additive category has weak cokernels, this topological space is an open subset of the Ziegler spectrum and this is a result of Kevin Schlegel. We also look at some module categories of rings where the Ziegler spectrum is known and calculate the global dimensions of the corresponding exact substructures. Second version contains minor changes to first version.
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https://arxiv.org/abs/2506.02304
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76d5c5e911b9e7909c829a3498c9cacea891c8f685bbaebe15529bce13eaa1d1
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2026-02-02T00:00:00-05:00
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On some results of Harish-Chandra for representations of p-adic groups, extended to their central extensions
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arXiv:2506.21334v2 Announce Type: replace Abstract: The aim of this article is to give a complete proof of results of Harish-Chandra linking the irreducibility of parabolic induction of a supercuspidal representation of a p-adic group to the analytic behavior of the mu-function of Harish-Chandra and to show that the proof remains valid in the case of a central extension.M
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https://arxiv.org/abs/2506.21334
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e072d939df577f0f7fba6d1e1e11f19084829f027aea71637f7b26e164d8883c
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2026-02-02T00:00:00-05:00
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Illumination number of 3-dimensional cap bodies
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arXiv:2507.08712v2 Announce Type: replace Abstract: The illumination conjecture asserts that any convex body in $n$-dimensional Euclidean space can be illuminated by at most $2^n$ external light sources or parallel beams of light. Despite recent progress on the illumination conjecture, it remains open in general, as well as for specific classes of bodies. Bezdek, Ivanov, and Strachan showed that the conjecture holds for symmetric cap bodies in sufficiently high dimensions. Further, Ivanov and Strachan calculated the illumination number for the class of 3-dimensional centrally symmetric cap bodies to be 6. In this paper, we show that even the broader class of all 3-dimensional cap bodies has the same illumination number 6, in particular, the illumination conjecture holds for this class. The illuminating directions can be taken to be vertices of a regular tetrahedron, together with two special directions depending on the body. The proof is based on probabilistic arguments and integer linear programming.
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https://arxiv.org/abs/2507.08712
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6e3c7aaa0ce018e2c5ce72cd2cd130bbca4102b575d8a9438e0f611aaf75ae18
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2026-02-02T00:00:00-05:00
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Twisted periods of modular forms
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arXiv:2507.17041v2 Announce Type: replace Abstract: Let $S_k$ denote the space of cusp forms of weight $k$ and level one. For $0\leq t\leq k-2$ and primitive Dirichlet character $\chi$ mod $D$, we introduce twisted periods $r_{t,\chi}$ on $S_k$. We show that for a fixed natural number $n$, if $k$ is sufficiently large relative to $n$ and $D$, then any $n$ periods with the same twist but different indices are linearly independent. We also prove that if $k$ is sufficiently large relative to $D$ then any $n$ periods with the same index but different twists mod $D$ are linearly independent. These results are achieved by studying the trace of the products and Rankin-Cohen brackets of Eisenstein series of level $D$ with nebentypus. Moreover, we give two applications of our method. First, we prove certain identities that evaluate convolution sums of twisted divisor functions. Second, we show that Maeda's conjecture implies a non-vanishing result on twisted central $L$-values of normalized Hecke eigenforms.
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https://arxiv.org/abs/2507.17041
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2026-02-02T00:00:00-05:00
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Self-Similar Solutions to the Hele-Shaw Problem with Surface Tension
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arXiv:2507.19443v2 Announce Type: replace Abstract: We consider the Hele-Shaw problem with surface tension in an infinite domain. We prove the existence of a family of self-similar solutions. At $t=0$, these solutions have a corner of angle $\theta$ with $ 0 0$, the solutions are smooth.
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https://arxiv.org/abs/2507.19443
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2026-02-02T00:00:00-05:00
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A Generalized Analytical Framework for the Nonlinear Best-Worst Method
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arXiv:2508.06048v2 Announce Type: replace Abstract: The nonlinear model of the best-worst method frequently produces multiple optimal weight sets, which are conventionally determined through optimization software. While an analytical approach exists that provides both a closed-form expression for the optimal interval-weights and a secondary objective function to determine the best optimal weight set, we demonstrate that this approach is only valid when preferences are quantified using the Saaty scale and only a single decision-maker is involved. To tackle this issue, we propose a framework compatible with any scale and any number of decision-makers. We first derive an analytical expression for optimal interval-weights and then select the best optimal weight set. After demonstrating that the values of consistency index for the Saaty scale in the existing literature are not well-defined, we derive a formula of consistency index. We also obtain an analytical expression for the consistency ratio, enabling its use as an input-based consistency indicator. Furthermore, we establish that when multiple best/worst criteria are present, weights may vary among best criteria and among the worst criteria. To address this limitation, we modify the original optimization model for weight computation in such instances, solve it analytically to obtain optimal interval-weights and then select the best optimal weight set using a secondary objective function. Finally, we demonstrate and validate the proposed approach using numerical examples and a real-world case study of ranking barriers to energy efficiency in buildings.
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https://arxiv.org/abs/2508.06048
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a99b650145a08cca2e62167b50492bcc89bed2562daf96e837af254320dc71d3
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A bottleneck model with shared autonomous vehicles: Scale economies and price regulations
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arXiv:2508.08848v2 Announce Type: replace Abstract: This study examines how scale economies in the operation of shared autonomous vehicles (SAVs) affect the efficiency of a transportation system where SAVs coexist with normal vehicles (NVs). We develop a bottleneck model where commuters choose their departure times and mode of travel between SAVs and NVs, and analyze equilibria under three SAV fare-setting scenarios: marginal cost pricing, average cost pricing, and unregulated monopoly pricing. Marginal cost pricing reduces commuting costs but results in financial deficits for the service provider. Average cost pricing ensures financial sustainability but has contrasting effects depending on the timing of implementation due to the existence of multiple equilibria: when implemented too early, it discourages adoption of SAVs and increases commuting costs; when introduced after SAV adoption reaches the monopoly equilibrium level, it promotes high adoption and achieves substantial cost reductions without a deficit. We also show that expanding road capacity may increase commuting costs under average cost pricing, demonstrating the Downs--Thomson paradox in transportation systems with SAVs. We next examine two optimal policies that improve social cost, including the operator's profit: the first-best policy that combines marginal cost pricing with congestion tolls, and the second-best policy that relies on fare regulation alone. Our analysis shows that these policies can limit excessive adoption by discouraging overuse of SAVs. This suggests that promoting SAV adoption does not always reduce social cost.
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https://arxiv.org/abs/2508.08848
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542146290d8b2167d1fb55160f00f6cdbbc9059db7354a5135be111870d29c97
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2026-02-02T00:00:00-05:00
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Relative braid group symmetries on modified iquantum groups and their modules
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arXiv:2508.12041v2 Announce Type: replace Abstract: We present a comprehensive generalization of Lusztig's braid group symmetries for quasi-split iquantum groups. Specifically, we give 3 explicit rank one formulas for symmetries acting on integrable modules over a quasi-split iquantum group of arbitrary Kac-Moody type with general parameters. These symmetries are formulated in terms of idivided powers and iweights of the vectors being acted upon. We show that these symmetries are consistent with the relative braid group symmetries on iquantum groups, and they are also related to Lusztig's symmetries via quasi $K$-matrices. Furthermore, through appropriate rescaling, we obtain compatible symmetries for the integral forms of modified iquantum groups and their integrable modules. We verify that these symmetries satisfy the relative braid group relations.
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https://arxiv.org/abs/2508.12041
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d0919ee975349cbe9ee7509b0e87a196bf1da897c57792c762b543bc0b2e8274
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2026-02-02T00:00:00-05:00
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Cycles of Length 4 or 8 in Graphs with Diameter 2 and Minimum Degree at Least 3
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arXiv:2508.19302v4 Announce Type: replace Abstract: In this short note it is shown that every graph of diameter 2 and minimum degree at least 3 contains a cycle of length 4 or 8. This result contributes to the study of the Erd\H{o}s-Gy\'arf\'as Conjecture by confirming it for the class of diameter-2 graphs.
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https://arxiv.org/abs/2508.19302
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87a0db0279e84acd030575ed9350bfc36cba7a20f90df0fa3ff78f0ca7e95483
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The Tautochrone of Huygens and Abel: From Constructive Geometry to Fractional Calculus
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arXiv:2509.05308v2 Announce Type: replace Abstract: In this paper, we explore the connections between Christiaan Huygens and Niels Henrik Abel through the tautochrone problem. The problem -- determining the curve along which a particle descends under gravity in the same time, regardless of its starting point -- has been a central topic at the intersection of physics, geometry, and analysis. Though these two major figures are separated by nearly two centuries, they approached the problem in radically different ways. While Huygens proposed a physical solution based on geometric construction, Abel approached the problem within the analytic framework of integral equations, employing a procedure that can be seen as anticipating and paving the way for the development of differential calculus of arbitrary order. This contrast highlights a broader historical narrative: the transformation of mathematical thinking from constructive geometry to abstract analysis.
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https://arxiv.org/abs/2509.05308
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20e5a7d4777c6eaba630c76fd975d6512e9c1274785fbda20683b5041a64d5a2
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2026-02-02T00:00:00-05:00
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Carryless Pairing: Additive Pairing in the Fibonacci Basis
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arXiv:2509.10382v4 Announce Type: replace Abstract: We define a pairing map $\pi : \mathbb{N}^2\to\mathbb{N}$ that encodes $x$ and $y$ into disjoint index bands inside the Zeckendorf support of a single integer. Evaluation and inversion use only addition, comparison, and bounded scans of supports; no multiplication, factorization, or digit interleaving is used. The device is carryless by construction: supports remain non-adjacent, so the output is already in Zeckendorf-normal form. The map is injective but not surjective; membership in its image is decidable by the same support machinery used for decoding. The core claims are mechanized in Rocq.
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https://arxiv.org/abs/2509.10382
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370c33304b1389fbf51560f63e5fbbea72d5bf172f7ec2bd451e3395d13d04f8
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2026-02-02T00:00:00-05:00
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Rough stochastic filtering
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arXiv:2509.11825v2 Announce Type: replace Abstract: This article is concerned with the well-posedness of the "filtering equations", due to Zakai and Kushner-Stratonovich, arising in nonlinear stochastic filtering. In general situations, notably in correlated diffusion models and when signal coefficients depend on the observation process, the well-posedness is a difficult problem, mainly due to conflicting martingale structures of the involved forward and backward equations. Crisan-Pardoux (2024) address this classical problem with BSPDE techniques, Du et al. (2013), a Sobolev-based approach that however requires increasingly strong regularity assumptions in high dimensions. In this work, we take a new mixed rough stochastic perspective which allows us to derive well-posed rough counterparts of the filtering equations. Importantly, the rough filtering equations are seen, upon randomization, to coincide with the classical filtering equations. Our framework yields well-posedness (existence, uniqueness, stability) under dimension-independent regularity assumptions, providing a robust and conceptually unified solution to a longstanding problem in stochastic filtering theory. To illustrate the flexibility of the method, we also treat rough versions of the classical Kalman-Bucy filter, with characteristics described by a new class of RDEs of rough Riccati type.
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https://arxiv.org/abs/2509.11825
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bdd44085ae3312f7af402375f6dd4d1c65b64f27dc8155c1fca5ed0aa2fb97ab
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2026-02-02T00:00:00-05:00
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Haussdorff consistency of MLE in folded normal and Gaussian mixtures
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arXiv:2509.12206v2 Announce Type: replace Abstract: We develop a constant-tracking likelihood theory for two nonregular models: the folded normal and finite Gaussian mixtures. For the folded normal, we prove boundary coercivity for the profiled likelihood, show that the profile path of the location parameter exists and is strictly decreasing by an implicit-function argument, and establish a unique profile maximizer in the scale parameter. Deterministic envelopes for the log-likelihood, the score, and the Hessian yield elementary uniform laws of large numbers with finite-sample bounds, avoiding covering numbers. Identification and Kullback-Leibler separation deliver consistency. A sixth-order expansion of the log hyperbolic cosine creates a quadratic-minus-quartic contrast around zero, leading to a nonstandard one-fourth-power rate for the location estimator at the kink and a standard square-root rate for the scale estimator, with a uniform remainder bound. For finite Gaussian mixtures with distinct components and positive weights, we give a short identifiability proof up to label permutations via Fourier and Vandermonde ideas, derive two-sided Gaussian envelopes and responsibility-based gradient bounds on compact sieves, and obtain almost-sure and high-probability uniform laws with explicit constants. Using a minimum-matching distance on permutation orbits, we prove Hausdorff consistency on fixed and growing sieves. We quantify variance-collapse spikes via an explicit spike-bonus bound and show that a quadratic penalty in location and log-scale dominates this bonus, making penalized likelihood coercive; when penalties shrink but sample size times penalty diverges, penalized estimators remain consistent. All proofs are constructive, track constants, verify measurability of maximizers, and provide practical guidance for tuning sieves, penalties, and EM-style optimization.
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https://arxiv.org/abs/2509.12206
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75b15bd62b939276d3d91faa35d9d0b4d89bd006ee8f9d2a8ff9e2f0ff121c5b
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2026-02-02T00:00:00-05:00
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Error Analysis of Discrete Flow with Generator Matching
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arXiv:2509.21906v2 Announce Type: replace Abstract: Discrete flow models offer a powerful framework for learning distributions over discrete state spaces and have demonstrated superior performance compared to the discrete diffusion models. However, their convergence properties and error analysis remain largely unexplored. In this work, we develop a unified framework grounded in stochastic calculus theory to systematically investigate the theoretical properties of discrete flow models. Specifically, by leveraging a Girsanov-type theorem for the path measures of two continuous-time Markov chains (CTMCs), we present a comprehensive error analysis that accounts for both transition rate estimation error and early stopping error. In fact, the estimation error of transition rates has received little attention in existing works. Unlike discrete diffusion models, discrete flow incurs no initialization error caused by truncating the time horizon in the noising process. Building on generator matching and uniformization, we establish non-asymptotic error bounds for distribution estimation without the boundedness condition on oracle transition rates. Furthermore, we derive a faster rate of total variation convergence for the estimated distribution with the boundedness condition, yielding a nearly optimal rate in terms of sample size. Our results provide the first error analysis for discrete flow models. We also investigate model performance under different settings based on simulation results.
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https://arxiv.org/abs/2509.21906
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717de269c2d9fa1eff65720ff5d9b681675e7d061155f3a007b235042d579b91
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2026-02-02T00:00:00-05:00
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Abstract Integration in Net Convergence Structures
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arXiv:2509.24430v2 Announce Type: replace Abstract: In this article, we propose a general theory of integration of the Riemann and Lebesgue types with respect to arbitrary measures and functions, connected by a continuous bilinear product, with values in abstract vector spaces endowed with a convergence structure given by nets. This covers both the topological and order based convergences in the literature. We then show that this integral satisfies most of the common properties of the objects that comprises integration theory. By establishing a generalized notion of summability on Riesz spaces and an integral built upon countable partitions of the base space, we then stablish some uniform, monotone and dominated convergence theorems for the refereed integrals, as well as a non-topological or order based Henstock Lemma and a general convergence theorem based on the notion of conjugated lattice seminorms. An application of these theorems is made to prove various equivalences concerning the Lebesgue, for which we give a brief survey, Saks and Riemann type integrals in partially ordered and topological vector spaces presented in the literature, for which we also make a thorough review. We finish the article with a possible way of classifying general integration procedures defined in abstract convergence structures, and pose some open problems based on them.
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https://arxiv.org/abs/2509.24430
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2026-02-02T00:00:00-05:00
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Multifractality in the Tree of Life: A Branching-Process RIFS Proof
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arXiv:2509.26637v2 Announce Type: replace Abstract: We study a branching-process random iterated function system (RIFS) defined by a recursive replacement of leaves by finite subtrees at strictly smaller contraction scales. This construction yields a tree-valued, infinite-depth random geometry that unifies classical branching processes and random iterated function systems while remaining distinct from both. We prove rigorously that the resulting branching-process RIFS is multifractal under explicit and mild assumptions. Two variants are analyzed: a non-anchored case with a nontrivial compact attractor, and a biologically motivated anchored case in which the invariant geometric set collapses to a point, while tangent measures obey the same multifractal law. The construction formalizes the foundational principles of nestedness, duality, and randomness in the living tree of life (Hudnall & D'Souza, 2025), yielding a minimal-condition theorem that explains the ubiquity of multifractal signatures in biological data.
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https://arxiv.org/abs/2509.26637
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2026-02-02T00:00:00-05:00
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The Magmoid of Normalized Stochastic Kernels
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arXiv:2510.01131v2 Announce Type: replace Abstract: Normalization, $D(X + 1) \to D(X) + 1$, is almost a distributive law; but because one of the distributive law axioms only holds up-to-idempotent, it yields a non-associative composition of normalized kernels. We introduce the Markov magmoid of normalized stochastic kernels: a normalized-by-construction semantics for probabilistic inference, unifying exact Bayesian observations and interventions as two parenthesizations of the same composite. Front-door and back-door criteria follow from the axioms of Markov magmoids; we implement these with non-associative monadic notation.
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https://arxiv.org/abs/2510.01131
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971c77c46d8f801d7b87668342bd137d689206aec4b77e888b3dbe65a4436099
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Absolutely Abelian Hilbert Class Fields and $\ell-$torsion conjecture
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arXiv:2510.10725v2 Announce Type: replace Abstract: There are several recent works where authors have shown that number fields $K$ with `sufficiently many' units and cyclic class group contain a Euclidean ideal class provided the Hilbert class field $H(K)$ is absolutely abelian. In this article, we explore the latter hypothesis: how often a number field $K$ has absolutely abelian Hilbert class field? For a number field $K$ to have absolutely abelian Hilbert class field, we obtain several criteria in terms of class number of $K$, P\'olya group of $K$, and genus number of $K$. We also show that for such number fields the $\ell-$torsion conjecture is true. Along with these, the article also reports some results on a theme to study class groups, developed by the authors, where primes of higher degree are used to study class groups.
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https://arxiv.org/abs/2510.10725
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7ba887ebde4f609b5c84f00455f3a3d00f86873d753b8c1d74e8247a409a3715
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2026-02-02T00:00:00-05:00
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On estimation of weighted cumulative residual Tsallis entropy for complete and censored samples
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arXiv:2510.12442v2 Announce Type: replace Abstract: Recently, weighted cumulative residual Tsallis entropy has been introduced in the literature as a generalization of weighted cumulative residual entropy. We study some new properties of weighted cumulative residual Tsallis entropy measure. Next, we propose some non-parametric estimators of this measure. Asymptotic properties of these estimators are discussed. Performance of these estimators are compared by mean squared error. Non-parametric estimators for weighted cumulative residual entropy measure are also discussed. Estimator for weighted cumulative residual Tsallis entropy for progressive type-II censored data is proposed and its performance is investigated by Monte-Carlo simulations for various censoring schemes. Two uniformity tests for complete samples are proposed based on an estimator of these two measures and power of the tests are compared with some popular tests. The tests perform reasonably well. Uniformity test under progressively type-II censored data is also developed. Some real datasets are analysed for illustration.
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https://arxiv.org/abs/2510.12442
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39668622dd497f24db4f6d77026e9c8a8a243218baf73be6edc1fc3d9b2399c2
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2026-02-02T00:00:00-05:00
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The spectrum of Dirichlet-to-Neumann maps for radial conductivities
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arXiv:2510.22585v2 Announce Type: replace Abstract: The problem of characterizing sequences of real numbers that arise as spectra of Dirichlet-to-Neumann (DtN) maps for elliptic operators has attracted considerable attention over the past fifty years. In this article, we address this question in the simple setting of DtN maps associated with a rotation-invariant elliptic operator $\nabla \cdot (\gamma\nabla \centerdot )$ in the ball in Euclidean space. We show that the spectrum of such a DtN operator can be expressed as a universal term, determined solely by the boundary values of the conductivity $\gamma$, plus a sequence of Hausdorff moments of an integrable function, which we call the Born approximation of $\gamma$. We also show that this object is locally determined from the boundary by the corresponding values of the conductivity, a property that implies a local uniqueness result for the Calder\'on Problem in this setting. We also give a stability result: the functional mapping the Born approximation to its conductivity is H\"older stable in suitable Sobolev spaces. Finally, in order to refine the characterization of the Born approximation, we analyze its regularity properties and their dependence on the conductivity.
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https://arxiv.org/abs/2510.22585
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81570ce28825421978d7f5295da48fcb782a7c760133649d93ede5a4b61ceef0
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Prime and Semiprime Ideals in Commutative Ternary $\Gamma$-Semirings: Quotients, Radicals, Spectrum
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arXiv:2510.23885v2 Announce Type: replace Abstract: The theory of ternary $\Gamma$-semirings extends classical ring and semiring frameworks by introducing a ternary product controlled by a parameter set $\Gamma$. Building on the foundational axioms recently established by Rao, Rani, and Kiran (2025), this paper develops the first systematic ideal-theoretic study within this setting. We define and characterize prime and semiprime ideals for commutative ternary $\Gamma$-semirings and prove a quotient characterization: an ideal $P$ is prime if and only if $T/P$ is free of nonzero zero-divisors under the induced ternary $\Gamma$-operation. Semiprime ideals are shown to be stable under arbitrary intersections and coincide with their radicals, providing a natural bridge to radical and Jacobson-type structures. A correspondence between prime ideals and prime congruences is established, leading to a Zariski-like spectral topology on $\mathrm{Spec}(T)$. Computational classification of all commutative ternary $\Gamma$-semirings of order $\leq 4$ confirms the theoretical predictions and reveals novel structural phenomena absent in binary semiring theory. The results lay a rigorous algebraic and computational foundation for subsequent categorical, geometric, and fuzzy extensions of ternary $\Gamma$-algebras.
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https://arxiv.org/abs/2510.23885
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7467bef8fdfa166b7d8e8cff4eebfafb5c9142ba65dfc13a1e5e9668bf7dfd20
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2026-02-02T00:00:00-05:00
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Uniqueness of the non-commutative divergence cocycle
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arXiv:2511.06903v2 Announce Type: replace Abstract: We show that, for $n \geq 3 $, 1-cocycles of degree zero on the Lie algebra of derivations of the free associative algebra $T(A_n)$ with values in $ \rvert T(A_n) \rvert \otimes \rvert T(A_n) \rvert $ are linear combinations of the non-commutative divergence and its switch, when restricted to finite-degree quotients. Here, $ \rvert T(A_n) \rvert $ denotes the space of cyclic words. Furthermore, we study 1-cocycles of degree zero on the Lie algebra of symplectic derivations of the free Lie algebra $ \mathfrak{L_{2n}}$, and prove the uniqueness of the Enomoto-Satoh trace.
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https://arxiv.org/abs/2511.06903
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2026-02-02T00:00:00-05:00
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Generalized ovals, 2.5-dimensional additive codes, and multispreads
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arXiv:2511.15843v2 Announce Type: replace Abstract: We present constructions and bounds for additive codes over a finite field in terms of their geometric counterpart, i.e., projective systems. It is known that the maximum number of $(h-1)$-spaces in PG$(2,q)$, such that no hyperplane contains three, is given by $q^h+1$ if $q$ is odd. Those geometric objects are called generalized ovals. We show that cardinality $q^h+2$ is possible if we decrease the dimension a bit. We completely determine the minimum possible lengths of additive codes over GF$(9)$ of dimension $2.5$ and give improved constructions for other small parameters, including codes outperforming the best linear codes. As an application, we consider multispreads in PG$(4,q)$, in particular, completing the characterization of parameters of GF$(4)$-linear $64$-ary one-weight codes. Keywords: additive code, projective system, generalized oval, multispread, one-weight code, two-weight code
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https://arxiv.org/abs/2511.15843
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5e828b135045414010f65c3dc121cfe2e47343b2f9e6eecbd212b53f1df96dfd
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2026-02-02T00:00:00-05:00
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On finiteness properties of separating semigroup of real curve
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arXiv:2511.18545v2 Announce Type: replace Abstract: A real morphism $f$ from a real algebraic curve $X$ to $\mathbb{P}^1$ is called separating if $f^{-1}(\mathbb{R} \mathbb{P}^1) = \mathbb{R} X$. A separating morphism defines a covering $\mathbb{R} X \to \mathbb{R} \mathbb{P}^1$. Let $X_1, \ldots, X_r$ denote the components of $\mathbb{R} X$. M. Kummer and K. Shaw~\cite{kummer_separating_2020} defined the separating semigroup of a curve $X$ as the set of all vectors $d(f) = (d_1(f), \ldots, d_r(f)) \in \mathbb{N}^{r}$ where $f$ is a separating morphism $X \to \mathbb{P}^1$ and $d_i(f)$ is the degree of the restriction of $f$ to $X_i$. In the present paper we prove that for a non-negative integer number $g$ the set of all separating semigroups of genus $g$ curves is finite.
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https://arxiv.org/abs/2511.18545
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2c331fe9318755d9ed322db9be69b0345d84232b68a548c0bd54808bb66022eb
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2026-02-02T00:00:00-05:00
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Covariance Estimation for Matrix-variate Data via Fixed-rank Core Covariance Geometry
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arXiv:2512.01070v3 Announce Type: replace Abstract: We study the geometry of the fixed-rank core covariance manifold and propose a novel covariance estimator for matrix-variate data leveraging this geometry. To generalize the separable covariance model, Hoff, McCormack, and Zhang (2023) showed that every covariance matrix $\Sigma$ of $p_1\times p_2$ matrix-variate data uniquely decomposes into a separable component $K$ and a core component $C$. Such a decomposition also exists for rank-$r$ $\Sigma$ if $p_1/p_2+p_2/p_1<r$, with $C$ sharing the same rank. They posed an open question on whether a partial-isotropy structure can be imposed on $C$ for high-dimensional covariance estimation. We address this question by showing that a partial-isotropy rank-$r$ core is a non-trivial convex combination of a rank-$r$ core and $I_p$ for $p:=p_1p_2$. This motivates studying the geometry of the space of rank-$r$ cores, $\mathcal{C}_{p_1,p_2,r}^+$. We show that $\mathcal{C}_{p_1,p_2,r}^+$ is a smooth manifold, except for a measure-zero subset, whereas $\mathcal{C}_{p_1,p_2}^{++}:=\mathcal{C}_{p_1,p_2,p}^+$ is itself a smooth manifold. The geometric properties, including smoothness of the positive definite cone via separability and the Riemannian gradient and Hessian operator relevant to $\mathcal{C}_{p_1,p_2,r}^+$, are also derived. Using this geometry, we propose a partial-isotropy core shrinkage estimator for matrix-variate data, supported by numerical illustrations.
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https://arxiv.org/abs/2512.01070
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7a1344e6baa208e33b0e851aa4711d1ff577892eae469cbdd278acd53419b681
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2026-02-02T00:00:00-05:00
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On the complex zeros and the computational complexity of approximating the reliability polynomial
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arXiv:2512.11504v2 Announce Type: replace Abstract: In this paper we relate the location of the complex zeros of the reliability polynomial to parameters at which a certain family of rational functions derived from the reliability polynomial exhibits chaotic behaviour. We use this connection to prove new results about the location of reliability zeros. In particular we show that there are zeros with modulus larger than $1$ with essentially any possible argument. We moreover use this connection to show that approximately evaluating the reliability polynomial for planar graphs at a non-positive algebraic number in the unit disk is #P-hard.
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https://arxiv.org/abs/2512.11504
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eadaf3f90cd3967a70d7bd1ce237f4f72bb70040676180cfe1d8437ae724b52f
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2026-02-02T00:00:00-05:00
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Taylor polynomials on left-quotients of Carnot groups
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arXiv:2512.12239v2 Announce Type: replace Abstract: We prove classical Taylor polynomial theorems for sub-Riemannian manifolds that are obtained as the submetric image of a Carnot group. For these theorems we also prove a sufficient condition for real analyticity and a result on L-harmonicity of Taylor polynomials.
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https://arxiv.org/abs/2512.12239
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b0219ef410d1c4af0b48e91dbbc06727be848159d3cd94981435c176428b8df7
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2026-02-02T00:00:00-05:00
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A Note on the Sum-Product Problem and the Convex Sumset Problem
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arXiv:2512.13849v2 Announce Type: replace Abstract: We provide a new exponent for the Sum-Product conjecture on $\mathbb{R} $. Namely for $A \subset \mathbb{R}$ finite, \[ \max \left\{ \left\lvert A+A \right\rvert , \left\lvert AA \right\rvert \right\} \gg_{\epsilon} \left\lvert A \right\rvert ^{\frac{4}{3} + \frac{10}{4407} - \epsilon} .\] We also provide new exponents for $A \subset \mathbb{R} $ finite and convex, namely \[ \left\lvert A+A \right\rvert \gg_{\epsilon} \left\lvert A \right\rvert ^{\frac{46}{29} - \epsilon}, \] and \[ \left\lvert A-A \right\rvert \gg_{\epsilon} \left\lvert A \right\rvert ^{\frac{8}{5} + \frac{1}{3440} -\epsilon} .\]
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https://arxiv.org/abs/2512.13849
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Academic Papers
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544a360deff76050aa77e5f276a299b7962b4872afc5ce3590cf75d786f170ee
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2026-02-02T00:00:00-05:00
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Higher Order Dualities between Prime Ideals
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arXiv:2512.22346v2 Announce Type: replace Abstract: Extending the works of Alladi and Sweeting and Woo, we state and prove the general higher order duality between prime ideals in number rings. We then use the second order duality to obtain the a new formula for the Chebotarev Density involving sums of the generalized M\"obius function and the prime ideal counting function. We also provide two estimates of such sums as an application of the duality identity. A discussion of the duality in a slightly more general setting is done at the end.
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https://arxiv.org/abs/2512.22346
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Academic Papers
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svg
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b4fbfd602a460fdcae5de106a9bd747ee0ac6d0184569f83ea4ee24b668c50d8
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2026-02-02T00:00:00-05:00
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Tameness of actions on finite rank median algebras
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arXiv:2601.01681v2 Announce Type: replace Abstract: We prove that for (compact) finite-rank median algebras the geometric rank equals the independence number of all (continuous) median-preserving functions to $[0,1]$. Combined with Rosenthal's dichotomy, this yields a generalized Helly selection principle: for finite-rank median algebras, the space of all median-preserving functions to $[0,1]$ is sequentially compact in the pointwise topology. Generalizing joint results with E. Glasner on dendrons (rank-1), we establish that every continuous action of a topological group $G$ by median automorphisms on a finite-rank compact median algebra is Rosenthal representable, hence dynamically tame. As an application, the Roller-Fioravanti compactification of finite-rank topological median $G$-algebras with compact intervals is often a dynamically tame $G$-system.
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https://arxiv.org/abs/2601.01681
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Academic Papers
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svg
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d13406e2113fe18c00dd5cc09a472810edb63bf896cbb7bfe19c94c6874dcf0e
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2026-02-02T00:00:00-05:00
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Singular basins in multiscale systems: tunneling between stable states
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arXiv:2601.02001v2 Announce Type: replace Abstract: Real-world systems often evolve on different timescales and possess multiple coexisting stable states. Whether or not a system returns to a given stable state after being perturbed away from it depends on the shape and extent of its basin of attraction. We show that basins of attraction in multiscale systems can exhibit special geometric properties in the form of singular funnels. Although singular funnels are narrow, they can extend to different regions of the phase space and, unexpectedly, impact the system's resilience to perturbations. Consequently, singular funnels may prevent common dimensionality reductions in the limit of large timescale separation, such as the quasi-static approximation, adiabatic elimination and time-averaging of the fast variables. We refer to basins of attraction with singular funnels as singular basins. We show that singular basins are universal and occur robustly in a range of multiscale systems: the normal form of a pitchfork bifurcation with a slowly adapting parameter, an adaptive active rotator, and an adaptive network of phase rotators.
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https://arxiv.org/abs/2601.02001
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Academic Papers
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