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今日/当前日期收录 183 信号源:cs.LG, q-bio, physics, cond-mat, math, stat.ML
2603.21283 2026-06-19 quant-ph math.CO 版本更新 70%

A Quantum Encoding of Traveling Salesperson Tours via Route Generation, Cost Phases, and a Reversible Valid-Permutation Oracle

旅行商问题路径的量子编码:路径生成、成本相位和可逆有效排列预言机

Alexander Johannes Stasik, Franz Georg Fuchs

专题命中 其他科学智能 :量子编码求解TSP,属于量子计算应用

AI总结 提出一种基于时间寄存器表示的旅行商问题量子编码,通过均匀路径生成、可逆有效排列检查及成本相位编码,将路径可行性及总长度信息嵌入量子态,使用O(n log n)量子比特和O(n^3 log n)门复杂度。

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AI中文摘要

针对n个城市的旅行商问题(TSP),我们提出了一种基于时间寄存器表示路径的紧凑量子编码。候选路径表示为离散时间步上n-1个城市标签的序列,其中包含一个固定的起始城市,其余城市以二进制寄存器编码。我们描述了构造的三个组成部分:路径寄存器上的均匀路径生成、可逆的有效性预言机以及编码总路径成本的相位预言机。有效性预言机检查非起始城市标签是否构成排列,并且对于不完全图,检查路径使用的每条有向边是否存在。然后,成本预言机将起始边、中间转移边和返回边的成本累积为有效路径的依赖路径的相位。这产生了候选路径的相干叠加,可行性和路径长度信息直接嵌入量子态中。完整构造使用O(n log n)个量子比特,而朴素实现的最坏情况基本门复杂度为O(n^3 log n)。该编码与振幅放大或谱过滤技术(如量子奇异值变换(QSVT)或Grover算法)兼容。然而,由于有效路径的比例呈指数级小,即使结合振幅放大,整体复杂度仍然是指数级的。

英文摘要

For a traveling salesperson problem (TSP) of n cities, we present a compact quantum encoding based on a time-register representation of tours. A candidate route is represented as a sequence of n-1 city labels over discrete time steps, with one fixed start city and the remaining cities encoded in binary registers. We describe three ingredients of the construction: uniform route generation over the route register, a reversible validity oracle, and a phase oracle that encodes the total tour cost. The validity oracle checks both that the non-start city labels form a permutation and, for incomplete graphs, that every directed edge used by the route exists. The cost oracle then accumulates the start-edge, intermediate-transition, and return-edge costs into a tour-dependent phase for valid routes. This yields a coherent superposition of candidate routes with feasibility and tour-length information embedded directly in the quantum state. The complete construction uses O(n log n) qubits, while a naive implementation has worst-case elementary-gate complexity O(n^3 log n). The encoding is compatible with amplitude amplification or spectral filtering techniques such as the quantum singular value transform (QSVT) or Grover's algorithm. However, due to the exponentially small fraction of valid tours, the overall complexity remains exponential even when combined with amplitude amplification.

2603.21868 2026-06-19 math.QA math.OA math.RT 版本更新 70%

Triangular Decomposition of the Crystal Lattice of Quantized Function Algebras: Revisited

量子函数代数晶体格的三角分解:再探

Ayan Dey

专题命中 其他科学智能 :量子函数代数晶体格的三角分解

AI总结 将三角分解定理从简单复李代数类型 $A_n$ 到 $E_7$ 推广到 $G_2$, $F_4$, $E_8$,证明了下晶体格 $\OAztG$ 的三角分解,并得到 Matassa-Yuncken 猜想及紧量子半群结果。

Comments 13 Pages

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AI中文摘要

设 $\g$ 是类型 $G_2$, $F_4$ 或 $E_8$ 的简单复李代数,$G$ 是满足 $\mathrm{Lie}(G)=\g$ 且紧实形式为 $K$ 的唯一连通单连通复李群。我们证明了量子函数代数 $\OtG$ 的下晶体格 $\OAztG$ 的三角分解定理,建立了 $\OAztG=A_0\text{-alg}<\RAzp \cup \RAzm>.$ 这将在~\cite{DDPa} 中最近对类型 $A_n, B_n, C_n, D_n, E_6$ 和 $E_7$ 得到的三角分解推广到所有简单复李代数。作为推论,我们得到:(i) Matassa-Yuncken 猜想的包含关系 $\OAztG\subseteq\OAztK$ 和 (ii) 晶体极限 $\CpKo$ 是一个具有唯一双不变 (Haar) 态的紧量子半群。

英文摘要

Let $\g$ be a simple complex Lie algebra of type $G_2$, $F_4$, or $E_8$, and let $G$ be the unique connected simply connected complex Lie group with $\mathrm{Lie}(G)=\g$ and compact real form $K$. We prove a triangular decomposition theorem for the lower crystal lattice $\OAztG$ of the quantized function algebra $\OtG$, establishing that $\OAztG=A_0\text{-alg}<\RAzp \cup \RAzm>.$ This extends the triangular decomposition recently obtained for types $A_n, B_n, C_n, D_n, E_6$, and $E_7$ in~\cite{DDPa} to all simple complex Lie algebras. As a consequence, we obtain: (i) the inclusion $\OAztG\subseteq\OAztK$ conjectured by Matassa-Yuncken and (ii) the crystal limit $\CpKo$ is a compact quantum semigroup with a unique bi-invariant (Haar) state.

2601.22107 2026-06-19 cs.LG 版本更新 70%

Prior-Informed Flow Matching for Graph Reconstruction

先验信息流匹配用于图重建

Harvey Chen, Nicolas Zilberstein, Santiago Segarra

发表机构 * Rice University(里士大学)

专题命中 其他科学智能 :流匹配用于图重建,通用方法

AI总结 提出先验信息流匹配(PIFM),一种结合嵌入先验与连续时间流匹配的条件流模型,用于从部分观测中重建图,在多个数据集上优于经典嵌入和生成基线。

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AI中文摘要

我们引入了\textit{先验信息流匹配(PIFM)},一种用于图重建的条件流模型。从部分观测中重建图仍然是一个关键挑战;经典嵌入方法通常缺乏全局一致性,而现代生成模型难以融入结构先验。PIFM通过将基于嵌入的先验与连续时间流匹配相结合来弥合这一差距。基于置换等变的失真-感知理论,我们的方法首先使用先验(如GraphSAGE或node2vec)根据局部信息形成邻接矩阵的信息化初始估计,然后应用校正流匹配来细化该估计,将其向干净图的真实分布传输并学习全局耦合。在不同数据集上的实验表明,PIFM持续增强经典嵌入,在重建精度上优于它们和最先进的生成基线。

英文摘要

We introduce \textit{Prior-Informed Flow Matching (PIFM)}, a conditional flow model for graph reconstruction. Reconstructing graphs from partial observations remains a key challenge; classical embedding methods often lack global consistency, while modern generative models struggle to incorporate structural priors. PIFM bridges this gap by integrating embedding-based priors with continuous-time flow matching. Grounded in a permutation equivariant version of the distortion-perception theory, our method first uses a prior, such as GraphSAGE or node2vec, to form an informed initial estimate of the adjacency matrix based on local information. It then applies rectified flow matching to refine this estimate, transporting it toward the true distribution of clean graphs and learning a global coupling. Experiments on different datasets demonstrate that PIFM consistently enhances classical embeddings, outperforming them and state-of-the-art generative baselines in reconstruction accuracy.

2601.20590 2026-06-19 math.CO 版本更新 70%

A bijection between edges of the Turán graph and irreducible elements in the dominance order lattice

Turán图的边与支配序格中不可约元素之间的双射

Nathanaël Hassler

专题命中 其他科学智能 :组合数学,Turán图与支配序格双射

AI总结 本文构建了部分在[1,p]内、和为n的整数分拆的支配序格中的交不可约元素与(n,p)-Turán图的边之间的双射,并利用该双射渐近计算了这些交不可约分拆上某些统计量的平均值。

Comments 10 pages, 2 figures, 1 table

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AI中文摘要

在本文中,我们构建了部分在$[1,p]$内、和为$n$的整数分拆的格(配备支配序)中的交不可约元素与$(n,p)$-Turán图的边之间的双射。利用这个双射,我们随后渐近计算了这些交不可约分拆上某些统计量的平均值。

英文摘要

In this paper we build a bijection between the meet-irreducible elements of the lattice of the compositions of $n$ with parts in $[1,p]$ equipped with the dominance order, and the edges of the $(n,p)$-Turán graph. Using this bijection, we then compute asymptotically the average value of some statistics on those meet-irreducible compositions.

2601.19443 2026-06-19 math.GR 版本更新 70%

On the maximal subgroups of almost simple and primitive perfect groups

关于几乎单群和本原完全群的最大子群

Patricia Medina Capilla, Luca Sabatini

专题命中 其他科学智能 :群论,几乎单群最大子群性质

AI总结 本文证明有限几乎单群的最大子群的导出列第10项是完全群,并推广到完全群的无核最大子群,且常数10是最优的。

Comments 18 pages, presentation improved, to appear in Math. Proc. Cambridge Philos. Soc

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AI中文摘要

我们证明,如果$G$是有限几乎单群且$H$是$G$的最大子群,那么$H$的导出列的第10项是完全群。如果$G$是完全群且$H$是无核的,同样成立。常数$10$是最优的。

英文摘要

We prove that, if $G$ is a finite almost simple group and $H$ is a maximal subgroup of $G$, then the $10$th term of the derived series of $H$ is perfect. The same is true if $G$ is perfect and $H$ is core-free. The constant $10$ is best possible.

2503.10816 2026-06-19 math.LO math.RA 70%

On the structure and theory of McCarthy algebras

关于麦卡锡代数的结构与理论

Stefano Bonzio, Gavin St. John

专题命中 其他科学智能 :研究McCarthy代数的结构,属于非经典逻辑的代数基础。

AI总结 本文研究麦卡锡代数的结构特性,提出其作为i-ubands子变种的代数框架,并给出简化公理化体系、半格分解定理及通过装饰偏序集的表示方法。

Comments This version incorporates a proper citation to the 1990 article of Guzman and Squier, as well the addition Section 6

Journal ref Semigroup Forum (2026)

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AI中文摘要

我们对麦卡锡代数进行了结构分析,该代数由定义麦卡锡逻辑(非交换版的克里逻辑)的三元代数生成。分析在非常一般的代数设定中进行,将麦卡锡代数作为带有反交换操作$'$的单位带(幂等单体)的子变种,即i-ubands。重要的(交换)子变种包括布尔代数、正交格、克里代数和反交换半格。本文的主要贡献是为麦卡锡代数提供简化的等价公理化体系、半格分解定理以及作为特定装饰偏序集的表示方法,从而唯一确定其代数结构。

英文摘要

We provide a structural analysis for McCarthy algebras, the variety generated by the three-element algebra defining the logic of McCarthy (the non-commutative version of Kleene three-valued logics). Our analysis will be conducted in a very general algebraic setting by introducing McCarthy algebras as a subvariety of unital bands (idempotent monoids) equipped with an involutive (unary) operation $'$ satisfying $x''\approx x$; herein referred to as i-ubands. Prominent (commutative) subvarieties of i-ubands include Boolean algebras, ortholattices, Kleene algebras, and involutive bisemilattices, hence i-ubands provides an algebraic common ground for several non-classical logics. Our main contributions consist in providing for McCarthy algebras: reduced and equivalent axiomatizations; a semilattice decomposition theorem; and representations as certain decorated posets from which the algebraic structure can be uniquely determined.

2109.14578 2026-06-19 math.GT 版本更新 70%

Milnor-type invariants for surface-links and cut-diagrams

曲面链与切割图的Milnor型不变量

Benjamin Audoux, Jean-Baptiste Meilhan, Akira Yasuhara

专题命中 其他科学智能 :将Milnor不变量推广到曲面链,属于低维拓扑。

AI总结 将Milnor链环不变量推广到4-空间中的曲面链(可能带边界),通过引入切割图(Gauss图的二维类比)构造群并提取Milnor型不变量,证明其为同痕不变量和链同伦不变量,并给出实现与分类结果。

Comments 40 pages. v.5: entirely revised version, with new organization; some applications and examples added

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AI中文摘要

我们将Milnor链环不变量推广到4-空间中的曲面链,可能带有边界。为此,我们引入了切割图的概念,它是Gauss图的二维类比。对于每个切割图,我们关联一个群,该群扩展了曲面链外部的基本群,并从其逐次幂零商中提取Milnor型不变量。我们证明这产生了曲面链的同痕不变量,并且其中一些甚至是链同伦不变量。我们给出了几个具体应用,包括实现和分类结果。进一步研究了切割图理论,旨在为4-空间中的曲面提供组合方法。

英文摘要

We generalize Milnor link invariants to surface-links in 4-space, possibly with boundary. To this end, we introduce the notion of cut-diagram, which is a 2-dimensional analogue of Gauss diagrams. To each cut-diagram, we associate a group extending the fundamental group of the exterior of a surface-link, and we extract Milnor-type invariants from its successive nilpotent quotients. We show that this yields concordance invariants for surface-links, and that some even are link-homotopy invariants. We give several concrete applications, including realization and classification results. The theory of cut-diagrams is further investigated, heading towards a combinatorial approach to surfaces in 4-space.

2511.21617 2026-06-19 math.NT 版本更新 70%

On efficient approximation of quadratic irrationals

关于二次无理数的高效逼近

Peter H. van der Kamp, Anthony Overmars, Marcel Jackson, Andrew N. W. Hone

专题命中 其他科学智能 :提供二次无理数收敛的高效算法,数论。

AI总结 本文提出高效计算二次无理数收敛的算法,证明在拉格朗日定理的伽罗瓦细化条件下,平方根的收敛序列的某些抽取是带符号的切比雪夫序列,并可通过Householder方法生成。

Comments 13 pages, 11 references, no figures, V2 contains two additional (multiplicative) algorithms (3.4 and 3.5) and an additional example (3.4)

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AI中文摘要

我们提供了高效算法来计算二次无理数的收敛。我们证明,对于平方根,在拉格朗日定理的伽罗瓦细化成立的情况下,收敛序列的某些抽取是带符号的切比雪夫序列,这些序列也可以通过Householder方法生成。

英文摘要

We provide efficient algorithms to compute convergents of quadratic irrationals. We show that for square roots, in settings where Galois' refinement of Lagrange's theorem holds, certain decimations of the sequence of convergents are signed Chebyshev sequences, which can be also be generated by a Householder method.

2509.19598 2026-06-19 cs.IT cs.DS math.IT 版本更新 70%

Efficient $\varepsilon$-approximate minimum-entropy couplings

高效的ε-近似最小熵耦合

Spencer Compton

专题命中 其他科学智能 :提出最小熵耦合近似算法,属于信息论

AI总结 针对离散概率分布的最小熵耦合问题,提出运行时间为n^{O(poly(1/ε)·exp(m))}的算法,实现H(ALG) ≤ H(OPT) + ε,证明对常数m存在多项式时间近似方案。

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AI中文摘要

给定m≥2个离散概率分布,每个分布有n个状态,最小熵耦合是边际分布与输入分布相同的最小熵联合分布。计算最小熵耦合是NP难的,但在设计近似算法方面取得了显著进展;在这项工作之前,最著名的多项式时间算法保证形式为H(ALG) ≤ H(OPT) + c,其中对于m=2,c≈0.53,对于一般m,c≈1.22 [CKQGK '23]。一个主要的开放问题是该任务是否是APX难的,或者是否存在多项式时间近似方案(PTAS)。在这项工作中,我们设计了一种算法,在运行时间n^{O(poly(1/ε)·exp(m))}内产生熵H(ALG) ≤ H(OPT) + ε的耦合:表明对于常数m存在PTAS。

英文摘要

Given $m \ge 2$ discrete probability distributions over $n$ states each, the minimum-entropy coupling is the minimum-entropy joint distribution whose marginals are the same as the input distributions. Computing the minimum-entropy coupling is NP-hard, but there has been significant progress in designing approximation algorithms; prior to this work, the best known polynomial-time algorithms attain guarantees of the form $H(\operatorname{ALG}) \le H(\operatorname{OPT}) + c$, where $c \approx 0.53$ for $m=2$, and $c \approx 1.22$ for general $m$ [CKQGK '23]. A main open question is whether this task is APX-hard, or whether there exists a polynomial-time approximation scheme (PTAS). In this work, we design an algorithm that produces a coupling with entropy $H(\operatorname{ALG}) \le H(\operatorname{OPT}) + \varepsilon$ in running time $n^{O(\operatorname{poly}(1/\varepsilon) \cdot \operatorname{exp}(m) )}$: showing a PTAS exists for constant $m$.

2507.17517 2026-06-19 math.LO math.GR 版本更新 70%

Minimal Banach-Tarski decompositions

最小 Banach-Tarski 分解

Cesare Straffelini, Kilian Zambanini

专题命中 其他科学智能 :Banach-Tarski分解的数学研究,属于数学。

AI总结 研究将三维球体或球分割并重组为 n 个全等副本所需的最小块数,推广了 Raphael Robinson 的已知结果。

Comments 22 pages. Online First version accepted for publication in Fundamenta Mathematicae

Journal ref Fundamenta Mathematicae 273 (2026), 177-198

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AI中文摘要

我们研究将三维球体或球分割并重组为原始物体的 $n$ 个全等副本所需的最小块数问题,推广了 Raphael Robinson 的一个已知结果。

英文摘要

We investigate the problem of finding the minimum number of pieces necessary for dividing a three-dimensional sphere or a ball and reassembling it to form $n$ congruent copies of the original object, generalising a known result by Raphael Robinson.

2505.18726 2026-06-19 cs.SD cs.LG eess.AS 版本更新 70%

Bioacoustic Geolocation: Species Sounds as Geographic Signals

生物声学地理定位:物种声音作为地理信号

Mustafa Chasmai, Wuao Liu, Subhransu Maji, Grant Van Horn

发表机构 * University of Massachusetts, Amherst(马萨诸塞大学阿姆赫斯特分校)

专题命中 其他科学智能 :生物声学地理定位,结合物种分布,属于科学智能。

AI总结 本文研究仅通过声音进行全球尺度地理定位,利用生物声学信号中的物种地理分布线索,提出结合物种范围预测与检索的地理定位方法,并验证多模态融合的潜力。

Comments Accepted to ICML 26

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AI中文摘要

我们能否仅通过听到的声音确定某人的地理位置?声学信号是否足以定位到国家、州甚至城市?在这项工作中,我们应对全球尺度音频地理定位的挑战,特别关注野生动物和自然声音。我们假设生物声学信号包含信息丰富的地理定位线索,因为物种具有明确的地理分布范围。为了验证这一假设,我们对图像地理定位和声景映射方法进行基准测试,设计预言机和以物种为中心的基线,并提出一种结合物种范围预测与基于检索的地理定位的混合方法。我们进一步探究地理定位是否随着物种多样性记录和跨邻近样本的时空聚合而改善。最后,我们将研究扩展到多模态地理定位,通过结合音频和视觉内容的电影案例研究。我们的结果突出了将生物声学信号纳入地理空间任务的潜力,为物种识别和音频地理定位的未来工作提供了动力。

英文摘要

Can we determine someone's geographic location solely from the sounds they hear? Are acoustic signals enough to localize within a country, state, or even city? In this work, we tackle the challenge of global-scale audio geolocation, with a particular focus on wildlife and natural sounds. We posit that bioacoustic signals contain informative geolocation cues because of well-defined geographic ranges of species. To test this hypothesis, we benchmark image geolocation and soundscape mapping methods, design oracles and species-centric baselines, and propose a hybrid approach that combines species range prediction with retrieval-based geolocation. We further ask whether geolocation improves with species-diverse recordings and spatiotemporal aggregation across neighboring samples. Finally, we extend our study to multimodal geolocation with case studies from movies that combine both audio and visual content. Our results highlight the potential of incorporating bioacoustic signals into geospatial tasks, motivating future work on species recognition and audio geolocation.

2507.14458 2026-06-19 math.DG math.CV 版本更新 70%

Spectral bundles on Abelian varieties, complex projective spaces and Grassmannians

阿贝尔簇、复射影空间和格拉斯曼流形上的谱丛

Ching-Hao Chang, Jih-Hsin Cheng, I-Hsun Tsai

专题命中 其他科学智能 :谱丛的数学研究,属于数学。

AI总结 通过模拟物理中的产生和湮灭算符,将高能级特征截面转化为全纯截面,赋予对偶阿贝尔簇上的谱丛自然全纯结构,并给出复射影空间上高能级特征截面维数的显式公式。

Comments 43 pages

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AI中文摘要

本文研究了阿贝尔簇、复射影空间$\mathbb{P}^{n}$和格拉斯曼流形上带有全纯线丛的Bochner-Kodaira拉普拉斯算子的谱分析。通过模拟物理中的产生和湮灭算符方法,我们将高能级特征截面转化为最低能级的全纯截面。这使得我们能够赋予定义在对偶阿贝尔簇上的这些谱丛以自然全纯结构。利用这种具体表达的转换,所有高能级特征截面都可以由theta函数形成的全纯截面显式表示。此外,通过消失定理和Hirzebruch-Riemann-Roch定理,我们给出了$\mathbb{P}^{n}$上高能级特征截面空间维数的显式公式。这些为弦理论学家最近通过数值分析讨论的一些问题提供了理论研究。我们还证明了格拉斯曼流形上的一些部分结果,并指出了未来研究的方向。

英文摘要

In this paper we study the spectral analysis of Bochner-Kodaira Laplacians on an Abelian variety, complex projective space $\mathbb{P}^{n}$ and a Grassmannian with a holomorphic line bundle. By imitating the method of creation and annihilation operators in physics, we convert those eigensections (of the \textquotedblleft higher energy" level) into holomorphic sections (of the \textquotedblleft lowest energy" level). This enables us to endow these spectral bundles, which are defined over the dual Abelian variety, with natural holomorphic structure. Using this conversion expressed in a concrete way, all the higher eigensections are explicitly expressible using holomorphic sections formed by theta functions. Moreover, we give an explicit formula for the dimension of the space of higher-level eigensections on $\mathbb{P}^{n}$ through vanishing theorems and the Hirzebruch-Riemann-Roch theorem. These give a theoretical study related to some problems newly discussed by string theorists using numerical analysis. Some partial results on Grassmannians are proved and some directions for future research are indicated.

2307.15130 2026-06-19 cs.CG math.GN 70%

Bounding the Interleaving Distance for Mapper Graphs with a Loss Function

对Mapper图的交织距离进行边界估计:带有损失函数的方法

Erin W. Chambers, Elizabeth Munch, Sarah Percival, Bei Wang

专题命中 其他科学智能 :Mapper图交织距离的数学研究,属于数学。

AI总结 本文研究了Mapper图的交织距离,提出通过损失函数衡量近似程度,实现多项式时间计算,为数据分析提供有效工具。

Comments Updating to fix some typos

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AI中文摘要

由图与映射到R^d的结构在数据应用中普遍存在,包括Reeb图、几何图和结嵌入。在数据分析流程中需要比较和聚类此类对象,因此需要它们之间的距离。本文研究了这些对象的离散化形式,称为Mapper图(当d=1时),通过寻找自然变换对来比较函数表示。然而,计算交织距离在许多情况下是NP难的。为此,我们借鉴Robinson最近的工作,为不构成自然变换的映射族定义质量度量,称为赋值。然后,我们给函数图像赋予度量空间的额外结构,并定义一个损失函数来衡量赋值与交织图示所需交换的接近程度。最后,我们证明在给定赋值下计算损失函数是多项式的。我们相信这一想法既强大又具有可移植性,有可能在广泛上下文中提供对交织的近似和边界估计。

英文摘要

Data consisting of a graph with a function mapping into $\mathbb{R}^d$ arise in many data applications, encompassing structures such as Reeb graphs, geometric graphs, and knot embeddings. As such, the ability to compare and cluster such objects is required in a data analysis pipeline, leading to a need for distances between them. In this work, we study the interleaving distance on discretization of these objects, called mapper graphs when $d=1$, where functor representations of the data can be compared by finding pairs of natural transformations between them. However, in many cases, computation of the interleaving distance is NP-hard. For this reason, we take inspiration from recent work by Robinson to find quality measures for families of maps that do not rise to the level of a natural transformation, called assignments. We then endow the functor images with the extra structure of a metric space and define a loss function which measures how far an assignment is from making the required diagrams of an interleaving commute. Finally we show that the computation of the loss function is polynomial with a given assignment. We believe this idea is both powerful and translatable, with the potential to provide approximations and bounds on interleavings in a broad array of contexts.

2507.09324 2026-06-19 math.RA cs.CC math.LO 版本更新 70%

The Network Satisfaction Problem for Relation Algebras with at most 4 Atoms

最多4个原子的关系代数的网络满足问题

Manuel Bodirsky, Moritz Jahn, Simon Knäuer, Matěj Konečný, Paul Winkler

专题命中 其他科学智能 :关系代数网络满足问题的数学研究。

AI总结 本文扩展了Cristiani和Hirsch的结果,证明最多4个原子的关系代数的网络满足问题要么在P中,要么是NP-hard。

Comments Full version of an ICALP 2026 paper, Article No. 167

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AI中文摘要

Andréka和Maddux分类了最多3个原子的关系代数,并特别证明了它们都是可表示的。Hirsch和Cristiani证明了这些代数中每一个的网络满足问题(NSP)要么在P中,要么是NP-hard。文献中包含了许多关于关系代数表示的结果;特别地,一些具有四个原子的关系代数不可表示。我们将Cristiani和Hirsch的结果扩展到最多4个原子的关系代数:NSP总是要么在P中,要么是NP-hard。为此,我们尽可能为这些代数构造了泛表示、完全泛表示甚至正规表示。

英文摘要

Andréka and Maddux classified the relation algebras with at most 3 atoms, and in particular they showed that all of them are representable. Hirsch and Cristiani showed that the network satisfaction problem (NSP) for each of these algebras is in P or NP-hard. The literature contains many results on representations of relation algebras; in particular, some relation algebras with four atoms are not representable. We extend the result of Cristiani and Hirsch to relation algebras with at most 4 atoms: the NSP is always either in P or NP-hard. To this end, we construct universal, fully universal, or even normal representations for these algebras, whenever possible.

2507.07606 2026-06-19 math.LO math.CO 版本更新 70%

Ramsey-like theorems for separable permutations

可分离排列的类Ramsey定理

Quentin Le Houérou, Ludovic Patey

专题命中 其他科学智能 :可分离排列的Ramsey定理,属于数学。

AI总结 研究无限团边着色中避免特定模式的无限制子团的存在性,证明可分离排列的避免性等价于标准模型中无限齐次集的存在,其他模式则不然。

Comments 49 pages

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AI中文摘要

我们对形如“无限团的每条边着色后存在一个无限子团避免某种模式”的类Ramsey定理进行了可计算性理论研究,特别关注传递模式。结果表明,对应于可分离排列的模式在该陈述的计算特征中扮演重要角色。我们证明,避免任何可分离排列等价于标准模型中无限齐次集的存在,而这一性质对任何其他模式均不成立。为此,我们发展了一种用于相对化对角非计算性的新论证。

英文摘要

We conduct a computability-theoretic study of Ramsey-like theorems of the form "Every coloring of the edges of an infinite clique admits an infinite sub-clique avoiding some pattern", with a particular focus on transitive patterns. As it turns out, the patterns corresponding to separable permutations play an important role in the computational features of the statement. We prove that the avoidance of any separable permutation is equivalent to the existence of an infinite homogeneous set in standard models, while this property fails for any other pattern. For this, we develop a novel argument for relativized diagonal non-computation.

2401.05158 2026-06-19 math.RT 版本更新 70%

On $τ$-tilting graphs for quasi-silted algebras

关于拟倾斜代数的$\ au$-倾斜图

Wei Dai, Changjian Fu, Shengfei Geng, Pin Liu

专题命中 其他科学智能 :τ-倾斜图的数学研究,属于数学。

AI总结 本文证明任意拟倾斜代数的$\ au$-倾斜图是连通的且具有可达面性质,通过$\ au$-约化与墙室结构给出商代数保持连通性的充分条件。

Comments In this revised version, the results previously established for quasi-tilted algebras are extended to the more general setting of quasi-silted algebras

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AI中文摘要

我们证明了任意拟倾斜代数的$\ au$-倾斜图是连通的且具有可达面性质。我们的方法利用了$\ au$-约化以及墙与室结构。特别地,我们观察到墙与室结构的一个充分条件,在该条件下$\ au$-倾斜图的连通性在取代数商时得以保持。作为直接推论,对于几类新的代数也建立了$\ au$-倾斜图的连通性。

英文摘要

We prove that the $τ$-tilting graph of any quasi-silted algebra is connected and has the reachable-in-face property. Our approach utilizes $τ$-reduction and wall and chamber structures. In particular, we observe a sufficient condition on the wall and chamber structure under which the connectivity of $τ$-tilting graphs is preserved under taking quotients of algebras. As an immediate consequence, the connectivity of $τ$-tilting graphs is also established for several new classes of algebras.

2506.15370 2026-06-19 math.MG 版本更新 70%

On polynomial inequalities for cone-volumes of polytopes

关于多面体锥体积的多项式不等式

Tom Baumbach, Martin Henk

专题命中 其他科学智能 :多面体锥体积的数学研究,属于数学。

AI总结 受离散对数Minkowski问题启发,研究给定矩阵U的锥体积集,证明其为路径连通的半代数集,并定义子空间浓度多胞体,为离散对数Minkowski问题提供新几何视角。

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AI中文摘要

受离散对数Minkowski问题启发,我们研究给定矩阵$U\in\mathbb{R}^{n\times m}$的锥体积集$C_{\tt cv}(U)$,它由多面体$P(U,b)=\{ x\in\mathbb{R}^n: U^\intercal x\leq b\}$, $b\in\mathbb{R}^n_{\geq 0}$的所有锥体积向量组成。我们将证明$C_{\tt cv}(U)$是一个路径连通的半代数集,这推广了平面情形或特定多面体的先前结果。此外,我们定义了一个子空间浓度多胞体$P_{\tt scc}(U)$,它几何地表示了球面上有限离散Borel测度的子空间浓度条件。这相当于$U$的基拟阵多胞体经过缩放,并且这两个集合$P_{\tt scc}(U)$和$C_{\tt cv}(U)$也为离散对数Minkowski问题提供了新的几何视角。

英文摘要

Motivated by the discrete logarithmic Minkowski problem we study for a given matrix $U\in\mathbb{R}^{n\times m}$ its cone-volume set $C_{\tt cv}(U)$ consisting of all the cone-volume vectors of polytopes $P(U,b)=\{ x\in\mathbb{R}^n : U^\intercal x\leq b\}$, $b\in\mathbb{R}^n_{\geq 0}$. We will show that $C_{\tt cv}(U)$ is a path-connected semialgebraic set which extends former results in the planar case or for particular polytopes. Moreover, we define a subspace concentration polytope $P_{\tt scc}(U)$ which represents geometrically the subspace concentration conditions for a finite discrete Borel measure on the sphere. This is up to a scaling the basis matroid polytope of $U$, and these two sets, $P_{\tt scc}(U)$ and $C_{\tt cv}(U)$, also offer a new geometric point of view to the discrete logarithmic Minkowski problem.

2412.04561 2026-06-19 math.CO math.AC 版本更新 70%

Differential operators, anisotropy, and simplicial spheres

微分算子、各向异性和单纯球面

Kalle Karu, Matt Larson, Alan Stapledon

专题命中 其他科学智能 :单纯球面微分算子的数学研究。

AI总结 本文在任意正特征下,针对单纯球面的Stanley-Reisner环的通用Artin约化,发现了涉及微分算子的恒等式,并利用这些恒等式证明了某些形式的各向异性及弱Lefschetz性质。

Comments To appear in IMRN

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AI中文摘要

我们在任意正特征下,在单纯球面的Stanley-Reisner环的通用Artin约化中发现了涉及微分算子的恒等式。这些恒等式推广了Papadakis和Petrotou用于证明代数g-猜想的特征2恒等式。我们证明这些恒等式是次数映射上某个恒等式的影子,并利用它们来证明通用Artin约化中某些形式的各向异性以及弱Lefschetz结果。

英文摘要

We find identities involving differential operators in the generic artinian reduction of the Stanley-Reisner ring of a simplicial sphere in any positive characteristic. These identities generalize the characteristic 2 identities used by Papadakis and Petrotou to give a proof of the algebraic g-conjecture. We show that these identities are a shadow of an identity on the degree map, and we use them to prove the anisotropy of certain forms on the generic artinian reduction of the Stanley--Reisner ring and to prove weak Lefschetz results.

2504.05406 2026-06-19 math.CO 版本更新 70%

Erdős-Ko-Rado Theorems for Paths in Graphs

图中路径的 Erdős-Ko-Rado 定理

Neal Bushaw, James Danielsson, Glenn Hurlbert

专题命中 其他科学智能 :研究图论中路径族的相交性质,属于数学组合领域

AI总结 本文研究图中长度-r路径族的1-相交和s-相交性质,证明太阳图和theta图满足EKR性质,并给出Hilton-Milner型结果和横贯数结论。

Comments 26 pages, revised

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AI中文摘要

一个集合族称为$s$-相交的,如果其中任意两个集合至少有$s$个公共元素。如果族中所有成员都包含某$s$个公共元素,则称为$s$-星。一个集合族称为$s$-EKR,如果它的所有$s$-相交子族的大小都不超过某个$s$-星的大小。例如,经典的1961年Erdős-Ko-Rado定理本质上表明,当$n$是$r$和$s$的足够大函数时,$\{1,2,\ldots,n\}$的所有$r$元子集族是$s$-EKR的,而1967年Hilton-Milner定理给出了这类集合中最大的非星相交族的近星结构。随后出现了两个重要猜想:1974年Chvátal猜想每个子集封闭的集合族是1-EKR的,以及2005年Holroyd和Talbot猜想,对于每个图,当每个极大独立集的大小至少为$2r$时,图的所有$r$元独立集族是1-EKR的。在本文中,我们给出了图中长度为$r$的路径族的类似1-EKR结果,特别是对于太阳图(即带有均匀悬挂边的环)和theta图(即共享相同两个端点的两两内部不相交路径的集合)。我们还证明了太阳图中此类路径的$s$-EKR结果,并给出了它们的Hilton-Milner型结果。一个集合称为集合族的横贯,如果它与族中每个成员相交,族的横贯数是最小横贯的大小。例如,星的横贯数为1,Hilton-Milner族的横贯数为2。我们以一些涉及所谓三角族的横贯结果结束本文,包括一些关于射影平面的结果。

英文摘要

A family of sets is $s$-intersecting if every pair of its sets has at least $s$ elements in common. It is an $s$-star if all its members have some $s$ elements in common. A family of sets is called $s$-EKR if all its $s$-intersecting subfamilies have size at most that of some $s$-star. For example, the classic 1961 Erdős-Ko-Rado theorem states essentially that the family of $r$-sized subsets of $\{1,2,\ldots,n\}$ is $s$-EKR when $n$ is a large enough function of $r$ and $s$, and the 1967 Hilton-Milner theorem provides the near-star structure of the largest non-star intersecting family of such sets. Two important conjectures along these lines followed: by Chvátal in 1974, that every subset-closed family of sets is 1-EKR, and by Holroyd and Talbot in 2005, that, for every graph, the family of all its $r$-sized independent sets is 1-EKR when every maximal independent set has size at least $2r$. In this paper we present similar 1-EKR results for families of length-$r$ paths in graphs, specifically for sun graphs, which are cycles with pendant edges attached in a uniform way, and theta graphs, which are collections of pairwise internally disjoint paths sharing the same two endpoints. We also prove $s$-EKR results for such paths in suns, and give a Hilton-Milner type result for them as well. A set is a transversal of a family of sets if it intersects each member of the family, and the transversal number of the family is the size of its smallest transversal. For example, stars have transversal number 1, and the Hilton-Milner family has transversal number 2. We conclude the paper with some transversal results involving what we call triangular families, including a few results for projective planes.

2408.14452 2026-06-19 math.OC 版本更新 70%

Taxicab distance based best-worst method for multi-criteria decision-making: An analytical approach

基于出租车距离的最优最劣多准则决策方法:一种解析方法

Harshit Ratandhara, Mohit Kumar

专题命中 其他科学智能 :研究多准则决策方法,属于运筹学领域

AI总结 本文对基于出租车距离的最优最劣方法进行解析分析,证明该模型可能导致多个最优权重集,并提出混合整数线性规划计算一致性指标及决策者辅助选择策略,显著提升计算效率。

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AI中文摘要

最优最劣方法是一种著名的基于距离的多准则决策方法,用于计算决策准则的权重。本文对该方法的出租车距离模型进行了全面的解析研究,旨在探讨这些解的唯一性,并进行严格的一致性分析。为此,首先构建了一个与原始问题等价的最优修正优化问题。然后解析求解该重构问题,并从其解中推导出最优权重集。与基于出租车模型数值实验的普遍理解相反,我们的解析框架证明该模型实际上可能导致多个最优权重集,并正式建立了这种情况发生的条件。随后采用混合整数线性规划模型计算一致性指标。还提出了一种决策者辅助的选择策略,以解决最优权重集的非唯一性问题。此外,建立了评估给定偏好可接受性的一致性比率阈值。该框架提供了坚实的数学基础,增强了对模型的理解,并消除了对优化软件的需求。通过显著提高模型的计算效率,使其能够应用于大规模、动态的现实场景,例如电力市场竞价策略和市场波动下的投资组合再平衡。通过数值示例展示了所提框架的有效性,并通过智能手机选择问题说明了其实用适用性。

英文摘要

The best-worst method is a well-known distance based multi-criteria decision-making method used for computing the weights of decision criteria. This article provides a comprehensive analytical examination of the taxicab distance based model of the method, with the objectives of investigating the uniqueness of these solutions, and performing a rigorous consistency analysis. To achieve this, an optimal modification based optimization problem, equivalent to the original one, is first formulated. This reformulated problem is then solved analytically, and the optimal weight sets are derived from its solutions. Contrary to the prevailing understanding derived from numerical experiments with the taxicab model, our analytical framework proves that the model can, in fact, lead to multiple optimal weight sets, and we formally establish the conditions for this occurrence. A mixed-integer linear programming model is then employed to compute the consistency index. A decision-maker-aided selection strategy is also proposed for addressing non-uniqueness of optimal weight sets. In addition, threshold values of the consistency ratio to assess the admissibility of given preferences are also established. This framework provides a solid mathematical foundation that enhances the understanding of the model and eliminates the requirement for optimization software. By significantly improving the model's computational efficiency, it enables implementation in large-scale, dynamic real-world applications such as electricity market bidding strategies and portfolio rebalancing under market volatility. The effectiveness of the proposed framework is demonstrated through numerical examples, and its practical applicability is illustrated via a smartphone selection problem.

2410.02248 2026-06-19 math.LO math.GR 版本更新 70%

Oligomorphic groups, their automorphism groups, and the complexity of their isomorphism

寡态群、其自同构群及其同构的复杂性

Gianluca Paolini, Andre Nies

专题命中 其他科学智能 :研究寡态群及其自同构群,属于数学逻辑

AI总结 本文研究Roelcke预紧的置换群子群,证明其内自同构群在自同构群中闭,且在外自同构群完全不连通局部紧;并给出方法证明两类寡态群的同构关系光滑,且其自同构群拓扑同构于寡态群,外自同构群为profinite。

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AI中文摘要

本文沿两个相互关联的方向建立了结果。1. 设$G$是自然数置换群$\mathrm{Sym}(\omega)$的Roelcke预紧闭子群。设$\mathrm{Aut}(G)$表示$G$的连续自同构群。则$\mathrm{Inn}(G)$在$\mathrm{Aut}(G)$中闭,其中$\mathrm{Aut}(G)$带有其(忠实)作用在开子群的陪集上的逐点收敛拓扑。在更强的假设$G$是寡态的条件下,$\+ N_G/G$是profinite的,其中$\+ N_G$表示$G$在$\mathrm{Sym}(\omega)$中的正规化子,且拓扑群$\mathrm{Out}(G)= \mathrm{Aut}(G)/\mathrm{Inn}(G)$是完全不连通、局部紧的。2a. 我们提供了一种一般方法,用于证明适当Borel类中寡态群的同构关系的光滑性。我们将其应用于两个这样的类:无代数性的寡态群,以及至多有限个本质子群共轭类的寡态群。2b. 利用该方法,我们还证明了如果$G$属于这样的Borel类,则$\mathrm{Aut}(G)$拓扑同构于一个寡态群,且$\mathrm{Out}(G)$是profinite的。

英文摘要

The paper establishes results following two interconnected directions. 1. Let $G$ be a Roelcke precompact closed subgroup of the group $\mathrm{Sym}(ω)$ of permutations of the natural numbers. Let $\mathrm{Aut}(G)$ denote the group of continuous automorphisms of $G$. Then $\mathrm{Inn}(G)$ is closed in $\mathrm{Aut}(G)$, where $\mathrm{Aut}(G)$ carries the topology of pointwise convergence for its (faithful) action on the cosets of open subgroups. Under the stronger hypothesis that~$G$ is oligomorphic, $\+ N_G/G$ is profinite, where $\+ N_G$ denotes the normaliser of~$G$ in $\mathrm{Sym}(ω)$, and the topological group $\mathrm{Out}(G)= \mathrm{Aut}(G)/\mathrm{Inn}(G)$ is totally disconnected, locally compact. 2a. We provide a general method to show smoothness of the isomorphism relation for appropriate Borel classes of oligomorphic groups. We apply it to two such classes: the oligomorphic groups with no algebraicity, and the oligomorphic groups with finitely many {essential} subgroups up to conjugacy. 2b. Using this method we also show that if $G$ is in such a Borel class, then $\mathrm{Aut}(G)$ is topologically isomorphic to an oligomorphic group, and $\mathrm{Out}(G)$ is profinite.

2503.13328 2026-06-19 q-fin.MF math.PR 版本更新 70%

Model-independent upper bounds for the prices of Bermudan options with convex payoffs

凸收益百慕大期权价格的无模型上界

David Hobson, Dominykas Norgilas

专题命中 其他科学智能 :研究百慕大期权定价,属于金融数学

AI总结 研究在给定欧式期权价格下,寻找具有凸收益的百慕大期权价格的无套利上界,通过刻画对偶问题并假设测度满足分散性条件完全求解,发现标准设定不足以定义最优模型,需要额外随机化。

Comments 55 pages, 6 figures. In the new version we work with arbitrary convex payoffs and marginal distributions that satisfy the Dispersion Assumption

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AI中文摘要

假设 $\mu$ 和 $\nu$ 是 $\mathbb{R}$ 上的概率测度,满足 $\mu \leq_{cx} \nu$。设 $a$ 和 $b$ 是 $\mathbb{R}$ 上的凸函数,且 $a \geq b \geq 0$。我们感兴趣的是寻找 $$\sup_{\mathbf{M}} \sup_{\tau} \mathbb{E}^{\mathbf{M}} \left[ a(X) I_{ \{ \tau = 1 \} } + b(Y) I_{ \{ \tau = 2 \} } \right] $$ 其中第一个上确界取遍所有一致模型 $\mathbf{M}$(即过滤概率空间 $(\Omega, \mathbf{F}, \mathbb{F}, \mathbb{P})$,使得 $Z=(z,Z_1,Z_2)=(\int_{\mathbb{R}} x \mu(dx) = \int_{\mathbb{R}} y \nu(dy), X, Y)$ 是一个 $(\mathbb{F},\mathbb{P})$ 鞅,且在 $\mathbb{P}$ 下 $X$ 服从分布 $\mu$,$Y$ 服从分布 $\nu$),第二个上确界中的 $\tau$ 是取值于 $\{1,2\}$ 的 $(\mathbb{F},\mathbb{P})$ 停时。我们的贡献首先是刻画并简化对偶问题,其次是在对测度 $\mu$ 和 $\nu$ 的一些结构假设(即 $\mu$ 和 $\nu$ 是绝对连续的概率测度且满足分散性假设)下完全求解该问题。一个关键发现是,由 $Z$ 生成的过滤的标准设定不足以定义最优模型,需要额外的随机化。即使边际分布 $\mu$ 和 $\nu$ 是无原子的,这一结论仍然成立。该问题可解释为:在给定同时到期的欧式期权价格的情况下,寻找具有两个可能行权日的百慕大期权价格的稳健或无模型无套利上界。

英文摘要

Suppose $μ$ and $ν$ are probability measures on $\mathbb{R}$ satisfying $μ\leq_{cx} ν$. Let $a$ and $b$ be convex functions on $\mathbb{R}$ with $a \geq b \geq 0$. We are interested in finding $$\sup_{\mathbf{M}} \sup_τ \mathbb{E}^{\mathbf{M}} \left[ a(X) I_{ \{ τ= 1 \} } + b(Y) I_{ \{ τ= 2 \} } \right] $$ where the first supremum is taken over consistent models $\mathbf{M}$ (i.e., filtered probability spaces $(Ω, \mathbf{F}, \mathbb{F}, \mathbb{P})$ such that $Z=(z,Z_1,Z_2)=(\int_{\mathbb{R}} x μ(dx) = \int_{\mathbb{R}} y ν(dy), X, Y)$ is a $(\mathbb{F},\mathbb{P})$ martingale, where $X$ has law $μ$ and $Y$ has law $ν$ under $\mathbb{P}$) and $τ$ in the second supremum is a $(\mathbb{F},\mathbb{P})$-stopping time taking values in $\{1,2\}$. Our contributions are first to characterise and simplify the dual problem, and second to completely solve the problem under some structural assumptions on the measures $μ$ and $ν$ (namely that $μ$ and $ν$ are absolutely continuous probability measures that satisfy the Dispersion Assumption). A key finding is that the canonical set-up in which the filtration is that generated by $Z$ is not rich enough to define an optimal model and additional randomisation is required. This holds even though the marginal laws $μ$ and $ν$ are atom-free. The problem has an interpretation of finding the robust, or model-free, no-arbitrage bound on the price of a Bermudan option with two possible exercise dates, given the prices of co-maturing European options.

2503.11479 2026-06-19 stat.CO math.PR math.ST stat.ME stat.TH 版本更新 70%

Towards practical PDMP sampling: Metropolis adjustments, locally adaptive step-sizes, and NUTS-based time lengths

走向实用的PDMP采样:Metropolis调整、局部自适应步长和基于NUTS的时间长度

Augustin Chevallier, Sam Power, Matthew Sutton

专题命中 其他科学智能 :改进PDMP采样方法,用于复杂分布采样

AI总结 针对PDMP采样需要计算模型特定界限的难题,提出Metropolis调整近似、自适应步长机制和NUTS启发的路径长度选择,集成得到双重自适应PDMP采样器,提升鲁棒性和效率。

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AI中文摘要

分段确定性马尔可夫过程(PDMP)在从复杂概率分布中采样方面具有重要前景。然而,其实践应用受到需要计算模型特定界限的限制。相反,虽然哈密顿蒙特卡洛(HMC)提供了一种普遍有效的采样方法,但其无法自适应调整步长,导致在采样漏斗形等复杂分布时性能受损。为解决这些限制,我们引入了三个创新概念:(a) 一种Metropolis调整的PDMP模拟近似,无需显式界限且不破坏不变测度;(b) 一种与Metropolis校正兼容的自适应步长机制;(c) 一种受无U型转弯采样器(NUTS)启发的方案,用于动态选择PDMP中的路径长度。这三个想法可以无缝集成到一个单一的“双重自适应”PDMP采样器中,具有良好的鲁棒性和效率特性。

英文摘要

Piecewise-Deterministic Markov Processes (PDMPs) hold significant promise for sampling from complex probability distributions. However, their practical implementation is hindered by the need to compute model-specific bounds. Conversely, while Hamiltonian Monte Carlo (HMC) offers a generally efficient approach to sampling, its inability to adaptively tune step sizes impedes its performance when sampling complex distributions like funnels. To address these limitations, we introduce three innovative concepts: (a) a Metropolis-adjusted approximation for PDMP simulation that eliminates the need for explicit bounds without compromising the invariant measure, (b) an adaptive step size mechanism compatible with the Metropolis correction, and (c) a No U-Turn Sampler (NUTS)-inspired scheme for dynamically selecting path lengths in PDMPs. These three ideas can be seamlessly integrated into a single, `doubly-adaptive' PDMP sampler with favourable robustness and efficiency properties.

2503.10236 2026-06-19 math.AG 版本更新 70%

Liftability and vanishing theorems for Fano threefolds in positive characteristic I

正特征Fano三维流形的可提升性与消失定理I

Tatsuro Kawakami, Hiromu Tanaka

专题命中 其他科学智能 :研究正特征Fano三维流形,属于代数几何

AI总结 本文证明正特征光滑Fano三维流形可提升到Witt向量环,并满足Akizuki-Nakano消失、Hodge到de Rham谱序列E1退化及晶体上同调无挠性,针对$|-K_X||$非常丰沛且Picard群由$\omega_X$生成的情形。

Comments 68 pages, v2: minor revisions

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AI中文摘要

在我们的系列论文中,我们证明了正特征光滑Fano三维流形可提升到Witt向量环。此外,我们证明了它们满足Akizuki-Nakano消失、Hodge到de Rham谱序列的$E_1$-退化以及晶体上同调的无挠性。在本文中,我们针对$|-K_X|$非常丰沛且Picard群由$\omega_X$生成的情形建立了这些结果。

英文摘要

In our series of papers, we prove that smooth Fano threefolds in positive characteristic lift to the ring of Witt vectors. Moreover, we show that they satisfy Akizuki-Nakano vanishing, $E_1$-degeneration of the Hodge to de Rham spectral sequence, and torsion-freeness of Crystalline cohomologies. In this paper, we establish these results for the case when $|-K_X|$ is very ample and the Picard group is generated by $ω_X$.

2404.02116 2026-06-19 math.FA 版本更新 70%

The lattice structure of negative Sobolev and extrapolation spaces

负Sobolev空间与外推空间的格结构

Sahiba Arora, Jochen Glück, Felix L. Schwenninger

专题命中 其他科学智能 :研究Sobolev空间格结构,属于泛函分析

AI总结 本文研究负指数Sobolev空间$W^{k,p}(\mathbb R^d)$中正锥的生成空间是向量格,并证明Banach格上正$C_0$-半群的外推空间中正锥生成空间也是向量格。

Comments 16 pages. This is version 4, contains minor corrections

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AI中文摘要

众所周知,若$k \in \{0,1\}$,则Sobolev空间$W^{k,p}(\mathbb R^d)$关于逐点几乎处处序是向量格,但若$k \ge 2$则不是。在本文中,我们考虑负$k$,并证明在这种情况下$W^{k,p}(\mathbb R^d)$中正锥的生成空间是向量格。我们还证明了一个相关的抽象结果:若$(T(t))_{t \in [0,\infty)}$是Banach格$X$上具有序连续范数的正$C_0$-半群,则外推空间$X_{-1}$中锥$X_{-1,+}$的生成空间是向量格。这补充了Bátkai、Jacob、Wintermayr和Voigt在扰动理论背景下得到的结果,并为无穷维正系统理论提供了额外背景。

英文摘要

It is well-known that the Sobolev spaces $W^{k,p}(\mathbb R^d)$ are vector lattices with respect to the pointwise almost everywhere order if $k \in \{0,1\}$, but not if $k \ge 2$. In this note, we consider negative $k$ and show that the span of the positive cone in $W^{k,p}(\mathbb R^d)$ is a vector lattice in this case. We also prove a related abstract result: if $(T(t))_{t \in [0,\infty)}$ is a positive $C_0$-semigroup on a Banach lattice $X$ with order continuous norm, then the span of the cone $X_{-1,+}$ in the extrapolation space $X_{-1}$ is a vector lattice. This complements results obtained by Bátkai, Jacob, Wintermayr, and Voigt in the context of perturbation theory and provides additional context for the theory of infinite-dimensional positive systems.

2503.01647 2026-06-19 math.CO math.AC 版本更新 70%

Volume Rigidity of Simplicial Manifolds

单纯流形的体积刚性

James Cruickshank, Bill Jackson, Shin-ichi Tanigawa

专题命中 其他科学智能 :证明单纯流形骨架的体积刚性,属于组合几何

AI总结 本文证明对于所有d≥4和1≤k≤d-3,单纯(d-1)-流形的k-骨架在R^d中的一般实现是体积刚性的,并猜想k=d-2时结论成立,验证了d=4,5,6的情况。

Comments 21 pages. Updated to match version published in Combinatorica DOI: https://doi-org.nuigalway.idm.oclc.org/10.1007/s00493-026-00218-x

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AI中文摘要

Cauchy和Dehn的经典结果意味着凸单纯多面体$P$的1-骨架是刚性的,即$P$的顶点在$\mathbb R^3$中保持边长不变的连续运动产生的多面体与$P$全等。Whiteley将此结果推广到$\mathbb R^d$($d\geq 3$)中的凸单纯多胞体,Kalai($d\geq 4$)和Fogelsanger($d\geq 3$)将其推广到单纯$(d-1)$-流形的1-骨架在$\mathbb R^{d}$中的一般实现。我们将推广Kalai的结果,证明对于所有$d\geq 4$和任意固定的$1\leq k\leq d-3$,单纯$(d-1)$-流形的$k$-骨架在$\mathbb R^{d}$中的每个一般实现都是体积刚性的,即其顶点在$\mathbb R^d$中保持$k$-面体积不变的连续运动产生的实现与原实现全等。此外,我们猜想该结果对$k=d-2$也成立,并验证了$d=4,5,6$时的情况。

英文摘要

Classical results of Cauchy and Dehn imply that the 1-skeleton of a convex simplicial polyhedron $P$ is rigid i.e. every continuous motion of the vertices of $P$ in $\mathbb R^3$ which preserves its edge lengths results in a polyhedron which is congruent to $P$. This result was extended to convex smplicial polytopes in $\mathbb R^d$ for all $d\geq 3$ by Whiteley, and to generic realisations of 1-skeletons of simplicial $(d-1)$-manifolds in $\mathbb R^{d}$ by Kalai for $d\geq 4$ and Fogelsanger for $d\geq 3$. We will generalise Kalai's result by showing that, for all $d\geq 4$ and any fixed $1\leq k\leq d-3$, every generic realisation of the $k$-skeleton of a simplicial $(d-1)$-manifold in $\mathbb R^{d}$ is volume rigid, i.e. every continuous motion of its vertices in $\mathbb R^d$ which preserves the volumes of its $k$-faces results in a congruent realisation. In addition, we conjecture that our result remains true for $k=d-2$ and verify this conjecture when $d=4,5,6$.

1408.5923 2026-06-19 math.HO 70%

Symmetric Matrices: Theory and Applications

对称矩阵:理论与应用

Helmut Kahl

专题命中 其他科学智能 :综述对称矩阵理论与应用,属于数学教育

AI总结 本文综述对称矩阵的理论与应用,为大学课程提供教学模块。

Comments 70 pages, 2 figures; ex. 39b) inserted; figure of title page transferred into corresponding chapter

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AI中文摘要

本文是对称矩阵的综述文本,旨在作为大学课程的教学模块脚本。

英文摘要

This text is a survey on symmetric matrices. It serves as a script for a module to be taught at university.

2502.06866 2026-06-19 cs.LG cs.AI econ.EM stat.AP stat.ML 版本更新 70%

Global Ease of Living Index: a machine learning framework for longitudinal analysis of major economies

全球生活便利指数:面向主要经济体纵向分析的机器学习框架

Arun Kumar Selvaraj, Tanay Panat, Rohitash Chandra

发表机构 * Transitional Artificial Intelligence Research Group, School of Mathematics and Statistics(过渡人工智能研究组,数学与统计学学院) Centre for Artificial Intelligence and Innovation(人工智能与创新中心) Pingla Institute(Pingla研究所)

专题命中 其他科学智能 :机器学习框架构建生活便利指数

AI总结 提出全球生活便利指数,结合社会经济和基础设施因素,利用机器学习处理缺失数据,并通过主成分分析和因子分析降维,为政策制定者提供改善生活质量的可操作工具。

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AI中文摘要

全球经济、地缘政治条件以及COVID-19疫情等破坏性事件对生活成本和生活质量产生了巨大影响。理解主要经济体中生活成本和生活质量的长期影响至关重要。一个透明且全面的生活指数必须包含生活条件的多个维度。在本研究中,我们提出了一种通过全球生活便利指数量化生活质量的方法,该指数将各种社会经济和基础设施因素整合为一个单一综合得分。我们的指数利用定义生活水平的经济指标,这有助于针对特定领域进行干预改进。我们提出了一个机器学习框架来处理特定国家某些经济指标的数据缺失问题。然后,我们整理并更新数据,并使用降维方法(主成分分析和因子分析)创建自1970年以来主要经济体的生活便利指数。我们的工作通过为政策制定者提供识别需要改进领域(如医疗系统、就业机会和公共安全)的实用工具,显著丰富了相关文献。我们的方法使用开放数据和代码,易于复现并适用于各种情境,为生活质量评估的持续研究和政策制定提供了透明度和可访问性。

英文摘要

The drastic changes in the global economy, geopolitical conditions, and disruptions such as the COVID-19 pandemic have impacted the cost of living and quality of life. It is essential to comprehend the long-term implications of the cost of living and quality of life in major economies. A transparent and comprehensive living index must include multiple dimensions of living conditions. In this study, we present an approach to quantifying the quality of life through the Global Ease of Living Index that combines various socio-economic and infrastructural factors into a single composite score. Our index utilises economic indicators that define living standards, which could help in targeted interventions to improve specific areas. We present a machine learning framework to address missing data for certain economic indicators in specific countries. We then curate and update the data and use a dimensionality reduction approach (Principal Component Analysis and Factor Analysis) to create the Ease of Living Index for major economies since 1970. Our work significantly adds to the literature by offering a practical tool for policymakers to identify areas needing improvement, such as healthcare systems, employment opportunities, and public safety. Our approach with open data and code can be easily reproduced and applied to various contexts, providing transparency and accessibility for ongoing research and policy development in quality-of-life assessment.

2501.08796 2026-06-19 math.CO 版本更新 70%

The Jacobian of a regular orthogonal matroid and torsor structures on spanning quasi-trees of ribbon graphs

正则正交拟阵的雅可比矩阵及带状图中生成拟树的托尔结构

Matthew Baker, Changxin Ding, Donggyu Kim

专题命中 其他科学智能 :研究拟阵雅可比群,属于组合数学

AI总结 将平面图中生成树构成雅可比群托尔的结构推广到任意亏格可定向曲面上的图(带状图),用生成拟树替代生成树,用正则正交拟阵的雅可比群替代原雅可比群。

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AI中文摘要

Chan--Church--Grochow 和 Baker--Wang 的前期工作表明,平面图 $G$ 中生成树的集合自然构成 $G$ 的雅可比群的一个托尔。非正式地说,这意味着 $G$ 的生成树集合自然形成一个群,只是没有指定的单位元。我们将这一事实推广到嵌入在任意亏格可定向曲面上的图,这些图可以等同于带状图。在此推广中,$G$ 的生成树集合被替换为带状图的生成拟树集合,而 $G$ 的雅可比群被替换为关联的正则正交拟阵 $M$(连同 $M$ 的一个关联正则表示)的雅可比群。我们的证明更一般地表明,由 Backman--Baker--Yuen 构造并后来由 Ding 推广的“BBY 托尔”族在正则正交拟阵(的正则表示)上具有自然的推广。除了揭示平面性在上述早期工作中的作用外,我们的结果还代表了正交拟阵(也称为“偶 delta-拟阵”或“拉格朗日正交拟阵”)在图的自然组合问题上的首批重要应用之一。

英文摘要

Previous work of Chan--Church--Grochow and Baker--Wang shows that the set of spanning trees in a plane graph $G$ is naturally a torsor for the Jacobian group of $G$. Informally, this means that the set of spanning trees of $G$ naturally forms a group, except that there is no distinguished identity element. We generalize this fact to graphs embedded on orientable surfaces of arbitrary genus, which can be identified with ribbon graphs. In this generalization, the set of spanning trees of $G$ is replaced by the set of spanning quasi-trees of the ribbon graph, and the Jacobian group of $G$ is replaced by the Jacobian group of the associated regular orthogonal matroid $M$ (along with an associated regular representation of $M$). Our proof shows, more generally, that the family of "BBY torsors" constructed by Backman--Baker--Yuen and later generalized by Ding admit natural generalizations to (regular representations of) regular orthogonal matroids. In addition to shedding light on the role of planarity in the earlier work mentioned above, our results represent one of the first substantial applications of orthogonal matroids (also called "even delta-matroids" or "Lagrangian orthogonal matroids") to a natural combinatorial problem about graphs.

2502.09968 2026-06-19 math.CO 70%

Minimum maximal matchings in permutahedra

排列体中的最小最大匹配

Sofia Brenner, Jiří Fink, Hung. P. Hoang, Arturo Merino, Vincent Pilaud

专题命中 其他科学智能 :研究排列体中最大匹配,属于图论

AI总结 研究排列体中最大匹配的最小规模,通过分析4-循环和Hall定理得出渐近下界和上界,构造出具体上界,并推导了排列体乘积中的最小最大匹配界。

Comments 11 pages, 2 figures

Journal ref Electron. J. Combin., vol. 33(2), #P2.50, 15 pp., 2026

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AI中文摘要

我们证明了排列体π_n中的最大匹配的最小规模M(π_n)渐近为n!/3。一方面,通过考虑排列体中的4-循环,得到下界M(π_n) ≥ n!(n-1)/(3n-2)。另一方面,通过多次应用Hall定理(类似于Forcade (1973)对超立方体的方法)和显式构造,得到渐近上界M(π_n) ≤ n!(1/3 + o(1))以及精确上界M(π_n) ≤ n!/3。我们还推导了排列体乘积中最小最大匹配的界。

英文摘要

We prove that the minimal size $M(π_n)$ of a maximal matching in the permutahedron $π_n$ is asymptotically $n!/3$. On the one hand, we obtain a lower bound $M(π_n) \ge n! (n-1) / (3n-2)$ by considering $4$-cycles in the permutahedron. On the other hand, we obtain an asymptotical upper bound $M(π_n) \le n!(1/3+o(1))$ by multiple applications of Hall's theorem (similar to the approach of Forcade (1973) for the hypercube) and an exact upper bound $M(π_n) \le n!/3$ by an explicit construction. We also derive bounds on minimum maximal matchings in products of permutahedra.