Advances in Mathematics最新文献

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Lax functorialities of the comma construction for ω-categories ω-范畴的逗号结构的松弛功能

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-03-01 Epub Date: 2026-01-15 DOI: 10.1016/j.aim.2025.110762

Dimitri Ara , Léonard Guetta

Motivated by the Grothendieck construction, we study the functorialities of the comma construction for strict ω-categories. To state the most general functorialities, we use the language of Gray ω-categories, that is, categories enriched in the category of strict ω-categories endowed with the oplax Gray tensor product. Our main result is that the comma construction of strict ω-categories defines a Gray ω-functor, that is, a morphism of Gray ω-categories. To makes sense of this statement, we prove that slices of Gray ω-categories exist. Coming back to the Grothendieck construction, we propose a definition in terms of the comma construction and, as a consequence, we get that the Grothendieck construction of strict ω-categories defines a Gray ω-functor. Finally, as a by-product, we get a notion of Grothendieck construction for Gray ω-functors, which we plan to investigate in future work.

在Grothendieck结构的启发下，我们研究了严格ω-范畴的逗号结构的泛函性。为了描述最一般的功能，我们使用Gray ω-范畴的语言，即在赋予oplax Gray张量积的严格ω-范畴中丰富的范畴。我们的主要结果是严格ω-范畴的逗号构造定义了一个Gray ω-函子，即Gray ω-范畴的态射。为了使这句话有意义，我们证明了Gray ω-范畴片的存在。回到格罗滕狄克构造，我们提出了一个关于逗号构造的定义，作为结果，我们得到严格ω-范畴的格罗滕狄克构造定义了一个Gray ω-函子。最后，作为一个副产品，我们得到了Gray ω-函子的Grothendieck构造的概念，我们计划在未来的工作中对此进行研究。

引用次数: 0

On the refined analyticity radius of 3-D generalized Navier-Stokes equations 三维广义Navier-Stokes方程的精细解析半径

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-03-01 Epub Date: 2026-01-07 DOI: 10.1016/j.aim.2025.110769

Dong Li , Ping Zhang

<div><div>We analyze the instantaneous growth of analyticity radius for three dimensional generalized Navier-Stokes equations. For the subcritical <math><msup><mrow><mi>H</mi></mrow><mrow><mi>γ</mi></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></math> case with <math><mi>γ</mi><mo>></mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math>, we prove that there exists a positive time <math><msub><mrow><mi>t</mi></mrow><mrow><mn>0</mn></mrow></msub></math> so that for any <math><mi>t</mi><mo>∈</mo><mo>]</mo><mn>0</mn><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>]</mo></math>, the radius of analyticity of the solution u satisfies the pointwise-in-time lower bound<math><mrow><mi>rad</mi></mrow><mo>(</mo><mi>u</mi><mo>)</mo><mo>(</mo><mi>t</mi><mo>)</mo><mo>≥</mo><msqrt><mrow><mo>(</mo><mn>2</mn><mi>γ</mi><mo>−</mo><mn>1</mn><mo>)</mo><mi>t</mi><mo>(</mo><mo>|</mo><mi>ln</mi><mo>⁡</mo><mi>t</mi><mo>|</mo><mo>+</mo><mi>ln</mi><mo>⁡</mo><mo>|</mo><mi>ln</mi><mo>⁡</mo><mi>t</mi><mo>|</mo><mo>+</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>)</mo></mrow></msqrt><mo>,</mo></math> where <math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>→</mo><mo>∞</mo></math> as <math><mi>t</mi><mo>→</mo><msup><mrow><mn>0</mn></mrow><mrow><mo>+</mo></mrow></msup></math>. This in particular gives a nontrivial improvement of the previous result by Herbst and Skibsted in [17] for the case <math><mi>γ</mi><mo>∈</mo><mo>]</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>/</mo><mn>2</mn><mo>[</mo></math> and also settles the decade-long open question in [17], namely, whether or not<math><munder><mrow><mrow><mi>lim</mi></mrow><mspace></mspace><mrow><mi>inf</mi></mrow></mrow><mrow><mi>t</mi><mo>→</mo><msup><mrow><mn>0</mn></mrow><mrow><mo>+</mo></mrow></msup></mrow></munder><mspace></mspace><mfrac><mrow><mrow><mi>rad</mi></mrow><mo>(</mo><mi>u</mi><mo>)</mo><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mrow><msqrt><mrow><mi>t</mi><mo>|</mo><mi>ln</mi><mo>⁡</mo><mi>t</mi><mo>|</mo></mrow></msqrt></mrow></mfrac><mo>≥</mo><msqrt><mrow><mn>2</mn><mi>γ</mi><mo>−</mo><mn>1</mn></mrow></msqrt></math> for all <math><mi>γ</mi><mo>≥</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math>. In the critical case <math><msup><mrow><mi>H</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></math> we prove that there exists <math><msub><mrow><mi>t</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>></mo><mn>0</mn></math> so that for any <math><mi>t</mi><mo>∈</mo><mo>

本文分析了三维广义Navier-Stokes方程解析半径的瞬时增长。对于含有γ>；12的亚临界Hγ(R3)情形，我们证明了存在一个正时间t0，使得对于任意t∈]0，t0]，解u的可解析性半径满足点向时间下界(u)(t)≥(2γ−1)t(|ln (t) |+ln (|) ln (t) |+Kt)，其中Kt→∞= t→0+。这特别给出了[17]中Herbst和Skibsted对γ∈]1/2,3/2[的非一般改进，并解决了[17]中十年来悬而未决的问题，即对于所有γ≥32，是否liminft→0+rad(u)(t)t|ln (t |)≥2γ−1。在临界情况H12（R3）下证明了存在t1>；0，使得对于任意t∈]0，t1], rad(u)(t)≥λ(t)t且λ(t)满足极限→0+ λ(t)=∞。

引用次数: 0

Stability of the favorable Falkner-Skan profiles for the stationary Prandtl equations 平稳Prandtl方程有利的Falkner-Skan型的稳定性

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-03-01 Epub Date: 2025-12-30 DOI: 10.1016/j.aim.2025.110749

Sameer Iyer

The (favorable) Falkner-Skan boundary layer profiles are a one parameter (

β \in [0, 2]

) family of self-similar solutions to the stationary Prandtl system which describes the flow over a wedge with angle

β \frac{π}{2}

. The most famous member of this family is the endpoint Blasius profile,

β = 0

, which exhibits pressureless flow over a flat plate. In contrast, the

β > 0

profiles are physically expected to exhibit a favorable pressure gradient, a common adage in the physics literature. In this work, we prove quantitative scattering estimates as

x \to \infty

which precisely captures the effect of this favorable gradient through the presence of new “CK” (Cauchy-Kovalevskaya) terms that appear in a quasilinear energy cascade.

（有利的）Falkner-Skan边界层轮廓是描述角为βπ2的楔形流动的平稳Prandtl系统的单参数（β∈[0,2]）自相似解族。这个家族中最著名的成员是终点Blasius剖面，β=0，它显示了平板上的无压流动。相反，β>；0剖面在物理上被期望表现出有利的压力梯度，这是物理文献中常见的格言。在这项工作中，我们证明了定量散射估计为x→∞，它通过出现在拟线性能量级联中的新“CK”（Cauchy-Kovalevskaya）项的存在精确地捕获了这种有利梯度的影响。

引用次数: 0

The defect of the F-pure threshold F-pure阈值的缺陷

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-03-01 Epub Date: 2026-01-20 DOI: 10.1016/j.aim.2026.110792

Alessandro De Stefani , Luis Núñez-Betancourt , Ilya Smirnov

Introduced by Takagi and Watanabe, F-pure thresholds are invariants defined in terms of the Frobenius homomorphism. While they find applications in various settings, they are primarily used as a local invariant. The purpose of this note is to start filling this gap by opening the study of its behavior on a scheme. To this end, we define the defect of the F-pure threshold of a local ring

(R, m)

by setting

dfpt (R) = \dim (R) - fpt (m)

. It turns out that this invariant defines an upper semi-continuous function on a scheme and satisfies Bertini-type theorems. We also study the behavior of the defect of the F-pure threshold under flat extensions and after blowing up the maximal ideal of a local ring.

由Takagi和Watanabe引入，f纯阈值是根据Frobenius同态定义的不变量。虽然它们在各种设置中都有应用，但它们主要用作局部不变量。本文的目的是通过打开其在方案上的行为研究来填补这一空白。为此，我们通过设dfpt(R)=dim (R)−fpt(m)来定义局部环（R,m）的f -纯阈值缺陷。结果表明，这个不变量定义了一个上半连续函数在一个方案上，并且满足bertini型定理。我们还研究了f -纯阈值在平面扩展和局部环极大理想爆破后的缺陷行为。

引用次数: 0

A relative Nash-Tognoli theorem over Q and application to the Q-algebraicity problem Q上的一个相对纳什-托格诺里定理及其在Q-代数问题中的应用

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-03-01 Epub Date: 2025-12-31 DOI: 10.1016/j.aim.2025.110757

Enrico Savi

We prove a relative version over

Q

of Nash-Tognoli theorem, that is: Let M be a compact smooth manifold with closed smooth submanifolds

M_{1}, \dots, M_{ℓ}

in general position, then there exists a nonsingular real algebraic set

M^{'} \subset R^{n}

with nonsingular algebraic subsets

M_{1}^{'}, \dots, M_{ℓ}^{'}

and a diffeomorphism

h : M \to M^{'}

such that

h (M_{i}) = M_{i}^{'}

for all

i = 1, \dots, ℓ

such that

M^{'}, M_{1}^{'}, \dots, M_{ℓ}^{'}

are described, both globally and locally, by polynomial equations with rational coefficients. In addition, if

M, M_{1}, \dots, M_{ℓ}

are nonsingular algebraic sets, then we prove the diffeomorphism

h : M \to M^{'}

can be chosen semialgebraic and the result can be extended to the noncompact case. In the proof we describe also the

Z / 2 Z

-homological cycles of real embedded Grassmannian manifolds by nonsingular algebraic representatives over

Q

via the Bott-Samelson resolution of Schubert varieties.

我们证明一个相对版本在问Nash-Tognoli定理,即:让M是一个紧凑的光滑流形与封闭光滑流形中的M1,…,Mℓ一般位置,那么存在一组非奇异的真正的代数M与非奇异的代数子集⊂Rn M1,…,Mℓ“和微分同胚映射h: M→M”这样h (Mi) = Mi的所有我= 1,…,ℓ这样M’,M1,…,Mℓ描述,全球和本地与有理系数多项式方程。此外，如果M，M1，…，M，M是非奇异代数集，则证明了微分同态h:M→M '可以是半代数的，并将结果推广到非紧情况。在证明中，我们还通过Schubert变分的bot - samelson解析，描述了Q上由非奇异代数表示的实嵌入Grassmannian流形的Z/ 2z同调循环。

引用次数: 0

Vertex algebras related to regular representations of SL2 与SL2正则表示相关的顶点代数

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-03-01 Epub Date: 2026-01-08 DOI: 10.1016/j.aim.2025.110763

Dražen Adamović , Antun Milas

We construct a family of potentially quasi-lisse (non-rational) vertex algebras, denoted by

C_{p}

p \geq 2

, which are closely related to the vertex algebra of chiral differential operators on

S L (2)

at level

- 2 + \frac{1}{p}

. We prove that for

p = 3

, there is an isomorphism between

C_{3}

and the affine vertex algebra

L_{- 5 / 3} (g_{2})

from the Cvitanović-Deligne series. Moreover, we also establish isomorphisms between

C_{4}

and

C_{5}

and certain affine W-algebras of types

F_{4}

and

E_{8}

, respectively. In this way, we resolve the problem of decomposing certain conformal embeddings of affine vertex algebras into affine W-algebras. An important feature is that

C_{p}

\frac{1}{2} Z_{\geq 0}

-graded with finite-dimensional graded subspaces and convergent characters. Therefore, for all

p \geq 2

, we show that the characters of

C_{p}

exhibit modularity, supporting the conjectural quasi-lisse property.

构造了一类潜在的拟无理数顶点代数，记为Cp， p≥2，它们与- 2+1p水平上SL(2)上的手性微分算子的顶点代数密切相关。证明了当p=3时，C3与Cvitanović-Deligne级数中的仿射顶点代数L−5/3（g2）之间存在同构。此外，我们还分别建立了C4和C5与F4和E8型仿射w代数之间的同构关系。这样，我们解决了若干仿射顶点代数的共形嵌入分解为仿射w代数的问题。一个重要的特征是Cp是12Z≥0级的，具有有限维的分级子空间和收敛特征。因此，对于所有p≥2，我们证明了Cp的性质呈现模性，支持了推测的拟lisse性质。

引用次数: 0

Topological Hochschild homology of the image of J 象J的拓扑Hochschild同调

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-03-01 Epub Date: 2026-01-02 DOI: 10.1016/j.aim.2025.110759

David Jongwon Lee , Ishan Levy

We compute the mod

(p, v_{1})

and mod

(2, η, v_{1})

THH of many variants of the image-of-J spectrum. In particular, we do this for

j_{ζ}

, whose TC is closely related to the K-theory of the

K (1)

-local sphere. We find in particular that the failure for THH to satisfy

Z_{p}

-Galois descent for the extension

j_{ζ} \to ℓ_{p}

corresponds to the failure of the p-adic circle to be its own free loop space. For

p > 2

, we also prove the Segal conjecture for

j_{ζ}

, and we compute the K-theory of the

K (1)

-local sphere in degrees

\leq 4 p - 6

我们计算了j光谱像的许多变体的mod （p,v1）和mod (2,η,v1) THH。特别地，我们对jζ这样做，它的TC与K(1)局部球的K理论密切相关。特别地，我们发现THH不能满足扩展jζ→∑p的zp -伽罗瓦下降，对应于p进圆不能成为它自己的自由环空间。对于p>；2，我们也证明了jζ的Segal猜想，并计算了K(1)-局部球在度≤4p−6的K-理论。

引用次数: 0

Corrigendum to “Notes on Plücker's relations in geometric algebra” [Advances in Mathematics 363 (2020) 106959] “几何代数中plencker关系注释”的勘误表[数学进展363 (2020)106959]

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-03-01 Epub Date: 2026-01-21 DOI: 10.1016/j.aim.2026.110801

Garret Sobczyk

引用次数: 0

Counting rationals and diophantine approximation in missing-digit Cantor sets 缺失数康托集的计数有理数和丢番图近似

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-03-01 Epub Date: 2026-01-22 DOI: 10.1016/j.aim.2026.110807

Sam Chow , Péter P. Varjú , Han Yu

We establish a new upper bound for the number of rationals up to a given height in a missing-digit set, making progress towards a conjecture of Broderick, Fishman, and Reich. This enables us to make novel progress towards another conjecture of those authors about the corresponding intrinsic diophantine approximation problem. Moreover, we make further progress towards conjectures of Bugeaud–Durand and Levesley–Salp–Velani on the distribution of diophantine exponents in missing-digit sets.

A key tool in our study is Fourier

ℓ^{1}

dimension introduced by the last named author in Yu (2021) [12]. An important technical contribution of the paper is a method to compute this quantity.

我们建立了缺失数集合中到给定高度的有理数的新上界，对Broderick、Fishman和Reich的猜想有了进一步的进展。这使我们能够在这些作者关于相应的本征丢番图近似问题的另一个猜想方面取得新的进展。此外，我们对Bugeaud-Durand和Levesley-Salp-Velani关于丢芬图指数在缺位数集合中的分布的猜想取得了进一步的进展。我们研究中的一个关键工具是傅里叶1维，由最后一位作者在Yu(2021)[12]中引入。本文的一个重要的技术贡献是计算这个量的方法。

引用次数: 0

Solution to Lawvere's first problem: A Grothendieck topos that has proper class many quotient topoi Lawvere第一个问题的解：一类具有适当类多商拓扑的Grothendieck拓扑

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics

Pub Date : 2026-03-01 Epub Date: 2025-12-30 DOI: 10.1016/j.aim.2025.110751

Yuhi Kamio, Ryuya Hora

This paper solves the first of the open problems in topos theory posted by William Lawvere, concerning the existence of a Grothendieck topos that has proper class many quotient topoi. This paper concretely constructs such Grothendieck topoi, including the presheaf topos on the free monoid generated by countably infinitely many elements

PSh (M_{ω})

. Utilizing the combinatorics of the classifying topos of the theory of inhabited objects and with the help of a system of pairing functions, the problem is reduced to a theorem of Vopěnka, Pultr, and Hedrlín, which states that any set admits a rigid relational structure.

本文解决了William Lawvere在拓扑理论中提出的第一个开放问题，即具有适当类多商拓扑的Grothendieck拓扑的存在性问题。本文具体构造了这样的Grothendieck拓扑，包括由可数无穷多元PSh（Mω）生成的自由模阵上的preshef拓扑。利用居住对象理论分类拓扑的组合学，在配对函数系统的帮助下，该问题被简化为vopovinka， Pultr和Hedrlín的定理，该定理指出任何集合都承认刚性关系结构。

引用次数: 0

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