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The main goal of this article is to connect some recent perspectives in the study of 4 4 -manifolds from the vantage point of singularity theory. We present explicit algorithms for simplifying the topology of various maps on 4 4 -manifolds, which include broken Lefschetz fibrations and indefinite Morse 2 2 -functions. The algorithms consist of sequences of moves, which modify indefinite fibrations in smooth 1 1 -parameter families. These algorithms allow us to give purely topological and constructive proofs of the existence of simplified broken Lefschetz fibrations and Morse 2 2 -functions on general 4 4 -manifolds, and a theorem of Auroux–Donaldson–Katzarkov on the existence of certain broken Lefschetz pencils on near-symplectic 4 4 -manifolds. We moreover establish a correspondence between broken Lefschetz fibrations and Gay–Kirby trisections of 4 4 -manifolds, and show the existence and stable uniqueness of simplified trisections on all 4 4 -manifolds. Building on this correspondence, we also provide several new constructions of trisections, including infinite families of genus- 3 3 trisections with homotopy inequivalent total spaces, and exotic same genera trisections of 4 4 -manifolds in the homeomorphism classes of complex rational surfaces.

We show that a A-linear map of Hilbert A-modules is induced by a unitary Hilbert module operator if and only if it extends to an ordinary unitary on appropriately defined enveloping Hilbert spaces. Applications to the theory of multiplicative unitaries let us to compute the equivalence classes of Hilbert modules over a class of C*-algebraic quantum groups. We, thus, develop a theory that, for example, could be used to show non-existence of certain co-actions. In particular, we show that the Cuntz semigroup functor takes a co-action to a multiplicative action.

The aim of this paper is to study, by using the mathematical tools developed by Chałupnik, Touzé, and Van der Kallen, the effect of the Frobenius twist on $\operatorname{Ext}$-group in the category of strict polynomial functors. As an application, we obtain explicit formulas of cohomology of the orthogonal groups and symplectic ones.

Let $\mathcal{M}$ be a $\sigma$-finite von Neumann algebra, equipped with a normal faithful state $\varphi$, and let $\mathcal{A}$ be maximal subdiagonal algebra of $\mathcal{M}$. We prove Stein-Weiss type interpolation theorem of Haagerup noncommutative $H^{p}$-spaces associated with $\A$., Comment: 16. arXiv admin note: text overlap with arXiv:2209.08542

The entropy power inequality (EPI) and the Brascamp-Lieb inequality (BLI) are fundamental inequalities concerning the differential entropies of linear transformations of random vectors. The EPI provides lower bounds for the differential entropy of linear transformations of random vectors with independent components. The BLI, on the other hand, provides upper bounds on the differential entropy of a random vector in terms of the differential entropies of some of its linear transformations. In this paper, we define a family of entropy functionals, which we show are subadditive. We then establish that Gaussians are extremal for these functionals by mimicking the idea in Geng and Nair (2014). As a consequence, we obtain a new entropy inequality that generalizes both the BLI and EPI. By considering a variety of independence relations among the components of the random vectors appearing in these functionals, we also obtain families of inequalities that lie between the EPI and the BLI., Comment: 38 pages, 1 figure. Submitted to the IEEE Transactions on Information Theory for possible publication

We determine the 2-fusion systems of J-component type in which the centralizer of some fully centralized involution has a maximal J-component that is the 2-fusion system of U 3 ( 3 ) .

We introduce an algorithmic framework to investigate spherical and geodesic growth series of braid groups relatively to the Artin's or Birman–Ko–Lee's generators. We present our experimentations in the case of three and four strands and conjecture rational expressions for the spherical growth series with respect to the Birman–Ko–Lee's generators.

We study a quotient of the group algebra of the braid group in which the Artin generators satisfy a cubic relation. This quotient is maximal among the ones satisfying such a cubic relation. We also investigate the proper quotients of it that appear in the realm of quantum groups, and describe another maximal quotient related to the usual Hecke algebras. Finally, we describe the connection between this algebra and a quotient of the algebra of horizontal chord diagrams introduced by Vogel. We prove that these two are isomorphic for n ≤ 5 .

We consider Artin-Tits groups of FC type and prove the two following results. If any two distinct elements of a standard parabolic subgroup are conjugated by a standard generator of the whole group, then this generator has to be in the subgroup. We also prove that classical transversals of standard parabolic subgroups of Artin-Tits groups of FC type are compatible with intersection with standard parabolic subgroups: If A X , A Y are two standard parabolic subgroups of an Artin-Tits groups A of FC type and T is the classical transversal of A X in A, then T ∩ A Y is a transversal of A X ∩ Y in A Y . So the two properties hold for Artin-Tits groups of spherical type and Right-angled Artin-Tits groups.

Motivated in part by representation theoretic questions, we prove that if G is a finite quasi-simple group, then there exists an elementary abelian subgroup of G that contains a member of each conjugacy class of involutions of G.

In this paper we introduce and study some geometric objects associated to Artin monoids. The Deligne complex for an Artin group is a cube complex that was introduced by the second author and Davis (1995) to study the K(\pi,1) conjecture for these groups. Using a notion of Artin monoid cosets, we construct a version of the Deligne complex for Artin monoids. We show that for any Artin monoid this cube complex is contractible. Furthermore, we study the embedding of the monoid Deligne complex into the Deligne complex for the corresponding Artin group. We show that for any Artin group this is a locally isometric embedding. In the case of FC-type Artin groups this result can be strengthened to a globally isometric embedding, and it follows that the monoid Deligne complex is CAT(0) and its image in the Deligne complex is convex. We also consider the Cayley graph of an Artin group, and investigate properties of the subgraph spanned by elements of the Artin monoid. Our final results show that for a finite type Artin group, the monoid Cayley graph embeds isometrically, but not quasi-convexly, into the group Cayley graph., Comment: 21 pages

Recently, a symmetric differential operator (SDO) is attracted to studying in the field of mathematical analysis. A new formal of SDO is presented to generalize some well known differential operators in a complex domain. According to this formulation, we shall exam the boundedness and compactness of this operator in complex spaces, such as Hilbert space and Sobolev space. For this purpose, we suggest new norms to solve the fractional Beltrami equation in the open unit disk. This operator has ability to depict the analytic geometric representation of the solution of second order differential equation utilizing the concept of Schwarzian derivative in the open unit disk. Applications in imagings are given the sequel.

There are two definitions of Einstein-Finsler spaces introduced by Akbar-Zadeh, which we will show is equal along the integral curves of $I$-invariant projective vector fields. The sub-algebra of the $C$-projective vector fields, leaving the $H$-curvature invariant, has been studied extensively. Here we show on a closed Finsler space with negative definite Ricci curvature reduces to that of Killing vector fields. Moreover, if an Einstein-Finsler space admits such a projective vector field then the flag curvature is constant. Finally, a classification of compact isotropic mean Landsberg manifolds admitting certain projective vector fields is obtained with respect to the sign of Ricci curvature., 25 pages

Let $k$ be a perfect field of positive characteristic and $Z$ an effective Cartier divisor in the projective line over $k$ with complement $U$. In this note, we establish some results about the formal deformation theory of overconvergent isocrystals on $U$ with fixed "local monodromy" along $Z$. En route, we show that a Hochschild cochain complex governs deformations of a module over an arbitrary associative algebra. We also relate this Hochschild cochain complex to a de Rham complex in order to understand the deformation theory of a differential module over a differential ring., 59 pages; fixed typos, improved exposition; comments welcome!

Let p = ( P ) be any prime of F q [ t ] , let m be any ideal of F q [ t ] not divisible by p and consider the space of Drinfeld cusp forms of level m p , i.e. for the modular group Γ 0 ( m p ) . Using degeneracy maps, traces and Fricke involutions we offer definitions for p -oldforms and p -newforms which turn out to be subspaces stable with respect to the action of the Atkin operator U P . We provide eigenvalues and/or slopes for p -oldforms and p -newforms and a condition to get the whole space of cusp forms as the direct sum between them.

This article is a continuation of our research on a one-to-one correspondence between n-dimensional spectral resolutions and n-dimensional observables on lexicographic types of quantum structures which started in Dvurecenskij and Lachman ( https://doi.org/10.1016/j.fss.2021.05.005 ). There we presented the main properties of n-dimensional spectral resolutions and observables, and we studied in depth characteristic points which are crucial for our study. Here we present the main body of our research. We investigate a one-to-one correspondence between n-dimensional observables and n-dimensional spectral resolutions with values in a lexicographic form of quantum structures such as perfect MV-algebras or perfect effect algebras. The multidimensional version of this problem is more complicated than a one-dimensional one because if our algebraic structure is k-perfect for k > 1 , then even for the two-dimensional case of spectral resolutions we have more characteristic points. The results obtained are applied to the existence of an n-dimensional meet joint observable of n one-dimensional observables on a perfect MV-algebra and a sum of n-dimensional observables.

In this paper, we prove that the adelic Galois representation ρ φ : Gal ( F q ( T ) sep / F q ( T ) ) ⟶ lim ← a Aut ( φ [ a ] ) ≅ G L 3 ( A ˆ ) associated to the Drinfeld module φ over F q ( T ) of rank 3, φ defined by φ T = T + τ 2 + T q − 1 τ 3 , is surjective.

We show that given an ideal I generated by regular functions f_1,...,f_r on the smooth complex variety X, the Bernstein-Sato polynomial of I is equal to the reduced Bernstein-Sato polynomial of the function g=\sum_{i=1}^rf_iy_i on the product of X with an r-dimensional affine space. By combining this with results from [BMS], we relate invariants and properties of I to those of g. We also use the result on Bernstein-Sato polynomials to show that the Strong Monodromy Conjecture for Igusa zeta functions of principal ideals implies a similar statement for arbitrary ideals., Comment: 7 pages; v.2: inaccuracy at the end of the proof of Theorem 1.4 corrected

We study the subconvexity problem for GL 3 ( R ) L-functions in the t-aspect using integral representations by combining techniques employed by Michel–Venkatesh in their study of the corresponding problem for GL 2 with ideas from recent works of Munshi, Holowinsky–Nelson and Lin. Our main objective is to give – from the perspective of integral representations of L-functions and automorphic representation theory – a possible explanation of the origin of the “key identity” arising in the latter series of works.

Let $E/F$ be a finite and Galois extension of non-archimedean local fields. Let $G$ be a connected reductive group defined over $E$ and let $M: = \mathfrak{R}_{E/F}\, G$ be the reductive group over $F$ obtained by Weil restriction of scalars. We investigate depth, and the enhanced local Langlands correspondence, in the transition from $G(E)$ to $M(F)$. We obtain a depth-comparison formula for Weil-restricted groups., 29 pages

In this paper, we introduce an algorithm for computing p-adic integrals on bad reduction hyperelliptic curves. For bad reduction curves, there are two notions of p-adic integration: Berkovich-Coleman integrals which can be performed locally; and abelian integrals with desirable number-theoretic properties. By covering a bad reduction hyperelliptic curve by annuli and basic wide open sets, we reduce the computation of Berkovich-Coleman integrals to the known algorithms on good reduction hyperelliptic curves. These are due to Balakrishnan, Bradshaw, and Kedlaya, and to Balakrishnan and Besser for regular and meromorphic 1-forms on good reduction curves, respectively. We then employ tropical geometric techniques due to the first-named author with Rabinoff and Zureick-Brown to convert the Berkovich-Coleman integrals into abelian integrals. We provide examples of our algorithm, verifying that certain abelian integrals between torsion points vanish., Comment: Comments Welcome! figure taken from arxiv::1606.09618; v2 minor revisions

Given Y a non-compact manifold or orbifold, we define a natural subspace of the cohomology of Y called the narrow cohomology. We show that despite Y being non-compact, there is a well-defined and non-degenerate pairing on this subspace. The narrow cohomology proves useful for the study of genus zero Gromov-Witten theory. When Y is a smooth complex variety or Deligne-Mumford stack, one can define a quantum D-module on the narrow cohomology of Y. This yields a new formulation of quantum Serre duality.

We construct a canonical basis for the space of weakly holomorphic Drinfeld modular forms. And we find that the basis elements satisfy a generating function and the duality among coefficients of the basis elements. Moreover we obtain the congruence properties of t-expansion coefficients of these functions under some conditions.

In a previous paper, we obtained several "compact versions" of Rubio de Francia's weighted extrapolation theorem, which allowed us to extrapolate the compactness of linear operators from just one space to the full range of weighted Lebesgue spaces, where these operators are bounded. In this paper, we study the extrapolation of compactness for bilinear operators in terms of bilinear Muckenhoupt weights. As applications, we easily recover and improve earlier results on the weighted compactness of commutators of bilinear Calder\'{o}n-Zygmund operators, bilinear fractional integrals and bilinear Fourier multipliers. More general versions of these results are recently due to Cao, Olivo and Yabuta (arXiv:2011.13191), whose approach depends on developing weighted versions of the Fr\'echet--Kolmogorov criterion of compactness, whereas we avoid this by relying on "softer" tools, which might have an independent interest in view of further extensions of the method., Comment: v3: final version, incorporated referee comments, to appear in Indagationes Mathematicae, 27 pages

We introduce multi-poly-Bernoulli–Carlitz numbers, function field analogues of multi-poly-Bernoulli numbers of Imatomi–Kaneko–Takeda. We explicitly describe multi-poly-Bernoulli Carlitz numbers in terms of the Carlitz factorial and the Stirling–Carlitz numbers of the second kind and also show their relationships with function field analogues of finite multiple zeta values.

We construct a generalisation of Bakry-Emery curvature to prove twisted intertwining relations for Markov semigroups. These relations are applied to Brascamp–Lieb type inequalities and spectral gap results. It extends the method of Arnaudon, Bonnefont and Joulin, to Riemannian manifolds and to a wider class of twists. These results are illustrated with several examples.

Let R be a commutative noetherian ring. The notion of n-wide subcategories of Mod R is introduced and studied in Matsui-Nam-Takahashi-Tri-Yen in relation to the cohomological dimension of a specialization-closed subset of Spec R. In this paper, we introduce the notions of n-coherent subsets of Spec R and n-uniform subcategories of D(Mod R), and explore their interactions with n-wide subcategories of Mod R. We obtain a commutative diagram which yields filtrations of subcategories of Mod R, D(Mod R) and subsets of Spec R and complements classification theorems of subcategories due to Gabriel, Krause, Neeman, Takahashi and Angeleri Hugel-Marks-Stovicek-Takahashi-Vitoria., Comment: 17 pages, to appear in IMRN

In this paper, we present ergodicity criteria for 1-Lipschitz functions on Z p , in terms of the van der Put coefficients as well as the inherent data associated with the function. These criteria are applied to provide sufficient conditions for ergodicity of the 1-Lipschitz p-adic functions with special features, such as everywhere/uniform differentiability with respect to the Mahler expansion. In particular, the ergodicity criteria are obtained for certain 1-Lipschitz functions on Z 2 and Z 3 , which are known as B -functions, in terms of the Mahler and van der Put expansions. These functions are locally analytic functions of order 1 (and therefore contain polynomials). For arbitrary primes p ≥ 5 , an ergodicity criterion of B -functions on Z p is introduced, which leads to an efficient and practical method of constructing ergodic polynomials on Z p that realize a given unicyclic permutation modulo p. Thus, a complete description of ergodic polynomials modulo p μ , which are reduced from all ergodic B -functions on Z p , is provided where μ = μ ( p ) = 3 for p ∈ { 2 , 3 } and μ = 2 for p ≥ 5 .

We present Chebyshev type cubature rules for the exact integration of rational symmetric functions with poles on prescribed coordinate hyperplanes. Here the integration is with respect to the densities of unitary Jacobi ensembles stemming from the Haar measures of the orthogonal and the compact symplectic Lie groups., Comment: 10 pages, 1 table

We give explicit isomorphisms between simple modules of degenerate cyclotomic Hecke algebras defined via various cellular bases. A special case gives a generalized Mullineux involution in the degenerate case., Comment: 21 pages

In 1946, Magnus presented an addition theorem for the confluent hypergeometric function of the second kind $U$ with argument $x+y$ expressed as an integral of a product of two $U$'s, one with argument $x$ and another with argument $y$. We take advantage of recently obtained asymptotics for $U$ with large complex first parameter to determine a domain of convergence for Magnus' result. Using well-known specializations of $U$, we obtain corresponding integral addition theorems with precise domains of convergence for modified parabolic cylinder functions, and Hankel, Macdonald, and Bessel functions of the first and second kind with order zero and one.

This paper focuses on the spherical fixed point set of a circle action on an ALE space endowed with Kronheimer's hyperkaehler metric. The induced metric on the sphere is described by using the algebraic geometry of rational curves on algebraic surfaces, in particular the lines on a cubic., Dedicated to the memory of Michael Atiyah

Let $\mathrm{k}_{0}$ be a field and $W$ a finite orthogonal reflection group over $\mathrm{k}_{0}$. We prove Serre's splitting principle for cohomological invariants of $W$ with values in Rost's cycle modules (over $\mathrm{k}_{0}$) if the characteristic of $\mathrm{k}_{0}$ is coprime to $|W|$. We then show that this principle for such groups holds also for Witt- and Milnor-Witt $K$-theory invariants., 20 pages

Weyl's unitary matrices, which were introduced in Weyl's 1927 paper [12] on group theory and quantum mechanics, are p × p unitary matrices given by the diagonal matrix whose entries are the p-th roots of unity and the cyclic shift matrix. Weyl's unitaries, which we denote by u and v , satisfy u p = v p = 1 p (the p × p identity matrix) and the commutation relation u v = ζ v u , where ζ is a primitive p-th root of unity. We prove that Weyl's unitary matrices are universal in the following sense: if u and v are any d × d unitary matrices such that u p = v p = 1 d and u v = ζ v u , then there exists a unital completely positive linear map ϕ : M p ( C ) → M d ( C ) such that ϕ ( u ) = u and ϕ ( v ) = v . We also show, moreover, that any two pairs of p-th order unitary matrices that satisfy the Weyl commutation relation are completely order equivalent, but that the assertion for three such unitaries fails. There is a standard tensor-product construction involving the Pauli matrices that produces irreducible sequences of anticommuting selfadjoint unitary matrices of arbitrary length. The matrices in this sequence are called Weyl-Brauer unitary matrices [11, Definition 6.63] . This standard construction is generalised herein to the case p ≥ 3 , producing a sequence of matrices that we also call Weyl-Brauer unitary matrices. We show that the Weyl-Brauer unitary matrices, as a g-tuple, are extremal in their matrix range, using recent ideas from noncommutative convexity theory.

The purpose of the present paper is to give an effective version of the noncritical $p$-tame Belyi theorem. That is to say, we compute explicitly an upper bound of the minimal degree of tamely ramified Belyi maps in positive characteristic which are unramified at a prescribed finite set of points., Comment: 16 pages

Let $\mathcal C$ be a class of Hausdorff topological semigroups which contains all zero-dimensional Hausdorff topological semigroups. A semigroup $X$ is called $\mathcal C$-$closed$ if $X$ is closed in each topological semigroup $Y\in \mathcal C$ containing $X$ as a discrete subsemigroup; $X$ is $projectively$ $\mathcal C$-$closed$ if for each congruence $\approx$ on $X$ the quotient semigroup $X/_\approx$ is $\mathcal C$-closed. A semigroup $X$ is called $chain$-$finite$ if for any infinite set $I\subseteq X$ there are elements $x,y\in I$ such that $xy\notin\{x,y\}$. We prove that a semigroup $X$ is $\mathcal C$-closed if it admits a homomorphism $h:X\to E$ to a chain-finite semilattice $E$ such that for every $e\in E$ the semigroup $h^{-1}(e)$ is $\mathcal C$-closed. Applying this theorem, we prove that a commutative semigroup $X$ is $\mathcal C$-closed if and only if $X$ is periodic, chain-finite, all subgroups of $X$ are bounded, and for any infinite set $A\subseteq X$ the product $AA$ is not a singleton. A commutative semigroup $X$ is projectively $\mathcal C$-closed if and only if $X$ is chain-finite, all subgroups of $X$ are bounded and the union $H(X)$ of all subgroups in $X$ has finite complement $X\setminus H(X)$., Comment: 19 pages

Let f be a polynomial in n variables over some number field and Z a subscheme of affine n-space. The notion of motivic oscillation index of f at Z was initiated by Cluckers in [7] and Cluckers-Mustaţǎ-Nguyen in [12] . In this paper we elaborate on this notion and raise several questions. The first one is stability under base field extension; this question is linked to a deep understanding of the density of non-archimedean local fields over which Igusa's local zeta function of f has a pole with given real part. The second one is around Igusa's conjecture for exponential sums with bounds in terms of the motivic oscillation index. Thirdly, we wonder if the above questions only depend on the analytic isomorphism class of singularities. By using various techniques as the GAGA theorem, resolution of singularities and model theory, we can answer the third question up to a base field extension. Next, by using a transfer principle between non-archimedean local fields of characteristic zero and positive characteristic, we can link all three questions with a conjecture on weights of l-adic cohomology groups of Artin-Schreier sheaves associated to jet polynomials. This way, we can answer all questions positively if f is a polynomial ‘of Thom-Sebastiani type’ with non-rational singularities. As a consequence, we prove Igusa's conjecture for arbitrary polynomials in three variables and polynomials with singularities of A − D − E type. In an appendix, we answer affirmatively a recent question of Cluckers-Mustaţǎ-Nguyen in [12] on poles of maximal order of twisted Igusa's local zeta functions.

O'Grady constructed a 6-dimensional irreducible holomorphic symplectic variety by taking a crepant resolution of some moduli space of stable sheaves on an abelian surface. In this paper, we naturally extend O'Grady's construction to fields of positive characteristic p greater than 2, called OG6 varieties. We show that a supersingular OG6 variety is unirational, its rational cohomology group is generated by algebraic classes, and its rational Chow motive is of Tate type. These results confirm in this case the generalized Artin--Shioda conjecture, the supersingular Tate conjecture and the supersingular Bloch conjecture proposed in our previous work, in analogy with the theory of supersingular K3 surfaces., Comment: Final version. To appear in I.M.R.N

Using the approach of so-called c-sequences introduced by the fifth author in his recent work, we give much simpler and shorter proofs of multivalued Perov's type results with respect to the ones presented in the recently published paper by M. Abbas et al. Our proofs improve, complement, unify and enrich the ones from the recent papers. Further, in the last section of this paper, we correct and generalize the well-known Perov's fixed point result. We show that this result is in fact equivalent to Banach's contraction principle.

We describe a procedure for classifying positively graded filiform nilpotent Lie algebras in low dimensions, over R , C , and Q . Using this procedure, we find all filiform nilpotent Lie algebras of dimension 15 and lower that admit a positive grading.

The goal of this paper is to provide a categorical framework that leads to the definition of shtukas a la Drinfeld and of excursion operators a la V. Lafforgue. We take as the point of departure the Hecke action of Rep ( G ˇ ) on the category Shv ( Bun G ) of sheaves on Bun G , and also the endofunctor of the latter category, given by the action of the geometric Frobenius. The shtuka construction will be obtained by applying (various versions of) categorical trace.

In this paper k always denotes a finite field of characteristic p > 0 . A variety over k means a reduced scheme separated and of finite type over k. An etale cover is always assumed to be finite (as in [SGA1, I, Def. 4.9] ).

In the first part of this article, we review a formalism of local zeta integrals attached to spherical reductive prehomogeneous vector spaces, which partially extends M. Sato's theory by incorporating the generalized matrix coefficients of admissible representations. We summarize the basic properties of these integrals such as the convergence, meromorphic continuation and an abstract functional equation. In the second part, we prove a generalization that accommodates certain non-spherical spaces with spherical quotients. As an application, the resulting theory applies to the prehomogeneous vector space underlying Bhargava's cubes, which is also considered by F. Sato and Suzuki–Wakatsuki in their study of toric periods.

In this note we derive a non-commutative version of the Wustholz analytic subgroup theorem in transcendence theory from the original (commutative) theorem and provide an application.

The Chabauty-Coleman method is a $p$-adic method for finding all rational points on curves of genus $g$ whose Jacobians have Mordell-Weil rank $r < g$. Recently, Edixhoven and Lido developed a geometric quadratic Chabauty method that was adapted by Spelier to cover the case of geometric linear Chabauty. We compare the geometric linear Chabauty method and the Chabauty-Coleman method and show that geometric linear Chabauty can outperform Chabauty-Coleman in certain cases. However, as Chabauty-Coleman remains more practical for general computations, we discuss how to strengthen Chabauty-Coleman to make it theoretically equivalent to geometric linear Chabauty. We apply these methods to genus 2 and genus 3 curves., fixed minor issues and updated exposition; to appear in Acta Arithmetica

In this note, we introduce the notion of almost unramified representations of quasi-split unitary groups of even ranks with respect to an unramified quadratic extension of local fields, and study their behavior under the local theta correspondence. These representations are in some sense the least ramified representations that have root number $-1$., 23 pages, to appear in Journal of Number Theory

Let D be a domain and let S be a torsion-free monoid such that D has characteristic 0 or the quotient group of S satisfies the ascending chain condition on cyclic subgroups. We give a characterization of when the monoid algebra D [ S ] is weakly Krull. As corollaries, we reobtain the results on when D [ S ] is Krull resp. weakly factorial, due to Chouinard resp. Chang. Furthermore, we deduce a characterization of generalized Krull monoid algebras analogous to our main result and we characterize the weakly Krull domains among the affine monoid algebras.

Given an autonomous first order algebraic ordinary differential equation F ( y , y ′ ) = 0 , we prove that every formal Puiseux series solution of F ( y , y ′ ) = 0 , expanded around any finite point or at infinity, is convergent. The proof is constructive and we provide an algorithm to describe all such Puiseux series solutions. Moreover, we show that for any point in the complex plane there exists a solution of the differential equation which defines an analytic curve passing through this point.

We improve bump conditions for the two-weight boundedness of Calder\'on-Zygmund operators introduced recently by R. Rahm and S. Spencer., Comment: v2: typos corrected

In this paper we make an initial study on type D moduli spaces in positive characteristic $p\neq 2$, where we allow $p$ ramified in the definite quaternion algebra. We classify the isogeny classes of $p$-divisible groups with additional structures in question. We also study the reduction of the type $D$ moduli spaces of minimal rank., Comment: 57 pages, to appear in Annales de l'Institut Fourier (Grenoble)