We show that the possible jump of the order in an 1-parameter deformation family of (possibly nonisolated) hypersurface singularities, with constant Lê numbers, is controlled by the powers of the deformation parameter. In particular, this applies to families of aligned singularities with constant topological type—a class for which the Lê numbers are “almost” constant. In the special case of families with isolated singularities—a case for which the constancy of the Lê numbers is equivalent to the constancy of the Milnor number—the result was proved by Greuel, Plénat, and Trotman.

As an application, we prove equimultiplicity for new families of nonisolated hypersurface singularities with constant topological type, answering partially the Zariski multiplicity conjecture.

We construct a discriminant-preserving map from the set of orbits in the space of quadruples of quinary alternating forms over the integers to the set of isomorphism classes of quintic rings. This map may be regarded as an analogue of the famous map from the set of equivalence classes of integral binary cubic forms to the set of isomorphism classes of cubic rings and may be expected to have similar applications. We show that the ring of integers of every quintic number field lies in the image of the map. These results have been used to establish an upper bound on the number of quintic number fields with bounded discriminant.

Let $\mathfrak{o}$ be a complete discrete valuation ring of mixed characteristic $(0,p)$ and $\mathfrak{X}_{0}$ a smooth formal $\mathfrak{o}$-scheme. Let $\mathfrak{X}\rightarrow \mathfrak{X}_{0}$ be an admissible blow-up. In the first part, we introduce sheaves of differential operators $\mathscr{D}_{\mathfrak{X},k}^{\dagger }$ on $\mathfrak{X}$, for every sufficiently large positive integer $k$, generalizing Berthelot’s arithmetic differential operators on the smooth formal scheme $\mathfrak{X}_{0}$. The coherence of these sheaves and several other basic properties are proven. In the second part, we study the projective limit sheaf $\mathscr{D}_{\mathfrak{X},\infty }=\mathop{\varprojlim }\nolimits_{k}\mathscr{D}_{\mathfrak{X},k}^{\dagger }$ and introduce its abelian category of coadmissible modules. The inductive limit of the sheaves $\mathscr{D}_{\mathfrak{X},\infty }$, over all admissible blow-ups $\mathfrak{X}$, is a sheaf $\mathscr{D}_{\langle \mathfrak{X}_{0}\rangle }$ on the Zariski–Riemann space of $\mathfrak{X}_{0}$, which gives rise to an abelian category of coadmissible modules. Analogues of Theorems A and B are shown to hold in each of these settings, that is, for $\mathscr{D}_{\mathfrak{X},k}^{\dagger }$, $\mathscr{D}_{\mathfrak{X},\infty }$, and $\mathscr{D}_{\langle \mathfrak{X}_{0}\rangle }$.

It is shown that if ϕ denotes a harmonic morphism of type Bl between suitable Brelot harmonic spaces X and Y, then a function f, defined on an open set V ⊂ Y, is superharmonic if and only if f ∘ ϕ is superharmonic on ϕ–1(V) ⊂ X. The “only if” part is due to Constantinescu and Cornea, with ϕ denoting any harmonic morphism between two Brelot spaces. A similar result is obtained for finely superharmonic functions defined on finely open sets. These results apply, for example, to the case where ϕ is the projection from ℝN to ℝn (N > n ≥ 1) or where ϕ is the radial projection from ℝN \ {0} to the unit sphere in ℝN (N ≥ 2).

We show that any Killing vector field on the unit tangent sphere bundle with Sasaki metric of a space of constant curvature k ≠ 1 is fiber preserving by studying some property of geodesies on the bundle. As a consequence, any Killing vector field on the unit tangent sphere bundle of a space of constant curvature k ≠ 1 can be extended to a Killing vector field on the tangent bundle.

The singularly perturbed Riccati equation is the first-order nonlinear ordinary differential equation $\hbar \partial _x f = af^2 + bf + c$ in the complex domain where $\hbar $ is a small complex parameter. We prove an existence and uniqueness theorem for exact solutions with prescribed asymptotics as $\hbar \to 0$ in a half-plane. These exact solutions are constructed using the Borel–Laplace method; that is, they are Borel summations of the formal divergent $\hbar $ -power series solutions. As an application, we prove existence and uniqueness of exact WKB solutions for the complex one-dimensional Schrödinger equation with a rational potential.

In this note we shall prove a theorem which is related to Montel’s theorem [1] on bounded regular functions. Let E be a measurable set on the positive y-axis in the z( = x + iy)-plane, E(a, b) be its part contained in 0 ≦ a ≦ y ≦ b, and ∣E(a, b)∣ be its measure. We define the lower density of E at y = 0 by

LEMMA, Let E be a set of positive lower density λ at y = 0. Then E contains a subset E1 of the same lower density at y = 0 such that E1 ∪ {0} is a closed set.

Given a smooth minimal surface F: Ω → ℝ3 defined on a simply connected region Ω in the complex plane ℂ, there is a regular SG circle pattern . By the Weierstrass representation of F and the existence theorem of SG circle patterns, there exists an associated SG circle pattern in ℂ with the combinatoric of . Based on the relationship between the circle pattern and the corresponding discrete minimal surface F∊: → ℝ3 defined on the vertex set of the graph of , we show that there exists a family of discrete minimal surface Γ∊: → ℝ3, which converges in C∞(Ω) to the minimal surface F: Ω → ℝ3 as ∊ → 0.

Given an ideal a on a smooth variety in characteristic zero, we estimate the F-jumping numbers of the reductions of a to positive characteristic in terms of the jumping numbers of a and the characteristic. We apply one of our estimates to bound the Hartshorne–Speiser–Lyubeznik invariant for the reduction to positive characteristic of a hypersurface singularity.

For a skew-symmetrizable cluster algebra ${\mathcal{A}}_{t_{0}}$ with principal coefficients at $t_{0}$, we prove that each seed $\unicode[STIX]{x1D6F4}_{t}$ of ${\mathcal{A}}_{t_{0}}$ is uniquely determined by its $C$-matrix, which was proposed by Fomin and Zelevinsky (Compos. Math. 143 (2007), 112–164) as a conjecture. Our proof is based on the fact that the positivity of cluster variables and sign coherence of $c$-vectors hold for ${\mathcal{A}}_{t_{0}}$, which was actually verified in Gross et al. (Canonical bases for cluster algebras, J. Amer. Math. Soc. 31(2) (2018), 497–608). Further discussion is provided in the sign-skew-symmetric case so as to obtain a weak version of the conjecture in this general case.

The purpose of this note is to write down the general form of Hankel kernels for the complex unit ball B in Cd. In the one dimensional case (unit disk Δ in C) this was done in [JP] and our treatment below has been guided by the insights gained there, and later, in a slightly different context, in [P]. We begin by summarizing the relevant facts in the case of the disk in a form convenient for us.

In [5] we defined the concepts of Frobenius and symmetric algebra for algebras of infinite vector space dimension over a field. It was shown there that with the introduction of a topology and the judicious use of the terms continuous and closed, many of the classical theorems of Nakayama [7, 8] on Frobenius and symmetric algebras could be generalized to the infinite dimensional case. In this paper we shall be concerned with showing certain algebras are (or are not) Frobenius or symmetric. In Section 3, we shall see that an algebra can be symmetric or Frobenius in “many ways”. This is a problem which did not arise in the finite dimensional case.

We consider the differential operator

in an n-dimensional (n≧2), orientiable, compact Riemann space R with the metric ds2 = gij(x)dxidxj. Here bij(x) is a contravariant tensor such that the quadratic form and ai(x) changes, by the coordininates transformation x → x̅, as follows:

Dans leurs mémoires classiques sur les groupes de transformations continus et infinis, MM. É. Cartan et E. Vessiot ont introduit la notion de prolongement des groupes de transformations pour définir l’isomorphisme et l’homomorphisme entre deux groupes de transformations.

Let $f\colon X\to B$ be a semistable fibration where X is a smooth variety of dimension $n\geq 2$ and B is a smooth curve. We give the structure theorem for the local system of the relative $1$ -forms and of the relative top forms. This gives a neat interpretation of the second Fujita decomposition of $f_*\omega _{X/B}$ . We apply our interpretation to show the existence, up to base change, of higher irrational pencils and on the finiteness of the associated monodromy representations under natural Castelnuovo-type hypothesis on local subsystems. Finally, we give a criterion to have that X is not of Albanese general type if $B=\mathbb {P}^1$ .

Following the work of Mustaţă and Bitoun, we recently developed a notion of Bernstein–Sato roots for arbitrary ideals, which is a prime characteristic analogue for the roots of the Bernstein–Sato polynomial. Here, we prove that for monomial ideals the roots of the Bernstein–Sato polynomial (over $\mathbb{C}$) agree with the Bernstein–Sato roots of the mod $p$ reductions of the ideal for $p$ large enough. We regard this as evidence that the characteristic-$p$ notion of Bernstein–Sato root is reasonable.

Let Q be a bounded open set of Rn (n ≧ 2) with a C∞ boundary Γ and assume that be functions in such that where s1, …, sN are integers ≧ and are n-tuples of non-negative integers with

We give a generators-and-relations description of differential graded algebras recently introduced by Ozsváth and Szabó for the computation of knot Floer homology. We also compute the homology of these algebras and determine when they are formal.

Fujii obtained a formula for the average number of Goldbach representations with lower-order terms expressed as a sum over the zeros of the Riemann zeta function and a smaller error term. This assumed the Riemann Hypothesis. We obtain an unconditional version of this result and obtain applications conditional on various conjectures on zeros of the Riemann zeta function.

Let Fp be the prime field of characteristic p (p: an odd prime), and put = Fp - {0, 1}. Then for λ ∈ we define

,

where denotes the Legendre symbol, and consider the sum