Let us consider the following two problems.

(A) Does either

or

hold for the not identically vanishing solution of the equation

for x ∈ Ω ⊂ Rn (n ≥ 3), where λ is a constant satisfying λ > E0 and V(x) is a 2-body or many-body potential?

Let $\ell $ be a prime number. The Iwasawa theory of multigraphs is the systematic study of growth patterns in the number of spanning trees in abelian $\ell $-towers of multigraphs. In this context, growth patterns are realized by certain analogs of Iwasawa invariants, which depend on the prime $\ell $ and the abelian $\ell $-tower of multigraphs. We formulate and study statistical questions about the behavior of the Iwasawa $\mu $ and $\lambda $ invariants.

Let be a stochastic process, where T is a finite interval. The ε-entropy is defined as the following quantity:

(1)

As a generalization of the conformal structure of type (2, 2), we study Grassmannian structures of type (n, m) for n, m ≥ 2. We develop their twistor theory by considering the complete integrability of the associated null distributions. The integrability corresponds to global solutions of the geometric structures.

A Grassmannian structure of type (n, m) on a manifold M is, by definition, an isomorphism from the tangent bundle TM of M to the tensor product V ⊗ W of two vector bundles V and W with rank n and m over M respectively. Because of the tensor product structure, we have two null plane bundles with fibres Pm-1(ℝ) and Pn-1(ℝ) over M. The tautological distribution is defined on each two bundles by a connection. We relate the integrability condition to the half flatness of the Grassmannian structures. Tanaka’s normal Cartan connections are fully used and the Spencer cohomology groups of graded Lie algebras play a fundamental role.

Besides the integrability conditions corrsponding to the twistor theory, the lifting theorems and the reduction theorems are derived. We also study twistor diagrams under Weyl connections.

The purpose of this paper is to prove the following three theorems which were announced in [2].

We give a formula for the class number of an arbitrary complex mutliplication (CM) algebraic torus over $\mathbb {Q}$ . This is proved based on results of Ono and Shyr. As applications, we give formulas for numbers of polarized CM abelian varieties, of connected components of unitary Shimura varieties and of certain polarized abelian varieties over finite fields. We also give a second proof of our main result.

The purpose of this note is to study regular actions of simple algebraic groups on projective threefolds as an application of the theory of algebraic threefolds, especially Mori Theory and the theory of Fano threefolds (cf. Mori [11], Iskovskih [7, 8]). The motivation for this study is as follows. In a series of papers, Umemura, in part jointly with Mukai, has classified maximal connected algebraic subgroups of the Cremona group of three variables and also constructed minimal rational threefolds which correspond to such subgroups (cf. Umemura [16-19], Mukai-Ume-mura [12]). In particular, Umemura and Mukai studied in [12] the SL(2, C)-equivariant smooth projectivization of SL(2, C)/G, where G is a binary icosahedral or octahedral subgroup of SL(2, C). The study of equivariant smooth projectivization of SL(2, C)/G for any finite subgroup G has been completed along their lines in Nakano [14]. The main trick of these studies is the investigation of equivariant contraction maps of extremal rays in the context of Mori Theory [11]. In this note, we apply a similar idea to projective threefolds with a regular action of a simple algebraic group and determine which simple algebraic groups can act regularly and nontrivially on projective threefolds and in which fashion. We also need some standard (but difficult) facts from the theory of Fano threefolds. For the precise statement, see Theorem 1 in the main text. For the proof of this theorem, we need a classification of closed subgroups of simple algebraic groups of codimension 1 and 2, which could be derived easily from the classical work of Dynkin [4]. However, we shall give a geometric proof independent of [4] which leads up directly to the proof of Theorem 1. On the whole, we shall establish by geometric methods the scarcity of closed subgroups of small codimension in simple algebraic groups, which is implied in Dynkin [4].

Let M be an n-dimensional hypersurface immersed in the (n + 1)-dimensional unit sphere Sn+1 with the standard metric by an immersion f. We denote by A the second fundamental form of the immersion / which is considered as a symmetric linear transformation of each tangent space TXM, i.e. for an arbitrary point x of M and the unit normal vector field ξ defined in a neighborhood of x, A is given by where is the covariant differentiation in Sn+i and Thus, A depends on the orientation of the unit normal vector field ξ and, in general, it is locally defined on M.

We prove that every smooth divisor in $\mathbb {P}^1\times \mathbb {P}^1\times \mathbb {P}^2$ of degree $(1,1,2)$ is K-stable.

This is a continuation of [5]. In the present part, we study irregular even surfaces of general type with K < 4χ, where K and χ denote respectively a canonical divisor and the holomorphic Euler-Poincaré characteristic. As the first main theorem, we show the following:

Let X be a non-compact Riemann surface and D ⊂ X an open subset. By a classical result due to Behnke and Stein [2] D is Runge in X (i.e. the restriction map has dense image) iff X\D has no compact connected components. In other words the obstruction to holomorphic approximation is purely topological. This result has been generalized to 1-dimensional Stein spaces by Mihalache in [11].

A protagonist here is a new-type invariant for type II degenerations of K3 surfaces, which is explicit piecewise linear convex function from the interval with at most $18$ nonlinear points. Forgetting its actual function behavior, it also classifies the type II degenerations into several combinatorial types, depending on the type of root lattices as appeared in classical examples.

From differential geometric viewpoint, the function is obtained as the density function of the limit measure on the collapsing hyper-Kähler metrics to conjectural segments, as in [HSZ19]. On the way, we also reconstruct a moduli compactification of elliptic K3 surfaces by [AB19], [ABE20], [Brun15] in a more elementary manner, and analyze the cusps more explicitly.

We also interpret the glued hyper-Kähler fibration of [HSVZ18] as a special case from our viewpoint, and discuss other cases, and possible relations with Landau–Ginzburg models in the mirror symmetry context.

Let A, B be integral symmetric positive definite matrices of degree m and two, respectively, and suppose that A[X] = B is soluble over Zp for every prime p. If m ≥ 7 and min B = min B[x] (Z2 ∋ x i≠ 0) is sufficiently large, then A[X] = B is soluble over Z. We gave a conditional result for m = 6 in [7] under an assumption on the estimate from above of a kind of generalized Weyl sums. Here we give an unconditional result for special sequences of B.

In der vorliegenden Arbeit wird untersucht, was für eine Struktur eine Untergruppe von endlicher p-Gruppe habe, wenn mit jeder Untergruppe von als ganzes vertauschbar ist. In diesen tritt natürlich ein nicht nur aus E bestehender Normalteiler von auf und er enthält ein Zentrumselement von der Ordnung p. Wir fragen: Gilt das Ähnliche auch, falls kein Normalteiler ist? Die gestellte Frage läßt sich für regülare beantworten, deren Klasse 2 ist, aber für andere bleibt sie noch offen. Bevor wir diese Frage untersuchen, stellen wir einige Definitionen und nötige Sätze und Hilfssätze zusammen.

We develop the formalism of universal torsors in equivariant birational geometry and apply it to produce new examples of nonbirational but stably birational actions of finite groups.

We give an asymptotically sharp lower bound for the slope λ(f) of a fibration f : S → B, where S is a surface and B is a curve, if there exists an involution on the general fibre F of f. We also construct a new lower bound of λ(f) depending increasingly on the irregularity of S; as an application of this new bound we have a criteria to control the existence of other fibrations on S.

Let us denote by η(z) the classical η-function of Dedekind defined by

For real x ≥ y ≥ 2 and positive integers a, q, let Φ(x, y; a, q) denote the number of positive integers ≤ x, free of prime factors ≤ y and satisfying n ≡ a (mod q). By the fundamental lemma of sieve, it follows that for (a,q) = 1, Φ(x,y;a,q) = φ(q)-1, Φ(x, y){1 + O(exp(-u(log u- log2 3u- 2))) + (u = log x log y) holds uniformly in a wider ranges of x, y and q.

Let χ be any character to the modulus q, and L(s, χ) be the corresponding L-function. Let be a (‘exceptional’) real character to the modulus q for which L(s, ) have a (‘exceptional’) real zero satisfying > 1 - c0/log q. In the paper, we prove that in a slightly short range of q the above first error term can be replaced by where ρ(u) is Dickman function, and ρ′(u) = dρ(u)/du.

The result is an analogue of the prime number theorem for arithmetic progressions. From the result can deduce that the above first error term can be omitted, if suppose that 1 < q < (log q)A.

For a complex function $F$ on $\mathbb{C}$, we study the associated composition operator $T_{F}(f):=F\circ f=F(f)$ on Wiener amalgam $W^{p,q}(\mathbb{R}^{d})\;(1\leqslant p<\infty ,1\leqslant q<2)$. We have shown $T_{F}$ maps $W^{p,1}(\mathbb{R}^{d})$ to $W^{p,q}(\mathbb{R}^{d})$ if and only if $F$ is real analytic on $\mathbb{R}^{2}$ and $F(0)=0$. Similar result is proved in the case of modulation spaces $M^{p,q}(\mathbb{R}^{d})$. In particular, this gives an affirmative answer to the open question proposed in Bhimani and Ratnakumar (J. Funct. Anal. 270(2) (2016), 621–648).

Let Ω be a bounded domain in Rl (l ≥ 2) with, C∞ boundary Γ of dimension l — 1 and let there be given a second order elliptic differential equation

(1) in Ω,

where ∂j = ∂/∂xi and all coefficients are assumed, for the sake of simplicity, to be real-valued and C∞ on .