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 .

Let Γ be a fuchsian group of the first kind and assume that Γ does not contain the element . Let S1(Γ) be the linear space of cusp forms of weight 1 on the group Γ and denote by d1 the dimension of the space S1(Γ). When the group Γ has a compact fundamental domain, we have obtained the following (Hiramatsu [3]):

(*) ,

where ς*(s) denotes the Selberg type zeta function defined by

.

Nakajima introduced a certain set of monomials realizing the irreducible highest weight crystals B(λ). The monomial set can be extended so that it contains crystal B(∞) in addition to B(λ). We present explicit descriptions of the crystals B(∞) and B(λ) over special linear Lie algebras in the language of extended Nakajima monomials. There is a natural correspondence between the monomial description and Young tableau realization, which is another realization of crystals B(∞) and B(λ).

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 introduce a log Picard variety over the complex number field by the method of log geometry in the sense of Fontaine-Illusie, and study its basic properties, especially, its relationship with the group of log version of m-torsors.

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.

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).

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.

It is well known [c.f. Kas] that every elliptic surface, with geometric genus 0, is given by a Weierstrass equation of the form

(relative to a suitable parameter, u, for the base) where the a’s and b’s are constants. For sufficiently general choices of a’s and b’s, the Mordell-Weil group (i.e., the group of solutions (x, y), with x and y rational functions of u) has rank 8.

For rank-one symmetric spaces M of the compact type all Kähler structures Fλ, defined on their punctured tangent bundles T0 M and invariant with respect to the normalized geodesic flow on T0 M, are constructed. It is shown that this class {Fλ} of Kähler structures is stable under the reduction procedure.

Let $k$ be a commutative ring, let ${\mathcal{C}}$ be a small, $k$-linear, Hom-finite, locally bounded category, and let ${\mathcal{B}}$ be a $k$-linear abelian category. We construct a Frobenius exact subcategory ${\mathcal{G}}{\mathcal{P}}({\mathcal{G}}{\mathcal{P}}_{P}({\mathcal{B}}^{{\mathcal{C}}}))$ of the functor category ${\mathcal{B}}^{{\mathcal{C}}}$, and we show that it is a subcategory of the Gorenstein projective objects ${\mathcal{G}}{\mathcal{P}}({\mathcal{B}}^{{\mathcal{C}}})$ in ${\mathcal{B}}^{{\mathcal{C}}}$. Furthermore, we obtain criteria for when ${\mathcal{G}}{\mathcal{P}}({\mathcal{G}}{\mathcal{P}}_{P}({\mathcal{B}}^{{\mathcal{C}}}))={\mathcal{G}}{\mathcal{P}}({\mathcal{B}}^{{\mathcal{C}}})$. We show in examples that this can be used to compute ${\mathcal{G}}{\mathcal{P}}({\mathcal{B}}^{{\mathcal{C}}})$ explicitly.

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

The Gaussian space (E*, μ) is a natural infinite dimensional analogue of Euclidean space with Lebesgue measure and a special choice of a Gelfand triple gives a fundamental framework of white noise calculus [2] as distribution theory on Gaussian space. It is proved in Kubo-Takenaka [7] that (E) is a topological algebra under pointwise multiplication. The main purpose of this paper is to answer the fundamental question: what are the derivations on the algebra (E)?

This is a contribution to the study of $\mathrm {Irr}(G)$ as an $\mathrm {Aut}(G)$-set for G a finite quasisimple group. Focusing on the last open case of groups of Lie type $\mathrm {D}$ and $^2\mathrm {D}$, a crucial property is the so-called $A'(\infty )$ condition expressing that diagonal automorphisms and graph-field automorphisms of G have transversal orbits in $\mathrm {Irr}(G)$. This is part of the stronger $A(\infty )$ condition introduced in the context of the reduction of the McKay conjecture to a question about quasisimple groups. Our main theorem is that a minimal counterexample to condition $A(\infty )$ for groups of type $\mathrm {D}$ would still satisfy $A'(\infty )$. This will be used in a second paper to fully establish $A(\infty )$ for any type and rank. The present paper uses Harish-Chandra induction as a parametrization tool. We give a new, more effective proof of the theorem of Geck and Lusztig ensuring that cuspidal characters of any standard Levi subgroup of $G=\mathrm {D}_{ l,\mathrm {sc}}(q)$ extend to their stabilizers in the normalizer of that Levi subgroup. This allows us to control the action of automorphisms on these extensions. From there, Harish-Chandra theory leads naturally to a detailed study of associated relative Weyl groups and other extendibility problems in that context.

For a pair $(R,I)$, where $R$ is a standard graded domain of dimension $d$ over an algebraically closed field of characteristic 0, and $I$ is a graded ideal of finite colength, we prove that the existence of $\lim _{p\rightarrow \infty }e_{HK}(R_{p},I_{p})$ is equivalent, for any fixed $m\geqslant d-1$, to the existence of $\lim _{p\rightarrow \infty }\ell (R_{p}/I_{p}^{[p^{m}]})/p^{md}$. This we get as a consequence of Theorem 1.1: as $p\longrightarrow \infty$, the convergence of the Hilbert–Kunz (HK) density function $f(R_{p},I_{p})$ is equivalent to the convergence of the truncated HK density functions $f_{m}(R_{p},I_{p})$ (in $L^{\infty }$ norm) of the mod $p$reductions$(R_{p},I_{p})$, for any fixed $m\geqslant d-1$. In particular, to define the HK density function $f_{R,I}^{\infty }$ in char 0, it is enough to prove the existence of $\lim _{p\rightarrow \infty }f_{m}(R_{p},I_{p})$, for any fixed $m\geqslant d-1$. This allows us to prove the existence of $e_{HK}^{\infty }(R,I)$ in many new cases, for example, when Proj R is a Segre product of curves.

This work is inspired by the recent paper of Mytilinaios and Slaman [9] on the infinite injury priority method. It may be considered to fall within the general program of the study of reverse recursion theory: What axioms of Peano arithmetic are required or sufficient to prove theorems in recursion theory? Previous contributions to this program, especially with respect to the finite and infinite injury priority methods, can be found in the works of Groszek and Mytilinaios [4], Groszek and Slaman [5], Mytilinaios [8], Slaman and Woodin [10]. Results of [4] and [9], for example, together pinpoint the existence of an incomplete, nonlow r.e. degree to be provable only within some fragment of Peano arithmetic at least as strong as P- + IΣ2. Indeed an abstract principle on infinite strategies, such as that on the construction of an incomplete high r.e. degree, was introduced in [4] and shown to be equivalent to Σ2 induction over the base theory P- + IΣ0 of Peano arithmetic.

Let G be a connected reductive linear algebraic group defined over an algebraically closed field of characteristic p. Assume that p is good for G. In this note we consider particular classes of connected reductive subgroups H of G and show that the cocharacters of H that are associated to a given nilpotent element e in the Lie algebra of H are precisely the cocharacters of G associated to e that take values in H. In particular, we show that this is the case provided H is a connected reductive subgroup of G of maximal rank; this answers a question posed by J. C. Jantzen.

In connection with the Shimura correspondence, Shintani [6] and Niwa [4] constructed a modular form by the integral with the theta kernel arising from the Weil representation. They treated the group Sp(1) × O(2, 1). Using the special isomorphism of O(2, 1) onto SL(2), Shintani constructed a modular form of half-integral weight from that of integral weight. We can write symbolically his case as “O(2, 1)→ Sp(1)” Then Niwa’s case is “Sp(l)→ O(2, 1)”, that is from the halfintegral to the integral. Their methods are generalized by many authors. In particular, Niwa’s are fully extended by Rallis-Schiffmann to “Sp(l)→O(p, q)”.

In this paper, we deal with the relation between the characteristic functions of meromorphic functions that share three values CM. As applications of our main results, we shall affirmatively settle two conjectures proposed by Mues and Osgood-Yang.

This paper deals with a decomposition problem for some classes of distributions. Let D be a given class of distribution on R1, which we are interested in. After showing that the class D is closed under convolution, our purpose is to give an answer to the inverse problem: if the convolution of two distributions μ1 and (μ2 belongs to D, then do μ1 and μ2 belong to D?