Let M and L be quadratic lattices over the maximal order of an algebraic number field. In case of dealing with representations of M by L, they sometimes assume certain indefiniteness and the condition rank L-rank M ≥ 3. In this case, representation problems are reduced not to global but to local problems by virtue of the strong approximation theorem for rotations and of the fact that for regular quadratic spaces U, V over a non-archimedian local field there is an isometry from U to V if dim V — dim U ≥ 3.

We continue our study of Yoshida’s lifting, which associates to a pair of automorphic forms on the adelic multiplicative group of a quaternion algebra a Siegel modular form of degree 2. We consider here the case that the automorphic forms on the quaternion algebra correspond to modular forms of arbitrary even weights and square free levels; in particular we obtain a construction of Siegel modular forms of weight 3 attached to a pair of elliptic modular forms of weights 2 and 4.

We introduce the Hurwitz-type spectral zeta functions for the quantum Rabi models, and give their meromorphic continuation to the whole complex plane with only one simple pole at $s=1$. As an application, we give the Weyl law for the quantum Rabi models. As a byproduct, we also give a rationality of Rabi–Bernoulli polynomials introduced in this paper.

In [5], Ikeda-Nagao-Nakayama gave a characterization of algebras of cohomological dimension ≦n In a subsequent paper [4] Eilenberg gave an alternative treatment of the same question. The present paper is devoted to the discussion of a number of questions suggested by the results of [4] and [5]. Among others it is shown that the conditions employed in stating the main results in [4] and [5] are equivalent, so that the main results of these two papers are in accord. Further, the cohomological dimension of a residue-algebra is studied in terms of that of the original algebra and the (module-) dimension of the associated ideal. The terminology and notation employed here are that of [3].

Einleitung: Die Ramanujan-Nagellshe Gleichung (Basis 2, Diskriminante—7).

Verallgemeinerung auf beliebige negative Diskriminanten D ≡ 1 mod. 23: Systematischer Ansatz zur Behandlung und erste notwendige Bedingungen für Lösbarkeit und Lösungen.

Endlichkeit der Lösungsanzahl.

Notwendige Bedingung für die Lösungen durch Kongruenzbetrachtung nach Potenzen von 2.

Notwendige Bedingung für die Lösungen durch Kongruenzbetrachtung nach Potenzen eines Primteilers von D.

Im Zusammenhang mit dem von Grothendieck in [2] Chap. IV (7.4.8) gestellten Problem “1st die ideal-adische Komplettierung eines ausgezeichneten Ringes wieder ausgezeichnet?” scheint aufgrund der Arbeiten von Marot [3] und Valabrega [8] im lokalen Fall die folgende Frage intéressant zu sein: “R sei ein lokaler universell japanischer Ring. 1st der singuläre Ort jeder endlich erzeugten R- Algebra A abgeschlossen in Spec (A)?” Diese Frage wird hier negativ beantwortet, d.h. wir werden einen lokalen universell japanischen Ring A konstruieren, dessen singulärer Ort nicht abgeschlossen in der Zariski-Topologie von Spec (A) ist.

Soit X un groupe abélien localement compact et dénombrable à l’infini; ζ sera sa mesure de Haar. Dans les articles précédents [10] et [11], pour un noyau de convolution de Hunt N sur X, nous avons défini la famille sous-ordonnée H(N; X) au noyau N, qui est une large classe de noyaux de convolution de Hunt sur X définie par N et la totalité des noyaux de convolution de Hunt bornés sur la droite réelle R portés par R+ = {t ∈ R; t ≧ 0}.

Many questions about free ideal rings ( = firs, cf. [5] and §2 below) which at present seem difficult become much easier when one restricts attention to local rings. One is then dealing with hereditary local rings, and any such ring is in fact a fir (§2). Our object thus is to describe hereditary local rings. The results on firs in [5] show that such a ring must be a unique factorization domain; in §3 we prove that it must also be rigid (cf. the definition in [3] and §3 below). More precisely, for a semifir R with prime factorization rigidity is necessary and sufficient for R to be a local ring.

Es ist wohlbekannt, dass die Frattini-Gruppe einer endlichen Gruppe G, mit anderen Worten, der Durchschnitt aller maximalen Untergruppen von G, nilpotent ist. Offen ist dagegen bis jetzt die Frage, ob die Faktorgruppe G/F*(G) von G nach dem Erzeugnis F*(G) aller minimalen Untergruppen eine besondere Struktur besitzt, z. B. ob sie auflösbar ist. Mehrere Schwierigkeiten sind der Grund hierfür: Einmal ist G/F*(G) im allgemeinen nicht nilpotent, zum anderen kennen wir die Struktur der einfachen und auflösbaren Gruppen (abgesehen von der Theorie von P. Hall) zu wenig. In der vorliegenden Note sollen teilweise Antworten auf die Frage nach der Auflösbarkeit von G/F*(G) gegeben werden.

In this paper we have given a biholomorphic classification of smooth, connected, protective, non-ruled surfaces X with a smooth, connected, hyperplane section C relative to L, where L is a very ample line bundle on X, such that g = g(C) = g(L) is less than or equal to six. For a similar classification of rational surfaces with the same conditions see [Li].

It is classical to characterize locally homogeneous Riemann manifolds by infinitesimal conditions. For example, [Si] asserts that the local-homogeneity is equivalent to the existence of linear isometries between tangent spaces which preserve the curvatures and their covariant derivatives up to certain orders. It is also known that the local homogeneity is equivalent to the existence of a certain tensor field of type (1, 2) (for this and a further study, see [TV]).

Let Λ be the Leech lattice which is an even unimodular lattice with no vectors of squared length 2 in 24-dimensional Euclidean space R24. Then the Mathieu Group M24 is a subgroup of the automorphism group .0 of Λ and the action on Λ of M24 induces a natural permutation representation of M24 on an orthogonal basis For , let Λm be the sublattice of vectors invariant under m:

A stochastic process X(A, ω) is called Brownian motion with an N-dimensional parameter when it satisfies the following conditions:

1) For any positive integer n and any set of points A1, A2, …, An in an N-dimensional Euclidian space EN, the joint variable is subject to an n-dimensional Gaussian distribution having the vector 0 as its mean vector.

We give an explicit determinant formula for a class of rational solutions of a q-analogue of the Painlevé V equation. The entries of the determinant are given by the continuous q-Laguerre polynomials.

All rings that occur in this note will be assumed to be commutative with unity and all modules will be finitely generated and unitary.

The grade of a module M over a noetherian local ring R is defined to be the length of a maximal R-sequence contained in the annihilator of M. If M has finite projective dimension it is well-known that grade M ≤ proj. dim M. We can say more when R is a regular local ring.

Let R be a ring with radical (all rings have a unit element, all modules are unital). Often, one wishes to lift modules modulo , that is, to a given, say, left R/-module U find a left R-module E with the property that E/E ≃ U. This is of course not always possible. Here I prove, roughly, that if a finitely generated projective U can be lifted at all, it can be lifted to a projective. Or rather, if U can be lifted to an E satisfying a certain mild condition, then E is projective (Lemma).

We establish here several “invariance of plurigenera type” theorems for twisted pluricanonical forms and metrics of adjoint ℝ-bundles.

We are studying some aspects of the action of Galois groups on the torsor of paths connecting two (possibly tangential) points on a projec-tive line minus a finite number of points. We obtain objects which formally behave like classical iterated integrals and polylogarithms. We formulate an analog of Zagier conjecture for these l-adic analogs of iterated integrals and polylogarithms.

As generalizations of the results in [5] and [4], the author gave some uniqueness theorems of meromorphic maps into PN(C) in previous papers [2] and [3].

Let G be a finite group. A(left) G-module A of G is said to be of trivial cohomology when Hn(H, A)= 0 for all rational integers n and for all subgroups H of G. The main purpose of the present note is to determine the structure of finitely generated G-modules of trivial cohomology, which turns out to be remarkably simple (See Theorem 1 and Corollary 3 below). We prove also an (easy) localization theorem for cohomological triviality.