We continue the work started in parts (I) and (II) of this series. In this paper, we classify which continuous quivers of type A are derived equivalent. Next, we define the new ${\mathcal {C}(A_{{\mathbb {R}},S})}$ , which we call weak continuous cluster category. It is a triangulated category, it does not have cluster structure but it has a new weaker notion of “cluster theory.” We show that the original continuous cluster category of [15] is a localization of this new weak continuous cluster category. We define cluster theories to be appropriate groupoids and we show that cluster structures satisfy the conditions for cluster theories. We describe the relationship between different cluster theories: some new and some obtained from cluster structures. The notion of continuous mutation which appears in cluster theories (but not in cluster structures) appears in the next paper [20].

In this paper we generalize results of the third paper of this series. As a corollary we can show the following: Let Li (1 ≤ i ≤ n) be a positive definite quadratic form which is equivalent to one of Cartan matrices of Lie algebras of type An (n ≥ 2), Dn (n ≥ 4), E6, E7, E8 and assume that is positive definite quadratic forms and satisfies that rk Mt ≥ 2 and implies rk K or rk L = 1. Then we have n = m and Lt is equivalent to a constant multiple of Ms(i) for some permutation s. Therefore we get the uniqueness of decompositions with respect to tensor products in this case.

The purpose of this paper is to define minimality of surfaces in an infinite dimensional space E by probabilistic methods with the description of the relation between minimal surfaces and harmonic functions on the space E, and to analyze purely analytic properties of a certain class of quadratic forms on the space E.

What we call the homological conjectures on commutative Noetherian local rings were first collected and partially settled by C. Peskine and L. Szpiro [PS1]. The subsequent remarkable progress was made by M. Hochster [H1] who conjectured the existence of big Cohen-Macaulay modules and solved it in the affirmative for equicharacteristic local rings. It is, however, still open in general setting.

We prove a functional equation for a vector valued real analytic Eisenstein series transforming with the Weil representation of $\operatorname{Sp}(n,\mathbb{Z})$ on $\mathbb{C}[(L^{\prime }/L)^{n}]$. By relating such an Eisenstein series with a real analytic Jacobi Eisenstein series of degree $n$, a functional equation for such an Eisenstein series is proved. Employing a doubling method for Jacobi forms of higher degree established by Arakawa, we transfer the aforementioned functional equation to a zeta function defined by the eigenvalues of a Jacobi eigenform. Finally, we obtain the analytic continuation and a functional equation of the standard $L$-function attached to a Jacobi eigenform, which was already proved by Murase, however in a different way.

This paper includes a proof, relative to the consistency of ZFC, of the consistency of ZFC, the continuum has singular cardinality and the extended plus-one hypothesis.

We establish explicit isomorphisms of two seemingly-different algebras, and their Schur algebras, arising from the centralizers of two different type B Weyl group actions in Schur-like dualities. We provide a presentation of the geometric counterpart of the above Schur algebras in [1] specialized at $q=1$ .

Let D be the open unit disk and Γ be the unit circle in the complex plane, and denote by Ω the Riemann sphere. If f(z) is a meromorphic function in D, and if ζ∈Г, then the principal cluster set of f at ζ is the set

Recently M. Ikeda [1] succeeded in determining the structure of absolutely segregated algebras, i.e. algebras whose 2-cohomology groups all vanish. His beautiful result reads : an algebra A, of finite rank over its ground field, is absolutely segregated if and only if i) the residue-algebra A/N modulo the radical N is separable and, moreover, ii) the A-left-module N is an (Mo)-module. A. simplification was given by H. Nagao [5], who obtained, besides an interesting-result on algebras with vanishing 3-(or higher) cohomology groups, an elegant short proof to the fact that under the assumption of i), the property ii) is necessary, and sufficient, for the absolute segregation of A.

Let K be the classical quaternion field over the field Q of rational numbers with the quaternion units 1, i, j, k, with relations i2 = j2 = − 1, k = ij = −ji. For a quaternion x ∈ K, we write its conjugate, trace and norm by and Nx, respectively. Put

and consider the map h: A → B defined by

Let G be a connected algebraic group, p a finite dimensional representation of G in a vector space V, all defined over a number field fc. To the pair (G, p) we can associate the family of theta series

Let Γ be a curve defined by y = A(x) in R2. The Cauchy transform on the curve Γ is a singular integral operator defined by the singular integral kernel

(1.1)

Let G be a p-adic connected reductive algebraic group and K a maximal compact subgroup of G. In [4], Casselman obtained the explicit formula of zonal spherical functions on G with respect to K on the assumption that K is special. It is known (Bruhat and Tits [3]) that the affine root system of algebraic group which has good but not special maximal compact subgroup is A1 C2, or Bn (n > 3), and all Bn-types can be realized by orthogonal groups. Here the assumption “good” is necessary for the Satake’s theory of spherical functions.

Let D be the unit disk | z | < 1 and let f be a complex-valued function continuous in D. The cluster set C(f, eiθ) of f at eiθ is defined by

The main object of this paper is to develop techniques for investigating the local properties of actions of an algebraic group on an algebraic variety. Our main tools are certain schemes which may be associated to Lie algebras.

Let Q be the set of symbols 1, 2,…, 3 p, where p is an prime number greater than 3. Let be a transitive permutation group on Ω, which is simple and in which the normalizer of a Sylow p-subgroup has order 2 p.

The metrics to which the title of the present paper refers are expressed in the form of elements of arc length as follows:

This paper is a sequel to Motohashi [4]. In [4], a series of theorems named “elimination theorems of uniqueness conditions” was shown to hold in the classical predicate calculus LK. But, these results have the following two defects : one is that they do not hold in the intuitionistic predicate calculus LJ, and the other is that they give no nice axiomatizations of some sets of sentences concerned. In order to explain these facts more explicitly, let us introduce some necessary notations and definitions. Let L be a first order classical predicate calculus LK or a first order intuitionistic predicate calculus LJ. n-ary formulas in L are formulas F(ā) in L with a sequence ā of distinct free variables of length n such that every free variable in F occurs in ā.