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Let k be an arbitrary even integer, and Ek(s;z) denote the non-holomorphic Eisenstein series (of weight k attached to SL2(ℤ)), defined by (1.1) below. In the present paper we first establish a complete asymptotic expansion of Ek(s;z) in the descending order of y as y → + ∞ (Theorem 2.1), upon transferring from the previously derived asymptotic expansion of E0(s;z) (due to the first author [16]) to that of Ek(s;z) through successive use of Maass' weight change operators. Theorem 2.1 yields various results on Ek(s;z), including its functional properties (Corollaries 2.1–2.3), its relevant specific values (Corollaries 2.4–2.7), and its asymptotic aspects as z → 0 (Corollary 2.8). We then apply the non-Euclidean Laplacian ΔH,k (of weight k attached to the upper-half plane) to the resulting expansion, in order to justify the eigenfunction equation for Ek(s;z) in (1.6), where the justification can be made uniformly in the whole s-plane (Theorem 2.2). [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study an analogue of Hilbert's seventeenth problem for functions over a valued field which are integral definite on some definable set; that is, that map the given set into the valuation ring. We use model theory to exhibit a uniform method, on various theories of valued fields, for deriving an algebraic characterization of such functions. As part of this method we refine the concept of a function being integral at a point, and make it dependent on the relevant class of valued fields. We apply our framework to algebraically closed valued fields, model complete theories of difference and differential valued fields, and real closed valued fields. [ABSTRACT FROM AUTHOR]

In this paper, we study triples of the form (E, θ, ϕ) over a compact Riemann surface, where (E, θ) is a Higgs bundle and ϕ is a global holomorphic section of the Higgs bundle. Our main result is an description of a birational equivalence which relates geometrically the moduli space of Higgs bundles of rank r and degree d to the moduli space of Higgs bundles of rank r-1 and degree d. [ABSTRACT FROM AUTHOR]

Intuitionistic fuzzy sets are a generalization of ordinary fuzzy sets which are characterized by a membership function and a non-membership function. In this paper we consider the problem of ranking a set of intuitionistic fuzzy numbers. We adopt a statistical viewpoint and interpret each intuitionistic fuzzy number as an ensemble of ordinary fuzzy numbers. This enables us to define a fuzzy rank and a characteristic vagueness factor for each intuitionistic fuzzy number. We show the reasonablesness of the results obtained by examining several test cases. [ABSTRACT FROM AUTHOR]