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We prove a rank 3 criterion for the simple connectedness of certain subsets of buildings and we give two applications of this criterion. The first generalizes a result of Tits for Chevalley groups to 3-spherical Kac-Moody groups. The second is the proof of the simple connectedness of certain flipflop geometries introduced in [Bennett C., Gramlich R., Hoffman C., Shpectorov S.: Curtis-Phan-Tits theory. In: Groups, combinatorics and geometry (Durham, 2001). World Sci. Publishing, River Edge, NJ, 2003, 13–29].

Let D be an integral domain with quotient field K . The Nagata ring D ( X ) and the Kronecker function ring Kr( D ) are both subrings of the field of rational functions K ( X ) containing as a subring the ring D [ X ] of polynomials in the variable X . Both of these function rings have been extensively studied and generalized. The principal interest in these two extensions of D lies in the reflection of various algebraic and spectral properties of D and Spec( D ) in algebraic and spectral properties of the function rings. Despite the obvious similarities in definitions and properties, these two kinds of domains of rational functions have been classically treated independently, when D is not a Prüfer domain. The purpose of this note is to study two different unified approaches to the Nagata rings and the Kronecker function rings, which yield these rings and their classical generalizations as special cases.

We give a characterization of cotilting modules over Prüfer domains, up to equivalence; moreover we show that tilting modules over Prüfer domains are of projective dimension at most one.

This article is about the derivation algebra of multi-loop algebras. Multi-loop algebras are algebras obtained by a generalization of a process known as twisting by automorphisms in the theory of Kac–Moody algebras. Multi-loop algebras are used in the realization of extended affine Lie algebras. Under certain conditions on an algebra 𝒜, we determine the derivation algebra of an n -step multi-loop algebra based on 𝒜 as the semidirect product of a multi-loop algebra based on the derivation algebra of 𝒜 and the derivation algebra of the Laurent polynomials in n -variables. This in particular determines the derivation algebras of the core modulo center of (almost all) extended affine Lie algebras.

Let R be an integral domain with quotient field Q . For R -submodules X and Y of Q denote by [ Y : X ] the R -module . An ideal I of R is a colon-splitting ideal of R if for all ideals J and K of R . We examine colon-splitting ideals and use these results to characterize a special class of colon-splitting ideals, the ideals of injective dimension 1. We show that every nonzero ideal of a domain is a colon-splitting ideal (has injective dimension 1) if and only if every maximal ideal is a colon-splitting ideal (resp., has injective dimension 1). From this we deduce new characterizations of h -local Prüfer domains and almost maximal Prüfer domains.

We extend the Lewis Correspondence between Maaß wave forms and period functions to subgroups of the modular group of finite index. The period functions then become vector valued functions.

To any spread 𝒮 of PG (3, q ) corresponds a family of locally hermitian ovoids of the Hermitian surface H (3, q 2 ), and conversely; if in addition 𝒮 is a semifield spread, then each associated ovoid is a translation ovoid, and conversely. In this paper we calculate the translation group of the locally hermitian ovoids of H (3, q 2 ) arising from a given semifield spread, and we characterize the p-semiclassical ovoid constructed in [Cossidente A., Ebert G., Marino G., Siciliano A.: Shult Sets and Translation Ovoids of the Hermitian Surface. Adv. Geom. 6 (2006), 475–494] as the only translation ovoid of H (3, q 2 ) whose translation group is abelian. If 𝒮 is a spread of PG (3, q ) and 𝒪(𝒮) is one of the associated ovoids of H (3, q 2 ), then using the duality between H (3, q 2 ) and Q − (5, q ), another spread of PG (3, q ), say 𝒮 2 , can be constructed. On the other hand, using the Barlotti-Cofman representation of H (3, q 2 ), one more spread of a 3-dimensional projective space, say 𝒮 1 , arises from the ovoid 𝒪(𝒮). In [Lunardon G.: Blocking sets and semifields. J. Comb. Theory Ser. A to appear] some questions are posed on the relations among 𝒮, 𝒮 1 and 𝒮 2 ; here we prove that 𝒮 and 𝒮 2 are isomorphic and the ovoids 𝒪(𝒮) and 𝒪(𝒮 1 ), corresponding to 𝒮 and 𝒮 1 respectively under the Plücker map, are isomorphic.

This article extends classical one variable results about Euler products, defined by integral valued polynomial or analytic functions, to several variables. We show there exists a meromorphic continuation up to a presumed natural boundary, and give a criterion, à la Estermann-Dahlquist, for the existence of a meromorphic extension to . In addition, we precisely describe the boundaries of analyticity and meromorphy for a multivariable Euler product determined by any toric variety (split over ). Using our method, we are also able to calculate a precise asymptotic for the number of n -fold products of integers that equal the n th power of an integer, for any n ≥ 3.