1. Introduction. As is well known, the functions

(1.1)

have various applications in theoretical physics and engineering. It is thus worthwhile to study their behaviour for real and complex values of the argument.

Many authors have made interesting and important contributions to the study of vector fields or infinitesimal transformations on compact orientable Riemannian manifolds and Hermitian manifolds without boundary. Recently, Hsiung (6, 7, 8) has extended some of these results to compact orientable Riemannian manifolds with boundary. The purpose of this paper is to continue Hsiung's work by studying vector fields and infinitesimal transformations on almost-Hermitian manifolds with boundary.

For a maximal torus in a quasi-split semi-simple simply-connected group over a local field of characteristic 0, Langlands and Shelstad constructed a cohomological invariant called the splitting invariant, which is an important component of their endoscopic transfer factors. We study this invariant in the case of a split real group and prove a decomposition theorem which expresses this invariant for a general torus as a product of the corresponding invariants for simple tori. We also show how this reduction formula allows for the comparison of splitting invariants between different tori in the given real group.

The set of all points in the Euclidean plane E2, the ratio of whose distances from two fixed points is a constant ƛ, is known as the circle of Apollonius [7, p. 62]. This ‘'Apollonius” set is a circle except for the degenerate cases where ƛ = 1 or ƛ = 0. In more general metric spaces the same definition applies to select certain Apollonius sets (or “ ƛ-sets” in our terminology), but of course these sets are not always circles. For example, all ƛ-sets (ƛ > 0) relative to a circle in E2 are two-point sets, and all ƛ-sets relative to E1 are either singletons or two-point sets. This paper deals with the topological structure of a metric space when certain cardinality conditions have been imposed on its ƛ-sets.

It is proved that if $G\,=\,{{G}_{1}}\,{{*}_{{{G}_{3}}}}\,{{G}_{2}}$ is free product of probability measure preserving $s$–regular ergodic discrete groupoids amalgamated over an amenable subgroupoid ${{G}_{3}}$, then the sofic dimension $s(G)$ satisfies the equality

$$s(G)=\mathfrak{h}(G_{1}^{0})s({{G}_{1}})+\mathfrak{h}(G_{2}^{0})s\left( {{G}_{2}} \right)-\,\mathfrak{h}(G_{3}^{0})s({{G}_{3}}),$$

where $\mathfrak{h}$ is the normalized Haar measure on $G$.

We provide a correspondence between one-sided coideal subrings and one-sided ideal two-sided coideals in an arbitrary bialgebroid. We prove that, under some expected additional conditions, this correspondence becomes bijective for Hopf algebroids. As an application, we investigate normal Hopf ideals in commutative Hopf algebroids (affine groupoid schemes) in connection with the study of normal affine subgroupoids.

Let $J$ be a Jacobian variety with toric reduction over a local field $K$. Let $J\,\to \,E$ be an optimal quotient defined over $K$, where $E$ is an elliptic curve. We give examples in which the functorially induced map ${{\Phi }_{J}}\,\to \,{{\Phi }_{E}}$ on component groups of the Néron models is not surjective. This answers a question of Ribet and Takahashi. We also give various criteria under which ${{\Phi }_{J}}\,\to \,{{\Phi }_{E}}$ is surjective and discuss when these criteria hold for the Jacobians of modular curves.

We strengthen, in various directions, the theorem of Garnett that every $\unicode[STIX]{x1D70E}$-compact, completely regular space $X$ occurs as a Gleason part for some uniform algebra. In particular, we show that the uniform algebra can always be chosen so that its maximal ideal space contains no analytic discs. We show that when the space $X$ is metrizable, the uniform algebra can be chosen so that its maximal ideal space is metrizable as well. We also show that for every locally compact subspace $X$ of a Euclidean space, there is a compact set $K$ in some $\mathbb{C}^{N}$ so that $\widehat{K}\backslash K$ contains a Gleason part homeomorphic to $X$, and $\widehat{K}$ contains no analytic discs.

A class of graphs is bridge-addable if given a graph $G$ in the class, any graph obtained by adding an edge between two connected components of $G$ is also in the class. The authors recently proved a conjecture of McDiarmid, Steger, and Welsh stating that if ${\mathcal{G}}$ is bridge-addable and $G_{n}$ is a uniform $n$-vertex graph from ${\mathcal{G}}$, then $G_{n}$ is connected with probability at least $(1+o_{n}(1))e^{-1/2}$. The constant $e^{-1/2}$ is best possible, since it is reached for the class of all forests.

In this paper, we prove a form of uniqueness in this statement: if ${\mathcal{G}}$ is a bridge-addable class and the random graph $G_{n}$ is connected with probability close to $e^{-1/2}$, then $G_{n}$ is asymptotically close to a uniform $n$-vertex random forest in a local sense. For example, if the probability converges to $e^{-1/2}$, then $G_{n}$ converges in the sense of Benjamini–Schramm to the uniformly infinite random forest $F_{\infty }$. This result is reminiscent of so-called “stability results” in extremal graph theory, the difference being that here the stable extremum is not a graph but a graph class.

Given an action ${\varphi }$ of inverse semigroup S on a ring A (with domain of ${\varphi }(s)$ denoted by $D_{s^*}$ ), we show that if the ideals $D_e$ , with e an idempotent, are unital, then the skew inverse semigroup ring $A\rtimes S$ can be realized as the convolution algebra of an ample groupoid with coefficients in a sheaf of (unital) rings. Conversely, we show that the convolution algebra of an ample groupoid with coefficients in a sheaf of rings is isomorphic to a skew inverse semigroup ring of this sort. We recover known results in the literature for Steinberg algebras over a field as special cases.

Throughout, we consider a finite Galois extension L|K of non-archimedian valued fields which are maximally complete [2, Chapter 2], Let v denote the valuation on L and let L* denote the group of non-zero elements of L. We mayidentify the value group v(L*) of L with a subgroup of D, where D denotes the minimal divisible ordered group containing v(K*). We denote the residue field of L by , and will always assume that the field extension is separable. The characteristic of will invariably be denoted by p ; much of what follows is trivial in case p = 0.

Finite extensions of complex commutative Banach algebras are naturally related to corresponding finite covering maps between the carrier spaces for the algebras. In the case of function rings, the finite extensions are induced by the corresponding finite covering maps, and the topological properties of the coverings are strongly reflected in the algebraic properties of the extensions and conversely. Of particular interest to us is the class of finite covering maps for which the induced extensions of function rings admit primitive generators. This is exactly the class of polynomial covering maps and the extensions are algebraic extensions defined by the underlying Weierstrass polynomials.

The purpose of this paper is to develop a suitable Galois theory for finite extensions of function rings induced by finite covering maps and to apply it in the case of Weierstrass polynomials and polynomial covering maps.

We calculate the group of homotopy classes of homotopy self-equivalences of 4-manifolds with free fundamental group and obtain a classification of such 4-manifolds up to s-cobordism.

Let $k$ be a global field, $\bar{k}$ a separable closure of $k$, and ${{G}_{k}}$ the absolute Galois group Gal$(\bar{k}/k)$ of $\bar{k}$ over $k$. For every $\sigma \,\in \,{{G}_{K}}$, let ${{\bar{k}}^{\sigma }}$ be the fixed subfield of $\bar{k}$ under $\sigma$. Let $E/k$ be an elliptic curve over $k$. It is known that the Mordell–Weil group $E({{\bar{k}}^{\sigma }})$ has infinite rank. We present a new proof of this fact in the following two cases. First, when $k$ is a global function field of odd characteristic and $E$ is parametrized by a Drinfeld modular curve, and secondly when $k$ is a totally real number field and $E/k$ is parametrized by a Shimura curve. In both cases our approach uses the non-triviality of a sequence of Heegner points on $E$ defined over ring class fields.

Decompositions of spectral type are obtained for closed Hilbert space operators with empty resolvent set, but whose square has closure which is spectral. Krein space situations are also discussed.

We characterize the Local Langlands Correspondence $\left( \text{LLC} \right)$ for inner forms of $\text{G}{{\text{L}}_{n}}$ via the Jacquet–Langlands Correspondence $\left( \text{JLC} \right)$ and compatibility with the Langlands Classification. We show that $\text{LLC}$ satisfies a natural compatibility with parabolic induction and characterize $\text{LLC}$ for inner forms as a unique family of bijections $\prod \left( \text{G}{{\text{L}}_{r}}\left( D \right) \right)\,\to \,\Phi \left( \text{G}{{\text{L}}_{r}}\left( D \right) \right)$ for each $r$, (for a fixed $D$), satisfying certain properties. We construct a surjective map of Bernstein centers $\mathfrak{Z}\left( \text{G}{{\text{L}}_{n}}\left( F \right) \right)\,\to \,\mathfrak{Z}\left( \text{G}{{\text{L}}_{r}}\left( D \right) \right)$ and show this produces pairs of matching distributions in the sense of Haines. Finally, we construct explicit Iwahori-biinvariant matching functions for unit elements in the parahoric Hecke algebras of $\text{G}{{\text{L}}_{r}}\left( D \right)$, and thereby produce many explicit pairs of matching functions.

The notion of shape fibration with the near lifting of near multivalued paths property is studied. The relation of these maps—which agree with shape fibrations having totally disconnected fibers—with Hurewicz fibrations with the unique path lifting property is completely settled. Some results concerning homotopy and shape groups are presented for shape fibrations with the near lifting of near multivalued paths property. It is shown that for this class of shape fibrations the existence of liftings of a fine multivalued map is equivalent to an algebraic problem relative to the homotopy, shape or strong shape groups associated.

In this paper we give an extension of (6, Theorem 1), using a similar method of proof, to every homogeneous Siegel domain of second kind which can be mapped biholomorphically into a Kâhler manifold of a certain class (Theorem 1). Then by a well-known result of Vinberg, Gindikin, and Pjateckiï-Sapiro (10) that every bounded homogeneous domain D,contained in a complex euclidean space CN,can be mapped biholomorphically onto an affinely homogeneous Siegel domain of second kind, the theorem follows for D(Theorem 2). (6, Theorem 1) is a generalization of the Ahlfors version of the Schwarz-Pick lemma in C1(1) to invariant volume for a star-like homogeneous bounded domain in CN;see also (4). In § 3 we give the inequality for a special non-symmetric Siegel domain of second kind using an explicit form of TD(z, )due to Lu (7).

A quasi-Frobenius ring is a ring with minimum condition satisfying the conditions r(l)H)) = H and l(r(L)) = L for right ideals H and left ideals L where r(S) (l(S)) denotes the right (left) annihilator of a subset S of the ring. Nakayama first defined and studied such rings (8; 9) and they have been studied by a number of authors (2; 3; 4; 6). A dual ring is a topological ring satisfying the conditions r(l)H)) = H and l(r)H)) = L for closed right ideals H and closed left ideals L. Baer (1) and Kaplansky (7) introduced the notion of such rings, which is a natural generalization of that of quaso-Frobenius rings. Numakura studied the analogy between dual rings and quasi-Frobenius rings in (10).

We compute the generator rank of a subhomogeneous $C^*\!$-algebra in terms of the covering dimension of the pieces of its primitive ideal space corresponding to irreducible representations of a fixed dimension. We deduce that every $\mathcal {Z}$-stable approximately subhomogeneous algebra has generator rank one, which means that a generic element in such an algebra is a generator.

This leads to a strong solution of the generator problem for classifiable, simple, nuclear $C^*\!$-algebras: a generic element in each such algebra is a generator. Examples of Villadsen show that this is not the case for all separable, simple, nuclear $C^*\!$-algebras.