Eigenvalue problems of the form Af = λBf, where λ is a complex parameter and A and B are operators on a Hilbert Space, have been considered by a number of authors (e.g., [1; 3; 5; 7; 10]). In this paper, we shall be concerned with the existence and nature of eigenfunction expansions associated with such problems, with no assumptions of self-adjointness. The form of the theorems to be given here is: if the system (A, B) is spectral and complete (definitions below), and F and G are operators satisfying certain “smallness” conditions, then (A + F, B + G) is also spectral and complete. The hypotheses for these theorems are chosen with an eye to applying the results to boundary-value problems on a compact interval. Such applications, together with an examination of circumstances under which the system (Dn, Dm) (D denoting differentiation) is spectral and complete under a broad class of boundary conditions, will be made in a later paper.

Let $K$ be a complex reductive algebraic group and $V$ a representation of $K$. Let $S$ denote the ring of polynomials on $V$. Assume that the action of $K$ on $S$ is multiplicity-free. If $\lambda $ denotes the isomorphism class of an irreducible representation of $K$, let $\rho \lambda :K\to GL({{V}_{\lambda }})$ denote the corresponding irreducible representation and ${{S}_{\lambda }}$ the $\lambda $-isotypic component of $S$. Write ${{S}_{\lambda }}.{{S}_{\mu }}$ for the subspace of $S$ spanned by products of ${{S}_{\lambda }}$ and ${{S}_{\mu }}$. If ${{V}_{v}}$ occurs as an irreducible constituent of ${{V}_{\lambda }}\otimes {{V}_{\mu }}$, is it true that ${{S}_{v}}\subseteq {{S}_{\lambda }}.{{S}_{\mu }}?$ In this paper, the authors investigate this question for representations arising in the context of Hermitian symmetric pairs. It is shown that the answer is yes in some cases and, using an earlier result of Ruitenburg, that in the remaining classical cases, the answer is yes provided that a conjecture of Stanley on the multiplication of Jack polynomials is true. It is also shown how the conjecture connects multiplication in the ring $S$ to the usual Littlewood–Richardson rule.

It is known that minimally thin and semithin sets in R2 are preserved by conformai mappings (see [3]) but it is not known whether or not analogous results hold true for ordinary thin and semithin sets respectively. In an unpublished work, Brelot has shown that ordinary thin sets at the origin are preserved by mappings of the form f(z) = z∝ (∝ > 0) where one always considers the principal branch of the mapping. We shall prove this result along with an analogous one for ordinary semithin sets and will see that the implications established by Jackson (see [4, Theorem 4]), for ordinary and minimally thin sets and by Brelot (see [2, p. 152]) for semithin sets in a half plane hold true for any wedge shaped region with vertex at the origin.

Let $\unicode[STIX]{x1D6E4}\leqslant \text{Aut}(T_{d_{1}})\times \text{Aut}(T_{d_{2}})$ be a group acting freely and transitively on the product of two regular trees of degree $d_{1}$ and $d_{2}$. We develop an algorithm that computes the closure of the projection of $\unicode[STIX]{x1D6E4}$ on $\text{Aut}(T_{d_{t}})$ under the hypothesis that $d_{t}\geqslant 6$ is even and that the local action of $\unicode[STIX]{x1D6E4}$ on $T_{d_{t}}$ contains $\text{Alt}(d_{t})$. We show that if $\unicode[STIX]{x1D6E4}$ is torsion-free and $d_{1}=d_{2}=6$, exactly seven closed subgroups of $\text{Aut}(T_{6})$ arise in this way. We also construct two new infinite families of virtually simple lattices in $\text{Aut}(T_{6})\times \text{Aut}(T_{4n})$ and in $\text{Aut}(T_{2n})\times \text{Aut}(T_{2n+1})$, respectively, for all $n\geqslant 2$. In particular, we provide an explicit presentation of a torsion-free infinite simple group on 5 generators and 10 relations, that splits as an amalgamated free product of two copies of $F_{3}$ over $F_{11}$. We include information arising from computer-assisted exhaustive searches of lattices in products of trees of small degrees. In an appendix by Pierre-Emmanuel Caprace, some of our results are used to show that abstract and relative commensurator groups of free groups are almost simple, providing partial answers to questions of Lubotzky and Lubotzky–Mozes–Zimmer.

Wilson loop diagrams are an important tool in studying scattering amplitudes of SYM $N=4$ theory and are known by previous work to be associated to positroids. We characterize the conditions under which two Wilson loop diagrams give the same positroid, prove that an important subclass of subdiagrams (exact subdiagrams) corresponds to uniform matroids, and enumerate the number of different Wilson loop diagrams that correspond to each positroid cell. We also give a correspondence between those positroids which can arise from Wilson loop diagrams and directions in associahedra.

Higher dimensional analogues of the modular group $\mathit{PSL}(2,\mathbb{Z})$ are closely related to hyperbolic reflection groups and Coxeter polyhedra with big symmetry groups. In this context, we develop a theory and dissection properties of ideal hyperbolic $k$-rectified regular polyhedra, which is of independent interest. As an application, we can identify the covolumes of the quaternionic modular groups with certain explicit rational multiples of the Riemann zeta value $\unicode[STIX]{x1D701}(3)$.

The idea of considering the set of the elements of a group as a space, provided with a topology, measure, or metric, connected somehow with the group operation, has been used often in the work of E. Cartan and others. In the present paper we shall study a very special group whose space can be embedded naturally into a projective plane and where the straight lines have a very simple group-theoretical interpretation. It remains to be seen whether this geometrical embedding in a projective space can be extended to other classes of groups and whether the method could become an instrument of geometrical investigation, like co-ordinates or reflections. In the final section it is shown how a geometrical theorem may lead to relations within the group.

We compare crystal combinatorics of the level $2$ Fock space with the classification of unitary irreducible representations of type B rational Cherednik algebras to study how unitarity behaves under parabolic restriction. We show that the crystal operators that remove boxes preserve the combinatorial conditions for unitarity, and that the parabolic restriction functors categorifying the crystals send irreducible unitary representations to unitary representations. Furthermore, we find the supports of the unitary representations.

The purpose of this paper is to exploit an analogy between algebraic extensions of fields and essential extensions of modules, in which the role of the algebraic closure of a field F is played by the injective hull H(M) of a unitary left R-module M. (The notion of * ‘algebraic’ extensions of general algebraic systems has been studied by Shoda; see, for example [5].)

In this analogy, the role of a polynomial p(x) is played by a homomorphism of R-modules

(1)

which will be called an ideal homomorphism into M. The process of solving the equation p(x) = 0 in F, or in an algebraic extension of F, will be replaced by the process of extending an ideal homomorphism (1) to a homomorphism F* from R into M, or into an essential extension of M.

This viewpoint of studying projective planes was given in my previous paper (12). It is discussed in other papers: Hall (4, 6, 7), Maisano (16), Lombardo-Radice (14, 15), Wagner (19). In particular, we consider how to make identifications in the free plane, or how identifications are forced when one begins with a non-degenerate quadrangle and makes free extensions of this quadrangle of a known plane. We shall continue to develop this topic using the notations and definitions of the previous paper (12). We consider the number of subplanes of certain planes, finding exact values in the cases of the known order-nine planes, and deriving a lower bound in a general case. We prove a theorem concerning the structure of all singly generated planes. We give an example to show that this structure is not definitive. Finally, we prove that a specific Walker-Knuth plane is singly generated.

B. Banaschewski [1] has produced a counterexample to [2, Theorem 6]. As noted in [1, Remark 1], our error occurs in the final paragraph of the purported proof of [2, Theorem 6], for P need not be a subpresheaf of I. Accordingly, it remains an open problem to find an analogue of [2, Proposition 1] in the context of Boolean spaces. We hope that attacks on this problem will be facilitated by the (valid) initial three paragraphs of the argument given for [2, Theorem 6].

The following alterations to [2] are in order. Example 5, being a corollary of Theorem 6, remains doubtful, although the special case noted on pp. 1034-1035 is not affected. In Corollary 7, the assertion that j preserves infectives remains doubtful, although the proof for divisibility of j(M)(G) is valid.

The real line will be R, Euclidean n-space will be Rn, the unit ball in Rn will be En, the unit sphere in Rn+1 will be Sn, and real projective n-space will be Pn. The canonical line bundle associated with the double cover Sn → Pn will be ηn. If γ is a vector bundle, E(γ) will be its associated cell bundle, S(γ) its associated sphere bundle, P(γ) its associated projective space bundle (P(γ) = S(γ) / (-1)) and T(γ) = E(γ)/S(γ) its Thorn space.

In a previous paper [1], the authors introduced and studied the notion of special conformally flat spaces and quasi-umbilical hypersurfaces. In that paper, the authors proved that every conformally flat space of codimension one in a Euclidean space is special and, conversely, every special conformally flat space can be isometrically immersed in a Euclidean space as a quasi-umbilical hypersurface.

In the present paper, the authors study the conformally flat spaces of codimension 2 in a Euclidean space. (Manifolds, mappings, functions, etc. are assumed to be sufficiently differentiate and we shall restrict ourselves only to manifolds of dimension n > 3.)

In (1, §§ 2.3 and 2.4) we proved that certain distributive lattices are simultaneously lattice-embeddable in the degrees of recursive unsolvability and in the hyperdegrees. Let ℒ be the non-distributive lattice {0,1, a0, a1,…}, where ai ∪ aj = 1 and ai ∩ aj = 1 whenever i ≠ j. We shall prove the following theorem.

THEOREM. The lattice ℒ is simultaneously lattice-embeddable in the degrees and hyperdegrees.

For A ⊆ N, let deg(A) and hyp(A) be the degree and hyperdegree of A, respectively. To prove the theorem we must construct hyperarithmetically incomparable sets A0, A1, … such that for Δ = deg, hypand for all distinct i, j:

1

2

Now, if each 〈Ai, Aj〉 were a generic pair in the sense of (1), then (2) would hold.

Le but de cet article est de montrer qu'il n'y a pas de relation étroite entre ces trois concepts.

Soit K une partie compacte du plan complexe fini, et soit A(K) (resp. a(K)) l'ensemble des fonctions continues sur K et holomorphes (resp. harmoniques) à l'intérieur K0 de K. Nous dirons que K est un ensemble d'approximation holomorphe (resp. harmonique) si toute fonction de A(K) (resp. a(K)) peut être approchée uniformément sur K par des fonctions holomorphes (resp. harmoniques) sur K, i.e., dans un voisinage de K.

Pour un voisinage fixé Ω de K, le théorème de Mergelian nous donne des conditions nécessaires et suffisantes, purement topologiques, pour que toute fonction de A(K) puisse être approchée uniformément sur K par des fonctions holomorphes dans Ω. D'autre part, on sait (par exemple [4]) qu'il n'est pas possible de caractériser topologiquement les ensembles d'approximation holomorphe.

We show that over a field k of characteristic zero an affine (n — 1)- space embedded as a closed subvariety in affine n-space and homogeneous for a codimension two linear torus action on is defined by the vanishing of a variable.

Eigenvalue problems for the biharmonic operator (iterated Laplacian) L = ΔΔ will be studied on bounded plane domains. Our purpose is to obtain asymptotic variational formulae for eigenvalues and eigenfunctions under the deformation of removing an ∊-disk and adjoining additional boundary conditions on the new boundary component thereby introduced, valid on a positive interval 0 < ∊ < ∊0. Eigenvalue problems can be considered in connection with each of the following sets of homogeneous boundary conditions.

In this article, we are concerned with the tempered fractional parabolic problem

$$ \begin{align*}\frac{\partial u}{\partial t}(x, t)-\left(\Delta+\lambda\right)^{\frac{\alpha}{2}} u(x, t)=f(u(x, t)), \end{align*} $$

where $-\left (\Delta +\lambda \right )^{\frac {\alpha }{2}}$ is a tempered fractional operator with $\alpha \in (0,2)$ and $\lambda $ is a sufficiently small positive constant. We first establish maximum principle principles for problems involving tempered fractional parabolic operators. And then, we develop the direct sliding methods for the tempered fractional parabolic problem, and discuss how they can be used to establish monotonicity results of solutions to the tempered fractional parabolic problem in various domains. We believe that our theory and methods can be conveniently applied to study parabolic problems involving other nonlocal operators.

This paper provides a functional analogue of the recently initiated dual Orlicz–Brunn–Minkowski theory for star bodies. We first propose the Orlicz addition of measures, and establish the dual functional Orlicz–Brunn–Minkowski inequality. Based on a family of linear Orlicz additions of two measures, we provide an interpretation for the famous $f$-divergence. Jensen’s inequality for integrals is also proved to be equivalent to the newly established dual functional Orlicz–Brunn–Minkowski inequality. An optimization problem for the $f$-divergence is proposed, and related functional affine isoperimetric inequalities are established.