In this paper we study the dependence of the first eigenvalue of the infinity Laplace with respect to the domain. We prove that this first eigenvalue is continuous under some weak convergence conditions which are fulfilled when a sequence of domains converges in Hausdorff distance. Moreover, it is Lipschitz continuous but not differentiable when we consider deformations obtained via a vector field. Our results are illustrated with simple examples.

We prove that for all n ≡ 13 or 37 (mod 72), there exists a biembedding of a pair of Steiner triple systems of order n in an orientable pseudosurface having precisely one regular pinch point of multiplicity 2.

The geography and botany problems of irreducible non-spin symplectic 4-manifolds with a choice of fundamental group from $\{{\mathbb{Z}}_p, {\mathbb{Z}}_p\oplus {\mathbb{Z}}_q, {\mathbb{Z}}, {\mathbb{Z}}\oplus {\mathbb{Z}}_p, {\mathbb{Z}}\oplus {\mathbb{Z}}\}$ are studied by building upon the recent progress obtained on the simply connected realm. Results on the botany of simply connected 4-manifolds not available in the literature are extended.

We prove the CR version of the Obata's result for the first eigenvalue of the sub-Laplacian in the setting of a compact strictly pseudoconvex pseudohermitian three-dimensional manifold with non-negative CR-Paneitz operator which satisfies a Lichnerowicz-type condition. We show that if the first positive eigenvalue of the sub-Laplacian takes the smallest possible value, then, up to a homothety of the pseudohermitian structure, the manifold is the standard Sasakian three-dimensional unit sphere.

Let G and T be topological groups, α : T → Aut(G) a homomorphism defining a continuous action of T on G and G♯ := G ⋊αT the corresponding semidirect product group. In this paper, we address several issues concerning irreducible continuous unitary representations (π♯, ${\mathcal{H}}$) of G♯ whose restriction to G remains irreducible. First, we prove that, for T = ${\mathbb R}$, this is the case for any irreducible positive energy representation of G♯, i.e. for which the one-parameter group Ut := π♯(1,t) has non-negative spectrum. The passage from irreducible unitary representations of G to representations of G♯ requires that certain projective unitary representations are continuous. To facilitate this verification, we derive various effective criteria for the continuity of projective unitary representations. Based on results of Borchers for W*-dynamical systems, we also derive a characterization of the continuous positive definite functions on G that extend to G♯.

In this paper we are concerned with a class of p(x)-Kirchhoff equation where the non-linearity has non-standard growth and contains a bi-non-local term. We prove, by using variational methods (Mountain Pass Theorem and Ekeland Variational Principle), several results on the existence of positive solutions.

For an abelian extension L/K of number fields, the Equivariant Tamagawa Number Conjecture (ETNC) at s = 0, which is equivalent to the Lifted Root Number Conjecture (LRNC), implies Rubin's Conjecture by work of Burns [3]. We show that, for relative biquadratic extensions L/K satisfying a certain condition on the splitting of places, Rubin's Conjecture in turn implies the ETNC/LRNC. We conclude with some examples.

For a derivation δ of a commutative Noetherian ${\mathbb C}$-algebra A, a homeomorphism is established between the prime spectrum of the Ore extension A[z;δ] and the Poisson prime spectrum of the polynomial algebra A[z] endowed with the Poisson bracket such that {A,A}=0 and {z,a}=δ(a) for all a ∈ A.

Let n be any positive integer and p be any prime. Also, let X be any spectrum and let K(n) denote the nth Morava K-theory spectrum. Then we construct a descent spectral sequence with abutment π∗(LK(n)(X)) and E2-term equal to the continuous cohomology of Gn, the extended Morava stabilizer group, with coefficients in a certain discrete Gn-module that is built from various homotopy fixed point spectra of the Morava module of X. This spectral sequence can be contrasted with the K(n)-local En-Adams spectral sequence for π∗(LK(n)(X)), whose E2-term is not known to always be equal to a continuous cohomology group.

For a given (d−1)-dimensional simplicial complex Γ, we denote its h-vector by h(Γ)=(h0(Γ),h1(Γ),. . .,hd(Γ)) and set h−1(Γ)=0. The known Swartz equality implies that if Δ is a (d−1)-dimensional Buchsbaum simplicial complex over a field, then for every 0 ≤ i ≤ d, the inequality ihi(Δ)+(d−i+1)hi−1(Δ) ≥ 0 holds true. In this paper, by using these inequalities, we give a simple proof for a result of Terai (N. Terai, On h-vectors of Buchsbaum Stanley–Reisner rings, Hokkaido Math. J. 25(1) (1996), 137–148) on the h-vectors of Buchsbaum simplicial complexes. We then generalize the Swartz equality (E. Swartz, Lower bounds for h-vectors of k-CM, independence, and broken circuit complexes, SIAM J. Discrete Math. 18(3) (2004/05), 647–661), which in turn leads to a generalization of the above-mentioned inequalities for Cohen–Macaulay simplicial complexes in co-dimension t.

A group is called a T-group if all its subnormal subgroups are normal. It is proved here that if G is a periodic (generalized) soluble group in which all subnormal subgroups of infinite rank are normal, then either G is a T-group or it has finite rank. It follows that if G is an arbitrary group whose Fitting subgroup has infinite rank, then G has the property T if and only if all its subnormal subgroups of infinite rank are normal.

A natural class of ideals, almost isometric ideals, of Banach spaces is defined and studied. The motivation for working with this class of subspaces is our observation that they inherit diameter 2 properties and the Daugavet property. Lindenstrauss spaces are known to be the class of Banach spaces that are ideals in every superspace; we show that being an almost isometric ideal in every superspace characterizes the class of Gurariy spaces.

In this paper we investigate the nature of family of pairs of separable Banach spaces (X, Y) such that $\mathcal{K}(X,Y)$ is complemented in $\mathcal{L}(X,Y)$. It is proved that the family of pairs (X,Y) of separable Banach spaces such that $\mathcal{K}(X,Y)$ is complemented in $\mathcal{L}(X,Y)$ is not Borel, endowed with the Effros–Borel structure.

We introduce the concept of an E-valued function algebra, a type of Banach algebra that consists of continuous E-valued functions on some compact Hausdorff space, where E is a Banach algebra. We present some basic results about such algebras, having to do with the Shilov boundary and the set of peak points of some commutative E-valued function algebras. We give some specific examples.

Let 1 ≤ p < ∞. A sequence 〈 xn 〉 in a Banach space X is defined to be p-operator summable if for each 〈 fn 〉 ∈ lw*p(X*) we have 〈〈 fn(xk)〉k〉n ∈ lsp(lp). Every norm p-summable sequence in a Banach space is operator p-summable whereas in its turn every operator p-summable sequence is weakly p-summable. An operator T ∈ B(X, Y) is said to be p-limited if for every 〈 xn 〉 ∈ lpw(X), 〈 Txn 〉 is operator p-summable. The set of all p-limited operators forms a normed operator ideal. It is shown that every weakly p-summable sequence in X is operator p-summable if and only if every operator T ∈ B(X, lp) is p-absolutely summing. On the other hand, every operator p-summable sequence in X is norm p-summable if and only if every p-limited operator in B(lp', X) is absolutely p-summing. Moreover, this is the case if and only if X is a subspace of Lp(μ) for some Borel measure μ.

We develop a theory of group actions and coverings on Brauer graphs that parallels the theory of group actions and coverings of algebras. In particular, we show that any Brauer graph can be covered by a tower of coverings of Brauer graphs such that the topmost covering has multiplicity function identically one, no loops, and no multiple edges. Furthermore, we classify the coverings of Brauer graph algebras that are again Brauer graph algebras.

In response to questions by Kassabov, Nikolov and Shalev, we show that a given subset A of a finite simple group G is the image of some word map w : G × G → G if and only if (i) A contains the identity and (ii) A is invariant under Aut(G).

In this paper we generalise a result of Izuchi and Suárez (K. Izuchi and D. Suárez, Norm-closed invariant subspaces in L∞ and H∞, Glasgow Math. J. 46 (2004), 399–404) on the shift invariant subspaces of $L^\infty(\mathbb{T})$ to the non-commutative setting. Considering these subspaces as $C(\mathbb{T})$-modules contained in $L^\infty(\mathbb{T})$, we show that under some restrictions, a similar description can be given for the ${\mathfrak{B}}$-submodules of ${\mathfrak{A}}$, where ${\mathfrak{A}}$ is a C*-algebra and ${\mathfrak{B}}$ is a commutative C*-subalgebra of ${\mathfrak{A}}$. We use this to give a description of the $\mathbb{M}_n({\mathfrak{B}})$-submodules of $\mathbb{M}_n({\mathfrak{A}})$.