Immunity properties and strong positive reducibilities

We use certain strong Q-reducibilities, and their corresponding strong positive reducibilities, to characterize the hyperimmune sets and the hyperhyperimmune sets: if A is any infinite set then A is hyperimmune (respectively, hyperhyperimmune) if and only if for every infinite subset B of A, one has ${\overline{K}\not\le_{\rm ss} B}$ (respectively, ${\overline{K}\not\le_{\overline{\rm s}} B}$ ): here ${\le_{\overline{\rm s}}}$ is the finite-branch version of s-reducibility, ??_ss is the computably bounded version of ${\le_{\overline{\rm s}}}$ , and ${\overline{K}}$ is the complement of the halting set. Restriction to ${\Sigma^0_2}$ sets provides a similar characterization of the ${\Sigma^0_2}$ hyperhyperimmune sets in terms of s-reducibility. We also show that no ${A \geq_{\overline{\rm s}}\overline{K}}$ is hyperhyperimmune. As a consequence, ${\deg_{\rm s}(\overline{K})}$ is hyperhyperimmune-free, showing that the hyperhyperimmune s-degrees are not upwards closed.     

Estimate for index of hypersurfaces in spheres with null higher order mean curvature

In a recent paper (Barros, Sousa in: Kodai Math. J. 2009) the authors proved that closed oriented non-totally geodesic minimal hypersurfaces of the Euclidean unit sphere have index of stability greater than or equal to n + 3 with equality occurring at only Clifford tori provided their second fundamental forms A satisfy the pinching: |A|² ≥ n. The natural generalization for this pinching is ?(r + 2)S _r+2 ≥ (n ? r)S _r  > 0. Under this condition we shall extend such result for closed oriented hypersurface Σ ⁿ of the Euclidean unit sphere ${\mathbb{S}^{n+1}}$ with null S _r+1 mean curvature by showing that the index of r-stability, ${Ind_{\Sigma^n}^{r}}$ , also satisfies ${Ind_{\Sigma^n}^{r}\ge n+3}$ . Instead of the previous hypothesis if we consider ${\frac{S_{r+2}}{{S_r}}}$ constant we have the same conclusion. Moreover, we shall prove that, up to Clifford tori, closed oriented hypersurfaces ${\Sigma^{n}\subset \mathbb{S}^{n+1}}$ with S _r+1 = 0 and S _r+2 < 0 have index of r-stability greater than or equal to 2n + 5.     

K(π, 1) and word problems for infinite type Artin–Tits groups, and applications to virtual braid groups

Let Γ be a Coxeter graph, let (W, S) be its associated Coxeter system, and let (A, Σ) be its associated Artin–Tits system. We regard W as a reflection group acting on a real vector space V. Let I be the Tits cone, and let E _Γ be the complement in I + iV of the reflecting hyperplanes. Recall that Salvetti, Charney and Davis have constructed a simplicial complex Ω(Γ) having the same homotopy type as E _Γ. We observe that, if ${T \subset S}$ , then Ω(Γ _T ) naturally embeds into Ω (Γ). We prove that this embedding admits a retraction ${\pi_T: \Omega(\Gamma) \to \Omega (\Gamma_T)}$ , and we deduce several topological and combinatorial results on parabolic subgroups of A. From a family ${\mathcal{S}}$ of subsets of S having certain properties, we construct a cube complex Φ, we show that Φ has the same homotopy type as the universal cover of E _Γ, and we prove that Φ is CAT(0) if and only if ${\mathcal{S}}$ is a flag complex. We say that ${X \subset S}$ is free of infinity if Γ _X has no edge labeled by ∞. We show that, if ${E_{\Gamma_X}}$ is aspherical and A _X has a solution to the word problem for all ${X \subset S}$ free of infinity, then E _Γ is aspherical and A has a solution to the word problem. We apply these results to the virtual braid group VB _n . In particular, we give a solution to the word problem in VB _n , and we prove that the virtual cohomological dimension of VB _n is n?1.     

Hofer geometry of a subset of a symplectic manifold

To every closed subset X of a symplectic manifold (M, ω) we associate a natural group of Hamiltonian diffeomorphisms Ham (X, ω). We equip this group with a semi-norm ${\Vert\cdot\Vert^{X, \omega}}$ , generalizing the Hofer norm. We discuss Ham (X, ω) and ${\Vert\cdot\Vert^{X, \omega}}$ if X is a symplectic or isotropic submanifold. The main result involves the relative Hofer diameter of X in M. Its first part states that for the unit sphere in ${\mathbb{R}^{2n}}$ this diameter is bounded below by ${\frac{\pi}{2}}$ , if n ≥ 2. Its second part states that for n ≥ 2 and d ≥ n there exists a compact subset X of the closed unit ball in ${\mathbb{R}^{2n}}$ , such that X has Hausdorff dimension at most d + 1 and relative Hofer diameter bounded below by π / k(n, d), where k(n, d) is an explicitly defined integer.     

On Bipartite Distance-Regular Graphs with Intersection Numbers c i = (q i ? 1)/(q ? 1)

Let q denote an integer at least two. Let ?? denote a bipartite distance-regular graph with diameter D ?? 3 and intersection numbers c _i = (q ⁱ ? 1)/(q ? 1), 1 ?? i ?? D. Let X denote the vertex set of ?? and let ${V = \mathbb{C}^X}$ denote the vector space over ${\mathbb{C}}$ consisting of column vectors whose coordinates are indexed by X and whose entries are in ${\mathbb{C}}$ . For ${z \in X}$ , let ${{\hat z}}$ denote the vector in V with a 1 in the z-coordinate and 0 in all other coordinates. Fix ${x, y \in X}$ such that ?(x, y) = 2, where ? denotes the path-length distance function. For 0 ?? i, j ?? D define ${w_{ij} = \sum {\hat z}}$ , where the sum is over all ${z \in X}$ such that ?(x, z) = i and ?(y, z) = j. We define W?=?span{w _ij | 0 ?? i, j ?? D}. In this paper we consider the space ${MW={\rm span} \{mw \mid m \in M, w \in W\}}$ , where M is the Bose?CMesner algebra of ??. We observe that MW is the minimal A-invariant subspace of V which contains W, where A is the adjacency matrix of ??. We give a basis for MW that is orthogonal with respect to the Hermitean dot product. We compute the square-norm of each basis vector. We compute the action of A on the basis. For the case in which ?? is the dual polar graph D _D (q) we show that the basis consists of the characteristic vectors of the orbits of the stabilizer of x and y in the automorphism group of ??.     

A classification of terminal quartic 3-folds and applications to rationality questions

This paper studies the birational geometry of terminal Gorenstein Fano 3-folds. If Y is not ${\mathbb{Q}}$ -factorial, in most cases, it is possible to describe explicitly the divisor class group Cl Y by running a Minimal Model Program on X, a small ${\mathbb{Q}}$ -factorialization of Y. In this case, the generators of Cl Y/ Pic Y are ??topological traces?? of K-negative extremal contractions on X. One can show, as an application of these methods, that a number of families of non-factorial terminal Gorenstein Fano 3-folds are rational. In particular, I give some examples of rational quartic hypersurfaces ${Y_4 \subset \mathbb{P}^4}$ with rk Cl Y =?2 and show that when rk Cl Y ???6, Y is always rational.     

Description of Closed Maximal Ideals in Topological Algebras of Continuous Vector-Valued Functions

Let X be a completely regular Hausdorff space, A be a unital locally convex algebra with jointly continuous multiplication and C(X,A) be the algebra of all continuous A-valued functions on X equipped with the topology of \({\mathcal{K}(X)}\) -convergence. Moreover, let \({\mathfrak{M}_{\ell}(A)}\) and \({\mathfrak{M}(A)}\) denote the set of all closed maximal left and two-sided ideals in A, respectively. In this note, we describe all closed maximal left and two-sided ideals in C(X,A) and show that there exist bijections from \({\mathfrak{M}_{\ell}(C(X, A))}\) onto \({X \times \mathfrak{M}_{\ell}(A)}\) and \({\mathfrak{M}(C(X, A))}\) onto \({X \times \mathfrak{M}(A)}\) . We also present new characterizations of closed maximal ideals in C(X, A) when A is a unital commutative locally convex Gelfand–Mazur algebra with jointly continuous multiplication.     

Suitable families of boxes and kernels of staircase starshaped sets in {\mathbb{R}^d}

Let S be an orthogonal polytope in ${\mathbb{R}^d}$ . There exists a suitable family ${\mathcal{C}}$ of boxes with ${S = \cup \{C : C {\rm in} \mathcal{C}\}}$ such that the following properties hold:

The staircase kernel Ker S is a union of boxes in ${\mathcal{C}}$ . Let ${\mathcal{V}}$ be the family of vertices of boxes in ${\mathcal{C}}$ , and let ${v_o\, \epsilon \mathcal{V}}$ . Point v _o belongs to Ker S if and only if v _o sees via staircase paths in S every point w in ${\mathcal{V}}$ . Moreover, these staircase paths may be selected to consist of edges of boxes in ${\mathcal{C}}$ . Let B be a box in ${\mathcal{C}}$ with vertices of B in Ker S. Box B lies in Ker S if and only if, for some b in rel int B and for every translate H of a coordinate hyperplane at ${b, b \epsilon}$ Ker (H ∩ S). For point p in S, p belongs to Ker S if and only if, for every x in S, there exist some p ? x geodesic λ (p, x) and some corresponding ${\mathcal{C}}$ - chain D containing λ (p, x) such that D is staircase starshaped at p.

     

Projective Hausdorff gaps

Todor?evi? (Fund Math 150(1):55–66, 1996) shows that there is no Hausdorff gap (A, B) if A is analytic. In this note we extend the result by showing that the assertion “there is no Hausdorff gap (A, B) if A is coanalytic” is equivalent to “there is no Hausdorff gap (A, B) if A is ${{\bf \it{\Sigma}}^{1}_{2}}$ ”, and equivalent to ${\forall r \; (\aleph_1^{L[r]}\,< \aleph_1)}$ . We also consider real-valued games corresponding to Hausdorff gaps, and show that ${\mathsf{AD}_\mathbb{R}}$ for pointclasses Γ implies that there are no Hausdorff gaps (A, B) if ${{\it{A}} \in {\bf \it{\Gamma}}}$ .     

Inequivalent measures of noncompactness

Two homogeneous measures of noncompactness ?? and ?? on an infinite dimensional Banach space X are called ??equivalent?? if there exist positive constants b and c such that b ??(S)??? ??(S)??? c ??(S) for all bounded sets ${S\subset X}$ . If such constants do not exist, the measures of noncompactness are ??inequivalent.?? We ask a foundational question which apparently has not previously been considered: For what infinite dimensional Banach spaces do there exist inequivalent measures of noncompactness on X? We provide here the first examples of inequivalent measures of noncompactness. We prove that such inequivalent measures exist if X is a Hilbert space; or if (??, ??,???) is a general measure space, 1??? p??? ??, and X?=?L ^p (??, ??,???); or if K is a compact Hausdorff space and X?=?C(K); or if K is a compact metric space, 0?<??? ?? 1, and X?=?C ^0,??(K), the Banach space of H?lder continuous functions with H?lder exponent ??. We also prove the existence of such inequivalent measures of noncompactness if ?? is an open subset of ${\mathbb{R}^n}$ and X is the Sobolev space W ^m,p (??). Our motivation comes from questions about existence of eigenvectors of homogeneous, continuous, order-preserving cone maps f : C??C and from the closely related issue of giving the proper definition of the ??cone essential spectral radius?? of such maps. These questions are considered in the companion paper [28]; see, also, [27].     

On Sharing Values of Meromorphic Functions and Their Differences

For a meromorphic function f in the complex plane, we prove that if f is a finite order transcendental entire function which has a finite Borel exceptional value a, if ${f(z+\eta)\not\equiv f(z)}$ for some ${\eta\in \mathbb{C}}$ , and if f(z + η) ? f(z) and f(z) share the value a CM, then $$ a=0 \quad {\rm and} \quad \frac{f(z+\eta)-f(z)}{f(z)}=A, $$ where A is a nonzero constant. We also consider problems on sharing values of meromorphic functions and their differences when their orders are not an integer or infinite.     

Full regularity of a free boundary problem with two phases

Let ?? be a bounded domain in ${\mathbb{R}^{n}, n\geq2}$ . We use ${\mathcal{M}_{\Omega}}$ to denote the collection of all pairs of (A, u) such that ${A\subset\Omega}$ is a set of finite perimeter and ${u\in H^{1}\left( \Omega\right)}$ satisfies $$u\left( x\right) =0\quad\text{a.e.}x\in A.$$ We consider the energy functional $$E_{\Omega}\left( A,u\right) =\int\limits_{\Omega}\left\vert\triangledown u\right\vert ^{2}+P_{\Omega}\left( A\right)$$ defined on ${\mathcal{M}_{\Omega}}$ , where P _??(A) denotes the perimeter of A inside ??. Let ${\left( A,u\right)\in\mathcal{M}_{\Omega}}$ be a minimizer with volume constraint. Our main result is that when n????7, u is locally Lipschitz and the free boundary ?A is analytic in ??.     

Convergence rates in the law of large numbers for random variables on partially ordered sets

Let (A, ≤) be a partially ordered set, {X _α} a collection of i. i. d. random variables, indexed byA. Let \(S_\alpha = \sum _{\beta \leqslant \alpha } X_\beta \) , |α|=card {β∈A, β∈α}. We study the convergence rates ofS _α/|α|. We derive for a large class of partially ordered sets theorems, like the following one: For suitabler, t with 1/2< <r/t≤1:E|X| ^t M (|X| ^t/r )<∞ andEX=μ if and only if $$S_\alpha = \sum _{\beta \leqslant \alpha } X_\beta $$ for all ε>0, where \(M(x) = \sum _{j< x} d(j)\) withd(j)=card {α∈A, |α|=j}.     

Erratic behavior of CAT(0) geodesics under G-equivariant quasi-isometries

We show that, given any connected, compact space ${Z \subset \mathbb{R}^n}$ , there exists a group G acting geometrically on two CAT(0) spaces X and Y, a G-equivariant quasi-isometry ${f\colon X\rightarrow Y}$ , and a geodesic ray c in X such that the closure of f (c), intersected with ${\partial Y}$ , is homeomorphic to Z. This characterizes all homeomorphism types of ??geodesic boundary images?? that arise in this manner.     

Tameness of holomorphic closure dimension in a semialgebraic set

Given a semianalytic set S in ${\mathbb{C}^n}$ and a point ${p \in \bar{S}}$ , there is a unique smallest complex-analytic germ X _p which contains S _p , called the holomorphic closure of S _p . We show that if S is semialgebraic then X _p is a Nash germ, for every p, and S admits a semialgebraic filtration by the holomorphic closure dimension. As a consequence, every semialgebraic subset of a complex vector space admits a semialgebraic stratification into CR manifolds.     

Factorization of Analytic Functions and Operator Inequalities

If A and B are contraction operators on a Hilbert space ${\mathcal{H}}$ that commute with a shift operator S, it is shown that A = BC for some contraction operator C on ${\mathcal{H}}$ that commutes with S if and only if ${AA^{*} \leq BB^{*}}$ .     

Difference Inequalities for the Groups and Semigroups of Operators on Banach Spaces

Let ${\mathbf{T}=\{T(t)\} _{t\in\mathbb{R}}}$ be a ??(X, F)-continuous group of isometries on a Banach space X with generator A, where ??(X, F) is an appropriate local convex topology on X induced by functionals from ${ F\subset X^{\ast}}$ . Let ?? _A (x) be the local spectrum of A at ${x\in X}$ and ${r_{A}(x):=\sup\{\vert\lambda\vert :\lambda \in \sigma_{A}(x)\},}$ the local spectral radius of A at x. It is shown that for every ${x\in X}$ and ${\tau\in\mathbb{R},}$ $$\left\Vert T(\tau) x-x\right\Vert \leq \left\vert \tau \right\vert r_{A}(x)\left\Vert x\right\Vert.$$ Moreover if ${0\leq \tau r_{A}(x)\leq \frac{\pi}{2},}$ then it holds that $$\left\Vert T(\tau) x-T(-\tau)x\right\Vert \leq 2\sin \left(\tau r_{A}(x)\right)\left\Vert x\right\Vert.$$ Asymptotic versions of these results for C ₀-semigroup of contractions are also obtained. If ${\mathbf{T}=\{T(t)\}_{t\geq 0}}$ is a C ₀-semigroup of contractions, then for every ${x\in X}$ and ????? 0, $$\underset{t\rightarrow \infty }{\lim } \left\Vert T( t+\tau) x-T(t) x\right\Vert\leq\tau\sup\left\{ \left\vert \lambda \right\vert :\lambda \in\sigma_{A}(x)\cap i \mathbb{R} \right\} \left\Vert x\right\Vert. $$ Several applications are given.