Domesticity in Generalized Quadrangles

An automorphism of a generalized quadrangle is called domestic if it maps no chamber, which is here an incident point-line pair, to an opposite chamber. We call it point-domestic if it maps no point to an opposite one and line-domestic if it maps no line to an opposite one. It is clear that a duality in a generalized quadrangle is always point-domestic and linedomestic. In this paper, we classify all domestic automorphisms of generalized quadrangles. Besides three exceptional cases occurring in the small quadrangles with orders (2, 2), (2, 4), and (3, 5), all domestic collineations are either point-domestic or line-domestic. Up to duality, they fall into one of three classes: Either they are central collineations, or they fix an ovoid, or they fix a large full subquadrangle. Remarkably, the three exceptional domestic collineatons in the small quadrangles mentioned above all have order 4.     

Frames and bases of subspaces in Hilbert spaces

In this paper we develop the frame theory of subspaces for separable Hilbert spaces. We will show that for every Parseval frame of subspaces {Wi}_i∈I for a Hilbert space H, there exists a Hilbert space K⊇H and an orthonormal basis of subspaces {Ni}_i∈I for K such that Wi=P(Ni), where P is the orthogonal projection of K onto H. We introduce a new definition of atomic resolution of the identity in Hilbert spaces. In particular, we define an atomic resolution operator for an atomic resolution of the identity, which even yield a reconstruction formula.     

Root System of a Perturbation of a Selfadjoint Operator with Discrete Spectrum

We analyze the perturbations T?+?B of a selfadjoint operator T in a Hilbert space H with discrete spectrum ${\{ t_k\}, T \phi_k = t_k \phi_k}$ . In particular, if t _k+1 ? t _k ?? ck ^{?? - 1}, ?? > 1/2 and ${\| B \phi_k \| = o(k^{\alpha - 1})}$ then the system of root vectors of T?+?B, eventually eigenvectors of geometric multiplicity 1, is an unconditional basis in H (Theorem 6). Under the assumptions ${t_{k+p} - t_k \geq d > 0, \forall k}$ (with d and p fixed) and ${\| B \phi_k \| \rightarrow 0}$ a Riesz system {P _k } of projections on invariant subspaces of T?+?B, Rank P _k ?? p, is constructed (Theorem 3).     

Exact solution of the hypergraph Turán problem for k-uniform linear paths

A k-uniform linear path of length ?, denoted by ? _? ^(k) , is a family of k-sets {F ₁,...,F _? such that |F _i ∩ F _i+1|=1 for each i and F _i ∩ F _bj = \(\not 0\) whenever |i?j|>1. Given a k-uniform hypergraph H and a positive integer n, the k-uniform hypergraph Turán number of H, denoted by ex _k (n, H), is the maximum number of edges in a k-uniform hypergraph \(\mathcal{F}\) on n vertices that does not contain H as a subhypergraph. With an intensive use of the delta-system method, we determine ex _k (n, P _? ^(k) exactly for all fixed ? ≥1, k≥4, and sufficiently large n. We show that $ex_k (n,\mathbb{P}_{2t + 1}^{(k)} ) = (_{k - 1}^{n - 1} ) + (_{k - 1}^{n - 2} ) + \cdots + (_{k - 1}^{n - t} )$ . The only extremal family consists of all the k-sets in [n] that meet some fixed set of t vertices. We also show that $ex(n,\mathbb{P}_{2t + 2}^{(k)} ) = (_{k - 1}^{n - 1} ) + (_{k - 1}^{n - 2} ) + \cdots + (_{k - 1}^{n - t} ) + (_{k - 2}^{n - t - 2} )$ , and describe the unique extremal family. Stability results on these bounds and some related results are also established.     

Conjugations on 6-manifolds

Conjugation spaces are spaces with an involution such that the fixed point set of the involution has \({\mathbb{Z} _2}\)-cohomology ring isomorphic to the \({\mathbb{Z} _2}\)-cohomology of the space itself, with the difference that all degrees are divided by two (e.g. \({\mathbb{C} {\rm P}^n}\) with the complex conjugation has \({\mathbb{R} {\rm P}^n}\) as fixed point set). One also requires that a certain conjugation equation is fulfilled. We give a new characterisation of conjugation spaces and apply it to the following realization problem: given M, a closed orientable 3-manifold, does there exist a simply connected 6-manifold X and a conjugation on X with fixed point set M? We give an affirmative answer.     

Zeros of polynomials embedded in an orthogonal sequence

Let $\{x_{k,n}\}_{k=1}^n$ and $\{x_{k,n+1}\}_{k=1}^{n+1}$ , n?????, be two given sets of real distinct points with x _1,n?+?1?<?x _1,n ?<?x _2,n?+?1?<?...?<?x _n,n ?<?x _n?+?1,n?+?1. Wendroff (cf. Proc Am Math Soc 12:554?C555, 1961) proved that if $p_n(x)=\displaystyle{\prod\limits_{k=1}^n(x-x_{k,n})}$ and $p_{n+1}(x)=\displaystyle \prod\limits_{k=1}^{n+1}(x-x_{k,n+1})$ then p _n and p _n?+?1 can be embedded in a non-unique infinite monic orthogonal sequence $\{p_n\}_{n=0}^{\infty}$ . We investigate the connection between the zeros of p _n?+?2 and the two coefficients b _n?+?1????? and ?? _n?+?1?>?0, which are chosen arbitrarily, that define p _n?+?2 via the three term recurrence relation $$ p_{n+2}(x)=(x-b_{n+1})p_{n+1}(x)-\lambda_{n+1}p_n(x). $$      

Polygons in Minkowski three space and parabolic Higgs bundles of rank 2 on \mathbb{C}{{\mathbb{P}}^1}

Consider the moduli space of parabolic Higgs bundles (E, Φ) of rank two on ??¹ such that the underlying holomorphic vector bundle for the parabolic vector bundle E is trivial. It is equipped with the natural involution defined by $ \left( {E,\varPhi } \right)\mapsto \left( {E,-\varPhi } \right) $ . We study the fixed point locus of this involution. In [GM], this moduli space with involution was identified with the moduli space of hyperpolygons equipped with a certain natural involution. Here we identify the fixed point locus with the moduli spaces of polygons in Minkowski 3-space. This identification yields information on the connected components of the fixed point locus.     

Nevanlinna–Pick Interpolation and Factorization of Linear Functionals

If \mathfrakA{\mathfrak{A}} is a unital weak-* closed algebra of multiplication operators on a reproducing kernel Hilbert space which has the property \mathbbA₁(1){\mathbb{A}_1(1)}, then the cyclic invariant subspaces index a Nevanlinna–Pick family of kernels. This yields an NP interpolation theorem for a wide class of algebras. In particular, it applies to many function spaces over the unit disk including Bergman space. We also show that the multiplier algebra of a complete NP space has \mathbbA₁(1){\mathbb{A}_1(1)}, and thus this result applies to all of its subalgebras. A matrix version of this result is also established. It applies, in particular, to all unital weak-* closed subalgebras of H ^∞ acting on Hardy space or on Bergman space.     

Normal Projections in Krein Spaces

Given a complex Krein space ${\mathcal{H}}$ with fundamental symmetry J, the aim of this note is to characterize the set of J-normal projections $$\mathcal{Q}=\{Q \in L(\mathcal{H}) : Q^2=Q \,{\rm and}\, Q^{\#}Q=QQ^{\#}\}.$$ The ranges of the projections in ${\mathcal{Q}}$ are exactly those subspaces of ${\mathcal{H}}$ which are pseudo-regular. For a fixed pseudo-regular subspace ${\mathcal{S}}$ , there are infinitely many J-normal projections onto it, unless ${\mathcal{S}}$ is regular. Therefore, most of the material herein is devoted to parametrizing the set of J-normal projections onto a fixed pseudo-regular subspace ${\mathcal{S}}$ .     

Factorable surfaces in the three-dimensional Euclidean and Lorentzian spaces satisfying Δr i ?=?λ i r i

In this paper we classify the factorable surfaces in the three-dimensional Euclidean space ${\mathbb{E}^{3}}$ and Lorentzian ${\mathbb{E}_{1}^{3}}$ under the condition ??r _i ?=??? _i r _i , where ${\lambda_{i}\in\mathbb{R}}$ and ?? denotes the Laplace operator and we obtain the complete classification for those ones.     

On hyperovals of polar spaces

We derive lower and upper bounds for the size of a hyperoval of a finite polar space of rank 3. We give a computer-free proof for the uniqueness, up to isomorphism, of the hyperoval of size 126 of H(5, 4) and prove that the near hexagon ${\mathbb E_3}$ has up to isomorphism a unique full embedding into the dual polar space DH(5, 4).     

Approximation of Invariant Subspaces in Some Dirichlet-Type Spaces

The Hilbert space \(\mathcal {D}_{2}\) is the space of all holomorphic functions f defined on the open unit disc \(\mathbb {D}\) such that \({f}^{'}\) is in the Hardy Hilbert space \(\mathbf {H}^2.\) In this paper, we prove that the invariant subspaces of \(\mathcal {D}_{2}\) with respect to multiplication operator \(M_{z}\) can be approximated with finite co-dimensional invariant subspaces. We also obtain a partial result in this direction for the classical Dirichlet space.     

Wolff’s inequality in multi-parameter Morrey spaces

We prove Wolff inequalities for multi-parameter Riesz potentials and Wolff potentials in Lebesque spaces L ^p (R ^d ) and multi-parameter Morrey spaces ${L^p_\lambda (R^d)}$ , where ${R^d=R^{n_1} \times R^{n_2} \times \cdots \times R^{n_k},\, \lambda = (\lambda _1,\ldots ,\lambda _k})$ and 0?<?λ _i ≤ n _i , 1?≤ i ≤ k, in the dyadic case as well as in the non-dyadic (continuous) case.     

Self-approximation of Hurwitz zeta-functions with rational parameter

In this paper, we show the self-approximation property for Hurwitz zeta-functions with rational parameters. Namely, we prove that ζ(s?+?iατ, a/b) approximates uniformly ζ(s?+?iβτ, a/b) for infinitely many real τ , where α, β are arbitrary real numbers linearly independent over $ \mathbb{Q} $ , and s is in a compact set lying in the open right half of the critical strip.     

Hypercyclic Subspaces on Fréchet Spaces Without Continuous Norm

Known results about hypercyclic subspaces concern either Fréchet spaces with a continuous norm or the space ω. We fill the gap between these spaces by investigating Fréchet spaces without continuous norm. To this end, we divide hypercyclic subspaces into two types: the hypercyclic subspaces M for which there exists a continuous seminorm p such that ${M \cap {\rm ker} p = \{0\}}$ and the others. For each of these types of hypercyclic subspaces, we establish some criteria. This investigation permits us to generalize several results about hypercyclic subspaces on Fréchet spaces with a continuous norm and about hypercyclic subspaces on ω. In particular, we show that each infinite-dimensional separable Fréchet space supports a mixing operator with a hypercyclic subspace.     

Construction of a Jacobi matrix from spectral data

A proof is given for the existence and uniqueness of a correspondence between two pairs of sequences {a},{b} and {ω},{μ}, satisfying b_i>0 for i=1,…,n?1 and ω₁<μ₁<?<μ_n?1<ω_n, under which the symmetric Jacobi matrices J(n,a,b) and J(n?1,a,b) have eigenvalues {ω} and {μ} respectively. Here J(m,a,b) is symmetric and tridiagonal with diagonal elements a_i (i=1,…,m) and off diagonal elements b_i (i=1,…,m?1). A new concise proof is given for the known uniqueness result. The existence result is new.     

On the generic kernel filtration for modules of constant Jordan type

Let ${E \cong (\mathbb{Z}/p)^2}$ be an elementary abelian p-group of rank two k an algebraically closed field of characteristic p, and let J =?J(kE). We investigate finitely generated kE-modules M of constant Jordan type and their generic kernels ${\mathfrak{K}(M)}$ . In particular, we answer a question posed by Carlson, Friedlander, and Suslin regarding whether or not the submodules ${J^{-i} \mathfrak{K}(M)}$ have constant Jordan type for all i ≥ 0. We show that this question has an affirmative answer whenever p = 3 or ${J^2 \mathfrak{K}(M) = 0}$ . We also show that this question has a negative answer in general by constructing a kE-module M of constant Jordan type for p ≥ 5 such that ${J^{-1} \mathfrak{K}(M)}$ does not have constant Jordan type.     

The cardinality of β A (S J )

Let J be an infinite set and let $I=\mathcal{P}_{f}( J)$ . For i??I, define $\mathcal{B}_{J}( i) =\{ f\mid f:\mathcal{P}( i) \rightarrow \mathcal{P}( i) \} $ and let $$S_{J}=\{ ( i,f) \mid i\in I\text{ and } f\in \mathcal{B}_{J}( i) \}.$$ For (i,f), (k,g)??S _J , define $f\ast g:\mathcal{P}( i\cup k) \rightarrow \mathcal{P}( i\cup k) $ as follows. For $x\in \mathcal{P}( i\cup k) $ , let $$( f\ast g) ( x) =\left\{\begin{array}{l@{\quad }l}g( x) , & \text{if\ }x=\emptyset, \\g( x\cap k) , & \text{if\ }x\cap k\neq \emptyset, \\f( x) , & \text{if\ }x\in \mathcal{P}( i\backslash k)\text{ and }x\neq \emptyset.\end{array}\right.$$ Define (i,f)?(k,g)=(i??k,f?g). It is shown that (S _J ,?) is a semigroup. Let ??S _J denote the collection of all ultrafilters on the set S _J . We consider (??S _J ,?), the compact (Hausdorff) right topological semigroup that is the Stone?C?ech Compactification of the semigroup (S _J ,?) equipped with the discrete topology. Similar to the construction in Grainger (Semigroup Forum 73:234?C242, 2006), it is shown that there is an injective map A???? _A (S _J ) of $\mathcal{P}( J) $ into $\mathcal{P}( \beta S_{J}) $ such that each ?? _A (S _J ) is a closed subsemigroup of (??S _J ,?), the set ?? _J (S _J ) is the smallest ideal of (??S _J ,?) and the collection $\{ \beta_{A}( S_{J}) \mid A\in \mathcal{P}( J) \} $ is a partition of???S _J . The main result is establishing that the cardinality of??? _A (S _J ) is $2^{2^{\vert J\vert }}$ for any?A?J.