Connectivity in frame matroids

We discuss the relationship between the vertical connectivity of a biased graph Ω and the Tutte connectivity of the frame matroid of Ω (also known as the bias matroid of Ω).     

On the duality of fusion frames

The fusion frames were considered recently by P.G. Casazza, G. Kutyniok and S. Li in connection with distributed processing and are related to the construction of global frames from local frames. In this paper we give new results on the duality of fusion frames in Hilbert spaces.     

A bitopological point-free approach to compactifications

We study structures called d-frames which were developed by the last two authors for a bitopological treatment of Stone duality. These structures consist of a pair of frames thought of as the opens of two topologies, together with two relations which serve as abstractions of disjointness and covering of the space. With these relations, the topological separation axioms regularity and normality have natural analogues in d-frames. We develop a bitopological point-free notion of complete regularity and characterise all compactifications of completely regular d-frames. Given that normality of topological spaces does not behave well with respect to products and subspaces, probably the most surprising result is this: The category of d-frames has a normal coreflection, and the Stone-?ech compactification factors through it. Moreover, any compactification can be obtained by first producing a regular normal d-frame and then applying the Stone-?ech compactification to it. Our bitopological compactification subsumes all classical compactifications of frames as well as Smyth?s stable compactification.     

On Stiefel’s parallelizability of 3-manifolds

We give a new elementary proof of the parallelizability of closed orientable 3-manifolds. We use as the main tool the fact that any such manifold admits a Heegaard splitting.     

Localization of Frames

Motivated by Balan, Casazza, Heil, and Landau's results on localization of frames in a separable Hilbert space ?, we investigate the transitivity of localization relation of the frame ? = {f _i } _i∈I and sequence ? = {e _j } _j∈G in ? with respect to the associated map a: I → G, where G is a discrete abelian group and I is a index set. We also study some properties of ? _p -column decay, ? _p -row decay, weak homogeneous approximation property, and strong homogeneous approximation property of frame ? = {f _i } _i∈I , and sequence ? = {e _j } _j∈G , with respect to the associated map a.     

Recommendations for construction of a nonlinear international Terrestrial Reference Frame

The problems of ITRF2008,the latest International Terrestrial Reference Frame,are pointed out and analyzed as follows:(1) ITRF is not a mm-level Terrestrial Reference Frame;(2) the origin of ITRF is neither the Earth's center of mass (CM) nor the center of figure (CF);(3) the scale of ITRF is not a uniform system in the sense of the gravitational theory of relativity.These problems result from the linear hypothesis used in the establishment and maintenance of ITRF,which includes the linear hypothesis of the...     

Parseval Frame Wavelet Multipliers in L2（Rd）

Let A be a d × d real expansive matrix. An A-dilation Parseval frame wavelet is a function ?? ?? L ²(? ^d ), such that the set $ \left\{ {\left| {\det A} \right|^{\frac{n} {2}} \psi \left( {A^n t - \ell } \right):n \in \mathbb{Z},\ell \in \mathbb{Z}^d } \right\} $ forms a Parseval frame for L ²(? ^d ). A measurable function f is called an A-dilation Parseval frame wavelet multiplier if the inverse Fourier transform of d??? is an A-dilation Parseval frame wavelet whenever ?? is an A-dilation Parseval frame wavelet, where ??? denotes the Fourier transform of ??. In this paper, the authors completely characterize all A-dilation Parseval frame wavelet multipliers for any integral expansive matrix A with |det(A)| = 2. As an application, the path-connectivity of the set of all A-dilation Parseval frame wavelets with a frame MRA in L ²(? ^d ) is discussed.