Angle criteria for frame sequences and frames containing a Riesz basis

We obtain a condition implying that the union of two frame sequences is also a frame sequence. Christensen found a condition for this in terms of orthogonal projections. We phrase our condition by use of the angle between closed subspaces. Also a lower bound formula is obtained. We then apply the results to find conditions for a frame containing a Riesz basis to be a Riesz frame.     

Dilations for systems of imprimitivity acting on Banach spaces

Motivated by a general dilation theory for operator-valued measures, framings and bounded linear maps on operator algebras, we consider the dilation theory of the above objects with special structures. We show that every operator-valued system of imprimitivity has a dilation to a probability spectral system of imprimitivity acting on a Banach space. This completely generalizes a well-known result which states that every frame representation of a countable group on a Hilbert space is unitarily equivalent to a subrepresentation of the left regular representation of the group. We also prove that isometric group representation induced framings on a Banach space can be dilated to unconditional bases with the same structure for a larger Banach space. This extends several known results on the dilations of frames induced by unitary group representations on Hilbert spaces.     

Systematic remarks on objectivity and frame-indifference,liquid crystal theory as an example

In this paper objectivity and material frame-indifference are systematically discussed, because changing the observer and changing the motion of a material with respect to an observer independent standard frame of reference have to be distinguished carefully. Objectivity and observer invariance of the physical laws and of the constitutive mappings are introduced. Semi-objectivity and objectivity of different time derivative operators are investigated. As examples, changing the observer in liquid crystal theory and changing the motion in linear heat conducting materials is considered.     

Constructing and evaluating alternative frames of discernment

We construct alternative frames of discernment from input belief functions. We assume that the core of each belief function is a subset of a so far unconstructed frame of discernment. The alternative frames are constructed as different cross products of unions of different cores. With the frames constructed the belief functions are combined for each alternative frame. The appropriateness of each frame is evaluated in two ways: (i) we measure the aggregated uncertainty (an entropy measure) of the combined belief functions for that frame to find if the belief functions are interacting in interesting ways, (ii) we measure the conflict in Dempster’s rule when combining the belief functions to make sure they do not exhibit too much internal conflict. A small frame typically yields a small aggregated uncertainty but a large conflict, and vice versa. The most appropriate frame of discernment is that which minimizes a probabilistic sum of the conflict and a normalized aggregated uncertainty of all combined belief functions for that frame of discernment.     

Pointfree Aspects of the Td Axiom of Classical Topology

Abstract

Abstraction from the condition defining To-spaces leads to the following notion in an arbitrary frame L: a filter F in L is called slicing if it is prime and there exist a, b e L such that a £ F, b e F, and a is covered by 6. This paper deals with various aspects of these slicing filters. As a first step, we present several results about the original td condition. Next, concerning slicing filters, we show they are completely prime and characterize them in various ways. In addition, we prove for the frames £>X of open subsets of a space X that every slicing filter is an open neighbourhood filter U(x) and X is td iff every U(x) is slicing. Further, for Top_D and Prm_D the categories of td spaces and their continuous maps, and all frames and those homomorphisms whose associated spectral maps preserve the completely prime elements, respectively, we show that the usual contravariant functors between Top and Frm induce analogous functors here, providing a dual equivalence between Top_D and the subcategory of Prm_D given by the To-spatial frames (not coinciding with the spatial ones). In addition, we show that Top_D is mono-coreflective in a suitable subcategory of Top. Finally, we provide a comparison between Jo-separation and sobriety showing they may be viewed, in some sense, as mirror images of each other.     

Weakly Pseudocompact Frames

We define the notion of weak pseudocompactness for frames and present several characterizations. Among them are characterizations in terms of strong inclusions, uniformities and cozero bases. Noteworthy is the fact that these internal characterizations are not translated from characterizations of weakly pseudocompact spaces.      

Existence of incomplete canonical Kirkman packing designs

For $u,v$ positive integers with $u\equiv v\equiv 4\left(mod6\right)$, let ICKPD$(u,v)$ denote a canonical Kirkman packing of order $u$ missing one of order $v$. In this paper, it is shown that the necessary condition for existence of an ICKPD$(u,v)$, namely $u\ge 3v+4$, is sufficient with a definite exception $(u,v)=(16,4)$, and except possibly when $v>76$, $v\equiv 4\left(mod12\right)$ and $u\in \{3v+4,3v+10\}$.     

A new identity for Parseval frames

In this paper we establish a surprising new identity for Parseval frames in a Hilbert space. Several variations of this result are given, including an extension to general frames. Finally, we discuss the derived results.