Projective tensor products and Apq spaces

The aim of this paper is to extend the notion of Apq space from its historical context in the work of Herz and to recognise such spaces as preduals of spaces of intertwining operators of induced representations as suggested by the work of Rieffel. This generalisation of Apq spaces involves considering projective tensor products of Lp spaces of Banach space-valued functions (the spaces of induced representations) and constructing a convolution of functions of such spaces. Sufficient conditions for the existence of the integral of the convolution are established. Most of this analysis depends upon an identity we derive of Radon-Nikodym derivatives of measures on homogeneous spaces involved. The elements of the generalised Apq space are shown to be cross-sections of a Banach semi-bundle over the double coset space corresponding to the groups from which the representations are induced, and their properties are duly discussed. In particular, the generalised form of the classical result Lp*Lq is a subset of Lr; where 1/r = 1/p + 1/q - 1; is shown to be true in this situation. The result that the Apq space is the predual of the space of intertwining operators is then established, under the condition that the intertwining operators can be approximated, in the ultraweak operator topology, by integral operators.Comment: 38 page

Operator-Valued Frames for the Heisenberg Group

A classical result of Duffin and Schaeffer gives conditions under which a discrete collection of characters on $\mathbb{R}$, restricted to $E = (-1/2, 1/2)$, forms a Hilbert-space frame for $L^2(E)$. For the case of characters with period one, this is just the Poisson Summation Formula. Duffin and Schaeffer show that perturbations preserve the frame condition in this case. This paper gives analogous results for the real Heisenberg group $H_n$, where frames are replaced by operator-valued frames. The Selberg Trace Formula is used to show that perturbations of the orthogonal case continue to behave as operator-valued frames. This technique enables the construction of decompositions of elements of $L^2(E)$ for suitable subsets $E$ of $H_n$ in terms of representations of $H_n$

Coordinating Complementary Waveforms for Sidelobe Suppression

We present a general method for constructing radar transmit pulse trains and receive filters for which the radar point-spread function in delay and Doppler, given by the cross-ambiguity function of the transmit pulse train and the pulse train used in the receive filter, is essentially free of range sidelobes inside a Doppler interval around the zero-Doppler axis. The transmit pulse train is constructed by coordinating the transmission of a pair of Golay complementary waveforms across time according to zeros and ones in a binary sequence P. The pulse train used to filter the received signal is constructed in a similar way, in terms of sequencing the Golay waveforms, but each waveform in the pulse train is weighted by an element from another sequence Q. We show that a spectrum jointly determined by P and Q sequences controls the size of the range sidelobes of the cross-ambiguity function and by properly choosing P and Q we can clear out the range sidelobes inside a Doppler interval around the zero- Doppler axis. The joint design of P and Q enables a tradeoff between the order of the spectral null for range sidelobe suppression and the signal-to-noise ratio at the receiver output. We establish this trade-off and derive a necessary and sufficient condition for the construction of P and Q sequences that produce a null of a desired order

Hypothesis Testing in Feedforward Networks with Broadcast Failures

Consider a countably infinite set of nodes, which sequentially make decisions between two given hypotheses. Each node takes a measurement of the underlying truth, observes the decisions from some immediate predecessors, and makes a decision between the given hypotheses. We consider two classes of broadcast failures: 1) each node broadcasts a decision to the other nodes, subject to random erasure in the form of a binary erasure channel; 2) each node broadcasts a randomly flipped decision to the other nodes in the form of a binary symmetric channel. We are interested in whether there exists a decision strategy consisting of a sequence of likelihood ratio tests such that the node decisions converge in probability to the underlying truth. In both cases, we show that if each node only learns from a bounded number of immediate predecessors, then there does not exist a decision strategy such that the decisions converge in probability to the underlying truth. However, in case 1, we show that if each node learns from an unboundedly growing number of predecessors, then the decisions converge in probability to the underlying truth, even when the erasure probabilities converge to 1. We also derive the convergence rate of the error probability. In case 2, we show that if each node learns from all of its previous predecessors, then the decisions converge in probability to the underlying truth when the flipping probabilities of the binary symmetric channels are bounded away from 1/2. In the case where the flipping probabilities converge to 1/2, we derive a necessary condition on the convergence rate of the flipping probabilities such that the decisions still converge to the underlying truth. We also explicitly characterize the relationship between the convergence rate of the error probability and the convergence rate of the flipping probabilities

A fortune is near at hand: White land buyers on the Nemaha Half-Breed Tract, 1857-1860

Throughout the 19th century, the federal government promoted the assimilation of Native Americans as individuals within white society. Allotment of land in severalty, or the granting of land to individual Indians, was one means to achieve assimilation because it was believed that Indians would adopt the lifestyle of white farmers once they received land. Though the attempt generally failed, the government remailed undeterred in its efforts to achieve that end. In 1887, Congress passed the Dawes Severalty Act which made allotment in severalty the standard policy on most reservations throughout the United States. One clear failure of allotment in severalty occurred on the Nemaha Half-Breed Tract, a reservation established in Nebraska to benefit the mixed-bloods of several Great Plains tribes. Though Congress created the reservation by treaty in 11830, it did not begin to allot the land until 1857. Once the land became available to the mixed-bloods, most of them sold their allotments to whites. This thesis describes the major purchasers of mixed-blood land on the Half-Breed Tract, including James W. Denver, the Commissioner of Indian Affairs; Stephen F. Nuckolls, the founder of Nebraska City; a group of German immigrants who sought to establish a socialistic society at their settlement called Arago; and several prominent local land speculators

A Taxonomy and Process for Structured Innovation, Creative Problem Solving and Opportunity Creating

Myriad problem-solving techniques exist, but the literature indicates that people and organisations lack appreciation of the range and nature of the techniques available, and do not fully understand the use, value and potential of such techniques. A more profound understanding of the role that different types of problem-solving technique can play and how they can be deployed more effectively in creativity and innovation processes would form a sound basis for the improvement of creative practices and innovation processes within organisations. This research aims to provide the means to improve innovation and creative problem solving by using more effective matching of participants’ cognitive styles to the techniques available. In order to achieve synergy in the relationship between the techniques and their users, this research examined the contribution that techniques make to the creative problem solving cycle, and the degree of creativity they encourage was explored first through a review of the relevant literature. This resulted in a novel classification of the techniques and the cognitive skills involved in creative problem solving. The relationship between people and techniques was investigated through a set of experiments in which individuals and groups undertook problem-solving exercises and responded to a questionnaire to evaluate their experience of the exercise. Participants’ preferred cognitive styles were determined so that problem-solving techniques could be selectively assigned to align with or be opposed to their preferred cognitive styles. Results were analysed using both qualitative and quantitative approaches. The cognitive styles provided parameters for a taxonomic framework for the techniques. An improved approach to describing personalities based on a continuum of cognitive abilities instead of a set of discrete cognitive styles was a further outcome of this work. The results demonstrate that people show significant preference for problem-solving activities and techniques that are in accord with their preferred cognitive styles. A key conclusion is that people who follow such an approach will improve their ideation productivity in terms of quantity and novelty and will gain more satisfaction from their experience than those who do not. Analysis of the purpose of creative problem solving techniques and the cognitive styles that such techniques encourage, revealed synergy between paradigms used by psychologists and those used by technologists. The synergy between paradigms established a platform for a new creative problem-solving strategy