3,129 research outputs found

    Projective tensor products and Apq spaces

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    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

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    A classical result of Duffin and Schaeffer gives conditions under which a discrete collection of characters on R\mathbb{R}, restricted to E=(−1/2,1/2)E = (-1/2, 1/2), forms a Hilbert-space frame for L2(E)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 HnH_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 L2(E)L^2(E) for suitable subsets EE of HnH_n in terms of representations of HnH_n

    Coordinating Complementary Waveforms for Sidelobe Suppression

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    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

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    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

    The Training and Supervision of Family Therapists in the Republic of Ireland

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

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    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

    Dismantling the Mine Act

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