Two Weight Inequalities for Discrete Positive Operators

We characterize two weight inequalities for general positive dyadic operators. We consider both weak and strong type inequalities, and general (p,q) mapping properties. Special cases include Sawyers Fractional Integral operator results from 1988, and the bilinear embedding inequality of Nazarov-Treil-Volberg from 1999. The method of proof is an extension of Sawyer's argument.Comment: 20 pages, 3 figures. v2 correction of minor typos v3: Correction of typos. v4: Two references adde

A Two Weight Inequality for the Hilbert transform Assuming an Energy Hypothesis

Subject to a range of side conditions, the two weight inequality for the Hilbert transform is characterized in terms of (1) a Poisson A_2 condition on the weights (2) A forward testing condition, in which the two weight inequality is tested on intervals (3) and a backwards testing condition, dual to (2). A critical new concept in the proof is an Energy Condition, which incorporates information about the distribution of the weights in question inside intervals. This condition is a consequence of the three conditions above. The Side Conditions are termed 'Energy Hypotheses'. At one endpoint they are necessary for the two weight inequality, and at the other, they are the Pivotal Conditions of Nazarov-Treil-Volberg. This new concept is combined with a known proof strategy devised by Nazarov-Treil-Volberg. A counterexample shows that the Pivotal Condition are not necessary for the two weight inequality.Comment: 60 pages, 1 figure. v3. An important revision: The Energy Condition is reformulated, a key concept of the proof, is reformulated. The main result is unchanged. v4. important display corrected. v6: The earlier versions incorrectly claimed a characterization, as was pointed out to us by S. Treil v7. Corrections in Section

Descension: The Fanon Zone(s)

The two texts that serve as bookends to the writings of Frantz Fanon, Peau noire, masques blancs and Les Damnés de la Terre are often situated as taking up two different elements and approaches to decolonization. The former dismantling the colonized psyche with aggressive deconstruction of the individual and the latter the shattering of the coercive regime of empire. This edition affords us the opportunity to linger with Black Skin, White Masks and to consider its seismic resonance over the last 70 years. The thinking in this essay is preoccupied with the “zones” that appear in Black Skin, White Masks in two ways. The first means to ensure that the attention granted to the zone of nonbeing does not distract us from the existence of another zone of subject (re)creation found in the text, the zone of hachures. The ambition here is to do a bit more that present a taxonomy of Fanon’s zones but to demonstrate the manner in which they function as essential components in a chain of reasoning and activity that is aimed at decolonization

Aurora Volume 83

College formerly located at Olivet, Illinois and known as Olivet University, 1912-1923; Olivet College, 1923-1939, Olivet Nazarene College, 1940-1986, Olivet Nazarene University, 1986-https://digitalcommons.olivet.edu/arch_yrbks/1149/thumbnail.jp

Relaxation timescales and decay of correlations in a long-range interacting quantum simulator

We study the time evolution of correlation functions in long-range interacting quantum Ising models. For a large class of initial conditions, exact analytic results are obtained in arbitrary lattice dimension, both for ferromagnetic and antiferromagnetic coupling, and hence also in the presence of geometric frustration. In contrast to the nearest-neighbour case, we find that correlations decay like stretched or compressed exponentials in time. Provided the long-range character of the interactions is sufficiently strong, pronounced prethermalization plateaus are observed and relaxation timescales are widely separated. Specializing to a triangular lattice in two spatial dimensions, we propose to utilize these results for benchmarking of a recently developed ion-trap based quantum simulator.Comment: 19 pages, 6 figures; v2: one section removed, appendices added; v3: upper bound corrected + minor corrections; v4: as publishe