19,489 research outputs found
L-like Combinatorial Principles and Level by Level Equivalence
We force and construct a model in which GCH and level by level equivalence between strong compactness and supercompactness hold, along with certain additional âL-like â combinatorial principles. In particular, this model satisfies the following properties: 1. âŠÎŽ holds for every successor and Mahlo cardinal ÎŽ. 2. There is a stationary subset S of the least supercompact cardinal Îș0 such that for every ÎŽ â S, €Ύ holds and ÎŽ carries a gap 1 morass. 3. A weak version of €Ύ holds for every infinite cardinal ÎŽ. 4. There is a locally defined well-ordering of the universe W, i.e., for all Îș â„ â”2 a regular cardinal, W H(Îș+) is definable over the structure ăH(Îș+),â ă by a parameter free formula. â2000 Mathematics Subject Classifications: 03E35, 03E55. â Keywords: Supercompact cardinal, strongly compact cardinal, strong cardinal, level by level equivalence between strong compactness and supercompactness, diamond, square, morass, locally defined well-ordering. âĄThe authorâs research was partially supported by PSC-CUNY grants and CUNY Collaborative Incentive grants. §The author wishes to thank the referee for helpful comments, suggestions, and corrections which have been incorporated into the current version of the paper. 1 The model constructed amalgamates and synthesizes results due to the author, the author and Cummings, and Aspero Ì and Sy Friedman. It has no restrictions on the structure of its class of supercompact cardinals and may be considered as part of Friedmanâs âouter model programmeâ.
Combinatorial Properties and Dependent choice in symmetric extensions based on L\'{e}vy Collapse
We work with symmetric extensions based on L\'{e}vy Collapse and extend a few
results of Arthur Apter. We prove a conjecture of Ioanna Dimitriou from her
P.h.d. thesis. We also observe that if is a model of ZFC, then
can be preserved in the symmetric extension of in terms of
symmetric system , if
is -distributive and is -complete.
Further we observe that if is a model of ZF + , then
can be preserved in the symmetric extension of in terms of
symmetric system , if
is -strategically closed and is
-complete.Comment: Revised versio
Experimental demonstration of quantum effects in the operation of microscopic heat engines
The heat engine, a machine that extracts useful work from thermal sources, is
one of the basic theoretical constructs and fundamental applications of
classical thermodynamics. The classical description of a heat engine does not
include coherence in its microscopic degrees of freedom. By contrast, a quantum
heat engine might possess coherence between its internal states. Although the
Carnot efficiency cannot be surpassed, and coherence can be performance
degrading in certain conditions, it was recently predicted that even when using
only thermal resources, internal coherence can enable a quantum heat engine to
produce more power than any classical heat engine using the same resources.
Such a power boost therefore constitutes a quantum thermodynamic signature. It
has also been shown that the presence of coherence results in the thermodynamic
equivalence of different quantum heat engine types, an effect with no classical
counterpart. Microscopic heat machines have been recently implemented with
trapped ions, and proposals for heat machines using superconducting circuits
and optomechanics have been made. When operated with standard thermal baths,
however, the machines implemented so far have not demonstrated any inherently
quantum feature in their thermodynamic quantities. Here we implement two types
of quantum heat engines by use of an ensemble of nitrogen-vacancy centres in
diamond, and experimentally demonstrate both the coherence power boost and the
equivalence of different heat-engine types. This constitutes the first
observation of quantum thermodynamic signatures in heat machines
Differential KO-theory: constructions, computations, and applications
We provide a systematic and detailed treatment of differential refinements of
KO-theory. We explain how various flavors capture geometric aspects in
different but related ways, highlighting the utility of each. While general
axiomatics exist, no explicit constructions seem to have appeared before. This
fills a gap in the literature in which K-theory is usually worked out leaving
KO-theory essentially untouched, with only scattered partial information in
print. We compare to the complex case, highlighting which constructions follow
analogously and which are much more subtle. We construct a pushforward and
differential refinements of genera, leading to a Riemann-Roch theorem for
-theory. We also construct the corresponding
Atiyah-Hirzebruch spectral sequence (AHSS) and explicitly identify the
differentials, including ones which mix geometric and topological data. This
allows us to completely characterize the image of the Pontrjagin character.
Then we illustrate with examples and applications, including higher tangential
structures, Adams operations, and a differential Wu formula.Comment: 105 pages, very minor changes, comments welcom
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