38,869 research outputs found
Quantifying Homology Classes
We develop a method for measuring homology classes. This involves three
problems. First, we define the size of a homology class, using ideas from
relative homology. Second, we define an optimal basis of a homology group to be
the basis whose elements' size have the minimal sum. We provide a greedy
algorithm to compute the optimal basis and measure classes in it. The algorithm
runs in time, where is the size of the simplicial
complex and is the Betti number of the homology group. Third, we
discuss different ways of localizing homology classes and prove some hardness
results
Coulomb Correlations and Instability of Spinless Fermion Gas in 1D and 2D
We study the stability of the ordinary Landau Fermi liquid phase for
interacting, spinless electrons. We require causality and demand that the Pauli
principle be obeyed. We find a phase diagram determined by two parameters: the
particle density and the interaction strength. We find that the homogeneous,
constant density Fermi liquid phase of a spinless Fermion gas is {\em never}
stable in 1D, but that it may have a restricted domain of stability in 2D.Comment: 11 Latex pages + 3 PostScript figures (spin0_1D.PS, spin0_2D.PS,
spin0_PH.PS
Surreal Decisions
Although expected utility theory has proven a fruitful and elegant theory in the finite realm, attempts to generalize it to infinite values have resulted in many paradoxes. In this paper, we argue that the use of John Conway's surreal numbers shall provide a firm mathematical foundation for transfinite decision theory. To that end, we prove a surreal representation theorem and show that our surreal decision theory respects dominance reasoning even in the case of infinite values. We then bring our theory to bear on one of the more venerable decision problems in the literature: Pascal's Wager. Analyzing the wager showcases our theory's virtues and advantages. To that end, we analyze two objections against the wager: Mixed Strategies and Many Gods. After formulating the two objections in the framework of surreal utilities and probabilities, our theory correctly predicts that (1) the pure Pascalian strategy beats all mixed strategies, and (2) what one should do in a Pascalian decision problem depends on what one's credence function is like. Our analysis therefore suggests that although Pascal's Wager is mathematically coherent, it does not deliver what it purports to, a rationally compelling argument that people should lead a religious life regardless of how confident they are in theism and its alternatives
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