737,904 research outputs found

    From Economic Activity to Understanding Spaces

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    This paper constructs the probability space underlying the random variable of any time dependent econometric specification. The construction links concrete economic activity, both perceived and recorded, and econometric formulations. Furthermore, it is argued that the probability events belonging to this space are forms of understanding economic activity held by each agent. The model establishes two aspects of any econometric formulation. Mainly, that learning must be unique between any two ticks of the clock and that not all forms of understandings can indeed become events in the random variable’s probability space. Finally, a model of the dependencies based on agent-based understandings, and evolution thereof, is presented as well.Knowledge intuitions probability

    Probability Theory of Random Polygons from the Quaternionic Viewpoint

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    We build a new probability measure on closed space and plane polygons. The key construction is a map, given by Knutson and Hausmann using the Hopf map on quaternions, from the complex Stiefel manifold of 2-frames in n-space to the space of closed n-gons in 3-space of total length 2. Our probability measure on polygon space is defined by pushing forward Haar measure on the Stiefel manifold by this map. A similar construction yields a probability measure on plane polygons which comes from a real Stiefel manifold. The edgelengths of polygons sampled according to our measures obey beta distributions. This makes our polygon measures different from those usually studied, which have Gaussian or fixed edgelengths. One advantage of our measures is that we can explicitly compute expectations and moments for chordlengths and radii of gyration. Another is that direct sampling according to our measures is fast (linear in the number of edges) and easy to code. Some of our methods will be of independent interest in studying other probability measures on polygon spaces. We define an edge set ensemble (ESE) to be the set of polygons created by rearranging a given set of n edges. A key theorem gives a formula for the average over an ESE of the squared lengths of chords skipping k vertices in terms of k, n, and the edgelengths of the ensemble. This allows one to easily compute expected values of squared chordlengths and radii of gyration for any probability measure on polygon space invariant under rearrangements of edges.Comment: Some small typos fixed, added a calculation for the covariance of edgelengths, added pseudocode for the random polygon sampling algorithm. To appear in Communications on Pure and Applied Mathematics (CPAM

    Space proof complexity for random 3-CNFs

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    We investigate the space complexity of refuting 3-CNFs in Resolution and algebraic systems. We prove that every Polynomial Calculus with Resolution refutation of a random 3-CNF φ in n variables requires, with high probability, distinct monomials to be kept simultaneously in memory. The same construction also proves that every Resolution refutation of φ requires, with high probability, clauses each of width to be kept at the same time in memory. This gives a lower bound for the total space needed in Resolution to refute φ. These results are best possible (up to a constant factor) and answer questions about space complexity of 3-CNFs

    Maximally Realistic Causal Quantum Mechanics

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    We recently constructed a causal quantum mechanics in 2 dim. phase space which is more realistic than the de Broglie-Bohm mechanics as it reproduces not just the position but also the momentum probability density of ordinary quantum theory. Here we present an even more ambitious construction in 2n dim. phase space. We conjecture that the causal Hamiltonian quantum mechanics presented here is `maximally realistic'. The positive definite phase space density reproduces as marginals the correct quantum probability densities of n+1n+1 different complete commuting sets of observables (e.g. q⃗\vec q, p⃗\vec p and n−1n-1 other sets). In general the particle velocities do not coincide with the de Broglie-Bohm velocities.Comment: Published versio
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