737,904 research outputs found
From Economic Activity to Understanding Spaces
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
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
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
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
different complete commuting sets of observables (e.g. , and
other sets). In general the particle velocities do not coincide with the
de Broglie-Bohm velocities.Comment: Published versio
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