34,012 research outputs found
Inverse Decision Modeling: Learning Interpretable Representations of Behavior
Decision analysis deals with modeling and enhancing decision processes. A
principal challenge in improving behavior is in obtaining a transparent
description of existing behavior in the first place. In this paper, we develop
an expressive, unifying perspective on inverse decision modeling: a framework
for learning parameterized representations of sequential decision behavior.
First, we formalize the forward problem (as a normative standard), subsuming
common classes of control behavior. Second, we use this to formalize the
inverse problem (as a descriptive model), generalizing existing work on
imitation/reward learning -- while opening up a much broader class of research
problems in behavior representation. Finally, we instantiate this approach with
an example (inverse bounded rational control), illustrating how this structure
enables learning (interpretable) representations of (bounded) rationality --
while naturally capturing intuitive notions of suboptimal actions, biased
beliefs, and imperfect knowledge of environments
Connection Conditions and the Spectral Family under Singular Potentials
To describe a quantum system whose potential is divergent at one point, one
must provide proper connection conditions for the wave functions at the
singularity. Generalizing the scheme used for point interactions in one
dimension, we present a set of connection conditions which are well-defined
even if the wave functions and/or their derivatives are divergent at the
singularity. Our generalized scheme covers the entire U(2) family of
quantizations (self-adjoint Hamiltonians) admitted for the singular system. We
use this scheme to examine the spectra of the Coulomb potential and the harmonic oscillator with square inverse potential , and thereby provide a general perspective for these
models which have previously been treated with restrictive connection
conditions resulting in conflicting spectra. We further show that, for any
parity invariant singular potentials , the spectrum is determined
solely by the eigenvalues of the characteristic matrix .Comment: TeX, 18 page
A Fluid Generalization of Membranes
In a certain sense a perfect fluid is a generalization of a point particle.
This leads to the question as to what is the corresponding generalization for
extended objects. The lagrangian formulation of a perfect fluid is much
generalized and this has as a particular example a fluid which is a classical
generalization of a membrane, however there is as yet no indication of any
relationship between their quantum theories.Comment: To appear in CEJP, updated to coincide with published versio
Drazin Inverses in Categories
Drazin inverses are a fundamental algebraic structure which have been
extensively deployed in semigroup theory and ring theory. Drazin inverses can
also be defined for endomorphisms in any category. However, beyond a paper by
Puystjens and Robinson from 1987, not much has been done with Drazin inverses
in category theory. As such, here we provide a survey of the theory of Drazin
inverses from a categorical perspective. We introduce Drazin categories, in
which every endomorphism has a Drazin inverse, and provide various examples
including the category of matrices over a field, the category of finite length
modules over a ring, and finite set enriched categories. We also introduce the
notion of expressive rank and prove that a category with expressive rank is
Drazin. Moreover, we study Drazin inverses in mere categories, in additive
categories, and in dagger categories. In an arbitrary category, we show how a
Drazin inverse corresponds to an isomorphism in the idempotent splitting, as
well as explain how Drazin inverses relate to Leinster's notion of eventual
image duality. In additive categories, we explore the consequences of the
core-nilpotent decomposition and the image-kernel decomposition, which we
relate back to Fitting's famous results. We then develop the notion of Drazin
inverses for pairs of opposing maps, generalizing the usual notion of Drazin
inverse for endomorphisms. As an application of this new kind of Drazin
inverse, for dagger categories, we provide a novel characterization of the
Moore-Penrose inverse in terms of being a Drazin inverse of the pair of a map
and its adjoint
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