15,832 research outputs found
Split structures in general relativity and the Kaluza-Klein theories
We construct a general approach to decomposition of the tangent bundle of
pseudo-Riemannian manifolds into direct sums of subbundles, and the associated
decomposition of geometric objects. An invariant structure {\cal H}^r defined
as a set of r projection operators is used to induce decomposition of the
geometric objects into those of the corresponding subbundles. We define the
main geometric objects characterizing decomposition. Invariant non-holonomic
generalizations of the Gauss-Codazzi-Ricci's relations have been obtained. All
the known types of decomposition (used in the theory of frames of reference, in
the Hamiltonian formulation for gravity, in the Cauchy problem, in the theory
of stationary spaces, and so on) follow from the present work as special cases
when fixing a basis and dimensions of subbundles, and parameterization of a
basis of decomposition. Various methods of decomposition have been applied here
for the Unified Multidimensional Kaluza-Klein Theory and for relativistic
configurations of a perfect fluid. Discussing an invariant form of the
equations of motion we have found the invariant equilibrium conditions and
their 3+1 decomposed form. The formulation of the conservation law for the curl
has been obtained in the invariant form.Comment: 30 pages, RevTeX, aps.sty, some additions and corrections, new
references adde
Categorical invariance and structural complexity in human concept learning
An alternative account of human concept learning based on an invariance measure of the categorical\ud
stimulus is proposed. The categorical invariance model (CIM) characterizes the degree of structural\ud
complexity of a Boolean category as a function of its inherent degree of invariance and its cardinality or\ud
size. To do this we introduce a mathematical framework based on the notion of a Boolean differential\ud
operator on Boolean categories that generates the degrees of invariance (i.e., logical manifold) of the\ud
category in respect to its dimensions. Using this framework, we propose that the structural complexity\ud
of a Boolean category is indirectly proportional to its degree of categorical invariance and directly\ud
proportional to its cardinality or size. Consequently, complexity and invariance notions are formally\ud
unified to account for concept learning difficulty. Beyond developing the above unifying mathematical\ud
framework, the CIM is significant in that: (1) it precisely predicts the key learning difficulty ordering of\ud
the SHJ [Shepard, R. N., Hovland, C. L.,&Jenkins, H. M. (1961). Learning and memorization of classifications.\ud
Psychological Monographs: General and Applied, 75(13), 1-42] Boolean category types consisting of three\ud
binary dimensions and four positive examples; (2) it is, in general, a good quantitative predictor of the\ud
degree of learning difficulty of a large class of categories (in particular, the 41 category types studied\ud
by Feldman [Feldman, J. (2000). Minimization of Boolean complexity in human concept learning. Nature,\ud
407, 630-633]); (3) it is, in general, a good quantitative predictor of parity effects for this large class of\ud
categories; (4) it does all of the above without free parameters; and (5) it is cognitively plausible (e.g.,\ud
cognitively tractable)
Tensor Regression with Applications in Neuroimaging Data Analysis
Classical regression methods treat covariates as a vector and estimate a
corresponding vector of regression coefficients. Modern applications in medical
imaging generate covariates of more complex form such as multidimensional
arrays (tensors). Traditional statistical and computational methods are proving
insufficient for analysis of these high-throughput data due to their ultrahigh
dimensionality as well as complex structure. In this article, we propose a new
family of tensor regression models that efficiently exploit the special
structure of tensor covariates. Under this framework, ultrahigh dimensionality
is reduced to a manageable level, resulting in efficient estimation and
prediction. A fast and highly scalable estimation algorithm is proposed for
maximum likelihood estimation and its associated asymptotic properties are
studied. Effectiveness of the new methods is demonstrated on both synthetic and
real MRI imaging data.Comment: 27 pages, 4 figure
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