37,341 research outputs found
Extending du Bois-Reymond's Infinitesimal and Infinitary Calculus Theory
The discovery of the infinite integer leads to a partition between finite and
infinite numbers. Construction of an infinitesimal and infinitary number
system, the Gossamer numbers. Du Bois-Reymond's much-greater-than relations and
little-o/big-O defined with the Gossamer number system, and the relations
algebra is explored. A comparison of function algebra is developed. A transfer
principle more general than Non-Standard-Analysis is developed, hence a
two-tier system of calculus is described. Non-reversible arithmetic is proved,
and found to be the key to this calculus and other theory. Finally sequences
are partitioned between finite and infinite intervals.Comment: Resubmission of 6 other submissions. 99 page
Kernel Belief Propagation
We propose a nonparametric generalization of belief propagation, Kernel
Belief Propagation (KBP), for pairwise Markov random fields. Messages are
represented as functions in a reproducing kernel Hilbert space (RKHS), and
message updates are simple linear operations in the RKHS. KBP makes none of the
assumptions commonly required in classical BP algorithms: the variables need
not arise from a finite domain or a Gaussian distribution, nor must their
relations take any particular parametric form. Rather, the relations between
variables are represented implicitly, and are learned nonparametrically from
training data. KBP has the advantage that it may be used on any domain where
kernels are defined (Rd, strings, groups), even where explicit parametric
models are not known, or closed form expressions for the BP updates do not
exist. The computational cost of message updates in KBP is polynomial in the
training data size. We also propose a constant time approximate message update
procedure by representing messages using a small number of basis functions. In
experiments, we apply KBP to image denoising, depth prediction from still
images, and protein configuration prediction: KBP is faster than competing
classical and nonparametric approaches (by orders of magnitude, in some cases),
while providing significantly more accurate results
The role of language in mathematical development: Evidence from children with specific language impairments
A sample (n=48) of eight year olds with Specific Language Impairments is compared with age-matched (n=55) and language matched controls (n=55) on a range of tasks designed to test the interdependence of language and mathematical development. Performance across tasks varies substantially in the SLI group, showing profound deficits in production of the count word sequence and basic calculation and significant deficits in understanding of the place-value principle in Hindu-Arabic notation. Only in understanding of arithmetic principles does SLI performance approximate that of age-matched-controls, indicating that principled understanding can develop even where number sequence production and other aspects of number processing are severely compromised
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