Regularization terms with sparse representations (for instance, the ℓ 1 norm regularizer) have been developed into an important tool in these applications, and a list of methods have been proposed [1 4].
Ritual elements were associated with the story even before Apuleius's version, as indicated in visual representations; for instance, a 1st-century BC sardonyx cameo depicting the wedding of Cupid and Psyche shows an attendant elevating a liknon (basket) used in Dionysiac initiation.
Other ideas have been mooted to boost Sunni representation, for instance by reserving some of the 275 seats in the new assembly for Sunni leaders.
Methods following this approach must keep a description of the background that in the simplest case can be just a reference image, but in more sophisticated algorithms becomes often a quite complex representation, for instance involving a probabilistic modeling of the background pixel values.
#Algebraic operations can be done readily on the power series representation; for instance, Euler's formula follows from Taylor series expansions for trigonometric and exponential functions.
For the last equality, we use the fact that for any g, h : [ a, b ] → R bounded functions for which the Riemann-Stieltjes integral ∫ a b h ( t ) d g ( t ) and the Riemann integral ∫ a b g ( t ) d t exist, we have the representation (see, for instance, [7]) D ( g ; h ) = ∫ a b Φ h ( t ) d g ( t ).
In terms of dynamics, the system is no longer viewed as having any such specific internal representation as, for instance, of the fact that an attached seaweed strip of a certain length has been ingested and must now be egested.
The representation of, for instance, three different states needs at least two bits.
The early figural representations of Christ, for instance, most often show him as the good shepherd by directly borrowing from a classical prototype.
Representations of villains, for instance, are riddled with clichés.
For this study, we maintain redundancy in the cluster representations such that for instance the term GO 0019480— l-alanine oxidation to pyruvate via d-alanine does not cluster with the terms in Example 2 after transformation but rather by itself in 111 {CTERM CTERM CTERM oxid-alanine doeshe occurrenot of three clustererms, rather than twithwithen termsGO term.
