Bijective function

As already mentioned, the sampling methods use the distributions in order to perform probability-weighted perturbations. For example, in the Monte Carlo approach, a random number e [0,1] is generated (probabihty threshold) and the CDF, corresponding to that probabihty, is inverted in order to retrieve the parameter value usable in the simulation. The existence of the inverse for univariate distributions is guaranteed by the monotonicity of the CDF. For N-D distributions this condition is not sufficient since the CDF(X) —> [0,1 ], Xe R and therefore it could not be a bijective function. From an application point of view, this means the inverse of a N-D CDF is not unique. 

A function cpis called bijective if it is both one-to-one and onto. 

A function cpis called a homeomorphism if it is bijective and both cpand its inverse cp -1 are continuous,... 

A function between topological spaces is continuous if inverse images of closed sets are closed A homeomorphism is a continuous bijection with continuous inverse. 

Lemma Bijective (one-to-one, onto) functions [15] exist between each of the above mentioned sets of graph invariants E,V,H,C- In other words we can write the relation as follows ... 

Proof. The easiest proof is that the function X2 g2 is bijective, and so is multiplication by a constant... 

If p = 2q + I, one can use the factor group H p = 2p / l, instead of a subgroup [ChAn90]. It can be represented by the numbers 1, q. An advantage of this group is that there is an efficiently computable bijection i from Hg p to Map q to 0 and the remaining numbers to themselves. (Of course, any discrete-logarithm function is a bijection, too, but hopefully not efficiently computable.) This means that exponentiation and similar functions from to Hgp can be iterated in a simple way. 

A function is invertible if it is bijective. This means that to every x there is a unique y and vice versa. For example, y = x is not bijective, because for a given y > 0, X = there are two solutions for x. 

Why Enumerate . - Clearly enumeration has played an important role in the history of chemistry. But does it still Are the noted enumerations just historical anachronisms Is enumeration irrelevant for modem interests in quantitative descriptions of different substances Indeed in all the areas we have noted, one may indeed argue that enumeration is but a first step towards a more comprehensive characterization and undertaking. Combinatorial formulae often merely identify two different enumerations to have equal values, with one of the enumerations being the easier to perform. We may note for instance that isomer enumeration in Polya theory identifies this enumeration to that of the enumeration of certain equivalence classes of functions. With the counts for two different sets of objects being equal, there often is a natural bijection i.e. a one-to-one correspondence) between the two sets, so that the objects of one set may be used to represent (or even name) those of the other. Thence for the case of chemical isomers again, the mathematical set of objects offers a nomenclature for the isomers. Conversely too, granted a nomenclature, a possibility for enumeration is offered one seeks to enumerate the names (which presumably exhibit some systematic structure). In some sense then a sensible nomenclature and enumera-... 

Even if the function / is bijective and simplicial, its inverse does not have to be simplicial. 

In Appendix A we described the numerical implementation of the Gaussian chain density functions that retains the bijectivity between the density fields and the external potential fields. The intrinsic chemical potentials /r/= SFj p that act as thermodynamic driving forces in the Ginzburg-Landau equations describing the dynamics, are functionals of the external potentials and the density fields. Together, the Gaussian chain density functional and the partial differential equations, describing the dynamics of the system, form a closed set. 

If/is a continuous function in an interval I, bijective on I and derivable into Xa and such... 

Let us denote two different wave functions by ir and 0, and their respective zero flux boundaries by df and 90. The condition that needs to be obeyed states that for a neighbourhood of 0, there exists 0 such that 90 can be deformed continuously by a bijective mapping into 90. The counterexample involves the ground state of the hydrogen atom. The authors then (wrongly) showed that no matter how small the neighbourhood of 0 is, 90 could not be deformed continuously into 90. Instead of reproducing their calculation... 

Let us consider the gradient of the phase field (j) of the oscillators. Due to the bijective mapping between the network space and the physical space, the phase (j) is also a function of the coordinates (ri,r2). The vector is perpendicular to the crests of waves and is computed easily by finite-difference approximation of the expression ... 

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