Rational points

A 25,000-pound annual production volume exemption is also a rational point at which to establish an exemption economically. Such an exemption would sharply reduce the economic impact of PMN requirements while protecting the public from unreasonable risk. 

Silverman/Tate Rational Points on Elliptic Curves. 

A morphism f X — S can always be considered as a family of schemes if we view it as the totality of its fibres, if f is flat we call it a fiat family of schemes parametrized bv 5 X is called the total space of the family f If 5 is connected and s e 5 is a k-rational point, f is also called a flat family of deformations of the fibre X(s). When S=Spec(A) with A a local k-algebra with residue field k, the flat morphism f is called a local family of deformations of %(q), the ribre over the closed point 5 if moreover the local k-algebra A is artinian, then f is called an infinitesimal family of deformations, or an infinitesimal deformation, of X(o). 

The family f is called complete (resp. effectively parametrized ) at the k-rational point s c S if the morphism g is smooth ( resp. etale ) at s If S is algebraic the completeness in s implies that g(S) contains a neighborhood of the point g(s) Hilb tn other words f parametrizes all "sufficiently small" deformations of (s) in Pr. "Effectively parametrized" implies that the classifying map g is f inite-to-one nearby s. Examples of... 

By taking a high enough power of the scheme, any rational point can be determined as a mark point. The power needed is just the Euler function of the quotient when the denominator has all powers of 2 (in general of the arity) divided out. 

Just as the Holder continuity can be measured exactly at rational points with a finite computation, but only bounds can be found for the curve as a whole, the Fourier decay rate can be measured exactly at rational frequencies with a finite computation, but only bounds found for the spectrum as a whole. The two measures turn out to be identical for the B-splines, but are typically different (by a small amount) for other basis functions. Both are changed by the same amount for every additional a factor in the mask. 

Here it should be noted that the function E( ) in the electrostatic approximation (1) is not a function in the usual sense and its continuity is broken up at any rational point =Q/p (where p and q are mutually simple numbers). This discontinuity appears by the following reason. At zero temperature the system of electrons and holes is arranged in the Wigner crystal with a cell consisting of TCNQ (or TTP) molecules. Let us study Ejj( ) with a little deviation of from Q p i.e. = q/P + where 1. Assume that /> 0. ... 

Y. Manin, Rational points on algebraic curves over function fields, (Izvestija Akad. Nauk, 27 (1963)). 

Since we felt at the beginning of the summer that there were enough unknown factors in chemical plant design that it should be possible to markedly reduce the cost of reprocessing by adopting a rational point of view, we set ourselves the problem of designing a completely rationalized reprocessing plant. This work has come to be known as Project Hope. 

Especially in UV-VIS spectroscopy, improved diode-array spectrophotometers or other fast spectral devices will be favorable for the fine resolution as well as the enhacement of sensitivity of spectra and other electric signals. By rapid accumulation of many scans, the best method for noise elimination can be employed from a rational point of view, and in the shortest amount of time, without altering the shape of the signals. 

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