You are given 100 stations and distance between each adjacent stations. Now you have to select 10 stations(means 10 hops) among those 100 stations in such a way that maximum of distance between any 2 hops will be minimised. By default 1 and 100 stations are selected , so you need to choose only 8 more stations.


Question: Various flavours of infinitesimals
Answer: I don't know of the Connes calculus, but the others (including nonstandard analysis &agrave; la Robinson) have been brought under a common framework using models of synthetic differential geometry. However: it is important to point out that the infinitesimals used in algebraic geometry (for jet bundles, etc.) are nilpotent infinitesimals, whereas the infinitesimals used in nonstandard analysis are invertible. So in a sense the answer to the question is, "yes and no", but I'm going to concentrate here on the "yes". 

This is explored in Models for Smooth Infinitesimal Analysis by Moerdijk and Reyes, which I recommend. This book can be read as "applied sheaf theory" or "applied Grothendieck topos theory", where the art is to choose a site (small category + covering sieves) judiciously to achieve several aims at once. In many of the models, one takes the underlying category of the site to be something like affine spectra of commutative rings, except one is not dealing with commutative rings exactly, but with richer algebraic structures called $C^\infty$-rings. The formal definition of these is in terms of a Lawvere algebraic theory which allows one to apply not just polynomial operations but more general operations based on $C^\infty$ functions. So the underlying category of the site in these models is the opposite of finitely generated $C^\infty$-rings, which Moerdijk and Reyes call $\mathbb{L}$ (for "locus"). 

The representing object which gives the locus of invertible infinitesimals is the spectrum of the $C^\infty$-ring given by $C^\infty$ functions $\mathbb{R} - \{0\} \to \mathbb{R}$, modulo the ideal of functions that vanish on some neighborhood of $0$, aka the $C^\infty$ ring of germs at $0$. This is really not much different from nonstandard infinitesimals: usually infinitesimal elements are thought of in some way as germs of functions at infinity, i.e., of functions $\mathbb{R} \to \mathbb{R}$ modulo those which vanish for sufficiently large $x$ (compare here the infinitesimals of Du Bois-Reymond and Hardy). Here Moerdijk and Reyes use $0$ instead of $\infty$. Either way, there are nonarchimedean elements, i.e., nonzero elements less than any $1/n$ in absolute value. 

[In nonstandard analysis, one typically refines this idea by considering germs of functions $\mathbb{R} \to \mathbb{R}$ at an "ideal point at infinity", i.e., at a non-principal ultrafilter $U$ on $\mathbb{R}$, or alternatively germs of functions $\mathbb{N} \to \mathbb{R}$ at a non-principal ultrafilter $U$ on $\mathbb{N}$. The more familiar buzzword here is "ultrapower", but see this MO answer by Fran&ccedil;ois Dorais, where the implicit message is that an ultrapower along $U$ is really the same as taking a stalk at $U$. (I call $U$ an "ideal point at infinity" because we can think of a non-principal ultrafilter $U$ on $\mathbb{N}$ as a point in the fiber over $\infty$ with respect to the canonical continuous map $\beta(\mathbb{N}) \to \mathbb{N} \cup \{\infty\}$, from the Stone-Cech compactification of $\mathbb{N}$ to the one-point compactification of $\mathbb{N}$.)] 

On the other hand, a typical representing object for nilpotent infinitesimals is the spectrum of the $C^\infty$-ring of functions $\mathbb{R} \to \mathbb{R}$ modulo the ideal of squares of elements which vanish at $0$. The "internal hom" represented by this spectrum gives the tangent bundle functor, and other jet bundles can be similarly represented, by using $C^\infty$-rings with different types of nilpotent elements. 

The tricky part of all this is to get the right notion of covering sieves, i.e., of sheaves w.r.t. a Grothendieck topology, to achieve disparate aims. One aim would be to embed the usual category of manifolds fully and faithfully in the category of sheaves, so as to preserve "good colimits", such as a manifold $M$ obtained as a colimit along an open covering of $M$. A different aim would be to arrange the topology so that the locus of invertible infinitesimals, as a presheaf on the site category $\mathbb{L}$, is a sheaf w.r.t. the topology. In summary, both aims can be achieved simultaneously so as to accommodate both nilpotent and invertible infinitesimals.