Calculus and Its Origins (Spectrum)
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The usual calculus is a special or limiting case. One approach is to look for structures which are close to known notions of calculus. An ideal generalization of traditional calculus to the discrete does not change language. It changes the meaning of the operations, possibly after extending language. But this flavor of calculus is not easy to manage and teach.
While intuitive, it is also more risky: it is easier to make mistakes. One can imagine historical circumstances, in which quantum calculus could been first developed and the calculus we know later been derived and have the status of generalized function theory or geometric measure theory. In the same way that we could in principle work with rational numbers alone, one could look exclusively at discrete geometries and look at continuum geometries as limiting cases. Computer game implementations are quite realistic but most matrix computations are kept with integers as it is faster than floating number arithmetic.
Any computation on a device is by nature finite as there is only a finite amount of memory available.
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In some sense, already numerical analysis is already a quantized calculus. However, the theory as told in numerical analysis books is much less elegant than the actual theory which is in the nature of things as engineering is often not elegant if it wants to be fast and efficient.
This idea is everywhere: look at the sequence 4, 15, 40, 85, , , , for example. How does it continue? In order to figure it out, we take derivatives 11, 25, 45, 71, , , , then again derivatives 14, 20, 26, 32, 38, 44 and again derivatives 6,6,6,6,6,6. We see now a pattern and can integrate the three times starting with 6,6,6,6,6,6,6 always adjusting the constant. This gives us the next term We can predict the future by analyzing the past.
A nice example is Riemannian geometry, which generalizes calculus in flat space. The frame work allows then to define notions like curvature or geodesics which are so central in modern theories of gravity. These notions can be carried over to discrete spaces. But it is not only physics which motivates to look at calculus.
All movies using some kind of CGI make heavy use of calculus ideas. The power, richness and applicability of calculus are all reasons why we teach it. Calculus is a wonderful and classical construct, rich of historical connections and related with many other fields. Here is a 30 second spot. We hardly have to mention all the applications of calculus the last 2 minutes of this 15 minute review for single variable calculus give a few.
We have barely scratched the surface of what is possible when extending calculus and how it can be applied in the future. The traditional exterior derivatives like div,curl grad or curvature notions based on differential forms define a traditional calculus. In classical calculus, the basic building blocks of space are simplices. Differential forms are functions of these simplices. In the continuum, one can not see these infinitesimal simplices. To remedy this, sheaf theoretical constructs like tensor calculus have been developed, notably by Cartan.
In the discrete, when looking at graphs, the structure is simple and transparent. The theorems become easy. Here , here and here are some write ups. Calculus on graphs is probably the simplest quantum calculus with no limits: everything is finite and combinatorial.
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I prefer to work on graphs but one can work also with finite abstract simplicial complexes. It turns out that the category of complexes and especially graphs is quite powerful, despite its simplicity. Graphs are quite an adequate language because the Barycentric refinement of an abstract finite simplicial complex is always the Whitney complex of a finite simple graph.
Most topological considerations therefore can be done on graphs. Exterior derivatives again lead to cohomology. The chain complex is bigger. The cohomology groups more interesting. The analogue of Euler characteristic is the more general Wu characteristic. There are generalized versions of the just mentioned theorems. At the moment , only a glimpse of the power of interaction calculus is visible. There are indications that it could be powerful: it allows to distinguish topological spaces, which traditional calculus does not: here is a case study.
And here is a preprint on Interaction cohomology.
Its not very helpful to look only for analogies to the continuum. Very general principles numerical and computer science demonstrate constantly how finite machines mode things show that the continuum can be emulated very well in the discrete. We also have to look out for new things.
A surprise for example is that there is a Laplacian for discrete geometries which is always invertible: discovered in February and proven in the fall , it leads to invariants and potential theory different from the usual Hodge Laplacian. Unlike the Hodge Laplacian which is the square of the Dirac Laplacian, this new Laplacian has a quantized and completely finite potential theory, no super symmetry. The total energy of the geometry is the Euler characteristic. For more, see the project page on my personal web page. The entries here are more like drafts, sometimes expository, sometimes research logs which I leave as it is for me also interesting to see later, where and how I was stuck.
So, it can happen that initially in an entry, the story has not yet been clear but at the end been understood. With limited time at hand, comments are currently not turned on as this would require time for moderation. This might change at some later when more material has been added, maybe when a substancial number of entires are posted.
At the moment there is not much to see yet. The simplest construct in mathematics is probably a finite set of sets. Unlike a simple set alone, it has natural algebraic, geometric, analytic and order structures built in. In the finite case, there is lot of overlap but still, there is a rich variety of structure. Algebraic structure An example of a an algebraic structure is to have the symmetric … …. If a set of set is equipped with an energy function, one can define integer matrices for which the determinant, the eigenvalue signs are known.
For constant energy the matrix is conjugated to its inverse and defines two isospectral multi-graphs. The counting matrix of a simplicial complex has determinant 1 and is isospectral to its inverse. The sum of the matrix entries of the inverse is the number of elements in the complex. The parametrized poincare-hopf theorem allows to see the f-vector of a graph in terms of the f-vector s of parts of the unit spheres of the graph.
Dehn-Sommerville spaces are generalized spheres as they share many properties of spheres: Euler characteristic and more generally Dehn-Sommerville symmetries. In all of these cases, the addition of graphs is the disjoint union which serves a nice monoid like the natural numbers. The f-function of a graph minus 1 is the sum of the antiderivatives of the f-function anti-derivatives evaluated on the unit spheres. Dehn-Sommerville relations are a symmetry for a class of geometries which are of Euclidean nature. Here is the obituary from the university of Washington.
Calculus and Its Origins | Mathematical Association of America
The well illustrated book is considered the bible on Tilings. Here is a page from that book: Links: Personal website at Washington. Wikipedia entry … …. Some update about recent activities: a new calculus course, the Cartan magic formula and some programming about the coloring algorithm. Quantum Calculus Calculus without limits. The joy of sets of sets The simplest construct in mathematics is probably a finite set of sets Energized Simplicial Complexes If a set of set is equipped with an energy function, one The counting matrix of a simplicial complex The counting matrix of a simplicial complex has determinant 1 and is Poincare-Hopf and the Clique Problem The parametrized poincare-hopf theorem allows to see the f-vector of a graph Small Dehn-Sommerville Spaces Dehn-Sommerville spaces are generalized spheres as they share many properties of spheres On Numbers and Graphs We have calculated with graphs from the very beginning: Humans computed with Commercial use is prohibited.