## Boundary operators

Consider the vector space of polynomials with coefficients on a field , with the obvious sum of functions and scalar multiplication. For each , consider the subspaces spanned by polynomials of order ,

These subspaces have dimension Consider now for each the maps defined in the following way:

where if and otherwise.

Schematically, this can be written as follows

and it is not hard to prove that these maps are homeomorphisms of vector spaces over

Notice this interesting relationship between and

The kernel of and the image of are isomorphic!

The reader will surely have no trouble to show that this property is satisfied at all levels: As a consequence, for all

We say that a family of homomorphisms are *boundary operators* if for all If this is the case, then trivially The example above is a bit stronger, because of the isomorphism of both subspaces.

So this is the question I pose as today’s challenge:

Describe **all** boundary operators

Include a precise relationship between kernels and images of consecutive maps.

### Leave a Reply Cancel reply

### We have moved!

### In the news:

### Recent Posts

- Migration
- Computational Geometry in Python
- Searching (again!?) for the SS Central America
- Jotto (5-letter Mastermind) in the NAO robot
- Robot stories
- Advanced Problem #18
- Book presentation at the USC Python Users Group
- Areas of Mathematics
- More on Lindenmayer Systems
- Some results related to the Feuerbach Point
- An Automatic Geometric Proof
- Sympy should suffice
- A nice application of Fatou’s Lemma
- Have a child, plant a tree, write a book
- Project Euler with Julia
- Seked
- Nezumi San
- Ruthless Thieves Stealing a Roll of Cloth
- Which one is the fake?
- Stones, balances, matrices
- Buy my book!
- Trigonometry
- Naïve Bayes
- Math still not the answer
- Sometimes Math is not the answer
- What if?
- Edge detection: The Convolution Approach
- OpArt
- So you want to be an Applied Mathematician
- Smallest Groups with Two Eyes
- The ultimate metapuzzle
- Where are the powers of two?
- Geolocation
- Boundary operators
- The Cantor Pairing Function
- El País’ weekly challenge
- Math Genealogy Project
- Basic Statistics in sage
- A Homework on the Web System
- Apollonian gaskets and circle inversion fractals
- Toying with basic fractals
- Unusual dice
- Wavelets in sage
- Edge detection: The Scale Space Theory
- Bertrand Paradox
- Voronoi mosaics
- Image Processing with numpy, scipy and matplotlibs in sage
- Super-Resolution Micrograph Reconstruction by Nonlocal-Means Applied to HAADF-STEM
- The Nonlocal-means Algorithm
- The hunt for a Bellman Function.
- Presentation: Hilbert Transform Pairs of Wavelets
- Presentation: The Dual-Tree Complex Wavelet Transform
- Presentation: Curvelets and Approximation Theory
- Poster: Curvelets vs. Wavelets (Mathematical Models of Natural Images)
- Wavelet Coefficients
- Modeling the Impact of Ebola and Bushmeat Hunting on Western Lowland Gorillas
- Triangulations
- Mechanical Geometry Theorem Proving

### Pages

- About me
- Books
- Curriculum Vitae
- Research
- Teaching
- Mathematical Imaging
- Introduction to the Theory of Distributions
- An Introduction to Algebraic Topology
- The Basic Practice of Statistics
- MA598R: Measure Theory
- MA122—Fall 2014
- MA141—Fall 2014
- MA142—Summer II 2012
- MA241—Spring 2014
- MA242—Fall 2013
- Past Sections
- MA122—Spring 2012
- MA122—Spring 2013
- Lesson Plan—section 007
- Lesson Plan—section 008
- Review for First part (section 007)
- Review for First part (section 008)
- Review for Second part (section 007)
- Review for Third part (section 007)
- Review for the Second part (section 008)
- Review for the Fourth part (section 007)
- Review for Third and Fourth parts (section 008)

- MA122—Fall 2013
- MA141—Spring 2010
- MA141—Fall 2012
- MA141—Spring 2013
- MA141—Fall 2013
- MA141—Spring 2014
- MA141—Summer 2014
- MA142—Fall 2011
- MA142—Spring 2012
- MA241—Fall 2011
- MA241—Fall 2012
- MA241—Spring 2013
- MA242—Fall 2012
- MA242—Spring 2012
- First Midterm Practice Test
- Second Midterm-Practice Test
- Third Midterm—Practice Test
- Review for the fourth part of the course
- Blake Rollins’ code in Java
- Ronen Rappaport’s project: messing with strings
- Sam Somani’s project: Understanding Black-Scholes
- Christina Papadimitriou’s project: Diffusion and Reaction in Catalysts

- Problem Solving
- Borsuk-Ulam and Fixed Point Theorems
- The Cantor Set
- The Jordan Curve Theorem
- My oldest plays the piano!
- How many hands did Ernie shake?
- A geometric fallacy
- What is the next number?
- Remainders
- Probability and Divisibility by 11
- Convex triangle-square polygons
- Thieves!
- Metapuzzles
- What day of the week?
- Exact Expression
- Chess puzzles
- Points on a plane
- Sequence of right triangles
- Sums of terms from Fibonacci
- Alleys
- Arithmetic Expressions
- Three circles
- Pick a point
- Bertrand Paradox
- Unusual dice
- El País’ weekly challenge
- Project Euler with Julia

- LaTeX

### Categories

### Archives

- November 2014
- September 2014
- August 2014
- July 2014
- June 2014
- March 2014
- December 2013
- October 2013
- September 2013
- July 2013
- June 2013
- April 2013
- January 2013
- December 2012
- August 2012
- July 2012
- June 2012
- May 2012
- April 2012
- November 2011
- September 2011
- August 2011
- June 2011
- May 2011
- April 2011
- February 2011
- January 2011
- December 2010
- May 2010
- April 2010
- September 2008
- September 2007
- August 2007

### @eseprimo

Error: Twitter did not respond. Please wait a few minutes and refresh this page.

### Math updates on arXiv.org

- Phase-Retrieval as a Regularization Problem. (arXiv:1702.05092v1 [math.NA])
- Spatial Adaptation in Trend Filtering. (arXiv:1702.05113v1 [math.ST])
- Entropy, noncollapsing, and a gap theorem for ancient solutions to the Ricci flow. (arXiv:1702.05118v1 [math.DG])
- Hyporeductive and Pseudoreductive Hopf algebras. (arXiv:1702.05120v1 [math.RA])
- Exact Diffusion for Distributed Optimization and Learning --- Part I: Algorithm Development. (arXiv:1702.05122v1 [math.OC])
- Construction and Analysis of an HDG Method for Incompressible Magnetohydrodynamics. (arXiv:1702.05124v1 [math.NA])
- L-infinity optimization to linear spaces and phylogenetic trees. (arXiv:1702.05127v1 [math.CO])
- Weierstrass points on $X_0^+(p)$ and supersingular $j$-invariants. (arXiv:1702.05131v1 [math.NT])
- Three natural subgroups of the Brauer-Picard group of a Hopf algebra with applications. (arXiv:1702.05133v1 [math.QA])
- L-Infinity optimization to Bergman fans of matroids with an application to phylogenetics. (arXiv:1702.05141v1 [math.CO])

### sagemath

- An error has occurred; the feed is probably down. Try again later.

I probably shouldn’t be answering this (out of ignorance) but …

In the abstract I would just reverse the process using subset operators and equivalence operations: 0-> f_n/M((x,y) ->f_n+1/M(x,y) where M(x,y) is set membership contraction by an equivalence relation defined by having the same object after the mapping.

A good example is for linear vector space and linear operators. Then the forward operation is reflected into a inverse/pullback operator on the covector space. Then presuming that the forward map is from R^(n+1)->R^n for all n we have the inverse mapping on the covector space: 0->C^1->C^2 . The C^2 covector space then separates the original R^(n+1) vector space into a orthogonal component and it’s complement.

Sorry I didn’t define the operator. This will take a few hours/minutes/days ; although I should know it!

Okay, although I should be able to do better. The mappings M_(n+1)^(n) don’t really map R^(n+1)->R^(n) when consider as automorphisms. They map to R^m, m0;

Then M_(n+1)^n: R^(n+1)->R^m .

Of course the template is antisymmetric products like the wedge product where the operator is: V/\ then V/\V/\ (any vector/element of exterior product algebra) is zero. But successive operations alter the dimensions: (M,n) then (M,n+1) even if the subspaces shrink in dimension.

I hope this makes a modicum of sense! It’s been years since I studied it.

I think I have a much better set of conclusions.

Given: a set of maps/transforms M^(n-1)_n,M^n_(n+1): R^(n+1) ->R^n->0 that extends down to n=2; probably =1, but that gets touchy on the boundary case.

1) Up to isomorphisms there is only one constructable transform chain. This is made by “pulling back” the canonical n=2 case.

2) Ignoring the isomorphisms there is a simple wedge/minor test to verify and qualify maps that satisfy the given. That is the rank of each M^(n-1)_n has to be n-2 ; and match (in a particular way) the previous and successive transforms.

And in some manner I don’t understand it matches the differentiation transform. This can’t be quite right; probably a transform aligning exterior products or some such.

If your interested I will try to work it out formally. There are some holes but everything seems to fit around the holes.

I have found a simple way to illustrate the basic construction. No vectors, covectors, or linearity (more or less).

Starting from 2 dimensions:

(x,y)->(x,0) Which can be forced by a rotation

(x,y,z)->(0,y,z)

(x,y,z,w)->(x,0,0,w)

(x,y,z,w,u)->(0,y,z,0,u)

(x,y,z,w,u,r)->(x,0,0,w,0,r)

From which the alternating pattern for each dimension is obvious.

Your (-1)^k applied to even/odd n/cases does this alternation.

The above of course implicitly has the requirement that each mapping is maximal.

You could always put in extra zeros but that trivializes it..

I should mention that the last entries were a canonical form. In fact only the zeros matter; the x,y… on the right can be any combination (up to some general restriction) of letters on the left.

More deeply one can drop in G*G^-1 (G unitary) between transforms and get “new” transforms that satisfy the requirements. It’s more interesting to see if all trees are isomorphic via. these unitary transforms; pushing and pulling back vectors and covectors.

I presume that untangling the generality takes us into category theory. Which I started to read but didn’t get too far. As one gets more abstract and general; one needs more abstract tools.

Revise the structure of the first computed maps:

Since all the boundary operators are homomorphisms from to we represent them as matrices in

Note that can only be the zero homomorphism. As for this can be identified with

anymatrix Since we are looking for non-zero homomorphisms, we have then for any choice ofNow, once we have chosen the next homomorphism can only be of the form

with Go from there, if you want.

Yes but for the record (future readers): the reason I went to canonical form is to make the process more obvious. For instance we want for merged transforms (a column vector):

Since it’s obvious that all of the columns are orthogonal to we have:

The reason I emphasized the canonical form is to avoid a proliferation of constants. Thus I inserted between the matrices where:

Which works as long as and real.

This yields:

Which in turn can be rendered canonical via

So:

Which in terms of my equations would be (using standard basis):

Of course picking the preserved basis was arbitrary and these could have been chosen.

and so on.