Using SageMath in Research - Krystal Guo
Transcript of Using SageMath in Research - Krystal Guo
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Krystal Guo
Universite libre de Bruxelles
CMS mini-course 2019, Regina
Using SageMath in Research
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What is sage?
SageMath is a free open-source mathematicssoftware system.
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What is sage?
SageMath is a free open-source mathematicssoftware system.
It’s an alternative to:
Magma
Maple
Mathematica
MATLAB
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What is sage?
SageMath is a free open-source mathematicssoftware system.
It’s an alternative to:
Magma
Maple
Mathematica
MATLAB
algebracombinatoricsgraph theorynumericalanalysisnumber theorycalculusstatistics
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What is sage?
SageMath is a free open-source mathematicssoftware system.
It’s an alternative to:
Magma
Maple
Mathematica
MATLAB
SageMath was started by William Stein.
algebracombinatoricsgraph theorynumericalanalysisnumber theorycalculusstatistics
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Guess this graph
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Goals for this mini-course
give you an idea of how ”casual” uses of sage inresearch
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Goals for this mini-course
give you an idea of how ”casual” uses of sage inresearch
For those who don’t program much: a window intowhat it can do, so you can use it, if you want to.
![Page 9: Using SageMath in Research - Krystal Guo](https://reader033.fdocuments.net/reader033/viewer/2022042421/6260e48f11ddf97faa40f5ec/html5/thumbnails/9.jpg)
Goals for this mini-course
give you an idea of how ”casual” uses of sage inresearch
For those who don’t program much: a window intowhat it can do, so you can use it, if you want to.
For those with programming background:suggest new contexts for using sage
![Page 10: Using SageMath in Research - Krystal Guo](https://reader033.fdocuments.net/reader033/viewer/2022042421/6260e48f11ddf97faa40f5ec/html5/thumbnails/10.jpg)
Goals for this mini-course
give you an idea of how ”casual” uses of sage inresearch
For those who don’t program much: a window intowhat it can do, so you can use it, if you want to.
For those with programming background:suggest new contexts for using sage
To introduce everyone to my favourite graph
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The When/Where/Why
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Checking conjectures and counterexamples
Theorem (Mohar 2013)
Every connected subcubic bipartite graph thatis not isomorphic to the Heawood graph has atleast one (in fact a positive proportion) of itseigenvalues in the interval [−1, 1].
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Checking conjectures and counterexamples
Theorem (Mohar 2013)
Every connected subcubic bipartite graph thatis not isomorphic to the Heawood graph has atleast one (in fact a positive proportion) of itseigenvalues in the interval [−1, 1].
Question: what about (−1, 1)?
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Checking conjectures and counterexamples
Theorem (Mohar 2013)
Every connected subcubic bipartite graph thatis not isomorphic to the Heawood graph has atleast one (in fact a positive proportion) of itseigenvalues in the interval [−1, 1].
Question: what about (−1, 1)?We construct an infinite family of connected cubicbipartite graphs which have no eigenvalues in theopen interval (−1, 1).
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Checking conjectures and counterexamples
Theorem (Mohar 2013)
Every connected subcubic bipartite graph thatis not isomorphic to the Heawood graph has atleast one (in fact a positive proportion) of itseigenvalues in the interval [−1, 1].
Question: what about (−1, 1)?We construct an infinite family of connected cubicbipartite graphs which have no eigenvalues in theopen interval (−1, 1).
Found by searching the census of cubicvertex-transitive graphs.
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Visualizing new mathematical objects
Continous quantum walk
Transition matrix of the quantum walk
U(t) = eitA
where A is the adjacency matrix of agraph.
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Visualizing new mathematical objects
Continous quantum walk
Transition matrix of the quantum walk
U(t) = eitA
where A is the adjacency matrix of agraph.
U(t) =
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Visualizing new mathematical objects
Continous quantum walk
Transition matrix of the quantum walk
U(t) = eitA
where A is the adjacency matrix of agraph.
U(t) =u
v
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Visualizing new mathematical objects
Continous quantum walk
Transition matrix of the quantum walk
U(t) = eitA
where A is the adjacency matrix of agraph.
U(t) =u
v
![Page 20: Using SageMath in Research - Krystal Guo](https://reader033.fdocuments.net/reader033/viewer/2022042421/6260e48f11ddf97faa40f5ec/html5/thumbnails/20.jpg)
Visualizing new mathematical objects
Continous quantum walk
Transition matrix of the quantum walk
U(t) = eitA
where A is the adjacency matrix of agraph.
U(t) =u
v
t 7→ (xt, yt)U(t)u,v = xt + iyt
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Plotting
U(t)a,b for t ∈ [0, 100].
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Plotting
U(t)a,b for t ∈ [0, 100].
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Plotting
U(t)a,b for t ∈ [0, 100].500].
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Plotting
U(t)a,b for t ∈ [0, 100].500].
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Other reasons to use SageMath
Build intuition by looking at small examples.
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Other reasons to use SageMath
Build intuition by looking at small examples.
Dry run for serious programming.
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Other reasons to use SageMath
Build intuition by looking at small examples.
Dry run for serious programming.
Resource: cython
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Other reasons to use SageMath
Build intuition by looking at small examples.
Dry run for serious programming.
Resource: cython
Calculator
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Other reasons to use SageMath
Build intuition by looking at small examples.
Dry run for serious programming.
Resource: cython
Calculator
.... many other uses!
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Get to know SageMath!
sage: runfile cms-guessthisgraph.sage
g1, g2, g3 have been defined
sage: g1.
clique_number()
degree()
diameter()
girth()
complement()
is_regular()
is_bipartite()
is_clique()
is_chordal()
is_planar()
is_connected()
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Constructing an awesome graph
We will demonstrate SageMath by constructing aspecific graph. We will use the following ingredients:
Projective geometry, hyperovals, finite fields.
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Strongly regular graphs
A strongly regular graph with parameters (n, k, a, c) is
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Strongly regular graphs
A strongly regular graph with parameters (n, k, a, c) is
a graph on n vertices such that
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Strongly regular graphs
A strongly regular graph with parameters (n, k, a, c) is
a graph on n vertices such that
k
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Strongly regular graphs
A strongly regular graph with parameters (n, k, a, c) is
a graph on n vertices such that
k a
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Strongly regular graphs
A strongly regular graph with parameters (n, k, a, c) is
a graph on n vertices such that
k a c
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Strongly regular graphs
Example: (25, 8, 3, 2).
A strongly regular graph with parameters (n, k, a, c) is
a graph on n vertices such that
k a c
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How do we construct strongly regular graphs?
One way is as the point (or line) graphs of anincidence structure.
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How do we construct strongly regular graphs?
One way is as the point (or line) graphs of anincidence structure.
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How do we construct strongly regular graphs?
One way is as the point (or line) graphs of anincidence structure.
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How do we construct strongly regular graphs?
One way is as the point (or line) graphs of anincidence structure.
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How do we construct strongly regular graphs?
One way is as the point (or line) graphs of anincidence structure.
Point graphvertices:adjacent if on thesame block
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How do we construct strongly regular graphs?
One way is as the point (or line) graphs of anincidence structure.
Point graphvertices:adjacent if on thesame block
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How do we construct strongly regular graphs?
One way is as the point (or line) graphs of anincidence structure.
Point graphvertices:adjacent if on thesame block
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How do we construct strongly regular graphs?
One way is as the point (or line) graphs of anincidence structure.
Point graphvertices:adjacent if on thesame block
Block graph
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How do we construct strongly regular graphs?
One way is as the point (or line) graphs of anincidence structure.
Point graphvertices:adjacent if on thesame block
Block graph
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How do we construct strongly regular graphs?
One way is as the point (or line) graphs of anincidence structure.
Point graphvertices:adjacent if on thesame block
Block graph
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How do we construct strongly regular graphs?
One way is as the point (or line) graphs of anincidence structure.
Point graphvertices:adjacent if on thesame block
Block graph
We get a strongly regular graph as a point (orblock) graph if we pick a ”nice” incidence structure.
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A ”nice” incidence structure
A generalized quadrangle is a pair (P,B)satisfying the following two axioms:
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A ”nice” incidence structure
A generalized quadrangle is a pair (P,B)satisfying the following two axioms:
any two points of P are on at most one linein B;
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A ”nice” incidence structure
A generalized quadrangle is a pair (P,B)satisfying the following two axioms:
any two points of P are on at most one linein B;
if point p ∈ P is not on line ` ∈ B, thenthere exists a unique point q ∈ ` such that pand q are collinear.
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A ”nice” incidence structure
A generalized quadrangle is a pair (P,B)satisfying the following two axioms:
any two points of P are on at most one linein B;
if point p ∈ P is not on line ` ∈ B, thenthere exists a unique point q ∈ ` such that pand q are collinear.
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A ”nice” incidence structure
A generalized quadrangle is a pair (P,B)satisfying the following two axioms:
any two points of P are on at most one linein B;
if point p ∈ P is not on line ` ∈ B, thenthere exists a unique point q ∈ ` such that pand q are collinear.
![Page 54: Using SageMath in Research - Krystal Guo](https://reader033.fdocuments.net/reader033/viewer/2022042421/6260e48f11ddf97faa40f5ec/html5/thumbnails/54.jpg)
A ”nice” incidence structure
A generalized quadrangle is a pair (P,B)satisfying the following two axioms:
any two points of P are on at most one linein B;
if point p ∈ P is not on line ` ∈ B, thenthere exists a unique point q ∈ ` such that pand q are collinear.
![Page 55: Using SageMath in Research - Krystal Guo](https://reader033.fdocuments.net/reader033/viewer/2022042421/6260e48f11ddf97faa40f5ec/html5/thumbnails/55.jpg)
A ”nice” incidence structure
A generalized quadrangle is a pair (P,B)satisfying the following two axioms:
any two points of P are on at most one linein B;
if point p ∈ P is not on line ` ∈ B, thenthere exists a unique point q ∈ ` such that pand q are collinear.
There will be parameters (s, t) such that:
each point is on t+ 1 lines;each line is on s+ 1 points
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How do we get a generalized quadrangle?
We will construct a particular generalizedquadrangle, denoted in the literature asT ∗2 (O).
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How do we get a generalized quadrangle?
We will construct a particular generalizedquadrangle, denoted in the literature asT ∗2 (O).
For this, we will need the following:
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How do we get a generalized quadrangle?
We will construct a particular generalizedquadrangle, denoted in the literature asT ∗2 (O).
For this, we will need the following:
finite field, GF (q), where q iseven;
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How do we get a generalized quadrangle?
We will construct a particular generalizedquadrangle, denoted in the literature asT ∗2 (O).
For this, we will need the following:
finite field, GF (q), where q iseven;
projective space over GF (q); and
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How do we get a generalized quadrangle?
We will construct a particular generalizedquadrangle, denoted in the literature asT ∗2 (O).
For this, we will need the following:
finite field, GF (q), where q iseven;
projective space over GF (q); and
a hyperoval in projective plane overGF (2h).
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Projective space
(Finite) projective space PG(n, q) is anincidence structure in the n+ 1-dimensionalspace over GF (q), where q is a prime power,where:
points: 1-dimensional subspaces
lines: 2-dimensional subspaces
planes: 3-dimensional subspaces
...
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Hyperovals
In PG(2, q), where q = 2h, we can have q + 2point where no points are collinear.
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Hyperovals
In PG(2, q), where q = 2h, we can have q + 2point where no points are collinear.
Such a set of q+2 points is called a hyperoval.
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Hyperovals
In PG(2, q), where q = 2h, we can have q + 2point where no points are collinear.
Such a set of q+2 points is called a hyperoval.
Fact:
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Hyperovals
In PG(2, q), where q = 2h, we can have q + 2point where no points are collinear.
Such a set of q+2 points is called a hyperoval.
Fact: Every hyperoval has the following form:
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Hyperovals
In PG(2, q), where q = 2h, we can have q + 2point where no points are collinear.
Such a set of q+2 points is called a hyperoval.
Fact: Every hyperoval has the following form: 1
tf(t)
, t ∈ GF (q)
∪010
,
001
.
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Hyperovals
In PG(2, q), where q = 2h, we can have q + 2point where no points are collinear.
Such a set of q+2 points is called a hyperoval.
Fact: Every hyperoval has the following form: 1
tf(t)
, t ∈ GF (q)
∪010
,
001
.
Open problem: classify all f(t) which give hyperovals.
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Hyperovals
In PG(2, q), where q = 2h, we can have q + 2point where no points are collinear.
Such a set of q+2 points is called a hyperoval.
Fact: Every hyperoval has the following form: 1
tf(t)
, t ∈ GF (q)
∪010
,
001
.
Open problem: classify all f(t) which give hyperovals.
Some f(t) which work:
t2, t6, t2k
where k, h are coprime
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The construction
PG(3, 2h)
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The construction
PG(3, 2h)
AG(3, 2h)
PG(2, 2h)
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The construction
PG(3, 2h)
AG(3, 2h)
PG(2, 2h)
hyperoval
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The construction
PG(3, 2h)
AG(3, 2h)
PG(2, 2h)
hyperoval
points
blocks
This will give a generalized quadrangle of order(q − 1, q + 1), where q = 2h.
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Give it a go!
sage: gq4 = GQTstar(GF(4),"hyperconic")
sage: h = pointsGraph(gq4)
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Try some stuff in SageMath!
lsq is a list of graphs, each of which is a Latinsquare graph of order 7, on 49 vertices.
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Try some stuff in SageMath!
lsq is a list of graphs, each of which is a Latinsquare graph of order 7, on 49 vertices.
If you can colour G with 7 colours, thenadding a complete graph on each colourclass will give another strongly regulargraph.
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Try some stuff in SageMath!
lsq is a list of graphs, each of which is a Latinsquare graph of order 7, on 49 vertices.
If you can colour G with 7 colours, thenadding a complete graph on each colourclass will give another strongly regulargraph.
G
![Page 77: Using SageMath in Research - Krystal Guo](https://reader033.fdocuments.net/reader033/viewer/2022042421/6260e48f11ddf97faa40f5ec/html5/thumbnails/77.jpg)
Try some stuff in SageMath!
lsq is a list of graphs, each of which is a Latinsquare graph of order 7, on 49 vertices.
If you can colour G with 7 colours, thenadding a complete graph on each colourclass will give another strongly regulargraph.
G
. . .
![Page 78: Using SageMath in Research - Krystal Guo](https://reader033.fdocuments.net/reader033/viewer/2022042421/6260e48f11ddf97faa40f5ec/html5/thumbnails/78.jpg)
Try some stuff in SageMath!
lsq is a list of graphs, each of which is a Latinsquare graph of order 7, on 49 vertices.
If you can colour G with 7 colours, thenadding a complete graph on each colourclass will give another strongly regulargraph.
G
. . .
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Or ...
Construct the following graphs. Are theyisomorphic?
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Installing sage
https://doc.sagemath.org/html/en/installation/