# SymPy Python library for symbolic mathematics computer algebra system (CAS)

SymPy is a Python library for symbolic mathematics. It aims to become a full-featured computer algebra system (CAS) while keeping the code as simple as possible in order to be comprehensible and easily extensible. SymPy is written entirely in Python.

via SymPy

# SQL database that is stored in a single file on disk. SQLite is built into Python

SQLite is a database that is stored in a single file on disk. SQLite is built into Python

# IPython qtconsole

in IPython qtconsole (which one could embed inside a custom PyQt/PySide application, and keep track of all the defined symbols

# math – Mathematical equation manipulation in Python – Stack Overflow

Using SymPy, your example would go something like this:

# Putting Expressions into Different Forms

Putting Expressions into Different Forms

# Can I check if an expression is positive using assumptions?

Can I check if an expression is positive using assumptions?

# Solve the system of equalities and inequalities

down vote

Backsubstitution is an option often ignored when people use Reduce. In the op’s case it gives:

Reduce[a == b + c && a >= 2 && b <= 10 && c == 5, {a, b, c}, Backsubstitution -> True]
2 <= a <= 15 && b == -5 + a && c == 5

# torus is a surface having genus one

torus is a surface having genus one

# mathematica graphing a 2 by 2 matrix as an object

is there a possibility to “transform” a matrix (inheriting the input-output “weights”) into a graph object inheriting the matrix relations?

you can convert it to an edge-weighted graph using `WeightedAdjacencyGraph`. This will give you a complete graph (a `Graph` expression) in which each vertex is also connected to itself. I am not sure how much sense it makes treat this matrix as a graph, given the full connectivity.

Since you have a fully connected graph, it does not matter in your case, but be aware that `WeightedAdjacencyGraph` represents missing edges with `Infinity`, not with zero. This is somewhat annoying because it is inconsistent with `WeightedAdjacencyMatrix`, which uses `0`. While `Infinity` does make sense in some (not all) applications, it is much more common to see data that uses 0.

# Surface Intersection: New in Mathematica 10

Surface Intersection

# Highlight the Intersection of Two Surfaces

Highlight the Intersection of Two Surfaces

# SliceDensityPlot3D—Wolfram Language Documentation

SliceDensityPlot3D[f, surf, {x, xmin, xmax}, {y, ymin, ymax}, {z, zmin, zmax}] generates a density plot of f over the slice surface surf as a function of x, y, and z. SliceDensityPlot3D[f, surf, {x, y, z} \[Element] reg] restricts the surface to be within region reg. SliceDensityPlot3D[f, {surf1, surf2, …}, …] generates density plots over several slices.

# Geometry of Linear Transformations of the Plane – HMC Calculus Tutorial

If V=R2 and W=R2, then T:R2R2 is a linear transformation if and only if there exists a 22 matrix A such that T(v)=Av for all vR2. Matrix A is called the standard matrix for T. The columns of A are T01 and T10 , respectively. Since each linear transformation of the plane has a unique standard matrix, we will identify linear transformations of the plane by their standard matrices. It can be shown that if A is invertible, then the linear transformation defined by A maps parollelograms to parallelograms. We will often illustrate the action of a linear transformation T:R2R2 by looking at the image of a unit square under T

# Affine transformation – Wikipedia

the image T(P) of any point P is determined by noting that T(A) = A′, T applied to the line segment AB is A′B′, T applied to the line segment AC is A′C′, and T respects scalar multiples of vectors based at A. [If A, E, F are collinear then the ratio length(AF)/length(AE) is equal to length(A′F′)/length(A′E′).] Geometrically T transforms the grid based on ABCD to that based in A′B′C′D′.Affine transformations do not respect lengths or angles; they multiply area by a constant factor

# Ehrhart polynomials pick’s theorem – Google Search

Search ResultsIn mathematics, an integral polytope has an associated Ehrhart polynomial that encodes the relationship between the volume of a polytope and the number of integer points the polytope contains. The theory of Ehrhart polynomials can be seen as a higher-dimensional generalization of Pick’s theorem in the Euclidean plane.

# On similarity classes of well-rounded sublattices of Z2 (PDF Download Available)

good circle packing density and converges to the hexagonal lattice

# On the number of lattice points in convex symmetric bodies and their duals (PDF Download Available)

On the number of lattice points in convex symmetric bodies and their duals

# On the number of lattice points in convex symmetric bodies and their duals | SpringerLink

H. Gillet and C. Soulé