co.combinatorics – total unimodularity of a matrix

Let G be the node-arc incidence matrix of a given directed network (rows of $ G $ correspond to nodes and their columns correspond to arcs. To let $ B_1, dots, B_K $ designate a partition of the nodes of the network. Suppose the network is such that a directed arc can go in from a node $ B_k $ to another node in $ B_ ell $ only if $ k < ell $, To let $ H $ denote a matrix with $ K $ Rows and assume that its columns are indexed by the arcs of the underlying network. We assume that $ H $ is that his $ (k, e) $-th entry is equal to one $ e in B_k $and nothing else.

Is the matrix (H; G) (obtained by the concatenation of rows of $ H $ and $ G $) completely unimodular? If not, can you give a counterexample?

I numerically examined some examples and confirmed the complete unimodularity for these examples. I thought it might be possible to exploit the structure of $ H $ (Note the special line structure) to formally prove the result. I tried to use the Ghouila Houri condition (see https://en.wikipedia.org/wiki/Unimodular_matrix), which appears to be a good candidate for exploiting the line structure. So far I have not been successful.

ordinary differential equations – Calculate the determinant of a solution matrix for the following linear system.

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analytic geometry – semiaxes of the ellipsoid from the square matrix

In three-dimensional Euclidean space, a quadratic surface can be defined by the following analytic equation:

$$ e_1 x ^ 2 + e_5 y ^ 2 + e_8 z ^ 2 + 2 e_2 y z + 2 e_3 x z + 2 e_4 xy + 2 e_6 x + 2 e_7 y + 2 e_9 z + e_ {10} = 0 $$

or in homogeneous matrix form:

$$ texttt {x} ^ T cdot texttt {E} cdot texttt {x} =
begin {bmatrix}
x_1 & x_2 & x_3 & 1
end {bmatrix}
begin {bmatrix}
e_1 & e_2 & e_3 & e_4 \
e_2 & e_5 & e_6 & e_7 \
e_3 & e_6 & e_8 & e_9 \
e_4 & e_7 & e_9 & e_ {10}
end {bmatrix}
begin {bmatrix}
x_1 \
x_2 \
x_3 \
1
end {bmatrix}
= 0
$$

Given only the matrix $ texttt {E} $ and to know that the surface is an ellipsoid, the center $ texttt {x} _ texttt {c} $ Ellipsoid is won by
$$ texttt {x} _ texttt {c} = texttt {E} _ {33} ^ {- 1} begin {bmatrix}
e_4 \
e_7 \
e_9
end {bmatrix},
Where ,
texttt {E} _ {33} =
begin {bmatrix}
e_1 & e_2 & e_3 \
e_2 & e_5 & e_6 \
e_3 & e_6 & e_8
end {bmatrix} $$

and the rotation matrix can be extracted from normalized eigenvectors of $ texttt {E} _ {33} $,

But I do not know how to get the semi-axes of the ellipsoid. Any help grateful.

Coordinate transformation – distance matrix within a neural network

So I want to create a NetGraph, which is a $ n times $ 3 (With $ n $ varying length) List of 3D coordinates as input and creates one $ n times n $ Distance matrix or one $ frac {n (n-1)} {2} $ List of distances between these points. I would run this net for two lists of 3D coordinates and feed the results into a MeanSquaredLossLayer. This would allow the neural network to learn to reproduce an array of points with the loss that is invariant with respect to shifts and rotations in 3D. However, I have only vague ideas about which modules to use. Maybe NetMapOperators and NetMapThreadOperators? If yes how? Thank you for the reply in advance!

Linear Algebra – What will be the third column of the given matrix?

That's the problemEnter image description here

My attempt: $ frac {1} { sqrt 2} x + 0.y + frac {1} { sqrt2} z = 0, x + z = 0, x = -z $ .

And $ frac {-1} { sqrt 3} x + frac {-1} { sqrt 3} y + frac {1} { sqrt 3} z = 0 $Now put $ x = -z $ we have $ frac {-2} { sqrt 3} x + frac {-1} { sqrt 3} y = 0 $ .$ -2x -y = 0 $. $ x = -y / 2 $

Now I take $ x = c $, then third column becomes $ begin {bmatrix} -2 \ -1 \ 2 end {bmatrix} $

Is it true?

Matrix – How do I solve a large system of differential equations with indexed functions?

I firmly believe that matrix methods are the right answer here, but I can not imagine how to set them up. Imagine a large number of coupled functions:

AB(m,n)(t)

Where m and n are integer indices that can be up to hundreds of thousands or thousands of thousands, and t is a continuous time variable. Imagine AB (m, n) as the concentration of AB of type (m, n), and these concentrations of different types can evolve over time. In addition, we have a single additional feature:

B(t)

it is also a concentration that evolves over time, but for which there is only one type. Initial conditions are:

AB(a,0)(0) = A0
AB(anythingelse)(0) = 0
B(0) = B0

Here a is a constant integer in the hundreds to thousands.

AB (m, n) can be converted into other types by two processes:

AB(m,n) + B --> AB(m-1,n+b)
AB(m,n) --> AB(m-1,n-1)

Here b is a constant integer that is significantly smaller than a. That is, the differential equations that determine the evolution of the system are:

D(AB(m, n)(t), t) == -k1(m, n) AB(m, n)(t) B(t) 
                     +k1(m + 1, n - b) AB(m + 1, n - b)(t) B(t) 
                     -k2(m, n) AB(m, n)(t) 
                     +k2(m + 1, n + 1) AB(m + 1, n + 1)(t)
D(B(t), t) == -k1(m, n) AB(m, n)(t) B(t)

Good assumptions for the forms for k1 and k2 that we can use for testing are:

k1(m_) = (m/a) k3
k2(m_, n_) = k4 n/(n + k5/m) + n k6 k1(m)

Here k3, k4, k5 and k6 are positive real numbers.

How on earth do I organize this system to solve it numerically? NDSolve (or ParametricNDSolve) or similar? The ultimate goal will be to have a measurable function that looks something like this:

Sum(m AB(m,n),{m,0,a},{n,0,infinity})

This function would then be suitable for experimental data that varies k3-k6 and possibly a and b. Later generalizations may be to have a broader distribution of starting concentrations, rather than all being identical, a range of different possible bs, etc.

Control Systems – Problem Solving the Kalman Filter in Mathematica: How to Define a Spectral Density Matrix and Calculate the Covariance Matrix?

I read the classic book on the theory of space control by Bernard Friedland.

To deepen my understanding of the Kalman filter, I would like to illustrate Example 11A Inverted Pendulum on page 418.

However, I can not find a way to define the spectral density matrices (of excitation noise and observation in equation (11A.2)) in Mathematica. How can I also calculate the covariance matrix of noise from a spectral density matrix?

Thank you very much.


Example 11A

numerical analysis – How to solve a quadratic matrix equation with a positive semidefinite constant?

I have the following quadratic matrix equation:

$ XAX + X = B $

Where $ A $ and $ B $ are all positive definite matrix.

The limitation here is that $ X $ is actually a covariance matrix and should definitely be positive.

All I have is that if there is no restriction, the equation can be solved by Bernoulli iteration in the following form:

$ X_ {k + 1} = -A ^ {- 1} (I-BX_k ^ {- 1}) $

However, this does not seem to uphold the restriction.

Any guides would be appreciated, thank you.