## Linear programs with strict inequalities and supremum objectives

Linear programming can only solve problems with low inequalities, such as "maximize $$c x$$ such as $$A x leq b$$This makes sense because strict inequality problems often have no solution, for example, "maximize $$x$$ such as $$x < 5$$"There is no solution.

But suppose we seek to find the supremum instead of the maximum. In this case, the above program has a solution – the supremum is $$5$$.

With a linear program with strict inequalities and a supreme objective or infimum, is it possible to solve it by reducing to a standard linear program?

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## What is the eigenvalue and eigenvector of a composition of two arbitrary linear transformations

Let $$S$$ and $$T$$ to be two linear transformation on $$mathbb {R} ^ n$$. One can find the eigenvalue and the eigenvector of $$S$$ and $$T$$. I'm trying to discover the reality between the value and the eigan vector of $$S$$, $$T$$ and $$S circ T$$. Is there a relationship like this?

Thank you.

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## Linear algebra – Scope of the null space of A

I apologize for the brief nature of this question, but I do not think it has been clarified in a previous post on this topic – Find a Covering Set for Null Space.

When we say that the null space of the matrix $$A$$ is equal to the duration of a set of vectors:

$$N (A) = span ( v with v_1, vec v_2,$$ $$… vec v_n)$$

Are we actually saying that:

$$A ( vec v_1) + A ( vec v_2) +$$ $$… A ( vec v_n) = vec 0$$

## reference request – ODE almost linear

Let $$A, B$$ be $$n times n$$ matrices. I am interested in the following ODE in $$mathbb {R} ^ n$$

$$frac {dx_t} {dt} = Ax_t + Bx ^ + _ t$$

or $$x ^ + = (x ^ + _ {1, t}, …, x ^ + _ {n, t})$$ and $$( cdot) ^ +$$ is the rectifier: $$r ^ + = max {0, r }.$$

Does this type of EDE have a name? And are there any known stability criteria? Has it been studied by anyone in general?

The closest I've found is the "linear threshold networks" studied here for example. I appreciate any reference similar to this system.

## linear dependence and independence [on hold]

What does it mean if we say that the functions e ^ x, e ^ 2x, e ^ -x are linearly dependent? What is the intuition of linear dependence and independence?

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## linear transformations – Understanding the linearity of derivatives

I am a little confused as to the assertion that the derivative of a function $$f$$ (whether it is a partial derivative, the total derivative or simply a derivative in $$mathbb {R}$$) is a linear transformation.

1.) By definition, the derivative is always defined at a given moment, for example $$a$$, so let $$Df (a)$$ to be the derivative at this point $$a$$. But if you consider the derivative according to any point $$x$$ from the domain of $$Df$$ then $$Df (x)$$ is not necessarily linear.

2.) If you consider a derivative at a given time $$a$$, $$Df (a)$$ and then you plug in another variable or vector $$v$$ then $$Df (a) (v)$$ is a linear transformation with respect to $$v$$.

Are these thoughts about derivatives correct?

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## linear algebra – Retrieves the translational component of a particular matrix multiplication

Let's say that I have a rotation matrix R 4x and a 4×4 translation matrix T. If I multiply the matrices in this order T * R, the translation component of T will not be affected. But if you multiply them in the R * T order, it will be affected by R.'s rotation values.

How can I describe the position vector of the resulting matrix from the multiplication R * T, in terms of the rotational components of R and the translational component of the matrix T? Can I describe it as the scalar product of the respective orientation component of the matrix R and the translation component of the matrix T?

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## Linear algebra – Conditions necessary for the consistency of a system of inequalities.

I was trying to find the answer to the next question I had when solving a more complex problem:

What are the conditions necessary for the coherence of a system of inequalities, that is to say that there exists at least one set of values ​​for the unknowns which satisfies all the inequalities of the system?

If the above question needs to be answered in a general way, I would make an exception to the answer to the following more specific question:

What are the necessary conditions for the following
begin {align} a_j> f_ {i, j} (x) a_i forall x text {and} forall i, j = 1, …, m, i not = j end {align}
or $$f_ {i, j}$$ is a smooth continuous function, and $$a_i, a_j$$ are constants, to be consistent (there is at least one set of constants $$a_1, …, a_m$$ which satisfy each inequality).

If you have any questions, do not hesitate to ask them.

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## Edge coloring in dense linear hypergraphs

Let $$H = (V, E)$$ to be a hypergraph. Yes $$kappa$$ is a cardinal, we say that a map $$c: E to kappa$$ is a edge coloring if every time $$e_1, e_2 in E$$ with $$e_1 cape_2 neq emptyset$$ then $$c (e_1) neq c (e_2)$$. The smallest cardinal $$kappa$$ such as there is a coloring of the edges $$c: E to kappa$$ is called the chromatic number of edge of $$H$$, noted by $$chi_e (H)$$.

We say that $$H = (V, E)$$ is a dense linear hypergraph if

1. $$bigcup E = V$$,
2. does not matter when $$e_1 neq e_2 in E$$ then $$| e_1 cap e_2 | 1$$, and
3. given $$a neq b in V$$ there is $$e in E$$ with $${a, b } in e$$.

Given a positive integer $$k$$, is there a dense linear hypergraph $$H = (V, E)$$ with $$V$$ finished and $$chi_e (H) <1 / k cdot | V |$$?

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## linear programming – The simplex method does not solve the problem of assignment?

The problem:

I'm trying to solve http://acm.timus.ru/problem.aspx?space=1&num=1076, it's an online judge for programming problems. The problem could be solved by simply applying an assignment problem solving algorithm such as mincost-maxflow or a Hungarian algorithm. The dice are small (150 out of 150) and I try to put forward a solution with the simplex method – it does not exceed the time limit and I do not think it's my fault.

Problem of assignment

$$min sum c_ {i, j} x_ {i, j} \ sum_ {i} x_ {i, j} = 1 \ sum_ {j} x_ {i, j} = 1 \ x_ {i, j} geq 0$$

What I did:

• I added the cycling check – indeed, cycling was present. I've used some schemes to eliminate cycling: Bland's rule, lexicographic rule and Zadeh's rule. None of them triggered the cycle check, but the solution still has not exceeded the time limit.
• I've removed all the artificial variables and pants. My initialization is $$x_ {i, i} = 1$$ and the other n-1 basic variables are 0 (I use variables that do not have a non-zero value after a Gaussian elimination with $$x_ {i, i}$$)

My hypothesis

I am not an expert in the areas of linear programming, but here are some of my thoughts: I think the problem is very degenerate, the rank matrix of constraints is $$2n-1$$ with n variables having a value equal to 1 and rest n-1 having a value equal to 0. During a step of the algorithm, we enter a situation in which many variables with value 0 are exchanged, which makes the simplex method exponential. I print the cost of the function and the discount ceases after a while, although the cost reduction indicates that an improvement is still possible (that is, the criterion of the ## EQU1 ## Stop of the simplex method is not satisfied).

What I do not want

• I have an accepted verdict on this problem with mincost-maxflow, so please do not suggest any Hungarian algorithm or similar, I want to know if simplex method can solve this problem
• Please do not suggest changes for the simplex method that make assumptions about the graphical model of the problem. suggestions such as "make the edges of value 0 an alternative path". I want to know if the purely simplex method can solve the problem (with possible modifications, but that makes no assumptions about the problem).

What I want

• I want to know if my hypothesis is correct and that the simplex method simply does not apply to this kind of problem.
• If not, what kind of system can be used to solve this problem.
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