# All Questions

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### Integration against divergence free vector fields

Let $\chi:\Omega\to \mathbb{R}^n$ be a vector field on a bounded, smooth domain $\Omega \subset \mathbb{R}^n$. Assume that for any divergence free vector field $\eta:\Omega \to \mathbb{R}^n$ we have ...
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### CDF of smallest eigenvalue of non-central Wishart matrix - how to evaluate the integral.

Does anybody know how to derive the distribution of the smallest root of a non-central Wishart matrix? I have got an integral expression that would give me the desired answer but cannot solve the ...
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### Casio fx-83gt - help! Linear Programming

Hi guys I'm studying a module called Operational Research and in particular linear programming. I am doing the simplex method and as anyone who studies linear programming would know, you need to ...
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### How to prove that all primitive polynomials are irreducible

Let $F$ be a finite field, and $F[X]$ set of all polynomials in $F$, how to prove that: why all primitive polynomials $\;$ $f \in F[X]$ $\;$ must be an irreducible. Note: Polynomial primitive is an ...
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### Finding the Formula of the Product of $e_{i,j}$ and $e_{k,l}$ to Return Zero Matrix

My teacher for calculus this year gave a handout on the first day with an excerpt from Rings, Fields, and Vector Spaces by B.A. Sethuraman. The reason for this is in the beginning of Sethuraman's book ...
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### Lie algebra: If ad(g) is solvable then g solvable?

I'm trying to prove that if the image of the adjoint representation of a Lie algebra g is solvable then g is solvable, ie, if for some n (ad(g))^(n) = 0 then there exists a m such that g^(m) = 0 My ...
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### Deriving equation in vector notation

I had some trouble deriving an equation from the book 'Elements of statistical Learning' p. 108 equation 4.9. This heavily relies on linear algebra, so I was wondering how the author came to his final ...
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### Does half-affine imply affine?

Let $V$ and $W$ denote real vector spaces, and consider a function $f : V \rightarrow W.$ Call $f$: Half-linear iff for all real $a,b \geq 0,$ we have $f(ax+by) = af(x)+bf(y)$. Half-affine iff for ...
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### Can the natural embedding $K\to K[X]/(f)$ be extended to form an isomorphism $L/K\to K[X]/(f)$?

I'm studying for an abstract algebra exam (covering commutative rings and Galois theory). As an exercise, I'm trying to work out on my own a proof of the theorem that, given a field $K$ and a ...
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### Given a block-form contraction operator $X$, can we write $I-XX^*$ as $PP^*$ with a nice block form of $P$?

Suppose the operator $$X = \begin{pmatrix} A & B \\ C & D \end{pmatrix}$$ is contractive, where $A, B, C$ and $D$ are themselves bounded operators, then we know that $I - XX^* = PP^*$ for ...
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### Plug in numbers to find relationship between numbers

A total of k passengers went on a bus trip. Each of the n buses that were used to transport the passengers could seat a maximum of x passengers. If one bus had 3 empty seats and the remaining buses ...
Given an operator $\rho$ on a Hilbert space $H$, is there a notion of nearest self-adjoint (positive) approximation of $\rho$ for a suitable norm? More specifically, does there exist an algebraic ...
Do I need to ensure I don't stray from the transform in the table? $\frac{-2}{s-1}$ this looks like $-2*\frac{a}{s^2-a^2},$ for $a=1$ Does this yield $-2\sinh(t)$, or should it fit perfectly to ...