Intereting Posts

Dimension of subspace of commuting matrices
Examples of statements which are true but not provable
$(a+b)^2+4ab$ and $a^2+b^2$ are both squares
Integral equal $0$ for all $x$ implies $f=0$ a.e.
Testing Zeros Of The Riemann Hypothesis
How did we find the solution?
$4$-digit positive integers that does not contain the digits $3$ and $4$ plus other properties
Formula for the series $f(x):=\sum\limits_{n=1}^\infty\displaystyle\frac{x}{x^2+n^2}$
Subgroups of $S_n$ of index $n$ are isomorphic to $ S_{n-1}$
solution of differential equation $\left(\frac{dy}{dx}\right)^2-x\frac{dy}{dx}+y=0$
Alternative definitions of stochastic processes?
Prove that $\lim \limits_{n \to \infty} \frac{x^n}{n!} = 0$, $x \in \Bbb R$.
Cubes of binomial coefficients $\sum_{n=0}^{\infty}{{2n\choose n}^3\over 2^{6n}}={\pi\over \Gamma^4\left({3\over 4}\right)}$
If $a_n=n^2+20$ and $d_n$ denotes the greatest common divisor of $a_n$ and $a_{n+1}$ then show that $d_n$ divides 81
Proving $\lim_{n\to \infty}\frac{n^\alpha}{2^n}=0, \alpha>1$

Definitions:

$$v(x)\equiv\{g_1(x),g_2(x),\ldots,g_n(x)\}^T$$

$$C\equiv \operatorname{cov}(v)=\langle vv^T \rangle -\langle v\rangle \langle v^T \rangle =\int f(x)v(x)v(x)^T \, dx-\int f(x)v(x) \, dx \int f(x’)v(x’)^T \, dx’$$

- Calculation of covariances $cov(x_i^{2},x_j)$ and $cov(x_i^{2},x_j^{2})$ for multinomial distribution
- What does Determinant of Covariance Matrix give?
- A question on conditional expectation leading to zero covariance and vice versa
- Minimum / Maximum and other Advanced Properties of the Covariance of Two Random Variables
- 3-sigma Ellipse, why axis length scales with square root of eigenvalues of covariance-matrix
- Uncorrelating random variables.

$$R(x)\equiv v(x)^T C^\dagger v(x)$$

z is an implicit parameter of f(x) and of all the g(x)’s.

How can one go about optimizing R wrt z if C is singular?

For an non-singular D, then we’d simply use $\frac{d}{dz}D^{-1}=-D^{-1}\frac{dD}{dz}D^{-1}$.

I found this helpful EQUATION for the derivative of a pseudoinverse, but it only applies when the change in z does not change the rank of C, because an $\epsilon$-order variation in the null space of C creates a $\epsilon^{-1}$-order variation in the null space of $C^{\dagger}$, inducing an $\epsilon^{0}$-order change in R.

So R has possible discontinuities in z whenever the change in z induces a change the integer rank of C. I gather that I need to check the values of R at each of those discontinuities and also values of z where $\frac{dR}{dz}=0$ between the discontinuities. But I have no idea how to do either of those. Testing for when C changes from full rank to rank deficient is easy, just check if the determinate is zero or non-zero. But I don’t what tests would tell us when the rank simply changes.

It’s important in this problem to note that v(x) will always have 0 component in the null space of C, so that simplifies the problem a little and assures us that R and its discontinuities will always be finite.

So how can I go about optimizing R(x)?

- A question on conditional expectation leading to zero covariance and vice versa
- Intuitive geometric explanation: existence of eigenvalue in odd dimension real vector space.
- Show $\exp(A)=\cos(\sqrt{\det(A)})I+\frac{\sin(\sqrt{\det(A)})}{\sqrt{\det(A)}}A,A\in M(2,\mathbb{C})$
- Need verification - Prove a Hermitian matrix $(\textbf{A}^\ast = \textbf{A})$ has only real eigenvalues
- Maximize the value of $v^{T}Av$
- Let $T,S$ be linear transformations, $T:\mathbb R^4 \rightarrow \mathbb R^4$, such that $T^3+3T^2=4I, S=T^4+3T^3-4I$. Comment on S.
- If both $A-\frac{1}{2}I$ and $A + \frac{1}{2}I$ are orthogonal matrices, then…
- Inner product for vector space over arbitrary field
- Determinant of transpose?
- Prove that a polytope is closed

- For which topological spaces $X$ can one write $X \approx Y \times Y$? Is $Y$ unique?
- Archimedean Proof?
- A 3-minute algebra problem
- Show that $ \mathbb{E} < \infty \Longleftrightarrow \sum_{n=1}^\infty \mathbb{P} < \infty $ for random variable $X$
- Is the group isomorphism $\exp(\alpha x)$ from the group $(\mathbb{R},+)$ to $(\mathbb{R}_{>0},\times)$ unique?
- Proof that $\mathbb{R}$ is not a finite dimensional vector space
- Is it true that $0.999999999\dots=1$?
- Recurrence relationship
- Irrationality of sum of two logarithms: $\log_2 5 +\log_3 5$
- DNF form ( $ \neg ((p \wedge q) \equiv (r \vee s)) $ )
- Closed form for improper definite integral involving trig functions and exponentials?
- Counting set partitions of $\{1,2,…,n\}$ into exactly $k$ non-empty subsets with max size $m$
- If $a$ is a quadratic residue modulo every prime $p$, it is a square – without using quadratic reciprocity.
- Fourier Analysis
- Average distance between two random points in a square