Articles of soft question

Measuring the set-theoretical complexity of sets/spaces encountered in general analysis

In analysis, it is common to encounter subsets of $\mathbb R$ (or even $\mathbb R^n$) which appear to be “well-behaved”, especially with regard to properties like being measurable, compactness, etc. It is core to descriptive set theory (DST) that one is able to impose a classification of such subsets by closure properties of varying degree. […]

Generalization of Cayley-Hamilton for non-diagonal coefficients related to other properties than eigenvalues?

The famous Cayley-Hamilton theorem states that a matrix $\bf A$ with eigenvalues $$\{\lambda_k({\bf A})\} \hspace{0.5cm} \text{ s.t. } \hspace{0.5cm} \det({\bf A}-\lambda_k({\bf A}){\bf I}) =0, \forall k$$ must be a root to it’s own characteristic polynomial: $$p(x) = \prod_{\forall_i}^n(x-\lambda_i({\bf A})) = c_0+c_1x+\cdots x^n$$ so that $$p({\bf A}) = c_0{\bf I}+c_1{\bf A} + \cdots +{\bf A}^n = […]

Basis of a basis

I’m having troubles to understand the concept of coordinates in Linear Algebra. Let me give an example: Consider the following basis of $\mathbb R^2$: $S_1=\{u_1=(1,-2),u_2=(3,-4)\}$ and $S_2=\{v_1=(1,3),v_2=(3,8)\}$ Let $w=(2,3)$ be a vector with coordinates in $S_1$, then $w=2u_1+3u_2=2(1,-2)+3(3,-4)=(11,-16)$. When I tried to found the coordinates of $w$ in $S_2$, I found the following problem: Which […]

What are some areas of research/industry involving stochastic processes that aren't finance-related?

I’ve always enjoyed probability and stochastic processes (took two courses in stochastic models in undergrad, and a PhD level intro to stochastic processes course for my master’s). Someday I’d like to go back to school, and most likely I’ll study applied probability/stochastic processes. Now, the vast majority of work in this area these days seems […]

Multiply fraction by $-1/-1$

Sometimes in algebraic exercises where the solution is a ratio I get the solution multiplied by $(-1/-1)$. For example I get $$ (2zy-y) / (3-z) = x $$ If I multiply the fraction by $(-1/-1)$ I get $$ (y-2zy) / (z-3) = x $$ This can be quite confusing If I have large algebraic terms. […]

What is second Bartlett identity?

I came across the term second Bartlett identity in the wikipedia link: https://en.wikipedia.org/wiki/Variance_function However could not find detail about it. Can anyone help me to understand what this identity is about. More importantly I want to know under what assumption this identity is true.

Does the Einstein summation convention still work for infinite sums?

One reason the Einstein summation convention seems to be useful, at least in my extremely limited experience, is that in calculations involving a chain rule of some sort and changes of coordinates, one can essentially skip many steps in a computation involving changing the order of summation, all appealing essentially to the distributive and commutative […]

How can I improve my explanation that the ratio $\theta=\frac{s}{r}$ that defines the radian measure holds for all circles?

I’m trying to demonstrate why the ratio $\theta=\frac{s}{r}$ (where $s$ is an arc measuring some $s$-units in length and $r$ is the radius of the circle) which defines the radian measure holds for all circles. I’m attempting to show this by using two circles, one of radius $1$ and another radius $r$. I would like […]

Which quadrant is the “first quadrant”?

In the coordinate plane split into four quadrants by the $x$- and $y$-axes, I learned (educated in a public school in the U.S.) that the “first quadrant” was the one with both $x$ and $y$ positive, almost always drawn to be the top right region. However, my friend (who is from the U.K.) recently told […]

Supplemental reference request-Graduate level PDE problems and solutions book

I have been able to find these two but I don’t know how valuable they are as a reference, “Problems and examples in differential equations” By Biler, and, “Partial Differential Equations through Examples and Exercises” by Pap. However, I don’t know if these are any good, or the ones that I’m looking for, maybe you […]