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Articles by Steven Gregory
Lysosomal Degradation Is Required for Sustained Phagocytosis of Bacteria by Macrophages
Publication HistoryPublished: June 1, 2017Accepted: May 8, 2017; Received in revised form: April 6, 2017; Received: November 30, 2016;
mathematical physics - why does the "separation of variables" method work? - Mathematics Stack Exchange
If we have $$\frac{dy(x)}{dx}=f(x)g(y(x)),$$ then the careful way of solving this equation (assuming $g(y) eq 0$) is to divide by $g(y)$ in order to get Replace $x$ with $t$ and integrate each side from an arbitrary lower bound $a$ to $x$: Now note that on the LHS we can do the substitution $w=y(t)$, which gives $dw=y'(t) dt$ so that the above becomes $$\int^{y(x)}_{y(a)} \frac{1}{g(w)} dw =\int_a^x f(t) dt.$$ Now be sloppy with notation and replace $t$ with $x$, $w$ with $y$ and write $y(x)$...
mathematical physics - Why does the "separation of variables" method for DEs work? - Mathematics Stack Exchange
If we have $$\frac{dy(x)}{dx}=f(x)g(y(x)),$$ then the careful way of solving this equation (assuming $g(y) eq 0$) is to divide by $g(y)$ in order to get Replace $x$ with $t$ and integrate each side from an arbitrary lower bound $a$ to $x$: Now note that on the LHS we can do the substitution $w=y(t)$, which gives $dw=y'(t) dt$ so that the above becomes $$\int^{y(x)}_{y(a)} \frac{1}{g(w)} dw =\int_a^x f(t) dt.$$ Now be sloppy with notation and replace $t$ with $x$, $w$ with $y$ and write $y(x)$...
How to think of a set?
You're not alone in being confused about how to think of a set. Working out a sensible language for talking about sets cost mathematicians a great deal of sweat and tears. It's now considered one of the major achievements of 20th-century mathematics, and it's still being built upon today. If you set out to explore our universe of sets, or the universes just beyond it, you'll quickly encounter sets that are dizzyingly, terrifyingly, hilariously counterintuitive.
Tricky Rectangle Problem
Here is a more combinatorial/algebraic approach: Let the Horizontal lines from bottom to top be $h_{0}, h_{1}, ..., h_{6}$ and Vertical lines from left to right be $v_{0}, v_{1}, ..., v_{6}$ Rectangles that contain bottom left yellow square is all rectangles that can be formed by choosing one horizontal line from $\{h_{0}, h_{1}\}$, another horizontal line from $\{h_{2}, ..., h_{6}\}$ and one vertical line from $\{v_{0}, v_{1}\}$, another vertical line from $\{v_{2}, ..., v_{6}\}$.
Naive Set Theory by Halmos is confusing to a layman like me
I think Halmos' Naive Set Theory is primarily concerned with set theory as a foundation on top of which mathematics is built, but the word "naive", if I understand correctly, just means he's viewing the concept of a set concretely as a collection of things rather than axiomatically as being whatever satisfies the axioms. As to set theory applied to machine learning, it may be that what is needed differs from the content of Halmos' book. I like the book by E.
Algebraic Solving and Uniqueness Proofs
The following issue came up in my Intro to Proofs course and I wasn't sure how to explain my distaste of the student proof. Prove that the solution for $x$ in $ax+b=c$ is unique ($a eq 0$). Student Proof: Solving gives \begin{align*} ax+b &= c\\ ax &=c-b\\ x &= \frac{c-b}{a}. \end{align*} Hence the solution must be $x=\frac{c-b}{a}.$ The Proof I Wanted: Suppose both $x$ and $y$ solve the equation, then \begin{align*} ax+b &= ay+b\\ ax&=ay\\ x&=y. \end{align*} Thus the solution is unique.
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