Articles of reference request

Is there a statistical measure of bitwise entropy?

(Somewhat inspired by this website, particularly Section III. Also, I might be using a different definition of entropy than usual; what I am using is closest to the physics definition (the one I encountered first) of the amount of disorder in a system.) Consider the following two “random” 256-bit strings: 1110001011010010101000001111001100001100011111000111011011101000000000000001111110010110010100011101010010111110000010010101001001101100111110011000000110111111000111101111000011010100001001100010010010011000000011101110000000110001101100000110111001100011 …created via RNG, and: […]

Cryptography textbook

Might come as a rather strange request but does anyone know a textbook on cryptography that is small and short, say around 300 pages max. I am tired of having a sore shoulder from carrying 5 heavy math textbooks. Want to carry around a cryptography textbook for anytime reading that is concise but I won’t […]

$f(x)$ irreducible over $\mathbb Q$ of prime degree ; then Galois group of $f$ is solvable iff $L=\mathbb Q(a,b)$ for any two roots $a,b$ of $f$?

Let $f(x)\in \mathbb Q[x]$ be irreducible polynomial of prime degree , $L$ be its splitting field , then how to show that $f(x)$ is solvable by radicals over $\mathbb Q$ iff $L=\mathbb Q(a,b)$ for any two roots $a,b$ of $f(x)$ ? i.e. how to show that $Gal(L/\mathbb Q)$ is solvable iff $L=\mathbb Q(a,b)$ for any […]

Simultaneous Maximization and Minimization

I have a function with two variables say $$g(x,y)=f(x)−h(x,y)\ $$ where $$f(x)= ax-bx^2\ $$ and $$h(x,y)=(x+y)^2\ $$ and $$ y\ge0, x+y\ge0.$$ My purpose is to maximize $g(x,y)$ for $x$, simultaneously minimizing $h(x,y)$ for $y$. How I can do this? Is there any literature available related to this problem? Waiting your expert response.

Pigeonhole principle exercises

I have an exam in combinatorics on Friday and the pigeonhole principle is a part of the material. Can someone give me a reference to a book with the hardest(!) questions on this material? Thank you very much, it can help me a lot!

Is following system of nonlinear ODEs recognized?

The following system of ODEs – is it recognized as distinct system, with meaningful background and uses? $$\frac{dx}{dt} = – [x(t)]^2 – x(t)y(t)$$ $$\frac{dy}{dt} = – [y(t)]^2 – x(t)y(t)$$ This is probably not needed, but initial conditions: $x(t=0) = x_0, \space y(t=0) = y_0$

Introductory text for Group Rings

Is there any other text books on Group rings except The algebraic structures of Group Rings by D.Passman. This book is really good but it will help if I know about other books on the topic too. Thanks!

$H^s(\mathbb{T})$ space is a Banach algebra with pointwise product

I have ran across the following theorem but the given proof does not convince me. Theorem Let $u, v \in H^s(\mathbb{T})$ with $s>1/2$. Then the pointwise product $uv$ is in $H^s(\mathbb{T})$ and $\lVert uv\rVert_s \le C\lVert u\rVert_s\lVert v \rVert_s$ for a constant $C$ not depending on $u$ and $v$. In functional terms, this theorem is […]

Statistics resources with examples for a C.S. student

I’m a computer science student and is fairly familiar with basic probability (calculating the probability of a event occurring, pmfs and pdfs) but I find it very difficult to grasp the concepts of advanced probability like principles of data reduction (sufficiency, likelihood principle, etc), point and interval estimation, Hypothesis testing, etc. I think it is […]

Looking for online matlab-based differential equations course/text.

I am looking for an online ODE course that would be matlab/project-oriented. A full online text/course in the spirit of this linear algebra text is preferred. I know about the following CODEE and its odetoolkit IDE the Interactive Differential Equations and its labs