Given a degree 1 rational function, in the form $$f(x) = \frac{ax+b}{cx+d}$$ I am trying to come up with a method to determine whether or not any integer coordinate points (i.e., $(x, f(x))$ such that $x \in \mathbb{Z}$ and $f(x) \in \mathbb{Z}$) exist within a finite domain from some lower bound...

Consider a $n$ by $n$ square subdivided into unit squares. What is the shortest path you can take through the square that touches every unit square? Touching the edges/vertices of the squares is sufficient, the path can be of any shape, with any start and end point. I have a solution but I don't ...

Do there exist non-constant integer sequences $a_n,b_n,c_n,d_n$ such that for all $n\geq 1$, $1.$ $a_n c_n-b_n d_n>0$ $2.$ $a_n,d_n>0$ and $a_n<-b_n$ and $d_n<-c_n$ $3.$ $\lim_{n\to\infty} \frac{a_nc_n-b_nd_n}{a_n}=0$ $4.$ $\lim_{n\to\infty} \frac{a_nc_n-b_nd_n}{b_n}=0$ I tried as follows: Since ...

Divide the plane into unit squares. I would like to know the largest number of grid squares that can be touched by a path of (Euclidean) length $l$. Touching an edge or vertex o the square counts; e.g. the path in the diagram below has length $1 + \sqrt{5}$ and touches $10$ squares. The start and...

The problem Consider the real numbers $x$ and $m$ that satisfy the equality $x^3+3x^2-(m^2+3m-1)x+m+3=0$ $a)$ If $x$ is fixed, determine the values ​​of m for which the equality is true $b)$ Determine the real numbers $m$ for which there are exactly three distinct values ​​of $x$ for which the...

The Weierstrass transform $W$ of $f:\mathbb{R}\to\mathbb{R}$ can be defined as the convolution of $f$ with a Gaussian: $$W[f](x) = \frac1{\sqrt{4\pi} } \int_{\mathbb{R}} dy\, f(x-y) e^{-y^2/4}.$$ It's not hard to show that the operator $W$ can also be written formally as $W=e^{D^2}$ with $D$ the ...

This might be a bit of a soft question, but it seems to be widely accepted among those familiar with PDEs that closed-form solutions rarely exist. Instead, much of the focus is on proving existence, uniqueness, boundedness, and other properties. My question is: How do we actually know that a PDE ...

Lemma 1.8 in Farb and Margalit's primer says that if transverse simple closed curves $\alpha$ and $\beta$ in a surface $S$ do not form any bigons, then in the universal cover of $S$, any pair of lifts $\tilde{\alpha}$ and $\tilde{\beta}$ of $\alpha$ and $\beta$ intersect in at most one point. Th...

Let $K$ be a field and let $A$ be a finitely generated commutative $K$-algebra which is also a local ring. Must $A$ be Artinian? Geometrically, I belive that this should be true: A local ring is "essentially a point" and an Artinian ring is "essentially a finite union of points". At Atiyah-Macdo...

I was constructing a figure using the below description. The question asks to prove something. But, when I constructed the figure and drew the circumcircle of $\triangle ANK$, the point $Q$ was not on the circle. Which means that I am missing something in the construction. A key point is that the...

I have a family of functions $P_{n,j}(x)$ defined by the following recursive relation: for all $0 \leq j\leq n,$ $$P_{n,j}(x)=P_{n,j-1}(x)-\sum_{i=1}^{j-1}P_{i,i}(x)P_{n-i,j-i}(x);\qquad P_{n,0}(x)=P_{n,n}(x-1);\qquad P_{n,n}(1)=\mathbb{1}_{n=1}.$$ This gives: $$P_{n,n}(x)=P_{n,n}(x-1)-\sum_{j=2}...

I came across the following statement about Sobolev spaces: $$ H^m(\Omega) \cong H^m(\Omega_1) \otimes \cdots \otimes H^m(\Omega_d) $$ where $ \Omega = \Omega_1 \times \cdots \times \Omega_d $. I'd like to understand: How exactly is this isomorphism defined? Can this result be generalized to $ L...

Consider the sum $$ S(x) = \frac{1}{2}+\sum_{k=1}^{\infty}\frac{x^k}{k!(2^k+1)}. $$ I have strong reasons to believe (it would take long time to explain them) that for $x \to -\infty$ the function $S(x)$ oscillates (does not have a limit) and that the amplitude of oscillations is finite (nonzero,...

The relation "has the same order of" is of equivalence in the set of all the groups. The relation "is isomorphic to" is of equivalence in each set made of the groups of one same order (so, in a sense, it gets a finer partition of the set of all the groups). The equivalence relation "is affinely i...

I am currently studying wave propagation in elastic materials. Working with the nonlinear case, in some approximation of high nonlinearity, I was able to get the following ODE $$ y(x)y'(x)y''(x)=-\omega^2. $$ This equation can be solved by a CAS or Wolfram Alpha using the incomplete $\Gamma$ func...