Assume $X$ is a compact metric space and $Y$ is a metric space, where $X$ and $Y$ are both connected. Assume $f:X\rightarrow Y$ is continuous. Define the map $g:X\rightarrow \mathbb{R}$ by $g(x)=d(f(x),Y\backslash f(B_{r}(x))$. Is this map continuous? Assuming $f(B_r(x))\neq Y$. Due to comments, ...

In Ahlfors's complex analysis $$ R(z)=\frac{P(z)}{Q(z)} $$ given as the quotient of two polynomials. We assume, and this is essential, that $P(z)$ and $Q(z)$ have no common factors and hence no common zeros. $R(z)$ will be given the value $\infty$ at the zeros of $Q(z)$. It must therefore be con...

Non-Euclidean geometries are generally presented as models for the first four of Euclid's axioms, which do not satisfy the fifth axiom (parallel's postulate). Most texts present two models of non-Euclidean geometries: hyperbolic geometry and spherical geometry. These two models can be considered ...

Assume I have any triangle $\triangle ABC$. I know that given the lengths two sides of the triangle and angle between them, I can find the length of the third side. In other words, given values of $AB$, $AC$, and $\angle BAC$, I can find $BC$. What if instead of the angle, I know the length of $...

Find all matrices $A$ such that $$A=\displaystyle{\sqrt{\pmatrix{1 & 3 & -3 \\ 0 & 4 & 5 \\ 0 & 0 & 9}}} $$ This is a UC Berkeley qualifying exam question. Now I know one method to solve this. Let $$A=\pmatrix{a & b & c \\ d & e & f \\ g & h & i} $$ Now on squaring we'll get $9$ equations and c...

How to compute the rational approximation of square root of a fraction, i.e. I'd like to find $ \frac{a}{b} \approx \sqrt{\frac{m}{n}}$, where $a$, $b$, $m$ and $n$ are integers. Ideally, the approximation should only use additions, subtractions and multiplications. Thanks.

For context this questions is "caused" by the proof by Thomason of the Gillet-Waldhausen theorem (which is proposition 1.11.7 in this paper). Let $\mathcal{C}$ be an exact category, we have a map $\alpha: \prod_{a+1}^b K(\mathcal{C})\to\prod_a^b K(\mathcal{C})$ induced by the functor sending $(...

As title states, I'm interested in figuring out how to construct a model of $\mathsf{ZFC} + \neg\mathsf{Reg}$, preferably off any regular model $\langle M, \in_M \rangle$ of $\mathsf{ZFC}$ without any forcing shenanigans involved. As I'm still working on the problem and don't want to spoil myself...

Is there a term for the relationship between two integers that have the same prime factors? For example, $6=(2)(3)$ and $12=(2)(2)(3)$. Can one describe this with something along the lines of "$6$ and $12$ are <term>" or "$6$ is <term> to $12$"? ETA: "Multiples" is indeed not what I'm after. $...

It's a well known result that the Gaussian integral $$\int_0^\infty e^{-x^2} dx = \frac{\sqrt{\pi}}{2}$$ evaluates to $\frac{\sqrt{\pi}}{2}$. This result can be obtained using double integrals with polar coordinates, among other things, but I'm particularly interested in evaluating this integral ...

Let $\sigma=(1,2,3,\dots,n)$ be an odd $n-$cycle in $S_n$ (so $n$ is even). It is known that the size of its conjugacy class is $|cl_{S_n}(\sigma)|=(n-1)!$. I am interested in the size of the subset $S=cl_{cl_{S_n}(\sigma)}(\sigma)$, that is, the set of all the $n-$cycles that we can obtain by co...

Find the basis of the image of a matrix: $$ A = \begin{pmatrix} 1 & 0 & 1\\ -1 & 1 & -2 \\ 2 & -1 & 4 \end{pmatrix} $$ I'm somewhat confused about the concept of the basis of the image, and I find it perplexing. I find the row-reduced echelon form of the matrix: $$ \begin{pmatrix} 1 & 0 & 1\\ 0 &...

I was just wondering if PEMDAS is realy needed in Math. I think that instead of trying to interpret math by using conventions, we just need to write math in the right order of operation from left to right. PEMDAS does not reflect the real life order of doing things. I am no Math expert, but if I ...

The question is to show that the function $ f(x) = 2x + |\cos x|$ is both one-one and onto. I have managed to show that it is onto but got stuck at proving that it is also one-to-one. To prove this I thought of using the first derivative and showing that the function is always increasing. For bo...

Suppose $a_n$ as the least positive integer $k>1$ such that there exactly exists n primes of the form $m!-k$,Find $ a_{100}$. I firmly believe that $a_n$ must exist for any $n$,is there any clever way to quickly find it? For example: $a_5=13$ because if and only if $m \in\{{4, 5, 9, 11, 12}\}$,...

Lets say we have a linear map $T:L^{1}(\lambda)\rightarrow L^{1}(\lambda)$ where $\lambda$ is some arbitrary measure. Lets say we have some function $F(x):=\int_{-\infty}^{\infty}f(t,x)d\lambda(t)$ where we assume that $f\in L^{1}(\lambda,\mathbb{R}^{2})$. By Fubini we know that $F(x),f_{t}(x):=f...

I am trying to understand a statement about symmetries of a triangle. The set of lecture notes I am working through assert that (1) every permutation corresponds to a symmetry and (2) every symmetry corresponds to exactly one permutation. As there are only six symmetries, I can write out each of ...

In Chapter $5$, Section $4$ of the book Continuum theory by Sam B. Nadler, Jr., the auther defines $\sigma$-connectedness (Definition $5.15$) to be the property of not being a disjoint union of countably many (including finitely many) and more than one nonempty closed subsets. It is well known (f...

Kait rolls a fair 6-sided die until she rolls a 6. if she rolls a 6 on the nth roll, she then rolls the die n more times. what is the probability that she rolls a 6 during these next $n$ times? This can be treated as two events, event $A_n$ she rolls a 6 on the $n$th roll and event $B_n$ she roll...