Intereting Posts

Find the equation of the tangent line to $y=x^4-4x^3-5x+7$
If $\operatorname{ord}_ma=10$, find $\operatorname{ord}_ma^6$
How to interpret “computable real numbers are not countable, and are complete”?
Does the golden angle produce maximally distant divisions of a circle?
which odd integers $n$ divides $3^{n}+1$?
Sum of cosines of primes
Fourier series for $\sin^2(x)$
Prove that $ f$ is uniformly continuous
Is every ideal of $R$ a sum of a nilpotent ideal and an idempotent ideal?
Ring with four solutions to $x^2-1=0$
Permuting 15 books about 2 shelves, with at least one book on each shelf.
Determining if a quadratic polynomial is always positive
condition for cones to be reciprocal
Integers in biquadratic extensions
Solutions of autonomous ODEs are monotonic

Take $S^3$ to be the three-sphere, that is, $S^3=\lbrace (x_1,x_2,x_3,x_4):x_1^2+x_2^2+x_3^2+x_4^4=1\rbrace$. Using the stereographic projection, $S^3=\mathbb{R}^3\cup \lbrace \infty \rbrace.$ Can someone explain how the complement of the solid torus (centered at the origin) $S^1\times D^2$, where $D^2$ is a 2-disk, is also a torus? I am reading Milnor’s paper “On Manifolds Homeomorphic to the 7-Sphere,” and this is a prerequisite to understand how Milnor glues the surfaces of two tori of the form $S^3 \times D^4$ in $S^6$ to create an exotic $7$-sphere.

- Proving that the set of limit points of a set is closed
- How do you rebuild your Math skills after college?
- Does evaluating hyperreal $f(H)$ boil down to $f(±∞)$ in the standard theory of limits?
- Is it morally right and pedagogically right to google answers to homework?
- Expanding problem solving skill
- $I^2$ does not retract into comb space
- “Truncated” metric equivalence
- Density of positive multiples of an irrational number
- Dilemma for Studying Probability Theory while Waiting to Learn Measure Theory
- Why are these two definitions of a perfectly normal space equivalent?

One solid torus is $x_1^2 + x_2^2 \leq \frac{1}{2}.$ The other solid torus is $x_3^2 + x_4^2 \leq \frac{1}{2}.$ The intersection, a torus, is…

The boundary of $x_1^2+x_2^2 < \frac{1}{2}$ (in the $x_1x_2$-plane) is a circle, with equation $x_1^2+x_2^2 = \frac{1}{2}$. Likewise for the boundary of $x_3^2+x_4^2 < \frac{1}{2}$. The boundary of either region is thus the cartesian product of two circles, a torus.

- Compute this factor group: $\mathbb Z_4\times\mathbb Z_6/\langle (0,2) \rangle$
- Book recommendation for network theory
- Distance between the product of marginal distributions and the joint distribution
- Elementary Applications of Cayley's Theorem in Group Theory
- Sufficient condition for a *-homomorphism between C*-algebras being isometric
- Is a non-negative random variable with zero mean almost surely zero?
- Markov Chain – method of generating function -to find nth power
- Proof for: $(a+b)^{p} \equiv a^p + b^p \pmod p$
- $R = \mathbb{Z}\ /\ (1+3i)$ – Ring homomorphisms
- Is “imposing” one function onto another ever used in mathematics?
- Using substitution while using taylor expansion
- Isomorphic subfields of $\mathbb C$
- A generalization of a divisibility relation for Fibonacci numbers
- Consider $u_t – \Delta u = f(u)$ and $u=0$ on $\partial\Omega \times (0,\infty)$. Show if $u(x,0) \geq 0$, then $u(x,t) \geq 0$
- Consecutive composite numbers