Recent Questions - Mathematics Stack Exchange |
- Using the properties of a group (G,.) to get certain conclusions, but unsure is my approach is acceptable
- Given the vector field $F$ compute the flux of the curl of $F$ through the surface $\Sigma$
- Why are there two results of the proof of the formula $\displaystyle 2tan^{-1} x-tan^{-1}\frac{x^{2} -1}{2x} =\frac{\Pi }{2}$(x>0)?
- Growth of polynomial functions and trigonometric functions
- What is the expected value of an Exponential r.v. raised to a Poisson r.v.?
- Let $x,y,z >0$ and $xy+yz+xz=2022xyz$
- Calculate conditional distribution of sum of normal
- Why growth function little-omega $\omega(m)$ equals $\Theta(\log{m})$
- Eigenvalue of A and A*
- The combined usage of min and max() in a graph definition.
- Given $n$ arbitrary points, generate a function that passes through all points, with the line (or arc) between two points resembling a straight line
- mapping the circle $|z|=3$ into $|z-1|=1$, the point $3+3i$ into $1$ and the point $3$ into $0$.
- Definition of a reflection as an isometric, involutory, linear map
- Prove o disprove that $X$ is connected?
- Can two hikers on different trails always reach the peak of a mountain while matching altitude?
- How do I find eigenspace bases in a generalized way?
- Does the summation property $\sum a_{k} x^{i+k}=x^{i} \sum a_{k} x^{k}$ have a name?
- Understanding the usage of a theorem about cohomology of a compact Lie group.
- Is there a symbol for embedded manifold?
- If $U_1,U_2,\dots$ are open subsets of $[0,1]$, then either prove or disprove the statements
- Uniform convergence of a sequence of operators
- Using linear approximation to show $P_0 \left( 1 + \frac{k - 1}{2}M \right)^{k/(k-1)} \approx P_0 \left(1 + \frac{k}{2} M \right)$
- Validation of Proof that Nested Interval Property implies Axiom of Completeness
- Why is $n(F,D)$ odd, when $F$ is odd?
- A problem arises when computing the integral of the sinc function using the Fourier transform.
- Quotients of equivalence relations
- Explicit description of $\mathbb{Z}\otimes_{\mathbb{N}}\mathbb{Z}$
- What is a direct proof of the isomorphism $\mathfrak{so}(3)_{\mathbb C}\simeq\mathfrak{sl}(2,\mathbb C)$?
- Proving a version of a local limit theorem
- Finding the Upper and Lower Bounds for Rational Zeros of a Polynomial
| Posted: 07 Aug 2021 08:19 PM PDT If a and b are any two elements of a group $(G,.)$ which are commute. Show that a and $b^{-1}$ are commute. To approach this proof, I started by assuming that the given statement "a and $b^{-1}$ are commute" is true and subsequently, used associative, identity and inverse properties to reach to the conclusion where L.H.S=R.H.S But in a few book solutions, they have generally started from the given case that $ab=ba$, so is my approach an acceptable way to prove such questions.  |
| Given the vector field $F$ compute the flux of the curl of $F$ through the surface $\Sigma$ Posted: 07 Aug 2021 08:14 PM PDT I was having some problems understanding how he found $\gamma(t)$ from the given $\Sigma$ and i was hoping someone could explain to me how if that is ok So the problem goes like this: Given the vector field $F(x, y, z) = (z, x, y)$, compute the flux of the curl of F through the surface $ Σ = (x, y, z) ∈ R^ 3 : z = xy, x^2 + y^ 2 ≤ 1 $ oriented so that the normal versor points upward So what the professor did was first he computed $\gamma(t)$ using parametrization and he immediately writes $\gamma(t)=(cos(t),sin (t), cos(t)sin(t) )$ with $t\in[0,2\pi]$ and from here he finds $\gamma$' and from there he computes $\int _\Sigma rot F *nd\sigma$=$\int_0^{2\pi} F(\gamma(t))\gamma'(t)dt$ and from there is history i can do it myself But what i couldn't understand is how did he get the $\gamma$  |
| Posted: 07 Aug 2021 08:10 PM PDT Here is the process of my consideration: The proof process from (1) $\displaystyle \therefore $The left and right sides of the equation are not equal, the original formula does not hold The proof process from (2) $\displaystyle \therefore $The left and right sides of the equation are equal, the original formula is established The results above are confusing me, I guess I missed some details. The formula is proven by the second option from the reference answer. But I can't understand why the first option doesn't work. By the way, except "α" is the first quadrant angle, is there another else hint from the condition "x>0"? I'm not an English speaker, so If there is any problem with the grammar to confuse you, please let me know. Thanks for any help.  |
| Growth of polynomial functions and trigonometric functions Posted: 07 Aug 2021 08:10 PM PDT I found an interesting property, which is the tangent function may be equal to the sixth degree polynomial as below:
This is the graph of both the functions $$\lim_{n\rightarrow \infty}(nx\prod_{i=1}^{\infty}(1-\frac {x^6}{i^6\pi^6}))$$ and $$\lim_{n\rightarrow \infty}n\tan x$$ The problem is that to have them equal, the growth of $n$ have to be different, for instance $9999\cdot 100\prod_{i=1}^{9999}(1-\frac {100^6}{i^6\pi^6})=-2.1104\cdot 10^{74}$ whereas $9999\tan {100}=-5871$. I may assume there are two cases, for which the value $n$ on the RHS be $f(n)$, such that $f(n)\geq n$ on the interval $(-\infty, 0)\cup(0, \infty)$ and $f(n)=n$ for when $n=0$: Case 1: $f(n)=cn$ for some constant $c$, which shows no significant changes. Case 2: $f(n)$ is some function of $n$, and may increase exponentially, this may include $a^n, n^n, n^b, a\wedge b\in \mathbb{Z}$. If there is any function fulfilling the case, then we can find the closed form of $\zeta(3)$ as follows: Set $\zeta(3)=\omega$, thus $$\lim_{n\rightarrow \infty}(nx\prod_{i=1}^{\infty}(1-\frac {x^6}{i^6\pi^6}))=\lim_{n\rightarrow \infty}f(n)\tan x$$. The coefficients of $x^7$ are $$\lim_{n\rightarrow \infty}\frac n {\pi^6}\omega ^2=\lim_{n\rightarrow \infty}f(n)\frac {17} {315}$$ The question is: what may be that function?  |
| What is the expected value of an Exponential r.v. raised to a Poisson r.v.? Posted: 07 Aug 2021 08:21 PM PDT Let $X \sim \operatorname{Exp}(\lambda)$ and $Y \sim \operatorname{Poisson}(\mu)$, where $X$ and $Y$ are independent and $\mu < \lambda$. Compute $\mathop{\mathbb{E}}[X^Y]$ using conditional expectation. We are given that the expectation of the moment generation function ($\mathop{\mathbb{E}}[e^{tX}]$) of $X$ is $\frac{\lambda}{\lambda - t}$, though I'm not sure where to use this. So far, what I've been able to do is $\mathop{\mathbb{E}}[X^Y] = \mathop{\mathbb{E}}[X^Y|Y=y]$ for all $y$. But I'm not sure how to proceed as there are both continuous and discrete distributions here.  |
| Let $x,y,z >0$ and $xy+yz+xz=2022xyz$ Posted: 07 Aug 2021 08:07 PM PDT Let $x,y,z>0$ and $xy+yz+xz=2022xyz$, Prove that: $$\frac{y^2}{x+2022y^2}+\frac{z^2}{y+2022z^2}+\frac{x^2}{z+2022x^2} \leq \frac{x+y+z}{4}$$ That problem from Viet Nam :)  |
| Calculate conditional distribution of sum of normal Posted: 07 Aug 2021 07:53 PM PDT I think this is very easy question, but I stucked. Can help me? Let $Y$ and $Z$ an independent standard normal and $X = Y+Z$. What I want to solve is
For the case of $p_{X|Y}$, I think it is a density of Gaussian whose mean is y and variance 1, because $y+Z$ is a translation of Gaussian. But I can't solve it in logical word.  |
| Why growth function little-omega $\omega(m)$ equals $\Theta(\log{m})$ Posted: 07 Aug 2021 07:51 PM PDT I was trying to prove that $\log(\log^*{n}) = \log^*(\log{n})$, where $\log^*$ is the iterative logarithm that calculates the number of times before we reach 0. Definition: We define $\omega(g(n))$ ("little-omega of $g$ of $n$") as the set $\omega(g(n))$ = {$f(n)$ : for any positive constant $c>0$, there exists a constant $n \gt n_0$ such that $0≤cg(n)<f(n)$ for all $n ≥ n_0$}. Definition: We define $\Theta(g(n))$ = { $f (n) $: there exist positive constants $c_1, c_2$, and $n_0$ such that $0 ≤c_1g(n) ≤ f(n) ≤ c_2g(n)$ for all $n ≥ n_0$}. Why growth function little-omega $\omega(m)$ equals $\Theta(\log{m})$ please, which is not clear based on respective definitions? $m$ is not defined as well, but I guess it's an integer. So, presumably, $m$ is an integer.  |
| Posted: 07 Aug 2021 08:14 PM PDT Why if $Ax=\lambda x$ and $A^*x=\mu x$ then $\lambda = \overline{\mu}$? I tried to say $(Ax,x)=(\lambda x,x)=\lambda(x,x)$ and $(Ax,x)=(x,A^*x)=(x,\mu x)=\overline{\mu}(x,x)$. This I think shows why is the case. But I don't understand why $(x,\mu x)=\overline{\mu}(x,x)$, since I think $(\lambda x,x) = \lambda(x,x)$. I might be completely wrong but I dont understand this. Especially because the () notation is just the inner product? So I like to think more in this notation $$(Ax,x)=(Ax)^Tx=(\lambda x)^Tx=\lambda x^Tx$$ But isn't it not T in this case but *??? I don't understand though with this notation where $(Ax,x)=(x,A^*x)$ comes from since $$(Ax,x)=(Ax)^Tx=x^TA^Tx=(x,A^Tx)=x^TA^*x=x^T\mu x=\mu x^Tx$$ So here $\mu=\lambda$? But I learned * is pretty much just T but with the conjugate taken on every entry. Im obviously missing something, can someone please explain this how the first connects to the second notation? Whenever I read the general inner product I translate it back to just matrix multiplication and stuff, so when I don't understand that one I'm just using the rules like a parrot without any understanding of what's happening. Also, sorry for the big post, I honestly don't know how to formulate my question so I'm hoping someone can see where things are going wrong for me and help. Thanks in advance.  |
| The combined usage of min and max() in a graph definition. Posted: 07 Aug 2021 07:46 PM PDT Mpq(z) is defined as $$M_{pq} (z) = \large \min_{x,y\in N(z)\backslash {p,q}} \max \left(\begin{array}{c}l(xz)+l(zp)-l(xp), \\ l(yz)+l(zp)-l(yp), \\ l(xz)+l(zq)-l(xq), \\ l(yz)+l(zq)-l(yq) \end{array} \right)$$ The first equation in max() is Where l(xp) is the Euclidean length of edge xp rounded to the nearest integer. The question is, why is min used in the above definition, and how does it affect max()? Note: This definition is located in the paper, Edge Elimination in TSP Instances, by Stefan Hougardy and Rasmus T. Schroeder (at https://arxiv.org/abs/1402.7301), in section 4 on page 5.  |
| Posted: 07 Aug 2021 07:43 PM PDT I was wondering one day, say there were $n$ arbitrary points. Label the points $\{(x_1, y_1), (x_2, y_2), (x_3, y_3),…, (x_n, y_n)\}$ All the points have a unique x-value. For simplicity, let's say that the x-values aren't too spaced out from each other, and that the differences between x-values are the same. Let the spacing between each x-value be $v_x$. I would like to generate a function that passes through all the points, with the line (or arc) between two points most resembling a straight line. This means that, between points, there shouldn't be any excessive wiggles that 'mess up' the graph in any way. It should be 'smooth'. As an addition, I want this function to wiggle up and down smoothly forever, not like a polynomial that zooms off to infinity, so some trigonomic waves should be used instead of polynomials. Unless, there's another type of function that wiggles up and down that I don't know about. I first thought of using Taylor and Maclaurin series, but realized that I didn't have any function to take the derivative of. Also, the result would be polynomials, which zoomed off to infinity. I then tried combining a few trigonomic functions in Desmos. That didn't work too well either, since these were approximations and didn't quite go through the points exactly. I also read a few other questions on this topic, but only found either:
Write the function as f(x). Use the variables $v_x$ and $n$. If sigma notation is used, write the x and y coordinates of the points as $x_i$.  |
| mapping the circle $|z|=3$ into $|z-1|=1$, the point $3+3i$ into $1$ and the point $3$ into $0$. Posted: 07 Aug 2021 07:47 PM PDT Question: Find the linear transformation which carries the circle $|z|=3$ into $|z-1|=1$, the point $3+3i$ into $1$ and the point $3$ into $0$. My Attempt: First, I've done problems like this before, but I ran into something at the beginning, so I want to first write my proof, then mention what I ran into and hope to have my solution verified and to see if I can get an explanation on why I had that problem and what it means. So here goes: Let $f(z)=w$ we such a transformation. We have that $f(3+3i)=1$ and $f(3)=0$. Now, we see that, using $z^*=\frac{R^2}{\bar z-\bar a}+a$, that $3+3i$ is symmetric to $3-3i$ with respect to $|z|=1$ and that $1$ is symmetric to $\infty$ with respect to $|z-1|=1$. Thus, we have $f(3-3i)=\infty$. So, using the cross ratio, we have $(w,1,0,\infty)=(z,3+3i,3,3-3i)$, and so we have $\frac{w-0}{w-\infty}\frac{1-\infty}{1-0}=\frac{z-3}{z-(3-3i)}\frac{3+3i-(3-3i)}{3+3i-3}$. Going through the calculation, we get that $w=\frac{(z-3)2i}{z-3+3i}$. So, first, does this solution look correct? Next, when I was trying to find that "last point", I was running into an issue using $f(3)=0$. I kept getting $3$ is symmetric to $3$ with respect to $|z|=3$ and $0$ is symmetric to $0$ with respect to $|z-1|=1$, so I was just getting a transformation value that I already knew. Why? Or, should I, of course, have used the point with the nonzero imaginary part? Any help is greatly apprecaited! Thank you.  |
| Definition of a reflection as an isometric, involutory, linear map Posted: 07 Aug 2021 07:35 PM PDT In $V = \mathbb{R}^n$, a reflection through a linear subspace $S \subset V$ of dimension $k \leq n$ can be defined to be the linear map $R : V \rightarrow V$ that satisfies \begin{equation} v \in S \implies Rv = v, \quad v \in S^\perp \implies Rv = -v. \qquad (1) \end{equation} (This is a generalization of the usual notion of a reflection, in which $k = n - 1$.) According to the second paragraph of the Wikipedia article on reflections, this definition seems to be equivalent to defining $R$ to be a linear map that is isometric ($\forall v, ||Rv|| = ||v||$) and involutory ($R^2 = 1$). However, I am having trouble coming up with a proof of this statement. The forward direction is easy, since any map that satisfies $(1)$ has a full set of eigenvalues that are $\pm 1$, so the involutory and isometry properties follow trivially. However, how does one prove the converse?  |
| Prove o disprove that $X$ is connected? Posted: 07 Aug 2021 07:32 PM PDT Given $ r> 0 $, let $ C_r $ be the circumference in the plane that has center at $ (0,0) $ and radius $ r $. Let $ X $ be a subset of $ \Bbb R ^ 2 $ that has the following properties,
Is $X$ connected? I affirm that it is connected. And to demonstrate, I do it by contradiction and I assume that $ X $ is not connected, so there are two separate sets $ A $ and $ B $, such that $ X = A \cup B $. I have a minimal idea, and that is that for example, $ C_1 $ being a connected set, being contained in $ X $, it must be completely contained in $ A $ or completely contained in $ B $, since it is $ 1 \in \Bbb Q $. Without loss of generality let's say that it is completely contained in $ A $. So somehow show that all other $ C_r $ is contained in $ A $. And then use some density argument to show that the points that are in $ X \cap C_r $ with $ r \in \Bbb R \setminus \Bbb Q $, cannot be in $ B $. So $ B $ has to be empty and I would have my absurdity. Any help for the exercise or how I can develop my idea. Thanks a lot.  |
| Can two hikers on different trails always reach the peak of a mountain while matching altitude? Posted: 07 Aug 2021 07:32 PM PDT Formally, suppose we have two continuous functions $f,g:[0,1]\to [0,1]$ with $f(0)=g(0)=0$ and $f(1)=g(1)=1$. Is it always possible to find continuous parameter functions $\phi_1,\phi_2$ both starting at $0$ and ending at $1$ such that $f(\phi_1(t))=g(\phi_2(t))$ for all times $t$? Equivalently, if we let $h:[0,1]^2\to [0,1]$ be defined by $h(x,y)=f(x)-g(y)$, is it always possible to find an implicit curve $\phi:[0,1]\to[0,1]^2$ connecting $(0,0)$ to $(1,1)$ satisfying $h(\phi(t))=0$ for all $t$? One proof method I attempted was based on this conjecture I had, which ended up being false. If the conjecture were true, taking the connected component of $(1,0)$ for the open set $\{(x,y) : h(x,y)\neq 0\}$ and path connecting its boundary would solve the problem, but such path connections are not guaranteed to exist, as demonstrated in my counter example. In the simpler case that $f,g$ have finitely many critical points (local extrema), the answer is in the affirmative. In this simpler case, the critical points are isolated, so both paths start out as strictly increasing and both hikers can move forward matching pace to stay at the same altitude. Whenever one hiker reaches a critical point, that hiker continues their direction while the other hiker reverses direction and starts backtracking. So long as they do not reach a critical point simultaneously, this method can be formalized to construct $\phi$ in such a way that it never backtracks on itself (even if the individual hikers do), and since there are only finitely many critical points, eventually $\phi$ must reach the corner $(1,1)$. This can be extended to the case where there are simultaneous critical points by employing a maze solving algorithm where $\phi$ always takes right-hand turns when possible, but the formalization is tedious. This method seems to be impossible in general case however, which could possibly be very pathological and have infinitely many critical points.  |
| How do I find eigenspace bases in a generalized way? Posted: 07 Aug 2021 07:30 PM PDT I have a 4x4 real matrix $A$ with three unknown eigenvalues $𝜆_{1}$, $𝜆_{2}$, $𝜆_{3}$ and the eigenspace $E_{𝜆_{3}}$ has $dim(E_{𝜆_{3}})=2$ and I have to prove that $A$ is diagonalizable but I have no idea how to do that. I was thinking that if I could show that $𝜆_{1}$ and $𝜆_{2}$ both have 1 dimensional eigenspaces but I don't know how to do that generally. Can I just assume it? I'm not sure where to go from here.  |
| Does the summation property $\sum a_{k} x^{i+k}=x^{i} \sum a_{k} x^{k}$ have a name? Posted: 07 Aug 2021 07:28 PM PDT Does this particular summation property have a name? I would like to see a proof for it but I am having difficulty finding one and I am not sure what to search for. I came across it in a document describing a listing of "summation facts".  |
| Understanding the usage of a theorem about cohomology of a compact Lie group. Posted: 07 Aug 2021 07:52 PM PDT We know that for $0\leq k\leq n$, the $k$-th de Rham cohomology group of the $n$-torus $\mathbb{T}^n = S^1 \times \dots \times S^1$ is $$H^k_{dR}(\mathbb{T}^n) \cong \mathbb{R}^{n\choose k}.$$ I just saw a theorem:
Now I am trying to understand the following example (on page 138 of Morita's "Geometry of Differential Forms"):
This is confusing to me: when the Lie algebra is $\mathbb{R}^n$, the subcomplex of left-invariant differential forms on $\mathbb{T}^n$ should be the complex of differential forms on $\mathbb{R}^n$, because a left-invariant differential form is determined by its action on the Lie algebra. Then when we compute the cohomology of this complex, we should get the de Rham cohomology of $\mathbb{R}^n$, which is very different from that of $\mathbb{T}^n$. Could anybody point out my mistakes in the above reasoning? Thank you! Update: I just realized that to be left-invariant, the differential $k$-form must be an $\mathbb{R}$-linear combination of $dx_{i_1}\wedge\dots\wedge dx_{i_k}$. So, forms like $f(x) dx_{i_1}\wedge\dots\wedge dx_{i_k}$ is not allowed if $f$ is non-constant. Then, everything is in the kernel while only $0$ is in the image. Hence the result follows.  |
| Is there a symbol for embedded manifold? Posted: 07 Aug 2021 08:19 PM PDT If we say the symbol (or degrees of freedom) for an 0-dimensional manifold (a point) embedded in $n$ dimensional space is $\mathbb{R}^n$. Since it takes $n$ coordinates to describe such a point. How would one write the degrees of freddom of an n-dimensional [surface topologically equivalent to a] sphere or n-dimensional ball embedded in m-dimensional space?  |
| If $U_1,U_2,\dots$ are open subsets of $[0,1]$, then either prove or disprove the statements Posted: 07 Aug 2021 07:59 PM PDT Suppose $U_1,U_2,\dots$ are open subsets of $[0,1]$. In each case, either prove the statement or disprove it. (a) If $m(\cap_{n=1}^\infty U_n)=0$, then for some $n ≥ 1$, we have $m(\overline{U_n})<1$, where $m$ is Lebesgue measure and $\overline{U_n}$ is the closure of $U_n$ in the usual topology on $[0, 1]$. (b) If $\cap_{n=1}^\infty U_n = \emptyset$, then for some $n\geq 1$, the set $[0,1] \setminus U_n$ contains a nonempty open interval. Thoughts. I think (a) is FALSE. i.e we can produce some open subsets $U_n$ such that $m(\cap_{n=1}^\infty U_n)=0$ and $m(\overline{U_n})=1$ For (b) I think it is TRUE. Suppose not. i.e. For each $n\geq 1$ if the closed set $[0,1]\setminus U_n$ does not contain a nonempty open interval then $m([0,1]\setminus U_n)=0$, so $m(U_n)=m([0,1])=1$ for all $n\geq 1$. So $\cap_{n=1}^\infty U_n\neq \emptyset$, contradiction. I feel that I skip some details. Thanks for any comments/ideas/answers.  |
| Uniform convergence of a sequence of operators Posted: 07 Aug 2021 07:40 PM PDT We are given a sequence of operators $A_nx(t)=x\left(t^{1+\frac{1}{n}}\right):C[0,1]\to C[0,1]$. It is easy to show that it converges pointwise to the unit operator by the Banach-Steinhaus theorem. Judging by the answer, there is no uniform convergence. But I can't show it. It is necessary to find an example of such a sequence $x_n(t)$, for which $\frac{||A_nx_n-Ix_n||}{||x_n||}$ does not tend to zero. I just can't find such a sequence. This task stuck in my head, there is either something very elementary, or vice versa. I've tried like this: Let $0<a<b<1$. Consider a continuous function equal to $0$ on $[0,a]$, linear on $[a,b]$, and equal to $1$ on $[b,1]$. Its norm is $1$. For a fixed $n$, put $a=(1/2)^{1+1/n}$ and $b=1/2$. We will take this as $x_n(t)$.  |
| Posted: 07 Aug 2021 08:15 PM PDT After conducting a series of experiments, a physicist concluded that the pressure around an object placed in a moving fluid is given by $P(M) = P_0 \left( 1 + \frac{k - 1}{2}M \right)^{k/(k-1)},$ where $M$ is the square of the ratio of the speed of the fluid to the speed of sound, $P_0$ is a positive constant, and $k$ is a positive integer greater than 1. Use linear approximation to prove that the pressure is approximately $P_0 \left(1 + \frac{k}{2} M \right)$ for small values of $M$.  |
| Validation of Proof that Nested Interval Property implies Axiom of Completeness Posted: 07 Aug 2021 08:06 PM PDT I know this is is about the millionth "proof that N.I.P implies A.o.C" questions, but I am wondering where my proof fails (I'm asking where because I'm quite confident it is wrong somewhere). Proof: Let there be some arbitrary set $A$. Let $a_n$ symbolize some element from this set $A$. Let $b$ be any upper bound on the set $A$. Pick some arbitrary element from the set $A$, and label this element $a_1$. Pick an arbitrary upper bound and label this $b_1$. These two create an interval $[a_1, b_1]$. The distance between these points is $|a_1-b_1|$. Now, pick some new $a_2$ and $b_2$ that creates an interval $[a_2, b_2]$ such that the $|a_2-b_2| = \frac{|a_1-b_1|}{2}$ (half the size of the old interval) nested inside the interval $[a_1, b_1]$. Keep on making smaller and smaller nested intervals $[a_n, b_n]$ where $a_n$ and $b_n$ get larger, and the distance between them approaches 0 (since the sequence $\frac{1}{2^n}$ has a limit of 0). Because of nested interval property, there exists some point $x$ such that $a_n \le x \le b_n$ for all n. The definition of $s = \text{sup} A$ is that $s \le b_n$ for all b, and $s \ge a_n$ for all a. This $x$ clearly satisfies both of these properties, and thus is the least upper bound of A. Since least upper bounds are unique, there is no different least upper bound. --- end proof It feels... not rigorous enough, but I can't exactly put my finger on where or why. I think I'm close-ish, though. Thanks!  |
| Why is $n(F,D)$ odd, when $F$ is odd? Posted: 07 Aug 2021 08:08 PM PDT This is a question from Do Carmo's Differential Forms and Applications (question 8, chapter 2). Actually, this question was made and answered here. The problem is: The answer redirects the OP to here ($f:\mathbb{S}^1\rightarrow\mathbb{S}^1$ odd $\Rightarrow$ $\mathrm{deg}(f)$ odd (Borsuk-Ulam theorem)), but in this Differential Forms course we didn't had contact with algebraic topology whatsoever, mainly on those firsts chapters. So, there is some way, probably not so hard to prove this statement. FYI, the problem is:
Is good to state that the book defines: $\displaystyle{ n(F;D) =\frac{1}{2 \pi} \int_{F \ \circ \ c} \theta _0}$ where $\displaystyle{ \theta_0 = \frac{fdg - g df}{f^2 + g^2}}$. I know that $n(F,D)$ is the number of counterclock-wise travels around the point, in this case, the origin. I just can't use any argument. If someone could help with this, it will be very appreciated. Thanks in advance.  |
| A problem arises when computing the integral of the sinc function using the Fourier transform. Posted: 07 Aug 2021 07:57 PM PDT It is well-known that the integral of sinc equals $\pi$: $$ \int_{-\infty}^\infty \frac{\sin x}{x}\ dx = \lim_{A \to \infty} \int_{-A}^A \frac{\sin x}{x}\ dx = \pi. $$ Here is a way to get this, but I think this argument is a nonsense: The above integral equals the value at $0$ of the Fourier transform of sinc function: $$ \sqrt{2\pi}\cdot \mathcal F_x \left[ \frac{\sin x}{x} \right](0), $$ where the normalization convention is $$ \mathcal F[f](t) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(x) e^{-itx}\ dx. $$ But we know that the sinc function is an even function and $$ \mathcal F \left[ \mathbf 1_{[-1,\ 1]} \right](t) = \sqrt{\frac{2}{\pi}} \cdot \frac{\sin t}{t}, $$ having the inverse transform $$ \mathcal F_x \left[ \frac{\sin x}{x} \right] = \sqrt{\frac{\pi}{2}} \cdot \mathbf 1_{[-1,\ 1]}. \ \ \ \ \ \ \ \ ......[*] $$ Hence we have $$ \int_{-\infty}^\infty \frac{\sin x}{x}\ dx = \sqrt{2\pi} \cdot \mathcal F_x \left[ \frac{\sin x}{x} \right] (0) = \pi.\ \ \ \ \ \ \ \ ......[**] $$ So what's wrong? The above argument uses the inverse transform. But the Fourier inversion theorem applies only if both $f$ and $\hat f$ are of $L^1$, which is not the case we are dealing. So we have to rely on the Plancherel theorem, which says that the Fourier transform gives a Hilbert space isomorphism of $L^2(\mathbb R)$. Then we do have the equation $[*]$, but the equality in $[*]$ is not a usual one; it only means the two funcitons are equal almost everywhere. So the specific value $\mathcal F [\text{sinc}](0)$ is not uniquely-deterined. Question: Yet many people uses the above argument to compute the sinc integral. Can this argument be justified by another way?  |
| Quotients of equivalence relations Posted: 07 Aug 2021 07:42 PM PDT Let $R,S$ and $T$ relations of equivalence in $A$ and suppose that $R\subset S\subset T$ prove that: If $R \circ T$ is a relation of equivalence in $A$, then $(S/R)\circ (T/R) =(S\circ T)/R$ This is my work that I have done. $$\begin{array}{crl} ([x]_R,[y]_R)\in (S/R)\circ (T/R) &\Rightarrow &\exists_z (x,z)\in(T/R) \wedge(z,y)\in (S/R) \\ &\Rightarrow &\ (x,z) \in T \wedge (z,y) \in S \\ &\Rightarrow&\ (x,y) \in S \circ T\\ &\Rightarrow & ([x]_R,[y]_R) \in (S \circ T)/R \end{array}$$ I appreciate your contributions with the definitions used.  |
| Explicit description of $\mathbb{Z}\otimes_{\mathbb{N}}\mathbb{Z}$ Posted: 07 Aug 2021 07:42 PM PDT $\newcommand{\Q}{\mathbb{Q}}\newcommand{\N}{\mathbb{N}}\newcommand{\Z}{\mathbb{Z}}$Recall the definition of the tensor product $\otimes_{\mathbb{N}}$ of commutative monoids (see also this note by Harold Simmons). Question. Is there an explicit/"nice" description of the tensor product $\Z\otimes_{\N}\Z$, where $\Z=(\Z,\cdot,1)$ is the multiplicative monoid of integers? A partial description. Eric Wofsey mentioned a really nice way to partly describe $\Z\otimes_\N\Z$ here: since $\Z\cong\N^{\oplus\N}\oplus\Z_2$ via the map sending an integer $k=2^{n_1}3^{n_2}5^{n_3}\cdots$ to $((n_1,n_2,n_3,\ldots),\mathrm{sgn}(k))$, we may use distributivity of tensor products along direct sums to write \begin{align*} (\Z\setminus\{0\})\otimes_{\N}(\Z\setminus\{0\}) &\cong (\N^{\oplus\N}\oplus\Z_2)\otimes_\N(\N^{\oplus\N}\oplus\Z_2)\\ &\cong (\N^{\oplus\N}\otimes_\N\N^{\oplus\N})\oplus(\N^{\oplus\N}\otimes_\N\Z_2)\oplus(\Z_2\otimes_\N\N^{\oplus\N})\oplus(\Z_2\otimes_\N\Z_2)\\ &\cong \N^{\oplus\N}\oplus\Z_2^{\oplus\N}. \end{align*} One could then add back $0$ to each factor of $\Z\setminus\{0\}$ above, adding new elements and relations to $\N^{\oplus\N}\oplus\Z_2^{\oplus\N}$. What is the result of this?  |
| Posted: 07 Aug 2021 08:08 PM PDT It is well known that $\mathfrak{su}(2)$, the real Lie algebra of traceless skew-Hermitian $2\times 2$ complex matrices, satisfies $\mathfrak{su}(2)_{\mathbb C}\simeq \mathfrak{sl}(2,\mathbb C)$. To see this, it is sufficient to observe that any traceless matrix $A$ can be written as $$A = i\left(\frac{A+A^\dagger}{2i}\right) + \left(\frac{A-A^\dagger}{2}\right),$$ where both components are traceless and skew-Hermitian, and the decomposition is unique. We also know that $\mathfrak{so}(3)\simeq\mathfrak{su}(2)$, where $\mathfrak{so}(3)$ is the real Lie algebra of traceless skew-orthogonal $3\times 3$ real matrices. This follows from observing that both Lie algebras satisfy the same commutation relations, $[T_i,T_j]=\epsilon_{ijk}T_k$ (or rather, we can always find bases for both spaces satisfying such relations). This should imply that also $\mathfrak{so}(3)_{\mathbb C}\simeq\mathfrak{sl}(2,\mathbb C)$ (as also mentioned in passing in this answer), but how can I show that this is the case more directly? As far as I understand, this statement should mean that, given any traceless $2\times 2$ complex matrix $A$, there is a bijection sending $A$ to two $3\times 3$ real skew-orthogonal matrices. What is this decomposition?  |
| Proving a version of a local limit theorem Posted: 07 Aug 2021 08:05 PM PDT Let $\{X_n \}$ be a sequence of integer valued i.i.d random variables that are symmetric around $0$, and $\mathbb{E}|X_1|^3<\infty, P[X_1 = 1]>0, P[X_1=0]>0$. Let $S_n = X_1+\dots+X_n$. Show that $$\lim_{n\rightarrow\infty}\sqrt{2\pi\sigma^2n}P[S_n=0]=1.$$ I know that I can write $$P[S_n = 0] =\frac{1}{2\pi} \int_{-\pi}^{\pi}\phi^n(t)dt=\frac{1}{2\pi\sqrt{n}}\int_{-\pi\sqrt{n}}^{\pi\sqrt{n}}\phi(t/\sqrt{n})^ndt.$$ Where $\phi$ is a characteristic function of $X_1$. But apart from that I am stuck.  |
| Finding the Upper and Lower Bounds for Rational Zeros of a Polynomial Posted: 07 Aug 2021 08:04 PM PDT UPDATE 9/29/18: The solution to this problem is that the statement is an "if-then" situation. Unfortunately I was interpreting the theorem in the converse: I thought that any upper bound will satisfy the criteria. However, it only says that IF the selected point satisfies the test, then we can surely say that it will be an upper bound for the zeros. It doesn't mean that any upper bound will satisfy the test. While trying to understand the upper and lower bounds of real zeros of a polynomial, I have come across something that seems to go against the logic in the textbook.
I'm trying to find the real zeros of $f(x) = 4x^4 - 20x^3 + 37x^2 - 24x + 5$. According to WolframAlpha, there is only one real zero at $x = {1\over2}$ (with multiplicity $2$). This would mean that anything after that would not be a zero according to the Rational Zero Theorem. For example, if I use synthetic division on one of the possible rational zeros, ${5\over4}$, then clearly ${1\over2} < {5\over 4}$ and $$\begin{array}{cccccc}\boxed{5\over4} & 4 & -20 & 37 & -24 & 5\\ & & 5 & -{75\over4} & {365\over16}& -{95\over64}\\\hline & 4 & -15 & {73\over4} & -{19\over16} & {225\over64}\end{array}$$ The signs alternate instead of being all positive or zero. Did I make a mistake somewhere in the synthetic division? Even if, immediately at the start, I know we end up with $-15$ which kills the concept of the upper bound right then and there. Or is there a slight subtlety I'm missing in the Upper and Lower Bound rules? Even if I apply synthetic division to the only zero: $$\begin{array}{cccccc}\boxed{1\over2} & 4 & -20 & 37 & -24 & 5\\ & & 2 & -9 & 14& -5\\\hline & 4 & -18 & 28 & -10 & \boxed{0}\end{array}$$ The numbers in the bottom row are not all positive or zero, which isn't telling me that ${1\over2}$ is an upper bound like I'd expect it to.  |
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