Recent Questions - Mathematics Stack Exchange |
- Property of Sets
- isometries of a compact semi-simple lie group other than left translations
- Slope Problem - slope is known
- Intuition behind the max-min inequality
- Finding CDF of Y when Y=I/X
- If a normal subgroup $N$ of $A_n$ contains any $3$-cycle, then $N = A_n$
- From a deck of 52 cards, a 5-card hand is dealt. Find the number of hands containing exactly one pair
- Proving that $\sqrt{13+\sqrt{52}} - \sqrt{13}$ is irrational.
- BIg-O notation true/false
- How can I show that the product from k=1 to n of (k+1/3) = gamma(n+4/3)/gamma(4/3)?
- prove if limit ${{x_n}/{y_n}}=1$,then limit ${{{x_n}^{1/n}}/{{y_n}^{1/n}}}=1$,
- Integral over the unit interval of the inverse regularized incomplete beta function. Simple integral needs a non-integral form. Closed form optional.
- Local degree of $z\mapsto z^n$
- Confusion about the definition of a manifold
- What is the factorization on $\mathbb{F}_4$ of an irreducible polynomial of $\mathbb{F}_2 [X]$ of degree $4$?
- Equivalences involving Quantifiers problem
- Prove by definition that $\displaystyle \lim_{z \to i} \dfrac{4z+i}{z+1} = \dfrac{5i}{i+1}$
- Other ways to find $x$ in $\frac{x}{x+7}=\frac49$
- How would I solve the last two portions of this parametric and vector question?
- Riemann stieltjes integral : Show that $\int_{a}^{b} fdg=0$ if only if $f(x) = 0$ for all $x \in [a,b]$.
- Random walk on $n$-dimensional cube
- Show $B=\bigcup \mathcal{A}$ is well-ordered and $A\leq B, \forall A\in \mathcal{A}$
- Taylor theorem with general remainder formula
- What are the conditions that a quartic equation roots are all complex [closed]
- Explicit formula for square root of a $3\times3$ positive definite matrix
- What can be the maximum and minimum number of leaves in free tree and binary tree? Along with the explanation?
- Discrepancy of answers between differing computations of $E[e^{W_s}e^{W_t}]$ ($W_t$ being the Wiener process)
- Find a line that crosses multiple line segments
- Dimension of affine variety mod $p$ can only increase
- How to calculate rigorously the degree of the map $f: S^1 \to S^1, f(z) = z^n$.
| Posted: 16 May 2021 07:50 PM PDT Let $(K_n)_{n=1}^\infty$ be a sequence of convex, compact sets in the space $\mathbb{R}^k$ such that $$K_1\subsetneqq K_2\cdots \subsetneqq K_n\cdots.$$ Put $K=\cup_{n=1}^\infty K_n$. Assume that $K$ is bounded. Can we deduce that $K$ is always not closed?  |
| isometries of a compact semi-simple lie group other than left translations Posted: 16 May 2021 07:41 PM PDT My understanding is that a left-invariant metric on a compact semi-simple lie group is defined in such a way that left translations are isometries. Bi-invariant metrics do not concern me for the moment. Are there isometries that are not left translations? Infinitesimally, are there killing vector fields that are not left-invariant? In particular, how to find all isometries on the 3-sphere (i.e. SU(2)), for example? I can write three killing vector fields in terms of coordinate frame field of the ambient 4-dimensional euclidean space; but how to write them intrinsically on the sphere? Thank you!  |
| Slope Problem - slope is known Posted: 16 May 2021 07:40 PM PDT Assume I have the following points:
The slope is 252.7216, right? I'm struggling to find out what the x-axis value is if y = 2000.  |
| Intuition behind the max-min inequality Posted: 16 May 2021 07:34 PM PDT Here is the statement of the max-min inequality:
I understand the proof, but I'm struggling to visualize this inequality. Here is my attempt at intuition. Let $f: \mathbb{R} \times \mathbb{R} \to \mathbb{R}$. The first slot of $f$ is the $z$-coordinate and the second is the $w$-coordinate. Then to find the value of $\sup_{z \in Z} \inf_{w \in W} f(z, w)$, we first walk along the $z$-axis. At each point along the $z$-axis, we stop and look out over the line that is parallel to the $w$-axis and intersects the point where we are currently standing. We record the elevation of the lowest valley created by $f$ along this line. We continue this process until we've walked along the entirety of the $z$-axis. At the end, we scour our notebook and return the largest number we recorded. In other words, $\sup_{z \in Z} \inf_{w \in W} f(z, w)$ is the highest, lowest valley found through this process. $\inf_{w \in W} \sup_{z \in Z} f(z, w)$ can be found through an analogous process. This time we walk along the $w$-axis and look out over the lines parallel to the $z$-axis. We note the highest peak along each line parallel to the $z$-axis. At the end, we return the lowest of the numbers we recorded, i.e., the lowest, highest peak. I am struggling to intuitively see why the highest, lowest valley will always be lower than the lowest, highest peak. Is there a better interpretation of this inequality? Thanks!  |
| Posted: 16 May 2021 07:40 PM PDT X be a uniform random variable on [1,3]. Y=I/X. What is the CDF of Y? My attempt so far: CDF of Y by definition: = P(Y<=y) = P (1/X <= y) = P (X>= 1/y) I don't know how to proceed.  |
| If a normal subgroup $N$ of $A_n$ contains any $3$-cycle, then $N = A_n$ Posted: 16 May 2021 07:29 PM PDT
What I have done If $n\geq 5$ the result follows by these lemmas: Let $n\geq3$, then every element of $A_n$ is a product of $3$-cycles and If $n\geq 5 $, then any $3$-cycle are conjugate in $A_n$. Now for $n=3$, we have that $A_3$ is a cyclic group generated by $(1 2 3)$ or $(1 3 2)$. Thus, if $(1 2 3)$ or $(1 3 2)\in N$, then $N=A_3$ My problem is when $n=4$. I know that $M=\{(1),(12)(34),(13)(24),(14)(23)\}\cong \mathbb{Z}_2\times \mathbb{Z}_2$ is a normal subgroup of $A_4$, but $M$ does't contain a $3$-cycle. How can I go on to show that if $N$ contains a $3$-cycle then $N=A_4$?  |
| Posted: 16 May 2021 07:48 PM PDT From a deck of 52 cards, a 5-card hand is dealt. Find the number of hands containing exactly one pair without considering the variation of the suits? my attempt to solve it : split the 52 into 13 subset of 4 identical items now I will choose one pair then I will choose one from each subset and then removing duplicates? $$ 0.5 \times \ ^{13}C_1 \times (\frac{^{12}C_1 \times ^{11}C_1 \times ^{11}C_1}{3!})$$  |
| Proving that $\sqrt{13+\sqrt{52}} - \sqrt{13}$ is irrational. Posted: 16 May 2021 07:48 PM PDT I'm trying to prove that a certain number is irrational. I've taken a number theory class, so I'm familiar with the proofs that $\sqrt{2}$ and $\sqrt{3}$ are irrational (assume it is rational, square both sides, and then arrive at a contradiction). However, I've tried this method with this number to no avail. This is the number in question: $\frac{p}{q} = \sqrt{13+\sqrt{52}} - \sqrt{13}$ $\frac{{p}^{2}}{{q}^{2}} = (13+\sqrt{52}) - 2(\sqrt{13+\sqrt{52}})(\sqrt{13}) + 13$ $\frac{{p}^{2}}{{q}^{2}} = 26 + 2\sqrt{13} - 2\sqrt{169 + 13\sqrt{52}}$ Once I get here, it just seems that I've made the question more complicated, or that I'm taking the wrong approach, and have no clue how to move on. Any help would be appreciated. Many thanks.  |
| Posted: 16 May 2021 07:20 PM PDT I am working on question about big-o notation, I think I understand the idea of big-O but the following notation make me confuse $$ n! = 2^{O(nlogn)}$$ $$ n^{\frac{loglogn}{logn}} = O(1)$$ $$ n^{log^4 n} = 2^{log^{O(1)}n} $$ My though referencing Show that $3^n = 2^{O(n)}$ for the first one id take log base 2 on both side $$ lg(n!) = O(nlogn)lg(2)$$ I know that lg(n!) is O(nlogn) $$ lg(n!) = O(nlogn)$$ Does it mean the answer is true? but How can I deal with the others  |
| How can I show that the product from k=1 to n of (k+1/3) = gamma(n+4/3)/gamma(4/3)? Posted: 16 May 2021 07:34 PM PDT This is not an homework, it is a simple curiosity.  |
| prove if limit ${{x_n}/{y_n}}=1$,then limit ${{{x_n}^{1/n}}/{{y_n}^{1/n}}}=1$, Posted: 16 May 2021 07:19 PM PDT I want to prove if limit ${x_n}$=limit ${y_n}$=${\infty}$,and limit ${{x_n}/{y_n}}=1$,then limit ${{{x_n}^{1/n}}/{{y_n}^{1/n}}}=1$, after taking log function, it is enough to shou limit ${{ln(x_n)}/{ln(y_n)}}=1$. how is it connected to limit ${ln((x_n)/(y_n))}=0$?  |
| Posted: 16 May 2021 07:34 PM PDT In the spirit of
With this said here is the integration defined constant. Please find any non-integral form of this. A closed form would be best, but it would probably be too hard to find: $$\mathrm{\mathcal I=\int_0^1I^{-1}_x(x,x)dx=\int_I I^{-1}xdx=.45697...}$$ The area is the area over the function's domain for $x\in\Bbb R$ Here is the graph of this function which has the new name for simplicity of $$\mathrm{I^{-1}_x(x,x)= I^{-1}x= I^{-1}(x)}:$$
What I have tried: This may not or may help. Here is the inverse of this inverse function which in itself is not $$\mathrm{I_x(x,x)}$$: Here is the description of the function and its properties: $$\mathrm{x=I_{I^{-1}_x(y,z)}(y,z)= I_{I^{-1}_x(x,x)}(x,x)= I_{I^{-1}x}(x,x)}$$ Possible closed forms with short decimal form here. Even though the first time I tried this I got more digits, Wolfram Aplha decided to give me around 5 digits when I asked it to evaluate the integral. Related problem The integral is not worth writing because if we assume the variables are not equal, we get a product of a hypergeometric function and a beta function as the result. Assuming the aforementioned condition is true gives an integral which cannot be evaluated by mathematica, but the same is true for the indefinite sophomore dream integrals which uses a sum of gamma functions. The non-integral forms of this constant is wanted because the goal of this question is to find an alternate form and not to rewrite the integral. The closed form is wanted, but not needed. The sum form of this function is only an approximation, but you are welcome to find the series representation tediously if you wish. Simply click the "functions" link above. The best answer will get the "check mark". Any insights and help is wanted too in the comments. The answers go in the answers and not in the comments. Please give me feedback and correct me please as always!  |
| Local degree of $z\mapsto z^n$ Posted: 16 May 2021 07:06 PM PDT I'm currently computing the local degree of $f:S^1\to S^1$ by $z\mapsto z^n$ with $n>0$. To do this, I tried to compute the local degree $\deg f|x_i$ where $x_i\in f^{-1}(1)$. Since $f$ is a local homeomorphism, $\deg f|x_i =\pm 1$ for each $i$. But from this, how can I show $\deg f|x_i = 1$?  |
| Confusion about the definition of a manifold Posted: 16 May 2021 07:06 PM PDT In many books I have seen the definition of a topological manifold as follows A topological manifold is a topological space that is Hausdorff, has a countable basis and is locally Euclidean. But in other definitions the countable basis part is omitted, so which of the two is true or "correct"?  |
| Posted: 16 May 2021 07:24 PM PDT Show that all irreducible polynomial of $\mathbb{F}_2 [X]$ of degree $4$ factored on $\mathbb{F}_4$ into a product of $2$ irreducible polynomials of degree $2$: I see that any polynomial $P$ of $\mathbb{F}_2 [X]$ of degree $4$, irreducible on $\mathbb{F}_2$, has a root in $\mathbb{F}_{2^4}$ which is an extension of degree $2$ of $\mathbb{F}_4$. Then $P$ is reducible on $\mathbb{F}_4$. It remains to show that $P$ does not admit a root in $\mathbb{F}_4$. This is my Problem.  |
| Equivalences involving Quantifiers problem Posted: 16 May 2021 07:30 PM PDT
Hello, I need some help here. I recently solved a similiar problem where Instead of interception I had Union. I assumed this would also be true, but apparently it is not. I do not see why.  |
| Prove by definition that $\displaystyle \lim_{z \to i} \dfrac{4z+i}{z+1} = \dfrac{5i}{i+1}$ Posted: 16 May 2021 07:49 PM PDT I'm having trouble to prove this limit using $\epsilon - \delta$. I know that, for every $\epsilon > 0$, exists $\delta > 0$, such that $0 < |z-i|<\delta \implies \left|\dfrac{4z+i}{z+1} - \dfrac{5i}{1+i}\right| < \epsilon$ we have $\left|\dfrac{4z+i}{z+1} - \dfrac{5i}{1+i}\right| = \left|\dfrac{4(z-i)-iz-1}{(z+1)(1+i)} \right|$ Now we can multiply by $i/i$ $\hspace{3.5cm} = \left|\dfrac{4i(z-i)+z-i}{i(z+1)(1+i)} \right|$ $\hspace{3.5cm} = \left|\dfrac{(z-i)(4i+1)}{i(z+1)(1+i)} \right| $ and I don't know what to do from here, I was trying to make appear the therm $(z-i)$ to find $\delta$ in terms of $\epsilon$, but I got stuck. Is this the right way to solve this problem? any tips will be helpful.  |
| Other ways to find $x$ in $\frac{x}{x+7}=\frac49$ Posted: 16 May 2021 07:30 PM PDT During solving a problem I got this equation and I want to solve for $x$: $$\frac{x}{x+7}=\frac49$$ It is equivalent to solving $9x=4(x+7)$ hence $x=\frac{28}5$. But is it possible to solve it other ways? I mean we have $x$ in both numerator and denominator of the fraction so how we can write it so that we have only one $x$ and solve it faster (I'm preparing for a timed exam) ?  |
| How would I solve the last two portions of this parametric and vector question? Posted: 16 May 2021 07:33 PM PDT A particle moves along the curve defined by the parametric equations $x\left(t\right)=2t$ and $y\left(t\right)=36-t^2$ for time $0\le t\le 6$. A laser light on the particle points in the direction of motion and shines on the x-axis. a) What is the velocity vector of the particle? $s=<2t,\:36-t^2>$, so $v=<2,\:-2t>$ My answer: $v=<2,\:-2t>$ b) Write an equation of the line tangent to the graph at $\left(2t,\:36-t^2\right)$ in terms of t and x. $\frac{dy}{dt}=-2t$ and $\frac{dx}{dt}=2$, so $\frac{dy}{dx}=\frac{-2t}{2}=-t$ My answer: $y-\left(36-t^2\right)=-t\left(x-2t\right)$ or $y=t^2-xt+36$ c) Express the x-coordinate of the point on the x-axis that the light hits as a function of t. Would I set $y=t^2-xt+36$ to 0 in order to get $x=\frac{t^2+36}{t}$? d) At what time t is the light moving along the x-axis with the slowest speed? Justify your answer. I'm not sure what to do here?  |
| Posted: 16 May 2021 07:47 PM PDT
I thought I could solve the problem using the Mean-Value Theorem for Riemann-Stieltjes integrals. We know that $ f,g$ are functions on $[a,b] $ with $ f$ continuous and $ g $ increasing. And let $m$ and $M$ be respectively the inf and sup of $f$ on $ [a, b]$. Then there exists $c \in [m, M] $ such that $$\int_{a}^{b} f(x)dg(x) = c(g(b)-g(a)).$$ So, suppose that $\int_{a}^{b} fdg=0$, therefore $c(g(b)-g(a))=0$. Then $c = 0$ or $(g(b)-g(a))=0$, but the second one can´t happen because $ g $ is an increasing function. Therefore, $c=0$. Also, we know that $f$ is continuous, therefore $c = f(x)$ for some $x \in [a,b]$. In particular, $c=0=f(x) $ for some $x \in [a,b]$. Is this a valid proof? Is there an other way to prove it?  |
| Random walk on $n$-dimensional cube Posted: 16 May 2021 07:37 PM PDT Consider a symmetric random walk along the edges of an $n$-dimensional unit cube. At each time step, a particle located at a particular vertex $(a_1, \ldots, a_n)$ moves to an adjacent neighbor each with equal probability. For an arbitrary starting point $(a_1, \ldots, a_n)$, how can I compute the probability that we will hit $(1, 1, \ldots, 1)$ before hitting $(0, 0, \ldots, 0)$? This will be a function of $n$. I thought about making a Markov chain with $2^{n}$ states, but I'm not entirely sure if this is the right approach. Under this representation, I'm pretty sure some state $u = (a_1, \ldots, a_n)$ can move to some other state if and only if we can retrieve the second state by turning a bit off in $u$ or turning a bit on (that is not already on) in $u$. Any help is appreciated. A solution is provided here https://arxiv.org/pdf/0711.2675.pdf, but it uses circuits and a less mathematical approach to derive the answer.  |
| Show $B=\bigcup \mathcal{A}$ is well-ordered and $A\leq B, \forall A\in \mathcal{A}$ Posted: 16 May 2021 07:41 PM PDT The relation $\leq$ here means, $A\leq B$ iff \begin{align*} &\bullet A\subseteq B\\ &\bullet \forall x\in A, \forall y\in B, y\leq x \Rightarrow y\in A \end{align*} Let $X$ be a partially ordered set and $\mathcal{A}$ a set of well-ordered subsets of $X$ where for all $A_1, A_2\in \mathcal{A}, A_1\leq A_2$ or $A_2\leq A_1$. Show that $B=\bigcup \mathcal{A}$ is well-ordered and $\forall A\in \mathcal{A},A\leq B$. My first doubt comes from the definition of the set $B$, which I suppose means $\bigcup_{A\in \mathcal{A}}A$. Using this $B$, is this argument enough: Each $A_i$ is well ordered, since $A_i\leq A_j, \forall i,j$ and $i\neq j$ then $A_i\cap A_j\neq \emptyset$ for $i\neq j$. Now every element of $A_i$ is comparable pair-wise, because every set $A_i$ has a non-empty intersection, there exists a well-order over $\bigcap_{A\in \mathcal{A}} A$. I haven't reached the result, what am I doing wrong? Do I have to use Zorn's Lemma or even the Axiom of Choice directly?  |
| Taylor theorem with general remainder formula Posted: 16 May 2021 07:17 PM PDT I came across Taylor theorem as following: Let the function $f(x)$ have $n+1$ derivatives in $(a- \delta , a+\delta )$ of the point $a$ and $p>0$.Then for every $x \in (a- \delta , a+\delta )$ there exist $c \in (a,x)$ so that: $$\begin{align}f(x) = &\sum_{k=0}^{n} \frac{(x-a)^k}{k!}f^{(k)}(a) \\&+ \left(\frac{x-a}{x-c}\right)^p\frac{(x-c)^{n+1}}{n!p}f^{(n+1)}(c) \end{align} $$ I am curious , what is the proof of this theorem? if you plug in for example $p=n+1$ you get the Lagrange remainder.But I didn't quite find anywhere a proof with $p$ where the remainder is expressed like that. Can someone share full proof of this theorem or link where this is proved ?  |
| What are the conditions that a quartic equation roots are all complex [closed] Posted: 16 May 2021 07:13 PM PDT Having a quartic equation in the form of: $$ ax^4+bx^3+cx^2+dx+e=0 $$ What would be the condition/s that makes all the roots complex?  |
| Explicit formula for square root of a $3\times3$ positive definite matrix Posted: 16 May 2021 07:38 PM PDT I have an algorithm for calibrating a vector magnetometer. The input is $N$ readings of the $x$, $y$, $z$ axes: $(x_1, x_2, \dots, x_N)$, $(y_1, y_2, \dots, y_n)$, and $(z_1, z_2, \dots, z_N)$. The algorithm fits an ellipsoid to the data by estimating a symmetric $3\times3$ matrix $A$. In order to calibrate the system, it needs to calculate $\sqrt{A}$. I am adapting the algorithm for a microcontroller with very little memory, so cannot load standard matrix manipulation libraries. Is there an explicit formula for calculating the square root of $3\times3$ positive definite matrix?  |
| Posted: 16 May 2021 07:11 PM PDT The minimum and the maximum number of leaves in a binary tree with n total nodes is 1 and n/2, is it correct? But how can we explain it?  |
| Posted: 16 May 2021 07:14 PM PDT I was looking at another thread, and the following two distinct solutions to $E[e^{W_s}e^{W_t}]$ (assume that $W_0 = 0$ and $t>s$) were given, with both giving identical answers (I have slightly re-written the notation in the second solution for the reader's sake): 1.
On the other hand, I took quite a different approach, getting a different answer (note that I write $\exp(x)$ instead of $e^x$):
Since $t > s$, we can write $t = a s$ for some $a > 1$. Hence, \begin{align*} E[\exp\left({W_s}\right)\exp\left({W_t}\right)] &= \exp\left( \frac{s + t + 2\sqrt{s(as)}}{2} \right) = \exp\left( \frac{s + t + 2s\sqrt{a}}{2} \right) \\ &> \exp\left( \frac{s + t + 2s \cdot \sqrt{1}}{2} \right) = \exp\left( \frac{t + 3s}{2} \right), \end{align*} and hence the discrepancy is clear (with the two answers being equal only if $s = t$). Naturally, my question is where my mistake is. Thanks!  |
| Find a line that crosses multiple line segments Posted: 16 May 2021 07:29 PM PDT I'm looking for a formula, or algorithm, that would allow me to figure out if there's a line that cross multiple line segments (those are always parallel to the y axis), and if there is, the equation for that line. For example, in the following diagram, I'm looking for any line that touches the 4 vertical bar, such as the red line shows. It can touch the bars anywhere.
I can figure out how to do it with 3 line segments, but anything over that I have no idea. I'm also not a mathematician so I probably got most terminology wrong. EDIT: a possible approach is recursive, by drawing the bow-tie of possible lines for the left-most two segments, and seeing if the third segment touches the bow-tie. If so, create a narrower bow-tie of acceptable lines for the first three, and so on.
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| Dimension of affine variety mod $p$ can only increase Posted: 16 May 2021 07:39 PM PDT Let $X$ be an affine variety defined by polynomials over $\mathbb{Z}$. Reducing the polynomials modulo $p$, we obtain a variety $X_p$ defined over the finite field $\mathbb{F}_p$.
Note: It seems likely to me that the dimension stays the same for all but finitely many primes $p$, and this answers says that it is the case for projective varieties. My question is different: I'm asking about affine varieties, I'm asking about all primes, and I'm only asking for an inequality. EDIT: To make the second part of the question explicit:
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| How to calculate rigorously the degree of the map $f: S^1 \to S^1, f(z) = z^n$. Posted: 16 May 2021 07:14 PM PDT This was done in Hatcher's algebraic topology example 2.32. However I do not understand it at all. An alternative approach I know of is to show that the degree for homology matches the degree for the fundamental group which requires Hurwitz map. For Hatcher's proof, I know that the map is locally a homeomorphism because by inverse function theorem. Then I need to determine the sign of it. How to do this mathematically instead of by words? Does the sign have anything to do with whether the map is orientation-preserving or not?  |
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