Recent Questions - Mathematics Stack Exchange |
- Probability that an element of one set is in a different set, both within a finite universal set
- Proving unboundedness of the natural numbers via the Axiom of Completeness
- Stochastic Processes: Markov Chains
- Question on U-Substitution (Calculus)
- Catalan Number- Permutations and combination
- Is it true that for a Group $G$ with Normal Group $N: G/N = GN/N$?
- Limit problem. L'Hospital's Rule not allowed.
- locally free resolution for computation of Ext sheaves in Hartshorne Proposition 6.5
- What kind of differentiation is used in the sequence d(xy) =xdy+ydx?
- How to factor $4d^2 -16w^2 -4cd+c^2$ ( from Hart William L., Intermediate Algebra, 1948)
- Solution verification $\left(A \cap B\right) \cup \left(A \cap B^c\right)$
- On $\ker\chi $ , $\: \chi :{\rm Gal}(E,F)\rightarrow S_n$
- Suppose $ \sum a_k x^k $ uniformly converges at $ [0,R) $, then it converges pointwise at $ R $.
- What is the probability that the sum of the squares of $3$ positive real numbers whose sum is less than $10$ is less than $16$?
- Limit construction process that avoids constructing a decimal like $x.99999999\ldots$
- Homotopy equivalence induces Deformation Retraction if... (Hatcher's AT)
- Compact cardinal cannot be successor?
- Optimization of 3 variable function without lagrange
- Sequence of Infinitely Differentiable Functions Has Uniformly Converging Subsequence
- Probabilities above $1$
- Parametrization of homogenized curve
- Exist $a,b$ such that $-1<a<0<b<1$ and satisfy $|P(a)| \geq 1$, $P(b) \geq 1$
- what is the formal notation about the CDF or PDF of a function with limited interval.
- Limit without hospital's rule $\lim_{x\rightarrow 0} (\frac{1}{x^2} - \frac{x}{\sin^3(x)})$
- Quasi-isometry of finitely generated group
- Proving ${\mathbb{P}}^n$ is Hausdorff
- Formulating list sorting as a pure math problem
- Non-measurable sets and sigma-algebra definition
- Finding a vector equation for a trajectory
- Finding generalized eigenvalues with linear constraints
| Probability that an element of one set is in a different set, both within a finite universal set Posted: 01 Dec 2021 10:27 AM PST Definitions: $$ x \in \{1,...,z\} \subseteq \Bbb N^n \supseteq\{1,...,y\}$$ Theorem statement: $$P(x \in \{1,...,y\}) = \begin{cases} \displaystyle \frac{(z + n)y}{nz}, & (z + n)y \le nz \\[2ex] 1, & (z + n)y > nz \end{cases}$$ First lemma: $$P(x \in \{1,...,y\}) = 1 \iff z \le y \iff x\in \{1,...,z\} \subset \{1,...,y\}$$ Second lemma: $$P(x \in \{1,...,y\}) \neq 1 \iff z > y \iff x\in \{1,...,z\} \supset \{1,...,y\}$$ $$x \notin \{1,...,y\} \iff x \in \{1,...,z\} - \{1,...,y\} \iff y <x \le z$$ $$\therefore z > y \iff P(x \notin \{1,...,y\}) = \frac{z-y}{z} \iff P(x \in \{1,...,y \}) = \frac yz$$ Third lemma: $$P(z \le y) = \frac yn$$ Thus: $$\begin{align} P(x \in \{1,...,y\}) &= \frac yn + \frac yz \lor 1 \\[2ex] &= \frac{yz + yn}{nz} \lor 1 \\[2ex] &= \frac{(z + n)y}{nz} \lor 1 \end{align}$$ The reason "$\lor \ 1$" is written is due to the fact that the fraction is sometimes equal to a number greater than one. A probability exists in the interval $[0,1]$, so if the fraction is greater than one, then the probability is equal to one, hence the piecewise expression. In this case, if the fraction is equal to one, it just means that the conditions for certainty are more than sufficient, but that obviously doesn't equate to an above-one probability, as that is nonsensical. Notes: The Definitions section could be removed by altering the theorem statement: $$P(x \le z \in (\{1,...,y\} \subseteq \Bbb N^n) \cap (\{1,...,z\} \subseteq \Bbb N^n)) = \begin{cases} \displaystyle \frac{(z + n)y}{nz}, & (z + n)y \le nz \\[2ex] 1, & (z + n)y > nz \end{cases}$$ This approach removes a section, but makes the theorem statement more cluttered. Also, I don't think the parentheses are necessary, because I'm pretty sure the intersection is to be computed last (see p. 26 of this article). Questions: $(1)$ Is my proof correct? $(2)$ There is currently no writing (except for a paragraph in the last section), because I found the expressions self-explanatory; should I change this? $(3)$ Is the altered theorem statement shown in the Notes section better than the one used inside the theorem? |
| Proving unboundedness of the natural numbers via the Axiom of Completeness Posted: 01 Dec 2021 10:21 AM PST In the book "Understanding Analysis, second edition" by Stephen Abbot, the unboundedness of the set of natural number $\mathbb{N}$ is proven as the following proof:
What I do not understand is the following: How can we assume AoC holds for natural numbers? Especially in this book, we start by defining $\mathbb{N}$, and then say that $\mathbb{Q}$ (the set of rational numbers) is an extension of $\mathbb{N}$. Then, it is shown that $\sup A$ may not exist for bounded $A \subset \mathbb{Q}$. Then, we finally define AoC to "close the holes" of $\mathbb{Q}$, which is an axiomatic way of defining $\mathbb{R}$. So, how can we now go back to $\mathbb{N}$, and assume AoC holds to prove $\mathbb{N}$ is unbounded. In summary, I believe (probably I am mistaken) the way we prove this is some sort of paradox since AoC comes after defining $\mathbb{N}$ as a property of real sets. What if $\mathbb{N}$ was indeed bounded, but AoC does not hold for $\mathbb{N}$, hence this proof is wrong? In my view, we should define an AoC argument for natural numbers (that any bounded of $\mathbb{N}$ admits a $\sup$, and indeed this is a member of the set, hence we have a $\max$), then use this in our proof. |
| Stochastic Processes: Markov Chains Posted: 01 Dec 2021 10:21 AM PST I would like to know how to solve question 3 enter image description here |
| Question on U-Substitution (Calculus) Posted: 01 Dec 2021 10:16 AM PST I am currently learning u-substitution of integrals. I am kind of confused of what should I choose for u value. Can someone tell me tricks of how to do it? Thanks. |
| Catalan Number- Permutations and combination Posted: 01 Dec 2021 10:28 AM PST A person is decorating his baby's room. He draws points $A_1, A_2, . . . , A_{2n}$ on a line. Then he paints $n$ circumferences such that:
Finally, he uses two colors, green and orange, to paint the points such that pairs of point on the same circumference get the same color. In how many ways can he decorates the room? How do we proceed, the answer is the Catalan number but how do we use induction to prove that? |
| Is it true that for a Group $G$ with Normal Group $N: G/N = GN/N$? Posted: 01 Dec 2021 10:30 AM PST Is it true that for a Group $G$ with Normal Group $N: G/N = GN/N$? I think the statement is correct. But why do we have to write: $[G,G]N/N$ here instead of just $[G,G]/N$? Thanks! |
| Limit problem. L'Hospital's Rule not allowed. Posted: 01 Dec 2021 10:30 AM PST I'm lost on the following problem. I don't know which method to use. $$\lim_{x \to \infty} \frac{\sin({\frac{2}{x}})+\frac{2}{x}}{\sin({\frac{1}{x}})}$$ |
| locally free resolution for computation of Ext sheaves in Hartshorne Proposition 6.5 Posted: 01 Dec 2021 10:06 AM PST I have a question about Proposition 6.5., Chap III (on page 234) from Hartshorne's Algebraic Geometry. Let $(X, \mathcal{O}_X$ be ringed space. Suppose there is an exact sequence $$ ... \to \mathcal{L}_1 \to \mathcal{L}_0 \to \mathcal{F} \to 0$$ in $\mathcal{Mod}(X)$, the category of $\mathcal{O}_X$ modules, where the $\mathcal{L}_i$ are locally free sheaves of finite rank (in this case we say $\mathcal{L}_{\bullet}$. is a locally free resolution of $\mathcal{F}$). Then for any $\mathcal{G} \in \mathcal{Mod}(X)$ we have $$ \mathcal{Ext}^i(\mathcal{F}, \mathcal{G}) \cong h^i(\mathcal{Hom}(\mathcal{L}_{\bullet}, \mathcal{G})). $$ here the $\mathcal{Ext}^i(\mathcal{F}, -)$ are the right derived functors of $\mathcal{Hom}(\mathcal{F}, -)$ and the right hand side the cohomology sheaves of complex $\mathcal{Hom}(\mathcal{L}_{\bullet}, \mathcal{G})$. The proof says that both sides are $\delta$-functors from $\mathcal{G}$ to $\mathcal{Mod}(X)$. For $i = 0$ they are equal, because then $\mathcal{Hom}(-,\mathcal{G})$ is contravariant and left exact. Both sides vanish for $i > 0$ when is $\mathcal{F}$ injective, because then $\mathcal{Hom}(-,\mathcal{G})$ is exact. So by (1.3A) they are equal. Theorem 1.3A. on page 206 states: Let $T = (T^i)_{i \ge 0}$ be a covariant $\delta$-functor from $\mathcal{A}$ to $\mathcal{B}$ ($\mathcal{A}$, $\mathcal{B}$ are abelian categories). If $T^i$ is effaceable for each $i > 0$, then $T$ is universal. The definitions of effaceable and universal $\delta$-functor can be looked up on the sage page. My question is: Why is it neccessary for the proof to assume that the $\mathcal{L}_i$ are locally free? I see no reason why the same proof shouldn't go through verbatim if we start with an arbitrary resolution of $\mathcal{F}$, ie a family $(\mathcal{A}_i)_{i \ge 0}$ in $\mathcal{Mod}(X)$ there is an exact sequence |
| What kind of differentiation is used in the sequence d(xy) =xdy+ydx? Posted: 01 Dec 2021 09:58 AM PST What kind of differentiation is used in the sequence d(xy) =xdy+ydx? How we differentiate functions this way are they using implicit differentiation,? |
| How to factor $4d^2 -16w^2 -4cd+c^2$ ( from Hart William L., Intermediate Algebra, 1948) Posted: 01 Dec 2021 10:29 AM PST I found this factoring problem in Hart, Intermediate Algebra, 1948, page $97$ , problem $n° 60$, ( at Archive.org). Link :https://archive.org/details/intermediatealge030183mbp/page/n110/mode/1up I tried to manipulate the formula using: $m^2n^2 = (mn)^2$ and also : $n^2 - m^2 = (m-n) ( m+n)$; I made use of addition's associativity and commutativity in order to exchange the order of the terms; but it led nowhere. Any hint will be appreciated. Thanks in advance
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| Solution verification $\left(A \cap B\right) \cup \left(A \cap B^c\right)$ Posted: 01 Dec 2021 10:26 AM PST Prove that: $\left(A \cap B\right) \cup \left(A \cap B^c\right)=A$ Attempt: $$\left[\left(A \cap B\right) \cup A\right] \cup \left[\left(A \cap B\right)\cup B^c\right] \iff \left[A \cup \left(A \cap B \right)\right] \cup A= A$$ I'm right? |
| On $\ker\chi $ , $\: \chi :{\rm Gal}(E,F)\rightarrow S_n$ Posted: 01 Dec 2021 09:58 AM PST Assume $E$ is a splitting field of a separable polynomial of degree $n$,so $E$ must have $n$ district roots $(a_1,...,a_n)$ of the polynomial, if $\phi\in{\rm Gal}(E,F)$ then $\phi (a_i)$ must be one of the $a_j$, now Consider the homomorphism $\: \chi :{\rm Gal}(E,F)\rightarrow S_n$ $\chi(\phi_i)=j$ Question1 : what's the $\ker\chi $? The kernel is consisting of all those $\phi_i$ that send $a_i$ to $a_1$ and so $i\rightarrow 1$? Now if I apply the $1st$ isomorphism theorem, I get that ${\rm Gal}(E,F)/\ker\chi $ is isomorphic to a subgroup of $S_n$ Question2: can I get that ${\rm Gal}(E,F) $ is isomorphic to a subgroup of $S_n$, with the $\chi$ homorphism? (This maybe simple but I am a bit confused right now) |
| Suppose $ \sum a_k x^k $ uniformly converges at $ [0,R) $, then it converges pointwise at $ R $. Posted: 01 Dec 2021 10:01 AM PST Prove the following theorem: Look at the series $ \sum a_k x^k $ and denote its radius of convergence as $ R $. Suppose the series uniformly converges at $ [0,R) $, then it converges pointwise at $ R $. One can prove the contrapositive, that is: If $ \sum a_k x^k $ does not converge pointwise at $ x_0 = R $. Then there is no uniform convergence of the series at $ [0,R) $ Proof from lecture notes: Suppose $ \sum a_k x^k $ does not converge pointwise at $ x_0 = R $. Suppose for the sake of contradiction that there is a uniform convergence at $ [0,R) $, then by Cauchy's criterion for uniform convergence we have that for all $ \epsilon>0 $ there exists $ n_0 $ s.t. for all $ m,n > n_0 $ it occurs that $ | \sup_{x\in [ 0,R) } \sum_{k=n}^{m} a_k x^k | < \epsilon $ and from continuity ( we talk about about a finite sum of continuous functions ) we get that $ | \sum_{k=n}^m a_k R^k | \leq \epsilon $. I didn't understand the proof above from the line where it says " and from continuity ... ", and specifically, how they jumped to " $ | \sum_{k=n}^m a_k R^k | \leq \epsilon $ ", if you have any idea, can you please explain? |
| Posted: 01 Dec 2021 10:15 AM PST
This is how I understood the question: Let $a,b,c\in\mathbb R^+$ with $$a+b+c<10$$ Then find the probability such that $$a^2+b^2+c^2<16$$ There are infinitely many positive real numbers. Know how to calculate probability? I would like to draw a circle or triangle area. But I can't establish a connection with the triangle area or the circle. The situations are also infinite. This question sounds as if it will be solved from the area of a figure. Do you think I am on the right track? Nothing comes to mind. |
| Limit construction process that avoids constructing a decimal like $x.99999999\ldots$ Posted: 01 Dec 2021 10:28 AM PST
Below is a constructive proof of the statement above. Let $(a_n)_{n≥1}$ be a monotonically increasing, bounded sequence where $a_n = A^{(n)}.\alpha_1^{(n)}\alpha_2^{(n)}\alpha_3^{(n)}\ldots$ with $A^{(n)} \in \mathbb N \cup \{0\}$ and $\alpha_j^{(n)} \in \{0, 1, 2, 3, \ldots, 9\}.$ By assumption, there's $T \ge 0$ s.t. $A^{(n)} < a_n \le T$ for all $n$. Now, $(A^{(n)})_{n\ge 1}$ is monotonically increasing, but it cannot fly off to infinity as it is bounded by $T$ so there must be some $n_0 \in \mathbb N$ s.t. $A^{(n)} = A$ for all $n \ge n_0.$ Similarly, there's some $n_1 \in \mathbb N$ s.t. $\alpha_1^{(n)} = \alpha_1$ for all $n \ge n_1$. Using the algorithm above, we construct a number $a = A.\alpha_1\alpha_2\alpha_3\ldots$ The last step is to show $a = \lim_{n \to \infty}a_n$. My question: Consider $(\alpha_1^{(n)})$. One possible $(\alpha_1^{(n)})$ is $1, 1, 1, 1,\ldots, 1, 2, 3, 6, 8, 8, 8, \ldots, 9, \ldots$. Since $(\alpha_1^{(n)})$ is monotonically increasing, if/when it reaches, say, $8$, it never turns back and takes a value $< 8$, so it has to keep growing and eventually must reach $9$ and stay there. So, how does this construction avoid the limit $A.999999\ldots?$ |
| Homotopy equivalence induces Deformation Retraction if... (Hatcher's AT) Posted: 01 Dec 2021 10:28 AM PST I'm currently studying Allen Hatcher's book Algebraic Topology and I've come across argument in a proof about which I'm not sure why it works. It can be looked up on page 353.
note a homotopy $H: Z \times [0,1] \to X$ is $\operatorname{rel} A$ for $A \subset Z$ if $h_t= H(-\times \{t\})\vert_A$ is independent of $t$.
My question is about the last claim in the proof: Why $W$ deformation retracts onto $Z$ if $\pi_i(W,Z) = 0$ for all $i$? In explicite terms why there exist a family of maps $h_t: W \to W, t \in [0,1]$, such that $h_0= id_X $(the identity map), $h_1(W)= Z$ and $h_t \vert _Z= id_Z$ for all $t$? also note the construction of the mapping cylinder as the quotient of the disjoint union $(Z \times [0,1]) \coprod X$ obtained by identifying each $(z, 1) \in X \times [0,1]$ with $f(z) \in X$.) |
| Compact cardinal cannot be successor? Posted: 01 Dec 2021 09:59 AM PST This is a follow-up question to $\kappa$ is compact $\implies$ $\kappa$ is regular. The definition I'm using for "compact" is the same as there. I am trying to show if $\kappa$ is compact, then $\kappa$ cannot be a successor cardinal. My first attempt was as follows: Assume $\kappa=\lambda^+$. Let $\{p_\alpha\}_{\alpha \in \kappa}$ be propositional variables, and let $p$ be $\bigvee_{\beta \in \lambda} p_\alpha$. Then the set of sentences $\Sigma=\{\neg p_\alpha\}_{\alpha \in \kappa} \cup \{p\}$ is obviously unsatisfiable, because if $p$ is true then at least some $p_\alpha$ must be true. Then, trying to reach the desired contradiction, I tried to show that any subset of $\Sigma$ having cardinality $< \kappa$ is satisfiable, but I realized this is false. (For example, $\{\neg p_\alpha\}_{\alpha \in \lambda} \cup \{p\}$ has size $\lambda < \kappa$ but it is not satisfiable.) Perhaps there's a more creative way to pick $\Sigma$ which actually works? |
| Optimization of 3 variable function without lagrange Posted: 01 Dec 2021 10:09 AM PST The temperature in space given by $T(x,y,z)=200xyz^2$ . Find the hottest temperature on a unit sphere centered at the origin.(without lagrange multipliers method) I took $x^2+y^2+z^2=1$ then using $z^2=1-x^2-y^2$, substituted into equation, then found partial derivatives of $T$ wrt $x$ and $y$ and equated to $0$. What do I do next? Stuck with two equations containing $x$ and $y$. |
| Sequence of Infinitely Differentiable Functions Has Uniformly Converging Subsequence Posted: 01 Dec 2021 10:04 AM PST Let $\{f_n\}$ be a sequence of $C^\infty$ functions on a compact interval such that for each $k$ there exists $M_k$ such that $|f_n^{(k)}(x)|\leq M_k$ for all $n$ and $x.$ Prove that there exists a subsequence converging uniformly, together with the derivatives of all orders, to a $C^\infty$ function. Could somebody provide a proof? |
| Posted: 01 Dec 2021 10:28 AM PST Is it okay for a probability to be above one, where all probabilities above one are equivalent to a probability of one? If not, is it okay to write something like this: $$P(Q) = \begin{cases} \displaystyle \frac ab, & \text{if} \ \ \displaystyle \frac ab \le 1 \\[2ex] 1, &\text{if} \ \ \displaystyle \frac ab > 1 \end{cases} $$ I posted the theorem I was working on here. In it you can see what caused this conundrum. According to Ethan Bolker's answer, there was no problem in using the piecewise expression, so I did that. |
| Parametrization of homogenized curve Posted: 01 Dec 2021 10:19 AM PST I am trying to parametrize the following curves: $F(z,x,y) = (x-y)^7 - z^5x^2, \quad G(z,x,y) = (x-y)^3 - zx^2$. I first try to find a parametrization for $g = G(1,x,y)$. I am trying to do it along the lines through the origin. Hence $y = xt$. For $g$ I get the following: $(x-xt)^3 - x^2 = x^3(1-t)^3 - x^2 \implies x^3(1-t) = x^2 \iff x = \frac{1}{(1-t)^3} \implies y = \frac{t}{(1-t)^3}$, which will give me the parametrization $[t_0,t_1] \mapsto [(t_1-t_0)^3 : t_0^3 :t_0^2t_1]$. However, doing the same with $f = F(1,x,y)$ I will get $x^7(1-t) = x^2 \implies x = \frac{1}{(1-t)^{7/5}} \implies y = \frac{t}{(1-t)^{7/5}} $. But here I don't know how to get the parametrization from that. How can I proceed from there? |
| Exist $a,b$ such that $-1<a<0<b<1$ and satisfy $|P(a)| \geq 1$, $P(b) \geq 1$ Posted: 01 Dec 2021 10:01 AM PST Problem: Let $n\geq 2$ be an even integer. Consider a monic polynomial P(x) ($\deg P=n$) with real coefficients which has $n$ real roots $x_1,x_2,...,x_n$ (no need to be distinct)and $-1\leq x_i \leq 1$ for all $i=1,2,...,n$. Prove that there doesn't exist two real number $a,b$ such that $-1<a<0<b<1$ and satisfy $|P(a)|\geq 1$ and $P(b)\geq 1$ *) My work so far is just prove that the statement is true for $n=2$. For general, i've checked for $n=4,6$ with some unique number and it's true. But i've not done for $n=4,6$ and no idea for general case Someone can help me with this problem ? Thanks :) |
| what is the formal notation about the CDF or PDF of a function with limited interval. Posted: 01 Dec 2021 10:30 AM PST Given an arbitrary function, like $d(t)$, within some finite interval $[t_1,t_2]$, we can get its PDF and CDF by sampling, as shown like My question is where there is any formal notation to describe them. The reason why I do not call them CDF and PDF is because $d(t)$ is deterministic. My final aim is to use some formal notation to describe the following question $$ \int_{t_1}^{t_2}g\left(d\left(t\right)\right)\cdot \mathbb{I}\left\{b(t)<\bar{B}\right\} \ dt=(t_2-t_1)\int_{0}^{\bar{B}}g\left(u\right)\ dF_{d(t)}(u) $$ where $F_{d(t)}(u)$ is the CDF of $d(t)$ within $[t_1,t_2]$. |
| Limit without hospital's rule $\lim_{x\rightarrow 0} (\frac{1}{x^2} - \frac{x}{\sin^3(x)})$ Posted: 01 Dec 2021 10:23 AM PST
I first was trying to replace $x = \frac{\pi}{2} + u$ but I can't find a rule or eliminate anything. After that I try using cubes: $\frac{\sin^3(x) - x^3}{x^2\sin^3(x)} = \frac{(\sin(x)-x)(\sin^2(x)+x^2+x\sin(x))}{x^2\sin^3(x)}$ but again i couldn't find any rule or formula. How can I solve this problem without using L'Hospital? Thank you very much in advance |
| Quasi-isometry of finitely generated group Posted: 01 Dec 2021 10:19 AM PST
Now my idea is as follows - we know that $(\Gamma, d_{S_1})$ and $Cay(\Gamma,S_1$) are quasi isometric. The same goes for $(\Gamma, d_{S_2})$ and $Cay(\Gamma,S_2)$. So I'm trying to show that $(\Gamma, d_{S_1})$ and $(\Gamma, d_{S_2})$ are quasi-isometric. But how do I get this from Milnor-Svarc? I understand that $\Gamma$ acts on $(\Gamma, d_{S_2})$ geometrically and all, so $(\Gamma, d_{S_2})$ and $\Gamma$ are quasi-isometric, but I'm not sure w.r.t to what metric (on $\Gamma$ - the one that acts). As I understand it, Milnor - Svarc states that if Γ acts geometrically on a geodesic proper metric space (X,d) then Γ is finitely generated by a subset S⊆Γ and (Γ,dS) and (X,d) are quasi-isometric Any help would be appreciated. |
| Proving ${\mathbb{P}}^n$ is Hausdorff Posted: 01 Dec 2021 10:21 AM PST I am trying to understand and complete the proof that the real projective space ${\mathbb{P}}^n$ is Hausdorff.In my notes it is modeled as${\mathbb{R}}^{n+1}\setminus \{0\}/\sim $ and it goes like this: It is enough to construct, given two different points $[a]$ and $[b]$ a continuous function $f:{\mathbb{P}}^n \rightarrow {\mathbb{R}} $ such that $f[a] \neq f[b] $ (why ?.......(1)) We fix $\omega$ in ${\mathbb{R}}^{n+1}\setminus \{0\}$ and define $f[\nu]$ as the squared of the distance from $\omega$ to the vector line $R\nu$ generated by $\nu$. Since $f\circ \pi(\nu) = f[\nu] = |\omega|^2-(\omega . \nu)^2/|\nu|^2$, it follows that $f \circ \pi $ is continuous and hence $f$ is continuous...(why?........(2)) ($\pi$ is the canonical projection $\pi : {\mathbb{R}}^{n+1}\setminus \{0\} \rightarrow {\mathbb{P}}^{n} $) It is then enough to take $w \sim a$ to have $f[a] = 0 \neq f[b]$ ...(*) I have two questions: 1)why is (1) true ? At first I thought I could justify (1) like this: Since ${\mathbb{R}}$ is Hausdorff, $\exists $ open sets $A,B$ such that for $f[a] \neq f[b] $, $f[a] \in A$ and $f[b] \in B$, with $A \cap B = \emptyset$. By hypothesis then $[a] \neq [b] $ and since $f$ is continuous, $f^{-1}(A) $ and $f^{-1}(B) $ are open sets of ${\mathbb{P}}^n$ containing $[a]$ and $[b]$ respectively such that $f^{-1}(A) \cap f^{-1}(B) = f^{-1}(A\cap B)=f^{-1}(\emptyset)=\emptyset $. Then ${\mathbb{P}}^n$ is Hausdorff But I don't think my proof is correct since for it to work we need $f$ to have the property stated at (1) but the function $f$ that they propose later has that property only when taking one of the points fixed, say $a$ as $a \sim w$ as done in (*), and not for any two points as needed for the definition of Hausdorff space. Besides if w is the center of a circle (in the 2-dimensional case for instance there are two lines that have the same distance to w, that is the tangent lines, so (1) does not hold for any two different $[a] $and $[b]$) 2)why $f \circ \pi $ continuous implies $f$ is continuous? don't think I can compose with a continuous $\pi^{-1}$, since that function is not well defined since $\pi$ is not injective
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| Formulating list sorting as a pure math problem Posted: 01 Dec 2021 10:05 AM PST Sorting a list is a classic problem in computer science, and many interesting algorithms such as merge sort and heap sort have been discovered. I'd like to have a precise formulation of the list sorting problem as a pure math problem, so that a hypothetical pure mathematician who has never heard of computers (such as Gauss, let's say) would be able to read the problem, understand it, and then invent algorithms such as merge sort and heap sort. A statement such as "devise an algorithm for sorting a list of numbers into ascending order" is inadequate, because it doesn't state which basic operations are allowed to be used by the algorithm. And it also doesn't say precisely what an "algorithm" is. Yes, it should be something that can be implemented in assembly code, but we are posing the problem for a mathematician who has never heard of a computer. Once the problem has been formulated precisely, it should be possible to state and prove theorems such as "The worst-case running time of the quicksort algorithm is $O(n^2)$" with a level of precision and rigor that would meet the standards of, say, baby Rudin. Question: How would you precisely state the problem of devising a list sorting algorithm? Bonus question: Do you know of a reference that presents list sorting algorithms as a topic in pure math, with a level of precision that would satisfy an author such as Rudin? Edit: This might make the question more specific. The beginning of chapter 8 of Introduction to Algorithms by Cormen et al [p. 191 in the third edition] states:
More specific question: How would you formulate that statement precisely as a math theorem -- the fact that any comparison sort must make $\Omega(n \lg n)$ comparisons in the worst case. In order to make this rigorous, we must either define precisely what a "comparison sort" is, or else avoid using the term entirely. |
| Non-measurable sets and sigma-algebra definition Posted: 01 Dec 2021 10:24 AM PST I´m starting to study about measure theory, but I have problems regarding the definition of measure space. In my class we saw that there exists sets that are not measurable(Vitali sets in $\mathbb R$) and in order to avoid this problem we define a sigma-algebra on a set $X$ and we call the elements of $\Sigma$ "measurable sets" But the problem I have is that how can we guarantee that with this "definition" we cannot construct a non-measurable set. Is there a theorem that says that there can´t be non-measurable sets on a sigma-algebra over $X$ with that definition? Or I just need to assume that we cannot find such sets over these circumstances? |
| Finding a vector equation for a trajectory Posted: 01 Dec 2021 10:04 AM PST
I'm not looking for the precise answers but what should i do with this problem? Am i supposed to get the velocity components of x and y? Pls any help would be greatly appreciated |
| Finding generalized eigenvalues with linear constraints Posted: 01 Dec 2021 10:02 AM PST I have a generalized eigenvalue problem $$Mx = \lambda Bx$$ with the additional constraint that $Cx=0$, where $M$ and $B$ are positive-definite and $C$ is a sparse and rectangular. Is there a simple way of solving for the generalized spectrum $(\lambda,x)$? Conceptually, if I had a way of computing a basis for the nullspace $N$ of $C$, I could simply solve instead the unconstrained generalized eigenvalue problem $$N^TMNy = \lambda N^TBNy$$ for which I have code. However, since $C$ is sparse, computing this $N$ (using, e.g., a SVD) is expensive and I would like to avoid it if a different, more efficient reformulation of the problem is possible. |
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