Recent Questions - Mathematics Stack Exchange |
- Is the pair of pants (3 punctured sphere) surface an incompressible surface?
- Fourier transform with an additional term in the power
- Tensor Product $\mathbb{Q} / \mathbb{Z} \otimes \mathbb{Z} / 2\mathbb{Z}$
- Would this be a correct cokernal
- family of functions with radial like behaviuor
- How can I visualize two multi-dimension(>>4) joint distributions and compare them？Is PCA or T-SNE suitable?
- How to compute minimum sample size of a simple linear regression model with given statistics values
- Prove the independence of a certain segment in a special triangle
- Limit evaluation of a function
- Finding the supremum, infimum, and bounds of $f(x) = x^2$ for $x \le a$ and $f(x) = a + 2$ for $x > a$ in the interval $(-a-1,a+1)$, where $a > -1$.
- Show that $\underset{N\to\infty}{\lim}\int_{-\pi}^\pi [g(t)\cos(\frac{t}{2})]\sin(Nt)dt=0$, where $g(t)=\frac{f(x-t)-f(x)}{\sin(t/2)}$.
- Question regarding a problem in set theory
- Three equations with 3 variables and one variable is an exponent
- 10 good and 3 bad batteries are mixed and then 5 are chosen. What is the probability of the fifth one being dead given that the first 4 aren't?
- Convex and Lipchitz continuity of functions
- elasticity calculation question
- Hessian matrix and global minumum
- Can we say anything about the eigenvalues/eigenvectors of a matrix composed by submatrices in a 'circulant' way?
- Show the system has one equilibrium point
- Show that $\sin47^\circ+\sin61^\circ-\sin11^\circ-\sin25^\circ=\cos7^\circ$
- How to show that $(A \cap B') \cup (B \cap A') = (A \cup B) - (A \cap B)$
- ~∃(X)(H(X) ∧ I(X)) - No human is immortal predicate logic
- Definition of Schauder basis
- Solving polynomial systems on MATLAB
- Count the number of ordered triples of positive integers whose product is not greater than a given number?
- Join in Lattice of Subobjects
- Null Space of fix point equation where coefficient matrix is a Kronecker Product
- Is there a guarantee that nonzero/nontrivial vector space will have a subspace that is not the zero vector space or itself?
- Integrability of a periodic function based on $\int_0^1 |f(a+t)-f(b+t)| dt$
| Is the pair of pants (3 punctured sphere) surface an incompressible surface? Posted: 10 May 2022 09:37 AM PDT Is the pair of pants (3 punctured sphere) surface an incompressible surface? My intuition says no since a simple closed curve around a boundary component will not bound a disk on the surface. |
| Fourier transform with an additional term in the power Posted: 10 May 2022 09:37 AM PDT Consider this relation $$\sigma(\omega_{I})=\int_{-\infty}^{+\infty} <\psi(0)|\psi(t)>e^{i(E_i/\hbar+\omega_I)t}dt$$ I know, by inverse Fourier transform, I can compute for $$\int_{-\infty}^{+\infty} <\psi(0)|\psi(t)>e^{i(E_i/\hbar)t}dt$$ But I really have no idea how to take the $\omega_I$ power into account in order to determine $\sigma(\omega_{I})$; I just couldn't see the relation between the two integrals. Can someone help me? |
| Tensor Product $\mathbb{Q} / \mathbb{Z} \otimes \mathbb{Z} / 2\mathbb{Z}$ Posted: 10 May 2022 09:35 AM PDT I believe that the tensor product $\mathbb{Q} / \mathbb{Z} \otimes \mathbb{Z} / 2\mathbb{Z}$ is trivial, i.e. any element is 0, but apparently there are in fact 2 elements. Why is this? |
| Would this be a correct cokernal Posted: 10 May 2022 09:34 AM PDT I'm a little confused on cokernels so I wanted to create an example and was hoping someone could check it to make sure I'm right. Definition of cokernel: If $ f: M → N $ is a homomorphism of A-modules, then the cokernel of $f$, is the quotient $N/$image$(f)$ My 2 examples $f:ℤ_3 → ℤ_6$ such that $f(2n) ∀ n ∈ ℤ$ so the image is $[0,2,4]$ so would the cokernel be $ℤ_2$ and if we had $g:ℤ_3 → ℤ_6$ such that $f(2n-1) ∀ n ∈ ℤ$ what would the cokernel be there |
| family of functions with radial like behaviuor Posted: 10 May 2022 09:33 AM PDT I am trying to find a generalized expression for a family of smooth radial functions that have the same global minimum, are finite asymptotically, and diverge at the origin: $$ f(0)=\infty $$ $$ f(r_0)=0 $$ $$ f(r_1) =a_1 $$ $$ f(\infty) = a_2>a_1 $$ For example, a minimal number variables function that obey that is: $$f(r)=(1-\exp(-c(r-r_0))^2$$ How can I further generalize this without changing the conditions the function satisfy? I thought about adding a polynomial series to that but that wont satisfy the finitness at $\infty$. |
| Posted: 10 May 2022 09:31 AM PDT How can I visualize two multi-dimension(>>4) joint distributions and compare them？Is PCA(Principal component analysis) or T-SNE suitable? |
| How to compute minimum sample size of a simple linear regression model with given statistics values Posted: 10 May 2022 09:28 AM PDT Suppose the statistics values are given as follows: $\sum_{i=1}^{n}x_i, \sum_{i=1}^{n}y_i, \sum_{i=1}^{n}x_iy_i, \sum_{i=1}^{n}x_i^2,\sum_{i=1}^{n}y_i^2$ Firstly, we can compute the regression coefficient $\beta_0, \beta_1$ such that: $Y =\beta_0 + \beta_1 X$ An ANOVA (F-test) can also be carried out to see whether X and Y are dependent. Suppose $F<F_{critical}$ , p-value $> 0.05$ such that there is no strong evidence that $\beta_1 \ne 0$. i.e. X and Y are independent. However, what if we were told that $\sum_{i=1}^{n}(x_i-\bar x)^2, \sum_{i=1}^{n}(y_i-\bar y)^2, \sum_{i=1}^{n}(x_i-\bar x)(y_i-\bar y)$ were doubled, X and Y are dependent by ANOVA, and find the minimum sample size $n$. My approach: I am new to ANOVA and statistics test, the only I can think of is to start it with F-test formula , and if X and Y are dependent, it should fulfill $F>F_{critical}$ but $F_{critical}$ is unknown due to unknown sample size $n$. Besides, the new model has the same $\beta_1$ as before. from doubling up $\sum_{i=1}^{n}(y_i-\bar y)^2$. We can also find out the new $SS_{TOTAL}$ Apologies for the unorganised ideas above. I have no idea of the right direction to solve this problem. I would like to know how to solve this, please. Thank you very much. |
| Prove the independence of a certain segment in a special triangle Posted: 10 May 2022 09:39 AM PDT Let $\triangle ABC$ be a triangle with obtuse angle $\angle A$ and $\overline{AB} = 1$. Also, let $\angle C = \gamma$ and $\angle B = 2\gamma$. If $E$ and $F$ are intersection points of perpendicular bisector of $\overline{BC}$ and circle $(A, \overline{AB})$, prove that $\overline{EF}$ is constant (not dependent of $\gamma$). Note that circle $(A, r)$ means a circle with point $A$ as its center and $r$ as its radius. It's not hard to see that $\angle FAE = 120^\circ$ but I don't know how to prove it either. Hope someone can help. |
| Limit evaluation of a function Posted: 10 May 2022 09:23 AM PDT $Limit_{x \rightarrow \infty \frac{arctan (x^\frac{3}{2})}{√x}}$
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| Posted: 10 May 2022 09:22 AM PDT My current progress; $x^2$ is increasing, so it will always have a minimum of $0$. Furthermore, if $a > -1$, then $a + 2 > 1$ and $-a -1 < 0$. Therefore, $f(x) \ge 0$. I tried to split $f$ and find the $\inf / \sup$ of $x^2$ on $(-a-1,a]$ and $a + 2$ on $(a,a+1)$ with the restriction that $a > -1$. Now clearly $(-a-1)^2 = a^2 + 2a + 1$ and $(a)^2 = a^2$, and $a + (a + 2) = 2a + 2$ and $a + 1 + (a + 2) = 2a + 3$. At this point, I am kind of stuck. What can I do? |
| Posted: 10 May 2022 09:20 AM PDT Problem: $\underset{N\to\infty}{\lim}\int_{-\pi}^\pi [g(t)\cos(\frac{t}{2})]\sin(Nt)dt=0$, where $g(t)=\frac{f(x-t)-f(x)}{\sin(t/2)}$. The Problem arises from Walter Rudin's PMA: For the texts highlighted in blue: the boundedness of $g(t)\sin(t/2)$ is trivial; and to show the boundedness of $g(t)\cos(t/2)$, I suppose we show that $\underset{t\to 0}{\lim}\frac{t}{cos(t/2)}=2$, then combine it with (79) and (81). But the texts highlighted in yellow really trouble me. (74) states that "$\underset{n\to\infty}{\lim}c_n=0$", where $c_n$ is the n^th Fourier coefficient of $f$ relative to an orthonormal series ${\phi_n(x)}$. Here I believe $c_n=\int_{-\pi}^\pi f(t)\overline{\phi_n(t)}dt$ by (66). But then how does that lead to "$\int_{-\pi}^\pi [g(t)\cos\frac{t}{2}]\sin Nt dt$ and $\int_{-\pi}^\pi [g(t)\sin\frac{t}{2}]\cos Nt dt$ tend to $0$ as $N\to\infty$"? Any hint would be greatly appreciated. |
| Question regarding a problem in set theory Posted: 10 May 2022 09:34 AM PDT See: Gelson Iezzi, Matemática (volume unico), page 11, exercise 10 item e): P = {r, {r}, {r,s}, t} Why {r, s} ⊂ P is false? Why is not a subset? |
| Three equations with 3 variables and one variable is an exponent Posted: 10 May 2022 09:31 AM PDT Can anyone help me in solving this ? Can you show me the steps I need to make. $$\frac{P}{T} = \frac{c}{T^a+f}$$ where, if T=6, P = 94; T=12, P =109; T=24, P=116.5 find a, c ,f Thanks and Regards, |
| Posted: 10 May 2022 09:31 AM PDT I approached this as follows: $A$: First four are good, $B$: Fifth one is bad. Then $$|A|={10\choose4}$$ because this is the number of ways we can choose 4 good batteries from the available 10. Then $$P(A)=\frac{10\choose4}{13\choose4}$$ I want to find $P(B|A)=\frac{P(A\cap B)}{P(A)}$. For this I need: $A\cap B$: First four are good and fifth one is dead. Then $|A\cap B|={10\choose4}{3\choose1}$ because this is the number of ways we can choose 4 good batteries from the available 10 followed by choosing 1 of the available 3 dead ones. Then $$P(A\cap B)=\frac{{10\choose4}\cdot{3\choose1}}{13\choose5}$$ However, once a substitute these values into the conitional probability expression I get $\frac53$. |
| Convex and Lipchitz continuity of functions Posted: 10 May 2022 09:33 AM PDT If a function is convex, is it true its gradient satisfies Lipchitz continuity? |
| elasticity calculation question Posted: 10 May 2022 09:22 AM PDT I calculate a time series data set and some variable weren't stationary at the level so I took the first differences. At the end of the study,the price variable is stationary at the level but the quantity of supply is stationary at the first differences. I use SUR to estimation, then got the β estimation results. The problem is So β=(%change of Qantity in differences)/(% change in price) This is the excess supply of the product that I dont know how to calculate elasticity from here and how to interpret. |
| Hessian matrix and global minumum Posted: 10 May 2022 09:32 AM PDT Given a vector field $F(x_1,\cdots,x_N)$ which all trajectories starting in it will converge to exactly one point. Does its Hessian matrix must be positive semidefinite? |
| Posted: 10 May 2022 09:30 AM PDT Circulant matrices, that is matrices in $\mathbb{R}^{n \times n}$ of the form $$\begin{pmatrix} c_0 & c_1 & c_2 & ... & c_n \\ c_n & c_0 & c_1 &... & c_{n-1} \\ c_{n-1} & c_n & c_0 &... & c_{n-2} \\ ... & ... &... &... &... \end{pmatrix}$$ have a particularly easy eigenvalue/eigenvector structure - in particular, the eigenvectors are of the form $(\omega^0,\omega^{1},...,\omega^{n-1})$ for $\omega \in \mathbb{C}$ a $n-th$ root of unity. I was wondering if we still can gain some insight into the spectrum/eigenvectors, if we relax the circulant condition to only hold for submatrices? That is, if we have a matrix of the form $$ C = \begin{pmatrix} C_0 & C_1 & C_2 & ... & C_m \\ C_m & C_0 & C_1 &... & C_{m-1} \\ C_{m-1} & C_m & C_0 &... & C_{m-2} \\ ... & ... &... &... &... \end{pmatrix}$$ where $C_i \in \mathbb{R}^{k \times k}$ are square matrices (and $k$ divides $n$), what are eigenvectors/values of $C$ and how do they relate to the $C_i$? The only trivial thing that one could immediately see is the case where $v \in \mathbb{R}^{k}$ is an eigenvector of all the $C_i$. Then of course also $(v,v,v,...,v)$ is an eigenvector of $C$ with eigenvalue $\sum_i \lambda_i$, where $v$ is an eigenvector of $C_i$ for eigenvalue $\lambda_i$. But this is of course not a very interesting special case. It would also be interesting to have some result under some additional assumptions, e.g. symmetry of $C$. |
| Show the system has one equilibrium point Posted: 10 May 2022 09:25 AM PDT I was wondering how we would show that the system: has only one equilibrium point. I have seen cases where the system is, for example: which I understand more how to calculate the equilibria, as you can simplify it better. But in the case above I am stuck. I found using Wolfram alpha that the equilibria point of would be (0,0) but I don't know how to show this. I thought maybe about writing the system as: but I don't know where I could go from here. I also seen that I could maybe use polar coordinates and started doing this: but Im not sure where to go from here either. Any help would be appreciated! |
| Show that $\sin47^\circ+\sin61^\circ-\sin11^\circ-\sin25^\circ=\cos7^\circ$ Posted: 10 May 2022 09:36 AM PDT Show that $$\sin47^\circ+\sin61^\circ-\sin11^\circ-\sin25^\circ=\cos7^\circ$$ NOTE: I have seen the other questions and solutions for this problem. I have a particular question if my idea has a potential. I decided to rearrange the LHS (without a particular reason, it just felt right to me) as follows $$(\sin47^\circ-\sin11^\circ)+(\sin61^\circ-\sin25^\circ)=\\2\cos\frac{47^\circ+11^\circ}{2}\sin\frac{47^\circ-11^\circ}{2}+2\cos\dfrac{61^\circ+25^\circ}{2}\sin\dfrac{61^\circ-25^\circ}{2}\\=2\cos29^\circ\sin18^\circ+2\cos43^\circ\sin18^\circ=2\sin18^\circ(\cos29^\circ+\cos43^\circ)=\\4\sin18^\circ\cos36^\circ\cos7^\circ$$ If this won't work, what is the intuition that leads to the appropriate rearranging? |
| How to show that $(A \cap B') \cup (B \cap A') = (A \cup B) - (A \cap B)$ Posted: 10 May 2022 09:32 AM PDT How to show that $(A \cap B') \cup (B \cap A') = (A \cup B) - (A \cap B)$ Let $C=(A \cap B')$ $$(A \cap B') \cup (B \cap A') = C \cup (B \cap A')$$ $$=(C \cup B) \cap (C \cup A')$$ $$(A \cup B) \cap (A \cap B)'$$ $$[(A \cap B') \cup B] \cap [(A \cap B') \cup A']$$ $$[(B \cup A) \cap (B \cup B')] \cap [(A' \cup A) \cap (A' \cup B')]$$ $$[(B \cup A) \cap U] \cap [U \cap (A \cap B)']$$ $$[(A \cup B) \cap U] \cap [(A \cap B)' \cap U]$$ $$(A \cup B) \cap (A \cap B)'$$ $$(A \cup B) - (A \cap B)$$ Is the process after introducing the new set $C$ logical? Is there a better algebraic solution of the problem? |
| ~∃(X)(H(X) ∧ I(X)) - No human is immortal predicate logic Posted: 10 May 2022 09:38 AM PDT When watching the Veritasium video "Math's final flaw", at 12:27, he quickly mentioned a logic statement that said "No human is immortal" I did some research on predicate logic (because I knew absolute nothing about this subject). And deciphered it as: EDIT: I misread the statement, I thought it was an | (not and) but it was I for Immortal, which makes much more sense. The correct statement is: $$ \sim\exists(x)(H(x)∧I(x)) $$ Meaning: There does not exist an x such that x is in both set Human and Immortal. I think the culprit was this image:
Which I looked at more than this:
I would love to know when this becomes useful for actual mathematics (except for Gödel's theorem). For trivial things like this it seems easier to write it as text, but when is that no longer true? Also, is there a way to convey the same, or almost the same meaning with math using less notation? |
| Posted: 10 May 2022 09:22 AM PDT I have a definition of a Schauder basis but I'm unsure of it. The definition I have is A sequence $\{e_k : k \in \mathbb{N} \}$ in a normed space $(V, \| \cdot \| )$ is a Schauder basis if
This definition to me seems to mean that the Schauder basis is countable. However a theorem is 'an infinite dimensional Hilbert space is separable if and only if it has a countable orthonormal basis' . This theorem seems to contradict that a Schauder basis has to be countable. What's the deal here? |
| Solving polynomial systems on MATLAB Posted: 10 May 2022 09:25 AM PDT I'm working on a large polynomial system with 40+ variables and even more equations. I am aware that SINGULAR can ``solve'' it by finding its Groebner basis. However, I have to solve a sequence of such systems with different coefficients generated in MATLAB. How should I proceed with SINGULAR, or, if possible, other softwares that allow configuration in MATLAB? Any suggestion is appreciated, thank you! |
| Posted: 10 May 2022 09:27 AM PDT Given N, count the number of ordered triplets(a,b,c) whose product $abc \leq N$. I have found the series here . But I am not sure I understand Benoit Cloitre work which proposed an efficent way to compute. $O(N)$ computation is simply not good enough, I need something sublinear . Possibly between $O(N^{8/9})$ and $O(N^{2/3})$. EDIT: As discussed with @henry here is my code from my broken interpretation of Andrew Lelechenko's formula |
| Posted: 10 May 2022 09:26 AM PDT In an elementary topos $\mathcal E$, the join $A \vee B$ of two subjects $A \to X$ and $B \to X$ is defined to be the image of the induced morphism $A \sqcup B \to X$. For sets it holds, that this is the same as the pushout of $A$ and $B$ along their intersection $A \cap B = A \wedge B = A \times_X B$. Does this hold in general? Here the image of a morphism $f\colon M \to X$ is defined to be the equalizer of the two inclusions $X \to X \cup_M X$. |
| Null Space of fix point equation where coefficient matrix is a Kronecker Product Posted: 10 May 2022 09:20 AM PDT I am interested in finding the null-space of the fix point equation: $$ Pw = w \\ (P_{out} \otimes P_{in}^T)w = w $$ Where $P \in \mathcal{R}^{nm \times nm}$ is the Kronecker product of two permutation/reflection matrices $P_{in} \in \mathcal{R}^{n\times n}, P_{out} \in \mathcal{R}^{m\times m}$ (with a single $\pm 1$ per column). Such an equation is also equivalent to: $P_{out}W = WP_{in}$, where $w$ is the vectorization of matrix $W \in \mathcal{R}^{m \times n}$. The problem with the introduction of the Kronecker product is that the dimensionality of $P$ easily scales beyond any tractable algorithm for eigendecomposition (I am interested in the nullspace $Q$ of the equation so I can parameterize $W$ to comply with the fix point equation): $$ (P - I) w = 0 \\ U \left[\begin{array}{ll} \Sigma & 0 \\ 0 & 0 \end{array}\right]\left[\begin{array}{l} B^{\top} \\ Q^{\top} \end{array}\right] w = 0 $$ Is there a possibility to compute the eigendecomposition of $(P - I)$ as a function of the eigendecomposition of the smaller and more tractable matrices $P_{in}$, $P_{out}$?. More General Problem: Things get a lot more challenging when there are more than a single symmetry constraints: $(^iP,\,\dots,\,^lP)= ((^iP_{in},\, ^iP_{out}),\dots, (^lP_{in}, \,^lP_{out}))$, and the fixed point equation turns to: $$ \begin{bmatrix} \,^iP - I \\ \vdots \\ \,^lP - I \end{bmatrix} w = 0 \\ $$ |
| Posted: 10 May 2022 09:35 AM PDT I know that the trivial vector space and the vector space itself are subspaces of any vector space. But, what I do not understand is that, given that the vector space is nonzero, how do we know that there is a subspace that is not the trivial vector space or itself? |
| Integrability of a periodic function based on $\int_0^1 |f(a+t)-f(b+t)| dt$ Posted: 10 May 2022 09:37 AM PDT
First of all what does it is bounded uniformly for all $a, b \in \mathbb{R}$? Does it mean that for all $a, b \in \mathbb{R}$, $\int_0^1 |f(a+t)-f(b+t)| dt \le M$ for a single $M$? And how does it help to solve the exercise? How the hint is useful when nothing cancels out to reach $\int_0^1 |f(t)| dt$ with the use of the hint? |
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