Recent Questions - Mathematics Stack Exchange |
- prove that subsets are fields (or not) of $\mathbb{C}$
- I don’t know how to start with the integral $ \int_{0}^{\frac{\pi}{2}} \frac{x \sin x}{1+\sin ^{2} x} d x ? $ It seems quite difficult. Who can help?
- SU(2) representations are conjugate iff they have same character.
- How to find the conditional probability density function $f_{X|X\geq 0}$ where $X\sim N(\mu , {\sigma}^2)$?
- Maximum/Minimum value of square of sums over the sphere?
- In the absence of CH, there are just $2^{2^{\aleph_0}}$ ultrapowers of $\mathbb{R}$ of length $\omega$ up to isomorphism.
- Confusion on the definition of a complex structure
- show that $\mathcal{N}(A) \subseteq \mathcal{N}(B)$ iff $BA^+A = B$.
- Linear Regression;Re-parameterization
- Index laws and modular arithmetic
- Matrices 3x3 with rank 1 and rank 2 are manifolds.
- $k$ different balls are taken randomly and put in n different boxes. what is the probability that no box contains more than one ball?
- Plotting tight bounds for simple Wiener Brownian motion - problems with classic definitions
- Determine the percentile of the score distribution
- Exercise for Wiener algebra
- the associated polynomial of circulant matrix
- Inverse Function involving the natural exponential
- Ultrafilters form a base for a topology
- Irreducible constituents of a faithful representation
- LIe algebras and the exponential map on sheaves?
- Decomposition of one $L^p$ space into two other .
- About the inequality $f(-x)<\left(\Gamma(1-x)\right)^{\frac{1}{1-x}}$
- How do I solve this limit with an absolute value?
- Solve a system of two equations in three unknowns locking the param to integer number
- Where does the linear equation in the proof of solvability in elementary numbers in [Chow 1999] come from?
- How to find AB?
- Quadratic formula in differential equations
- Are these "finite-ish" sets closed under union?
- Comparing the cardinalities of generic $\mathbb{R}$s
- Solutions to the Laplace Equation $\Delta u =0$, where $u= \log p$
| prove that subsets are fields (or not) of $\mathbb{C}$ Posted: 26 Sep 2021 08:11 PM PDT I'm in a course in fields/galois theory using DJH Garling's book but just starting I'm struggling with this question. Which of the following subsets are subfields of $\mathbb{C}$. i) $$\{a+bi: a,b\in \mathbb{Q}\}$$ ii)$$\{a+\omega b: a,b \in \mathbb{Q}. \omega=1/2 (-1+\sqrt{3}i) \}$$ iii) $$\{a+2^{1/3}b: a,b \in \mathbb{Q} \}$$ iv) $$\{a+2^{1/3}b+4^{1/3}c: a,b,c \in \mathbb{Q} \}$$ I proved the first is a field but the third is not. The issue with the second and last sets is that I proved all requirements (axioms of groups under + and *) but the multiplicative inverse for nonzero elements. In ii) and iv), I assumed that the inverse must be the same as in $\mathbb{C}$ conjugate divided by norm but I couldn't figure out the algebra to write the inverse as an element of the set. I found similar questions here but not exactly and some of them use tools that are yet to be learnt in the course I assume. This is the first class and the very beginning of the book so I assume that a pretty elementary proof is required. Any help is appreciated, thank you.  |
| Posted: 26 Sep 2021 08:14 PM PDT How do we deal with the integral $$ \int_{0}^{\frac{\pi}{2}} \frac{x \sin x}{1+\sin ^{2} x} d x ? $$  |
| SU(2) representations are conjugate iff they have same character. Posted: 26 Sep 2021 08:10 PM PDT Let $G$ be a group with a finite presentation. Let $\alpha: G \to SU(2)$ be a representation, its character is $G \to^{\alpha} SU(2) \to^{tr} \mathbb{R}$.
The $\implies$ direction is pretty easy. Any help/hint is appreciated.  |
| Posted: 26 Sep 2021 08:09 PM PDT Problem: Assume $X\sim N(\mu , {\sigma}^2)$, please find (1) The conditional probability density function of $X$ given $X \geq 0$; (2) $E(X|X\geq 0)$ when $\mu = 2, \sigma=1$. I tried to first write out the conditional distribution function: $F(x|D)=P(X\leq 0| Y\in D)=P(X\leq x|X\geq0)=\frac{P(X\leq x, X\geq 0)}{P(X\geq 0)}=\frac{P(0\leq X\leq x)}{P(X\geq 0)}=\frac{\int_{0}^{x}f(x)dx}{\int_{0}^{+\infty}f(x)dx}$, if $x\geq0$. $F(x|D)=0$, if $x <0$. However, I can't extract a $\int_{-\infty}^{x}$ from $F(x|D)=\frac{\int_{0}^{x}f(x)dx}{\int_{0}^{+\infty}f(x)dx}$ in order to get the density function out of the distribution function. I don't know whether my trial is on the right way or not, please help me, thanks a lot!  |
| Maximum/Minimum value of square of sums over the sphere? Posted: 26 Sep 2021 08:07 PM PDT Let $f : \mathbb{S}^{n-1} \to \mathbb{R}$ be the function $$f(x) = \left( \sum_{k=1}^n x_k \right)^2.$$ Can someone point me to a reference which estimates $\max_{x \in \mathbb{S}^{n-1}} f(x)$ and $\min_{x \in \mathbb{S}^{n-1}} f(x)$?  |
| Posted: 26 Sep 2021 08:07 PM PDT We consider structures of the language of ordered fields. In the absence of CH, there are just $2^{2^{\aleph_0}}$ ultrapowers of $\mathbb{R}$ of length $\omega$ up to isomorphism. This seems to be a corollary of the following paper: Linus Kramer, Saharon Shelah, Katrin Tent, Simon Thomas "Asymptotic cones of finitely presented groups" arXiv:math/0306420 [math.GT] https://arxiv.org/abs/math/0306420 But the paper deals with a more general setting, which is unfamiliar to me. Can a easier argument prove the first statement?  |
| Confusion on the definition of a complex structure Posted: 26 Sep 2021 08:05 PM PDT I am reading the notes of Moroianu's Lectures on Kahler Geometry: https://moroianu.perso.math.cnrs.fr/tex/kg.pdf. On page 30 we want to prove that the Levi-Civita connection and Chern connection coincide iff the manifold is Kahler. It was first note that $TM$ can be viewed as a complex vector bundle, as the complex structure $J$ allow us to identify $iX:= JX$. In the first line of the proof of Lemma 5.7, we have $\bar{\partial}f(X) = \frac{1}{2}(X+iJX)(f)$. My question is that, is this not just $0$ using the identification we made? I believe that I have probably understood something wrongly here, but I cannot warp my head around what is going on.  |
| show that $\mathcal{N}(A) \subseteq \mathcal{N}(B)$ iff $BA^+A = B$. Posted: 26 Sep 2021 08:01 PM PDT For $A\in\mathbb{R}^{p\times n}$ and $B\in\mathbb{R}^{m\times n}$, show that $\mathcal{N}(A) \subseteq \mathcal{N}(B)$ iff $BA^+A = B$. Note "$^+$" indicates Moore-Penrose pseudo-inverse. My attempt: Suppose $BA^+A = B$ and $x \in\mathcal{N}(A)$. Then $Ax = 0$. We have $$Bx = (BA^+A)x = BA^+(Ax) = 0.$$ This shows that $x\in \mathcal{N}(B)$. Suppose $\mathcal{N}(A)\subseteq\mathcal{N}(B)$. Let $x\in\mathbb{R}^n$ where $x\in\mathcal{N}(A)\subseteq\mathcal{N}(B)$. Thus we have $Ax = 0$. But note that $KAx = 0$ for all $K$ matrices with $p$ columns. Thus assuming $BA^+$ has $p$ columns we have $$Bx = Ax = BA^+(Ax) = BA^+Ax.$$ The second half I am honestly not sure if I was able to do that or not. Let me know what I need to fix! Feedback is great, thanks!  |
| Linear Regression;Re-parameterization Posted: 26 Sep 2021 08:01 PM PDT I have two functions that I am trying to re-parameterize in the linear-regression context. The following is the progress I've made on each, but am hesitant about my approach and am looking for some feedback. (i) $$t = \begin{cases} b_1 + b_2y, & y < 2 \\ b_3 + b_4y, & y \geq 2\\ \end{cases} $$ In order to express in the form of $t = \textbf{A}\alpha(x)$ I defined the following: $$ \textbf{A} = (b_1,b_2,b_3,b_4)\\ \alpha(x) = \begin{cases} (1,y,0,0)^T & y < 2 \\ (0,0,1,y)^T, & y \geq 2\\ \end{cases}$$ (ii) $$t = (1+b_1x_1) e^{-x_2+b_2} $$ In order to express in the form of $t = \textbf{A}\alpha(x)$ I defined the following: $$\textbf{A} = (e^{b_2},b_1e^{b_2})\\ \alpha(x) = (\alpha_1(x),a_2(x))^T\\ \text{where } \alpha_i(x) = (x_1)^{i-1}e^{-x_2} $$ As an extension to (ii), I need to formulate a function that takes $\textbf{A}$ back to $(b_1, b_2)$ and came up with the following: $(b_1, b_2) = \begin{pmatrix} 0 & e^{-b_2}\\ b_2e^{-b_2} & 0 \end{pmatrix}\textbf{A}^T$ I am skeptical about the logic of this approach because I feel like you would need to already know the coefficients $b_i$ to define the above matrix. Some feedback here would be very helpful.  |
| Index laws and modular arithmetic Posted: 26 Sep 2021 08:00 PM PDT Suppose r is a primitive root mod m, is it true that r^(a-b) = 1 mod m imply r^a = r^b mod m?  |
| Matrices 3x3 with rank 1 and rank 2 are manifolds. Posted: 26 Sep 2021 07:57 PM PDT Let $M,N \subset \mathbb{R}^{n}$ sets of matrices $3x3$ with rank 1 and rank 2 respectively. Show that $M$ and $N$ are manifolds such that dim $M=5$ and dim $N=8$. I have thought about using the determinant function $ det: \mathbb{R}^{n^{2}} \rightarrow \mathbb{R}$ but i have problems when i deal with $f^{-1}(0)$. Thanks for your answers.  |
| Posted: 26 Sep 2021 07:58 PM PDT The sample space is $r^n$ but I can't seem to find the combination for the second part of the question of computing the probability.  |
| Plotting tight bounds for simple Wiener Brownian motion - problems with classic definitions Posted: 26 Sep 2021 07:53 PM PDT I am trying to plot the standard bounds of simple Brownian motion (implemented as a Wiener process), but I have found some difficulties when drawing the typical equations:
Example of tested bounds and Brownian realizations: https://i.stack.imgur.com/Q7Lec.png Actually it works so good, that I don't know if it is just a coincidence (maybe I made a mistake when defining the Brownian paths), but I don't found this bound in any website, so if right, it could be useful for everybody so I left it here, but certainly, I don't have the ability to probe anything related to it: if is "mathematically" right, if is "tight" as an "almost-sure" true limit, if it is going to be surpassed infinitely many times or not, etc... I hope you can help to tell me if is right, or just a mistake that has a beauty plot. Beforehand, thanks you very much. I left the Matlab code I use:  |
| Determine the percentile of the score distribution Posted: 26 Sep 2021 07:51 PM PDT The score of a student on a certain exam is represented by a number between 0 and 1. Suppose that the student passes the exam if this number is at least 0.55. Suppose we model this experiment by a continuous random variable S, the score, whose probability density function is given by: $$\ f(x) = \begin{cases} 4x & 0 \le x \le .5\\ 4-4x & .5 \le x \le 1\\ 0 & otherwise \end{cases}$$ 1)What is the probability that the student fails the exam? 2)What is the score that he will obtain with a 50% chance, in other words, what is the 50th percentile of the score distribution? For question #1 I got: $$f(.55) = P(X\le.55) = \int_{-\infty}^{.55} f(x) \,dx $$ Which gave me $$\int_{0}^{.5} 4x \,dx +\int_{.5}^{.55} (4-4x) \,dx$$ Which when worked out I got $$P(X\lt .55) = 0.595 $$ My question is regarding #2 is it as simple as Or do I need to use:  |
| Posted: 26 Sep 2021 07:55 PM PDT Let $\mathcal{M}(\Bbb{T})$ be the space of finite complex Borel measure.Let a measure $\mu \in \mathcal{M}(\Bbb{T})$ has the following property: $$\sum_{n\in \Bbb{Z}}|\hat{\mu}(n)|<\infty$$ Prove $\mu(dx) = fdx$ for some $f\in C(\Bbb{T})$,where $\hat{\mu}(n) = \int e^{-2\pi inx}\mu(dx)$ .Moreover the representative density associated to the measure $\mu$ is unique. My attempt: First if $\mu(dx) = fdx$ then we see $f = \sum_{n\in \Bbb{Z}}\hat{\mu}(n)e^{2\pi i n x}$ easy to check $f(x)$ is well defined since the series converge absolutely and uniformly.Only need to check that for all the measureable $E$ we have $$\int_E \mu(dy) = \int_E\sum_{n\in \Bbb{Z}}\hat{\mu}(n)e^{2\pi i n y} dy = \sum_{n\in \Bbb{Z}}\int_E\hat{\mu}(n)e^{2\pi i n y}dy$$ Where the second equality due to uniformaly convergence of the sequence. we may use Fubini write the integral RHS as $$\int_E\hat{\mu}(n)e^{2\pi i n y}dy = \int\int\chi_E(y) e^{-2\pi i n x}e^{2\pi i n y}\mu(dx)dy= \int (\int \chi_E(x+z)e^{2\pi i n z}dz) \mu(dx) $$ The Fubini holds since $\mu$ is finite measure I don't know how to preceed,the Fourier coefficient for indicator appears,it does not have "nice" convergence property,So I guess may need to add some decaying factor into it ?  |
| the associated polynomial of circulant matrix Posted: 26 Sep 2021 07:50 PM PDT I tried understanding the circulant matrix with a little more knowledge behind it. In some documents, there is an associated polynomial of the ciculant matrix. $$ g(x) = a_1 + a_2 x+ a_3 x^2 + \cdots + a_n x^{n-1} $$ However, in general, the polynomial is given without introduction. I want to get how it is derived.  |
| Inverse Function involving the natural exponential Posted: 26 Sep 2021 08:11 PM PDT My question: If $g(x)=e^{f(x)}$, show that $g^{-1}(x)=f^{-1}(\ln{x})$. It is also given that $g(x)$ and $f(x)$ have an inverse for $x>0$. My efforts so far: $y=e^{f(x)}$ $\ln{y}=f(x)$ $f^{-1}(\ln{y})=f^{-1}(f(x))$ $f^{-1}(\ln{y})=x$ I am not quite sure where to go from here.  |
| Ultrafilters form a base for a topology Posted: 26 Sep 2021 08:11 PM PDT I am trying to prove the set of ultrafilters over a subset A of $\mathbb{N}$, given by $B=\{b \in UF(\mathbb{N}): A \in b\}$ forms a base for a topology over $UF(\mathbb{N})$. I think I should use the fact that the intersection of two ultrafilters over $\mathbb{N}$ is an ultrafilter, however, I dont know if this is true.  |
| Irreducible constituents of a faithful representation Posted: 26 Sep 2021 07:54 PM PDT I am new to representation theory and have a doubt:
My thoughts: If the Group action is faithful in the whole vector space, it must be so in any subspace. So the answer should be yes. Am I missing something?  |
| LIe algebras and the exponential map on sheaves? Posted: 26 Sep 2021 08:00 PM PDT I've been thinking a lot about the very simple differential equation $f\in C^1(\mathbb R,\mathbb R^n), A \in M_n; Df=Af$. There is an informal idea present: $$f(x+t) = e^{Dt}f(x) = e^{At}f(x)$$ My question is hopefully not too vague: How can we make the idea of $e^{Dt}$ both rigorous and general? These ideas are floating around everywhere but I've never seen it explicitly written out! We can directly calculate $e^{Dt} f(x)$: for analytic functions this is explicitly $f(x+t)$, but in $C^1$ we have no hope! Of course we could by fiat define $e^{Dt}$ on all functions (even on generalized functions) but this feels arbitrary, since it has no relationship to the spectrum of the derivative and moreover it does not help us generalize to difference equations and other generalized differential equations! My post is very long because as far as I know all of this is just mathematical apocrypha and so showing what I tried before coming to SE requires a lot of description. I'm flying by a lot here, sorry if I made some obvious mistakes! This idea can be made rigorous by noting that $\mathbb{R}$ is a lie group via translation and that $\langle At \rangle$ is also a lie group. Since the exponential map acts as a functor and these groups are simply connected we must have $e^{Dt}f = e^{At}f$. We generalize $e^{Dt}$ using one-parameter families and this gives us $e^{Dt}=S_t$, where $S_t$ is translation by $t$ for $C^1$. For a general manifold this is also not too bad, nontrivial maps $\phi_x: T_x \mathbb R \to T_x M$ is a choice of tangent vector. This (locally) lifts to a map $\mathbb R \to M$, the one parameter group associated to that vector field. For some $U_x \subseteq \mathbb R$, we have $e^{Dt}: U_x \to M$ by $e^{Dt}f = e^{t(df)}$ where $d$ is just the common pushforward and $f$ is a one parameter group. This is in analogy with mild solutions from semigroups. We can now ask for which $f$ we have $e^{\phi_x}f = e^{D_x t}f$. This only makes sense locally in general. For non-groups we can do something silly and just write $e^{Dt}f = He^{Dt}gH^{-1}$ by a homotopy that keeps the images constant. As an example, on $M=\mathbb R^2$ we can locally take $f(t) = (t,1)$ to $g(t) = (e^t,1)$ which goes to $g(t+1) = (e^{t+1},1)$ and then maps back to $f(t+1) = (t+1,1)$. Now we've actually made sense of our formula in all of $C^1(\mathbb R, M)$ by choosing a good family of homotopies to glue together. We can also extend $e^\phi$ in the same way but our equation only makes sense globally if we make the same choices of homotopies for both maps which in general may not exist. Do these choices only work in the case where $\phi$ is complete, i.e. do we recover the normal theory of ODEs on manifolds when we try to use this exponential flows? This theory bothers me in a few ways: (1) There is no use of the spectral calculus of the derivative. If we think about "matrix-valued eigenvalues $\lambda$" the fact that $e^{\lambda t}f(t_0)$ is an eigenfunction for $D$ with eigenvalue $\lambda$ solves our equation. Moreover in our invariant $\lambda$-subspace we have $e^{\lambda t} f(x) - f(x+t) = 0$. An incredibly nonrigorous and vague application of spectral calculus gets us that $e^{Dt} f(x) - f(x+t) = 0$. This is all just gesturing because $M_n$ and $C_1$ are not fields! Moreover $D$ is not a normal operator and so it shouldn't seem to have a nice spectral theory? (2) This does not hold for difference equations! $S(f(n)) = Af(n)$ is solved by $A^n$, and in the same vague sense as above the fact that the eigensequences of S look like $\lambda^n$ suggests the fact $S^n f(x) = f(x+n)$. But $\mathbb Z$ is not a lie group, so clearly there must be something more general going on here! Additionally, in both these examples the "matrix-valued spectrum" comes from the complex-valued spectrum. (3) The use of homotopies here seems arbitrary and annoying, why should I need to make all of these choices for what seems to be an incredibly natural equivalence between infinitesimal generation and propagation? They just get canceled out anyway? More to the point $e^{Dt}$ is weird in that even if it can explicitly be calculated on only very few functions it seems to have a natural extension to any map out of any monoidal object in some space of nice categories. Both of these suggest the use of a sort of "geometric reasoning". This is also talked about apocryphally, in random chats over coffee: Our differential equation asks us to calculate an invariant subspace of $C^1$ as a $M_n[x]$-module given by $p(D)$ action. In particular the support of $C^1$ on the prime ideals of $M_n[x]$ is given precisely by the spectrum of the operator $D$. Localizing to the prime ideal $I(U)$ we get a sheaf of local functions on the spectrum. Note that in finite dimensional linear algebra terms if our module structure on $V$ is given by $T$-action, the stalks of these sheaves $M[x]_{(x-\lambda)} \otimes V$ are precisely the eigenspaces associated to $\lambda$. The cotangent bundle at each stalk are the vectors that have multiplicity 2. In terms of banach algebras the spectrum of an element $\sigma(a)$ also matches up but with a continuity condition: the characters of the algebra are continuous maps to the spectrum of the ring by the ideal generated by $\lambda$, each prime ideal acts on an element $a$ sending $V$ to its localization by $a$-action. (In the noncommutative setting this also works since irreps of V correspond to the primitive ideals). Now back on the example of $C^1$, on each stalk of this sheaf we actually have a space of bounded operators which does contain the map $e^{Dt}$. However the space of bounded operators on $C^1$ is not a sheaf! But if we sheafify the presheaf we get a global operator "$e^{Dt}$". Note that this seems to create objects that are in $M_n[[x]]$, which makes sense since it's a more "local" object than the polynomial ring. The fact that this matches up with a sort of spectral theory makes sense, because the operators we create in this sheafification are precisely those that only match up locally with the spectral theory! Moreover, this construction seems to actually work in a some space of abstract categories where $C^1$ is replaced with maps $[X,Y]$ where $X$ is a monoidal object and $D$ is given by the action of $x\in X$ on $f$. If we can put some sort of polynomial structure on the multiplication map we can now write out this task in full generality. My question is whether this sheafification map matches up above with a choice of homotopy for a Lie Algebra? It should, but it's not clear to me that it does. Sheafification is natural as the adjunct to the presheaf inclusion, but it's not clear to me that the manifold construction works in the same way since we make so many choices. Also, in the case of an ODE, are there any other differential operators added in the sheafification process? It seems like I should be able to make sense of any analytic function of D in this way but that seems too general? It's not clear to me that the sheaf we construct should actually have any bearing on our original ODE, but clearly $e^{Dt}$ actually does help make a solution. Lastly, is there an easier way to do this, this seems like way too much for a simple idea! These ideas come up vaguely in discussions all the time - there must have been serious attempts at formalizing them, are there any references? I assume this is related to constructions in synthetic geometry but I'm not sure! I don't know who to ask about this! Addendum: for $Df(t) = A(t)f(t)$ we can note that the $A(t)$ spectrum of $D$ does not satisfy $e^{A(t)}f(x) = f(x+t)$, instead it satisfies $e^{\Omega(t,x,A)}f(x) = f(x+t)$ via magnus formula, so our "global theory" tells us that we should be able to solve $Df = Af$ by $e^{Dt}f = e^{\Omega(t,x,A)}f$.  |
| Decomposition of one $L^p$ space into two other . Posted: 26 Sep 2021 08:04 PM PDT $\mathbf {The \ Problem \ is}:$ If $1 \leq p\lt q \lt r \lt \infty,$ and $f \in L^q(\mathbb T)$ ; does there exist $g$ in $L^p$ and $h \in L^r$ such that $f = g+h ?$ $\mathbf {My \ approach}:$ Note, as $\mathbb T$ has a finite measure, then $L^1$ is the biggest space . Then, if $f \in L^q$ then trivially, $f = g+h$ where $g=f \in L^p$ and $h=0.$ Does there exist a non-trivial decomposion of functions in $L^q(\mathbb T)$ space into that of $L^p(\mathbb T)$ and $L^r(\mathbb T)?$ A small hint is warmly appreciated, thanks ij advance .  |
| About the inequality $f(-x)<\left(\Gamma(1-x)\right)^{\frac{1}{1-x}}$ Posted: 26 Sep 2021 08:14 PM PDT Prove or disprove that $-1< x<0$ then we have : $$f(-x)<\left(\Gamma(1-x)\right)^{\frac{1}{1-x}}$$ Where : $$f(x)=2-x^{-\frac{212}{1000}x^{x^{\frac{1}{5}x^{2x^{\frac{1}{5}x}}}}}$$ My motivation : To describes it I just use words . Can we approximate a curve using power towers and also somes coefficient ? In my example (if true) the two functions seems to have no link .So is it feasible in the exotic maths world ? So let's try this example ! My attempt : I have tried basic manipulation .We can delete the coefficient $2$ wich are not in the power tower using a new and positive coefficient (here $-\frac{212}{1000}$) because there is a symmetry with $y=1$.Next I have tried power series but it becomes very complicated because the inequality is tight and we needs a sufficiently large ordrer wich is a negative point . Question : How to (dis)prove it ? Thanks in advance ! Ps:feel free to copy/paste it in Desmos .  |
| How do I solve this limit with an absolute value? Posted: 26 Sep 2021 07:57 PM PDT $$\lim_{x\to 2}\frac{|x-2|}{2x-x^2}$$ I know the answer of the left hand limit is $1/2$; while the right hand limit is $-1/2$. But I don't understand how do you get that? If I factor $-x$ from the denominator, I'll get $(-2+x)$ which cancels out with the numerator. Then I'll get $1/-x$. If I plug in the limit of 2 from the left hand, it would be 1/2. Wouldn't it also be 1/2 from the right hand, as well? I'm getting confused on how to work this out. Please help, thank you!  |
| Solve a system of two equations in three unknowns locking the param to integer number Posted: 26 Sep 2021 08:14 PM PDT I have a system as follows $$6 + 4n = x,\quad 8 + 3m = x.$$ now I know I can get to the point where: $$n = (2 + 3m)/4,$$ and I can parametrize $m$ as $z$ to get all possible values in $R$. What I want to do instead is to lock down $n$ and $m$ to be integer numbers instead of real numbers. And I'd like to get the very first number if that exist. In the example above $m = 1$ wouldn't work since $n$ would be $1.25$. However $m = 2$ would do the trick and would lead to $n = 2$. I wonder, how do I set in the system this constraint? Edit: I guess one idea there could be to round up to the next integer by doing something like $$m = 1,\quad n = (2 + 3 * 1 + 4 - 1)/4.$$ Would that be correct?  |
| Posted: 26 Sep 2021 08:08 PM PDT How do one get the equation below in the proof in [Chow 1999]? Chow writes on page 444 - 445: Definition. A tower is a finite sequence $A=(\alpha_1,\alpha_2,...,\alpha_n)$ of nonzero complex numbers such that for all $i\in\{1,2,...,n\}$, there exists some integer $m_i>0$ such that $\alpha_i^{m_i}\in A_{i-1}$ or $e^{\alpha_im_i}\in A_{i-1}$ (or both). A tower is reduced if the set $\{\alpha_i\}$ is linearly independent over $\mathbb{Q}$. If $\beta\in\mathbb{C}$, then a tower for $\beta$ is a tower $A=(\alpha_1,\alpha_2,...,\alpha_n)$ such that $\beta\in A_n$. until now, I only see: Where does the above linear equation come from? [Chow 1999] Chow, T.: What is a closed-form number. Am. Math. Monthly 106 (1999) (5) 440-448  |
| Posted: 26 Sep 2021 07:57 PM PDT [Problem]
I am in high school, but my foundation in geometry is a little shaky due to Quarantine cutting off the semester. This is a question from my states mathematics competition test. I already know the answer, but would greatly appreciate it if someone could explain how to do this question. Also, if someone could link to some online materials or physical textbooks that I could use to strengthen my geometry?  |
| Quadratic formula in differential equations Posted: 26 Sep 2021 08:13 PM PDT
The solution of this differential equation involves using the quadratic formula for a quadratic in terms of $y'$ but I'm a bit bothered that we get a $ \pm$ when we do that: $$y' =- \frac12 \pm \sqrt{\frac{4y}{x} +1} \tag{1}$$ And then we could do $y=xt$ and solve but how exactly do we understand the plus or minus quantity which we get in step-1? It seems that the procedure of completing the quadratic formula generates two differential equation which solves the one in (0). So, should I solve both ones and the actual solution for (0) is a linear combination of both?  |
| Are these "finite-ish" sets closed under union? Posted: 26 Sep 2021 07:57 PM PDT Now asked at MO. Throughout, we work in $\mathsf{ZF}$. Say that a set $X$ is $\Pi^1_1$-pseudofinite if for every first-order sentence $\varphi$, if $\varphi$ has a model with underlying set $X$ then $\varphi$ has a finite model. (See here, and the answer and comments, for background.) Every $\Pi^1_1$-pseudofinite set is Dedekind-finite basically trivially, and with some model theory we can show that every amorphous set is $\Pi^1_1$-pseudofinite. Beyond that, however, things are less clear. In particular, I noticed that I can't seem to prove a very basic property of this notion:
I'm probably missing something simple, but I don't see a good way to get a handle on this. A structure on $X=A\sqcup B$ might not "see" that partition at all, and so none of the simple tricks I can think of work.  |
| Comparing the cardinalities of generic $\mathbb{R}$s Posted: 26 Sep 2021 07:59 PM PDT This is yet another question about cardinalities in forcing extensions of models of $\mathsf{ZF+\neg AC}$ (see also here). Specificially, I think I've isolated the simplest question which I can't yet answer. I suspect this is actually quite easy and I'm just having a silly moment, but currently I don't see the argument:
Note that a negative answer to the above-linked question in the case of Cohen forcing would give a positive answer to this question. "Obviously" the answer should be yes, but I don't see how to prove that. We have a set $\mathcal{X}\in V$ of names for reals such that every real in a Cohen extension is named by some element of $\mathcal{X}$, and each Cohen real $a$ induces an equivalence relation $\sim_a$ on $\mathcal{X}$ as $\nu\sim_a\mu\leftrightarrow\nu[a]=\mu[a]$. So really this question is asking for a bijection in $V[c,d]$ between $\mathcal{X}/\sim_c$ and $\mathcal{X}/\sim_d$. Intuitively this should exist since Cohen forcing is as homogeneous as one could hope; however, in the absence of choice in $V[c,d]$ I don't actually see how to build one. EDIT: While it's not my main question, I'd also be interested in a weak negative result: are there $M\models\mathsf{ZF}$ and mutually-Cohen-over-$M$ reals $c,d$ such that $M[c,d]\models \mathbb{R}^{M[c]}\not\equiv\mathbb{R}^{M[d]}$? I am really interested in the determinacy case, but any sort of pathology like this would be quite cool. To preempt one natural attempt, note that Cohen forcing kills determinacy so we can't use determinacy in $V[c,d]$ even though we have it in $V$. While at first glance this might appear to contradict (say) the generic absoluteness of the theory of $L(\mathbb{R})$ given large cardinals, there is no discrepancy since $(L(\mathbb{R}))^V[G]\not=(L(\mathbb{R}))^{V[G]}$ in general.  |
| Solutions to the Laplace Equation $\Delta u =0$, where $u= \log p$ Posted: 26 Sep 2021 08:08 PM PDT Find all real solutions to the two dimensional Laplace equation $U_{xx} + U_{yy} =0$ of the form $u=\log p(x,y)$, where $p$ is a quadratic polynomial. Solution: Let $p(x,y) = Ax^2 + By^2 +Cxy + D$ be a quadratic polynomial such that $A, B \not= 0$. Then $$U_{x} = \frac{2Ax + Cy}{\ln(10)(Ax^2 + By^2 +Cxy + D},$$ $$U_{xx} = \frac{2A ln(10)(Ax^2 + By^2 +Cxy + D) - \ln(10)(2Ax + Cy)^2}{\ln(10) (Ax^2 + By^2 +Cxy + D)^2},$$ $$U_{y} = \frac{2By + Cx}{\ln(10)(Ax^2 + By^2 +Cxy + D)},$$ $$U_{yy} = \frac{2B \ln(10)(Ax^2 + By^2 +Cxy + D) - \ln(10)(2By + Cx)^2}{\ln(10) (Ax^2 + By^2 +Cxy + D)^2}.$$ This implies $$U_{xx} + U_{yy} = \frac{2A \ln(10)(Ax^2 + By^2 +Cxy + D) - \ln(10)(2Ax + Cy)^2}{\ln(10) (Ax^2 + By^2 +Cxy + D)^2} + \frac{2B \ln(10)(Ax^2 + By^2 +Cxy + D) - \ln(10)(2By + Cx)^2}{\ln(10) (Ax^2 + By^2 +Cxy + D)^2} = 0. $$ I feel like I am not doing this right. Is there a simpler way? Thanks. And also how do I find such solutions.  |
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