Recent Questions - Mathematics Stack Exchange |
- Draw a Graph from a Given Degree Sequence
- How to prove that the graphs of continuous mappings are homeomorphic to each other (set of subsets of topological space / homeomorphic)
- What does the symbol ⩫ stand for in the context of "similar to a distribution"?
- If X is connected, then show that no two points with disjoint open neighborhoods can have a retract of X.
- Norm equivalence in a Banachspace
- Lower bound for the "normalized scaling coefficient" of a matrix
- Determining vector that has known dot product with another vector, with a max on the taxicab distance
- $f$ is a square-integrable function, $ \|f(x+h)-f(x)\|_{L^2}=O(h^{1+\alpha}), h\rightarrow 0. $
- Existence and Uniqueness of Nonlinear ODE
- Prove that sequence converges
- Applying the formalism of common knowledge to a simple example: Is "$Y=3$" common knowledge among the boy, the robot and the girl?
- Time and Distance Problem : Circular Track
- is $det(ABA)$ $= det(B)$?
- $L_p$ and measure convergence
- Using value outside galois field and do calculations inside
- Measurability of the integral of Brownian motion
- About the minimum of the Gamma function on $(0,1)$
- Expected number of tosses until showing differently from first result
- Calculus : integration of inverse function
- How to compactly write a simple set for 2 variables when second variable may have same properties as first or it may optionally be positive infinity?
- Taylor expansion of $\sin \pi z$ at $z = -1$.
- Find the perimeter of a polygon $ABCDEF$
- When is the projection of an ellipsoid a circle?
- Stone–Čech compactification of a Tychonoff space can be taken as closure of diagonal mapping image in Tychonoff cube
- Can we figure out a distribution function for how long does it take for the following system to terminate? [closed]
- Solution to Nonlinear ODE $s''( t) s( t) =( s'( t))^{2} +B( s( t))^{2} s'( t) -g\cdot s( t) s'( t)$
- Valuations with the same valuation ring
- If $A$ is a $5 \times 5$ complex matrix with $A^4=A^2 \neq A$, then what are its possible minimal polynomials?
- Seeking elegant proof why 0 divided by 0 does not equal 1
- tensor product of sheaves commutes with inverse image
| Draw a Graph from a Given Degree Sequence Posted: 11 Dec 2021 03:52 AM PST I want to prove that this degree sequence $(5,5,5,2,2,2,1)$ isn't valid to draw a graph from it, the graph doesn't need to be simple. I am looking for a Theroem or a way to contradict the assumption that we can make a graph from it. My solution was the following, for the given nodes:degrees => $(A:5; B:5; C:3; D:2; E:2; F:2; G:1)$ Note that the vertex $C$ is the one that makes the contradiction, since we should have another 2 extra edges, but we can't add them to the previous nodes. So my question is: Is there any theorem which I can use to prove this contradiction? Because I feel like my solution isn't enough. |
| Posted: 11 Dec 2021 03:52 AM PST I need to prove that the graphs of all continuous mappings of the same space are homeomorphic to each other: |
| What does the symbol ⩫ stand for in the context of "similar to a distribution"? Posted: 11 Dec 2021 03:46 AM PST I came across this symbol (⩫) in a set of notes about constructing a confidence interval for the mean difference $\mu_X - \mu_Y$ for two Normal distributions $X$ and $Y$ with unknown population means, unknown population variances and no assumption about the equality of said variances.
$t_v$ denotes is the t distribution with $v$ degrees of freedom. This "result" was given to me without any explanation and my lecturer dismissed the meaning behind the symbol, stating that I can simply treat it as having the same meaning as $\sim$. Surely the meaning of the two notations must be slightly different? |
| Posted: 11 Dec 2021 03:35 AM PST Let $X$ be a connected space and $x_{0} , x_{1} \in X$ be two points of $X$ which have disjoint open neighbourhoods in $X$. Show that $A$={ $x_{0}, x_{1}$ } can never be a retract of X. We know that the Reverse image of any open set is open under the continuous mapping. But I need a proof process. (Question source: Algebraic Topology A Primer -Satya Deo - Chapter 2 (Section 3) - Exercise 2) |
| Norm equivalence in a Banachspace Posted: 11 Dec 2021 03:49 AM PST Let $E$ be a Banachspace with two norm on it: $\|.\|_1, \|.\|_2$ such that $\forall x\in E: \|x\|_1 \le \|x\|_2$. Prove that the norms are equivalent. I was working on some functional analysis exercises and couldn't solve this one. The exercise gave a hint to use Banach-Schauder open mapping theorem:
I don't see how I can use this hint for this exercise. Any help would be appreciated. |
| Lower bound for the "normalized scaling coefficient" of a matrix Posted: 11 Dec 2021 03:30 AM PST Let $A:X\rightarrow Y$ be a linear operator between real, finite dimensional vector spaces. Are there any known non-zero lower bounds for the expression, in case $A$ has some "structure"? $$ \inf_{x\neq 0}\frac{||Ax||_2^2}{||x||_2^2} \quad ? $$ If we don't assume any structure on $A$, obviously one could pick $A$ to be the zero matrix in which the infimum would be $0$, so being non-zero is some "structure" that $A$ should have. Perhaps also there are some dual results to the $\sup$-version of this expression, which is the operator norm? |
| Posted: 11 Dec 2021 03:30 AM PST Is there a way to determine a vector $\vec{x}$ such that its dot product with a known vector is a certain value, when constricted that the taxicab distance is a certain value? Find $\vec{x}$ such that $\vec{x}\cdot\vec{y} = a$ Another way to look at it: Let $c_i \in N$ for $i \in [1, 2, ..., n]$ Let $total \in N$ Let $b \in N$ Find $x_i \in N$ for $i \in [1, 2, ..., n]$ Such that: $$ \sum_{i=0}^{n}{x_i} = b \\ \sum_{i=0}^{n}{x_i \cdot c_i} = total $$ Lets say you are at a restaurant that sells different pretzels in boxes. Due to the difference in the kind of pretzel, all boxes are equally sized, but with a different amount of pretzels inside. You want exactly $total$ pretzels, divided over $b$ boxes. How many boxes of which kind of pretzel should you take? Example: $$ c_1 = 604 \\ c_2 = 4500 \\ c_3 = 8111 \\ c_4 = 58516 \\ c_5 = 213 \\ c_6 = 4343 \\ c_7 = 8398 \\ c_8 = 7029 \\ b= 20 \\ total = 268974 $$ Answer: $$ \begin{align} \sum_{i=0}^{n}{x_i} =&\space b \\ 2 + 3 + 1 + 3 + 1 + 2 + 4 + 4 =&\space 20 \\ \sum_{i=0}^{n}{x_i \cdot c_i} =&\space total \\ 2 \cdot 604 + 3 \cdot 4500 + 1 \cdot 8111 + 3 \cdot 58516 \quad & \\ + 1 \cdot 213 + 2 \cdot 4343 + 4 \cdot 8398 + 4 \cdot 7029 =&\space 268974 \end{align} \\ $$ Therefore $$ x_1 = 2, x_2 = 3, x_3 = 1, x_4 = 3, x_5 = 1, x_6 = 2, x_7 = 4, x_8 = 4 $$ Is a solution Note: I found this solution using brute force. I'm looking for an analytical way. |
| $f$ is a square-integrable function, $ \|f(x+h)-f(x)\|_{L^2}=O(h^{1+\alpha}), h\rightarrow 0. $ Posted: 11 Dec 2021 03:28 AM PST
Let $F(x)=\int_0^x f(t)\mathrm{d}t$, I can prove that $$ F(x+h)-F(x)=o(h^{1/2}), h\rightarrow 0, $$ but it seems have no relation with the problem. I have no idea how to start. Any help! Thanks. |
| Existence and Uniqueness of Nonlinear ODE Posted: 11 Dec 2021 03:25 AM PST What are the conditions on the non-linear functions $f$ and $g$, for the system $$\ddot{x}+(1+g(\dot{x},t))\dot{x}+f(x) = 0,$$ to have global solutions for all $t\geq 0$? |
| Posted: 11 Dec 2021 03:50 AM PST I have the following sequence: $u_n = \frac{1}{9} \cdot (10 - 0.1^n)$ and I want to prove that it converges to $\frac{10}{9}$. I know that I must show For an arbitrary $\epsilon > 0$ there is an $N > 0$ such that if $n > N$ then $|u_n - \frac{10}{9}| < \epsilon$ And I have tried this multiple times, however I always get a result where $N$ ends up being $< 0$ for some $\epsilon > 0$. Can someone show how to correctly prove it? Edit: I get to $0.1^n < 9 \epsilon$ but after taking the log, I get $n > -log_{10}(9 \epsilon)$ which is negative for certain values for $\epsilon$ |
| Posted: 11 Dec 2021 03:39 AM PST I am trying to understand the formalism of common knowledge. The sense behind the concept is already well explained here, however I struggle with linking the formalism to a practical use case. When reading the definitions it first appears straight forward: We have a set of states $S$, an event $E\subseteq S$ as a subset of these states and a partition $P_i=\{\{\ldots\},\{\ldots\},\ldots,\{\ldots\}\}$ of $S$ representing the knowledge of a decision maker $i$ in a state. Here I am not quite clear how I should practically understand the partition's elements (i.e. these subsets of $S$). The puzzle pieces seem to fall into place when reading on: In state $s\in S$, decision maker $i$ knows that one of the states in $P_i(s)$ occurs, but he doesn't know which one. Here $P_i(s)$ is the unique element in $P_i$ that contains $s$. The knowledge function $K_i(e)=\{s\in S|P_i(s)\subset e\}$ is the set of states (a subset of the event $e$), where the decision maker knows that event $e$ occurs. In other literature, instead of the proper subset, the improper subset is also used: $K_i(e)=\{s\in S|P_i(s)\subseteq e\}$. The operator for the idea "everyone knows $e$" is defined by intersecting the knowledge of all decision makers $i$ as follows: $E(e)=\bigcap_{i}K_i(e)$. Iterating the function $E$ is understood as the well known function composition $E^1(e)=E(e)$ and $E^{n+1}=E(E^n(e))$. Finally the Common Knowledge function is given by: $$C(e)=\bigcap_{n=1}^{\infty}E^n(e)=E^1(e)\cap E^2(e)\cap E^3(e)\ldots$$ So good so far, but when trying to applicate this formalism to a practicla use case, I'm missing the right impetus. I found a very simple and illustrative example in this YT video "Is It Common Knowledge?" (by James Miller):
I would really appreciate it if someone can help me how to use the (admittedly simple) formalism to describe this use case. My rough ideas are:
I would be grateful if you could help me put the puzzle pieces together. |
| Time and Distance Problem : Circular Track Posted: 11 Dec 2021 03:32 AM PST
Now I have found that $A$ and $B$ will meet at the starting point $O$ after $LCM(\frac{L}{6x},\frac{L}{x})=\frac{L}{x}$ hours where $L$ is the length of the track. Thanks in advance !!! |
| Posted: 11 Dec 2021 03:49 AM PST Square matrix A, B A is inventible. if $det(ABA)=0$ it can't be also $det(B)=0$ is this right? Because, only $\begin{align}\det(ABA^{-1}) &= \det(B) \end{align}$ and $A \neq {A}^-1$ So then, is $\begin{align}\det(ABA) &= \det(B) \end{align}$ a false statement? |
| Posted: 11 Dec 2021 03:45 AM PST Consider the Lebesgue measure. I'm asked to study the $L_p$ and measure convergence of the following functions:
Therefore, $g_n$ doesn't converge to $0$ in $L_p$. But I don't know how to study the convergence in measure. Any hint? Thanks! |
| Using value outside galois field and do calculations inside Posted: 11 Dec 2021 03:48 AM PST currently, I'm working Shamir Secret Sharing algorithm and for not so big numbers (long passwords more than 7 chars) my calculations are broken because of overflows. The case is this: I have a big number: suppose 49584309583 and if I use it in calculations eventually be lost of precision so I've heard that calculations can be done in Galois field for example in GF(251^1) So the question is: can I get my number to be converted back from GF to outside GF? |
| Measurability of the integral of Brownian motion Posted: 11 Dec 2021 03:33 AM PST Hi once again a measurability question about Brownian motion. I'm not that familiar in how to prove that a function is measurable rigorously, so I'm stuck at the following exercise: Show that for $T>0$ the mapping $$\omega \rightarrow X_T(\omega) = \int_{0}^{T} B_t(\omega) dt$$ is measurable. The Brownian motion is therefore assumed to be jointly measurable with continuous paths. How can I prove this? I know the definition of measurability that the preimage of measurable sets are measureable. But what are here the measurable sets in the image space. And how to proceed from there? Any help is really appreciated. |
| About the minimum of the Gamma function on $(0,1)$ Posted: 11 Dec 2021 03:46 AM PST Problem :Denotes by $x_{min}=k$ the minimum of the Gamma function $x!$ on $(0,1)$ then prove or disprove that : $$\left(e^{-\frac{k^{2}}{C^2}}\right)!>k$$ Where $C=-1+\frac{1}{\ln(3)}+\ln(3)$I cannot find an attempt because the problem of the minimum of $x!$ is really hard . In consequence I have tried numerical tools and it seems true for the first five digit ($0.8856$) Question : How to (dis)prove it ? Have you seen this equation before in the litterature ? Thanks ! |
| Expected number of tosses until showing differently from first result Posted: 11 Dec 2021 03:48 AM PST What is the expected number of coin tosses (heads comes with probability 0.5) until we get a result which is different from the first toss? My first intuition is 3: we need to toss once, and then we have a geometric distribution with expectation of 2. But, there are 2 possible scenarios: one scenario is when the first toss is heads and the other scenario is when the first toss is tails. So, this might suggest that the expectation is less than 3.... A bit confusing... PMF I got:
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| Calculus : integration of inverse function Posted: 11 Dec 2021 03:49 AM PST In this problem $\frac pq = \frac {139}4 $, $p+q=143$ But what if, there is another case that satisfies conditions but function f(x) is not differentiable at somewhere or all in [1,8]? ( this problem is Korea's KSAT number 30 problem for the selection area of calculus.) |
| Posted: 11 Dec 2021 03:47 AM PST It's been about 40 years since learning higher math like partial differential equation and linear algebra including simple tensors but never needed much of it. Now I'm working on a Wolfram Mathematica programming project that also requires describing solution sets in proper math notation. Below is what I have come up with and I am not even sure if I used all math symbols correctly. For example I'm not certain I wrote And correctly (in programming $$\{n,h \in \mathbb{Z} \geq 0 \lor n \in \mathbb{Z} \geq 0 \land h \to +\infty\}$$ Please help this old geezer out... (1.) I need most compact form (less is more) for this set making certain all math symbols are written correctly by most current standards. If set is written correctly and cannot be compacted more without losing meaning please comment below validating this. Thank you for sharing your expertise. Item 2 may be ignored but I leave it as a reference to myself. (2.) Why isn't my fat Z (written as $\Z$) for the set of all Integers rendering correctly? I used a Latex source for that. Should I have instead referenced a Tex source? Nevermind I figured out how to write $\mathbb{Z}$ and found the greatest tutorial right here in SE Math Meta MathJax |
| Taylor expansion of $\sin \pi z$ at $z = -1$. Posted: 11 Dec 2021 03:32 AM PST Taylor expansion of $\sin \pi z$ at $z = -1$ is $$\sin\pi z = -\sin(\pi(z+1)) = -\sum_{n=0}^\infty \frac{(-1)^n\pi^{2n+1}}{(2n+1)!}(z+1)^{2n+1}$$ so that $$\sin\pi z = \sum_{n=0}^\infty \frac{(-1)^{n+1}\pi^{2n+1}}{(2n+1)!}(z+1)^{2n+1}. \tag{$\dagger$}$$ But if I try this \begin{align} \sin\pi z & = \sum_{n=0}^\infty \frac{(-1)^n\pi^{2n+1}}{(2n+1)!}z^{2n+1} \\& = \sum_{n=0}^\infty \frac{(-1)^n\pi^{2n+1}}{(2n+1)!}(z+1-1)^{2n+1} \\ & = \sum_{n=0}^\infty \frac{(-1)^n\pi^{2n+1}}{(2n+1)!}\left(\sum_{k=0}^{2n+1}\binom{2n+1}{k}(z+1)^k(-1)^{2n-k+1}\right).\tag{$\dagger^*$} \end{align} In this case, how can I reduce $(\dagger^*)$ as the above form $(\dagger)$? Just direct calculation? |
| Find the perimeter of a polygon $ABCDEF$ Posted: 11 Dec 2021 03:36 AM PST A circle, with a radius of $12$ cm and with the center coinciding with the center of an equilateral triangle with a side of $36$ cm, intersects the sides of the triangle at points $A, B, C, D, E$ and $F$. Find the perimeter of the polygon $ABCDEF$. Image of the question:
I solved it as follows: I proved that this polygon is a regular hexagon, found its side, it is $12$ and then the perimeter is $6⋅12=72$. I was confused by the simplicity of the task. Did I solve the problem correctly? |
| When is the projection of an ellipsoid a circle? Posted: 11 Dec 2021 03:35 AM PST Consider an ellipsoid in the three dimensional Euclidean space, say $$\frac{x^2}{a^2}+\frac{y^2}{b^2} + \frac{z^2}{c^2} =1 $$ where $a$, $b$, $c$ are positive reals. I'm counting the number of planes through the origin so that the image is a perfect circle. There may be divergent cases if we consider the case that some of $a$, $b$, $c$ are coincide. But at first, let us focus on the case that $a$, $b$, $c$ are all different, say $a>b>c$. I guess the answer would be $4$. I have made many efforts but failed. What I have observed is the that at least two such planes exists and the radius of the circle is $b$. Just consider rotating plane possesses $y$ axis and apply intermediate value theorem. Causion! We are concerning projection, not intersection. PS. Now I guess there are infinitely many... |
| Posted: 11 Dec 2021 03:27 AM PST Let $X$ - Tychonoff topological space. Show that Stone–Čech compactification of $X$ can be obtained by taking the closure of the image of the space $X$ under the mapping $\Delta_{f\in C(X, I)}f$ in the space $\prod_{f\in C(X, I)}I_f$, where
$$\Delta_{f\in C(X, I)}f: X\to \prod_{f\in C(X, I)}I_f \;$$ $$\Delta_{f\in C(X, I)}f: x\mapsto \{f(x)\}_{f\in C(X, I)}\in\prod_{f\in C(X, I)}I_f \; .$$
I will be glad for any idea, comment, hint or advice. |
| Posted: 11 Dec 2021 03:23 AM PST Suppose we simulate the following system: In the beginning there are $N$ robots and a large reserve of fuel: $T$ units of fuel in total. Each robot takes some amount of fuel to start out. They consume fuel at a constant rate: 1 unit / second - regardless of what they are doing. If a robot runs out of fuel, it loses (dies in a sense) and it doesn't affect the system anymore. The robots can access the reserve and they can take some amount of fuel for themselves. The amount of elapsed time between refuels follows an exponential distribution with rate $\lambda$. The amount of fuel the robots take at any given occasion follows a uniform distribution ~ $U(0, F)$. If there is only 1 robot remaining then it gets all the remaining fuel. For the sake of simplicity, let's assume that taking/transfering fuel happens pretty much instantly. Let's also assume that the robots can store and use any amount of fuel. No additional fuel or robots are added during the simulation, no parameters are changed, only the initial values are given: $T, N, \lambda, F$. Is it possible to state anything about the distribution of the termination time in terms of these parameters? The main motivation was to figure out for how long would it take to live up some sort of resource that's nonrenewable and is necessary for life. I've tried to define a function for the remaining fuel based on the time points when robots die. Let $d_k$ denote the times of deaths where $k \in [1;n-1]$ and $0 \le d_1 \le d_2 \le ... \le d_{n-1}$. Following $r(t)$ shows the remaining fuel in the system at time point $t$: $r(t)=\begin{cases}T-Nt & 0 \le t \le d_1\\ T-(N-1)t - d_1 & d_1 \lt t \le d_2 \\ ...\\ T-2t - d_1 - ... - d_{n-2} & d_{n-2} \lt t \le d_{n-1} \\ T-t - d_1 - ... - d_{n-2} - d_{n-1} & d_{n-1} \lt t \le T - d_1 - ... - d_{n-1} \\ \end{cases}$ However, this didn't really bring me closer to anything. |
| Solution to Nonlinear ODE $s''( t) s( t) =( s'( t))^{2} +B( s( t))^{2} s'( t) -g\cdot s( t) s'( t)$ Posted: 11 Dec 2021 03:49 AM PST I've solved linear ODEs before. This however is something completely new to me. I want to solve it without using approximations or anything. $s''( t) s( t) =( s'( t))^{2} +B( s( t))^{2} s'( t) -g\cdot s( t) s'( t)$ These are the equations I started with $ \begin{array}{l} s'( t) =-Bs( t) i( t)\\ r'( t) =g\cdot i( t)\\ i'( t) =i( t)( Bs( t) -g) \end{array}$ The reason I'm solving this particular equation is because I want to solve for all of the functions $(i(t),r(t),s(t))$ from above. I just figured $s(t)$ might be the place to start. I really don't have any clue where to start here. Any help would be awesome! Thanks! --edit-- I made a mistake above. In order to fix it I changed $i'( t) =i( t)( Bs( t) -1)$ to this $i'( t) =i( t)( Bs( t) -g)$. My bad... ---Edit--- I might have a step towards the answer. $ \begin{array}{l} v( s) =s'( t)\\ \Longrightarrow \\ s''( t) =\frac{dv( s)}{dt} =\frac{dv}{ds}\frac{ds}{dt} =v'( s) \cdot \frac{ds}{dt} =v'( s) \cdot v( s)\\ \Longrightarrow \\ v'( s) \cdot v( s) s=v( t)^{2} +Bs^{2} v( t) -g\cdot s\cdot v( t)\\ \Longrightarrow \\ v'( s) \cdot s=v( t) +Bs^{2} -g\cdot s \end{array}$ Wolfram Alpha says that the solution to this differential equation is $v(s)=Bs^2+c_1s-gsln(s)$ So that means that $s'(t)=Bs(t)^2+c_1s(t)-gs(t)\ln(s(t))$ This is certainly better than before |
| Valuations with the same valuation ring Posted: 11 Dec 2021 03:20 AM PST Given a field $k$, a valuation map $v$ is a surjective group homomorphism $k^\times\twoheadrightarrow H$, where $H$ is an ordered group (called the valuation group), such that $v(a+b)\ge \operatorname{min}(v(a),v(b))$. I must prove that if I have another valuation $v':k\twoheadrightarrow H'$, with the property that $v(a)\ge 0\iff v'(a)\ge 0$, then there is an order-preserving isomorphism $f:H\to H'$ such that $v'=f\circ v$. Firstly, observe that, with $a,b\in k$, $v(a)\ge v(b)$ if and only if $v'(a)\ge v'(b)$: $$v(a)\ge v(b)\iff v(ab^{-1})\ge 0\iff v'(ab^{-1})\ge 0 \iff v'(a)\ge v'(b).$$ From $v'=f\circ v$ we see that on an element $h\in H$ (using that $h=v(a_h)$ for some $a_h\in k$) we must set $f(h):=v'(a_h)$. This shows that if such $f$ is a preserving order isomorphism, it will be the unique one. The observation above guarantees that $f(h)$ doesn't depend on the choice of $a_h$, and that $f$ is order-preserving. If $h,l\in H$, in order to prove that $f$ is an homomorphism: $$f(h+l)=f(v(a_{h})+v(a_{l}))=f(v(a_ha_l))=v'(a_ha_l)=v'(a_{h})+v'(a_{l})=f(h)+f(l).$$ Plus, $f$ is bijective because the map $f':H'\to H$ defined $h'\mapsto v(a_{h'})$, for $h'\in H'$ and $a_{h'}$ defined analogously to $a_h$, is the inverse of $f$. Does this outline of the proof make sense? I'm a bit suspicious because I don't find where one should use that $v(a+b)\ge \operatorname{min}(v(a),v(b))$; did I use this hypothesis without seeing it? |
| Posted: 11 Dec 2021 03:43 AM PST For this I have that $A^4-A^2=0$ so any possible minimal polynomial must divide $t^4-t^2=t^2(t-1)(t+1)$. Using the given condition I've been able to check that e.g. $A=-I$ works for $t+1$ so $t+1$ is a possible minimal polynomial, but $t-1$ is not possible since $A=I \implies A=A^2$. Similarly for $(t-1)(t+1)=t^2-1$ we can find a matrix $A$ satisfying $A^2=I$ but $A \neq -I$ so this is also possible. But there are many other possible polynomials including $t^2+t$, $t^3-t$, $t^2$, $t^3+t^2$, $t^3-t^2$ and $t^4-t^2$ that also divide $t^4-t^2$ and it seems either quite hard or very tedious to construct a $5 \times 5$ matrix satisfying one of these polynomials and checking that it doesn't satisfy any of the lower degree ones (unless it's a simple case like $t^2-t=0 \implies A^2=A$ so that one doesn't work). How do I find which of these are possible minimal polynomials? |
| Seeking elegant proof why 0 divided by 0 does not equal 1 Posted: 11 Dec 2021 03:42 AM PST Several years ago I was bored and so for amusement I wrote out a proof that $\dfrac00$ does not equal $1$. I began by assuming that $\dfrac00$ does equal $1$ and then was eventually able to deduce that, based upon my assumption (which as we know was false) $0=1$. As this is clearly false and if all the steps in my proof were logically valid, the conclusion then is that my only assumption (that $\dfrac00=1$) must be false. Unfortunately, I can no longer recall the steps I used to arrive at the contradiction. If anyone could help me out I would appreciate it. |
| tensor product of sheaves commutes with inverse image Posted: 11 Dec 2021 03:44 AM PST Let $f : X \to Y$ be a morphism of ringed spaces and $\mathcal{M}$, $\mathcal{N}$ sheaves of $\mathcal{O}_Y$-modules. Then one has a canonical isomorphism $f^*(\mathcal{M} \otimes_{\mathcal{O}_Y} \mathcal{N}) \cong f^*\mathcal{M} \otimes_{\mathcal{O}_X} f^*\mathcal{N}$, but I cannot find a proof in any of the standard references. The problem is that the definitions of the functors $f^*$ and $\otimes$ are so cumbersome that I cannot even write down a map between these two sheaves. Surely there is a nice way to do this: to give you an idea of what I mean by "nice," I am the type of person who likes to define such functors as adjoints to some less complicated functor, prove that they exist, and then forget the construction. |
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