Recent Questions - Mathematics Stack Exchange |
- Number of unique solutions for binary vector
- Orthogonality of joint probability and conditional probability measures
- Is the cardinality of a set of natural numbers is the same of the cardinality of a set of natural numbers raise to some power?
- Alternative expression for Riemann curvature tensor
- Cesaro convergence for $\{|a_n|\}$
- Example of Galois Extension E/F such that $[E:F]=2$ and $\text{char}(F)=2$
- Dominated Convergence Theorem for Weak Derivative of Heaviside Function
- Why does a particular approximation to relate epsilon and n to find the limit of a sequence work and others do not?
- an interesting question related to sequence of function wnd arzela ascoli..
- Line graph Hamilton equivalent to Eulerian in digraph
- A question on integration in a BVP
- How to prove that $f$ is a closed map?
- Idempotent relations in a ring
- Analytic representation of Euler's totient $\varphi (n)$
- Is the ratio of the perimeter of any shape with circular curves to its diameter result in an irrational number?
- In group-theory, are the elements in a set other sets or are they precise numbers?
- If $H \subset gHg^{-1}$, what can be implied from this relation?
- Why exactly do we write $\mathcal{L}(x(t),x'(t),t)$ instead of simply $\mathcal{L}(x(t),t)$?
- How we should prove that Spec(R)= Spec(C[X]) for polynomials with 1 variable for function: h(0) = h(1)? [closed]
- The certainty behind the spanning set of solution for a second order linear homogeneous ODE with constant coefficients
- Help needed for $\lim_{x→0}[(a_1^x+..a_n^x)/n]^{(n/x)}$
- Number of permutations in $[2n]$ such that every cycle has exactly one even number
- Determine the value of $\mathbb{E}[X]\mathbb{E}[1/X]$ with $X$ a random variable such that $0<a\leq X\leq b$
- A problem related to Kelvin transform from Harmonic Function Theory
- Limit problem (limit of a sum as a definite integral)
- Converting an integral equation to a differential equation
- The limit as $\epsilon \rightarrow 0+$ of $\frac{1}{\epsilon}P\big(\frac{u}{\epsilon}\big)$
- Integral $\int^\infty_0 \frac{\cos(\omega x)}{\cosh(ax)-\cos\beta}\text{d}x$
- Why $f(0)\neq 0$ where $f $ is a polynomial over the field $\mathbb F_q$ and $\deg(f)=m > 0$?
- Going down theorem fails
| Number of unique solutions for binary vector Posted: 19 Mar 2022 10:39 AM PDT This type of problem seems to be a bang-bang control type, problem, where a vector $q$ takes on a different value based on the value of a variable $t$ at a certain index. So the problem I would like some on how to solve most efficiently (so preferably not a brute force iterative method) is the following. $t \in \{[t_{0},t_{f}]| \mathbb{R}^{1 \times N}\}$ $q \in \mathbb{Z}^{1 \times N} $ $q_i=\begin{cases} \mbox{0,} & \mbox{if }t_{1} < t_{i} < t_{2} \\ \mbox{test,} & \mbox{else} \end{cases}$ Find all unique $q$'s and the range for $t$ for which they are valid for. |
| Orthogonality of joint probability and conditional probability measures Posted: 19 Mar 2022 10:39 AM PDT Suppose that $(X,Y)$ are real valued random variables on some space $(\Omega, \mathcal{F})$. Let $P,Q$ be two possible joint probability measures for $(X,Y)$. Let $P_{Y|X}$ and $Q_{Y|X}$ be two possible regular conditional probability measures for $Y|X$. I understand the following definition:
Does the orthogonality of $P,Q$ relate in any way to the orthogonality of $P_{Y|X}$ and $Q_{Y|X}$? My guess is that $P \perp Q$ implies $P_{Y|X} \perp Q_{Y|X}$ since we seem to be restricting $P,Q$ to the smaller $\sigma$-algebra $\sigma(X)$ in this case. How about the converse? Is this question even well defined, because we should really be considering $P_{Y|X = x}$ and $Q_{Y|X = x}$ for a given $x \in \mathbb{R}$. My question is motivated by these lecture notes on information theory. Particularly, the part where they start talking about conditional divergence and entropy. I see expressions like $\frac{P_{Y|X}}{Q_{Y|X}}$ used, which I'm guessing represents a Radon-Nikodym derivative. But since we're conditioning, I'm not sure what this means exactly. I don't think there's a "conditional" version of the Lebesgue decomposition out there. |
| Posted: 19 Mar 2022 10:37 AM PDT Is is possible to prove $\left|\mathbb{N}\right| = \left|\mathbb{N}\right|^n$ where $\mathbb{N}$ is the set of all natural numbers, and $n$ is an positive integer? |
| Alternative expression for Riemann curvature tensor Posted: 19 Mar 2022 10:36 AM PDT There is the usual expression for the Riemann tensor $$R^a_{bcd}=\partial_c\Gamma^a_{db}-\partial_d\Gamma^a_{cb}+\Gamma^a_{ce}\Gamma^e_{db}-\Gamma^a_{de}\Gamma^e_{cb}.$$ However, in the last page of https://www.mathi.uni-heidelberg.de/~walcher/teaching/wise1516/geo_phys/SigmaAndLGModels.pdf, another expression is used $$R^a_{bcd}=\partial_c\Gamma^a_{db}-\partial_d\Gamma^a_{cb}+g^{am}g_{fe}\Gamma^f_{md}\Gamma^e_{cb}-g^{am}g_{fe}\Gamma^f_{mc}\Gamma^e_{db}.$$ How does one obtain the first expression from the second? I've never seen the second expression before. The first one is obvious from the expression $R=\text{d}\Gamma-[\Gamma\wedge\Gamma]$. However, the second one is less intuitive. In orthogonal coordinates it would be easy to obtain since $$\Gamma_{abc}=-\Gamma_{bac}.$$ However, is there a way to see this two expression are equivalent without using orthogonality? I think it will have to do with the metricity condition $$\partial_ag_{bc}=\Gamma_{bca}-\Gamma_{cba}.$$ Bonus question: How do you do index placement in stackexchange? |
| Cesaro convergence for $\{|a_n|\}$ Posted: 19 Mar 2022 10:35 AM PDT Does $\lim_n \frac{1}{n} \sum_{k=1}^n|a_k| = 0$ imply that $\lim_n |a_n| = 0$ |
| Example of Galois Extension E/F such that $[E:F]=2$ and $\text{char}(F)=2$ Posted: 19 Mar 2022 10:34 AM PDT As seen here, given a field extension $E/F$ such that $[E:F] = 2$ it not necessarily the case that the extension is Galois if $\text{char}(F) = 2$. What is an example of a degree 2 extension where $\text{char}(F) = 2$ and the extension is Galois? |
| Dominated Convergence Theorem for Weak Derivative of Heaviside Function Posted: 19 Mar 2022 10:33 AM PDT My question primarily stems from this post and Bressan's "Lecture Notes on Functional Analysis (With Applications to Linear Partial Differential Equations)". In it, to prove that the weak derivative of Heaviside function does not exist, they used Lebesgue's dominated convergence theorem to show that for some $g\in L^1_{loc}(\mathbb{R})$, $$ \lim_{\delta\to 0^+}\int_{-\delta}^\delta |g(x)|\,dx = 0 $$ I am not perplexed about how to use DCT, but rather why. Isn't this fact obvious for any function since the integral will be over a region of zero measure in the limit (i.e. $\int_a^a f(x)\,dx=0$ for any $a\in\mathbb{R}$)? Why do we need to invoke DCT to show this fact? I'm not sure what did I miss here. |
| Posted: 19 Mar 2022 10:32 AM PDT Consider the following: $$u_n= \frac{n^2+2}{2n^2+5n+4}$$ The limit of the given sequence $u_n$ is 1/2. To show this, we consider: $$\left|u_n-\frac12\right|=\frac{5n}{2n^2+5n+4}<\frac{5n}{2n^2}<\frac{5n}{5n}<\frac{5n}{7}$$ We use the approximation 5/2n to relate n with $\epsilon$, but why can't we use the last one (viz. 5n/7)? Please explain, I am a bit confused. Also please share any further reads, if possible. |
| an interesting question related to sequence of function wnd arzela ascoli.. Posted: 19 Mar 2022 10:21 AM PDT Let fn be a sequence of equicontinous function on a compact set with also given that fn converge pointwise to a continous function f then is it true that fn converge uniformly to f on the compact sets K?? i am stuck whether the statement is true or false?? Any hint will be appreciated.. |
| Line graph Hamilton equivalent to Eulerian in digraph Posted: 19 Mar 2022 10:27 AM PDT Define the line graph $L(D)$ of a digraph $D$ as follows. The vertices of $L(D)$ are the arcs of $D$, and there is an arc from vertex $a_1$ to vertex $a_2$ in $L(D)$ iff the head of arc $a_1$ is the tail of $a_2$ in $D$. Prove that $L(D)$ has a directed Hamilton cycle if and only if $D$ has a directed Euler tour. There are many similar questions about undirected graphs. But I don't know how to do this in digraphs. |
| A question on integration in a BVP Posted: 19 Mar 2022 10:19 AM PDT I have the following BVP: |
| How to prove that $f$ is a closed map? Posted: 19 Mar 2022 10:16 AM PDT Let $X=\{(x,y) \in \mathbb{R}^2 \mid y=0 \text{ or }x\ge0\}$ and $f:X \rightarrow \mathbb{R}$ defined by $f(x,y)=x$. I want to find $C \subset X$ closed such that $f(C) \subset \mathbb{R}$ isn't closed. In other words, I'm searching a closed set in $\mathbb{R}^2$ such that its intersection with $X$ is something "strange" that doesn't give me a closed set of $\mathbb{R}$ when projected onto the $x$ axes (we are taking the euclidean topology on $\mathbb{R}$ and the subspace topology inducted by the euclidean topology on $\mathbb{R}^2$). |
| Idempotent relations in a ring Posted: 19 Mar 2022 10:39 AM PDT Let $(A,+,.)$ be a ring such that, if $x \in A$ with $6x = 0$, then $x=0$. Let $a,b,c \in A$ such that $a-b$ , $b-c$ , $c-a$ are idempotent. Prove that $a=b=c$. Unfortunately, I haven't made any big progress on this one. I noticed that $(a+b+c)^2 = 3(a^2+b^2+c^2)$ and I tried finding an expression $E$ with $6E =0$, but with no success. I haven't encountered many problems with rings, so I am quite a beginner in this area. Can you help me on this? |
| Analytic representation of Euler's totient $\varphi (n)$ Posted: 19 Mar 2022 10:18 AM PDT Let $$\pi_0 (x)=\frac12 \lim_{h \to 0}[\pi (x+h)+\pi (x-h)]$$ where $\pi (x)$ is the prime counting function. Then $$\pi_0 (x)=\sum_{n=1}^\infty \frac{\mu (n)}{n}f(x^{1/n})$$ where $$f(x)=\operatorname{li}(x)-\sum_{\rho} \operatorname{li}(x^\rho)-\log (2)+\int_x^\infty \frac{dt}{t(t^2-2)\log t},$$ and the sum is over the non-trivial zeros $\rho$ of the Riemann zeta function. This is taken from https://en.wikipedia.org/wiki/Explicit_formulae_for_L-functions. Euler's totient function $\varphi (n)$ is a number-theoretic function just as $\pi (n)$. Is there an analogous, analytic representation of $\varphi (n)$? |
| Posted: 19 Mar 2022 10:31 AM PDT Is the ratio of the perimeter of any shape with circular curves to its diameter result in an irrational number? I suppose it would depend one what would be defined as "circular curves", as well as where you would measure the diameter on a shape that is not infinitely radially symmetrical like a circle. So I'd like to keep the question open to different possibilities compared with others. What brings me to ask to the question in the first place is imagining the irrational or transcendental property emerging from curves. One scenario to consider would be a square with rounded corners. If it was a ratio of the shape's perimeter to the the diameter measured from one corner to a corner diagonally opposite of it, would it result in an irrational number?
How about the same scenario, but it's the ratio of the shape's perimeter to the diameter measured from the center of one flat side to the opposite flat side?
The reason I have not figured this out for myself is that I don't know how to get the perimeter from a square with curved corners, or where to measure from in the case where the diameter is measured from the corners. (I don't know how to get the exact points on the curve.) |
| In group-theory, are the elements in a set other sets or are they precise numbers? Posted: 19 Mar 2022 10:38 AM PDT I am starting out with group theory for my computer science degree, it's part of the basic maths subject, it is covered in about 1.5 pages and then moves on with topology. From what I understood you have a set, eg. $M=\{a,b,c,d\}$ and with this set you can define an "internal composition law" or "operation" with this set so that $(M,*)$ is an abelian group. In my textbook I understood that an operation should return an element within the set $M$. So my question is,
I think I am missing something here... Someone said that I can start with examples like $a^2=b^2=c$, however would $a^2$ still be part of the original set? Because say $a = 86, b = 42, c = 0, d = 7$; then $a^2$ wouldn't be part of our M set from what I believe. Maybe don't know what they are asking me to do exactly, I also thought of supposing the "operation" is a sum, so if I sum $a + b$ the answer wouldn't be part of set $M$ either. I have studied set theory before, stuff like intersections, unions, complements, etc but I am completely new to group-theory. And to all the unfriendly people out there, you don't have to answer or comment if you don't want to help, maybe work on your own personal issues before insulting someone who is just asking a question. |
| If $H \subset gHg^{-1}$, what can be implied from this relation? Posted: 19 Mar 2022 10:31 AM PDT There are some well-answered questions in this site in regard to the relationship of $gHg^{-1}$ and $H$, for example: Conjugate subgroup strictly contained in the initial subgroup? and we have already known that $gHg^{-1}$ is not always contained in $H$, however, I am curious about what can we know of $H$ and $gHg^{-1}$ if $H \subset gHg^{-1}$? If $H$ is finite, then the two groups equal. What about $H$ being infinite? I have an intuitive feeling that $gHg^{-1}$ must have less elements than $H$ somehow, but without finiteness I don't have any tools to prove it. Also I note that some people will rewrite this condition as $H \cap gHg^{-1} = H$, but they seem equivalent to me. |
| Why exactly do we write $\mathcal{L}(x(t),x'(t),t)$ instead of simply $\mathcal{L}(x(t),t)$? Posted: 19 Mar 2022 10:20 AM PDT I have many times seen Lagrangian written as $\mathcal{L}(x(t),x'(t),t)$. I undestand that this is a function of $x(t)$, $x'(t)$ and $t$. So it can theoretically look something like $$\mathcal{L}(x(t),x'(t),t) = 5\cdot x(t)+(x'(t))^2+2\cdot t$$ (and therefore $\mathcal{L}(a,b,c) = 5\cdot a+b^2+2\cdot c$) But why don't we simply write $$\mathcal{L}(x(t),t) = 5\cdot x(t)+(x'(t))^2+2\cdot t$$ (and therefore $\mathcal{L}(a(t),b) = 5\cdot a(t)+(a'(t))^2+2\cdot b$) Is the $x'(t)$ there just to make it clear that Lagrangian has it or are there any other reasons? (sorry for my English, btw) |
| Posted: 19 Mar 2022 10:32 AM PDT How I should prove that Spec(R)= Spec(C[X]) for polynomials with 1 variable for function: h(0) = h(1)? I have:
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| Posted: 19 Mar 2022 10:35 AM PDT I observed that the solution space of ODE $$y''(t)+ay'(t)+by(t)=0, t\in [a,b],$$ with constant real coefficients $a$ and $b$, is always spanned by the functions $y_\lambda(t)=e^{\lambda t}$. But the characteristic equation $$\lambda^2+a\lambda+b=0$$ can only be obtained by substituting $e^{\lambda t}$ to the given ODE. So, how to say all possible solutions (other than polynomials, sinusoidal, exponential (if possible)) for the above ODE can be spanned by these basis vectors? OR how can we reject the possibility of such crazy functions beyond the spanning set as a solution for the same? |
| Help needed for $\lim_{x→0}[(a_1^x+..a_n^x)/n]^{(n/x)}$ Posted: 19 Mar 2022 10:23 AM PDT I'm running into this question prepping for an exam:
Here is how I was trying to solve it: Can anyone give it a look and let me know if it makes any sense? If not, could you walk me through? Thanks in advance! P.S It seems the layout is a bit messed up here with my latex. Symbols seem to be squeezing each other. |
| Number of permutations in $[2n]$ such that every cycle has exactly one even number Posted: 19 Mar 2022 10:23 AM PDT Since there are n even numbers in $[2n]$ we know we will have $n$ cycles. If length of all cycles is 2 we have $n!$ ways to make a permutation which satisfies given condition. We can also have some cycles of length 3(they contain one even and two odd numbers) and for each of those we have a fixed point and it must be even. Its easy to notice that the highest number of pairs of cycles of lengths 1 and 3 is whole part of the number $\frac{n}{2}$ (I don't know how to write whole part and this sentence is generally terrible but bear with me) So now I want to see what happens when we have $k$ pairs of cycles of length 1 and 3. We choose two even and two odd numbers in $n \choose 2$$n \choose 2$ ways.Now we can make 4 permutations out of those numbers(we choose cycle of length 1 in two ways because it must be even and then we choose a cycle of length 3 out of numbers that are left) We repeat this $k$ times (for last pair of cycles we choose two even and two odd numbers in $n-2(k-1) \choose 2$$n-2(k-1) \choose 2$ ways and make a permutation in 4 ways) Now we have $2(n-2k)$ elements left ($(n-2k)$ even and $(n-2k)$ add). Now we pair them even with odd in $((n-2k)!)^2$ ways. Ok, so now I want to divide all of this with something(since order is not important) but I cant figure out with what. Notice I didn't divide with anything when I was counting pairs of cycles of length 1 and 3 and when I was counting transpositions.I thought maybe I could divide it all by $n!$ at the end(now that is) and then multiply it by 2 since I was considering order when I was counting number of permutations for those pairs of cycles, but that just doesn't seem right. I would appreciate some help with finishing this.Also alternative solution would also be greatly appreciated since I have a feeling I'm doing this the worst possible way. |
| Posted: 19 Mar 2022 10:25 AM PDT
It is clear that we have $$\mathbb{E}[X]\mathbb{E}[1/X]\geq \big(\mathbb{E}[\sqrt{X}\sqrt{1/X}]\big)^2=1$$ by the Cauchy-Schwarz inequality, but I have no idea about how to determine the upper bound of the product, any help would be appreciated . |
| A problem related to Kelvin transform from Harmonic Function Theory Posted: 19 Mar 2022 10:32 AM PDT
The above picture is from Harmonic Function Theory written by Sheldon, Wade and Paul, I don't understand why polynomials are locally dense in $C^2$-norm and how can we conclude from this point that the result holds for arbitrary $C^2$ functions. I need more hints. |
| Limit problem (limit of a sum as a definite integral) Posted: 19 Mar 2022 10:28 AM PDT Could somebody please help me with this: $$\lim_{n\to \infty}\frac{(1^2+1)}{(1-n^3)}+\frac{(2^2+2)}{(2-n^3)}....\frac{(n^2+n)}{(n-n^3)}$$ I have an intuition that it can be expressed as an integral and as the general method prescribes, I am trying to get $\frac rn$ term in it so that I can replace it with $x$ and $1/n$ with $dx$ but I am unable to do so. I tried by representing a general term as $$\frac{(r^2+r)}{(r-n^3)}$$ but taking out $n^3$ or $r^2$ or $r$ ain't working for me (unable to manipulate it further). Thank you. |
| Converting an integral equation to a differential equation Posted: 19 Mar 2022 10:30 AM PDT I was recently working on a problem and ended up with an integral equation that I was hoping can be solved or at least be converted to a differential equation. I have no experience in integral equation so I have no Idea how to proceed. Consider the following integral equation: $$\phi(x,y) = \sigma(\int \phi(x',y')w(x,y,x',y') dx'dy')$$ Where the integration bounds and the functions are defined over a rectangle. The function $\sigma$ in general a continuous non-linear function. Assuming the function $w(x,y,x',y')$ is known, is there any general solution or differential equation form of this equation? Is it possible to convert this to a differential equation in case $\sigma$ is linear? Edit: As far as I can tell, it seems that there is no general method to approaching this problem except through numerical methods. In that case, are there any general theorems for equations of this form on the existence of solutions given boundary conditions? For instance, given a set of boundary conditions for the relevant function, does there exist a function that satisfies the above relation (unique -or not-solution for $\phi(x,y)$ given $w(x,y,x',y')$ and boundary conditions for $\phi$ and unique -or not- solution for $w(x,y,x',y')$ given $\phi(x,y)$ and the boundary conditions for $w$)? |
| The limit as $\epsilon \rightarrow 0+$ of $\frac{1}{\epsilon}P\big(\frac{u}{\epsilon}\big)$ Posted: 19 Mar 2022 10:25 AM PDT I need to evaluate the limit $$ \lim_{\epsilon \rightarrow 0+} \frac{1}{\epsilon}P\left(\frac{u}{\epsilon}\right),$$ where $P(u)$ is some continuous probability density function. Notably, this function and all of its derivatives should vanish as its argument goes to plus or minus infinity. For physical reasons I suspect the limit may end up involving $\frac{d}{du} \sigma(u)P(u), $ where $\sigma(u)$ is the sign (signum) function (a generalized function). This limit has a curious feature. Since $P(u)$ and all of its derivatives vanish at infinity, one can apply L'Hôpital's repeatedly, and the indeterminant form never resolves: $$ \frac{P(u/\epsilon)}{\epsilon} = -u \frac{P'(u/\epsilon)}{\epsilon^2} = (-u)^2 \frac{P''(u/\epsilon)}{2\epsilon^3}=\dots = (-u)^n \frac{P^{(n)}(u/\epsilon)}{n!\epsilon^{n+1}}$$ Is there some way to use this property to derive a signum function representation of the limit? Some exploratory stuff: The function $P(u/\epsilon - s u/\epsilon)$ where $s$ is some parameter near $0$ has Taylor expansion $$ P\left(\frac{u}{\epsilon}-s\frac{u}{\epsilon}\right) = \sum_{n=0}^\infty \frac{s^n}{n!}\Big(-\frac{u}{\epsilon}\Big)^nP^{(n)}\Big(\frac{u}{\epsilon}\Big), $$ which because of the above "curious feature" becomes $$ P\left(\frac{u}{\epsilon}-s\frac{u}{\epsilon}\right) = P(u/\epsilon)\sum_{n=0}^\infty s^n =\frac{P(u/\epsilon)}{1-s}. $$ This is odd, but I am not sure if it is useful. |
| Integral $\int^\infty_0 \frac{\cos(\omega x)}{\cosh(ax)-\cos\beta}\text{d}x$ Posted: 19 Mar 2022 10:34 AM PDT Maybe you can help me, understanding an arising ambiguity: Consider the integral, which is on page 30 integral (6) of the Bateman Project( see link below) $$\int^\infty_0 \frac{\cos(\omega x)}{\cosh(ax)+\cos(\beta)}\text{d}x=\frac{\pi}{a\sin(\beta)}\frac{\sinh(\frac{\beta \omega}{a})}{\sinh(\frac{\pi \omega}{a})}$$ which holds for $\text{Re}(a)\pi>\text{Im}(a^*\beta) $. Say I want to calculate the integral with a minus in the denominatorfor real $a, \beta$ (hence the above restriction on the parameters is always satisfied), which I have naively done by $\int^\infty_0 \frac{\cos(\omega x)}{\cosh(ax)-\cos(\beta)}\text{d}x=\int^\infty_0 \frac{\cos(\omega x)}{\cosh(ax)+\cos(\beta\pm\pi)}\text{d}x= \frac{\pi}{a\sin(\beta\pm \pi)}\frac{\sinh(\frac{(\beta\pm\pi) \omega}{a})}{\sinh(\frac{\pi \omega}{a})}=-\frac{\pi}{a\sin(\beta)}\frac{\sinh(\frac{(\beta\pm\pi) \omega}{a})}{\sinh(\frac{\pi \omega}{a})}$, then I get the ambiguity of the choosen sign infront of the $\pi$ in the $\sinh(\frac{(\beta\pm\pi) \omega}{a})$. Can anyone explain to me what it is the correct way (sign of $\pi$) to solve this integral with the minus in the denominator? Link for formula: https://authors.library.caltech.edu/43489/1/Volume%201.pdf |
| Why $f(0)\neq 0$ where $f $ is a polynomial over the field $\mathbb F_q$ and $\deg(f)=m > 0$? Posted: 19 Mar 2022 10:22 AM PDT
I have worked with different examples such as $x^3+x=f \in F_2[x]$ but could not find why $f(0) \neq 0$ is necessary for residue class ring $F_q[x]/(f)$. |
| Posted: 19 Mar 2022 10:28 AM PDT Maybe this exercise comes from some textbook, but I do not know.
I observe that $$k[x(x-1),x^2(x-1)]=\{f(x)\in k[x]\mid f(0)=f(1)\},$$ and $$k[x(x-1),x^2(x-1),z]\simeq k[x,y,z]/(y^2-xy-x^3).$$ We have a morphism from $\mathrm{Spec}\ k[x,y,z]/(y^2-xy-x^3)$ to $\mathbb{A}^2$. But I still have not solved the exercise. And I do not know how one found this counterexample. Why did he consider this? Thanks. |
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