Recent Questions - Mathematics Stack Exchange |
- Top dimensional Cech cohomology of stratified space
- Given $S,T:V \to V$ , 1)if $Im(ST)=\{0\}$ then $Ker(S)=V$ or $Ker(T)=V$ ,2) if $Im(ST)=\{0\}$ then $Im(S) \subseteq Ker(T)$
- Probability of two sets of points occuring at a distance x along a line of length L
- Finding the inverse Fourier transform of $\hat{f}(\omega)=2\pi e^{-|\omega-c|}$
- Weak limit of a certain measure sequence is stationary
- How to solve $\int_0^1 \int_0^1 \sqrt{1 - x} \sqrt{1 - y}\,\, dx dy$?
- Positive limit set of dynamic systems
- Clarification on the limit superior and inferior in topological spaces
- Calculate achievable points for vehicle with turning circle
- How to define an NTU game in set builder notation?
- Is this integral positive?
- Non-orientable genus of Unknot and it's uniqueness
- Solving $x^{-x^{1-x}}=\sqrt[\sqrt2]{2}$
- Solutions for the composition of homogeneous ODE
- example of $L/\Bbb{Q}_p$ such that there is no prime element $π$ of ring of integers $L$ such that $p＝π^e$.
- Is the interval $[0,2] \subset \Bbb R$ compact in the topology generated by half open intervals $[a,b)$?
- Question on Lemma 1.4.9 in Atiyah
- Pell's equation fundamental solution vs. an earlier convergent of $\sqrt{d}$
- Number of equilateral triangles formed by connecting dots on a circle.
- Probability with number and values of desired outcomes variable
- How many ways you arrange $12$ marbles where two marbles of the same color are considered the same?
- Solution of Bessel equation when the parameter is not rational
- When is the compact-open topology locally compact?
- Finding the values of 9 variables given 4 expressions and a constraint on the values of the variables
- Is this a known factorial approximation?
- Finding a differential operator C that satisfies AB=CA
- Determine coordinates of right corner of a triangle when angles and coordinates of the other points are known
- Looking for a direct way to evaluate $\int_0^1\frac{\ln(x)\ln(2+x)}{1+x}dx$
- How to compute SSR with just residuals and Xi?
- When a smooth curve is an immersion (John Lee's Smooth manifold book p 156) and Example 7.3
| Top dimensional Cech cohomology of stratified space Posted: 25 Feb 2022 03:38 AM PST I have a compact space $X$ that is a manifold of dimension $n$ outside of a singular set $S$ (which is closed in $X$). Suppose that $X\setminus S$ is also oriented.
This question is inspired by $X=$ Hawaiian earring, and $S=$ origin, in this case, we have that $\check H^1(X,S)\simeq \check H^1(X) \simeq \bigoplus_{i\in \mathbb{N}} \mathbb Z$ (apparently the last isomorphism can be proved noting that $X$ is an inverse limit see Hawaiian earring as inverse and direct limit (even though I do not understand why, if you can explain please leave a comment) |
| Posted: 25 Feb 2022 03:35 AM PST
I've asked this question before and as a comment suggested in my previous post I wanted to open a post about these 2 statements I am confused on why the first one is not true. If $Im(ST)=\{0\}$ then $ST=0$ , and if $Ker(T)=V$ then $T(v)=0$ for all $v\in V $ therefore $Im(T)=\{0\}$ so $dimIm(T)=0$ and $dimKer(T)=dim(V)$ from here $Ker(T)=V$ in our case the statement is not true because we will get $Ker(ST)=V$? and not $Ker(S)=V$ or $Ker(T)=0$? As a comment suggested in the post above that I need to understand first what $Ker(S)=V$ means and as I mentioned above I believe it means that $Im(S)=0$? because we get $S(v)=0$ for all $v \in V$? for the second one , I would appreciate some hints and tips .. what does it mean for $Im(S) \subseteq Ker(T)$ Thank you |
| Probability of two sets of points occuring at a distance x along a line of length L Posted: 25 Feb 2022 03:34 AM PST Consider two sets of points : A and B.[ Number of points in A= a, Number of points in B=b]. A and B are randomly placed along a line of fixed length L. What is the probability of some A occuring next to some B at a distance less than x. (x is also a fixed length, x<L)? Also in the same circumstances how many points in A will be occuring next to B at a distance less than x? an image describing the sitaution |
| Finding the inverse Fourier transform of $\hat{f}(\omega)=2\pi e^{-|\omega-c|}$ Posted: 25 Feb 2022 03:32 AM PST Finding the inverse Fourier transform of $\hat{f}(\omega)=2\pi e^{-|\omega-c|}$ is not as easy as I thought. There are two ways I investigated, and it was the table-based way that was the intended way to solve this. I wanted to use the following rules of inverse transforms: Symmetry rule: $2\pi f(-\omega)\longrightarrow \hat{f}(t)$ where I put $f(\omega)=e^{-i\omega T}\hat{f}$ But the problem is that shift of the variable with the constant c, in the original function, $\hat{f}(\omega)=2\pi e^{-|\omega-c|}$ . Alternatively, I tried to find it by using the definition of the inverse Fourier transform, but got: \begin{equation} \int_{-\infty}^0 e^{\omega(1+it)+c}d\omega+\int_{0}^\infty e^{\omega(it-1)-c}d\omega=\lim_{B\rightarrow \infty}\frac{e^c}{1+it}+\frac{1}{it-1}(e^c-e^B) \end{equation} but as can be seen, that B to infinity does not complete the integral. Any ideas? Thanks |
| Weak limit of a certain measure sequence is stationary Posted: 25 Feb 2022 03:26 AM PST I am not very conversant about this topic and was given this question as an assignment. This is my first post so I do apologize if the question is very silly. Let $\mu$ be any probability measure on a locally compact topological group $G$. Let $\sigma$ be any probability measure on $G$. Then if the sequence of measures $$\frac{1}{n}(\sigma+\mu*\sigma+\cdots+\mu^{n-1}*\sigma)$$ converges to $\nu$ weakly, show that $\nu$ is $\mu$-stationary, i.e. $\mu*\nu=\nu$. |
| How to solve $\int_0^1 \int_0^1 \sqrt{1 - x} \sqrt{1 - y}\,\, dx dy$? Posted: 25 Feb 2022 03:40 AM PST $$\int_0^1 \int_0^1 \sqrt{1 - x} \sqrt{1 - y}\,\, dx dy $$ I don't really know what to do with this integral. Can anyone give some hints to start? Wolfram says the answer is 4/9 but I don't really know what to do with this instead of changing the integral to the product of two definite integrals.
$$\int_0^1 \int_0^1 \sqrt{1 - x} \sqrt{1 - y}\,\, dx dy = \left( \int_0^1 \sqrt{1 - x}\, dx \right)\left( \int_0^1 \sqrt{1 - y}\, dy\right)$$ What should I do? |
| Positive limit set of dynamic systems Posted: 25 Feb 2022 03:20 AM PST I am given a system dynamics in general form as so: $\dot{x}=f(t, x(t))$ and let $x\left(t, x_{0}, t_{0}\right)$ be a bounded solution. Given the limit set as defined: ${w_{\left(x_{0}, t_{0}\right)}} \triangleq\left\{\bar{x} \in \mathbb{R}^{\mathrm{n}}: \exists\left\{t_{k}\right\}_{k \geq 1}\right.$ with $t_{k} \rightarrow \infty$ as $k \rightarrow \infty$ S.T. $\left.x\left(t_{k}, x_{0}, t_{0}\right) \rightarrow \bar{x}\right\}$ I would like to prove the following proposition: Proposition $w_{\left(x_{0}, t_{0}\right)}$ is $(1)$ nonempty, $(2)$ closed, $(3)$ bounded, (4) $d\left(x\left(t, x_{0}, t_{0}\right), w_{\left(x_{0}, t_{0}\right)}\right) \rightarrow 0$ as $t \rightarrow \infty$ where $d\left(x\left(t, x_{0}, t_{0}\right), w_{\left(x_{0}, t_{0}\right)}\right)=\inf _{\bar{x} \in w}\left\|x\left(t, x_{0}, t_{0}\right)-\bar{x}\right\|$. Could you please provide help on how I go about proving the propositions? |
| Clarification on the limit superior and inferior in topological spaces Posted: 25 Feb 2022 03:19 AM PST Let $(X, \mathcal{T})$ be a topological space and $\left(A_n\right)_{n\in \mathbb{N}}$ a sequence of subsets of $X$. My reading material defines the limit superior as the set of points of $X$ such that every neighborhood of a point $x \in \lim\sup_nA_n$ intersects with infinitely many members of the sequence $\left(A_n\right)_{n\in \mathbb{N}}$. The limit inferior is defined similarly as a set of points of $X$, but such that any neighborhood of a $x \in \lim\inf_nA_n$ intersects with all except for finitely many members of the sequence $\left(A_n\right)_{n\in \mathbb{N}}$. Then with these definitions, are the complements of $ \lim\sup_nA_n$ and $\lim\inf_nA_n$ sets $S_1$ and $S_2$, respectively, such that 1.) any $x\in S_1$ has a neighborhood that intersects with at most finitely many members of the sequence $\left(A_n\right)_{n\in \mathbb{N}}$ and 2.) formulated logically $\forall x \in S_2:\forall x \in U \in \mathcal{T}:\forall n \in \mathbb{N}:\exists n_0 \geq n: U\cap A_n = \varnothing$? |
| Calculate achievable points for vehicle with turning circle Posted: 25 Feb 2022 03:15 AM PST I'm trying to determine the set of achievable positions for a vehicle that can travel a fixed distance. In the following diagram the blue marker is the vehicle, and the pink shape is the set of achievable positions. I know the turning circle radius and the maximum distance that can be travelled. From looking at the problem the "travel distance" is the distance from the origin the point directly ahead on the curve. It is also the distance along the circumference of the curve. But, I cannot think of the maths to determine the shape of the outer edge of the pink shape. Any tips?
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| How to define an NTU game in set builder notation? Posted: 25 Feb 2022 03:13 AM PST I am trying to construct a mathematical object using set-builder notation, but I am having some trouble. Let $N$ be an arbitrary set. The object I need to construct is a map $\nu$ that links each set $S\subseteq N$ with a set $\nu(S)\subset\mathbb{R}^S$ (if $N$ is a player set, this is the notion of a non-transferable utility game in game theory). This object is well-known in my field, and most authors just define it in words (like I just did but adding some regularity assumptions). However, I have found no paper that constructs this object in set builder notation, and I can't come up with a reasonable way to do so. One of my attempts at defining this object is this one: \begin{gather} \nu=\{(\forall S\subseteq N)(\nu(S)\subset\mathbb{R}^S)\} \end{gather} Another of my attempts at defining this object is this one: \begin{gather} \nu:2^N\to\mathbb{R}^S \end{gather} Or perhaps even this one: \begin{gather} \nu:2^N\to\mathbb{R}^N \end{gather} But, for one reason or another, they are all inaccurate depictions of the object that I want to construct. Can anybody help me construct the object I described (i.e, an NTU game) in set builder notation? Thank you all very much for your time. |
| Posted: 25 Feb 2022 03:39 AM PST Today I was discussing with a classmate about the sign of the integral $$\int_{B(0,r)} \int_{B(0,r)} \ln(\|x-y\|)dxdy,$$ where $B(0,r)$ denotes the ball of center $0$ and radius $r$ in $\mathbb{R^2}$. My friend said that this integral is negative for every $r>0$ because the function $\ln |x-y|$ is "very negative" at $x=y$. I don't agree and I told him that I think there exists a critical $r$ from which the integral is positive. However, I don't know how to prove it. I rewrite it by using polar coordinates as $$\int_0^r\int_0^r\int_0^{2\pi}\int_0^{2\pi} \ln(\sqrt{r_1^2+r_2^2-2r_1r_2\cos(t_1-t_2)})r_1r_2dt_1dt_2dr_1dr_2.$$ I computed this integral with the software Mathematica and I obtained positive values with, for example, $r=2$. However, this proof is not valid for my friend. Does anyone know how to prove it rigorously? |
| Non-orientable genus of Unknot and it's uniqueness Posted: 25 Feb 2022 03:12 AM PST In the book "Topology Now!" by Robert Messe one of the practice problem suggests, " One could define the nonorientable genus of a knot to be zero for the trivial knot, and for any other knot K to be the smallest number p such that the surface formed by taking; the connected sum of p projective plans and removing one disk will span K " Is unknot the only knot that bounds a mobius band?? |
| Solving $x^{-x^{1-x}}=\sqrt[\sqrt2]{2}$ Posted: 25 Feb 2022 03:03 AM PST
Here is what I've tried: Since RHS is $2^{\frac1{\sqrt2}}$, base of LHS should be in the form $2^a$. I checked $x=\frac12$ is an answer but not sure if there are other answers. Using $x=2^a$ left side will be equal to, $$\large(2^a)^{\large-(2^a)^{\large(1-2^a)}}=\large(2^a)^{\large-2^{\large(a-a2^a)}}=\large2^{\large-a\times2^{\large(a-a2^a)}}$$ Hence we have $$a.2^{a(1-2^a)}={\frac{1}2}$$But don't know how to continue further. |
| Solutions for the composition of homogeneous ODE Posted: 25 Feb 2022 02:57 AM PST Suppose we have a second-order homogeneous ODE represented by $$ F(x,y,y',y'')=0 \quad (1), $$ and then consider the $4th$ order homogeneous ODE \begin{equation} F(t,F(x,y,y,y''),F(x,y,y',y'')',F(x,y,y',y'')'')=0. \quad (2) \end{equation} I want to know about the relation between the solutions of (1) and (2). For example, since they are homogeneous, I know that solutions of (1) are all solutions of $(2)$. Particularly, I want to know in which cases there are other solutions of $(2)$ that are not solutions of $(1)$. |
| Posted: 25 Feb 2022 03:13 AM PST Let $L/\Bbb{Q}_p$ be ramification index $e$ extension. Let $π$ be prime element of $L$. Then, $p＝π^eu$ ($u$: unit element of ring of integers of $L$)。 But I have met the remark that we can choose prime element $π$ such that $p＝π^e$ (I cannot find the reference now, sorry・・・). I think this is true in the case $e＝1$, but when $e≧2$, I think we cannot take $π$ in general・・・ Is the remark correct statement ? If not, I want to know counterexample, that is, example of $L/\Bbb{Q}_p$ such that there is no prime element $π$ such that $p＝π^e$. |
| Posted: 25 Feb 2022 03:23 AM PST
I think not, but I cannot find a cover that wouldn't have a finite sub-cover. If I let $A_n=[0, 2+ \frac{1}{n})$, then I would have that $[0,2] \subseteq \bigcup_{n \in \Bbb N} A_n$, but I don't know how to check that if this cover has a finite sub-cover? |
| Question on Lemma 1.4.9 in Atiyah Posted: 25 Feb 2022 03:15 AM PST The text can be found here: https://www.maths.ed.ac.uk/~v1ranick/papers/atiyahk.pdf I am having two struggles on the proof of Lemma 1.4.9. First, why is $C^+(X) \cap C^-(X) = X$? To me, it would appear that it should be $\{1/2\} \cup X/(\{0\} \cup \{1\}) \cup X$. Secondly, I am confused why $E|C^+$ and $E|C^-$ are trivial, or how to prove that $E$ is a vector bundle. Any help would be greatly appreciated! |
| Pell's equation fundamental solution vs. an earlier convergent of $\sqrt{d}$ Posted: 25 Feb 2022 03:14 AM PST Let $(p,q)$ be the fundamental solution to Pell's equation $x^2-dy^2=1$, which makes $\frac{q}{p}$ a convergent of $\sqrt{d}$, per the theorem underlying the continued fraction algorithm for solving Pell's equation. Is it always (or with finitely many exceptions) the case that at least one of the reduced fractions $\frac{q}{p-1},\frac{q}{p+1}$ is also a convergent of $\sqrt{d}$? Examples: $$ d=73\\ 2281249^2-267000^2d=1\\ \frac{267000}{2281249-1}=\frac{125}{1068},\ a\ convergent\ of\ \sqrt{d} $$ $$ d=95\\ 39^2-4^2d=1\\ \frac{4}{39-1}=\frac{2}{17},\ a\ convergent\ of\ \sqrt{d} $$ $$ d=96\\ 49^2-5^2d=1\\ \frac{5}{49+1}=\frac{1}{10},\ a\ convergent\ of\ \sqrt{d} $$ |
| Number of equilateral triangles formed by connecting dots on a circle. Posted: 25 Feb 2022 03:20 AM PST Fifteen dots are evenly spaced on the circumference of a circle. Three line segments are drawn to connect the dots such that they do not intersect in the circle. Find the number of equilateral triangles that can be formed. My thoughts: it is easy to select 3 line segments to get an isosceles triangle but how do I know if it is going to be equilateral? The answer is 5. |
| Probability with number and values of desired outcomes variable Posted: 25 Feb 2022 03:31 AM PST Two fair dice are randomly rolled. Let p be the sum of their face values and subset P is randomly formed of set X={2,3,4,5,6,7,8,9,10,11,12}. The probalility that p belongs to P is? |
| How many ways you arrange $12$ marbles where two marbles of the same color are considered the same? Posted: 25 Feb 2022 03:27 AM PST Problem: Note: Two marbles of the same color are considered identical. Answer: |
| Solution of Bessel equation when the parameter is not rational Posted: 25 Feb 2022 03:07 AM PST Is the general solution of the Bessel equation valid if the parameter is a positive non rational number such as $1/\sqrt 2$? Or is it that just the Bessel functions of 1st species can solve the equation? |
| When is the compact-open topology locally compact? Posted: 25 Feb 2022 03:01 AM PST Let $X$ and $Y$ be topological spaces, and consider the compact-open topology on $C(X,Y)$, which is generated by open sets of the form $$\{\text{continuous }f\colon X\to Y:f(K)\subseteq U\}\text{ for compact }K\subseteq X\text{ and open }U\subseteq Y.$$
One special case of particular importance is the fact that $C(G,T)$ is locally compact, where $G$ is a locally compact Hausdorff abelian group, and $T$ is the circle group. This is important for Pontryagin duality, because it is one part of knowing that the Pontryagin dual of a locally compact Hausdorff abelian group is still a locally compact Hausdorff abelian group. For this special case, the standard proof uses equicontinuity and Arzela-Ascoli. But this approach only works if $Y$ is a metric space. |
| Posted: 25 Feb 2022 03:40 AM PST Let's say I have $9$ variables $(a, b, c, d, e, f, g, h, i)$ and I know they are all different and they all have values between $1$ and $9$ included. Basically these variables will each have a different value between 1 and 9. So: $$a, b, c, d, e, f, g, h, i \in [1, 9]$$ $$a \neq b \neq c \neq d \neq e \neq f \neq g \neq h \neq i$$ I also know that: $$a + b + c + d + e = 22$$ $$a + b + f + g + h = 22$$ $$d + e + g + h + i = 22$$ How do I determine the values of $a, b, c, d, e, f, g, h, i$? |
| Is this a known factorial approximation? Posted: 25 Feb 2022 03:31 AM PST While playing with the fixed points (i.e. $e^{\pm\pi/3}$) of the iterated composition of $(1-x)^{-1}$ and a kind of unitary transform, I stumbled across what I believed to be an identity that connects the factorial and the geometric series. For $n=1$, it converges to $(1-e)^{-1}$ and always seems to point the nearest integer as the proper value of $n!$ $$ n! \approx \frac{-1}{1-\sum_{k=0}^{\infty} \frac{1}{(n*k)!}} |n \in Z^{+} $$ So my question is twofold, is it a known identity? And how does one manage to prove this since the sum diverges at $n=0$ and $\forall\ {}n \in Z, n!\ge1$. It seems to make it a difficult task. |
| Finding a differential operator C that satisfies AB=CA Posted: 25 Feb 2022 03:32 AM PST Let $D=\frac{d}{dx}$ , $A=\sum_{i=0}^{i=n} a_n(x)D^{n}$ and $B=b(x)D$, where $a_n(x)$ and $b(x)$ are sufficiently smooth functions and $n$ is an arbitrary positive integer. $A$ may not be invertible. Assume that $AB=CA$. What can be said about operator $C$? Is it possible to write $C$ as a function of $A$, $D$ and $B$? Note that if $A$ is invertible, $C=ABA^{-1}$. Additionally, if $A,B$ and $C$ are matrices, this problem is similar to the homogeneous Sylvester matrix equation Sylvester matrix equation . |
| Posted: 25 Feb 2022 03:21 AM PST
Given: Ax,y Bx,y angle B in radians To determine: Cx,y on both sides of AB Context: I need the answer to this question to determine the startpoint and the end point of a linear gradient line. In SVG, the linearGradient element specifies the gradient line (a vector) by specifying the startpoint and the endpoint as a percentage of the size of the bounding box of the image to be filled. I need to convert these points to coordinates in units as used by the image. The mozilla developer site https://developer.mozilla.org/en-US/docs/Web/CSS/gradient/linear-gradient() explains under the title "Composition of a linear gradient": The gradient line is defined by the center of the box containing the gradient image and by an angle. The end points are defined by the intersection of the gradient line with a perpendicular line passing from the box corner which is in the same quadrant. Composition of a linear gradient My convertion routine takes the following steps (see drawing):
My question is about point 5: "Determine coordinates of right corner of a triangle when angles and coordinates of the other points are known". I repost a rejected answer to this question, but it is complex and only handles quadrant IV. A simpler answer would be nice. |
| Looking for a direct way to evaluate $\int_0^1\frac{\ln(x)\ln(2+x)}{1+x}dx$ Posted: 25 Feb 2022 03:04 AM PST The following integral $$I=\int_0^1\frac{\ln(x)\ln(2+x)}{1+x}dx=-\frac{13}{24}\zeta(3)$$ has been evaluated in different ways ( see here, here, here, and here) but these four solutions involve many arguments of polylogs and much simplifications are required to reach the final closed form. I am wondering if there is a simpler direct way to prove it. I tried some tricks but none worked out. Here is my best try: Following @Zacky's idea, lets denote: $$ I(a)=\int_0^1\frac{\ln(x)\ln(1+a(1+x))}{1+x}dx$$ and note that $I(1)=I$ and $I(0)=0,$ $$I'(a)=\int_0^1\frac{\ln(x)}{1+a(1+x)}dx=\frac{\text{Li}_2\left(-\frac{a}{1+a}\right)}{a}.$$ Integrate both sides from $a=0$ to $1$, $$ \int_0^1 I'(a)=I(a)|_0^1=I(1)-I(0)=I-0=I=\int_0^1\frac{\text{Li}_2\left(-\frac{a}{1+a}\right)}{a}da$$ Integrate by parts, $$I=\int_0^1\frac{\ln(a)\ln(1+2a)}{a(1+a)}da-\int_0^1\frac{\ln(a)\ln(1+a)}{a(1+a)}da$$ $$=\int_0^1\frac{\ln(x)\ln(1+2x)}{x(1+x)}dx+\frac{5}{8}\zeta(3).$$ To finish the proof, we need to show $$\int_0^1\frac{\ln(x)\ln(1+2x)}{x(1+x)}dx=-\frac76\zeta(3)$$ and i am stuck here. I also added $I$ to both sides: $$2I=\int_0^1\frac{\ln(x)\ln\left(\frac{2+x}{1+2x}\right)}{1+x}dx+\int_0^1\frac{\ln(x)\ln(1+2x)}{x}dx$$ then used the subbing $x\to (1-x)/(1+x)$ for the first integral: $$2I=\int_0^1\frac{\ln\left(\frac{1-x}{1+x}\right)\ln\left(\frac{3+x}{3-x}\right)}{1+x}dx+\int_0^1\frac{\ln(x)\ln(1+2x)}{x}dx$$ and I made it even worse. Any other thoughts ( without complicated arguments of polylogs) ? thanks, |
| How to compute SSR with just residuals and Xi? Posted: 25 Feb 2022 03:04 AM PST
How do we calculate SSR? I know SSE is the square of residuals all added together, but SSR is a subtraction between prediction for each observation and the population mean. Not sure how calculate SSR. For SSE, I got 59.960. |
| When a smooth curve is an immersion (John Lee's Smooth manifold book p 156) and Example 7.3 Posted: 25 Feb 2022 03:35 AM PST In John Lee's Intro to Smooth Manifold book (2003 Springer) , I need some help with an example of an immersion. On page 156 Example 7.1 c), If $\gamma(t): J \to M$ is a smooth curve ...then $\gamma$ is an immersion if and only if $\gamma'(t)\neq 0 $ for all $t \in J$. Can someone provide a general explanation for what this $\gamma'(t)\neq 0 $ means? The curve does not "stop"? How is the $\gamma'(t)\neq 0 $ condition related to the fact that a map $F$ is an immersion if its pushforward $F_*$ is injective? Is $\gamma'(t)$ suppose to be injective (one to one)? Would this mean $\gamma'(t)$ have to be different for each $t$ ? Seems like if $\gamma'(t)=a$ a constant, then it will not be one to one? In example 7.3 on p157, how is it that: $$| e^{-2\pi i c n_2}(e^{2\pi ic n_1}-e^{2\pi ic n_2})|=|e^{2\pi ic n_1}-e^{2\pi ic n_2}| $$ I can't figure out what happened to $e^{-2\pi i c n_2}$, am I missing something obvious? |
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