I'm not an expert on the subject, but I was wondering what's the difference between "Riemannian Geometry" and "Differential topology". To me the names suggest they're different things, but looking through some of the books preview they seem to have similar content.  Can anyone clarify? Is maybe one a subset of the other? What sorts of problems do they allow to solve that the other doesn't etc.
I have just finished my undergraduate studies. During last two semesters I've taken two subjects dealing with manifolds:   Analysis on manifolds, containing: definition of manifold, tangent space (as derivations and classes of curves), vector fields, vector bundles, flows, Lie derivatives, integration on manifolds (Stokes theorem), forms, Hodge decomposition theorem Introduction to differential geometry (for me it should be called introduction to Riemannian manifolds) containing: tensor calculus introduction, Riemannian manifold definition, connections, curvatures, geodesic, normal coordinates, geodesic completness theorem, classification throught curvature, jacobi fields, harmonic maps.    Now, for me differential geometry was/is a theory about manifolds, so anything dealing with manifolds is branch of differential geometry. On the other hand I am preparing for taking part in local conference called: "algebraic and differential topology".  I have read: ,  ,  , yet I still don't really know if differential topology is subtheory of differential geometry or is it seperate theory and how is it located between differential geometry and algebraic topology. For example I expect that studies on Riemmanian manifolds are part of differential geometry but would problem of classification manifolds up to diffeomorphism be a part of differential topology or geometry ?  Request: I would be grateful for your characterisation of differential topology and differential geometry possibly with examples of problems, theorems present at them.
Need a matrix to seventh power which is the identity but the original is not or the negative identity The question on the homework wants a $2\times2$ matrix $R$, but $R$ not equal to positive or negative identity, that when you raise to seventh power you get the identity.  $R^7=\operatorname{Id}$.  I thought it would have to be some ones and negative ones,  it no luck, then tried some ones and some $.5$.  Just using an online matrix calculator to do brute force.  We just started talking about eigenvalues with visual example, so assuming wouldn't need that since we have not talked about how to find them yet.  Any hints or ideas to get me back on track? Why is ATP the most prevalent form of chemical energy storage and utilization in most cells? Intuitive/direct proof that a rectangle partitioned into rectangles each with at least one integer side must itself have an integer side A challenge problem asked to show that if rectangle $R$ with axis-parallel sides is partitioned into finitely many subrectangles $R_1,R_2,\ldots,R_n$ (also with axis-parallel sides), such that each $R_i$ has at least one integer side length, then $R$ must also have at least one integer side length. The only proof I've seen, although simple, is quite unintuitive. Namely you look at the integral  $$ \int_R e^{2 \pi i (x+y)} dx dy $$ and note that the integral is $0$ if and only if $R$ has an integer side length, and then note that by assumption the integral is $0$ over all the $R_i$, hence summing the integral over all $R_i$ gives that the integral is also $0$ over $R$. Can someone give a more direct/intuitive argument that doesn't use voodoo like a complex-valued integral? Question Tag problem! Question:     A man like you should not do this, _______?   Which question tag should I use in the blank space?   should he? should you?   Which would be grammatically correct? I think 1 is true. I am an undergrad in economics and really want to get a helping hand as I get from math and stats from a Stack Exchange site for economics. Where can I post my questions on economics topics? How bias arises in using maximum likelihood to determine the variance of a Gaussian? While reading , I ran into the following explanation for why there is an error in estimating the variance of Gaussian distribution using Maximum Likelihood. On ,      Illustration of how bias arises in using maximum likelihood to   determine the variance of a Gaussian. The green curve shows the true   Gaussian distribution from which data is generated, and the three red   curves show the Gaussian distributions obtained by fitting to three   data sets, each consisting of two data points shown in blue, using the   maximum likelihood results (1.55) and (1.56). Averaged across the   three data sets, the mean is correct, but the variance is   systematically under-estimated because it is measured relative to the   sample mean and not relative to the true mean.   I'm pasting figure 1.15 and the required equations (1.55, 1.56) here for convenience.  $\mu_{ML} = \frac{1}{N}\sum_{n=1}^{N}x_n$   $\sigma_{ML}^2 = \frac{1}{N}\sum_{n=1}^{N}(x_n - \mu_{ML})^2$    I can imagine how maximum likelihood estimation would look pictorially. However, I'm not able to understand the figure or the explanation. I would appreciate if someone could label the figure or explain what points are considered in each part of the figure and why the curve could look like that in each case.  I looked up , and I'm able to understand why there is a factor of $\frac{N-1}{N}$ between the true $\sigma^2$ and $\sigma_{MLE}^2$.  ${\rm I\!E}[\sigma_{MLE}^2]= (\frac{N-1}{N})\sigma^2 $ A map is continuous if and only if for every set, the image of closure is contained in the closure of image Why might one also use a blank constructor? I was reading some Java recently and came across something (an idiom?) new to me: in the program, classes with multiple constructors would also always include a blank constructor. For example:   public class Genotype {   private boolean bits[];   private int rating;   private int length;   private Random random;    public Genotype() {              //  &lt;= THIS is the bandit, this one right here     random = new Random();   }    /* creates a Random genetoype */   public Genotype(int length, Random r) {     random = r;     this.length = length;     bits = new boolean[length];      for(int i=0;i&lt;length;i++) {         bits[i] =random.nextBoolean();     }   }    /* copy constructor */   public Genotype(Genotype g,Random r) {     random = r;     bits = new boolean[g.length];     rating = g.rating;     length = g.length;      for(int i=0;i&lt;length;i++) {         bits[i] = g.bits[i];     }    } }   The first constructor doesn't seem to be a "real" constructor, it seems as though in every case one of the other constructors will be used. So why is that constructor defined at all? Measures: Atom Definitions Let $\Omega$ be a measure space with measure $\mu$. (Here, a measure is only meant to be countable additive!)  Consider a subset $A\in\Sigma$.  Then according to the  it is an atom if: $$(1)\quad\forall E\subseteq A:\quad\mu(E)&lt;\mu(A)\implies\mu(E)=0\quad(\mu(A)&gt;0)$$ and according to the  it is an atom if: $$(2)\quad\forall E\in\Sigma:\quad\mu(E\cap A)=0\lor\mu(E^c\cap A)=0\quad(\mu(A)&gt;0)$$  Now, these definitions agree for the really trivial cases: $$\mu\equiv0$$ $$\mu\equiv\infty$$ but they differ for the less trivial case: $$\quad\mu(E\neq\varnothing):=\infty,\,\mu(\varnothing):=0$$ namely the atoms are w.r.t. (1) all nonempty subsets whereas w.r.t. (2) only the singletons.  Excluding this pathological case, are the definitions equivalent? (More precisely, assume there exists a measurable subset $0&lt;\mu(F)&lt;\infty$.)  So far I checked that: $$(2)\implies(1):\quad\mu(E)&lt;\mu(A)\implies\mu(A\setminus E)&gt;0\implies\mu(E)=0$$ but what about the other direction? Render not showing UV texture I have added a texture to a cube in the UV editor and it shows up fine in textured view  But when I render it there is no texture  What have I done wrong? Should I remove Adult Dependent from my 2019 tax return so they will receive a COVID Stimulus Check? On my 2019 Tax Return I claimed my 28 year old girlfriend as a dependent. This makes her ineligible to receive a COVID Stimulus Check. My return was already accepted and I have already received my refund. Should I amend my return, remove her as a dependent? Would she then be eligible to receive the stimulus check? Find the volume common to two circular cylinders, each with radius r, if the axes of the cylinders intersect at right angles. (using disk/washer)  I saw no example of this problem anywhere.. I saw an example how to solve it without calculus but I'm afraid that'll be deductions on my test.. I have the volume of a cylinder and the method of area between two curves but this is out of my reach.... I don't know what I can try... Which graphics cards are supported by the new AMDGPU driver in Ubuntu 16.04? How do I get a platform-dependent new line character? Ideals in the ring of Gaussian integers What are the proper ideals, prime ideals, maximal ideals of $\mathbb{Z}[i]$, the ring of Gaussian integers.    Check whether $(1+i)$ is prime or maximal ideal.     Can someone help me please. I have no idea how to crack this problem. Thanks for your time. Can't do reviews after failing 1 audit How do I create variable variables? Number of solutions of an arithmetic function's equation Say, an equation is given below \begin{equation} 2\pi(x) - \pi(2x)=\omega(x) \end{equation}  where $x$ is a positive integer, $\pi(x)$ is the prime-counting function, and $\omega(x)$ is the number of distinct prime factors of $x$.  I would like to know if this equation holds for finitely many values of $x$ or infinitely many.  If one uses asymptotic formula (from number theory) it can be written that \begin{equation} \omega(x) \sim \ln x / \ln \ln(x) \end{equation} (from )  and  \begin{equation} \pi(x) \sim x/\ln(x), \end{equation} so the above equation can be rewritten (after simplification) as \begin{equation} 2x\ln(2)=\ln(x)\ln\ln(x)\ln(2x) +f(x) \end{equation}  where $f(x)$ is an error term.  If  I plot left side (which is linear, supposed to be a straight line) and right side(a curve),assuming $ f(x) $ is 100% accurate, then the equation implies(?!) that the right will intersect left side. Since line can intersect a curve ($ln  $ function) finite times, so there are finite intersection point. So, can I say that there are only finitely many values of $x$?  graph example-   here f(x) is assumed as x + sinx  without the f(x)- Exclusion in formation of comparative statements I was going through the topic "formation of comparative and superlative".       The rule says     When two objects are compared with each other, the latter term of comparison must exclude the former, as in "Iron is more useful than any other metal". If we say "Iron is more useful than any metal", that is the same thing as saying    "Iron is more useful than iron", since iron is itself a metal.   I am finding difficulty to relate the rule with example. Horizontal line inside a matrix I tried to make a horizontal line inside a matrix by the attached code. However, it didnot work the compiling process gives a time out message. It work normally without the \hline command. Any help!  \documentclass[journal,draftcls,onecolumn,12pt,twoside]{IEEEtran} \usepackage{array} \usepackage{blkarray} \begin{document} \begin{equation}  \centering W_b(1)=\begin{blockarray}{cccccc} \begin{block}{(ccccc)c}  a      &amp; b     &amp;     &amp;  &amp;  &amp; \matindex{\leftarrow d=0} \\ \ddots &amp; \ddots&amp;     &amp;  &amp;  &amp; \matindex{\vdots} \\    &amp;     a &amp; b   &amp;  &amp;  &amp; \matindex{\ \ \ \   \leftarrow d=d_b-1} \\   %\hline  c      &amp; d     &amp;     &amp;  &amp;  &amp; \matindex{\leftarrow d=d_b} \\   \ddots &amp;\ddots &amp;     &amp;  &amp;  &amp; \matindex{\vdots} \\    &amp;   c   &amp;  d  &amp;  &amp;  &amp; \matindex{\leftarrow d=D}  \end{block} \end{blockarray}, \label{eq:eq4}  \end{equation}  \end{document}