I've been trying to understand the difference between the notion of smooth (which I understand well) and Riemannian (which I am newly acquainted with).   The definition in the tag for 'riemannian-geometry' is as follows: Riemannian manifolds are smooth manifolds with an inner product smoothly attached to the tangent space of each point.  If $X$ is a a smooth $n$ manifold, then we always have $T_p(X)\cong\mathbb{R}^n$ (as vector spaces) which therefore, by pulling back pushing forward and borrowing the dot product on $\mathbb{R}^n$, it follows that $T_p(X)$ has perfectly good inner product. But, I am supposed to believe that this does not always vary smoothly with $p$. I'm having trouble believing this. Can someone show me a relatively simple manifold which is smooth but not Riemannian?
I am taking a course in Differential Topology right now, but I know of another Subject called "Riemannian Geometry" which studies Riemannian Manifolds. The definition of a real smooth manifold and a Riemannian Manifold are very similar except for the detail that makes them Riemannian (having a inner product on the tangent space of every element which varies smoothly with respect to this element of the R. Manifold).  So my question is: is there any example of a Smooth Manifold that is NOT a Riemannian Manifold?  Thanks.
Does being granted a Schengen visa and not going have any consequences? I'm granted a one-entry Schengen visa and now I can't go. Will having an unused visa in my travel passport have any consequences like future visas refusal? Do I need to do anything special? A 2x2 matrix $M$ exists. Suppose $M^3=0$ show that (I want proof) $M^2=0$ Grub rescue - error: unknown filesystem I've had at least one question featured as a "hot network question" and during that time it got thousands of views in the course of a few days. But if you look at that question now you won't have any way of knowing that that's where the bulk of its view count came from.  Is there a way to find out if an arbitrary question has been a "hot network question" or not? Reading grep patterns from a file Is there a continuous bijection from $[0,1]$ onto $[0,1] \times [0,1]$? That is with $I=[0,1]$ and $S=[0,1] \times [0,1]$, is there a continuous bijection $$ f: I \to S? $$  I know there is a continuous bijection $g:C \to I$ from the Cantor set $C$ to $[0,1]$. The square $S$ is compact so there is a continuous function  $$ h: C \to S. $$ But this leads nowhere. Is there a way to construct such an $f$?  I ask because I have a continuous functional $F:S \to \mathbb R$. For numerical reason, I would like to convert it into the functional $$ G: I \to \mathbb R, \\ G = F \circ f , $$ so that $G$ is continuous. What are the various physical mechanisms for energy transfer to the photon during blackbody emission? By conservation of energy, the solid is left in a lower energy state following emission of a photon.  Clearly absorption and emission balance at thermal equilibrium, however, thermodynamic equilibrium is a statement of the mean behaviour of the system, not a statement that the internal energy is constant on arbitrarily short timescales.  The energy has to come from somewhere during emission, and go somewhere during absorption.  Energy in a solid can be stored as kinetic and potential energy of electrons and nuclei, either individually or in collective modes such as phonons and plasmons.  In thermal equilibrium energy will be stored more or less in various forms depending on the temperature and material.  However, even if most of the thermal energy in a particular solid at temperature $T$ is stored in the form of phonons, it could be that phonons primarily interact with light indirectly via electrons, e.g. a phonon excites an electron in a phonon-electron interaction, which can interact with light via the EM field.  Given that light is an EM field, it makes sense to me that it is emitted and absorbed by charged particles.  The electron-photon interaction is probably dominant for visible and ultraviolet light, given that metals are opaque, while semiconductors and insulators are transparent to (visible and UV) light with energy lower than their bandgap. However, once you get into energies in the IR and below, or X-rays and above, other mechanisms apparently take over.  For example, on the high-energy end of the spectrum I've heard that gamma rays can interact directly with nuclear degrees of freedom, which is reasonable considering that gamma rays are emitted during a lot of nuclear reactions.  A review of  might give clues to important light-matter interactions over a broad range of wavelengths.  Whether all of the these processes are involved in blackbody emission is a somewhat different question.  What physical processes mediate energy transfer during blackbody emission, and in which energy ranges are the various processes dominant? I have a collection that contains linked assets. I would like to override their materials and give them a unique material. Is it possible to assign a material to a whole collection, and how ?  Thanks ! I'm looking for a true 360 degree camera, so that's both vertical and horizontal 360 degrees. I've seen some clip-ons for the iPhone, but am really not wanting to by an iPhone 4S for this.  What I would like to see: - decent resolution: at least 1920x1080. - and 30fps  I can't settle for an alternative that's not vertically 360 degrees (or at least much more than 180 degrees).  I've seen the ball camera, which isn't out yet, but would be really well suited for my case, given it does video as well. Linear congruence fill in the missing step? Currently working on this problem and I'm having trouble seeing how it goes from one line to the next.    $45x \equiv 63\mod 11$ goes to $x \equiv 8\mod 11$     Any help would be awesome thanks.  Please try to explain how it works to help me see thanks. Give permission to debian-transmission to write into a www-data directory How can you accelerate without moving? How do you configure tomcat to bind to a single ip address (localhost) instead of all addresses? Humanity meeting another similar intelligent life form deep into space (or not)? I'm looking for a plot line (it should be a short story) that depicts Earth population (or their close descendants) establishing contact with another intelligent life form deep into space.   After careful studying of DNA samples and language / communication symbols they then realize that the civilization in question is also from Earth but that they departed very long ago and travelled throughout the galaxy in the completely opposite direction so that they are now meeting practically at the point of the circle opposite to Earth. There might have been some anatomical differences as well.  As far as I can remember, the story concludes shortly after that.  This should be a work of a well-known SF author, so I'm hoping for a quick response (if time didn't corrupt my memory too much).  I don't think it was part of an anthology, but it's entirely possible it was in a magazine.  As for the timeline, I've read it about 2-3 years ago, but since I have a passion for older SF works it might not be of much use.  Edit: Not sure if this could help but for what it's worth, it might be contained in . What is a reliable method to locate floating islands in Terraria? JavaScript variable definition: Commas vs. Semicolons How to find out if an item is present in a std::vector? All I want to do is to check whether an element exists in the vector or not, so I can deal with each case.  if ( item_present )    do_this(); else    do_that(); I have a Dell XPS 12 running Ubuntu 13.10 and it doesn't appear to find it's wireless.  I have tried the suggestions  but they made no difference  peter@xps:/lib/firmware$ ls -l /lib/firmware/iwlwifi-7260-7.ucode  -rw-r--r-- 1 root root 682892 Aug  4 12:50 /lib/firmware/iwlwifi-7260-7.ucode peter@xps:/lib/firmware$ sudo lshw -c network   *-network UNCLAIMED             description: Network controller        product: Wireless 7260        vendor: Intel Corporation        physical id: 0        bus info: pci@0000:06:00.0        version: 63        width: 64 bits        clock: 33MHz        capabilities: pm msi pciexpress cap_list        configuration: latency=0        resources: memory:f7c00000-f7c01fff  peter@xps:~/backports-3.10-2$ grep IWL7260_UCODE_API_MAX drivers/net/wireless/iwlwifi/iwl-7000.c  #define IWL7260_UCODE_API_MAX   7     .ucode_api_max = IWL7260_UCODE_API_MAX,         \ peter@xps:~/backports-3.10-2$ dmesg | grep iwl peter@xps:~/backports-3.10-2$ iwconfig lo        no wireless extensions.  peter@xps:~$ find / -name *7260-6* peter@xps:~$ find / -name *7260-7* /lib/firmware/iwlwifi-7260-7.ucode   peter@xps:~$ dmesg | grep -i -A 20 intel [ 1318.013278] Intel(R) Wireless WiFi driver for Linux, in-tree:d [ 1318.013281] Copyright(c) 2003-2013 Intel Corporation [ 1318.015154] cfg80211: World regulatory domain updated: [ 1318.015157] cfg80211:   (start_freq - end_freq @ bandwidth), (max_antenna_gain, max_eirp) [ 1318.015159] cfg80211:   (2402000 KHz - 2472000 KHz @ 40000 KHz), (300 mBi, 2000 mBm) [ 1318.015161] cfg80211:   (2457000 KHz - 2482000 KHz @ 40000 KHz), (300 mBi, 2000 mBm) [ 1318.015162] cfg80211:   (2474000 KHz - 2494000 KHz @ 20000 KHz), (300 mBi, 2000 mBm) [ 1318.015164] cfg80211:   (5170000 KHz - 5250000 KHz @ 40000 KHz), (300 mBi, 2000 mBm) [ 1318.015165] cfg80211:   (5735000 KHz - 5835000 KHz @ 40000 KHz), (300 mBi, 2000 mBm)     What I have found is that it appears to be loading an old version of the firmware but I don't know how or why  peter@xps:/lib/firmware$ modinfo iwlwifi | grep 7260 firmware:       iwlwifi-7260-6.ucode peter@xps:/lib/firmware$ ls *7260-6* ls: cannot access *7260-6*: No such file or directory   Any suggestions of how to get this setup very welcome. How do I assign the output of a command into an array? The number of (non-equal) forests on the vertex set V = {1, 2, ...,n} that contains exactly 2 connected components is given by The number of (non-equal) forests on the vertex set V = {1, 2, ...,n} that contains exactly 2 connected components is given by $\sum_{k=1}^{n-1} {n-1 \choose k-1} k^{k-2} (n-k)^{n-k-2}$.  I am unsure how to approach this question. Could someone guide me through the proof? If G is a spanning tree and it is disconnected then I know it is a spanning forest. Also, I know by Cayley's thm that there are $n^{n-2}$ nonequal tree on a vertex set V = {1, 2, ...,n} which is where i'm assuming the $k^{k-2}$ comes from but I do not understand where the rest of it is coming from.