How do I prove what's written in the title? I was able to get an incomplete proof for the case $x&gt;2$. Here's my try: Use $e^x = \sum_{j=1}^{\infty} \frac{x^j}{j!}$.  Now we can see that if $x$ is a real number, then: $$e^{x^2/2}-e^x/2-e^{-x}/2=\sum_{j=1}^\infty \frac{\frac{x^{2j}}{2^{j-1}}-x^j-(-x)^j}{2j!}$$ and we need to show this is positive. Now, if $j$ is odd, then the numerator is just $x^{2j}/2^{j-1}$ which is always greater than zero. But if $j$ is even, multiply the numerator by $2^{j-1}$, and check if the result is positive. The result is $x^{2j} -2^{j} \cdot x^j=x^{j}(x^j-2^j)$ So if $x$ is greater then $2$, we get a positive result, but otherwise, we don't. So how do I  continue for other values of $x$?
Prove the following inequality:   $$\dfrac{e^x + e^{-x}}2 \le e^{\frac{x^2}{2}}$$   This should be solved using Taylor series.  I tried expanding the left to the 5th degree and the right site to the 3rd degree, but it didn't help.  Any tips?
Let $f$ be a holomorphic function defined on $\Bbb C$ such that $f^{-1}(\Bbb R)=\Bbb R$. Prove that there exists $a,b \in \Bbb R, \; a \neq 0 \;$ such that $f(z)=az+b \;$ for every $z \in \Bbb C$.  I was given a hint: define the function $g(z)=\frac{f(z)-f(0)}{z}$. such $g$ is also holomorphic in $\Bbb C$ since the only singularity is in $0$, and it is a removable singularity since we can define $g(0)=f'(0)$. Next I thought about proving $g$ is bounded, and thus constant (using Liouville's theorem), but pretty much out of ideas. How can I finish the proof/solve in another way? Run a command without making me wait iOS Launching Settings -> Restrictions URL Scheme How do you join on the same table, twice, in mysql? I have 2 tables. One (domains) has domain ids, and domain names (dom_id, dom_url).  the other contains actual data, 2 of which columns require a TO and FROM domain names. So I have 2 columns rev_dom_from and rev_dom_for, both of which store the domain name id, from the domains table.  Simple.  Now I need to actually display both domain names on the webpage. I know how to display one or the other, via the LEFT JOIN domains ON reviews.rev_dom_for = domains.dom_url query, and then you echo out the dom_url, which would echo out the domain name in the rev_dom_for column.  But how would I make it echo out the 2nd domain name, in the dom_rev_from column? Ubuntu 17.10 Keyboard not working after login following upgrade from 17.04 (Gnome version) How did half of Darth Maul survive? I searched around, and currently there are two methods suggested; installing an enhancement for Windows Subsystem for Linux and installing an XServer.  I want to know which method is the most hassle-free (easy to install AND to use), and which one is less memory-heavy.  I just want Synaptic and CMake. Why couldn't that be a builtin feature? I need to host several Apache virtual hosts with SSL from a single IP.  Now - I understand that because SSL wraps around the HTTP request, there's no way to know which host is being requested until a public key has been sent to the client first. This essentially breaks the possibility of SSL virtual hosts using a standard SSL certificate.  I have obtained a Unified Communications Certificate (UCC), otherwise known as a Subject Alternative Name (SAN) certificate. This allows me to serve the same certificate for multiple domains.  I would like this to be the certificate served by Apache for any SSL request - and then have Apache resolve the virtual host as usual, once the encryption has been established.  How should I configure Apache for this? I have tried to research how this can be done, but all I can find are quotes which say that it is possible, but no specifics:    wiki.apache.org/httpd/NameBasedSSLVHostsWithSNI     While Apache can renegotiate the SSL   connection later after seeing the   hostname in the request (and does),   that's too late to pick the right   server certificate to use to match the   request hostname during the initial   handshake, resulting in browser   warnings/errors about certificates   having the wrong hostname in them.   serverfault.com/questions/48334/apache-virtual-hosts-with-ssl     Incidentally, it is possible to have   multiple SSL-secured named virtual   hosts on a single IP address - I do it   on my website - but it produces all   sorts of warnings in the Apache logs,   and certificate warnings in the   browser. I certainly wouldn't   recommend it for a production site   that needs to look clean.   -David Jul 31 at 4:58   www.digicert.com/subject-alternative-name.htm     Virtual Host Multiple SSL sites on a single   IP address. Hosting multiple   SSL-enabled sites on a single server   typically requires a unique IP address   per site, but a certificate with   Subject Alternative Names can solve   this problem. Microsoft IIS 6 and   Apache are both able to Virtual Host   HTTPS sites using Unified   Communications SSL, also known as SAN   certificates.     Please help. Creating and configuring .bash_profile I was setting path for anaconda. an answer to a question  has the following solution. I searched for .bash_profile, but I can't seem to find it. How do I setup this file and get it to run on startup? I'm very new to GIS concepts. Is there a way or procedure of finding out whether the co-ordinate system is in UTM, latitude/longitude or in stateplane using .prj file?   My .prj file contains the following text:      GEOGCS["WGS 84",DATUM["WGS_1984",SPHEROID["WGS   84",6378137,298.257223563,AUTHORITY["EPSG","7030"]],AUTHORITY["EPSG","6326"]],PRIMEM["Greenwich",0,AUTHORITY["EPSG","8901"]],UNIT["degree",0.01745329251994328,AUTHORITY["EPSG","9122"]],AUTHORITY["EPSG","4326"]] On June 28, 2017, Apple  that     macOS High Sierra will be the last macOS release to support 32-bit apps without compromise.   How can I find out which apps on my Mac are 32-bit so that I can see if I need to upgrade them? Wikipedia introduces the vector product for two vectors $\vec a$ and  $\vec b$ as $$ \vec a \times\vec b=(\| \vec a\| \|\vec b\|\sin\Theta)\vec n $$ It then mentions that $\vec n$ is the vector normal to the plane made by $\vec a$ and  $\vec b$, implying that $\vec a$ and  $\vec b$ are 3D vectors. Wikipedia mentions something about a 7D cross product, but I'm not going to pretend I understand that.  My idea, which remains unconfirmed with any source, is that a cross product can be thought of a vector which is orthogonal to all vectors which you are crossing.  If, and that's a big IF, this is right over all dimensions, we know that for a set of $n-1$ $n$-dimensional vectors, there exists a vector which is orthogonal to all of them.  The magnitude would have something to do with the area/volume/hypervolume/etc. made by the vectors we are crossing.  Am I right to guess that this multidimensional aspect of cross vectors exists or is that last part utter rubbish? Why does k-means clustering algorithm use only Euclidean distance metric? Is there a specific purpose in terms of efficiency or functionality why the k-means algorithm does not use for example cosine (dis)similarity as a distance metric, but can only use the Euclidean norm? In general, will K-means method comply and be correct when other distances than Euclidean are considered or used?  [Addition by @ttnphns. The question is two-fold. "(Non)Euclidean distance" may concern distance between two data points or distance between a data point and a cluster centre. Both ways have been attempted to address in the answers so far.] How to assign vertex groups and bone weights through Blender's Python API? I'm making a script to import custom objects, it is still at early stage but so far vertices, faces and uv are created correctly. I want to do weight painting through Blender's Python API but I don't know how to do it properly.  Below an extract of the code that create the objects, how I could add vertex group and relative weights? I know how to extract vertex indices and weights from the input model but don't know how to use them,   (Blender version is 2.72)      for i in range(0,self. subsetCount):         subset = self.subset[i]         coords = subset.getV3(0)         faces = subset.getIndices()         me = bpy.data.meshes.new(subset.name+"_Mesh")          ob = bpy.data.objects.new(str(i).zfill(2)+'_'+subset.name, me)         ob.location = (0, 0, 0)         bpy.context.scene.objects.link(ob)          me.from_pydata(coords, [], faces)         me.update(calc_edges=True)          mat = bpy.data.materials.new(subset.name)         mat.diffuse_color = (random.uniform(0.0, 1.0), \                              random.uniform(0.0, 1.0), \                              random.uniform(0.0, 1.0))         me.materials.append(mat)          for f in me.polygons:             f.material_index = 0             f.use_smooth = 1          # vert_uvs = subset.getV2(0x28)         # if len(vert_uvs) &gt; 0:         #     me.uv_textures.new("UV0")         #     uvlist = [uv for pair in [vert_uvs[l.vertex_index] for l in me.loops] for uv in pair]         #     me.uv_layers[-1].data.foreach_set("uv", uvlist)   edit  found this to add the group:          vxgroup = ob.vertex_groups.new("group")   still don't know how to set weights for vertices (best would be at once)  edit  I was hoping to use something like a foreach_set but can't figure out how to use it, instead solved this way:          for grp in ob.vertex_groups:             grp.add(range(0,len(me.vertices)), 1.0, 'REPLACE')   is just an example and set all vertices weight in all groups of the object ob (if you try it in console select another mesh then back to see the weights applied). How to increase battery life on Ubuntu 20.04 and what power saving software should I install, if any? I recently installed Ubuntu 20.04 LTS on my Lenovo Ideapad Flex 5 and realized that battery life in Ubuntu is not as good as Windows. The battery drains faster in Ubuntu. I have seen the questions in Ask Ubuntu about this, but I haven't found anything that regards Ubuntu 20.04 LTS. I've done some research and found out that the battery drain is because most vendors optimize their drivers for Windows and with Linux, you have to hope that the kernel drivers do the job (which they usually do). After that, I researched how to increase my battery life and almost all sources I found recommended installing tlp or powertop or something similar to these. But I have a few questions.  Does tlp, powertop, and similar software actually affect battery life?  If the answer to 1 is yes (in a positive way), what would you recommend I install (such as whether I should  install tlp or powertop by itself or in a combination)?  Are there any other ways to increase battery life in Ubuntu 20.04?   Note: I am a beginner to Linux and Linux distros and do not want to mess with anything advanced that may negatively affect the system. I may not be able to immediately know how to tweak any of the aforementioned software, so please suggest any tweaks if they are needed to get results. And if it helps to know, here are my system specs: OS: Ubuntu 20.04.1 LTS x86_64 Host: 81X2 IdeaPad Flex 5 14ARE05 Kernel: 5.7.17-050717-generic CPU: AMD Ryzen 7 4700U with Radeon Graphics (8) @ 2.000GHz GPU: AMD ATI 04:00.0 Renoir (These were given by neofetch) P.S: If you need anything else, just ask. Thanks in advance! Recording claimable expenses in GnuCash Why did Bilbo physically age faster than Gollum? due to versus as a result of I already have cast iron skillet but I can't find a decent "normal" pan, for eggs and stuff, sometimes steaks, more or less general purpose.   Here are some more features: high quality, durable, non stick, oven safe, dish washer safe, extremely flat surface - not to be higher in the middle, this is very important.  Of course pricier usually means better quality, but is it worth it based on the criteria I'm after? Is it likely to be worth going from pans below $60, to pans from $60 to $130, or to pans over $130? Explanation on when the possessive should be used instead of an attributive noun How would you explain to a person who is learning English, and whose native language does not have attributive nouns, when the possessive should be used instead of an attributive noun?  In particular, how would you explain it referring to the following list of sentences?   Today's news is bad This week's schedule has been changed Tomorrow's lunch will be at the French restaurant This year's report will be communicated in the reunion room Last summer's vacations were memorable Last year's meetings were a complete disaster