Examples of Series that fail Ratio Test But Succeeds Root Test
Give a sequence such that root test works while ratio test fails
Is Checksum to file back decoding possible? So, I was having this weird thought. If the checksums(MD5 or SHA...) are unique to any file and are modified if any part of the file is edited, why cant the checksums be decoded back to get the original file? That is if I want to send a large(say 700MB) video, instead I could send just the hash and the file be decoded locally in my friend's computer. I know the thought is stupid, but is it possible by any chance? Or am I missing some thing here? And forgive me if I'm wrong, I ain't a professional. I met this equation frequently in Guass-Newton optimizations. But I dont understand why the left and right side of the equation can be equal.  Lets say the Jacobian is $2$ by $2$   and Hessian is $$\begin{bmatrix}\frac{\partial^2f_1}{\partial^2 x_1 } &amp; \frac{\partial^2f_1}{\partial^2 x_2  } \\ \frac{\partial^2f_2}{\partial^2 x_1 } &amp; \frac{\partial^2f_2}{\partial^2 x_2  }\end{bmatrix}$$   But the right hand side of the equation is  $$J^TJ =\begin{bmatrix} \left( \frac{\partial f_1}{\partial x_1} \right)^2 + \left( \frac{\partial f_2}{\partial x_1} \right)^2  &amp;  \frac{\partial f_1}{\partial x_1}\frac{\partial f_1}{\partial x_2} +\frac{\partial f_2}{\partial x_1}\frac{\partial f_2}{\partial x_1}\\ \frac{\partial f_1}{\partial x_1}\frac{\partial f_1}{\partial x_2} +\frac{\partial f_2}{\partial x_1}\frac{\partial f_2}{\partial x_1} &amp;\left( \frac{\partial f_1}{\partial x_2} \right)^2 + \left( \frac{\partial f_2}{\partial x_2} \right)^2\end{bmatrix} $$  Why can these two be equal as presented in papers? Prove if $|z| < 1$ and $ |w| < 1$, then $|1-zw^*| \neq 0$ and $| {{z-w} \over {1-zw^*}}| < 1$ Prove if $|z| &lt; 1$ and  $ |w| &lt; 1$, then $|1-zw^*| \neq 0$ and $| {{z-w} \over {1-zw^*}}| &lt; 1$Given that $|1-zw^*|^2 - |z-w|^2 = (1-|z|^2)(1-|w|^2)$I think the first part can be proven by saying $|1-zw^*| = 0$ if and only if $zw^*$ = 1.  And given the conditions that cannot be true. However I don't know if this part is right. I was trying to understand the internal structure of AES (Advanced Encryption Standard)  The Standard order of steps within a round:   Substitute Bytes Shift Rows Mix Columns Add Round Key   Substitute Bytes, Shift Rows &amp; Mix Columns satisfies the two most important properties required for a secure cipher i.e confusion and diffusion   Substitute Bytes provides Confusion i.e. that each binary digit (bit) of the ciphertext should depend on several parts of the key.  Shift Rows along with Mix Column provides Diffusion i.e. if we change a single bit of the plaintext, then (statistically) half of the bits in the ciphertext should change  Changing the order of Step 1 and Step 2 won't have any effect. But to what extent can we change the order of other steps? Does it weakens the security of AES, and if so, how? What's that ship in the opening credits of 'Star Trek: Enterprise'? Inductive proof of the closed formula for the Fibonacci sequence Fibonacci numbers $f(n)$ are defined recursively:  $f(n) = f(n-1) +f(n-2)$ for $n &gt; 2$ and $f(1) = 1$, $f(2) = 1$. They also admit a simple closed form: $$\sqrt 5 f( n ) = \left(\dfrac{1+ \sqrt 5}{2}\right)^n- \left(\dfrac{1 - \sqrt 5}{2}\right)^n \tag1$$  How to prove (1) using induction?  Remarks One could get (1) by the general method of solving recurrences: look for solutions of the form $f(n)=r^n$, then fit them to the initial values. But there should be a more concrete proof for this specific sequence, using the principle of mathematical induction. UK visit visa: What documents prove genuine relationship with the visitor? How can I create launcher icon for PhpStorm? I have PhpStorm on my PC, to launch it I have to go to phpstorm/bin/ and launch phpstorm.sh, how can I pin this to launcher so that I dont have to go into folders every time? Call blocking for iPhone? How do I block unwanted callers without jailbreaking? Custom post type - posts list - white screen of death i am getting a weird error - white screen in the list of posts for a specific Custom post type (just for that one)   tried deactivating all the plugins tried checking for error (debugging = true)   Still nothing the page just doesnt echos anything... (nothing in source too)  I am talking about such a url in the admin:     Here is register_post_type part i am using:    function register_submodelcpt() {     $labels = array(         'name'                  =&gt; __('Sub Models', THEME_NAME),         'singular_name'         =&gt; __('Sub Models', THEME_NAME),         'add_new'               =&gt; __('New Model', THEME_NAME),         'add_new_item'          =&gt; __('Add new Model', THEME_NAME),         'edit_item'             =&gt; __('Edit Model', THEME_NAME),         'new_item'              =&gt; __('New Model', THEME_NAME),         'all_items'             =&gt; __('All Sub Models', THEME_NAME),         'view_item'             =&gt; __('Watch Model', THEME_NAME),         'search_items'          =&gt; __('Search Models', THEME_NAME),         'not_found'             =&gt;  __('No Models found', THEME_NAME),         'not_found_in_trash'    =&gt; __('No Models found in trash', THEME_NAME),          'parent_item_colon'     =&gt; '',         'menu_name'             =&gt; __('Sub Models', THEME_NAME),      );      $args = array(         'labels'                =&gt; $labels,         'public'                =&gt; true,         'publicly_queryable'    =&gt; true,         'show_ui'               =&gt; true,          'show_in_menu'          =&gt; true,          'query_var'             =&gt; true,         'rewrite'               =&gt; array('slug' =&gt; 'submodels'),         'capability_type'       =&gt; 'post',         'has_archive'           =&gt; true,          'hierarchical'          =&gt; true,         'menu_position'         =&gt; 5,         'menu_icon'             =&gt; get_stylesheet_directory_uri().'/images/cpt/subcars.png',                     'supports'              =&gt; array('title', 'thumbnail', 'revisions', 'page-attributes')     );      register_post_type('submodelscpt',$args); } add_action('init', 'register_submodelcpt');   Did anyone encounter such an issuse? can you think of a reason this might happen?   Another weird thing when i change this:    To this:     The posts list loads correctly... What is the advantage of accepting the axiom of choice over other axioms (for e.g. axiom of determinacy)?  It seems that there is no clear reason to prefer over other axioms..  Thanks for help. When will my device get the Android 4.4 update (KitKat)? Principled way of collapsing categorical variables with many levels? What techniques are available for collapsing (or pooling) many categories to a few, for the purpose of using them as an input (predictor) in a statistical model?    Consider a variable like  (discipline chosen by an undergraduate student). It is unordered and categorical, but it can potentially have dozens of distinct levels. Let's say I want to use major as a predictor in a regression model.  Using these levels as-is for modeling leads to all sorts of issues because there are just so many. A lot of statistical precision would be thrown away to use them, and the results are hard to interpret. We're rarely interested in specific majors -- we're much more likely to be interested in broad categories (subgroups) of majors. But it isn't always clear how to divide up the levels into such higher-level categories, or even how many higher-level categories to use.  For typical data I would be happy to use factor analysis, matrix factorization, or a discrete latent modeling technique. But majors are mutually exclusive categories, so I'm hesitant to exploit their covariance for anything.  Furthermore I don't care about the major categories on their own. I care about producing higher-level categories that are coherent with respect to my regression outcome. In the binary outcome case, that suggests to me something like linear discriminant analysis (LDA) to generate higher-level categories that maximize discriminative performance. But LDA is a limited technique and that feels like dirty data dredging to me. Moreover any continuous solution will be hard to interpret.  Meanwhile something based on covariances, like multiple correspondence analysis (MCA), seems suspect to me in this case because of the inherent dependence among mutually exclusive dummy variables -- they're better suited for studying multiple categorical variables, rather than multiple categories of the same variable.  edit: to be clear, this is about collapsing categories (not selecting them), and the categories are predictors or independent variables. In hindsight, this problem seems like an appropriate time to "regularize 'em all and let God sort 'em out". Glad to see this question is interesting to so many people! Add a time limit before you can vote on a post this works in viewport but not in render.  import bpy from math import *  def frame_handler(scene):     z = scene.objects['Lift'].location[2]     scene.objects['Text'].data.body = str(floor(z / 4.0) - 2)  bpy.app.handlers.frame_change_pre.clear() bpy.app.handlers.frame_change_pre.append(frame_handler)   How to get it working in render? I've read  and changed from "bpy.data.objects" to "scene.objects" as already seen in the script. What am I missing? Blender v. 2.81.16  Edit. Tried that depsgraph thing mentioned in comments and changed to frame_change_post:  import bpy from bpy.app.handlers import persistent from math import *  @persistent  # maybe not relevant but doesn't harm anything? def frame_handler(scene, depsgraph):     z = scene.objects['Lift'].evaluated_get(depsgraph).location[2]     scene.objects['Text'].data.body = str(floor(z / 4.0) - 2)  bpy.app.handlers.frame_change_pre.clear() bpy.app.handlers.frame_change_post.clear() bpy.app.handlers.frame_change_post.append(frame_handler)   Now the Text doesn't show up at all in rendered (but works in viewport). In electromagnetism, why does nature prefer the right-hand rule over the left-hand rule? Evaluating $\int_{I^n} \left( \min_{1\le i \le n}x_i \right)^{\alpha}\,\, dx$ Let $\alpha \in \mathbb R$ and let's call $I:=[0,1]$. Evaluate   $$  \int_{I^n} \left( \min_{1\le i \le n}x_i \right)^{\alpha}\,\, dx.  $$   Well, the case $n=1$ is easy and the integral equals $\frac{1}{\alpha+1}$, for every $\mathbb R \ni \alpha \ne - 1$.  I've done also the case $n=2$ and, if I'm not wrong, it's $\displaystyle \frac{2}{(\alpha+1)(\alpha+2)}$.  My big problem is that I cannot understand how to deal with the general case. Any ideas?   Thanks in advance. How to change SQL Server collation In D&amp;D 4E as with many RPGs, characters have a fixed armour value or AC. When attacking, the attacker must roll a die to overcome the target's AC. In 4E offensive magic works the same way: the target has a fixed Fortitude/Reflex/Will defence determined by their stats, and in order to hit, the attacker rolls to overcome whichever defence is relevant.   However it was not always like this. Some spells in 3E and now again in 5E work the opposite way. In those editions the target is the one who rolls against a fixed value determined by the caster's stats, in order to avoid spell-damage.  Both methods compare the attacker and target's stats, and have an element of randomness. I personally like the 4E system as it simplifies combat without leaving anything out. Why was the decision made, when designing the early D&amp;D editions (and still when iterating upwards into (A)D&amp;D 1e&ndash;3e), to use saving throws to control spells?  Answers should consist of designer quotes and not personal opinions. Why couldn't the Jedi detect Senator Palpatine as having powers? In Return of the Jedi, when trying to sneak into the base on a shuttle, Luke said he senses Darth Vader's presence and said he shouldn't have come lest he jeopardize the mission. Darth Vader also sensed Luke, but let him in anyways. Also, Yoda and Obi-Wan could detect the Force in young Anakin.  So if Jedi and Sith can detect each other, why couldn't they detect Senator Palpatine to begin with? It seems like a giant plot hole.