Dirichlet's Approximation Theorem shows how it is possible to find a rational approximation as close as required to any real number. Has it been proved whether there always exists a ratio of two primes as close as required to any real number?
Consider the closed interval $[0,1]$, there is $\frac{2}{3} \in [0,1]$ where $p=2$ and $q=3$. Similarly consider $[2,3]$, one can have $\frac{5}{2} \in [2,3]$ where $p=5$ and $q=2$. Does every interval of the form $[a,b]$, where $a,b \in \mathbb{R}$ contain a rational of this kind. If yes, how can we prove it?
What does "sudo apt-get update" do? Why is Buoyant Force $V\rho g$? For a submerged object, buoyant force ($F_b$) is defined as:  $$F_b = V_{\text{submerged}} \times \rho \text{ (density)} \times g \text{ (gravitational constant)}$$   Conceptually, the buoyant force equation says that buoyant force exerted is equal to the weight of the volume of water a given object displaced. Why? I went online to   and found the following explanation, but it seems like non-sequitur to me:      Explanation: When an object is removed, the volume that the object occupied will fill with fluid. This volume of fluid must be supported by the pressure of the surrounding liquid since a fluid can not support itself. When no object is present, the net upward force on this volume of fluid must equal to its weight, i.e. the weight of the fluid displaced. When the object is present, this same upward force will act on the object. Short story or novel from the 1970s or before about two warring alien tribes This is my first time posting on Math exchange. I have been self-teaching myself Mathematics and recently started learning some Algebra. I was posed the following questions, in which I would like to answer before I proceed, but having some trouble finding proper solutions. So the format will be question, and my answer.      What name do we give an element $a \in \mathbb{Z}_n$ that has a multiplicative inverse?   A modular multiplicative inverse of an integer $a$ is an integer $x$ such that $ax \cong 1 (\mod m)$. This means that the remainder after dividing $ax$ by integer $m$ is equal to $1$.      Give an example, for some $n \in \mathbb{N}$, of an element of $\mathbb{Z}_n$, that does not have a multiplicative inverse?   This question is throwing me off because - does the $n$ we speak of matter for both $n \in \mathbb{N}$ and $\mathbb{Z}_n$ ot solely $n$? My guess here would be $0$ because we cannot have an inverse of $0$.      Prove that multiplicative inverse are unique; an element $a \in \mathbb{Z}_n$ cannot have two multiplicative inverses.    Suppose that an element $a\in\mathbb{Z}_n$ has an inverse $a'$. Next, suppose that $a^*$ is also an inverse of $a$. Then, $a'$ is a solution to $a \cdot_n x=1$, and similarly $a^*$ has a solution to $a \cdot_n x=1$.   Lemma: Suppose $a'$ is multiplicative inverse of $a$ in $\mathbb{Z}_n$, then for any $b \in \mathbb{Z}_n$ the equation $a \cdot_n x=b$ has a unique solution, such that $x = a' \cdot_n b$.   By Lemma, equation $a \cdot_n x = 1$ has a unique solution, namely $x=a'=a^*$. Therefore, an element $a\in \mathbb{Z}_n$ cannot have two multiplicative inverses.    Thank you for the help. I would love any constructive feedback for these. If you can help me elaborate on some ideas, that would be really helpful also.  Thank you very much! Are the XBLA and PC versions of Minecraft compatible? Area is calculating wrong using $area in field calculator (QGIS 2.8.1-Wien) When should a verb be followed by a gerund instead of an infinitive? Some verbs are followed by ing, e.g. I enjoy swimming. We can't say I enjoy to swim. Likewise, some verbs are followed by to, e.g. I decided to make a plan.   Which particular verbs are followed by ing and to? Can you please provide a list for that?  Moreover, which verbs can be followed by both without having the meaning changed? I am kind of confused on the terminology of &quot;space&quot;. From  I am getting that  In mathematics, a space is a set (sometimes called a universe) with some added structure.  And from topological and metric spaces, I know that we a space is a a tuple of a set and a structure, e.g. $(X, \tau)$, $(X, d)$, where $\tau$ and $d$ are a topology and a metric respectively. On the other hand, in machine learning the term &quot;features space&quot; is used a lot for sets like $\mathbb R^n$, e.g.  and although it often refers to the set underlaying set itself, not the tuple of set plus the added structure. In fact, I have done the same in a previous publication (which the peer-reviewers accepted), but I would like to be both correct and precise in what I research, write, and submit. Is it just that machine learning researchers are imprecise in their terminology? Is it just an ? I think, what some people are doing, might be they consider a space to be a set, which is somewhat structured, instead considering the set with the structure (so, the tuple), to be the space.  I am not sure how to think about this. How do I deal with this in my writing? Especially since, what happens a lot is that I need subsets and element out of the underlying sets of all kinds of spaces.  Of course it would greatly help if there was a general name for a set that is the underlying set of a space, , but it seems that there is no dedicated name for such a set. Let's say there was a features space $(X, \cdot)$, (where I am not even sure, what structure we would add). Then it would be great to have a name for the set, let's call it an asdf. So we could say the feature asdf.  Afterthoughts: What makes it worse for me is that it seems ( and ) I am not even using the word &quot;structure&quot;, right, since it is about operations, so we are talking about an &quot;algebraic structure&quot;, while &quot;geometric space&quot; is... something else...? So apparently a &quot;vector space&quot; is actually not a &quot;geometric space&quot;, but an &quot;algebraic structure&quot;. I can understand that point, but the language gets even more confusing. Failed upgrade from Ubuntu 19.10 to 20.04 "Please update the microcode" While installing the Ubuntu 20.04 updates during upgrade from 19.10 to 20.04, my screen suddenly went completely white and displayed a message saying  Please logout and try again.  But I could not locate my cursor and so I switched off my laptop. After restarting it again, it was showing an error message saying  Please update the microcode  Why did this happen? How can I solve this problem? I have a spreadsheet in Google Docs with two data rows and lots of columns like so:    Is it possible to easily convert these rows into two columns and lots of rows (obviously retaining all information and formulas)? In other words, to something like this:   19.9.2009   967,15 €  28.9.2009   950,78 € 12.11.2009   942,03 € ..           ..   There's probably a neat mathematical term for such a conversion, but I don't remember which (matrix something?). Feel free to suggest a better name for the question.  Edit: To clarify, ultimately I want to get rid of the original (two row) version and continue working on the converted one. TRANSPOSE(), , got me very close but not quite there. I want the raw data in the transposed version, not cells that mirror the original ones. How to change Function (Fn) Key behavior on wireless Logitech keyboard What is the non-parametric equivalent of a two-way ANOVA that can include interactions? Hi I am trying to find the non-parametric equivalent of a two-way ANOVA (3x4 design) which is capable of including interactions. From my reading in Zar 1984 "Biostatistical analysis" this is possible using a method put forth in Scheirer, Ray, and Hare (1976), however, according to other posts online it was inferred that this method is no longer appropriate (if it ever was).  Does anyone know what method would be appropriate for doing so, and if so the corresponding functions in R or Stata? How to change Power button shutdown action to run a script under systemd How do I make a bootable Ubuntu USB? Why do I get TypeError: can't multiply sequence by non-int of type 'float'? Ever since mods gained the ability to undelete comments, deleting them has become easier as the confirmation dialog was removed.  However, this is not the case in the mod queue &mdash; I still get a confirmation dialog for each comment I want to delete in the queue under a custom-flagged post (single-comment flags can be deleted just fine). And since this is the place from where I delete comments the most, it can be annoying.  In addition, the "purge comments" button, when invoked from the queue, refreshes it, so I have to scroll back down, remember which post I just purged, and mark the flag as helpful. Usually not a big issue but if there are multiple "obsolete comments" flags, then it can get confusing.  Could the workflow for comment thread moderation from within the queue be improved? Ideally, make comment deletion work like everywhere else on the site, and make the purge button AJAX-based like the comment deletes. Equivalent Resistance? How do you find the equivalent resistance here? (I realized it's a Wheatstone bridge, but it's unbalanced, so that doesn't help.) I tried using Kirchhoff's laws, but there is no cell given. I need to test this code, but when writing a trigger, I get an invalid type I need help. I am currently trying to test this code before uploading  public class CoreValuesExt { ApexPages.StandardController sc; public boolean readonly {get; set;} public Core_Values__c cv {get; set;} public integer countEdit = 1; public string buttonValue {get; set;} public integer recordScan;  public CoreValuesExt(ApexPages.StandardController sc) {     this.sc = sc;     cv = new Core_Values__c();     cv.Account__c = sc.getid();      readonly = true;     buttonValue = 'Edit';     recordScan = [SELECT COUNT() FROM Core_Values__c WHERE Account__c = :cv.Account__c];    if (recordScan != 0){    cv = [SELECT id, HSE__c, Professionalism_Integrity__c, Quality_Accuracy__c, Technology_Innovation__c, Timeliness__c, Value_for_money__c FROM Core_Values__c WHERE Account__c = :cv.Account__c];     }  }  public ApexPages.PageReference Save() {     return sc.Save(); }  public void ApplyAction() {     countEdit++;     Integer modolus = Math.mod(countEdit, 2);     String total = cv.HSE__c + cv.Professionalism_Integrity__c + cv.Quality_Accuracy__c + cv.Technology_Innovation__c + cv.Value_for_Money__c + cv.Timeliness__c;     if(modolus == 0)     {         readonly = false;         buttonValue = 'Save';     }     else     {         if(total != 'nullnullnullnullnullnull')            {         if (recordScan == 0){         insert cv;         readonly = true;         buttonValue = 'Edit';         }         else         {         update cv;         readonly = true;         buttonValue = 'Edit';         }         }     }    }   }  With this trigger.  trigger CoreValuesTrigger on Core_Values__c (before insert) {  Core_Values__c cvs = Trigger.new; CoreValuesExt.ApplyAction(cvs);  }   But when I try to save, I get a message saying:      Illegal assignment from List to   CoreValuesExt__Core_Values__c at line 3 column 1   "  How do I go through with this? I need help. Thanks Using global variables in a function Proving that kernels of evaluation maps are generated by the $x_i - a_i$ I want to prove that the kernel of the evaluation map $s_a : \mathbb{C}[x_1,\dots,x_n] \rightarrow \mathbb{C}, x_i \mapsto a_i$ where $a = (a_1,\dots, a_n) \in \mathbb{C}^n$ is the ideal generated by $\{x_1 - a_1, \dots, x_n - a_n\}$.    The proof in the book first shows it easily for the case $a = 0$, and says that by substitutions of variables $x_i' = x_i - a_i$, you can show it for the rest of the cases.  I know that $f \in \ker s_a$ iff $f(a) = 0$.  And $\phi : \mathbb{C}[x_1,\dots, x_n] \rightarrow \mathbb{C}[x_1 - a_1, \dots, x_n-a_n], x_i \mapsto x_i - a_i$ is a surjective ring homomorphism.  Please give a hint and not the full answer.  Thanks.