Prove that the curvature of $\alpha$ is greater than $1/R^2$ for $\alpha$ lying on a sphere
Curvature of a curve on a sphere
The "Spell Storing" ability of the Shield Guardian says:     To do so, the wearer must cast the spell on the guardian.  The spell has no effect but is stored within the guardian.   This seems to indicate you cannot cast, for instance, Fire Shield on the Guardian, because that has a range of Self.  This seems different than what is intended, as the wording for a ring of spell storing is:     This ring stores spells cast into it, holding them until the attuned   wearer uses them....Any creature can cast a spell of 1st through 5th   level into the ring by touching the ring as the spell is cast.  The spell has no effect, other than to be stored in the   ring.   The use of the word "into," which does not seem to indicate any range, versus the use of the word "on," which seems to mean the guardian is a target of the spell.    Which is the correct interpretation, grammatically, versus what is the RAI? I tried Office 2013, but now wish to uninstall it. I tried to uninstall it but it always hangs at step  2 out of 4. It uses no CPU and no disk activity. So basically it's just idle.   Does anyone have a solution to this problem? Need to proof by induction (calculus1 n and n+1) 'AND' vs '&&' as operator Cannot locate Ubuntu Software Center via Firefox to open APT links Why are news headlines in present tense if they refer to past events? In news papers we see headings like this  &quot;India signs a pact with Russia&quot; &quot;Sachin hits another century&quot; &quot;Obama wins presidential election&quot;  These are completed events, aren't they? Then, why are these sentences not mentioned in past tense? Is there an upper limit to temperature in thermodynamics or statistical mechanics In many presentations of statistical mechanics where we have a system of particles having mass, such as the molecules of an ideal gas, the temperature is often equated to the average relative velocity of the particles. However no such velocity can exceed that of light. Is there therefore an absolute upper bound to temperature as well as an absolute zero? How do I call a Scala Object method using reflection? How many numbers of 6 digits, that can be formed with digits 2,3,9. And also divided by 3? How many numbers of 6 digits, that can be formed with digits 2,3,9. And also divided by 3?  $$$$ I was trying to to add 2^6 (when there is no 2)+ C(6,2)2^2 (when 2 can be in two places)+C(6,3)2^3 (when 2 can be in three places)... Converting from DMS to DD using Python in Field Calculator? I just watched a TV show and the guy says:     I ain't trying to fight with nobody with a baby.   So here, I can't understand the "nobody" since it make no sense to me, but it sounds like anybody, and if it's "I ain't trying to fight with anybody with a baby." that makes sense to me. How do I reset my Unity configuration? I've been messing around with Unity and broke something, how do I "start over"? How do I run an application with arguments from the command line without losing the terminal? I use dwm as a window manager on Ubuntu, I like tiled windows and I like them as few as possible. I sometimes need to run an application (let's say it's the pdf viewer mupdf in x11) from the terminal, the application will be launched in a new window but the terminal from which I launched becomes this prompt-less console for that application to log whatever.  I use dmenu to launch applications, but sometimes an application needs arguments (file names) to start. I use uxrvt within x11 managed by dwm. Is there any Android XML documentation? How to us possessive apostrophe with words in quotes? For example,   ...a few days later I discovered that those five little boys were not that well-behaved (as I firstly thought). In fact they were quite mischievous and conniving. The broken tree next to my house and the cat with its legs tied were all their "inventions". And Lilly, as it later turned out, was able to get to Denver so fast not without those "little inventors'" help. LINQ to SQL Where Clause Optional Criteria I am working with a LINQ to SQL query and have run into an issue where I have 4 optional fields to filter the data result on.  By optional, I mean has the choice to enter a value or not.  Specifically, a few text boxes that could have a value or have an empty string and a few drop down lists that could have had a value selected or maybe not...  For example:      using (TagsModelDataContext db = new TagsModelDataContext())      {         var query = from tags in db.TagsHeaders                     where tags.CST.Equals(this.SelectedCust.CustCode.ToUpper())                      &amp;&amp; Utility.GetDate(DateTime.Parse(this.txtOrderDateFrom.Text)) &lt;= tags.ORDDTE                     &amp;&amp; Utility.GetDate(DateTime.Parse(this.txtOrderDateTo.Text)) &gt;= tags.ORDDTE                     select tags;         this.Results = query.ToADOTable(rec =&gt; new object[] { query });     }   Now I need to add the following fields/filters, but only if they are supplied by the user.   Product Number - Comes from another table that can be joined to TagsHeaders. PO Number - a field within the TagsHeaders table. Order Number - Similar to PO #, just different column. Product Status - If the user selected this from a drop down, need to apply selected value here.   The query I already have is working great, but to complete the function, need to be able to add these 4 other items in the where clause, just don't know how! Multiplicative inverse questions 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! How long does it take for police 'presence' to clear when they are alerted? I engaged with some enemy NPCs, and accidentally shot a civilian in the head (though not sure it would've mattered if I hit them somewhere else or not) as they ran across my screen. As a result of this, I was alerted by the game that the NCPD (the police) were automatically en route to 'investigate', and that I could avoid them by leaving the area and &quot;lying low for a while&quot;. How long does it take for the police to stop searching for me, and is there any kind of visual information on the HUD that alerts me to when NCPD is or is no longer looking for me? Could we send a man safely to the Moon in a rocket without knowledge of general relativity? My images are blurry! Why isn't WPF's SnapsToDevicePixels working? I'm using some Images in my WPF applcation.  XAML:  &lt;Image Name="ImageOrderedList"        Source="images/OrderedList.png"        ToolTip="Ordered List"        Margin="0,0,5,5"        Width="20"        Height="20"        SnapsToDevicePixels="True"        MouseUp="Image_MouseUp"        MouseEnter="Image_MouseEnter"        MouseLeave="Image_MouseLeave" /&gt;   But, they appear fuzzy.  Why doesn't that SnapsToDevicePixels="True" line prevent this problem?