Can a multiclassed Sorcerer/Tempest Cleric use Charisma instead of Wisdom to determine save DC for the Wrath of the Storm feature?
What is the DC of the Tempest Cleric's Wrath of the Storm feature?
I'm trying to perfectly overlap two images, one complementary of the other, but apperas a white stripe in the conjunction of the path, all points are perfectly aligned how can I fix? Proof that the One-Dimensional Simple Harmonic Oscillator is Non-Degenerate? The standard treatment of the one-dimensional  (SHO) using the raising and lowering operators arrives at the countable basis of eigenstates $\{\vert n \rangle\}_{n = 0}^{\infty}$ each with corresponding eigenvalue $E_n = \omega \left(n + \frac{1}{2}\right)$. Refer to this construction as the abstract solution.  How does the abstract solution also prove uniqueness? Why is there only one unique sequence of countable eigenstates? In particular, can one prove the state $\vert 0\rangle$ is the unique ground state without resorting to coordinate representation? (It would then follow that the set $\{\vert n \rangle\}_{n = 0}^{\infty}$ is also unique.)  The uniqueness condition is obvious if one solves the problem in coordinate representation since then one works in the realm of differential equations where uniqueness theorems abound. Most textbooks ignore this detail (especially since they often solve the problem both in coordinate representation and abstractly), however I have found two exceptions:   Shankar appeals to a theorem which proves one-dimensional systems are non-degenerate, however this is unsatisfactory for two reasons:   Not every one-dimensional system is non-degenerate, however a general result can be proven for a large class of potentials (the SHO potential is in such a class). The proof requires a departure from the abstract solution since it classifies the potentials according to their functional properties.  Griffiths addresses this concern in a footnote stating that the equation $a \vert 0\rangle = 0$ uniquely determines the state $\vert 0\rangle$. Perhaps this follows from the abstract solution, however I do not see how. Testing random number generation I have a program which generates random numbers. How can I test if the random number generator is generating actual random numbers? Available space not changing when resizing a partition how to set static ip from bash scriptlinux I want to set static IP using bash script what I would to achieve I want to set the same IP to static which is come from DHCP, for example, my eth0 set to DHCP and it picks IP 51.89.84.82 for eth1 it pick 10.0.0.1 I want to set the same IP to static how can I achieve this in centos7. 1 ) find interface name and its ip , subnet mask , gateway. 2 ) loop each interface and set the same ip , subnet and gateway. Is there a way to "rescue" mesh that has very bad topology? Resizing Controls in MFC I am writing a program which has two panes (via CSplitter), however I am having problems figuring out out to resize the controls in each frame. For simplicity, can someone tell me how I would do it for a basic frame with a single CEdit control?   I'm fairly sure it is to do with the CEdit::OnSize() function... But I'm not really getting anywhere...  Thanks! :) Is it possible to mount a virtual hard disk (VHD, HDD, VDI, VMDK) on Ubuntu? How can this be done? Prove by induction that $n! > n^2$ How does one prove by induction that  $n! &gt; n^2$  for $n \geq 4$ I'm trying to install bang and olufsen drivers, or HP hd audio driver, as only 2 of the 4 speakers work and the 2 amps don't work in ubuntu  lspci -v | grep -i audio 00:03.0 Audio device: Intel Corporation Broadwell-U Audio Controller (rev 09)     Subsystem: Hewlett-Packard Company Broadwell-U Audio Controller 00:1b.0 Audio device: Intel Corporation Wildcat Point-LP High Definition Audio Controller (rev 03)     Subsystem: Hewlett-Packard Company Wildcat Point-LP High Definition Audio Controller   The volume is terrible. Is there any way to fix this?  The speakers work perfectly on the pre-installed Windows 8. I have the following equation, where there is a lot of stuff under the sum symbol:  \begin{equation} d(\vec{x},\vec{y})=   \sum_{Z_{xy}\in\vec{Z}_{xy},\forall x\in\vec{x},\forall y\in\vec{y}}      f(Z_{xy}) \end{equation}   In the resulting document, I find it kind of hard to read. Is there a way to write the equation to make the result more readable, e.g. putting the stuff under the sum symbol on different lines? Does the force of kinetic friction increase with the relative speed of the objects involved? If not, why not? Does the force of kinetic friction increase with the relative speed of the objects involved?    I have heard and read that the answer is no.  This is counter intuitive, and is a big part of why the "Airplane on a Treadmill" question is so interesting.  What phenomena are at work with kinetic friction, and why does it not increase with relative speed? My question is specifically for badges in a particular tag. I've heard that the rules have been changed.  How and when can one get bronze, silver and gold badges for a specific tag, say ? What are the new rules governing these badges? Should I remove Adult Dependent from my 2019 tax return so they will receive a COVID Stimulus Check? On my 2019 Tax Return I claimed my 28 year old girlfriend as a dependent. This makes her ineligible to receive a COVID Stimulus Check. My return was already accepted and I have already received my refund. Should I amend my return, remove her as a dependent? Would she then be eligible to receive the stimulus check? Finding plane equation given two lines Determine an equation of the plane containing the lines $\frac{x-1}{2}=\frac{y+1}{-1}=\frac{z-5}{6}$; $r=\lt1,-1,5\gt+t\lt1,1,-3\gt$.   I calculated the cross product between the directional vector of both lines to find the normal vector $n$, but when I looked for an intersection point $r_0$ to apply the formula: $\lt r-r_0\gt \bullet \ \ n$, I did not find any.  Can I use the point $\lt1,-1,5\gt$ given in the line $r$? Or is it not possible to find an equation of the plane containing two lines that do not intersect? I want to create buffer polygon(with curved edges) from  points like this using ArcGIS for Desktop.   distance toward North from a point 5, toward South 1.5, toward west 1, toward East 2.   all this distance is attached as attribute in point in different fields. Proof of the First Isomorphism Theorem for Groups The statement is the First Isomorphism Theorem for Groups from Abstract Algebra by Dummit and Foote. This proof was left as a exercise, so I'd like to check if all is ok. In particular, I'm a bit worried about the (*) line. It looks a bit awkward, but is it too bad? Any suggestions?    Theorem: If $\varphi : G \rightarrow H$ is a homomorphism of groups, then $ker \varphi \trianglelefteq G$ and $G/ker \varphi \cong \varphi (G)$.  Proof: Let $\varphi:G \rightarrow H$ be a group homomorphism. If $g \in G$ and $h \in ker \varphi$, then we have that $ghg^{-1} \in ker \varphi$ because $$\varphi (ghg^{-1})= \varphi (g) \varphi (h) \varphi (g^{-1})= \varphi (g) \varphi (g^{-1}) = \varphi (gg^{-1}) = \varphi (1) =1,$$ so that $ker \varphi \trianglelefteq G$. Now, define $\phi: G/ker \varphi \rightarrow \varphi (G)$ by $$\phi (g(ker \varphi))=\varphi(g),$$ for each $g(ker \varphi) \in G/ker \varphi$, for some $g \in G$. This is well defined because if $g' \in g(ker \varphi),$ then $$\phi (g'(ker \varphi))= \varphi(g')= \varphi(g)= \phi (g(ker \varphi)).$$ Also, this is a group isomorphism because for each $\varphi (h) \in \varphi (G),$ for some $h \in G$, we have $$(*) \ \phi^{-1}\{\varphi(h)\}= \phi^{-1} \varphi[g(ker \varphi)]= \phi^{-1} \varphi(\pi^{-1}\{g(ker \varphi)\})= \{g(ker \varphi)\},$$ a set with a single element of $G/ker \varphi$, so that it is a bijection, and for every $g(ker \varphi), g'(ker \varphi) \in G/ker \varphi$, for some $g,g' \in G$, we have $$\phi [g(ker \varphi) g'(ker \varphi)]=\phi [(gg')(ker \varphi)]=\varphi(gg')=\varphi(g) \varphi(g')=\phi [g(ker \varphi)] \phi [g'(ker \varphi)],$$ so that it is a group homomorphism. Support for hardware components How can I find out if Ubuntu supports a particular hardware component, that is, if it works in Ubuntu?  I am going to put together a computer and I am choosing components.   How do I know if Ubuntu supports this or that motherboard, or this or that sound card, or this or that graphics card, etc? How do I know that there is no conflict in Ubuntu between this and that hardware component?   What resources are there to help with such situations?  I know that there are computers with Ubuntu pre-installed and that there are  for whole computers but this is not what I'm looking for. I'm looking for resources to find out support for particular hardware components and conflicts among particular hardware components. Can you retrain the Magic Initiate spell using the Spellcasting feature? got me thinking about the mechanics of the Magic Initiate feat when you have class levels in the class into which you were inducted.    Among the classes you can pick with this feat are the Bard, Sorcerer, and Warlock, which know a limited number of spells and all have the same statement under Spells Known.     Additionally, when you gain a level in this class, you can choose one of the [class] spells you know and replace it with another spell from the [class] spell list , which must also be of a level for which you have spell slots.   The term "learning" is used both to describe new spells acquired on level-up and the spell acquired through the feat. This would seem to indicate your Magic Initiate spell counts among the spells you know, and is thus eligible for retraining if you learned it as a spell of your class.  An odd change in wording occurs during the Magic Initiate feat description (emphasis mine):     In addition, choose one 1st-level spell from the same list. You learn that spell and can cast it at its lowest level.   This generalization in describing the learned spell's level seems to imply that there exists a scenario where it would not be appropriate to restrict the spell to being cast as a 1st-level spell.  Could one of the charisma casters retrain Magic Initiate into a higher level spell, which they could then cast once per long rest without using a slot?  This would turn an otherwise unexciting feat into, effectively, a Mystic Arcanum for Bards and Sorcerers. This seems too powerful to be intentional,  but the feat description appears to have gone out of its way to permit it. This occurs in the  released recently to the public. When someone posts a code block in their question or answer, the code block is cut off on the right hand side, and I can't scroll to the right to view the rest of it. How can I view these code blocks?