Faced the following problem:  Let $p$ be prime, and $\bar a$ be a reduced residue in $\mathbb{Z}_p$. Let $b \in \mathbb{N}$ and consider equation $\bar x^m = \bar a$ in $\mathbb{Z}_p$. Show that the number of solutions to this equation is either $0$ or $gcd(m, p -1)$.  While I can show the similar result for equation $\bar x^m = \bar{-1}$ in $\mathbb{Z}_p$, the above one is a trouble for me. I can see some cases when there are no solutions for $\bar x^m = \bar a$, like if $m$ is even and $\bar a$ is not a quadratic residue modulo $p$. I can also show that if $gcd(m, p -1) = 1$ then $\bar x^m = \bar a$ has exactly $1$ solution in $\mathbb{Z}_p$. But if I assume $gcd(m, p - 1) = d &gt; 1$, then I can olny show that $\bar x^m = \bar 1$ has $d$ solutions and don't have an idea how to prove it for $\bar a$. Please give me a hint, if any.
Good evening! I'm studing something about finite fields and I didn't find anything that can answer my question.  I have a finite field $\mathbf{F}_q$ and the equation to solve is $x^n=a$. Which are the condition to resolve this equation? For which values of $q, n, a$ there exist a solution?  I will be very happy if anyone can answer this question, possibly with a reference of a book (or an article) that explai this with more details.  Thank in advance to everyone who can answer this one, I really need to understand this.
What is a catchy word that means (non-)self-descriptive I'm having fun with paradoxes, and I'm trying to come up with a catchy word for non-self-descriptive.  Maybe "autosemantic" fits the bill for self-descriptive (definitely open to other suggestions!), but I'm having a hard time negating it in a natural way.  For an example of the sentence I'm trying to construct:     Is "inautosemantic" inautosemantic?   It just isn't flowing. I would like to be able to execute a SOQL query that will allow me to see which users have access to a certain record type for a certain object.  For example, if I have a case record type called Phone Team, I would like to make a query that returns the names of all the users that have been assigned the Phone Team record type, as can be seen in the native user interface in a profile's Object Settings page.  So far I have managed to get a query that can get the names of all the users that have, for example, read access to the case object:  SELECT Assignee.Name FROM PermissionSetAssignment      WHERE PermissionSetId IN          (SELECT ParentId FROM ObjectPermissions WHERE SObjectType = 'Case'               AND PermissionsRead = true AND Parent.IsOwnedByProfile = true)   However, there doesn't seem to be any way to access record type assignment in the same way that you can object permissions.  I see from  that the info can be obtained via the metadata API, but I would much prefer to do it by SOQL if possible.  Thanks "1 out of 100 chickens is" or "1 out of 100 chickens are"? I'm in an argument. To me "are" makes more sense. I understand the rationale for is because it's only one chicken, but chickens itself is plural. Help? Let $m \in \mathbb{Z^+} , n \in \mathbb{Z^+}$ and let $d=\gcd(m,n)$. Prove that  $$ m\mathbb{Z}+n\mathbb{Z}=d\mathbb{Z}. $$  My attempt is use inclusion to show. Let $a \in m\mathbb{Z}+n\mathbb{Z}$. Then we have $a=ms+nt$. Since $d=\gcd(m,n)$, by Bézout's Lemma, we have $d=ms+nt$ for some integers $s$, $n$. Hence, we have $a=ms+nt=d \in d\mathbb{Z}$. Let $a \in d \mathbb{Z}$. Then we have $a=db=msb+ntb \in m\mathbb{Z}+n\mathbb{Z}$. Is my proof valid?  Remark: the following is the part 2 of the question. Prove that $m\mathbb{Z}\cap n\mathbb{Z}=\frac{mn}{d}\mathbb{Z}$. This one I have no idea how to start Find the asymptotic tight bound in  $$ T(n) = 4T\left(\frac{n}{2}\right) + n^{2}\log n. $$  where $ \log n= \log _{2}n $ and $T(1) = 1$.  I should solve this using all three common methods: iteration, master theorem and substitution.  It is highly likely that this kind of recurrence equation will be in my test in two days. Thank you very much in advance! According to general relativity, inertial mass and gravitational mass are the same, and all accelerated reference frames (such as a uniformly rotating reference frame with its proper time dilation) are physically equivalent to a gravitational field of the same strength.  Refer: Einstein's thought experiment of a Physicist in an Accelerated Box.  I find that the 'How' part of this question is somewhat easy to understand Mathematically but hard to visualize physically and on the 'Why' part I am totally blank. I've built a current sense differential amplifier that measures the voltage across a 1 Ohm resistor, to give me a reading for the current through it.  I've connected it as follows, all resistors are matched 10k to give a gain of 1:    (Image source: )  I am powering it from a single rail 12 volt source. The problem is even though this is a rail to rail opamp (Output Low swing from 5-20mV) I'm getting a constant error of 0.62V and it seems the opamp is not able to go below this value.   I have tried to resolve this by connected Vout to ground using a 10k resistor, but that makes no difference whatsoever.  I tried it with dual supply configuration (+8V, -8V) and that seems to resolve the issue. However I want to run this off a single supply rail, so this is not a viable option.  How can I resolve this? I'm trying to measure the voltage drop across a 1 ohm resistor to give me a reading for current through the circuit (0-1V represents 0-1 A) so I can set up a current limit. A better way (or practice) to doing this is also welcomed. The initial (but not ideal) fix I can think of, is to offset the readings by a few volts?  Edit:  Changed R1-R4 from 10k to 100k and I'm down to about 30-50mV! File sync error (AUTH_FAILED) How do I create a public view from a private view It seems like I can't modify a private view to be public, or duplicate a private view to be public.  Am I restricted to copying by hand all the settings of my private view, then creating a fresh public view and selecting all the right columns in the same order with all the sort and filter options? at least one element fixed by all the group $G$ is a p-group and $S$ is a set that $G$ acts on. p does not divide $|S|$. Why is there at least one element $a\in S$ such that $|O(a)|=1$, or in other words, $G_a=G$?  I tried to ask this question yesterday but did not word it right, so the one who helped me claim that this part is clearly true and I really do not understand why, and I couldn't get a straight answer for the past day.   Any of you are welcome to vote to delete this question (after some discussion with me about the problem I hope), but I just really want to know why it is "clearly true, because I still couldn't reason it right. permission error when installing ttf-mscorefonts-installer/Can't drop privileges as file couldn't be accessed by user '_apt' Possible Duplicate:          I received two answers to a question I asked. One solution suggested I do things in a completely different manner than I have been. Another acknowledged that the first solution was correct but went on to explain how to do what I was asking.  Should I accept the first answer because it provides the better way of doing things or should I accept the second one because it answers the precise question that was asked? What is the probability that $x_1+x_2+...+x_n \le n$? When a Java VM crashes with an EXCEPTION_ACCESS_VIOLATION and produces an hs_err_pidXXX.log file, what does that indicate? The error itself is basically a null pointer exception. Is it always caused by a bug in the JVM, or are there other causes like malfunctioning hardware or software conflicts?  Edit: there is a native component, this is an SWT application on win32. My earliest PhD graduation date is 10 months away. Funding permits me to stay 1–3 months longer, so the final date is still uncertain.  It is never too early to make contacts for possible post-docs. But how long in advance is too early to formally apply to post-doc job openings or funding opportunities? Depending on the bureaucracy, the path from applying to starting can probably be anything from a week up to a year — someone I know is applying with  which had the opening advertised in August, application deadline in October, will have interviews in December/January, and the commencement of work only in September, 13 months after the initial job advertisement. My personal guess is that the long time between interviews and starting is to create a level playing field between those who do and those who don't need to go through a visa application procedure.  If I'm too early in applying for a post-doc, a hosting institution might prefer someone who can start earlier and where the graduation date is more certain. For example, in my case the graduation date is still uncertain, so there is a risk in accepting even if they are willing to wait. Project funding might also require the money to be spent rather soon.  If I wait too long, there might be a considerable period where I'm between jobs. Having a long (>6 months) period between jobs is a disadvantage for me. Maybe it's also a disadvantage for a hosting institution (I'm not sure).  How do I find the balance? For example, considering the duration of funding and visa applications, does it make sense to apply e.g. ? Or is it in any case virtually inevitable to have at least several months before my PhD graduation and the start of a post-doc? Should the results of a calculation happen automatically, or require a button push? This is a problem I've ran into several times, and I imagine that others have ran across it too.    Basically, I have a complicated equation, and I want the user to type in the values needed to calculate the results.  (For what it's worth, there are good default values for the variables in the equation.)  I've constructed a GUI that shows the user the names of the variables/fields that need to be provided, along with a control (like a numeric up/down, slider, etc.) to enter the values of the field.  Below that, I have an area that shows the output of the equation.  So my question is this.  Should my results area automatically change the calculated results whenever the user makes any changes to any of the input fields, or should they push a "Calculate" button to generate the results?  One other thing that probably applies is that the user is not trying to "get the right answer" out of the results.  They shouldn't be using the output to then go back and modify the input to get a better answer.  Are there any benefits to requiring the button press?  Is the extra click to see the results worth those benefits? If $\alpha$ is an algebraic element and $L \subset K$ are both field, does the polynomial ring $L[\alpha]$ is also a field?  I am trying to prove that the ring of fraction $L(\alpha)$ is equal to $L[\alpha]$. To do this, I use an exercise done in class : Let $K$ a field, $\alpha \in K$ and $L$ a subfield of $K$. Then $L(\alpha)$ is the smallest subfield of $K$ containing $L$ and $\alpha$.  Is anyone could help me? What are some of the rule changes that Pathfinder brings in comparison to D&amp;D 3.5?  I am curious what changes were made so I can explain to my group what it has to offer. How can I keep track of a decimal number that indicates the current paragraph's place in a logical hierarchy and print it in the margin of each paragraph?  (To yield a numbering in the style of Tractatus Logico-Philosophicus).  \def\on{} \def\down{} \def\up{}  \on This is the first paragraph, paragraph number 1.  \on This is paragraph number 2.  \down This is paragraph number 2.1  \down 2 This is 2.101  \up This is 2.11  \down 2.111  \up 3 3  \bye   If \down is starred, begin with zero.  If \up is followed by an integer, it should take it as an argument, unless there is a newline between \up and the integer.  An indefinite number of decimals should be possible.  PS.  I apologize for changing the question after (2) answers have been given, but I realize I failed to appreciate a feature of Wittgenstein's numbering: the last decimal is never 0.  Therefore, \down &lt;n&gt; is better than \down*, as now demonstrated above.  I think the 0 is supposed to indicate an additional level of unimportance. Baking cloth simulation use only 20% of cpu ressources, how to get 100%? Baking a cloth use only 20% of CPU, inactive process of cpu is 78%, is there a way to use those extra 78% ? Blender 2.75 Windows 10