Equivalent definitions of primary ideals
Primary ideals in Noetherian rings
What is the meaning of the beta-coefficient for an interaction term in a crossover study? I have asked a similar question here:  I am puzzled by the interpretation for an interaction term. In my data my Y is an interval variable with the health outcome of an experiment. I have used an interaction term in which I have interacted the condition with the predisposition of the subject considering health status at base level. They are both categorical variables (factor variables in R).  Now it gets complicated because the Condition was two treatments: in treatment A subjects got the placebo first and then the real medicine whereas in treatment B they go the real medicine first and the placebo second. All it changes is the order.   Health outcome = a + Condition * Health.Base   I have the worst state of health at the base level as my reference category I find that interaction with the Condition is statistically significant but I am not sure how to interpret this.  I use the lm() function of R (although my design looks more like an ANOVA) and in the output I get the b coefficient in an output that looks like this:  ConditionB:Health.Base.So.and.So         (Beta and p-value) ConditionB:Health.Base.Excellent         (Beta and p-value)   A statistically significant interaction term for those in Excellent health at baseline would mean that they are affected by the Condition B more than Condition A compared to the reference category people (Poor health at baseline). Is this right? What does the beta-coefficient represent?   If I would like to examine for each Health category at the base line separately without comparing to a reference category I would have to code each category as a dummy variable. However, in this case I would compare whether membership to a specific health status at the base line significantly changes between conditions compared to those who belong to the other health statuses at the base line. Is this right? Again, what does the beta-coefficient represent?  Would it be right to assume that the choice of the interaction between the Condition and the dummy variables is easier to interpret?  --- EDIT --- R output:  Call: lm(formula = HealthOutcome ~ Condition * HealthStatus,      data = datA)  Residuals:     Min      1Q  Median      3Q     Max  -1.5957 -0.5942 -0.2640  0.4423  2.4423   Coefficients:                                        Estimate Std. Error t value     Pr(&gt;|t|)     (Intercept)                            2.595652   0.053044  48.934  &lt; 2e-16 *** ConditionCondB                        -0.001449   0.077071  -0.019    0.985     HealthStatusSo.and.So                 -0.331693   0.078094  -4.247  2.35e-05 *** HealthStatusExcellent                 -0.836724   0.092692  -9.027  &lt; 2e-16 *** ConditionCondB:HealthStatusSo.and.So   0.137490   0.110612   1.243    0.214     ConditionCondB:HealthStatusExcellent  -0.199787   0.133943  -1.492    0.136     --- Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1  Residual standard error: 0.8045 on 1059 degrees of freedom   (68 observations deleted due to missingness) Multiple R-squared:   0.16, Adjusted R-squared:  0.156  F-statistic: 40.33 on 5 and 1059 DF,  p-value: &lt; 2.2e-16 Table cells larger than they are meant to be I've created a map system for a game that runs on the principle of drawing the picture of the map from tiles. There are many reasons for this which I won't go into here but if you really want to know then I'm sure you can find out how to contact me ;)  I have  so you can see exactly where the problem lies and the source. The issue is the line between the top 2 tiles and the bottom 2 tiles, I can't figure out why it's gone like this and any help would be appreciated.  In the source is a marker called "stackoverflow", if you search for "stackoverflow" when viewing source then it should take you to the table in question.  I have also uploaded an . Order of a element on a Symmetric Group Where can I ask questions regarding Electronics? Objective-C for Windows What would be the best way to write Objective-C on the Windows platform?  Cygwin and gcc? Is there a way I can somehow integrate this into Visual Studio?  Along those lines - are there any suggestions as to how to link in and use the Windows SDK for something like this. Its a different beast but I know I can write assembly and link in the Windows DLLs giving me accessibility to those calls but I don't know how to do this without googling and getting piecemeal directions.  Is anyone aware of a good online or book resource to do or explain these kinds of things? From what I know, it is unknown whether $x_n=x_{n-1}^2 + 1$ has a closed form. Is there a recurrence relation which is known to have no closed form with a proof of inexistence?  Assuming a closed form is a non recursive description using the elementary operations of addition multiplication and power, or, assuming any other good definition of "closed form".  Edit  There are similar questions out there, but the answers are a little bit going around the question.   So is there one with proof or is it unknown? And if it depends on the definition of a closed form, then what are the (or some) options? How strong can a closed form definition be to still have a recursive relation that can be proved to not being able to have its form?  Update  Because it is a big question, and I don't want to open a new small one, I think a good representative small question about this subject is:  Is there a recurrence relation with a domain of the natural numbers, described by elementary functions, which has no closed form defined as:  A closed form of a recurrence relation with a domain of natural numbers is a function $f$ that can be described as a composition of a constant number of elementary functions without reccursion. i.e, $f(n) = f_1(f_2( \dots (f_k(n))) \dots) $ where $f_i$ is elementary for all $1 \leq i \leq k$   I think this definition is the most intuitive and basic.  For example, the relation $x_n = x_{n-1}+1$ has a closed form in this sense: $$f(n) = f_1(f_2(f_3(f_4(n))))$$  Where $f_1$ is division, $f_2(n)=(n,2)$, $f_3$ is multiplication and $f_4(n) = (n,n+1)$   Is there a recurrence relation which has no closed form in this sense? Creating Inky-looking characters in Blender Can I use two 7805 ICs in parallel to get double current capacity? I use a 7805 for a project where the circuit needs a higher current (~2.8&nbsp;A) at 5&nbsp;V. So I assume that if I use both ICs in parallel I can increase the maximum current capacity. Would it work? Some common problems with displaying routes created from street data, such as bus routes, include:   Routes may overlap themselves, such as when a route takes the same road in opposite directions Routes may overlap other routes that take the same street. Routes are often overly detailed, or "noisy", owing to their derivation from street data. Indicating a route's direction unambiguously without overloading it with arrows can be a challenge.   What are some techniques for dealing with these issues in order to create a practical and usable route map, such as this ? Do you know of any such public transport maps that were produced by automatic techniques rather than manual illustration/digitization?  Specifically I am using ArcGIS 10, but solutions involving other software are welcome as well.  One technique I have had some success with for dealing with 1) is using ArcMap's  with a negative offset -- assuming the lines are oriented in the direction of travel, this will offset the lines such that the right and left-hand sides are distinct and oriented as you would expect (at least for  countries, for left-hand traffic countries one would use a positive offset). However the result tends to be a bit wonky around turns, interchanges and and U-turns, and is inflexible when dealing with multiple routes.  Another possibility that I am not very familiar with but which seems promising is  -- however, I have not had much luck in finding examples of using representations for transportation maps and whether it might fit the bill. Learning Regular Expressions I don't really understand regular expressions.  Can you explain them to me in an easy-to-follow manner?  If there are any online tools or books, could you also link to them? Distribution of Product of Random Variables with one being the normal distribution. Edit record from site.com In site.com I have a page which iterates records of an object with data repeater.  I want to edit these records from this page ,but I didn't find anything to achieve this.  If there is any option to do this please help me to implement this.  Thanks Correct plural form of a noun preceded by "zero" When using zero as a quantifier, is it correct to use the singular form on the object of the quantifier, or the plural form?  It sounds confusing when I put it that way, but what I mean is: Which is correct?   Your password expires in 0 days. Your password expires in 0 day.   Essentially I suppose I'm asking, does "singular" mean "one" or "the opposite of more than one", as zero is not "plural" in the traditional "more than one" sense?  I'm pretty sure "days" sounds correct, but I can't be sure. The Download CSV link seems broken in SEDE query result window I've created   which works nice. I would really like to download those results as an CSV file. Therefor I click on the provided link on the right hand side of the screen called Download CSV. The result is:  The webpage cannot be found  https://data.stackexchange.com/stackoverflow/csv/202355?UserId=187606 I expected: A file in CSV format. It looks like a regression of  Queries without a parameter work as advertised. Can this be fixed? how many semantically different boolean functions are there for n boolean variables? In short, this is an assignment question for a course I am taking - the exact wording is this:  "Given n Boolean variables, how many 'semantically' different Boolean functions can you construct?"  Now, I had a crack at this myself - and got pretty stuck. The question doesnt state how many boolean operators there are (and, or, xor, nand, nor, iff, implies, not) nor does it state whether brackets should be used, i.e. a ^ (b v c) is different from (a ^ b) v c.  So, my question for you is - is this question possible given the limited information available?   Is it going to be something like ${n^x}$ where x is the number of boolean operators.  Any direction here would be greatly appreciated. Most efficient way to randomly "sort" (Shuffle) a list of integers in C# I need to randomly 'sort' a list of integers (0-1999) in the most efficient way possible.  Any ideas?  Currently, I am doing something like this:  bool[] bIndexSet = new bool[iItemCount];  for (int iCurIndex = 0; iCurIndex &lt; iItemCount; iCurIndex++) {     int iSwapIndex = random.Next(iItemCount);     if (!bIndexSet[iSwapIndex] &amp;&amp; iSwapIndex != iCurIndex)     {         int iTemp = values[iSwapIndex];         values[iSwapIndex] = values[iCurIndex];         values[iCurIndex] = values[iSwapIndex];         bIndexSet[iCurIndex] = true;         bIndexSet[iSwapIndex] = true;     } } Show that Statistic is complete Few application icons not appearing in Ubuntu Dock after upgrading to Ubuntu 17.10 How do I remove a malformed line from my sources.list? While learning about Random variables I came across the mean of random variable X.  The definition says that the expected value of random variable E(X) = Mean of Random variable X  I am not able to understand why is that so. Can any one please help me with it