uncountable noun + and + uncountable noun Which sounds better?    There is water and butter in my fridge. There are water and butter in my fridge.   I think it should be: is.  But what if we said:    How much flour and butter is needed to make a pizza? How much flour and butter are needed to make a pizza?   In that case, I think the plural verb: are, is the correct choice, which means (I think) there is a contradiction between both sentences.
"There Is"/"There are" depends on plurality of the first list element or not? It seems I put a   Bounty assigned by outside party, two lengthy, reference-citing answers, one "-1" (awarded the bounty), one "-2", two others scored "0" and "-2" respectively, the answers suggesting one or the other is correct, 73 comments and no consensus so far - and me, as the asker, lost without a clue what to think of the answers anymore -  this is no longer "Learners' Level" question, so I thought I'd bring it here and hear what the experts have to say.    Original question:  Plurality of verb depending on plurality of list elements  An edit was suggested to my sentence.     There was were an orange, some grapes, two apples and a small pile of cherries on the plate.   In my native language plurality of the verb always follows plurality of the first element on the list. There were an orange,... sounds awkward to me, no matter what follows up. My simple solution was reordering:     There were some grapes, an orange, two apples and a small pile of cherries on the plate.   But that's not the first time I faced this situation and I'd like to know what the rules of grammar say about that - was my editor overzealous or am I trying to copy rules of my language that don't apply in English?    Someone linked a  for negatives, where the situation is more clear-cut ("There is no..."). Same applies to connection with "or" apparently, per accepted answer there. Still, nothing for "and".  It seems there is a consensus that in that if the verb goes after the list, it will be plural.     an orange, some grapes, two apples and a small pile of cherries were on the plate.   There doesn't seem to be clear consensus for:     On the plate { was | were } an orange, some grapes, two apples and a small pile of cherries   Adding more to the confusion is the abbreviated form: "There's"     One last note. In North America, at least, there is a widespread use of using a singular form like "there is" and "there was", without regard for the subject item or items, and this "there is" is often shortened to "there's":           There's three apples on the table!      Could you please clarify this mess?
Measures of residuals heteroscedasticity This wikipedia  lists a number of techniques to detect OLS residuals heteroscedasticity. I would like to learn which hands-on technique is more efficient in detecting regions affected by heteroscedasticity.  For example, here the central region in the OLS 'Residuals vs Fitted' plot seen to have higher variance than the sides of the plot (I am not entirely sure in facts, but let's assume it's the case for the sake of the question). To confirm, looking at the error labels in the QQ plot we can see that they match the error labels in the center of the Residuals plot.  But how can we quantify the residuals region that has significantly higher variance? How do I change NEI to cheat mode? sudo credential caching on by default What is a good complex analysis textbook, barring Ahlfors's? I'm out of college, and trying to self-learn complex analysis. I'm finding  text difficult. Any recommendations? I'm probably at an intermediate sophistication level for an undergrad. (Bonus points if the text has a section on the Riemann Zeta function and the Prime Number Theorem.) Why don't Death Eaters always use Avada Kedavra? When the Death Eaters attacked Hogwarts, why did the Death Eaters not just use Avada Kedavra and kill everyone? How to calculate Area Under the Curve (AUC), or the c-statistic, by hand I am interested in calculating area under the curve (AUC), or the c-statistic, by hand for a binary logistic regression model.  For example, in the validation dataset, I have the true value for the dependent variable, retention (1 = retained; 0 = not retained), as well as a predicted retention status for each observation generated by my regression analysis using a model that was built using the training set (this will range from 0 to 1).  My initial thoughts were to identify the "correct" number of model classifications and simply divide the number of "correct" observations by the number of total observations to calculate the c-statistic. By "correct", if the true retention status of an observation = 1 and the predicted retention status is > 0.5 then that is a "correct" classification. Additionally, if the true retention status of an observation = 0 and the predicted retention status is &lt; 0.5 then that is also a "correct" classification. I assume a "tie" would occur when the predicted value = 0.5, but that phenomenon does not occur in my validation dataset. On the other hand, "incorrect" classifications would be if the true retention status of an observation = 1 and the predicted retention status is &lt; 0.5 or if the true retention status for an outcome = 0 and the predicted retention status is > 0.5. I am aware of TP, FP, FN, TN, but not aware of how to calculate the c-statistic given this information. Construct a Borel set on R such that it intersect every open interval with non-zero non-"full" measure This is from problem $8$, Chapter II of Rudin's Real and Complex Analysis.  The problem asks for a Borel set $M$ on $R$, such that for any interval $I$, $M \cap I$ has measure greater than $0$ and less than $m(I)$.  I was thinking of taking the Cantor approach: taking $R$ to be the union of $[a,b]$ with $a$ and $b$ rationals, and for each $[a,b]$ we construct Cantor sets inside it. During theconstruction of each Cantor set, in order to have positive measure on it, we need to take off smaller and smaller intervals from it, namely the proportion goes to $0$. As a result, these Cantor sets are extremely "dense" on their ends. If for an interval $I$ it intersects with the Cantor set on $[a,b]$ while $b-a&gt;&gt;m(I)$, we shall expect the measure of intersection to be rather close to $m(I)$ and then we lose control on these cases.  Is there any way to fix this or shall I consider other approaches?  Thank you Setting a document in MS Word-12pt (12bp) Warning  If you're looking for advice how to make your document look like it's been written in Word, this is most likely not the question you're looking for. This question is mostly of theoretical nature, as it results in tiny differences, which will most likely not be noticed by someone who doesn't allow the use of LaTeX.  Questions that might be more helpful for this matter are:           Question  I learned that MS Word uses a slightly different version of the unit "point" (pt) than TeX does:     The 12 point of Word will be PostScript point, which in TeX would be called 12bp. A TeX pt is slightly smaller: it's 1/72.27 inch, while a bp/PostScript point is 1/72 inch. See also    ( in )  I'm writing a paper that would usually be expected to be "typeset" in MS Word, thus I want to use the same font size as Word would.  How do I set a document e.g. in the "12pt" font size that MS Word would use?  In case it matters, I'm using the article document class, Latin Modern (lmodern) as a font with the T1 font encoding and compile with pdfLaTeX, but input on different set-ups is more than welcome. How does conservation work in wormholes? My questions might seem naive as I haven't plunged into the mathematical descriptions, but wormholes are interesting even if I don't actually know them :)  First of all, is wormhole making two possible paths betwwen points (like the tunnel in most visualisations) or is it just making two points beside each other like in the game Portal? Of course, second case can be bent into a tunnel, but there would be nothing special about tunnel part and no preferred way to divide into "normal" and "tunnel" part of spacetime.  My main questions are these:   Is momentum conserved? Is kinetic energy conserved? How does potential energy work?   To elaborate... In the game Portal you can fall into floor and fly out of the wall or floor elsewhere with the same speed, but another direction (normal to the portal). So momentum is not conserved, but kinetic energy is. Is that how it would actually work? Where does the momentum go then?  And does it really allow you to increase your potential energy while retaining kinetic? It would seem fair to take away some kinetic energy if you are put in a region where you have more potential energy.  Either way is the change in energy non-continuous like in the game and discussion that's linked below? Or are there just more than one way to move through but the change in potential energy is smooth in any of those? That would mean more potential equilibrium points though...  Similar questions has been discussed , is it correct about how the falling guys gets a lot of energy (in form of a potential one) by draining it from the mouths? A common mistake for beginning group theory students is the belief that a quotient of a group $G$ is necessarily isomorphic to a subgroup of $G$.  Is there a characterization of the groups in which this property holds?  If this question is too broad, I might ask if such a characterization exists for $p$-groups.    History: I originally posed the opposite question, regarding groups for which $\exists N\unlhd G\,:\, \not\exists H \unlhd G\, \text{  s.t. } H \cong G/N$, and crossposted this to .  I received an answer there to the (now omitted) peripheral question about probability, which shows that most finite groups probably have this property.  After this, I changed the question to its current state, as this smaller collection of groups is more likely to be characterizable. how to make winforms webbrowser control to work very similar to targeted Internet explorer My intent is to show that a composition of bijections is also a bijection by showing the existence of an inverse.  But my approach requires the associativity of function composition.  Let $f:  X \rightarrow Y, g:  Y \rightarrow Z, h:  Z \rightarrow W$ be functions.  $((f \circ g) \circ h)(x) = h((f \circ g)(x)) = h(g(f(x)))$, and $(f \circ (g \circ h))(x) = (g \circ h)(f(x)) = h(g(f(x)))$.  However, I am having problems in justifying that the two compositions, $(f \circ g) \circ h$ and $f \circ (g \circ h)$, have the same domain and range.  When I consulted ProofWiki, whose link is at the bottom, I got even more confused.  Specifically, for $(f \circ g) \circ h = f \circ (g \circ h)$ to be defined, ProofWiki requires that dom$g =$ codom$f$ and dom$h =$ codom$g$.  First of all, I think that it should be dom$g =$ range$f$ ....  Moreover, as you can see in the example below, you actually have to adjust domains and ranges of $f, g, h$ for the requirement to hold true.  Let $f: \mathbb R \rightarrow \mathbb R$ be $f(x) = 2x$, $g: \mathbb R^+ \rightarrow \mathbb R$ be $g(y) = ln(y)$, $h: \mathbb R \rightarrow \mathbb R$ be $h(z) = z - 10$.   Then $((f \circ g) \circ h)(x) = ln(2x) - 10 = (f \circ (g \circ h))(x)$, with dom$((f \circ g) \circ h) = \mathbb R^+$ = dom$(f \circ (g \circ h))$.  As a result, we need to set dom$f = \mathbb R^+$, range$f = \mathbb R^+$; dom$g$, range$g$, dom$h$, and range$h$ remain the same.  Am I allowed to do that?  This adjustment implies that when we say dom$f = X$, $f$ must be defined for all elements in $X$, but $X$ may not be the entire set of elements for which $f$ is defined. In which situations is it OK to omit articles in short sentences? Recently, in many new online software I often see that developers omit articles in short sentences, especially on action buttons or tooltips.     For instance: "Add new task" (not "Add a new task"), "Create project" (not "Create a project"), but often in the same software I can see "Make a copy". What are situations it can be justified? Why the first moment is standardized before computing higher moments, but higher moments are not? I am creating table in LaTeX and I need to get the text to perform line break automaticaly when it's too long. And if possible, I'd like to get the text alignment to the left. Is it possible to handle this situation? How would one prove that $$\lim_{n\rightarrow \infty} \frac{F_{n+1}}{F_n}=\frac{\sqrt{5}+1}{2}=\varphi$$  where $F_n$ is the nth Fibonacci number and $\varphi$ is the Golden Ratio? Martingale formulation of Bellman's Optimality Principle From David Williams' Probability with Martingales     Related question:   What I tried: For no 2, if $\ln Z_n - n \alpha$ is a supermartingale, then for $m &lt; n$, $$E[\ln Z_n - n \alpha | \mathscr F_m] \le \ln Z_m - m \alpha$$ $$ \to E[\ln Z_N - N \alpha | \mathscr F_0] \le \ln Z_0 - 0 \alpha$$ $$ \to E[\ln Z_N - N \alpha] \le \ln Z_0$$ $$ \to E[\ln \frac{Z_N}{Z_0}] \le N \alpha$$ Is that right?  For no 1, for $m &lt; n$, $$E[\ln Z_n - n \alpha | \mathscr F_m] = E[\ln Z_n | \mathscr F_m] - n \alpha$$ $$ = E[\ln \frac{Z_n}{Z_0} | \mathscr F_m] - n \alpha + \ln Z_0$$ $$ = E[\ln \frac{Z_n}{Z_{n-1}} | \mathscr F_m] + ... + E[\ln \frac{Z_{m+1}}{Z_{m}} | \mathscr F_m]$$ $$+ \ln \frac{Z_m}{Z_{m-1}} + ... + \ln \frac{Z_{1}}{Z_{0}} - n \alpha + \ln Z_0$$ Note that $$Z_n - Z_{n-1} \le |Z_n - Z_{n-1}| \le C_n |\epsilon_n - \epsilon_{n-1}| \le 2C_n \le 2Z_{n-1}$$ Also, I think $$\ln \frac{Z_n}{Z_{n-1}} \le |Z_n - Z_{n-1}|$$ Hence we have $$E[\ln \frac{Z_n}{Z_{n-1}} | \mathscr F_m] + ... + E[\ln \frac{Z_{m+1}}{Z_{m}} | \mathscr F_m]$$ $$+ \ln Z_m - n \alpha$$ $$\le E[2Z_{n-1} | \mathscr F_m] + ... + E[2Z_{m} | \mathscr F_m]$$ $$+ \ln Z_m - n \alpha$$ $$= 2(E[Z_{n-1} | \mathscr F_m] + ... + E[Z_{m} | \mathscr F_m])$$ $$+ \ln Z_m - n \alpha$$ One thing to do would be to show that $$2E[Z_{n-1} | \mathscr F_m] + ... + 2E[Z_{m} | \mathscr F_m] \le (n-m) \alpha$$ possibly by showing that $2E[Z_{\{\cdot\}} | \mathscr F_m] \le \alpha$ How would I do that? Something else: $$E[\ln \frac{Z_n}{Z_m} | \mathscr F_m] + \ln Z_m - n \alpha$$ $$ \le E[|Z_n - Z_m| | \mathscr F_m] + \ln Z_m - n \alpha$$ $$ \le E[2Z_m | \mathscr F_m] + \ln Z_m - n \alpha$$ $$ \le 2Z_m + \ln Z_m - n \alpha$$ Now $2Z_m \le (n-m) \alpha?$ We have $$2Z_m = 2\sum_{k=1}^{m} C_k(\epsilon_k - \epsilon_{k-1})$$ $$\le 2\sum_{k=1}^{m} Z_{k-1}$$ I'm not quite sure how to show that $$2\sum_{k=1}^{m} Z_{k-1} \le (n-m)\alpha$$ if that's even true. How can/else can I approach this problem?  For no 3, no idea. Any hints? I think it has something to do with a stopping time. Might we have to use the ff lemma (red box):    ? Bold name in bib when having a prefix in last name How can I change an element's class with JavaScript? LINK : fatal error LNK1104: cannot open file 'D:\...\MyProj.exe'