The phrase running short of something means when you don't have enough of something, but I want to mention how much of something I am running short of for example I want to buy a pair of shoes the price was around 100$ but I found out that I had only 90$ so needed 10$ now how would I describe it using running short of? &quot;I am running short by 10$ or &quot;I am running short of 10$.
In another  regarding a difference between short on and short of, the  says the following:     ... But this is not always the case. A person can be short on/of money,   but lacking a specific amount, someone can only be €10 short or short   €10. This construction never uses a preposition.   but is that really true? I can see sentences like "I am short of 10 dollars" quite often.
Find $\,\lim_{x\to 1}\frac{\sqrt[n]{x}-1}{\sqrt[m]{x}-1}$ XML serialization without null XMLArray Why is a person's location exclusively tied to their communicator? In Star Trek it is frequently seen that when a crew member wants to avoid being tracked, they'll leave their communicator somewhere (quarters, corridor etc) which throws the computer and ship's security off their trail.  However, the entire interior of the ship is under constant surveillance by the internal sensors. Barring damage to the ship or sensors, or some space phenomenon causing radiation X to create interference, why aren't these internal sensors able to continue tracking every person on board regardless of where their comm-badge is? In the worst case, they should be able deduce who the errant individual is by simple subtraction: "I can see where all the other comm-badges are, so subtract them and the remaining life form must be the person I am looking for." But this never seems to be done either.  Is Star Fleet security really so inept or is there some other limitation in play that is never mentioned? How did Rutherford conclude that most of the mass (as well as the positive charge) was concentrated in the nucleus? Geiger and Marsden's experiment led Rutherford to believe that the positive charge and most of the mass of the atom was concentrated in a small region. I understand what led him to conclude the way the positive charge is positioned in the atom. But how did he conclude that most of the mass was in a small region (the nucleus)?  How did the distribution of the mass matter after all? Given that the electric force is greater than the gravitational force by many magnitudes, the force between the positice charge and the electrons was predominantly electric.   So how did Rutherford conclude that most of the mass is in the nucleus? How to shrink Windows 7 boot partition with unmovable files I have just bought an HP laptop with Windows&nbsp;7 (64 bit). It has a 500&nbsp;GB HDD with three partitions: a small hidden system partition, a 12&nbsp;GiB HP recovery partition, and a 450&nbsp;GiB C: boot partition.  I would like to split this large C: partition into two partitions, leaving only 100&nbsp;GiB for the system, and giving the rest to a new data partition.  Although the Windows built-in Disk Management utility has an option to shrink the bootable partition, it only allows me to shrink it roughly by half, even though only 20&nbsp;GiB on the partition is used. As far as I understand, system unmovable files lie in the middle of the partition, preventing Disk Management utility to do what I want.    And since new HP laptops don't come with OS installation disks (they only allow you to create recovery disks yourself), I can't just repartition HDD and then reinstall OS.  So, is there a way to shrink the C: bootable partition and preserve Windows&nbsp;7 working?  P.S.: I have tried to use the third-party  utility, and after shrinking the partition, Windows&nbsp;7 stopped booting, with a . System recovery didn't work, and I had to do factory recover. Since this is a long process, I would like to avoid doing it again :) So, please, suggest only proven solutions. Excel function that evaluates a string as if it were a formula? 'AND' vs '&&' as operator Can you use an alias in the WHERE clause in mysql? I need to use an alias in the WHERE clause, but It keeps telling me that its an unknown column. Is there any way to get around this issue? I need to select records that have a rating higher than x. Rating is calculated as the following alias:  sum(reviews.rev_rating)/count(reviews.rev_id) as avg_rating Why are there two tensions on my free body diagram? Equivalent Rotation using Baker-Campbell-Hausdorff relation Is there a way in which one can use the BCH relation to find the equivalent angle and the axis for two rotations? I am aware that one can do it in a precise way using Euler Angles but I was wondering whether we can use just the algebra of the rotation group to perform the same computation? Function is Continuous When Restricted to Closed Sets Which Cover the Space Suppose that $\{A_i\}$ is a finite collection of closed sets in $X$ such that $X = \bigcup A_i$, and $f : X \to Y$ restricted to each $A_i$ is continuous. Then $f$ is continuous.   Okay. The base case ($n=2$) is just the pasting lemma. I am having a little trouble with the inductive case, specifically with what the hypothesis is and how to deal with continuity with respect to subspaces. I believe the inductive hypothesis is to assume that any function over a topological space that is the union of $n-1$ closed sets such that the function restricted to each of these $n-1$ closed set is continuous must be continuous over the whole space. Then I assume I have a topological space $X = \bigcup_{i=1}^n A_i$, where each $A_i$ is closed, and a function $f : X \to Y$ such that $f|_{A_i}$ is continuous for each $i$. Thus $X = \bigcup_{i=1}^{n-1} A_i \cup A_n$, where $C = \bigcup_{i=1}^{n-1} A_i$ is closed since it is a finite union of closed sets. If I can show $f|_C$ is continuous, then I simply apply the pasting lemma to finish the problem. Give $C$ the subspace topology, and notice that each $A_i$ is closed in $C$. Thus we have a topological space $C$ that is the union of $n-1$ closed. But do I still have continuity $f$ restricted to $A_i$ since I am no longer considering them subspaces of $X$ but of $C$? When one says $f|_{A_i}$ is continuous, is one saying that $f$ is continuous when restricted to $A_i$ and $A_i$ is given the subspace topology?  EDIT:  I think I figured out the lemma I need. Claim: Let $A$ and $C$ be subspaces of $X$ such that $A \subseteq C$. Topologize $C$ with subspace topology with respect to $X$. Then the subspace topology on $A$ as a subspace of $X$ (call it $\tau_X$) is the same as the subspace topology of $A$ as a subspace of $C$ (call it $\tau_C$).   Proof: Let $U$ be open in $X$. Then $A \cap U$ is some open set in $A$ as a subspace of $X$. Since $C$ is a subspace of $X$, $C \cap U$ is open in $C$ as a subspace of $X$. Then $A \cap (C \cap U) = (A \cap C) \cap U = A \cap U$ is open in $A$ as a subspace of $C$, thereby proving $\tau_X \subseteq \tau_C$. Now let $O$ be some open set in $C$ as a subspace of $X$. Then $A \cap O$ is some open subset of $A$ as a subspace of $C$. But $O = C \cap U$ for some open set in $X$. Therefore $A \cap O = A \cap (C \cap U)= A \cap U$, which is also open in $A$ as a subspace of $X$. Hence $\tau_C \subseteq \tau_X$  Kind of obvious. But it was hard to spot the need for this lemma when I was considering subspace of subspaces and continuous functions, etc. Am I right in thinking this is the lemma I need? Prove linear combinations of logarithms of primes over $\mathbb{Q}$ is independent Suppose we have a set of primes $p_1,\dots,p_t$. Prove that  $\log p_1,\dots,\log p_t$ is linear independent over $\mathbb{Q}$. Now, this implies $ \sum_{j=1}^{t}x_j\log(p_j)=0 \iff x_1=\dots=x_t=0$.  I think I have to use that fact that every $q\in\mathbb{Q}$ can be written as $\prod_{\mathcal{P}}$, where $n_p$ is a unique sequence ($n_2$,$n_3$,$\dots$) with domain $\mathbb{Z}$. Here, $\mathcal{P}$ denotes the set of all integers.  Now how can I use this to prove the linear independency? Can you switch to the other Unarmored Defense by multiclassing? PHB p164 states:      If you already have the Unarmored Defense feature, you can't gain it again from another class.   So if I start as a Monk and gain the UD(Wis) feature, and then at next level up, take a level in Barbarian, can I switch to the alternate UD(Con)? Or am I stuck with UD(Wis), even if I take 19 levels in Barbarian?  I'm unclear if the rule is stating this or stating that you can't use both at the same time, which I know. What is Reflection in Java Can anyone explain? A type-I superconductor can expel almost all magnetic flux (below some critical value $H_c$) from its interior when superconducting. Light as we know is an electromagnetic wave. So what would happen if we shine a light deep into a type-I superconductor? Suppose the magnetic field of the light is significantly lower than $H_c$.  And what about type-II superconductors? How to prove that the set of rational numbers are countable? What font is used in this image? Suppose the terms of the sequence of real numbers $\{a_n\}$ satisfy $|a_{n+1} - a_n| \le 2^{-n}$ for all $n$. Prove that $\{a_n\}$ is Cauchy.   My Work  So by the definition of a Cauchy sequence, for all $\varepsilon &gt; 0$ $\exists N$ so that for $n,m \ge N$ we have $|a_m - a_n| \le \varepsilon$. However, questions like  make me understand that the $2^{-n}$ condition is necessary for this to be a true statement.   So I am wondering how to appeal to the Cauchy definition for this proof. Do I prove that every convergent sequence is therefore Cauchy, and then try to prove convergence? If the sum of $n$ cubes is zero, then the sum must be no larger than $\frac n3$. Assume that $a_1,...a_n$ are real numbers and $-1 \leq a_i\leq 1$ for $1\leq i\leq  n$. If $$a_1^3+\ldots +a_n^3=0$$ Then show that $$a_1+a_2+\ldots+a_n\le n/3$$   I just came cross this problem the other day. I totally have no idea how to prove it. Are there any hints about which theorem, principle or method to apply? Embed .mov file via 'Add Media' not working