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Vector Formalism in Introductory Physics I: Taking the Magnitude of Both Sides

TL;DR: I don’t like the way vectors are presented in calculus-based and algebra-based introductory physics. I think a more formal approach is warranted. This post addresses the problem of taking the magnitude of both sides of simple vector equations. If you want the details, read on.

This is the first post in a new series in which I will present a more formal approach to vectors in introductory physics. It will not have the same flavor as my recently begun series on angular quantities; that series serves a rather different purpose. However, there may be some slight overlap between the two series if it is appropriate.

I am also using this series to commit to paper (screen, really) some thoughts and ideas I have had for some time with the hope of turning them into papers for submission to The Physics Teacher. I’d appreciate any feedback on how useful this may be to the community.

To begin with, I want to address issues in the algebraic manipulation of vectors with an emphasis on coordinate-free methods. I feel that in current introductory physics courses, vectors are not exploited to their full potential. Instead of learning coordinate-free methods, students almost always learn to manipulate coordinate representations of vectors in orthonormal, Cartesian coordinate systems and I think that is unfortunate because it doesn’t always convey the physics in a powerful way. Physics is independent of one’s choice of coordinate system, and I think students should learn to manipulate vectors in a similar way.

Let’s begin by looking at a presumably simple vector equation:

$a\mathbf{A} = -5\mathbf{A}$

The object is to solve for $a$ given $\mathbf{A}$. Don’t be fooled; it’s more difficult than it looks. In my experience, students invariably try to divide both sides by $\mathbf{A}$ but of course this won’t work because vector division isn’t defined in Gibbsian vector analysis. Don’t let students get away with this if they try it! The reasons for not defining vector division will be the topic of a future post.

(UPDATE: Mathematics colleague Drew Lewis asked about solving this equation by combining like terms and factoring, leading to $(a+5)=0$ and then to $a = -5$. This is a perfectly valid way of solving the equation and it completely avoids the “division by a vector” issue. I want to call attention to that issue though, because when I show students this problem, they always (at least in my experience) try to solve it by dividing. Also, in future posts I will demonstrate how to solve other kinds of vector equations that must be solved by manipulating both dot products and cross products, each of which carries different geometric information, and I want to get students used to seeing and manipulating dot products. Thanks for asking Drew!)

One could simply say to “take the absolute value of both sides” like this:

$\left| a\mathbf{A} \right| = \left| -5\mathbf{A}\right|$

but this is problematic for two reasons. First, it destroys the sign on the righthand side. Second, a vector doesn’t have an absolute value because it’s not a number. Vectors have magnitude, not absolute value, which is an entirely different concept from that of absolute value and warrants separate consideration and a separate symbol.

We need to do something to each side to turn it into a scalar because we can divide by a scalar. Let’s try taking the dot product of both sides with the same vector, $\mathbf{A}$, and proceed as follows:

\begin{aligned} a\mathbf{A}\bullet\mathbf{A} &= -5\mathbf{A}\bullet\mathbf{A} && \text{dot both sides with the same vector} \\ a\lVert\mathbf{A}\rVert^2 &= -5\lVert\mathbf{A}\rVert^2 && \text{dot products become scalars} \\ \therefore a &= -5 && \text{solve} \end{aligned}

This is a better way to proceed. It’s formal, and indeed even pedantic, but I dare say it’s the best way to go if one wants to include dot products. Of course in this simple example, one can see the solution by inspection, but my goals here are to get students to stop thinking about the concept of dividing by a vector and to manipulate vectors algebraically without referring to a coordinate system.

Let’s now look at another example with a different vector on each side of the equation.

$a\mathbf{A} = -5\mathbf{B}$

Once again the object is to solve for $a$ given $\mathbf{A}$ and $\mathbf{B}$. Note that solving for either $\mathbf{A}$ or $\mathbf{B}$ is obviously trivial so I won’t address it; it’s simply a matter of scalar division. Solving for $a$ is more challenging because we must again suppress the urge to divide by a vector. I will show two possible solutions. Make sure you understand what’s being done in each step.

\begin{aligned} a\mathbf{A} &= -5\mathbf{B} && \text{given equation} \\ a\mathbf{A}\bullet\mathbf{A} &= -5\mathbf{B}\bullet\mathbf{A} && \text{dot both sides with the same vector} \\ a\mathbf{A}\bullet\mathbf{A} &= -5\mathbf{B}\bullet\left(\dfrac{-5}{\hphantom{-}a}\mathbf{B}\right) && \text{substitute from original equality} \\ a\lVert\mathbf{A}\rVert^2 &= \dfrac{25}{a}\lVert\mathbf{B}\rVert^2 && \text{dot products become scalars} \\ a^2 &= 25\dfrac{\lVert\mathbf{B}\rVert^2}{\lVert\mathbf{A}\rVert^2} && \text{rearrange} \\ \therefore a &= \pm 5\dfrac{\lVert\mathbf{B}\rVert}{\lVert\mathbf{A}\rVert} && \text{solve} \end{aligned}

We get two solutions, and they are geometrically opposite each other; that’s the physical implication of the signs. (I suppose we could argue over whether or not to just take the principal square root, but I don’t think we should do that here because it would throw away potentially useful geometric information.) We can find a cleaner solution that accounts for this. Consider the following solution which exploits the concepts of “factoring” a vector into a magnitude and a direction and the properties of the dot product.

\begin{aligned} a\mathbf{A} &= -5\mathbf{B} && \text{given equation} \\ a\mathbf{A}\bullet\mathbf{A} &= -5\mathbf{B}\bullet\mathbf{A} && \text{dot both sides with \textbf{A}} \\ a\lVert\mathbf{A}\rVert^2 &= -5\lVert\mathbf{B}\rVert\widehat{\mathbf{B}}\bullet\lVert\mathbf{A}\rVert\widehat{\mathbf{A}} && \text{factor each vector into magnitude and direction} \\ a\lVert\mathbf{A}\rVert^2 &= -5\lVert\mathbf{B}\rVert\lVert\mathbf{A}\rVert\,\widehat{\mathbf{B}}\bullet\widehat{\mathbf{A}} && \text{push magnitude through the dot product} \\ \therefore a &= -5\dfrac{\lVert\mathbf{B}\rVert}{\lVert\mathbf{A}\rVert}\,\widehat{\mathbf{B}}\bullet\widehat{\mathbf{A}} && \text{solve} \end{aligned}

See the geometry? It’s in the factor $\widehat{\mathbf B}\bullet\widehat{\mathbf A}$. If $\mathbf{A}$ and $\mathbf{B}$ are parallel, this factor is $+1$ and if they are antiparallel it is $-1$. Convince yourself that those are the only two options in this case. (HINT: Show that each vector’s direction is a scalar multiple of the other vector’s direction.) This solution won’t work if the two vectors aren’t collinear. If we’re solving for $a$ then both vectors are assumed given and we know their relative geometry.

Let’s look at another example from first semester mechanics, Newton’s law of gravitation,

$\mathbf{F} = G\dfrac{M_1 M_2}{\lVert\mathbf{r}_{12}\rVert^2}\left( -\widehat{\mathbf r}_{12}\right)$

where $\mathbf{r}_{12} = \mathbf{r}_1 - \mathbf{r}_2$ and should be read as “the position of 1 relative to 2.” Let’s “take the magnitude of both sides” by first writing $\mathbf{F}$ in terms of its magnitude and direction, dotting each side with a vector, and dividing both sides by the resulting common factor.

\begin{aligned} \lVert\mathbf{F}\rVert\left(-\widehat{\mathbf{r}}_{12}\right) &= G\dfrac{M_1 M_2}{\lVert\mathbf{r}_{12}\rVert^2}\left( -\widehat{\mathbf{r}}_{12}\right) && \text{given equation} \\ \lVert\mathbf{F}\rVert\left(-\widehat{\mathbf{r}}_{12}\bullet\widehat{\mathbf{r}}_{12}\right) &= G\dfrac{M_1 M_2}{\lVert\mathbf{r}_{12}\rVert^2}\left(-\widehat{\mathbf{r}}_{12}\bullet\widehat{\mathbf{r}}_{12}\right) && \text{dot both sides with the same vector} \\ \lVert\mathbf{F}\rVert\left(-\lVert\widehat{\mathbf{r}}_{12}\rVert^2\right) &= G\dfrac{M_1 M_2}{\lVert\mathbf{r}_{12}\rVert^2}\left(-\lVert\widehat{\mathbf{r}}_{12}\rVert^2 \right) && \text{dot products become scalars} \\ \therefore \lVert\mathbf{F}\rVert &= G\dfrac{M_1 M_2}{\lVert\mathbf{r}_{12}\rVert^2} && \text{divide both sides by the same scalar} \end{aligned}

Okay, this isn’t an Earth-shattering result becuase we knew in advance it has to be the answer, but my point is how we formally went about getting this answer. More specifically, the point is how we went about it without dividing by a vector.

Let’s now consider a final example from introductory electromagnetic theory, and this was the example that got me thinking about this entire process of “taking the magnitude of both sides” about a year ago. It’s the expression for the electric force experienced by a charged particle in the presence of an electric field (obviously not its own electric field).

$\mathbf{F} = q\mathbf{E}$

That one vector is a scalar multiple of another means the two must be collinear, so they must either be parallel or antiparallel. An issue here is that $q$ is a signed quantity. Again, we have a choice about which vector with which to dot both sides; we could use $\mathbf{F}$ or we could use $\mathbf{E}$. If we use the former, we will eventually need to take the square root of the square of a signed quantity, which may lead us astray. Therefore, I suggest using the latter.

\begin{aligned} \mathbf{F} &= q\mathbf{E} && \text{given equation} \\ \mathbf{F}\bullet\mathbf{E} &= q\mathbf{E}\bullet\mathbf{E} && \text{dot both sides with the same vector} \\ \lVert\mathbf{F}\rVert\widehat{\mathbf{F}}\bullet\lVert\mathbf{E}\rVert\widehat{\mathbf{E}} &= q\lVert\mathbf{E}\rVert^2 && \text{factor LHS, simplify RHS} \\ \lVert\mathbf{F}\rVert\lVert\mathbf{E}\rVert\,\widehat{\mathbf{F}}\bullet\widehat{\mathbf{E}} &= q\lVert\mathbf{E}\rVert^2 && \text{push the magnitude through the dot product} \\ \therefore \lVert\mathbf{F}\rVert &= \dfrac{q}{\widehat{\mathbf{F}}\bullet\widehat{\mathbf{E}}}\lVert\mathbf{E}\rVert && \text{solve} \end{aligned}

This may look overly complicated, but it’s quite logical, and it reflects goemetry. If $q$ is negative, then the dot product will also be negative and the entire quantity will be positive. If $q$ is positive, then the dot product will also be positive and again the entire quantity will be positive. Geometry rescues us again, as it should in physics. We can also rearrange this expression to solve for either $q$ or $\lVert\mathbf{E}\rVert$ with the sign of $q$ properly accounted for by the dot product. By the way, $\widehat{\mathbf{F}}$ and $\widehat{\mathbf{E}}$ can’t be orthogonal becuase then their dot product would vanish and the above expression would blow up. Geometry and symmetry, particularly the latter, preclude this from happening.

In summary, “taking the magnitude of both sides” of a simple vector equation presents some challenges that are mitigated by exploiting geometry, something that is neglected in introductory calculus-based and algebra-based physics courses. I suggest we try to overcome this by showing students how to formally manipulate such equations. One advantage of doing this is students will see how vector algebra works in more detail than usual. Another advantage is that students will learn to exploit geometry in the absence of coordinate systems, which is one of the original purposes of using vectors after all.

Do you think this would make a good paper for The Physics Teacher? Feedback welcome!


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# Fixing a broken correlation matrix

I'm trying to work through the process talked about in the article "Fixing a broken correlation matrix" by Phil Joubert and Stephen Langdell (found here https://www.nag.co.uk/IndustryArticles/fixing-a-broken-correlation-matrix.pdf), but I'm having trouble replicating the results in the 3x3 example they give. I'm attempting the "quick and dirty" method of patching up the broken matrix $C$ given by

\begin{bmatrix}1&0.95&0\\0.95&1&0.95\\0&0.95&1\end{bmatrix}

It says $C$ can be decomposed as $C = Q\Lambda Q^T$ where $Q$ is the matrix whose columns are eigenvectors of $C$ and $\Lambda$ is a matrix with the corresponding eigenvalues on the diagonal and zeros elsewhere. The article claims that we can get the "patched up" matrix $C'= Q\Lambda' Q^T$ where $\Lambda'$ is $\Lambda$, but with negative entries replaced by zero. However, I do not get the $C'$ they get given by

\begin{bmatrix}1&0.73454&0.07908\\0.73454&1&0.73454\\0.07908&0.73454&1\end{bmatrix}

The eigenvalues I get for $C$ are $\Lambda = \{-\frac{19\sqrt{2}-20}{20}, 1, \frac{19\sqrt{2}+20}{20}\}$ (these match the paper) and the corresponding eigenvectors (in matrix form, i.e. $Q$) are

\begin{bmatrix}1&1&1\\-\sqrt{2}&0&\sqrt{2}\\1&-1&1\end{bmatrix}

From what it sounds like in the article, I now need to replace the first eigenvalue with 0, and come up with $C'$ as described earlier where $\Lambda'$ is given by

\begin{bmatrix}0&0&0\\0&1&0\\0&0&\frac{19\sqrt{2}+20}{20}\end{bmatrix}

However, $C'= Q\Lambda' Q^T$ with these matrices does not give the $C'$ found in the article. Did I calculate my eigenvalues and eigenvectors incorrectly? At first I thought I needed to calculate a different eigenvector with $\Lambda_1 = 0$, but using that eigenvector did not give me their $C'$ either. Any help would be greatly appreciated!

I do not yet see exactly, what happened here, but I'm able to reproduce the given matrix $C'$ First, I get the same eigenvalues by similarity rotations on C. But by rotations, the matrix $Q$ in my setting is a scaling of your $Q$. If you take $Q \cdot Q^t$ with your matrices, you don't get the identity matrix - but which is expected if $Q$ is a rotation-matrix/is orthonormal. So my matrix $R = \operatorname{norm_columns}(Q)$ which has the sum-of-squares over a column equalling $1$ does the job better. (my earlier remark $R=Q/2$ was wrong and based only on a too short view on the two matrices)
The next step is not yet sufficient: doing $S = R^t \cdot \Lambda' \cdot R$ gives now a covariance-matrix: $$\text{ S =} \left[ \begin{array} {rrr} 1.09& 0.83& 0.09\\ 0.83& 1.17& 0.83\\ 0.09& 0.83& 1.09 \end{array} \right]$$ The final step could be the (standard) norming of a covariance to a corrleation-matrix: you put the diagonal elements $s_{k,k}$ of $S$ into the diagonal matrix $D = \operatorname{diag}(\{1/\sqrt{s_{k,k}} \})$ and multiply $C' = D \cdot S \cdot D$ getting $$\text{ C' =} \left[ \begin{array} {rrr} 1.000& 0.735& 0.079\\ 0.735& 1.000& 0.735\\ 0.079& 0.735& 1.000 \end{array} \right]$$ which looks trustworthy. So the final step in the described method was possibly forgotten to be written down...
• @mjswartz : (my earlier comment and my answer had a mistake) Your matrix Q does not fit the requirement of a rotation-matrix $R$ that $R \cdot R^t = I$ But it can be reached by norming the columns such that the sum-of-squares along a column should equal 1. If you do that norming you get the correct result (notice, that eigenvectors at different eigenvalues are unique only up to scaling of the columns by some arbitrary scalar factor) I'll correct the faulty remark in my answer, too. – Gottfried Helms Sep 7 '16 at 21:16

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#### Russia/USSR

###### back to index

Find any positive integer solution to $x^2 - 51y^2 = 1$.

Solve in integers $\frac{x+y}{x^2-xy+y^2}=\frac{3}{7}$

For any arithmetic sequence whose terms are all positive integers, show that if one term is a perfect square, this sequence must have infinite number of terms which are perfect squares.

Find all positive integer $n$ such that $(3^{2n+1} -2^{2n+1}- 6^n)$ is a composite number.

Find $n$ different positive integers such that any two of them are relatively prime, but the sum of any $k$ ($k < n$) of them is a composite number.

Solve this equation in positive integers $$x^3 - y^3 = xy + 61$$

Show that the following inequality holds for any positive integer $n$: $$(2n+1)^n \ge (2n)^n + (2n-1)^n$$

Let $a, b, c, m, n, p, k$ be positive real numbers that satisfy $a+m = b+n = c+p=k$. Show that $an+bp+cm < k^2$.

Let $a_1=a_2=1$ and $a_{n}=(a_{n-1}^2+2)/a_{n-2}$ for $n=3, 4, \cdots$. Show that $a_n$ is an integer for $n=3, 4, \cdots$.

Show that if a polynomial $P(x)$ satisfies $P(2x^2-1)=(P(x))^2/2$, then it must be a constant.

Is it possible to construct 12 geometric sequences to contain all the prime between 1 and 100?

Is function $f(x)=\lg(x+\sqrt{x^2+1})$ an odd or even function?

Let $a$ and $b$ be two positive real numbers not exceeding $1$. Prove $$\frac{1}{\sqrt{a^2 + 1}}+\frac{1}{\sqrt{b^2 +1}}\le\frac{2}{\sqrt{1+ab}}$$

Given two grids shown below, is it possible to transform (a) to (b) after a series of operations? In each operation, one can change all the signs in either one entire row or one entire column.


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## Abstract

The numerical simulations are used to conduct the comparative study of pin-fins cooling channel and multi-impingement cooling channel on the heat transfer and flow, and to design the multi-impingement channel through the parameters of impinging distance and impingement-jet-plate thickness. The Reynolds number ranges from 1e4 to 6e4. The dimensionless impinging distance is 0.60, 1.68, 2.76, respectively, and the dimensionless impinging-jet-thickness is 0.5, 1.0, 1.5, respectively. The endwall surface, pin-fins surface, impinging-jet-plate surface are the three object surfaces to investigate the channel heat transfer performance. The heat transfer coefficient $h$ and augmentation factor $Nu/Nu0$ are selected to measure the surface heat transfer, and the friction coefficient $f$ is chosen to evaluate the channel flow characteristics. The impinging-jet-plate surface owns higher heat transfer coefficient and larger area than pin-fins surface, which are the main reasons to improve the heat transfer performance of multi-impingement cooling channel. Reducing the impinging distance can improve the endwall surface heat transfer obviously and enhance impingement plate surface heat transfer to some extent, decreasing the thickness of impinging-jet-plate can significantly increase its own heat transfer coefficient, which both all increase the cooling air flow loss.

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Ceramic Matrix Composites (CMCs) are primary candidates for advanced gas turbine engine application that require intense high temperature tests and validations. Before CMCs used in engine hot sections, a lot of tests need to be done, especially thermal test. A thermal test rig has been set up to simulate the engine turbine thermal environment. Propane gas is used to simulate the practical aviation fuel and compressed air with flow regulator is used as cooling media. The capabilities and limitations of the test facility have been calibrated and discussed in this paper. A CMC turbine vane with internal cooling path was tested on this burning rig. The results showed that the CMC vane could withstand the 1200 ℃ thermal cycling test but the coating was disappeared. It has been proved that such test rig and method could simulate the thermal boundary conditions of turbine vanes and blades.

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Control of Mach 1.5 elliptic jet with ventilated triangular tabs is studied experimentally, in the presence of different levels of pressure gradient at the nozzle exit. Three different sets of ventilated tabs with circular, triangular and trapezoidal ventilations were studied. Two tabs were placed, at the ends of major and minor axes, at the exit of the elliptic nozzle of aspect ratio 3.37. The mixing enhancement caused by these tabs was studied in the presence of adverse and favorable pressure gradients, corresponding to nozzle pressure ratio (NPR) from 3 to 8. For Mach 1.5 jet NPR 3 corresponds to 18 % adverse pressure gradient and NPR 8 corresponds to 118 % favorable pressure gradient. The results of ventilated tabs are compared with unventilated truncated triangular tabs of identical geometry. The difference between the mixing promoting efficiency of the unventilated and ventilated tabs is only marginal (around 5–6 %). All tabs cause jet bifurcation and weaken the waves in the jet core. The tab with trapezoidal ventilation, at NPR 3, promotes mixing to an extent of reducing the core to about 92 %. At higher NPRs the mixing caused by unventilated tab is slightly better than the ventilated tabs.


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# Noun Modifiers can Modify slightly far away noun

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24 Mar 2013, 17:55
is the answer for the second question a. as it is right.
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03 Apr 2013, 08:12
Great post by E-Gmat.

I have got confused after reading this post. Here is why.

a) (GMAT Prep question) Visitors to the park have often looked up into the leafy canopy and saw monkeys sleeping on the branches, whose arms and legs hang like socks on a clothesline.

(A) saw monkeys sleeping on the branches, whose arms and legs hang
(B) saw monkeys sleeping on the branches, whose arms and legs were hanging
(C) saw monkeys sleeping on the branches, with arms and legs hanging
(D) seen monkeys sleeping on the branches, with arms and legs hanging
(E) seen monkeys sleeping on the branches, whose arms and legs have hung

OA-D
GMAT Prep explanation for 'E'- whose illogically refers to branches.
1) If we ignore the tense error in 'E', is it still an incorrect choice based on noun modified by 'whose'? Can't 'whose' refer to 'monkeys' as 'sleeping on the branches modifies 'monkeys'?
2) How 'with arms and legs...' in 'D' refers to the action 'sleeping' and not 'branches'? As per my knowledge, there is no rule for 'COMMA+Prep phrase'. 'with arms and legs..' can grammatically refer to 'branches'. Prep phrases play role of an adverbial modifier or adjectival modifier. I am confused. According to me, if we ignore tense error for E, then 'E" and 'D' both have logical modifiers. (** I am not questioning correctness of OG answers but trying to understand the concept)

b) (OG 12th edition) The nephew of Pliny the Elder wrote the only eyewitness account of the great eruption of Vesuvius in two letters to the historian Tacitus.

(A) The nephew of Pliny the Elder wrote the only eyewitness account of the great eruption of Vesuvius in two letters to the historian Tacitus.
(B) To the historian Tacitus, the nephew of Pliny the Elder wrote two letters, being the only eyewitness accounts of the great eruption of Vesuvius.
(C) The only eyewitness account is in two letters by the nephew of Pliny the Elder writing to the historian Tacitus an account of the great eruption of Vesuvius.
(D) Writing the only eyewitness account, Pliny the Elder’s nephew accounted for the great eruption of Vesuvius in two letters to the historian Tacitus.
(E) In two letters to the historian Tacitus, the nephew of Pliny the Elder wrote the only eyewitness account of the great eruption of Vesuvius

OA: E

OG explanation for 'A'- 'in two letters to the historian Tacitus' illogically refers to eruption.
How can it be? When we know that it is illogical for 'in letters' to refer to 'eruption'. Prep phrase can also modify verb. Then why can prep phrase in 'A', modify 'wrote'??

Can you please explain how prepositional modifiers function-when they are at end of sentence, at start of the sentence, with comma, without comma?
Any help is deeply appreciated.

Regards,
Heman2727
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16 Apr 2013, 01:18
egmat wrote:

3: Like the great navigators who first sailed around the Earth gathering information about its size and the curvature of its surface, astronomers have made new observations that show with startling directness the large-scale geometry of the universe. (Choice D)

After reading this one, almost all of you will say that “gathering” is a verb-ing modifier that is placed after “Earth” and is not preceded by a comma. Hence, it must modify “Earth”. This modification makes no sense because Earth dis not gather information. The great navigators did. This sentence is incorrect. BUT IN REALITY, this sentence is absolutely correct. Here is why.

Structurally, “who first sailed around the Earth” is a clause. Here “who” stands for “the great navigators”. Now together “the great navigators who first sailed around the Earth” is a big noun phrase (refer to the mini article on noun phrases and Noun modifiers).

Many of you may argue that this entity contains a “who clause”. How can we classify as a noun phrase. We can classify this as a noun phrase since it has a noun at its head. It is of the construction – Noun + Clause.

The head of this big noun phrase is “the great investigators”. Now the “who” clause that modifies “the great investigators” cannot be placed anywhere else in the sentence. This gives “gathering”, a noun modifier, the liberty to jump over the preceding modifier and modify the head – “the great investigators”. Hence, “gathering” in this sentence is correctly modifying “the great investigators”.

Futhermore, logically “earth” cannot gather information.

Hi eGMAT/Experts,

I would like to build my understanding more on the above Red colored part.

Example 1
When we say that the earth cannot gather information, it means that it is illogical and hence we should move further to see if we have any other noun that could have been modified by the entity "gathering information.....". By doing so, we found the noun - great navigators and hence the phrase "gathering..." must be modifying "great navigators"

Now, taking the same concept forward to the below example--

Example 2
The nephew of Pliny the Elder wrote the only eyewitness account of the great eruption of Vesuvius in two letters to the historian Tacitus

As the above poster correctly mentions that the eruption of Vesuvius cannot happen in the two letters, hence the modification is illogical and we should move further to see if there is any valid verb/noun to whom it can modify. Here, "Wrote" is the verb that is getting modified correctly by the phrase - in two letters to the historian Tacitus.

Question is - Why we are saying that choice 2 is illogical, however going by the same framework and logic makes the first example correct. Please respond.

Thanks
H
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16 Apr 2013, 23:37
egmat wrote:
Hi Simba2012,
I'm glad that you liked the article. But what makes me really happy that now you understand in which scenario "which" can modify a slightly far away noun.
Let me also congratulate for solving the execise correctly. Great job there.

You can also check out a foundation concept on Noun phrases and Noun modifiers.

noun-modifiers-can-modify-slightly-far-away-noun-prequel-135910.html#p1105284

Thanks.

Can you please confirm if my understanding is right on Sunspots question in OG 13.
Sunspots,vortices of gas associated with strong electromagnetic activity, appear as dark spots on the suns surface, which have never been sighted on the suns poles or equators.

can we say that which here points to dark spots ? because on the suns surface is a noun modifier ?
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17 Apr 2013, 00:12
vigneshceg wrote:
Can you please confirm if my understanding is right on Sunspots question in OG 13.
Sunspots,vortices of gas associated with strong electromagnetic activity, appear as dark spots on the suns surface, which have never been sighted on the suns poles or equators.

can we say that which here points to dark spots ? because on the suns surface is a noun modifier ?

Hi Vigneshceg,

Your understanding is correct. You can do a double check with the verb present in the Relative Clause. The Verb is Plural, so it must refer to plural noun. Hence, Dark spots. More often than not(90% cases), you have the helping verb that will clarify the noun that the relative pronoun is referring to. I have given an analysis in this thread only, you may take a look on it.

Thanks
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17 Apr 2013, 02:01
imhimanshu wrote:
vigneshceg wrote:
Can you please confirm if my understanding is right on Sunspots question in OG 13.
Sunspots,vortices of gas associated with strong electromagnetic activity, appear as dark spots on the suns surface, which have never been sighted on the suns poles or equators.

can we say that which here points to dark spots ? because on the sun`s surface is a noun modifier ?

Hi Vigneshceg,

Your understanding is correct. You can do a double check with the verb present in the Relative Clause. The Verb is Plural, so it must refer to plural noun. Hence, Dark spots. More often than not(90% cases), you have the helping verb that will clarify the noun that the relative pronoun is referring to. I have given an analysis in this thread only, you may take a look on it.

Thanks
H

Thanks for the reply H.
I saw your previous posts and was informative.
To be honest, I am confused. I thought in the blah blah modified the verb wrote.
We must have more clarity and direction on the thought.
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Re: Noun Modifiers can Modify slightly far away noun [#permalink]

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17 Apr 2013, 02:56
vigneshceg wrote:

Thanks for the reply H.
I saw your previous posts and was informative.
To be honest, I am confused. I thought in the blah blah modified the verb wrote.
We must have more clarity and direction on the thought.

Please refer to the one written on "Feb 25, 2013 8:38 am". I think, you have seen the post written yesterday by me, that is a doubt posted by me not a learning...
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Re: Noun Modifiers can Modify slightly far away noun [#permalink]

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28 Apr 2013, 14:11
The decision of the European leaders to use the Continent’s bailout funds to recapitalize struggling banks would provide help to banks without directly adding to the sovereign debt in the coming months, which has been a problem for Spain and potentially for Italy.

It is said that the above sentence is incorrect because modifier "in the coming months" modifies "help to banks" and not "sovereign debt". I agree to this but,
I also read a "comma which rule" that states the below. Applying this rule it is clear that WHICH is modifying the nearest singular noun "sovereign debt". Please help me understand what is it that I am missing here??

Regarding the use of COMMA + which on the GMAT:
If which has a SINGULAR verb, then it should refer to the nearest preceding SINGULAR noun.
If which has a PLURAL verb, then it should refer to the nearest preceding PLURAL noun.

Cheers,
Raghu.
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Re: Noun Modifiers can Modify slightly far away noun [#permalink]

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30 Apr 2013, 11:03
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Hi Heman2727,

The first thing we should keep in mind is that the rule discussed in this article pertains ONLY to Noun Modifiers.

Another thing I would like to bring to your attention is the nature of prepositional phrases. They can modify Nouns as well as Verb. So we must be very careful to ascertain what it should logically modify and then make sure that its placement is correct.

If you study the GMAT sentence that you have provided, the prepositional phrase “on the branches” modifies the action denoted by “sleeping”. Where do monkeys sleep? On the branches. So this prepositional phrase is not part of “monkeys”. This is the reason why, all the relative pronouns after “branches” refer to “branches”. The relative pronoun cannot jump over those prepositional phrases that do not modify the same noun entity as the relative pronoun modifier modifies.

Study these official examples:
1. Although she had been known as an effective legislator first in the Texas Senate and later in the United States House of Representatives, Barbara Jordan did not become a nationally recognized figure until 1974, when she participated in the hearings on the impeachment of President Richard Nixon, which were televised nationwide.

2. Emily Dickinson’s letters to Susan Huntington Dickinson, which were written over a period beginning a few years before Susan’s marriage to Emily’s brother and ending shortly before Emily’s death in 1886, outnumber her letters to anyone else.

This is the reason why choice D is the correct answer here because “with arms and legs hanging” can logically only refer to monkeys.

Now let’s come to your next question.
The nephew of Pliny the Elder wrote the only eyewitness account of the great eruption of Vesuvius in two letters to the historian Tacitus.

In this sentence, we must first ascertain what “in two letters” is supposed to modify. It should modify “wrote the only eyewitness account”. Since prepositional phrases can refer to nouns also, incorrect placements make the modification ambiguous. Here, “wrote” is separated by “in two letter” by so many noun entities. This makes it appropriate to place this prepositional phrase closer to the entity it is supposed to modify to avoid any room for ambiguous.

So the original sentence still leaves a little room for ambiguous, but not the correct answer where “in two letters” has been placed in the beginning of the sentence.

Hope this helps.
Thanks.
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Re: Noun Modifiers can Modify slightly far away noun [#permalink]

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30 Apr 2013, 11:10
Expert's post
imhimanshu wrote:
egmat wrote:

3: Like the great navigators who first sailed around the Earth gathering information about its size and the curvature of its surface, astronomers have made new observations that show with startling directness the large-scale geometry of the universe. (Choice D)

After reading this one, almost all of you will say that “gathering” is a verb-ing modifier that is placed after “Earth” and is not preceded by a comma. Hence, it must modify “Earth”. This modification makes no sense because Earth dis not gather information. The great navigators did. This sentence is incorrect. BUT IN REALITY, this sentence is absolutely correct. Here is why.

Structurally, “who first sailed around the Earth” is a clause. Here “who” stands for “the great navigators”. Now together “the great navigators who first sailed around the Earth” is a big noun phrase (refer to the mini article on noun phrases and Noun modifiers).

Many of you may argue that this entity contains a “who clause”. How can we classify as a noun phrase. We can classify this as a noun phrase since it has a noun at its head. It is of the construction – Noun + Clause.

The head of this big noun phrase is “the great investigators”. Now the “who” clause that modifies “the great investigators” cannot be placed anywhere else in the sentence. This gives “gathering”, a noun modifier, the liberty to jump over the preceding modifier and modify the head – “the great investigators”. Hence, “gathering” in this sentence is correctly modifying “the great investigators”.

Futhermore, logically “earth” cannot gather information.

Hi eGMAT/Experts,

I would like to build my understanding more on the above Red colored part.

Example 1
When we say that the earth cannot gather information, it means that it is illogical and hence we should move further to see if we have any other noun that could have been modified by the entity "gathering information.....". By doing so, we found the noun - great navigators and hence the phrase "gathering..." must be modifying "great navigators"

Now, taking the same concept forward to the below example--

Example 2
The nephew of Pliny the Elder wrote the only eyewitness account of the great eruption of Vesuvius in two letters to the historian Tacitus

As the above poster correctly mentions that the eruption of Vesuvius cannot happen in the two letters, hence the modification is illogical and we should move further to see if there is any valid verb/noun to whom it can modify. Here, "Wrote" is the verb that is getting modified correctly by the phrase - in two letters to the historian Tacitus.

Question is - Why we are saying that choice 2 is illogical, however going by the same framework and logic makes the first example correct. Please respond.

Thanks
H

Hi Himanshu,

If you read my explanation to Heman2727 on the OG 12 question, you should get your answer. As far as the GMAT Prep question of "gathering..." is concerned, it is really an uncommon question where we go by the logic and kind of overlook the grammar. However, this is not a common practice.

Hope this helps.
Thanks.
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Re: Noun Modifiers can Modify slightly far away noun [#permalink]

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30 Apr 2013, 11:57
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pkonduri wrote:
The decision of the European leaders to use the Continent’s bailout funds to recapitalize struggling banks would provide help to banks without directly adding to the sovereign debt in the coming months, which has been a problem for Spain and potentially for Italy.

Hi Raghu,

Noun Modifiers, in this case the relative pronoun "which", can only jump over prepositional phrases preceding them when those prepositional phrases modify the same entity as the noun modifier after the prepositional phrase does. If the preceding prepositional phrase modifies any other entity, then the noun modifier cannot jump over that prepositional. In that case it will modify the immediate preceding noun.

This is what is happening in the sentence that you have provided. "in the coming months" refers to the verb "would provide". This prepositional phrase is not part of "sovereign debts". So, "which" cannot jump over "in the coming months" and modify "sovereign debts".

Now the rules that you have cited also stand correct. But that rule applies to those sentence where the modification is correct. In the above sentence, the modification at the first is incorrect. So there is no question of application of those rules here.

Let's study a set of correct example sentences to see the application of these rules:

1. The box of nails that is made up of wood is in the shelf.
2. The box of nails that are rusted is is in the shelf.

In sentence 1, since the verb "is" after "that" is singular, "that" refers to "box" even if it is preceded by another noun entity "nails". The noun modifier "that" can jump over "of nails" because "of nails" also refers to the "the box." In sentence 2, the verb after "that" is "are", a plural verb. Hence, it needs a plural subject. So that refers to the immediate preceding plural noun "nails".

Hope this helps.
Thanks.
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Re: Noun Modifiers can Modify slightly far away noun [#permalink]

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07 May 2013, 22:58
Little late to reply, but thank you for your time. I understand the rule now.

egmat wrote:
pkonduri wrote:
The decision of the European leaders to use the Continent’s bailout funds to recapitalize struggling banks would provide help to banks without directly adding to the sovereign debt in the coming months, which has been a problem for Spain and potentially for Italy.

Hi Raghu,

Noun Modifiers, in this case the relative pronoun "which", can only jump over prepositional phrases preceding them when those prepositional phrases modify the same entity as the noun modifier after the prepositional phrase does. If the preceding prepositional phrase modifies any other entity, then the noun modifier cannot jump over that prepositional. In that case it will modify the immediate preceding noun.

This is what is happening in the sentence that you have provided. "in the coming months" refers to the verb "would provide". This prepositional phrase is not part of "sovereign debts". So, "which" cannot jump over "in the coming months" and modify "sovereign debts".

Now the rules that you have cited also stand correct. But that rule applies to those sentence where the modification is correct. In the above sentence, the modification at the first is incorrect. So there is no question of application of those rules here.

Let's study a set of correct example sentences to see the application of these rules:

1. The box of nails that is made up of wood is in the shelf.
2. The box of nails that are rusted is is in the shelf.

In sentence 1, since the verb "is" after "that" is singular, "that" refers to "box" even if it is preceded by another noun entity "nails". The noun modifier "that" can jump over "of nails" because "of nails" also refers to the "the box." In sentence 2, the verb after "that" is "are", a plural verb. Hence, it needs a plural subject. So that refers to the immediate preceding plural noun "nails".

Hope this helps.
Thanks.
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Re: Noun Modifiers can Modify slightly far away noun [#permalink]

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02 Jun 2013, 09:14
Firstly,I would like to thank you for the article shared.I have another doubt with the question where signifcance of which is specified.

Letters are not the doer of the verb-ing 'outnumbering'

Is this another reason for the original sentence being wrong.

Also please answer to the below question.

John,a resident of Lake City and an employee of The Bell Company,is currently working on a new project that will revolutionize the telecommunication industry,which will lead to a lowering of rates for making international calls.

A)is currently working on a new project that will revolutionize the telecommunication industry,which will lead to a lowering of rates for

B)is currently working on a new project that will revolutionize the telecommunication industry,leading to a lowering of rates for

C)which is currently working on a new project to revolutionize the telecommunication industry and also lead to a lower rates for

D)who is currently working on a new project in the telecommunication industry,which will lead to a lowering of rates for

E)has been currently working on a new project that will revolutionize the telecommunication industry as well as lower the rates for.
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Re: Noun Modifiers can Modify slightly far away noun [#permalink]

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10 Jun 2013, 13:54
Amazing article Shraddha. I cannot believe how you could simplify this matter so well. It definitely did clear up the doubts that I had. I have read many articles on this topic but none provides the article as this one does.

I must also thank the community for posting their doubts about official questions. You know apart from the questions in the article, I could apply the fundas to these official questions. This is simply great!!

Do you have one place where you have all your articles. I think these are articles worthy of bookmarking for any GMAT aspirant!

Thanks once again. Really heart felt thanks...
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Re: Noun Modifiers can Modify slightly far away noun [#permalink]

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11 Jul 2013, 10:36
Simba2012 wrote:
Excellent article!

1) The decision of the European leaders to use the Continent’s bailout funds to recapitalize struggling banks would provide help to banks without directly adding to the sovereign debt of countries, which has been a problem for Spain and potentially for Italy.

In this sentence "which" is referring to sovereign debt and even though there is prepositional phrase " of countries" between which and the noun- sovereign debt, the sentence is correct because " of countries" is correctly modifying the noun sovereign debt.

2)The decision of the European leaders to use the Continent’s bailout funds to recapitalize struggling banks would provide help to banks without directly adding to the sovereign debt in the coming months, which has been a problem for Spain and potentially for Italy.

In this sentence "which" is referring to sovereign debt as well, however the prepositional phrase -"in the coming months" that is between "which" and the noun- sovereign debt is not modifying the noun but instead modifying "provide help"

Simba2012 plz check
pkonduri wrote:
The decision of the European leaders to use the Continent’s bailout funds to recapitalize struggling banks would provide help to banks without directly adding to the sovereign debt in the coming months, which has been a problem for Spain and potentially for Italy.

Hi Raghu,

Noun Modifiers, in this case the relative pronoun "which", can only jump over prepositional phrases preceding them when those prepositional phrases modify the same entity as the noun modifier after the prepositional phrase does. If the preceding prepositional phrase modifies any other entity, then the noun modifier cannot jump over that prepositional. In that case it will modify the immediate preceding noun.

This is what is happening in the sentence that you have provided. "in the coming months" refers to the verb "would provide". This prepositional phrase is not part of "sovereign debts". So, "which" cannot jump over "in the coming months" and modify "sovereign debts".

Now the rules that you have cited also stand correct. But that rule applies to those sentence where the modification is correct. In the above sentence, the modification at the first is incorrect. So there is no question of application of those rules here.

Let's study a set of correct example sentences to see the application of these rules:

1. The box of nails that is made up of wood is in the shelf.
2. The box of nails that are rusted is is in the shelf.

In sentence 1, since the verb "is" after "that" is singular, "that" refers to "box" even if it is preceded by another noun entity "nails". The noun modifier "that" can jump over "of nails" because "of nails" also refers to the "the box." In sentence 2, the verb after "that" is "are", a plural verb. Hence, it needs a plural subject. So that refers to the immediate preceding plural noun "nails".

Hope this helps.
Thanks.
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Re: Noun Modifiers can Modify slightly far away noun [#permalink]

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16 Jul 2013, 07:10
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shirish87 wrote:
Firstly,I would like to thank you for the article shared.I have another doubt with the question where signifcance of which is specified.

Letters are not the doer of the verb-ing 'outnumbering'

Is this another reason for the original sentence being wrong.

Also please answer to the below question.

John,a resident of Lake City and an employee of The Bell Company,is currently working on a new project that will revolutionize the telecommunication industry,which will lead to a lowering of rates for making international calls.

A)is currently working on a new project that will revolutionize the telecommunication industry,which will lead to a lowering of rates for

B)is currently working on a new project that will revolutionize the telecommunication industry,leading to a lowering of rates for

C)which is currently working on a new project to revolutionize the telecommunication industry and also lead to a lower rates for

D)who is currently working on a new project in the telecommunication industry,which will lead to a lowering of rates for

E)has been currently working on a new project that will revolutionize the telecommunication industry as well as lower the rates for.

Hi Sirish,

Comma + verb-ing modifiers present either the HOW aspect of the preceding action or the result. The reason why “outnumbering” is not correct is that Emily did not wrote the letters to SHD by outnumbering” her letters. Also, “outnumbering” is not the effect of Emily’s writing to SHD. Hence, use of “outnumbering” is not correct.

Now let’s look at the sentence that you have presented:

John, a resident of Lake City and an employee of The Bell Company, is currently working on a new project that will revolutionize the telecommunication industry, which will lead to a lowering of rates for making international calls.

MEANING ANALYSIS:

John is working on a project. This project will revolutionize the telecommunication industry. This industry will lead to decrease in rates for international calls. But it does not make sense to say that the industry will lead to decrease in the rates for international calls.

ERROR ANALYSIS:

From our meaning analysis, we know that it does not make sense to say that the industry will lead to decrease in the rates for international calls. Hence, use of “which” is incorrect in this sentence.

PoE:

A) is currently working on a new project that will revolutionize the telecommunication industry, which will lead to a lowering of rates for: Incorrect for the reason stated above.

B) is currently working on a new project that will revolutionize the telecommunication industry, leading to a lowering of rates for: Correct. Comma + Verb-ing modifiers correctly presents how the new project will revolutionize the telecommunication industry. It will do so by lowering the rates for international calls.

C) which is currently working on a new project to revolutionize the telecommunication industry and also lead to a lower rates for: Incorrect.
1. “which” illogically refers to “The Bell Company”.
2. Absence of verb for the subject “John” leads to fragment error.

D) who is currently working on a new project in the telecommunication industry, which will lead to a lowering of rates for: Incorrect. Same “which” modification error as in choice A and fragment error as in choice C.

E) has been currently working on a new project that will revolutionize the telecommunication industry as well as lower the rates for: Incorrect.
1. Use of present perfect tense is incorrect.
2. The new project will do just one thing. It will revolutionize the telecommunication industry. The forced parallel list changes the meaning of the sentence.

Hope this helps.
Thanks.
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Re: Noun Modifiers can Modify slightly far away noun [#permalink]

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16 Jul 2013, 10:32

An economic recession can result from a lowering of employment rates triggered by a drop in investment, which causes people to cut consumer spending and starts a cycle of layoffs leading back to even lower employment rates.

(a) a lowering of employment rates triggered by a drop in investment, which causes people to cut consumer spending and start a cycle of layoffs leading back to even lower employment rates.

(b) a lowering of employment rates triggered by dropping investment, which causes people to cut consumer spending and starts a cycle of layoffs leading back to even lower employment rates.

(c) falling employment rates triggered by a drop in investment, which cause cutbacks in consumer spending, starting a cycle of layoffs that lead to even lower employment rates.

(d) falling employment rates that are triggered by a drop in investment, causing people to cut consumer spending and starting a cycle of layoffs that lead back to even lower employment rates.

(e) falling employment rates that are triggered by a drop in investment, causing cutbacks in consumer spending and starting a cycle of layoffs leading to even lower employment rates.

Can you please explain the usage of modifiers in this question

Regards,
Veenu
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Re: Noun Modifiers can Modify slightly far away noun [#permalink]

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18 Jul 2013, 11:37
egmat wrote:
Hi Veenu,

An economic recession can result from a lowering of employment rates triggered by a drop in investment, which causes people to cut consumer spending and starts a cycle of layoffs leading back to even lower employment rates.

(a) a lowering of employment rates triggered by a drop in investment, which causes people to cut consumer spending and start a cycle of layoffs leading back to even lower employment rates.

(b) a lowering of employment rates triggered by dropping investment, which causes people to cut consumer spending and starts a cycle of layoffs leading back to even lower employment rates.

(c) falling employment rates triggered by a drop in investment, which cause cutbacks in consumer spending, starting a cycle of layoffs that lead to even lower employment rates.

(d) falling employment rates that are triggered by a drop in investment, causing people to cut consumer spending and starting a cycle of layoffs that lead back to even lower employment rates.

(e) falling employment rates that are triggered by a drop in investment, causing cutbacks in consumer spending and starting a cycle of layoffs leading to even lower employment rates.

This one is very interesting. The key to solve this one is to understand the intended logical meaning of the sentence. So let’s do that.

MEANING ANALYSIS:

Economic recession can result from falling employment rates. By the way, the employment rates
fall is triggered by a drop in investment. Coming back to what we were talking about, the falling investment rates cause cuts in consumer spending and start a cycle of layoffs that lead to even lower employment rates.

Now that we understand the meaning of the question, I would like you to take a stab at this question. Analyze yourself the role of various modifiers in each answer choice and find the correct answer. Post your analysis here so that I can understand your analysis.

So all the very best.

Looking forward to your analysis.
Thanks.

An economic recession can result from a lowering of employment rates triggered by a drop in investment, which causes people to cut consumer spending and starts a cycle of layoffs leading back to even lower employment rates.

(a) a lowering of employment rates triggered by a drop in investment, which causes people to cut consumer spending and start a cycle of layoffs leading back to even lower employment rates.--causes vs starts, and usage of people and consumer together leading to redundancy.

(b) a lowering of employment rates triggered by dropping investment, which causes people to cut consumer spending and starts a cycle of layoffs leading back to even lower employment rates.

(c) falling employment rates triggered by a drop in investment, which cause cutbacks in consumer spending, starting a cycle of layoffs that lead to even lower employment rates.-- which is modifying falling employment rates, but cutbacks in consumer spending and layoffs are parallel activities.

(d) falling employment rates that are triggered by a drop in investment, causing people to cut consumer spending and starting a cycle of layoffs that lead back to even lower employment rates.

(e) falling employment rates that are triggered by a drop in investment, causing cutbacks in consumer spending and starting a cycle of layoffs leading to even lower employment rates.-- here verb-ing form is the result of the preceding clause-- correct answer

Kindly review it

Regards,
Veenu
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Re: Noun Modifiers can Modify slightly far away noun [#permalink]

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18 Jul 2013, 13:06
Expert's post
egmat wrote:
Hi Veenu,

An economic recession can result from a lowering of employment rates triggered by a drop in investment, which causes people to cut consumer spending and starts a cycle of layoffs leading back to even lower employment rates.

(a) a lowering of employment rates triggered by a drop in investment, which causes people to cut consumer spending and start a cycle of layoffs leading back to even lower employment rates.

(b) a lowering of employment rates triggered by dropping investment, which causes people to cut consumer spending and starts a cycle of layoffs leading back to even lower employment rates.

(c) falling employment rates triggered by a drop in investment, which cause cutbacks in consumer spending, starting a cycle of layoffs that lead to even lower employment rates.

(d) falling employment rates that are triggered by a drop in investment, causing people to cut consumer spending and starting a cycle of layoffs that lead back to even lower employment rates.

(e) falling employment rates that are triggered by a drop in investment, causing cutbacks in consumer spending and starting a cycle of layoffs leading to even lower employment rates.

This one is very interesting. The key to solve this one is to understand the intended logical meaning of the sentence. So let’s do that.

MEANING ANALYSIS:

Economic recession can result from falling employment rates. By the way, the employment rates
fall is triggered by a drop in investment. Coming back to what we were talking about, the falling investment rates cause cuts in consumer spending and start a cycle of layoffs that lead to even lower employment rates.

Now that we understand the meaning of the question, I would like you to take a stab at this question. Analyze yourself the role of various modifiers in each answer choice and find the correct answer. Post your analysis here so that I can understand your analysis.

So all the very best.

Looking forward to your analysis.
Thanks.

Shraddha- I'd go with E (although I haven't seen your analysis )...
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Re: Noun Modifiers can Modify slightly far away noun [#permalink]

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23 Aug 2013, 09:32
egmat wrote:
Hi Veenu,

An economic recession can result from a lowering of employment rates triggered by a drop in investment, which causes people to cut consumer spending and starts a cycle of layoffs leading back to even lower employment rates.

(a) a lowering of employment rates triggered by a drop in investment, which causes people to cut consumer spending and start a cycle of layoffs leading back to even lower employment rates.

(b) a lowering of employment rates triggered by dropping investment, which causes people to cut consumer spending and starts a cycle of layoffs leading back to even lower employment rates.

(c) falling employment rates triggered by a drop in investment, which cause cutbacks in consumer spending, starting a cycle of layoffs that lead to even lower employment rates.

(d) falling employment rates that are triggered by a drop in investment, causing people to cut consumer spending and starting a cycle of layoffs that lead back to even lower employment rates.

(e) falling employment rates that are triggered by a drop in investment, causing cutbacks in consumer spending and starting a cycle of layoffs leading to even lower employment rates.

This one is very interesting. The key to solve this one is to understand the intended logical meaning of the sentence. So let’s do that.

MEANING ANALYSIS:

Economic recession can result from falling employment rates. By the way, the employment rates
fall is triggered by a drop in investment. Coming back to what we were talking about, the falling investment rates cause cuts in consumer spending and start a cycle of layoffs that lead to even lower employment rates.

Now that we understand the meaning of the question, I would like you to take a stab at this question. Analyze yourself the role of various modifiers in each answer choice and find the correct answer. Post your analysis here so that I can understand your analysis.

So all the very best.

Looking forward to your analysis.
Thanks.

E looks good.

I am not sure If I am right in saying this but to me it sounds as if the phrase " lowering of unemployment rate " would cause something like a cutback in spending.

I don't think it is right to say it causes people to cut spending.

A better option would be it forces people to cut the spending.

Based on this I eliminated A , B ,and C.

Of course i can be wrong. :D

Could you please review my analysis.

And thanks for the original article. It is pretty good.

Thanks again,
Sandeep.
Re: Noun Modifiers can Modify slightly far away noun   [#permalink] 23 Aug 2013, 09:32

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# Noun Modifiers can Modify slightly far away noun

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text:
Main point: Let an adversary add realistic data to your analysis to see if it is resilient.

At the heart of statistics is the desire to make what we could call question / answer equilibrium. The question has a sound relation to that answer via the observation. The observations should ideally be monotonically increasing but ever true and the question / answer congruence should be resilient to additions to the data. This requires us to be careful about the questions in particular and sometimes we need to amend the question to make the answer more resilient.

Imagine a question like "what is the life expectancy". We can calculate that by taking the mean of all ages of all deceased people we have registered and report that as the answer. However, this is not resilient as new data might alter the answer because people get older. To mitigate this we can reiterate on the question and ask "what is the life expectancy in 2020" which would have the resilience to new data.

A formal method is to apply an adversarial model to statistics. To do this we make a system where we can prove that the addition of new data will change the question / answer relation. A such formalism can be based on typed graphs where we model all that entities and their relations. We prove that adding new entities will not change already published results.

We can imagine a system that has the following entities: Person, Country, Year. These entities can be linked by Person -- livesIn --> Country, Person -- deceasedIn --> Year. An example dataset to adhere to that ontology could be laid out as follows

To answer questions in the above-pictured dataset we can use the Monte Carlo simulation technique. For the first question, "what is the life expectancy," we draw a deceased person 10.000 times and average their ages. To answer the other question, "what is the life expectancy in 2020" we draw a person that has a deceasedIn relation to 2020 10.000 times and average over their ages.

For each of the questions, we invite the meanest adversary we know and allow her to put in new entities with one rule: The entities are realistic. Ie. as we are in 2021 now it is not possible to add more people deceased in 2020. The adversary quickly figures out that she can merely decease one of the still-alive people to change the answer to the first question. However, for the second question no new relations within the specified scope would

The above network would obviously be wildly bigger in reality and also include many more classes of data. Data might be gathered from several different data sources which can be modeled using multi-layer networks. Furthermore, When communicating in real life one should also take context into account. Ie. If we ask "what is the life expectancy" it can reasonably be assumed that it is in the year 2020 (and not eg. 1820).

I am currently building Fog that is a Haskell library to experiment with formalisms for skeptic reasoning. If you are also curious about this field and want to talk, please reach out!


text:
rev: Reverse of a Clifford object In clifford: Arbitrary Dimensional Clifford Algebras

Description

The “reverse” of a term is simply the basis vectors written in reverse order; this changes the sign of the term if the number of basis vectors is 2 or 3 (modulo 4). Taking the reverse is a linear operation.

Both Hestenes and Chisholm use a dagger to denote the reverse of A, as in omitted. But both Perwass and Dorst use a tilde, as in omitted.

omitted

where <A> is the grade operator; and it is easy to prove that

omitted

We can also show that

≤ft<AB\right>_r=(-1)^{r(r-1)/2}≤ft<B^\dag A^\dag\right>_r

Usage

 1 2 ## S3 method for class 'clifford' rev(x)

Arguments

 x Clifford object

Author(s)

Robin K. S. Hankin

grade,Conj
 1 2 3 4 5 6 7 8 x <- rcliff() rev(x) A <- rblade(g=3) B <- rblade(g=4) rev(A %^% B) == rev(B) %^% rev(A) # should be small rev(A * B) == rev(B) * rev(A) # should be small