1.

A whimsical clock maker fashioned both of the hands on his clock to be the same length. Here are some pictures of the clock. What time is it in each picture?

a)

b)

c)

d)

e)

2.

Is the clock maker's clock display ever ambiguous? If so, at what time(s)?

## Thursday, June 22, 2017

## Monday, June 19, 2017

### Harmonic Half-Pipe

When a skateboarder skates back and forth in a half-pipe, executing the motion the same way every time, the motion is periodic in both the horizontal sense and the vertical sense. We can make a simple version of this problem by allowing a bead to slide without resistance on a wire under the influence of gravity:

For most wire shapes, solving for the motion of the bead will require numerical methods. In any case, the specific time dependence of the bead's motion will depend on the shape of the wire. I wondered what wire shape would cause the bead's

I did a minute or two of searching and didn't see the problem solved anywhere online, so I worked it out. Here's an animation of the solution:

At all times, the height of the "skateboarder" is the same as the height of the mass that bobs on the spring. (This is called "simple harmonic motion.")

The shape of the wire is given by the graph of \(x = \pm p(y)\), where \[p(y) = \sqrt{y_0\,y-y^2}+y_0\sin^{-1}\!\!\sqrt{y/y_0}\,.\] Here, the constant \(y_0 =\frac{2g}{\omega^2}\) is the height of the half-pipe, which is also double the amplitude of the spring motion; and \(\omega\) is the angular frequency of the simple harmonic motion you want to produce. (There are also more complicated solutions that don't turn vertical at the ends.)

To find this solution, I worked in the first quadrant with the wire given by \(x = p(y)\) and \(p(y)\) unknown. Let the bead begin from rest at \((x,y) = (p(y_0), y_0)\). From energy conservation,

\[\frac{1}{2}mv^2 + mgy = mgy_0\,.\]

Express the speed as \(v^2 = ((p^\prime)^2 + 1)\dot{y}^2\). Now if we specify the vertical motion as \(y(t) = y_0\cos^2\left(\frac{\omega t}{2}\right)\), then we can express the time derivative in terms of the coordinate itself: \(\dot{y}^2 = \omega^2(y_0\,y - y^2)\). This eventually leads to a separable first-order nonlinear differential equation for \(p(y)\), \[\frac{dp}{dy} = \sqrt{\gamma\frac{y_0}{y}-1}\,\] where \(\gamma = \frac{2g}{\omega^2y_0}\). Note that \(\gamma \geq 1\), because the vertical acceleration can't possibly exceed \(g\), so that \(\frac{1}{2}\omega^2y_0 \leq g\) or in other words \(\gamma \geq 1\). Also note that \(\frac{dp}{dy}\rightarrow\infty\) as \(y\rightarrow 0\), which makes sense because the half-pipe must be horizontal where the vertical motion reverses direction.

Integrating with the boundary condition \(p(0) = 0\) so that the half-pipe goes through the origin, we get, with the help of Dwight 194.21 and 192.11, \[p = \int_0^y{\sqrt{\gamma \frac{y_0}{\xi}-1}\,d\xi} = \sqrt{\gamma\,y\,y_0 - y^2} + \gamma y_0 \left(\frac{\pi}{2}-\tan^{-1}\!\!\sqrt{\gamma\frac{y_0}{y}-1}\right)\,.\] If we take \(y_0 = \frac{2g}{\omega^2}\) then \(\gamma = 1\) and this simplifies to \[p = \sqrt{y\,y_0-y^2} + y_0\left(\frac{\pi}{2}-\tan^{-1}\!\!\sqrt{\frac{y_0}{y}-1}\right)\,.\] The arctangent can be expressed more simply as \(\cos^{-1}\!\!\sqrt{\frac{y}{y_0}}\) (to see this, draw a right triangle with legs \(y\) and \(\sqrt{y\,y_0-y^2}\)), whereupon we also notice that \(\frac{\pi}{2} - \cos^{-1}(\cdots) = \sin^{-1}(\cdots)\), so in the end \[p(y) = \sqrt{y_0\,y-y^2}+y_0\sin^{-1}\!\!\sqrt{y/y_0}\] as above.

For most wire shapes, solving for the motion of the bead will require numerical methods. In any case, the specific time dependence of the bead's motion will depend on the shape of the wire. I wondered what wire shape would cause the bead's

*vertical*motion to be the same as if the bead were simply bobbing on a spring.I did a minute or two of searching and didn't see the problem solved anywhere online, so I worked it out. Here's an animation of the solution:

At all times, the height of the "skateboarder" is the same as the height of the mass that bobs on the spring. (This is called "simple harmonic motion.")

The shape of the wire is given by the graph of \(x = \pm p(y)\), where \[p(y) = \sqrt{y_0\,y-y^2}+y_0\sin^{-1}\!\!\sqrt{y/y_0}\,.\] Here, the constant \(y_0 =\frac{2g}{\omega^2}\) is the height of the half-pipe, which is also double the amplitude of the spring motion; and \(\omega\) is the angular frequency of the simple harmonic motion you want to produce. (There are also more complicated solutions that don't turn vertical at the ends.)

***

To find this solution, I worked in the first quadrant with the wire given by \(x = p(y)\) and \(p(y)\) unknown. Let the bead begin from rest at \((x,y) = (p(y_0), y_0)\). From energy conservation,

\[\frac{1}{2}mv^2 + mgy = mgy_0\,.\]

Express the speed as \(v^2 = ((p^\prime)^2 + 1)\dot{y}^2\). Now if we specify the vertical motion as \(y(t) = y_0\cos^2\left(\frac{\omega t}{2}\right)\), then we can express the time derivative in terms of the coordinate itself: \(\dot{y}^2 = \omega^2(y_0\,y - y^2)\). This eventually leads to a separable first-order nonlinear differential equation for \(p(y)\), \[\frac{dp}{dy} = \sqrt{\gamma\frac{y_0}{y}-1}\,\] where \(\gamma = \frac{2g}{\omega^2y_0}\). Note that \(\gamma \geq 1\), because the vertical acceleration can't possibly exceed \(g\), so that \(\frac{1}{2}\omega^2y_0 \leq g\) or in other words \(\gamma \geq 1\). Also note that \(\frac{dp}{dy}\rightarrow\infty\) as \(y\rightarrow 0\), which makes sense because the half-pipe must be horizontal where the vertical motion reverses direction.

Integrating with the boundary condition \(p(0) = 0\) so that the half-pipe goes through the origin, we get, with the help of Dwight 194.21 and 192.11, \[p = \int_0^y{\sqrt{\gamma \frac{y_0}{\xi}-1}\,d\xi} = \sqrt{\gamma\,y\,y_0 - y^2} + \gamma y_0 \left(\frac{\pi}{2}-\tan^{-1}\!\!\sqrt{\gamma\frac{y_0}{y}-1}\right)\,.\] If we take \(y_0 = \frac{2g}{\omega^2}\) then \(\gamma = 1\) and this simplifies to \[p = \sqrt{y\,y_0-y^2} + y_0\left(\frac{\pi}{2}-\tan^{-1}\!\!\sqrt{\frac{y_0}{y}-1}\right)\,.\] The arctangent can be expressed more simply as \(\cos^{-1}\!\!\sqrt{\frac{y}{y_0}}\) (to see this, draw a right triangle with legs \(y\) and \(\sqrt{y\,y_0-y^2}\)), whereupon we also notice that \(\frac{\pi}{2} - \cos^{-1}(\cdots) = \sin^{-1}(\cdots)\), so in the end \[p(y) = \sqrt{y_0\,y-y^2}+y_0\sin^{-1}\!\!\sqrt{y/y_0}\] as above.

## Monday, June 12, 2017

### Ninth Circuit: The Law Forbids Setting Immigration Policy Via Press Release

To date, President Trump's travel bans have generated a whole catalog of district court and circuit court opinions. Reading some of these opinions, I've been surprised to see how often the judges have bypassed the statutory interpretation issues in the case and reached the constitutional questions pertaining to the Establishment Clause of the First Amendment and the Due Process clause of the Fifth Amendment.

It's surprising because the accepted practice in Federal courts is that you only reach constitutional questions if you must. When there's a way to decide a case short of deciding a constitutional question, you should take that route. And here it seems clear that there's at least a statutory question to be settled, because there are two specific statutes in the Immigration and Naturalization Act (INA) that contradict one another:

Recently, when the Fourth Circuit upheld a temporary restraining order on the travel ban, it too bypassed this statutory question and reached constitutional questions, finding against the Government on First Amendment grounds. However, one of the concurring judges, Barbara Milano Keenan, did say that she would have arrived at the same conclusion by the simpler expedient of statutory interpretation. Keenan's argument wasn't that 1152(a)(1)(A) overrules 1182(f); indeed, she indicated that thought the two were compatible on their face. Instead, her statutory reading was confined to 1182(f) itself, and in particular the burdens it places on the President by its text. Here is the crux of Keenan's argument for why the travel ban exceeds the powers granted to the President by Congress in the Immigration and Naturalization Act:

Keenan's argument appears to have been influential beyond her own circuit, because today, in upholding yet again the temporary restraining order against the Government, the Ninth Circuit court backed away from its prior constitutional reasoning and embraced Kennan's approach. Like Keenan, the Ninth Circuit held that (here I give a bunch of excerpts):

Logically speaking, I don't think the Ninth Circuit's decision hinges on resolving the contradiction between 1182(f) and 1152(a)(1)(A). But drawing a line between what Trump did and what Carter, Reagan, and Obama did becomes easier for the Ninth Circuit if 1152(a)(1)(A) limits 1182(f), and this unsurprisingly is what the Ninth Circuit finds. By applying several canons of statutory interpretations, the opinion imposes this reading on them:

Having found the Executive Order to be in conflict with immigration law as written, the Ninth Circuit reminds everyone that if the legislative and executive branches don't like the law, they're perfectly entitled to come together and rewrite the law to their mutual satisfaction:

In a series of footnotes, the Ninth Circuit did make several down-payments on constitutional questions, in case those questions arise later in the Supreme Court. Probably the juiciest footnote in the opinion is the one that goes out of its way to rule Trump's tweets in-bounds according to the Federal Rules of Evidence:

It's surprising because the accepted practice in Federal courts is that you only reach constitutional questions if you must. When there's a way to decide a case short of deciding a constitutional question, you should take that route. And here it seems clear that there's at least a statutory question to be settled, because there are two specific statutes in the Immigration and Naturalization Act (INA) that contradict one another:

1182(f). Whenever the President finds that the entry of any aliens or of any class of aliens into the United States would be detrimental to the interests of the United States, he may by proclamation, and for such period as he shall deem necessary, suspend the entry of all aliens or any class of aliens as immigrants or nonimmigrants, or impose on the entry of aliens any restrictions he may deem to be appropriate.

1152(a)(1)(A). [N]o person shall receive any preference or priority or be discriminated against in the issuance of an immigrant visa because of the person’s race, sex, nationality, place of birth, or place of residence.

So which is it? Can the President ban everyone from Country X, or may the President do no such thing? Or is there no contradiction here after all, because 1182(f) refers to "entry" while 1152(A)(1)(a) refers to 'issuance of a visa'?

Recently, when the Fourth Circuit upheld a temporary restraining order on the travel ban, it too bypassed this statutory question and reached constitutional questions, finding against the Government on First Amendment grounds. However, one of the concurring judges, Barbara Milano Keenan, did say that she would have arrived at the same conclusion by the simpler expedient of statutory interpretation. Keenan's argument wasn't that 1152(a)(1)(A) overrules 1182(f); indeed, she indicated that thought the two were compatible on their face. Instead, her statutory reading was confined to 1182(f) itself, and in particular the burdens it places on the President by its text. Here is the crux of Keenan's argument for why the travel ban exceeds the powers granted to the President by Congress in the Immigration and Naturalization Act:

The plain language of Section 1182(f) permits a president to act only if he "finds" that entry of the aliens in question "In other words, awould bedetrimental to the interests of the United States" (emphasis added). In my view, an unsupported conclusion will not satisfy this "finding" requirement. Otherwise, a president could act in total disregard of other material provisions of the INA, thereby effectively nullifying that complex body of law enacted by Congress.

Here, the President's "finding" in Section 2(c) is, in essence, a non sequitur because the "finding" does not follow from the four corners of the Order’s text. In particular, the text fails to articulate a basis for the President’s conclusion that entry by any of the approximately 180 million individuals subject to the ban "would be detrimental to the interests of the United States."

*finding*must be more than a bare assertion—and what's more, the Executive Order, read carefully, doesn't even make a bare assertion. It includes some rhetoric about the*countries*and how dangerous those countries are, but it never even*claims*that the__aliens__are dangerous, let alone*finds*them to be so, with any of the due diligence or rationality that is connoted by the notion of a "finding" in law.Keenan's argument appears to have been influential beyond her own circuit, because today, in upholding yet again the temporary restraining order against the Government, the Ninth Circuit court backed away from its prior constitutional reasoning and embraced Kennan's approach. Like Keenan, the Ninth Circuit held that (here I give a bunch of excerpts):

the President did not meet the essential precondition to exercising his delegated authority: The President must make a sufficient finding that the entry of these classes of people would be "detrimental to the interests of the United States."

There is no finding that present vetting standards are inadequate, and no finding that absent the improved vetting procedures there likely will be harm to our national interests.

The Order makes no finding that nationality alone renders entry of this broad class of individuals a heightened security risk to the United States

The Order does not tie these nationals in any way to terrorist organizations within the six designated countries. It does not identify these nationals as contributors to active conflict or as those responsible for insecure country conditions. It does not provide any link between an individual's nationality and their propensity to commit terrorism or their inherent dangerousness.

the Order does not provide a rationale explaining why permitting entry of nationals from the six designated countries under current protocols would be detrimental to the interests of the United States.

The Order's discussion of country conditions fails to bridge the gap. Indeed, its use of nationality as the sole basis for suspending entry means that nationals without significant ties to the six designated countries, such as those who left as children or those whose nationality is based on parentage alone, should be suspended from entry. Yet, nationals of other countries who do have meaningful ties to the six designated countries—and may be contributing to the very country conditions discussed—fall outside the scope of Section 2(c). Consequently, EO2’s focus on nationality "could have the paradoxical effect of barring entry by a Syrian national who has lived in Switzerland for decades, but not a Swiss national who has immigrated to Syria during its civil war."

the Order specifically avoids making any finding that the current screening processes are inadequate. As the law stands, a visa applicant bears the burden of showing that the applicant is eligible to receive a visa or other document for entry and is not inadmissible. See 8 U.S.C. § 1361. The Government already can exclude individuals who do not meet that burden. Seeid. The Order offers no further reason explaining how this individualized adjudication process is flawed such that permitting entry of an entire class of nationals is injurious to the interests of the United States.

Former Presidents have invoked § 1182(f) under non-exigent circumstances to address compromised security conditions abroad but have tied exclusions to the culpable conduct of barred aliens, such as aliens who contributed to a country's situation in a specified way or were members of particular narrowly defined and/or dangerous groups.

President Obama's Executive Order 13726, for example, suspended the entry into the United States of persons who were responsible or complicit in particular actions or policies that threaten the stability of Libya

In two instances, former Presidents have distinguished classes of aliens on the basis of nationality. But these distinctions were made not because of a particular concern that entry of the individuals themselves would be detrimental, but rather, as retaliatory diplomatic measures responsive to government conduct directed at the United States. For example, President Carter's proclamation barring the future entry of Iranians occurred during the exigent circumstance of the Iranian hostage crisis.

President Reagan's suspension of entry of certain Cuban nationals as immigrants came as a response to the Cuban government's own suspension of "all types of procedures regarding the execution" of an immigration agreement between the United States and Cuba, which had "disrupt[ed] normal migration procedures between the two countries."

Section 1182(f) requires that the President exercise his authority only after meeting the precondition of finding that entry of an alien or class of alienswould bedetrimental to the interests of the United States. Here, the President has not done so.

***

Logically speaking, I don't think the Ninth Circuit's decision hinges on resolving the contradiction between 1182(f) and 1152(a)(1)(A). But drawing a line between what Trump did and what Carter, Reagan, and Obama did becomes easier for the Ninth Circuit if 1152(a)(1)(A) limits 1182(f), and this unsurprisingly is what the Ninth Circuit finds. By applying several canons of statutory interpretations, the opinion imposes this reading on them:

In prohibiting nationality-based discrimination in the issuance of immigrant visas, Congress also in effect prohibited nationality-based discrimination in the admission of aliens.The argument, basically, is that in a reasonable set of immigration rules, the rules for entry can't be meaningfully different from the rules for visas. Remember, though, that Judge Keenan opined differently in her concurrence for the Fourth Circuit. On the other hand, her fellow Judge Stephanie Thacker didn't join that part of her opinion…so it seems the basic meaning of the INA as a whole remains somewhat unsettled.

***

Having found the Executive Order to be in conflict with immigration law as written, the Ninth Circuit reminds everyone that if the legislative and executive branches don't like the law, they're perfectly entitled to come together and rewrite the law to their mutual satisfaction:

We have based our decision holding the entry ban unlawful on statutory considerations, and nothing said herein precludes Congress and the President from reaching a new understanding and confirming it by statute.Indeed, it is Congress's inability (or unwillingness) to write legislation over the past ten years that has encouraged Presidents to write so much policy via Executive Order. In this trend Trump represents a new low, because thanks to his administration's overall incompetence, his Executive Orders aren't much better than press releases. Another word for policy-by-press-release is

*authoritarianism*.
***

In a series of footnotes, the Ninth Circuit did make several down-payments on constitutional questions, in case those questions arise later in the Supreme Court. Probably the juiciest footnote in the opinion is the one that goes out of its way to rule Trump's tweets in-bounds according to the Federal Rules of Evidence:

See Donald J. Trump (@realDonaldTrump), Twitter (June 5, 2017, 6:20 PM), https://twitter.com/realDonaldTrump/status/871899511525961728 ("That’s right, we need a TRAVEL BAN for certain DANGEROUS countries, not some politically correct term that won’t help us protect our people!") (emphasis in original); see also Elizabeth Landers, White House: Trump's tweets are "official statements", CNN (June 6, 2017, 4:37 PM), http://www.cnn.com/2017/06/06/politics/trump-tweets-official-statements/ (reporting the White House Press Secretary's confirmation that the President's tweets are "considered official statements by the President of the United States"). We take judicial notice of President Trump’s statement as the veracity of this statement "can be accurately and readily determined from sources whose accuracy cannot reasonably be questioned." Fed. R. Evid. 201(b)(2).

I had to laugh when I read that parenthetical "emphasis in original." Translation: GRANDPA WRITES IN ALL-CAPS.

## Friday, June 9, 2017

### i.e. and e.g.: I'm Done

Instead of trying to explain the difference between

*i.e.*and*e.g.*, I'm going to propose today that we, all of us, strike these abbreviations from our writing entirely.
Reading a Supreme Court decision last week, I was amazed to see that

*e.g.*was used where*i.e.*was meant.
Unless I'm mistaken on the substance—always possible, since I'm not a lawyer—Justice Sotomayor wanted the sense of

*in other words*here; but she wrote*e.g.,*which has the sense of*for example*.
Folks, written English doesn't get any higher-status than a published decision of the U.S. Supreme Court. If a distinction isn't observed there, then it's hard to maintain that the distinction exists.

This week at work, I was sent a memo that used

*i.e.*for*e.g.*, or maybe*e.g.*for*i.e.*, or maybe both (it happened more than once). The writer was a Harvard graduate.
I'm not criticizing Harvard here, or Yale for that matter (Sonia Sotomayor's alma mater). Rather, what I take from these two examples is that

**.***i.e.*and*e.g.*are strictly meaningless
From now on, if I want to say "for example," then that's what I'll write. Likewise for "that is," or "in other words."

Abbreviations mar prose anyway. I hereby purge

*i.e.*and*e.g.*from my written lexicon.
Subscribe to:
Posts (Atom)