Two (Relatively) Easy Mensa Math Brain Busters

 Two relatively easy math problems to challenge the gray matter of energetic Brane Space readers:

1)  Find the smallest integer that:

- When divided by 3 has a remainder of 1.

- When divided by 4 has a remainder of 2.

- When divided by 5 has a reminder of 3.

- When divided by 6 has a remainder of 4

- When divided by 7 has a remainder of 5

2) You run an enterprise that makes cone-shaped paper cups. You start with a round piece of paper (radius = 1).  Then remove one sector, after which you connect the two straight edges to form a cone.

Find the maximum ratio of volume to surface area.

(Hint: You will need to use some basic differential calculus)

Solutions to Galaxy Density Wave Problem

 The Problem again:

The number c of epicycle oscillations per orbit about a galactic center, is given by the ratio of the star’s epicycle frequency (k_o)  to its orbital angular speed, W.

Or:   c =  k_o / W  

a) Compute W for the Sun given that the Sun’s period around the galactic center is: 2 x 10 ⁸ yrs.

  

b) Then find the Sun’s epicyclic frequency (k_o) if we know c = 1.35. (Note: if the ratio c is integral, or a non-decimal number, we say the orbit is “closed”. If non-integral it is not closed). Because of differential rotation rates and the local angular velocity (W _L) differing from the inertial value, the solar region is open.

c) Compute the local angular velocity (W _L) for the Sun if:  

m(W  -  W _L) = n k_o   where m = 2 and n = 1

Solutions:

a) Assume that the Sun moves in a nearly circular orbit about the galactic center with the radius of orbit R = 10⁴ parsecs. It then makes an entire orbital revolution in 2 x 10⁸ years. We need to obtain this period in seconds:

T(s) = (2 x 10 ⁸ yrs.) (365.25 days/yr.) (86,400 s/ day)

T(s) = 6.3 x 10 ¹⁵ s

 Then: W  = (2 π rad)/ T(s) = 2 π rad/ 6.3 x 10 ¹⁵ s  

W  » 10 ^-15 rad s^-1

(b) We know c = 1.35 and we have: c =  k_o / W .  Thus the epicyclic frequency:

k_o = 1.35  W    = 1.35 (10 ^-15 rad s^-1) = 1.35 x 10 ^-15 rad s^-1

 (c) Re-arrange the given expression:

m(W  -  W _L) = n k_o

 By deliberately choosing to have the Sun complete n orbits in the rotating frame of reference (while executing m epicycle oscillations) we can write:

W _L =  W  - n k_o / m   =   W  - k_o / 2

Therefore:

W _L =  10 ^-15 rad s^-1   - ½(1.35 x 10 ^-15 rad s^-1 ) =

3.2 x 10 ^-16 rad s^-1

Modeling Density Wave Galaxy Formation And Problems Associated With It

 

                                 Density wave formed spiral galaxy

Interestingly, the use of density wave model development is largely contingent on the Boltzmann equation which is also used in space plasma physics. Thus:

¶f/ ¶t + v grad f + F/m ¶f/¶t = (¶f/¶t)_C

where the ¶f/ ¶t are partial derivatives, and (¶f/¶t)_C

is the time rate of change in f (the velocity distribution function) due to collisions, i.e. between masses within the system. Technically, the Boltzmann eqn. is applied to FLUIDS and for that purpose the galaxies to which density wave approaches or models are applied are modeled firstly in the fluid format. (It is easier when dealing with an agglomeration of some 100 or 200 billion separate stars and associated orbits to think of them as comprising a "fluid" as opposed to say, 100 billion separate bodies to be treated in a 100 billion -body problem of celestial mechanics!)

We see from this that the question as to why the spiral pattern is not affected by stars much further out cannot really be properly answered unless a full vetting of the assumed density waves for the particular galaxy is presented. In this sense, one recognizes that a full analysis of density waves for a galaxy - call it "Barred G1"- is needed before one can say stars in a given G1 region (e.g. inner or outer) "move faster or more slowly" than the waves at that place. We need to know then: the physical conditions for the establishment of the density waves at location r1 in G1 and r10 in G1 where the r's denote radial distances from the center with r10 = 10 (r1).

Another problem in dealing with density wave models is the fact they are  mainly  based  on the mode chosen for particular dynamical  wave equations  that  can be applied to the fluid framework. (Note: A "mode" is a standing wave that can be supported by a disk of given dimensions, mass.) More broadly, most astronomers who work in this narrow specialist area use the term interchangeably with Fourier m-component. (And it should be understood here that one of the main tools is Fourier analysis of the waves, but alas Fourier analysis is only taught usually to those who take advanced Calculus or analysis courses).

As an example, a particular Fourier coefficient, call it a_n, applicable to a wave - may be defined:

a_n = 1/π ò _-π ^π f(x) cos mx dx


where m is the Fourier m-component.

    What types of modes can one have in these models? One is the "global" or m= 1 mode. Then there are the unstable (m= 2) modes.

    Whether one mode or another appears (or is used in a spiral galaxy modeling) is critical since it may well determine at what stage a barred spiral develops, if at all. Alas, another complexity enters here since mode analysis is not simply a stand alone but also incorporates a subtle aspect called "marginal stability analysis" wherein one will solve for a quantity Q and if it is very close to 1 one has the case of marginal stability and tightly wound modes or in the case of spirals, around the Sa class. The trouble is that when one seriously incorporates any heating of the disk for whatever reason (say a massive central black hole sucking up matter and generating much radiation) then the desired values of Q are soon out of range, making it impossible for a given spiral structure to sustain itself.

     Lastly, whenever one considers density waves in galaxies, it's important to bear in mind there remain enormous stumbling blocks even when applied to the simplest models of galactic disks (e.g. "zero thickness" disks). One of these arises from potential theory. Thus, the perturbed gravitational field at one location depends on the density perturbation at every other location. How will you know, ab initio, that the density perturbation at location r, φ, z say, does not accelerate the associated wave (in the fluid rest frame) to a higher velocity than any stars at the same or near location? You don't unless you investigate! 

Still, an amazing number of relatively simple problems pertaining to spiral galaxies can be solved using just basic information.  One such problem is given below for the interested (and industrious) reader to work out:

Problem:

The number c of epicycle oscillations per orbit about a galactic center, is given by the ratio of the star’s epicycle frequency (k_o)  to its orbital angular speed, W.

Or:   c =  k_o / W  

a) Compute W for the Sun given that the Sun’s period around the galactic center is: 2 x 10 ⁸ yrs.

  

b) Then find the Sun’s epicycle frequency (k_o) if we know c = 1.35. (Note: if the ratio c is integral, or a non-decimal number, we say the orbit is “closed”. If non-integral it is not closed). Because of differential rotation rates and the local angular velocity (W _L) differing from the inertial value, the solar region is open.

c) Compute the local angular velocity (W _L) for the Sun if:  

m(W  -  W _L) = n k_o   where m = 2 and n = 1

See Also:

Selected Questions - Answers From All Experts Astronomy Forum (Spiral Galaxy Shape)

And:

Is This The Most Distant Galaxy - Or Is The Freakish (Z-Shift) Object HD1 Something Else?

And:

ESA's "Gaia" Telescope Reveals Bounty Of Data On Our Galaxy (brane-space.blogspot.com)

And:

- https://www.msn.com/en-us/news/technology/mond-vs-dark-matter-research-suggests-that-rotation-curves-of-galaxies-stay-flat-indefinitely/ar-BB1onwMa?rc=1&ocid=socialshare&cvid=b2d65aab8fdf49bcb6b4bebde6e94f54&ei=9

WSJ CORE Econ Model Proves Americans' Economic Perceptions Aren't Based On Hard Data - But Delusions

 

         The CORE Map and relative scores (color coded)

                                         CORE Table Results (2014-21)

"The latest poll from Harris  released  by the Guardian suggests that well over half the country is delusional. Polling shows that 55% of Americans believe the economy is shrinking, and 56% think we are in a recession. Neither of those things are remotely true. Meanwhile 49% believe the stock market is down for the year even though it's at a record high. Even more bizarre, 49% believe that unemployment is at a 50-year high, though the unemployment rate has been under 4%, a near 50-year low." - Heather Digby Parton

 In my April 5th post, I noted half of the respondents in an April 4th WSJ survey:

What’s Wrong With the Economy? It’s You, Not the Data

Defined inflation incorrectly, conflating high prices with high inflation. But the sobering fact is that once prices rise they seldom go back down to what normal plebes believe to be 'proper' levels, in this case pre-pandemic.

The true guide for high inflation is whether the overall year's average rate is higher than the long term average inflation rate.   For example, the U.S. Inflation Rate is currently at 3.15%, compared to 3.09% last month and 6.04% last year, and 8.3 percent in 2022

This is lower than the long term average of 3.28%.

Hence, inflation is going down, receding - irrespective of what you paid for eggs, milk and bread in 2022, 2023.  This is given you do not base inflation rate on specific prices of specific groceries or other goods.  To reinforce that take, by more than 2-to-1 (56% to 25%) WSJ respondents "said the economy had gotten worse, rather than gotten better over the past two years."   But as the WSJ writers pointed out:

"That is difficult to square with the robust employment growth, unemployment near its lowest in a half century, and in gross domestic product (GDP) which actually accelerated last year."

 These sorts of confirmation bias errors combined with the continued dismal rating of Biden on inflation and the economy means a more objective perspective is desperately needed to wake our fellow Americans up. They simply can't be allowed to go into this momentous general election in a sleepwalking condition. (Much of which I also attribute to a carryover from the pandemic).

In a follow up piece (April 25th) the WSJ "talked with hundreds of Americans in nearly every corner of the country" and found a unifying mental glitch: Over and over again they insisted their own economy wasn't thriving regardless of what the economic indices and numbers looked like.  But as the earlier WSJ piece noted:

"As the saying goes you can't eat gross domestic product, but what you eat is part of GDP and while many people say they are cutting back on groceries and eating out, the data show that collectively they are not."  

No they are not, wonder of wonders. They're taking expensive plane trips, going to Swiftie concerts and as the WSJ adds:

"Americans as a whole are still spending more on groceries and restaurant meals, even after adjusting for inflation, than in 2019."

This disconnect led to the introduction of the CORE score (WSJ, ibid.) to try to assist 'Muricans getting a grip on reality - because their actions don't conform to it.  The motivation (April 25th, ibid.):

 "It is important to know who is benefiting from the economic growth and whose lives are being improved.  It is also important to know which places are benefiting.  The American map has become so fractured between rich and poor communities that a more granular analysis is needed.  

To that end, the CORE score is rooted in county level measurements, making it possible to see differences within counties along lines of race and ethnicity, age, education, income and sex.  To that end the CORE score is broken down into four central elements: economic security, economic opportunity, health and political voice."

The graphic shown (as a map) at the top of this post, as well as the table given, bear this out and show how detached too many of our countrymen have become - likely from watching too many FOX news propaganda efforts.  For example, the District of Columbia (depicted as a bastion of thugs and poverty by the Right)  has the highest state level CORE score for 2021 -the most recent year for which the statistics were available.  Meanwhile, South Texas features some of the lowest scores. But why is that?  Perhaps because this state like too many other red ones, refused to expand Medicaid access during the pandemic when it had the chance.  Readers can use the color-coded legend at the top of the map with associated CORE scores to parse out the differences for other areas. 

 Another intriguing feature is the graph showing a radically changing GDP over 2014-21 but scarcely any variation in CORE score. How explain this?  As  Financial Times contributor David Pilling wrote in 2018  article ('Why GDP Is A Faulty Measure Of Success',  TIME, Feb. 5, p. 41):

"Invented in the 1930s, the figure is a child of the manufacturing age - good at measuring physical production but not the services that dominate modern economies. How would GDP measure the quality of mental health care or the availability of day care centers and parks in your area?  Even Simon Kuznets, the Belarussian economist who practically invented GDP, had doubts about his creation."

GDP is supposed to measure the total production and consumption of goods and services in the United States. But the numbers that make up the Gross Domestic Product by and large only capture the monetary transactions we can put a dollar value on. Almost everything else is left out: old growth forests that maintain cooling and act as CO2 repositories, watersheds, animal habitats, e.g. the Everglades, and costs of infrastructure maintenance. But ALL of these count toward  the physical security and welfare of a society. If bridges collapse owing to maintenance failure and hundreds or thousands of drivers are inconvenienced, delayed  - then that has a cost and hence an economic impact!

In addition, there are hundreds of other contributions not registered that arguably have  major economic repercussions. For example, a 2015 Forbes article highlighted how 40 million family caregivers in the U.S. are putting their own careers on hold to provide unpaid care — sometimes for decades.   The estimated  total value of the care has been put at nearly $1 trillion. This isn't reckoned into the GDP.

Thus, removing these aberrations, as the CORE score does, discloses a relatively stable, almost straight line graph, as we see.  This is the basis for the authors saying they "see little improvement in American well being since our data began in 2005."  They agree that at least health scores have improved, but that is because - thanks to government - health coverage has expanded. If it were up to the Righties and Trumpers the misery index would be vastly greater, I can assure you!  The authors nonetheless go on to point out the enhanced health factors are "offset by a decrease in measures of economic security". (Based on credit bureau data on household finances.)  This may also explain why so many Americans are so dissatisfied with an economy that "looks strong according to traditional metrics."

See Also:

• Trump Allies' Plan To Sideline Fed & Devalue U.S. Dollar...

And:

Our Economic Reality: Do You Prefer Higher Unemployment - Or Higher Inflation? There Is No 3rd Option 

And:

Home » blogs » Robert Freeman's blog

Debunking the Myth of “The Greatest Economy Ever”

Robert Freeman

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Donald Trump | Economic Policy

by Robert Freeman | September 28, 2020 - 5:39am

---

Donald Trump routinely claims he “built the greatest economy in the history of the world.” It is a claim rarely challenged in the mainstream media. If true, it would be at least part of a possible claim for re-election. But is it true?

There are five things we need to know to evaluate Trump’s claim to economic wizardry.

The first is that rather than “building” a strong economy, Trump hitched a ride on the one that was handed to him by Barak Obama and Joe Biden. When Trump took office, the economy had already been growing for 90 straight months. It would go on to grow for another 38 months before Trump drove it into the ditch. It was the longest U.S. economic expansion since data was first collected, in 1854. But Trump didn’t “build” it. He inherited it.

The second thing to know is that Trump’s stewardship of the economy has been far less effective than that of many other presidents over the past 60 years. In 2018, Trump’s best year for growth, the economy expanded 3.18%. That is the 29^th best year for growth since JFK took office in 1961. For 2017, growth clocked in at 2.22%, 44^th best in the last 59 years.

Best ever? It’s not even close to being close.

What about median income? That is a reliable measure of economic well-being. According to the House Joint Economic Committee, “During the last two years of the Obama administration, annual median household income increased $4,800. This is three times more than the $1,400 increase during the first two years of the Trump administration.” Best ever? Hardly.

Job growth? That, too, is a reliable indicator of a strong economy. The same Joint Economic Committee reported, “During the last 33 months of the Obama administration, nonfarm job growth averaged 224,000 per month. During the first 33 months of the Trump administration, the average was 34,000 jobs per month less.” Best ever? Please.

Trump-Biden Debates? Sure, Bring 'Em On - But Use the Format Of the 1960 JFK-Nixon Debates

                  Biden vs. Trump debate? It ought to be a gimme - for Joe

Yesterday, alarmed by Joe’s continued awful polls in swing states (at -13 vs. Trump in Nevada for example), the Biden campaign “shook up the race”- in the parlance of the NY Times- offering to have the first presidential debate June 27^th.   This would bring up the debate scheduled by three months, while also taking it out of the province of the Presidential Debate Commission – which I for one think is an excellent idea. Because that Commission, for whatever reason, has just turned the debates into a 3 ring circus, what with catcalls, peanut gallery hoots and hollers, and boorish interruptions – namely by Trump which we beheld in 2020, e.g.

Don't Be Fooled - Last Night Was Another Shit Show...

According to the Times, the “move was meant to jolt Americans to attention sooner than later about their consequential choice in 2024.”  But that is doubtful given too many of my countrymen appear to be in  state of more or less permanent miasma.  They’ve lost the ability to reason, to think critically and especially to see Donald Trump as the existential threat he is.  

A perfect example was the Times’ surveying 12 women voters two days ago, the majority of whom said they’d vote for Trump again as they did in 2020. This despite conceding Trump was "abhorrent", "crude", "bigoted", "politically unstable", "mentally unstable", "narcissistic".

Why then pull the lever for the orange imp? Because they dismissed Joe as “an idiot”, “incompetent”, and “ineffective” while they still see Trump as “funny”, “energetic”, a “strong leader” and “former businessman”.  Which they think cancels out his negatives.  All of which caused Janice’s head to spin whereupon she dismissed the lot as “boneheaded Trumper nitwits".

Which may be right, and that moron level of function may be due to excessive exposure to FOX TV or maybe too much TikTok. Who knows? What I do know is the absence of critical perception regarding the principals in this campaign is leading us toward a political catastrophe in November. People - too many - are mistaking Trump's feral and psychotic outbursts for "energy" and even "youth", just as they can't see that the economic indices are actually much better than they were under Trump, i.e.

Donald Trump & GOP "Better For Economy" ? This Delusion Belied By The Facts...

While Biden’s advisers believe a Trump-Biden rematch will mark a jump start for Joe’s dismal approval ratings, I am not so sure. I suspect the problem may be a lot more systemic and require more competent (read 'Aggressive') campaign strategists and thinkers, akin to what Bill Clinton had in his “war room” in the 1992 and 1996 campaigns.  Think James Carville and George Stephanopoulos.  I myself suggested recruiting Rahm Emmanuel, to head the campaign.  Get him back from Tokyo and put him to work... now!

Having said that, if the Biden team can hold the Trump bunch to the format suggested:  in a closed studio, no studio audience, one single moderator and the two candidates, period. 

Like the JFK- Nixon 1960 debates, e.g. the first one moderated by Howard K. Smith:

Kennedy vs. Nixon: The first 1960 presidential debate (youtube.com)

I believe Biden will have a much higher chance of success.  The sedate aura and somber environment alone will deter a madman sociopath like Trump from unseemly outbursts which will merely paint him as unhinged and psychotic.  Finally showing viewers the kind of unfit hyena he really is and why he should not be presiding over the cleaning of porto- potties, far less  attain the presidency of the U.S. 

Will it be a spectacle? Sexy? Armchair grabbing and pulse pounding? Likely no.  But that's not the point. The point is the conveyance of information and the background knowledge and class of the two men - just as the 1960 presidential debates between John F. Kennedy and Richard Nixon. Click on the link above and watch the full Kennedy-Nixon debate to see what a genuine presidential debate can aspire to.  No name calling, no feral rhetoric, no nicknames or call outs to crowds (mainly because there weren't any.)

Basically, we need to return the upcoming debates to an atmosphere of dignity and mutual respect as opposed to chaos and incivility

The fact the Trump team swiftly accepted the Biden team's challenge means that at least on paper they are amenable to their candidate not behaving like "a mad monkey with a gun" for once. (A phrase I owe to Jesse MacIntosh from one 2016 interview with Chris Hayes on MSNBC)

Another plus for the debates is keeping them between the two principals. No 'bullfrog with a bullhorn' voiced Bobby Kennedy Jr. spouting his anti-vaxx BS or conspiracy ideations. Also, taking the focus off the binary nature of this campaign - in which Americans need to grasp RFK Jr. can never win, only toss the election to Trump by siphoning off votes for Biden.

We will see what happens, but so far the odds look good for a genuine debate, devoid of theatrics, bombast and chaos - between Joe Biden and Dotard Trump.

 See Also:

2024 Elections | Donald Trump | Joe Biden

‘Trump Will Hit Below the Belt’: Here’s How Biden Is Preparing for June 27 Debate

by Carl Gibson | June 16, 2024 - 5:40am | permalink

— from Alternet

President Joe Biden and former President Donald Trump are set to face off in their first televised debate of the 2024 cycle later this month on CNN. Both campaigns are scrambling to get their candidates prepared for their first debate in four years.

The New York Times reported that Biden's advisors are preparing for a feisty 90 minutes of intense back-and-forth with the 45th president of the United States, who they don't expect to play fair despite the strict rules both campaigns agreed to. The debate, which will not include a live audience, has a format in which neither candidate gives an opening statement but rather goes right into answering moderators' questions with a two-minute time limit. A flashing red light will blink when the candidate speaking has five seconds remaining.

Both candidates will also get one minute to rebut their opponent after each question. And notably, the speaker's microphone will be cut off after the time limit has elapsed for each answer or rebuttal. This is a stark departure from their lone 2020 debate, in which Trump constantly interjected, prompting Biden to famously say at one point, "will you shut up, man?"

And:

Odds these debates will happen? I’m answering all your questions.

And:

Donald Trump

No Wonder Republicans Are Afraid of a Debate — Donald Trump Is Barely Holding It Together at Trial

by Amanda Marcotte | May 18, 2024 - 6:25am | permalink

— from Salon

Donald Trump has tried to taunt President Joe Biden by claiming he's ready for a presidential debate "anytime, anywhere." The gambit left him little choice but to immediately accept when the Biden campaign offered debate dates in June and September, and terms like not having an audience and allowing real journalists to moderate. But within mere hours it became clear that Trump and the rest of the GOP already regretted the decision.

Pretty quickly, Trump tried to change the terms of the debate, pretending that it will be held on Oct. 2 on Fox News. The Biden campaign swiftly rejected this lie, accusing Trump of "playing games," and pointing out that Trump frequently talks big but then ends up "pulling out at the last minute, or not showing up at all."

It's true, of course. Trump has a habit of promising that he'll do bold things and then backing out, whether it's his empty promises to testify at his various trials or his false claims he'll release policy proposals in a week or two. (It's been over a month of silence, for instance, since he promised he would release an abortion platform in "14 days.") In 2020, still burned by his terrible first debate with Biden, Trump refused to show up at the second and held an ego-flattering rally instead.

And:

Donald Trump

That Was No Debate, It Was a 90-Minute Insult to America

by Michael Winship | October 1, 2020 - 7:18am | permalink

Just minutes in, Trump turned it into a travesty; bellowing, repeatedly interrupting and talking out of turn, lying, pigheadedly violating the agreements carefully worked out by the official Commission on Presidential Debates and both campaign staffs for a proper discussion of policy. At a time when we’re desperate for rational solutions, Trump was the proverbial bull in the china shop of democracy, and when he was done there was barely a piece of porcelain left intact. He acted as if this was one of his rallies, God help us, playing to our lowest animal instincts and insulting our intelligence.

Throughout his life and so-called career, Trump has always insisted that rules are for suckers and don’t apply to him—he can shoot a person on Fifth Avenue, grab a woman by her genitals, refuse to pay his millions of debt—and last night was no exception. By all accounts, what happened Tuesday is what he planned all along. Maybe it satisfied his bloodthirsty, delusional base, stewing in a toxic soup of Fox News and QAnon conspiracies, but for most of us it was one more manifestation of the tragedy imposed on the nation—and the world—by a megalomaniac with nary a scruple or a scintilla of human kindness.

And:

Trump Is Scared Shitless: He Can't Handle the Truth, so He Wouldn't Let Biden Talk

by Amanda Marcotte | October 1, 2020 - 8:44am | permalink

— from Salon

Forget the euphemistic language being used by much of the mainstream press to describe Tuesday night's presidential debate between Donald Trump and former Vice President Joe Biden. That wasn't a "contentious debate," nor were the candidates "bickering." No, Biden showed up to do normal politician things, only to find himself trapped on stage with Trump, a shit-flinging monkey whose only purpose was to prevent anything resembling truth from slipping through and touching the ears of the public.

For an hour and a half, the same thing kept happening over and over. (And over, and over.) Biden would try to talk, about literally anything — Trump's failures, his own policy ideas, the state of the nation, hell, possibly the weather — and Trump would immediately interrupt the second he heard a truth about to be stated, ranting over Biden and silencing any factual information or expression of decency under a blanket of lies and vitriol.

And:

Donald Trump

Trump's Debate Performance Showcased the Only Thing His Cult Cares About

by Jeremy Sherman | October 1, 2020 - 8:30am | permalink

— from Alternet

The kid is in his room playing video games. Mom calls to him that it's time to do chores and homework.

idAnd:en

“That Was a Hot Mess, Inside a Dumpster Fire, Inside a Train Wreck”

The first presidential debate of 2020 was unlike anything we have seen before. CNN’s Jake Tapper said: “That was a hot mess, inside a dumpster fire, inside a train wreck.” “He was his own tweets come to life.” “We’ll talk about who won the debate, who lost the debate … One thing for sure, the American people lost.” Conservative pundit William Kristol called it “a spectacle… an embarrassment… a disgrace… because of the behavior of one man, Donald Trump. The interrupting and the bullying, the absence of both decency and dignity — those were Donald Trump’s distinctive contributions to the evening, and they gave the affair the rare and sickening character of a national humiliation.”