Wednesday, July 1, 2026

A Math Treat: D.E. Littlewood's Elegant Introduction to Non-Commutative Algebras (Pt. 1)

  In his timeless monograph 'The Skeleton Key Of Mathematics' (CHapter XIII, p. 101, Algebras) D.E. Littlewood gives us one of the best introductions to that domain of mathematics.  He starts in a logical place, Hamiton's 19th century development of the quaternions.  He begins by looking at Hamilton's approach to vector and scalar products, i.e. in 3 dimensions:


Then noting(p. 102) :

 "A vector (e.g. R) with components x, y,z would be denoted by:

 xi + yj + zk"

Adding:

"If for the law of multiplication it is assumed (for the unit vectors):

2 = - I,  j 2 = - I,  k 2 = - I, 

i j =k jk = i, ki = j

ji = -k,  kj = -i, ik = - j

Then the product of two vectors (xi, yj, zk), (x' i, y'j, z' k) will have a scalar part which is equal to the scalar part with the sign reversed, and a vector part equal to the vector product. Thus:

(xi, yj, zk) (x' i, y'j, z' k) =  - (xx' + yy' + zz') + i(y z' - y'z) 

+ j (zx' - z' x)   + k( xy' - x' y)

The quaternions so formed will obey all the laws of numbers except the commutative law for multiplication. Most importantly, they are associative, i.e. three quaternions A, B, C satisfy:

AB (C) =  A (BC)

"If then:

 Qx 0 +  ix 1   jx 2   kx 3

is any quaternion, the quaternion obtained by changing the sign of the vector part Q = x 0 -  ix 1  -  jx 2  -   kx 3 ) is called the conjugate quaternion.  The product of a quaternion with its conjugate is a positive non-zero scalar, i.e.

  ( x 0 +  ix 1   +  jx 2   kx 3 )  (x 0 -  ix 1  -  jx 2  -   kx 3 ) = 

(x 0 2+  x 1 2   x 2 2 +  x 32)

which is called the norm of the quaternion.  It is thus always possible to divide by a quaternion, except of course by zero, e.g.

I/ (x 1 +  ix 1   +  jx 2  +  kx 3 ) =

 (x 0 -  ix 1  -  jx 2  -   kx 3 / (x 0 2+  x 1 2   x 2 2 +  x 32)

Quaternions with real coefficients, or real quaternions, are thus said to form a division algebra.

If complex coefficients are allowed, however, division may fail. This is given that for complex numbers it is possible that:

 x 0 2+  x 1 2   x 2 2 +  x 32   = 0

without all the quantities (x 0 ,  x 1 ,  x 2 ,  x 3  ) being zero.

Therefore complex quaternions do not forma a division algebra.  It can be shown, however, they are equivalent to the algebra of two rowed matrices, if we set:



Thence we have here one of the most elegant - and simplest- presentations of non-commutative algebras. As well as the basis for what we call 'algebras' in general.

Next: Higher dimensional algebras

Tuesday, June 30, 2026

The Untold Story Of The Reagan Tax Cuts? Conservo Parrots Still Remain Dishonest In Disclosure

 

The Laffer curve: David Stockman ate it up

Arthur Laffer in 1974 had a new theory on why tax rates were inefficient and high, or one might say "inefficiently high".  Meeting with Dick Cheney, and eager to please then President Gerald Ford,  Laffer  proceeded to sketch his infamous diagram for his "Laffer curve" on a napkin in a DC restaurant on why the rich could be said to be "over taxed".

As drawn, it was totally convincing, especially for a guy like Cheney with minimal math skills. Note the line defining the highest marginal tax rate of 70% for Gerald Ford's presidency. What Laffer's curve sought to show is that by cutting that rate down, say to 50%, one could increase the revenues by nearly 35%! Of course, the 50% turned out to be wholly arbitrary and in fact after Reagan became President in 1980 the rates were cut down to 50% by 1981, then to 28% (by 1988). After all, if one could increase revenues by cutting taxes 20%, imagine what one could do by cutting them more than 40%!

Thus did Laffer's curve become the basis of Reagan's tax cuts 1981 and the whole tax cut meme ever since. This despite the fact that in reality no community or even human body has managed to GROW by virtue of starving! But try to tell the bulk of Americans, who continue to buy into this codswallop at a mind-boggling rate! Despite the fact there's never been evidence it's actually worked!  

Enter now Phil Gramm and Michael Solon ('The Surprising Truth About Reagan's Tax Cut, WSJ,  June 20-21, p. A11) who continue to spread Reagan tax cut fantasies. They insist that a 
"bracket creep" (because of inflation at the time)  was actually responsible for expanding deficits, not the Reagan tax cuts.  But is this true, or are these two conservo experts (one at American Enterprise Inst. the other at Hudson Inst.) just playing word games?

Note that when Laffer's crappola was first presented to Reagan's budget maven David Stockman there was no mention of tax bracket influence. So how did Laffer manage to convince Stockman that HIGHER tax rates cause revenues to decrease?  Well, he  argued that higher tax rates on the rich would only cause them to work fewer hours, or if REALLY rich, invest in fewer projects, enterprises, hence create fewer jobs. Thus, revenues over all would decline, first from the working rich because Uncle Sam would get less taxes by virtue of their reduced work, and also from the investing rich because they'd create fewer jobs and thus no workers would be around to pay the taxes Uncle Sam wants. Thus, Laffer argued, the higher tax rates were inefficient!  No mention at all of how abandoning them would explode deficits.

Fortunately, enough economic real indicators existed - even then -  to test Laffer's curve on an empirical basis. Thus, given the progressive Reagan cut - from 70% to 50% by 1981, then down to 28% by 1988, it should be possible to match the claims against economic reality. Especially, given one would expect to find lower debt % of GDP if Laffer's claims were true.

A first full examination of the empirical effects arrived in a text  (The Indebted Society, 1995) by  James Medofff and Andrew Harless, wherein they found, p. 23:

"For the health of the economy, Reagan's policies turned out to be just about the worst thing that could have happened: investment did not increase, growth continued to stagnate, and the federal deficit ballooned to new dimensions....

In 1981, the year Reagan took office, the public debt was 26.5 % of the gross domestic product (GDP)....In 1993, the year that Bush left office, the public debt was a staggering 51.9 percent of the GDP."

(Gramm and Solon cite a recorded deficit of 2.6% in 1980 - but NOT the deficit for the year 1981- after Reagan took office)

Thus, we have the first evidence that Laffer was plying smoke and mirrors, not sound economic policy! If the debt as percentage of GDP nearly doubled by the end of the Bush Sr. presidency (and recall he kept Reagan's rates for most of his tenure) then we see what a disaster they were.

More impressive yet was Medoff and Harless' analysis in their chapter 'Let The Eat Cake' (p. 84) which looked at actual data(p. 87)and found that: 

"high tax rates are associated with higher productivity growth. There is a consistent and strong relationship."

This was written barely a year into Bill Clinton's imposition of a marginally higher tax rate on the wealthiest, and we saw after the fact more than 20 million jobs created, even as the deficits decreased. Plus a healthy ($600m) surplus was left for Bush Jr. - which was then promptly pissed away in his tax cuts!

Less well known, but which I can document since I lived there, is that supply side economics was tried in Barbados, in FY 1987. The usually democratic socialist state had just elected a new government (in 1986) that was determined to experiment for the first time with the 'trickle down' supply side bunkum adopted by Reagan. They totally eliminated all taxes totally for those earning under $15,000/ year and also cut marginal rates in most income categories from 20-30%. 

They were warned by the country's top economists it would lead to economic disaster, but they took no heed. Finding it more to their liking to pander to a naive (then!) populace to garner votes, they couldn't renege once in power, especially if they wanted re-election.

The supply side idiocy was implemented for tax year 1987 and beyond when the chickens came to roost.  Five years later (and with cost of the losing of 35,000 out of 105,000 jobs, with reserves barely at $11 million, the island had to go to the IMF for loans as its cash flow had evaporated. 

The IMF injected nasty medicine - in the form of across the board pay cuts of 8 percent for all civil servants and higher taxes- though devaluation of the currency was avoided. To escape the deficit pit created succeeding governments had to also impose VAT or value added taxes, including on foodstuffs. This also serves as an object lesson to all who still believe supply side baloney can work in any venue, or that tax cuts are an answer.

What was the biggest irony of all? That one of Reagan's top professional economists, Martin Feldstein, actually pooh-poohed Laffer's curve. In a 1986 econ article Feldstein admitted he "never believed Laffer" and referred to his "curve proposition" as the "height of supply side hyperbole". The tragedy is that Fedlstein's article was snuffed by the Reaganites, and Feldstein himself never broached it, especially after being given an office in the administration. A pity, because decades of foolishness and pain might have been avoided.

Because of that, the belief that tax cuts for the rich are the best way to apply them remains dogma, especially among Republicans. But let's never forget the underside of their credo: that tax cuts are pushed not merely to benefit the rich, but to "starve the beast". In other words, dissolve and destroy social welfare protections if at all possible.  True to form, Gramm and Solon in their WSJ piece write:

"It was the explosion of social welfare spending that became the driving force in icreasing the federal budget deficit and has been ever since."

Adding:

"The legacy of the Reagan program is that by reversing the growth of the welfare state and cutting tax rates Reagan gave the nation 25 years of prosperity."

No mention that during his two terms in office, the national debt nearly tripled, rising from under $1 trillion to $2.6 trillion, owing to out of control defense spending.  
Nor that the Reagan  indirect Social Security cuts in 1983 (that raised the retirement age, delayed COLAs and increased taxes on S.S. to 50%) had any effect on slowing the deficits.

See Also:


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Monday, June 29, 2026

Scorching European Heat Wave Shocks American Tourists & Bears Out Forecast Of A Climate Tipping Point

 

Current European heat wave relative to earlier periods
Parisians desperate to cool off on a 46 C day
Londoners cope with a 35 C day in UK heat wave

Thanks to a new climate analysis we now know the heatwave scorching western Europe is the most severe and widespread ever and is only possible due to the climate crisis driven by fossil fuel burning. This, of course, introduces the role of tipping points.

Prof. Gunter Weller (formerly of the Univ. of Alaska-Fairbanks Geophysical Institute) as long ago as 1985-86 forecast warming of the Arctic leading to a global tipping point in temperatures. He predicted that by the 2010s heat waves occurring in limited planetary regions (say Europe, India or western U.S.) would be 3-4 C warmer than heat waves experienced by those regions in the 1970s. In fact this has been found to be the case.

                                              Prof. Gunther Weller (UAF), 1987

 A new analysis by scientists from the World Weather Attribution (WWA) consortium shows how rapidly extreme heat is worsening regional heat waves exactly as Weller predicted. Weller's forecasts in 1986 were based on the increase rate of CO2 concentration build up in the atmosphere (it was 347 ppm in 1986). Warning that concentrations in excess of 410 ppm would unleash heatwaves 3-4C hotter than those in 70s.

 Coincidentally, as recently as 2003, a heatwave like the current one in Europe would have been 2C cooler due to the lower level of global heating at the time - and the lower CO2 concentration (376 ppm). In 1976, another famous heatwave year, it would have been 3.5C cooler. (I.e. 3.5 C warmer now).  All this would have led Weller to remark: "I told you so!".

In line with Weller's tying CO2 concentrations to more significant regional heat waves half of Europe’s 850 largest cities are enduring their worst ever heat stress.  This stress represents a combination of temperature and humidity. Such muggier conditions mean sweating is less effective at cooling the body, making heatwaves even more dangerous.

The analysis comes as the UK recorded its hottest ever June temperature on Thursday, 36.7C (98.06F) in Somerset, and much of western Europe recorded a sharp rise in medical emergencies, including some deaths. In summer 2022, more than 60,000 people died due to a heat wave in Europe. The statistical analysis needed to assess the impact of the current heatwave will take time to complete. Nonetheless, the heatwave is certain to exact a heavy toll and is also disrupting lives and livelihoods, with schools closed, hospitals struggling and rail and air journeys cancelled across the continent.

Just as concerning are the sweltering night-time temperatures currently about 100 times more likely today than in 2003. Prof. Weller, in his 1986 forecasts warning a significant component of his tipping point would be increasingly higher night time temperatures coinciding with higher daytime temperatures. A recipe for increasing heat deaths in regions like Europe where barely 25% have air conditioning.

The WWA scientists, meanwhile, have warned that without urgent climate action, future heat conditions would get even more extreme and the current summer could seem relatively cool in retrospect. This again aligns with Weller's predictions of 3-4 week long heatwaves by the 2030s with temperatures in the 40- 50C range.

Dr. Theodore Keeping, an extreme weather research associate at Imperial College London and part of the WWA team, has noted:

This is the most severe and widespread heatwave to have ever affected this large a region of Europe. We found that in the last 50 years, during which time the planet has warmed by 1.1C, the chance of a heatwave like this has changed immensely. This event would not have been possible in June without climate change. "

He added that many capital cities were experiencing not only their hottest recorded three-day period in June but the hottest three-day period at any time of year. At least 100 million people in Europe were expected to face temperatures above 35C on Thursday.

The scientists used wet bulb globe temperatures to assess the additional impact of high humidity. Prof. Keeping noting:

 “It accounts for the ability of the human body to cool itself down. With the worst conditions ever experienced in 45% of cities over 50,000 people, the health impacts of this heatwave are likely to be extremely high. The speed of change is startling.

Commenting on the WWA analysis, Simon Stiell, the UN’s climate chief - quoted in The Guardian, said: 

Climate change is running rampant, caused by the world’s addiction to burning coal, oil and gas. But the solutions are equally clear: a faster shift to clean energy – which is now much cheaper than fossil fuels – as well as protecting forests and building climate resilience.”

This is even more critical given that wider use of air conditioning can't be depended upon  - either in the U.S. or Europe. How will communities adapt to constant power grid breakdowns arising from electricity overuse - say during prolonged 3 to 4 week 35- 40C  heat waves? The answer isn't clear apart from utility companies rationing the 'juice'. This may mean doing without electricity for extended periods each day in 40C + heat waves, perhaps up to 12 hours or more. Can Americans adapt to such rationing? Most power grids as currently designed will no longer work 24/7 but have to be shut down because of the lack of capacity as millions try to stay cool -  overtaxing the system.

And we haven't yet gone to the threat of transportation and infrastructure damage such as expansion of rails for rails systems. This led Deutsche Bahn and other rail companies to warn against all nonessential travel over the weekend. Meanwhile, as temperatures hit 104 F outside Berlin, the concrete of the A2 highway burst because of the insanely high temperatures.

Let's be clear the WWA team research is not fanciful or 'scare mongering' any more than Dr. Weller's 1986 work at the Geophysical Institute.  Both used observed and reliable forecast temperature data (ice cores in the case of Weller) to render predictions, In the WWA case it meant analyzing the hottest three-day period across a large area of western Europe, sitting under a “heat dome”. 

Using peer-reviewed methods, they unequivocally found that climate change was the driving force behind the severity of the heat. In Prof. Weller's case, he unequivocally found from the increased CO2 concentrations found in ice cores that climate change was definitively behind future heat waves in disparate regions - from India, to Europe to the USA.

Both the WWA and Weller ruled out natural variability of the weather.  In Weller's case from solar variability (thanks to inputs from the Space Physics Dept.) In the WWA's case discounting particular influences from the El Niño event that began in the Pacific Ocean 2 months ago. The current weather pattern, a blocked high-pressure system trapping hot air over Europe and drawing warm air up from the Sahara, is not unusual in summer, the scientists said. Instead, the level of heat has been supercharged by global heating. Prof. Gunther Weller would concur.

The UK government’s official adviser, the Climate Change Committee, said in May that the country’s infrastructure was “built for a climate that no longer exists”.

That advisor is spot on correct. Most illuminating has been a weekend (June 27-28) WSJ article ('Americans Get Humbled In Heat Wave', p. 1A), noting: 

"Americans in Europe have been getting the shock of their lives this summer as temperatures soar to new records across the continent. Highs have approached 100 degrees Fahrenheit in the UK this weekFor the tourists from some of the muggiest U.S. states there is bewilderment: How has a lifetime of sweltering heat not prepared them for this?"

With a common refrain quoted from one American from Texas (similar to others from FL, NC):

"Being a Texan, it kind of feels embarrassing to be in London, and be like, 'I can't handle the heat here'"

Or as another Yank from NC put it:

"These heat waves are different. You can't get any reprieve from it. Once you do get hot, you stay hot."

Even more sobering, as the article puts it:

"Heat waves are becoming more common in Europe - this is the second in two months - and deadly."

So one almost felt sorry for the Americans visiting London, many for the first time, who expected "mostly overcast skies and cool temperatures in the mid-60s or low 70s."

Well, they would have had to have visited London in July, 1978 when we did, having to wear light jackets (or sweaters) most of the time, e.g.

Near River Thames, London, July 1978
           Me in Hyde Park, Janice at Greenwich Observatory, July 1978

For now these Yank tourists can take some solace in the observation (WSJ, ibid.):

"But extreme temperatures aren't yet a fact of every day life."

However, if Prof. Weller's forecasts hold they soon will be. Needless to say, we are glad we visited London (10 days) when we did.

See Also:



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