Category Archive 'Mathematics'
21 Mar 2024

# Harvard Snobs Can’t Spell

### Harvard, Home Schooling, Mathematics, Spelling

From 2020, but impressive! Harvard Magazine’s May-June 2020 issue published an article that warned of the inherent dangers of homeschooling — and included an image that contained a spectacular spelling error.

Clearly, some Harvard weenies would be better off had they been home-schooled.

HT: Daily Caller.

09 Nov 2020

# Statistical Analysis and the 2020 Election

### 2020 Election, Benford's Law, Mathematics, Voter Fraud

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Himalaya Australia was one of a number of sources applying Benford’s Law to the 2020 Election results.

Facebook has banned references to Benford’s Law.

As the vote counting for the 2020 Presidential Election continues, various facts suggest rampant frauds in Joe Bidenâ€™s votes. So does mathematics in terms of the votes from precincts.

Benfordâ€™s law or the first-digit law, is used to check if a set of numbers are naturally occurring or manually fabricated. It has been applied to detect the voting frauds in Iranian 2009 election and various other applications including forensic investigations.

This is what described by Wikipedia:

â€œBenfordâ€™s law, or the first-digit law, is an observation about the frequency distribution of leading digits in many real-life sets of numerical data. The law states that in many naturally occurring collections of numbers, the leading digit is likely to be small.

For example, in sets that obey the law, the number 1 appears as the leading significant digit about 30% of the time, while 9 appears as the leading significant digit less than 5% of the time. If the digits were distributed uniformly, they would each occur about 11.1% of the time. Benfordâ€™s law also makes predictions about the distribution of second digits, third digits, digit combinations, and so on.â€

Home
Joe Bidenâ€™s votes violate Benfordâ€™s Law (Mathematics)

Joe Bidenâ€™s votes violate Benfordâ€™s Law (Mathematics)
2020 Presidential ElectionDonald TrumpJoe BidenVoter fraud
Himalaya Australia
Himalaya Australia Nov. 07

As the vote counting for the 2020 Presidential Election continues, various facts suggest rampant frauds in Joe Bidenâ€™s votes. So does mathematics in terms of the votes from precincts.

Benfordâ€™s law or the first-digit law, is used to check if a set of numbers are naturally occurring or manually fabricated. It has been applied to detect the voting frauds in Iranian 2009 election and various other applications including forensic investigations.

This is what described by Wikipedia:

â€œBenfordâ€™s law, or the first-digit law, is an observation about the frequency distribution of leading digits in many real-life sets of numerical data. The law states that in many naturally occurring collections of numbers, the leading digit is likely to be small.

For example, in sets that obey the law, the number 1 appears as the leading significant digit about 30% of the time, while 9 appears as the leading significant digit less than 5% of the time. If the digits were distributed uniformly, they would each occur about 11.1% of the time. Benfordâ€™s law also makes predictions about the distribution of second digits, third digits, digit combinations, and so on.â€

One of the examples is the population of the world, which are naturally occurring numbers.
Distribution of first-digit (in %) of population numbers in 237 countries in 2010.
Source: wikipedia.org

A number of people on the internet have checked the votes (precinct by precinct) of Joe Biden, Donald Trump as well as other candidates for their legitimacy in terms of the Benfordâ€™s Law.

According a Reddit user, r/dataisbeautifulâ€™s calculation, the â€˜normalâ€™ distribution of first digits for the different candidates based on Benfordâ€™s law is illustrated below.
Source of image: https://bit.ly/3l7mUE5

Youtuber Nyar has shared the observations on a number of counties, concluding that Trump and othersâ€™ votes have natural distribution but not for Joe Bidenâ€™s.

In Fulton County, Georgia, which overlaps with the Atlantic metropolitan where Joe Biden is expected to win, all of the three candidates have normal distributions for their votes. (Joe Biden 72.6%, Donald Trump 26.2%, Jo Jorgensen 1.2%. Source: .theguardian.com)

In Miami-Dade County of Florida, which includes the Miami metropolitan where Joe Biden is expected to win, all candidatesâ€™ votes obey Benfordâ€™s Law. (Joe Biden 53.4%, Donald Trump 46.1%, Jo Jorgensen 0.3%. Source: theguardian.com)

However, in the Milwaukee County of Wisconsin, which is in one of the key swing states, Joe Bidenâ€™s votes violate Benfordâ€™s Law while other candidatesâ€™ donâ€™t. (Joe Biden 69.4%, Donald Trump 29.4%, Jo Jorgensen 0.9%. Source: theguardian.com)

Home
Joe Bidenâ€™s votes violate Benfordâ€™s Law (Mathematics)

Joe Bidenâ€™s votes violate Benfordâ€™s Law (Mathematics)
2020 Presidential ElectionDonald TrumpJoe BidenVoter fraud
Himalaya Australia
Himalaya Australia Nov. 07

As the vote counting for the 2020 Presidential Election continues, various facts suggest rampant frauds in Joe Bidenâ€™s votes. So does mathematics in terms of the votes from precincts.

Benfordâ€™s law or the first-digit law, is used to check if a set of numbers are naturally occurring or manually fabricated. It has been applied to detect the voting frauds in Iranian 2009 election and various other applications including forensic investigations.

This is what described by Wikipedia:

â€œBenfordâ€™s law, or the first-digit law, is an observation about the frequency distribution of leading digits in many real-life sets of numerical data. The law states that in many naturally occurring collections of numbers, the leading digit is likely to be small.

For example, in sets that obey the law, the number 1 appears as the leading significant digit about 30% of the time, while 9 appears as the leading significant digit less than 5% of the time. If the digits were distributed uniformly, they would each occur about 11.1% of the time. Benfordâ€™s law also makes predictions about the distribution of second digits, third digits, digit combinations, and so on.â€

One of the examples is the population of the world, which are naturally occurring numbers.
Distribution of first-digit (in %) of population numbers in 237 countries in 2010.
Source: wikipedia.org

A number of people on the internet have checked the votes (precinct by precinct) of Joe Biden, Donald Trump as well as other candidates for their legitimacy in terms of the Benfordâ€™s Law.

According a Reddit user, r/dataisbeautifulâ€™s calculation, the â€˜normalâ€™ distribution of first digits for the different candidates based on Benfordâ€™s law is illustrated below.
Source of image: https://bit.ly/3l7mUE5

Youtuber Nyar has shared the observations on a number of counties, concluding that Trump and othersâ€™ votes have natural distribution but not for Joe Bidenâ€™s.

In Fulton County, Georgia, which overlaps with the Atlantic metropolitan where Joe Biden is expected to win, all of the three candidates have normal distributions for their votes. (Joe Biden 72.6%, Donald Trump 26.2%, Jo Jorgensen 1.2%. Source: .theguardian.com)
Image from github.com/ (https://bit.ly/2GGTXjq)

In Miami-Dade County of Florida, which includes the Miami metropolitan where Joe Biden is expected to win, all candidatesâ€™ votes obey Benfordâ€™s Law. (Joe Biden 53.4%, Donald Trump 46.1%, Jo Jorgensen 0.3%. Source: theguardian.com)
Image from github.com/ (https://bit.ly/2GGTXjq)

However, in the Milwaukee County of Wisconsin, which is in one of the key swing states, Joe Bidenâ€™s votes violate Benfordâ€™s Law while other candidatesâ€™ donâ€™t. (Joe Biden 69.4%, Donald Trump 29.4%, Jo Jorgensen 0.9%. Source: theguardian.com)
Image from github.com/ (https://bit.ly/2GGTXjq)

And in Chicago of Illinois, Joe Bidenâ€™s votes are abnormal.

So does that of Allegheny of Pennsylvania which includes Pittsburg. (Joe Biden 59.0%, Donald Trump 39.9%, Jo Jorgensen 1.2%. Source: theguardian.com)

It looks like maybe Biden had lost big cities like Chicago and Pittsburgh, which is why the fraudulent votes need to be brought in, which skew his curve away from a normal looking one.

04 Oct 2020

# Mathematics Association Declares Math is Racist

### Inadvertent Humor, Mathematical Association of America, Mathematics, Racial Politics

The Mathematical Association of America released a statement Friday claiming both that mathematicians should engage in â€œuncomfortable conversationsâ€ about race, and that policies of from the Trump administration, like the lack of a mask mandate in the United States, are somehow an affront to mathematics. The group concludes with a call for a â€œpursuit of justiceâ€ within math. …

“It is time for all members of our profession to acknowledge that mathematics is created by humans and therefore inherently carries human biases. Until this occurs, our community and our students cannot reach full potential,” wrote the group. “Reaching this potential in mathematics relies upon the academy and higher education engaging in critical, challenging, sometimes uncomfortable conversations about the detrimental effects of race and racism on our community. The time is now to move mathematics and education forward in pursuit of justice.”

RTWT

12 Feb 2019

# Penrose Tiles

### Mathematics, Penrose Tiles, Roger Penrose

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In 1974, Roger Penrose, a British mathematician, created a revolutionary set of tiles that could be used to cover an infinite plane in a pattern that never repeats. In 1982, Daniel Shechtman, an Israeli crystallographer, discovered a metallic alloy whose atoms were organized unlike anything ever observed in materials science. Penrose garnered public renown on a scale rarely seen in mathematics. Shechtman won the Nobel Prize. Both scientists defied human intuition and changed our basic understanding of natureâ€™s design, revealing how infinite variation could emerge within a highly ordered environment.

At the heart of their breakthroughs is â€œforbidden symmetry,â€ so-called because it flies in the face of a deeply ingrained association between symmetry and repetition. Symmetry is based on axes of reflectionâ€”whatever appears on one side of a line is duplicated on the other. In math, that relationship is reflected in tiling patterns. Symmetrical shapes such as rectangles and triangles can cover a plane with neither gap nor overlap, and in an ever-repeating pattern. Repeated patterns are called â€œperiodicâ€ and are said to have â€œtranslational symmetry.â€ If you move a pattern from place to place, it looks the same.

Penrose, a bold, ambitious scientist, was interested less in identical patterns and repetition, and more in infinite variation. To be precise, he was interested in â€œaperiodicâ€ tiling, or sets of tiles that can cover an infinite plane with neither gap nor overlap, without the tiling pattern ever repeating itself. That was a challenge because he couldnâ€™t use tiles with two, three, four, or six axes of symmetryâ€”rectangles, triangles, squares, and hexagonsâ€”because on an infinite plane they would result in periodic or repeated patterns. That meant he had to rely on shapes believed to leave gaps in the tiling of a planeâ€”those with forbidden symmetries.

Penrose turned to five-axis symmetry, the pentagon, to create his plane of non-repeating patterns, in part, he has said, because pentagons â€œare just nice to look at.â€ What was remarkable about Penrose tiles was that even though he derived his tiles from the lines and angles of pentagons, his shapes left no awkward gaps. They snugged together perfectly, twisting and turning across the plane, always coming close to repetition, but never quite getting there.

Penrose tiling captured public attention for two major reasons. First, he found a way to generate infinitely changing patterns using just two types of tiles. Second, and even more spectacular, his tiles were simple, symmetrical shapes that on their own betrayed no sign of their unusual properties.

Penrose made several versions of his aperiodic tile sets. One of his most famous is known as the â€œkiteâ€ and the â€œdart.â€ The kite looks like the kidsâ€™ toy of the same name, and the dart looks like a simplified outline of a stealth bomber. Both divide cleanly along axes of symmetry and each has two simple, symmetrical arcs on their surface. Penrose established one placing rule: for a â€œlegalâ€ tile placement these arcs must match up, creating contiguous curves. Without this rule, kites and darts can be placed together in repeating patterns. With this rule, repetition never comes. The kite and the dart tile forever, dancing around their five axes, creating starbursts and decagons, winding curves, butterflies and flowers. Shapes recur but new variations keep creeping in.

Edmund Harriss, an assistant clinical professor in mathematical studies at the University of Arkansas, who wrote his Ph.D. thesis on Penrose tiles, offers a comparison. â€œImagine youâ€™re on a world that is just made up of squares,â€ Harriss says. â€œYou start walking, and when you get to the edge of the square, and the next square is exactly the same, you know what youâ€™re going to see if you walk forever.â€ Penrose tiling has the exact opposite nature. â€œNo matter how much information you have, how much youâ€™ve seen of the tiling, youâ€™ll never be able to predict what happens next. It will be something that youâ€™ve never seen before.â€

RTWT

10 Jul 2018

22 Jun 2018

# Exeter Prof: “Mathematics to Blame for Global Disparities in Wealth”

### Egalitarianism, Left Think, Mathematics, Racial Politics

Campus Reform shares the latest breakthrough in thought from today’s Academy.

In a chapter for a new textbook, University of Exeter professor Paul Ernest warns that mathematics education can cause “collateral damage” to society by training students in “ethics-free thought.”

He even argues that since money involves mathematics, math is “implicated in the global disparities of wealth” because math students are taught to value “detached” and “calculative” reasoning.

RTWT

10 Jul 2017

# Calculating the Odds at Gettysburg

### Battle of Gettysburg, GIGO, History, Mathematics, Pickett's Charge

Military historians have long debated whether Lee’s decision to attack the Union center on the third day of the Battle of Gettysburg with Pickett’s Division ever had any chance of succeeding.

Up at Norwich University in Vermont, Michael J. Armstrong (with Steve Sondergen) was the most recent to have a try at settling the dispute.

We used computer software to build a mathematical model of the charge. The model estimated the casualties and survivors on each side, given their starting strengths.

We used data from the actual conflict to calibrate the modelâ€™s equations. This ensured they initially recreated the historical results. We then adjusted the equations to represent changes in the charge, to see how those affected the outcome. This allowed us to experiment mathematically with several different alternatives.

The first factor we examined was the Confederate retreat. About half the charging infantry had become casualties before the rest pulled back. Should they have kept fighting instead? If they had, our model calculated that they all would have become casualties too. By contrast, the defending Union soldiers would have suffered only slightly higher losses. The charge simply didnâ€™t include enough Confederate soldiers to win. They were wise to retreat when they did.

We next evaluated how many soldiers the Confederate charge would have needed to succeed. Lee put nine infantry brigades, more than 10,000 men, in the charge. He kept five more brigades back in reserve. If he had put most of those reserves into the charge, our model estimated it would have captured the Union position. But then Lee would have had insufficient fresh troops left to take advantage of that success.
Ammunition ran out

We also looked at the Confederate artillery barrage. Contrary to plans, their cannons ran short of ammunition due to a mix-up with their supply wagons. If their generals had better coordinated those supplies, the cannons could have fired twice as much. Our model calculated that this improved barrage would have been like adding one more infantry brigade to the charge. That is, the supply mix-up hurt the Confederate attack, but was not decisive by itself.

Finally, we considered the Union Army. After the battle, critics complained that Meade had focused too much on preparing his defences. This made it harder to launch a counter-attack later. However, our model estimated that if he had put even one less infantry brigade in his defensive line, the Confederate charge probably would have succeeded. This suggests Meade was correct to emphasize his defense.

Pickettâ€™s Charge was not the only controversial part of Gettysburg. Two days earlier, Confederate Gen. Richard Ewell decided against attacking Union soldiers on Culpâ€™s Hill. He instead waited for his infantry and artillery reinforcements. By the time they arrived, however, it was too late to attack the hill.
Was Ewellâ€™s Gettysburg decision actually wise?

Ewell was on the receiving end of a lot of criticism for missing that opportunity. Capturing the hill would have given the Confederates a much stronger position on the battlefield. However, a failed attack could have crippled Ewellâ€™s units. Either result could have altered the rest of the battle.

A study at the U.S. Military Academy used a more complex computer simulation to estimate the outcome if Ewell had attacked. The simulation indicated that an assault using only his existing infantry would have failed with heavy casualties. By contrast, an assault that also included his later-arriving artillery would have succeeded. Thus, Ewell made a wise decision for his situation.

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I’m afraid I do not buy the analysis on Ewell’s decision one bit. The Union 1st and XIth Corps were retiring in disorder late in afternoon having been beaten in a hard fight west of Gettysburg. Ewell’s Corps was arriving from the North, on the right flank of the collapsing Union line. What do you suppose would have happened to the Union Army if Stonewall Jackson had survived the Battle of Chancellorsville, six weeks earlier, and been commanding that corps instead of Ewell? Those computer simulations up at Norwich are clearly not accurately calculating for momentum and morale.

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For every Southern boy fourteen years old, not once but whenever he wants it, there is the instant when itâ€™s still not yet two oâ€™clock on that July afternoon in 1863, the brigades are in position behind the rail fence, the guns are laid and ready in the woods and the furled flags are already loosened to break out and Pickett himself with his long oiled ringlets and his hat in one hand probably and his sword in the other looking up the hill waiting for Longstreet to give the word and itâ€™s all in the balance, it hasnâ€™t happened yet, it hasnâ€™t even begun yet, it not only hasnâ€™t begun yet but there is still time for it not to begin against that position and those circumstance which made more men than Garnett and Kemper and Armistead and Wilcox look grave yet itâ€™s going to begin, we all know that, we have come too far with too much at stake and that moment doesnâ€™t need even a fourteen-year-old boy to think This time. Maybe this time with all this much to lose than all this much to gain: Pennsylvania, Maryland, the world, the golden dome of Washington itself to crown with desperate and unbelievable victory the desperate gamble, the cast made two years ago.

â€”William Faulkner, Intruder in the Dust, 1948.

29 Jun 2017

# xkcd

### Cartoon, Humor, Mathematics, xkcd

Hat tip to Karen L. Myers.

14 Mar 2017

26 Feb 2017

# Math for the Social Justice Major

### Left Think, Mathematics, Social Justice, Yale

Yale Classmate Seattle Sam writes:

I created a course that I think will be in next yearâ€™s Yale course catalog.

Math for the Social Justice Major

Mathematics was devised by old white men who sought to oppress the uneducated masses. In this course we will explore a more empathetic approach to the subject.

The course will explore questions such as:

How does the number 6 make you feel?

If John has 6 marbles and Sue has 2, isnâ€™t that unfair?

How can there be any â€œincorrectâ€ answers?

Isnâ€™t identifying a number as â€œpositiveâ€ or negativeâ€ stereotyping?

If you identify with 5 more than 4, why shouldnâ€™t that be a solution to 2+2=?

What did Euclid know and when did he know it?

Isnâ€™t a null set non-inclusive?

What should you do if the solution to an equation make you feel unsafe?

Shouldnâ€™t we just deem the Parallel Postulate proved?

Whatâ€™s the point of carrying pi out to more than two decimals?

Arenâ€™t < and > judgmental symbols?

Who are you to determine that a fraction is improper?

Why do you think prime numbers have only a token even member?

Why shouldnâ€™t an inverse tangent have the same value as a cosine?

Arenâ€™t right angles reactionary?

Are there really any absolute values?

Why should binomials and polynomials be considered deviants?

Isnâ€™t a Real Number just your perception?

Just because a number canâ€™t be expressed as a ratio of integers, why should it be called irrational?

16 Jan 2016

# Music and Math

### Classical Music, Ludwig van Beethoven, Mathematics

How is it that Beethoven, who is celebrated as one of the most significant composers of all time, wrote many of his most beloved songs while going deaf? The answer lies in the math behind his music. Using the â€œMoonlight Sonataâ€, we can begin to understand the way Beethoven was able to convey emotion and creativity using the certainty of mathematics.

The standard piano octave consists of 13 keys, each separated by a half step. A standard major or minor scale uses 8 of these keys with 5 whole step intervals and 2 half step ones.

The first half of measure 50 of â€œMoonlight Sonataâ€ consists of three notes in D major, separated by intervals called thirds that skip over the next note in the scale. By stacking the notes first, third, and fifth notes – D, F sharp, and A – we get a harmonic pattern known as a triad.

But, these arenâ€™t just arbitrary magic numbers. Rather, they represent the mathematical relationship between the pitch frequencies of different notes, which form a geometric series. The stacking of these three frequencies creates â€˜consonanceâ€™, which sounds naturally pleasant to our ears. Examining Beethovenâ€™s use of both consonance and dissonance can help us begin to understand how he added the unquantifiable elements of emotion and creativity to the certainty of mathematics.

For a deeper dive into the mathematics of the â€œMoonlight Sonataâ€, watch the TED-Ed Lesson Music and math: The genius of Beethoven – Natalya St. Clair

Animation by Qaâ€™ed Mai

Via Ratak Monodosico.

31 Oct 2015

# The Mathematics of Vampire-Human Relations

### Mathematics, Vampires

Ella Morton reviews the academic literature addressing the possibility of vampire-humanity peaceful coexistence and the alternative hypothesis which argues that hungry vamps will simply eat themselves out of human prey.

A surprisingly large number of academic studiesâ€”as in, more than oneâ€”have applied mathematical modeling to the concept of human-vampire co-existence. … these papers look at whether Earth’s vampire population would inevitably annihilate humanity, and, if so, how long it would take.

the Archives of Never Yet Melted in the 'Mathematics' Category.
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