PotW: Sum of Prime Factors [Factorization]

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Who was Sofya Kovalevskaya?

Some people, like Daniel Bernoulli, seem destined for a life of mathematics. Bernoulli, for example, was born to a wealthy family of mathematicians, and had easy access to the finest mathematics institutions in the world. Some people, however, became renowned mathematicians against the odds. Sofya Kovalevskaya was a mathematician who became well known for her work around the world, despite facing an uphill battle to study mathematics!

The Life of a Mathematician

Kovalevskaya was born in 1850 in Moscow. Her first exposure to mathematics came at a young age. Instead of using wallpaper, her parents covered the walls of her bedroom with sheets of calculus problems, which she said spurred an early interest in mathematics. However, after her childhood, paths to mathematics for women weren't easy - at the time, women were forbidden from attending universities in Russia, and the universities that accepted women were thousands of miles away. However, a friend of Sofia, a man named Vladimir Kovalevskij, pretended to be her husband so she could travel to Germany and study mathematics. 

Studying with Karl Weierstrass, a mathematician whose specialization was calculus, Sofia was able to graduate from the University of Heidelberg in 1874. However, owing to the discrimination against women at the time, she was barred from teaching at a university, despite being a doctor of mathematics. Thus, she spent the next few years away from mathematics, writing literature and helping Vladimir, who she decided to actually marry, with his business ventures. However, when Vladimir died, Sofia took up mathematics again.

In 1880, she wrote a paper on Abelian integrals that was well received. Following this, she wrote three papers on the refraction of light. This led to her appointment as a temporary professor at the University of Stockholm. Five years later, in 1889, she was granted the title of full professor, the first woman to earn the title in the 19th century at a European university (two women had been professors in the previous century). However, her life was cut tragically short, as she died two years later at the age of 41.

Kovalevskaya's Accomplishments

In addition to being a pioneer for women in the field of mathematics, Kovalevskaya made several important contributions to mathematics, many of which are still being studied today! Though she only wrote a few papers on mathematics, they contained several groundbreaking discoveries.

Her graduate school work on partial differential equations led to the creation of the Cauchy-Kovalevskaya Theorem. This theorem described a set of solutions for a set of differential equations.

Arguably Kovalevskaya's most notable contribution to mathematics was the idea of a Kovalevskaya Top. The 'top' isn't something that can be spun on a table, but rather deals with mathematical questions about rotation. Previously, Euler and Lagrange had studied the motion of rotating objects. However, Euler had only examined objects based on the center of gravity, while Lagrange had only looked at symmetrical objects. Kovalevskaya examined objects that weren't symmetrical, which led to the idea of the Kovalevskaya Top! This discovery won Kovalevskaya the Prix Bordin from the French Academy of Science, one of the world's most prestigious institutions at the time!

Problem of the Week: Ratio of Areas of Parallelograms [Geometry]

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What are Narcissistic Numbers?

Numbers can be many things: they can be odd, they can be irrational, and they can be friendly. Today on the Center of Math Blog, we'll be talking about a different type of number: the narcissistic number!

What is a Narcissistic Number?

A narcissistic number is a number that, as its name might suggest, is full of itself. An n-digit number is narcissistic if the number is equal to the sum of each of its digits to the nth power. 153, for example is a narcissistic number. This is because it is a 3-digit number, and we can observe that 1³ + 5³ + 3³ = 153. We can very easily name 9 narcissistic numbers: the first nine positive integers are all narcissistic numbers - any number to the first power is equal to itself. There are no two digit narcissistic numbers, but there are a few three digit narcissistic numbers. In addition to 153, the three narcissistic numbers are 370 (3³ + 7³ + 0³ = 370), 371 (3³ + 7³ +1 ³ = 371), and 407 (4³ + 0³ + 7³ = 407).

How many Narcissistic Numbers are there?

You might expect there to be an infinite number of narcissistic numbers. However, there is a finite number of narcissistic numbers, and it might not be as many as you think there are. As proven by D. Winter, and verified by D. Hoey, there exist only 88 Narcissistic Numbers. How were they able to prove it? They were able to prove that a narcissistic number could not exist even in theory for a number with greater than 60 digits, as n*9ⁿ will be less than 10ⁿ  for any n > 60. Thus, there has to be a finite number of narcissistic numbers. From there, it was simply a matter of plugging and chugging. The largest of the narcissistic numbers is 39 digits long, and narcissistic numbers with 38, 37, and 35 digits!

Are there Narcissistic Numbers in different bases?

The counting system we've talked about above is base-10 (where there are only 10 digits), but narcissistic numbers exist in other bases as well. For example in base 3, 5 is a narcissistic number. 5 written in base 3 is 12, and 1+4=5, making it a narcissistic number. For a full list of narcissistic numbers in base 10 and different bases, check out this link from Wolfram MathWorld: http://mathworld.wolfram.com/NarcissisticNumber.html

PotW: Sequences that sum to zero [Combinatorics]

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History of Indian Mathematics

Check out this new series all about Indian Mathematics!

In this five part series, you'll learn everything from the origins of zero and the numeral system, to the infinite series used today in algorithms to calculate the value of pi!

Part I: The Base 10 Numeral System
Part II: Brahmagupta
Part III: Bhaskara II
Part IV: Madhava
Part V: Ramanujan
 

PotW: 5 integers whose sum and product are less than 10 [Combinatorics]

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Who was Daniel Bernoulli?

Curious about how energy is transferred in a liquid? Or fluid dynamics? If so, keep reading about Daniel Bernoulli - a Swiss mathematician who served as the head of mathematics at Russia's most prestigious university and studied with Leonhard Euler!

The Life of a Mathematician

Daniel Bernoulli was born on January 29, 1700. He was the son of Johann Bernoulli, a mathematician who contributed to calculus in its early days who also educated Leonhard Euler. His uncle was Jacob Bernoulli, who worked on the Bernoulli distribution, a principle used in discrete probability. Daniel was encouraged to become a merchant or study medicine, but he was able to follow in his father's footsteps and pursue mathematics! He would largely succeed: achieving prestigious positions in Italy, Germany, France, England, and Russia!

Bernoulli started in Venice, where he pioneered an hourglass that would tell the time consistently on a ship regardless of how turbulent the weather was. This creation was well received and recognized through the mathematical community. The Russian head of state Catherine the Great was so impressed that she offered Bernoulli a position as a chair of mathematics in St. Petersburg. When he was hesitant at first, Catherine offered another another position to Bernoulli's brother, Nicolaus. 

At St. Petersburg, Bernoulli accomplished his finest work. After his brother Nicolaus died shortly after going to St. Petersburg, Johann Bernoulli sent his pupil, Euler, to study with Daniel Bernoulli. While in St. Petersburg, Euler and Bernoulli worked on several different fields of mathematics, including probability, oscillations, and hydrodynamics, a field Bernoulli helped pioneer. Shortly after his return to Switzerland, he transitioned to a life more focused on medicine.

Bernoulli's Accomplishments

As stated above, the creation of the hourglass earned Bernoulli much praise in the scientific world, as it allowed sailors to keep track of time when normally the ship being tossed and blow would affect that rate at which the sand moved in the hourglass. 

Additionally, he studied oscillations and energy. He proved that strings on musical instruments like violins are an infinite series of harmonic vibrations superimposed on the string. And while it had been known that kinetic energy turned into potential energy (and vice versa) for solids, Bernoulli proved that for liquids, kinetic energy is turned into pressure.

What Bernoulli is most well known for, however, is his work with hydrodynamics, a term he created. Bernoulli published his findings in his most acclaimed work, Hydrodynamica. In this work, he explored energy as it relates to gases and fluids. He wrote down the basic laws of kinetic theory of gases and, a century before Van der Wals discovered it, wrote down the foundation of the equation of state. This work would influence several different fields, including mathematics, physics, and chemistry!