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Thursday, September 13, 2018

Problem of the Week: 09-13-18: Two Independent Events [Probability]

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Solution:


Video:

Thursday, September 6, 2018

Thursday, August 30, 2018

Problem of the Week 08-30-18: Solving Equation for Variables [Algebra]

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Video:

Thursday, August 23, 2018

Problem of the Week 08-23-18: Solving a Trigonometric Equation [Trig]

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Video:

Thursday, August 16, 2018

Problem of the Week 08-16-18: Nullspace of a Matrix [Linear Algebra]

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Video:

Thursday, August 9, 2018

Problem of the Week 08-09-18: Logarithmic Differentiation [Calculus]

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Video:

Thursday, August 2, 2018

Problem of the Week 8-02-18: Double Integral [Calculus]

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Solution below:



Video:
*Edit: there is not suppose to be a constant of integration at the end since this is a definite integral.

Wednesday, August 1, 2018

The Fields Medal

Today, Peter Scholze, Caucher Birkar, Alessio Figalli, and Akshay Venkates won the Fields Medal- the highest honor a mathematician can receive

To celebrate their victory, we compiled a list of important facts about the award:

1. The Fields Medal is awarded to two, three, or four mathematicians under the age of 40.
Image result for fields medal
The Fields Medal

2. It has been awarded every four years since 1950 at the International Congress of Mathematicians. 

3. The award “recognizes outstanding mathematical achievement for existing work and for the promise of future achievement.” 

4. An individual cannot win a Fields Medal more than once.

5. Many describe the Fields Medal as the mathematician's "Nobel Prize."

6. There is a monetary award of about $15,000 Canadian dollars.

7. The award is named after Canadian mathematician John Charles Fields.
Image result for john charles fields
John Charles Fields

8. Finnish mathematician Lars Ahlfors and American mathematician Jesse Douglas were the first recipients in 1936.

9. Maryam Mirzakhani became the first woman to win the Fields Medal in 2014

10. Jean-Pierre Serre became the youngest recipient at the age of 27. 

11. In 2006, Grigori Perelman, refused his Fields Medal. He said, "I'm not interested in money or fame; I don't want to be on display like an animal in a zoo.”

12. So far, 4 mathematicians under the age of 30 have won.

Source: 

Thursday, July 26, 2018

Problem of the Week 7-26-18: 2nd Order Nonhomogenous Linear Differential Equation [Differential Equations]

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Solution below:


Video:

Thursday, July 19, 2018

Problem of the Week 7-19-18: Invertible Matrix [Linear Algebra]

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Solution below:


Video:
*Note: In the video, k- 1 - 6(k - 1) reduces to k2- 6k + 5 NOT k2 - 6k - 5

Friday, July 13, 2018

Unlucky Numbers Around the World



Friday the 13th is upon us!  Across the nation, there are people avoiding black cats, spilled salt, ladders, and broken mirrors. Not everyone, however, is scared of 13 because they have other numbers to fear. 


In honor of today, here are 6 unlucky numbers from around the world.


4 - China
In China, the number 4 is considered unlucky because it is homophonous to the word "death." Similar to how the 13th floor is absent in many buildings in the West, the 4th floor is often left out in China.


9 - Japan
Just like the number 4 in China, the number 9 is a bad omen in Japan because it sounds similar to the Japanese word for “torture.”


17 - Italy
The Roman numeral for 17 is XVII, and an anagram for XVII is "VIXI.” This translates to “I lived," aka “my life is over."


26  - India 
For India, tragedies and disasters have the pattern of striking on the 26th (the 2008 Mumbai terror attack, the 2001 Gujarat earthquake, the 2004 Indian Ocean tsunami, 2008 Ahmedabad bomb blasts). For this reason, India doesn't really like the number 26. Can you blame them?


39 - Afghanistan
The number 39 is considered cursed in Afghanistan due to the fact that 39 translates to  ‘morda gow’ which means ‘dead cow” - which is slang for pimp. Since 39 is linked to prostitution, the number is largely avoided.


0888 888 888 - Bulgaria
After three people in Bulgaria with the phone number 0888 888 888 died within ten years, a Bulgarian mobile company suspended the phone number because they believed it was cursed. If you are in Bulgaria, try calling the number. Surely, no one will pick up. 




Thursday, July 12, 2018

Tuesday, July 10, 2018

Thursday, July 5, 2018

Tuesday, July 3, 2018

Problem of the Week 7-3-18: Invariant Sums of Digits [Number Theory]

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Solution below.

Thursday, June 28, 2018

Advanced Knowledge Problem of the Week 6-28-18: Conjugates of a Subgroup [Group Theory]

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Solution below.

Tuesday, June 26, 2018

Problem of the Week 6-26-18: Integers and Quartics [Number Theory]

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Solution below.

Thursday, June 21, 2018

Tuesday, June 19, 2018

Problem of the Week 6-19-18: A Polynomial Inequality [Inequalities]

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Solution below.

Thursday, June 14, 2018

Tuesday, June 12, 2018

New Video Series on Integration Tricks

Integration is a vast topic with many diverse techniques meant to help find the integrals of functions. Many delightful and elegant methods are used to tackle difficult integrals. This video series talks about a few of the less common, but still very useful, techniques. These videos cover topics such as the tangent half-angle substitution, integration with a parameter, and how symmetry in integrals can be useful. 


Problem of the Week 6-12-18: Making Polynomials Prime [Number Theory]

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Solution below.

Thursday, June 7, 2018

Advanced Knowledge Problem of the Week 6-7-18: Order of an Element [Abstract Algebra]

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Solution below.

Tuesday, June 5, 2018

Problem of the Week 6-5-18: A Composite Polynomial [Number Theory]

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Solution below.

Thursday, May 31, 2018

Tuesday, May 29, 2018

Problem of the Week 5-29-18: Numbers with Certain Divisors [Number Theory]

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Solution below.

Thursday, May 24, 2018

New Video Series on Complex Analysis

Interested in learning a bit more about complex variables and some theorems about them? Then check out this new video series covering some of the basics of complex analysis! Complex analysis is an immensely useful subject that is found in many branches of mathematics. The residue theorem is especially well known as being an extremely important method for the integration of functions (real or complex!). Some other topics discussed include complex differentiation, Cauchy's Integral Formula, Liouville's theorem, the Möbius transformation, and much more!


Advanced Knowledge Problem of the Week 5-24-18: A Metric Space [Topology]

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Solution below.

Monday, May 21, 2018

Problem of the Week 5-22-18: Integer Function [Number Theory]

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Solution below.

Thursday, May 17, 2018

Advanced Knowledge Problem of the Week 5-17-18: Alternating Group Generators [Group Theory]

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Solution below.

Tuesday, May 15, 2018

Problem of the Week 5-15-18: A Set of Integers [Number Theory]

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Solution below.

Thursday, May 10, 2018

Advanced Knowledge Problem of the Week 5-10-18: How many prime factors? [Number Theory]

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Solution below.

Tuesday, May 8, 2018

Some Notable Mathematicians/Teachers


Many mathematicians have been notable not just for their contributions to mathematics, but also for helping teach the next generation of mathematicians. A number of important mathematicians have tutored and helped other important mathematicians progress in their careers. Here are a few such mathematicians.

Paul Erdős - Paul Erdős was a notable mathematician who lived an eccentric and mathematically productive lifestyle. He was known for collaborating with many other mathematicians, so many that his collaborating spawned the concept of the Erdős number, which measures how far away mathematicians are from citing some of Erdős's work. Erdős also influenced younger mathematicians, such as the Fields Medalist Terence Tao.

Ernst Kummer - Ernst Kummer was a notable mathematician who made important contributions to mathematics in fields such as geometry and number theory. He also helped educate people in mathematics, one example being that he trained german officers in ballistics. He also was the advisor of several notable students and helped inspire Leopold Kronecker to pursue a career in mathematics.

David Hilbert - David Hilbert was a very prominent mathematician who created the list of Hilbert's problems; a list of 23 unsolved (at the time) and important problems in mathematics. Hilbert helped make the University of Göttingen a leading institution in mathematics. Hilbert also had many doctoral students; at the University of Göttingen Hilbert had 69 doctoral students.

George Pólya - George Pólya was another prominent Hungarian mathematician like Paul Erdős. He wrote numerous books about problem solving, which served to help many students problem solve in mathematics. One of his more popular books, How to Solve It, has sold more than one million copies and discusses strategies for approaching/solving problems.

Monday, May 7, 2018

Problem of the Week 5-8-18: Sum of Digits [Number Theory]

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Solution below.

Thursday, May 3, 2018

Advanced Knowledge Problem of the Week 5-3-18: A Lack of Primitive Roots [Number Theory]

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Solution below.

Tuesday, May 1, 2018

Problem of the Week 5-1-18: Problem on Primality [Number Theory]

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Solution below.

Tuesday, April 24, 2018

Some History and Unsolved Problems in Number Theory

Number theory is a very old subject, which is concerned with the set of integers. Number theory started with the concept of integers and simple operations on the integers such as addition, subtraction, etc. Number theory of the greeks is primarily found in the works of Euclid and Plato. Indian mathematicians of antiquity such as Brahmagupta also made significant contributions (one of Brahmagupta's contributions was work on what is now known as Pell's equation). Pierre de Fermat was an important figure in number theory as well, he is responsible for Fermat's theorem as well as Fermat's Last Theorem (a problem which is now solved). Some additional major figures in early number theory were Leonhard Euler, Joseph-Louis Lagrange, and Carl Friedrich Gauss.

Eventually number theory itself started to split into recognizable subbranches, two major ones being algebraic number theory and analytical number theory. Analytical number theory is concerned with the use of real and complex analysis to number theory and can be said to have started with the Dirichlet prime number theorem. Algebraic number theory is concerned with the use of abstract algebra in number theory, and has it origins in reciprocity and cyclotomy.

Despite the significant work done in number theory, there are still plenty of unsolved problems in the field. Many of these problems are easy to understand (although clearly not so easy to solve). Here are a few of them.




Problem of the Week 4-24-18: Board Covering [Number Theory]

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Solution below.

Thursday, April 19, 2018

Advanced Knowledge Problem of the Week 4-19-18: Contour Integral [Complex Analysis]

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Solution below.

Tuesday, April 17, 2018

Problem of the Week 4-17-18: Fraction Representations [Algebra]

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Solution below.

Thursday, April 12, 2018

A Week of Singularities: April 17-22, 2018: An AMS Special Session + satellite events


Upcoming research lectures at the Center
Tuesday-Thursday, April 17-19: Lecture series by Bernard Teissier and Paolo Aluffi at the Worldwide Center of Mathematics

  • Tuesday, April 17, 3-4pm, Bernard Teissier: "A geometric introduction to valuation theory via infinite dimensional algebraic geometry, Part 1"
  • Tuesday, April 17, 4:30-5:30pm, Bernard Teissier: "A geometric introduction to valuation theory via infinite dimensional algebraic geometry, Part 2"
  • Wednesday, April 18, 3-4pm, Bernard Teissier: "A geometric introduction to valuation theory via infinite dimensional algebraic geometry, Part 3"
  • Wednesday, April 18, 4:30-5:30pm, Paolo Aluffi: "Segre classes and other intersection-theoretic invariants of singular schemes, Part 1"
  • Thursday, April 19, 3-4pm, Paolo Aluffi: "Segre classes and other intersection-theoretic invariants of singular schemes, Part 2"
  • Thursday, April 19, 4:30-5:30pm, Paolo Aluffi: "Segre classes and other intersection-theoretic invariants of singular schemes, Part 3"

Links: 
A Week of Singularities:

Center of Math Research:


Advanced Knowledge Problem of the Week 4-12-18: Cofinite Topology Compactness [Topology]

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Solution below.

Tuesday, April 10, 2018

Problem of the Week 4-10-18: Polynomial Zero's [Polynomials]

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Solution below.

Thursday, April 5, 2018