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Samples of My Work in Mathematics & Closely Related Areas

 If you want your Statement of Purpose or Personal Statement to be successful, you have to write it in such a way as to make those in charge of the selection process curious about you and to look forward to meeting you. You need to portray yourself in your statement as the kind of person that they want to have in their program. I am a practiced master at drafting your story in the best, most eloquent fashion possible, in the way that is most appealing to those who make the selection. I am so certain of my ability that I draft the first paragraph of your statement free of charge and at no further obligation.  If you really like the first paragraph that I produce, then I would then be honored to finish the statement on your behalf. 

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Statements of Excellence in Mathematics

For more than 20 years, I have helped hundreds of applicants from all over the world to get accepted into the finest English-speaking universities, graduate programs, fellowships, scholarships, internships, and residency positions. I provide my clients with uniquely creative, state-of-the-art statements of purpose, personal statements, and letters of motivation, intent, interest, goals, objectives and mission. As a courtesty service for those applicants who decide to use my service for their statement, I am happy to edit your resume or CV.

I edit and enhance cover letters and letters of recommendation. In short, I am your one-stop shop for all of your paperwork needs: so you can focus on your march to success with full paperwork support.

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Great Accomplishments in Mathematics

March 14th is Pi Day, and in celebration of 3.14 and the irrational transcendent number that is pi. Take a moment to grab some dessert and hunker down to this highly subjective list of the five greatest mathematical discoveries in history.

  1.  e + 1 = 0

This equation, “Euler's Identity”, is both stunning in its beauty and deceiving in its simplicity.

For those unfamiliar with the symbols, e here refers to Euler's Number, the base of the natural logarithm and equal to approximately 2.718, and refers to the imaginary number.

Even if this equation looks like nonsense, take a moment to revel in the fact that with just seven symbols, Euler's identity manages to link five of the most important yet seemingly disparate constants in mathematics.

It is beyond us to adequately describe in words the profundity of the connections within mathematics that this identity encapsulates, so we’ll leave it to those much more qualified mathematicians who have gone before.

The great physicist Richard Feynman called the identity "one of the most remarkable, almost astounding, formulas in all of mathematics". Mathematics professor Keith Devlin said: "like a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's Equation reaches down into the very depths of existence”.  

  1.  Fast Fourier Transform

To understand the importance of the discovery of the fast Fourier transform (FFT) on the modern computing age, you need to first understand the purpose of the discrete Fourier Transform (DFT).

The DFT is a transform. First introduced by Fourier in the early 19th century, it has the capability to break down signals (think sound waves or wireless signals) into their component frequencies. Once a signal is transformed into its various frequencies, it can often be manipulated in a much easier fashion.

For example, a sound decomposed into frequencies can have its high-frequency noises (which should be unnoticeable) filtered out thereby decreasing the noise and size of the signal without harming the quality.

This is just one of a large amount of DFT applications that range from data and image compression (by being able to discard the least noticeable frequencies) to Magnetic Resonance Imaging and many fields in between. Now this is obviously all well and good from a theoretical standpoint, but the DFT and its inverse suffer from requiring a largely impractical amount of time to compute.

Were it not for the invention of a FFT by J.W. Tukey and John Cooley in the ‘60s, the DFT might have remained a mere footnote in history. However, their algorithm drastically reduced the time needed to compute the DFT. It led to the ubiquity of its application across engineering and mathematical fields.

3. Gödel's Incompleteness Theorems

At the beginning of the 20th century, mathematician David Hilbert presented a list of twenty-three of the most important unsolved problems in mathematics.

Second on his list was to prove that the axioms of arithmetic are actually consistent i.e. free from internal contradictions.

It may appear true in an obvious way, yet in a groundbreaking 1931 paper that unified logic, mathematics and philosophy, Gödel is widely believed to have proven Hilbert's problem in the negative.

In fact, his incompleteness theorems went so far as to show that in any axiomatic system, such as arithmetic, there exist statements that are totally undecidable. An imperfect but useful analogy is found within the liar paradox, for example.

In this paradox we begin with a machine that can be fed any statement and outputs whether that statement is true or false with truly unfailing accuracy.

But now consider inputting the statement: "this statement is false". The machine could output either "true" nor "false" , but a contradiction would be produced - the equivalent of an undecidable proposition.

The results from Gödel 's theorems still resonate within many areas of mathematics and philosophy and have been extended to speculate on the philosophical limitations of computational systems and even the human mind itself.

5. Euclid's Elements

No list of mathematical achievements would be complete without the inclusion of the most seminal and influential mathematical work to come out of ancient Greece.

Written around 300BC, Euclid's work built the foundation for modern mathematics. He introducing a set of axioms and proceeded to demonstrate by mathematical rigor a collection of theorems that naturally followed.

Covering subjects ranging from algebra to plane geometry (also now known as Euclidean Geometry), Elements remained a cornerstone of mathematical teaching for more than 2,000 years following its creation. 

Elements influenced the thinking of great minds, including Dostoevsky and Einstein. Abraham Lincoln's inclusion of the phrase "dedicated to the proposition" in his Gettysburg address is often attributed to his readings of Euclid.

These are great accomplishments, but math scholars around the world continue to do great work. Would you like to join them? You’ll be well on your way there with a great personal statement. It just so happens that we write them, so don’t hesitate to get in touch if you’d like our help.