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The Hopf Fibration: Homotopy Groups of Spheres
Created for Stanford Mathematics Camp 2016. A brief introduction to the Hopf Fibration for introductory topology students.
Trey Connelly
Is e + $\pi$ irrational?
In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number. The set of all rational numbers, often referred to as ”the rationals”, is usually denoted by a boldface Q (or blackboard bold , Unicode ); it was thus denoted in 1895 by Giuseppe Peano after quoziente, Italian for ”quotient”. The decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number. These statements hold true not just for base 10, but also for any other integer base (e.g. binary, hexadecimal). A real number that is not rational is called irrational. Irrational numbers include √2, , e, and . The decimal expansion of an irrational number continues without repeating. Since the set of rational numbers is countable, and the set of real numbers is uncountable, almost allreal numbers are irrational.
jackson
La revelación de los sentimientos
La revelación de los sentimientos
ana isabel
almokri
Difficulty Adjustment Algorithms in Cryptocurrency Protocols
As of this writing, the algorithm employed for difficulty adjustment in the CryptoNote reference code is known by the Monero Research Lab to be flawed. We describe and illustrate the nature of the flaw and recommend a solution. By dishonestly reporting timestamps, attackers can gain disproportionate control over network difficulty. We verify this route of attack by auditing the CryptoNote reference difficulty adjustment code, which, we reimplement in the Python programming language. We use a stochastic model of blockchain growth to test the CryptoNote reference difficulty formula against the more traditional Bitcoin difficulty formula. This allows us to test our difficulty formula against various hash rate scenarios. This research bulletin has not undergone peer review, and reflects only the results of internal investigation.
Surae Noether
The addition formulas for the hyperbolic sine and cosine functions via linear algebra
We present a geometric proof of the addition formulas for the hyperbolic sine and cosine functions, using elementary properties of linear transformations.