An Introduction to Discrete Mathematics: how to
... are usually supplied at the ends of each section or, at least, at the end of each chapter. You should spend much more time doing mathematics, rather than just reading about it. That is to say, you should spend much more time on exercises than on studying text. When you find an exercise easy, you mig ...
... are usually supplied at the ends of each section or, at least, at the end of each chapter. You should spend much more time doing mathematics, rather than just reading about it. That is to say, you should spend much more time on exercises than on studying text. When you find an exercise easy, you mig ...
DOMINO TILINGS AND DETERMINANTS V. Aksenov and K. Kokhas
... counts all pairs of matchings (τ1 , τ2 ) taken with the sign sgn(τ1 ) sgn(τ2 ). The Proof. The expression (Pf product of the signs is equal to 1 if one matching can be transformed into the other one by an even permutation, and −1, if this permutation is odd. To calculate sgn(τ1 ) sgn(τ2 ), let us dr ...
... counts all pairs of matchings (τ1 , τ2 ) taken with the sign sgn(τ1 ) sgn(τ2 ). The Proof. The expression (Pf product of the signs is equal to 1 if one matching can be transformed into the other one by an even permutation, and −1, if this permutation is odd. To calculate sgn(τ1 ) sgn(τ2 ), let us dr ...
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... cryptosystem can be cracked. The simplest way to factor a large number is divide it by every integer starting with 1, hoping to get an integer. The numbers that are used in cryptosystems are sufficiently large to make this approach impossible with modern computing power. There are other methods to c ...
... cryptosystem can be cracked. The simplest way to factor a large number is divide it by every integer starting with 1, hoping to get an integer. The numbers that are used in cryptosystems are sufficiently large to make this approach impossible with modern computing power. There are other methods to c ...
manembu - William Stein
... Euler, Johan Heinrich Lambert, and Joseph Louis Lagrange. Euler developed much of the modern theory with his work De Fractionlous Continious. He showed that every rational can be expressed as a simple continued fraction. He also provided an expression for e as a continued fraction, which he used to ...
... Euler, Johan Heinrich Lambert, and Joseph Louis Lagrange. Euler developed much of the modern theory with his work De Fractionlous Continious. He showed that every rational can be expressed as a simple continued fraction. He also provided an expression for e as a continued fraction, which he used to ...
A bijection between bargraphs and Dyck paths
... number of peaks. Corollary 3.3 states that the diagonal sums of this table (which is the triangular sequence [9, A271940]) are the Catalan numbers. This observation for small values of n led to the results in this paper. We end this section by describing other properties of φ, which can be used to g ...
... number of peaks. Corollary 3.3 states that the diagonal sums of this table (which is the triangular sequence [9, A271940]) are the Catalan numbers. This observation for small values of n led to the results in this paper. We end this section by describing other properties of φ, which can be used to g ...
Collatz conjecture
The Collatz conjecture is a conjecture in mathematics named after Lothar Collatz, who first proposed it in 1937. The conjecture is also known as the 3n + 1 conjecture, the Ulam conjecture (after Stanisław Ulam), Kakutani's problem (after Shizuo Kakutani), the Thwaites conjecture (after Sir Bryan Thwaites), Hasse's algorithm (after Helmut Hasse), or the Syracuse problem; the sequence of numbers involved is referred to as the hailstone sequence or hailstone numbers (because the values are usually subject to multiple descents and ascents like hailstones in a cloud), or as wondrous numbers.Take any natural number n. If n is even, divide it by 2 to get n / 2. If n is odd, multiply it by 3 and add 1 to obtain 3n + 1. Repeat the process (which has been called ""Half Or Triple Plus One"", or HOTPO) indefinitely. The conjecture is that no matter what number you start with, you will always eventually reach 1. The property has also been called oneness.Paul Erdős said about the Collatz conjecture: ""Mathematics may not be ready for such problems."" He also offered $500 for its solution.