Total recursive functions that are not primitive recursive
... In the late 1920s, the mathematicians Gabriel Sudan and Wilhelm Ackermann, students of David Hilbert, were studying the foundations of computation. Both Sudan and Ackermann are credited [4] with discovering total computable functions (termed simply “recursive” in some references) that are not primit ...
... In the late 1920s, the mathematicians Gabriel Sudan and Wilhelm Ackermann, students of David Hilbert, were studying the foundations of computation. Both Sudan and Ackermann are credited [4] with discovering total computable functions (termed simply “recursive” in some references) that are not primit ...
Finite and Infinite Sets
... the Pigeonhole Principle. The “pigeonhole” version of this property says, “If m pigeons go into r pigeonholes and m > r, then at least one pigeonhole has more than one pigeon.” In this situation, we can think of the set of pigeons as being equivalent to a set P with cardinality m and the set of pige ...
... the Pigeonhole Principle. The “pigeonhole” version of this property says, “If m pigeons go into r pigeonholes and m > r, then at least one pigeonhole has more than one pigeon.” In this situation, we can think of the set of pigeons as being equivalent to a set P with cardinality m and the set of pige ...
a la Finance University Paris 1 Stochastic Calculus 2 Annie Millet
... integrable. For each n ≥ 1, let Sn = inf{t ≥ 0 : |Mt | ≥ n}. Then Sn is a stopping time and if the local martingale M is continuous, replacing τn by τn ∧ n ∧ Sn , we may impose that the martingale M τn remains bounded. (2) An integrable local martingale M need not be a martingale. Such an counter-ex ...
... integrable. For each n ≥ 1, let Sn = inf{t ≥ 0 : |Mt | ≥ n}. Then Sn is a stopping time and if the local martingale M is continuous, replacing τn by τn ∧ n ∧ Sn , we may impose that the martingale M τn remains bounded. (2) An integrable local martingale M need not be a martingale. Such an counter-ex ...
Elliptic Curves and The Congruent Number Problem
... proved many facts about congruent numbers. For example, he showed that the product of any square h2 and 24 is congruent and that multiplying 24 by a sum of squares 12 +32 +52 · · · or the sum h2 + 2h2 + 3h2 + · · · yields a congruent number. He stated (without proof) that no square can be a congruen ...
... proved many facts about congruent numbers. For example, he showed that the product of any square h2 and 24 is congruent and that multiplying 24 by a sum of squares 12 +32 +52 · · · or the sum h2 + 2h2 + 3h2 + · · · yields a congruent number. He stated (without proof) that no square can be a congruen ...
Chapter Seven Real Numbers and the Pythagorean Theorem
... In this lesson, you will ● provide geometric proof of the Pythagorean Theorem. ● use the Pythagorean Theorem to find missing side lengths of right ...
... In this lesson, you will ● provide geometric proof of the Pythagorean Theorem. ● use the Pythagorean Theorem to find missing side lengths of right ...
Real Numbers and the Pythagorean Theorem
... In this lesson, you will ● provide geometric proof of the Pythagorean Theorem. ● use the Pythagorean Theorem to find missing side lengths of right ...
... In this lesson, you will ● provide geometric proof of the Pythagorean Theorem. ● use the Pythagorean Theorem to find missing side lengths of right ...
Solutions to HW 6 - Dartmouth Math Home
... Therefore, π is a homomorphism. Now we just need to identify its kernel. Suppose that x ∈ ker π, so that π(x) = N e. Since π(x) = N x by definition, this tells us that N x = N e. These two cosets are equal if and only if xe−1 ∈ N , or x ∈ N . Therefore, x ∈ ker π if and only if x ∈ N , so ker π = N ...
... Therefore, π is a homomorphism. Now we just need to identify its kernel. Suppose that x ∈ ker π, so that π(x) = N e. Since π(x) = N x by definition, this tells us that N x = N e. These two cosets are equal if and only if xe−1 ∈ N , or x ∈ N . Therefore, x ∈ ker π if and only if x ∈ N , so ker π = N ...
S-parts of terms of integer linear recurrence sequences Yann
... In the present note, we investigate the following related problem. Let S = {q1 , . . . , qs } be a finite, non-empty set of distinct prime numbers. For a non-zero integer m, write m = q1r1 . . . qsrs M , where r1 , . . . , rs are non-negative integers and M is an integer relatively prime to q1 . . . ...
... In the present note, we investigate the following related problem. Let S = {q1 , . . . , qs } be a finite, non-empty set of distinct prime numbers. For a non-zero integer m, write m = q1r1 . . . qsrs M , where r1 , . . . , rs are non-negative integers and M is an integer relatively prime to q1 . . . ...
High Dive Concept
... Q1 - all (sin +, cos +, tan +) Q2 - students (sin +, cos -, tan -) Q3 - take (sin -, cos -, tan +) Q4 - calculus (sin -, cos +, tan -) When sin x = # then inverse sin of # = x (Inverse sin and cos only have solutions when the ratio # is between -1 and 1) (Inverse tan has solutions for all real #) ...
... Q1 - all (sin +, cos +, tan +) Q2 - students (sin +, cos -, tan -) Q3 - take (sin -, cos -, tan +) Q4 - calculus (sin -, cos +, tan -) When sin x = # then inverse sin of # = x (Inverse sin and cos only have solutions when the ratio # is between -1 and 1) (Inverse tan has solutions for all real #) ...
Fundamental theorem of calculus
The fundamental theorem of calculus is a theorem that links the concept of the derivative of a function with the concept of the function's integral.The first part of the theorem, sometimes called the first fundamental theorem of calculus, is that the definite integration of a function is related to its antiderivative, and can be reversed by differentiation. This part of the theorem is also important because it guarantees the existence of antiderivatives for continuous functions.The second part of the theorem, sometimes called the second fundamental theorem of calculus, is that the definite integral of a function can be computed by using any one of its infinitely-many antiderivatives. This part of the theorem has key practical applications because it markedly simplifies the computation of definite integrals.