x - El Camino College
... As you may have noticed from the examples so far, the complex zeros of polynomials with real coefficients come in pairs. • Whenever a + bi is a zero, its complex conjugate a – bi is also a zero. ...
... As you may have noticed from the examples so far, the complex zeros of polynomials with real coefficients come in pairs. • Whenever a + bi is a zero, its complex conjugate a – bi is also a zero. ...
CS 2336 Discrete Mathematics
... Remark • Mathematical induction is a very powerful technique, because we show just two statements, but this can imply infinite number of cases to be correct • However, the technique does not help us find new theorems. In fact, we have to obtain the theorem (by guessing) in the first place, and indu ...
... Remark • Mathematical induction is a very powerful technique, because we show just two statements, but this can imply infinite number of cases to be correct • However, the technique does not help us find new theorems. In fact, we have to obtain the theorem (by guessing) in the first place, and indu ...
1346492101.
... 12cm 12cm , and of slant length of 9cm, given that the upper rim of the cylinder touches the slant faces of the pyramid, show that its volume is (7 marks) 4h3 2h . Hence find the maximum volume. x 2 3x 4 ...
... 12cm 12cm , and of slant length of 9cm, given that the upper rim of the cylinder touches the slant faces of the pyramid, show that its volume is (7 marks) 4h3 2h . Hence find the maximum volume. x 2 3x 4 ...
Full text
... As was observed by Kung [9] or Sved [17], results (1. i)-(l. iii f ) follow from a simple glance at the binomial array mod 2. However, the reader should note that all six of these properties can also be obtained as immediate consequences of certain well-known facts about binomial coefficients. For i ...
... As was observed by Kung [9] or Sved [17], results (1. i)-(l. iii f ) follow from a simple glance at the binomial array mod 2. However, the reader should note that all six of these properties can also be obtained as immediate consequences of certain well-known facts about binomial coefficients. For i ...
Gergen Lecture I
... motives. They form a bridge between geometry and arithmetic; pure mathematics and high-energy physics. The final goal of these lectures is to provide evidence for a Galois theory of periods. There should be a large (pro)-algebraic group acting on the space of periods. This is the first step towards ...
... motives. They form a bridge between geometry and arithmetic; pure mathematics and high-energy physics. The final goal of these lectures is to provide evidence for a Galois theory of periods. There should be a large (pro)-algebraic group acting on the space of periods. This is the first step towards ...
From highly composite numbers to transcendental
... Schanuel’s Conjecture [7] state that Let x1 , . . . , xn be Q–linearly independent complex numbers. Then at least n of the 2n numbers x1 , . . . , xn , ex1 , . . . , exn are algebraically independent. One of the most important and open special cases of Schanuel’s conjecture is the conjecture on alge ...
... Schanuel’s Conjecture [7] state that Let x1 , . . . , xn be Q–linearly independent complex numbers. Then at least n of the 2n numbers x1 , . . . , xn , ex1 , . . . , exn are algebraically independent. One of the most important and open special cases of Schanuel’s conjecture is the conjecture on alge ...
Full text
... Lemma 2J: If c is an odd cube greater than unity, then neither (c + l)/2 nor ( c - l ) / 2 can be perfect cubes. Proof: To demonstrate the result, it is equivalent to show that the diophantine equations X - 2Y3 = 1 and X3 - 27 3 = -1 have no solutions (X, 7) with X > 1. By Theorem 5 of [2], we have ...
... Lemma 2J: If c is an odd cube greater than unity, then neither (c + l)/2 nor ( c - l ) / 2 can be perfect cubes. Proof: To demonstrate the result, it is equivalent to show that the diophantine equations X - 2Y3 = 1 and X3 - 27 3 = -1 have no solutions (X, 7) with X > 1. By Theorem 5 of [2], we have ...
(pdf)
... discovered a revolutionary type of cipher: his cipher incorporated an asymmetric key. In all the other cryptosystems, decryption is simply the opposite of encryption; these systems employ a symmetric key, because decryption and encryption are symmetrical. In an asymmetrical cipher, there are two dis ...
... discovered a revolutionary type of cipher: his cipher incorporated an asymmetric key. In all the other cryptosystems, decryption is simply the opposite of encryption; these systems employ a symmetric key, because decryption and encryption are symmetrical. In an asymmetrical cipher, there are two dis ...
SINGULAR CONTINUOUS SPECTRUM OF HALF
... the behaviour of a non-relativistic charged particle in a one-dimensional lattice. Periodic models of such type were considered first by Kronig and Penney in [17]. The classical results and a detailed list of references on the theory of one-dimensional Schrödinger operators with δ- and δ 0 -interact ...
... the behaviour of a non-relativistic charged particle in a one-dimensional lattice. Periodic models of such type were considered first by Kronig and Penney in [17]. The classical results and a detailed list of references on the theory of one-dimensional Schrödinger operators with δ- and δ 0 -interact ...
on the nonexistence of odd perfect numbers
... Proposition 2.6 (Bang, Sylvester [23], Birkhoff-Vandiver [1]). If m is an integer, and m ≥ 2 then Φd (m) is divisible by a prime q with oq (m) = d, except when m = 2 and d = 1 or 6 and when m is a Mersenne number and d = 2. In our case, m will always be an odd prime. Whenever d = 2 we have m ≡ 1 (mo ...
... Proposition 2.6 (Bang, Sylvester [23], Birkhoff-Vandiver [1]). If m is an integer, and m ≥ 2 then Φd (m) is divisible by a prime q with oq (m) = d, except when m = 2 and d = 1 or 6 and when m is a Mersenne number and d = 2. In our case, m will always be an odd prime. Whenever d = 2 we have m ≡ 1 (mo ...
Lecture 3 - People @ EECS at UC Berkeley
... The above ideas are generally attributed to both Miller [M76] and Rabin [R76]. More accurately, the randomized algorithm is due to Rabin, while Miller gave a deterministic version that runs in polynomial time assuming the Extended Riemann Hypothesis (ERH): specifically, Miller proved under the ERH t ...
... The above ideas are generally attributed to both Miller [M76] and Rabin [R76]. More accurately, the randomized algorithm is due to Rabin, while Miller gave a deterministic version that runs in polynomial time assuming the Extended Riemann Hypothesis (ERH): specifically, Miller proved under the ERH t ...
EVALUATING DETERMINANTS OF CONVOLUTION
... Note: There are an infinite number of ways to choose the generating functions and end up with the determinants being the desired sequence. The only restriction is that the relationship among all the generating functions described in Theorem 3.1 remains true. This example could have just as easily be ...
... Note: There are an infinite number of ways to choose the generating functions and end up with the determinants being the desired sequence. The only restriction is that the relationship among all the generating functions described in Theorem 3.1 remains true. This example could have just as easily be ...
MULTIPLICATIVE SEMIGROUPS RELATED TO THE 3x + 1
... conjecture is equivalent to the Weak 3x + 1 Conjecture. Applegate and Lagarias [1] subsequently proved both of these conjectures. Their result gave a complete characterization of the elements of the semigroup W , showing that it consisted of all positive rationals whose denominator is not divisible ...
... conjecture is equivalent to the Weak 3x + 1 Conjecture. Applegate and Lagarias [1] subsequently proved both of these conjectures. Their result gave a complete characterization of the elements of the semigroup W , showing that it consisted of all positive rationals whose denominator is not divisible ...
Downloadable PDF - Rose
... time in the history of the Josephus Problem. There are some mathematicians who have studied the variants of the Josephus Problem. See [8]. Our teacher Dr.Miyadera and his students have also studied the Josephus Problem and its variants, and they have talked at the conference [7]. They have published ...
... time in the history of the Josephus Problem. There are some mathematicians who have studied the variants of the Josephus Problem. See [8]. Our teacher Dr.Miyadera and his students have also studied the Josephus Problem and its variants, and they have talked at the conference [7]. They have published ...
on unramified galois extensions of real quadratic
... It is natural to expect that there exist infinitely many real quadratic number fields each having a strictly unramified ^-extension for larger n. However, it seems difficult to construct such fields in the same way as above, because the number of terms of the polynomial whose roots are all real incr ...
... It is natural to expect that there exist infinitely many real quadratic number fields each having a strictly unramified ^-extension for larger n. However, it seems difficult to construct such fields in the same way as above, because the number of terms of the polynomial whose roots are all real incr ...
Complex Numbers - Mathematical Institute Course Management BETA
... The complex number a+bi may be identified with the point (a, b) of the euclidean plane R2 . Just as we speak of ‘the real line’ when we represent real numbers geometrically on the number line, so we speak of ‘the complex plane’ (also known as ‘the Argand diagram’) when we picture C as a 2-dimensiona ...
... The complex number a+bi may be identified with the point (a, b) of the euclidean plane R2 . Just as we speak of ‘the real line’ when we represent real numbers geometrically on the number line, so we speak of ‘the complex plane’ (also known as ‘the Argand diagram’) when we picture C as a 2-dimensiona ...
New Generalized Cyclotomy and Its Applications
... numbers are related to some cyclic codes [13, 15]. Cyclotomic numbers of order up to 24 with respect to primes have been calculated with various kinds of character sums. Generalized cyclotomic numbers of order 2 with respect to p2 were presented in [7] for cryptographic purpose, but no proof was giv ...
... numbers are related to some cyclic codes [13, 15]. Cyclotomic numbers of order up to 24 with respect to primes have been calculated with various kinds of character sums. Generalized cyclotomic numbers of order 2 with respect to p2 were presented in [7] for cryptographic purpose, but no proof was giv ...