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... acquired the status of epic collaboration in the annals of mathematics. Legend has it that they had four axioms for this collaborative work. The first was that when one wrote to the other it did not matter whether what was said was right or wrong, for otherwise, they could not write freely and openly ...
... acquired the status of epic collaboration in the annals of mathematics. Legend has it that they had four axioms for this collaborative work. The first was that when one wrote to the other it did not matter whether what was said was right or wrong, for otherwise, they could not write freely and openly ...
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... (R-nodes and <£-nodes implies that the type ((R or «£) determination within each of the left and right subtrees of any uniform Fibonacci tree gives the correct type determination in the whole tree. the induction hypothesis, Uk has Fk_2 ...
... (R-nodes and <£-nodes implies that the type ((R or «£) determination within each of the left and right subtrees of any uniform Fibonacci tree gives the correct type determination in the whole tree. the induction hypothesis, Uk has Fk_2 ...
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... annotation because its key results follow those of Selberg [2]. My sources for the rather unfortunate history of this proof are Andrews [1] and Shapiro [3]; the basic ideas in this paper are due to Landau, Chebyshev, Möbius, and Selberg; the proofs that I give are usually syntheses of proofs in [1] ...
... annotation because its key results follow those of Selberg [2]. My sources for the rather unfortunate history of this proof are Andrews [1] and Shapiro [3]; the basic ideas in this paper are due to Landau, Chebyshev, Möbius, and Selberg; the proofs that I give are usually syntheses of proofs in [1] ...
Elementary Evaluation of Convolution Sums
... αβ has the above form and is such that ν ≥ 2 or 0 ≡ 0 (mod 3). As an example, we determine formulae for the number of representations of a positive integer n by octonary quadratic forms Equation 1.4 and Equation 1.5 using the convolution sums for αβ = 33 = 3 · 11, αβ = 40 = 23 · 5 and αβ = 56 = 23 · ...
... αβ has the above form and is such that ν ≥ 2 or 0 ≡ 0 (mod 3). As an example, we determine formulae for the number of representations of a positive integer n by octonary quadratic forms Equation 1.4 and Equation 1.5 using the convolution sums for αβ = 33 = 3 · 11, αβ = 40 = 23 · 5 and αβ = 56 = 23 · ...
22, 2012 From highly composite numbers to t - IMJ-PRG
... Proceedings a paper of Ramanujan entitled “Highly composite numbers” [Proc. Lond. Math. Soc. (2) 14, 347–409 (1915 ; JFM 45.0286.02)]. But it was not the whole work on the subject, and in “The lost notebook and other unpublished papers”, one can find a manuscript, handwritten by Ramanujan, which is ...
... Proceedings a paper of Ramanujan entitled “Highly composite numbers” [Proc. Lond. Math. Soc. (2) 14, 347–409 (1915 ; JFM 45.0286.02)]. But it was not the whole work on the subject, and in “The lost notebook and other unpublished papers”, one can find a manuscript, handwritten by Ramanujan, which is ...
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... The elements of Pascal’s triangle are binomial coefficients, thus its elements reduced modulo 2 are the coefficients of the successive powers n = 0, 1, 2, 3, . . . of the Z2 [X] polynomial x + 1 in the case of triangle 1, and of the polynomial x2 + 1 in the case of triangle 2. As noted by Wolfram [5 ...
... The elements of Pascal’s triangle are binomial coefficients, thus its elements reduced modulo 2 are the coefficients of the successive powers n = 0, 1, 2, 3, . . . of the Z2 [X] polynomial x + 1 in the case of triangle 1, and of the polynomial x2 + 1 in the case of triangle 2. As noted by Wolfram [5 ...
3.6 The Real Zeros of a Polynomial Function
... worth noting now. If a polynomial (with real coefficients) is of odd degree, then it must contain at least one linear factor. (Do you see why?) This means that it must have at least one real zero. ...
... worth noting now. If a polynomial (with real coefficients) is of odd degree, then it must contain at least one linear factor. (Do you see why?) This means that it must have at least one real zero. ...
On Angles Whose Squared Trigonometric Functions are Rational
... √ 1 √ dn ) and those of β1 , β2 , · · · are in dQ( d1 , · · · , dn ), where d ∈ / Q( d1 , · · · , dn ). Then, by the addition and substraction formulas for tangents, tan(α1 + α2√+ · · · + β1 +√β2 + · · ·) and tan(α · · · − β1 − β2 − · · ·) will be of the√form a + b d and a − b d where ...
... √ 1 √ dn ) and those of β1 , β2 , · · · are in dQ( d1 , · · · , dn ), where d ∈ / Q( d1 , · · · , dn ). Then, by the addition and substraction formulas for tangents, tan(α1 + α2√+ · · · + β1 +√β2 + · · ·) and tan(α · · · − β1 − β2 − · · ·) will be of the√form a + b d and a − b d where ...
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... into Eq. (2) and using the initial condition l2 — 1 yields c = ^ , giving the equation for ln for n > 3. (Actually, the formula also holds for n > 1). 2. It is easy to see that rn = d n , since for each nonpalindromic composition there is one which has the summands in reverse order. For palindromic ...
... into Eq. (2) and using the initial condition l2 — 1 yields c = ^ , giving the equation for ln for n > 3. (Actually, the formula also holds for n > 1). 2. It is easy to see that rn = d n , since for each nonpalindromic composition there is one which has the summands in reverse order. For palindromic ...
- ScholarWorks@GVSU
... The closure properties of the number systems discussed in Section 1.1 and the properties of the number systems in Table 1.2 on page 18 are being used as axioms in this text. A definition is simply an agreement as to the meaning of a particular term. For example, in this text, we have defined the ter ...
... The closure properties of the number systems discussed in Section 1.1 and the properties of the number systems in Table 1.2 on page 18 are being used as axioms in this text. A definition is simply an agreement as to the meaning of a particular term. For example, in this text, we have defined the ter ...
Y n - Bulletin of the Iranian Mathematical Society
... explicit, the cardinality of the exceptional set in (1.9) is rather small since it is contained within the cardinality of the exceptional set given by Brünner, Perelli, and Pintz [9] (see our §1.1). Moreover, under the likely assumption that the Siegel zero β does not exist, the upper bound in (1.8 ...
... explicit, the cardinality of the exceptional set in (1.9) is rather small since it is contained within the cardinality of the exceptional set given by Brünner, Perelli, and Pintz [9] (see our §1.1). Moreover, under the likely assumption that the Siegel zero β does not exist, the upper bound in (1.8 ...
Sieve Methods
... converges. This was the first result of its kind, regarding the Twin-prime problem. A slew of sieve methods were developed over the years — Selberg’s upper bound sieve, Rosser’s Sieve, the Large Sieve, the Asymptotic sieve, to name a few. Many beautiful results have been proved using these sieves. T ...
... converges. This was the first result of its kind, regarding the Twin-prime problem. A slew of sieve methods were developed over the years — Selberg’s upper bound sieve, Rosser’s Sieve, the Large Sieve, the Asymptotic sieve, to name a few. Many beautiful results have been proved using these sieves. T ...
Distinguishing Cartesian powers of graphs
... number of hypercubes was determined: D(Q2 ) = D(Q3 ) = 3 and D(Qd ) = 2 for d ≥ 4. Now, hypercubes are the simplest instances of Cartesian product graphs, that is, Qd = K2d , where Gr stands for the rth power of G with respect to the Cartesian product. Then Albertson [1] proved that for a connected ...
... number of hypercubes was determined: D(Q2 ) = D(Q3 ) = 3 and D(Qd ) = 2 for d ≥ 4. Now, hypercubes are the simplest instances of Cartesian product graphs, that is, Qd = K2d , where Gr stands for the rth power of G with respect to the Cartesian product. Then Albertson [1] proved that for a connected ...
- ScholarWorks@GVSU
... The closure properties of the number systems discussed in Section 1.1 and the properties of the number systems in Table 1.2 on page 18 are being used as axioms in this text. A definition is simply an agreement as to the meaning of a particular term. For example, in this text, we have defined the ter ...
... The closure properties of the number systems discussed in Section 1.1 and the properties of the number systems in Table 1.2 on page 18 are being used as axioms in this text. A definition is simply an agreement as to the meaning of a particular term. For example, in this text, we have defined the ter ...
The zeros of random polynomials cluster uniformly near the unit circle
... The aim of this paper is to show that the phenomenon of uniform concentration of zeros around the unit circle is universal, in the sense that no independence or equidistribution on the coefficients is required, but only conditions on their size. Our method, based on elementary complex analysis, reduce ...
... The aim of this paper is to show that the phenomenon of uniform concentration of zeros around the unit circle is universal, in the sense that no independence or equidistribution on the coefficients is required, but only conditions on their size. Our method, based on elementary complex analysis, reduce ...
ANALYSIS OF CASINO SHELF SHUFFLING MACHINES 1
... Isolated but serious work on shuffling was reported in a 1955 Bell Laboratories report by Edgar Gilbert. He used information theory to attack the problems and gave some tools for riffle shuffling developed jointly with Claude Shannon. They proposed what has come to be called the Gilbert–Shannon–Reed ...
... Isolated but serious work on shuffling was reported in a 1955 Bell Laboratories report by Edgar Gilbert. He used information theory to attack the problems and gave some tools for riffle shuffling developed jointly with Claude Shannon. They proposed what has come to be called the Gilbert–Shannon–Reed ...
Section 2.1: Shift Ciphers and Modular Arithmetic
... – Note: In ASCII the letter A has an ASCII number of 65. We could write a program that does the shift cipher for us. The basic formula would be y = (x – 65 + 3) MOD 26 (where x is the numerical ASCII value of the letter). Everything works great except for the letters X, Y, and Z. We would need a con ...
... – Note: In ASCII the letter A has an ASCII number of 65. We could write a program that does the shift cipher for us. The basic formula would be y = (x – 65 + 3) MOD 26 (where x is the numerical ASCII value of the letter). Everything works great except for the letters X, Y, and Z. We would need a con ...
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... with a - 1 there exists only one cycle of maximum length. This conclusion is false, as the counterexample (1, 0, 0, 1, 0, 0, 0, 0, 0) demonstrates; this tuple is not part of the basic Ducci sequence (which has maximum length) but is part of another Ducci sequence that also has maximum length. (This ...
... with a - 1 there exists only one cycle of maximum length. This conclusion is false, as the counterexample (1, 0, 0, 1, 0, 0, 0, 0, 0) demonstrates; this tuple is not part of the basic Ducci sequence (which has maximum length) but is part of another Ducci sequence that also has maximum length. (This ...
21(2)
... a Fibonacci identity gives one or several hyperbolic identities and conversely, provided that the indices or arguments have the form kntk'm kx±kfy. The indices may be null or negative. ...
... a Fibonacci identity gives one or several hyperbolic identities and conversely, provided that the indices or arguments have the form kntk'm kx±kfy. The indices may be null or negative. ...