Series: Infinite Sums
... Series: Infinite Sums Series are a way to make sense of certain types of infinitely long sums. We will need to be able to do this if we are to attain our goal of approximating transcendental functions by using ‘infinite degree’ polynomials. But before we try to add together an infinite number of pol ...
... Series: Infinite Sums Series are a way to make sense of certain types of infinitely long sums. We will need to be able to do this if we are to attain our goal of approximating transcendental functions by using ‘infinite degree’ polynomials. But before we try to add together an infinite number of pol ...
The Remainder Theorem
... So we get x2 + 3x + 3 on top (this is q(x)), with a remainder of 16. As you recall, from long division of regular numbers from elementary school, that your remainder (if there is one) has to be less than whatever you divided by. In polynomial terms, since we're dividing by a linear factor (that is, ...
... So we get x2 + 3x + 3 on top (this is q(x)), with a remainder of 16. As you recall, from long division of regular numbers from elementary school, that your remainder (if there is one) has to be less than whatever you divided by. In polynomial terms, since we're dividing by a linear factor (that is, ...
Full text
... In the above list, we may find a one-to-one correspondence between the partitions in (a) and the partitions in (c). Given a partition beginning with 1 0 0, we may replace these three digits with O i l . Both strings will have equal value because Fn + 2 = Fn + 1 + Fn. However, out of each of these pa ...
... In the above list, we may find a one-to-one correspondence between the partitions in (a) and the partitions in (c). Given a partition beginning with 1 0 0, we may replace these three digits with O i l . Both strings will have equal value because Fn + 2 = Fn + 1 + Fn. However, out of each of these pa ...
Full text
... Tk2.OK.2m 61 The Fn+2 compositions of (n + 1) using lfs and 2fs when put into the nested greatest integer function with 1 and 2 the exponents on a can be arranged so that the results are the integers 1, 2, . .., Fn + 2 i-n sequence. VtlOO^: We have illustrated Theorem 6 for n = 1, 2, . .., 5. Assume ...
... Tk2.OK.2m 61 The Fn+2 compositions of (n + 1) using lfs and 2fs when put into the nested greatest integer function with 1 and 2 the exponents on a can be arranged so that the results are the integers 1, 2, . .., Fn + 2 i-n sequence. VtlOO^: We have illustrated Theorem 6 for n = 1, 2, . .., 5. Assume ...
![[50] Vertex Coverings by monochromatic Cycles and Trees.](http://s1.studyres.com/store/data/017263913_1-610cbf1ff480871337fed479f820c171-300x300.png)
[50] Vertex Coverings by monochromatic Cycles and Trees.
... Lemma 2 to provide an explicit f(r) in Theorem 1. Proof of Lemma 2. It is well known that Kr contains a monochromatic triangle in every r-coloring if t = 3r! (see, for example, [9, p. 127] ). This fact implies that in every r-coloring of the edges of Kn there exist at least (';)/(';=j) ~ cn 3 /t 3 m ...
... Lemma 2 to provide an explicit f(r) in Theorem 1. Proof of Lemma 2. It is well known that Kr contains a monochromatic triangle in every r-coloring if t = 3r! (see, for example, [9, p. 127] ). This fact implies that in every r-coloring of the edges of Kn there exist at least (';)/(';=j) ~ cn 3 /t 3 m ...
Section 4 Notes - University of Nebraska–Lincoln
... although each box has a name – the least residue element – there are many numbers in each box.) Definition: a congruent to b modulo m, written a b mod m Theorem 4.1: a b mod m if and only if there exists an integer k such that a mk b . Prove. Theorem 4.2: Every integer is congruent mod m to ...
... although each box has a name – the least residue element – there are many numbers in each box.) Definition: a congruent to b modulo m, written a b mod m Theorem 4.1: a b mod m if and only if there exists an integer k such that a mk b . Prove. Theorem 4.2: Every integer is congruent mod m to ...
Postscript (PS)
... E. With this definition of addition, it can be shown that E is an abelian group with identity element O. Note that inverses are very easy to compute. The inverse of (x,y) (which we write as –(x,y) since the group operation is additive) is (x,-y) for all (x,y) ε E. The following ECDSA protocol is bas ...
... E. With this definition of addition, it can be shown that E is an abelian group with identity element O. Note that inverses are very easy to compute. The inverse of (x,y) (which we write as –(x,y) since the group operation is additive) is (x,-y) for all (x,y) ε E. The following ECDSA protocol is bas ...
weak laws of large numbers for arrays of rowwise negatively
... implying that the second term of (3.12) goes to 0 as ncx. Hence, (3.10) follows from (3.11)and (3.12). The exclusion of p- 1 (cf., (3.7)) in Theorem 3.2 is interesting and relates to the proof of the sequence of centering constants. Inequalities (3.8), (3.9) and (3.13)in the proof of Theorem 3.2 dep ...
... implying that the second term of (3.12) goes to 0 as ncx. Hence, (3.10) follows from (3.11)and (3.12). The exclusion of p- 1 (cf., (3.7)) in Theorem 3.2 is interesting and relates to the proof of the sequence of centering constants. Inequalities (3.8), (3.9) and (3.13)in the proof of Theorem 3.2 dep ...
Carmichael numbers with three prime factors
... The first two Carmichael numbers generated by (5, 1) are 7 × 73 × 103 and 17 × 53 × 1201. There are no others with p < 100. We now return to the question of bounds for q. This is rather more tricky, because q(h, ∆) increases with h. Recall that (4) gives at once q ≤ 2p(p − 1). The optimal bound, due ...
... The first two Carmichael numbers generated by (5, 1) are 7 × 73 × 103 and 17 × 53 × 1201. There are no others with p < 100. We now return to the question of bounds for q. This is rather more tricky, because q(h, ∆) increases with h. Recall that (4) gives at once q ≤ 2p(p − 1). The optimal bound, due ...
1.1 Patterns and Inductive Reasoning
... 2 can be written as the sum of two primes. This is called Goldbach’s Conjecture. No one has ever proven this conjecture is true or found a counterexample to show that it is false. As of the writing of this text, it is unknown if this conjecture is true or false. It is known; however, that all even n ...
... 2 can be written as the sum of two primes. This is called Goldbach’s Conjecture. No one has ever proven this conjecture is true or found a counterexample to show that it is false. As of the writing of this text, it is unknown if this conjecture is true or false. It is known; however, that all even n ...
NATURAL BOUNDARIES OF DIRICHLET SERIES Gautami
... question. In general, it does not suffice to prove that each point is a limit point of poles or zeros of the single factors, since poles and zeros might cancel. There are, of course, many examples of special cases where precise information was obtained, as was Q done by Estermann [2] who proved that ...
... question. In general, it does not suffice to prove that each point is a limit point of poles or zeros of the single factors, since poles and zeros might cancel. There are, of course, many examples of special cases where precise information was obtained, as was Q done by Estermann [2] who proved that ...
i(k-1)
... order, merge them into a single sorted list. Merge(a,b): 1. If a is empty, return b 2. If b is empty, return a 3. Otherwise, if the first element of b comes before the first element of a, return a list with the first element of b at the front and the result of Merge(a, rest of b) as the rest 4. Othe ...
... order, merge them into a single sorted list. Merge(a,b): 1. If a is empty, return b 2. If b is empty, return a 3. Otherwise, if the first element of b comes before the first element of a, return a list with the first element of b at the front and the result of Merge(a, rest of b) as the rest 4. Othe ...
BPS states of curves in Calabi–Yau 3–folds
... developed, Equation (1) should be proven as the basic result relating Gromov– Witten theory to the BPS invariants. The correct mathematical definition of the D–brane moduli space is unknown at present, although there has been recent progress in case the curves move in a surface S ⊂ X (see [12], [13] ...
... developed, Equation (1) should be proven as the basic result relating Gromov– Witten theory to the BPS invariants. The correct mathematical definition of the D–brane moduli space is unknown at present, although there has been recent progress in case the curves move in a surface S ⊂ X (see [12], [13] ...
Elementary methods in the study of the distribution of prime numbers
... Here p runs through the prime numbers. The product formula is the analytic expression of the unique factorization theorem. We can formally verify (1.4) by expanding each factor 1/(1 — p~s) as a geometric series and multiplying together the factors. We remark that the zeta function appears first to h ...
... Here p runs through the prime numbers. The product formula is the analytic expression of the unique factorization theorem. We can formally verify (1.4) by expanding each factor 1/(1 — p~s) as a geometric series and multiplying together the factors. We remark that the zeta function appears first to h ...
MATHEMATICAL DIVERSIONS 2011
... A Bologna sandwich The cubic formula was one the first discoveries by European mathematicians that went further than what the Greeks had done centuries before. Tartaglia claimed to have it (he did) and Cardano just needed to know. After he swore to never divulge it, Tartaglia revealed the secret. S ...
... A Bologna sandwich The cubic formula was one the first discoveries by European mathematicians that went further than what the Greeks had done centuries before. Tartaglia claimed to have it (he did) and Cardano just needed to know. After he swore to never divulge it, Tartaglia revealed the secret. S ...
Structure and Randomness in the prime numbers
... News flash: On 31 Jan 2012, I was able to show that every odd number greater than 1 can be written as the sum of five primes or less, by modifying Vinogradov’s argument. In 1742, Christian Goldbach conjectured that in fact every odd number n greater than 5 should be the sum of three primes. This is ...
... News flash: On 31 Jan 2012, I was able to show that every odd number greater than 1 can be written as the sum of five primes or less, by modifying Vinogradov’s argument. In 1742, Christian Goldbach conjectured that in fact every odd number n greater than 5 should be the sum of three primes. This is ...
Mathematical induction Elad Aigner-Horev
... So at this point we might be inclined to believe that replacing 2 with 1 is possible. However, the fact that we can prove the induction step is meaningless unless we can find P some n from which we can start. Here, there cannot be such an n. Indeed, for every n ∈ Z+ we have ni=1 i12 ≥ 1 as the first ...
... So at this point we might be inclined to believe that replacing 2 with 1 is possible. However, the fact that we can prove the induction step is meaningless unless we can find P some n from which we can start. Here, there cannot be such an n. Indeed, for every n ∈ Z+ we have ni=1 i12 ≥ 1 as the first ...
Lecture 6
... outputs the correct result, with high probability, for every input! Randomness is a very useful algorithmic tool. Until the year 2002, there were no efficient deterministic primality testing algorithms. ...
... outputs the correct result, with high probability, for every input! Randomness is a very useful algorithmic tool. Until the year 2002, there were no efficient deterministic primality testing algorithms. ...
iNumbers A Practice Understanding Task – Sample Answers
... A Practice Understanding Task – Sample Answers In order to find solutions to all quadratic equations, we have had to extend the number system to include complex numbers. ...
... A Practice Understanding Task – Sample Answers In order to find solutions to all quadratic equations, we have had to extend the number system to include complex numbers. ...