FIBONACCI SEQUENCE / NUMBER: One sequence that is famous is the fibonacci sequence; or fibonacci number. His sequence is defined as F*n = F*n-1 + F*n-2. Which was the linear recurrance equation. This sequence is found in manmade, mathematics, and ideals of persons. It is used in the reproduction of bunny rabbits (which were his first findings), sunflower pedals spiraling, and rationalized squares. This number is also used by architects in structures and other artist. The squence was discovered in the 12th century by a man named Leonardo Fibonacci. He discovered this sequence that seemed to go along with today's phi. Each number in the sequence (starting with 0 and 1) is just the 2 numbers before it added up's total. This number sequence is all about ratios.
--http : // mathworld . wolfram . com / FibonacciNumber . html -- http : // answers . yahoo . com / question / index?qid = 20080224134015AAj7qZA -- http : // goldennumber . net / fibonser . htm
In mathematics, an elliptic divisibility sequence (EDS) is a sequence of integers satisfying a nonlinear recursion relation arising from division polynomials on elliptic curves. EDS were first defined, and their arithmetic properties studied, by Morgan Ward in the 1940s. Morgan Ward (1901–1963) was an American mathematician, a professor of mathematics at the California Institute of Technology. Ward received his Ph.D. from Caltech in 1928, with a dissertation entitled The Foundations of General Arithmetic; his advisor was Eric Temple Bell. He became a research fellow at Caltech, and then in 1929 a member of the faculty; he remained at Caltech until his death in 1963. Among his doctoral students was Robert P. Dilworth, who also became a Caltech professor. Ward is the academic ancestor of over 450 mathematicians and computer scientists through Dilworth and another of his students, Donald Darling. Ward's research interests included the study of recurrence relations and the divisibility properties of their solutions, diophantine equations including Euler's sum of powers conjecture and equations between monomials, abstract algebra, lattice theory and residuated lattices, functional equations and functional iteration, and numerical analysis. He also worked with the National Science Foundation on the reform of the elementary school mathematics curriculum, and with Clarence Ethel Hardgrove he wrote the textbook Modern Elementary Mathematics.Ward's works are collected in the Caltech library. A symposium in his memory was held at Caltech on November 21-22, 1963.
HEY TAYLOR HERE AND HERE IS MY RESPONSE TO THIS WEEKS PROMPT!!
(I hope that this is right.))
I guess one of them is the one made up by some guy when he was a kid in his classroom. His name was Gauss and he was or is a famous mathematician. He basically came up with a sequence on the spot. The sequence was called finding the sum of an arithmetic sequence. This sequence is used to find all the sums of an integer to another integer. To use this sequence you most 'Add the first term to the last term and multiply by half the number of terms.' (http://jwbales.us/precal/part8/part8.3.html) .
So say if you wanted to find all of the sums of the number between 1 and 100.
'You would do this 1+ 100, 2+99, 3+98,.... You would do this until you get the answer 101 50 times or (5050."http://www.newton.dep.anl.gov/askasci/math99/math99155.htm)
(REMEBER I DO NOT KNOW HOW TO CITE SO...) http://goldennumber.net/fibonser.htm http://www.newton.dep.anl.gov/askasci/math99/math99155.htm
SO THAT IS ALL I THINK!!! NOW I AM GOING TO RELAX BECAUSE THIS BEE STING HURTS!!!! PEACE OUT MY PEOPLE!!!
"In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field K of real or complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in K, and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are linear subspaces of this space. Sequence spaces are typically equipped with a norm, or at least the structure of a topological vector space.
The most important sequences spaces in analysis are the ℓp spaces, consisting of the p-power summable sequences, with the p-norm. These are special cases of Lp spaces for the counting measure on the set of natural numbers. Other important classes of sequences like convergent sequences or null sequences form sequence spaces, respectively denoted c and c0, with the sup norm. Any sequence space can also be equipped with the topology of pointwise convergence, under which it becomes a special kind of Fréchet space called FK-space."
This is Nathan with the response on the week 5 blog prompt.
A famous sequence is the Morris number sequence, or the look-and-say sequence. It is the sequence of integers that begins like this: 1,11,21,1211,111221,312211,13112221,1113213211. To get the next number of the sequence from the previous one, read the digits of the previous member, counting the number of digits in groups of the same digit. 1 is read off as "one 1" or 11. 11 is read off as "two 1's" or 21. 21 is read off as "one 2, then one 1" or 1211. 1211 is read off as "one 1, then one 2, then two 1's" or 111221. 111221 is read off as "three 1's, then two 2's, then one 1" or 312211. This sequence grows indefinitely. No other digits other than 1,2,and 3 appear in the sequence. This sequence was introduced by John Conway in his paper "The Weird and Wonderful Chemistry of Audioactive Decay." It's known as the Morris number sequence after cryptographer Robert Morris, and the puzzle is sometimes referred to as the Cuckoo's Egg, from a description of Morris in a book.
"The Padovan sequence is named after Richard Padovan who attributed its discovery to Dutch architect Hans van der Laan in his 1994 essay Dom. Hans van der Laan : Modern Primitive. The sequence was described by Ian Stewart in his Scientific American column Mathematical Recreations in June 1996."
Juggler sequences were publicised by American mathematician and author Clifford A. Pickover. The name is derived from the rising and falling nature of the sequences, like balls in the hands of a juggler.
If a juggler sequence reaches 1, then all subsequent terms are equal to 1. It is conjectured that all juggler sequences eventually reach 1. This conjecture has been verifed for initial terms up to 10^6, but has not been proved. Juggler sequences therefore present a problem that is similar to the Collatz conjecture, about which Paul Erdős stated that "mathematics is not yet ready for such problems".
For a given initial term n we define l(n) to be the number of steps which the juggler sequence starting at n takes to first reach 1, and h(n) to be the maximum value in the juggler sequence starting at n.
the lucas sequences! finally found one! its really complicated too! this is a blog i had to makeup over the holidays so here it iss!
In mathematics, the Lucas sequences Un(P,Q) and Vn(P,Q) are certain integer sequences that satisfy the recurrence relation
xn = Pxn-1 - Qxn-2, where P and Q are fixed integers. Any other sequence satisfying this recurrence relation can be represented as a linear combination of the Lucas sequences Un(P,Q) and Vn(P,Q).
Famous examples of Lucas sequences include the Fibonacci numbers, Pell numbers, Lucas numbers and Jacobsthal numbers. Lucas sequences are named after the French mathematician Édouard Lucas.
Contents [hide] 1 Recurrence relations 2 Examples 3 Algebraic relations 3.1 Distinct roots 3.2 Repeated root 4 Other relations 5 Specific names 6 Applications 7 References 8 See also
[edit] Recurrence relations Given two integer parameters P and Q, the Lucas sequences of the first kind Un(P,Q) and of the second kind Vn(P,Q) are defined by the recurrence relations:
and
It is not hard to show that for n > 0,
[edit] Examples
[edit] Algebraic relations The characteristic equation of the recurrence relation for Lucas sequences Un(P,Q) and Vn(P,Q) is:
It has the discriminant D = P2 − 4Q and the roots:
Thus:
Note that the sequence an and the sequence bn also satisfy the recurrence relation. However these might not be integer sequences.
[edit] Distinct roots When , a and b are distinct and one quickly verifies that
.
It follows that the terms of Lucas sequences can be expressed in terms of a and b as follows
[edit] Repeated root The case D = 0 occurs exactly when P = 2S and Q = S2 for some integer S so that a = b = S. In this case one easily finds that
. [edit] Other relations The terms of Lucas sequences satisfy relations that are generalizations of those between Fibonacci numbers Fn = Un(1, − 1) and Lucas numbers Ln = Vn(1, − 1). For example:
General P = 1, Q = -1
Among the consequences is that Ukm is a multiple of Um, implying that Un can be prime only when n is prime. Another consequence is an analog of exponentiation by squaring that allows fast computation of Un for large values of n. These facts are used in the Lucas–Lehmer primality test.
[edit] Specific names The Lucas sequences for some values of P and Q have specific names:
Un(1,−1) : Fibonacci numbers Vn(1,−1) : Lucas numbers Un(2,−1) : Pell numbers Vn(2,−1) : Companion Pell numbers or Pell-Lucas numbers Un(1,−2) : Jacobsthal numbers Un(3, 2) : Mersenne numbers 2n − 1 [edit] Applications LUC is a public-key cryptosystem based on Lucas sequences[1] that implements the analogs of ElGamal (LUCELG), Diffie-Hellman (LUCDIF), and RSA (LUCRSA). The encryption of the message in LUC is computed as a term of certain Lucas sequence, instead of using modular exponentiation as in RSA or Diffie-Hellman. However, a paper by Bleichenbacher et al.[2] shows that many of the supposed security advantages of LUC over cryptosystems based on modular exponentiation are either not present, or not as substantial as claimed.
Charlie Marie Bourgeois's...
ReplyDeleteFIBONACCI SEQUENCE / NUMBER:
One sequence that is famous is the fibonacci sequence; or fibonacci number.
His sequence is defined as F*n = F*n-1 + F*n-2.
Which was the linear recurrance equation.
This sequence is found in manmade, mathematics, and ideals of persons.
It is used in the reproduction of bunny rabbits (which were his first findings), sunflower pedals spiraling, and rationalized squares.
This number is also used by architects in structures and other artist.
The squence was discovered in the 12th century by a man named Leonardo Fibonacci. He discovered this sequence that seemed to go along with today's phi.
Each number in the sequence (starting with 0 and 1) is just the 2 numbers before it added up's total.
This number sequence is all about ratios.
--http : // mathworld . wolfram . com / FibonacciNumber . html
-- http : // answers . yahoo . com / question / index?qid = 20080224134015AAj7qZA
-- http : // goldennumber . net / fibonser . htm
Elliptic divisibility sequence
ReplyDeleteIn mathematics, an elliptic divisibility sequence (EDS) is a sequence of integers satisfying a nonlinear recursion relation arising from division polynomials on elliptic curves. EDS were first defined, and their arithmetic properties studied, by Morgan Ward in the 1940s. Morgan Ward (1901–1963) was an American mathematician, a professor of mathematics at the California Institute of Technology. Ward received his Ph.D. from Caltech in 1928, with a dissertation entitled The Foundations of General Arithmetic; his advisor was Eric Temple Bell. He became a research fellow at Caltech, and then in 1929 a member of the faculty; he remained at Caltech until his death in 1963. Among his doctoral students was Robert P. Dilworth, who also became a Caltech professor. Ward is the academic ancestor of over 450 mathematicians and computer scientists through Dilworth and another of his students, Donald Darling. Ward's research interests included the study of recurrence relations and the divisibility properties of their solutions, diophantine equations including Euler's sum of powers conjecture and equations between monomials, abstract algebra, lattice theory and residuated lattices, functional equations and functional iteration, and numerical analysis. He also worked with the National Science Foundation on the reform of the elementary school mathematics curriculum, and with Clarence Ethel Hardgrove he wrote the textbook Modern Elementary Mathematics.Ward's works are collected in the Caltech library. A symposium in his memory was held at Caltech on November 21-22, 1963.
"Morgan Ward." Wikipedia, the Free Encyclopedia. Web. 16 Nov. 2010. .
"Elliptic Divisibility Sequence." Wikipedia, the Free Encyclopedia. Web. 16 Nov. 2010. .
HEY TAYLOR HERE AND HERE IS MY RESPONSE TO THIS WEEKS PROMPT!!
ReplyDelete(I hope that this is right.))
I guess one of them is the one made up by some guy when he was a kid in his classroom. His name was Gauss and he was or is a famous mathematician. He basically came up with a sequence on the spot. The sequence was called finding the sum of an arithmetic sequence. This sequence is used to find all the sums of an integer to another integer. To use this sequence you most 'Add the first term to the last term and multiply by half the number of terms.' (http://jwbales.us/precal/part8/part8.3.html) .
So say if you wanted to find all of the sums of the number between 1 and 100.
'You would do this 1+ 100, 2+99, 3+98,.... You would do this until you get the answer 101 50 times or (5050."http://www.newton.dep.anl.gov/askasci/math99/math99155.htm)
(REMEBER I DO NOT KNOW HOW TO CITE SO...)
http://goldennumber.net/fibonser.htm
http://www.newton.dep.anl.gov/askasci/math99/math99155.htm
SO THAT IS ALL I THINK!!! NOW I AM GOING TO RELAX BECAUSE THIS BEE STING HURTS!!!! PEACE OUT MY PEOPLE!!!
this is lawrence
ReplyDelete"In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field K of real or complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in K, and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are linear subspaces of this space. Sequence spaces are typically equipped with a norm, or at least the structure of a topological vector space.
The most important sequences spaces in analysis are the ℓp spaces, consisting of the p-power summable sequences, with the p-norm. These are special cases of Lp spaces for the counting measure on the set of natural numbers. Other important classes of sequences like convergent sequences or null sequences form sequence spaces, respectively denoted c and c0, with the sup norm. Any sequence space can also be equipped with the topology of pointwise convergence, under which it becomes a special kind of Fréchet space called FK-space."
citation
http://en.wikipedia.org/wiki/Sequence_space
This is Nathan with the response on the week 5 blog prompt.
ReplyDeleteA famous sequence is the Morris number sequence, or the look-and-say sequence. It is the sequence of integers that begins like this:
1,11,21,1211,111221,312211,13112221,1113213211.
To get the next number of the sequence from the previous one, read the digits of the previous member, counting the number of digits in groups of the same digit.
1 is read off as "one 1" or 11.
11 is read off as "two 1's" or 21.
21 is read off as "one 2, then one 1" or 1211.
1211 is read off as "one 1, then one 2, then two 1's" or 111221.
111221 is read off as "three 1's, then two 2's, then one 1" or 312211.
This sequence grows indefinitely. No other digits other than 1,2,and 3 appear in the sequence.
This sequence was introduced by John Conway in his paper "The Weird and Wonderful Chemistry of Audioactive Decay." It's known as the Morris number sequence after cryptographer Robert Morris, and the puzzle is sometimes referred to as the Cuckoo's Egg, from a description of Morris in a book.
Cite.
http://en.wikipedia.org/wiki/Morris_number_sequence
Feroz doing the prompt thing.
ReplyDeletePadovan Sequence (used for recurrence relation)
"The Padovan sequence is named after Richard Padovan who attributed its discovery to Dutch architect Hans van der Laan in his 1994 essay Dom. Hans van der Laan : Modern Primitive. The sequence was described by Ian Stewart in his Scientific American column Mathematical Recreations in June 1996."
The first few values of P(n) are
1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, ...
BAM.
woops.
ReplyDeleteSource:
http://en.wikipedia.org/wiki/Padovan_sequence
This is Toriii! :O
ReplyDeleteJuggler sequences were publicised by American mathematician and author Clifford A. Pickover. The name is derived from the rising and falling nature of the sequences, like balls in the hands of a juggler.
If a juggler sequence reaches 1, then all subsequent terms are equal to 1. It is conjectured that all juggler sequences eventually reach 1. This conjecture has been verifed for initial terms up to 10^6, but has not been proved. Juggler sequences therefore present a problem that is similar to the Collatz conjecture, about which Paul Erdős stated that "mathematics is not yet ready for such problems".
For a given initial term n we define l(n) to be the number of steps which the juggler sequence starting at n takes to first reach 1, and h(n) to be the maximum value in the juggler sequence starting at n.
dangitt!
ReplyDeleteCITEE!::http://en.wikipedia.org/wiki/Juggler_sequence
the lucas sequences!
ReplyDeletefinally found one!
its really complicated too!
this is a blog i had to makeup over the holidays so here it iss!
In mathematics, the Lucas sequences Un(P,Q) and Vn(P,Q) are certain integer sequences that satisfy the recurrence relation
xn = Pxn-1 - Qxn-2,
where P and Q are fixed integers. Any other sequence satisfying this recurrence relation can be represented as a linear combination of the Lucas sequences Un(P,Q) and Vn(P,Q).
Famous examples of Lucas sequences include the Fibonacci numbers, Pell numbers, Lucas numbers and Jacobsthal numbers. Lucas sequences are named after the French mathematician Édouard Lucas.
Contents [hide]
1 Recurrence relations
2 Examples
3 Algebraic relations
3.1 Distinct roots
3.2 Repeated root
4 Other relations
5 Specific names
6 Applications
7 References
8 See also
[edit] Recurrence relations
Given two integer parameters P and Q, the Lucas sequences of the first kind Un(P,Q) and of the second kind Vn(P,Q) are defined by the recurrence relations:
and
It is not hard to show that for n > 0,
[edit] Examples
[edit] Algebraic relations
The characteristic equation of the recurrence relation for Lucas sequences Un(P,Q) and Vn(P,Q) is:
It has the discriminant D = P2 − 4Q and the roots:
Thus:
Note that the sequence an and the sequence bn also satisfy the recurrence relation. However these might not be integer sequences.
[edit] Distinct roots
When , a and b are distinct and one quickly verifies that
.
It follows that the terms of Lucas sequences can be expressed in terms of a and b as follows
[edit] Repeated root
The case D = 0 occurs exactly when P = 2S and Q = S2 for some integer S so that a = b = S. In this case one easily finds that
.
[edit] Other relations
The terms of Lucas sequences satisfy relations that are generalizations of those between Fibonacci numbers Fn = Un(1, − 1) and Lucas numbers Ln = Vn(1, − 1). For example:
General P = 1, Q = -1
Among the consequences is that Ukm is a multiple of Um, implying that Un can be prime only when n is prime. Another consequence is an analog of exponentiation by squaring that allows fast computation of Un for large values of n. These facts are used in the Lucas–Lehmer primality test.
[edit] Specific names
The Lucas sequences for some values of P and Q have specific names:
Un(1,−1) : Fibonacci numbers
Vn(1,−1) : Lucas numbers
Un(2,−1) : Pell numbers
Vn(2,−1) : Companion Pell numbers or Pell-Lucas numbers
Un(1,−2) : Jacobsthal numbers
Un(3, 2) : Mersenne numbers 2n − 1
[edit] Applications
LUC is a public-key cryptosystem based on Lucas sequences[1] that implements the analogs of ElGamal (LUCELG), Diffie-Hellman (LUCDIF), and RSA (LUCRSA). The encryption of the message in LUC is computed as a term of certain Lucas sequence, instead of using modular exponentiation as in RSA or Diffie-Hellman. However, a paper by Bleichenbacher et al.[2] shows that many of the supposed security advantages of LUC over cryptosystems based on modular exponentiation are either not present, or not as substantial as claimed.
hah site
ReplyDeletehttp://en.wikipedia.org/wiki/Lucas_sequence