The Wedderburn–Etherington numbers are an integer sequence named for Ivor Malcolm Haddon Etherington and Joseph Wedderburn that can be used to count certain kinds of binary trees. The first few numbers in the sequence are
0, 1, 1, 1, 2, 3, 6, 11, 23, 46, 98, 207, 451, 983, 2179, 4850, 10905, 24631, 56011, ...
Combinatorial interpretation
These numbers can be used to solve several problems in combinatorial enumeration. The nth number in the sequence (starting with the number 0 for n = 0) counts
* The number of unordered rooted trees with n leaves in which all nodes including the root have either zero or exactly two children. These trees have been called Otter trees, after the work of Richard Otter on their combinatorial enumeration. They can also be interpreted as unlabeled and unranked dendrograms with the given number of leaves.
* The number of unordered rooted trees with n nodes in which the root has degree zero or one and all other nodes have at most two children. Trees in which the root has at most one child are called planted trees, and the additional condition that the other nodes have at most two children defines the weakly binary trees. In chemical graph theory, these trees can be interpreted as isomers of polyenes with a designated leaf atom chosen as the root.
* The number of different ways of organizing a single-elimination tournament for n players (with the player names left blank, prior to seeding players into the tournament). The pairings of such a tournament may be described by an Otter tree.
* The number of different results that could be generated by different ways of grouping the expression
Formula
The Wedderburn–Etherington numbers may be calculated using the recurrence relation
beginning with the base case
In terms of the interpretation of these numbers as counting rooted binary trees with n leaves, the summation in the recurrence counts the different ways of partitioning these leaves into two subsets, and of forming a subtree having each subset as its leaves. The formula for even values of n is slightly more complicated than the formula for odd values in order to avoid double counting trees with the same number of leaves in both subtrees.
]]>The best -
The three Is - Intuition, Intelligence, and Improvement - I think that is sufficient in Mathematics.
]]>I'm seeing a lot of LaTex errors in this. I hope you don't mind but I've had a go at editing some but I'm worried I might have changed your proof in the process. Please have a look and see if it's ok. Also ask if you are unsure of the Latex that works for MIF. Not all feaures of the coding do, I'm sorry. What did you want list to do? Some frac errors towards the end.
Bob
]]>In mathematics, tetration (or hyper-4) is an operation based on iterated, or repeated, exponentiation. There is no standard notation for tetration, though
and the left-exponent are common.Under the definition as repeated exponentiation,
means , where n copies of a are iterated via exponentiation, right-to-left, I.e. the application of exponentiation n-1 times. n is called the "height" of the function, while a is called the "base," analogous to exponentiation. It would be read as "the nth tetration of a".It is the next hyperoperation after exponentiation, but before pentation. The word was coined by Reuben Louis Goodstein from tetra- (four) and iteration.
Tetration is also defined recursively as
allowing for attempts to extend tetration to non-natural numbers such as real and complex numbers.
The two inverses of tetration are called super-root and super-logarithm, analogous to the nth root and the logarithmic functions. None of the three functions are elementary.
Tetration is used for the notation of very large numbers.
Introduction
The first four hyperoperations are shown here, with tetration being considered the fourth in the series. The unary operation succession, defined as
, is considered to be the zeroth operation.Addition
n copies of 1 added to a.
Multiplication
n copies of a combined by addition.
Exponentiation
n copies of a combined by multiplication.
Tetration
n copies of a combined by exponentiation, right-to-left.
Succession, (a′ = a + 1), is the most basic operation; while addition (a + n) is a primary operation, for addition of natural numbers it can be thought of as a chained succession of n successors of a; multiplication (a × n) is also a primary operation, though for natural numbers it can analogously be thought of as a chained addition involving n numbers of a. Exponentiation can be thought of as a chained multiplication involving n numbers of a and tetration
as a chained power involving n numbers a. Each of the operations above are defined by iterating the previous one; however, unlike the operations before it, tetration is not an elementary function.The parameter a is referred to as the base, while the parameter n may be referred to as the height. In the original definition of tetration, the height parameter must be a natural number; for instance, it would be illogical to say "three raised to itself negative five times" or "four raised to itself one half of a time." However, just as addition, multiplication, and exponentiation can be defined in ways that allow for extensions to real and complex numbers, several attempts have been made to generalize tetration to negative numbers, real numbers, and complex numbers. One such way for doing so is using a recursive definition for tetration; for any positive real
and non-negative integer , we can define recursively as:The recursive definition is equivalent to repeated exponentiation for natural heights; however, this definition allows for extensions to the other heights such as
, , and as well – many of these extensions are areas of active research.]]>b) 12 is the first sublime number.
In number theory, a sublime number is a positive integer which has a perfect number of positive factors (including itself), and whose positive factors add up to another perfect number.
The number 12, for example, is a sublime number. It has a perfect number of positive factors (6): 1, 2, 3, 4, 6, and 12, and the sum of these is again a perfect number: 1 + 2 + 3 + 4 + 6 + 12 = 28.
There are only two known sublime numbers: 12 and
. The second of these has 76 decimal digits:6,086,555,670,238,378,989,670,371,734,243,169,622,657,830,773,351,885,970,528,324,860,512,791,691,264.
c) We know 6 is the first perfect number : Sum of the factors whose proper factors sum to the number itself.
(1 + 2 + 3 = 6).
28 is the second perfect number.
496 is the third perfect number.
8128 is the fourth perfect number.
d) 17 is the sum of the first 4 prime numbers, and the only prime which is the sum of 4 consecutive primes.
e) 25 is the first centered square number besides 1 that is also a square number.
In elementary number theory, a centered square number is a centered figurate number that gives the number of dots in a square with a dot in the center and all other dots surrounding the center dot in successive square layers. That is, each centered square number equals the number of dots within a given city block distance of the center dot on a regular square lattice. While centered square numbers, like figurate numbers in general, have few if any direct practical applications, they are sometimes studied in recreational mathematics for their elegant geometric and arithmetic properties.
f) 30 is the smallest sphenic number.
In number theory, a sphenic number is a positive integer that is the product of three distinct prime numbers.
Definition : A sphenic number is a product pqr where p, q, and r are three distinct prime numbers. In other words, the sphenic numbers are the square-free 3-almost primes.
Examples : The smallest sphenic number is 30 = 2 × 3 × 5, the product of the smallest three primes. The first few sphenic numbers are
30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, ...
g) 142857 is the smallest base 10 cyclic number.
A cyclic number is an integer in which cyclic permutations of the digits are successive integer multiples of the number. The most widely known is the six-digit number 142857, whose first six integer multiples are
142857 × 1 = 142857
142857 × 2 = 285714
142857 × 3 = 428571
142857 × 4 = 571428
142857 × 5 = 714285
142857 × 6 = 857142.
h) 9814072356 is the largest perfect power that contains no repeated digits in base ten.
i) Pandigital number: In mathematics, a pandigital number is an integer that in a given base has among its significant digits each digit used in the base at least once. For example, 1234567890 is a pandigital number in base 10. The first few pandigital base 10 numbers are given by :
1023456789, 1023456798, 1023456879, 1023456897, 1023456978, 1023456987, 1023457689.
]]>To be sure I understand I'm going to apply what you said first to the related case of
Taking logs of both sides we get:
Rearranging:
If we now make the y=x^t substitution:
And since y=x^t:
We now have a way to generate solutions. When I try to use this method to find those more complicated answers I began listing earlier, however, I still don't know how, other than by graphing it (which is very difficult for negative answers). For example, graphically it seems there must be some t the real part of which is around -2.56 for which
, yielding the solution to the original problem y=2, x=-0.767... . Is this what happens when you consider t over all the complex numbers? I can see now the solution space for positive t is rather simple (for both equations), but it's still tricky to assess when t is negative. I suppose I am still uninformed, and will come back in a little bit. ]]>In his 1947 paper, R. L. Goodstein introduced the specific sequence of operations that are now called hyperoperations. Goodstein also suggested the Greek names tetration, pentation, etc., for the extended operations beyond exponentiation. The sequence starts with a unary operation (the successor function with n = 0), and continues with the binary operations of addition (n = 1), multiplication (n = 2), exponentiation (n = 3), tetration (n = 4), pentation (n = 5), etc.
Various notations have been used to represent hyperoperations. One such notation is
. Another notation is , an infix notation which is convenient for ASCII. The notation is known as 'square bracket notation'.Knuth's up-arrow notation
is an alternative notation. It is obtained by replacing in the square bracket notation by arrows.For example:
the single arrow
represents exponentiation (iterated multiplication)So I want something from you first:
Using a graph plotter, try a = 4 then 5 then 1 then 1/2 then 1/3.
Comment on what you are observing in these cases. Make three conclusions about how the value of a affects the graph.
Then use the series expansion for e^x with x = 1 to compute e to 8 decimal places .
Post the steps in your working and your final result. Pay particular attention to explaining how you know your answer is accurate to 8 dp (without just comparing with a published result).
Bob
]]>the impossible cube and the hypercube are different examples, right?
]]>If r and θ are modulus and amplitude of a complex number,
then z=r(cosθ+isinθ).
Argument of Z and Amplitude of Z mean the same thing and are used interchangeably when we talk about complex numbers. When we plot the point of complex number on graph, and join it to the origin, the angle it makes with the x-axis is the argument or amplitude of complex number Z.
See this link:
Amplitude(or Argument) of a complex number:
Let z=x+iy where x,y are real,
and ; then the value of for which the equations: …(1) and …(2)are simultaneously satisfied is called the Argument(or Amplitude) of z and is denoted by Arg z (or,
) .Clearly, equations (1) and (2) are satisfied for infinite values of \theta ; any of these values of
is the value of Amp z . However, the unique value of lying in the interval and satisfying equations (1) and (2) is called the principal value of Arg z and we denote this principal value by arg z or amp z .Unless otherwise mentioned, by argument of a complex number we mean its principal value.Since,
and (where n=any integer), it follows that, where .]]>