The history of every major galactic civilization tends to pass through three distinct and recognizable phases, those of Survival, Inquiry and Sophistication, otherwise known as the How, Why, and Where phases. For instance, the first phase is characterized by the question ‘How can we eat?’, the second by the question ‘Why do we eat?’ There’s more to mathematics than rigour and proofs
This is from a physicist point of view. I've found that mathematicians and physicists think in *very* different ways. Basically physicists think in terms of analogies, whereas mathematicians think in terms of proofs, because they are often exploring "strange new worlds" in which analogies don't work. Snowflakes. If you look at a glass of water, you just see water. However, once you know some advanced math, whenever you look at water, you can imagine snowflakes. (1) What is it like to have an understanding of very advanced mathematics
www.math.ualberta.ca/~mss/misc/A Mathematician's Apology.pdf
This is an attempt to give a quick guide to the top few levels of the Tricki. It may cease to be feasible when the Tricki gets bigger, but we might perhaps be able to automate additions to it. Clicking on arrows just to the right of the name of an article reveals its subarticles. If you want to hide the subarticles again, then you should click to the right of them rather than clicking on the name of one of the subarticles themselves, since otherwise you will follow a link to that subarticle. What kind of problem am I trying to solve?
Glass Klein Bottles for sale - inquire within At last, Acme has conquered topological and engineering frontiers to manufacture genuine glass Klein Bottles. These are the finest closed, non-orientable, boundary-free manifolds sold anywhere in our three spatial dimensions. These elegant bottles make splendid gifts, outstanding classroom displays, and inferior mouse-traps. Acme Klein Bottle
Specifications for ACME Klein Bottles Specifications for ACME Klein Bottles No two Acme Klein Bottles are alike! Since Acme glassblowers individually hand craft each one, dimensions will vary and you may find occasional bubbles or streaks in the glass. These prove that your Acme Klein Bottle was made by real, 3-dimensional humans, adding to its unique status in a world of identical, machine made commodities. WARNING! Acme constructs each Klein Bottle from genuine Baryonic matter.
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Time complexity Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, where an elementary operation takes a fixed amount of time to perform. Thus the amount of time taken and the number of elementary operations performed by the algorithm differ by at most a constant factor. Since an algorithm's performance time may vary with different inputs of the same size, one commonly uses the worst-case time complexity of an algorithm, denoted as T(n), which is defined as the maximum amount of time taken on any input of size n. Time complexities are classified by the nature of the function T(n).
P versus NP problem Diagram of complexity classes provided that P≠NP. The existence of problems within NP but outside both P and NP-complete, under that assumption, was established by Ladner's theorem. The P versus NP problem is a major unsolved problem in computer science. Informally, it asks whether every problem whose solution can be quickly verified by a computer can also be quickly solved by a computer. It was introduced in 1971 by Stephen Cook in his seminal paper "The complexity of theorem proving procedures" and is considered by many to be the most important open problem in the field. It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute to carry a US$1,000,000 prize for the first correct solution.
Computational complexity theory is a branch of the theory of computation in theoretical computer science and mathematics that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other. A computational problem is understood to be a task that is in principle amenable to being solved by a computer, which is equivalent to stating that the problem may be solved by mechanical application of mathematical steps, such as an algorithm. A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used.
The problem states that any investment at r-percent interest per annum will be doubled in approximately years. a) Using simple compound interests Since the initial principal is to be doubled we have the equation (formula for compound interests): 2 = (1 + r/100)n Taking the log of both sides: log 2 = n · log(1 + r/100) or n = log 2 / [log(100 + r) - 2] Next we seek to find a number, x, such that when divided by r, the result is approximately n. Thus x / r = log 2 / [log(100 + r) - 2] Then x ~ 0.301r / [log(100 + r) - 2] Previous monthly puzzles: May 2004
Previous monthly puzzles: June-July 2004
Numbers: Facts, Figures & Fiction Click on cover for larger image Numbers: Facts, Figures & Fiction by Richard Phillips. Published by Badsey Publications. See sample pages: 24, 82, 103. Order the book direct from Badsey Publications price £12.
Primality Proving 2.1: Finding very small primes For finding all the small primes, say all those less than 10,000,000,000; one of the most efficient ways is by using the Sieve of Eratosthenes (ca 240 BC): Make a list of all the integers less than or equal to n (greater than one) and strike out the multiples of all primes less than or equal to the square root of n, then the numbers that are left are the primes. (See also our glossary page.) For example, to find all the odd primes less than or equal to 100 we first list the odd numbers from 3 to 100 (why even list the evens?) The first number is 3 so it is the first odd prime--cross out all of its multiples.
Prime number checker
People who know a lot about a lot have long been an exclusive club, but now they are an endangered species. Edward Carr tracks some down ... From INTELLIGENT LIFE Magazine, Autumn 2009 CARL DJERASSI can remember the moment when he became a writer. It was 1993, he was a professor of chemistry at Stanford University in California and he had already written books about science and about his life as one of the inventors of the Pill. Now he wanted to write a literary novel about writers’ insecurities, with a central character loosely modelled on Norman Mailer, Philip Roth and Gore Vidal.
An Introduction to Wavelets: What Do Some Wavelets Look Like? W hat do S ome W avelets L ook L ike? Wavelet transforms comprise an infinite set. The different wavelet families make different trade-offs between how compactly the basis functions are localized in space and how smooth they are.
When reading the list, it is important to recognize that a mathematical concept is independent of the symbol chosen to represent it. For many of the symbols below, the symbol is usually synonymous with the corresponding concept (ultimately an arbitrary choice made as a result of the cumulative history of mathematics), but in some situations a different convention may be used. For example, depending on context, "≡" may represent congruence or a definition. Further, in mathematical logic, numerical equality is sometimes represented by "≡" instead of "=", with the latter representing equality of well-formed formulas. In short, convention dictates the meaning. Each symbol is shown both in HTML, whose display depends on the browser's access to an appropriate font installed on the particular device, and in TeX, as an image.
Power law An example power-law graph, being used to demonstrate ranking of popularity. To the right is the long tail, and to the left are the few that dominate (also known as the 80–20 rule). In statistics, a power law is a functional relationship between two quantities, where one quantity varies as a power of another. For instance, the number of cities having a certain population size is found to vary as a power of the size of the population. Empirical power-law distributions hold only approximately or over a limited range. Empirical examples of power laws