e^(i theta) Consider the function on the right hand side (RHS) f(x) = cos( x ) + i sin( x )Differentiate this function f ' (x) = -sin( x ) + i cos( x) = i f(x)So, this function has the property that its derivative is i times the original function. What other type of function has this property?A function g(x) will have this property if dg / dx = i g This is a differential equation that can be solved with seperation of variables (1/g) dg = i dx (1/g) dg = i dx ln| g | = i x + C | g | = ei x + C = eC ei x | g | = C2 ei x g = C3 ei xSo we need to determine what value (if any) of the constant C3 makes g(x) = f(x). (This is the usual justification given in textbooks.)By use of Taylors Theorem, we can show the following to be true for all real numbers: sin x = x - x3/3! Knowing that, we have a mechanism to determine the value of ei, because we can express it in terms of the above series: e^(i) = 1 + (i) + (i)2/2! i1 = i i2 = -1 terms repeat every fouri3 = -i i4 = 1 i5 = i i6 = -1 etc...
Entropy where T is the absolute temperature of the system, dividing an incremental reversible transfer of heat into that system (dQ). (If heat is transferred out the sign would be reversed giving a decrease in entropy of the system.) The above definition is sometimes called the macroscopic definition of entropy because it can be used without regard to any microscopic description of the contents of a system. The concept of entropy has been found to be generally useful and has several other formulations. Entropy was discovered when it was noticed to be a quantity that behaves as a function of state, as a consequence of the second law of thermodynamics. The absolute entropy (S rather than ΔS) was defined later, using either statistical mechanics or the third law of thermodynamics. In the modern microscopic interpretation of entropy in statistical mechanics, entropy is the amount of additional information needed to specify the exact physical state of a system, given its thermodynamic specification.
Ada Lovelace Augusta Ada King, Countess of Lovelace (10 December 1815 – 27 November 1852), born Augusta Ada Byron and now commonly known as Ada Lovelace, was an English mathematician and writer chiefly known for her work on Charles Babbage's early mechanical general-purpose computer, the Analytical Engine. Her notes on the engine include what is recognised as the first algorithm intended to be carried out by a machine. Because of this, she is often described as the world's first computer programmer.[1][2][3] Ada described her approach as "poetical science" and herself as an "Analyst (& Metaphysician)". As a young adult, her mathematical talents led her to an ongoing working relationship and friendship with fellow British mathematician Charles Babbage, and in particular Babbage's work on the Analytical Engine. Biography[edit] Childhood[edit] Ada, aged four On 16 January 1816, Annabella, at George's behest, left for her parents' home at Kirkby Mallory taking one-month-old Ada with her. Adult years[edit]
Hammack Home This book is an introduction to the standard methods of proving mathematical theorems. It has been approved by the American Institute of Mathematics' Open Textbook Initiative. Also see the Mathematical Association of America Math DL review (of the 1st edition), and the Amazon reviews. The second edition is identical to the first edition, except some mistakes have been corrected, new exercises have been added, and Chapter 13 has been extended. Order a copy from Amazon or Barnes & Noble for $13.75 or download a pdf for free here. Part I: Fundamentals Part II: How to Prove Conditional Statements Part III: More on Proof Part IV: Relations, Functions and Cardinality Thanks to readers around the world who wrote to report mistakes and typos! Instructors: Click here for my page for VCU's MATH 300, a course based on this book. I will always offer the book for free on my web page, and for the lowest possible price through on-demand publishing.
Entropy Figure 1: In a naive analogy, energy in a physical system may be compared to water in lakes, rivers and the sea. Only the water that is above the sea level can be used to do work (e.g. propagate a turbine). Entropy represents the water contained in the sea. In classical physics, the entropy of a physical system is proportional to the quantity of energy no longer available to do physical work. Entropy is central to the second law of thermodynamics, which states that in an isolated system any activity increases the entropy. History The term entropy was coined in 1865 [Cl] by the German physicist Rudolf Clausius from Greek en- = in + trope = a turning (point). The Austrian physicist Ludwig Boltzmann [B] and the American scientist Willard Gibbs [G] put entropy into the probabilistic setup of statistical mechanics (around 1875). The formulation of Maxwell's paradox by James C. Entropy in physics Thermodynamical entropy - macroscopic approach Entropy in quantum mechanics Black hole entropy
Top 50 Free Open Source Classes on Computer Science : Comtechtor Computer science is an interesting field to go into. There are a number of opportunities in computer science that you can take advantage of. With computers increasingly becoming a regular part of life, those who can work with computers have good opportunities. You can find a good salary with a program in computer science, and as long as you are careful to keep up your skills. Here are 50 free opencourseware classes that can help you learn more about computer science: Introduction to Computer Science Learn the basics of computer science, and get a foundation in how computer science works. Introduction to Computer Science: Learn about the history of computing, as well as the development of computer languages. Comprehensive Computer Science Collections If you are interested in courses that are a little more comprehensive in nature, you can get a good feel for computer science from the following collections: Programming and Languages Computer Software Computer Systems and Information Technology
8. Reductio ad Absurdum – A Concise Introduction to Logic 8.1 A historical example In his book, The Two New Sciences,[10] Galileo Galilea (1564-1642) gives several arguments meant to demonstrate that there can be no such thing as actual infinities or actual infinitesimals. One of his arguments can be reconstructed in the following way. Galileo proposes that we take as a premise that there is an actual infinity of natural numbers (the natural numbers are the positive whole numbers from 1 on): He also proposes that we take as a premise that there is an actual infinity of the squares of the natural numbers. Now, Galileo reasons, note that these two groups (today we would call them “sets”) have the same size. If we can associate every natural number with one and only one square number, and if we can associate every square number with one and only one natural number, then these sets must be the same size. But wait a moment, Galileo says. Galileo argues that the reason we reached a contradiction is because we assumed that there are actual infinities.
Gödel's incompleteness theorems Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The two results are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible, giving a negative answer to Hilbert's second problem. The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an "effective procedure" (i.e., any sort of algorithm) is capable of proving all truths about the relations of the natural numbers (arithmetic). For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. Background[edit] First incompleteness theorem[edit] Diagonalization[edit] B.
untitled Chapter 4: Music I preface the following by the admission that I have developed little to no musical aptitude as yet and have never studied music theory. However, that does not seem to have stopped me from uncovering what looks to be some very interesting observations having applied Mod 9 to the frequencies generated by the black and white keys of the musical scale. 1955 saw the introduction of the International Standard Tuning of 440 Hz on the A of Middle C Octave. The C's were all 3 & 6, same for C sharp, then D's are all 9's and on until I came to F which revealed the 1 2 4 8 7 5 sequence, in order. You will notice that using this tuning at 440 Hz we see that: 5 sections of the octave are 1 2 4 8 7 5 4 sections are 3 & 6 3 sections are 9 Immediately the Pythagorean 3 4 5 triangle springs to mind. Above, we can clearly see that, as with the numbers, the octaves are paired up symmetrically and reflected, separated by 3 octaves. 2 sections that are 3 6, versus 4, using the 440 Hz tuning.
Image evolution What is this? A simulated annealing like optimization algorithm, a reimplementation of Roger Alsing's excellent idea. The goal is to get an image represented as a collection of overlapping polygons of various colors and transparencies. We start from random 50 polygons that are invisible. Fitness is a sum of pixel-by-pixel differences from the original image. This implementation is based on Roger Alsing's description, though not on his code. How does it look after some time? 50 polygons (4-vertex) ~15 minutes 644 benefitial mutations 6,120 candidates 88.74% fitness 50 polygons (6-vertex) ~15 minutes 646 benefitial mutations 6,024 candidates 89.04% fitness 50 polygons (10-vertex) ~15 minutes 645 benefitial mutations 5,367 candidates 87.01% fitness 50 polygons (6-vertex) ~45 minutes 1,476 benefitial mutations 23,694 candidates 93.35% fitness 50 polygons (6-vertex) ~60 minutes 1,595 benefitial mutations 28,888 candidates 93.46% fitness Does it work on all images? It depends, success varies.
The Zero Point Field: How Thoughts Become Matter? | HuffPost Life Since I have mentioned the zero point field (ZPF) so much in my past HuffPost articles, and seeing as how it is a vital component to what is going on, it only makes sense to provide a more detailed analysis for all those Quantum buffs who struggle with my theory that thoughts equal matter. So, let's start with the basics and show what is known about the ZPF, and how its discovery come about? ZPF Basics In quantum field theory, the vacuum state is the quantum state with the lowest possible energy; it contains no physical particles, and is the energy of the ground state. Liquid helium-4 is a great example: Under atmospheric pressure, even at absolute zero, it does not freeze solid and will remain a liquid. This would seem to imply that a vacuum state -- or simply vacuum -- is not empty at all, but the ground state energy of all fields in space, and may collectively be called the zero point field. In physics, there is something called the Casmir effect.
Antikythera mechanism The Antikythera mechanism (Fragment A – front) The Antikythera mechanism (Fragment A – back) The Antikythera mechanism (/ˌæntɨkɨˈθɪərə/ ANT-i-ki-THEER-ə or /ˌæntɨˈkɪθərə/ ANT-i-KITH-ə-rə) is an ancient analog computer[1][2][3][4] designed to predict astronomical positions and eclipses. Professor Michael Edmunds of Cardiff University, who led a 2006 study of the mechanism, described the device as "just extraordinary, the only thing of its kind", and said that its astronomy was "exactly right". The mechanism was housed in a wooden box approximately 340 × 180 × 90 mm in size and comprised 30 bronze gears (although more could have been lost). The Antikythera mechanism is kept at the National Archaeological Museum of Athens. Origins and discovery[edit] The mechanism was discovered in a shipwreck off Point Glyphadia on the Greek island of Antikythera. Mechanism[edit] Schematic of the artifact's known mechanism Operation[edit] Gearing[edit] Known gear scheme[edit] Wright proposal.