Abstract Algebra: Groups of Permutations | Physics Forums - The Fusion of Science and Community 1. The problem statement, all variables and given/known data List the elements of the cyclic subgroup of [tex]S_6[/tex] generated by [tex]f = \left(\begin{array}{llllll} 1 & 2 & 3 & 4 & 5 & 6\\ 2 & 3 & 4 & 1 & 6 & 5\\ \end{array}\right)[/tex]2. Relevant equations3. The attempt at a solution I really do not understand what the elements of a permutation really is. I know if I write this as the product of disjoint cycles I get (2341)(65) but other than that I have no idea what the elements could be.

set theory | Neural Outlet.. This post is continuing on from my discussion on large numbers and notation. I am going to assume some knowledge of Set Theory for this post as there really is a lot to write about and I have found it hard to cut it down to a nice size. 2. Infinities Infinity as a concept has scared mathemeticians for a long time, not only was it hard to get your head around, it wasn’t really usable. Leibniz was facinated by infinity and it’s counterpart infintesimal The Isha Upanishad, a Hindu scripture, gives what I think is the polite version of infinity: “if you remove a part from infinity or add a part to infinity, still what remains is infinity” – this philosophical statement explains a mathematical proposition, cardinal arithmetic, and it comes from a branch of mathematics called set theory developed by Georg Cantor and Richard Dedekind in the 1870s. Cardinal infinites (Aleph numbers): Say we have a set that holds all the natural numbers: what is the size of such a set? It turns out but how? etc.. and

Demography “Demo-” from Ancient Greek δῆμος dēmos, means “the people” and “-graphy” from γράφω graphō, means “measurement.”[1] Demographic analysis can be applied to whole societies or to groups defined by criteria such as education, nationality, religion and ethnicity. Institutionally, demography is usually considered a field of sociology, though there are a number of independent demography departments.[2] Formal demography limits its object of study to the measurement of populations processes, while the broader field of social demography population studies also analyze the relationships between economic, social, cultural and biological processes influencing a population.[3] The term demographics refers to characteristics of a population. Methods[edit] There are two types of data collection — direct and indirect — with several different methods of each type. Direct methods[edit] A census is the other common direct method of collecting demographic data. Censuses do more than just count people.

How to Capitalize and Format Reference Titles in APA Style by Chelsea Lee APA Style has special formatting rules for the titles of the sources you use in your paper, such as the titles of books, articles, book chapters, reports, and webpages. The different formats that might be applied are capitalization (see Publication Manual, section 4.15), italics (see section 4.21), and quotation marks (see section 4.07), and they are used in different combinations for different kinds of sources in different contexts. The formatting of the titles of sources you use in your paper depends on two factors: (a) the independence of the source (stands alone vs. part of a greater whole) and (b) the location of the title (in the text of the paper vs. in the reference list entry). More on Italics Versus Nonitalics As you can see in the table above, the titles of works that stand alone (such as a book or a report) are italicized in both the text and the reference list. More on Capitalization: Title Case Versus Sentence Case Text Examples Reference List Entry Examples

Chapter 14: Mathematical Foundations - SWEBOK Introduction Software professionals live with programs. In a very simple language, one can program only for something that follows a well-understood, nonambiguous logic. The Mathematical Foundations knowledge area (KA) helps software engineers comprehend this logic, which in turn is translated into programming language code. The mathematics that is the primary focus in this KA is quite different from typical arithmetic, where numbers are dealt with and discussed. Mathematics, in a sense, is the study of formal systems. The SWEBOK Guide’s Mathematical Foundations KA covers basic techniques to identify a set of rules for reasoning in the context of the system under study. Figure 14.1: Breakdown of Topics for the Mathematical Foundations KA Breakdown of Topics for Mathematical Foundations The breakdown of topics for the Mathematical Foundations KA is shown in Figure 14.1. 1 Set, Relations, Functions Set. N = {0, 1, 2, 3, …} = the set of nonnegative integers. Finite and Infinite Set. Subset.

Logical symbols | JD2718 I finally got to the math-y part of my logic elective. On the first day we symbolized simple English language statements, and learned about “not,” “or,” “and,” “if… then… ,” and logical equivalence. But there are so many symbols floating around for those five items! Later today or tomorrow I will put up a (prettified) version of the table I put on the board: it included multiple symbols, and the corresponding symbols from arithmetic or set theory, as appropriate. (This table is actually an image that comes from a site promoting totally new notation (the far right column). Like this: Like Loading...

Edmond Halley Edmond Halley, FRS (commonly misspelt as Edmund,[2] pronounced /ˈɛdmənd ˈhæli/;[3][4] 8 November 1656 – 14 January 1742) was an English astronomer, geophysicist, mathematician, meteorologist, and physicist who is best known for computing the orbit of the eponymous Halley's Comet. He was the second Astronomer Royal in Britain, succeeding John Flamsteed. Early life[edit] Halley was born in Haggerston, Shoreditch, England. His father, Edmond Halley Sr., came from a Derbyshire family and was a wealthy soap-maker in London. As a child, Halley was very interested in mathematics. Career[edit] Publications and inventions[edit] Halley became an assistant to John Flamsteed, the Astronomer Royal at the Greenwich Observatory, in 1675, and among other things, had the job of assigning what is now called Flamsteed numbers to stars. In 1686, Halley published the second part of the results from his Helenian expedition, being a paper and chart on trade winds and monsoons. A 1702 printing of Halley's chart

Guidelines on Reading Philosophy It will be difficult for you to make sense of some of the articles we'll be reading. This is partly because they discuss abstract ideas that you're not accustomed to thinking about. They may also use technical vocabulary which is new to you. Sometimes it won't be obvious what the overall argument of the paper is supposed to be. The prose may be complicated, and you may need to pick the article apart sentence by sentence. Here are some tips to make the process easier and more effective. Contents Skim the Article to Find its Conclusion and Get a Sense of its Structure A good way to begin when you're trying to read a difficult article is to first skim the article to identify what the author's main conclusion is. When you're skimming the article, try also to get a general sense of what's going on in each part of the discussion. The articles we read won't always have a straightforward structure. This is the conclusion I want you to accept. The conclusion I want you to accept is A. and so on.

permutations - Prove that if the identity is written as the product of $r$ transpositions, then $r$ is an even number Mind–body problem Different approaches toward resolving the mind–body problem. The mind–body problem in philosophy examines the relationship between mind and matter, and in particular the relationship between consciousness and the brain. Each of these categories itself contains numerous variants. The two main forms of dualism are substance dualism, which holds that the mind is formed of a distinct type of substance not governed by the laws of physics, and property dualism, which holds that mental properties involving conscious experience are fundamental properties, alongside the fundamental properties identified by a completed physics. Several philosophical perspectives have been developed which reject the mind–body dichotomy. Mind–body interaction and mental causation[edit] Philosophers David L. Mind–body interaction has a central place in our pretheoretic conception of agency... Elizabeth is expressing the prevailing mechanistic view as to how causation of bodies works... Neural correlates[edit]

The Analyst The Analyst, subtitled "A DISCOURSE Addressed to an Infidel MATHEMATICIAN. WHEREIN It is examined whether the Object, Principles, and Inferences of the modern Analysis are more distinctly conceived, or more evidently deduced, than Religious Mysteries and Points of Faith", is a book published by George Berkeley in 1734. The "infidel mathematician" is believed to have been Edmond Halley, though others have suggested Sir Isaac Newton was intended.(Burton 1997, 477) Background and purpose[edit] Alciphron was widely read and caused a bit of a stir. Berkeley sought to take mathematics apart, claimed to uncover numerous gaps in proof, attacked the use of infinitesimals, the diagonal of the unit square, the very existence of numbers, etc. Content[edit] The Analyst was a direct attack on the foundations and principles of the infinitesimal calculus, specifically on Newton's notion of fluxions and on Leibniz's notion of infinitesimal change. Its most frequently quoted passage: Analysis[edit] D.

Guidelines on Writing a Philosophy Paper Philosophical writing is different from the writing you'll be asked to do in other courses. Most of the strategies described below will also serve you well when writing for other courses, but don't automatically assume that they all will. Nor should you assume that every writing guideline you've been given by other teachers is important when you're writing a philosophy paper. Contents What Does One Do in a Philosophy Paper? A philosophy paper consists of the reasoned defense of some claim Your paper must offer an argument. Three Stages of Writing 1. The early stages of writing a philosophy paper include everything you do before you sit down and write your first draft. Discuss the issues with others As I said above, your papers are supposed to demonstrate that you understand and can think critically about the material we discuss in class. It's even more valuable to talk to each other about what you want to argue in your paper. Make an outline Give your outline your full attention. 2. 3. No.

GroupTheory Note: You are looking at a static copy of the former PineWiki site, used for class notes by James Aspnes from 2003 to 2012. Many mathematical formulas are broken, and there are likely to be other bugs as well. These will most likely not be fixed. You may be able to find more up-to-date versions of some of these notes at A group is an algebra (see AlgebraicStructures) with an associative binary operation (usually written as either multiplication or addition), a constant identity element e such that ex = xe = x for all x, and an inverse operation x -> x-1 such that xx-1 = x-1x = e for all x. More formally, a group is a set G together with an operation *:S×S→S that satisfies: Closure ∀x,y ∈ G, x*y ∈ G. Associativity ∀x,y,z ∈ G, (x*y)*z = x*(y*z) Identity ∃e∈G ∀x∈G e*x = x*e = x. Inverses ∀x∈G ∃x-1∈G x*x-1 = x-1*x = e. An abelian group also satisfies: Commutativity ∀x,y∈G xy=yx. Here are some useful facts that apply in any group: (xy)-1 = y-1x-1. Lemma 1

Introduction to Embodiments of Mind by Warren S. McCulloch Introduction to Embodiments of Mind by Warren S. McCulloch By Seymour Papert Embodiments of Mind was published in Cambridge, MA by the M.I.T. Press in 1965. When McCulloch's essays are hard to understand, the trouble lies less often in the internal logic of the individual arguments than in the perception of a unifying theme that runs, sometimes with exuberant clarity, sometimes in a tantalizingly elusive way, through the whole work. Embodiments of Mind must not be read as a more pleasing name for the set of puzzles sometimes called "the mind-body problem." It would be futile to discuss whether this enterprise belongs to philosophy, to neurology, or to psychology except that each of these disciplines has established traditions and modes of thought that preclude progress by deformation of the problem. We need no longer be trapped in this dilemma. Jean Piaget, by origin a zoologist, set out to elucidate the mechanisms of knowledge by studying their development in small children.