What is Sigma: Definition and 228 Discussions

Sigma (uppercase Σ, lowercase σ, lowercase in word-final position ς; Greek: σίγμα) is the eighteenth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 200. In general mathematics, uppercase ∑ is used as an operator for summation. When used at the end of a letter-case word (one that does not use all caps), the final form (ς) is used. In Ὀδυσσεύς (Odysseus), for example, the two lowercase sigmas (σ) in the center of the name are distinct from the word-final sigma (ς) at the end.

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  1. G

    What are the chances of a 5 sigma confidence rating being wrong?

    I'm trying to figure out what the odds of 5 sigma confidence rating has of being wrong. According to one website it is 0.000028% which is 1 in 35,000 but I've seen so many divergent answers as to what the odds of 5 sigma being wrong are that I want to be sure. I've seen people say it is as high...
  2. rustynail

    Help with capital sigma notation please.

    I was playing a bit with the Riemann Zeta function, and have been struggling with some notation problems. The function is defined as follows \zeta (s) = \sum_{n=1}^{\infty} \frac{1}{n^s} where s \in \mathbb{C} we know that n^s = exp(s\;ln\;n) so I can write \zeta (s) =...
  3. W

    Understanding Sigma Algebra: Example 3 Explanation & B=(3/4,6) Definition

    at http://www.theponytail.net/wiki/pmwiki.php/Main/SigmaAlgebra in example 3, in information terms, why G is equivalent as observing only the first coin toss. Secondly, what does B=(3/4,6) means?
  4. K

    How to interpret this Sigma notation.

    Homework Statement Taken from Wikipedia's Palindromic number entry, under formal definition header: Although palindromic numbers are most often considered in the decimal system, the concept of palindromicity can be applied to the natural numbers in any numeral system. Consider a number n >...
  5. P

    Topological sigma model, Euler Lagrange equations

    Homework Statement My question refers to the paper "Topological Sigma Models" by Edward Witten, which is available on the web after a quick google search. I am not allowed to include links in my posts, yet. I want to know how to get from equation (2.14) to (2.15). We consider a theory of maps...
  6. S

    Useful pi systems generating the borel sigma algebra

    Homework Statement For the purpose of my module, we define the lebesgue measure over Int(a,b) where Int(a,b) = (a,b] where -∞≤a≤b<∞ (a,∞) else And l((Int(a,b) = b-a a≤b<∞ ∞ a≤b =∞ And thus we say the borel sigma algebra is generated by...
  7. Y

    How do I simplify sigma notation to find the sum in terms of n only?

    Homework Statement I'm actually asked to calculate the area under the curve 5x + x2 over the interval [0,1] using Riemann Sums. I found the formula for the Riemann sum over the interval, it being the following: \sum^{n}_{k=1} (\frac{5k}{n}+\frac{k^{2}}{n^{2}})(\frac{1}{n}) However, I...
  8. V

    Expressing a Sum in Sigma Notation: 1 + (2/3) + (3/5) + (4/7) + (5/9)

    Homework Statement "Express the following sum in sigma notation: 1 + (2/3) + (3/5) + (4/7) + (5/9)" Homework Equations The Attempt at a Solution I've figured out what they all have in common (1+2=3, 2+3=5, 3+4=7, 4+5=9) but I've been searching through the book and on the...
  9. Andre

    The X3 sensor and the Sigma XD1

    Andy has elaborated about photo sensors https://www.physicsforums.com/showpost.php?p=3091595&postcount=2. The essential part: This gives the impression that a lot is wasted. If a "red" photon happens to hit a blue filter, it's lost. So every "pixel" containing two green and a red and a blue...
  10. N

    IR terms from massless modes in 2D sigma model

    Hi All, My first post here. As far as I've understood there are problems with massless states in two dimensions. In the problem I am working on I have a 2D (non linear) sigma model with a lot of symmetries. The worldsheet fields come as heavy and massless. My main interest is to...
  11. B

    Ln sigma vs 1/T curve for p-type semiconductor

    Homework Statement a) Sketch a schematic band structure diagram for a p-type semiconductor. Mark the energy of the top of the valence band, the bottom of the conduction band and the impurity level on the diagram. b) Sketch the ln (σ) versus 1/T curve for a p-type semiconductor, where σ is...
  12. K

    Quick question about sigma algebra proof.

    If A is a sigma algebra on a set S. if Ai E A. i = 1,...,n prove that the intersection of all Ai E A. now isn't the intersection just the empty set in this case? Which is already proved as the empty set is always contained in A?
  13. K

    Proving Sigma Field Properties on Set S

    I have the following to prove: Given a sigma field/sigma algebra B on a set S. Prove: i) 0 E B ii) if B1,..,Bk E B then UBi E B for i = 1 to n and nBi E B for i = 1 to n iii) if B1,B2... E B then nBi for i = 1 to infinity E B so this is what I have so far. i) A sigma algebra is...
  14. J

    How Do You Prove a Sigma Notation Equality Using Induction?

    Homework Statement 2n-1 Sigma (3i+1) = n(6n-1) i=0 prove for all positive n Homework Equations The Attempt at a Solution It holds true for n=1 5=5 then P: m+1 2m+1 Sigma(3i+1) = (m+1)(6(m+1)-1) or 6m^2 + 11m + 5 i=0 then 2m+1 Sigma(3i+1) = m(6m-1) +...
  15. Z

    Six sigma in neutrino article?

    "six sigma" in neutrino article? I was reading this article... http://www.physicstoday.org/resource/1/phtoad/v64/i12/p8_s1 in the first paragraph they say "...notwithstanding the assertion that it is a six-sigma effect". What does this mean? I tried to google, but didn't find that very...
  16. A

    Independence of sigma algebras

    I am trying to establish whether the following is true (my intuition tells me it is), more importantly if it is true, I need to establish a proof. If $X_1, X_2$ and $X_3$ are pairwise independent random variables, then if $Y=X_2+X_3$, is $X_1$ independent to $Y$? (One can think of an...
  17. S

    Pi+ + p -> Sigma+ + pi+ + K0

    pi+ + p --> Sigma+ + pi+ + K0 Dear forum readers; I have a simple question. Could someone tell me the isospin is conserved in the above reaction? This is the way i understood this. LHS possible Isospin values are 1/2, 3/2 RHS possible Isospin value is 5/2 5/2 is not in LHS. so...
  18. S

    Feynman diagram of secondary Sigma plus decay

    Can someone describe the Feynman diagram for the Sigma plus decay to a positive pion and a neutron? I'm fine with the primary decay to the neutral pion and a proton. There appears to be insufficient down quarks created.
  19. Saitama

    Do f-orbitals form sigma bonds?

    Homework Statement This is not a homework question. I found a question in my book that which orbitals form sigma bonds. I am wondering if any of the f-orbitals form sigma bonds? Homework Equations The Attempt at a Solution
  20. N

    What is application of nonlinear sigma model?

    Please teach me this: I hear that linear sigma model has application in pion,nuclear physics.But do not understand what is the application of nonlinear sigma model in QFT? Thank you very much in advance.
  21. E

    Random Variable Measurability w.r.t. Sigma Fields

    Hello everyone, I'm having a little trouble with a probability problem with three parts; I think I'm having trouble wrapping my head around just what's going on here. If anyone could give me a starting point, I'd appreciate it. Here's the problem (Billingsley 5.1) (X a random variable)...
  22. M

    How to Solve a Double Summation Problem?

    Homework Statement \sum_{k=1}^{3} \sum_{j=0}^{4}k^{j} The attempt at a solution so this above means: (1+1^{1}+1^{2}+1^{3}+1^{4})*(1+2^{1}+2^{2}...)*(1+3...)*... or (1*1^{1}*1^{2}*1^{3}*1^{4})+(1*2^{1}*2^{2}...)+(1*3...)+...
  23. E

    Mastering Partial Fractions: A Guide for Solving Advanced Summation Problems

    Homework Statement Homework Equations The Attempt at a Solution Obviously I don't need a solution because it's right there. What I need to understand is what happened after the third equation sign and more importantly, how would I learn to solve these kinds of problems on my own. I looked at...
  24. N

    Why dimensionless coupling in nonlinear sigma model corresponding to temperature?

    Please teach me this: In QTF theory book of Schoeder say: In nonlinear sigma model L=\intdx\frac{1}{2g^{2}}(\delta _{\mu}n^{\rightarrow})(consider2<d<4 where d is the dimension number of spacetime).If we consider the Lagrangian as the Boltzman weight of a partition function,then the...
  25. N

    Is Hidden Symmetry Broken In Linear and Nonlinear Sigma Model?

    Please teach me this: In linear and nonlinear sigma model, the spontaneous symmetry broken is inevitable happened or it may be happened and it may be not happened?Because the expectation value of field may be zero or may be not zero,then it seem that sometimes the hidden symmetry broken not...
  26. J

    Forming a sigma field from a countable infinite set

    The professor did this problem in class but I need help with understanding it a little more. For any countably infinite set, the collection of its finite subsets and their complements form a field F. Prove that this conjecture.
  27. N

    Why we call ''nonlinear'' sigma model?

    Please teach me this: The linear sigma model L(Lagrangian)=\frac{1}{2}(\delta_{\mu}\Phi^{i})^{2} + \frac{1}{2}\mu^{2}(\Phi^{i})^{2} - \frac{\lambda}{4}((\Phi^{i})^{2})^{2}. The nonlinear sigma model: L=f_{ij}({\Phi^{i}})\delta_{\mu} \Phi^{i}\delta^{\mu}\Phi{j}. After put condition O(N)...
  28. Z

    Difference between sigma algebra and topological space

    What is the difference between sigma algebra and topological space topological space?also what is the meaning of algebra on a set? the definitions are very similar except that in the case of sigma algebra the union is taken to include infinite number of sets .right?
  29. R

    Help with sigma notation where the top value is x?

    Homework Statement In each case, x is an integer between -6 and 6 inclusive. Homework Equations x Σ 2i =12 i=1 The Attempt at a Solution 2x1 = 12 + 2x2 = 12 +...+2x(x) = 12
  30. B

    Sigma Notation: Definition and Examples

    What does this mean? (see attachment) "Where b is a block defined by the contiguity condition c that may exist between elements of s, and n is the number of elements in that block" I know is not possible to get a solution without the actual function, but how does this reads?
  31. Saitama

    What is the value of n in the identity sin^3xsin3x=\sum^n_{m=0}{}^nC_mcosmx?

    Homework Statement Suppose sin^3xsin3x=\sum^n_{m=0}{}^nC_mcosmx is an identity in x, where C0, C1, ...Cn are constants, and Cn \neq0, then what is the value of n? Homework Equations The Attempt at a Solution I expanded the sigma notation and got:-...
  32. Saitama

    Converting Sigma Notation: Simplifying Binomial Theorem

    Converting Sigma notation... Homework Statement (Not a homework question) Hi! I have been encountering problems in Binomial Theorem which includes converting the sigma notation. Like \sum_{k=0}^n \frac{n!}{(n-k)!k!} a^kb^{n-k}=(a+b)^n I got many questions in my exam of this type with four...
  33. T

    Calculating Width of Gaussian PDF(x) Given Sigma & N

    Hi Experts, I'm working in industry and have an application requiring some expert knowledge on statistics/probability. I have a probability distribution function (PDF) for a Gaussian random variable. I know the standard deviation of the PDF. I also know total number of experiments conducted...
  34. B

    Sigma Algebras generated by Sets

    Hi, All: A simple question: If A is the sigma- algebra generated by a collection C of subsets of an ambient set X. Isn't it trivial that the sigma-algebra generated by A is A itself? One definition is that the sigma algebra generated by a collection S of subsets is the...
  35. T

    Is the set of rational numbers in [0,1] a set in the algebra?

    Hello, first I’d like to clarify that the only difference between an algebra and a sigma-algebra, is that we have A,B \in \mathcal{A} \Rightarrow A \cup B \in \mathcal{A} \text{ (1) for } \mathcal{A} \text{ algebra}A_1, A_2, A_3, \ldots\in\mathcal{A} \Rightarrow \bigcup_{i=1}^{\infty}A_i...
  36. R

    Why are sigma fields significant in probability theory?

    As the title. Why are sigma fields important in probability? The only one reason I can think of is that sigma fields are used as domain, e.g. borel fields uses sigma fields instead of power set. However, are there any other significances of sigma fields in probability theory? Thanks for your...
  37. N

    What is a application of linear sigma model?

    Please teach me this: Which interaction is the result of linear sigma model.Please teach me what purpose the linear sigma model implies? Thank you very much in advanced.
  38. H

    Six Sigma in the Aerospace Industry

    I'm trying to finish a report on the how effective six sigma is on Aerospace organization. Can you take 30 sec of your time to complete the following survey? Thank you. http://www.surveymonkey.com/s/9BCWDJW
  39. R

    Sigma algebra proof in measure theory

    Homework Statement Let \mathcal{A} be σ-algebra over a set X, and μ a measure in \mathcal{A}. Let A_{n} \in \mathcal{A} with \sum_{n=1}^{\inf} \mu(A_{n})< \inf Show that this implies μ ({x \in X : x \in A_n for infinitely many n}) = 0 . The Attempt at a Solution I don't even see how is the...
  40. M

    Sigma Notation: Learn How to Evaluate Expressions

    Express this in sigma notation? 3^3 - 3^4 + 3^5 - ... - 3^100 Evaluate these two sigmas? n ∑ (i-2)^2 i =1 n ∑ (4-i^2) i =1 I don't really understand sigma notation so I'm really interested in the process and explanation of how to do it. Any help would be greatly appreciated!
  41. T

    Extending Sigma Sums to Real Numbers

    I mean an extension analogous to the extension of factorial to gamma(ignoring the shift). I want to extend the iterative number, usually written below the sigma, to be a real number, rather than an exclusively natural number. The easiest way I can think of is to convert the sum into a...
  42. E

    Sigma algebra generated by a piecewise function

    I have this homework question and I'm a little bit stuck. The question is: Let (Ω, F , P ) be the unit interval probability space, and X be the random variable defined by (See attatchment) Find the σ-algebra generated by X. From my understanding of sigma-algebras I want to look...
  43. R

    Sigma field generated by singletons of real line

    Suppose set A=\{\{\omega\}:\omega\in\mathbb{R}\}. What can you say more about it? In particular, on the \sigma(A) the smallest sigma field generated by A, i.e. it is closed under complements/countable intersections or unions and the whole space is in the sigma field. Clearly, if here...
  44. R

    Probability sigma field

    Homework Statement Let (\Omega,\mathcal{F},P) be a probability space. Let \mathcal{N}\subseteq\mathcal{P}(\Omega) where \mathcal{N}:=\{N\subseteq\Omega:\exists M\in\mathcal{F},P(M)=0 and N\subseteq M\}. (a) Want to show that \mathcal{F}'=\{F\cup N:F\in\mathcal{F}, N\in\mathcal{N}\} defines a...
  45. D

    Is Sigma Xi membership useful?

    A couple years ago I was made an associate member of Sigma Xi, an honor society for science and engineering researchers. But after that time, the only thing I've gotten out of the membership is a magazine subscription, which I could do without. Are there other "hidden" benefits to being a member...
  46. J

    Sigma Notation for Λ1 + Λ2 + Λ3 +...+ Λn = 1

    How do I notate this: \Lambda_{1}+\Lambda_{2}+\Lambda_{3}+\ldots+\Lambda_{n}=1 using sigma notation?
  47. A

    Simplfication of a sigma notation

    Homework Statement 1/1(4) + 1/4(7) + 1/7(10)+...+ 1/(3n-2)(3n+1) and the sigma notation is pretty obvious Homework Equations nothing really.. The Attempt at a Solution I can see an obvious pattern its n(n+3) but n cannot be 2,3,5,6 etc.. and the second digit in the first term...
  48. A

    Left endpoint approximation & Riemann Sums (Sigma)

    1. The problem statement, all variables and givennown data 1)FInd the nth left endpoint approximation Ln for f(x) = 3x^2-2 on [0,2]. What is the limit as n approaches infinity Ln in this case? 2)Evaluate: \sum45i=5 (2i-5) Homework Equations Ln = \sumNj=1 f(cj)(xj-xj-1) The...
  49. A

    Understanding Product Sigma Algebras: Definition, Notation, and Example

    Homework Statement I should mention beforehand that I do not come from a math background so I may ask some trivial questions. I am reading the book "Real Analysis" by Folland for a course I am taking and am attempting to understand a definition of product sigma algebra. It is stated in the...
  50. A

    References for Nonlinear Sigma Model.

    Can anybody please suggest some references (preferably review articles or lecture notes etc, freely available online) for learning linear and nonlinear sigma models and their applications in particle physics?
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