Geometric Definition and 813 Threads

  1. Z

    Geometric measure of entanglement for fermions or bosons?

    For a system consisting of multiple components, say, a spin chain consisting ofN≥3spins, people sometimes use the so-called geometric measure of entanglement. It is related to the inner product between the wave function and a simple tensor product wave function. But it seems that none used this...
  2. AntSC

    S1 Probability - Binomial & Geometric Distribution

    Homework Statement Four players play a board game which requires them to take it in turns to throw two fair dice. Each player throws the two dice once in each round. When a double is thrown the player moves forward six squares. Otherwise the player moves forward one square Homework Equations...
  3. E

    Prove that for a,b,c > 0, geometric mean <= arithmetic mean

    Homework Statement Let ## a,b,c \in \mathbb{R}^{+} ##. Prove that $$ \sqrt[3]{abc} \leq \frac{a+b+c}{3}. $$ Note: ## a,b,c ## can be expressed as ## a = r^3, b = s^3, c = t^3 ## for ## r,s,t > 0##. Homework Equations ## P(a,b,c): a,b,c \in \mathbb{R}^{+} ## ## Q(a,b,c): \sqrt[3]{abc} \leq...
  4. S

    How Does Snell's Law Help Calculate Distance Between Parallel Lines?

    I can't seem to find the proof for the distance between the two parallel lines. Homework Equations : Snells law: μ1sinθ1=μ2sinθ2 Sin (A+B)= sinAcosB + sinBcosA[/B]The Attempt at a Solution : tried using the parallel lines to get a result in terms of the initial angle of incidence ϑ, as the...
  5. M

    MHB Geometric Meaning of Cylindrical & Spherical Mappings

    Hey! :o What is the geometric meaning of the following mappings, that are written in cylindrical coordinates?? (Wondering) The mappings are: $$(r, \theta, z) \rightarrow(r, \theta , -z) \\ (r, \theta , z) \rightarrow (r, \theta +\pi , -z)$$ And what is the geometric meaning of the following...
  6. S

    Question about geometric algebra -- Can any one help?

    I was given the following as a proof that the inertial tensor was symmetric. I won't write the tensor itself but I will write the form of it below in the proof. I am confused about the steps taken in the proof. It involves grade projections. A \cdot (x \wedge (x \cdot B)) = \langle Ax(x...
  7. W

    Sum of a geometric series of complex numbers

    Homework Statement Given an integer n and an angle θ let Sn(θ) = ∑(eikθ) from k=-n to k=n And show that this sum = sinα / sinβ Homework Equations Sum from 0 to n of xk is (xk+1-1)/(x-1) The Attempt at a Solution The series can be rewritten by taking out a factor of e-iθ as e-iθ∑(eiθ)k from...
  8. C

    Perpendicular geometric objects

    Trying to solve a drawing task. The following has to be achieved: - Produce a Square (vertices of ABCD) - Cut line BC into two equal parts, label the point in between as M. - Draw a line between point M and A. - Divide angle of BMA into two equal parts. Label the location where the bisector...
  9. Shackleford

    What is geometric interpretation of this equality?

    Homework Statement Prove that for any a, b ∈ ℂ, |a - b|2 + |a + b|2 = 2(|a|2 + |b|2). Homework Equations |a|2 = aa* (a - b)* = (a* - b*) (a + b)* = (a* + b*) * = complex conjugate The Attempt at a Solution I've already shown that the relation is true. I'm not quite sure what the...
  10. G

    How Does Lens Focal Length Affect Microscope Magnification and Field Diameter?

    Homework Statement First, thanks in advace. Let us consider a microscope where the objective L1 has f1=20mm and magnification 10x. In the image plane is located a diafragm M with diameter 19mm (see fig). The size of the CCD is 4,8mm (vertical) x 5,6mm (horizontal). 20mm before of the CCD...
  11. G

    Finding the height of a ball with a geometric series

    Homework Statement A ball is dropped from one yard and come backs up ##\dfrac{2}{3}## of the way up and then back down. It comes back and ##\dfrac{4}{9}## of the way. It continues this such that the sum of the vertical distance traveled by the ball is is given by the series...
  12. M

    Sum of Geometric Series by Differentiation

    Homework Statement Find the sum of the following series: Σ n*(1/2)^n (from n = 1 to n = inf). Homework Equations I know that Σ r^n (from n = 0 to n = inf) = 1 / (1 - r) if |r| < 1. The Attempt at a Solution [/B] I began by rescaling the sum, i.e. Σ (n+1)*(1/2)^(n+1) (from n = 0 to n =...
  13. R

    Finding Value of Sum of Geometric Series

    Homework Statement Let ## S_k , k = 1,2,3,…,100 ## denote the sum of the infinite geometric series whose first term is ## \frac{k-1}{k!} ## and the common ratio is ##\frac {1}{k}##. Then value of ##\frac {100^2}{100!} + \sum\limits_{k=1}^{100} | (k^2 - 3k + 1)S_k | ## is Homework Equations...
  14. K

    The geometric shape of parametric equations

    Hello everyone, I have another question mark buzzing inside my head. After the elimination steps of a matrix, I'm having some problems about imagining in 3D. For example, x=t , y=2t, z=3t what it shows us? Or, x=t+2, y=t,,z=t ? Or another examples you can think of. ( Complicated ones of...
  15. N

    MHB Geometric Distributions Anyone?

    I'm struggling with question 10. I'm not sure how to account for the different time? I'm probably just overthinking it. I have attached a picture of the answer as well. Again, trying to figure out question #10
  16. L

    Simulate a reality by exploiting geometric solutions

    It will essentially be The Matrix. Except every one who plays will be God's and they get to build the world up from atomic structure. The main mechanism of action is people finding stable formations can publish them. You get to name an object and it becomes available as a clip board item to...
  17. L

    Binomial vs Geometric form for Taylor Series

    Homework Statement Sorry if this is a dumb question, but say you have 1/(1-x) This is the form of the geometric series, and is simply, sum of, from n = 0 to infiniti, X^n. I am also trying to think in terms of Binomial Series (i.e. 1 + px + p(p-1)x/2!...p(p-1)(p-2)(p-(n-1) / n!). 1/(1-x) is...
  18. H

    Complex numbers: Find the Geometric image

    Homework Statement Find the Geometric image of; 1. ## | z - 2 | - | z + 2| < 2; ## 2. ## 0 < Re(iz) < 1 ## Homework EquationsThe Attempt at a Solution In both cases i really am struggling to begin these questions, complex numbers are not my best field. There are problems before this one...
  19. Imager

    Geometric Magnetic Pole vs Magnetic North Pole

    The article below is an excerpt from Discover magazine. What I don’t understand is the difference between the geometric magnetic and the magnetic north poles. From the article the North and South magnetic poles dips are not opposite of each other, so how is the geomagnetic pole calculated? Is...
  20. RJLiberator

    Can Variable Coefficients Be Used in Geometric Series Sums?

    Homework Statement I am giving the sum: k=1 to infinity Σ(n(-1)^n)/(2^(n+1)Homework Equations first term/(1-r) = sum for a geometric series The Attempt at a Solution [/B] With some manipulation of the denominator 2^(n+1) = 2*2^n I get the common ratio to be (-1/2)^n while the coefficient is...
  21. C

    Explain geometric constraints solver for CAD to a newbie

    I have seen the following said on a forum: This makes good sense to me. To me, as a programmer, this sounds like he is asking for a better API to look up the coordinate at runtime; perhaps that would force the engine to evaluate expressions in a particular order, but then that's what functional...
  22. I

    Geometric series vs. future value computation based on geometric series

    Homework Statement Hello! Revising geometric series, I have understood that I have the following issue - I have read again about these series and, please, take a look at what I have gotten as a result (picture attached). If I calculate a future cash flow, that is I take, for example, a = 0,5...
  23. throneoo

    Geometric distribution Problem

    Homework Statement a man draws balls from an infinitely large box containing either white and black balls , assume statistical independence. the man draws 1 ball each time and stops once he has at least 1 ball of each color . if the probability of drawing a white ball is p , and and q=1-p is...
  24. T

    Understanding an expansion into a geometric series

    Hi all, I am reading through Riley, Hobson, and Bence's Mathematical Methods for Phyisics and Engineering, and on page 854 of my edition they describe (I am replacing variables for ease of typing) "expanding 1/(a-z) in (z-z0)/(a-z0) as a geometric series 1/(a-z0)*Sum[((z-z0)/(a-z0))^n] for n...
  25. P

    What is the inverse of infinity in geometry?

    I seem to recall reading a geometry method that showed zero to be the inverse of infinity. Can you give me a reference for that?
  26. L

    Geometric Optics - Magnification

    Homework Statement A concave mirror forms an image on a screen twice as large as an object. Both object and mirror are then moved such that the new image is 3x the size of the object. If the screen is moved 75cm, how far did the object move? Homework Equations m = image distance / object...
  27. A

    What is the Proof for the Sum to Infinity of an Infinite Geometric Progression?

    Homework Statement An infinite geometric progression is such that the sum of all the terms after the nth is equal to twice the nth term. Show that the sum to infinity of the whole progression is three times the first term. Homework Equations [/B] S_{n} = \frac{a(1-r^n)}{1-r}\\ S_{\infty} =...
  28. Dethrone

    MHB How Is $\delta = \sqrt{9+\epsilon}-3$ the Largest Choice in a Limit Proof?

    Verify, by a geometric argument, that the largest possible choice of $\delta$ for showing that $\lim_{{x}\to{3}}x^2=9$ is $\delta = \sqrt{9+\epsilon}-3$ I have no clue, hints?
  29. JonnyMaddox

    How can I calculate a rotation using geometric algebra?

    Hi, I want to calculate a rotation of a vector GA style with this formula e^{-B \frac{\pi}{2}}(2e_{1}+3e_{2}+e_{3})e^{B\frac{\pi}{2}}. Now since no book/pdf on GA exists where a calculation is explicitly done with numbers, I wounder how to calculate this. Should I substitude e^{-B...
  30. C

    Steps to differentiate a geometric sum

    Can someone guide me with the steps to differentiate a geometric sum, x? ^{n}_{i=0}\sumx^{i}=\frac{1-x^{n+i}}{1-x} If I'm not wrong, the summation means: = x^0 + x^1 + x^2 + x^3 + ... + n^i Problem is: I have basic knowledge on differentiating a normal numbers but how do I apply...
  31. JonnyMaddox

    Is Associativity Key in Simplifying Multivector Products in Geometric Algebra?

    Hi, I just want to see if I understood this. Since the geometric product is associative and so on we can write for two multivectors A and B given by A= \alpha_{0}+\alpha_{1}e_{1}+\alpha_{2}e_{2}+\alpha_{3}e_{1}\wedge e_{2} B= \beta_{0}+ \beta_{1}e_{1}+\beta_{2}e_{2}+\beta_{3}e_{1}\wedge e_{2}...
  32. X

    Geometric Ratio Pipe Problem: How to Find D2/D1?

    Homework Statement Find D2/D1. See attachment. Homework Equations The Attempt at a Solution Ans: D2/D1 = 0.68 I can't figure this one out. Any special trigonometric identities that might help here? Thanks.
  33. JonnyMaddox

    Geometric Product: Definition and Calculation

    Hey JO, I'm reading a book on geometric algebra and in the beginning (there was light, jk) a simple calculation is shown: Geometric product is defined as: ab = a \cdot b + a \wedge b or ba = a \cdot b - a\wedge b Now (a\wedge b)(a \wedge b)=(ab-a \cdot b)(a\cdot b - ba) =-ab^{2}a-(a...
  34. S

    MHB How to Find the Sum of a Geometric Series with Variables?

    Hey, Sorry if I am in the wrong part of the forums not sure where this question goes. I am having trouble with a geometric series that has letters involved. I understand the forumla for finding the sum of first n elements with just numbers. However the series i have is .. a1 = -5, a2 = -5x, a3...
  35. S

    What are some methods for calculating point positions in a geometric chassis?

    Hi, I'm trying to calculate the position of the points from a struture like the picture attached. This is a mechanic structure that consist on 3 triangles (blue, orange and green), i know all the triangles sides lengths and also their angles (and some more colored in green). My objective is...
  36. S

    Variance of Geometric Brownian motion?

    I am trying to derive the Probability distribution of Geometric Brownian motion, and I don't know how to find the variance. start with geometric brownian motion dX=\mu X dt + \sigma X dB I use ito's lemma working towards the solution, and I get this. \ln X = (\mu - \frac{\sigma...
  37. X

    Good Books or Free Resources for Geometric Graph Theory

    I've been doing some light reading on Geometric Graph Theory, and it seem interesting to me. However, at the moment I've only managed to find a few Wikipedia articles and one .PDF of lecture notes. I'm looking for something which is more complete, such as a book or a website for example...
  38. A

    Yarman Geometric Norm or Impedence Normalization

    I'm working with some old software to optimize antenna networks, and I've come across some stuff that I don't understand. For the software to run, all impedence values entered must be normalized to 1 ohm. What does it mean to normalize an impedence and how do I do it? Also, the manual for the...
  39. F

    Simple geometric series question

    Take the case for the mean: \bar{x} = \frac{1}{N} \Big( \sum_{i=1}^Ni \Big) If we simply use the formula for the sum of a geometric series, we get \bar{x} = \frac{N}{2} (2a + (N - 1)d) where a and d both equal 1, so we simply get the result \bar{x} = \frac{1}{2} (N + 1)...
  40. J

    Geometric interpretation for d²f/dxdy

    If the following integral: $$\\ \iint\limits_{a\;c}^{b\;d} f(x,y) dxdy$$ represents: So which is the geometric interpretation for ##f_{xy}(x_0, y_0)## ?
  41. A

    Geometric tolerances Concentricity

    Hello, I need help. I do not know what to give values of the geometrical tolerance for Concentricity. They are two parts connected by a pin. Both parts are moving relative to each. Thanks for any advice.
  42. T

    Geometric, Exponential and Poisson Distributions - How did they arise?

    I'm going through the Degroot book on probability and statistics for the Nth time and I always have trouble 'getting it'. I guess I would feel much better if I understood how the various distribution arose to begin with rather than being presented with them in all there dryness without context...
  43. G

    Solving Series: Calculate ##\sum\frac{4^{n+1}}{5^n}##

    Homework Statement Calculate ##\sum\frac{4^{n+1}}{5^n}## (where n begins at 0 and approaches infinity). Homework Equations The Attempt at a Solution I could easily solve this if the numerator were just ##4^n## instead of ##4^{n+1}##, because then it would be a geometric series with...
  44. M

    Understanding Special Relativity on a Geometric and Intuitive Level

    So here's the deal guys: I have a bachelor's in physics and have gotten an A in an undergraduate special relativity course but I do not feel that I fully understand the subject. I can do the problems in special relativity which require the various formulas involved in the subject and I even...
  45. M

    Understanding the Formula for the Sum of a Geometric Series

    For the question, shouldn't the sum be a(1/1-r) since we know lrl < 1 then that rn → 0 as n → ∞? I just don't quite understand why they wrote the sum is a(r/1-r). Is there a specific reason they did this? This is just a regular geometric series right? Is there any difference since the sum starts...
  46. S

    Geometric Mean Radius of Hollow Conductor

    Homework Statement GMR_{hollow cylinder}=Re^{-Kμ} where K=\frac{AR^4-R^2r^2+Br^4+r^4ln(R/r)}{(R^2-r^2)^2}, where R is the outer radius and r is the inner radius, and mu is the relative permeability. We are to determine the numerical values of A and B. I am stumped on how to begin attempting...
  47. J

    Is There a Function That Always Results in a Positive Sign?

    "Geometric absolute" If exist a function called absolute that ensures that the result have always the posite sign, so, exist some function that ensures that the sign of the result is always ×? ##f(x) = x## ##f(\frac{1}{x}) = x##
  48. B

    Geometric Derivation of the Complex D-Bar Operator

    This picture from https://www.amazon.com/dp/0198534469/?tag=pfamazon01-20 is all you need to derive the Cauchy-Riemann equations, i.e. from the picture we see i \frac{\partial f}{\partial x} = \frac{\partial f}{\partial y} should hold so we have i \frac{\partial f}{\partial x} = i...
  49. kq6up

    How do you use vectors to prove the theorem about parallelogram diagonals?

    Homework Statement This problem is from Mary Boas' "Mathematical Methods in the Physical Sciences" 3rd Ed. Capter 3 Section 4 Problem 3 Use vectors to prove the the following theorems from geometry: 3. The diagonals of a parallelogram bisect each other. Homework Equations Just...
  50. I

    How Does Hypergeometric Distribution Calculate Equal Feathered Arrows Remaining?

    Homework Statement (a) At the start of the competition, Shirley has 20 arrows in her quiver (a quiver is a container which holds arrows). 13 of Shirley’s arrows have red feathers, and 7 have green feathers. Arrows are not replaced when they are shot at the target. (i) At the end of the...
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