What is Unity: Definition and 126 Discussions

Unity is a cross-platform game engine developed by Unity Technologies, first announced and released in June 2005 at Apple Inc.'s Worldwide Developers Conference as a Mac OS X-exclusive game engine. The engine has since been gradually extended to support a variety of desktop, mobile, console and virtual reality platforms. It is particularly popular for iOS and Android mobile game development and used for games such as Pokémon Go, Monument Valley, Call of Duty: Mobile, Beat Saber and Cuphead. It is cited to be easy to use for beginner developers and is popular for Indie game development.The engine can be used to create three-dimensional (3D) and two-dimensional (2D) games, as well as interactive simulations and other experiences. The engine has been adopted by industries outside video gaming, such as film, automotive, architecture, engineering and construction.
Several major versions of Unity have been released since its launch. The latest stable version, 2021.1.0 , was released on 23 March 2021 (23 March 2021).

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  1. STEMucator

    Ideals and commutative rings with unity

    Homework Statement From contemporary abstract algebra : http://gyazo.com/08def13b62b0512a23505811bcc1e37e Homework Equations "A subring A of a ring R is called a (two-sided) ideal of R if for every r in R and every a in A both ra and ar are in A." So I know that since A and B are ideals of...
  2. Fantini

    MHB Is R/N a Ring with Unity?

    Good afternoon. Here is the problem: Show that if $R$ is a ring with unity and $N$ is an ideal of $R$ such that $N \neq R$, then $R/N$ is a ring with unity. My answer: Consider the homomorphism $\phi: R \to R/N$. Given $r \in R$ we have that $\phi(r) = r + N = \phi(1 \cdot r) = \phi(r \cdot 1)...
  3. C

    What is the Sum of nth Roots of Unity and How Can It Be Proven?

    i'm trying to prove the sum of nth roots of unity = 0, but I don't really know how to proceed: suppose z^n = 1 where z ε ℂ, suppose the roots of unity for z are 1, ω, ω^2, ω^3 ... ω^n the sum of these would be S = 1 + ω, ω^w, ω^3 +...+ ω^(n-1) + ω^n from here I had an idea to do some...
  4. M

    Complex number equation and roots of unity

    I have some math problems What is the solution to this equation : z dash(complex conjugate) = z^3 Z is complex number I try to multiply both sides by Z in the left i get Z dash Z => |Z| but i don't see the solution ---- P is primitive 9th root of unity. Calculate the sum 1 + 2P...
  5. S

    MHB Nth Roots of Unity Challenge Problem

    Challenge Problem $1,a_1,a_2,a_3, \cdots ,a_{n-1}$ are the $n^{\text{th}}$ roots of unity. Find the value of i) $(1-a_1)(1-a_2)(1-a_3) \cdots (1-a_{n-1})$ ii)$\displaystyle \frac{1}{2-a_1}+\frac{1}{2-a_2}+\frac{1}{2-a_3}+\cdots +\frac{1}{2-a_{n-1}}$
  6. U

    Proving the Group Properties of M, the Set of Nth Roots of Unity

    Hello, Please help in solving the four set of problems, i will be very happy explaining comment as really want to understand. The problem will spread to the extent of understanding preduduschey. 1 Problems: The set M, M = {e^(j*2*pi*k/n) , k= 0,1,2...n-1} denotes the set of the nth...
  7. I

    Gain bandwidth product of unity gain opamp

    i have attached a bode plot and the opamp dc second order lpf circuit. the opamp Tl082 has a gain bandwidth product of 4Mhz according to the datasheet. this circuit has a bandwidth of 414.25khz. can we determine the opamp' individual gain-bandwidth product from this plot ?? if yes , how do we...
  8. T

    Relation between open loop and unity feedback close loop response

    hi! can someone please help me with this problem...i have to match the open loop respone to the close loop unity feedback system of a control system. what is the relation between the two? thanx in advance
  9. T

    Find 5th roots of unity solving x^5 -1=0 and use the result for sin18 and cos18

    Homework Statement Find 5th roots of unity solving algebraically x^5-1=0. Using the result, find sin18 and cos18The Attempt at a Solution x^5 = 1\\ x = \sqrt[5]{1} since we have 5 roots: x_k, k = 0,1,2,3,4 \\ \\ x_k = e^{i\frac{2k\pi}{n}}, n=5 \\ x_0 = e^{i0} = 1\\ x_1 =...
  10. O

    MHB Showing that Q(sqrt(p)) is in Q adjoined the pth root of unity

    i am having trouble showing that \mathbb{Q}(\sqrt{p*}) \subset \mathbb{Q}(\zeta_{p}) where p* = (-1)^{\frac{p-1}{2}}p . in other words, if p = 1 (mod 4) then p* = p and if p = 3 (mod 4) then p* = -p. i encountered this in the context of galois theory and i have no idea how to start. it seems...
  11. denjay

    What is Unity in Physics?

    I came across this term in Elements of Modern X-ray Physics by Nielsen. I'm assuming this term isn't specific to the book (because that would be ridiculous). I've always taken math courses geared toward Physics/Engineering so some math terms were never used so this is probably one of them...
  12. N

    Roots of unity form a cyclic group

    In a lot of places, I can read that the roots of unity form a cyclic group, however I can find no proofs. Is the reasoning as follows: Let's work in a field of characteristic zero (I think that's necessary). Let's look at the nth roots of unity, i.e. the solutions of x^n - 1. There are n...
  13. C

    Ambiguity about roots of unity in discrete Fourier transform

    Hi everyone, I have a question on the discrete Fourier transform. I already know its a change of basis operator on C^N between the usual orthonormal basis and the "Fourier" basis, which are vectors consisting of powers of the N roots of unity. But if i recall correctly from complex...
  14. C

    Plotting the roots of unity on the complex plane

    Homework Statement Find the 6th complex roots of √3 + i. Homework Equations z^6=2(cos(π/6)+isin(π/6)) r^6=2, r=2^1/6 6θ=π/6+2kπ, θ=π/36+kπ/3 The Attempt at a Solution When k=0, z = 2^1/6(cos(π/36)+isin(π/36)), When k=1, z = 2^1/6(cos(13π/36)+isin(13π/36)), When k=2, z =...
  15. L

    Galois Groups of Extensions by Roots of Unity

    Consider field extensions of the form Q(u) where Q is the field of rational numbers and u=e^{\frac{2\pi i}{n}}, the principal nth root of unity. For what values of n is the Galois group of Q(u) over Q cyclic? It seems to at least hold when n is prime or twice an odd prime, but what else...
  16. Pattonias

    Ubuntu Unity has forced me to find a new distro.

    I have become so frustrated with the Unity GUI that I my never use Ubuntu again. Not long ago, Ubuntu was the go to distro for user friendly Linux. Now it is so difficult to do even the most basic tasks and overloaded with so much useless software that I don't think I'll ever use it again...
  17. T

    If z is one of the roots of unity with index n, find the sum

    Homework Statement Given the fact that z is one of the n-th roots of unity, find the sum below: 1 + 2z + 3z2 + ... + nzn-1Homework Equations (1-x)(1+x+...+xn-1) = 1 - xn The Attempt at a Solution honestly I don't know how to do this. any help is appreciated
  18. K

    Unity means just plain regular 1 ?

    "unity" means just plain regular "1"? Im not sure whether this is a physics or math question, but in many physics problems, instead of saying "1", the problem will say "unity", like "the sine of theta is unity" or the "index of refraction is unity". I am assuming "unity" means just plain...
  19. J

    Solve Un Subseteq U2n | Roots of Unity Proof

    Hey everyone! I would really appreciate some help with this problem. I have been racking my brain for hours now, and nothing seems to work/convince me. Homework Statement Show that Un \subseteq U2n for every positive integer, n. Homework Equations [1] Un = {z ε ℂ, zn = 1} [2] Un =...
  20. D

    Help with proof (Roots of unity question)

    1. The problem statement, all variables and given/kno(It n data I need to prove that \prod_{k=1}^{n-1}2\sin\tfrac{k\pi}{n}=n I just don't know where to start and what type of proof to do. Is there any help to get? :D Homework Equations I know this has nothing to do with an equation...
  21. S

    Can Polynomials be Factored if Numerical Approximations to Roots are Used?

    I need to factor x5 - 1.I know (x-1) is a factor and have gotten: (x-1)(x4+x3+x2+x +1) I'm not sure where to go from here. Thanks in Advance.
  22. A

    Proving N-th Root of Unity: e^{\frac{2k\pi i}{n}}

    Homework Statement Prove that z=e^{\frac{2k\pi i}{n}},n\in\mathbb{N},k\in\mathbb{Z}, 0\leq k\leq n-1 is an n-th root of unity. The Attempt at a Solution So I know I have to come to the conclusion that z^{n}=1. I'm thinking of using the property e^{i\theta}=cos\theta+isin\theta, but...
  23. M

    Unity Power Factor: Impact on Motor Performance

    hi, iam wondering about the unity power factor .. for example if we have a motor that will cause the power factor to be lagging by 0.3 what will happen if we add capacitor to make the power factor 1 (unity) now we will not have any reactive power and we reduced the current to the minimum...
  24. S

    Complex Numbers - Complex Roots of Unity

    Need help with this please: Homework Statement (1 + cosθ + isinθ) / (1 - cosθ - isinθ) = icotθ/2 The first step in the solutions shows: (2cos^2θ/2 + i2sinθ/2cosθ/2) / (2sin^2θ/2 - i2sinθ/2cosθ/2) Homework Equations I can't get there. The Attempt at a Solution I tried multiplying by: (1 -...
  25. K

    Roots of Unity - is this correct?

    I'm going over some things I didn't do too well on in my latest Algebra test. One question was: List all of the roots of x^{8}\:-1\:=0, and write them in the form a+bi. So I knew I had to list all the 8th roots of unity. In other places in the test they used the notation e^{iθ} and this was a...
  26. V

    What is minimum value of |a+bw+cw^2|? whrere w is cube root of unity?

    a,b,c are integers not all equal and w is the cube root of unity then minimum value of |a+bw+cw2|(w is not equals one). My answer |a+bw+cw2|<=|a|+|bw|+|cw2| |a|+|bw|+|cw2|=a+b+c. so at lest one value of |a+bw+cw2| will smaller than the minimum value of a+b+c. for integers this minimum...
  27. W

    Principal ideals of rings without unity

    Both my book and lecturer have in the definition a ring omitted the requirement of a unity. I was reading in my book about ideals, more specifically principal ideals. I stumbled over a formula that differed by whether or not the ring had a unity. As an example I state the two for principal left...
  28. Z

    Show the roots of unity add up to zero.

    Homework Statement Prove that \Sigma^{n}_{k=1} wk = 0 and there has to be at least two phasors/exponentials Homework Equations complex analysis The Attempt at a Solution I tried writing out the sigma on the first line. Then I tried writing the same thing with n+1 on the...
  29. lisab

    Happy Eid to PFers: Wishing You Joy and Unity!

    Happy Eid to PFers who celebrate it! (I'm not sure if it's http://www.npr.org/blogs/thetwo-way/2011/08/30/140056443/when-is-eid-muslims-cant-seem-to-agree", there seems to be a bit of confusion :smile:)
  30. D

    Proving Primitive Root of Unity: z = cos(2pi/n) + isin(2pi/n)

    Homework Statement show that cos(2pi/n) + isin(2pi/n) is a primitive root of unity Homework Equations The Attempt at a Solution if i know z = cos(2pi/n) + isin(2pi/n) is an nth root and I'm trying to prove that z is a primitive nth root. is it correct to assume that z^k is not...
  31. D

    What is the proof for cos(2pi/n) + isin(2pi/n) being a primitive root of unity?

    Homework Statement I must show that cos(2pi/n) + isin(2pi/n) is a primitive root of unity Homework Equations a primitive root of unity is an nth root of unity that does not equal 1 when raised to the kth power for k less than n and great than or equal to 1 The Attempt at a Solution...
  32. J

    Defining Mathcad Variables to be equal to Unity

    While attempting to solve quadratic equation application in MATHCAD (pertaining to the Markowitz Portfolio theory), I am now stumped at how to perfom what would seem to be a trivial operation, namely: setting a sum of unknown variables (i.e., a constraint) equal to '1.' For example, how can...
  33. D

    Proving Q/Z isomorphic to U∗: Roots of Unity in C

    Show that Q/Z is isomorphic to the multiplicative group U∗ consisting of all roots of unity in C. (That is, U∗ = {z ∈ C|zn= 1 for some n ∈ Z+}.) I don't really understand how to prove this isomorphism
  34. W

    How to construct a map from S^2 to RP^2 with covering time being unity?

    it is easy to construct a map from S^2 to S^2, with covering time being unity but how to do the similar task on the projected manifold RP^2=S^2/Z_2? i tried to use the stereographical trick the points on the lower half semisphere are projected onto the plane the problem is that the...
  35. E

    A commutative ring with unity and Ideals

    Let R be a commutative ring with unity. I and J are ideals of R. Show that If I + J = R, then I∩J=IJ. I know that IJ⊆(I∩J). But I can't do inverse.
  36. C

    Solving a complex equation with roots of unity

    Homework Statement z is a complex number. Find all the solutions of (z+1)^5 = z^5 The Attempt at a Solution Of course one could expand (z+1)^5, but I remeber our professor solving this with roots of unity. Can anyone help?
  37. O

    Roots of unity of matrices

    hi. i have recently become very interested in the idea of the nth roots of unity. i have discovered how to calculate them (using eigenvalues), and i find it very fascinating that there are not n many nth roots of unity(unlike scalars). aparently in the case where the matrix is 2x2, there are...
  38. G

    Prove Nth Roots of Unity: \omega, \overline{\omega}, \omega^{r}

    Homework Statement Show that, if \omega is an nth root of unity, then so are \overline{\omega} and \omega^{r} for every integer r. Homework Equations \omega=r^{1/n}e^{i((\theta+2\pi)/n)} The Attempt at a Solution I got the first part and for \omega^{r} I have it equals...
  39. H

    Imaginary parts of roots of unity

    Hi all, What happens when we take the product of the imaginary parts of all the n-roots of unity (excluding 1)? I read somewhere that we get n/(2^(n-1)). How can we prove this? Thanks!
  40. T

    Trigonometric derivatives and roots of unity

    sin x. d(sin x)/dx = cos x. d(cos x)/dx = -sin x. d(-sin x)/dx = - cos x. d(-cos x)/dx = sin x. i. i^2 = -1. i^3 = -i i^4 = 1 i^5 = i. I know there is a relationship between trig, the complex numbers, and exponential functions. Is there a relationship between the pattern shown here?
  41. E

    Is there a new solution for Cardano's formula?

    Just asking: is it likely that there is any connection between roots as \sqrt[3]{1} and \sqrt[7]{1}?
  42. M

    Root Locus of Unity Feedback Transfer Function: Find K Range for Stability

    I want a MATLAB code to draw the root locus for a characteristic equation of a transfer function of unity feedback, also what is the range of K that keeps the system stable , here is the characteristic equation: s^5+600s^4+50000s^3+ks^2+24ks+80k=0 please help, thanks in previous
  43. K

    Abstract Algebra - roots of unity

    Homework Statement I want to find out if the sixth root of unity is a subgroup of the complex numbers with multiplication. Homework Equations The Attempt at a Solution I know it's true but my problem is getting there. I know the sixth root of unity must be closed under the...
  44. G

    Engineering Unity gain in lowpass STC circuit

    One of the question asks to find the frequency at which the gain becomes 0dB for a low pass Single Time Constant circuit. In the solution manual after finding the 3dB frequency = 10^6, it states that since the gain falls off at a rate of -20dB/dec (see attatchment for graph) starting at...
  45. M

    Can 1/(1 + x) be simplified to 1-x for a very small value of x?

    If you have a fraction, for example, \frac{1}{{1.0091532\times10^{-12}} + 1} Is there a simple way to convert it to a more easily calculated form, specifically, 1-x (where x is a very small number)
  46. O

    Is e^ix multivalued, roots of unity, etc

    questions: why is the sum of all the roots of unity equal to zero? z^(1/n)=z1,z2,...zn z1+z2+...+zn=0 It's obviously true when there's an even number of roots, (because each root has a partner that is pi radians away and therefore the negative of the other root). but i can't figure out...
  47. X

    Partitions of unity: support of a function

    in my readings, spivak or elsewhere, I've come across this several times but i don't have the formal training (maturity) to know how to use it. intuitively: by the atlas maps on the manifold, we can chop up a manifold into patchs. for each patch, by smoothness or something, there is a smooth...
  48. B

    C10 Group, 10th roots unity with complex number multiplication

    This is for 10th root unity with complex number multiplication. I am working on closure. I have multiplied 2 elements of my set and I have so far that cos[(n+k)360/10] + isin[(n+k)360/10]. Thus I know that if n+k<=9 then there is an element in the set. Now I need to show for if n+k>9 and if...
  49. baywax

    Consilience: The Unity of Knowledge

    I was recently made aware of the book, "Consilience: The Unity of Knowledge". The author's name is E.O. Wilson, a well known and established biologist. http://en.wikipedia.org/wiki/Consilience:_The_Unity_of_Knowledge I saw the opportunity to bring the idea of consilience to this forum...
  50. M

    Primitive 5th root of unity extension

    Hi, Let E = Q(a), where a is a primitive fifth root of unity. Find a basis for E as a vector space over Q, and express a/a-3 in terms of this basis. I can find a basis for E: {1, a, a^2, a^3}, but am not sure how to express a/a-3 in terms of this basis. Any help much appreciated.
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