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### If $\sin A+\sin B =a,\cos A+\cos B=b$, find $\cos(A+B),\cos(A-B),\sin(A+B)$

If $\sin A+\sin B =a,\cos A+\cos B=b$, find $\cos(A+B),\cos(A-B),\sin(A+B)$ Prove that $\tan A+\tan B= 8ab/((a^2+b^2)^2-4a^2)$
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I was working through a physics problem related to magnetic flux, but was confused at the math the solution uses. I understand up till the last line: $c=1.65-.12t\\ A=c^2/4\pi\\ \Phi_B=BA=(\frac{B}{... 1answer 92 views ### Why it is true for rapid decreasing function$g$that:$\sup_{x\in\mathbb{R}}|x|^{l\geq 0}|g^{(k\geq 0)}(x-y)|\leq A_{l,k}(1+|y|)^{l}$If$g$is of rapid decrease, that is$\displaystyle\sup_{x\in\mathbb{R}}|x|^{l\geq 0}|g^{(k\geq 0)}(x)|<\infty$, then we have: $$\displaystyle\sup_{x\in\mathbb{R}}|x|^{l\geq 0}|g^{(k\geq 0)}(x-y)|\... 1answer 99 views ### On the existence of a positive fundamental period A function f has period t if for all x in the domain it is true that f(x+t) = f(x). A function is called periodic if it has (at least one) period. Take any periodic function f -periodicity ... 2answers 162 views ### Factoring numbers of the form 11111111 Why 11111111 is divisible by 73? How can we get all the prime factors? It is clear that it is divisible by 11. Is there any formulae for 1111...11 (n times)? Give me some idea. Thanks in ... 1answer 83 views ### Find all conformal automorphisms Let G be a non-empty simply connected and bounded domain in \mathbb{C} and let a, b \in G with a \neq b. Find all conformal automorphisms of G such that a dn b are two fixed points. Moreover, ... 1answer 85 views ### Is there a proof of \sum_{n=0}^x {{(-1)^n(x-n+1)^x}\over{n!(x-n)!}} = 1 using induction? Can someone prove (or disprove) this equality?$$\sum_{n=0}^x {{(-1)^n(x-n+1)^x}\over{n!(x-n)!}} = 1$$where the value of x can vary. This is a pattern found in derivatives and stuff but I'm not ... 2answers 46 views ### a question about abstract algebra, a question related to permutation Given X=\{1,2,......n\}, let us call a permutation p of X an adjacency if it is a transposition of the form (i,i+1) for i\le n-1. Prove that (i,j) is a product of an odd number of ... 1answer 109 views ### Proof of why the partition function Z in probabilistic graphical models (PGM) is NP-complete I was wondering if someone knew why computing the partition function for probabilistic graphical models is NP-Hard? I would like to see a full blown rigorous proof, however, I am as happy to get a ... 1answer 146 views ### Corollaries of Green-Tao Theorem? there is already a good thread which discusses some corollaries of the Green-Tao Theorem, here: Constructing arithmetic progressions The question I was wondering about is of a similar flavor but isn'... 1answer 44 views ### How to show using Cauchy integral formula? Let f(z) be an entire function such that \;|f(z)| \le M|z|^n\; for any \;z ∈ \Bbb C\; , \;\text{and}\;M >0\; is a fixed number. Using Cauchy integral formula, show that for any$$\;k\ge n+1\;... 1answer 43 views ### Calculation of the Fourier transform of$x/(x^2+1)^2$using the properties of the transform I am trying to calculate the Fourier transform of $$f(x)=\frac{x}{(x^2+1)^2}$$ using the property of Fourier transform. So I am trying to use$$\widehat{g_1(x)g_2(x)}=\frac{1}{2\pi}\widehat g_1 *\... 0answers 26 views ### Suffix string starting at i S is the string of characters:TACGCGGT For string S and each of the positions i=1,2,\dots,9 write down the suffix string starting at position i. What is ... 1answer 92 views ### Matrices similar only to themselves Find all matrices similar only to themselves, i.e., PTP^{-1}=T for any invertible P. My attempt: PT = TP. Am I going about this correctly? If so, how do I find all matrices that are ... 3answers 77 views ### Let x be any non zero real number. Show that x^8-x^5-\frac{1}{x}+\frac{1}{x^4}\ge0 Let x be any non zero real number.Show that x^8-x^5-\frac{1}{x}+\frac{1}{x^4}\ge0 This question is from the Regional Olympiad Materials.This is trivial proof, but I am stuck designing the ... 2answers 80 views ### Proving a Combinatorial Theorem The Theorem My Problem I don't really understand how the RHS counts the number of final positions for a 1. I understand how summing all of these cases would be the same as counting all the ... 0answers 171 views ### Calculating centre of mass of a cylindrical wedge I'm sort of stuck with a problem. Don't know if I'm applying the theory in a wrong way. Here's the problem: Let S be the solid enclosed by the cylinder y^2+z^2=9 and the planes x=0, y=3x and ... 1answer 140 views ### Probability that no two teams in a tournament win the same number of games Six teams play a tournament in which every team plays every other team exactly once. No ties occur, and each team has a \dfrac{1}{2} probability of winning any game it plays. Find the probability ... 2answers 46 views ### Inner product on the k-tensor space This is homework so no answers please. The problem is "Given inner product vector space V, define an inner product on T^{k}(V) by declaring the standard basis \{e^{*}_{i_{1}}\otimes...\otimes e^{*... 0answers 130 views ### Splitting of short exact sequence of sheaves Let X be a smooth projective variety over a field, say k. Consider the short exact sequence of k-modules,$$0 \to A_1 \to A \to A_2 \to 0$$where A and A_2 are k-algebras. Since these can ... 1answer 40 views ### How do I perform taylor expansion of the following? Taylor expansion about (x,y) of f(x + a,\; y + k\; f(x + b,\; y + c)) I do not understand what happens to the second f inside. The inspiration for this question is Runge-Kutta methods. 1answer 232 views ### What's the minimal structure needed to define a notion of derivative? I know that, for example, to define a limit all you need is the notion of "closeness" generated by a topology; and to define an integral you need a measure function and a sigma-algebra on which it is ... 1answer 54 views ### On Prime and Maximal Ideals in a Commutative Ring with Unity Let R be a commutative ring with 1 \neq 0, I and P are ideals of R. If P is prime and I \cap P \neq 0, does it follows that either I \subseteq P or I is also a prime ideal ... 1answer 386 views ### Limit cycle of dynamical system x'=xy^2-x-y, y'=y^3+x-y Consider a planar ODE system z'=f(z) with z=(x,y),$$ f(x,y)=(xy^2-x-y,y^3+x-y). $$Using polar coordinates, one can get$$ r'=r(r^2\sin^2\theta-1),\quad \theta'=1. $$With Mathematica one can ... 1answer 91 views ### Method for solving \frac{dy}{dx} = 2xy+2 I tried using the integrating factor method, since the equation is first order linear: suppose R is the integrating factor. We have \log(R) = \displaystyle \int -2x dx = -x^2+c_0 so R = c_1 e^{... 1answer 69 views ### How to calculate the residue of the fourier transform? I have been struggling calculating the Fourier transform of f(x)=\frac{x}{(x^2+1)^2}. I tried to calculate f(t)=\int\frac{x}{(x^2+1)^2}e^{-ixt}\,dx directly by integration by parts, but it is not ... 2answers 133 views ### Can the Kahler differentials of a “good” local ring R be free of rank not equal to dim(R)? Let R be a local ring containing a field isomorphic to its residue field k. Assume R is a localization of a finitely-generated k-algebra. Can \Omega_{R/k} be free of rank r\neq\dim{R}? ... 1answer 63 views ### Determine whether each function is one-to-one, onto, or both g:\mathbb Z \times \mathbb Z where g is defined by g(x)=x-1 My guess is that this is onto and one-to-one. But is the correct interpretation of this problem that g is a function of an ... 1answer 499 views ### Find the inverse of a Trig Matrix I don't have a clue of what's going on. We haven't learn this in class so I need all the help possible. The more detailed of an explanation, the better. Thanks in advance. The only info I have is that ... 1answer 145 views ### Product measures and \sigma- finite measures Problem similar to folland chapter 2 problem 51. The actual problem in Folland mentions that X,Y are not necessarily \sigma-finite. Then how can I use Fubini-Tonelli theorem? 0answers 361 views ### understanding and visualizing the span of sets I've been researching for a while and trying to wrap my head around spanning of vector spaces completely (by visualizing them in R3) before moving on to Linear Independence, Basis' and anything else ... 0answers 35 views ### lines that only intersect a curve at 2 points I have some data points that define a curve and what i need to find is the slope of the lines definedby line 1 = p1&p2 line 2 = p1&p3 line 3 = p1&p4 . . . line29= p1&p30 line30= p2&... 1answer 82 views ### Is the following statement true ? If L is a decidable language and L' \subseteq \; L, then L' is also decidable ? Prove your answer is correct [closed] Is the following statement true ? If L is a decidable language and L' \subseteq \; L, then L' is also decidable ? Prove your answer is correct I can't figure out this question. Any tips ? 2answers 88 views ### Derivative of integral question Given f is function with continous derivatives, how do I obtain f(x) in terms of x from the equation below? Thanks in advance.$$ f(x)=\lim_{t\to 0} \frac{1}{2t} \int_{x-t}^{x+t} s f'(s) ds $$1answer 157 views ### How to FIND the limit of a sequence with epsilon definition? So I have this sequence in hand: x_n=\frac{\sqrt{n}}{n+1}, and I can intuitively see that its limit as n\to\infty is 0, and I can verify it with the \varepsilon definition of the limit. But I am ... 1answer 86 views ### The limit of the powers of an primitive non-negative matrix over its spectral radius Let A be a non-negative primitive matrix. Then$$\lim_{n\to\infty}\left[\frac{A}{\rho(A)}\right]^n=xy^T,$$where x, y are the Perron roots of A and A^T respectively, they satisfy x^Ty=1. ... 2answers 43 views ### Is there a simpler way to rewrite this binomial chain? Consider some binomial chain that looks like this:$$\binom{N}{k_1}\binom{N-k_1}{k_2}\binom{N-k_1-k_2}{k_3}\binom{N-k_1-k_2-k_3}{k_4} \cdots \binom{N-k_1-k_2-\cdots -k_{t-1}}{k_t}$$Where all ... 1answer 180 views ### The spectral radius of A and its transpose Let A be a non-negative irreducible n\times n matrix. Then the function$$f(t)=\rho(tA+(1-t)A^T)$$is increasing on$[0,1/2]$, and is decreasing on$[1/2,1]$. Here are the notations.$A$is non-... 1answer 101 views ### Best approximation for a closed set in a finite dimensional normed space First of all I'd like to mention that it is a part of my home work so I'd like if you won't give the answer itself, but try to guide me into it. I've been losing my mind for the last couple of hours ... 1answer 117 views ### Segment ordered density conjecture. I have a set$S\subset\mathbb {R}^2$with the following property (P)$\forall x,y\in S$,$\forall\mathscr{C}$a convex set that contains$x$in its interior,$bd\mathscr{C}\cap [x,y]\subset \overline{...
Define a sequence of r.v.'s {$X_n$}$_{n\ge 1}$ iteratively, such that $X_1\sim\text{Unif}(0,1]$ and $X_{n+1}\sim\text{Unif}(0,X_n]$. Could someone please explain why this is equivalent to: Let a ...