# All Questions

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### Linear transformation (Ax) and the solution set

given that $$A \in \mathbb{R}^{mxn} , S \subseteq \mathbb{R}^n, b \in \mathbb{R}^m$$ Where all elements (x) in S satisfy the inequality $$Ax \le b$$ Then S must be a convex set I was tasked to show ...
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### Finding $\sum_{i = 1}^{n} \frac{n} { \gcd(i, n)}$

I am trying to solve this problem, the most important part of this problem is to find ?$$\sum_{i = 1}^{n} \frac{n}{\gcd(i, n)}$$ Where $n$ could be $10^{12}$
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### Branch Wr of Lambert's fct, with r<0

Please help me solve $x$ exp($x$) + r $x$ = a, for a real $x$, with -1 << r < 0 and a > 4. That equation has been solved (i) for r > 0, by the way, by means of branch Wr of Lambert's ...
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### conditional probability. $P(E_1E_2|E_3) = P(E_1|E_2)P(E_2)$?

$$P(E_1E_2|E_3) = P(E_1|E_2)P(E_2)$$ I don't understand why the probability of $E_1E_2$ given that $E_3$ has happened is $P(E_1|E_2)P(E_2)$. Could someone explain this?
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### Generate Uniformly Random Points on a Transformed Sphere

I have a sphere transformed by an affine transformation (represented as a 4x4 matrix). How should I get uniformly distributed points on the transformed sphere's surface? Note that the obvious ...
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### On the existence of the essential supremum

Let $(X, \mathcal{M}, \mu)$ be a $\sigma$-finite measure space. If $f$ is a real-valued, measurable, and a.e. finite function, does it have a finite essential supremum? My intuition says not ...
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### Functions (fourier transformation)

Can someone tell me how to determine whether this function will be odd or even?. Thanks
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### Torsion and curvature of a curve

A regular curve $\textbf{$\gamma$}$ in $\mathbb{R}^3$ with curvature $> 0$ is called a generalized helix if its tangent vector makes a fixed angle $\theta$ with a fixed unit vector $\textbf{a}$. ...
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### Finding the derivative for a double polynomial function?

In my problem, I am attempting to find $f'(x)$ when $f(x)=(5x^2-2x+8)(4x^2+7x-3)$. For my work I have: \begin{align} & \frac{d}{dx} (uv) = u\frac {dv}{dx} + v\frac {du}{dx} \\[8pt] = {} & ...
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### Derivative of a Linear Transformation

I am having trouble grasping the concept of derivatives in relation to linear transformations. If I have a function f(x)=Tx which is the linear transformations given by the matrix T. Then what is the ...
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### Finding the Normal & Tangent Vector and write an Equation.

Find a normal vector and a tangent vector at the point P. Write an equation for the tangent line and an equation for the normal line. $x^2 + xy + y^2 = 3; P(−1,−1).$ So what I did first was find the ...
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### Finding the equation of the tangent (in slope-intercept form) at a particular point?

One of the practice problems in my Calculus book is as follows: The graph of y=$8/(x^2-4)$ is called the Witch of Agnesi. (a) Find y' d/dx (u/v) = (v du/dx - u dv/dx)/($v^2$) ...
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### The Developing Map

I am looking for resources to learn about the developing map, but the only source I know is Thurston's book. Could anyone direct me to other sources? Lecture notes, articles, books, etc? Thanks in ...
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### Prove it is not a closed Curve

I wanna prove that $(cos(t^3+t),sin(t^3+t))=γ(t)$ which is a reparametrization of a circle .Is not a closed curve like the circle. WHat i did is take this problem to the Complex plane so ...
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### Trace the curve $r=33(1 -\sin \theta)$ [on hold]

Please can someone give me exact step by step solution of this question along with the graph?
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Let $K$ be a field and $G$ be a finite group, and denote by $K[G]$ its group ring, then Maschke's theorem is: Suppose $\mbox{char}(K)$ does not divide $|G|$. Let $V$ be a $K[G]$-module. If $W ... 0answers 16 views ### Question about creating$2\times 2$covariance matrix with call option? I'm completely stuck on how to do this problem. How can you go about calculating the variance of$Y$and the covariance between$X$and$Y$? I'm not sure how to use the information given to solve this ... 0answers 18 views ### Calabi's theorem I've just heard about Calabi's theorem (Minimal immersions of surfaces in Euclidean spheres). Theorem Let$\phi : \mathbb{C}\mathbb{P}^1 \longrightarrow (S^n,g_{S^n})$be a full harmonic map. ... 4answers 50 views ### If$w$be nth root of unity , then$1+2w+3w^2+\dots+nw^{n-1}$is equal to? I tried it by letting expression$1+2w+3w^2+\dots+nw^{n-1}= x$and then multiplying$w$both sides . I subtracted equation 1 from 2 but it does not seems to help me because i have just started ... 1answer 17 views ### Name for a set of coset representatives which contains a transversal A transversal for a quotient group$G/N$is a subset of$G$which contains precisely one element for each coset of$G/N$. This can be defined analogously for any collection of sets and not just cosets ... 0answers 30 views ### Constructing the Lebesgue Measure I am studying for an exam and I am trying to work out a concise construction of the Lebesgue measure. Here is what I have: If$\mathcal{A}$is an algebra of open intervals in$\mathbb{R}$with a ... 0answers 11 views ### Higgs to fermion amplitude To compute the amplitude for the process bottom-antibottom to Higgs in quantum field theory. This amplitude is a scalar, i.e. Lorentz invariant. So far I have the S -matrix which is S=Texp[−i∫d 4 ... 0answers 20 views ### Does the graph have an Euler's circuit? Each of the following describes a graph. In each case answer yes, no , or not necessary to this question. Does the graph have an Euler's circuit? Justify your answer. a) G is a connected graph with ... 0answers 21 views ### Does the Look and Say sequence have terms with odd number of digits? (also for other bases) I had just put up a question on Puzzling SE about the Look and Say sequence. I was just curious. In the base-2 sequence, some terms had an odd number of digits. Does the normal base-10 sequence have ... 3answers 20 views ### Finding all points (x,y) on a graph with tangent lines passing through a particular point? In this particular case, I am trying to Find all points (x,y) on the graph of$f(x)=x^2$with tangent lines passing through the point (3,8). Now then, I know the graph of$x^2$. What now? 0answers 5 views ### Nested Expectations and Bernoulli RV sum I have some questions regarding the use of law of total expectation. I'm not sure whether the following terms are correct in general: Given Random variables X, Y, Z, which of the following is true? ... 0answers 20 views ### Dirichlet eigenfunction cannot be extended to a continuous function on the closure I need to show that there exist a bounded domain$ \Omega \subset \mathbb{R}^2 $, and a Dirichlet eigenfunction$u$on$ \Omega$such that u cannot be extended to a continuous function on$ ...
Let $f(x)=0$ if $x=0$ and $f(x)=\frac{sinx}{x}$ otherwise I am trying to prove that this is uniformly continuous on Real numbers. I know I need to separate the intervals and show that ...