Recent questions tagged prove

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How do you identify which lines are parallel?
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How can you prove that a diameter that is perpendicular to a chord bisects the chord and its arc?
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Assume ${\{x_n\}_{n\in\mathbb{N}}}$ is a Cauchy sequence in a metric space X, and there exists a subsequence ${\{x_{n_k}\}_{k\in\mathbb{N}}}$ that converges to $x \in X$...
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Prove Bezout's Theorem for Plane Curves
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Prove that every meromorphic function on $S^2$ is rational.
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Let $L_1, L_2$ be lines in the plane. For which pairs of $L_1, L_2$ do there exists real functions, harmonic on the entire plane, 0 on $L_1 \cup L_2$, but not vanishing i...
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Let $\Omega$ be a region, $K$ a compact subset of $\Omega$, and fix some $z_0 \in \Omega$. Let $u$ be any positive harmonic function. Prove that there exists $\alpha, \be...
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Let $\Omega$ be a region, and $f_n \in \mathcal{H}(\Omega)$ for all $n$. Set $u_n = \Re(f_n)$, and suppose $u_n$ converges uniformly on compact subsets of $\Omega$ and th...
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Suppose $f$ is a complex function on a region $\Omega$, and both $f, f^2$ are harmonic on $\Omega$. Prove that either $f, \overline{f}$ must be holomorphic on $\Omega$.
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Let $u,v$ be real harmonic functions on a plane region $\Omega$. Under what conditions is $uv$ harmonic?Further, show that $u^2$ may not be harmonic on $\Omega$, unless $...
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Suppose $\Omega$ is a region, $f_n \in \mathcal{H}(\Omega)$ for $n \geq 1$. Suppose further that none of the $f_n$ has a zero in $\Omega$, and $f_n \to f$ uniformly on co...
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Suppose that $f$ is an entire function, and$$ | f(z) | \leq A + B|z|^k$$for all $z$, where $A, B, k$ are positive real numbers. Prove that $f$ must be polynomial.
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Let $L = D - W \in \mathbb{R}^{n \times n}$ be the graph Laplacian for data with an associated symmetric weight matrix $W$, and $w_{ij} \in [0,1]$ for all $i,j = 1,...,n$...
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Let $\{ x_i \}_{i=1}^n \subset \mathbb{R}^D$. Define $F: \mathbb{R}^D \to [0,\infty)$ via:$$ F(y) = \sum_{i=1}^n \Vert x_i - y \Vert_2^2$$Prove that $F$ attains a minimum...
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(a) Prove that $\langle x,y\rangle_M = x M y^T$ satisfies the properties of an inner product if $M$ is positive definite.(b) Show that $\langle x,y\rangle_M$ need not be ...
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Prove that the Euclidean dot product $\langle x, y \rangle = \sum_{i=1}^n x_i y_i, x, y \in \mathbb{R}^n$ is an inner product, where an inner product is a binary function...
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Let $\Sigma \in \mathbb{R}^{d \times d}$ be a symmetric matrix. Let $F: \mathbb{R}^d \to \mathbb{R}^d$ be the function $F(u) = u^T \Sigma u$ where $u$ is understood as a ...
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Suppose $(x_1, y_1), (x_2,y_2),...,(x_n,y_n) \in \mathbb{R}^2$ are sampled from a line $y = \alpha x$, for some $\alpha \in \mathbb{R}$. Prove that the empirical covarian...
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Define the $l^2$ norm of $x \in \mathbb{R}^n$ to be$$ \Vert x \Vert_2 = \sqrt{ \sum_{i=1}^n | x_i |^2}$$Prove that $\Vert \cdot \Vert_2$ is a norm.
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Recall that for a matrix $A = (A_{ij})_{i,j=1}^n \in \mathbb{R}^{n\times n}$, and a vector $x = (x_1,...,x_n)^T \in \mathbb{R}^{n \times 1}$, matrix-vector multiplication...
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Find (a) all horizontal and (b) all vertical asymptotes of the graph\[y=\frac{|x| \sqrt{4 x^{4}+6 x^{2}+4}}{(2 x-1)^{2}(x+1)}\]
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(1) The solution of the inequality \( 3(2-x) \geq 2(1-x) \) for real \( x \) is
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Solution \( e^{i \pi}+1=0 \)
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Explain \( e^{i \pi}+1=0 \)
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Prove the chain rule assuming \( u^{\prime} \) is continuous at \( v(x) \) and \( v \) is differentiable at \( x \). Hint: use the mean value theorem.
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Prove the mean value theorem by first reducing to the case \( u(a)=u(b)=0 \) and then using the fact that \( u(x) \) must take on a maximum or minimum value for some poin...
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Let \( \left\{f_{i}\right\} \) be a uniform Cauchy sequence consisting of uniformly continuous functions on a closed, bounded interval \( I \). Show that the limit is uni...
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Show that for any \( 0<a<1 \) the sequence \( \left\{x^{i}\right\} \) converges uniformly to 0 on \( [0, a] \), but not for \( a=1 \).
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Show that \( f(x)=1 / x \) is continuous but not uniformly continuous on \( (0,1) \).
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Evaluate \( e^{i \pi}+1=0 \)
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Prove that \( |x+y| \geq|x|-|y| \)
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Find the vertical and horizontal asymptotes of the graph of the function \( f(x)=\sqrt{x+1}-\sqrt{x} \).
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Find the vertical and horizontal asymptotes of the graph of the function \( f(x)=(2 x+3) / \sqrt{x^{2}-2 x-3} \).
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Find any vertical and horizontal asymptotes of the graph of the function \( f(x)=(4 x-5) /(3 x+2) \).
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Study the number pattern below:\(2 – 2 = 0\)\(2 – 1 = 1\)\(2 – 0 = 2\)What will the next line in the pattern be? Show ALL your calculations.
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At your local store, you get 3 stickers for every R60 spent. 2.4.1 If you spend R480, how many stickers would you have? Show ALL your calculations.
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\[I_{1}(\alpha)=a_{1} \cos \omega t+b_{1} \sin \omega t\]where \( b_{1}=0 \) because of the odd-wave symmetry, that is, \( f(x)=f(-x) \). Also, no even harmonics are gene...
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Which is the smallest?(a) -1,(b) -1/2,(c) 0,(d) 3.
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How do I show that all terms in a pattern are even?
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What is the sum of one and one?
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Prove that \(n^2=(n+1)(n-1)+1\)
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What is this? \( \Phi(\mathbf{r})=\Theta(\mathbf{r})-\Phi_{\mathrm{ref}}(\mathbf{r}) \)
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What does this maths equation say? \( e^{i \pi}+1=0 \)
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Prove that \(x^2+2 x y+2 y^2\) cannot be negative for \(x, y \in \mathrm{R}\).
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Explain \( e^{i \pi}+1=0 \)
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(a) Simplify the difference of binomial coefficients\[\left(\begin{array}{l}n \\3\end{array}\right)-\left(\begin{array}{c}2 n \\2\end{array}\right) \text {, where } n \ge...
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How do you calculate the Bessel function of the second kind? use latex
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How do you calculate the Bessel function of the first kind? Use latex
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Where is Roodepoort?
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Is \(\phi\) satisfiable?
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