# Recent questions tagged prove

72
views
How do you identify which lines are parallel?
135
views
How can you prove that a diameter that is perpendicular to a chord bisects the chord and its arc?
84
views
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 $x \in X$...
65
views
Prove Bezout's Theorem for Plane Curves
102
views
Prove that every meromorphic function on $S^2$ is rational.
106
views
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...
82
views
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 $\alpha, \be... 77 views 1 answers Let$\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$and th... 169 views 1 answers Suppose$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$. 93 views 1 answers Let$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$, unless$...
135
views
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...
66
views
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.
58
views
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 $i,j = 1,...,n$...
75
views
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...
59
views
(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 ...
59
views
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...
65
views
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 ...
75
views
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...
72
views
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.
82
views
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...
91
views
Find (a) all horizontal and (b) all vertical asymptotes of the graph$y=\frac{|x| \sqrt{4 x^{4}+6 x^{2}+4}}{(2 x-1)^{2}(x+1)}$
126
views
(1) The solution of the inequality $$3(2-x) \geq 2(1-x)$$ for real $$x$$ is
99
views
Solution $$e^{i \pi}+1=0$$
94
views
Explain $$e^{i \pi}+1=0$$
145
views
Prove the chain rule assuming $$u^{\prime}$$ is continuous at $$v(x)$$ and $$v$$ is differentiable at $$x$$. Hint: use the mean value theorem.
122
views
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...
70
views
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...
129
views
Show that for any $$0<a<1$$ the sequence $$\left\{x^{i}\right\}$$ converges uniformly to 0 on $$[0, a]$$, but not for $$a=1$$.
106
views
Show that $$f(x)=1 / x$$ is continuous but not uniformly continuous on $$(0,1)$$.
81
views
Evaluate $$e^{i \pi}+1=0$$
95
views
Prove that $$|x+y| \geq|x|-|y|$$
113
views
Find the vertical and horizontal asymptotes of the graph of the function $$f(x)=\sqrt{x+1}-\sqrt{x}$$.
158
views
Find the vertical and horizontal asymptotes of the graph of the function $$f(x)=(2 x+3) / \sqrt{x^{2}-2 x-3}$$.
128
views
Find any vertical and horizontal asymptotes of the graph of the function $$f(x)=(4 x-5) /(3 x+2)$$.
155
views
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.
157
views
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.
206
views
$I_{1}(\alpha)=a_{1} \cos \omega t+b_{1} \sin \omega t$where $$b_{1}=0$$ because of the odd-wave symmetry, that is, $$f(x)=f(-x)$$. Also, no even harmonics are gene...
136
views
Which is the smallest?(a) -1,(b) -1/2,(c) 0,(d) 3.
276
views
How do I show that all terms in a pattern are even?
148
views
What is the sum of one and one?
597
views
Prove that $$n^2=(n+1)(n-1)+1$$
210
views
What is this? $$\Phi(\mathbf{r})=\Theta(\mathbf{r})-\Phi_{\mathrm{ref}}(\mathbf{r})$$
233
views
What does this maths equation say? $$e^{i \pi}+1=0$$
517
views
Prove that $$x^2+2 x y+2 y^2$$ cannot be negative for $$x, y \in \mathrm{R}$$.
322
views
Explain $$e^{i \pi}+1=0$$
358
views
(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...
780
views
Is $$\phi$$ satisfiable?