Recent questions tagged set

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Define an open set in a metric space.
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Given a sequence ${\{x_n\}_{n\in\mathbb{N}}}$ in a metric space X, prove the following statements:(a) If $d(x_n,x_{n+1}) < 2^{-n}$ for every $n \in \mathbb{N}$, then ${\{...
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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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if $A, B$ are disjoint subsets of the plane, $A$ is compact, $B$ is closed, then there exists a $\delta 0$ such that, for all $\alpha \in A$, $\beta \in B$, $| \alpha - ...
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Let $\{ x_i \}_{i=1}^n \subset \mathbb{R}^D$ be a discrete set on unique points. Recall that the DBSCAN algorithm depends on two parameters: $\epsilon$ and MinPts.(a) Des...
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Suppose $x_1,...,x_n \in \mathbb{R}^D$ are data points, and we introduce an outlier $x^o$ with the property that, for some $\delta 0$, $\Vert x_i - x^o \Vert_2 \delta$...
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Let $x_1,...,x_n \in \mathbb{R}^d$. Fix some positive integer $K$. Let $C_1,...,C_K$ be a partition of the data with centroids $\mu_1,...,\mu_K$. Let$$ F(C_1,...,C_k) \s...
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When dimension reducing data in $\mathbb{R}^D$ using PCA, the choice of embedding dimension is crucial. Many heuristics exist to estimate a good dimension. One is to choo...
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Recall that the variance of a set of numbers $x_1,...,x_n \in \mathbb{R}$ is defined as $\sigma^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \mu)^2$, where we define the mean as $...
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Solution \( e^{i \pi}+1=0 \)
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Explain \( e^{i \pi}+1=0 \)
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Can I perform multiple hypothesis tests on the same data set?
118
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if \( 2^{m+n}=32 \) and \( 3^{3 m-2 n}=243 \) find \( m: n= \) ?
70
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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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How do I test whether the function \(y=x^2-3x+12\) is continuous from the left and from the right and to identify any discontinuities
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Prove that \(n^2=(n+1)(n-1)+1\)
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(Lael and Nourouzi 2008) Let \((V, \mu, \nu)\) be an \(I F\)-normed space. Assume further that \(\mu(x, t)>0\) for all \(t>0\) implies \(x=0\). Define\[\|x\|_\alpha=\inf ...
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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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What is a non-central t-distribution?
322
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Why is pi important in mathematics?
548
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Define an uncountable set
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What is an unbounded set?
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Which sets of numbers are part of the set of real numbers?
260
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Can mathematics teaching be seen as a messy set of functions?
605
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What structures does the set of real numbers carry?
315
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What is set theory?
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How are equivalence relations used in mathematics?
303
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What is the cardinality of the set of real numbers?
538
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What is the cardinality of the set of natural numbers?
527
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In the figure below, DC is the diameter of a circle with centre O. CE is a tangent to the circle at \(C\). The diagonals of cyclic quadrilateral \(A B C D\) intersect at ...
467
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Prove that the polynomial \(x^5 - x^4 + x^3 - x^2 + x - 1\) is reducible over the integers.
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Prove the following:(a) \((\cos \theta+\sin \theta)^2=1+\sin 2 \theta\)(b) \(\frac{\cos 2 x}{\cos x-\sin x}=\cos x+\sin x\)(c) \(\cos ^4 \alpha-\sin ^4 \alpha=\cos 2 \alp...
217
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Evaluate \(\sum_{k=0}^8 2 k=\)
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Prove that the function $f(x) = \frac{1}{x}$ is uniformly continuous on the interval $(1,\infty)$.
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Prove that if $(x_n)$ is a sequence of points in a metric space $(X,d)$, then $(x_n)$ has a convergent subsequence if and only if it is a Cauchy sequence.
331
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How important are aesthetics in mathematics?
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The model developed from sample data that has the form of \(\hat{y}=b_0+b_1 x\) is known asa. regression equationb. correlation equationc. estimated regression equationd....
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Define a function \(f\) on the unit interval \(0 \leq x \leq 1\) by the rule\[f(x)= \begin{cases}1-3 x & \text { if } 0 \leq x<1 / 3 \\ 3 x-1 & \text { if } 1 / 3 \leq x<...
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Let \(A B C D E F\) be a regular hexagon. Let \(P\) be the intersection point of \(\overline{A C}\) and \(\overline{B D}\). Suppose that the area of triangle \(E F P\) is...
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If \(x\) is a real number such that \((x-3)(x-1)(x+1)(x+3)+16=116^2\), what is the largest possible value of \(x\) ?
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Let \(P\) be a right prism whose two bases are equilateral triangles with side length 2. The height of \(P\) is \(2 \sqrt{3}\). Let \(l\) be the line connecting the centr...
389
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Mr. Jones teaches algebra. He has a whiteboard with a pre-drawn coordinate grid that runs from \(-10\) to 10 in both the \(x\) and \(y\) coordinates. Consequently, when h...
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There are 300 points in space. Four planes \(A, B, C\), and \(D\) each have the property that they split the 300 points into two equal sets. (No plane contains one of the...
365
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Say that a sequence \(a_1, a_2, a_3, a_4, a_5, a_6, a_7, a_8\) is cool if- the sequence contains each of the integers 1 through 8 exactly once, and- every pair of consecu...
401
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Let \(H\) be a regular hexagon with area 360 . Three distinct vertices \(X, Y\), and \(Z\) are picked randomly, with all possible triples of distinct vertices equally lik...
326
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Let \(m\) and \(n\) be positive integers such that \(m^4-n^4=3439\). What is the value of \(m n\) ?
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Let \(a_n=n(2 n+1)\). Evaluate\[\left|\sum_{1 \leq j<k \leq 36} \sin \left(\frac{\pi}{6}\left(a_k-a_j\right)\right)\right| .\]
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Let \(O\) be the set of odd numbers between 0 and 100 . Let \(T\) be the set of subsets of \(O\) of size 25 . For any finite subset of integers \(S\), let \(P(S)\) be the...
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