# Recent questions tagged set

29
views
Define an open set in a metric space.
116
views
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 ${\{... 77 views 1 answers Let$\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$and th... 93 views 1 answers if$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 - ...
143
views
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...
100
views
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 $\delta 0$, $\Vert x_i - x^o \Vert_2 \delta$...
72
views
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...
66
views
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...
65
views
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 \$...
99
views
Solution $$e^{i \pi}+1=0$$
94
views
Explain $$e^{i \pi}+1=0$$
168
views
Can I perform multiple hypothesis tests on the same data set?
118
views
if $$2^{m+n}=32$$ and $$3^{3 m-2 n}=243$$ find $$m: n=$$ ?
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 $$a=1$$.
86
views
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
99
views
Normalize the set of functions $$\phi_{n}(\theta)=e^{i n \theta}$$, $$0 \leq \theta \leq 2 \pi$$. To do so, you need to multiply the functions by a so-called normaliz...
Prove that $$n^2=(n+1)(n-1)+1$$
(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 ... 517 views 1 answers Prove that $$x^2+2 x y+2 y^2$$ cannot be negative for $$x, y \in \mathrm{R}$$. 387 views 1 answers What is a non-central t-distribution? 322 views 1 answers Why is pi important in mathematics? 548 views 2 answers Define an uncountable set 380 views 2 answers What is an unbounded set? 515 views 1 answers Which sets of numbers are part of the set of real numbers? 260 views 1 answers Can mathematics teaching be seen as a messy set of functions? 605 views 1 answers What structures does the set of real numbers carry? 315 views 1 answers What is set theory? 479 views 1 answers How are equivalence relations used in mathematics? 303 views 1 answers What is the cardinality of the set of real numbers? 538 views 1 answers What is the cardinality of the set of natural numbers? 527 views 1 answers 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 views 1 answers Prove that the polynomial $$x^5 - x^4 + x^3 - x^2 + x - 1$$ is reducible over the integers. 200 views 0 answers 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 views 1 answers Evaluate \(\sum_{k=0}^8 2 k=$$ 398 views 0 answers Prove that the function f(x) = \frac{1}{x} is uniformly continuous on the interval (1,\infty). 544 views 0 answers Prove that if (x_n) is a sequence of points in a metric space (X,d), then (x_n) has a convergent subsequence if and only if it is a Cauchy sequence. 331 views 1 answers How important are aesthetics in mathematics? 267 views 1 answers The model developed from sample data that has the form of $$\hat{y}=b_0+b_1 x$$ is known asa. regression equationb. correlation equationc. estimated regression equationd.... 579 views 0 answers Define a function $$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<... 501 views 0 answers 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... 342 views 0 answers 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 $$x$$ ? 294 views 0 answers 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 views 0 answers 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... 424 views 0 answers 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 views 0 answers 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 views 0 answers 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 views 0 answers Let $$m$$ and $$n$$ be positive integers such that $$m^4-n^4=3439$$. What is the value of $$m n$$ ? 229 views 0 answers 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| .$
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...