## Evaluating limits with square roots February 11

Evaluate the limit step-by step:$\lim_{t \to 0}$ $(3−\sqrt{9 + 9t}−\sqrt{9−t} +\sqrt{9 + 8t−t2}) \over t2$

## Is an open map $f \mathbb{R} \to \mathbb{R}$ implies that $f$ is continuous February 11

Let $f : \mathbb{R} \to \mathbb{R}$ such that $f$ is an open map. Is it true that $f$ is continuous??I am unable to find a counterexample (I think it is false)Thanks for the help!!

If $f_1=O(g_1)$ and $f_2=O(g_2)$ then $$\frac{f_1}{f_2}=\frac{O(g_{1})}{O(g_{2})}=O(\frac{g_1}{g_2})$$ I want to show this relation is true or not.ThanksNow, you would need $\Omega(\cdot)$ for $f_2$. Intuitively: you need an upperbound (i.e., $O(\cdo ## Chromatic Number of Circulant Graph February 11 Consider the Circulant Graph$Ci_{2n}(1,n-1,n)$as described here:http://mathworld.wolfram.com/MusicalGraph.htmlAnother way to describe$Ci_{2n}$would be$2n$vertices with vertex set$V=\{a_0, \dots , a_{n-1}, b_0, \dots , b_{n-1}\}$and with edge se ## Resolution calculus converting into set of clauses February 11 Here is T:T = a v ¬b, ¬a v (c^d), b I am suppose to use resolution calculus to prove that T |= d ^ b holds.As in the first step, we translate T into a set of clauses, all clauses being in CNF.My lecturer converts T into the following clauses:Clause 1 ## What are the recent real life use or applications of the Cauchy Random Variable February 11 We have a short assignment on the described question and I already have gone through a lot of trash results from Google. I can't seem to find any.I don't know where else to post this question. Please guide me! It will be a great help. ## Prove that$cos^2\theta. sin^4\theta=…$February 11 Prove that: $$cos^2\theta.sin^4\theta=\frac{1}{32}(cos6\theta-cos2\theta+cos4\theta)$$Attempt: $$L.H.S=cos^2\theta.sin^4\theta$$ $$=cos^2\theta.sin^2\theta.sin^2\theta$$ $$=\frac{1+cos2\theta}{2}.\frac{1-cos2\theta}{2}.\frac{1-cos2\theta}{2}$$ $$= ## With what probability is this polynomial equal to zero (mod a prime p) February 11 1 If we suppose that we have a polynomial q(x) of the following form:q(x) = \sum_{i=0}^N{c_i x^i} \text{ where } c_i=0 \text{ or } c_i=1In other words, if we are given a polynomial with binary coefficients (either 0 or 1), what is the probability t ## Probability that a given function is prime… February 11 If we have a set of primes p_1, p_2, ... , p_n, we can easily construct a function of their product:$$f(\alpha) = \alpha \left( \prod_{k=1}^n{p_k} \right) + 1$$I'm wondering what the probability is that this function equals a prime. I want to s ## About O big notation February 11 If f_1=O(g_1) and f_2=O(g_2) then$$\frac{f_1}{f_2}=\frac{O(g_{1})}{O(g_{2})}=O(\frac{g_1}{g_2})$$I want to show this relation is true or not.Thanks ## What is product of 1\cdot2\cdot 3\cdot…\cdot n February 11 Suppose that F is the required function.I need the value of this function till n natural numbers with a direct mathematical expression. ## Use the definition of convergence of a sequence to show \lim \frac {2n^2}{n^3+3}= 0 February 11 Use the definition of convergence of a sequence to show$$\lim \frac {2n^2}{n^3+3}= 0$$I understand that to do this we must show \frac {2n^2}{n^3+3} \leq \varepsilon, but I'm not sure how to do that. ## modern analysis: metric spaces and \varepsilon-neighborhoods February 11 3 Prove or disprove that d(f,g) = ({\int_0^1 |f(x)-g(x)|^{2}dx})^{1/2}, on C[0,1] is a metric. If so, describe the \varepsilon-neighborhood.Do you know that for f\in C[0,1]$$||f||_2=\left(\int_0^1|f(x)|^2dx\right)^{1/2}$$is it's euclidean nor ## A small doubt in metric space. February 11 1 Is \mathbb{R} an open ball in \mathbb{R}?If we write B(0,\infty) as the open ball then \mathbb{R} is an open ball in \mathbb{R}. Is it correct?Thanks.No, balls have finite radius. It is an open set however.1 4 ## Expected payment of a roll of dice with rerolls February 11 We've got the following game:You roll two dices. You get paid equal to the number rolled.Additionally, if you roll doubles, you reroll (Same rules apply to that roll. That means there's not limit to how much you can win).What's the expected payment o ## Is there a specific name for these methods of summation February 11 When calculating summation of series I use these methods ;Ex: Method One$$U_r=\frac{1}{(r-1)r(r+1)(r+2)}U_r=\frac{1}{(r-1)r(r+1)(r+2)}\left[\frac{(r+2)-(r-1)}{3}\right]$$Then$$U_r=\frac{1}{3(r-1)r(r+1)}-\frac{1}{3r(r+1)(r+2)}$$Then clearly$$U ## A reduction of$10%$… February 11 A reduction of$10%$% in the price of sugar would enable a man to buy$2$kgs of sugar more for Rs.$125$. Find the reduced price per kg.My attempt: Let the initial price of sugar be Rs.$x$kg.Price after$10%$% reduction $$=x-10 percent of x$$ $$= ## How to fit sum of products of sine waves February 11 System Model:Y(t_1, t_2, t_3) = A*[2+k_1*cos(w_1t_1+\phi_1)+k_2*cos(w_2t_2+\phi_2)+4*k_3*cos(\dfrac{w_1t_1+\phi_1}{2})*cos(\dfrac{w_2t_2+\phi_2}{2})*cos(\dfrac{w_1t_1+\phi_1-w_2t_2-\phi_2}{2}-w_3t_3-\phi_3)]Conditions:A>0 -1 \leq k_1,k_2 \leq 1$$0 ## Linear Algebra. Is this question realte to combination and factorials February 11 I am not able to understand this question and what is the entries of matrix A exactly. Question Thanks. ## Show that$st$,$s^2-t^2/2$and$s^2+t^2/2$are relatively prime. February 11 Let$s$and$t$be odd integers. Show that$st$,$s^2-t^2/2$and$s^2+t^2/2$are relatively prime. I've seen this question on here, but unfortunately some of the cases were not covered, and I want to confirm that the way I'm proving it is correct bec You Might Also Like • So I have a polynomial$p$of$n$'th degree and q given by$q(x)=p[x,x_1,x_2,...,x_k]$, meaning that for$x$... • In Edwards' Galois Theory, in the chapter on Cyclotomic polynomials, the author devotes a lot of effort to p ... • In Gelfand and Shilov Vol I (of Generalized Function), on page 257, they write down the following equation t ... • This is what I understand about conics being projectively equivalent.Two conics$C1=V(F)$and$C2=V(G)$are ... •$X, X_1, X_2,\ldots $are real random variables with$\mathbb{P}(X_n\leq x)\to \mathbb{P}(X\leq x)$whenever ... • This question is an exact duplicate of: Trace Class: Definition [closed] 1 answer Given a Hilbert space$\ma ...
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