0
votes
1answer
171 views

Eliminate the parameter and sketch the curve

The question is (in parametric equations): $$x = 2\sin(t)$$ $$y = \cos(t)$$ for $0 \le t \le \pi/2$ I need to eliminate the parameter and the sketch the curve... any ideas?
2
votes
3answers
392 views

How to find the $\arcsin 2$?

How would I find $\arcsin 2$? I'm helping my little sister with her calculus "pre-test" before classes begin, and I don't remember how to do it in order to explain to her. Help?
7
votes
0answers
1k views

Using distance formula to find slope, any reason to use the concluding equation?

So, today I was observing a class that I will be a TA for this semester and the professor started to talk about the distance formula $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Well, my mind wandered a little ...
1
vote
1answer
899 views

How to find ALL cluster points of a sequence?

The task of finding a limit point of a sequence is 'simple'; if the sequence converges, we locate the limit point by inspection and proved that it the sequence converges to that point. By uniqueness, ...
2
votes
0answers
435 views

Derivative of a matrix norm

Consider the function $V:\mathbb{R}\to\mathbb{R}$ given by $$ V(t)=\|I - e^{At}\|^2 $$ where $I$ is the identity matrix and $A$ is a square matrix. The norm is the Euclidean norm on ...
3
votes
4answers
183 views

How to prove $t^{23}+1$ irreducible in $F_p$?

I have tried to prove that $t^2+1$ is irreducible over $F_3$ by supposing to the contrary $t^2+1=(t+\alpha)(t+\beta)=t^2+(\alpha+\beta)t+\alpha\beta$. Then, $\alpha+\beta\equiv 0 \pmod 3, ...
2
votes
2answers
365 views

Cauchy nets in a metric space

Say that a net $a_i$ in a metric space is cauchy if for every $\epsilon > 0$ there exists $I$ such that for all $i, j \geq I$ one has $d(a_i,a_j) \leq \epsilon$. If the metric space is complete, ...
0
votes
0answers
104 views

Prove that a group of exponent three is abelian [duplicate]

Possible Duplicate: If a group satisfies $x^3=1$ for all $x$, is it necessarily abelian? I tried to prove the following. I don't know if the result is true or if I need more hypotheses. ...
7
votes
1answer
230 views

Proving XY is perpendicular on :CD

In cyclic quadrilateral $ABCD$ the point $E$ is in the middle of $BC$, the perpendicular on $BC$ pass the point $E$ and intersect $AB$ in $X$, and the perpendicular on $AD$ pass the point $E$ and ...
0
votes
2answers
181 views

Reduced cost zero for the two-phase Simplex?

I cannot understand the line -12, -4, -5, 1, 1, -1, 0, 0, 0. Now the formula $\bf c - \bf A ^t \bf y$ when $c=0$ will result into the line. It is just many times a ...
1
vote
1answer
96 views

On the set closure operator, S*

The closure of a set S, which is S*, is defined as S0 ∪ S1 ∪ S2 ∪ ... So if S contained the English alphabet, S* = {empty string} ∪ {a,b,c,..} ∪ {aa, ab, ac, ... , ba, ...
-4
votes
1answer
54k views

exponential function or any number raised to the power infinity [closed]

please what are the answers and proofs to the following questions e raised to the power 1 (e^1) e raised to the power infinity (e^infinity) e raised to the power of minus infinity ...
3
votes
1answer
521 views

Subsequential limit of sequence

I'm trying to determine all subsequential limit points of the following sequence X_n = cos(n) Not sure how to decompose this into subsequences. Anyone know how ...
2
votes
1answer
162 views

How to solve this literal equation

I have a literal equation that needs to be solved for $\theta$: $$mg \sin(\theta) = \mu mg \cos(\theta)\left({ M+m \over m}\right) $$
2
votes
2answers
703 views

Ebook resources for math learning (10th-12th grade)

After searching the site, i found some questions like mine, but none fits my case. I am a 11th grade student, and i have an ebook reader(kindle). I would like to know where can i find(free) content ...
3
votes
1answer
104 views

Partitioning a Set so that the Sum is the Same

Let $M = \{1, 2, \dots , n\}$. What would be necessary and sufficient condition(s) for the number $m$, so that $M$ can be expressed as the disjoint union of $m$ subsets $A_i$, $(i = 1, 2, \dots, m$), ...
3
votes
1answer
122 views

Name or notation for $\mathbb Z/2\mathbb Z\ast\mathbb Z/2\mathbb Z\ast\cdots\ast \mathbb Z/2\mathbb Z$

Is there a standard notation for the n-fold free product of a group with itself? In particular, I'd like to know a nice name or notation for the the $n$-fold free product of $\mathbb Z/2\mathbb Z$ ...
1
vote
2answers
497 views

get value at a point on an exponential curve

I'm not super with math but I need to make a function in my web app to get the value of a point on a curve when I know the curve points that are set. Here is what I did, I put a set of point with the ...
3
votes
3answers
200 views

Does the series $\sum_{k=1}^\infty\left(\frac{1}{k}-\frac{1}{2^k}\right)$ diverge

Is there a handy way to tell if $\sum_{k=1}^\infty\left(\frac{1}{k}-\frac{1}{2^k}\right)$ diverges or not? I have a hunch that it diverges, since it looks like the sum is just $\zeta(1)-1=\infty$. But ...
1
vote
2answers
43 views

found a function from equations and inequality?

I'm Software engineer and I'm having little issue solving this problem let's called H. Well I'm looking for the mathematical expression of the function f(x) based on 3 equations and one inequality. ...
2
votes
2answers
144 views

Prove inequality $2\le\frac{a+b}{c+1} +\frac{b+c}{a+1} +\frac{c+a}{b+1} \le3$

Assume that $a,b,c$ are real numbers from the interval $(\frac{1}{2},1)$. What is the proof that $$2\le\frac{a+b}{c+1} +\frac{b+c}{a+1} +\frac{c+a}{b+1} \le3$$ holds?
4
votes
1answer
268 views

Matrix Chain Multiplication?

The following are questions about using dynamic programming for matrix chain multiplication. Pseudocode can be found in the Wikipedia article on matrix chain multiplication. 1) Why is the time ...
1
vote
1answer
127 views

An analytic function whose successive derivatives at a point converge in series

Qual season... Let $\Omega$ be a region in which $g$ is analytic and suppose that for a $z_0\in\Omega$, $$\sum_{n=0}^\infty g^{(n)}(z_0)<\infty.$$ Prove that $g$ is entire and that the series ...
1
vote
1answer
154 views

Entropy of Zipf and Zeta Distributions

I was wondering how to show entropy of the zeta distribution. It is: $$ H_\mathrm{zeta}(X) = \sum_{k=1}^\infty \frac{1/k^s}{\zeta(s)} \log(k^s \zeta(s))$$ The entropy of the zipf distribution is: ...
3
votes
2answers
636 views

Need Help Understanding Universal Quantifier Introduction in First-Order Logic

I'm reading Mathematical Logic by Stephen Cole Kleene and I am stuck trying to understand the logic behind his rule of universal quantifier introduction. In the text, for propositional and ...
2
votes
1answer
93 views

solution for ODE problem

I was trying to simulate a physical system which lead me to this equation. I don't know if it has any solution or not, but I guess you guys can help me find the answer. $$v'(t) = a + s * ...
2
votes
1answer
306 views

Cauchy Integral Formula Confusion

According to Cauchy's Integral Formula, we have: Let $U$ be an open subset of the complex plane. Let $f: U \rightarrow \mathbf{C}$ be a holomorphic function. Let $\gamma$ be the boundary of some ...
1
vote
1answer
140 views

Finding the necessary diameter for a pulley to match the ratio of other pulleys

I have an analog odometer for a bicycle. It works with two pulleys connected by a small rubber drive belt. One pulley is affixed to the hub, and the other pulley is on the odometer unit itself. To ...
1
vote
2answers
765 views

Optimality conditions and Directions in Simplex method

I am trying to understand the optimality conditions in Simplex -method, more in the chat here -- more precisely the terms such as "reduced cost" i.e. $\bar{c}_j=c_j-\bf{c}'_B \bf{B}^{-1} \bf{A}_j$ and ...
2
votes
2answers
109 views

If value halves every $5$ years, when will the dollar be worth $1/1,000,000$ its current value?

This was a GRE multiple choice question. At a $15$ percent annual inflation rate, the dollar would decrease by approximately one-half every $5$ years. At this inflation rate, in approximately how ...
3
votes
1answer
142 views

Study the convergence of the following series

I have to study the convergence of the following series: $$\sum_{n\ge1} \frac{ n! } { p (p+1) \cdots (p + n - 1) }\text{ where }p > 0.$$ I tried d'Alembert criterion but $\lim_{n\to\infty} \frac ...
3
votes
1answer
62 views

Is independence preserved by conditioning?

$X_1$ and $X_2$ are independent. $Y_1|X_1\sim\mathrm{Ber}\left(X_1\right)$, $Y_2|X_2\sim\mathrm{Ber}\left(X_2\right)$. Are $Y_1$ and $Y_2$ necessarily independent? (Assume ...
0
votes
1answer
126 views

Dimension of disjoint union of manifolds

While it is clear that a disjoint union of two $d$-manifolds is a $d$-manifold, it is not clear to me if the disjoint union of a $d_1$-manifold and a $d_2$-manifold is still a manifold and if yes ...
5
votes
2answers
529 views

Finding the congruence with a large power modulo a large integer

How would I reduce the following modulus: $$3^{2146483648} \pmod {4294967297}$$ It's supposed to end up being congruent to 10,324,303 but I'm having some difficulty getting there via Euler's phi ...
4
votes
2answers
129 views

Quick Julia/Mandelbrot Testing

I have successfully implemented a realtime Julia/Mandelbrot set generator on the GPU. Primarily out of curiosity, what I'm looking for now is a faster test algorithm. Ideally, I want a boolean ...
1
vote
1answer
101 views

Basis and Extension

I have 2 questions - If $B = \{x(i) : i \in I \}$ be a basis of a subspace $S$ ( finite or infinite dimensional ) of a vector space $V$ and $C = \{ y(i) + S : i = 1 \text{ to } n \}$ is a basis ...
0
votes
0answers
67 views

Isoperimetric Area Function

I'm reading the definition of isoperimetric function area and appeared the following notation: $Ł^{n}_{g}(\Omega)$, where n is dimension of Riemannian manifold $(M,g)$. What is the meaning of ...
3
votes
1answer
292 views

Blow-up of ODE solution

I am a newcomer to ODE. The relevant theorem that I can think of is about the maximum open interval of existence of the solution. But I have not learned to find the interval on which the solution ...
1
vote
2answers
152 views

Lift a variety over a number field to the complex numbers

Let's say I have a number field $k$ and a curve $C$ over $k$, can we lift $C$ to a variety $V$ over $\mathbb{C}$? It is like "complexifying" the curve. I read somewhere that it is possible to lift an ...
1
vote
2answers
324 views

Finding combined time to repair two machines where time is exponentially distributed

I am trying to solve the following problem. The time $T$ required to repair a machine is an exponentially distributed random variable with mean 10 hours. a) What is the probability that a repair ...
2
votes
4answers
705 views

Continuous Deformation Of Punctured Torus

This is problem 11 (b) from the first chapter of "Basic Topology" by M.A. Armstrong. The author hasn't had time to develop many theorems or mathematical machinery, so this problem should be able to ...
0
votes
1answer
73 views

Integral on a surface

Let $D$ be the parallelogram on the $(x,y)$ plane ($z=0$) that comes from the intersection of the lines: $y=x$, $x = \pi$, $y = x + \pi$ and $x=0$. Compute the following integral: $$ \int_S \sqrt{ 1 ...
7
votes
4answers
4k views

How to convert radicals to decimals without a calculator

How can one convert radicals to decimals(approximate value) when the number is not perfect such as $\sqrt2$, $\sqrt3$, $\sqrt5$, etc. Without the use of calculators.
1
vote
0answers
226 views

Derivative of a function evaluated at a certain point with respect to the original function

Given a function $c(r)$, where $r \in [0,1]$ and another function $$ x = \alpha \times c(0) $$ where $\alpha$ is a constant and $c(0)$ is the function $c$ evaluated at $r = 0$, my question is - what ...
0
votes
4answers
117 views

A trigonometric limit

The software maple 12 has calculated that: $ \displaystyle \lim_{x\to 0}\Bigg( \frac {\cos(\pi x)}{\sin(\pi x)}\;\;-\;\frac{\pi x}{\sin^2 (\pi x)}\bigg)=0$ How can I prove this equality? I have ...
1
vote
2answers
83 views

Directional derivatives - $t \mapsto f(x+tv)$

First I will write what is written in the book and then I will ask the question: Suppose $f:\mathbb{R}^3 \rightarrow \mathbb{R}$ is a real-valued function. Let $v$ and $x \in \mathbb{R}^3$ be ...
10
votes
2answers
390 views

$\{(x,y)\!\in\!\mathbb{B}^n; -\varepsilon\leq-\|x\|^2\!+\!\|y\|^2\leq\varepsilon\}\approx\mathbb{B}^k\!\times\!\mathbb{B}^{n-k}$

The question is motivated by the notion of handle attachment, Morse theory, critical points of index $k$, Morse lemma, sublevel sets, etc. For $0\!\leq\!k\!\leq\!n$ and ...
0
votes
1answer
214 views

find all continuous function $f:\mathbb R^+\to\mathbb R^+$ satisfy: $f(x)=f(\frac{x+1}{x+2})$

Find all continuous function $f:\mathbb R^+\to\mathbb R^+$ satisfy: $f(x)=f(\frac{x+1}{x+2})$ Find all continuous function $f:\mathbb R\to\mathbb R$ satisfy $\forall a<b, \exists c \in (a,b): ...
0
votes
2answers
90 views

canonical divisor of $B=\left \{ w_{0}^{3}+w_{1}^{3}+w_{2}^{2}w_0 =0\right \}$?

Help me please! How can I find any canonical divisor of $B=\left \{ w_{0}^{3}+w_{1}^{3}+w_{2}^{2}w_0=0 \right \}\subseteq \mathbb{P}^{2}$? Thanks in advance.
4
votes
1answer
144 views

Limit of sub-sequence and limit of sequence

Suppose we are given a sequence of nonnegative real numbers: $$a_1\geq a_2\geq a_3\geq\dots\geq 0$$ such that $$\lim_{n \to \infty} a_n=0.$$ Assume that $$\lim_{k\to\infty} (a_{n_k}\ln ...

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