2
votes
3answers
71 views

Integrate $\displaystyle \int \sin(\sqrt{at})dt$

Integrate $\displaystyle \int \sin(\sqrt{at})dt$ Here is what I tried. Let $u=\sqrt{at}$, then $\displaystyle\ du=\frac{a}{2\sqrt{at}}dt=\frac{a}{2u}dt\implies \frac{2udu}{a}=dt.$ So by subsitution, ...
1
vote
0answers
26 views

a question about Fenchel's theorem(differential geometry)

I am an undergraduate student studying differential geometry right now. I am just finishing reading how to prove Fenchel's theorem:The total curvature of a smooth closed curve in 3-dimensional space ...
3
votes
2answers
51 views

$\sum_{n=1}^{\infty} \frac{1}{n+1!} \prod_{k=1}^{n} f(k)$ Prove the divergence of a series [on hold]

How can I prove the divergence of the series $$\sum_{n=1}^{\infty} \left(\frac{1}{(n+1)!} \prod_{k=1}^{n} f(k)\right) $$ if $f:\mathbb{N} \rightarrow \mathbb{N}$ is injective? $ $
1
vote
3answers
49 views

Slopes of inverse functions

I have a question that states if $f(x) = x^3+3x-1$ from $(-\infty,\infty)$ calculate $g'(3)$using the formula $$ g'(x)= \left(\frac1{f'(g(x))}\right )$$ If I am thinking about this correctly does ...
0
votes
1answer
63 views

How to show this fraction is not an integer

Suppose $k\geq 2$ is an integer. I want to show $$\frac{1+k+k(k-2)}{1+\frac{k-1}{k}+\frac{(-1-\sqrt{k-1} )^2}{k(k-2)}}$$ is not an integer. It is equal to $$\frac{(k-2) k (k^2-k+1)}{2 (k^2-2 ...
-2
votes
2answers
46 views

Prove that to any three numbers positive integers [on hold]

Prove that for any three positive integers, following equality holds $$\operatorname{lcm}(ab , bc , ca ) \cdot \gcd(a , b, c )=abc$$
0
votes
2answers
43 views

Lebesgue differentiation

Some days back I was doing the lebsegue integration. I was amazed by the integration's ability to maximize the potential of Reinman integration. Are there any new differentiation (except the metric ...
1
vote
2answers
25 views

Compact sets are bounded: shape of the cover matters?

To prove a compact sets is bounded, we assume there's a "open ball cover" (each with R=1) that covers the set. And take maximum distance over the center of the balls +2 as the boundary. Why could we ...
1
vote
3answers
54 views

$x^2+y^2<1, x+y<3$ is open or closed?

I'm trying to figure out if $$\{x^2+y^2<1, x+y<3|(x,y)\in \mathbb R^2\}$$ is open or closed. I tried to imagine this set. It looks, for me, as a 'pizza', or a circular sector, which have two ...
2
votes
2answers
23 views

Choosing the right sign for inverse functions?

If I have to find an inverse function and through the algebra I get a $\pm$ sign how do I know which one to choose from if its in a given interval? For example a question asks: The function ...
-2
votes
0answers
41 views

prove function is well-defined and integrable [on hold]

$1)$Let (x)=x-m(x), m(x) unique integer minimizing |x-m(x)|, for x different from $\frac{n}{2}$, n odd. And (x)=0 when x equals $\frac{n}{2}$, n odd. Now $f(x) = (x) + \frac{(2x)}{2^2} + ...
4
votes
3answers
305 views

Students in a class, girls sitting with boys and boys sitting with girls

This is a very interesting word problem that I came across in an old textbook of mine. So I mused over this problem for a while and tried to look at the different ways to approach it but unfortunately ...
0
votes
0answers
14 views

How to show remaining time in timer clock matlab on pressing ESC? [migrated]

I have a program given i want to show the time remains when pressing ESC in stopwatch so anyone help me in this program i want to show the time remains after pausing stopwatch in dialog on pressing ...
1
vote
1answer
37 views

Inequality using integrals and absolute values

Let $u,v$ be continous functions in $[a,b]$ a compact interval and let $c > 0$. Suppose that $\forall x\in [a,b]$, the following inequality is true: $$|u(x)-v(x)|\leq c\int^x_a|u(t)-v(t)|dt$$ ...
1
vote
3answers
23 views

Constructing Polynomial Function from Set of Points and Slopes

I only have a basic knowledge of calculus but I would like to know if it's possible to, given a set of points each with their own slopes, construct the simplest (or any) polynomial function that ...
-4
votes
0answers
34 views

For a group epimorphism $f : G \to H$ with kernel $K$, prove that $G \simeq K \rtimes H$. Why is $G \simeq K\times H$ if $G$ is abelian?

For a group epimorphism $f : G \to H$ with kernel $K$, prove that $G \simeq K \rtimes H$. Why is $G \simeq K\times H$ if $G$ is abelian? This question is from group theory in Abstract Algebra and no ...
0
votes
1answer
44 views

Number of combinations where the sum of values must be the same

My question is as follows: let there be $n$ different numbers $a_i$ in a set $A$, where each $a_i$ is a number between 0 and 1. How many different sets of values can I have that fulfill the condition ...
1
vote
1answer
24 views

Lines in the plane from Concrete Mathematics. How many bounded regions are there?

I'm studying Concrete Mathematics by Donald Knuth. While doing the warmup questions, from the recurrence chapter, I stumbled upon an interesting problem. I have an intuition about the solution but I ...
2
votes
3answers
55 views

Complex numbers $z$ satisfying $|z−a|+|z+a| = 2|b|\Leftrightarrow |a|\le |b|$.

could you help me to prove the next statement? Show that there are complex numbers $z$ satisfying $$|z−a|+|z+a| = 2|b| $$ if and only if $|a| \le |b|$. I did the first implication using the ...
1
vote
0answers
25 views

natural map to the homotopy fibre

In the paper Homology fibrations and group completion theorem, McDuff-Segal, page 280, paragraph 4, line 2-line 3 and Configuration spaces of positive and negative particles, McDuff, page 105, line ...
1
vote
3answers
55 views

Some questions about proofs of irrational numbers

I have some questions about some things I want to clarify in regard to basic questions that ask to show that roots are irrational, for example $\sqrt{3}$, $\sqrt{5}$ and $\sqrt{6}$. To me, I think ...
-2
votes
2answers
28 views

Show $\lim_{m \rightarrow \infty} \frac{1}{m^2+1} = 0$ using the $\varepsilon$-$N$ definition of convergence

Show that $$\lim_{m \rightarrow \infty} \frac{1}{m^2+1} = 0$$ using the $\varepsilon$-$N$ definition of convergence.
1
vote
1answer
16 views

Simulation of interacting Ornstein-Uhlenbeck processes

I would like to simulate the following system of interacting OU processes on $[0,T]$: $$dX_t^1=(X_t^2-X_t^1)\,dt+\sigma_1 \,dW_t^1,\quad X_0^1=x_1$$ $$dX_t^2=(X_t^1-X_t^2)\,dt+\sigma_2 ...
1
vote
0answers
15 views

Mixed vs Equal Characteristic Local Rings

This is more of a vague, intuitive type of question, so perhaps there isn't anything too concrete anyone can offer. I am trying to get a sense of precisely why working with local rings of equal ...
0
votes
0answers
16 views

Uniform Sampling on Intersection of Simplices

I'm trying to sample uniformly on the intersections of several simplicies, with all coordinates being non-negative. That is, given $$A\vec{x}=\vec{b} \ \ and \ \ \vec{x} \geq \vec{0},$$ I want to ...
1
vote
2answers
27 views

How to solve the system of equations $\{10^{-4}x_1+x_2=1, x_1+x_2=2\}$ using finite precision arithmetic with three significant figures?

Consider the following two equations: $10^{-4}x_1+x_2=1$ $x_1+x_2=2$ Solve using Gaussian Elimination using finite precision arithmetic with three significant figures. I'm a little ...
0
votes
0answers
31 views

Interpretations of divergent series

I have already bumped into this question and found it not quite useful for my own purposes. The usual way the sum of the infinite series is taught and defined in 1st year undergrad course is that its ...
0
votes
2answers
61 views

Summation of special series

Does anybody know how to evaluate $$\sum_{i=2}^n(i^2)\cdot{i\choose2}$$ How about the general case of $(i^k)*{i\choose2}$? A nice formula would be great!
-1
votes
1answer
47 views

stuck on an function question [on hold]

I'm studying CompSci, While I'm having fun with that, I haven't had a higher level math class since 2009. Unfortunately, I'm required to take calculus in order to pursue something I'm passionate ...
0
votes
0answers
17 views

Differentiate functional with delta function when calculating Euler-Lagrange equation

The paper "active contours without edges" by Chan and Vese http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=902291, My goal is to understand how to derive the corresponding euler-lagrange ...
2
votes
2answers
42 views

Probability of a group of people voting yes or no

I am in need of some explanation as for whatever reason I just can't wrap my head around a problem. The question basically breaks down like this: There are $8$ people on a jury ($3$ men and $5$ ...
1
vote
3answers
38 views

What is the number of mappings?

It is given that there are two sets of real numbers $A = \{a_1, a_2, ..., a_{100}\}$ and $B= \{b_1, b_2, ..., b_{50}\}.$ If there is a mapping $f$ from $A$ to $B$ such that every element in $B$ has an ...
0
votes
0answers
5 views

Find $\alpha$ which makes the problem have an optimal solution or none at all + dual problem LP

min $x_1 + \alpha x_2 $ subject to $4x_1+3x_2\leq29$ $x_1+x_2\geq4$ $x_1\leq5$ $x_2\leq7$ Find for which $\alpha$ the given problem has an optimal solution or no solution at all. Provide the ...
2
votes
1answer
55 views

Proving a trigonometric identity with tangents [on hold]

Prove that: $$\tan^227^\circ +2 \tan27^\circ \tan36^\circ=1$$ any help, I appreciate it.
2
votes
0answers
23 views

Counting points on the Klein quartic

In Moreno's book "Algebraic Curves over Finite Fields", he mentions the following in passing with no further comments ($K$ denotes the Klein quartic defined by $X^3 Y + Y^3 Z + Z^3 X = 0$): The ...
0
votes
2answers
29 views

Summation of “products”

Ok, I am pretty sure I won't get an answer because the question is somewhat hard, and I have done research to no prevail but how can I find the summation of ab if I know the summation of a and the ...
4
votes
3answers
426 views

Example of a ring with infinitely many zero divisors and finitely many invertible elements

I am preparing to my abstract algebra exam and I try to find an example of a ring with infinitely many zero divisors and finitely many invertible elements (rather simple if possible). Does it even ...
6
votes
3answers
137 views

Prove or disprove that $\sqrt{2+\frac{1}{n}}$ is irrational for $n \in \mathbb{Z}^+$

I have good reason to suspect that $\sqrt{2+\frac{1}{n}}$ is irrational for all $n \in \mathbb{Z}^+$ but a proof of this eludes me. I've tried proof by contradiction have had no success. I've also ...
-6
votes
0answers
31 views

One-to-One functions help [duplicate]

The one-to-one functions $g$ and $h$ are defined as follows: $$g=\{(9,8), (5,9), (8,-9), (9,-2)\}$$ $$h(x)=4x-9$$ SOLVE $$(g^{-1})(-9)=?$$ $$(h^{-1})(x)=?$$ $$(H o H^{-1})(7)=?$$
1
vote
2answers
55 views

If $AB$ and $BA$ are defined, then $AB$ and $BA$ are square matrices.

So as self-practice, I'm going over some proofs from Linear Algebra. I came across the following proof: $$\text{Prove that if both products $AB$ and $BA$ are defined, then $AB,BA \in M_{n,n}$.}$$ I ...
0
votes
1answer
17 views

Derive $E(X^k)$ I need help with the substitution piece.

If $X\sim\mathrm{WEI}(\theta,\beta)$, derive $E(X^k)$ assuming $k > -\beta$. Note that $X\sim\mathrm{WEI}(\theta,\beta)=\dfrac{\beta}{\theta^\beta}x^{\beta -1}e^{-(x/\theta)^\beta}$ I know to ...
2
votes
1answer
14 views

What does s=jω actually mean in terms of the complex plane and Laplace transforms?

I was trying to solve a problem on RC circuits. The current source was of the form $\cos (\omega t)$ which transforms in the manner of Laplace to $\frac{s}{s^2+\omega^2}$. I thought I’d use the ...
8
votes
2answers
124 views

How to find $I=\int_{-4}^4\int_{-3}^3 \int_{-2}^2 \int_{-1}^1 \frac{x_1-x_2+x_3{-}x_4}{x_1+x_2+x_3+x_4} \, dx_1 \, dx_2 \, dx_3 \, dx_4$

How can I find this integral $$I=\int_{-4}^4\int_{-3}^3 \int_{-2}^2 \int_{-1}^1 \frac{x_1-x_2+x_3{-}x_4}{x_1+x_2+x_3+x_4} \, dx_1 \, dx_2 \, dx_3 \, dx_4$$
1
vote
1answer
35 views

Prove that $\prod_1^{\infty}(1-a_i) > 0$ iff $\sum_1^{\infty}{a_i} < \infty$

Suppose $\{a_i\}_1^{\infty} \subset (0,1)$ a) $\prod_1^{\infty}(1-a_i) > 0$ iff $\sum_1^{\infty}{a_i} < \infty$ b) Given $\beta \in (0,1)$, exhibit a sequence $\{a_i\}$ such that ...
0
votes
1answer
17 views

Derivative of log determinant of triangular matrix

It is known that $$\frac{\partial\log|A|}{\partial A}=A^{-T}$$ However, if $L$ is a lower triangular positive definite matrix and take the log determinant, $\log |L|=\sum_i\log L_{ii}$. Question is ...
0
votes
1answer
12 views

Determine Orthogonal and non orthogonal using Coordinates

Can we identify using coordinates that if Polygon is orthogonal or non orthogonal. data = [(100, 100), (100, 200), (300, 200), (600, 400), (1150, 400), (1150,300), (600,300), (300,100)](These ...
0
votes
1answer
18 views

Characteristic Function and Convergence in Distribution of Sequence of R.V.

I am trying to solve the following: Let $X_1,X_2,...$ be a sequence of random variables with $P(X_n=\frac{k}{n})=\frac{1}{n}, k=0,1,2,...,n$. Find the characteristic function of $X_n$ and show that ...
0
votes
3answers
61 views

Polynomial whose one of its roots is $\cos(\pi/7)$

Let $P(x)$ be a 3rd-degree polynomial with integer coefficients, one of whose roots is $\cos(\pi/7)$. Compute $\frac{P(1)}{P(-1)}$ I saw this question in a contest math problem, and I know that it ...
3
votes
1answer
44 views

Open sets in the product topology

Let $\{X_{\alpha}\}_{\alpha\in A}$ be a family of topological spaces. The product topology on $X=\prod_{\alpha\in A}X_{\alpha}$ is the weak topology generated by the coordinate maps ...
1
vote
0answers
23 views

Where I can find the proof that- for every knot there is a Conway Notation?

At the end of The Knot Book - Collin Adams there is a list of knots. He has given a Conway Notation for each of those knots, from which I have assumed that every knot has a Conway Notation. Or for ...

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