2
votes
3answers
37 views

Showing that $\int_{-n}^{n}{x+\tan{x}\over A +B(x+\tan{x})^{2n}}dx=0$

Where n is an integer, $n\ge1$ and $(A,B)$ just constants $$I=\int_{-n}^{n}{x+\tan{x}\over A +B(x+\tan{x})^{2n}}dx=0$$ It is obvious that $$\int_{-n}^{n}x+\tan{x}dx=0$$ Let make a ...
1
vote
1answer
33 views

equation of a circle that has a radius of $\sqrt{10}$

I am supposed to find the equation of a circle that has a center $(\sqrt{5},2\sqrt{2})$ and a radius $\sqrt{10}$. I have no idea of solving an equation which the given center ($\sqrt{5},2\sqrt{2})$. ...
0
votes
1answer
12 views

Metric hyperbolic polar

The metric tensor hyperbolic is given by $$g=4\frac{\mathrm d x^2+\mathrm d y^2}{(1-x^2-y^2)^2}.$$ I have to right it in polar. I know that the euclidien metric $\mathrm d x^2+\mathrm d y^2$ is given ...
1
vote
0answers
16 views

Transformation of the gradient

For a function $f\in C^2$, $f:\mathbb{R}^n\to\mathbb{R}$ and a point $x\in\mathbb{R}^n$ with $\nabla^2f(x)$ positive definit one can calculate the new point $x^+=x+s$ as follows: Change the ...
0
votes
0answers
17 views

An integral inequality $\int_0^{π/2} t\left(\frac{\sin(nt)}{\sin t}\right)^4 dx \lt \frac{π^2n^2}{4}$ [duplicate]

QUESTION: $$\int_0^{π/2} t\left(\frac{\sin(nt)}{\sin t}\right)^4 dx \lt \frac{π^2n^2}{4}$$ I guess it should be divided, but I couldn't find out the way to prove it.
-4
votes
5answers
48 views

On the equation $|x|^2+|x|-6=0$

Which of the following are true for $$|x|^2+|x|-6=0$$ 1. It has $4$ roots 2. The sum of the roots is $-1$ 3. The product of the roots is $-4$ 4. The product of the roots is $-6$ Only one of the ...
1
vote
1answer
24 views

How does the following graph have an Euler tour and not every node has degree that is even?

The theorem states: A connected graph has an Euler tour if and only if every vertex has even degree. But this graph has node 'A' with degree = 3. Graph image. ...
0
votes
1answer
19 views

Probability: Application Of “Expected Value”

So, I was learning expected value today and I'm trying to understand the significance of calculating this term "Expected value". In this simple example, What is the expected value when we roll a ...
1
vote
1answer
12 views

$nD$ rotation around a general $(n-2)$-dimensional subspace

According the Rodrigues' Rotation Formula $3D$ rotation matrix $\in$ $SO(3)$ corresponding to a rotation by an angle $\theta$ about a fixed axis specified by the unit vector $\hat{\omega}=(\omega_x,\...
-3
votes
0answers
20 views

Tuition service for math CBSE [on hold]

Can anyone suggest best tuition service for CBSE 12th grade math in Kuwait?
1
vote
0answers
17 views

Measurable function and the Mean Value Theorem

Let $\,f:[a,b]\to \mathbb{R}\,$ be continuous on $[a,b]$ and derivable on $(a,b)$. By the mean value property, for all $\,x\in (a,b)\,$ there exists $\,\xi_x\in (a,x)\,$ such that $\,f(x)-f(a)=f'\left(...
-6
votes
0answers
23 views

In $\triangle ABC$ $\overline{XY}$ is parallel to $\overline{AC}$ and divide the triangle into two parts of equal area. Find $\frac{AX}{AB}$ [on hold]

In $\triangle ABC$ $\overline{XY}$ is parallel to $\overline{AC}$ and divide the triangle into two parts of equal area. Then the $\frac{AX}{AB}$ equals $\cfrac{\sqrt{2}+1}{2}$ $\cfrac{2- \sqrt{2}}{...
1
vote
0answers
19 views

minimize quadratic form

In question Minimize Energy using Gauss-Seidel method with successive over- relaxation., when $$ E = \sum_i \|I_i - \mathbf N_i^T\mathbf L\|^2 + \lambda\sum_{i,j}\|\mathbf N_i - \mathbf N_j\|^2 = \...
1
vote
3answers
41 views

How to prove that $\lim_{k\to+\infty}\frac{\sin(kx)}{\pi x}=\delta(x)$

It is well-known that: $$\lim_{k\to+\infty}\frac{\sin(kx)}{\pi x}=\delta(x).$$ This can also be written as $$ 2\pi\delta(x)=\int^{+\infty}_{-\infty}e^{ikx}\,\mathrm dk.$$ However, I don't know how to ...
0
votes
0answers
8 views

Vectorizing equations in MATLAB

I am working on collaborative filtering using matrix factorization in MATLAB. I am using Gradient Descent for parameter learning. The cost function to optimize is : $ J = \vert I \odot (R - U V') \...
0
votes
0answers
20 views

Future learning for a math graduate in applied mathematics references

As a mathematics graduate with focus on programming we did a whole lot of coding of some mathematical statements (as well as proving them), but yet rarely giving real life examples and applications ...
2
votes
0answers
16 views

How to reflect this phase plane?

This is the phase plane of $X'(t)=\bigl(\begin{smallmatrix} 3 &1 \\ -4 &-1 \end{smallmatrix}\bigr)X(t)$. Which ODE system correspond to a phase plane that is a reflected image along the ...
-2
votes
5answers
41 views

Compute $\lim_{x\to 0}\tan (\pi/4+x)^{1/x}$ [on hold]

$$\lim_{x\to 0}\tan (\pi/4+x)^{1/x}$$ How to solve this problem?
0
votes
2answers
24 views

Show that if $a\equiv b(modm)$, then $gcd(a,m)=gcd(b,m)$ [duplicate]

I still don't have a clear approach , but this is what I see. $m|b$ and $m|a$ or $m\nmid b$ and $m\nmid a$. I may think that the way is showing $gcd(a,m)\leq gcd(b,m)$ and $gcd(a,m)\geq gcd(b,m)$
0
votes
1answer
13 views

Find equal side lengths for isosceles triangle from middle angle and area?

I know that this is a really easy question, but I am looking for the answer to this question: The area of this isosceles triangle is 5cm squared. The angle ABC is 22 degrees. Work out ...
0
votes
0answers
29 views

What is the graph called? [on hold]

I want to recreate graph shown here. I cannot find on excel, please help me understand the maths behind this graph.
0
votes
2answers
45 views

Calculate $\log_{\frac{1}{2}}{(2\sqrt[3]{4})}$

I have this exercise:$$\log_{\frac{1}{2}}{(2\sqrt[3]{4})}$$ There is not equation or something else, how should I solve this type of exercise?
2
votes
2answers
36 views

Operation of permutations on functions

Let $P$ be the additive group of mappings from $\mathbf{Z}^n$ to $\mathbf{Z}$. For $f \in P$ and $\sigma \in \mathfrak{S}_n$ (the symmetric group of degree $n$) let $\sigma f$ be the element of $P$ ...
1
vote
1answer
27 views

If the diameters of ball bearings are normally distributed, determine the percentage with diameters between $0.610$ and $0.618$ inches.

If the diameters of ball bearings are normally distributed with mean $0.6140$ inches and standard deviation $0.0025$ inches, determine the percentage of ball bearings with diameters Between $0.610$ ...
0
votes
1answer
31 views

Residue of a trig function multiplied by a polynomial

can somebody help me to find the residue for: I tried to make two series centered at $(z - k\pi)$ for $\sin(z)$ and $1- \cos(2z)$ but I don't know what to do with the $(z+\pi)^2$....and obviously, i ...
3
votes
1answer
31 views

Show that every square n is congruent to $ 0$ or $1 \pmod{8}$

If $n$ is odd then $n$ is congruent to $1 \pmod{8}$, but if $n$ is even the we have to do it by cases. Let $n=2k$ so $n^2=4k^2$. When $k$ is even, then it's congruent to $0 \pmod{8}$, but when $k$...
1
vote
1answer
25 views

How can we fill in some missing details in this proof?

Let $(X,d)$ and $(Y,\rho)$ be metric spaces and $X$ is compact. Suppose $f:X\to Y$ and $f_n:X\to Y$ are continuous functions such that for every $x\in X$, $\rho(f_n(x),f(x))$ decreases to $0$ as $n\...
3
votes
4answers
90 views

Solve $2^x+4^x=2$

This is the equation, but the result is different from wolframalpha: $$2^x+4^x=2$$ $$2^x+2^{2x}=2^1$$ $$x+2x=1$$ $$x=\frac{1}{3}$$ WolframAlpha: $x=0$ Where is the error?
1
vote
1answer
31 views

Write $\,-4i\,$ in polar form

Write $\,-4i\,$ in polar form ${re}^{i\theta}$, with $r$, $\theta\in \mathbb R$, and $\,r\geq0,\;0\leq\theta<2\pi$. I let $\,z=-4i\,$ first, then get $\,r=\sqrt{0+{4^2}}=4$. However, $\,\tan\theta\...
1
vote
1answer
13 views

If $\gamma _1$ and $\gamma _2$ are two different geodesic from $x$ to $y$, then, there is no minimal geodesic from $x$ to $z$

Let $(M,g)$ a Riemanian manifold and $x,y\in M$. Suppose there is two different geodesic $\gamma _1,\gamma _2$ that connect $x$ and $y$. Show that no one of these two geodesic are minimizing after $y$....
2
votes
2answers
16 views

A simple group such that $[G:H]=n$ can be embedded into $A_n$

Let $G$ be a finite simple group and $H$ be a proper subgroup of $G$ such that $|G:H|=n$. Then, how do I prove that $G$ can be embeded into $A_n$? I can prove that $G$ can be embedded into $S_n$ ...
0
votes
4answers
59 views

Evaluation of $\int_{-1}^{0}\frac{x^2+2x}{\ln(x+1)}dx$

Evaluation of $\displaystyle \int_{-1}^{0}\frac{x^2+2x}{\ln(x+1)}dx$ $\bf{My\; Try::}$ Let $$I = \int_{-1}^{0}\frac{x^2+2x}{\ln(x+1)}dx\;,$$ Put $x+1=t\; $ Then $dx = dt$ and changing limits, we get ...
0
votes
1answer
13 views

If $\cos\alpha = \frac{2\cos\beta - 1}{2-\cos\beta}$ , $(0<\alpha , \beta< \pi)$, then $\tan\frac{\alpha}{2}\cot\frac{\beta}{2}$ is equal to?

If $\cos\alpha = \frac{2\cos\beta - 1}{2-\cos\beta}$ , $(0<\alpha , \beta< \pi)$, then $\tan\frac{\alpha}{2}\cot\frac{\beta}{2}$ is equal to? MY ATTEMPT: I tried simplifying the equation to ...
2
votes
2answers
53 views

Simplify $\arccos\left(2\cos x\right)$.

Let $x\in[\pi/3,2\pi/3]$. We know that $\arccos (\cos x)=x$ but what we can say about $\arccos\left(2\cos x\right)$? Are there, for example, any "half-angle formula" also for inverse trigonometric ...
0
votes
0answers
13 views

Matlab double integration result does not match with my self calculate

Here is a double integration, self learning $$ P=\int_{-w}^{w}\int_{l}^{\frac{y_h(x_b+w)}{x_h}+l}\frac{1}{2}erfc[\frac{\log{\frac{z_h(y_b-l)}{y_h}}-\mu}{\sigma\sqrt2}]\space dy_bdx_b$$ after derive ...
6
votes
4answers
85 views

Subtracting expressions with radicals

I want to subtract the expressions $20\sqrt{72a^3b^4c} - 14\sqrt{8a^3b^4c}$. I simplified this to $120ab^2\sqrt{2ac}-28ab^2\sqrt{2ac}$. My textbook says the answer is $92ab^2\sqrt{2ac}$. Why doesnt ...
0
votes
0answers
21 views

Lindeberg condition's counterexample (central limit theorem)

My aim is to find an example where CLT is true but not the (equivalent to Lindeberg's) condition: Find a sequence of indipendent $(X_k)\sim\mathcal{N}(0,\sigma^2_k)$, so that they respect the clt ...
3
votes
3answers
28 views

Pattern on last digits of numbers to a certain power

There are 4 one-digit numbers which when squared have a last digit equal to the first number. They are 0,1,5 and 6. There are 2 two-digit numbers which when squared have their last two digits equal ...
0
votes
1answer
22 views

The differential equation $-y''+(1+x)y=\lambda y,x\in (0,1).$

The problem $$-y''+(1+x)y=\lambda y,x\in (0,1), y(0)=y(1)=0$$ has a non zero solution $1.$ for all $\lambda <0.$ $2.$ for all $\lambda\in[0,1].$ $3.$ for some $\lambda\in (2,\infty).$ $4.$ for ...
3
votes
0answers
32 views

The number of positive integer solutions to the equation $x_1+2x_2+…+nx_n=n^2.$

Let $n \ge 2, n \in \mathbb N$. $A_n$ denotes the number of positive integer solutions to the equation $$x_1+2x_2+...+nx_n=n^2.$$ Prove inequality $$\frac{n^n(n-1)^{n-1}}{2^{n-1}\left(n!\right)^...
0
votes
1answer
22 views

A question about interval operation result in a game

A state is:$A_{q}=(A_{q}^{0},...,A_{q}^{E})$ where $A_{i}^{j}$ is interval, $q$ and $E$ are positive integer The initial state is $A_{m}=((0,1),\emptyset...,\emptyset)$ , $m>E$ Procedure: Every ...
5
votes
6answers
91 views

Find a six digit integer [on hold]

Find a integer with six different digits such that the six digit integer is divisible by each of its digits? For example, find ABCDEF such that A, B, C, D, E and F all can divide the number ABCDEF. ...
0
votes
1answer
16 views

square of polynomial still harmonic? [on hold]

Let $P(z)=\sum_{i=0}^n a_i z^i$ be a polynomials on $\mathbb{C}[z]$ such that $a_i$ are real numbers. $|P(z)|^2$ is a harmonic function ?
2
votes
0answers
21 views

The differential equation $y'(x)=\lambda sin(x+y(x)),y(0)=1.$

For $\lambda\in\mathbb{R},$ consider the differential equation $$y'(x)=\lambda \sin(x+y(x)),y(0)=1.$$ Then the initial value problem has: $1.$ no solution in any neighbourhood of $0.$ $2.$ a ...
0
votes
1answer
37 views

Conjecture about primes and the faculty

Given a prime $p>5$ there exist a prime $q<p$ such that $kp+q=m!$, for some $k,m\in\mathbb Z_+$ where $m>2$. I want help to prove the conjecture or to find a counter-example.
0
votes
0answers
38 views

No square has a decimal expansion ending in 79

Show that no square number has a decimal ending in 79. More generally, find all possible two-digit endings for squares. Let any digit number ending at 79 be represented as $$a_nx^n+.....+7x+9$$ Plug ...
0
votes
2answers
30 views

How to apply Chinese Reminder Theorem to this congruence system?

\begin{align*} 17x & \equiv -15 \pmod{5}\\ -11x & \equiv 5 \pmod{3}\\ 23x & \equiv 15 \pmod{7} \end{align*} $5$, $3$, $7$ are coprime, so the system has solution mod $105$. I'm not sure ...
1
vote
1answer
34 views

Unable to understand why gcd(bt+r,b)=gcd(b,r) [duplicate]

I am trying to understand greatest common divisor so If a=bt+r for integers t & r then why gcd(a,b)=gcd(b,r).I am unable to understand it.

15 30 50 per page