# All Questions

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### How prove this inequaliy $\sup_{x \in [a, b]} |f(x)| \leq \dfrac{1}{b-a} \int_{a}^{b} |f(t)| dt + \int_{a}^{b} |f'(t)| dt$

let $f \in C^1([a, b])$ with $a, b \in \mathbb{R}, a < b$ show that $$\sup_{x \in [a, b]} |f(x)| \leq \dfrac{1}{b-a} \int_{a}^{b} |f(t)| dt + \int_{a}^{b} |f'(t)| dt$$ I've tried to use the ...
2answers
23 views

### show that if $y$ is orthogonal to $x_n$ and $x_n$ converges to $x$ then $x$ is orthogonal to $y$

help me. someone who can help me? spaces is inner product. It is section 3.2, issue 4 introduction to functional analysis book author Kreyszig
1answer
21 views

### Proving question on Functions.

For a function $f:S\rightarrow S$ , if $f$ is injective, then ,$f\circ f\circ f$ is injective. Can I get hints on how I can prove that it is true or false?
0answers
19 views

### Where is the dominated convergence theorem being used? (crosspost).

I am cross-posting a question I asked on cross-validated here. It is a mathematical doubt on the application of the dominated convergence theorem in the time series setting. I leave the ...
0answers
18 views

### Determine the radius of convergence of the power series

I tried the ratio test to find where $a_n/a_{n+1} < 1$ but I ended up with $(2n+6)(x-8)^4/4(2n+4) < 1$ and I don't know where to go from there.
1answer
39 views

### What is the center of power series?

The power series is: $$\sum\limits_{n=1}^{\infty}\frac{(x+4)^n}{n+1}$$ Any help appreciated!
0answers
35 views

### totient function and inclusion-exclusion principle

How can one prove the property established by Gauss $$\sum_{d\mid n} \varphi(d)=n$$ using the inclusion–exclusion principle? I was thinking to use that with the same method one can prove that ...
2answers
22 views

### A non-homogeneous recurrence of Fibonacci sequence

I am working on Fibonacci sequence (recursively) But the hack is that I have the following non-homogeneous version: $$F_n =F_{n-1} +F_{n-2} +g(n)$$ Where: $F_1=g(1)$ $F_0=g(0)$ I have to use: ...
3answers
142 views

### double Integral calculus

I want to calculate the double integral: $$\int_0^t \int_0^s \frac{\min(u,v)}{uv} \, dv \, du$$ I don't know how to o that even if it seems simple. Thanks in advance for your help
1answer
18 views

### Question in regard to solving for inverse laplace transform

I am having some confusion when it comes to solving for the inverse laplace transform. ( We are allowed the tables with the common values by the way). Il give an example. Take, ...
0answers
33 views

### Try to show $f(x)=-x$ is an orientation preserving map from $S^n$ to itself

Consider a map $f:S^n \to S^n$ defined by $$f(x)=-x$$ and want to show that this map is orientation preserving iff $n$ is odd. What I have done is, consider the standard orientation n-form on ...
1answer
32 views

### Is my algorithm correct? (Polar decomposition)

I cant seem to find my mistake. Consider this matrix $T =$\begin{bmatrix} 2 & 1 & 1 \\[0.3em] -1 & 2 & 0 \\[0.3em] 0 & 1 & -1 \end{bmatrix} I need ...
3answers
22 views

### Probability of choosing two bulbs with the same rating given that one has a specific rating

I am trying to teach myself statistics, and working through Jay DeVore's excellent text of "Probability and Statistics for Engineering and the Sciences". The problem is as follows: We have box of the ...
3answers
47 views

### Is there any symbol for intersects?

Given two polygons $A$ and $B$ the intersection of them is represented by the notation $A \cap B$ that returns a geometry (more precisely the set of intersecting points) resulting from the ...
1answer
65 views

### Upperbound on the following logarithmic function with matrix

I am trying to find an upperbound the expression below with a function $f$ that is a function of the identity matrix $$\log(1+\mathbf{h}^* \mathbf{\Sigma} \mathbf{h}) \leq f( {\bf I},{\bf h })$$ ...
0answers
23 views

### vase with blue and red balls

At first I hope this is not a duplicate post. I tried to find it but I have not found it. I hope that someone could help me with understanding the exercise. This question is about a vase with r red ...
1answer
31 views

### Prove that $L=K[x]/\langle m(x)\rangle$ is an algebraic extension of $K$

Let $m(x)$ be an irreducible polynomial of degree $n$. How can I prove that $$L=\frac{K[x]}{\langle m(x)\rangle}$$ is: A vector space of dimension $n$, An algebraic extension of $K$?
1answer
39 views

### Complex stationary point of $\frac{z}{1-e^{-z}}+z$?

I apply the method of steepest descents I need to know the stationary points $z_0$ of the function $$p(z)=\frac{z}{1-e^{-z}}+z,$$ such that, $0 <\mathrm {Im} (z)<2 \pi$. That is, I want $z_0$ ...
0answers
6 views

### Uniqueness of terms

One usually defines first-order terms to be variable symbols, constant symbols and for terms $t_1,...,t_n$ and a function symbol $f$ also $ft_1...t_n$ to be a term, cf. Ebbinghaus et. al. Then one can ...
1answer
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1answer
54 views

### Can eight circles be constructed from three circles?

Given three sufficiently spaced circles in a plane, is it possible, using a straight edge and compass, to construct the eight circles that are tangent to all three given circles?
3answers
72 views

### Find the Derivative of $\frac{1}{\cos^2(2x)+\sin^2(2x)}$ [closed]

Calculate the derivative of: $$\frac{1}{\cos^2(2x)+\sin^2(2x)}.$$ How would I calculate such a derivative?
2answers
61 views

### How do ideas in differential geometry expand upon ideas from introductory calculus

I just went through first year in mathematics and used Stewart's book for calculus. I am trying to self study differential manifold and I find many concepts such as chart, atlas very similar to that ...
1answer
11 views

### Sequence from generating function with integral

So, let $A(x)$ be the generating function of $a_0,a_1,\dots$ then what would be the sequence of the generating function: $$\int^x_0 A(t)dt$$ Since I am not much acquainted with integrals any help ...
2answers
594 views

### Dirac delta convolution with function

I've come into a bit of a snag, and thought some more talented mathematicians could maybe help. I am trying to do the following integral: $$S(x,t) = \int I(z)\delta(x-G(z,t)) \mathrm{d}z,$$ where ...
1answer
33 views

### Real Analysis Question- Differentiability on an interval

So this is the question I am trying to answer... At $(-2,0)$ and $(0,2)$ we just differentiate using normal rules of calculus, yes? Here is my attempt for at $x=0$. Is this correct? For d)I think ...
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### Representing a 5 star rating weighted arithmetic mean

I have a working set of equations to find a "True Value" from a 5 star rating. For example, if I have 119(a) five-stars, 25(b) four-stars, 13(c) three-stars, 6(d) two-stars, and 3(e) one-stars. Then ...
0answers
11 views

### Infinite sums over integral of triple associated Legendre polynomials

I have a couple of integrals of triple infinite sums of associated Legendre polynomials, which I'd like to integrate using Gaunt's Formula. Any help would be very much appreciated, as I'm really ...
0answers
19 views

### Does every irreducible projective cubic curve have a nonsingular point of inflection?

Does every irreducible projective cubic curve necessarily have a nonsingular point of inflection? I've been trying to construct counterexamples, to no avail, which leads me to believe the question can ...
1answer
18 views

### Find $f,g$ such that $f \equiv g \mod 2i\pi$ has finitely many solutions

I'm interested by two holomorphics functions $f,g : \mathbb C \to \mathbb C$ such that the set $$E := \{z \in \mathbb C \mid e^{f(z)} = e^{g(z)} \}$$ is finite and non-empty. For example : $f,g$ ...
1answer
40 views

### Explicit solution of parametric solutions of an ODE

I need to find the explicit solution of the following ODE: $y'+\sin y'=x$, $y=y(x)$. I have found these two parametric solutions: $x=t+\sin t$ and $y=\frac{t^2}{2}+t\sin t+\cos t+c$, $c\in\Bbb R$. ...

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