0
votes
2answers
30 views

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 ...
1
vote
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
0
votes
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?
1
vote
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 ...
0
votes
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.
2
votes
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!
0
votes
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 ...
0
votes
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: ...
3
votes
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
1
vote
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, ...
2
votes
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 ...
2
votes
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 ...
1
vote
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 ...
0
votes
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 ...
0
votes
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 $$\log(1+\mathbf{h}^* \mathbf{\Sigma} \mathbf{h}) \leq f( {\bf I},{\bf h })$$ ...
1
vote
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 ...
-1
votes
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 $K$?
3
votes
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 $z_0$ ...
0
votes
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 ...
0
votes
1answer
34 views

Ito Formula (Poisson basic process)

Let, $N_t$ be a Poisson process and let $X_t$ solve the SDE $d{X_t}=a_t dt +J_t dN_t$. Then, Ito´s fórmula is: $$df(t,X_t)=(\frac{\partial f}{\partial t} + \frac{\partial f}{\partial x}a_t)dt + ...
1
vote
0answers
30 views
+50

Understanding stability of fixed points in 2D maps.

I'm trying to understand the stability analysis for a map of the form $$(x_{n+1}, y_{n+1}) = A(x_n,y_n)$$ Where A is a 2x2 matrix - assumed to be diagonalisable and with distinct eigenvalues. I ...
8
votes
2answers
180 views

Maximum number of Sylow subgroups

I've been studying Sylow-$p$ subgroups, and I've come across this problem. Let $G$ be a finite group. Show that the number of Sylow subgroups of $G$ is at most $\frac{2}{3}|G|$ . ($|G|$ is the ...
-1
votes
1answer
23 views

The minimum perimeter and maximum height of a triangle under constraints [duplicate]

I do not get the following formulas : The minimum perimeter of any triangle (abc), given the heights corresponding to the a and b-sides. The maximum height corresponding to the side b of any ...
0
votes
2answers
35 views

Field Extension Question for Polynomials

I cannot seem to find the answer to this question on the internet. It is a question about field extensions for an element $a,b \neq F$ but in some extension $K$. I am wondering if $F(a,b)= \lbrace ...
1
vote
2answers
441 views

Can we do long division in $\mathbb Z[\sqrt n]$, where $n$ is square free?

Can we do long division of elements $a + b\sqrt n$ and $c + d\sqrt n$ in $\mathbb Z[\sqrt n]$? I gave it a shot, but wasn't sure how to proceed... I'd like to use the gcd algorithm to show that two ...
0
votes
1answer
22 views

Intrinsic definition of divergence and curl

Are the intrinsic definitions of divergence and curl the theorems of Green-Ostrogradski and Stokes-Ampere respectively ? What is a rigorous derivation of their expression in a coordinate system ?
3
votes
2answers
33 views

Functions or relations stable under automorphism

Suppose we have a structure $M$, that is, a set $S$ with some designated functions and/or relations on that set. We can define automorphisms for this structure. What is the term in the standard logic ...
0
votes
0answers
15 views

$\int_{0}^{+\infty}\frac{x^2+x^3}{x^2\cdot \left | x-1 \right | \cdot \frac{3}{4} \left |x-4 \right |^{\frac{4}{3}}} dx$ convergence

Does $$\int_{0}^{+\infty}\frac{x^2+x^3}{x^2\cdot \left | x-1 \right | \cdot \frac{3}{4} \left |x-4 \right |^{\frac{4}{3}}} dx$$ converge? Domain of this integrand is $x \in \mathbb{R} : x\neq 0, ...
1
vote
1answer
38 views

Question on one point compactification

I was given the following question in my general topology class assignment which is multi parts - most of which I managed alright by myself some of which I need help on. We are given a non compact ...
0
votes
0answers
12 views

Universality of the Simplex Category. Proving Functoriality of the Map.

Let $B$ be a strict monoidal category, and $\left \langle c,\mu ',\eta ' \right \rangle$ a monoid in $B$. Now suppose we consider the simplex category $\left \langle \triangle ,+,0 \right \rangle$, ...
2
votes
4answers
52 views

Real Analysis - differentiable

$f:[0,\infty]\rightarrow \mathbb{R}$ is twice differentiable. If $f''$ is bounded and exists the limit of $f(x)$ at infinity, then $\lim_{x\rightarrow \infty}f'(x)=0$. I tried to use the Taylor's ...
1
vote
1answer
16 views

Identification of the Lie algebra of an isotropy group with the tangent space - stuck with a statement

I think I am stuck with the following statement that I read on the Encyclopedia of Mathematics website regarding Isotropy representations: "If $G$ is a Lie group acting smoothly and transitively on ...
1
vote
0answers
40 views

Can a matrix be similar to more than one matrix?

I have a little query about similar matrices I've been struggling with. Suppose I have a 5x5 diagonal matrix A with 5 distinct eigenvalues as entries in the main diagonal. The question is, to how ...
0
votes
1answer
20 views

determining distribution composed of uniform distributions

Let $X,Y,Z$ be i.i.d. $U(0,1)$ distributed. How can I determine the distribution of $$ \frac{X}{X+Y+Z}?$$ I have no idea how to go about this problem. Obviously this expression also has values ...
0
votes
1answer
8 views

Find the length of the intercept cut by the side $BC$ on the y-axis .

The equation of two equal sides $AB$ and $AC$ of isosceles triangle $\triangle ABC$ are $x+y=5$ and $7x-y=3$ respectively.What will be the length of the intercept cut by side $BC$ on the y-axis? ...
0
votes
0answers
30 views

some series about :On some strange summation formulas by R. William Gosper

I read the paper On some strange summation formulas by R. William Gosper and I looking the following series maple could sum any idea how to get it thanks $$\sum _{z=1}^{\infty } \frac{(-1)^z \cos ...
6
votes
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?
-5
votes
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?
4
votes
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 ...
1
vote
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 ...
3
votes
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 ...
0
votes
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 ...
3
votes
1answer
20 views

Connections between Cesaro summation and Borel summation of series

Let $\sum_{n=0}^\infty x_n$ be a given series of numbers, let $S_n=\sum_{k=0}^n x_k$, $n=0,1,2,...$, let $g\in \mathbb R$. We say that this series is convergent to $g$ in the sense of Cesaro if $$ ...
9
votes
1answer
116 views
+100

$p \in C - D$, inflection point for $C$ iff inflection point for $C \cup D$.

Show that if $C$ and $D$ are projective curves in $\mathbb{P}_2$ and $p \in C - D$ then $p$ is a point of inflection for the curve $C$ if and only if $p$ is a point of inflection for the curve $C \cup ...
3
votes
3answers
102 views

Easy question : $\int (xdy+ydx)$

I am ashamed to ask such an easy question but, well: Lets say I got a function $$ f(x,y)=xy $$ Now let's compute the total differential of the function $$ d(f(x,y))=xdy+ydx $$ Now if I do $$ \int ...
-2
votes
1answer
32 views

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 ...
1
vote
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 ...
3
votes
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 ...
1
vote
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 : $f,g$ ...
2
votes
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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