0
votes
0answers
10 views

Is there any reason to expect the Riemann sum over $[a,b]$ to converge to the definite integral $\int_{a}^{b} f(x) \, dx$?

When learning the definite integral 'rigorously', most first courses seem to follow the steps below. Sketch the function over $[a,b]$ Construct arbitrary left and right function value partitions, ...
0
votes
0answers
4 views

Cobordism Groups an the Pontryagin-Thom Construction

I am confused by the statement that $\Omega^\text{framed}_1(S^3) \cong \mathbb{Z}$ which I came across as an application of the Pontryagin-Thom construction for showing that $\pi_3(S^2) \cong ...
0
votes
0answers
3 views

Reference request to Prof. Wilhelm Blashke's DG book

Please suggest how can one get a copy of the book: TA Guess_on_W Blashke Is there an English translation of Professor Blashke's book on Differential Geometry? (Einführung in die Differential ...
0
votes
1answer
13 views

Determine the derivative $\frac{dy}{dx}$ of the integral

Determine the derivative of the integral $$ \,\int_{\sqrt x}^{0}\sin (t^2)dt $$ What does this question mean. I do not understand it and I think you can't integrate $\sin t^2\,$.
-1
votes
1answer
22 views

How does infinity have a measure?

Unit interval, [0,1], is uncountably infinite but has metric (not measure) of 1 in the metric space (X,d) for real number line. What does this mean?
0
votes
0answers
10 views

Cross products in non-standard basis

I have a problem where I am given a set of linearly independent but not orthonormal vectors {e1, e2, e3}. My task is to convert this to an orthonormal basis {u1, u2, u3} using Gram Schmidt method. The ...
0
votes
0answers
11 views

Computing probability of $X$ conditioning by the $U=\min(X,Y)=u$

Lets $X$ and $Y$ be independan random variables with $P(X\leq x)=F_x(x)$ and $P(Y\leq y)=F_y(y)$. Lets $U=\min(X,Y)$. I know that $F_u(u)=1-(1-F_x(u)(1-F_y(u)))$ By definition: $P(X \leq x |U=u)= ...
0
votes
0answers
4 views

Lagrange function on MATLAB

I'm trying to write the Lagrange function in Matlab and I need some help. This is what a friend and I have got so far, I am just not getting how to finish: function y = lagrange(X, Y, x) %lagrange ...
2
votes
0answers
12 views

System of Equations which can be solved by inequalities

I think I am smelling inequalities here. In the first equation I used Holder's inequality to show, $xyz \le 1$ , But in the second equation I used Titu's Lemma to get $x+y+z \le 3$ .But I think ...
0
votes
0answers
7 views

The Golden Ratio in a Circle and Equilateral Triangle. Geomertic/Trigonometric Proof?

Geogebra gave me 1.61 for the following Golden Ratio construction shown below. Firstoff, has anyone seen anything similar to this construction? Basically begin with an equilateral triangle. ...
0
votes
0answers
5 views

The bound of gap between zeros of this nonnegative trigonometric polynomial - a prime sieve function

I constructed the following cosine sum which zeros show the prime sieve by given prime set $p \le p_i$, where $p_i$ is $i^{th}$ prime. Zeros less than $p_{i+1}^2$ are all primes. So gap between the ...
1
vote
0answers
8 views

Quotient of Trigonometric functions

suppose we fix a constant $a\in (0,\pi)$ and consider the function $$f(z)=\frac{\cos(az)}{\sin(\pi z)},\quad z\in \mathbb{C}-\bigcup\limits_{k\in\mathbb{Z}}B_{\epsilon}(k)$$ where $\epsilon>0$. ...
-11
votes
0answers
16 views

How do I solve for A in the following systems of equation? [on hold]

A=S+S+H+O+L+E+S S=1 O=1 H=2 So then, does L must equal 0 for A to equal True?
1
vote
0answers
10 views

How to prove second order differentiation matrix is of the form..

Given that the matrix: $$D2 = \left[\begin{matrix}a11&a12&a13\\a21&a22&a23\\a31&a32&a33\end{matrix}\right]$$ is a second-order differentiation matrix in the sense, for a ...
2
votes
1answer
15 views

Finding the limit at infinity of $f(z) = \frac{\overline{z}}{|z|^2}$

I would like to make sure I'm doing everything right and not missing anything, since I know that some familiar functions do crazy things in the complex setting. Since $|z|^2 = \overline{z}z$ I ...
0
votes
0answers
10 views

precompact operators in a Hilbert space [functional analysis]

I've linked to a Theorem (from H&N's Applied Functional Analysis) whose proof I'm trying to understand (I asked a question about the previous chunk of the proof yesterday). The theorem is ...
-1
votes
0answers
18 views

Prove that $f$ is onto. [duplicate]

Let $X$ be a compact metric space and $f$ is an isometry on $X$ i.e. $d(f(x),f(y))=d(x,y)\forall x,y\in X$. Prove that $f$ is onto. In order to show that it is onto I choose one $y\in X$ . To ...
3
votes
1answer
22 views

Definition of convergence of a sequence

Can this be definition? For each $\epsilon>0$, there exists $K \in \Bbb N$ such that if $n\ge K$, then $|a_n-a|\le\epsilon$. Which one < or <=epsilon
1
vote
3answers
23 views

I want to find whether the expression $D = \sqrt{5t^2 - 40t+125}$ is increasing or decreasing when $t=5$.

I want to find whether the expression $D = \sqrt{5t^2 - 40t+125}$ is increasing or decreasing when $t=5$. My logic is I want to find whether is $f'(5)>0$ or $f'(5) < 0$. I need to use the ...
0
votes
0answers
3 views

Formula for calculating sliding markup

I am trying to come up with a formula for calculating markup for products that range in value from a few cents up to tens of Dollars. At 10c I would like the markup to be around 500%, and from 2 ...
1
vote
0answers
15 views

Why should $\phi'$ and $\phi''$ be $\mathcal O(1)$?

As Strogatz writes in his book Nonlinear Dynamics And Chaos (p. 64) There are often several ways to nondimensionalize an equation, and the best choice might not be clear at first. Therefore we ...
1
vote
0answers
8 views

Is this toric variety the blowup of $\mathbb C^2$ at some point?

Let $u_1=e_1,\quad u_0=e_1+2e_2,\quad u_2=e_2$. Consider the fan consisting of the following cones $\sigma_1= \langle u_1,u_0\rangle$, $\sigma_2=\langle u_0,u_2\rangle$ and their faces. Then the toric ...
2
votes
2answers
12 views

Finding the tangent of an ellipse that is perpendicular to a line

The books say's "Find the equations of the tangents to $x^2+3y^2=4$ which are perpendicular to the line $x-2y=7$" I've graphed them and found that the given line does not pass through the ...
2
votes
0answers
39 views

Proof of $\sum\limits_{k=1}^{\infty} \frac{1}{k^4} = \frac{\pi^4}{90}$ using the Fourier series of $|x|$

I'm sure easier proofs exist, but I have to specifically use the method in the picture: This is what my attempt is: First, I did some manipulation to figure out that $$ \sum_{k=1}^\infty \frac ...
0
votes
0answers
6 views

Some ideal property in a local ring

If we change the ideal $(X_1,X_2^2-X_1,...,X^2_{n+1}-X_n,...)$ to $({X_1}^2,X_2^2-X_1,...,X^2_{n+1}-X_n,...)$ in this problem, what is the answer to the raised question? Again, the new local ring $R$ ...
0
votes
2answers
18 views

Is there a faster way to do this?

Let $f(x)$, $g(x)$ and $h(x)$ be three functions. Given that $f(x)=2(x ^5) - 8(x ^2) + 1$ $f(x-3) = g(3x-2)$ $g(3x+1)=h(x+3)$ Prove $h(x)=f(x-5)$.
1
vote
0answers
14 views

Evaluation of $\lim_{n\rightarrow \infty}\frac{1}{n^{2m}}\left[(n^2+1^2)^{m}(n^2+2^2)^m(n^2+3^2)^m…(2n^2)^{m}\right]^{\frac{1}{n}}$

Evaluation of $$\lim_{n\rightarrow \infty}\frac{1}{n^{2m}}\left[(n^2+1^2)^{m}(n^2+2^2)^m(n^2+3^2)^m..............(2n^2)^{m}\right]^{\frac{1}{n}}$$ $\bf{My\; Try::}$ Let $$L=\lim_{n\rightarrow ...
0
votes
0answers
7 views

$\frac {2 \cdot 3^{m + 1}}{k} - \frac {2 \cdot 3^{m}}{k + 1} \in \Bbb N_+$, find all possible values of $k, m$.

If $$\frac {2 \cdot 3^{m + 1}}{k} - \frac {2 \cdot 3^{m}}{k + 1} \in \Bbb N_+$$ and $$\frac {2 \cdot 3^{m}}{k} - \frac {2 \cdot 3^{m}}{k + 1} \le 1$$ where $k, m \in \Bbb N_+$ and $k \ge 2$, find all ...
2
votes
0answers
13 views

Distance between incentre and orthocentre.

I want to prove that the distance between incentre and orthocentre is $$\sqrt{2r^2-4R^2\cos A\cos B\cos C} $$here $r$ is inradius and $R$ is circumradius. I considered $\triangle API$ ($P$ is ...
1
vote
1answer
18 views

Do all fields have a total cyclic order?

It is well known the finite commutative rings, $Z/nZ$, are not discretely ordered rings. The axiom $\forall x \forall y \forall z((0<z \land x<y) \rightarrow (x*z < y*z))$ is false for the ...
0
votes
2answers
16 views

How to identify an orthogonal(orthonormal matrix)?

This question was asked in an examination a while back. I was able to solve this question but the computation required was too much. The solution said that the trick to solving this lies in the fact ...
1
vote
1answer
7 views

Find the matrix of the given linear transformation $T$ with respect to a given basis.

How do you solve $T (f(t)) = f(2t - 1)$ from $P_2$ to $P_2$, with respect to basis $\beta = (1, t-1, (t-1)^2)$?
0
votes
0answers
4 views

Line Integrals and Contour Maps

What does a line integral on a contour map for a function represent? If I wanted to estimate the line integral on a contour map with a drawing of the line and of the contour map for f(x, y), how could ...
0
votes
0answers
3 views

Why does the square of the norm of the fourier bessel series $\|J_n(\alpha_ix) \|^2=\frac{b^2}{2}J^2_{n+1}(\alpha_ib)$ when $J_n(\alpha b)=0$.

The square norm in the coefficient of the fourier bessel series (where x=the weight function) was solved in a proof using: $$2\alpha^2\int_0^{b}xJ_n^2(\alpha x)dx=\alpha^2b^2[J_n'(\alpha ...
0
votes
2answers
11 views

Show that this limit yields $\gamma=\lim_{n \to \infty}\sum_{k=1}^{n-1}\frac{1}{k}-\sum_{k= n}^{(n-1)^2}\frac{1}{k}$

$\gamma =0.5772156...$ is Euler's constant Show that this limit yields $$\gamma=\lim_{n \to \infty}\sum_{k=1}^{n-1}\frac{1}{k}-\sum_{k=n}^{(n-1)^2}\frac{1}{k}$$
1
vote
0answers
21 views

Evaluate the integral $\iint \operatorname{curl}(yi+2j)\cdot n \, d\sigma $

Evaluate the integral $\iint \operatorname{curl}(yi+2j)\cdot n \, d\sigma $ where $\sigma$ is the surface in the first octant made up of part of the plane $2x+3y+4z=12$ and triangular in the ...
1
vote
0answers
21 views

Riemman-Stieltjes Integral Exercise

The truth is that I have no experience with the integral of Riemann-Stieltjes and developing a Bayesian inference problem in the book "Mathematical Statistics" by Shao, appears one of these steps, I ...
0
votes
0answers
15 views

Numerically stable version of calculation with cancellation

What's a numerically stable way to compute $$ \frac{2^{1/n}}{2^{1/n}-1} $$ for large (integer) $n$?
3
votes
2answers
26 views

Isometric embedding of $\ell ^ 1$ in $\ell ^\infty$ in finite dimensions

It is well known that $\mathbb{R}^n$ with $\ell ^1$ norm can be embedded into $\mathbb{R}^k$ with $\ell^\infty$ norm for some $k\in \mathbb{N}.$ But I guess, this is not true in complex case that is ...
0
votes
1answer
11 views

Calculating norm of matrix?

So i have two matrices $A$ (that fulfills $Ax=b_1$) and matrix $B$ (that fulfills $Ax=b_2$). Im trying to calculate how to obtain = $$\| b − \widehat b\|_\infty$$ would this be the $∞$-norm of the ...
0
votes
2answers
25 views

If $A,B$ are sets and $B$ is finite, and there is an injection $f:A \to B$, then $A$ is finite and $card(A) \leq card(B)$

I am stuck with proving this statement for a while. Prop. If $A,B$ are sets and $B$ is finite, and there is an injection $f:A \to B$, then $A$ is finite and $\operatorname{card}(A) \leq ...
0
votes
0answers
4 views

Inverse Laplace transform of $\sqrt{H(s)}$

In there any way to find inverse Laplace transform of a function in the following general form \begin{equation} F(s)=\sqrt{\dfrac{a_n s^n+a_{n-1}s^{n-1}+\cdots+a_1 s+a_0}{b_n ...
0
votes
1answer
12 views

Completing the square

$2x^2 - 6x - 5$ My workings are $2 ( x^2 - 3x + (3/2)^2 - (3/2)^2 - 2.5 ) = 0$ $ 2 ( x - 1.5)^2 - 4.75 = 0$ $ (x-1.5)^2 = 2.375 $ From here I go on to find X which is not the correct answer .. ...
1
vote
0answers
18 views

Quotient map on Real line

Let $X=\Bbb R$ and $q: X \rightarrow(\Bbb R, T_{std})$ be a function $q(x)= \begin{cases} 1-|x|, & \text{x $\le1$} \\ |x-1|, & \text{x $\ge$1} \end{cases}$ Q1 Suppose X is given the ...
0
votes
0answers
16 views

Problem in solving a question of conics.

Let $A$ and $B$ be fixed points in a plane such that the length of the line segment $AB$ is $d$. Let the point $P$ describe an ellipse by moving on the plane such that the sum of the lengths of the ...
1
vote
1answer
26 views

Proving the Axiom of Choice is equalivalent to the statement “If $A$ can be well-ordered, then so can $\mathcal{P}(A)$.”

I am still not completely confident in proving equivalence between the Axiom of Choice and statements such as the one posed in the title, so I want to make sure that I am on the right track. So ...
0
votes
0answers
11 views

Connceted componets of a complement of a union of hyperplanes

Let $(W,S)$ be an irreducible coxeter system with graph $\Gamma$ and associated bilinear form $B$. a) $B$ is nondegenerate, but not positive definite. b) For each $s \in S$, the coxeter graph ...
-3
votes
0answers
20 views

Prove an identity [on hold]

Anyone has any idea on how to prove $\sum \limits_{i=0}^{l} \sum\limits_{j=0}^i (-1)^j {m-i\choose m-l} {n \choose j}{m-n \choose i-j} = 2^l {m-n \choose l}?$
0
votes
1answer
11 views

When is the radius of convergence of the product of two complex power series twice the radius of convergence of the product of the radii

Proving that the product has a larger radius then the product isn't too bad using the nth root test, but another practice question I have asks for examples of power series $\sum a_kz^k$ with radius of ...
-2
votes
1answer
14 views

Find the matrix of the given linear transformation T with respect to the given basis.

Question I'm not sure where to start with number 6. Can someone help? Thanks!

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