# All Questions

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### 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, ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 $\epsilon>0$. ...
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### 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?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### $\frac {2 \cdot 3^{m + 1}}{k} - \frac {2 \cdot 3^{m}}{k + 1} \in \Bbb N_+$, 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $\beta = (1, t-1, (t-1)^2)$?
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### 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 ...
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### 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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