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Let R and S be reflexive relations on A. Suppose that R is also transitive. Prove S ⊆ R if and only if (S ◦ R) = R.

To prove this, I was going to assume S ⊆ R and prove (S ◦ R) = R, then do it the other way, assume (S ◦ R) = R and prove S ⊆ R. Then to do the first part I was going to do what my professor calls the “double subset strategy,” where, […]

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Find the source file causing a Mixed Content warning

Google Chrome has flagged a client’s WordPress site as Not Secure. So we are trying to remove a Mixed Content warning that we get using the Inspect feature of the browser. Mixed Content: The page at ‘https://www.CLIENT.com/‘ was loaded over HTTPS, but requested an insecure image ‘http://www.CLIENT.com/wp-content/uploads/2018/05/CompanyLogo.png‘. This content should also be served over HTTPS. […]

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On Pitt’s Inequality (Weighted Fourier Inequality)

One of Pitt’s Theorem (from “Theorems on Fourier Series” by H R Pitt, 1937) states that for an integrable periodic function $F$ over $[-\pi,\pi]$, $$ \sum_{n=1}^{\infty} |a_n|^q n^{-q\lambda} \leq K(p,q,\lambda) \int_{-\pi}^{\pi}|F(\theta)|^p|\theta|^{p\alpha}, $$ where, $a_n$‘s are the Fourier series coefficients, K is independent of $F$, $1<p\leq q <\infty$, ${1}/{p} + {1}/{p’}=1$, $0\leq \alpha < 1/p’$ and […]

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Prove that $\frac{[ABC]}{[XYZ]}=\frac{2R}{r}$

Prove that $$\frac{[ABC]}{[XYZ]}=\frac{2R}{r}\,,$$ where $[\,\_\,]$ represents area of triangle, $X,Y,Z$ are the points of contact of incircle with sides of triangle $ABC$, $R$ is circumradius, and $r$ is inradius. [Here is my textbook proof] (https://i.stack.imgur.com/C72Xw.jpg) Incase you are wondering what theorem 36 is. Look below Theorem 36: In two triangles $A_1B_1C_1$ and $A_2B_2C_2$ we have […]

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Computing Constant $C_2$ of Inequality $h_x([2]P) \geq 4h_x(P) − C_2 $

Let $E/Q$ be an elliptic curve given by a Weierstrass equation $E : y^2 = x^3 + Ax + B$ with $A, B ∈ Z$ and, $h_x(P) = \log H(x(P))$, where, $x(P) = p/q, H(x(P))=\max\{|p|, |q|\}$. There is a constant $C_2$ that depends on $A$ and $B$ such that $h_x([2]P) \geq 4h_x(P) − C_2 $ […]

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About the sum $S(p_n)=\sum_{1\le k\lt n}\,p_n\mod\;p_k$

For $\,p_n\gt2\,$ let’s define the sum $\,S(p_n)=\sum_{1\le k\lt n}\,p_n\;mod\;p_k$, where $\,p_k\,$ represents the $\,k$-th prime. The first terms of the sequence $\,S(p_n)\,$ (OEIS A033955 – sum of the remainders when the $\,n$-th prime is divided by primes up to the $\,(n-1)$-th prime) are: $S(3)=1\;\;\;(p_2=3)$ $S(5)=3\;\;\;(p_3=5)$ $S(7)=4\;\;\;(p_4=7)$ $S(11)=8\;\;\;(p_5=11)$ $S(13)=13\;\;\;(p_6=13)$ $S(17)=18\;\;\;(p_7=17)$ $S(19)=27\;\;\;(p_8=19)$ $S(23)=29\;\;\;(p_9=23)$ $S(29)=46\;\;\;(p_{10}=29)$ The graphical representation […]

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About the sum $S(p_n)=\sum_{1\le k\lt n}\,p_n\;mod\;p_k$

For $\,p_n\gt2\,$ let’s define the sum $\,S(p_n)=\sum_{1\le k\lt n}\,p_n\;mod\;p_k$, where $\,p_k\,$ represents the $\,k$-th prime. The first terms of the sequence $\,S(p_n)\,$ (OEIS A033955 – sum of the remainders when the $\,n$-th prime is divided by primes up to the $\,(n-1)$-th prime) are: $S(3)=1\;\;\;(p_2=3)$ $S(5)=3\;\;\;(p_3=5)$ $S(7)=4\;\;\;(p_4=7)$ $S(11)=8\;\;\;(p_5=11)$ $S(13)=13\;\;\;(p_6=13)$ $S(17)=18\;\;\;(p_7=17)$ $S(19)=27\;\;\;(p_8=19)$ $S(23)=29\;\;\;(p_9=23)$ $S(29)=46\;\;\;(p_{10}=29)$ The graphical representation […]

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About the sum $S(p_n)=\sum_{1\le k\lt n}\,p_n\;mod\;p_k$

For $\,p_n\gt2\,$ let’s define the sum $\,S(p_n)=\sum_{1\le k\lt n}\,p_n\;mod\;p_k$, where $\,p_k\,$ represents the $\,k$-th prime. The first terms of the sequence $\,S(p_n)\,$ (OEIS A033955 – sum of the remainders when the $\,n$-th prime is divided by primes up to the $\,(n-1)$-th prime) are: $S(3)=1\;\;\;(p_2=3)$ $S(5)=3\;\;\;(p_3=5)$ $S(7)=4\;\;\;(p_4=7)$ $S(11)=8\;\;\;(p_5=11)$ $S(13)=13\;\;\;(p_6=13)$ $S(17)=18\;\;\;(p_7=17)$ $S(19)=27\;\;\;(p_8=19)$ $S(23)=29\;\;\;(p_9=23)$ $S(29)=46\;\;\;(p_{10}=29)$ The graphical representation […]

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About the sum $S(p_n)=\sum_{1\le k\lt n}\,p_n\;mod\;p_k$

For $\,p_n\gt2\,$ let’s define the sum $\,S(p_n)=\sum_{1\le k\lt n}\,p_n\;mod\;p_k$, where $\,p_k\,$ represents the $\,k$-th prime. The first terms of the sequence $\,S(p_n)\,$ (OEIS A033955 – sum of the remainders when the $\,n$-th prime is divided by primes up to the $\,(n-1)$-th prime) are: $S(3)=1\;\;\;(p_2=3)$ $S(5)=3\;\;\;(p_3=5)$ $S(7)=4\;\;\;(p_4=7)$ $S(11)=8\;\;\;(p_5=11)$ $S(13)=13\;\;\;(p_6=13)$ $S(17)=18\;\;\;(p_7=17)$ $S(19)=27\;\;\;(p_8=19)$ $S(23)=29\;\;\;(p_9=23)$ $S(29)=46\;\;\;(p_{10}=29)$ The graphical representation […]

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A Tensor Calculation with Braids

I am trying to follow the derivation of the Jone’s Polynomial from a braid representation presented in chapter 2 of Ohtsuki’s Quantum Invariants. The representation of the braid $b$ with $n$ strands is a linear operator (which we treat as a matrix after choosing some basis) on $(\mathbb{C}^2)^{\otimes n}$ is $\psi_n(b)$. Let $\sigma_i$ denote the […]