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Trying to understand relationship between Dirichlet energy of graphs and discrete laplacian

I’m a noob cs masters student trying to understand how the laplacian and the Dirichlet sum are related. So there is this popular expression with graph adjacency matrix $A$ and the laplacian $L$, $$ \sum_{ij}A_{ij}(x_i -x_j)^2 = x^tLx. $$ I’m trying to find the proof for this (for specific problem of undirected graphs). I tried […]

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unique real solution to x + e^x = 0 has no special properties, right?

Let $a$ be the unique real number such that $a + e^a = 0$. I claim that (1) $a$ is irrational. (Easy enough: If $a$ were rational, then write $a = p/q$ for integers $p,q$. It follows that $e^a = -a$ is rational, and hence $e^p = (e^a)^q$ is also rational. But this contradicts the […]

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Matlab – Multiplication of matrices $A.B$, A being a triangular matrix

I have this code for the multiplication $A.B=C$, $A(m,n)$, $B(n,p):$ C = zeros(m,n) for j=1:n aux = zeros(m,1) for k=1:p for i=1:m aux(i) = aux(i)+A(i,k)*B(k,j) end end C(:,j) = aux end How can I rewrite this code for $A$, $a_{ij}=0$, $i<j$, in a way that these null elements will not be acessed? What I tried, […]

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Find solution of differential equation

Find the solution of differential equation y”+8y’+17y=f(t), y(0)=0, y'(0)=0, where f(t) is the periodic function f(t)=1 , 0 f(t)=0 , π and f(t+2π)=f(t)

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Verifying a continued fraction related to $\logφ$.

The continued fraction is the following, $${1+\cfrac{1\cdot 2}{3φ+\cfrac{1\cdot 2}{5+\cfrac{3\cdot 4}{7φ+\cfrac{3\cdot 4}{9+\ddots}}}}}=\frac{2}{3\logφ}\tag{1}$$ Where, $$φ=\frac{1+\sqrt{5}}{2}$$ Something with I found in an backside of my previous notes where I kept my recreational math works. So can the closed form be verified in any sort (most preferably by using already established identities)?

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Meagre sets have empty interior.

Suppose $A$ is a meagre set i.e. a set of first category.Then $A=\large\cup_{n\in \mathbb N} $$A_n$,where each $A_n$ is nowhere dense.Now $A^o=(\large \cup_n A_n)^o\subset\large \cup_n A_n^o=\phi$ as $A_n^o\subset (\bar A_n)^o=\phi$.Thus we are done.Is this proof correct?

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Proving the inequality using am gm theorem

I was asked to prove that $a^3+b^3 \le (a^2+b^2)(a^4+b^4)$ I expanded the rhs and used am gm and got $a^6+a^2b^4+a^4b^2+b^6 \ge 4a^3b^3$ and struck.i think I lack intuition or I don’t have enough experience.Any hints

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Well behaved colon between powers of ideals when the associated graded ring is Cohen-Macaulay.

I’m reading a paper: A formula for the core of an ideal, by Claudia Polini and Bernd Ulrich and I’m in trouble with the following problem: Let $R$ be a Cohen-Macaulay ring and $I$ be an ideal of $R$ with $\mathrm{ht}(I)>0$. If $\mathrm {gr}_I(R)$ is Cohen-Macaulay, then $I^{m+n}:_RI^n = I^m$ for all $n,m \in \mathbb{N}$. […]

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Determinants $3 \times 3$ matrix proofs

Suppose $A$ is a $3\times3$ matrix such that $\det(A)=\frac{1}{125}$. Find $\det(5A^{−1})$. I know that this can also be written as $\det(5/A)$ However, I am struggling to work out what $A$ is Please help

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is it ok to say “Please don’t unstick it” when you don’t want it that is stuck to something to become unstuck?

Ok, this is a picture of a diaper. It has a small strip on its back. After use, The trip will be used to fasten the diaper for easy disposal. Your child often tries to remove or unstick it. is it ok to say “Please don’t unstick it“. The oxford dictionary has the adjective “unstuck” […]