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### property of asymptotic rate of convergence of gradient decent algorithm on a quadratic form

Geven a quadratic form $f(x)=\frac{1}{2}x^TAx+x^Tb+a$, where $A$ is a symmetric positive definite matrix. We use gradient decent to compute the global min by $x_n=x_{n-1}-\triangledown f(x_{n-1})$ and ...
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### the tangent bundle of an $m$-dimensional manifold is $2m$

It is often taken as obvious that for $m$-dimensional manifold $M$ in $\mathbb{R}^n$, $TM$ is a $2m$-dimensional manifold in $T\mathbb{R}^n=\mathbb{R}^n\times\mathbb{R}^n$. But still is there an ...
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I must find the eigenvalues of the following system: $\hat H = V (|\alpha \rangle \langle \beta| +|\beta\rangle\langle\alpha|)$, where $\langle \alpha|\alpha \rangle=\langle\beta|\beta\rangle=1$, $\... 0answers 4 views ### a simple inequality on the simplex Could anyone help me to show: For any$v=(v_1,v_2, v_3), v_i\ge 0, \sum_{i=1}^{3} v_i=1$one has$\sum_{i=1}^{3} v_i|\bar p_i-p|\le\sum_{i=1}^{2}\sum_{j=i+1}^{3}\frac{v_i+v_j}{2} |\bar p_i-\bar p_j|$... 0answers 4 views ###$\mathrm{Supp}(\mathrm{Exc}(f))=\mathrm{Supp}(K_X-f^*K_Y)?$Let$f:X\to Y$be a log resolution of normal varieties, let$\mathrm{Exc}(f)$be the locus (the complement in$X$of the largest open where$f$is an isomorphism). Is there an example of$f$, such ... 0answers 2 views ### What do two ODEs with the same Lyapunov function have in common Assume I have two$N$-dimensional systems ofODEs$\frac{dX}{dt}=F(X)$with$F: \mathbb{R}^N\rightarrow\mathbb{R}^N$and$\frac{dY}{dt}=G(Y)$with$G: \mathbb{R}^N\rightarrow\mathbb{R}^N$. Also, ... 0answers 11 views ### I'm wondering if my work is right. Propositional Logic. So this is my works, I'm wondering if I can do step 7 or can't. If there is another way to solve this, please help me. Also sorry if my english is bad. 1answer 12 views ### Relation between roots and coefficients of an equation If$p,q,r $are the roots of the equation$x^3 - 3px^2 + 3q^2x - r^3 = 0$Hello! I hope everybody is doing well. Can anybody please help me with the above problem? My Solution: From Vieta’s$p+q+r ...
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Prove by induction, that a set with $n$ elements has $$\frac{n(n−1)(n−2)}6$$ subsets containing exactly three elements whenever $n$ is an integer greater than or equal to $3$.
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### How can I show $f(x) = x^3 - 6x^2 + 11x - 6$ is one-one or not?

I am stuck in showing whether the function mentioned above is one-one or not. I can clearly tell that it is not one-one by looking the graph, but I have to show it by proving that if $f(a) \ne f(b)$, ...
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### Integral of $(dx)^2$

What is the integral of $(dx)^2$, that is, $\int{(dx)^2}$
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### Be G an abelian finite group. If every element of G has at maximum order two then $G \cong (Z/2Z)^n$ [duplicate]

I'm trying to prove this proposition Be G an abelian finite group. If every element of G has at maximum order two then $G \cong (Z/2Z)^n$ I tried a lot, but my ideas are not so useful. Somebody ...
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### Finding orthogonal of a set in functional analysis

I am self studying functional analysis from Kreyszig Functional analysis and I need help in Ex 3.3 Question no. 5 which is ------> If X = $R^2$ . Then find orthogonal complement of M if M is {x} , ...
I know that there exists a $2\pi$ periodic continuous function whose Fourier series has a divergent subsequence at a point. However, I heard that for any $2\pi$ periodic continuous function $f(x)$, ...
### $\frac{1}{\sqrt1}+\frac{1}{\sqrt2}+\dots+\frac{1}{\sqrt{n}}<2\sqrt{n}$?
I'm trying to prove this by induction, but something doesn't add up. I see a solution given here, but it is actually proving that the expression is greater than $2\sqrt{n}$. I'd appreciate some ...