# All Questions

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### Existence of countable ordinal

Prove that there exist countable ordinal $\xi$ such that $\xi=\omega^\xi$. The $\xi= \sup \{b^i| b_1=w, b_{i+1}=w^{b^i}\}$ should work. But how to prove that $\xi$ is countable?
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### Count ways to express N as sum of Fibonacci Numbers

Find the number of different ways in which a number N could be represented as sum of Fibonacci numbers, without using K in the representation where K is also a Fibonacci number. Fibonacci series is ...
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### Singular Value Decomposition of Rank 1 matrix

I am trying to understand singular value decomposition. I get the general definition and how to solve for the singular values of form the SVD of a given matrix however, I came across the following ...
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### plotting directions in 3-space

Consider the point $(1,1,1)$ in $\mathbb{R}^3$. I have two directions emanating from this point which I would like to draw (arrows) using Maple. The directions are (-1.37, -0.25, 0) and (-0.88, -0.50, ...
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### SVM - Variable Input Dimension

Is it possible for a trained support vector machine (SVM) to take an input of a different length (say during the testing phase) than the length used when it was trained? e.g. training data input: ...
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### Perturbation theory for algebraic equations

I'm trying to find expansion (up to the 2nd non zero term) for the roots of: $x^5-x^2+\epsilon=0$ as $\epsilon\rightarrow0$ So I've assumed the solution may be written as a power series ...
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### A stronger concept than total boundness

A space, every proper principal filter of which is refined by a Cauchy filter, is called totally bounded. Is there a term (and theory) about a stronger concept: a space every proper filter of which ...
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### How does this equation create this chart?

I am trying to understand this formula from the chart above. For example, from the middle graph, How does h(x) = 0.5x get the coordinates ...
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### why the zero “eigenvector”

Find the eigenvectors of $G$: G=$\pmatrix{5/4 & \sqrt3/4 & 0 \\ \sqrt3/4 & 3/4 & 0 \\ 0 & 0 & 2}$ I compute the characteristic polynomial and find that the eigenvalues are ...
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### Basis for set of polynomials

I just did a problem where I think I am able to draw the following conlusion The set of all polynomials with complex coeff of degree $\le n$ has the same basis as The set of all ...
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### Weierstrass $\tanh \frac{\theta}{2}$ substitution confusion.

I'm already familiar with the trigonometric version of this substitution $t = \tan \frac{\theta}{2}$ and it's geometrical derivation involving the unit circle found here. However, I'm not sure how ...
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### how to integrate $\frac1{2-3x^2}\,dx$

How do we evaluate $$\int\frac{dx}{2-3x^2}$$
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### $M \otimes k \neq 0$?

Let $(A,m,k=A/m)$ be a commutative local ring and $M$ be a $A$ module (not necessarily finitely generated) , and Let $x=(x_{1},...,x_{n})$ be a $M$-regular sequence such that $ann(M/xM)$is proper ...
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### calculate svd - example with roots

Do an Singular Value Decomposition of $$\begin{bmatrix} 0 & \sqrt{2} & 0 & \sqrt{2} \\ \sqrt{2} & 0 & \sqrt{2} & 0 \end{bmatrix}$$ I have tried to find it following the ...
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### Proper equality sign for boundary value definition

Say I have a function that needs to define a boundary condition, like $f(0) = A$. In this case it is usually fairly self evident from context, that this is a requirement that $f(0)$ needs to satisfy. ...
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### A problem in graph transformations.

I was going through a paper on graph transformations: see here. I can't understand a concept introduced there. I am giving an extract from that paper below ($VG_1$ denotes the set of vertices of a ...
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### How do i justify integration by polar-coordinates for Riemann-integration?

I completely understand how to transform Lebesgue integration to integration by polar-coordinates using the surface measure. However, i wonder if there is a weaker version of this justifying ...
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### Let $f(z)=z^2\bar{z}^3$. Calculate $f_z(z)$ and $f_\bar{z}(z)$.

Let $f(z)=z^2\bar{z}^3$. Calculate $f_z(z)$ and $f_\bar{z}(z)$. I'm just wondering what the techniques are to solve this. Should I use the definition ? Or the product rule ? How this product rule ...
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### cost minimization

There are n cities $c_1,c_2,...c_n$(in decreasing order of popularity) where a company wants to open its N branches. There is cost $w_i$ for opening a branch in city $c_i$. If company has budget W , ...
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### Problem with convex function

In Papadimitriou book I found a problem. If I know that function $f$ is a convex function, and I have values $x_2,...,x_n$, is function $g(x_1) = f(x_1,x_2,...,x_n)$ also a convex function? I know ...
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### Prove that T is a countable set

Let $T$ be a nonempty subset of the interval $(\,0, 1)$. If every finite subset $\{x_1,x_2,…,x_n\}$ of $T$ (with no two of equal) has the property that $x_1^2+x_2^2+⋯+x_n^2<1$, then prove that $T$ ...
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### Require help with the convolution of two complex conjugates

I need to find the convolution of the following two functions: When rationalizing the denominator, the numerators become complex conjugates of each other. I have tried obtaining the Fourier ...
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### Characteristic functions and weak convergence

Prove the following statement: $X_n => 0$ (convergence in distribution) if and only if $(\exists\; \epsilon>0: |t|<\epsilon) \;\; \phi_n(t) \rightarrow 1$, where $\phi_n(t)$ is the ...
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### Sum of random numbers is divisible by $10$

Suppose that $15$ three-digit numbers have been randomly chosen and we are about to add them. What is the probability that the number would be divisible by $10$? If there were only two or three ...
### Prove that $f_n$ converges uniformly on $[a,b]$
Let $f_n$ be a sequence of functions defined on $[a,b]$. Suppose that for every $c \in [a,b]$, there exist an interval around $c$ in which $f_n$ converges uniformly. Prove that $f_n$ converges ...
Theory Number Problems After I saw that post i wanted to solve the first one which is $a\mid b^2,b^2\mid a^3,a^3\mid b^4,b^4\mid a^5\cdots$ Prove that $a=b$ Now i started by proving that $a$ and $b$ ...