# All Questions

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### Initial Value Problem to find non-zero terms and convergence

Consider the following initial value problem: $$e^{-x}y'' + ln(1 + x)y' - x^2y = 0$$ $$y(0) = 1$$ $$y'(0) = 2$$ Show that $x = 0$ is an ordinary point in the differential equation. Find the first $4$ ...
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### A set of numbers has an average of 100

A set of numbers has an average of 100. And the largest element is 5 greater than 3 times the smallest element. Which element cannot be in the set? i) 30 ii) 80 iii) 120 iv) 154 v) 50
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### Replace linear return with cosine return

Below you will find the position profile in time for a servo-device I am building. Currently the motion follows the orange profile moving away from zero to a maximum value and then quickly returning ...
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### T1 space in general topology

Let (X, τ) be a topological space where τ = {A ⊆ X : p ∈ A} ∪ {∅} (i.e. is p inclusion topology). Then show that (X, τ) is not T1 -space . I know that the definition Aspace X is called T1 space if for ...
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### Proof by Induction, any help would be greatly appreciated [closed]

the question is: $$2^m-7 > 10n, m=n, n\ge7$$ This has completely lost me and I can not use the calculators that work with less than or equal to. Thanks in advance!
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### B = {w | w = xbaby, where x, y ∈ Σ ∗}, Σ = {a, b} - DFA

I am having some trouble with this problem. I believe that the language that B recognizes must included a substring "bab" in-order to ever hit an acceptance state. Below is a screenshot of ...
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### A unitary operator is a closed operator

Let $\mathcal{H}$ be an infinite-dimensional Hilbert space, with $U$ a densely defined, unitary operator. I was wondering if such an operator is in fact a closed operator, which is equivalent to \...
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### Performing principal components analysis on the linear approximation of a time-series from another time-series

I was reading a recent paper and was trying to understand the novel factor analysis method that they introduce. I am not terrific at linear algebra so I was hoping to get some intuition behind what ...
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### Fourier Transform Commutes Derivatives

Consider two integrable functions $f(x)$ and $g(x)$, whose derivatives (and higher order derivatives) are also integrable. From the properties of the Fourier transform (and convolutions), the Fourier ...
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### Two versions of Kleene's recursion theorem - what's the relationship between them?

Below are two versions of Kleene's recursion theorem. How are they related? Are they equivalent? If not, does one of them (which one?) imply the other? Note that both $U(n,x)$ and $\phi_n(x)$ is the ...
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### Determine if the set $A=\{(x,y) \in \overline{B}\mid x \geqslant0 \}$ is open or closed.

Let $\overline{B}= \{(x,y) \in \Bbb{R}^2 \mid x^2+y^2 \leqslant 1 \}.$ Determine if the set $A=\{(x,y) \in \overline{B}\mid x \geqslant0 \}$ is open or closed. If I let $(x,y)\in A$ and $r >0$ ...
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### How to deal with irrational values with a bijective number system?

Having done some research on bijective numeration - that is, a number system in which every non-negative integer can be represented in exactly one way using a finite string using a finite set of ...
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### Find essential spectrum of an operator

Let $\mathcal{H}$ be a complex, separable, infinite-dimensional Hilbert space. Given an operator $T\in\mathcal{B(H)}$ and $\pi:\mathcal{B(H)}\rightarrow\mathcal{B(H)}/\mathcal{K(H)}$ the canonical ...
There are three ($A_1,A_2 and A_3)$ people sitting around the table, each with $a_1=2,a_2=3,a_3=4$ apples. To ensure each one has equal amount of apples, A3 just gives A1 one apple. In this case, ...