# All Questions

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### Solutions of a second order nonlinear differential equation

I am in trouble solving the following differential equation: $$\ddot{x}(t)=-\alpha\dot{x}(t)\dfrac{x(t)}{\left(\beta^2+x(t)^2\right)^{\frac{5}{2}}}$$ where $\alpha$ and $\beta$ are constant. How can I ...
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### Constructing a “one-way function” of two variables (a.k.a “stop my friend from hacking my game”)

This might be more of a computer science question than a mathematics one; I thought I'd start here but perhaps people might want to point me to a better forum, if this isn't the right one. ...
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### Inverse matrices properties.

I know about the properties of matrix multiplication for multiplication such as $A(BC)=(AB)C$. However I'm curious if $(AC)B$ would also have the same value. I'm asked to represent $A$ in terms of $B$ ...
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### Tangent space to $\mathbb{R}P^{n}$

I could not find any other question here related to this. If I have missed out, then this could be voted as a duplicate(Sorry if it is!). I was just trying to figure out the tangent space to the ...
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### Problem finding the number of r-element multi-subsets of the multi-set $M=\{ a_{1},a_{2},…,a_{n},m.b \}$

Let $m,n,r \in \mathbb{N}$. Find the number of $r$-element multi-subsets of the multi-set $$M= \{ a_{1},a_{2},...,a_{n},m.b \}$$ when $r \leq m,r\leq n$. Below is the given answer. ...
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### How to integrate $\int_{-\infty}^\infty e^{- \frac{1}{2} ax^2 } x^{2n}dx$

How can I approach this integral? ($0<a \in \mathbb{R}$ and $n \in \mathbb{N}$) $$\large\int_{-\infty}^\infty e^{- \frac{1}{2} ax^2 } x^{2n}\, dx$$ Integration by parts doesn't seem to make ...
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### Fourier Transform of mix partial derivative

I know FT{$\frac{\partial u}{\partial x}$} = (ik)FT{u}. Give a function $U(x,y)$. Is the following true? FT{ $\frac{\partial^2 U}{\partial y \partial x}$} = FT{$\frac{\partial U}{\partial y}$} ...
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### Natural numbers object via initial morphism

I assume that a natural number object (or see nLab) can be defined as an initial morphisms. (edit: as in the title, I ment initial morphism, not objects) $\hspace{1cm}$ Thoughts: Probably $X:=1$, ...
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### First order differential equation with time-varying parameter

If I have: $\dot{\sigma}(t) = -\gamma \sigma(t)$ where $\gamma$ is a constant, the solution is given by: $\sigma(t) = \sigma(0) e^{-\gamma t}$ Now what if I have the differential equation: ...
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### Derivative of SquareRoot with h-formula

I know the general formula for getting a derivative, and the formula for the derivative of the square root function, but I'm interested in how to do prove it using the formula for the definition of ...
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### Range of sum of Normal Distribution.

May be its silly question but I was just wondering is there any way to find out the absolute range of sum of values of Random normal distribution of N numbers with mu and sigma as mean and Std. Dev. ? ...
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### Why are “not bounded” operators not everywhere defined?

Let $X, Y$ be Banach spaces, $\mathcal{D}(T)$ a subspace of $X$, and $T\colon X\to Y$ a linear map. Such a $T$ is commonly called an unbounded linear operator, where unbounded just means that the ...
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### proof of convergence in probability [closed]

Let $(X_n)_{n\in\mathbb{N}}$ be i.i.d. random variables taking values in the set of natural number $\mathbb{N}$. Assume that $\mathbb{P}(X_1=i)=p_i>0$ for $i\in\mathbb{N}$. Let $D_n$ denote the ...
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### How to integrate $\int\limits_0^\infty{\frac{1}{{\sqrt {({y_1} - {y_2}){y_2}} }}d{y_2}}$?

I tried to do but it does not exist. $$\int\limits_0^\infty{\frac{1}{{\sqrt {({y_1} - {y_2}){y_2}} }}d{y_2}}$$ Thank you
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### Finding $f_Y$ such that $Z=Y\cos(X)\sim\mathcal{N}_{0,\sigma}$ for $X\sim\mathcal{U}[0,2\pi]$

I need to choose the probability distribution $f_Y(y)$ of a random variable $Y$ such that the variable $Z=Y\cos(X)$ is normally distributed with zero mean, i.e. ...
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### If a generated subgroup is cyclic

I would like to make a similar question to question "Exercise on generated subgroup": Let $G$ be a finite group and $H\leq G$, $H$ cyclic with $|H|=exp(G)$. If $x\in C_{G}(H)\smallsetminus H$, then ...
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### Non-differentiability of a function of two variables at a point

I have problems understanding this: Function $\;g(x,y)\;$ is given, for which a) $\;g_x(0,0)=7\;$ b) $\;g(t+2t^2,\sin3t+4t^2)=5e^t\;$ c) $\;\lim_{t\to 0}\frac{g(t,2t)-g(3t,4t)}t=10\;$ They ask ...
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### A derivative which is not Lebesgue integrable on any interval?

If $f=x^2\sin(x^{-2})$, then $f'$ exists everywhere (including $x=0$) but $f'$ is not Lebesgue integrable on $[0,1]$ (precisely because of the singularity at $x=0).$ I'm trying to find a function $f$ ...
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### Tate's thesis - continuous map from a local field to circle group

I am currently reading Decomposition of Unitary Representations defined by a discrete subgroups of nilpotent groups, by C.C. Moore. It is metioned that if $\mathbb{K}$ is a $p$-adic field in his ...
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### Power series coefficients

I've been trying for days now to find a closed form for the coefficients of the power series about $x=0$ of the function $$f(x)=\exp\left(r^2\frac{x(n-2)-x^2(n-1)+x^n}{(x-1)^2}\right),$$ but I ...
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### Solving Coin Toss Problem

If a coin is tossed 3 times,there are possible 8 outcomes. HHH HHT HTH HTT THH THT TTH TTT In the above experiment we see 1 sequnce has 3 consecutive H, 3 sequence has 2 consecutive H and 7 sequence ...
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### Proving the differentiablity of a function.

Consider the differentiablity of the following function: $$f(x)=x\left(x+3\right)e^{-\frac{x}{2}}$$ My text proves the differentiability by taking 'Left Hand Derivative' and 'Right Hand Derivative' ...
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### Characteristic polynomial factor over the real numbers

Ve=the set of symmetric 2x2 matrices I'm trying to show that any element of Ve has a characteristic polynomial that factors over the real numbers and has two distinct eigenvalues unless the matrix ...
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### Solving using separation of variable

First of all, I'm learning how to apply separation of variable method, and the first question I came across is this. Solve $y'= 2x + y$ using separation of variables with substitution $u = 2x + y$. ...
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### Bochner: Lebesgue Obsolete?

Bochner's notion of integral: $$F\text{ Bochner integrable}:\iff \exists S_n\in\mathcal{S}:\quad \int\|S_m-S_n\|\mathrm{d}\mu\to 0\quad(S_n\to F)$$ This version totally circumvents Lebesgue's notion ...
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### arranging the variables around when using inverse

If I want to show that $a$*$x$*$a^{-1}$ = $y$, is it acceptable to show that $x$*$a$*$a^{-1}$ = $y$ which then simplifies to $x$*$1$ so $x$=$y$? If not, how could I reorder them using what property?
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### Krull dimension in finite ring extensions

Let $K$ be a field and $R=K[a_1, \dots, a_n]$ a finite ring extension. Suppose that the degree of transcendence of $R$ over $K$ is $r$. Then the Krull dimension of $R$ is at most $r$. I would like to ...
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### Triple Integral in Spherical Coordinates.

$\newcommand{\de}{\operatorname{d}}$A little stuck on this one. $$\iiint_V ye^{-(x^2+y^2+z^2)^2}\,{\rm d} V$$ Use Spherical Coordinates to evaluate where V is the solid that lies between y=0 and the ...
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### Elias Stein : Real Analysis

I cannot understand why this particular line in the text is true: " Moreover, there are $O(k^{d-1})$ cubes in $\cal{Q}\ '$ " For the text see ...
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### divisibility on prime and expression

This site is amazing and got good answer. This is my last one. If $4|(p-3)$ for some prime $p$, then $p|(x^2-2x+4)$. can you justify my statement? High regards to one and all.
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### Alternative hash table analysis

Let us say that we have to hash $n$ elements to $m$ hash slots. Now what could be the average length of a chain. We can assume that prob. that 2 elements will map to a particular location will be ...
Is Mean Value Theorem (Rolle's Theorem) applicable for the following function: $$\log \frac{x^2 + ab}{(a+b)x}$$ in the interval $(a,b)$ My text says that it's applicable. But isn't the function ...