# All Questions

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### Are these definitions of a continuous random variable equivalent?

In a textbook I'm reading: A random variable $X$ is continuous if there exists a function $f_X$ such that $f_X(x) \le 0$ for all $x$, $\int_\infty^\infty f_X(x) dx = 1$ and for every $a \le b$, ...
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### subtraction method

I have studied long ago different method for subtracting two numbers, for instance the borrow method or the Austrian method (I hope I am using the right names; I am not an english native speaker and ...
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### Explication on how obtaining $\int \langle \nabla w, \nabla w\rangle = \lambda \int \langle w, w\rangle$

Could anyone is able to explain to me how to obtain $\int \langle \nabla w, \nabla w\rangle = \lambda \int \langle w, w\rangle$ related to user7530's comment in the question : Rayleigh quotient ...
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### Separability and reduced tensor fields

Let $L/K$ be a finite field extension. I need to prove that if $L/K$ is separable, then $E\otimes_KL$ is reduced for every algebraic field extension $E/K$. I've readed that I need to use the ...
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### Unable to remove the indeterminacy of a limit.

Evaluate $$\lim_{x\to 1} \frac{p}{1-x^p}-\frac{q}{1-x^q}$$ where $p,q$ are Natural Numbers.  I tried rationalization, but I wasn't able to get anywhere. I'm not being able to remove the ...
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### Constructible $n$-gons

Let $\xi$ be the primitive root of unity. If $n=5$, then the minimal polynomial of $xi$ over rationals would have degree $5-1=2^2$ which is a fermat prime so $5$-gon is constructible. If $n=8$, ...
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### Derivative of convolution type integral equation with respect to time

Consider the equation: $$x(t) = \int\limits_{0}^t \gamma^{t-s} u(s)ds$$ where $x, u$ are functions and $\gamma \in \mathbb{R}_{>0}$ I am trying to obtain the derivative of this equation with ...
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### Total derivative using $(x,y)\mapsto f(x,y)$ notation

Related to What is the difference between $\frac{\mathrm{d}}{\mathrm{d}x}$ and $\frac{\partial}{\partial x}$? Is it possible to make sense of $(x,y)\mapsto f(x,y)$ when taking the total derivative ...
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### Evaluating the integral of $\frac{\cos(x) - e^{-x}}{x}$ using contour integration

I am trying to evaluate the value of $$\int_0^\infty\frac{\cos(x) - e^{-x}}{x}dx$$. I am assuming I am supposed to use contour integration, as I was required just before to calculate the value of ...
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### Probability of stopping a production line even when the failed ratio is not more than 6 percent

Imagine a company has someone who controls the production line and every time he does that, he choses 15 products and makes sure that no more than 1 product has failed according to the rules. If more ...
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### Can someone help me with this question of finding x as exponent?

The equation is: $$6^{x+1} - 6^x = 3^{x+4} - 3^x$$ I need to find x. I forgot how to use logarithm laws. Help would be appreciated. Thanks.
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### Set of marginals is convex

Let $[n] = \{1,2,\cdot,n\}$ and $[m] = \{1,2,\cdot,m\}$. Let $Z_{1,2}$ denote the set of all probability distribution on the Cartesian product $[m]\times [n]$. Let $S_{1,2}$ denote a convex closed ...
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### How to create a function for residual sum of squares in matlab

I need to write a matlab loss function for the residual sum of squares function. in particular: $$G(\beta)=(\sum_{i=0}^n y_i-X\beta)^2$$ where y is a 60 x 1 vector x= matrix of 5 vectors ...
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### Solving Binary Linear Programming Problem Using KKT

Execuse me, I know that if I searched a lot I could find the answer, However I have already did my research and I am running out of time. I need the detailed solution of the following linear problem ...
Let $\mathcal{C}$ be the category of abelian groups endowed with tensor product. It is a monoidal category with unit object the integers. Given a monoid $(R, \mu_R , \iota_R)= R$ in this category ...
### Find all positive integers that solve Mordell's equation $y^2=x^3+37$
Find all Mordell's equation: $$y^2=x^3+k$$ where $k=37$ positive integer numbers,I can't find the when $k=37$ the mordell equation solution with some result,and we can known this equation have ...