# All Questions

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### Covariance of m-fold integrated Wiener process

The problem I'm trying to perform a Bayesian approach to the Maximum Likelihood Estimation procedure of Wecker and Ansley (1983). To this end, I need to compute the full likelihood of the data given ...
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### Prove that there are no integer solutions to a given equation

Prove that $4x = y^2 + 1$ has no integers solutions for (x,y) By rules of divisibility: $$a | b \implies \frac {b}{a} = n$$ for $a,b,n \in \mathbb{Z}$ So let, $a=4x$, $b=y^2$,and $c =1$ ...
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### Thoughts on Mac Lane's Homology for learning basic cohomology theory related to algebraic topology

I apologize, this may be a duplicate question as many reference requests here are. For those who have read or have experience with the content covered in Mac Lane 's Homology, what are your thoughts ...
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### Curvature of curve

$r(t) = (-3sint)i + (-3sint)j + (cost)k$ I got as far as:$$||r'(u)|| = sqrt{(18cos^2u + sin^2u)}$$ But I cannot evaluate $\int_0^t||r'(u)||dt$
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### Calling all genius: $p_1^{e_1} p_2^{e_2}…p_k^{e_k}=e_1^{p_1} e_2^{p_2}…e_k^{p_k}$

Find all positive integer $e_i$ and prime number $p_1^{e_1} p_2^{e_2}...p_k^{e_k}=e_1^{p_1} e_2^{p_2}...e_k^{p_k}$ for $k\ge 1$. Is this impossible or what? I've been trying for at least a week. I've ...
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### Solving a simple Distance Geometry problem

I'm trying to solve the following problem: Given the absolute positions of four points in 3D space, and the distances from these four points to a fifth point, find the position of the fifth point. ...
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### dimension of direct products

Suppose $\{V_i\}_{i\in I}$ is a family of $k$ vector spaces. Is it possible to calculate $\dim\oplus_{i\in I} V_i$ and $\dim\prod_{i\in I}V_i$?
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### Frobenius theorem for singular foliations , what are hypotheses impose over an endomorphism $P:TM\to TM$ that it span a singular foliation in M?

Let $M$ a smooth manifold. Given a morphism $P:TM\to TM$, i.e, $\pi\circ P\equiv Id$ and is linear over the fibers. If we suppose that $P^2=P$, then for each $x\in M$, $P_{x}:T_xM\to T_xM$ is a ...
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### showing 2 separate basis for a Vector space

Suppose $V$ is a vector space over the field $F$ and $V$ consists of more than the zero vector Suppose $V$ is spanned by a set $S$ of $s$ vectors and $V$ contains a linearly independent set $B$. ...
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### Geometry , gre problem

Smallest distance from a point P to any point on the circle C is 5 and the largest distance from the the point P to the circle is C is 11 . If point P is situated outside the circle C , then what is ...
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### How to prove which number is bigger??

Prove which number is larger: a) $10^{100}$ or $10^{10^{100}}$ b) $e^\pi$ or $\pi^e$ I know we all know how to plug these into the calculator and check, but how someone mathematically prove which ...
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### Very basic question about the definition of the derivative

Why is the definition of the derivative shown here as $\dfrac{\Delta x}{\Delta y}$ if immediately above the slope (derivative) is defined as $\dfrac{\Delta y}{\Delta x}$? Why is $\Delta x$ in the ...
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### I do not know how to slove it

If a child can run 10 meters while a car travels 30 meters, how many meters can the child run while the car travels 66 meters
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Using Newton's binomial theorem to argue that: $n \ge 1$
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### What is this function of 2 variables?

Can you tell me the function f(K,N) that has the following values? For my education, please also explain how you tackled the problem. ...
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### Finding derivative for multiple chain rules in one problem

Find the derivative of $$y=\sqrt{e^{-3t^2}+5}$$ There seems to be several layers to this. I'm not quite sure how to go from one to the next.
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### Maclaurin series accuracy

Find an $n_1$ such that the $n_1$th-order Taylor polynomial for $\sin(x)$ about $x=0$ gives an approximation of $\sin(x)$ with an error of less than $5\cdot 10^{-10}$, for all $x$ between $0$ and ...
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### When a symmetric densely defined operator is an adjoint operator?

I am wondering if it is possible to say that if a symmetric differential operator is densely defined then the operator is self-adjoint indeed? More Precisely, Let $A:D(A)(\subset H)\to H$ a densely ...
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### If $g(x) = \sqrt{10 − x}$ What is the Domain for $(g\circ g)(x) = g(g(x))$?

What is the domain for $10 − \sqrt{10 − x} ≥$ ?
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### Least sum of power of distances

Let $n$ points in a $3$-dimensional space. Are there any general intuitive methods to tackle the problem of finding the point $X$ such that the sum of distances $A_1X^x+ A_2X^x + ... +A_nX^x$ (where ...