# All Questions

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### Intersection of perpendicular tangent lines - generalization of directrix?

This is a funny little problem that I came up with. For a differentiable function $f$, define a locus of points $P$ as follows: Let $m$ be an arbitrary tangent line to $f$, and let $n$ be another ...
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### What is the generalized solution to $M = \int_0^1 (1-x^m)^n$?

What is the generalized solution to $$M = \int_0^1 (1-x^m)^n \,dx$$ (where it can be expressed in terms of n and m) and m and n are rational numbers if considering m and n as natural numbers makes ...
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### If $gcd(a,b)+LCM(a,b)=gcd(a,c)+LCM(a,c)$, then are b and c equal?

Is it true that, if $gcd(a,b)+LCM(a,b)=gcd(a,c)+LCM(a,c)$, then are $b$ and $c$ equal? For coprime $(a,b)$ and $(a,c)$, it is trivial. It is also easy when $a|b$ and $a|c$. I am unable to construct ...
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### Christoffel Symbol in terms of determinant of the metric

It is generally known that, if g is the metric, $$Γ^𝜇_{\mu\alpha}= \partial_{\alpha}(ln\sqrt{|g|}).$$ It is also known that Christoffels are symmetric on its second and third components, meaning ...
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### How do we evaluate this REALLY tricky integral?

$$\int{\frac{\cos 9x + \cos 6x}{2 \cos 5x-1} dx}$$ The objective is to find the answer in terms of $\sin 4x$ and $\sin x$ I would like to share my attempt and then ask a conceptual doubt as usual but ...
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### Showing that $\{x \in X: \sup_n\sum_{i=0}^n f(T^i(x)) = \infty\}$ is a $T$ invariant set

Let $(X, \sigma, \mu)$ be a probability space, $T:X\to X$ a $\mu$-invariant map and $f\in L^1(\mu)$. Define $A := \{x \in X: \sup_n\sum_{i=0}^n f(T^i(x)) = \infty\}$. I would like to conclude that $A$ ...
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### Markov chains: Prove vector of mean hitting times is minimal non-negative solution

Definitions: Let $H^A = \inf \{n \geq 0 : X_n \in A \}$ be the first hitting-time of the set $A$. Let $k_i^A = \mathbb{E}_i(H^A)$ be the mean hitting time of $A$ from $i$. Theorem: The vector of mean ...
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### Hadamard Gap Theorem and Lacunary Functions

Does anyone know any good reference or even a simple proof for Hadamard Gap Theorem or even just the fact that a lacunary function diverges at 1 (I mean the limit not just the evaluation). In fact, I ...
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### Vector Newton's method without using matrix inverse?

For a set of non liner equations $f_i(\vec x)$ were $i\in \mathbb{N}$ was the index, one can construct a vector F(\vec x)= \begin{pmatrix} f_1{\vec x} \\ f_2(\vec x) \end{pmatrix} \...
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• 3,544
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### Rate of change of depth of a liquid in a spherical container

I'm just wondering about how you would do this question but I am unable to type it using LaTeX as for some reason it is not properly covnerting my code to the correct equation but here it is as an ...
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### Fractional Brownian Motion and Infinite Crossings

It's well known that regular Brownian motion crosses any point infinitely many times. Is this fact also true for fractional brownian motion? In particular when the hurst parameter $H>1/2$.
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### Constructing Smooth Manifolds

I’ve been trying to teach myself differential geometry using Will Merry’s notes (found here) and am struggling to prove Proposition 1.17 on page 7. Here’s the statement… Let $M$ be a set. Suppose we ...
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### Existence of a non-trivial linear combination of k+1 orthonormal linearly indep. vectors in the left nullpace of a matrix with k columns

Suppose $v_{1}, ..., v_{k+1}$ is a linearly independent, orthonormal set of vectors in $R^{n}$. Prove that for $Y \in R^{n \times k}$ there exists a non-trivial linear combination of these vectors $w$ ...
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1 vote
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### Advice on proof-based exercises and more theoretical math subjects

i am a soon to be graduate in physics and i am considering to do a master in the theoretical physics area, but i find myself a little bit "unfaithfull" of my mathematical skills. Let me ...
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### Maximization optimization problem of possibly log-concave function?

I'm trying to solve a non-convex optimization problem, trying to figure if their are any tricks to transform it into an approximate convex form. here's a simpler form of the problem I'm trying to ...
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1 vote
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### How to prove that $\sum_{i=1}^{n} i^{-2} \leq 2 - \frac{1}{n}$ for all $n \geq 2$?
Prove that $$\sum_{i=1}^{n} i^{-2} \leq 2 - \frac{1}{n}$$ for all $n \geq 2$ The above series is coming to be like $1+1/4+1/9+1/16+...+1/(n^2)$. But how to prove the above inequality? I also tried to ...