# All Questions

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### Are multi-valued functions a rigorous concept or simply a conversational shorthand?

In Brown and Churchill's book, the concept of multivalued functions is not discussed in a very rigorous way (if at all). But I can see that branch cuts have importance in complex analysis, so I want ...
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### Function whose power series coefficients contain logarithms

Is there a function that can be expressed as a power series $$f(x)=\sum_{n=0}^\infty a_n x^n$$ whose coefficients $a_n$ are expressions containing $\log n$ or something similar?
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### Why can we make this integral change of limits? Is it obvious?

When deriving the equation for the impulse-momentum theorem, the following occurs: $...=\int\limits_{t_1}^{t_2}\frac{d\vec p}{dt}dt = \int\limits_{\vec p_1}^{\vec p_2}d\vec p=...$ I know the $dt$s ...
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### Represent a prime number $p$ congruent to $1$ $\pmod{3}$ by a sum of a square and $3$ times a square

I want to have a proof of the fact that each prime number is the sum of a square and three times a square (Euler).
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### Finding the perimeter of the room

If the length and breadth of a room are increased by $1$ $m$, the area is increased by $21$ $m^2$. If the length is increased by $1$ $m$ and breadth is decreased by $1$ $m$ the area is decreased by ...
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### Proving independence of random variables

If $X$ and $Y$ are independent exponential random variables with parameter $\lambda$ and $\mu$. Let $Z=\min(X,Y)$, prove that $Z$ and $\mathbf 1_{\{X<Y\}}$ are independent. I don't know, how ...
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### An algebraic topology proof of a result from analysis

A colleague of mine recently brought up the following result from real analysis: Theorem: If $f:\mathbb{R}^2\to\mathbb{R}^2$ is continuous and $|f(x)-f(y)|\geq |x-y|$ for all $x,y$, then $f$ is onto. ...
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### Solutions to functional equation $f(a_1t+x_1,a_2t+x_2,x_1,x_2)=g(b_1t+x_1,b_2t+x_2,x_1,x_2)+h(t)$.

Find all functions $f$, $g$, and $h$ such that $$f(a_1t+x_1,a_2t+x_2,x_1,x_2)=g(b_1t+x_1,b_2t+x_2,x_1,x_2)+h(t)$$ Is it possible (and what I was hoping for) to express the solution as ...
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### Vectors with given angle and magnitude

Give an example of vectors $\mathbf{v}$ and $\mathbf{w}$ such that the angle between $\mathbf{v}$ and $\mathbf{w}$ is $\frac{2\pi}{3}$ and $||\mathbf{v} \text{ x } \mathbf{w}||=\sqrt{3}$. Should I ...
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### Weak convergence plus strong convergence

Let $H$ be a Hilbert space and let $(x_n), (y_n)$ be sequences in $H$ such that $(x_n)$ converges strongly to $x$ and $(y_n-x_n)$ converges weakly to 0. I can show that $(y_n)$ converges weakly to ...
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### Paraboloid Curvature calculation methods

If we have a paraboloid generated as a surface of revolution of the 2d function $f(x)=ax^2+b$, the equation of the 3d graph is $f(x,y)=ax^2 + ay^2+b$. The gaussian curvature of a 3d graph $f(x,y)$ is ...
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### A certain two-step subgroup of a nilpotent group

Let $\Gamma$ be a finitely-generated, torsion-free, nilpotent group, of nilpotency class $n\ge 2$. Is there an $N \lhd \Gamma$, such that (i) $N$ is two-step nilpotent, (ii) $\Gamma / N$ is torsion ...
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### How to Self-Study Higher Math Without Solutions

I've been lurking on this site for several months, and as someone studying higher mathematics independently (i.e., outside of a college/institutional setting), this forum has probably been the best ...
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### Exponential Growth Differential Equation

A population of buffalo grows exponentially (the rate of growth is determined by the population itself) but has a carrying capacity. Its population (in tens of thousands) at a time t ( in years ) is ...
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### range of one increasing computation function? [on hold]

We know that that the range of any recursive partial function is recursively enumerable. I see in page 154 in books "Problems in Set Theory, Mathematical Logic and the Theory of Algorithms" that the ...