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Let $\delta >0$ and consider $j\in \mathbb{N}$ such that $\delta 2^j >\frac{1}{2\sqrt{2}}.$ I am looking for an upper bound of
$$f(x) := e^{-\delta x^2 }\frac{\sinh\left(\sqrt{\delta^2 x^4 -x^2}\right)}{\sqrt{\delta^2 x^4 -x^2}}$$
on the interval $\left[2^{j-1}, 2^{j+1}\right]$ of the form
$$f(x)...
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I am self-studying measure theory using Measure Theory by Donald Cohn. The book makes the following definition:
Definition$\quad$ Suppose that $f:X\to[-\infty,+\infty]$ is $\mathscr{A}$-measurable and that $A\in\mathscr{A}$. Then $f$ is integrable over $A$ if the function $f\chi_A$ is integrable...
I am an engineering student and am trying to prove the following combinatorics identity in math:
$$\sum_{{m=k}}^{N} C(m,k) = C(N+1, K+1)$$
It was suggested to me to use Proof By Induction so I tried to do this problem.
Step 1: Show this identity is true for a specific choice of $N=K$
LHS:
$$\s...1
I want to prove the following theorem:
Let $S$ be a compact surface, and $N:S\rightarrow \Bbb{S}^2$ the Gauss map, then we have
$$ \int_{S} |K| \,dA = \int_{\Bbb{S}^2} \#N^{-1} \,dA $$
where $K$ is the Gaussian curvature of the surface $S$, and $\#N^{-1}$ is the number of the preimages of the Ga...
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This is a question I have had for many years and have always wondered about it.
In my home country, French Fries (https://en.wikipedia.org/wiki/French_fries) are a very tasty snack. However, we don't always have access to deep fryers (https://en.wikipedia.org/wiki/Deep_fryer) and end up cooking F...
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A space is realcompact if its a closed subspace of an arbitrary product of real lines, with product topology.
A cardinal $\kappa$ is called measurable if there exists a (countably additive) $\{0, 1\}$-valued measure $\mu:\kappa\to \{0, 1\}$ with $\mu(\kappa) = 1$ and $\mu(\{x\}) = 0$ for $x\in \k...
In wikipedia, the notion of straight line is described as a basic notion, a primitive that is not defined. I wonder if there're any formal definition for a straight line in any specificular context so far, for instance in $\mathbb R^n$, in differential geometry .. ? Thanks.
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Let $((-\pi,\pi], \mathcal{B}((-\pi,\pi]), m)$ be a measure space with Lebesgue measure $m$. Show that
$$
\int_{-\pi}^\pi |\lim_{m\to\infty}\lim_{d\to \infty} \sqrt{d}\cos(m!x)^{2d}|dx=0
$$
and
$$
\lim_{m\to\infty}\lim_{d\to \infty}\int_{-\pi}^\pi | \sqrt{d}\cos(m!x)^{2d}|dx=1/\sqrt{\pi}
$$
It s...
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My question relates to the theorem and proof given in Erwin Kreyszig's Advanced engineering mathematics:
Theorem 1 is as follows:
My question is as follows: In the proof of Theorem 2, what is the justification for concluding $a_{m+1}$ = $b_{m+1}$, given that $a_{m+1}$ = $b_{m+1}$ only holds whe...1
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