on the necessity of gluing conditions March 23

Suppose we are given a family of schemes $\left\{X_i\right\}_i$, with $U_{ij}$ open in $X_i$ such that there exists isomoprhism $\phi_{ij}: U_{ij} \rightarrow U_{ji}$. Why do we need the condition $\phi_{ij}(U_{ij} \cap U_{ik}) = U_{ji} \cap U_{jk}$

Proving an integration with a modified Bessel function and an exponential March 23    1

Proving an integration with a modified Bessel function and an exponential
I am trying to prove the following identity:where $\mu, h, H$, and $\tilde{\gamma}$ are real constants. The only hint that I have is use the relation between the modified bessel function of ...

Verification of Fourier transformation of Io-sinh function March 23

Verification of Fourier transformation of Io-sinh function
I try to match, but it could not match $I_o-\sinh$ Practical Fourier Transform pair developed by Ben Logan, transform pair also published in The Practical Application of the Fourier Integral ...

Mathieu function rescale problem March 23    1

Mathieu function rescale problem
The Mathieu functions are the solutions for the equation $$ y''+(a-2q\cos(2z))y=0 $$If we require the solution has the form $$ y(z) = e^{i r z}f(z) $$where $f(z)$ is a periodic function with ...

How to prove that $a-b \ge \sqrt3{ab}$ March 23

Let $a$ and $b$ be distinct positive integers such that $ab(a+b)$ is divisible by $(a^2+ab+b^2)$.How to prove that $|a-b| \ge \sqrt[3]{ab}$? I have no idea how to do this. Can this be proved with simple way?

Initial value problem for a system of ODEs with two parameters March 23    2

Initial value problem for a system of ODEs with two parameters
Consider the following initial value problem for the autonomous system of ODEs \begin{equation}%%\label{eqn: } \begin{cases} %\vspace{3mm} x'(t)=y(t),\; t>0,\\ %\vspace{3mm} y'(t)=-\frac{ ...

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