Some more tips on writing an SOP:

– Your qualifications are already in your your CV. So, do not include numbers and dates and other details.

– Tell a story rather than having scattered facts. Try to weave all you want to say into a unified piece that has flow. If your paragraphs can be reordered freely, then it is a sign your essay lacks cohesion.

– Decide on an impression you want to make and adjust your language and choice of details accordingly. For instance, you may want to present yourself as an organized and hardworking person than, say, a smart/genius one. Deciding on the personality image you want to convey will really give your essay a soul!

– Include specific details not just about yourself, but also about the department. Let them know that you know them, and that they are not “just another school on your list.”

Good luck đź™‚

]]>I really wish I could remember the original problem where I encountered this. I think the odds of success approach 1-ln 2 as the number of hats and boxes approaches infinity, which works out to just over 30%. Can anyone point me to the original?

]]>https://youtu.be/R1rzI0Y_d3I

The 5-cell is an analog of the tetrahedron.

https://youtu.be/BjvdrhK8yws

Tesseract is a four-dimensional hypercube – an analog of a cube.

https://youtu.be/Pa0c7M4lZv0

The 16-cell is an analog of the octahedron.

https://youtu.be/np0ZxC1wXqc

The 24-cell is one of the regular polytope.

https://youtu.be/T01qw0_qitI

A hypersphere is a hypersurface in an n-dimensional Euclidean space formed by points equidistant from a given point, called the center of the sphere.

The “good part” is linked as Part 1 in the first paragraph.

]]>$A = mathbb{Z}$, $mathfrak{p} = 2mathbb{Z}$, $I_j = p_j mathbb{Z}, j > 1$.

The intersection of the $I_j$ is ${ 0 } subset mathfrak{p}$, but no $I_j$ is a subset of the initial prime ideal. ]]>