day 106: the topic of the night is graphs

Salut, students.

The content below is courtesy of the venerable numb3rs.wolfram.com. It’s a little peek into the geometric realm of mathers, graphers, and pathers. (Not panthers.)

A regular graph of degree r is a graph all of whose vertices have the same number of edges incident on them. Regular graphs therefore have a high degree of symmetry, and include a number of particularly beautiful examples, such as those illustrated above.

There is a class of graphs known as strongly regular graphs that are even more symmetric than regular graphs. In a strongly regular graph, not only does each vertex have the same number of neighbors (usually denoted k), but every adjacent pair of vertices has the same number (usually denoted lambda) of common neighbors, and every nonadjacent pair of vertices has the same number (usually denoted mu) of common neighbors. A strongly regular graph on nu vertices is therefore said to have parameters (nu, k, lambda, mu). Strongly regular graphs are very special mathematical objects that have connections to other areas of mathematics, including group theory and statistics. One class of strongly regular graphs is the so-called lattice graphs Ln, whose vertices can be thought of as squares on an n×n chessboard, and whose edges can be thought of as pairs of squares that are connected by a possible move by a rook chess piece. It turns out that the lattice graphs are strongly regular, in particular L4, which is strongly regular with parameters (16, 6, 2, 2). Shrikhande established that there exists exactly one other strongly regular graph with those parameters, and that graph, illustrated below in another embedding, now bears his name.

Good night math lovers, graph lovers, and path lovers. (And laugh lovers.) Pursue symmetry relentlessly, in life and in dreams. Keep on keeping on, with a fire and fight.

KT out

day 10: mobility

“A structure’s mobility is the amount it vibrates when driven by a fluctuating force.” –Stephen A. Hambric

 

KStar’s mobility is continually on my mind. (If only I could put it into my body so painlessly.)  It’s driven me to earn a degree in medicine, to develop a practice in orthopedic surgery, to work toward becoming a competitive CrossFit athlete and Olympic weightlifter, and to carry a lacrosse ball on my person at all times.  The mobility discussed by structural acousticians is not the mobility I care to study.

 

My task: explain, in layman’s terms, the transmittal of structural vibrations through a surrounding medium, focusing specifically on the mechanism for the resulting far-field acoustic radiation pattern.  This is a well-studied problem, so papers exist which lead me through the differential equations and transforms that explain the physics of propagation.  To complete my task it seems essential I explain why the structure vibrates as it does at a given frequency for a particular force; this is the first link between the structural vibration to the near-field radiation.  But nobody explains it! Instead they say “and we call this mobility.” I know, I know, its vibration is a certain amplitude for a given force because of the atoms and their crystalline/non-crystalline structure and the conditions of the environment and so on and so forth but settling on defining something that can be derived seems a bit incomplete.  Shamefully, I suspect my bitterness derives less from principle than from the thought of the work I need to do.  Oh, if I could use equations, this would all fit on one little line.  Math is lovely because it forces brevity, which I clearly lack.

 

Math, like one of KStar’s elastic bands, is a distractor.  Both allow me to do fantastical things, such as being brief or squatting with my knees out.  Yayyy math! Yayyy bands!

 

Also, I just got a paper cut.  CrossFit, Oly and mobility WOD are critical dampers to the simple harmonic motion of my wimpiness, but Jesus Christ this cut is throbbing.

 

Whoa–reallllll classy, KT. JC was crucified and you’re calling on Him for a paper cut.  

day 1: categories

“Combinations” tops the list of my favorite mathematical formalisms. Their symmetry is delightful; their count of possibilities, calming and quick. Moreover they address categories of a whole; exclusive categories are exact, clean, pleasing. Understanding why C(n,k) gives the number of k-combinations from a given set of size n is a proof I must reprove each time I approach combinations; it slips from my mind’s grasp so easily.

Their elegance drew me to statistical mechanics and that fascinating world of quantum particles. What truly boggled me were those most basic categories of particles: bosons and fermions. I wrote a thesis on these. Dear dad routinely called my bosons “bison.” He is the funniest dad I know. Just hilarious. “Today I am going to speak to you about bison, who can occupy multiple states at the same time, such as Minnesota and Montana, even when they are identical.”

Categories must be mutually exclusive to be viable for combination techniques. The following categories are not so incongruous

Goals

Physical

• 10 dead hang pullups
• 250 lb deadlift
• incredible squat form: upright torso, toes forward, knees out, solid balance against wall
• ability to do all CrossFit movements including handstand pushups, pistols, consecutive double-unders, and consecutive toes-to-bars
• 30 burpees in less than two minutes, 50 burpees in less than five minutes
• splits

Career

• prepare for MCAT each day
• find and begin research
• be selected for full-time scribing
• complete all pre-requisite courses

Personal

• blog once a day; purpose may range from accountability to drills to comprehension
• explore
• become a lululemon ambassador
• become mobility certified (and a supple leopard)
• correspond once a week
• every time I have a mean thought, generate a kind and positive one
• reconnect with my faith: pray morning and night

And Aby’s guest appearance:

Aby’s Goals

Physical

• complete shoulder rehabilitation
• enroll in Olympic lifting class
• consecutive muscle-ups
• eat clean for a six pack
• eat clean for a strong immune system
• eat clean to reduce LDL
• attain and practice perfect form in all movements

Career

• red shirt
• expand business knowledge-base through reading and conversations with entrepreneurs and business-owners

Personal

• monthly date with Kristina (Kristina’s reaction: woohoo!!!)
• make our house our home: paint, fill with physical symbols of our passions, our family, and our love

A pinch of salt

WHEN a dream is born in you

With a sudden clamorous pain,

When you know the dream is true

And lovely, with no flaw nor strain,

O then, be careful, or with sudden clutch

You’ll hurt the delicate thing you prize so much. 

Dreams are like a bird that mocks,

Flirting the feathers of his tail.

When you seize at the salt box,

Over the hedge you’ll see him sail.

Old birds are neither caught with salt nor chaff:

They watch you from the apple bough and laugh. 

Poet, never chase the dream.

Laugh yourself and turn away.

Mask your hunger; let it seem

Small matter if he come or stay;

But when he nestles in your hand at last,

Close up your fingers tight and hold him fast.

Robert Graves

Aby and I are looking at townhouses. (!!)