Even existence requires triangle inequalities

Regularity theorems for compound bubbles or immiscible fluids require that the weights satisfy triangle inequalities. For example, if the interface between two regions of prescribed planar areas has high cost, two tangent disks minimizes weighted perimeter.

Here we note that even existence requires triangle inequalities, even for three fluids. Consider weight 3 for the 01, 02, and 12 interfaces (and weight 1 fo the rest), with say unit areas for 1 and 2 and small area for 3. Now the minimizer should be some kind of 12 double bubble inside a 3 bubble, with 13 and 23 interfaces coinciding with 30 interfaces. At a singularity, 13 and 30 interfaces meeting tangentially and then coincide, conjecturally the only new type of singularity.

Inspired by a question from Gary Lawlor.

Christmas Bridge at the Shore

Much enjoyed the festive Christmas party bridge games at Karen Sylvester’s Cape May County Duplicate Bridge Club December 9 and Martin de Bruin’s Shore Bridge Club December 10 (thanks to Terri U’Prichard). Great refreshments included Jane Havighurst’s cinnamon cake. Roger Read and I had a fortunate 76% game.

With Michael Berkowitz at Fall 2021 Austin Bridge Nationals

After partners had to cancel, I paid to play in Nail Pairs with Michael Berkowitz, son of David and Lisa Berkowitz, a clear and friendly teacher, a member of the board of the Educational Foundation. With a 63.86% game in the second session, we qualified for the finals and earned 7. 27 Platinum points. One highlight was this squeeze at 3N for an overtrick for a tie for top. (See below; actual bidding was 1N 3N. Double dummy makes just 3N.) East led the H6 and then overtook his partner’s Q to return a third Heart to my Ace. After three rounds of Clubs ending in my hand, I advanced the S10, East failing to cover, to isolate the Spade menace to East who was also guarding the Hearts. In the end position I was on the board with SA7 and D2, with HA5 and DA in my hand. East could not guard both suits and had actually chosen to unguard the Hearts, so I played the Aces of Spades and Hearts and won the last trick with the H5 in my hand.

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Here are some poems of mine, the first written in college, though it sounds as if I could have written it recently. Continue reading ‘Poems’ »

Wulff Shape Isoperimetric

The following old post-publication correspondence provides some information and questions on the original proof that the Wulff shape minimizes anisotropic surface energy for fixed volume, as treated in our paper:

MR1297699 (95g:49080)
Brothers, John E. (1-IN); Morgan, Frank (1-WLMS)
The isoperimetric theorem for general integrands.
Michigan Math. J. 41 (1994), no. 3, 419–431.
49Q20 Continue reading ‘Wulff Shape Isoperimetric’ »

Trained Hawks, Falcons, and Owls

Ocean City, New Jersey, hires hawks, falcons, and owls to ward off the seagulls. Recently seagulls attacked me and my dinner at the Ocean Cafe on the Boardwalk, and they called in a hawk to chase them away. May 25 I found a dead hawk on my beach at 17th Street:

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Books and Movies 2021


Verdi’s Un Ballo in Maschera, Met Opera Stream https://www.metopera.org/

Verdi’s Requiem https://m.youtube.com/watch?v=fjUQ8siaEj4

Hans Christian Andersen https://m.youtube.com/watch?v=DTYd0wWBd5o&has_verified=1

The Professor and the Madman (Netflix)

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Books and Movies 2020

Moonlight (gay black, Netflix)
The First Temptation of Christ (Netflix)
Love Lettering by Kate Clayborn (Meg and Reid)
Star Trek Picard U

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Sea Creatures (Ocean City)

[See also iNaturalist and some more professional, California PHOTOS by my friend Abby Thompson.]

Blue Crab, Laughing Gull









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Ocean City

After four itinerant years since retirement, on September 28, 2020, I bought a small condo on the Boardwalk in Ocean City, New Jersey. Living on the ocean has been a life-long dream. I have daily sunrise walks, bike rides on the Boardwalk, and body surfing in the ocean (even now in November in my wet suit). I’m also playing lots of bridge online with my favorite partners from across the country and beyond. Live cam of my beach.  More  photos  at  SeeingHappy.com.BodySurf3 Continue reading ‘Ocean City’ »