Watson-Crick pairing, the Heisenberg group and Milnor invariants

We study the secondary structure of RNA determined by Watson-Crick pairing without pseudo-knots using Milnor invariants of links. We focus on the first non-trivial invariant, which we call the Heisenber invariant. The Heisenberg invariant, which is an integer, can be interpreted in terms of the Heisenberg group as well as in terms of lattice paths. We show that the Heisenberg invariant gives a lower bound on the number of unpaired bases in an RNA secondary structure. We also show that the Heisenberg invariant can predict allosteric structures for RNA. Namely, if the Heisenberg invariant is large, then there are widely separated local maxima (i.e., allosteric structures) for the number of Watson-Crick pairs found.

Portrait of Blanche and Richard Stoppelman (Stoppelmann?) Portraits Couples

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An improved technique for estimating short-term survival of released line-caught fish, and an application comparing barotrauma-relief methods in red emperor (Lutjanus sebae Cuvier 1816).

Promotion of better procedures for releasing undersize fish, advocacy of catch-and-release angling, and changing minimum legal sizes are increasingly being used as tools for sustainable management of fish stocks. However without knowing the proportion of released fish that survive, the conservation value of any of these measures is uncertain. We developed a floating vertical enclosure to estimate short-term survival of released line-caught tropical and subtropical reef-associated species, and used it to compare the effectiveness of two barotrauma-relief procedures (venting and shotline releasing) on red emperor (Lutjanus sebae). Barotrauma signs varied with capture depth, but not with the size of the fish. Fish from the greatest depths (40-52 m) exhibited extreme signs less frequently than did those from intermediate depths (30-40 m), possibly as a result of swim bladder gas being vented externally through a rupture in the body wall. All but two fish survived the experiment, and as neither release technique significantly improved short-term survival of the red emperor over non-treatment we see little benefit in promoting either venting or shotline releasing for this comparatively resilient species. Floating vertical enclosures can improve short-term post-release mortality estimates as they overcome many problems encountered when constraining fish in submerged cages.

Reproduction of Richard Dehmel reading with his wife Ida Dehmel nee Auerbach sitting across from him, early in their marriage; Parkstrasse, Hamburg Portraits; Couples

Verso: Erste Wohnung Parkstrasse Fruehzeit der Ehe 1901

Therese Godshaw (Gottschalk) nee Molling holding her great-grandson, Richard Godshaw; Buffalo, New York.

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Studio portrait of Richard Molling, Margot Molling, Adolf Molling, Ilse Molling, Lena Molling nee Marx; Schoenau am Koenigssee, Germany.

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House of Karl Gottschalk on Richard-Wagner-Strasse; Hannover, Germany.

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House of Karl Gottschalk on Richard-Wagner-Strasse; Hannover, Germany.

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Therese Godshaw (Gottschalk) nee Molling holding her great-grandson, Richard Godshaw; Buffalo, New York.

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Dr. Simberg, Rudolf Mosse, Richard Israel and other unidentified men hunting with Count Czarnecki Portraits; Children

Jagd beim Grafen Czarnecki

Portrait of Richard Beer-Hofmann Portraits Men

Signed by photographer bottom right

Studio portrait of Richard Molling, Margot Molling, Adolf Molling, Ilse Molling, Lena Molling nee Marx; Schoenau am Koenigssee, Germany.

Text on the back: Zur Erinnerung an den Sommer in Schoenau-Koenigsee 1921. Mollings.