Uniform random sampling of planar graphs in linear time

This article introduces new algorithms for the uniform random generation of labelled planar graphs. Its principles rely on Boltzmann samplers, as recently developed by Duchon, Flajolet, Louchard, and Schaeffer. It combines the Boltzmann framework, a suitable use of rejection, a new combinatorial bijection found by Fusy, Poulalhon and Schaeffer, as well as a precise analytic description of the generating functions counting planar graphs, which was recently obtained by Gim\'enez and Noy. This gives rise to an extremely efficient algorithm for the random generation of planar graphs. There is a preprocessing step of some fixed small cost. Then, the expected time complexity of generation is quadratic for exact-size uniform sampling and linear for approximate-size sampling. This greatly improves on the best previously known time complexity for exact-size uniform sampling of planar graphs with $n$ vertices, which was a little over $O(n^7)$.Comment: 55 page

An enumeration of equilateral triangle dissections

We enumerate all dissections of an equilateral triangle into smaller equilateral triangles up to size 20, where each triangle has integer side lengths. A perfect dissection has no two triangles of the same side, counting up- and down-oriented triangles as different. We computationally prove W. T. Tutte's conjecture that the smallest perfect dissection has size 15 and we find all perfect dissections up to size 20.Comment: Final version sent to journal

Seven Staggering Sequences

When my "Handbook of Integer Sequences" came out in 1973, Philip Morrison gave it an enthusiastic review in the Scientific American and Martin Gardner was kind enough to say in his Mathematical Games column that "every recreational mathematician should buy a copy forthwith." That book contained 2372 sequences. Today the "On-Line Encyclopedia of Integer Sequences" contains 117000 sequences. This paper will describe seven that I find especially interesting. These are the EKG sequence, Gijswijt's sequence, a numerical analog of Aronson's sequence, approximate squaring, the integrality of n-th roots of generating functions, dissections, and the kissing number problem. (Paper for conference in honor of Martin Gardner's 91st birthday.)Comment: 12 pages. A somewhat different version appeared in "Homage to a Pied Puzzler", E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, Wellesley, MA, 2009, pp. 93-11

Two-divisibility of the coefficients of certain weakly holomorphic modular forms

We study a canonical basis for spaces of weakly holomorphic modular forms of weights 12, 16, 18, 20, 22, and 26 on the full modular group. We prove a relation between the Fourier coefficients of modular forms in this canonical basis and a generalized Ramanujan tau-function, and use this to prove that these Fourier coefficients are often highly divisible by 2.Comment: Corrected typos. To appear in the Ramanujan Journa

Study of the correlation between bicuspid aortic valve and the development of aortic dissection

La disección aórtica (AD) es la condición letal más comúnmente diagnosticada de la arteria aorta y consiste en el redireccionamiento del flujo sanguíneo desde el lumen de la aorta hasta la media de la pared de la aorta a través de una pequeña fisura en la intima. Las causas específicas de la formación de esta fisura, y de la subsecuente dilatación de la pared, todavía no han sido completamente determinadas aunque diversos estudios muestran que puede ser debida o bien a cambios químicos o bien a efectos mecánicos en la pared de la aorta. Este trabajo se centra en el estudio de posibles efectos mecánicos, inducidos por cambios en la hemodinámica de la arteria, que puedan haber conducido al debilitamiento de la pared de la aorta. Válvula aórtica bicúspide (BAV) es la enfermedad congénita del corazón más común y se ha demostrado su importante contribución en el desarrollo de numerosas condiciones cardiovasculares. Esta enfermedad modifica el orificio de salida del corazón, y por tanto el perfil hemodinámico de eyección, del flujo de sangre, lo que podría tener consecuencias en el comportamiento mecánico de la pared de la aorta. Este estudio tiene como objetivo determinar que existe una correlación entre los cambios en la hemodinámica producidos por la presencia de BAV y la formación de AD usando técnicas de análisis de dinámica de fluidos computacional (CFD). Para determinar dicha relación, análisis CFD se han realizado en tres geometrías diferentes: un caso de válvula aórtica tricúspide (TAV) y dos casos distintos de BAV. Todas las geometrías son idealizadas y contemplan la raíz de la aorta, la aorta ascendente y el comienzo del cayado aórtico. Los resultados de los análisis muestran un incremento en la velocidad de eyección de la sangre para ambos casos de BAV debido a la reducción en el área efectiva del orificio. Además, el estudio muestra un incremento en las fuerzas de rozamiento de la pared y en la presión de la pared externa de la aorta. Estos resultados nos llevan a la conclusión de que BAV podría causar hipertensión en la pared externa de la aorta, la cual es una causa mecánica conocida del debilitamiento de vasos sanguíneos

The Neuroanatomic Basis of the Acupuncture Principal Meridians

Acupuncture involves treating illness by inserting needles at specified body locations (acupoints). The Principal meridians are pathways that join acupoints with related physiologic effects. Despite nearly 5000 years of continuous clinical study, an accepted anatomic or physiologic basis for acupuncture's clinical effects has remained elusive. Some acupoints overlie peripheral nerves, and fMRI studies demonstrate that acupoints have specific effects on central nervous system processing. Traditional Chinese Medicine (TCM) founders described the body's viscera based on anatomic dissections yet not a discrete nervous system. By applying computer graphics and virtual human imaging techniques to human developmental neuroanatomy, this paradox may potentially be explained: acupuncture Principal meridians likely are TCM's representation of the nervous system. This neuroanatomic model of the Principal meridians is consistent with acupuncture's known neurophysiologic effects, and may allow 5 millennia of accumulated TCM observations regarding human health and illness to be understood in modern anatomic and physiologic terms

Endogent: Centre for Anatomy and Invasive Techniques

The invention of new endoscopical techniques for surgery and interventional radiology demand improved training at postgraduate level. The Endogent Centre for Anatomy and Invasive Techniques support these requirements by establishing hands-on practical training courses by using new procedures for cadaver embalming. Cadavers fixed by conventional procedures using formalin for conservation, are of limited use for practical surgical courses due to the profound changes of colour, strength and fragility of organs and tissues. The new Thiel embalming technique is based on the use of 4-chloro-3- methylenphenol, various salts for fixation, boric acid for disinfecting, and ethylene glycol for preservation of tissue plasticity, while the concentration of formalin is kept to the strict minimum (0.8%). This results in well preserved organs and tissues concerning colour, consistency, flexibility and plasticity. The articular joints remain freely movable and the peritoneal cavity can be inflated for laparoscopic procedures. Up to now this cadaver model was used in our institute for laparoscopic bariatric surgery, colon surgery, arthroscopy and thorax surgery. Another feature is that the lungs can be ventilated during surgical procedures. Preliminary findings seem to indicate that the corpses also serve as a suitable phantom for assessing thorax radiological equipment. Expert clinicians work as tutors and give intensive instructions before the participants start with hands-on surgery. We intend to expose also our undergraduate medical students to demonstrations of surgical approaches on Thiel embalmed corpses, in order to reveal the need for detailed anatomical knowledge in the clinic at an early stage in the medical curriculum