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Archive for the ‘MATHEMATICS’ Category

How (and why) to raise e to the power of a matrix

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April 1, 2021 at 6:49 pm

Posted in MATHEMATICS

From Ramanujan to renormalization: the art of doing away with divergences and arriving at physical results

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Wolfgang Bietenholz
A century ago Srinivasa Ramanujan – the great self-taught Indian genius of mathematics – died, shortly after returning from Cambridge, UK, where he had collaborated with Godfrey Hardy. Ramanujan contributed numerous outstanding results to different branches of mathematics, like analysis and number theory, with a focus on special functions and series. Here we refer to apparently weird values which he assigned to two simple divergent series, \sum_{n\geq 1}{n} and \sum_{n\geq 1}{n^{3}}. These values are sensible, however, as analytic continuations, which correspond to Riemann’s ζ-function. Moreover, they have applications in physics: we discuss the vacuum energy of the photon field, from which one can derive the Casimir force, which has been experimentally measured. We also discuss its interpretation, which remains controversial. This is a simple way to illustrate the concept of renormalization, which is vital in quantum field theory.
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Click to access 2102.09371.pdf

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February 19, 2021 at 11:45 am

Posted in MATHEMATICS

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Roger Penrose: Is Mathematics Invented or Discovered?

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October 18, 2020 at 9:40 am

Posted in MATHEMATICS

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Packing Moons Inside the Earth

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Sunil K. Chebolu
Using ideas of sphere packing problem we estimate the number of solid moons that can be packed inside the Earth, assuming that both the Moon and the Earth are perfect sphere.
Read more at https://arxiv.org/abs/2006.00603

Click to access 2006.00603.pdf

Written by physicsgg

June 2, 2020 at 9:01 am

Posted in MATHEMATICS

Joe Polchinski Memorial Lecture: A Brief History of Branes

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Paul Townsend (University of Cambridge, Department of Applied Mathematics and Theoretical Physics, UK)
Abstract of the memorial lecture “A Brief History of Branes”: Joe Polchinski made many groundbreaking discoveries in theoretical physics. This talk will focus on his contributions to the circle of ideas that led to M-theory in the late 1990s, especially his work of the 1980s on supermembranes (’86) and D-branes and T-duality (’89). This will be part of a survey of the changing role of branes in physics, with personal commentary on various related topics (such as M-branes, U-dualities, black branes) in supergravity and string theory.

Polchinski was a professor at the Kavli Institute for Theoretical Physics at the University of California, Santa Barbara. His great contributions to theoretical physics, including the discovery of D-branes –– a type of membrane in string theory –– have led to advances in the understanding of string theory and quantum gravity. In 2008, he shared ICTP’s Dirac Medal with Juan Maldacena and Cumrun Vafa for their fundamental contributions to superstring theory. The three scientists’ profound achievements have helped to address outstanding questions like confinement of quarks and QCD mass spectrum from a new perspective and have found applications in practical calculations. In addition to the Dirac Medal, Polchinski was awarded the American Physical Society’s 2007 Dannie Heineman Prize for Mathematical Physics, the Milner Foundation’s Physics Frontiers Prize in 2013 and 2014, as well as the 2017 Breakthrough Prize in Fundamental Physics. His work touched the lives of many ICTP scientists, from the hundreds who attended his lectures to those who worked directly with him.

Written by physicsgg

April 5, 2019 at 9:33 pm

Is The Starry Night Turbulent?

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The Starry Night
Vincent van Gogh (1889)


James Beattie, Neco Kriel
Vincent van Gogh’s painting, The Starry Night, is an iconic piece of art and cultural history. The painting portrays a night sky full of stars, with eddies (spirals) both large and small. Kolmogorov1941’s description of subsonic, incompressible turbulence gives a model for turbulence that involves eddies interacting on many length scales, and so the question has been asked: is The Starry Night turbulent? To answer this question, we calculate the azimuthally averaged power spectrum of a square region (1165×1165 pixels) of night sky in The Starry Night. We find a power spectrum, P(k), where k is the wavevector, that shares the same features as supersonic turbulence. It has a power-law P(k)∝k2.1±0.3 in the scaling range, 34≤k≤80. We identify a driving scale, kD=3, dissipation scale, kν=220 and a bottleneck. This leads us to believe that van Gogh’s depiction of the starry night closely resembles the turbulence found in real molecular clouds, the birthplace of stars in the Universe.

Read more at https://arxiv.org/pdf/1902.03381.pdf

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February 12, 2019 at 10:59 pm

Posted in ASTRONOMY, MATHEMATICS

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A crowd that flows like water

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Dynamic response and hydrodynamics of polarized crowds
Nicolas Bain, Denis Bartolo
Modeling crowd motion is central to situations as diverse as risk prevention in mass events and visual effects rendering in the motion picture industry. The difficulty of performing quantitative measurements in model experiments has limited our ability to model pedestrian flows. We use tens of thousands of road-race participants in starting corrals to elucidate the flowing behavior of polarized crowds by probing its response to boundary motion. We establish that speed information propagates over system-spanning scales through polarized crowds, whereas orientational fluctuations are locally suppressed. Building on these observations, we lay out a hydrodynamic theory of polarized crowds and demonstrate its predictive power. We expect this description of human groups as active continua to provide quantitative guidelines for crowd management.

Read more at http://science.sciencemag.org/content/363/6422/46

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January 4, 2019 at 6:13 pm

Noether’s Theorem and Symmetry

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A.K. Halder, Andronikos Paliathanasis, P.G.L. Leach
In Noether’s original presentation of her celebrated theorm of 1918 allowance was made for the dependence of the coefficient functions of the differential operator which generated the infinitesimal transformation of the Action Integral upon the derivatives of the depenent variable(s), the so-called generalised, or dynamical, symmetries. A similar allowance is to be found in the variables of the boundary function, often termed a gauge function by those who have not read the original paper. This generality was lost after texts such as those of Courant and Hilbert or Lovelock and Rund confined attention to point transformations only. In recent decades this dimunition of the power of Noether’s Theorem has been partly countered, in particular in the review of Sarlet and Cantrijn. In this special issue we emphasise the generality of Noether’s Theorem in its original form and explore the applicability of even more general coefficient functions by alowing for nonlocal terms. We also look for the application of these more general symmetries to problems in which parameters or parametric functions have a more general dependence upon the independent variables
Read more at https://arxiv.org/pdf/1812.03682.pdf

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December 23, 2018 at 7:54 am

Posted in MATHEMATICS, mechanics

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