scalar-field

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Stone Fields, Generative Sculpture by Giuseppe Randazzo

This project has started from a search for a 3d-objects optimal packing algorithm over a surface, but evolved in something rather different. I love the work by Richard Long, from which this project takes its cue. The way he fills lonely landscapes with arcaic stones patterns and its eroic artistic practice, in his monumental vision, is in strong contrast with this computational approach that – ironically – allows virtual stones creation and sorting in a non phisical, mental way, a ‘lazy’ version, so to speak. The virtual stones created from several fractal subdivision strategies, find their proper position within the circle, with a trial and error hierarchical algorythm. A mix of attractors and scalar fields (some with Perlin noise) drives the density and size of the stones. The code is a C++ console application that outputs a OBJ 3d file.

Selected and Posted to Cross-Connect by Andrew

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Interesting stuff.

This project has started from a search for a 3d-objects optimal packing algorithm over a surface, but evolved in something rather different. […] The virtual stones created from several fractal subdivision strategies, find their proper position within the circle, with a trial and error hierarchical algorythm. A mix of attractors and scalar fields (some with Perlin noise) drives the density and size of the stones. The code is a C++ console application that outputs a OBJ 3d file.

Source&Credit: Stone Fields (novastructura byGiuseppe Randazzo )

↑ The line integral over a scalar field f can be thought of as the area under the curve C along a surface z = f(x,y), described by the field.

Line integral

In mathematics, a line integral (sometimes called a path integral, contour integral, or curve integral; not to be confused with calculating arc length using integration) is an integral where the function to be integrated is evaluated along a curve.

The function to be integrated may be a scalar field or a vector field. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve). This weighting distinguishes the line integral from simpler integrals defined on intervals.

2

Higgs Boson

On the 4th of July 2012, ATLAS and CMS experiments both reported a particle with a mass of around 126GeV at CERN’s Large Hadron Collider. The particle is consistent with the Higgs boson predicted by the standard model.

The Higgs boson creates a Higgs field which theoretically exists everywhere in the universe and interacts with subatomic fundamental particles like quarks and leptons to give them mass. How much mass a particle has depends on how much interaction is has with the field, all particles are equal before they enter the Higgs field, it is the Higgs field that gives the particles mass depending on their interactions with it.

In the Standard Model, the higgs field is a scalar tachyonic field ( “scalar” meaning that it doesn’t transform under Lorentz transformations and “tachyonic” referring to the field as a whole having imaginary, or complex, mass). While tachyons are purely theoretical particles that move faster than the speed of light, fields with imaginary mass have an important role in modern physics.

A sudoku of linear functionals

Today’s glossary provoked by Michael Spivak.

Below I’ve drawn two vector spaces connected by a linear homomorphism ƒ, plus a linear functional λ going to ℝ. After seeing these pictures I hope it’s easier to understand how the pullback ƒ* works.

Here’s one of the main pictures for flavour:

Also, you can probably just skim the pictures and get the point (especially the Number Field one and the final one). That’s the fastest way to read this post.



Start with an abstract 𝓥ector space.

I’ll do some violence because I’ll need coordinates in a minute.

Keep reading

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Why cosmic inflation’s last great prediction may fail

“[I]f the measured value for n_s stays what it’s thought to be right now, and after a decade we’ve constrained r < 10-3, then the simplest models for inflation are all wrong. It doesn’t mean inflation is wrong, but it means inflation is something more complicated than we first thought, and perhaps not even a scalar field at all.

If nature is unkind to us, the last great prediction of cosmic inflation — the existence of primordial gravitational waves — will be elusive to us for many decades to come, and will continue to go unconfirmed.”

Cosmic inflation, our earliest theory of the Universe and the phenomenon that sets up the Big Bang, didn’t just explain a number of puzzles, but made a slew of new predictions for the Universe. In the subsequent 35 years, five of the six have been confirmed, with only primordial gravitational waves left to go. Inflation predicts that they could be large or small, but based on the simplest classes of models and the measured value of the density fluctuations, the gravitational waves must, according to cosmologist Mark Kamionkowski, be within the range of telescopes during the next decade. If we find them, either one of the two simplest models could be correct, but if we don’t, then the two simplest classes of inflationary models are all wrong, and gravitational waves from inflation may be invisible to us for the foreseeable future.

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Could dark energy be caused by frozen neutrinos?

“Since its discovery in 1998, the accelerated expansion has lacked a compelling, simple explanation that didn’t hypothesize a completely new set of forces, properties or interactions. If you wanted a scalar field — a quintessence model — it had to be finely tuned. But in a very clever paper just submitted yesterday by Fergus Simpson, Raul Jimenez, Carlos Pena-Garay, and Licia Verde, they note that if a generic scalar field couples to the neutrinos we have in our Universe, that fine-tuning goes away, and that scalar field will automatically begin behaving as a cosmological constant: as energy inherent to space itself.”

The accelerated expansion of our Universe was one of the biggest surprise discoveries of all-time, and something that still lacks a good physical explanation. While many models of dark energy exist, it remains a completely phenomenological study: everything appears consistent with a cosmological constant, but nothing appears to be a good motivator for why the Universe should have one. Until now, that is! In a new paper by Fergus Simpson, Raul Jimenez, Carlos Pena-Garay and Licia Verde, they note that any generic scalar field that couples to the neutrino sector would dynamically and stably give rise to a type of dark energy that’s indistinguishable from what we’ve observed. The huge advance is that this scenario doesn’t require any fine-tuning, thanks to this dark energy arising from neutrinos “freezing,” or becoming non-relativistic. In addition, there are experimental signatures to look for to confirm it, too, in the form of neutrinoless double-beta decay!

My string theorist friend posted on facebook that he has “a new favorite metric”

when I clicked the link to the paper:

We calculate the ground state energy of a massive scalar field in the background of a cosmic string of finite thickness (Gott-Hiscock metric).

….

SERIOUSLY