bilinear

3 ±ε   ×   7 ∓ε

Without using a calculator or even pen and paper–or even doing the arithmetic in your head–¿which is the larger product? of:

  • 3.01 × 6.99
  • 2.99 × 7.01

You know 3 × 7  =  21. But what happens if you gently place your finger on the teeter-totter?

 

Although this is “just arithmetic” and so doesn’t require a learnèd vocabulary of higher mathematics, it touches on a few “higher maths” topics:

  • bilinearity being the way I would answer it “conceptually rather than computationally”

  • wiggly numbers

  • ± ε
  • statistical robustness
  • tensor product of modules

  • isoperimetric inequality

    @zentree @isomorphisms what’s all that math nonsense? It’s just an instance of an isoperimetric inequality.

    — Messi is my Homeboy (@gappy3000)

    July 13, 2014

I’ll leave the question/reasoning unanswered with a space for you to answer which is larger and why.

3±ε  ×   7∓ε    =    21±¿?

Bilinear filtering

Bilinear filtering : In this method the four nearest texels to the pixel center are sampled (at the closest mipmap level), and their colors are combined by weighted average according to distance. This removes the ‘blockiness’ seen during magnification, as there is now a smooth gradient of color change from one texel to the next, instead of an abrupt jump as the pixel center crosses the texel boundary. Bilinear filtering is almost invariably used with mipmapping; though it can be used without, it would suffer the same aliasing and shimmering problems as its nearest neighbor.

vimeo

Talia Shuvalov ‘Bilinear’ capsule collection by Cara Stricker

Working drawing demo

Here’s a quick video of the drawing part working.
The blurry-ness is because of the low res brush texture with bilinear filtering disabled.

Next post will go over the details.

arxiv.org
[1508.06349] Maxwell-Dirac stress-energy tensor in terms of Fierz bilinear currents

[ Authors ]
Shaun Inglis, Peter Jarvis
[ Abstract ]
We analyze the stress-energy tensor for the self-coupled Maxwell-Dirac system in the bilinear current formalism, using two independent approaches. The first method used is that attributed to Belinfante: starting from the spinor form of the action, the well-known canonical stress-energy tensor is augmented, by extending the Noether symmetry current to include contributions from the Lorentz group, to a manifestly symmetric form. This form admits a transcription to bilinear current form. The second method used is the variational derivation based on the covariant coupling to general relativity. The starting point here at the outset is the transcription of the action using, as independent field variables, both the bilinear currents, together with a gauge invariant vector field (a proxy for the electromagnetic vector potential). A central feature of the two constructions is that they both involve the mapping of the Dirac contribution to the stress-energy from the spinor fields to the equivalent set of bilinear tensor currents, through the use of appropriate Fierz identities. Although this mapping is done at quite different stages, nonetheless we find that the two forms of the bilinear stress-energy tensor agree. Finally, as an application, we consider the reduction of the obtained stress-energy tensor in bilinear form, under the assumption of spherical symmetry.

arxiv.org
[1508.06278] Quantum Electrodynamics in d=3 from the epsilon-expansion

[ Authors ]
Lorenzo Di Pietro, Zohar Komargodski, Itamar Shamir, Emmanuel Stamou
[ Abstract ]
We study Quantum Electrodynamics in d=3 (QED_3) coupled to N_f flavors of fermions. The theory flows to an IR fixed point for N_f larger than some critical number N_f^c. For N_f<= N_f^c, chiral-symmetry breaking is believed to take place. In analogy with the Wilson-Fisher description of the critical O(N) models in d=3, we make use of the existence of a perturbative fixed point in d=4-2epsilon to study the three-dimensional conformal theory. We compute in perturbation theory the IR dimensions of fermion bilinear and quadrilinear operators. For small N_f, a quadrilinear operator can become relevant in the IR and destabilize the fixed point. Therefore, the epsilon-expansion can be used to estimate N_f^c. An interesting novelty compared to the O(N) models is that the theory in d=3 has an enhanced symmetry due to the structure of 3d spinors. We identify the operators in d=4-2epsilon that correspond to the additional conserved currents at d=3 and compute their infrared dimensions.

arxiv.org
[1508.06540] Soft Walls in Dynamic AdS/QCD and the Techni-dilaton

[ Authors ]
Nick Evans, Peter Jones, Marc Scott
[ Abstract ]
Dynamic AdS/QCD is a modification of AdS/QCD that includes the running of the anomalous dimension of the q-bar q quark bilinear and in which the generation of the constituent quark mass plays the role of an IR wall. The model allows one to move away smoothly from the controlled spectrum of the N=2 super Yang-Mills theory of the D3/probe-D7 system to more QCD-like theories with chiral symmetry breaking. We investigate soft wall behaviour in the model that gives Regge trajectories with M_{n,s}^2 ~ n,s. To achieve these behaviours requires the quark’s constituent mass to fall peculiarly sharply in the IR so that meson physics is sensitive to RG scales well below the quark’s on-shell mass. Including soft wall behaviour in models of walking gauge dynamics breaks the near conformal symmetry which is present above the quark on-shell mass which can generate a large mass for the techni-dilaton like state. We conclude that the meson spectrum is rather sensitive to the IR decoupling.

arxiv.org
[1508.04998] On some new forms of lattice integrable equations

[ Authors ]
Nicoleta-Corina Babalic, A. S. Carstea
[ Abstract ]
Inspired by the forms of delay-Painleve equations, we consider some new differential-discrete systems of KdV, mKdV and Sine-Gordon - type related by simple one way Miura transformations to classical ones. Using Hirota bilinear formalism we construct their new integrable discretizations, some of them having higher order. In particular, by this procedure, we show that the integrable discretization of intermediate sine-Gordon equation is exactly lattice mKdV and also we find a bilinear form of the recently proposed lattice Tzitzeica equation. Also the travelling wave reduction of these new lattice equations is studied and it is shown that all of them, including the higher order ones, can be integrated to Quispel-Roberts-Thomson (QRT) mappings.

In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allows one of the arguments to be “twisted” in a semilinear manner, thus the name; which originates from the Latin numerical prefix sesqui- meaning “one and a half”. The basic concept of the dot product – producing a scalar from a pair of vectors – can be generalized by allowing a broader range of scalar values and, perhaps simultaneously, by widening the definition of what a vector is.
arxiv.org
[1508.04167] Decays $Z\toγγ$ and $Z\to gg$ in the Standard Model Extension

[ Authors ]
J. Castro-Medina, H. Novales-Sánchez, J. J. Toscano, E. S. Tututi
[ Abstract ]
The $Z\to \gamma \gamma$ and $Z\to gg$ decays are studied in the context of the renormalizable version of the Standard Model Extension. The $CPT$–odd $\bar{\psi}\gamma_5 \gamma^\mu b_\mu\psi$ bilinear interaction, which involves the constant background field $b_\alpha$ and which has been a subject of interest in literature, is considered. It is shown that the $Z\to \gamma \gamma$ and $Z\to gg$ decays, which are strictly zero in the standard model, can be generated radiatively at the one-loop level. It is found that these decays are gauge invariant and free of ultraviolet divergences, and that they only are generated by a space-like four-vector $b$.

arxiv.org
[1508.04654] Alternative integrable discretisation of Korteweg de Vries equation

[ Authors ]
Nicoleta-Corina Babalic, A. S. Carstea
[ Abstract ]
We present an alternative integrable discretization of differential-difference KdV equation based on Hirota bilinear formalism. It is shown that using two tau functions the direct discretisation of the bilinear equations gives immediately the well known discrete KdV equation. We comment also on integrability and relation with the classical bilinear form involving only one tau function.

arxiv.org
[1508.04661] On various integrable discretisations of a general two component Volterra system

[ Authors ]
Nicoleta-Corina Babalic, A. S. Carstea
[ Abstract ]
We present two integrable discretisations of a general differential-difference bicomponent Volterra system. The results are obtained by discretising directly the corresponding Hirota bilinear equations in two different ways. Multisoliton solutions are presented together with a new discrete form of Lotka-Volterra equation obtained by an alternative bilinearisation.

bestessaycheap.com
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