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   Carlos Alberto Garay @ 5:08 pm on 29 Sep 2019

   Let´s begin by reading: "Artificial honeycomb lattices with Dirac cone dispersion provide a macroscopic platform to study the massless Dirac quasiparticles and their novel geometric phases. In this paper, a quadruple-degenerate state is achieved at the center of Brillouin zone (BZ) in a two-dimensional honeycomb lattice phononic crystal, which is a result of accidental degeneracy of two double-degenerate states. In the vicinity of the quadruple-degenerate state, the dispersion relation is linear. Such quadruple degeneracy is analyzed by rigorous representation theory of groups. Using Yan-Feng Chen1 1 National Laboratory of Solid State Microstructures & Department of Materials Science and Engineering, Nanjing University, Nanjing 210093, People’s Republic of China 2 Division of Mathematical and Computer Sciences and Engineering, King Abdullah University of Science and Technology (KAUST),Thuwal 23955-6900, Saudi Arabia k p⋅   method, a reduced Hamiltonian is obtained to describe the linear Dirac dispersion relations of such quadruple-degenerate state, which is well consistent with the simulation results. Near such accidental degeneracy, we observe some unique wave propagating properties, such as defect insensitive propagating character and Talbot effect."I. INTRODUCTION Many unique phenomena in graphene such as quantum Hall effect, Zitterbewegung, Klein paradox and pseudo-diffusion, are attributed to the unique dispersion relation of massless quasiparticles solved by Dirac equation.1-5 The eigen-energy E is linearly proportional to the wave vector k at the six corners of the hexagonal boundary of the BZ. The upper and lower bands near the K point act as two cones touching at one degenerate point, which is the so-called Dirac point and such conical dispersion is called Dirac cone. Compared to graphene or photonics and phononics with Dirac cone dispersion at the corner of the BZ, 6 the recent observation of Dirac cones at the center of BZ in photonic and phononic crystals has also attracted much attention. Under certain circumstances, those Dirac cones can be mapped into a zero-index material, whose parameters (e.g. permittivity and permeability in electromagnetics, effective mass density and reciprocal of bulk modulus in acoustics) are both vanishing. 7-11 It provides a new method to achieve zero-index materials with simple photonic and phononic crystals so that many interesting properties such as wave shaping and cloaking are easily demonstrated. 8, 912, 13

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   Carlos Alberto Garay @ 5:11 pm on 29 Sep 2019

   SECOND RADING: Cancer metabolism of cone photoreceptors

   Oncotarget. 2015 Oct 20; 6(32): 32285–32286.

   Published online 2015 Oct 4. doi: 10.18632/oncotarget.5963

   PMCID: PMC4741680

   PMID: 26450906

                            Cancer metabolism of cone photoreceptors

   Thierry Léveillard

   Author information Article notes Copyright and License information Disclaimer

   This article has been cited by other articles in PMC.

   Cancer cells divert cellular physiology for their own growth through activation of proto-oncogenes and inactivation of tumor suppressors by somatic mutations, amplifications, translocations and loss of alleles. The multistep tumor progression is a succession of clonal expansions of cells with mutant genotypes that acquire traits that enable them to become tumorigenic and ultimately malignant [1]. Cancer cells generate through genomic instability random mutations and among them rare genetic changes promoting tumor formation. The reprogramming of energetic metabolism as hallmark of cancer cells was recently revisited [2]. This feature of cancer metabolism has been discovered sixty years ago by Otto Warburg who showed that most cancer cells metabolize high amounts of glucose that is secreted as lactate even in the presence of oxygen, a phenomenon named aerobic glycolysis since then [3]. This aberrant metabolism appeared originally wasteful given how little ATP is produced by glycolysis compared to oxidative phosphorylation. It produces mainly carbohydrate metabolites that are essential for cell proliferation and is compensated by a dramatic increase in glucose uptake by cancer cell. This phenomenon is used clinically to visualize tumors by 2-fluoro-6-deoxyglucose positron emission tomography. Recent results indicate that for cancer cells, the redirection of glucose toward aerobic glycolysis is not a consequence of the activation of proto-oncogene or of the inactivation of tumor suppressor genes but the first stage of multistep tumorigenesis. This is illustrated by the fact that the inactivation of the NAD-dependent protein deacetylase SIRT6 results in cell transformation by rewiring energetic metabolism to aerobic glycolysis independently of any other oncogenic event [4]. SIRT6-dependent deacetylation of histone H3 limits the expression of genes encoding glycolytic enzymes. SIRT6 deletion increase the levels of acetylated histone H3 that results in an increase expression of genes involved in aerobic glycolysis. However aerobic glycolysis is not specific to cancer cells (Figure ​(Figure1B1B).

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   Carlos Alberto Garay @ 5:26 pm on 29 Sep 2019

   DEGENERATION OF A CONE EXPLAINED IN MATHEMATICAL TERMS

   Degenerate conic

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   Degenerate conics

   In geometry, a degenerate conic is a conic (a second-degree plane curve, defined by a polynomial equation of degree two) that fails to be an irreducible curve. This means that the defining equation is factorable over the complex numbers (or more generally over an algebraically closed field) as the product of two linear polynomials.^{[note 1]}

   Using the alternative definition of the conic as the intersection in three-dimensional space of a plane and a double cone, a conic is degenerate if the plane goes through the vertex of the cones.

   In the real plane, a degenerate conic can be two lines that may or may not be parallel, a single line (either two coinciding lines or the union of a line and the line at infinity), a single point (in fact, two complex conjugate lines), or the null set (twice the line at infinity or two parallel complex conjugate lines).

   All these degenerate conics may occur in pencils of conics. That is, if two real non-degenerated conics are defined by quadratic polynomial equations f = 0 and g = 0, the conics of equations af + bg = 0 form a pencil, which contains one or three degenerate conics. For any degenerate conic in the real plane, one may choose f and g so that the given degenerate conic belongs to the pencil they determine.

   Degeneration

   In the complex projective plane, all conics are equivalent, and can degenerate to either two different lines or one double line.

   In the real affine plane:

   • Hyperbolas can degenerate to two intersecting lines (the asymptotes), as in {\displaystyle x^{2}-y^{2}=a^{2},} or to two parallel lines: {\displaystyle x^{2}-a^{2}y^{2}=1,} or to the double line {\displaystyle x^{2}-a^{2}y^{2}=a^{2},} as a goes to 0.
   • Parabolas can degenerate to two parallel lines: {\displaystyle x^{2}-ay-1=0} or the double line {\displaystyle x^{2}-ay=0,} as a goes to 0; but, because parabolae have a double point at infinity, cannot degenerate to two intersecting lines.
   • Ellipses can degenerate to two parallel lines: {\displaystyle x^{2}+a^{2}y^{2}-1=0} or the double line {\displaystyle x^{2}+a^{2}y^{2}-a^{2}=0,} as a goes to 0; but, because they have conjugate complex points at infinity which become a double point on degeneration, cannot degenerate to two intersecting lines.

   Degenerate conics can degenerate further to more special degenerate conics, as indicated by the dimensions of the spaces and points at infinity.

   • Two intersecting lines can degenerate to two parallel lines, by rotating until parallel, as in {\displaystyle x^{2}-ay^{2}-1=0,} or to a double line by rotating into each other about a point, as in {\displaystyle x^{2}-ay^{2}=0,} in each case as a goes to 0.
   • Two parallel lines can degenerate to a double line by moving into each other, as in {\displaystyle x^{2}-a^{2}=0} as a goes to 0, but cannot degenerate to non-parallel lines.
   • A double line cannot degenerate to the other types.
   • Another type of degeneration occurs for an ellipse when the sum of the distances to the foci is mandated to equal the interfocal distance; thus it has semi-minor axis equal to zero and has eccentricity equal to one. The result is a line segment (degenerate because the ellipse is not differentiable at the endpoints) with its foci at the endpoints. As an orbit, this is a radial elliptic trajectory.

    

   Applications

   Degenerate conics, as with degenerate algebraic varieties generally, arise as limits of non-degenerate conics, and are important in compactification of moduli spaces of curves.

   For example, the pencil of curves (1-dimensional linear system of conics) defined by {\displaystyle x^{2}+ay^{2}=1} is non-degenerate for {\displaystyle a\neq 0} but is degenerate for {\displaystyle a=0;} concretely, it is an ellipse for {\displaystyle a>0,} two parallel lines for {\displaystyle a=0,} and a hyperbola with {\displaystyle a<0} – throughout, one axis has length 2 and the other has length {\displaystyle 1/{\sqrt {|a|}},} which is infinity for {\displaystyle a=0.}

   Such families arise naturally – given four points in general linear position (no three on a line), there is a pencil of conics through them (five points determine a conic, four points leave one parameter free), of which three are degenerate, each consisting of a pair of lines, corresponding to the {\displaystyle \textstyle {{\binom {4}{2,2}}=3}} ways of choosing 2 pairs of points from 4 points (counting via the multinomial coefficient).

   External video

   Type I linear system, (Coffman).

   For example, given the four points {\displaystyle (\pm 1,\pm 1),} the pencil of conics through them can be parameterized as {\displaystyle (1+a)x^{2}+(1-a)y^{2}=2,} yielding the following pencil; in all cases the center is at the origin:^{[note 2]}

   • {\displaystyle a>1:} hyperbolae opening left and right;
   • {\displaystyle a=1:} the parallel vertical lines {\displaystyle x=-1,x=1;}
   • {\displaystyle 0<a<1:} ellipses with a vertical major axis;
   • {\displaystyle a=0:} a circle (with radius {\displaystyle {\sqrt {2}}});
   • {\displaystyle -1<a<0:} ellipses with a horizontal major axis;
   • {\displaystyle a=-1:} the parallel horizontal lines {\displaystyle y=-1,y=1;}
   • {\displaystyle a<-1:} hyperbolae opening up and down,
   • {\displaystyle a=\infty :} the diagonal lines {\displaystyle y=x,y=-x;}

   (dividing by {\displaystyle a} and taking the limit as {\displaystyle a\to \infty } yields {\displaystyle x^{2}-y^{2}=0})

   • This then loops around to {\displaystyle a>1,} since pencils are a projective line.

   Note that this parametrization has a symmetry, where inverting the sign of a reverses x and y. In the terminology of (Levy 1964), this is a Type I linear system of conics, and is animated in the linked video.

   A striking application of such a family is in (Faucette 1996) which gives a geometric solution to a quartic equation by considering the pencil of conics through the four roots of the quartic, and identifying the three degenerate conics with the three roots of the resolvent cubic.

   Pappus's hexagon theorem is the special case of Pascal's theorem, when a conic degenerates to two lines.

    

   Source: https://en.wikipedia.org/wiki/Degenerate_conic

    

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