Turing Instability and Pattern Formation in an Activator-Inhibitor System with Nonlinear Diffusion

In this work we study the effect of density dependent nonlinear diffusion on pattern formation in the Lengyel--Epstein system. Via the linear stability analysis we determine both the Turing and the Hopf instability boundaries and we show how nonlinear diffusion intensifies the tendency to pattern formation; %favors the mechanism of pattern formation with respect to the classical linear diffusion case; in particular, unlike the case of classical linear diffusion, the Turing instability can occur even when diffusion of the inhibitor is significantly slower than activator's one. In the Turing pattern region we perform the WNL multiple scales analysis to derive the equations for the amplitude of the stationary pattern, both in the supercritical and in the subcritical case. Moreover, we compute the complex Ginzburg-Landau equation in the vicinity of the Hopf bifurcation point as it gives a slow spatio-temporal modulation of the phase and amplitude of the homogeneous oscillatory solution.Comment: Accepted for publication in Acta Applicandae Mathematica

Pattern formation driven by cross--diffusion in a 2D domain

In this work we investigate the process of pattern formation in a two dimensional domain for a reaction-diffusion system with nonlinear diffusion terms and the competitive Lotka-Volterra kinetics. The linear stability analysis shows that cross-diffusion, through Turing bifurcation, is the key mechanism for the formation of spatial patterns. We show that the bifurcation can be regular, degenerate non-resonant and resonant. We use multiple scales expansions to derive the amplitude equations appropriate for each case and show that the system supports patterns like rolls, squares, mixed-mode patterns, supersquares, hexagonal patterns

Monads in Double Categories

We extend the basic concepts of Street's formal theory of monads from the setting of 2-categories to that of double categories. In particular, we introduce the double category Mnd(C) of monads in a double category C and define what it means for a double category to admit the construction of free monads. Our main theorem shows that, under some mild conditions, a double category that is a framed bicategory admits the construction of free monads if its horizontal 2-category does. We apply this result to obtain double adjunctions which extend the adjunction between graphs and categories and the adjunction between polynomial endofunctors and polynomial monads.Comment: 30 pages; v2: accepted for publication in the Journal of Pure and Applied Algebra; added hypothesis in Theorem 3.7 that source and target functors preserve equalizers; on page 18, bottom, in the statement concerning the existence of a left adjoint, "if and only if" was replaced by "a sufficient condition"; acknowledgements expande

Double Adjunctions and Free Monads

We characterize double adjunctions in terms of presheaves and universal squares, and then apply these characterizations to free monads and Eilenberg--Moore objects in double categories. We improve upon our earlier result in "Monads in Double Categories", JPAA 215:6, pages 1174-1197, 2011, to conclude: if a double category with cofolding admits the construction of free monads in its horizontal 2-category, then it also admits the construction of free monads as a double category. We also prove that a double category admits Eilenberg--Moore objects if and only if a certain parameterized presheaf is representable. Along the way, we develop parameterized presheaves on double categories and prove a double-categorical Yoneda Lemma.Comment: 52 page

Studies of Radiative Penguin B Decays at BaBar

We summarize results on a number of observations of penguin dominated radiative decays of the B meson. Such decays are forbidden at tree level and proceed via electroweak loops. As such they may be sensitive to physics beyond the standard model. The observations have been made at the BaBar experiment at PEP-II, the asymmetric B factory at SLAC.Comment: 3 pages, 5 figure

Turing pattern formation in the Brusselator system with nonlinear diffusion

In this work we investigate the effect of density dependent nonlinear diffusion on pattern formation in the Brusselator system. Through linear stability analysis of the basic solution we determine the Turing and the oscillatory instability boundaries. A comparison with the classical linear diffusion shows how nonlinear diffusion favors the occurrence of Turing pattern formation. We study the process of pattern formation both in 1D and 2D spatial domains. Through a weakly nonlinear multiple scales analysis we derive the equations for the amplitude of the stationary patterns. The analysis of the amplitude equations shows the occurrence of a number of different phenomena, including stable supercritical and subcritical Turing patterns with multiple branches of stable solutions leading to hysteresis. Moreover we consider traveling patterning waves: when the domain size is large, the pattern forms sequentially and traveling wavefronts are the precursors to patterning. We derive the Ginzburg-Landau equation and describe the traveling front enveloping a pattern which invades the domain. We show the emergence of radially symmetric target patterns, and through a matching procedure we construct the outer amplitude equation and the inner core solution.Comment: Physical Review E, 201

On the numerical closeness of the effective phenomenological electroweak mixing angle θ\theta and the \MS parameter θ^\hat\theta

It happens that $s^2$ and $\hat s^2$ are equal with 0.1% accuracy, though they are split by radiative corrections and a natural estimate for their difference is 1%. This degeneracy occurs only for $m_t$ value close to 170\GeV, so no deep physical reason can be attributed to it. However, another puzzle of the Standard Model, the degeneracy of s_\Eff^2 and $s^2$, is not independent of the previous one since a good physical reason exists for s_\Eff^2 and $\hat s^2$ degeneracy. We present explicit formulas which relate these three angles.Comment: 10 pages, latex, 3 figures included; published on MPLA. A few numerical estimates are improved; journal-ref is adde