Acyclic edge coloring of 2‐degenerate graphs

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a′(G). A graph is called 2‐degenerate if any of its induced subgraph has a vertex of degree at most 2. The class of 2‐degenerate graphs properly contains series–parallel graphs, outerplanar graphs, non ? regular subcubic graphs, planar graphs of girth at least 6 and circle graphs of girth at least 5 as subclasses. It was conjectured by Alon, Sudakov and Zaks (and much earlier by Fiamcik) that a′(G)?Δ + 2, where Δ = Δ(G) denotes the maximum degree of the graph. We prove the conjecture for 2‐degenerate graphs. In fact we prove a stronger bound: we prove that if G is a 2‐degenerate graph with maximum degree Δ, then a′(G)?Δ + 1. © 2010 Wiley Periodicals, Inc. J Graph Theory 69: 1–27, 2012     

Criteria for Validity of the Maximum Norm Principle for Parabolic Systems

We consider systems of partial differential equations, which contain only second derivatives in the x variables and which are uniformly parabolic in the sense of Petrovskii. For such systems we obtain necessary and, separately, sufficient conditions for the maximum norm principle to hold in the layer Rn × ( 0,T ] and in the cylinder × ( 0,T], where is a bounded subdomain of Rn. In this paper the norm is understood in a generalized sense, i.e. as the Minkowski functional of a compact convex body in Rm containing the origin. The necessary and sufficient conditions coincide if the coefficients of the system do not depend on t. The criteria for validity of the maximum norm principle are formulated as a number of equivalent algebraic conditions describing the relation between the geometry of the unit sphere of the given norm and coefficients of the system under consideration. Simpler formulated criteria are given for certain classes of norms: for differentiable norms, p-norms ( 1 p ) in Rm, as well as for norms whose unit balls are m-pyramids, m-bipyramids, cylindrical bodies, m-parallelepipeds. The case m = 2 is studied separately.     

Influence of power-law index on transitional Reynolds numbers for flow over a semi-circular cylinder

In this work, the governing partial differential equations (continuity and Cauchy’s momentum equations) describing the flow of power-law type non-Newtonian fluids across a semi-circular cylinder (oriented with its curved surface in the upstream direction) have been solved numerically. In particular, consideration has been given to the delineation of the critical Reynolds numbers denoting the onset of flow separation from the surface of the cylinder and the onset of the laminar vortex shedding regime. This information is germane to establish the scaling of the macroscopic characteristics like drag coefficient and Strouhal number on the governing parameters, namely, Reynolds number and power-law index. The present results clearly suggest that the transitional Reynolds numbers show a strong dependence on the type (shear-thinning and shear-thickening) of fluid behavior as well as on the severity of the shear-dependence of the viscosity. With reference to the behavior seen in Newtonian fluids, the flow remains not only attached to the surface up to higher Reynolds numbers, but shear-thinning behavior also delays the onset of the laminar vortex shedding regime. As expected, shear-thickening fluids, of course, display the opposite characteristics.     

K3 surfaces,Picard numbers and Siegel disks

If a K3 surface admits an automorphism with a Siegel disk, then its Picard number is an even integer between 0 and 18. Conversely, using the method of hypergeometric groups, we are able to construct K3 surface automorphisms with Siegel disks that realize all possible Picard numbers. The constructions involve extensive computer searches for appropriate Salem numbers and computations of algebraic numbers arising from holomorphic Lefschetz-type fixed point formulas and related Grothendieck residues.     

A single oscillating bubble in liquids with high Mach number

The oscillation characteristics of a single bubble and its induced radiation pressure and the dissipated power are essential for a wide range of applications. For bubble oscillations with high Mach number, the influence of the liquid compressibility is significantly strong and should be fully considered. In the present paper, the bubble wall motion equation with the second-order Mach number is employed for investigating a free oscillating bubble in the liquid with numerical and experimental verifications. For the purpose of comparisons, the revised Keller-Miksis equation up to the first-order Mach number is solved with the same conditions (e.g. the initial conditions and the ambient pressure). Through our simulations, comparing with the predictions by the first-order equation, we find that: (1) The bubble radius, the bubble wall radial velocity and the bubble wall radial acceleration predicted by the second-order equation with high Mach number are significantly different respectively, and the dimensionless differences increase with the increase of the Mach number. (2) The valid prediction range of the second-order equation is much larger. (3) The dissipated power predicted by the second-order equation with high Mach number is smaller.     

Exploring the Origin of Maximum Entropy States Relevant to Resonant Modes in Modern Chladni Plates

The resonant modes generated from the modern Chladni experiment are systematically confirmed to intimately correspond to the maximum entropy states obtained from the inhomogeneous Helmholtz equation for the square and equilateral triangle plates. To investigate the origin of maximum entropy states, the inhomogeneous Helmholtz equation is modified to consider the point interaction coming from the driving oscillator. The coupling strength associated with the point interaction is characterized by a dimensionless factor α. The δ potential of the point interaction is numerically modelled by a truncated basis with an upper index N. The asymptotic behavior for the upper index N is thoroughly explored to verify that the coupling strength of α = 1.0 can make the theoretical resonant modes agree excellently with the maximum entropy states as N→∞. It is further authenticated that nearly the same resonant modes can be obtained by using a larger coupling strength α when a smaller upper index N is exploited in the calculation.     

Simulation input data modeling

Input data modeling is a critical component of a successful simulation application. A perspective of the area is given with an emphasis on available probability distributions as models, estimation methods, model selection and discrimination, and goodness of fit. Three specific distribution classes (lambda,S _B , TES processes) are discussed in some detail to illustrate characteristics that favor input models. Regarding estimation, we argue for maximum likelihood estimation over method of moments and other matching schemes due to intrinsic superior properties (presuming a specific model) and the capability of accommodating messy data types. We conclude with a list of specific research problems and areas warranting additional attention.     

Rotation Numbers of Discontinuous Orientation-Preserving Circle Maps

We extend a few well-known results about orientation preserving homeomorphisms of the circle to orientation preserving circle maps, allowing even an infinite number of discontinuities. We define a set-valued map associated to the lift by filling the gaps in the graph, that shares many properties with continuous functions. Using elementary set-valued analysis, we prove existence and uniqueness of the rotation number, periodic limit orbit in the case when the latter is rational, and Cantor structure of the unique limit set when the rotation number is irrational. Moreover, the rotation number is found to be continuous with respect to the set-valued extension if we endow the space of such maps with the Haussdorff topology on the graph. For increasing continuous families of such maps, the set of parameter values where the rotation number is irrational is a Cantor set (up to a countable number of points).     

Topology and geometry of nematic braids

Topological analysis of disclinations in nematic liquid crystals is an interesting and diverse topic that goes from strict mathematical theorems to applications in elaborate systems found in experiments and numerical simulations. The theory of nematic disclinations is shown from both the geometric and topological perspectives. Entangled disclination line networks are analyzed based on their shape and the behavior of their cross section. Methods of differential geometry are applied to derive topological results from reduced geometric information. For nematic braids, systems of −1/2 disclination loops, created by inclusion of homeotropic colloidal particles, a formalism of rewiring is constructed, allowing comparison and construction of an entire set of different conformations. The disclination lines are described as ribbons and a new topological invariant, the self-linking number, is introduced. The analysis is generalized from a constant −1/2 profile to general profile variations, while retaining the geometric treatment. The workings of presented topological statements are demonstrated on simple models of entangled nematic colloids, estimating the margins of theoretical assumptions made in the formal derivations, and reviewing the behavior of the disclinations not only under topological, but also under free-energy driven constraints.     

FINITE ELEMENT APPROXIMATION OF A NONLINEAR STEADY-STATE HEAT CONDUCTION PROBLEM

1. Introduction Anciellt Chinese mathematicians have done many fund~al discoveries, even thoughmad of them are now usually called by western names, e.g., the Pascal triangle, the HornerIcheme, the Gaussian elimination, see [231. In the modern era3 Chinese mathematiciare have.jalso got many priorities. For instance, the fast Proof Of convergence of the finite element methodfor a linear elliptic boundary vajue problem was done in the pioneering work [61 by K. Feng in1965 (for the English tra…