A high‐resolution scheme for low Mach number flows

A method for computing low Mach number flows using high‐resolution interpolation and difference formulas, within the framework of the Marker and Cell (MAC) scheme, is presented. This increases the range of wavenumbers that are properly resolved on a given grid so that a sufficiently accurate solution can be obtained without extensive grid refinement. Results using this scheme are presented for three problems. The first is the two‐dimensional Taylor–Green flow which has a closed form solution. The second is the evolution of perturbations to constant‐density, plane channel flow for which linear stability solutions are known. The third is the oscillatory instability of a variable density plane jet. In this case, unless the sharp density gradients are resolved, the calculations would breakdown. Under‐resolved calculations gave solutions containing vortices which grew in place rather than being convected out. With the present scheme, regular oscillations of this instability were obtained and vortices were convected out regularly. Stable computations were possible over a wider range of sensitive parameters such as density ratio and co‐flow velocity ratio. Copyright © 2004 John Wiley Sons, Ltd.     

On the Number of Topological Types Occurring in a Parameterized Family of Arrangements

Let be an o-minimal structure over ℝ, a closed definable set, and

Refined Bounds on the Number of Connected Components of Sign Conditions on a Variety

Let \({\textnormal {R}}\) be a real closed field, \(\mathcal{P},\mathcal{Q} \subset {\textnormal {R}}[X_{1},\ldots,X_{k}]\) finite subsets of polynomials, with the degrees of the polynomials in \(\mathcal{P}\) (resp., \(\mathcal{Q}\)) bounded by d (resp., d ₀). Let \(V \subset {\textnormal {R}}^{k}\) be the real algebraic variety defined by the polynomials in \(\mathcal{Q}\) and suppose that the real dimension of V is bounded by k′. We prove that the number of semi-algebraically connected components of the realizations of all realizable sign conditions of the family \(\mathcal{P}\) on V is bounded by

$\sum_{j=0}^{k'}4^j{s +1\choose j}F_{d,d_0,k,k'}(j),$

where \(s = \operatorname {card}\mathcal{P}\), and

$F_{d,d_0,k,k'}(j)=\binom{k+1}{k-k'+j+1} (2d_0)^{k-k'}d^j \max\{2d_0,d \}^{k'-j}+2(k-j+1).$

In case 2d ₀≤d, the above bound can be written simply as

$\sum_{j = 0}^{k'} {s+1 \choose j}d^{k'} d_0^{k-k'} O(1)^{k}= (sd)^{k'} d_0^{k-k'} O(1)^k$

(in this form the bound was suggested by Matousek 2011). Our result improves in certain cases (when d ₀?d) the best known bound of

$\sum_{1 \leq j \leq k'}\binom{s}{j} 4^{j} d(2d-1)^{k-1}$

on the same number proved in Basu et al. (Proc. Am. Math. Soc. 133(4):965–974, 2005) in the case d=d ₀.

The distinction between the bound d ₀ on the degrees of the polynomials defining the variety V and the bound d on the degrees of the polynomials in \(\mathcal{P}\) that appears in the new bound is motivated by several applications in discrete geometry (Guth and Katz in arXiv:1011.4105v1 [math.CO], 2011; Kaplan et al. in arXiv:1107.1077v1 [math.CO], 2011; Solymosi and Tao in arXiv:1103.2926v2 [math.CO], 2011; Zahl in arXiv:1104.4987v3 [math.CO], 2011).     

A Complexity Theory of Constructible Functions and Sheaves

Foundations of Computational Mathematics - In this paper we introduce constructible analogs of the discrete complexity classes $$mathbf {VP}$$ and $$mathbf {VNP}$$ of sequences of functions. The...     

Using flow to switch the valency of bacterial capture on engineered surfaces containing immobilized nanoparticles

Toward an understanding of nanoparticle-bacterial interactions and the development of sensors and other substrates for controlled bacterial adhesion, this article describes the influence of flow on the initial stages of bacterial capture (Staphylococcus aureus) on surfaces containing cationic nanoparticles. A PEG (poly(ethylene glycol)) brush on the surface around the nanoparticles sterically repels the bacteria. Variations in ionic strength tune the Debye length from 1 to 4 nm, increasing the strength and range of the nanoparticle attractions toward the bacteria. At relatively high ionic strengths (physiological conditions), bacterial capture requires several nanoparticle-bacterial contacts, termed "multivalent capture". At low ionic strength and gentle wall shear rates (on the order of 10 s(-1)), individual bacteria can be captured and held by single surface-immobilized nanoparticles. Increasing the flow rate to 50 s(-1) causes a shift from monovalent to divalent capture. A comparison of experimental capture efficiencies with statistically determined capture probabilities reveals the initial area of bacteria-surface interaction, here about 50 nm in diameter for a Debye length κ(-1) of 4 nm. Additionally, for κ(-1) = 4 nm, the net per nanoparticle binding energies are strong but highly shear-sensitive, as is the case for biological ligand-receptor interactions. Although these results have been obtained for a specific system, they represent a regime of behavior that could be achieved with different bacteria and different materials, presenting an opportunity for further tuning of selective interactions. These finding suggest the use of surface elements to manipulate individual bacteria and nonfouling designs with precise but finite bacterial interactions.     

A Helly-Type Theorem for Semi-monotone Sets and Monotone Maps

We consider sets and maps defined over an o-minimal structure over the reals, such as real semi-algebraic or globally subanalytic sets. A monotone map is a multi-dimensional generalization of a usual univariate monotone continuous function on an open interval, while the closure of the graph of a monotone map is a generalization of a compact convex set. In a particular case of an identically constant function, such a graph is called a semi-monotone set. Graphs of monotone maps are, generally, non-convex, and their intersections, unlike intersections of convex sets, can be topologically complicated. In particular, such an intersection is not necessarily the graph of a monotone map. Nevertheless, we prove a Helly-type theorem, which says that for a finite family of subsets of $\mathbb{R }^n$ , if all intersections of subfamilies, with cardinalities at most $n+1$ , are non-empty and graphs of monotone maps, then the intersection of the whole family is non-empty and the graph of a monotone map.     

Bounding the number of stable homotopy types of a parametrized family of semi-algebraic sets defined by quadratic inequalities

We prove a nearly optimal bound on the number of stable homotopytypes occurring in a k-parameter semi-algebraic family of setsin R, each defined in terms of m quadratic inequalities. Ourbound is exponential in k and m, but polynomial in . More precisely,we prove the following. Let R be a real closed field and let = {P₁, ... , P_m} R[Y₁, ... ,Y,X₁, ... ,X_k], with deg_Y(P_i) 2, deg_X(P_i) d, 1 i m. Let S R^+k be a semi-algebraic set,defined by a Boolean formula without negations, with atoms ofthe form P 0, P 0, P . Let : R^+k R^k be the projection onthe last k coordinates. Then the number of stable homotopy typesamongst the fibers S_x = ^–1(x) S is bounded by (2^mkd)^O(mk).