On flux terms in volume averaging

This note examines the modeling of non-convective fluxes (e.g., stress, heat flux and others) as they appear in the general, unclosed form of the volume-averaged equations of multiphase flows. By appealing to the difference between slowly and rapidly varying quantities, it is shown that the natural closure of these terms leads to the use of a single, slowly-varying combined average flux, common to both phases, plus rapidly-varying local contributions for each phase. The result is general and only rests on the hypothesis that the spatial variation of the combined average flux is adequately described by a linear function of position within the averaging volume. No further hypotheses on the nature of the flow (e.g., about specific flow regimes) prove necessary. The result agrees with earlier ones obtained by ensemble averaging, is illustrated with the example of disperse flows and discussed in the light of some earlier and current literature. A very concise derivation of the general averaged balance equation is also given.     

Estimation of the entropy of a multivariate normal distribution

Motivated by problems in molecular biosciences wherein the evaluation of entropy of a molecular system is important for understanding its thermodynamic properties, we consider the efficient estimation of entropy of a multivariate normal distribution having unknown mean vector and covariance matrix. Based on a random sample, we discuss the problem of estimating the entropy under the quadratic loss function. The best affine equivariant estimator is obtained and, interestingly, it also turns out to be an unbiased estimator and a generalized Bayes estimator. It is established that the best affine equivariant estimator is admissible in the class of estimators that depend on the determinant of the sample covariance matrix alone. The risk improvements of the best affine equivariant estimator over the maximum likelihood estimator (an estimator commonly used in molecular sciences) are obtained numerically and are found to be substantial in higher dimensions, which is commonly the case for atomic coordinates in macromolecules such as proteins. We further establish that even the best affine equivariant estimator is inadmissible and obtain Stein-type and Brewster–Zidek-type estimators dominating it. The Brewster–Zidek-type estimator is shown to be generalized Bayes.     

Sierpiński-Zygmund functions that are Darboux, almost continuous, or have a perfect road

In this paper we show that if the real line is not a union of less than continuum many of its meager subsets then there exists an almost continuous Sierpiński–Zygmund function having a perfect road at each point. We also prove that it is consistent with ZFC that every Darboux function is continuous on some set of cardinality continuum. In particular, both these results imply that the existence of a Sierpiński–Zygmund function which is either Darboux or almost continuous is independent of ZFC axioms. This gives a complete solution of a problem of Darji [4]. The paper contains also a construction (in ZFC) of an additive Sierpiński–Zygmund function with a perfect road at each point. Received February 28, 1996     

On the smoothness of orthonormal wavelets arising from HDAFs

In this paper, we give an estimation for the smoothness of a new family of symmetric orthonormal wavelets arising from Hermite Distributed Approximating Functionals (HDAFs). Their corresponding associated low pass filters are symmetric with respect to the origin.     

Kinetic study of the CN radical reaction with 2-methylfuran

The gas phase reaction of the ground state cyano-radical (CN (X²∑⁺)) with 2-methylfuran (2-MF) is investigated in a quasi-static reaction cell at pressures ranging from 2.2 to 7.6 Torr and temperatures ranging from 304 to 440 K. The CN radicals are generated in their ground electronic state by pulsed laser photolysis of gaseous cyanogen iodide (ICN) at 266 nm. Their concentration is monitored as a function of reaction time using laser-induced fluorescence at 387.3 nm on the B²∑⁺ (ν′ = 0) ← X²∑⁺ (ν″ = 0) vibronic band. The reaction rate coefficient is found to be rapid and independent of pressure and temperature. Over the investigated temperature and pressure ranges, the rate coefficient is measured to be 2.83 (± 0.18) × 10⁻¹⁰ cm³ molecules s⁻¹. The enthalpies of the stationary points and transition states on the CN + 2-MF potential energy surface are calculated using the CBS-QB3 computational method. The kinetic results suggest the lack of a prereactive complex on the reaction entrance channel with either a very small or nonexistent entrance energy barrier. In addition, the potential energy surface calculations reveal only submerged barriers along the minimum energy path. Based on comparisons between previous CN reactions with unsaturated hydrocarbons, the most likely reaction pathway is CN addition onto one of the unsaturated carbons followed by either H or methyl elimination. The implications for the transformation of biomass-derived fuels in nitrogen-rich flames is discussed.     

On Group Chromatic Number of Graphs

Let G be a graph and A an Abelian group. Denote by F(G, A) the set of all functions from E(G) to A. Denote by D an orientation of E(G). For f ∈ F(G,A), an (A,f)-coloring of G under the orientation D is a function c : V(G)↦A such that for every directed edge uv from u to v, c(u)−c(v) ≠ f(uv). G is A-colorable under the orientation D if for any function f ∈ F(G, A), G has an (A, f)-coloring. It is known that A-colorability is independent of the choice of the orientation. The group chromatic number of a graph G is defined to be the least positive integer m for which G is A-colorable for any Abelian group A of order ≥m, and is denoted by χ_g(G). In this note we will prove the following results. (1) Let H₁ and H₂ be two subgraphs of G such that V(H₁)∩V(H₂)=∅ and V(H₁)∪V(H₂)=V(G). Then χ_g(G)≤min{max{χ_g(H₁), max_v∈V(H2)deg(v,G)+1},max{χ_g(H₂), max_u∈V(H1) deg (u, G) + 1}}. We also show that this bound is best possible. (2) If G is a simple graph without a K_3,3-minor, then χ_g(G)≤5.     

Parameterizations of Masks for Tight Affine Frames with Two Symmetric/Antisymmetric Generators

Parameterizations of FIR orthogonal systems are of fundamental importance to the design of filters with desired properties. By constructing paraunitary matrices, one can construct tight affine frames. In this paper we discuss parameterizations of paraunitary matrices which generate tight affine frames with two symmetric/antisymmetric generators (framelets). Based on the parameterizations, several symmetric/antisymmetric framelets are constructed.     

Fuzzy symmetric groups

We study fuzzy symmetric subgroups and obtain some properties of fuzzy symmetric subgroups of symmetric groups.     

Adaptive Wavelet Methods for Elliptic Operator Equations with Nonlinear Terms

We present an adaptive wavelet method for the numerical solution of elliptic operator equations with nonlinear terms. This method is developed based on tree approximations for the solution of the equations and adaptive fast reconstruction of nonlinear functionals of wavelet expansions. We introduce a constructive greedy scheme for the construction of such tree approximations. Adaptive strategies of both continuous and discrete versions are proposed. We prove that these adaptive methods generate approximate solutions with optimal order in both of convergence and computational complexity when the solutions have certain degree of Besov regularity.     

Decomposition of the Flow Polynomial

The flow polynomials denote the number of nowhere-zero flows on graphs, and are related to the well-known Tutte polynomials and chromatic polynomials. We will show the decomposition of the flow polynomials by edge-cuts and vertex-cuts of size 2 or 3. Moreover by using this decomposition, we will consider what kind of graphs have the same flow polynomials. Another application of the decomposition results is that if a bridgeless graph G does not admit a nowhere-zero k-flow and G has a small vertex- or edge-cut, then a proper bridgeless subgraph of G (a graph minor) does not admit a nowhere-zero k-flow either.