Properties of the Realization of Inner Functions

 In this paper we investigate fundamental properties of state-space realizations for inner functions. We derive necessary and sufficient conditions for the inner function to have a realization such that the associated C ₀-semigroup is exponentially stable. Furthermore, we give necessary and sufficient conditions on the inner function such that the C ₀-semigroup is a group. Combining these results, we have that the C ₀-semigroup is an exponentially stable C ₀-group if and only if the inner function is the product of a constant of modulus one and a Blaschke product for which the zeros satisfy the Carleson–Newman condition and the zeros lie in a vertical strip bounded away from the imaginary axis. Date received: January 11, 1999. Date revised: May 16, 2002. RID="*" ID="*"This paper was supported by the Volkswagen Stiftung (RiP program at Oberwolfach) and by the Deutsche Forschungsgemeinschaft.     

On the Efficient Evaluation of Coalescence Integrals in Population Balance Models

The solution of population balance equations is a function f(t,r,x) describing a population density of particles of the property x at time t and space r. For instance, the additional independent variable x may denote the particle size. The describing partial differential equation contains additional sink and source terms involving integral operators. Since the coordinate x adds at least one further dimension to the spatial directions and time coordinate, an efficient numerical treatment of the integral terms is crucial. One of the more involved integral terms appearing in population balance models is the coalescence integral, which is of the form ∫ ₀ ^x κ(x–y, y) f(y) f(x–y)dy. In this paper, we describe an evaluation method of this integral which needs only operations, where n is the number of degrees of freedom with respect to the variable x. This cost can also be obtained in the case of a grid geometrically refined towards x=0.     

Effects of planar inlet plenums on the hydrodynamically developing flows in rectangular microchannels of complementary aspect ratios

The effects of planar inlet plenum geometry on the developing flow fields in two rectangular microchannels of reciprocal aspect ratios (H/W ∼2.75 and ∼0.40) were investigated for Re _D =  1–100 using micro particle image velocimetry (μPIV). These two microchannels were made by a precision sawing and silicon microfabrication techniques. Both the velocity profiles and centerline velocity developments were clearly resolved and extracted along the axial distance from μPIV results. The entrance lengths were found from the centerline velocities using a decaying exponential fitting function where the centerline velocity reaches 99% of the fully developed centerline velocity. The proposed fitting function showed excellent agreement with the experimental results. The planar plenum was shown to cause an upstream predevelopment resulting in the significant reductions in the entrance lengths. Two entrance length correlations were proposed in the forms of Atkinson et al.’s (AIChE J 15:548–553, 1969) and Chen’s (J Fluids Eng 95:153–158, 1973) correlations. The proposed entrance length correlations showed that acquired constant portion and slope of the entrance lengths showed 23–27 and 70–81% reductions for H/W =  2.75 while the entrance length correlation for H/W =  0.40 showed 69–73% increase and 41–63% decrease in the constant portion and slope, respectively.     

On the complexity of binary samples

Consider a class of binary functions h: X→{ − 1, + 1} on an interval . Define the sample width of h on a finite subset (a sample) S ⊂ X as ω _S (h) =  min _x ∈ S |ω _h (x)| where ω _h (x) = h(x) max {a ≥ 0: h(z) = h(x), x − a ≤ z ≤ x + a}. Let be the space of all samples in X of cardinality ℓ and consider sets of wide samples, i.e., hypersets which are defined as Through an application of the Sauer-Shelah result on the density of sets an upper estimate is obtained on the growth function (or trace) of the class , β > 0, i.e., on the number of possible dichotomies obtained by intersecting all hypersets with a fixed collection of samples of cardinality m. The estimate is .      

Polynomial algorithms for finding the asymptotically optimum plan of the multiindex axial assignment problem

To construct the asymptotically optimum plan of the p-index axial assignment problem of order n, p algorithms α₀, α₁, ..., α _p−1 with complexities equal to O(n^p+1), O(n^p), ..., O(n²) operations, respectively, are proposed and substantiated under some additional conditions imposed on the coefficients of the objective function. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 176–181, November–December 2005.     

The global avalanche characteristics of two Boolean functions and algebraic immunity

The global avalanche characteristics criterion of two Boolean functions was introduced by Zhou et al. [On the global avalanche characteristics criterion of two Boolean functions and the higher order nonlinearity, Inform. Sci. 180(2) (2010), pp. 256–265] to measure the cryptographic behaviour in a global characteristic. The two indicators σ _{f, g} and Δ _{f, g} of Boolean functions f and g were presented. In this paper, a new upper bound on σ _{f, g} is derived, and a technique on constructing Boolean functions to attain the lower bound on the sum-of-squares indicator is described by using the disjoint spectra method. Some new upper bounds on Δ _{f, g} and σ _{f, g} are deduced for two special Boolean functions. Two relationships between σ _{f, g} and algebraic immunity of the two Boolean functions are obtained. Finally, some links among different cryptographic indicators are shown.     

Reaction Function Based Dynamic Location Modeling in Stackelberg–Nash–Cournot Competition

We formulate a dynamic facility location model for a firm locating on a discrete network. It is assumed that this locating firm will act as the leader firm in an industry characterized by Stackelberg leader–follower competition. The firm’s I competitors are assumed to act as Cournot firms and are each assumed to operate under the assumption of zero conjectural variation with respect to their I–1 Cournot competitors. Using sensitivity analysis of variational inequalities within a hierachical mathematical programming approach, we develop reaction function based dynamic models to optimize the Stackelberg firm’s location decision. In the second half of this paper, we use these models to illustrate through a numerical example the insights yielded by our approach.