Characterisation of non-linear viscoelastic foods by the indentation technique

A method to determine the non-linear viscoelastic constitutive constants from indentation force–displacement data corresponding to different indentation speeds has been developed. The method consists of two parts. In the first part, the force–displacement data is expressed as two functions which represent the strain and the time-dependent responses, respectively. From these functions, the time-dependent constants and the instantaneous force–displacement response are obtained. In the second part, the strain-dependent variables are determined from the instantaneous force–displacement response through an inverse analysis based on the Levenberg–Marquardt method. The method was verified by numerical experiments using the properties of cheese as examples.     

On restricting Cauchy–Pexider functional equations to submanifolds

Sufficient geometric conditions are given which determine when the Cauchy–Pexider functional equation f(x)g(y) = h(x + y) restricted to x, y lying on a hypersurface in ${\mathbb{R}^d}$ has only solutions which extend uniquely to exponential affine functions ${\mathbb{R}^d \to \mathbb{C}}$ (when f, g, h are assumed to be measurable and non-trivial). The Cauchy–Pexider-type functional equations ${\prod_{j=0}^df_j(x_j)=F(\sum_{j=0}^dx_j)}$ for ${x_0, \ldots,x_d}$ lying on a curve and ${f_1(x_1)f_2(x_2)f_3(x_3)=F(x_1+x_2+x_3)}$ for x ₁, x ₂, x ₃ lying on a hypersurface are also considered.     

Large deformation extensional rheology of bread dough

The stress–strain curves of bread dough were derived under uniaxial compression, uniaxial tension and equi-biaxial tension loading conditions. In uniaxial compression, a lubricant was used to eliminate frictional effects between the loading platens and the sample. In uniaxial tension, cylindrical samples with thin flat discs at both ends (‘I’ samples) were tested. The discs at both ends were allowed to air-dry and were subsequently glued onto the loading platens. In equi-biaxial tension, a thin disc of dough was inflated into a bubble using pressurised air. The thickness at the top of the bubble was measured by shining a light through the walls of the bubble and recording the change in light intensity as the wall becomes thinner. All methods ensured that uniform deformation was obtained. Stress and strain were accurately evaluated using image analysis techniques. The tests were performed at various strain rates and speeds that defined the time dependence of the material. A non-linear viscoelastic model based on the Prony series and Van der Waals hyperelasticity was used to predict all test data. The model had a total of five material parameters and two time constants, which were set to represent the actual time scales of the experiments. A reasonable agreement between the experimental data and the chosen material model was observed.     

Biaxial deformation of dough using the bubble inflation technique. I. Experimental

The bubble inflation test has been used to determine the equi-biaxial stress-strain curve of flour/water dough. This was achieved by undertaking experimental measurements of strain, wall thickness and radius of curvature at the top of the bubble as well as applied pressure. It was observed that the bubble was spherical initially but changed to an elliptical shape at large strains. The analysis derived by Bloksma (1957) was also used to calculate stress and strain at the top of the inflated bubble from gauge pressure and bubble volume data. It was found that the analysis led to accurate bubble heights at moderate strains only, a consequence of the non-spherical bubble shape. In addition, at the top of the bubble, the analytical strain was larger and the thickness was much smaller than the experimental values. The bubble wall thickness distribution was more uniform than the analytical predictions. The discrepancies in bubble height, radius of curvature, strain and thickness had a major effect in the analytical stress-strain curve, as both stress and strain were overestimated, the stress being in error by a factor of four at large strains. Therefore, it is concluded that when the bubble inflation test is used, it is necessary that experimental readings of strain, thickness and radius of curvature as well as pressure should be made to ensure accurate stress-strain curves.     

Bipartitional polynomials and their applications in combinatorics and statistics

Bipartitional polynomials are multivariable polynomials Y_mn=Y_mn(cy₀₁,cy₁₀,cy₁₁,…,cy_mn), c^k≡c_k, defined by a sum over all partitions of the bipartite number (mn). Recurrence relations, generating functions and some basic properties of these polynomials are given. Applications in Combinatorics and Statistics are briefly indicated.     

Parallel queuing systems with and without decentralized server reallocation — A simulation study

This paper describes the results of an initial exploratory research effort on the subject of the dynamic reallocation of servers in a group of parallel queuing systems under decentralized control. Each queuing system is described as being (M/E_k/c_i,t): (FCFS/∞/∞) where t = 1,.., ∞ and i = 1,..., I. Server reallocation is handled by a (G^(b)/G/C): (FCS/C/K) system — where C = Σ_t=1^∞Σ_i=1^Ic_i,t and K is an unknown constant — located at a higher level than the group members. The customers of the latter system are requests for servers independently generated by identical heuristic event-triggered operating policies at each of the lower level systems. These local policies are of the stochastic review, relative control number type. Performance is measured by means of a variety of criteria from the viewpoint of the group as a whole (global) as well as an individual queuing system (local). Results of experiments using a simulation model confirmed the null hypothesis that the customer service behaviour of a group that shared servers was significantly different from the behaviour of a control group that did not share servers.     

Distributions of Random Partitions and Their Applications

Assume that a random sample of size m is selected from a population containing a countable number of classes (subpopulations) of elements (individuals). A partition of the set of sample elements into (unordered) subsets, with each subset containing the elements that belong to same class, induces a random partition of the sample size m, with part sizes {Z ₁,Z ₂,...,Z _N } being positive integer-valued random variables. Alternatively, if N _j is the number of different classes that are represented in the sample by j elements, for j=1,2,...,m, then (N ₁,N ₂,...,N _m ) represents the same random partition. The joint and the marginal distributions of (N ₁,N ₂,...,N _m ), as well as the distribution of are of particular interest in statistical inference. From the inference point of view, it is desirable that all the information about the population is contained in (N ₁,N ₂,...,N _m ). This requires that no physical, genetical or other kind of significance is attached to the actual labels of the population classes. In the present paper, combinatorial, probabilistic and compound sampling models are reviewed. Also, sampling models with population classes of random weights (proportions), and in particular the Ewens and Pitman sampling models, on which many publications are devoted, are extensively presented.      

A note on distinct distance subsets

It is shown that given a set of N points in the plane, sphere or hyperbolic plane, there is a subset of size ${\gtrsim (N/\log N)^{1/3}}$ with all pairwise distances between points distinct.