CONTINUOUS AND DISCRETE N-CONVEX APPROXIMATIONS

We show that the best L_p-approximant to continuous functions by n-convex functions is the limit of discrete n-convex approximations.The techniques of the proof are then used to show the existence of near interpolants to discrete n-convex data by continuous n-convex functions if the data points are close.

Continuous and Discrete Similarity Coefficient for Identifying Essential Proteins Using Gene Expression Data

Essential proteins play a vital role in biological processes,and the combination of gene expression profiles with Protein-Protein Interaction(PPI)networks can improve the identification of essential proteins.However,gene expression data are prone to significant fluctuations due to noise interference in topological networks.In this work,we discretized gene expression data and used the discrete similarities of the gene expression spectrum to eliminate noise fluctuation.We then proposed the Pearson Jaccard coefficient(PJC)that consisted of continuous and discrete similarities in the gene expression data.Using the graph theory as the basis,we fused the newly proposed similarity coefficient with the existing network topology prediction algorithm at each protein node to recognize essential proteins.This strategy exhibited a high recognition rate and good specificity.We validated the new similarity coefficient PJC on PPI datasets of Krogan,Gavin,and DIP of yeast species and evaluated the results by receiver operating characteristic analysis,jackknife analysis,top analysis,and accuracy analysis.Compared with that of node-based network topology centrality and fusion biological information centrality methods,the new similarity coefficient PJC showed a significantly improved prediction performance for essential proteins in DC,IC,Eigenvector centrality,subgraph centrality,betweenness centrality,closeness centrality,NC,PeC,and WDC.We also compared the PJC coefficient with other methods using the NF-PIN algorithm,which predicts proteins by constructing active PPI networks through dynamic gene expression.The experimental results proved that our newly proposed similarity coefficient PJC has superior advantages in predicting essential proteins.

A numerical investigation into the solids phase chromatography using a combined continuous and discrete approach

Solids phase chromatography for particle classification is based on different retention times of particles with different properties when they are elutriated through a confined geometry. This work aims at a fundamental understanding of such a technology by using the combined continuous and discrete method. A packed bed is employed as the model confined geometry. The numerical method is compared first with experimental observations, followed by a parametric analysis of the effects on the flow hydrodynamics and solids behaviour of various parameters including the number of injected particles, the superficial gas velocity, the contact stiffness and the diameter ratio of the packed column to the packed particles. The results show that the modelling captures some important features of the flow of an injected pulse of fine particles in a packed bed. An increase in the number of injected particles or the superficial gas velocity reduces the retention time, whereas the contact stiffness does not show much effect over the range of 5 × 10^2 to 5× 10^4 N/m. It is also found that the effect on the retention time of the diameter ratio of the packed column to the packed particles seems complex showing a non-monotonous dependence.

Continuous and Discrete Adjoint Approach Based on Lattice Boltzmann Method in Aerodynamic Optimization Part I:Mathematical Derivation of Adjoint Lattice Boltzmann Equations

The significance of flow optimization utilizing the lattice Boltzmann(LB)method becomes obvious regarding its advantages as a novel flow field solution method compared to the other conventional computational fluid dynamics techniques.These unique characteristics of the LB method form the main idea of its application to optimization problems.In this research,for the first time,both continuous and discrete adjoint equations were extracted based on the LB method using a general procedure with low implementation cost.The proposed approach could be performed similarly for any optimization problem with the corresponding cost function and design variables vector.Moreover,this approach was not limited to flow fields and could be employed for steady as well as unsteady flows.Initially,the continuous and discrete adjoint LB equations and the cost function gradient vector were derived mathematically in detail using the continuous and discrete LB equations in space and time,respectively.Meanwhile,new adjoint concepts in lattice space were introduced.Finally,the analytical evaluation of the adjoint distribution functions and the cost function gradients was carried out.

Revisiting novel symmetries in coupled N=2 supersymmetric quantum systems:examples and supervariable approach

We revisit the novel symmetries in N=2 supersymmetric quantum mechanical models by considering specific examples of coupled systems.Further,we extend our analysis to a general case and list out all the novel symmetries.In each case,we show the existence of two sets of discrete symmetries that correspond to the Hodge duality operator of differential geometry.Thus,we are able to provide a proof of the conjecture which points out the existence of more than one set of discrete symmetry transformations corresponding to the Hodge duality operator.Moreover,we derive on-shell nilpotent symmetries for a generalized superpotential within the framework of supervariable approach.

Numerical simulation of sedimentation of microparticles usingthe discrete particle method

A mathematical model has been formulated based on the combined continuous and discrete particle method for investigating the sedimentation behaviour of microparticles in aqueous suspensions, by treating the fluid phase as continuous and the particles phase as discrete, thus allowing the behaviour of individual particles to be followed and the evolution of the structure of the particle phase to be investigated as a function of time. The model takes into account most of the prevailing forces acting on individual particles including van der Waals attractive, electrostatic repulsive, gravitational, Brownian, depletion, steric, contact and drag forces. A code has also been developed based on the model. This paper reports some preliminary modelling results of mono-dispersed microparticles settling in aqueous suspensions under various conditions. The results show the short time dynamics of the fluid phase, which has a similar order of magnitude to the particle phase. Such short time dynamics could bear significance to processes such as particle aggregation when their size becomes very small. Preliminary analyses of the results have also been carried out on the evolution of particle settling based on a newly proposed parameter, local normalised volume fraction （LNVF）.