Symmetry-Breaking Phenomena in an Optimization Problem for some Nonlinear Elliptic Equation

Let $\Omega$ be a bounded domain in ${\bf R^n}$ with Lipschitz boundary, $\lambda >0,$ and $1\le p \le (n+2)/(n-2)$ if $n\ge 3$ and $1\le p< +\infty$ if $n=1,2$. Let $D$ be a measurable subset of $\Omega$ which belongs to the class $ {\cal C}_{\beta}=\{D\subset \Omega \quad | \quad |D|=\beta\} $ for the prescribed $\beta\in (0, |\Omega|).$ For any $D\in{\cal C}_{\beta}$, it is well known that there exists a unique global minimizer $u\in H^1_0(\Omega)$, which we denote by $u_D$, of the functional \[\quad J_{\Omega,D}(v)=\frac12\int_{\Omega}|\nabla v|^2\, dx+\frac{\lambda}{p+1}\int_{\Omega}|v|^{p+1}\, dx -\int_{\Omega}\chi_Dv\,dx \] on $H^1_0(\Omega)$. We consider the optimization problem $ E_{\beta,\Omega}=\inf_{D\in {\cal C}_{\beta}} J_D(u_D) $ and say that a subset $D^*\in {\cal C}_{\beta}$ which attains $E_{\beta,\Omega}$ is an optimal configuration to this problem. In this paper we show the existence, uniqueness and non-uniqueness, and symmetry-preserving and symmetry-breaking phenomena of the optimal configuration $D^*$ to this optimization problem in various settings.     

Electroencephalogram (EEG) time series classification: Applications in epilepsy

Epilepsy is among the most common brain disorders. Approximately 25–30% of epilepsy patients remain unresponsive to anti-epileptic drug treatment, which is the standard therapy for epilepsy. In this study, we apply optimization-based data mining techniques to classify the brain's normal and epilepsy activity using intracranial electroencephalogram (EEG), which is a tool for evaluating the physiological state of the brain. A statistical cross validation and support vector machines were implemented to classify the brain's normal and abnormal activities. The results of this study indicate that it may be possible to design and develop efficient seizure warning algorithms for diagnostic and therapeutic purposes. Research was partially supported by the Rutgers Research Council grant-202018, the NSF grants DBI-980821, CCF-0546574, IIS-0611998, and NIH grant R01-NS-39687-01A1.     

An evolutionary method for optimization of plate buckling resistance

Optimization of plate buckling resistance is very complicated, because the in-plane stress resultants in the prebuckled state of a plate are functions of thickness distribution. This paper discusses the problem of finding the optimum thickness distribution of isotropic plate structures, with a given volume and layout, that would maximise the buckling load. A simple numerical method using the finite-element analysis is presented to obtain the optimum thickness distribution. Optimum designs of compression-loaded rectangular plates with different boundary conditions and plate aspect ratios are obtained by using the proposed method. Optimum designs from earlier studies and the methods of buckling analysis used to attain these results are discussed and compared with the designs from the proposed method. This paper also examines the reliability of the optimality criterion generally used for plate buckling optimization, which is based on the uniform strain energy density.     

Optimization technique based on learning automata

Optimization techniques are finding increasingly numerous applications in process design, in parallel to the increase of computer sophistication. The process synthesis problem can be stated as a largescale constrained optimization problem involving numerous local optima and presenting a nonlinear and nonconvex character. To solve this kind of problem, the classical optimization methods can lead to analytical and numerical difficulties. This paper describes the feasibility of an optimization technique based on learning systems which can take into consideration all the prior information concerning the process to be optimized and improve their behavior with time. This information generally occurs in a very complex analytical, empirical, or know-how form. Computer simulations related to chemical engineering problems (benzene chlorination, distillation sequence) and numerical examples are presented. The results illustrate both the performance and the implementation simplicity of this method.Nomenclature c _i penalty probability - cp precision parameter on constraints - D variation domain of the variablex - f(·) objective function - g(·) constraints - i,j indexes - k iteration number - N number of actions - P probability distribution vector - p _i ith component of the vectorP as iterationk - r number of reactors in the flowsheet - u(k) discrete value or action chosen by the algorithm at iterationk - u _i discrete value of the optimization variable in [u _min,u _max] - u _min lowest value of the optimization variable - u _max largest value of the optimization variable - Z random number - x variable for the criterion function - xp precision parameter on criterion function - W(k) performance index unit output at iterationk - ₀, ₁ reinforcement scheme parameters - _p sum of the probability distribution vector components     

Algorithms for flows with parametric capacities

We consider the problem of finding maximal flows with respect to capacities which are linear functions of a parametert [0,T]. Since this problem is a special case of a parametric linear program the classichorizontal approach can be applied in which optimal solutions are computed for successive subintervals of [0,T]. We discuss an alternative algorithm which approximates in each iteration the optimal solution for allt [0,T]. Thisvertical algorithm is a labeling type algorithm where the flow variables are piecewise linear functions. Flow augmentations are done alongconditional flow augmenting paths which can be found by modified path algorithms. The vertical algorithm can be used to solve the parametric flow problem optimally as well as to compute a good approximation for allt if the computation of the optimal solution turns out to be too time consuming.Partially supported by NSF Grants ECS-8412926 and INT-8521433, and NATO Grant RG 85/0240.     

用模糊综合评价值作为单纯形优化指标

本文介绍了一个模糊综合评价方法,讨论了模糊运算方法对运算结果即综合评价值的影响,并将模糊综合评价值用作单纯形优化指标。     

Chemometric approach in optimization of micellar liquid chromatographic separation of some halogenated phenols

Simultaneous optimization of separation quality and analysis time of the micellar liquid chromatography of nine chlorophenol isomers was investigated. The effect on retention of three experimental parameters was studied using multivariate analysis. The factors studied were the concentration of sodium dodecyl sulfate, propanol content and pH of the mobile phase. The experiments were performed according to the face-centred cube central composite design and the inverse form of the experimental retention times for analytes were fitted to the polynomial models. The results of the analysis of variance showed that the models obtained explain greater than 99% of the variance observed in the chromatograms. Good predictive ability of the models was verified as high values of the statistics R² and F were obtained for the linear relationship between predicted cross-validated and experimental values of the dependent variable. The study showed that the use of Pareto-optimality method, an approach from multi-criteria decision-making, allows the selection of the best possible combinations of separation and analysis time in micellar liquid chromatography of chlorophenols.     

On-column injection onto capillary columns. Part 2: Study of sampling conditions; practical recommendations

During one year continuous use of on-column injection, the typical advantages described in our first report have fully been confirmed. In addition the analysis of large sample volumes has proved promising. Only minor modifications have been applied to the on-column injector device. Broad evidence has been gathered showing that full separation efficiency of the capillary columns after on-column injection is attained only when cold trapping or the solvent effect, as band shortening mechanisms, are working- While, under the conditions of on-column injection, cold trapping is less efficient than with other injection techniques, the opposite holds true for the solvent effect. Compared with splitless injection, the danger of excessive solvent condensation on the column is increased. A working rule is presented for establishing the optimal chromatographic conditions for handling large sample volumes while ensuring full separation efficiency yet avoiding harm to the column.     

Inductively coupled plasma atomic emission spectrometry — Optimization of the operating conditions in the determination of trace of elements in line-rich emission matrices

Radial viewing 40.68 MHz inductively coupled plasma atomic emission spectrometer was used in the determination of Y, Sc and rare earth elements in Eu₂O₃ or Lu₂O₃ as pure rare earth matrices. The Mg II 280.270 nm/Mg I 285.213 nm line intensity ratio was measured to evaluate the robustness of the operating conditions. The operating conditions were affected by varying the incident power and sheathing gas flow rate. The carrier gas flow rate remained a constant value. The relationship between the Mg II 280.270 nm/Mg I 285.213 nm ratio and the excitation temperature was obtained. A dependence of the magnesium ratio in the pure solvent and the corresponding values in the presence of the above matrices was established.     

Optimization of stepwise gradient elution in reversed-phase chromatography

Summary Equations describing multi-step gradient elution with a mobile phase of constant composition in each step were derived. These equations useful for calculating the retention volumes in both gradient HPLC and TLC were derived on the basis of the relationship between the isocratic capacity factor and the volume fraction of the organic modifier. The validity of the equations was experimentally verified in a LiChrosorbRP-18-water/methanol system for 11 methyl- and chlorobenzenes and phenols. A satisfactory agreement between the theoretical and experimental k′ values was found.