This is a novel attempt to produce a rigorous mathematical model of a complex system. The complex system under study is the relationship between therapists and their clients.The success of psychotherapy depends on the nature of the relationship between a therapist and a client. We use dynamical systems theory to model the dynamics of the emotional interaction between a therapist and client. We determine how the therapeutic endpoint and the dynamics of getting there depend on the parameters of the model.Previously Gottman et al. [26] used a very similar approach (physical-sciences paradigm) for modeling and making predictions about husband-wife relationships. They modeled interactions using difference equations and then compared the behavior of those equations to the experimental affect data coded from video of married couples in a 15-minute discussion. The parameters they determined in this way had high predictive value of whether the marriages were stable and also gave new insights into the dynamics of how couples interact. Since that novel approach shed light on the dyadic interaction between couples we thought that it also had the possibility to give us new insights into the relationship between therapist and client.We describe the emotional state of both therapist and client with coupled, first order, nonlinear ordinary differential equations (ODE's). The rate of change of the emotional state of the therapist and client is proportional to their previous state, their uninfluenced state when alone, and an influence function which depends on the state of the other person. We formulated influence functions based on the research literature on psychotherapy and the therapeutic alliance. We then determined the critical points from the intersection of the nullclines and used a numerical ODE solver (Matlab ODE113) to compute the trajectories from different initial conditions.To empirically validate this approach, 73 unique therapy sessions were video-recorded. Four of these interactions (chosen by our psychotherapy expert) were selected to be modeled and were coded using Gottman's Specific Affect Coding System. The results validate this prototypical approach to psychotherapy; we have shown that human interaction (in the context of psychotherapy) can be quantified and modeled using differential equations.

Vitamin B-1 (thiamine) was found to give one anodic peak and three anodic peaks by cyclic voltammetry at a planar glassy carbon electrode, when treated with 0.1 mol dm(-3) sodium hydroxide for 5 and 30 min, respectively. The reverse cyclic voltammograms did not show any cathodic peaks. The electro-active form of thiamine was subjected to cyclic voltammetric studies under a variety of solution conditions. The effect of pH was investigated over the range pH 7.0 to 12.5. The magnitudes of the an(a) values suggested that the initial oxidation reaction of thiamine involved the loss of one electron and the product of this reaction is likely to be a disulfide. The electrode reaction was found to be adsorption-controlled when only phosphate buffer was used. However, when acetonitrile was added to this supporting electrolyte at concentrations of 17.5 and 20% v/v the peak current became diffusion-controlled A convenient and rapid method of analysis for two different multivitamin tablet formulations was developed. After a simple pre-treatment procedure, extracts were analysed using HPLC with a wall-jet amperometric detector. The results of the analyses suggest that the proposed method has promise for the routine determination of vitamin B-1 in the products examined.

Cyclic voltammetry was used to study the oxidation of vitamin D2 (ergocalciferol) and vitamin D3 (cholecalciferol) at a planar glassy carbon electrode. The electrode reaction for cholecalciferol was found to be dependent on the apparent pH between 4.95 and 6.10, and pH independent between pH 6.10 and 8.65 when the solutions contained 90% methanol; this suggested a pK(a) value of 6.10 for vitamin D3. Similar behaviour was exhibited by ergocalciferol, and a pK(a) value of 6.35 was found. The peak currents for both vitamins were found to be dependent on the methanol concentration of the supporting electrolyte. The peak currents were also found to be dependent on the ionic strength of the acetate buffer (pH 6.0) over the range 0.1-1.0 mol dm-3. Both substances were oxidized-in one step, which was found to be an irreversible reaction; the final product for vitamin D2 Can undergo absorption at the electrode surface. The parent compounds could be undergoing oxidation at the triene moieties. The optimum mobile phase for liquid chromatography with amperometric detection was found to be 95% methanol-0.05 mol dm-3 acetate buffer (pH 6.0); the detector was operated at a potential of +1.3 V (versus Ag-AgCl), and a linear response was obtained for vitamin D3 over the range from 10 to 100 ng injected; for vitamin D2 the response was linear from 20 to 200 ng injected. Extracts of pharmaceutical products were separated on reversed-phase columns prior to amperometric detection of the vitamins. Cholecalciferol was successfully determined in a multivitamin tablet, and ergocalciferol in a multivitamin liquid preparation.