The use of Jacobians in chemical thermodynamics has focused chiefly on explicit functions. We generalize this powerful method to incorporate implicit Considering a slight deviation from equilibrium, governed by the Gibbs–Duhem equations, can simplify theories preferential solvation and adsorption significantly. Since these deviations are zero at function theorem be applied postulating their dif...

Summary Previous Mizar articles [7, 6, 5] formalized the implicit and inverse function theorems for Frechet continuously differentiable maps on Banach spaces. In this paper, using system [1], [2], we formalize these Euclidean spaces by specializing them. We referred to [4], [12], [10], [11] in formalization.

This talk deals with stability and sensitivity analysis for optimal control problems of an ordinary differential equation with a first-order state constraint. We consider the case when the Hamiltonian and the state constraint are regular. Malanowski (2) obtained Lipschitz continuity and directional differentiability of solutions in L, using generalized implicit functions theorems in infinite di...

The rendering of an implicit surface requires many function evaluations of the underlying implicit function. The time efficiency of the rendering is dominated by the number of function evaluations times the efficiency of these evaluations. If the implicit surface is created by means of a CSG model the function evaluations can be made more efficient if the underlying function is a Lipschitz func...

It is proved in this paper that the maximum number of limit cycles of system { dx dt = y, dy dt = kx− (k + 1)x2 + x3 + (α + βx + γx2)y is equal to two in the finite plane, where k > 11+ √ 33 4 , 0 < | | 1, |α| + |β| + |γ| = 0. This is partial answer to the seventh question in [2], posed by Arnold.

This paper aims to study the steady-state response of the practical sliding mode controlled system (SMC). The nonlinearity of the switching term is approximately characterized by using its equivalent describing function. The parasitic dynamics is modeled as a first order lag transfer function, and the possible transport delay is particularly considered. Subsequently, the frequency domain method...