Our aim in this work is to study the classical continuous boundary control vector  problem for triple nonlinear partial differential equations of elliptic type involving a Neumann boundary control. At first, we prove that the triple nonlinear partial differential equations of elliptic type with a given classical continuous boundary control vector have a unique "state" solution vector,  by using the Minty-Browder Theorem. In addition, we prove the existence of a classical continuous boundary optimal control vector ruled by the triple nonlinear partial differential equations of elliptic type with equality and inequality constraints. We study the existence of the unique solution for the triple adjoint equations

The aim of this research is to prove the idea of maximum m_X-N-open set, m-N-extremally disconnected with respect to t and provide some definitions by utilizing the idea of m_X-N-open sets. Some properties of these sets are studied.

In this work, the classical continuous mixed optimal control vector (CCMOPCV) problem of couple nonlinear partial differential equations of parabolic (CNLPPDEs) type with state constraints (STCO) is studied. The existence and uniqueness theorem (EXUNTh) of the state vector solution (SVES) of the CNLPPDEs for a given CCMCV is demonstrated via the method of Galerkin (MGA). The EXUNTh of the CCMOPCV ruled with the CNLPPDEs is proved. The Frechet derivative (FÉDE) is obtained. Finally, both the necessary and the sufficient theorem conditions for optimality (NOPC and SOPC) of the CCMOPCV with state constraints (STCOs) are proved through using the Kuhn-Tucker-Lagrange (KUTULA) multipliers theorem (KUTULATH).

A novel demountable shear connector for precast steel-concrete composite bridges is presented. The connector uses high-strength steel bolts, which are fastened to the top flange of the steel beam with the aid of a special locking nut configuration that prevents bolts from slipping within their holes. Moreover, the connector promotes accelerated construction and overcomes the typical construction tolerance issues of precast structures. Most importantly, the connector allows bridge disassembly. Therefore, it can address different bridge deterioration scenarios with minimum disturbance to traffic flow including the following: (1) precast deck panels can be rapidly uplifted and replaced; (2) connectors can be rapidly removed and replaced; and (

The investigation of determining solutions for the Diophantine equation  over the Gaussian integer ring for the specific case of  is discussed. The discussion includes various preliminary results later used to build the resolvent theory of the Diophantine equation studied. Our findings show the existence of infinitely many solutions. Since the analytical method used here is based on simple algebraic properties, it can be easily generalized to study the behavior and the conditions for the existence of solutions to other Diophantine equations, allowing a deeper understanding, even when no general solution is known.