Duality, triality and complementary extremum principles in non-convex parametric variational problems with applications

This paper presents a reasonably complete duality theory anda nonlinear dual transformation method for solving the fullynonlinear, non-convex parametric variational problem inf{W(u- µ) - F(u)}, and associated nonlinear boundary valueproblems, where is a nonlinear operator, W is either convexor concave functional of p = u, and µ is a given parameter.Detailed mathematical proofs are provided for the complementaryextremum principles proposed recently in finite deformationtheory. A method for obtaining truly dual variational principles(without a dual gap and involving the dual variable p* of uonly) in n-dimensional problems is proposed. It is proved thatfor convex W(p), the critical point of the associated LagrangianL_µ(u, p*) is a saddle point if and only if the so-calledcomplementary gap function is positive. In this case, the systemhas only one dual problem. However, if this gap function isnegative, the critical point of the Lagrangian is a so-calledsuper-critical point, which is equivalent to the Auchmuty'sanomalous critical point in geometrically linear systems. Wediscover that, in this case, the system may have more than oneprimal-dual set of problems. The critical point of the Lagrangianeither minimizes or maximizes both primal and dual problems.An interesting triality theorem in non-convex systems is proved,which contains a minimax complementary principle and a pairof minimum and maximum complementary principles. Applicationsin finite deformation theory are illustrated. An open problemleft by Hellinger and Reissner is solved completely and a purecomplementary energy principle is constructed. It is provedthat the dual Euler-Lagrange equation is an algebraic equation,and hence, a general analytic solution for non-convex variational-boundaryvalue problems is obtained. The connection between nonlineardifferential equations and algebraic geometry is revealed.     

Multiplexed in situ immunofluorescence using dynamic DNA complexes

DNA-based erasers: The designed strand-displacement reactions between dynamic DNA probes (carrying fluorescent dyes () and quencher domains (?)) and DNA-antibody conjugates generate detachable immunofluoresence-reporting complexes that can be assembled (first image) and disassembled (second image) reversibly within fixed cells. With this system, different protein targets can be imaged sequentially within the same cells (third image).     

Benchmark calculations of some molecular properties of O2, CN and other selected small radicals using the ROHF-CCSD(T) method

The capability of the ROHF-CCSD(T) method in obtaining accurate molecular properties in a defined and controlled way is analysed. Electron affinity, polarizability, and hyperpolarizability of the oxygen molecule in its ground state, electron affinity, electric dipole moment of the CN radical, and some other molecules serve as model cases for obtaining the ‘right result for the right reason’. Most calculated CCSD(T) data were extrapolated to the complete basis set (CBS) limit in order to minimize the basis set dependence of results. Some problems, specific to open shell systems include effects due to the spin adaptation, and details in the selection of the reference orbitals and related selection of denominators in non-iterative triples and other subtleties, which can affect the accuracy of the final ROHF-CCSD(T) results, are investigated.