Strict local minimum

Remark 1 If condition (ii) becomes H(jc ) is positive definite, then x is a strict local minimum. 

For arbitrary small perturbation vectors p, the first term on the right dominates the second consequently, the right-hand side is positive. By definition, x is a strict local minimum of f. ... 

Let be a smooth symplectic manifold (compact or noncompact) and let V = sgradfT be a Hamiltonian system that is Liouville-integrable on a certain nonsingular compact three-dimensional constant-energy surface Q by means of a Bott integral /. Let m by the number of such periodic solutions of the system v on the surface Q on which the integral f attains a strictly local minimum or maximum (then the solutions are stable). Next, let p be the number of two-dimensional... 

Two properties, in particular, make Feynman s approach superior to Benioff s (1) it is time independent, and (2) interactions between all logical variables are strictly local. It is also interesting to note that in Feynman s approach, quantum uncertainty (in the computation) resides not in the correctness of the final answer, but, effectively, in the time it takes for the computation to be completed. Peres [peres85] points out that quantum computers may be susceptible to a new kind of error since, in order to actually obtain the result of a computation, there must at some point be a macroscopic measurement of the quantum mechanical system to convert the data stored in the wave function into useful information, any imperfection in the measurement process would lead to an imperfect data readout. Peres overcomes this difficulty by constructing an error-correcting variant of Feynman s model. He also estimates the minimum amount of entropy that must be dissipated at a given noise level and tolerated error rate. 

The strict convexity of the function G in the reaction polyhedron D results in the following important property. In this polyhedron G has the unique local minimum. At the same time this local minimum is a global one. 

Extrema (minima or maxima) of a function can be examined by checking the eigenvalues of the Hessian at its stationary points. If all the eigenvalues of the Hessian are positive (negative) at a stationary point, then the function / is at a local minimum (maximum). Likewise, if all the eigenvalues of the Hessian are positive for all x, then /(x) is said to be strictly convex, with a global minimum at the stationary point. 

Asymptotically we start with Sc ( Dg M= l) + B( Pu M=0). As the two atoms approach each other a pure o-bond is formed by coupling into a singlet the 4s e on Sc and the 2p e on the B atom n) (4s3d )T.2pj), where the subscript T means coupled into a triplet . The 3dxz(7i) electron remains strictly localized on the metal, while 0.2 e are transferred from B to Sc via the CT-frame. As we move in, passing the /-minimum at 5.18 bohr, the g-minimum is formed at 3.90 bohr while the bond character changes drastically. The overall reaction, from infinity to / to can be represented with the following valence-bond-Lewis pictures... 

It is important to keep the amount of potentially hazardous waste in the laboratory to a minimum at all times. Periodically, contents of the laboratory waste containers will be transferred to appropriate containers for final disposal. These must be stored in a safe location, often outdoors, while awaiting pick-up by a disposal service. There will be local restrictions for such storage. A locked storage area may be needed, for example, to prevent unauthorized access to hazardous materials. The fire department niiay set strict limits as to how much flammable material may be present. All containers must be marked with contents, and the storage area will no doubt require warning signs. 

If the matrix Q is positive semidefinite (positive definite) when projected into the null space of the active constraints, then (3-98) is (strictly) convex and the QP is a global (and unique) minimum. Otherwise, local solutions exist for (3-98), and more extensive global optimization methods are needed to obtain the global solution. Like LPs, convex QPs can be solved in a finite number of steps. However, as seen in Fig. 3-57, these optimal solutions can lie on a vertex, on a constraint boundary, or in the interior. A number of active set strategies have been created that solve the KKT conditions of the QP and incorporate efficient updates of active constraints. Popular methods include null space algorithms, range space methods, and Schur complement methods. As with LPs, QP problems can also be solved with interior point methods [see Wright (1996)]. 

It is clearly seen that, at a vertex of the reaction polyhedron, G achieves its local maximum value (due to the strict convexity of G and the fact that its minimum point is positive). Therefore near each vertex, as well as in the vicinity of some faces, the G function can be used to construct a region that is unattainable from outside. Let us consider the case of one vertex and then a more awkward general situation. 

Thus the analysis of a local minlroum in a cormon function space shows it is an absolute minimum, given by (2), an answer which coincides with the result of duality theory. The naive discussion which admits two stable solutions can hold only in a very strict topology, like that of C (0,1) -... 

Proof of the equivalence the local and global minimum of a strictly convex potential... 

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