Extremum defined

Thus if T has at least one negative eigenvalue we can choose a set of coefficients cM so as to make the second variation negative. Expressed in another way this means that the extremum defined by (4.10) is unstable if the matrix T is not positive definite. In order to show that the extremum is stable one would have to prove that T is positive definite for all sets of spin orbitals This is a very difficult problem. It is easy, on the other hand, for a given set to find the eigenvalues of T and thus to find out whether that set can lead to an instability. 

Therefore function = f(d d J has an extremum defining determining the optimum attitude relation ... 

This means that there is a cross-over temperature defined by (1.7) at which tunneling switches off , because the quasiclassical trajectories that give the extremum to the integrand in (2.1) cease to exist. This change in the character of the semiclassical motion is universal for barriers of arbitrary shape. 

Frieden s theory is that any physical measurement induces a transformation of Fisher information J I connecting the phenomenon being measured to intrinsic data. What we call physics - i.e. our objective description of phenomenologically observed behavior - thus derives from the Extreme Physical Information (EPI) principle, which is a variational principle. EPI asserts that, if we define K = I — J as the net physical information, K is an extremum. If one accepts this EPI principle as the foundation, the status of a Lagrangian is immediately elevated from that of a largely ad-hoc construction that yields a desired differential equation to a measure of physical information density that has a definite prior significance. 

Optimisation may be used, for example, to minimise the cost of reactor operation or to maximise conversion. Having set up a mathematical model of a reactor system, it is only necessary to define a cost or profit functionOptimisation and then to minimise or maximise this by variation of the operational parameters, such as temperature, feed flow rate or coolant flow rate. The extremum can then be found either manually by trial and error or by the use of a numerical optimisation algorithms. The first method is easily applied with ISIM, or with any other simulation software, if only one operational parameter is allowed to vary at any one time. If two or more parameters are to be optimised this method however becomes extremely cumbersome. 

The surface BCDE represents a segment of the surface defined by the fundamental equation characteristic of a composite system with coordinate axes corresponding to the extensive parameters of all the subsystems. The plane Uo is a plane of constant internal energy that intersects the fundamental surface to produce a curve with extremum at A, corresponding to maximum entropy. Likewise So is a plane of constant entropy that produces a curve with extremum A that corresponds to minimum energy at equilibrium for the system of constant entropy. This relationship between maximum entropy... 

The validity discriminant is then defined by the extremum problem... 

According to Eq. (63), the 1-RDM and the generalized Fockian F commute at the extremum hence the NSOs are the solutions of the eigenproblem (60) with the nonlocal potential defined by the identity (64). One should note, however, that Eq. (64) does not completely define v e- In fact, the diagonal elements... 

In some problems the possible region of independent variables is defined by equality or inequality constraints. As you have seen in Section 1.2, such constrained extremum problems are easy to solve if both the constraints and the... 

To obtain a meaningful extremum problem the number of experiments and the set of feasible vectors of the independent variables T are fixed. In most cases T is defined by inequalities x1- < x < x, i = l,2,...,k. Though introducing penalty functions such constrained extremum problems can be solved by the methods and modules described in Section 2.4, this direct approach is usually very inefficient. In fact, experiment design is not easy. The dimensionality of the extremum problem is high, the extrema are partly on the boundaries of the feasible region T, and since the objective functions are... 

In earlier publications we have shown that the principle of selection can be deduced from the premises of a self-replicating system as an extremum principle. It states that inherent linear autocatalysis causes the relative population numbers to take on values that correspond to the highest reproductive efficiency of the system as a whole. The distribution of relative concentrations in the stationary population is, after a short induction period, independent of changes of the system as a whole. The population consists of a uniquely defined wild-type (or several equivalent,... 

Chemistry remains a mystery without understanding three basic attributes of matter cohesion, structure and affinity, each of them shaped by an extremum principle. The principle of minimum energy regulates chemical cohesion, which results from the interaction between atoms, also known as chemical bonding. Minimization of angular momentum dictates the three-dimensional arrangement of atoms in chemical substances, which defines their... 

The first derivative of L with respect to y, provided the set [ j=0] is satisfied, is the derivative of 0. This result is true, even if the set of decision variables does not define the extremum. Therefore, these derivatives may be used to point the way towards the extremum. Of course, the usual caveats apply with respect to the possibility of a saddle point or local extremum. 

Suppose you wish to find an extreme value of / (x) on a surface g (x) = c, with c a constant. Explain why the gradients of / and g must be parallel at the desired extrema. Suppose you wish to find an extreme value of / (x) on a surface for which (x) = Ci and g2 (x) = C2- Explain why the gradient of / must be contained in the plane defined by the gradients of gi and g2 at a desired extremum. Explain what this has to do with the Lagrange undetermined multiplier calculation. 

Can the exact ground-state energy, defined as the extremum of the variational functional [] in Eq. (4), be written as a functional of the one-particle density An affirmative answer to this question was given by Hohenberg and Kohn in 1964 [20]. It is, however, a very conditioned answer. In order to show the specific limitations that must be fulfilled in order to make this answer true, let us start by reformulating the variational principle for the energy as a functional of the wavefunction. 

From a study at the extremum of Eq. (8), it is seen that Im reaches a maximum value (Imax) integrated width defined by the followii relationship ... 

The first condition ensures a point of extremum (minimum or maximum), and the second condition ensures that the extremum is a minimum. The Gibbs free energy plays the role of a thermodynamic potential function at fixed temperature and pressure. By analogy to the mechanical system, the mathematical conditions that define a stable equilibrium state at constant pressure and temperature are ... 

Consider now a periodic orbit at u at energy Eju(Uq). By definition, p =0 so that at Uq Eq. (26) holds exactly. We have thus proved that classically, not only does a periodic orbit correspond to an adiabatic barrier or well but also that at the barrier or well the adiabatic approximation is exact. It is also easy to see that if the adiabatic approximation is exact at the extremum of Em(u) then at the extrenum Uq one must find a periodic orbit of energy Ein(uo). From Eq. (26) we find that Pu=0, and since dEni/du=0 at u necessarily p O. We have thus proved that a necessary and sufficient condition for the classical adiabatic approximation to be exact at an adiabatic barrier or well is the existence of a periodic orbit at the barrier or the well. Periodic orbits though are defined uniquely by the potential energy surface and the masses. Classically, we have removed the ambiguity in the definition of the extrema of Eni(u). 

Saddle point is defined in the stability theory of ODE as the point of a function or surface, which is a stationary point but not an extremum. Let... 

There may be situations when pjquations 7 and 9 are realized while Equations 8 and 10 are not realized, i.c. the system is stable toward infinitely small perturbations while being unstable toward finite ones (a local maximum of entropy or a minimum of internal energy with at least one additional extremum). In such cases it is generally agreed to speak of a metastable equilibrium (metastable state) of the system. Conditions 8 and 10 define a stable equilibrium. When simultaneously breaking conditions 7 10, the system proves to be absolutely unstable. 

---