Constrained variable

Constrained optimization refers to optimizations in which one or more variables (usually some internal parameter such as a bond distance or angle) are kept fixed. The best way to deal with constraints is by elimination, i.e., simply remove the constrained variable from the optimization space. Internal constraints have typically been handled in quantum chemistry by using Z matrices if a Z matrix can be constructed which contains all the desired constraints as individual Z-matrix variables, then it is straightforward to carry out a constrained optimization by elunination. 

Tolosana R (2005) Geostatistics for constrained variables positive data, compositions and probabilities. Application to environmental hazard monitoring. PhD Thesis, Universitat de Girona, Spain... 

Constrained food habits oi trophic position Spatio-temporal variation in trophic position can confound and complicate interpretation of trends in MeHg concentration in the indicator Highly constrained Variable and complex Variable and complex constrained in some filter feeders Variable and complex Constrained Ontogenetic shifts with increasing size less variable in adults... 

The second entropy then is a function of three constrained variables, S(2)(A°x, Arx,x x,Xr). The internal part, which accounts for the resistance to the flux, is as given earlier it characterizes the transition x —> x. The external part is entirely influenced by the reservoir, and it consists of two parts the equilibration with the reservoir for the initial state x, and the transition x —> x". 

Important, poorly-constrained variables in this carbon balance include temporal and spatial variations in precipitation and RO DIC, carbonate dust, and pedogenic carbonate and soil C02. In addition, ocean and lake response to river carbon input is not well known. Although marine and freshwater aquatic organisms can be fertilized by increases in N, P, Si, Fe, Zn and C02 (Cassar et al. 2004 Zondervan 2007), it is not clear how much of the... 

If all of the variables appeared in all of the equations, then the use of optimum seeking methods for the direction of the calculations would be impractical because the search dimensionality would become excessive. However, the opposite is true for these applications. The system of equations is sparse only a few variables are present in each equation. This requires that only a few variables need to be search variables with the rest being state variables. Search variables must be chosen carefully. Generally, the most constrained variables should be chosen as search variables, and the least constrained variables chosen as state variables. The opposite choice will often drive the highly constrained variables into the infeasible region causing computational difficulties. Also for the applications illustrated in this paper, minor equilibrium species should not be chosen as search variables. 

Selected entries from Methods in Enzymology [vol, page(s)] Add-base catalysis [with site-directed mutants, 249, 110-118 altered pH dependencies, 249, 110] commitment to [in determination of intrinsic isotope effects, 249, 343, 347-349 in interfacial catalysis, 249, 598-599 equilibrium isotope exchange in, 249, 443-479 hydrogen tunneling in, 249, 373-397] interfacial [competitive inhibitors, kinetic characterization, 249, 604-605 equilibrium parameters, 249, 587-594 forward commitment to, 249, 598-599 interpretation, 249, 578-586 (constraining variables for high processivity, 249, 582-586 kinetic variables at interface,... 

The innovative potential and the ingenuity which have gone into designing libraries, vectors and selection protocols are impressive. However, considerable input has come from structural modelling and design considerations, in the proliferation of phage-display scaffolds to provide constrained variable epitopes in different environments or contexts . In addition the method has been used to understand protein folding, protein-protein interactions and structure-function relationships in catalysis. I hope that, in this review,... 

Pb and Cs), commonly used for estimating sedimentation and particle-reworking rates (see chapter 7) in estuaries, with trace metals concentrations (Wen et al., 1999). These radionuclides can help constrain variability in the degree of reprocessing of trace metals across horizontal gradients as well as provide information on the historical accumulation of trace metals in estuaries (Ravichandran et al., 1995b). 

Again, the ideas illustrated in the above example may be generalized. Constraints placed on a given set of variables are depicted as hypersurfaces in the hyperspace spanned by the independent variables alone. These constrained variables can then be altered only in such a manner that the corresponding representative point and the path it traces out will always remain entirely on that surface. This important principle must be fully grasped to reach an understanding of the Second Law of Thermodynamics. 

The approach assumes that all the constrained variables can be measured or estimated on-line at a sampling period much smaller than the time constant of the controlled plant. Notice that the decision variables u in the RTO problem may very well be set points of feedback controllers acting directly on the plant manipulated variables. In this case, the constraint controller can be viewed as a primary controller in a cascade control configuration that corrects the set points produced at the RTO level. 

Clearly there can come a point when estimates of toxicity are so imprecise or subject to bias that any conclusions drawn from them are meaningless. At the same time, it would be inappropriate to devote resources to constraining variability unless it was going to benefit decision-making in risk assessment. 

A complete absence of any constraint on variability would seem to be irresponsible. The development of test protocols is an important contribution to constraining variability because it limits the latitude for systematic variation between operators. However, the resulting variability within or between laboratories... 

Growth rate, therefore, is a constrained variable and the correct solution to any new set of conditions involves a simultaneous knowledge of both the nucleation rate and the growth rate. 

Peptide aptamers, paptamers, thioredoxin-inserted proteins (TIPs), perturbagens, artificial proteins selected from combinatorial libraries that display conformationaUy constrained variable regions. Peptide aptamers are selected from randomized expression libraries based on their in-vivo binding capacity to the appropriate target... 

Internally, there are two thermally coupled junctions at the top and the bottom of the column, splitting liquid and vapor streams, respectively. As such, we will again use the constrained variables 4>v and 4> , to govern the manner in which material is distributed throughout the column. Mcx eover, the reflux in a single CS needs to be specified as a so-called reference reflux, and R i will again be used. 

This chapter provides an informal discussion of the basic concepts behind the minimization of a function F x) vith constrained variables x. The necessary and sufficient conditions to solve a constrained minimization problem are called KKT conditions (by Karush, Kuhn, and Tucker) or Fritz John conditions in certain specific situations. 

Jacobian at the last time interval J is adjusted by replacing the constrained variable with 4 Thus, the desired gradient is included in the last line of the matrix,... 

---