Total chain rule

This rule of thumb does not apply to all polymers. For certain polymers, such as poly (propylene), the relationship is complicated because the value of Tg itself is raised when some of the crystalline phase is present. This is because the morphology of poly(propylene) is such that the amorphous regions are relatively small and frequently interrupted by crystallites. In such a structure there are significant constraints on the freedom of rotation in an individual molecule which becomes effectively tied down in places by the crystalhtes. The reduction in total chain mobility as crystallisation develops has the effect of raising the of the amorphous regions. By contrast, in polymers that do not show this shift in T, the degree of freedom in the amorphous sections remains unaffected by the presence of crystallites, because they are more widely spaced. In these polymers the crystallites behave more like inert fillers in an otherwise unaffected matrix. 

Using the chain-rule of partial differentiation (H, p 335) the total differential of (HT) can be obtained and isolated on the left hand of the equation Division of both sides of the resulting equation by dx yields a first order differential equation of the form... 

The uncertainty of the measured lattice spacing is given by the total derivative Ad, which can be written according to the chain rule as ... 

Given an some orbital- and eigenvalue-dependent xc-functional the main task is to derive a relation with allows the calculation of die xc potential (65). Two equivalent approaches to this task are possible. One can either minimize the total energy with respect to the total RKS potential under the subsidiary condition that the orbitals satisfy the RKS equations [48, 49,55,36] or one can use the chain rule for functional differentiation to replace the derivative with respect to 7 by derivatives with respect to and... 

Thus the total entropy becomes stationary if the temperatures of both systems are equal. We emphasize that in more complicated systems we can handle the constraint of constant energy more conveniently by the method of undetermined multipliers. In our derivation we have used the chain rule for derivation. 

The situation becomes different if we want to remove some gas from the system, keeping the volume V constant, and not the total entropy S, but the molar entropy 5 = C3. Then we have to apply the chain rule for functions with multiple arguments... 

To determine the defining ODE, we write the total differential of the family using the chain rule ... 

This case describes the role of consolidation centers in the lean supply chain. The description here articulates the rules for designing a consolidation center, a supply chain middle man that makes the total chain more effective. Consolidation centers are playing an increased role in many supply chains, particularly as feeders to manufacturing operations that produce complex products with many components. Every supplier cannot or will not link its operations with its customers. The consolidation center enables this linkage — at least until better suppliers can be found. In some cases, consolidation centers are under the control of third-party logistics providers responsible for ordering material, its transportation, and its preparation for the manufacturing center. 

This makes intuitive sense in that you can cancel 3D in the first term and 3E in the second term, if the variable held constant is the same for both partials in each term. This chain rule is reminiscent of the definition of the total derivative for a function of many variables. 

Weeds constitute one of the major problems in agriculture. They are an important constraint on yield in most crops across the world. These days, growers spend much money on weed control. About 6 million ton of herbicides was sold in the world in 2006 and herbicides are the most used pesticides they form approximately 38% of the total amount of pesticides (FAO 2006). Nowadays, an effort to tighten rules for authorization and application of pesticides, research and develop products that are target-specific, degrade quickly and do not accumulate in the food chain, exists in... 

With a four-carbon chain either a CHjCHj or two CH, s must be added as branches to give a total of 6 C s. Placing CHXH, anywhere on the chain is ruled out because it lengthens the chain. Two CH, s are added, but only to central Cs to avoid extending the chain. If both CH, s are introduced on the same C, the isomer is... 

The Ufson-Roig matrix theory of the helix-coil transition In polyglycine is extended to situations where side-chain interactions (hydrophobic bonds) are present both In the helix and in the random coil. It is shown that the conditional probabilities of the occurrence of any number and size of hydrophobic pockets In the random coil can be adequately described by a 2x2 matrix. This is combined with the Ufson-Roig 3x3 matrix to produce a 4 x 4 matrix which represents all possible combinations of any amount and size sequence of a-helix with random coil containing all possible types of hydrophobic pockets In molecules of any given chain length. The total set of rules is 11) a state h preceded and followed by states h contributes a factor wo to the partition function 12) a state h preceded and followed by states c contributes a factor v to the partition function (3) a state h preceded or followed by one state c contributes a factor v to the partition function 14) a state c contributes a factor u to the partition function IS) a state d preceded by a state other than d contributes a factor s to the partition function 16) a state d preceded by a state d contributes a factor r to the partition function. 

Cyclic hydrocarbon systems consisting of three or more rings may be named in accordance with the principles stated in Rule A-31. The appropriate prefix tricyclo- , tetracyclo- , etc., is substituted for bicyclo- before the name of the open-chain hydrocarbon containing the same total number of carbon atoms. Radicals derived from these hydrocarbons are named according to the principles set forth in Rule A-31.4. 

Consider a molecular skeleton composed of n main-group atoms, M , which takes the form of a chain, ring, cage, or framework. Let g be the total number of valence electrons of the molecular skeleton. When a covalent bond is formed between two M atoms, each of them effectively gains one electron in its valence shell. In order to satisfy the octet rule for the whole skeleton, 5 (Sn—g) electron pairs must be involved in bonding between the M atoms. The number of these bonding electron pairs is defined as the bond valence b of the molecular skeleton ... 

---