Minimal crossing number

Knots are presented by their two dimensional projection indicating the over- and underpasses. Surely, every knot can be drawn in many different ways, with the different numbers of crossings, but the table uses for each knot the projection with minimal possible number of crossings. For instance, an unknot and a trefoil have minimal crossing numbers 0 and 3, respectively. 

Each knot in the table is denoted by its minimal crossing number. In most cases, there is more than one type of knot with any given minimal crossing number for instance, two knots with five crossings. These different... 

Minimize contamination via fewer cross-connections and fewer hose stations. Minimize the number of hoses required in loading/un-loading facilities. Cross-contamination, sometimes even from catalytic amounts of material, can result in undesired hazardous reactions. To prevent contamination due to rainwater and spills, consider storing a material that can react vigorously with water under a roof. 

In a controlled, cross-sectional comparison of 100 patients with mood disturbance who had taken lithium for at least 6 months and 100 psychiatrically normal controls, lithium did not increase the prevalence of thyroid autoimmunity a minimally larger number of control subjects had antithyroid peroxidase antibodies (11 controls versus 7 patients with mood disorders) and anti-thyroglo-bulin antibodies (15 versus 8) (259). 

Some caution needs to be exercised in using such a standard the label should ideally be nonexchangeable and the number of incorporated isotopes must be sufficient so that there is minimal cross talk from the naturally... 

All afferent inputs to the striatum that have been studied so far have formed axonal fields in which the individual axonal branches cross over the dendrites of individual spiny neurons, making synapses mostly en passant. This is the cruciform axodendritic pattern of innervation (Fox et al. 1971), which places each axon into position to contact the maximum number of neurons but minimizes the number of synapses possible with each postsynaptic cell. This is in contrast to the longitudinal axodendritic synaptic arrangement formed by striatopallidal fibers (Fox and Rafols 1976), in which individual axonal branches form multiple synaptic contacts on the dendrites of postsynaptic neurons. 

So, one way to achieve a crossed aldol is to make sure that one of the reagents has no alpha protons. That will minimize the number of potential produets. Your textbook may or may not show methods for achieving a crossed aldol where both starting ketones have alpha protons. You should look through your textbook and lecture notes to see if you are responsible for such methods. 

Batch deodorization is best suited to low capacity plants where the flexibility of the process can be exploited to handle a multiplicity of oil products with minimal cross-contamination. Vessel oil capacities range from 25,000 to 50,000 lb, and overall refinery throughput will typically be on the order of 250,000 Ib/day. The typical issues associated with this type of installation are the high steam usage occasioned by the continual need to reheat the equipment, the poor opportunities for heat recovery, and the peak demands on services that operating a number of batch units in parallel can develop. 

Both the membrane and the housing in such UF units are made of polysulfone material. Epoxy-based adhesives are commonly used in such units. Such materials have enough heat resistance to be processed by hot water sterilization. External pressurized UF is used to avoid particle contamination in the filtered water. However, higher cross flow inside the hollow fibers cannot be created in the external pressurized-type UF units. Higher cross flow inside the fibers minimizes the number of particles adhering to the inside surface of the fibers. Fortunately, clogging of fibers due to the presence of particles in external pressurized ultrafiltration units is not a serious problem with the level of particle contamination found in conventional makeup systems used in the industry. 

Where space permitted, additional formulae have been added to existing pages in Volume 1 to minimize the number of cross-references between the two volumes. The arrangement and content of chapters is the same as in the first Edition. 

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