text:
Dominant group/Metagenome

The dominant metagenome is commonly a member of a dominant group. "Often a metagenome comprises a few dominant organisms". "Random community genomes (metagenomes) are now commonly used to study microbes in different environments."

Universals
Def.
 * 1) consisting of many different elements or
 * 2) capable "of various forms" is called diverse.

Def. the quality of consisting of many different elements or capable of various forms is called diversity.

Def. "the diversity (number and variety of species) of plant and animal life within some [or a] region" is called biodiversity.

Def. a frequency of occurrence or extent of existence is called distribution.

Def. (German "ein Maß für die Vielfalt der biologischen Arten innerhalb eines Lebensraumes oder geographischen Gebietes und somit für die Vielfalt von Flora und Fauna") is called species richness (or German Artenvielfalt).

Def. "the number of species in a site, habitat or clade" is called species richness.

Dominant group
Examples from primary sources are to be used to prove or disprove each hypothesis. These can be collected per subject or in general.

 * 1) Accident hypothesis: dominant group is an accident of whatever processes are operating.
 * 2) Artifact hypothesis: dominant group may be an artifact of human endeavor or may have preceded humanity.
 * 3) Association hypothesis: dominant group is associated in some way with the original research.
 * 4) Bad group hypothesis: dominant group is the group that engages in discrimination, abuse, punishment, and additional criminal activity against other groups. It often has an unfair advantage and uses it to express monopolistic practices.
 * 5) Control group hypothesis: there is a control group that can be used to study dominant group.
 * 6) Entity hypothesis: dominant group is an entity within each field where a primary author of original research uses the term.
 * 7) Evolution hypothesis: dominant group is a product of evolutionary processes, such groups are the evolutionary process, produce evolutionary processes, or are independent of evolutionary processes.
 * 8) Identifier hypothesis: dominant group is an identifier used by primary source authors of original research to identify an observation in the process of analysis.
 * 9) Importance hypothesis: dominant group signifies original research results that usually need to be explained by theory and interpretation of experiments.
 * 10) Indicator hypothesis: dominant group may be an indicator of something as yet not understood by the primary author of original research.
 * 11) Influence hypothesis: dominant group is included in a primary source article containing original research to indicate influence or an influential phenomenon.
 * 12) Interest hypothesis: dominant group is a theoretical entity used by scholarly authors of primary sources for phenomena of interest.
 * 13) Metadefinition hypothesis: all uses of dominant group by all primary source authors of original research are included in the metadefinition for dominant group.
 * 14) Null hypothesis: there is no significant or special meaning of dominant group in any sentence or figure caption in any refereed journal article.
 * 15) Object hypothesis: dominant group is an object within each field where a primary author of original research uses the term.
 * 16) Obvious hypothesis: the only meaning of dominant group is the one found in Mosby's Medical Dictionary.
 * 17) Original research hypothesis: dominant group is included in a primary source article by the author to indicate that the article contains original research.
 * 18) Primordial hypothesis: dominant group is a primordial concept inherent to humans such that every language or other form of communication no matter how old or whether extinct, on the verge of extinction, or not, has at least a synonym for dominant group.
 * 19) Purpose hypothesis: dominant group is written into articles by authors for a purpose.
 * 20) Regional hypothesis: dominant group, when it occurs, is only a manifestation of the limitations within a region. Variation of those limitations may result in the loss of a dominant group with the eventual appearance of a new one or none at all.
 * 21) Source hypothesis: dominant group is a source within each field where a primary author of original research uses the term.
 * 22) Term hypothesis: dominant group is a significant term that may require a 'rigorous definition' or application and verification of an empirical definition.

Genera differentia
The genera differentia for possible definitions of "dominant group" fall into the following set of orderable pairs:

'Orderable' means that any synonym from within the first category can be ordered with any synonym from the second category to form an alternate term for "dominant group"; for example, "superior class", "influential sect", "master assembly", "most important group", and "dominant painting". "Dominant" falls into category 171. "Group" is in category 61. Further, any word which has its most or much more common usage within these categories may also form an alternate term, such as "ruling group", where "ruling" has its most common usage in category 739, or "dominant party", where "party" is in category 74. "Taxon" or "taxa" are like "species" in category 61. "Society" is in category 786 so there is a "dominant society".

"A related, but separate, definition relies on a linguistic identity that differs from that of the dominant society [5]."

The relative synonyms for "dominant group" configured as genera (any two) differentia (any two) could be as many as 2400 depending on the number of words that may be reasonably exact to each genus and differentia. The total number of occurrences of genera (up to two) differentia (up to two) relative synonyms for "dominant group" whether the term occurs in a specific article or not may be on the order of 10 % of the scholarly articles searched by Google scholar, for example. Finding all such occurrences may be less important than understanding the intent of the authors in using the entity as part of any article.

Synonyms
Unless otherwise stated, the small group study designation such as "Adaptation", from "Metagenome", is placed at the beginning of the search stream, e.g., "adaptation class sect superior rules", without the quotes.

As the above table demonstrates, the popularity of the genera differentia in the order listed is high. Two "dominant" relative synonyms and two "group" relative synonyms likely constitutes the concepts of a definition of "dominant group". Placing the field of interest "adaptation" in the concept string reduces the popularity of the definition to nearly that of "dominant group" itself (73,800 as of 4/21/12). This suggests a multitude of possible definitions such as

Def. a ruling class that arranges a superior adaptation is called a dominant group,

Def. superior adaptations that arrange the ruling class is called a dominant group, or

Def. a superior class adapts a ruling arrangement is also called a dominant group.

Adaptations
Individual change in order to adapt to the environment (some form of the adjustment strategy of adaptation) is often the only realistic alternative. "It can take place by way of absorption of a non-dominant group into an established dominant group".

Analysis of the first relative synonym for "dominant group" is for the citation following: "class", "The healing of wounds, which takes place in all animals, forms another class of adaptive processes.", "Aside from the class of adaptations that are for the good of the individual, there is another class connected solely with the preservation of the race.", class of mammals, "class of phenomena", "class of cases", "class of variation", "class of facts", "class of external influences", "class of speculation"; "arrangement", "arrangement of the series [of horses]", "The necessity then arises and creates another arrangement, perfecting or altering the existing structures.", "arrangement of hard [plant] parts", "the arrangement of [individuals] into species, genera, families, etc., is only a scheme invented by man for purposes of classification."; "superior", "a living optical instrument might thus be formed as superior to one of glass,", "the superior perceptiveness of the forefinger-tip might have arisen through selection"; "The periodically repeated development of the many-celled individual out of the single cell, representative of the species (or the individual ontogeny), which in general follows the same rules as the preceding ontogeny, but is each time somewhat modified according to the amount to which the species-cell has itself been changed in the phylogeny.", "On this theory we also understand the exceptions to these rules.", "We have had, in respect to the influence of the outer world on organisms, the same experience as with the rules for the weather, - when we come to examine the facts critically there are found to be as many exceptions as confirmations of the rule.".

Def. a superior class of adaptive processes that rules the arrangement of individuals, species, genera, etc. is called a dominant group.

Animal behaviors
"If dominant group members do not have complete control over subordinate reproduction, this conflict may favour the evolution of infanticidal behaviour (by either subordinates or dominants or both)."

Biodiversity
Species richness is "the number of species in a site, habitat or clade."

"Subtaxa within taxa are often distributed unevenly: Uneven distribution of species among: a, eutherian orders, with rodents being the dominant group; b, rodent families, with murids being dominant; and c, murid genera."

"Rates of morphological and taxic diversification are often incongruent, or even uncoupled61, again highlighting that there is more to biodiversity than numbers of taxa."

"At present, it is hard to tell under what circumstances disparity precedes, or perhaps drives, species richness, and when the reverse applies."

"One serious shortcoming of the "dominant group" concept is that the rhizosphere of the "dominants" is hard to define within the matrix of subdominants. Another weakness is that the existence of the dominants may depend totally on what happens to their seedlings among the subdominants in their early life phase. Of course this could, at least partially, be accounted for by considering all life phases of the dominant group of species. A further drawback is the ignorance with respect to the ecological and conservational significance of rare species (or of species that are rare only in their visible life phase but may be present in large numbers in the seed bank or in subterranean clonal elements). These "hidden" elements may bear the potential of becoming dominants in the case of a sudden disturbance."

Dispersals
"Dispersal is a fundamental process affecting the genetic structure of populations, speciation, and extinction." "Studies in group-living fish and mammals have shown that dominant group members can control group membership of subordinates, and failure to be accepted into a group or being evicted from a group can be costly for subordinates". "We removed only male breeders because they are social dominant group members and to control for sex effects."

Ecological dominance
Ecological dominance is the degree to which a species is more numerous than its competitors in an ecological community, or makes up more of the biomass.

"In many paleontological studies, taxonomic diversity" carries with it "[t]he implicit assumption ... that more species in a lineage indicate greater biomass in the lineage as well." "Although this generalization may apply in animals, there are strong reasons to think that among plants there is no strong correlation between biomass dominance and species diversity".

"In the intertidal zone, the biomass was dominated by the bivalves, but in the subtidal region the polychaetes were the dominant group."

"Numerically, polychaetes (30%) were the dominant group intertidally, and crustaceans (40%) subtidally. Porifera formed 71% of the biomass of associated intertidal organisms, while subtidal biomass was dominated by the bivalves Striostrea margaritacea and Perna perna (87%)."

Natural selections
"We have seen that it is the common, the widely-diffused, and the widely-ranged species, belonging to the larger genera, which vary most; and these will tend to transmit to their modified offspring that superiority which now makes them dominant in their own countries." "[A]ll plants throughout all time and space should be related to each other in group subordinate to group, in the manner which we everywhere behold -- namely, varieties of the same species most closely related together, species of the same genus less closely and unequally related together, forming sections and sub-genera, species of distinct genera much less closely related, and genera related in different degrees, forming sub-families, families, orders, sub-classes, and classes."

The term “dominant group” does not appear in Darwin’s 1859 book, “On the Origin of Species by Means of Natural Selection”. But the plural term “dominant groups” appears thirteen times. The earliest use of the phrase is on page 343, “The dominant species of the larger dominant groups tend to leave many modified descendants, and thus new sub-groups and groups are formed.”

From 1871 is the statement by Alfred Russel Wallace, “The third law is, that the species which resemble or “mimic” these dominant groups, are comparatively less abundant in individuals, and are often very rare.” “The Nemophas grayi is the larger, stronger, and better armed insect, and belongs to a more widely spread and dominant group, very rich in species and individuals, and is therefore most probably the subject of mimicry by the other species.” The third law statement from 1871 also occurs in 1867.

Phylogenetics
"Caenogastropoda is the dominant group of marine gastropods in terms of species numbers, diversity of habit and habitat and ecological importance." The molecular genetics of this group should provide insight into why they became dominant.

"This paper reports the first comprehensive multi-gene phylogenetic study of the group." "Despite the addition of large amounts of new molecular data, many caenogastropod lineages remain poorly resolved or unresolved in the present analyses, possibly due to a rapid radiation of the Hypsogastropoda following the Permian–Triassic extinction during the early Mesozoic."

Phylogeography
"[M]odern haplochromines gave rise to several major adaptive radiations; the most prominent ones are those of [Lake Malawi] LM and [Lake Victoria] LV." The radiation of the Tropheini from Lake Tanganyika (LT) "must now be considered as an additional radiation of the modern haplochromines, corroborating the much older perception that LT accommodates several independent species flocks".

"This implies that the ancestor of the Tropheini successfully re-entered the lake habitat and evolved into the presently dominant group in the rocky littoral zone of LT."

"Thus, this lineage of modern haplochromines managed to occupy "empty niches" in an apparently "full" ecosystem, as all remaining tribes, which now account for about 200 species, had already been established when the ancestor of the Tropheini secondarily entered LT". "The observation that these fish underwent an independent adaptive radiation in LT underlines the haplochromines' propensity for speciation."

Agriculture
Def. the art or science of cultivating the ground, including the harvesting of crops, and the rearing and management of livestock; tillage; husbandry; farming is called agriculture.

"Dairy heifers were superovulated in the presence (dominant group, N = 8) or absence (non-dominant group, N = 6) of a dominant follicle". "Dominant group" occurs twenty-five times in the article. CL stands for corpora lutea.

"Between Days 7 and 12 of the oestrous cycle, heifers were assigned to two groups according to the presence (dominant group; N = 9) or the absence (non-dominant group; N = 8) of a dominant follicle and were then superovulated."

"Annual broadleaf (fifteen species) was clearly the dominant group under all the tillage systems, compared with the perennial (six species), and annual grassy (one species) weeds."

Freshwater ecosystems
In a freshwater ecosystem such as the large shallow Chinese lake, Lake Taihu, depending upon the time of year (summer or winter), location (Meiliang Bay or lake center), wind speed and direction, "temperature, underwater light climate, nutrients and grazing by zooplankton and" fish, the lake has a dominant species, dominant genera, and/or a dominant group. Nutrient-phytoplankton stages are identified in the lake:

 * 1) "an oligo-mesotrophic stage with low algal biomass until 1981,"
 * 2) "a eutrophic situation with blooms of Microcystis during 1988–1995,"
 * 3) "hypertrophic condition with the dominance of Planctonema and total phosphorus up to 200 mg m-3 from 1996 to 1997", and
 * 4) "the restoration period after 1997."

The wax and wane of phytoplankton assemblages, involving changes in dominant species, or dominant genera, are "mainly controlled by temperature, wind and turbidity while the long-term biomass dynamics" of dominant groups are influenced by nutrient level. "The long-term development of eutrophication in Lake Taihu" with respect to the nutrient-phytoplankton stages consist of

 * 1) a rapid increase in total phosphorus (from < 10 up to about 20 mg m-3) and phytoplankton biovolume during the early years (1960 - 1981) with the dominant group being diatoms of two specific dominant species;
 * 2) change in phytoplankton from diatoms to the dominant group cyanobacteria by 1988 to 1995, including Microcystis, where total phosphorus fluctuated between 50 mg m-3 and 150 mg m-3, usually near 100 mg m-3;
 * 3) phosphorus increase again, peaking in 1996 or 1997 throughout the lake, up to 200 mg m-3, with the dominant group becoming green algae of three dominant genera, including Planctonema; and
 * 4) total phosphorus declines to around 70 mg m-3, biovolume reaches levels similar to those of 1991, and the dominant group becomes cyanobacteria again.

"The bacterial members within the dominant group of our clone library belonged to unclassified taxa, while the second predominant group consisted of members of the phylum Proteobacteria." "The third most dominant group of the clone library was classified into the phylum Cyanobacteria."

"Only 42·3 % of the clones found with ≥97 % similarity to a named genus were also present in adjacent oligotrophic lakes, including three of the dominant groups."

Subseafloors
"However, at depth, the subseafloor metagenome becomes a unique dataset, with Archaea becoming a dominant group based on ribosomal sequence analysis."

"A variety of phylotypes belonging to green nonsulfur bacteria were detected, primarily in the topmost clay layer, where they were the dominant group".

"Among them Marine Crenarchaeotic Group I (MG I) was the most dominant group (49.2% of the archaeal library)".

Arachnology
"Since spiders are probably a dominant group of predators of insects (Bristowe, 1941; Riechert, 1974; Turnbull, 1973), there is considerable interest in their feeding ecology."

"Formicids comprised the dominant group at all sites."

"In the middle and lower litter layers, those spider families recognized as web-building foragers were the numerically dominant group."

Bacteriology
Bacteria may be divided into antigenic and probiotic forms.

Antigenic bacteria
"Together the two strains, A-17 and A-18, accounted for approximately 73 per cent of all cultures examined during the entire period."

“It was clearly the dominant group during the short period of study.”

"Another 0 group, A-18, which had been found as a single culture prior to the interruption in the study, became the dominant group and persisted throughout the rest of the period covered by the table, 574 days."

Probiotic bacteria
"Molecular Diversity, Cultivation, and Improved Detection by Fluorescent In Situ Hybridization of a Dominant Group of Human Gut Bacteria Related to Roseburia spp. or Eubacterium rectale" is an article (title) describing the use of clusters of gene sequences to identify and determine the likely population of bacteria.

"The human colonic microbiota consists of at least 500 bacterial species". A phylotype is an operational taxonomic unit (OTU). "The most numerous OTU in the Roseburia and E. rectale cluster" groups with "cultured E. rectale strains". E. rectale may have a "widespread presence ... in the human gut." "The largest single cluster of cultured strains was found to center on the species E. rectale, which has long been considered one of the most abundant species in the human large intestine".

Batrachology
"Anurans are the dominant group in terms of numbers and diversity in most of the tropics, while salamanders and caecilians are only occasionally abundant."

"The fauna characteristics of amphibians, appeared Center-South China Region, and Megophryids and Ranids were dominant groups while the fauna characteristics of reptiles appeared South China Region, and Colubrids was dominant group."

"Generally speaking the hylids are the dominant group in the cleared areas along the valley floor while the microhylids are restricted to the forests."

"Plethodontid salamanders are the dominant group of salamanders inhabiting deciduous Ozark forests ( USA) ( Johnson 1992)."

"Amphibian is the dominant group of aquatic vertebrates other than fish in terms of their numbers and distribution, particularly the frogs and toads."

"In the FZ Sound Collection, the dominant group is amphibians, due to the fact that it is the group studied by several researchers at the FZ, all of them interested in bioacoustics."

Carcinology
"The taxonomy of the Sesarminae, the largest subfamily of the Grapsidae and the dominant group in most mangroves, is still very unstable, with the identities of many ecologically important genera and species still unclear."

"Macrophthalmus belongs to the family Ocypodidae, subfamily Macrophthalminae, a dominant group of crabs inhabiting the intertidal zone along the coast of China, including the Yellow-Bo Sea, East China Sea, and South China Sea (Shen, 1932, 1936, 1940; Lin, 1949)."

"Among these, the most dominant group is Calanoida, with the Cyclopoida and Harpacticoida being relatively less important."

Entomology
"Sub-dominant group, in entom., in Kirby's system, those groups of insects which either never enter the tropics or those tropical insects whose range does not exceed 50° in the Old World of 43° in the New : in contradistinction to his predominant groups and dominant groups, in the first of which he includes groups extending from the arctic region, where vegetation ceases, to the equator, and in the second those which reach to the tropics but fall short of the polar circles."

Ants
"Leaf-cutter ants evolved remarkably recently (≈8–12 million years ago) to become the dominant herbivores of the New World tropics."

"Attine agriculture achieves its evolutionary apex in the leaf-cutting ants of the genera Acromyrmex and Atta, the dominant herbivores of the New World tropics".

"Such a recent origin for this ecologically dominant group explains their conspicuous absence from Dominican amber (15–20 mya) and may help to explain why, so far as is known, most leaf-cutting ants cultivate the same cultivar species".

Bees
With respect to "[f]oraging stratification in the Tropical Atlantic Rainforest (TAF)", "stingless bees (Meliponinae) were numerically dominant in all strata".

"Nowadays, the ecological dominance of stingless bees among insects visiting flowers in the Neotropical lowlands has been attributed to their high eusociality, perennial colony activity, and generalized food habits".

"Stingless bees are the dominant group of insects that visit flowers in the TAF".

Butterflies
According to JSTOR (url=http://www.jstor.org/), the earliest use of the term “dominant group” occurs in an article on the classification of butterflies, where overall the author discusses "[t]he family groups into which butterflies should be primarily divided".

Alfred Russel Wallace in Natural Selection Am. Ed. 135 writes: “When we consider this singular apparatus, which in some species is nearly half an inch long, the arrangement of muscles for its protrusion and retraction, its perfect concealment during repose, its blood-red color, and the suddenness with which it can be thrown out, we must, I think, be led to the conclusion that it serves as a protection to the larva, by startling and frightening away some enemy when about to seize it, and is thus one of the causes which has led to the wide extension and maintained the permanence of this now dominant group”.

Insect wings
“That the wings are exceedingly important organs has never been questioned, indeed, it is a generally accepted view that they are one of the most important factors in making the Insecta such a dominant group, and the causes which induce their atrophy must be undoubtedly be deep-seated.”

Ichthyology
"This group [Neopterygii] contains most of the modern bony fishes".

"Early neopterygian fishes (Fig. 13.9) shared the seas and fresh waters with the chondrosteans through most of the Triassic, Jurassic, and Cretaceous periods but gradually became the dominant group." "They presumably became dominant because of gradual but significant changes in structure that gave them a competitive edge over the chondrosteans. These changes culminated in the subdivision Teleostei, the dominant bony fishes today."

"[I]t was suggested that 'the actinopterygians (which is the dominant group of fish at the present time with more than 20 000 species) responded to selection pressures by selective enlargement of parts of the brain that enabled a species to occupy an adaptive niche with special success' (Jerison 1973)."

Dominant individuals in a social group are often assumed to be the exclusive parents of young in that group. "However, easy access to genetic parentage data has challenged this view, and both subordinate group members and individuals from other social groups have been identified as the true parents of offspring in many social groups". Mixed parentage within social groups can occur as a result of

 * 1) reproductive concessions by dominant individuals,
 * 2) tug-of-war over reproduction within the group, or
 * 3) parasitic spawning by other sexually mature individuals.

The sizes of dominant males and females and size range of subordinates allow comparison and assignment as a parent based on genotypes, parentage is assigned to the dominant group member except when the other potential parent is significant and probability exceeds that of the dominant group member.

Mammalogy
"Just as is the case with humans, in at least some species of nonhuman primates such as chimpanzees, dominant group members regularly adopt postures that make them appear as large as possible".

“The subfamily Leporinae is the dominant group of rabbits and hares of today, being native to all the continents except Australia.”

“The double dominant group is slightly but not significantly below expectation.”

"Reproductive suppression in carnivore species is usually associated with co-operative breeding: subordinate females that are prevented from breeding themselves will help to raise the offspring of more dominant group members (Creel & Creel 1991), providing dominant females with a direct benefit from reproductive suppression."

"Subordinate baboons voluntarily follow the dominant group member to foraging patches where they themselves starve."

"In terrestrial ecosystems from temperate to polar latitudes, mammals are the dominant group of herbivores during winter."

Mollusks
"The second category, mollusks, are a dominant group of animals found in wet meadows as well as in deepwater habitats."

“The group spatted in 1926 was the dominant group at all stations, especially at Stns. 21 and 8 in the middle of the beds where density is greatest.”

Fungi
"Marr et al (1986) categorized four physiological groups into which fungi could be placed: Group I (negative group) where neither laccase nor tyrosinase was detectable; Group II (tyrosinase-dominant group) where the tyrosinase re?"

"Lipomyces was present in nearly all soils but was the dominant group of yeasts only in those soils from which other yeasts were virtually absent."

Geese
“The order of feeding and the formation in moving toward food followed the sequence of the interspecific peck order in that the most dominant group fed first, followed by the next in dominance, down the scale.”

Diatoms
Def. one of the Diatomaceae, a family of minute unicellular algae having a siliceous covering of great delicacy is called a diatom.

"Diatoms were the dominant group in most lakes at most times, comprising over 70% of total periphyton biomass."

"In the NO3 + PO4 and NH4 + PO4 treatments, the dominant group shifted to diatoms."

"Diatomeae was the dominant group in both the periods when minima were recorded."

"Chlorophytes were the dominant group throughout the watershed with 19 species."

“The blue-greens are the dominant group.”

Protistology
"The dinoflagellates appropriated plastids from both red and green lineages; however, the dominant group in the contemporary ocean is overwhelmingly red." "[T]he ensemble of organisms responsible for primary production on land is green, while the ecologically dominant groups of eukaryotic photoautotrophs in the contemporary oceans are red."

Hypotheses

 * 1) There is no dominant group gene or genome.

title: Dominant group/Metagenome


text:
User:Marshallsumter/Dominant group (chemistry)

The term "dominant group" appears to serve as an identifier used by authors of chemistry refereed journal articles to indicate an observation of phenomena in the course of analysis. At the more general level it may be a chemical entity such as a chemist, chemical species, or laboratory.

Functional group
Per the Wikipedia article functional group: "In organic chemistry, functional groups are specific groups of atoms within molecules that are responsible for the characteristic chemical reactions of those molecules. The same functional group will undergo the same or similar chemical reaction(s) regardless of the size of the molecule it is a part of. However, its relative reactivity can be modified by nearby functional groups." "In diffuse clouds the dominant group appears to be -CH3."

Ions
From the Wikipedia article ion: "An ion is an atom or molecule in which the total number of electrons is not equal to the total number of protons, giving it a net positive or negative electrical charge."

The hydrogen anion (shown in the diagram at right), with its loosely held two-electron cloud, has a larger radius than the neutral atom, which in turn is much larger than the bare proton of the cation. Hydrogen forms the only cation that has no electrons, but even cations that (unlike hydrogen) retain one or more electrons, are smaller than the neutral atoms or molecules from which they are derived.

"In the second dominant group of ions we generally see more discrepancies in the model and the HIS data".

Organic acids
"An organic acid is an organic compound with acidic properties. The most common organic acids are the carboxylic acids, whose acidity is associated with their carboxyl group –COOH." per the Wikipedia article organic acid.

"That good correlation can be obtained if one focuses attention on the dominant group can be nicely illustrated with the benzoic acids."

Volatile organic compound
From the Wikipedia article volatile organic compound: "Volatile organic compounds (VOCs) are organic chemicals that have a high vapor pressure at ordinary, room-temperature conditions. Their high vapor pressure results from a low boiling point, which causes large numbers of molecules to evaporate or sublimate from the liquid or solid form of the compound and enter the surrounding air."

"As such alcohols became the dominant group of volatiles in the overripe mango, the relative concentrations of other groups of compounds were, of course, correspondingly reduced."

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/


text:
Dominant group/Economics

The term dominant group is a theoretical entity that is used within the social science of economics to indicate phenomena of importance.

In the social sciences such as sociology, it has been defined more precisely.

"Dominant group" may be used at least three fundamental ways:
 * 1) identification of an economic entity,
 * 2) in association with economics in some way, or
 * 3) in reference to a specific social group.

Dominant group
Examples from primary sources are to be used to prove or disprove each hypothesis. These can be collected per subject or in general.
 * Accident hypothesis: dominant group is an accident of whatever processes are operating.
 * Artifact hypothesis: dominant group may be an artifact of human endeavor or may have preceded humanity.
 * Association hypothesis: dominant group is associated in some way with the original research.
 * Bad group hypothesis: dominant group is the group that engages in discrimination, abuse, punishment, and additional criminal activity against other groups. It often has an unfair advantage and uses it to express monopolistic practices.
 * Control group hypothesis: there is a control group that can be used to study dominant group.
 * Entity hypothesis: dominant group is an entity within each field where a primary author of original research uses the term.
 * Evolution hypothesis: dominant group is a product of evolutionary processes, such groups are the evolutionary process, produce evolutionary processes, or are independent of evolutionary processes.
 * Identifier hypothesis: dominant group is an identifier used by primary source authors of original research to identify an observation in the process of analysis.
 * Importance hypothesis: dominant group signifies original research results that usually need to be explained by theory and interpretation of experiments.
 * Indicator hypothesis: dominant group may be an indicator of something as yet not understood by the primary author of original research.
 * Influence hypothesis: dominant group is included in a primary source article containing original research to indicate influence or an influential phenomenon.
 * Interest hypothesis: dominant group is a theoretical entity used by scholarly authors of primary sources for phenomena of interest.
 * Metadefinition hypothesis: all uses of dominant group by all primary source authors of original research are included in the metadefinition for dominant group.
 * Null hypothesis: there is no significant or special meaning of dominant group in any sentence or figure caption in any refereed journal article.
 * Object hypothesis: dominant group is an object within each field where a primary author of original research uses the term.
 * Obvious hypothesis: the only meaning of dominant group is the one found in Mosby's Medical Dictionary.
 * Original research hypothesis: dominant group is included in a primary source article by the author to indicate that the article contains original research.
 * Primordial hypothesis: dominant group is a primordial concept inherent to humans such that every language or other form of communication no matter how old or whether extinct, on the verge of extinction, or not, has at least a synonym for dominant group.
 * Purpose hypothesis: dominant group is written into articles by authors for a purpose.
 * Regional hypothesis: dominant group, when it occurs, is only a manifestation of the limitations within a region. Variation of those limitations may result in the loss of a dominant group with the eventual appearance of a new one or none at all.
 * Source hypothesis: dominant group is a source within each field where a primary author of original research uses the term.
 * Term hypothesis: dominant group is a significant term that may require a 'rigorous definition' or application and verification of an empirical definition.

Values
Def. "a social group that controls the value system and rewards in a particular society" is called a dominant group.

Def. "[a] hierarchy of values that all moral agents possess, demonstrated by their choices", per Wiktionary value system, is called a value system.

A value system is a set of consistent ethic values (more specifically the personal and cultural values) and measures used for the purpose of ethical or ideological integrity.

Def. "[s]omething of value given in return for an act" is called a reward.

A process that reinforces behavior — something that, when offered, causes a behavior to increase in intensity is called a psychological reward.

Def. "the ethnic group in a society that exercises power to create and maintain a pattern of economic, political, and institutional advantage, which in turn results in the unequal (disproportionately beneficial to the dominant group) distribution of resources" is called a dominant ethnic group.

A comparative advantage is when "one nation's [or individual's] opportunity cost of producing an item is less than another nation's opportunity cost of producing that item. A good or service with which a nation has the largest absolute advantage (or smallest absolute disadvantage) is the item for which they have a comparative advantage."

Def. a group that "possesses a disproportionate share of societal resources, privileges, and power" is called a dominant group.

The term economics may refer to a variety of phenomena.

Def. "[t]he study of resource allocation, distribution and consumption; of capital and investment; and of management of the factors of production", from economics, is called economics.

From the Wikipedia article definitions of economics:

Def. "a science which studies human behaviour as a relationship between ends and scarce means which have alternative uses" is called economics.

Alternately,

Def. a science that studies the production, consumption, and transfer of wealth is called economics.

Acquisitions
"[S]ectors are identified as dominant firm, dominant group of firms, or fragmented sectors on the basis of 'outside' information about the sectors."

Architectures
Notation: let the symbol g be a network of firms.

"The dominant group architecture, gdk, is characterized by one complete non-singleton component with k ≥ 2 firms and n − k singleton firms."

Banks
"In continental Europe, one dominant group are regional or local savings banks."

Capital and labor
"The dominant group exports capital to the minority group and the latter exports labor to the dominant group."

"In his model, the government weighs up inflation, unemployment and labour market reforms, all of which are disliked by the majority in the electorate, who are also the dominant group in the labour market."

"In Continental Europe the majority of executive managers are members of the dominant group, with ownership majority."

Cartels
"Cartels may fail when members socially distant from the dominant group defect."

Def. "[a] group of businesses [or nations] that collude to limit competition within an industry or market" is called a cartel.

Discriminations
In an article about "[a] general equilibrium model of statistical discrimination" by economists Andrea Moro and Peter Norman, the term 'dominant group' occurs fourteen times.

"The dominant group gains from discrimination, rationalizing why a majority may be reluctant to eliminate discrimination." "While discrimination is always preferable for the dominant group it may simply not be sustainable in equilibrium."

With respect to "Gains for the dominant group", "The monotone spillover effects also create an incentive to discriminate. If a certain group could choose between a symmetric equilibrium and an equilibrium where the other group is discriminated against they would always choose to discriminate the other group."

Economists
“It might well repay the students of this subject to pass in review the voluminous evidence that could be brought to prove that these ideas were held almost unquestioningly by the dominant group of economists of that period.”

"From the beginnings of political economy as a separate branch of study, two more or less conflicting conceptions of the subject have been present in the minds of men. According to one conception the aim of economics is to study prices, profits, and trade; according to the other, it is to study the relations between wealth and human welfare."

Investors
"As Friedman shifts focus away from government and its regulation of capital, he focuses the limelight on a new dominant group: investors."

Macroenonomics
"Unfavourable changes of the macroeconomic environment, which are caused not by any individual dominant group, but by big insiders taken as a social class, we call the external insider rent effect."

Microeconomics
"What we say on macroeconomic policy (of which we know little) receives much more attention than what we say on microeconomic policy (on which we know considerable). Stigler (1982) suggests that "society" (viewed as a collection of interest groups) is not stupid; it merely accepts that part of our policy advice that is in some dominant group's interest to accept."

Monopolies
"If such a fringe existed and became a threat to the dominant firm then we would expect the dominant group, by predatory action, to do something about it." Dominant firm is a relative synonym for "dominant group".

"Monopolistic market structures and their consequences have for long been part of established economic analysis ... However, real-world monopolies do not always fit the textbook categories ... The matter has been brought to a head in recent years by cases initiated by antitrust authorities against allegedly monopolistic firms ... These cases have served to highlight the inadequacies of traditional monopoly analysis ... From these cases has emerged a much better understanding of dominant firms."

Def. “to exercise a dominant influence over the production, acquisition, supply or price of goods or services in that market”, from New Zealand's Commerce Act, Section 3(8), is called a dominant position in that market.

"Any abuse by one or more undertakings of a dominant position within the common market or in a substantial part of it shall be prohibited as incompatible with the common market insofar as it may affect trade between member states.", Article 86 of the Treaty of Rome.

"Dominance is a measure of market power. Being in a ‘dominant position’ is interpreted ... as having sufficient market power (economic strength) ... to behave to an appreciable extent in a discretionary manner without suffering detrimental effects in the relevant market(s). This interpretation stresses independence of behaviour, i.e., conduct that is pursued independently of the presence, actions or reactions of existing or potential competitors, purchasers or suppliers.", in Magnum Corporation Ltd./Dominion Breweries (1987).

"Many African countries attempted to undertake institutional reforms after independence. Unfortunately, a few urban elites still dominated the process; the masses were not enfranchised and provided with opportunities to participate effectively in the design of the rules. As a result, many of the institutional arrangements produced by these latter-day reform efforts were inefficient and produced Leviathan states whose redistributive powers were regularly used by the politically dominant groups to amass wealth for themselves."

"In each instance, the politically dominant group is attempting to design a constitution that will legitimize its hold on power and allow it to continue to monopolize resource allocation."

"In 1972, the ruling elites in Cameroon hurriedly put together a constitution that abolished the country's federal system and made Cameroon a one-party dictatorship." Bold added.

Oligopolies
The phrase 'dominant group' occurs three times in an article on 'Networks of collaboration in oligopoly'.

 * 1) "Specifically, we show that networks with the dominant group architecture, stars, and inter-linked stars are stable.",
 * 2) "In addition, networks with a large dominant group and several isolated firms are also stable.", and
 * 3) "In particular, the complete network, where every firm has a collaboration link with every other firm, and the network with a dominant group, which contains a large number of completely connected firms and several isolated firms, appear to be stable under different competitive environments."

"[U]nder moderate competition, spillovers may have the effect of lowering the level of collaborations."

For evaluating the effects of entry regulation on oligopolistic interaction, "[l]et there be two groups of firms, call them D (for the dominant group) and F (for the fringe)." The results from an analysis of "the effects of entry restrictions on conduct ... are consistent with a [Heinrich Freiherr von Stackelberg] von Stackelberg type of industry where the degree of oligopolistic interaction among the leading firms is reduced as a consequence of the relaxation of the legal entry barriers."

Producer networks
"When only the lowest-cost producer(s) make positive profits, then the network takes on a dominant group architecture: there is a (small) complete subnetwork of the low-cost producers, accompanied by the remainder of firms all of which are isolated. This network structure also emerges when costs of link formation are high, and competition is Cournot."

Consumers
"Accordingly, in any voluntary agency where authority is vested in a non-consumer group, the nature and degree of consumer participation will depend, amongst other factors, upon the way in which the dominant group defines and applies the term itself. Will the semantics of participation be used to provide consumers with access to powerful positions within voluntary agencies or will the very definition attributed to the term serve to re-inforce a role of passivity and powerlessness?"

Power tools
"In turn, electric tools, the dominant group, can be further broken down by specific products (drills, saws, other, and parts and accessories) and by power source (plug-in and cordless)."

Relative dominant groups
The subject designation such as "economics" is placed at the beginning of the search stream, e.g., "economics class sect superior rules".

The first test in economics indicates that the "dominant group" definition using the genera differentia "class sect superior rules", not necessarily in any order, may not have usage close to that of "dominant group".

In the article cited following: "sect" is used as "section", sections of the article itself, "class" usually refers to a "class of test statistics", "superior" refers to a "superior predictive ability", and "rules" refers to "decision rules" or "trading rules". A definition of "dominant group" per preceding cited article may be

Def. a superior class [of test statistics] that [influences] [decision or trading rules] in a specific section [of the article] is called a dominant group.

Other definitions are likely. In the above article, the word "dominant" and the term "dominant group" did not occur.

A second article suggests some other situations. In the citation following, "class" refers to "A majority group, say, the workers, who control the policy might rationally choose to have a constitution which limits their power, say, to expropriate the wealth of the capitalist class.", "the class of feedback policy rules", and "[t]he most general class of decision policies"; "sect" refers to article sections again; "rules" refers to "policy rules" or "decision rules"; and "superior" refers to "Typically the iterative process of the policy rule change inducing investment function change inducing policy rule change, etc., did converge. Given that it converges, the limiting policy rule is consistent in the sense described in Sections I1 and IV. In all cases for which it did converge, we searched for and found linear feedback policy rules which were superior to this consistent rule, typically by a substantial amount." In this article "dominant group" does not occur but "dominant" does: "dominant player", "dominant firm", and "[f]or policy selection, the policymaker is dominant".

Here a "dominant group" definition is very similar to the above yet may be

Def. a superior class of [policy or decision] rules described in a section of the article is called a dominant group.

These two examples are closer to a genus differentia definition.

A third article is not too different from the first two. Per the following citation: "class" occurs over 100 times in the article and refers to class of stock, as in "superior class of stock", class A shares owned by corporate insiders of top management, class B are the inferior class of stock; "sect" refers again to "section" of the article; "superior" as already indicated; and "rules" refers to takeover rules benefitting the insiders and "voting rules" also benefitting insiders.

Def. a superior class [of stock owned by the corporate insiders] rules [takeover gains and voting for the insiders] is called a dominant group of stock.

Each of the "dominant group" definitions requires additional words to clarify what's happening; i.e., to make the definition more precise to the specific situations.

Synonyms
The genera differentia for possible definitions of "dominant group" fall into the following set of orderable pairs:

'Orderable' means that any synonym from within the first category can be ordered with any synonym from the second category to form an alternate term for "dominant group"; for example, "superior class", "influential sect", "master assembly", "most important group", and "dominant painting". "Dominant" falls into category 171. "Group" is in category 61. Further, any word which has its most or much more common usage within these categories may also form an alternate term, such as "ruling group", where "ruling" has its most common usage in category 739, or "dominant party", where "party" is in category 74. "Taxon" or "taxa" are like "species" in category 61. "Society" is in category 786 so there is a "dominant society".

Hypotheses

 * 1) When money, greed or its equivalent is involved there are always dominant groups.

title: Dominant group/Economics


text:
Engineering Projects/Remotes

Idea
Remotes are needed to control robots. Remotes could be tethered, infared, bluetooth, wifi, adhoc wifi, or radio controlled. Remotes could have in them gryos, accelerometers, video and buttons. Remotes could be old cell phones, old mutlimedia pc parts, old vhs-dvd remotes, modern Wii motes and knumchucks. Most of these remotes can be connected through usb dongles to a PC. The remote codes can be captured by a PC. The trick is to substitute a robot, or the robot control system for the PC. This requires understanding the USB protocol and a USB host port on the robot.

Subsystems

 * ATI Wonder RF Remote
 * Wii Nunchuk
 * Wii Remote Plus
 * RC control Systems
 * Blue Tooth Adapters

Tutorials
USB Protocol Analyzer

title: Engineering Projects/Remotes


text:
User:Egm6321.f11.team2/HW2

=R*2.1 - Homogenous Solutions of the Second-Order Legendre Equation =

Given
The Legendre Differential Equation is a second-order ordinary differential equation given as follows:

For this problem, we take the specific case n = 1, in which the Legendre Differential Equation takes the form

It is asserted that the following functions satisfy 2.1.2:

Find
Show that $$y_h^1(x)$$ and $$y_h^2(x)$$ satisfy 2.1.2.

Solution
This problem was solved without referring to past solutions.

$$y_h^1$$ is by far the simpler of the two functions, so we prove that it satisfies 2.1.2 first in order to make the process clear.

First, we find the first and second derivatives of $$\displaystyle y_h^1$$:

Substituting 2.1.5 and 2.1.6 into 2.1.2 yields

which proves that $$y_h^1$$ satisfies equation 2.1.2.

The process for showing that $$y_h^2$$ satisfies 2.1.2 is exactly the same as above.

We begin by carefully taking the first two derivatives of $$y_h^2$$:

$$\displaystyle y_h^2(x) = \frac{x}{2}\log\left(\frac{1+x}{1-x}\right)-1 $$

Simplifying the last term yields

For the second derivative, we can easily differentiate the first term by using the results from taking the first derivative.

simplifying yields

Finally, substituting 2.1.7 and 2.1.8 into 2.1.2 yields

which proves that $$y_h^2$$ satisfies 2.1.2.

Since the Legendre Differential Equation is second-order, and we now possess two linearly independent solutions to the homogenous equation (see R2.6), every solution of the homogenous form of the equation can be expressed as a linear combination of $$y_h^1$$ and $$y_h^2$$.

=R*2.2- Reduced-Order Solution of an ODE with Missing Dependent Variable=

Given
$$\displaystyle y''+ y' = x $$ $$\displaystyle p(x):=y'(x)$$ $$\displaystyle p' + p = x$$(2.2.1) Solution is that $$\displaystyle p(x)= k_1 e^{-x}+x-1 $$(2.2.2)

Find
Verify that eq (2.2.2) is indeed solution for eq (2.2.1).

Solution
Solved on my own

Equation 2.2.1 is that $$\displaystyle p' + p = x$$(2.2.1) And now Let $$\displaystyle T = x - p$$ $$\displaystyle \frac {dT}{dx}= 1 - p' $$ $$\displaystyle p'=1- \frac {dT}{dx}$$ Plug in eq (2.2.1) The equation 2.2.1 is following that $$\displaystyle 1- \frac{dT}{dx} = t$$ $$\displaystyle 1 - T = \frac {dT}{dx} $$ $$\displaystyle dx = -\frac {dT}{(T-1)}$$ $$\displaystyle -k - x = ln(T-1)$$ $$\displaystyle T - 1 = e^{-k-x}$$ $$ \displaystyle T= 1 - ke^{-x}$$ $$ \displaystyle p(x)= x - T = ke^{-x} + x - 1$$(2.2.2) It shows that eq (2.2.2) is indeed solution for eq (2.2.1). If you want double check, $$\displaystyle p=ke^{-x} + x - 1$$ $$\displaystyle p'= -ke^{x}+1$$ Thus, $$\displaystyle p'+p=ke^{-x} + x - 1-ke^{x}+1=x$$

=R*2.3- Linearity and Nonlinearity in General First-Order ODEs= Solved on my own

Given
An ODE of 1st order is read as

$$\displaystyle M(x,y)+N(x,y)\,y'=0 $$ (2.3.1)

Find
show that the eqn (2.3.1) is linear in $$\displaystyle y'$$, and that it is in general an N1-ODE. But it is not the most general N1-ODE as represented by $$ \displaystyle G(y',y,x)=0 $$ (2.3.2) Give an example of a more general N1-ODE.

Definition of Linear
If $$\displaystyle L_{1} $$ is a linear operator, then it must satisfy $$\displaystyle L_{1}(\alpha u+\beta v)=\alpha L_{2}(u)+\beta L_{2}(v)$$ Where $$\displaystyle u,v $$ are the independent variables, and $$\displaystyle \alpha, \beta \in \mathbb{R}.$$ $$\displaystyle L_{1}(\alpha y'_{1}+\beta y'_{2})=-[\alpha(\frac{M}{N})_{1}+\beta(\frac{M}{N})_{2}]$$ $$\displaystyle L_{1}(\alpha y'_{1}+\beta y'_{2})=-\alpha(\frac{M}{N})_{1}-\beta(\frac{M}{N})_{2}$$ $$\displaystyle L_{1}(\alpha y'_{1}+\beta y'_{2})=\alpha L_{1}(y'i_{1})+\beta L_{1}(y'_{2})$$

$$\displaystyle \therefore M(x,y)\,dx+N(x,y)\,y'=0 $$ is  linear in $$\displaystyle y' $$.

And it is easy to proof that $$\displaystyle L_{1}(\alpha y_{1}+\beta y_{2})\neq\alpha L_{1}(y_1)+\beta L_{1}(y_2)$$ So, is in general an N1-ODE.

An example of the general N1-ODE
$$ \displaystyle M(x,y)+N(x,y)f(y')=0 $$. (2.3.3) where $$ \displaystyle f(y') $$ is an arbitrary function of $$ \displaystyle y' $$.

=R*2.4- Verification of Nonlinearity in a Particular First-Order ODE=

Find
Verify that (2.4.1) is a N1-ODE.

Solution
Solved on my own

[[media:pea1.f11.mtg8.djvu|General nonlinear ODE of order 1 (N1-ODE)]] is

(2.4.1) is a statement of (2.4.2) showing that the highest order is one. Thus, (2.4.1) is a fist order differential equation.

[[media:pea1.f11.mtg4.djvu|Linearty]] has to satisfy the following condition:

[[media:pea1.f11.mtg5.djvu|Nonlinearty ]] is the condition not satisfing the above statement as follows:

(2.4.1) can be rewritten as follows:

And then, the nonlinearity can be checked with the following two functions:

From (2.4.6) and (2.4.7), $$\displaystyle F(\alpha y)\neq\alpha F(y)$$ is found so that (2.4.1) is verified to be a nonlinear equation.

Finally it is concluded that (2.4.1) is a N1-ODE (Nonlinear First Oder Differential Equation).

=R*2.5- Converting a General First-Order ODE into Particular Form =

Given
Equation 2.5.1 satisfies the most general form $$\displaystyle G\left(y',y,x \right)= 0 $$, but can be converted into [[media:pea1.f11.mtg8.djvu|particular form]] $$ \displaystyle M(x,y)+N(x,y)\,y'=0 $$.

Find
Using the hint: and Illustrate a way to convert equation 2.5.1 into particular form $$ \displaystyle M(x,y)+N(x,y)\,y'=0 $$.

Solution
Solved on my own

$$\displaystyle \bar M(x,y)+ \bar N(x,y)(y')^3=0 $$

$$\displaystyle \bar N(x,y)(y')^3=-\bar M(x,y) $$

$$\displaystyle y'=\left[-\frac{\bar M(x,y)}{\bar N(x,y)}\right]^{1/3} $$

Let

$$\displaystyle M(x,y):=\left[\bar M(x,y)\right]^{1/3} $$

And

$$\displaystyle N(x,y):=\left[\bar N(x,y)\right]^{1/3} $$

Equation 2.5.4 becomes $$ \displaystyle M(x,y)+N(x,y)\,y'=0 $$ which is a particular form of Equation 2.5.1.

=R*2.6 - Linear Independence of Solutions to the Second-Order Legendre ODE=

Given
The Legendre Differential Operator/Equation takes the form: $$\displaystyle L_2(y) := (1-x^2)y''-2xy'+n(n+1)y=0$$ (1) $$\displaystyle n=1 \Rightarrow (1-x^2)y''-2xy'+2y=0$$ (2) Two homogeneous solutions for this equation are: $$\displaystyle y^1_{H}(x) = x$$ (3) $$\displaystyle y^2_{H}(x) = \frac{x}{2} \log \left(\frac{1+x}{1-x} \right) - 1$$ (4)

Find
Show that equations (3) and (4) are linearly independent, i.e., show that $$\forall \alpha \in \mathbb R, y^1_{H}(\cdot) \ne \alpha \, y^2_{H}(\cdot)$$ i.e., for any given $$\alpha$$, show that $$\exists \hat x \rm \ such \ that \ y^1_{H}(\hat x) \ne \alpha y^2_{H}(\hat x)$$ Plot $$y^1_{H}(x)$$ and $$y^2_{H}(x)$$.

Solution
Solved on my own Since we know there exists an $$\alpha$$ and only one $$\alpha$$ such that for $$\hat x_1$$ $$y^1_{H}(\hat x_1)=\alpha_1 y^2_{H}(\hat x_1)$$ We also know that for any other point $$\hat x_2$$ there exists an $$\alpha_2$$ such that $$y^1_{H}(\hat x_2)=\alpha_2 y^2_{H}(\hat x_2)$$ The two homogeneous solutions are linearly independent if $$\alpha_1 \ne \alpha_2$$ For $$\hat x_1 = \frac{1}{2}$$ $$\displaystyle \frac{1}{2} = \alpha_1 \left[\frac{\frac{1}{2}}{2} \log \left(\frac{1+\frac{1}{2}}{1-\frac{1}{2}} \right) - 1 \right]$$ Solving for $$\alpha_1$$ we determine $$\alpha_1$$ = -0.5677175. For $$\hat x_2 = \frac{1}{4}$$ $$\displaystyle \frac{1}{4} = \alpha_2 \left[\frac{\frac{1}{4}}{2} \log \left(\frac{1+\frac{1}{4}}{1-\frac{1}{4}} \right) - 1 \right]$$ Solving for $$\alpha_2$$ we determine $$\alpha_2$$ = -0.2571305. Since $$\alpha_1 \ne \alpha_2$$ the homogeneous solutions must be linearly independent. This can be seen by looking at their graph.

=R*2.7 - Deriving Exact First Order Differential Equations from Level Sets=

Given
The family of level sets

Find
Find

$$G(y',y,x) = \frac{d}{dx}\phi(x,y) = 0$$

and show that it is an N1-ODE.

Solution
This solution was found without referring to past solutions.

We know from the chain-rule that

$$\frac{d\phi}{dx} = \frac{\partial\phi}{\partial x} + \frac{\partial\phi}{\partial y}\frac{\partial y}{\partial x}$$

Applying this directly yields

$$ \begin{align} \frac{\partial\phi}{\partial x} &= 2xy^{\frac{3}{2}} + 3x^{-1} \\ \frac{\partial\phi}{\partial y} &= \frac{3}{2}x^2y^{\frac{1}{2}} + 2y^{-1} \end{align} $$

Thus

which, as can be seen, is already in the form $$M(x,y) + N(x,y)y' = 0$$.

Due to the way that it was derived, it is expected that 2.7.2 is exact, and indeed

$$\displaystyle M_{,y} = 3xy^{\frac{1}{2}} = N_{,x} \qquad M_{,y} := \frac{\partial M}{\partial y}$$

It is obvious that G is an ordinary differential equation and that it is first-order.

To prove that it is nonlinear, suppose that $$y_1$$ and $$y_2$$ are functions of x. Then, slightly changing our notation to G(y) for clarity, we see that

$$ \begin{align} \displaystyle G(\alpha\cdot y_1 + \beta\cdot y_2) &= 2x\left(\alpha\cdot y_1 + \beta\cdot y_2\right)^{\frac{3}{2}} + 3x^{-1} \\    &\qquad + \left(\frac{3}{2}x^2\left(\alpha\cdot y_1 + \beta\cdot y_2\right)^{\frac{1}{2}} + 2\left(\alpha\cdot y_1 + \beta\cdot y_2\right)^{-1}\right)\left(\alpha\cdot y_1 + \beta\cdot y_2\right)' \\ &\neq \alpha\cdot G(y_1) + \beta\cdot G(y_2) \end{align} $$

The powers of y in G make it nonlinear.

=R*2.8 - First Exactness Condition=

Given
First exactness condition is that an ODE can be written in the form $$\displaystyle M(x,y) + N(x,y)\frac {dy}{dx} = 0$$

Find
Does the following N1-ODE satisfy the first exactness condition? $$\displaystyle M(x,y)cosy' + N(x,y)logy' = 0$$(2.8.1)

Solution
Solved on my own

In order to satisfy the first exactness condition, it must be in the form of: $$\displaystyle M(x,y) + N(x,y)\frac {dy}{dx} = 0$$ $$\displaystyle y'= -\frac{M(x,y)}{N(x,y)}$$(2.8.2) So rearrange eq (2.8.1) to fit eq (2.8.2) $$\displaystyle M(x,y)cosy' + N(x,y)logy' = 0$$ $$\displaystyle M(x,y) + N(x,y)\frac{logy'}{cosy'}=0$$ $$\displaystyle f(y')=\frac{logy'}{cosy'}$$ $$\displaystyle f(y')$$ is any function of y'. If the function $$\displaystyle f(y')$$ has no explicit inverse $$\displaystyle f^{-1}(y'),$$ then it can not satisfy in the form of eq (2.8.2) In this case, $$\displaystyle f(y')$$ does not have exact inverse $$\displaystyle f^{-1}(y')$$ Thus, This does not satisfy the first exactness condition.

=R*2.9- Commutative Partial Derivatives= Solved on my own

Given
Review calculus, and differentiation $$\displaystyle \frac{\partial^{2}\phi(x,y)}{\partial x\partial y}=\frac{\partial^{2}\phi(x,y)}{\partial y\partial x}$$ (2.9.1)

Find
Find the minimum degree of differentiability of the function $$\displaystyle \phi(x,y) $$ such that the above equation is satisfied, State the full theorem and provide a proof.

Clairaut's theorem
According to the Schwarz's theorem,, if

 * $$\displaystyle f \colon \mathbb{R}^n \to \mathbb{R}$$

has continuous second partial derivatives at any given point in $$ \displaystyle \mathbb{R}^n $$, then for $$ \displaystyle 1 \leq i,j \leq n,$$

 * $$ \displaystyle \frac{\partial^2 f}{\partial x_i\, \partial x_j}(a_1, \dots, a_n) = \frac{\partial^2 f}{\partial x_j\, \partial x_i}(a_1, \dots, a_n).\,\!$$

And we then know that the minimum degree of differentiability of the function $$\displaystyle \phi(x,y) $$ must be three when it satisfies eqn (2.9.1)

Proof of Clairaut's theorem
The result was first discovered by L.Euler around 1734 in connection with problems in hydrodynamics. In order to prove this theorem, we shall first introduce the Mean Value theorem as follows: Let $$ \displaystyle f:[a,b]\rightarrow\mathbb{R} $$ be a continuous function on the closed interval $$ \displaystyle [a,b]$$, and derivative on the open interval $$ \displaystyle (a,b)$$, where $$ \displaystyle a(2.9.2)

Suppose Q is a closed rectangle with sides parallel to the coordinate axes, put
 * $$\triangle(f,Q)=f(a+h,b+k)-f(a+h,b)-f(a,b+k)+f(a,b)$$

(2.9.3)

 * $$\ u(t)=f(t,b+k)-f(t,b)$$

 * $$\triangle(f,Q)=u(a+h)-u(a)$$

 * $$\ =hu'(x)$$

 * $$ =h(\frac{\partial f}{\partial x}(x,b+k)-\frac{\partial f}{\partial x}(x,b)$$

 * $$=hk\frac{\partial^{2}}{\partial y\partial x}f(x,y) $$

(2.9.4) So, we put

 * $$A=\frac{\partial^{2}}{\partial y\partial x}f(a,b)$$,

Chose $$ \displaystyle \epsilon>0 $$. if $$ \displaystyle h $$ and $$ \displaystyle k $$ are sufficiently small, we have

 * $$|\frac{\triangle(f,Q)}{hk}-A|<\epsilon$$

(2.9.5) Fix $$ \displaystyle h $$, and let $$ \displaystyle k\rightarrow0 $$, then

 * $$|\frac{(\frac{\partial f}{\partial y}(a+h,b)-\frac{\partial f}{\partial y}(a,b)}{h}-A|\leq\epsilon$$

(2.9.6) Since $$ \displaystyle \epsilon$$ was is arbitrary, and (2.9.6) holds for all sufficiently small $$ \displaystyle h\neq0$$

 * $$\frac{\partial^{2}}{\partial y\partial x}f(a,b)=\frac{\partial^{2}}{\partial x\partial y}f(a,b)$$

(2.9.7) Since $$ \displaystyle (a,b)$$ is an arbitrary point in $$ \displaystyle \mathbb{R}^2$$, so
 * $$\frac{\partial^{2}}{\partial y\partial x}f(x,y)=\frac{\partial^{2}}{\partial x\partial y}f(x,y)$$

=R*2.10-Verification of solution for N1-ODE=

Given
[[media:pea1.f11.mtg7.djvu|Particular N1-ODE]] is

where $$\displaystyle M(x,y)=75x^4$$ and $$\displaystyle N(x,y)=cos y$$

Find
Verify that (2.10.1) is the solution for the (2.10.2).

Solution
Solved on my own

(2.10.1) would be rewritten as follows:

Differentiating the both sides of (2.10.3), (2.10.3) results in

Thus, (2.10.1) is verified to be the solution for (2.10.2.)

=R 2.11-Solving PDE for Integrating Factor=

Given
For

Find
Why solving equation 2.11.1 for the [[media:pea1.f11.mtg11.djvu|integrating factor]] $$\displaystyle h(x,y) $$ is usually not easy ?

Solution
Solving equation 2.11.1 for the integrating factor $$\displaystyle h(x,y) $$ is usually not easy because in general equation 2.11.1 is a non-linear partial differential equation for unknown function $$\displaystyle h$$ and it is having terms $$\displaystyle h_x, h_y, N_x, M_y $$ which makes it very difficult to solve for $$\displaystyle h(x,y)$$.

=R*2.12 - Case 2: Finding $$h$$ as a Function of y=

Given
A N1-ODE satisfies the first exactness condition if it takes the particular form: $$\displaystyle M(x,y) + N(x,y)y'$$ (1) The second exactness condition states that $$\displaystyle M_{y}(x,y) = N_{x}(x,y) \ i.e., \frac{\partial M(x,y)}{\partial y} = \frac{\partial N(x,y)}{\partial x}$$ (2) If a N1-ODE satisfies the first exactness condition but not the second exactness condition the Euler Integrating factor method (IFM) can be used. This requires an integrating factor $$h(x,y)$$ be found such that the following N1-ODE is exact: $$\displaystyle h(x,y) \left[M(x,y)+N(x,y) \, y' \right]=0$$ (3) This can be written as: $$\displaystyle \underbrace{(hM)}_{\displaystyle\color{blue}{\bar M}} + \underbrace{(hN)}_{\displaystyle\color{blue}{\bar N}} \, y' = 0$$ (4) To find $$h$$ we must apply the second exactness condition to obtain: $$\displaystyle h_{x}N-h_{y}M+h(N_{x}-M_{y})=0$$ (5)

Find
Suppose $$h_{x}(x,y)=0$$, thus $$h$$ is a function of y only, then equation (5) becomes: $$\displaystyle \frac{h_{y}}{h}=\frac{1}{M}(N_{x}-M{y})=: m(y)$$ (6) Find $$h$$ using equation (6).

Solution
Solved on my own

To find $$h$$ we can integrate equation (6) to obtain: $$\displaystyle \int \frac{h_{y}}{h}= \int \frac{1}{M}(N_{x}-M{y})$$ (7) $$\displaystyle \ln(h(y)) = \int^y m(s)ds + k$$ $$\displaystyle h(y) = exp \left[\int^y m(s)ds + k \right]$$

=R*2.13 - Using an Integrating Factor to Render an Inexact L1-ODE-VC Exact=

Find
Show that h(x) = x is an integrating factor for 2.13.1, and that

$$y(x) = \frac{x^3}{4} + \frac{k}{x}$$

is its solution.

Solution
This solution was found without reference to past solutions.

We first show that 2.13.1 is not already exact, or else its integrating factor would be $$1$$!

Re-arranging 2.13.1, we arrive at the standard form

$$\displaystyle \underbrace{x^{-1}y-x^2}_M + \underbrace{1}_N y' = 0$$

Now,

$$\displaystyle M_{,y} = x^{-1} \quad\neq\quad 0 = N_{,x}, \qquad M_{,y} := \frac{\partial M}{\partial y}$$

so 2.13.1 is not exact.

We thus proceed to multiply 2.13.2 by h(x) = x which yields

Now,

$$\displaystyle \bar{M}_{,y} = 1 = \bar{N}_{,x}$$

so 2.13.2 is exact and h(x) = x is the integrating factor that makes 2.13.1 exact.

Since every exact differential equation may be produced from a first integral $$\phi(x,y) = k$$ such that

$$M(x,y) = \frac{\partial\phi}{\partial x}$$

and

$$N(x,y) = \frac{\partial\phi}{\partial y}$$

we may integrate $$\bar{M}$$ and $$\bar{N}$$ to find $$\phi$$ as follows:

Combining 2.13.3 and 2.13.4 yields

$$\displaystyle \phi(x,y) = k = xy - \frac{x^4}{4}$$

which upon solving for $$y$$ yields

$$\displaystyle y = \frac{x^3}{4} + \frac{k}{x}$$

While this already proves the assertion, we can also show that this is the correct expression for $$y$$ by plugging it back in to equation 2.13.1:

$$\displaystyle \begin{align} y' &= \frac{3}{4}x^2 - \frac{k}{x^2} \\y' + x^{-1}y &= \left(\frac{3}{4}x^2 - \frac{k}{x^2}\right) + x^{-1}\left(\frac{x^3}{4} + \frac{k}{x}\right) \\ &= x^2 \end{align} $$

=R*2.14 - General L1-ODE-VC=

Given
$$\displaystyle h(x)=exp\left[\int^x a_0(s)ds + k_1\right] $$
(2.14.1) $$\displaystyle y(x)=\frac {1}{h(x)}\left[\int^x h(s)b(s)ds + k_2 \right]$$
(2.14.2) $$\displaystyle a_1(x)y'+a_0(x)y= b(x)$$
(2.14.3)

Find
Solve the general L1-ODE-VC 1. $$\displaystyle a_1(x)=1$$ $$\displaystyle a_0(x)=x$$ $$\displaystyle b(x)=2x + 3 $$ 2. Find $$\displaystyle y(x)$$ in term of $$\displaystyle a_1(x), a_0(x), b(x)$$ 3. $$\displaystyle a_1(x)=x^2+1$$ $$\displaystyle a_0(x)=x$$ $$\displaystyle b(x)=2x$$

Solution
Solved on my own

an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve ordinary differential equations, but is also used within multivariable calculus, in this case often multiplying through by an integrating factor allows an inexact differential to be made into an exact differential. From lecture note, $$\underbrace{\left( hM \right)}_+\underbrace{\left( hN \right)}_{y}'=0$$ And applying second exactness condition, the equation is following that $$\underbrace{\left( h\frac{\partial M}{\partial y}+M\frac{\partial h}{\partial y} \right)}_{\frac{\partial \bar{M}}{\partial y}}=\underbrace{\left( h\frac{\partial N}{\partial x}+N\frac{\partial h}{\partial x} \right)}_{\frac{\partial \bar{N}}{\partial x}}$$ If we assume that $$\displaystyle h_y(x,y)=0$$, then h is function of x only and the equation is following that $$\left( h\frac{\partial M}{\partial y} \right)=\left( h\frac{\partial N}{\partial x}+N\frac{\partial h}{\partial x} \right)$$ And we obtain some equations through rearranging equation above $$\displaystyle h{{N}_{x}}+N{{h}_{x}}-h{{M}_{y}}=0$$ $$\displaystyle {{N}_{x}}=\frac{\partial N}{\partial x}$$ $$\displaystyle {{M}_{y}}=\frac{\partial M}{\partial y}$$ $$\displaystyle {{h}_{x}}=\frac{\partial h}{\partial x}$$ And we can obtain h(x) through dividing by h

$$\displaystyle \frac {h_x}{h}=-\frac{1}{N}(\underbrace{N_x}_{0}-\underbrace{M_y}_{a_0(x)})=a_0(x)$$ $$\displaystyle h(x)={{e}^{-\int{\frac{1}{N}\left( {{N}_{x}}-{{M}_{y}} \right)dx}}}=exp\left[\int^x a_0(s)ds + k_1\right] $$
(2.14.1) And in order to obtain y(x), we can apply the integrating factor to obtain the following equations $$\displaystyle y'+a_0y=b \rightarrow h(y'+a_0y)=hb$$ and $$\displaystyle h_x/h=a_0 \rightarrow ha_0=h_x=h'$$ $$\displaystyle hy'+h'y=hb$$ Thus, $$\displaystyle y(x)=\frac {1}{h(x)}\left[\int^x h(s)b(s)ds + k_2 \right]$$
(2.14.2)

part 1
1. In order to solve this problems, plug given information in eq (2.14.3) the equation is following that $$\displaystyle y'+ xy = 2x + 3 $$ Also, eq (2.14.1) is following that $$\displaystyle h(x)=exp\left[\int^x sds+k_1\right] $$ $$\displaystyle h(x)=exp\left[\frac {1}{2}s^2\right]$$ And eq (2.14.2) is following that $$\displaystyle y(x)=\frac {1}{h(x)}\left[\int^x h(s)b(s)ds + k_2 \right]$$ $$\displaystyle = \frac {1}{exp\left[\frac {1}{2}x^2\right]}\left[\int^x exp\left[\frac {1}{2}s^2\right]*(2s+3)ds+k_2\right]$$ $$\displaystyle =\frac {1}{exp\left[\frac {1}{2}x^2\right]}\left[\int^x e^{\frac {1}{2}s^2}*2sds+\int^x e^{\frac {1}{2}s^2}*3ds\right]$$ Let $$\displaystyle t= \frac {s^2}{2}$$ $$\displaystyle \frac {dt}{s}=ds$$ Finally, the equation is that $$\displaystyle y(x) =\frac {1}{exp\left[\frac {1}{2}x^2\right]}\left[\int^x 2e^tdt+\int^x e^{\frac {1}{2}s^2}*3ds\right]=\frac {1}{exp\left[\frac {1}{2}x^2\right]}\left[2e^{\frac {x^2}{2}}+\int^x e^{\frac {1}{2}s^2}*3ds\right]$$ In order to solve more, numerical method must be used.

part 2
From eq 2.14.3 $$\displaystyle a_1(x)y'+a_0(x)y= b(x)$$ $$\displaystyle y'+ \frac {a_0(x)}{a_1(x)}y=\frac {b(x)}{a_1(x)}$$ and plug in eq 2.14.1 $$\displaystyle h(x)=exp\left[\int^x \frac {a_0(s)}{a_1(s)}ds\right] $$ $$\displaystyle y(x)=\frac {1}{exp\left[\int^x \frac {a_0(s)}{a_1(s)}ds\right]}\left[\int^x exp \left[ \int^x \frac {a_0(s)}{a_1(s)}ds \right] \frac{b(s)}{a_1(s)}ds\right]$$

part 3
$$\displaystyle (x^2+1)y'+xy=2x$$ $$\displaystyle y'+\frac {x}{x^2+1}y=\frac {2x}{x^2+1} $$ At this point, $$\displaystyle h(x)=exp\left[ \int^x \frac {s}{s^2+1}ds \right] $$ Let $$\displaystyle t=s^2+1 $$ $$\displaystyle ds = \frac {dt}{2s} $$ $$\displaystyle h(x)=exp \left[ \int^x \frac {dt}{2t} \right] $$ $$\displaystyle h(x)=exp \left[ \frac {1}{2}log(x^2+1) \right] = \sqrt{{x^2}+{1}} $$
(2.14.4) Then plug in eq 2.14.2 $$ \displaystyle y(x)=\frac {1}{\sqrt{{x^2}+{1}}}\left[ \int^x \frac {2s}{\sqrt{{s^2}+{1}}}ds\right] $$ Let $$\displaystyle u= s^2 + 1 $$ $$\displaystyle du = 2sds \rightarrow ds= \frac {du}{2s} $$ $$\displaystyle y(x)=\frac {1}{\sqrt{{x^2}+{1}}}\left[ \int^x u^{-\frac {1}{2}}du \right] = \frac {1}{\sqrt{{x^2}+{1}}}\left[ 2 \sqrt{{x^2}+{1}}+ C \right] $$ Final solution is that $$\displaystyle y(x)= 2 + \frac {C}{\sqrt{{x^2}+{1}}} $$

=R*2.15- Integration constant= Solved on my own

Given
Since $$\displaystyle P(x)y'+Q(x)y=R(x)$$
(2.15.1)

$$\displaystyle y'+\frac {Q(x)}{P(x)}y=\frac{R(x)}{P(x)}$$
(2.15.2)

is an L1-ODE_VC, there should be only one integration constant, not two.

Find
Show that the integration constant $$\displaystyle k_{1} $$ in $$\displaystyle h(x)=exp[\int^{x}a_{0}(s)ds+k_{1}]  $$
(2.15.3)

is not necessary, i.e., only $$ \displaystyle k_{2} $$ in $$ \displaystyle y(x)=\frac{1}{h(x)}[\int^{x}h(s)b(s)ds+k_{2}] $$
(2.15.4)

is necessary.

Solution
Substituting the eqn (2.15.3) to eqn (2.15.4), we obtain the following expression $$\displaystyle y(x)=\frac{1}{e^{k_{1}}\cdot e^{\int^{x}a_{0}(s)ds}}[e^{k_{1}}\cdot\int^{x}e^{\int^{x}a_{0}(s)ds}b(s)ds+k_{2}]$$ $$\displaystyle y(x)=\frac{1}{e^{\int^{x}a_{0}(s)ds}}[\int^{x}e^{\int^{x}a_{0}(s)ds}b(s)ds+\frac{k_{2}}{e^{k_{1}}}]$$ Then, we can define a new constant $$\displaystyle \bar{k}_{2} $$ such that $$\displaystyle \bar{k}_{2}=\frac{k_{2}}{e^{k_{1}}} $$ hence, this new constant contains $$\displaystyle k_{1} $$. In other words, $$\displaystyle k_{1} $$ is unnecessary.

=R*2.16-Solution of L1-ODE-VC=

Given
[[media:pea1.f11.mtg11.djvu|Integrating factor]] is

The derivated result in King 2003 p.512 is

Find
Show that (2.16.1) agrees with (2.16.3).

Solution
Solved on my own

[[media:pea1.f11.mtg11.djvu|General L1-ODE-VC]] is

In King 2003 p.512, the L1-ODE-VC is presented as follows:

From (2.16.3) and (2.164), the following relations can be derived.

From the relations, in King 2003 P512, the integrating factor is shwon as follows:

When (2.16.1) is substituted by (2.16.9), (2.16.1) becomes

$$\displaystyle \Rightarrow y(x)= k_2\exp\left(-\int^x P(t)dt\right)+\exp\left(-\int^x P(t)dt\right)\int^x \exp\left(\int^x P(t)dt\right)b(s)ds$$

Substituting the relation (2.16.8) for (2.16.10), (2.16.10) can be rewritten as follows:

(2.16.10) is the solution presented in King 2003 p.512 identifying A, $$\displaystyle y_H(x)$$ and $$\displaystyle y_p(x)$$ as follows:

Thus, it is verified that (2.16.1) agrees with (2.16.3).

=R*2.17-Solving for homogenous counterpart =

Given
Instead of identifying $$\displaystyle y_H(x) $$ from $$\displaystyle y(x)=\frac{1}{h(x)}\left[\int^x h(s)b(s)ds + k_2\right]$$.

Find
Solve the [[media:pea1.f11.mtg12.djvu|homogenous counterpart]] of $$\displaystyle y' + a_0(x)\,y=0 $$.

Solution
Solved on my own

$$\displaystyle y' + a_0(x)\,y=0 $$ $$\displaystyle y'= -a_0(x)y $$ $$\displaystyle \frac{dy}{y}=-a_0(x)dx $$ $$\displaystyle \int^y \frac{dy}{y}=-\int^s a_0(s)ds $$ $$\displaystyle y(x)=exp \left[-\int^x a_0(s)ds + k \right]$$ $$\Rightarrow y_H(x)=exp \left[-\int^x a_0(s)ds + k \right]$$

=R*2.18 - Integrating Factor Method=

Given
A L1-ODE-VC satisfies the first exactness condition if it takes the particular form: $$\displaystyle M(x,y) + N(x,y)y'$$ (1) The second exactness condition states that $$\displaystyle M_{y}(x,y) = N_{x}(x,y) \ i.e., \frac{\partial M(x,y)}{\partial y} = \frac{\partial N(x,y)}{\partial x}$$ (2) If a L1-ODE-VC satisfies the first exactness condition but not the second exactness condition the Euler Integrating factor method (IFM) can be used. This requires an integrating factor $$h(x,y)$$ be found such that the following N1-ODE is exact: $$\displaystyle h(x,y) \left[M(x,y)+N(x,y) \, y' \right]=0$$ (3)

Find
$$\displaystyle \left [x^4 y + 10 \right ] + (\frac{1}{2} x^2)y'=0$$ (4) $$\displaystyle M(x,y) = x^4 y + 10$$ $$\displaystyle N(x) = \frac{1}{2} x^2$$ Find out whether equation (4) is exact. If it is not exact find the integrating factor $$h$$ to make it exact.

Solution
Solved on my own

Since equation (4) satisfies the first exactness condition we must test the second exactness condition. $$\displaystyle M_{y} = x^4$$ $$\displaystyle N_{x} = x$$ $$\displaystyle x^4 \ne x$$ Therefore, equation (4) is not exact. To make it exact an integration factor $$h$$ must be found such that: $$\displaystyle \frac{h_{x}}{h}= -\frac{1}{N(x)}(N_{x}-M{y})=: n(x)$$ Solving for h(x) produces: $$\displaystyle h(x) = exp \left[\int^x n(s)ds + k \right]$$ $$\displaystyle h(x) = exp \left[2 \int^x (s^2 - \frac{1}{s})ds + k \right]$$ $$\displaystyle h(x) = exp \left[2 (\frac{1}{3} x^3 - \ln x) + k \right]$$

=References=

=Contributing Members= Egm6321.f11.team2.rho 20:41, 21 September 2011 (UTC)

title: User:Egm6321.f11.team2/HW2