In mathematics, Hodge theory, named after W. V. D. Hodge, is one aspect of the study of differential forms of a smooth manifold M. More specifically, it works out the consequences for the cohomology groups of M, with real coefficients, of the partial differential equation theory of generalised Laplacian operators associated to a Riemannian metric on M.

More than half also have proteinuria ( protein in the urine ), which in one eighth is severe enough to cause nephrotic syndrome ( generalised swelling due to low protein content of the blood ).

More advanced questions involve the topology of the curve and relations between the curves given by different equations.

He reported that, " More and more of the ' new historians ' are now trying to discover what was going on inside people's heads in the past, and what it was like to live in the past, questions which inevitably lead back to the use of narrative.

More, relying on legal precedent and the maxim " qui tacet consentire videtur " ( literally, who ( is ) silent is seen to consent ), understood that he could not be convicted as long as he did not explicitly deny that the King was Supreme Head of the Church, and he therefore refused to answer all questions regarding his opinions on the subject.

More fundamentally, Christopher Green argues that Douglas Cooper's terms were " later undermined by interpretations of the work of Picasso, Braque, Gris and Léger that stress iconographic and ideological questions rather than methods of representation.

The site, however, also questions whether Earhart could have been so far from Howland and points out that the R-1340-S3H1 may have been from a USAAF B-17E ( 41-2429 ; nicknamed " Why Don't We Do This More Often ") reported to have exploded in mid-air over the area on August 7, 1942.

More recently, questions have arisen about Neoplatonism as the dominant intellectual system of late 15th-century Florence, and scholars have indicated that there might be other ways to interpret Botticelli's mythological paintings.

More research seems warranted, but at any rate the Cape Verde population is effectively extinct since 2000, all surviving birds being hybrids with Black Kites ( which merely raises further questions about their taxonomic status ).

He reported that, " More and more of the ' new historians ' are now trying to discover what was going on inside people's heads in the past, and what it was like to live in the past, questions which inevitably lead back to the use of narrative.

More detailed studies are needed to resolve these questions, namely reevaluation of morphological data in the light of the molecular findings, and new molecular studies which thoroughly sample the questionable genera.

More recently the company has enjoyed success with Robert H Frank's The Economic Naturalist ( 2008, ISBN 978-0-7535-1338-5 ), where the author had his economics students pose interesting questions from everyday life and explain them through economics.

More survey questions were also available for download.

More than a year later, Marvel attempted to conclusively settle in-print ( but not in-story ) all lingering questions and complications regarding the character ; stating as confirmation that the Pryor that appeared in Uncanny X-Men, from 2008 to 2009, was indeed the original from the 1980s stories.

More generally, Iwasawa theory asks questions about the structure of Galois modules over extensions with Galois group a p-adic Lie group.

More general questions of this kind on an arbitrary differentiable manifold are the subject of de Rham cohomology, that allows one to obtain purely topological information using differential methods.

Robert Todd Carroll writes, " More objectionable than the questionable scientific validity of the device, however, are the moral and legal questions its use raises.

( More discussion of Paul Lazarsfeld's experimental questions.

More questions arose as to how they plan to conquer the world.

Sir Thomas More would have been an extraordinary play, apart from the questions involving its revision.

More thorough knowledge of basic theories is required, and often there are questions on less-emphasized portions of normal high school chemistry curricula, such as organic chemistry and coordination chemistry.

More discursive were the responsa of the later geonim after the first half of the 9th century, when questions began to be sent from more distant regions, where the inhabitants were less familiar with the Talmud, and were less able to visit the Babylonian academies, then the only seats of Talmudic learning.

More specifically, the term usually refers either to the various theories about dealing with national questions ( e. g. how does ideology XYZ solve the national question?

More importantly, the studies tend to pose two different sets of questions.

More common, however, ( though not the most formal style ) is the syntax that treats used as a past tense of an ordinary verb, and forms questions and negatives using did: Did he use to come here?

In their second book, More Trivial Trivia, the authors criticized practitioners who were " indiscriminate enough to confuse the flower of Trivia with the weed of minutiae "; Trivia, they wrote, " is concerned with tugging at heartstrings ," while minutiae deals with such unevocative questions as " Which state is the largest consumer of Jell-O?

More detailed instructions may be obtained from your local building supply houses and craftsmen.

More rarely, a script may have separate letters for tones, as is the case for Hmong and Zhuang.

More complex alkenes may be named using the E-Z notation, used to describe molecules having three or four different substituents ( side groups ).

More specialized bicycle components may require more complex tools, including proprietary tools specific for a given manufacturer.

More complexly, an economy may be unable to export enough goods to pay for its imports, but is able to find funds elsewhere.

More recent research into the Pygmy subspecies suggests this may be an underestimate.

More direct government involvement can lead to specific areas of crony capitalism, even if the economy as a whole may be healthy.

More complex computations may involve many operations and data elements together.

( More rarely, crystals may be deposited directly from gas ; see thin-film deposition and epitaxy.

* More than one assignment may be performed in a single statement.

More conceptually, this rule expresses the fact that a change in the x < sub > i </ sub > direction may change all of g < sub > 1 </ sub > through g < sub > k </ sub >, and any of these changes may affect f.

More generally, unspecific coercion may be the form taken by disciplinary coercion, and this appears to be in fact the case within the most effective command systems of the modern world.

More generally among Baptists, a variety of parachurch agencies and evangelical educational institutions may be supported generously or not at all, depending entirely upon the local congregation's customs and predilections.

More broadly speaking, Chinese classic texts may refer to texts, be they written in vernacular Chinese or in classical Chinese, that existed before 1912, when the last imperial Chinese dynasty, the Qing Dynasty, fell.

" In addition to the various works of Brewster already mentioned, the following may be added: Notes and Introduction to Carlyle's translation of Legendre's Elements of Geometry ( 1824 ); Treatise on Optics ( 1831 ); Letters on Natural Magic, addressed to Sir Walter Scott ( 1832 ); The Martyrs of Science, or the Lives of Galileo, Tycho Brahe, and Kepler ( 1841 ); More Worlds than One ( 1854 ).

More generally, values may be of any of the kinds described as a level of measurement.

More rarely, other beaters such as cartwheel mallets ( known to kit drummers as " soft sticks ") may be used.

More generally, a function may map equivalent arguments ( under an equivalence relation ~< sub > A </ sub >) to equivalent values ( under an equivalence relation ~< sub > B </ sub >).

More generally, a function may map equivalent arguments ( under an equivalence relation on ) to equivalent values ( under an equivalence relation on ).

More broadly, eschatology may encompass related concepts such as the Messiah or Messianic Age, the end time, and the end of days.

More recent experiments have suggested that these morphological descriptions of endocytic events may be inadequate, and a more appropriate method of classification may be based upon the clathrin-dependence of particular pathways, with multiple subtypes of clathrin-dependent and clathrin-independent endocytosis.

More generally, one may use the fact that the operators do not commute,

More generally, the observed IQ improvement may be the result of the slow reduction in environmental neurotoxins, including lead, cadmium, mercury, unfiltered coal smoke, tobacco, and ethanol.