His generalization of the classical Riemann-Roch theorem launched the study of algebraic and topological K-theory.

He shifted attention from the study of individual varieties to the relative point of view ( pairs of varieties related by a morphism ), allowing a broad generalization of many classical theorems.

Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms.

In a review Alexander Macfarlane wrote: " The main idea of the work is not unification of the several methods, nor generalization of ordinary algebra so as to include them, but rather the comparative study of their several structures.

Another broad generalization of curvature comes from the study of parallel transport on a surface.

Fields are important objects of study in algebra since they provide a useful generalization of many number systems, such as the rational numbers, real numbers, and complex numbers.

* Supergroup ( physics ), a generalization of groups, used in the study of supersymmetry

In geometry, inversive geometry is the study of those properties of figures that are preserved by a generalization of a type of transformation of the Euclidean plane, called inversion.

In the spirit of generalization to higher dimensions, inversive geometry is the study of transformations generated by the Euclidean transformations together with inversion in an n-sphere:

In mathematics, a moving frame is a flexible generalization of the notion of an ordered basis of a vector space often used to study the extrinsic differential geometry of smooth manifolds embedded in a homogeneous space.

This study was also an example of stimulus generalization.

Another generalization defines a complex Hadamard matrix to be a matrix in which the entries are complex numbers of unit modulus and which satisfies H H < sup >*</ sup >= n I < sub > n </ sub > where H < sup >*</ sup > is the conjugate transpose of H. Complex Hadamard matrices arise in the study of operator algebras and the theory of quantum computation.

In the mid 1990s, in the study of non-equilibrium dynamics of spin glass models, a generalization of the fluctuation-dissipation theorem was discovered that holds for asymptotic non-stationary states, where the temperature appearing in the equilibrium relation is substituted by an effective temperature with a non-trivial dependence on the time scales.

According to Richard Flathman, Singer's 1961 book, Generalization in Ethics-An essay in the Logic of Ethics, with the Rudiments of a System of Moral Philosophy, was, at its publication, the " most detailed study of the topic " of generalization of the universalizability principle.

In abstract algebra, the superreal numbers are a class of extensions of the real numbers, introduced by H. Garth Dales and W. Hugh Woodin as a generalization of the hyperreal numbers and primarily of interest in non-standard analysis, model theory, and the study of Banach algebras.

Humans had long distinguished ourselves from the rest of the animal kingdom as “ Man the Toolmaker .” In response to Goodall ’ s revolutionary findings, Louis Leaky wrote, “ We must now redefine man, redefine tool, or accept chimpanzees as human !” Over the course of her study, Goodall found evidence of mental traits in chimpanzees such as reasoned thought, abstraction, generalization, symbolic representation, and even the concept of self, all previously thought to be uniquely human abilities.

Later on, Guido Stampacchia proved his generalization to the Lax – Milgram theorem in in order to study the regularity problem for partial differential equations and coined the name " variational inequality " for all the problems involving inequalities of this kind.

Research compiled from two sources ( a Cornell University nation-wide study, and a Harvard University study on youth ) has shown that social stratification between age groups causes stereotyping and generalization ; for instance, the media-perpetuated myth that all adolescents are immature, violent and rebellious.

Coincidence theory ( the study of coincidence points ) is, in most settings, a generalization of fixed point theory, the study of points x with f ( x )

In order to avoid this generalization, Benhabib suggests that we should study ordinary people who live in communities instead of performing a study in an unfamiliar environment.

Another public interest group is the commission of laymen or educators which is appointed to study an educational problem and to make recommendations.

The quest for holism leads most anthropologists to study a particular place, problem or phenomenon in detail, using a variety of methods, over a more extensive period than normal in many parts of academia.

The Annalistes, especially Lucien Febvre, advocated a histoire totale, or histoire tout court, a complete study of a historic problem.

The comparative study of the various legal systems may show how different legal regulations for the same problem function in practice.

An overview of econometric methods used to study this problem can be found in Card ( 1999 ).

It also awards Master and Grandmaster titles for achievement in problem and study composing and solving, and periodically publishes FIDE Albums of the best problems.

A major problem confronting the study of the early history of gunpowder is ready access to sources close to the events described.

The study of human problem solving required new kinds of human measurements and, with Anders Ericsson, Simon developed the experimental technique of verbal protocol analysis.

A 1998 study of the Stony Brook University Algorithm Repository showed that, out of 75 algorithmic problems, the knapsack problem was the 18th most popular and the 4th most needed after kd-trees, suffix trees, and the bin packing problem.

As professionals focused on problem solving, applied mathematicians look into the formulation, study, and use of mathematical models in science, engineering, business, and other areas of mathematical practice.

Some of the more important results in the study of monoids are the Krohn-Rhodes theorem and the star height problem.

As a result significant funding was made available to study various aspects of the problem and to confirm the initial findings.

# The experiment should be so designed and based on the results of animal experimentation and a knowledge of the natural history of the disease or other problem under study that the anticipated results will justify the performance of the experiment.

The problem of how this is produced, known as the binding problem, is the subject of considerable study.

Selective reporting is sometimes referred to as a " file drawer " problem, which arises when only positive study results are made public, while studies with negative or null results are not made public.

However, the numerical sign problem makes it difficult to use lattice methods to study QCD at high density and low temperature ( e. g. nuclear matter or the interior of neutron stars ).

Most researches begin with a general statement of the problem, or rather, the purpose for engaging in the study.

John Bardeen solved this problem in his study of the metal-insulator-metal junction.

As their space logistical abilities improved through the 1960s they continued to study the problem, but in each case the problem of increasing ICBM numbers meant the numbers of interceptors needed grew to overwhelm any possible launch capability.

For example, in one study, experienced users showed no problem using any design, from the first to the last, while naive user and self-identified power users both failed repeatedly.

John Adams took to heart the advice given him by his legal mentor, Jeremiah Gridley, to `` pursue the study of the law, rather than the gain of it ''.

In the following year her father undertook to give a course in Hebrew theology to Johns Hopkins students, and this brought to the Szold house a group of bright young Jews who had come to Baltimore to study, and who enjoyed being fed and mothered by Mamma and entertained by Henrietta and Rachel, who played and sang for them in the upstairs sitting room on Sunday evenings.

Let us not confuse the issue by labeling the objective or the method `` psychoanalytic '', for this is a well established term of art for the specific ideas and procedures initiated by Sigmund Freud and his followers for the study and treatment of disordered personalities.

I am not aware of great attention by any of these authors or by the psychotherapeutic profession to the role of literary study in the development of conscience -- most of their attention is to a pre-literate period of life, or, for the theologians of course, to the influence of religion.

In his study Samuel Johnson, Joseph Wood Krutch takes this line when he says that what Aristotle really means by his theory of catharsis is that our evil passions may be so purged by the dramatic ritual that it is `` less likely that we shall indulge them through our own acts ''.

According to William Ringler's study, Stephen Gosson, the theater business in London had become a thriving enterprise by 1577, and, in the opinion of many, a thoroughly bad business.

All areas of history were either favorably or adversely affected by the geographical environment, and no respectable historian could pursue the study of history without a thorough knowledge of geography.

Nor does Sen. Jackson discuss the delicate situation created by the presence in the White House of a corps of presidential assistants engaged in the study of foreign policy.

A recent study on radiation exposure by the AEC's division of biology and medicine stated: `` The question of the biological effect of ( radiation ) doses is not considered '' herein.

The urban land use study carried out by the planning division staff has consisted of identifying and mapping all urban land uses which are of significance to statewide planning.

The rural land use study is being carried out under contract by the University of Rhode Island and identifies all agricultural land uses in the state by type of use.

Be it enacted by the Senate and House of Representatives of the United States of America in Congress assembled, That the Secretary of the Interior is hereby authorized and directed to make or cause to be made a study covering -- ( 1 )

The first step is a comprehensive self study made by faculty, by outside consultants, or by a combination of the two.

A visit to the site by a group of several persons can usually bring out new ideas or verify opinions most helpful to the planning study of any recreation area.

The medical title of `` Lobar Ventilation In Man '' by Drs. C. J. Martin and A. C. Young, covers a brief paper which is one part of a much larger effort to apply electronics to the study of the respiratory process.

Already this study has suggested ways to cut his foundation manhours from 170 to 105 by eliminating idle time and wasted motion.

The study of the radio emission of the moon and planets began with the detection of the thermal radiation of the moon at 1.25-cm wave length by Dicke and Beringer ( 1946 ).

Those antisera shown by immunoelectrophoresis to be of the `` broad spectrum '' type were selected for use in the present study.