A similar type of investigation occurs in many mathematical theories, such as the study of continuous maps ( morphisms ) between topological spaces in topology ( the associated category is called Top ), and the study of smooth functions ( morphisms ) in manifold theory.

One of the main topics in differential topology is the study of special kinds of smooth mappings between manifolds, namely immersions and submersions, and the intersections of submanifolds via transversality.

Moreover, differential topology does not restrict itself necessarily to the study of diffeomorphism.

Differential topology is the study of the ( infinitesimal, local, and global ) properties of structures on manifolds having no non-trivial local moduli, whereas differential geometry is the study of the ( infinitesimal, local, and global ) properties of structures on manifolds having non-trivial local moduli.

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure ( e. g. inner product, norm, topology, etc.

In modern introductory texts to functional analysis, the subject is seen as the study of vector spaces endowed with a topology, in particular infinite dimensional spaces.

More than one century after Euler's paper on the bridges of Königsberg and while Listing introduced topology, Cayley was led by the study of particular analytical forms arising from differential calculus to study a particular class of graphs, the trees.

The study of symplectic manifolds is called symplectic geometry or symplectic topology.

Other " miscellaneous " research areas in theoretical chemistry include the mathematical characterization of bulk chemistry in various phases ( e. g. the study of chemical kinetics ) and the study of the applicability of more recent math developments to the basic areas of study ( e. g. for instance the possible application of principles of topology to the study of elaborate electronic structure ).

To understand the concept, it is useful to study several examples first, of which there are many: all free objects, direct product and direct sum, free group, free lattice, Grothendieck group, product topology, Stone – Čech compactification, tensor product, inverse limit and direct limit, kernel and cokernel, pullback, pushout and equalizer.

Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces.

Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible.

After having made great strides in topology, he then turned to the study of dynamical systems, where he made significant advances as well.

The study of network topology recognizes eight basic topologies:

He stressed training in awareness of abstracting, using techniques that he had derived from his study of mathematics and science.

Karpov won a gold medal for academic excellence in high school, and entered Moscow State University in 1968 to study mathematics.

The technique has been applied in the study of mathematics and logic since before Aristotle ( 384 – 322 B. C.

It can be applied in the study of classical concepts of mathematics, such as real numbers, complex variables, trigonometric functions, and algorithms, or of non-classical concepts like constructivism, harmonics, infinity, and vectors.

Two aspects of this attitude deserve to be mentioned: 1 ) he did not only study science from books, as other academics did in his day, but actually observed and experimented with nature ( the rumours starting by those who did not understand this are probably at the source of Albert's supposed connections with alchemy and witchcraft ), 2 ) he took from Aristotle the view that scientific method had to be appropriate to the objects of the scientific discipline at hand ( in discussions with Roger Bacon, who, like many 20th century academics, thought that all science should be based on mathematics ).

Initially a study of systems of polynomial equations in several variables, the subject of algebraic geometry starts where equation solving leaves off, and it becomes even more important to understand the intrinsic properties of the totality of solutions of a system of equations, than to find a specific solution ; this leads into some of the deepest areas in all of mathematics, both conceptually and in terms of technique.

In classical mathematics, analytic geometry, also known as coordinate geometry, or Cartesian geometry, is the study of geometry using a coordinate system and the principles of algebra and analysis.

It is useful to study sets naively at an early stage of mathematics in order to develop facility for working with them.

Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures.

A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis.

Their cognitive science of mathematics was a study of the embodiment of basic symbols and properties including those studied in the philosophy of mathematics, via embodied philosophy, using cognitive science.

* Theoretical chemistry is the study of chemistry via theoretical reasoning ( usually within mathematics or physics ).

Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows ( also called morphisms, although this term also has a specific, non category-theoretical meaning ), where these collections satisfy some basic conditions.

In 1879, Peirce was appointed Lecturer in logic at the new Johns Hopkins University, which was strong in a number of areas that interested him, such as philosophy ( Royce and Dewey did their PhDs at Hopkins ), psychology ( taught by G. Stanley Hall and studied by Joseph Jastrow, who coauthored a landmark empirical study with Peirce ), and mathematics ( taught by J. J. Sylvester, who came to admire Peirce's work on mathematics and logic ).

* Theoretical chemistry – study of chemistry via fundamental theoretical reasoning ( usually within mathematics or physics ).

Note that unlike Hypatia he did not study ' mathematics, philosophy and astronomy ', thus he and his followers came into conflict with the ancient University of Alexandria which pursued all forms of knowledge including science and human anatomy, politics and history according to the model inaugurated by Alexander the Great, the founder of Alexandria.

Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous.

Although the main objects of study in discrete mathematics are discrete objects, analytic methods from continuous mathematics are often employed as well.

Much of mathematics is grounded in the study of equivalences, and order relations.

Then he moved to the Humboldt University of Berlin ( then called the Friedrich William University ) in 1878 where he continued his study of mathematics under Leopold Kronecker and the renowned Karl Weierstrass.

Hence for both logic and mathematics, the different formal categories are the objects of study, not the sensible objects themselves.

As Freda Easton explained in her study of Waldorf schools, " Whether one accepts anthroposophy as a science depends upon whether one accepts Steiner's interpretation of a science that extends the consciousness and capacity of human beings to experience their inner spiritual world.

The term is sometimes popularly taken to mean " knowledge meant only for certain people " or " knowledge that must be kept hidden ", but for most practicing occultists it is simply the study of a deeper spiritual reality that extends beyond pure reason and the physical sciences.

Family role theory extends this to study paternalistic, maternalistic and sibling roles, and postulates that one's later relationships are formed largely in order to fill the roles one has grown to find comfortable as part of one's family environment-the family of origin thus setting pattern for the family of choice.

One major approach to economic freedom comes from classical liberal and Right-libertarian traditions emphasizing free markets and private property, while another extends the welfare economics study of individual choice, with greater economic freedom coming from a " larger " ( in some technical sense ) set of possible choices.

Klein's aim was then to study objects living on the homogeneous space which were congruent by the action of G. A Cartan geometry extends the notion of a Klein geometry by attaching to each point of a manifold a copy of a Klein geometry, and to regard this copy as tangent to the manifold.

The study of numbers, called algebra at the beginning undergraduate level, extends to abstract algebra at a more advanced level ; and the study of functions, called calculus at the college freshman level becomes mathematical analysis and functional analysis at a more advanced level.

The study of tribology is commonly applied in bearing design but extends into almost all other aspects of modern technology, even to such unlikely areas as hair conditioners and cosmetics such as lipstick, powders and lipgloss.

His work extends from laboratory-based cellular-molecular neurobiology to the study of semiotic processes underlying animal and human communication, especially language and language origins.

For the first time, a Guadalupe Mountains cave extends deep enough that scientists may study five separate geologic formations from the inside.

Another study also suggested that large medial temporal lobe lesions, that extends laterally to include other regions produces more extensive RA, covering 40 to 50 years.

The Bergen meeting subsequently refined the second point, and produced a three-cycle framework of qualifications, which in the UK terminology ( adopted, at least partially, by many European countries ) would be Bachelor for a first degree of three years, Master for subsequent study, and Doctor for a degree which has " made a contribution through original research that extends the frontier of knowledge by developing a substantial body of work ".

For example, the learning sciences extends beyond psychology, in that it also accounts for, as well as contributes to computational, sociological and anthropological approaches to the study of learning.

A small prospective study ( n = 322 ) however, suggests that varicocele correction aimed at restoring fertility appears to be most appropriate for men whose infertility extends beyond 2 years.