The authors set about answering this fundamental question through a detailed investigation of the patient's ability, tactually, ( 1 ) to perceive figure and ( 2 ) to locate objects in space, with his eyes closed ( or turned away from the object concerned ).

Functors were first considered in algebraic topology, where algebraic objects ( like the fundamental group ) are associated to topological spaces, and algebraic homomorphisms are associated to continuous maps.

In abstract algebra, the fundamental theorem on homomorphisms, also known as the fundamental homomorphism theorem, relates the structure of two objects between which a homomorphism is given, and of the kernel and image of the homomorphism.

The fundamental insight here is that, whilst parallel arrangements of anisotropic objects lead to a decrease in orientational entropy, there is an increase in positional entropy.

The metaphysician attempts to clarify the fundamental notions by which people understand the world, e. g., existence, objects and their properties, space and time, cause and effect, and possibility.

This may have given early humans the impression that weight is an unchanging, fundamental property of objects in the material world.

Analogous to the quantum versus classical reformation, Einstein's general and special theories of relativity have expanded the scope of mechanics beyond the mechanics of Newton and Galileo, and made fundamental corrections to them, that become significant and even dominant as speeds of material objects approach the speed of light, which cannot be exceeded.

The concept of momentum plays a fundamental role in explaining the behavior of variable-mass objects such as a rocket ejecting fuel or a star accreting gas.

These fundamental ontological categories provide the basis for communication in an age: a horizon of unspoken and seemingly unquestionable background meanings, such as human beings understood unquestioningly as subjects and other entities understood unquestioningly as objects.

It describes the relationship between the fundamental objects of physical reality and those of everyday experience as well as those of a more abstract social nature.

Principal ideal domains are thus mathematical objects which behave somewhat like the integers, with respect to divisibility: any element of a PID has a unique decomposition into prime elements ( so an analogue of the fundamental theorem of arithmetic holds ); any two elements of a PID have a greatest common divisor ( although it may not be possible to find it using the Euclidean algorithm ).

Mass was identified as a fundamental property of objects connected to their inertia, while weight became identified with the force of gravity on an object and therefore dependent on the context of the object.

Although Dirac's original intentions were satisfied, his equation had far deeper implications for the structure of matter, and introduced new mathematical classes of objects that are now essential elements of fundamental physics.

However, defining physical objects in terms of fundamental particles ( e. g. quarks ) leaves open the question of what is the nature of a fundamental particle and thus asks what categories of being can be used to explain physical objects.

The traditional goal of ontological inquiry in particular is to divide the world " at its joints " to discover those fundamental categories or kinds into which the world ’ s objects naturally fall.

In general, the objects and arrows may be abstract entities of any kind, and the notion of category provides a fundamental and abstract way to describe mathematical entities and their relationships.

Winding numbers are fundamental objects of study in algebraic topology, and they play an important role in vector calculus, complex analysis, geometric topology, differential geometry, and physics, including string theory.

Amongst their notable achievements are objects capable of manipulating fundamental forces of the universe, such as the Mask of Life and the Mask of Creation.

These texts were all based on the same fundamental ideas: ( 1 ) learning foreign languages through the vernacular ; ( 2 ) obtaining ideas through objects rather than words ; ( 3 ) starting with objects most familiar to the child to introduce him to both the new language and the more remote world of objects: ( 4 ) giving the child a comprehensive knowledge of his environment, physical and social, as well as instruction in religious, moral, and classical subjects ; ( 5 ) making this acquisition of a compendium of knowledge a pleasure rather than a task ; and ( 6 ) making instruction universal.

After this essay was circulated in samizdat and then published outside the Soviet Union ( initially on July 6, 1968, in the Dutch newspaper Het Parool through intermediary of the Dutch academic and writer Karel van het Reve, followed by The New York Times ), Sakharov was banned from all military-related research and returned to FIAN to study fundamental theoretical physics.

The study of plants is vital because they are a fundamental part of life on Earth, which generates the oxygen and food that allow humans and other life forms to exist.

Physical cosmology, as a branch of astronomy, is the study of the largest-scale structures and dynamics of the universe and is concerned with fundamental questions about its formation and evolution.

* Physical chemistry is the study of the physical fundamental basis of chemical systems and processes.

However, a different notion of compactness altogether had also slowly emerged at the end of the 19th century from the study of the continuum, which was seen as fundamental for the rigorous formulation of analysis.

The field is a new breadth of study in graduate programs, and it integrates elements from all classical areas of chemistry with a focus on fundamental issues that are unique to materials.

* Physical chemistry – study of the physical and fundamental basis of chemical systems and processes.

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

These fields of study were essentially created by Claude Shannon, who published fundamental papers on the topic in the late 1940s and early 1950s.

Recent study of the determinants of aggregate economic growth have stressed the importance of fundamental economic institutions and the role of cognitive skills.

He ended the paper by calling for a " very extended study of all the various existing stocks of languages, in order to determine the most fundamental properties of language "-almost a program statement for the modern study of linguistic typology, and a very Boasian approach.

Such methods are useful tools for biologists in many areas of research, including those who study the mechanisms of human and other diseases or fundamental biological processes in eukaryotic or prokaryotic cells.

The mid-infrared, approximately 4000 – 400 cm < sup >− 1 </ sup > ( 2. 5 – 25 μm ) may be used to study the fundamental vibrations and associated rotational-vibrational structure.

The Leyden jar was used to conduct many early experiments in electricity, and its discovery was of fundamental importance in the study of electricity.

suggests that there is a broad agreement among such sources that philosophy involves the study of fundamental or general topics ; e. g. " the most fundamental and general concepts and principles involved in thought, action and reality ", " the most general questions about our universe and our place in it ", the " absolutely fundamental reason of everything it investigates ", or " the fundamental reasons or causes of all things ".

Because of the likelihood of errors being introduced each time a manuscript was copied, the filiation of different version of the same text is a fundamental part of the study and criticism of all texts that have been transmitted in manuscript.

The fundamental laws of fluid dynamics, thermodynamics, and motion are used to study the atmosphere.

For this reason the most fundamental science is generally understood to be " physics " – the name for which is still recognizable as meaning that it is the study of nature.

Whereas games of chance provided the impetus for the mathematical study of probability, fundamental issues are still obscured by the superstitions of gamblers.

* John Rawls: Revitalized the study of normative political philosophy in Anglo-American universities with his 1971 book A Theory of Justice, which uses a version of social contract theory to answer fundamental questions about justice and to criticise utilitarianism.

He also provided an algebraic definition of fundamental groups of schemes and more generally the main structures of a categorical Galois theory.

* The fundamental theorem of algebra states that the algebraic closure of the field of real numbers is the field of complex numbers.

This is a fundamental idea, which first surfaced in algebraic topology.

In his 1799 doctorate in absentia, A new proof of the theorem that every integral rational algebraic function of one variable can be resolved into real factors of the first or second degree, Gauss proved the fundamental theorem of algebra which states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.

In mathematics, more specifically algebraic topology, the fundamental group ( defined by Henri Poincaré in his article Analysis Situs, published in 1895 ) is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other.

He used étale coverings to define an algebraic analogue of the fundamental group of a topological space.

* The fundamental groups considered in algebraic geometry are also profinite groups, roughly speaking because the algebra can only ' see ' finite coverings of an algebraic variety.

The fundamental groups of algebraic topology, however, are in general not profinite.

The following definitions are also fundamental to algebraic topology, differential topology and geometric topology.

This approach is also called nonabelian algebraic topology, and generalises to higher dimensions ideas coming from the fundamental group.

The fundamental theorem of Galois theory provides a link between algebraic field extensions and group theory.

The term generic programming was originally coined by David Musser and Alexander Stepanov in a more specific sense than the above, to describe an approach to software decomposition whereby fundamental requirements on types are abstracted from across concrete examples of algorithms and data structures and formalised as concepts, analogously to the abstraction of algebraic theories in abstract algebra.

A common fundamental assumption of many specification approaches is that programs are modelled as algebraic or model-theoretic structures that include a collection of sets of data values together with functions over those sets.

In simple cases, it relates l-adic representations of the étale fundamental group of an algebraic curve to objects of the derived category of l-adic sheaves on the moduli stack of vector bundles over the curve.

Although cohomology is fundamental to modern algebraic topology, its importance was not seen for some 40 years after the development of homology.

He has made fundamental contributions to the fields of algebraic geometry, number theory, and topology.

Based on his research of the structure of the unit group of quadratic fields, he proved the Dirichlet unit theorem, a fundamental result in algebraic number theory.

John Torrence Tate Jr. ( born March 13, 1925 ) is an American mathematician, distinguished for many fundamental contributions in algebraic number theory, arithmetic geometry and related areas in algebraic geometry.