HomotopyIn topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from ὁμός "same, similar" and τόπος "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy (həˈmɒtəpiː, ; ˈhoʊmoʊˌtoʊpiː, ) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology. In practice, there are technical difficulties in using homotopies with certain spaces.
Homotopy groupIn mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space. To define the n-th homotopy group, the base-point-preserving maps from an n-dimensional sphere (with base point) into a given space (with base point) are collected into equivalence classes, called homotopy classes.
Homotopy categoryIn mathematics, the homotopy category is a built from the category of topological spaces which in a sense identifies two spaces that have the same shape. The phrase is in fact used for two different (but related) categories, as discussed below. More generally, instead of starting with the category of topological spaces, one may start with any and define its associated homotopy category, with a construction introduced by Quillen in 1967. In this way, homotopy theory can be applied to many other categories in geometry and algebra.
Homotopy theoryIn mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topology, the theory has also been used in other areas of mathematics such as algebraic geometry (e.g., A1 homotopy theory) and (specifically the study of ). In homotopy theory and algebraic topology, the word "space" denotes a topological space.
Homotopy groups of spheresIn the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure of spheres viewed as topological spaces, forgetting about their precise geometry. Unlike homology groups, which are also topological invariants, the homotopy groups are surprisingly complex and difficult to compute.
Spectrum (topology)In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory. Every such cohomology theory is representable, as follows from Brown's representability theorem. This means that, given a cohomology theory,there exist spaces such that evaluating the cohomology theory in degree on a space is equivalent to computing the homotopy classes of maps to the space , that is.Note there are several different of spectra leading to many technical difficulties, but they all determine the same , known as the stable homotopy category.
CW complexA CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better properties than simplicial complexes, but still retains a combinatorial nature that allows for computation (often with a much smaller complex). The C stands for "closure-finite", and the W for "weak" topology.
Cell junctionCell junctions or junctional complexes, are a class of cellular structures consisting of multiprotein complexes that provide contact or adhesion between neighboring cells or between a cell and the extracellular matrix in animals. They also maintain the paracellular barrier of epithelia and control paracellular transport. Cell junctions are especially abundant in epithelial tissues. Combined with cell adhesion molecules and extracellular matrix, cell junctions help hold animal cells together.
Whitehead torsionIn geometric topology, a field within mathematics, the obstruction to a homotopy equivalence of finite CW-complexes being a simple homotopy equivalence is its Whitehead torsion which is an element in the Whitehead group . These concepts are named after the mathematician J. H. C. Whitehead. The Whitehead torsion is important in applying surgery theory to non-simply connected manifolds of dimension > 4: for simply-connected manifolds, the Whitehead group vanishes, and thus homotopy equivalences and simple homotopy equivalences are the same.
Proof by contradictionIn logic, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction. Although it is quite freely used in mathematical proofs, not every school of mathematical thought accepts this kind of nonconstructive proof as universally valid. More broadly, proof by contradiction is any form of argument that establishes a statement by arriving at a contradiction, even when the initial assumption is not the negation of the statement to be proved.
Mathematical proofA mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation".
Cell adhesionCell adhesion is the process by which cells interact and attach to neighbouring cells through specialised molecules of the cell surface. This process can occur either through direct contact between cell surfaces such as cell junctions or indirect interaction, where cells attach to surrounding extracellular matrix, a gel-like structure containing molecules released by cells into spaces between them. Cells adhesion occurs from the interactions between cell-adhesion molecules (CAMs), transmembrane proteins located on the cell surface.
Internet memeAn Internet meme, commonly known simply as a meme (miːm, ), is a cultural item (such as an idea, behaviour, or style) that is spread via the Internet, often through social media platforms. Inspired by the concept of memes proposed by Richard Dawkins in 1972, Internet memes can take various forms, such as images, videos, GIFs, and various other viral sensations. Characteristics of memes include their susceptibility to parody, their use of intertextuality, their propagation in a viral pattern, and their evolution over time.
Application securityApplication security (short AppSec) includes all tasks that introduce a secure software development life cycle to development teams. Its final goal is to improve security practices and, through that, to find, fix and preferably prevent security issues within applications. It encompasses the whole application life cycle from requirements analysis, design, implementation, verification as well as maintenance. Different approaches will find different subsets of the security vulnerabilities lurking in an application and are most effective at different times in the software lifecycle.
Pesticide applicationPesticide application refers to the practical way in which pesticides (including herbicides, fungicides, insecticides, or nematode control agents) are delivered to their biological targets (e.g. pest organism, crop or other plant). Public concern about the use of pesticides has highlighted the need to make this process as efficient as possible, in order to minimise their release into the environment and human exposure (including operators, bystanders and consumers of produce).
Background checkA background check is a process a person or company uses to verify that an individual is who they claim to be, and this provides an opportunity to check and confirm the validity of someone's criminal record, education, employment history, and other activities from their past. The frequency, purpose, and legitimacy of background checks vary among countries, industries, and individuals. An employment background check typically takes place when someone applies for a job, but it can also happen at any time the employer deems necessary.
SabinesThe Sabines (ˈseɪbaɪnz, , ˈsæbaɪnz, ; Sabini; Sabini, all exonyms) were an Italic people who lived in the central Apennine Mountains of the ancient Italian Peninsula, also inhabiting Latium north of the Anio before the founding of Rome. The Sabines divided into two populations just after the founding of Rome, which is described by Roman legend. The division, however it came about, is not legendary. The population closer to Rome transplanted itself to the new city and united with the preexisting citizenry, beginning a new heritage that descended from the Sabines but was also Latinized.
Sabina (region)Sabina (Latin: Sabinum), also called the Sabine Hills, is a region in central Italy. It is named after Sabina, the territory of the ancient Sabines, which was once bordered by Latium to the south, Picenum to the east, ancient Umbria to the north and Etruria to the west. It was separated from Umbria by the River Nar, today's Nera, and from Etruria by the River Tiber. Today, Sabina is mainly northeast of Rome in the regions Lazio, Umbria and Abruzzo.
Marcus Terentius VarroMarcus Terentius Varro (116–27 BC) was a Roman polymath and a prolific author. He is regarded as ancient Rome's greatest scholar, and was described by Petrarch as "the third great light of Rome" (after Virgil and Cicero). He is sometimes called Varro Reatinus ("Varro of Rieti") to distinguish him from his younger contemporary Varro Atacinus. Varro was born in or near Reate (now Rieti) to a family thought to be of equestrian rank, and always remained close to his roots in the area, owning a large farm in the Reatine plain, reported as near Lago di Ripasottile, until his old age.
