- Axiom: it is a sentence or proposition that is not proved or demonstrated and is considered as self-evident or as an initial necessary consensus for a theory building or acceptation. Therefore, it is taken for granted as true, and serves as a starting point for deducing and inferencing other truths. In mathematics, an axiom is any starting assumption from which other statements are logically derived. It can be a sentence, a proposition, a statement or a rule that enables the construction of a formal system. In many contexts, "axiom," "postulate," and "assumption" are used interchangeably. Mathematics distinguishes two types of axioms: logical axioms and non-logical axioms.
- Bayes' theorem (or Bayes' rule or Bayes' law) is a result in probability theory that relates conditional probabilities. If A and B denote two events, P(A|B) denotes the conditional probability of A occurring, given that B occurs. The two conditional probabilities P(A|B) and P(B|A) are in general different. Bayes theorem gives a relation between P(A|B) and P(B|A). Bayes' theorem is that it gives a rule how to update or revise the strengths of evidence-based beliefs in light of new evidence at posteriori.
- Causality: It denotes a necessary relationship between one event (called cause) and another event (called effect) which is the direct consequence (result) of the first.
- Column vector: In linear algebra, a column vector is an m × 1 matrix,
i.e. a matrix consisting of a single column of m elements.
The transpose of a column vector is a row vector and vice versa. The set
of all column vectors forms a vector space which is the dual space to the
set of all row vectors.
- Coordinate space Fn: In mathematics and Linear Algebra, it is the prototypical example of an n-dimensional vector space over a field F. It can be defined as the product space of F over a finite index set.
- Coordinate vector: In Linear Algebra it is an explicit representation of a vector in an abstract vector space as an ordered list of numbers or, equivalently, as an element of the coordinate space Fn. Coordinate vectors allow calculations with abstract objects to be transformed into calculations with blocks of numbers (matrices and column vectors), which we know how to do explicitly.
- Euclidean space: In modern mathematical language, distance and angle can be generalized easily to 4-dimensional, 5-dimensional, and even higher-dimensional spaces. An n-dimensional space with notions of distance and angle that obey the Euclidean relationships is called an n-dimensional Euclidean space. An essential property of a Euclidean space is its flatness. Other spaces exist in geometry that are not Euclidean. For example, the surface of a sphere is not; a triangle on a sphere will have angles that sum to something greater than 180 degrees. There is only one Euclidean space of each dimension, while there are many non-Euclidean spaces of each dimension. Often these other spaces are constructed by systematically deforming Euclidean space.
- Formal system: In Logic and Mathematics it consists of two components, a formal language plus a set of inference rules or transformation rules. A formal system may be formulated and studied for its intrinsic value, or it may be intended as a description (a model) of external phenomena.
- Formal language: In mathematics, logic, and computer science, it is a language that is defined by precise mathematical or machine processable formulas.
- Hilbert space (named after the German mathematician David Hilbert): It generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the two-dimensional plane and three-dimensional space to infinite-dimensional spaces.
- Index set: In mathematics, the elements of a set A may be indexed or labelled by means of a set J that is on that account called an index set. The indexing consists of a surjective function from J onto A and the indexed collection is typically called an (indexed) family.
- Klein bottle: It is a certain non-orientable surface with no distinct
"inner" and "outer" sides. Other related non-orientable
objects include the Möbius strip and the real projective plane. Whereas
a Möbius strip is a two dimensional object with one side and one edge,
a Klein bottle is a two dimensional object with one side and no edges. (For
comparison, a sphere is a two dimensional object with no edges and two sides.)
- Lagrange multipliers are a method for finding the extrema of a function of several variables subject to one or more constraints; it is the basic tool in nonlinear constrained optimization. The technique is able to determine where on a particular set of points (such as a circle, sphere, or plane) a particular function is the smallest (or largest). Lagrange multipliers compute the stationary points of the constrained function.
- Laplace's equation: It is a partial differential equation. The solutions of Laplace's equation are important in many fields of science, notably the fields of electromagnetism, astronomy, and fluid dynamics, because they describe the behaviour of electric, gravitational, and fluid potentials. The general theory of solutions to Laplace's equation is known as potential theory.
- Linear algebra: It is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. Vector spaces are a central theme in modern mathematics; thus, linear algebra is widely used in both abstract algebra and functional analysis. Linear algebra also has a concrete representation in analytic geometry and it is generalized in operator theory.
- Maxwell's equations: in Electromagnetism they are a set of four equations describing the interrelationship between electric field, magnetic field, electric charge, and electric current. Although Maxwell himself was the originator of only one of these equations (by virtue of modifying an already existing equation), he derived them all again independently.
- Möbius strip or Möbius band: It is a surface with only one
side and only one boundary component. It has the mathematical property of
being non-orientable. It is also a ruled surface. A model can easily be
created by taking a paper strip and giving it a half-twist, and then joining
the ends of the strip together to form a single strip. In Euclidean space
there are in fact two types of Möbius strips depending on the direction
of the half-twist: clockwise and counterclockwise.
- Model: It is a pattern, plan, representation, or description designed
to show the structure or workings of an object, system, or concept.
- Model (abstrac), an abstraction or conceptual object used in the creation
of a predictive formula.
- Causal model, an abstract model that uses cause and effect logic.
- Mathematical model, an abstract model that uses mathematical language.
Computer model, a computer program which attempts to simulate an abstract model of a particular system.
- Parallelogram law: In mathematics it states that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the squares of the lengths of the two diagonals. In case the parallelogram is a rectangle, the two diagonals are of equal lengths and the statement reduces to the Pythagorean Theorem.
- Poisson's equation: It is a partial differential equation with broad
utility in electrostatics, mechanical engineering and theoretical physics.
The Poisson equation is
where ? is the Laplace operator, and f and ? are real or complex-valued
functions on a manifold. When the manifold is Euclidean space, the Laplace
operator is often denoted as and so Poisson's equation is frequently written
as
In three-dimensional Cartesian coordinates, it takes the form
For vanishing f, this equation becomes Laplace's equation
- Polynomial: In mathematics it is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ( is a polynomial). Division by an expression containing a variable is not generally allowed in polynomials.
- Postulate, or axiom: indicates a starting assumption from which other statements are logically derived. It does not have to be self-evident (constancy of the speed of light in a vacuum is not self-evident, however it was used as a postulate in the special theory of relativity). Some axioms are experimental facts, but some are just assumptions not based on anything.
- Potential theory: The term "potential theory" arises from the fact that, in 19th century, the fundamental forces of nature were believed to be derived from potentials which satisfied Laplace's equation. Hence, potential theory was the study of functions which could serve as potentials. Nowadays the equations which describe forces are systems of non-linear partial differential equations such as the Einstein equations and the Yang-Mills equations, and the Laplace equation is only valid as a limiting case. Nevertheless, the term "potential theory" has remained as a convenient term for describing the study of functions which satisfy the Laplace equation.
- Real numbers: In mathematics they may be described informally as numbers that can be given by an infinite decimal representation, such as 2.4871773339…. The real numbers include both rational numbers, such as 42 and ?23/129, and irrational numbers, such as the square root of 2, and can be represented as points along an infinitely long number line.
- Ricci curvature tensor: It provides one way of measuring the degree to which the geometry determined by a given Riemannian metric might differ from that of ordinary Euclidean n-space. Like the metric itself, the Ricci tensor is a symmetric bilinear form on the tangent space of the Riemannian manifold. Roughly speaking, the Ricci tensor is a measure of volume distortion; that is, it encapsulates the degree to which n-dimensional volumes of regions in the given n-dimensional manifold differ from the volumes of comparable regions in Euclidean n-space.
- Rule of inference: In Logic it is a function from sets of formulae to formulae. The argument is called the premise set and the value the conclusion. They can also be viewed as relations holding between premises and conclusions, whereby the conclusion is said to be inferable (or derivable or deducible) from the premises. If the premise set is empty, the conclusion is said to be a theorem or axiom of the logic.
- Row vector: In linear algebra, a row vector is a 1 × n matrix,
that is, a matrix consisting of a single row:
The transpose of a row vector is a column vector. The set of all row vectors
forms a vector space which is the dual space to the set of all column vectors.
- Scalar multiplication: In mathematics it is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). Note that scalar multiplication is different from scalar product which is an inner product between two vectors. More specifically, if K is a field and V is a vector space over K, then scalar multiplication is a function from K × V to V. The result of applying this function to c in K and v in V is cv.
- Singularity: In mathematics:
- Essential singularity, a singularity near which a function exhibits extreme
behaviour.
- Isolated singularity, a mathematical singularity that has no other singularities
close to it.
- Mathematical singularity, a point at which a given mathematical object
is not defined.
- Spinors: In the theory of the orthogonal groups they are elements of a complex vector space introduced to expand the notion of spatial vector. They are needed because the full structure of the group of rotations in a given number of dimensions requires some extra number of dimensions to exhibit it. More formally, spinors can be defined as geometrical objects constructed from a given vector space endowed with a quadratic form by means of an algebraic or quantization procedure. Spinors thus form a projective representation of the rotation group.
- Surjective function: in Mathematics it is a function f if its values span its whole co-domain; that is, for every y in the co-domain, there is at least one x in the domain such that f(x) = y. Said another way, a function f: X ? Y is surjective if and only if its range f(X) is equal to its co-domain Y.
- Universality: In statistical mechanics it is the observation that there are properties for a large class of systems that are independent of the dynamical details of the system. Systems that display universality tend to be chaotic and often have a large number of interacting parts.
- Vector: This is in mathematics and physics:
- An element in a vector space, often represented as a coordinate vector
- An object defined by both magnitude and direction; in contrast to a scalar,
an object with magnitude only.
- Probability vector, a coordinate vector with nonnegative entries summing
to one, representing a discrete probability distribution
- A one-dimensional, directional matrix: a row vector or column vector
- Vector space (or linear space): In mathematics it is a collection of
objects (called vectors) that may be scaled and added. More formally, a
vector space is a set on which two operations, called (vector) addition
and (scalar) multiplication, are defined and satisfy certain natural axioms
which are listed below. Vector spaces are the basic objects of study in
linear algebra, and are used throughout mathematics, science, and engineering.
The most familiar vector spaces are two- and three-dimensional Euclidean
spaces. Vectors in these spaces are ordered pairs or triples of real numbers,
and are often represented as geometric vectors which are quantities with
a magnitude and a direction, usually depicted as arrows. These vectors may
be added together using the parallelogram rule (vector addition) or multiplied
by real numbers (scalar multiplication). The behaviour of geometric vectors
under these operations provides a good intuitive model for the behaviour
of vectors in more abstract vector spaces, which need not have a geometric
interpretation. For example, the set of (real) polynomials forms a vector
space.