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Calculus for temporal reasoning (relating to time instances) of events
Allen's interval algebra is a calculus for temporal reasoning that was introduced by James F. Allen in 1983. The calculus defines possible relations between
Allen's_interval_algebra
Area of artificial intelligence
representations. Examples of temporal calculi include Allen's interval algebra, and Vilain's & Kautz's point algebra. The most prominent spatial calculi are mereotopological
Spatial–temporal_reasoning
Topics referred to by the same term
held investment bank Allens of Mayfair, a butcher shop in London from 1830 to 2015 Allens Boots, a retail store in Austin, Texas Allens, Inc., a brand of
Allen
American computer scientist and linguist (born 1950)
Frederick Allen (born 1950) is an American computational linguist recognized for his contributions to temporal logic, in particular Allen's interval algebra. He
James F. Allen (computer scientist)
James_F._Allen_(computer_scientist)
Relative location of objects in space
Dimensionally Extended nine-Intersection Model (DE-9IM) Water-level task Allen's interval algebra (temporal analog) Commonsense reasoning J Freeman (1975), "The
Spatial_relation
for qualitative reasoning over networks of relation algebras, such as RCC-8, Allen's interval algebra and more. Spatial relation DE-9IM Randell, Cui & Cohn
Region_connection_calculus
Database that stores information relating to past, present and future time
keys, temporal referential integrity, temporal predicates with Allen's interval algebra and time-sliced and sequenced queries. For illustration, consider
Temporal_database
sub-task of figuring out all the temporal information in a document. Allen's interval algebra is one scheme for types of temporal relations. Rule-engineering
Temporal_annotation
Reals with an extra square root of +1 adjoined
In algebra, a split-complex number (or hyperbolic number, also perplex number, double number) is based on a hyperbolic unit j satisfying j 2 = 1 {\displaystyle
Split-complex_number
Concept in topology
compactification. For example, the open interval ( 0 , 1 ) {\displaystyle (0,1)} is usually compactified into the closed interval [ 0 , 1 ] {\displaystyle [0,1]}
Stone–Čech_compactification
Mathematical function, inverse of an exponential function
Sprott, Julien Clinton (2010), "Elegant Chaos: Algebraically Simple Chaotic Flows", Elegant Chaos: Algebraically Simple Chaotic Flows. Edited by Sprott Julien
Logarithm
1969 non-fiction book by G. Spencer-Brown
include Boolean arithmetic; The primary algebra (Chapter 6 of LoF), whose models include the two-element Boolean algebra (hereinafter abbreviated 2), Boolean
Laws_of_Form
Form in relational database normalization
normalization which extends the relational algebra and generalizes relational operators (such as join) to support interval data, which can be useful in temporal
Sixth_normal_form
Branch of mathematics
Calculus provides a generalization of concepts from elementary geometry and algebra. For example, instead of describing the slope of a straight line, calculus
Calculus
Topological construction on a map between spaces
{\displaystyle (x,1)\sim f(x)} . Here I {\displaystyle I} denotes the unit interval [0, 1] with its standard topology. Note that some authors (like J. Peter
Mapping_cone_(topology)
Mathematics glossary
glossary of properties and concepts in algebraic topology in mathematics. See also: glossary of topology, list of algebraic topology topics, glossary of category
Glossary of algebraic topology
Glossary_of_algebraic_topology
Branch of mathematics
of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, a
Geometry
Continuous deformation between two continuous functions
important invariants in algebraic topology. In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work
Homotopy
Branch of mathematics
It originated as a topic in algebraic topology, but nowadays is learned as an independent discipline. Besides algebraic topology, the theory has also
Homotopy_theory
Complement of an open subset
set that contains all of its boundary points. An example is the closed interval [ a , b ] {\displaystyle [a,b]} , which is closed in the real line because
Closed_set
Topological space arising from a usual graph
each edge e = x y ∈ E {\displaystyle e=xy\in E} by a copy of the unit interval I = [ 0 , 1 ] {\displaystyle I=[0,1]} , where 0 {\displaystyle 0} is identified
Graph_(topology)
Branch of music theory
such as Allen Forte, have emphasized the Z-relation, which obtains between two sets that share the same total interval content, or interval vector—but
Set_theory_(music)
Type of topological space
manifolds and simplicial complexes and has particular significance for algebraic topology. It was initially introduced by J. H. C. Whitehead to meet the
CW_complex
Functions of an angle
interval where the function is monotonic, and is thus bijective from this interval to its image by the function. The common choice for this interval,
Trigonometric_functions
Computer modeling of time-varying behavior of a dynamical system
time interval and calculating the integral of the derivatives through numerical integration. Some methods use a fixed step through the interval, and others
Dynamical_system_simulation
Number used for counting
(2nd ed.). London: George Allen & Unwin. Szczepanski, Amy F.; Kositsky, Andrew P. (2008). The Complete Idiot's Guide to Pre-algebra. Penguin Group. ISBN 978-1-59257-772-9
Natural_number
Algebraic structure associated with a topological space
In mathematics, the term homology, originally introduced in algebraic topology, has three primary, closely related usages. First, there is the homology
Homology_(mathematics)
Analysis of datasets using techniques from topology
The initial motivation is to study the shape of data. TDA has combined algebraic topology and other tools from pure mathematics to allow mathematically
Topological_data_analysis
Typeface style used in mathematics
in Unicode or amsmath LaTeX) is sometimes used by number theorists and algebraic geometers to designate the group scheme of n-th roots of unity. Latin
Blackboard_bold
Exponent of a power of two
then the measure of the interval from x to y plus the measure of the interval from y to z should equal the measure of the interval from x to z. Such a measure
Binary_logarithm
Describes the fundamental group in terms of a cover by two open path-connected subspaces
In mathematics, the Seifert–Van Kampen theorem of algebraic topology (named after Herbert Seifert and Egbert van Kampen), sometimes just called Van Kampen's
Seifert–Van_Kampen_theorem
Integral transform useful in probability theory, physics, and engineering
converting differentiation and integration in the time domain into the algebraic operations multiplication and division in the Laplace domain (analogous
Laplace_transform
Theorem in queueing theory
the average time that a customer spends in the system (W). Expressed algebraically the law is L = λ W . {\displaystyle L=\lambda W.} The relationship is
Little's_law
Theorem in quantum mechanics
{\displaystyle f} be a function from projection operators to the unit interval with the property that, if a set { Π i } {\displaystyle \{\Pi _{i}\}} of
Gleason's_theorem
Period of cultural flourishing from 786 to 1258
role in the development of algebra, arithmetic and Hindu–Arabic numerals. He has been described as the father of algebra by some. Another Persian mathematician
Islamic_Golden_Age
Ancient Greek analogue astronomical computer
the Greeks had enough mathematical knowledge beyond that of Babylonian algebra, to model the elliptical nature of planetary motion. Of special delight
Antikythera_mechanism
Australian and American mathematician (born 1975)
includes topics in harmonic analysis, partial differential equations, algebraic combinatorics, arithmetic combinatorics, geometric combinatorics, probability
Terence_Tao
American computer scientist (1932–2020)
mathematics in 1954 and began teaching school in Peru, New York. Allen taught math from algebra to trigonometry. She needed a Master's degree to earn a teaching
Frances_Allen
Process in mathematics of decomposing a topological space
decomposition theorem for irreducible sufficiently-large 3-manifolds. Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford
JSJ_decomposition
work on multiplicative functions over short intervals Gretchen Matthews (born 1973), American algebraic coding theorist Lilian Matthiesen, applies Fourier
List_of_women_in_mathematics
Mathematician (1845–1918)
"corrupter of youth". Kronecker objected to Cantor's proofs that the algebraic numbers are countable, and that the transcendental numbers are uncountable
Georg_Cantor
Typographical symbol (*)
and mathematicians (as, for example, in the A* search algorithm or C*-algebra). The English term asterisk is from Late Latin asteriscus, from Ancient
Asterisk
Study of the practices and possibilities of music
preserve the intervals between tones in a set. Expanding on the methods of musical set theory, some theorists have used abstract algebra to analyze music
Music_theory
Propagation of information or matter faster than the speed of light
with Hubble's law, defined as the rate of increase in proper distance per interval of cosmological time, is not a velocity in a relativistic sense. Moreover
Faster-than-light
Random process independent of past history
\Sigma } is the set of all states for the Markov chain. Let the sigma-algebra on the probability space be generated by the cylinder sets. Let the probability
Markov_chain
Country in South Asia
of the concept of zero as a number, negative numbers, arithmetic, and algebra. Trigonometry was further advanced in India, and the modern definitions
India
Speculative mechanism for achieving faster-than-light travel
path of the spaceship always have an ordered timelike separation interval (in algebraic terms, c2dt2 is always larger than dx2 + dy2 + dz2). This means
Krasnikov_tube
Polyphonic music with separate melodies
his own, custom-built terminology), by means of linking them to simple algebraic procedures. In counterpoint, the functional independence of voices is
Counterpoint
U.S. naval officer and computer scientist (1906–1992)
the Laning and Zierler system, which was the first compiler to accept algebraic notation as input. Her department released some of the first compiler-based
Grace_Hopper
Concept in Hlibert spaces mathematics
for simplicity. For any real-valued function f {\displaystyle f} on an interval I ⊆ R , {\displaystyle I\subseteq \mathbb {R} ,} one may define a matrix
Trace_inequality
Transformation of a topological space
In topology, especially algebraic topology, the cone of a topological space X {\displaystyle X} is intuitively obtained by stretching X into a cylinder
Cone_(topology)
Method of logical reasoning
Russell, Bertrand (1927). An Outline of Philosophy. London and New York: Allen and Unwin. reprinted in Bertrand Russell, The Basic Writings of Bertrand
Inductive_reasoning
Mathematical group of the homotopy classes of loops in a topological space
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of
Fundamental_group
School in Auckland, New Zealand
Form IV with Theorem One in Geometry and Lesson One in French, Latin, Algebra, etc. – all the start of Form III work. The object was to get through two
St_Peter's_College,_Auckland
Later, in 1850, it was used in America by Henry O'Reilly. 1847 Boolean algebra, the basis for digital logic, is introduced by George Boole in his book
List of British innovations and discoveries
List_of_British_innovations_and_discoveries
Methods in artificial intelligence research
or other details, such as atmospheric pressure. Similarly, Allen's temporal interval algebra is a simplification of reasoning about time and Region Connection
Symbolic artificial intelligence
Symbolic_artificial_intelligence
American mathematician (1879–1964)
for his derivation of the eponymously named Wilson score interval, which is a confidence interval used widely in statistics. He was also the first managing
Edwin_Bidwell_Wilson
Abstraction useful in the construction and triangulation of topological spaces
triangulation of topological spaces, and also in the computation of related algebraic invariants of such spaces. A Δ-set is somewhat more general than a simplicial
Delta_set
Number measuring the chance an event occurs
probability for some zero-probability events, for example by using a σ-algebra of such events (such as those arising from a continuous random variable)
Probability
Number line and triangular tiling's symmetry mathematical structure
−1, 0, 1, 2, ...) that are periodic in a certain sense, or in purely algebraic terms as a group with certain generators and relations. They are studied
Affine_symmetric_group
affirmation/denial, which are not permitted in Aristotelian logic or standard Boolean algebra. Buddhist logicians such as Dignaga and Dharmakirti criticized the Jaina
Jaina_seven-valued_logic
Interpretation of probability
Halpern found a counterexample based on his observation that the Boolean algebra of statements may be finite. Other axiomatizations have been suggested
Bayesian_probability
Theorem in set-theoretic geometry
decomposition, based on earlier work by Giuseppe Vitali concerning the unit interval and on the paradoxical decompositions of the sphere by Felix Hausdorff
Banach–Tarski_paradox
State of being real
Karatsuba, A. A. (1986). "The Method of Trigonometric Sums in Number Theory". Algebra, Mathematical Logic, Number Theory, Topology. American Mathematical Society
Existence
Self-awareness about thinking, higher-order thinking skills
Dushkin Pub., 2002. Print. Barell, J. (1992), "Like an incredibly hard algebra problem: Teaching for metacognition" In A. L. Costa, J. A. Bellanca, &
Metacognition
Types of special mathematical functions
The incomplete gamma functions are available in various of the computer algebra systems. Even if unavailable directly, however, incomplete function values
Incomplete_gamma_function
In general, exponentiation fails to be commutative
Putnam Competition, which prompted Alvin Hausner to extend results to algebraic number fields. Main source: An infinite set of trivial solutions in positive
Equation_xy_=_yx
Italian mathematician and glottologist (1858–1932)
first example of a space-filling curve which demonstrated that the unit interval and the unit square have the same cardinality. Today it is understood to
Giuseppe_Peano
sophomore Darryl Williams was shot in the neck by a sniper during the halftime interval while standing with teammates and coach in the end zone of the Charlestown
List of school shootings in the United States (before 2000)
List_of_school_shootings_in_the_United_States_(before_2000)
Theory in algebraic topology
In the mathematical disciplines of algebraic topology and homotopy theory, Eckmann–Hilton duality in its most basic form, consists of taking a given diagram
Eckmann–Hilton_duality
Calculus using a logically rigorous notion of infinitesimal numbers
A real-valued function f on the interval [a, b] is continuous if and only if for every hyperreal x in the interval *[a, b], we have: *f(x) ≅ *f(st(x))
Nonstandard_analysis
Hungarian mathematician (1802–1860)
only condition that he should be allowed to play on his violin for an interval between meeting each opponent. He disarmed or wounded all his antagonists
János_Bolyai
Distance function defined between probability distributions
distance between two random vectors with given dispersion matrices". Linear Algebra and Its Applications. 48: 257–263. doi:10.1016/0024-3795(82)90112-4. ISSN 0024-3795
Wasserstein_metric
Form of radar used to create images of landscapes
multidimensional discrete Fourier transform. Computational Kronecker-core array algebra is a popular algorithm used as new variant of FFT algorithms for the processing
Synthetic-aperture_radar
1929 book by Alfred North Whitehead
471. Whitehead published A Treatise on Universal Algebra in 1898 which profiled various algebraic structures for comparison. He joined Bertrand Russell
Process_and_Reality
Prefix applied to microcontrollers made by Toshiba
Holzman, Albert G.; Kent, Allen (1978). Encyclopedia of Computer Science and Technology: Volume 10 - Linear and Matrix Algebra to Microorganisms: Computer-Assisted
Toshiba_TLCS
Attraction of masses and energy
method was a form of graphical numerical integration since concepts of algebra and calculus were unknown at the time. This was later confirmed by Italian
Gravity
Mathematical concept
producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us the extended real numbers. We can also treat
Infinity
Development of mathematics in South Asia
of the concept of zero as a number, negative numbers, arithmetic, and algebra. In addition, trigonometry was further advanced in India, and, in particular
Indian_mathematics
at the House of Wisdom and continued his studies in Greek geometry and algebra under the caliph's patronage. Al-Wathiq succeeded his father. Al-Wathiq
History_of_Islam
sources of al-Khwarizmi's algebra, Osiris I, p. 263–277: "In a sense, Khwarizmi is more entitled to be called the father of algebra than Diophantus, because
List of people considered father or mother of a scientific field
List_of_people_considered_father_or_mother_of_a_scientific_field
Pathological embedding of the sphere in 3D space
doi:10.1090/mbk/046. ISBN 978-0-8218-4316-1. MR 2350979. Hatcher, Allen, Algebraic Topology Hocking, John Gilbert; Young, Gail Sellers (1988) [1961].
Alexander_horned_sphere
Type of topological space
embeddings of real projective spaces". Retrieved 22 Sep 2011. Hatcher, Allen (2001). Algebraic Topology. Cambridge University Press. ISBN 978-0-521-79160-1.
Real_projective_space
Systematic approach to understanding the behavior of humans and other animals
(Compare: if we find out how a computer program solves problems in linear algebra, we don't say it's not really solving them, we just say we know how it
Behaviorism
American mathematician and philosopher (1894–1964)
for integrals, or using the language of functional analysis and Banach algebras, is however a relatively routine process. The Paley–Wiener theorem relates
Norbert_Wiener
Operation in topology
(2016). Homotopical Topology (2nd ed.). Springer. p. 20. Hatcher, Allen, Algebraic topology. Cambridge University Press, Cambridge, 2002. xii+544 pp.
Join_(topology)
American statistician
first computer-intensive statistical techniques, replacing traditional algebraic derivations with data-based computer simulations. Efron was born in St
Bradley_Efron
word ‘equals’, Diophantus took a fundamental step from verbal algebra towards symbolic algebra. Struik, Dirk J. (1987). A Concise History of Mathematics.
Timeline of scientific discoveries
Timeline_of_scientific_discoveries
Concept in algebraic topology
generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics. Fibrations are used, for example, in
Fibration
Computing Research, National Academy Press, retrieved 30 August 2007 Newell, Allen; Simon, H. A. (1963), "GPS: A Program that Simulates Human Thought", in
Timeline of artificial intelligence
Timeline_of_artificial_intelligence
Philosophical system based on the teachings of Pythagoras
dots, the combination of Babylonian algebra and Pythagorean arithmetic provided the basis for Greek geometric algebra. By attempting to establish a system
Pythagoreanism
Inverse of a finite difference
indefinite sums of hypergeometric terms and are implemented in computer algebra systems. The Gregory-Laplace summation formula expresses the indefinite
Indefinite_sum
Records of Earth's people
contributions in various fields, such as Al-Khwarizmi's development of algebra and Avicenna's comprehensive philosophical system. The folktales One Thousand
Human_history
Property of topological spaces
topological space at the nLab Strickland 2009, Lemma 1.4(c). Hatcher, Allen (2001). Algebraic Topology (PDF). (See the Appendix) Brown 2006, section 5.9. Booth
Compactly_generated_space
History of radiation-based medical imaging
significant computing power and applied an algebraic reconstruction technique, using the Kaczmarz method from numerical algebra. Encouraged by the promising results
History of computed tomography
History_of_computed_tomography
Indian inventions
Seshadri constant – In algebraic geometry, a Seshadri constant is an invariant of an ample line bundle L at a point P on an algebraic variety.The name is
List of Indian inventions and discoveries
List_of_Indian_inventions_and_discoveries
3rd-century BCE inscriptions in South Asia
Victor J.; Parshall, Karen Hunger (2014). Taming the Unknown: A History of Algebra from Antiquity to the Early Twentieth Century. Princeton University Press
Edicts_of_Ashoka
In algebraic topology, the nth symmetric product of a topological space consists of the unordered n-tuples of its elements. If one fixes a basepoint,
Symmetric_product_(topology)
List of concepts in artificial intelligence
described with the aid of a membership function valued in the real unit interval [0, 1]. Fuzzy sets generalize classical sets, since the indicator functions
Glossary of artificial intelligence
Glossary_of_artificial_intelligence
Boolean algebra, developed by the British mathematician George Boole in his work The Laws of Thought, published in 1854. His Boolean algebra was further
History_of_computing_hardware
ALLENS INTERVAL-ALGEBRA
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ALLENS INTERVAL-ALGEBRA
ALLENS INTERVAL-ALGEBRA
ALLENS INTERVAL-ALGEBRA
ALLENS INTERVAL-ALGEBRA
ALLENS INTERVAL-ALGEBRA
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ALLENS INTERVAL-ALGEBRA