Macaulay2
»
Documentation
Packages
»
Macaulay2Doc
>
Function
next |
previous
|
forward
|
backward
|
up
|
index
|
toc
Function -- the class of all functions
See also
using functions
using functions with optional inputs
making functions
local variables in a function
making functions with a variable number of arguments
making functions with multiple return values
making new functions with optional arguments
Types of
function
:
CompiledFunction
-- the class of all compiled functions
CompiledFunctionClosure
-- the class of all compiled function closures
FunctionClosure
-- the class of all function closures
Functions and methods returning a function :
Function @@ Function
-- composition of functions
Function _ Thing
-- attach the first argument to a function of two or more arguments
Type %= Function
-- see
installing augmented assignment methods
Type &= Function
-- see
installing augmented assignment methods
Type **= Function
-- see
installing augmented assignment methods
Type *= Function
-- see
installing augmented assignment methods
Type ++= Function
-- see
installing augmented assignment methods
Type += Function
-- see
installing augmented assignment methods
Type -= Function
-- see
installing augmented assignment methods
Type ..<= Function
-- see
installing augmented assignment methods
Type ..= Function
-- see
installing augmented assignment methods
Type //= Function
-- see
installing augmented assignment methods
Type /= Function
-- see
installing augmented assignment methods
Type <<= Function
-- see
installing augmented assignment methods
Type <==>= Function
-- see
installing augmented assignment methods
Type ===>= Function
-- see
installing augmented assignment methods
Type ==>= Function
-- see
installing augmented assignment methods
Type >>= Function
-- see
installing augmented assignment methods
Type @= Function
-- see
installing augmented assignment methods
Type @@= Function
-- see
installing augmented assignment methods
Type \= Function
-- see
installing augmented assignment methods
Type \\= Function
-- see
installing augmented assignment methods
Type ^**= Function
-- see
installing augmented assignment methods
Type ^= Function
-- see
installing augmented assignment methods
Type ^^= Function
-- see
installing augmented assignment methods
Type _= Function
-- see
installing augmented assignment methods
Type |-= Function
-- see
installing augmented assignment methods
Type |= Function
-- see
installing augmented assignment methods
Type ||= Function
-- see
installing augmented assignment methods
Boolean >> Function
-- see
OptionTable >> Function
-- attaching options to a function
List >> Function
-- see
OptionTable >> Function
-- attaching options to a function
OptionTable >> Function
-- attaching options to a function
profile(Function)
-- see
profile
-- profile a function
Type ??= Function
(missing documentation)
Methods that use a function :
? Function
-- see
? Symbol
-- view brief documentation of a symbol
about(Function)
-- see
about
-- search the documentation
accumulate(Function,Thing)
-- see
accumulate
-- apply a binary operator repeatedly
accumulate(Function,Thing,Thing)
-- see
accumulate
-- apply a binary operator repeatedly
accumulate(Function,Thing,VisibleList)
-- see
accumulate
-- apply a binary operator repeatedly
accumulate(Function,VisibleList)
-- see
accumulate
-- apply a binary operator repeatedly
accumulate(VisibleList,Function)
-- see
accumulate
-- apply a binary operator repeatedly
accumulate(VisibleList,Thing,Function)
-- see
accumulate
-- apply a binary operator repeatedly
addHook(MutableHashTable,Thing,Function)
-- see
addHook
-- add a hook function to an object for later processing
addHook(Sequence,Function)
-- see
addHook
-- add a hook function to an object for later processing
addHook(Symbol,Function)
-- see
addHook
-- add a hook function to an object for later processing
all(BasicList,BasicList,Function)
-- whether all corresponding elements of a pair of lists satisfy a condition
all(BasicList,Function)
-- whether all elements of a list satisfy a specified condition
all(ZZ,Function)
-- see
all(BasicList,Function)
-- whether all elements of a list satisfy a specified condition
all(HashTable,Function)
-- whether all key/value pairs in a hash table satisfy a specified condition
any(BasicList,BasicList,Function)
-- whether any corresponding elements of a pair of lists satisfy a condition
any(BasicList,Function)
-- whether any elements of a list satisfy a specified condition
any(ZZ,Function)
-- see
any(BasicList,Function)
-- whether any elements of a list satisfy a specified condition
any(HashTable,Function)
-- whether all key/value pairs in a hash table satisfy a specified condition
apply(BasicList,BasicList,Function)
-- apply a function to pairs of elements, one from each list
apply(BasicList,String,Function)
-- see
apply(BasicList,BasicList,Function)
-- apply a function to pairs of elements, one from each list
apply(String,BasicList,Function)
-- see
apply(BasicList,BasicList,Function)
-- apply a function to pairs of elements, one from each list
apply(String,String,Function)
-- see
apply(BasicList,BasicList,Function)
-- apply a function to pairs of elements, one from each list
apply(BasicList,Function)
-- apply a function to each element of a list
apply(String,Function)
-- see
apply(BasicList,Function)
-- apply a function to each element of a list
apply(Thing,Function)
-- apply a function to an object with an iterator
apply(ZZ,Function)
-- apply a function to {0,..., n-1}
applyKeys(HashTable,Function)
-- see
applyKeys
-- apply a function to each key in a hash table
applyKeys(HashTable,Function,Function)
-- see
applyKeys
-- apply a function to each key in a hash table
applyPairs(HashTable,Function)
-- see
applyPairs
-- apply a function to each pair in a hash table
applyValues(HashTable,Function)
-- see
applyValues
-- apply a function to each value in a hash table
code(Function)
-- see
code
-- display source code
createTask(Function)
-- see
createTask
-- create a task
createTask(Function,Thing)
-- see
createTask
-- create a task
fold(Function,Thing)
-- see
fold
-- apply a binary operator repeatedly
fold(Function,Thing,Thing)
-- see
fold
-- apply a binary operator repeatedly
fold(Function,VisibleList)
-- see
fold
-- apply a binary operator repeatedly
fold(VisibleList,Function)
-- see
fold
-- apply a binary operator repeatedly
fold(VisibleList,Thing,Function)
-- see
fold
-- apply a binary operator repeatedly
frames(Function)
-- see
frames
-- get the frames associated to a closure
Function \ Set
(missing documentation)
Function Thing
-- function application
hashTable(Function,List)
(missing documentation)
Function \ Ideal
-- see
Ideal / Function
-- apply a function to generators of an ideal
Ideal / Function
-- apply a function to generators of an ideal
installAssignmentMethod(Symbol,HashTable,Function)
-- see
installAssignmentMethod
-- install methods assignment to the value of an operator
installAssignmentMethod(Symbol,HashTable,HashTable,Function)
-- see
installAssignmentMethod
-- install methods assignment to the value of an operator
integrate(Function,Constant,Constant)
-- see
integrate
-- numerical integration
integrate(Function,Constant,InfiniteNumber)
-- see
integrate
-- numerical integration
integrate(Function,Constant,Number)
-- see
integrate
-- numerical integration
integrate(Function,InfiniteNumber,Constant)
-- see
integrate
-- numerical integration
integrate(Function,InfiniteNumber,InfiniteNumber)
-- see
integrate
-- numerical integration
integrate(Function,InfiniteNumber,Number)
-- see
integrate
-- numerical integration
integrate(Function,Number,Constant)
-- see
integrate
-- numerical integration
integrate(Function,Number,InfiniteNumber)
-- see
integrate
-- numerical integration
integrate(Function,Number,Number)
-- see
integrate
-- numerical integration
localDictionaries(Function)
-- see
localDictionaries
-- get local dictionaries
locate(Function)
-- see
locate
-- locate source code
new Manipulator from Function
-- see
Manipulator
-- the class of all file manipulators
map(ChainComplex,ChainComplex,Function)
-- make a map of chain complexes
map(GradedModule,GradedModule,Function)
-- see
map(ChainComplex,ChainComplex,Function)
-- make a map of chain complexes
map(Module,Module,Function)
-- create a matrix by specifying a function that gives each entry
map(Module,ZZ,Function)
-- create a matrix from a free module by specifying a function that gives each entry
memoize(Function)
-- see
memoize
-- record results of function evaluation for future use
memoize(Function,List)
-- see
memoize
-- record results of function evaluation for future use
merge(HashTable,HashTable,Function)
-- see
merge
-- merge two hash tables
mergePairs(BasicList,BasicList,Function)
-- see
mergePairs
-- merge sorted lists of pairs
methodOptions(Function)
-- recover the options used when a method function was created
options(Function)
-- see
options(Command)
-- get optional arguments and default values of functions and methods
package(Function)
-- see
package
-- get containing package
partition(Function,VirtualTally)
-- see
partition
-- partition a set or list by values of a function
partition(Function,VirtualTally,VisibleList)
-- see
partition
-- partition a set or list by values of a function
partition(Function,VisibleList)
-- see
partition
-- partition a set or list by values of a function
partition(Function,VisibleList,VisibleList)
-- see
partition
-- partition a set or list by values of a function
position(VisibleList,Function)
-- see
position
-- the first element of a list satisfying a condition
position(VisibleList,VisibleList,Function)
-- see
position
-- the first element of a list satisfying a condition
positions(MutableList,Function)
-- see
positions
-- which elements of a list satisfy a condition
positions(VisibleList,Function)
-- see
positions
-- which elements of a list satisfy a condition
product(VisibleList,Function)
-- product of values of a function
product(VisibleList,VisibleList,Function)
-- product of results of applying a function pairwise
product(ZZ,Function)
-- product of consecutive values of a function
profile(String,Function)
-- see
profile
-- profile a function
scan(BasicList,BasicList,Function)
-- see
scan
-- apply a function to each element in a list or sequence
scan(BasicList,Function)
-- see
scan
-- apply a function to each element in a list or sequence
scan(Thing,Function)
-- see
scan
-- apply a function to each element in a list or sequence
scan(ZZ,Function)
-- see
scan
-- apply a function to each element in a list or sequence
scanKeys(Database,Function)
-- see
scanKeys
-- apply a function to each key in a hash table or database
scanKeys(HashTable,Function)
-- see
scanKeys
-- apply a function to each key in a hash table or database
scanLines(Function,List)
-- see
scanLines
-- apply a function to each line of a file
scanLines(Function,String)
-- see
scanLines
-- apply a function to each line of a file
scanPairs(HashTable,Function)
-- see
scanPairs
-- apply a function to the pairs in a hash table
scanValues(HashTable,Function)
-- see
scanValues
-- apply a function to each value in a hash table or database
schedule(Function)
-- see
schedule
-- schedule a task for execution
schedule(Function,Thing)
-- see
schedule
-- schedule a task for execution
select(BasicList,Function)
-- select elements from a list
select(HashTable,Function)
-- select part of a hash table
select(Thing,Function)
-- select elements from an object with an iterator
select(ZZ,BasicList,Function)
-- select a limited number of elements from a list
select(ZZ,Function)
-- select integers
select(ZZ,HashTable,Function)
-- select a limited number of pairs from a hash table
Set / Function
(missing documentation)
sort(List,Function)
(missing documentation)
sublists(VisibleList,Function)
-- see
sublists
-- process interspersed subsequences of a visible list
sublists(VisibleList,Function,Function)
-- see
sublists
-- process interspersed subsequences of a visible list
sublists(VisibleList,Function,Function,Function)
-- see
sublists
-- process interspersed subsequences of a visible list
sublists(VisibleList,Function,Function,Nothing)
-- see
sublists
-- process interspersed subsequences of a visible list
sublists(VisibleList,Function,Nothing)
-- see
sublists
-- process interspersed subsequences of a visible list
sublists(VisibleList,Function,Nothing,Function)
-- see
sublists
-- process interspersed subsequences of a visible list
sublists(VisibleList,Function,Nothing,Nothing)
-- see
sublists
-- process interspersed subsequences of a visible list
sum(VisibleList,Function)
-- sum results of applying a function
sum(VisibleList,VisibleList,Function)
-- sum results of applying a function pairwise
sum(ZZ,Function)
-- sum consecutive values of a function
Function \\ Thing
-- see
Thing // Function
-- apply a function
Thing // Function
-- apply a function
ultimate(Function,Thing)
-- see
ultimate
-- ultimate value for an iteration
Function \ String
-- see
VisibleList / Function
-- apply a function to elements of a list
Function \ VirtualTally
-- see
VisibleList / Function
-- apply a function to elements of a list
Function \ VisibleList
-- see
VisibleList / Function
-- apply a function to elements of a list
List / Function
-- see
VisibleList / Function
-- apply a function to elements of a list
String / Function
-- see
VisibleList / Function
-- apply a function to elements of a list
VirtualTally / Function
-- see
VisibleList / Function
-- apply a function to elements of a list
VisibleList / Function
-- apply a function to elements of a list
Fixed objects of class
Function
:
about
-- search the documentation
abs
-- absolute value function
accumulate
-- apply a binary operator repeatedly
acos
-- arccosine
acosh
-- inverse hyperbolic cosine
acot
-- compute the inverse cotangent
acoth
-- compute the hyperbolic arctangent of a number
addCancelTask
-- see
addCancelTask(Task,Task)
-- specify that the completion of one task triggers the cancellation of another
addDependencyTask
-- see
addDependencyTask(Task,Task)
-- schedule a task, but ensure that it will not run until another task finishes
addEndFunction
-- add an ending function
addHook
-- add a hook function to an object for later processing
addStartFunction
-- add a startup function
addStartTask
-- see
addStartTask(Task,Task)
-- schedule a task upon completion of another
adjoint
-- the tensor-Hom adjunction maps
adjoint'
-- see
adjoint
-- the tensor-Hom adjunction maps
agm
-- arithmetic-geometric mean
alarm
-- set an alarm
all
-- whether all elements satisfy a specified condition
ambient
-- ambient free module of a subquotient, or ambient ring
analyticSpread
-- Compute the analytic spread of a module or ideal
ancestor
-- see
ancestor(Type,Type)
-- whether one type is an ancestor of another
ancestors
-- the ancestors of something
annihilator
-- the annihilator ideal
antipode
-- antipode for skew commuting polynomial rings
any
-- whether any elements satisfy a specified condition
append
-- add an element to the end of a list
applicationDirectory
-- the path to the user's application directory
apply
-- apply a function to each element
applyKeys
-- apply a function to each key in a hash table
applyPairs
-- apply a function to each pair in a hash table
applyTable
-- apply a function to each element of a table
applyValues
-- apply a function to each value in a hash table
apropos
-- symbols matching a pattern
arXiv
-- link to an arXiv article
ascii
-- ASCII character conversion
asin
-- arcsine
asinh
-- inverse hyperbolic sine
assert
-- check whether something is true, raise an error if not
associatedPrimes
-- find associated primes
atan
-- compute the arctangent of a number
atan2
-- compute an angle of a certain triangle
atanh
-- compute the hyperbolic arctangent of a number
atEndOfFile
-- see
atEndOfFile(File)
-- test for end of file
autoload
-- arrange for a function to be loaded automatically
baseFilename
-- the base part of a filename or path
baseName
-- the base name of a generator
baseRing
-- produce the ring from which a ring was formed
basis
-- basis or generating set of all or part of a ring, ideal or module
beginDocumentation
-- start documentation section of a package
benchmark
-- accurate timing of execution
BesselJ
-- Bessel function of the first kind
BesselY
-- Bessel function of the second kind
Beta
-- Beta function
betti
-- display or modify a Betti diagram
between
-- insert something between elements of a list
binomial
-- binomial coefficient
borel
-- see
borel(Matrix)
-- make a Borel fixed submodule
cacheValue
-- cache values of functions in their arguments
cancelTask
-- see
cancelTask(Task)
-- stop a task
canonicalBundle
(missing documentation)
capture
-- evaluate Macaulay2 code and capture the output
ceiling
-- ceiling function
centerString
-- center a string or net
chainComplex
-- make a chain complex
changeBase
-- change integer base
char
-- computes the characteristic of the ring or field
characters
-- get characters from a string
check
-- perform tests of a package
checkDegrees
-- compares the degrees of generators of two modules
chi
class
-- class of an object
clean
-- Set to zero elements that are approximately zero
clearEcho
-- turn off echoing
code
-- display source code
codim
-- compute the codimension
coefficient
-- coefficient of a monomial
coefficientRing
-- get the coefficient ring
coefficients
-- monomials and their coefficients
cohomology
-- general cohomology functor
coimage
-- coimage of a map
cokernel
-- cokernel of a map of modules, graded modules, or chaincomplexes
collectGarbage
-- collect the garbage in memory
columnAdd
-- add a multiple of one column to another
columnate
-- arrange strings in columns
columnMult
-- multiply a column by a ring element
columnPermute
-- permute some columns
columnRankProfile
-- find the column rank profile of a mutable matrix
columnSwap
-- interchange columns
combine
-- combine hash tables
commandInterpreter
-- the top level command interpreter
commonest
-- the most common elements of a list or tally
commonRing
-- find smallest containing ring
comodule
-- submodule to quotient module
compareExchange
-- compares and possibly exchanges the value of an atomic integer
complement
-- see
complement(Matrix)
-- find the minimal generators for cokernel of a matrix (low level form)
complete
components
-- list the components of a direct sum
compose
-- composition as a pairing on Hom-modules
compositions
-- list the compositions of an integer
compress
-- extract nonzero columns from a matrix
concatenate
-- join strings
conductor
-- the conductor of a finite ring map
cone
-- mapping cone or polyhedral cone
conjugate
-- complex conjugate
connectionCount
-- the number of connections
content
-- the content of a polynomial
contract
-- contract one matrix by another
contract'
-- see
contract'(Matrix,Matrix)
-- contract a matrix by a matrix, the dual notion
conwayPolynomial
-- provide a Conway polynomial
copy
-- copy an object
copyDirectory
-- see
copyDirectory(String,String)
copyFile
-- see
copyFile(String,String)
cos
-- compute the cosine
cosh
-- compute the hyperbolic cosine
cot
-- cotangent
cotangentSheaf
-- cotangent sheaf of a projective variety
coth
-- hyperbolic cotangent
cover
-- get the covering free module
coverMap
-- see
coverMap(Module)
-- the surjective map from a free module to a module corresponding to the generators
cpuTime
-- seconds of cpu time used since Macaulay2 began
createTask
-- create a task
csc
-- cosecant
csch
-- hyperbolic cosecant
currentColumnNumber
-- current column number of the current input file
currentDirectory
-- current working directory
currentPosition
-- the current position in the current source file
currentRowNumber
-- current row number of the current input file
currentTime
-- get the current time
debug
-- open the private dictionary of a package
decompose
deepSplice
-- remove all levels of subsequences from a sequence or list
default
-- default member of a family
degree
degreeGroup
(missing documentation)
degreeLength
-- the length of the degree vector
degrees
-- degrees of generators
degreesMonoid
-- see
degreesRing
-- the ring or monoid of degrees
degreesRing
-- the ring or monoid of degrees
delete
-- delete some elements of a list
demark
-- insert a string between elements of a list of strings
denominator
-- denominator of a fraction
depth
-- depth of a net
describe
-- real description
determinant
-- determinant of a matrix
diagonalMatrix
-- make a diagonal matrix
diameter
-- diameter of an interval
dictionary
-- determine the dictionary to which a symbol belongs
diff
-- differentiate or take difference
diff'
-- see
diff'(Matrix,Matrix)
-- differentiate a matrix by a matrix, the dual notion
difference
-- difference
Digamma
-- Digamma function
dim
-- compute the Krull dimension
directSum
-- direct sum of modules or maps
disassemble
-- disassemble pseudocode or a function
discriminant
-- see
discriminant(RingElement,RingElement)
dismiss
-- dismiss a package
distinguished
-- Compute the distinguished subvarieties of a pullback, intersection or cone
divideByVariable
-- divide all columns by a (power of a) variable
doc
-- a simple documentation function
multidoc
-- see
doc
-- a simple documentation function
document
-- create a documentation node
makeDocumentTag
-- see
DocumentTag
-- the class of all document tags
drop
-- drop some elements from a list or sequence.
dual
-- dual module or map
eagonNorthcott
-- see
eagonNorthcott(Matrix)
-- Eagon-Northcott complex of a matrix of linear forms
echoOff
-- turn off echoing
echoOn
-- turn on echoing
effCone
(missing documentation)
effGenerators
(missing documentation)
eigenvalues
-- find eigenvalues of a matrix
eigenvectors
-- find eigenvectors of a matrix over RR or CC
eint
-- exponential integral
elements
-- list of elements
eliminate
End
-- module of endomorphisms
endPackage
-- end a new package
entries
-- lists the entries of a matrix
erase
-- remove a global symbol
erf
-- error function
erfc
-- complementary error function
error
-- deliver error message
euler
-- Euler characteristic
eulers
-- list the sectional Euler characteristics
even
-- tell whether an integer is even
EXAMPLE
-- hypertext EXAMPLE item
examples
-- list the examples in documentation
exchange
-- exchange the value of an atomic integer
exec
-- execute another program
exp
-- exponential function
expectedReesIdeal
-- symmetric algebra ideal plus jacobian dual
expm1
-- exponential minus 1
exponents
-- the exponents of a polynomial
export
-- export symbols from a package
exportFrom
-- export symbols from a package's private dictionary
exportMutable
-- export mutable symbols from a package
expression
-- convert to an expression
extend
-- extend a module map to a chain map, if possible
exteriorPower
-- exterior power
factor
-- factor a ring element or a ZZ-module
Fano
-- Fano scheme
fileExecutable
fileExists
-- whether a file exists
fileLength
-- the length of a file
fileMode
-- set or get file mode
fileReadable
fileTime
-- get or set file modified time
fileWritable
fillMatrix
-- fill a mutable matrix with random numbers
findFiles
-- find files recursively
findHeft
-- find a heft vector for a list of degrees
findProgram
-- load external program
findSynonyms
-- see
findSynonyms(Symbol)
-- find synonyms of symbols
first
-- first element of a list
firstkey
-- get the first key
fittingIdeal
-- Fitting ideal of a module
flagLookup
-- flag a symbol
flatten
-- flatten a nested list or a matrix
flattenRing
-- write a ring as a (quotient of a) polynomial ring
flip
-- isomorphism map of commutativity of tensor product
floor
-- floor function
fold
-- apply a binary operator repeatedly
forceGB
-- declare that the columns of a matrix are a Gröbner basis
fork
-- fork the process
format
-- format a string or a real number
formation
-- recover the methods used to make a module
frac
-- construct a fraction field
fraction
frames
-- get the frames associated to a closure
fromDividedPowers
-- Translates from divided power monomial basis to ordinary monomial basis
fromDual
-- Ideal from inverse system
functionBody
-- get the body of a function
Gamma
-- Gamma function
gb
-- compute a Gröbner basis
gbRemove
-- remove Gröbner basis
gbSnapshot
-- the Gröbner basis matrix as so far computed
gcd
-- greatest common divisor
gcdCoefficients
-- gcd with coefficients
gcdLLL
-- compute the gcd of integers, and small multipliers
GCstats
-- information about the status of the garbage collector
genera
-- list of the successive linear sectional arithmetic genera
generateAssertions
-- generate assert statements from experimental input
generator
-- provide a single generator
generators
-- provide matrix or list of generators
genericMatrix
-- make a generic matrix of variables
genericSkewMatrix
-- make a generic skew symmetric matrix of variables
genericSymmetricMatrix
-- make a generic symmetric matrix
genus
-- arithmetic genus
get
-- get the contents of a file
getc
-- get a byte
getChangeMatrix
-- get the change of basis matrix
getenv
-- get value of environment variable
getGlobalSymbol
-- create a global symbol in a global dictionary
getNetFile
-- get the sequence of completed lines (nets) from a net file
getNonUnit
-- retrieve a previously discovered non-unit
getPrimeWithRootOfUnity
-- find a prime p with a primitive n-th root of unity r in ZZ/p
getSymbol
-- make a global user symbol from a string
getWWW
-- get a web page
GF
-- make a finite field
globalAssign
-- global assignment, in function form
globalAssignFunction
-- the standard method for the global assignment hook
globalAssignment
-- install standard global assignment method
globalReleaseFunction
-- the standard method for the global variable release hook
gradedModule
-- make a graded module
gradedModuleMap
-- make a map of graded modules
gramm
(missing documentation)
graphIdeal
-- see
graphIdeal(RingMap)
-- the ideal of the graph of the regular map corresponding to a ring map
graphRing
-- see
graphRing(RingMap)
-- the coordinate ring of the graph of the regular map corresponding to a ring map
Grassmannian
-- see
Grassmannian(ZZ,ZZ)
-- the Grassmannian of linear subspaces of a vector space
groebnerBasis
-- Gröbner basis, as a matrix
groupID
-- the process group identifier
hash
-- hash code of an object
hashTable
-- make a hash table
heft
-- heft vector of ring or monoid
height
-- height of a net
hermite
(missing documentation)
hilbertFunction
-- the Hilbert function
hilbertPolynomial
-- compute the Hilbert polynomial
hilbertSeries
-- compute the Hilbert series
hold
-- hold something in a holder expression
Hom
-- module of homomorphisms
homogenize
-- homogenize with respect to a variable
homology
-- general homology functor
homomorphism
-- get the homomorphism from element of Hom
homomorphism'
-- get the element of Hom from a homomorphism
hooks
-- list hooks attached to a key
horizontalJoin
-- join nets or strings horizontally
html
-- convert to HTML format
httpHeaders
-- see
httpHeaders(String)
-- prepend http headers to a string
hypertext
-- prepare hypertext for display
icFracP
-- compute the integral closure in prime characteristic
icFractions
-- fractions integral over an affine domain
icMap
-- natural map from an affine domain into its integral closure
icPIdeal
-- compute the integral closure in prime characteristic of a principal ideal
ideal
-- make an ideal
idealizer
-- compute Hom(I,I) as a quotient ring
identity
-- the identity function
image
-- image of a map
imaginaryPart
-- imaginary part
importFrom
-- import symbols to the current private dictionary
independentSets
-- some size-maximal independent subsets of variables modulo an ideal
index
-- numeric index of a ring variable
indices
-- indices of a polynomial; also components for a direct sum
inducedMap
-- compute an induced map
inducesWellDefinedMap
-- whether a map is well defined
info
-- convert hypertext to info format
input
-- read Macaulay2 commands and echo
insert
-- copy a list, inserting an element
installAssignmentMethod
-- install methods assignment to the value of an operator
installedPackages
installMethod
-- install methods
installPackage
-- load and install a package and its documentation
instance
-- whether something has a certain type
instances
-- see
instances(Type)
integralClosure
-- integral closure of an ideal or a domain
integrate
-- numerical integration
intersect
-- compute an intersection
intersectInP
-- Compute distinguished varieties for an intersection in A^n or P^n
intersection
-- compute an intersection
interval
-- construct an interval
inverse
-- compute the inverse
inverseErf
-- inverse error function
inversePermutation
-- inverse permutation
inverseRegularizedBeta
-- inverse of the regularized beta function
inverseRegularizedGamma
-- inverse of the upper regularized gamma function
inverseSystem
-- Inverse systems with equivariance
irreducibleCharacteristicSeries
-- irreducible characteristic series of an ideal
irreducibleDecomposition
-- express a monomial ideal as an intersection of irreducible monomial ideals
isAffineRing
-- whether something is an affine ring
isANumber
-- whether a number is not not-a-number
isBorel
-- whether an ideal is fixed by upper triangular changes of coordinates
isc
-- ISC lookup
isCanceled
-- see
isCanceled(Task)
-- whether a task has been canceled
isCommutative
-- whether a ring is commutative
isConstant
-- whether a ring element is constant
isDirectory
-- whether a file is a directory
isDirectSum
-- whether something is a direct sum
isEmpty
isField
-- whether something is a field
isFinite
-- whether a number is finite
isFinitePrimeField
-- whether a ring is a finite prime field
isFreeModule
-- whether something is a free module
isGlobalSymbol
-- whether a global symbol with a given name exists
isHomogeneous
-- whether something is homogeneous (graded)
isIdeal
-- whether something is an ideal
isInfinite
-- whether a number is infinite
isInjective
-- whether a map is injective
isInputFile
-- see
isInputFile(File)
-- whether a file is open for input
isIsomorphic
-- Probabilistic test for isomorphism of modules
isIsomorphism
-- whether a map is an isomorphism
isLiftable
(missing documentation)
isLinearType
-- Determine whether module has linear type
isListener
-- see
isListener(File)
-- whether a file is open for listening
isLLL
-- is a basis an LLL basis?
isMember
-- test membership in a list or set
isModule
-- whether something is a module
isMonomialIdeal
-- whether something is a monomial ideal
isMutable
-- whether something may be modified
isNormal
-- determine if a reduced ring is normal
isOpen
-- whether a file or database is open
isOutputFile
-- see
isOutputFile(File)
-- whether a file is open for output
isPolynomialRing
-- whether something is a polynomial ring
isPrimary
-- determine whether a submodule is primary
isPrime
-- whether a integer or polynomial is prime
isPrimitive
-- whether an element is a primitive element of a finite field
isProjective
(missing documentation)
isPseudoprime
-- see
isPseudoprime(ZZ)
-- whether an integer is probably prime
isQuotientModule
-- whether something is evidently a quotient of a free module
isQuotientOf
-- whether one thing is a quotient of another
isQuotientRing
-- whether something is a quotient ring
isReady
-- see
isReady(File)
-- whether a file has data available for reading
isReal
-- whether a number is real
isReduction
-- Determine whether an ideal is a reduction
isRegularFile
-- whether a file is a regular file
isRing
-- whether something is a ring
isSkewCommutative
-- whether a ring has skew commuting variables
isSorted
-- whether a list is sorted
isSquareFree
-- whether something is square free monomial ideal
isStandardGradedPolynomialRing
-- Checks whether a ring is a polynomial ring over a field with variables of degree 1
isSubmodule
-- whether a module is evidently a submodule of a free module
isSubquotient
-- see
isSubquotient(Module,Module)
-- check whether a module is a subquotient of another
isSubset
-- whether one object is a subset of another
isSupportedInZeroLocus
-- whether support of a module is contained in the zero locus of the (irrelevant) ideal
isSurjective
-- whether a map is surjective
isTable
-- whether something is a list of lists of equal length
isUnit
-- whether a ring element is a unit
isWellDefined
-- whether a map is well defined
iterator
-- get an iterator
jacobian
-- the Jacobian matrix of partial derivatives
jacobianDual
-- Computes the 'jacobian dual', part of a method of finding generators for Rees Algebra ideals
join
-- join lists, sequences, and iterable objects
kernel
-- kernel of a ringmap, matrix, or chain complex
kernelLLL
(missing documentation)
kernelOfLocalization
-- the kernel of the localization map
keys
-- keys used in a hash table, dictionary, or database
kill
-- kill a process
koszul
-- Koszul complex or specific matrix in the Koszul complex
last
-- last element of a list
lcm
-- least common multiple
leadCoefficient
-- the coefficient of the leading term
leadComponent
-- the leading component(s) of a vector or matrix
leadMonomial
-- the leading monomial of a ring element
leadTerm
-- get the greatest term
left
-- left endpoint of an interval
length
-- length of an object
lift
-- lift to another ring
liftable
-- whether lifting to another ring is possible
limitFiles
limitProcesses
lines
-- split a string into lines
linkFile
-- see
linkFile(String,String)
-- make a new link to a file
listForm
-- convert to list form
listSymbols
-- compact display of symbols and their values
LLL
-- compute an LLL basis
lngamma
-- logarithm of the Gamma function
load
-- read Macaulay2 commands
loadPackage
-- load a package
localDictionaries
-- get local dictionaries
localize
-- localize an ideal at a prime ideal
locate
-- locate source code
log
-- logarithm function
log1p
-- logarithm of 1+x
lookup
-- look up methods
lookupCount
-- reference count for a symbol
LUdecomposition
-- LU decomposition
makeDirectory
-- see
makeDirectory(String)
-- make a directory
makePackageIndex
-- create an index of installed packages
makeS2
-- compute the S2ification of a reduced ring
map
-- make a map
markedGB
-- see
markedGB(Matrix,Matrix)
-- make a marked Gröbner basis
match
-- regular expression matching
mathML
-- convert to MathML format
matrix
-- make a matrix
max
-- yields the maximum element in a list or sequence
maxPosition
-- position of the largest element
memoize
-- record results of function evaluation for future use
memoizeClear
-- see
memoize
-- record results of function evaluation for future use
memoizeValues
-- see
memoize
-- record results of function evaluation for future use
merge
-- merge two hash tables
mergePairs
-- merge sorted lists of pairs
method
-- make a new method function
methodOptions
-- see
methodOptions(Function)
-- recover the options used when a method function was created
methods
-- list methods
midpoint
-- midpoint of an interval
min
-- yields the minimum element in a list or sequence
mingens
-- minimal generator matrix
mingle
-- mingle elements of several lists
minimalBetti
-- minimal betti numbers of (the minimal free resolution of) a homogeneous ideal or module
minimalPresentation
-- compute a minimal presentation
minimalPrimes
-- minimal primes of an ideal
minimalReduction
-- Find a minimal reduction of an ideal
minimize
minimizeFilename
-- minimize a file name
minors
-- see
minors(ZZ,Matrix)
-- ideal generated by minors
minPosition
-- position of the smallest element
minus
-- additive inverse
mkdir
mod
-- reduce modulo an integer
module
-- make or get a module
modulo
-- see
modulo(Matrix,Matrix)
-- find the pre-image (pullback) of image of a map (low level version)
monoid
-- make or retrieve a monoid
monomialCurveIdeal
-- make the ideal of a monomial curve
monomialIdeal
-- make a monomial ideal
monomials
-- matrix of monomials in a ring element or matrix
monomialSubideal
-- find the largest monomial ideal in an ideal
moveFile
-- see
moveFile(String,String)
multidegree
-- multidegree
multigraded
-- convert a Betti tally into a multigraded Betti tally
multiplicity
-- Compute the Hilbert-Samuel multiplicity of an ideal
mutableIdentity
-- see
mutableIdentity(Ring,ZZ)
-- make a mutable identity matrix
mutableMatrix
-- make a mutable matrix
nanosleep
-- sleep for a given number of nanoseconds
needs
-- read Macaulay2 commands if necessary
needsPackage
-- load a package if not already loaded
nefCone
(missing documentation)
nefGenerators
(missing documentation)
net
-- format for printing, as a net
netList
-- see
netList(VisibleList)
-- a table of boxes
newClass
-- set the class and parent of an object
newCoordinateSystem
-- change variables
newNetFile
-- create a new net file
newPackage
-- the preamble of a package
newRing
-- make a copy of a ring, with some features changed
next
-- get the next object from an iterator
nextkey
-- the next key in a database
nextPrime
-- compute the smallest prime greater than or equal to a given number
norm
normalCone
-- The normal cone of a subscheme
notImplemented
-- print an 'not implemented' error message
nullhomotopy
-- make a null homotopy
nullSpace
-- find the null space of a mutable matrix
number
-- count how many elements of a list satisfy a condition
numColumns
-- see
numColumns(Matrix)
-- number of columns in a matrix or mutable matrix
numerator
-- numerator of a fraction
numeric
-- convert to floating point
numericInterval
-- converts to an interval
numgens
-- the number of generators
numRows
-- see
numRows(Matrix)
-- number of rows in a matrix or mutable matrix
odd
-- tell whether an integer is odd
oeis
-- OEIS lookup
ofClass
-- English phrases for types
on
-- trace a function each time it's run
openDatabase
-- open a database file
openDatabaseOut
-- open a database file for writing
openFiles
-- list the open files
openIn
-- see
openIn(String)
-- open an input file
openInOut
-- open an input output file
openListener
-- see
openListener(String)
-- open a port for listening
openOut
-- see
openOut(String)
-- open an output file
openOutAppend
-- see
openOutAppend(String)
-- open an output file for appending
options
-- get options
override
-- override default values for optional arguments
pack
-- pack elements of a list or string into several shorter lists or strings
package
-- get containing package
packageTemplate
-- a template for a package
pad
-- pad a string or net with spaces
pager
-- display with paging
pairs
-- list the pairs in a hash table, dictionary, or basic list
parent
-- parent type of an object
part
-- select terms of a polynomial by degree(s) or weight(s)
partition
-- partition a set or list by values of a function
partitions
-- list the partitions of an integer
parts
-- display terms of a polynomial degree by degree
pdim
-- calculate the projective dimension
peek
-- examine contents of an object
peek'
-- examine contents of an object
permanents
-- ideal generated by square permanents of a matrix
permutations
-- produce all permutations of a list
pfaffians
-- ideal generated by Pfaffians
pivots
-- see
pivots(Matrix)
-- list of pivot locations of a matrix
plus
-- addition
installHilbertFunction
-- see
poincare
-- assemble degrees of a ring, module, or ideal into a polynomial
poincare
-- assemble degrees of a ring, module, or ideal into a polynomial
poincareN
-- assemble degrees into polynomial
polarize
-- given a monomial ideal, computes the squarefree monomial ideal obtained via polarization
poly
-- see
poly(String)
-- make a polynomial using classic Macaulay syntax
position
-- the first element of a list satisfying a condition
positions
-- which elements of a list satisfy a condition
power
-- power
powermod
-- powers of integers mod N
precision
preimage
-- preimage of a map
prepend
-- add an element to the beginning of a list
presentation
-- presentation of a module or ring
pretty
-- a pretty printer
primaryComponent
-- find a primary component corresponding to an associated prime
primaryDecomposition
-- irredundant primary decomposition of an ideal
print
-- print something
printerr
-- print something to stderr
printString
-- lowlevel function to print a string, net, or symbol
processID
-- the process identifier
product
profile
-- profile a function
Proj
-- make a projective variety
projectiveHilbertPolynomial
-- Hilbert polynomial of projective space
promote
-- promote to another ring
protect
-- protect a symbol
prune
-- prune, e.g., compute a minimal presentation
pseudocode
-- produce the pseudocode for a function
pseudoRemainder
-- compute the pseudo-remainder
pullback
-- compute the pullback morphism
pushForward
-- see
pushForward(RingMap,Module)
pushout
-- compute the pushout morphism
QRDecomposition
-- compute a QR decomposition of a real matrix
quotient
-- quotient or division
quotient'
-- matrix quotient (opposite)
quotientRemainder
-- matrix quotient and remainder
quotientRemainder'
-- matrix quotient and remainder (opposite)
radical
-- the radical of an ideal
radicalContainment
-- whether an element is contained in the radical of an ideal
random
-- get a random object
randomKRationalPoint
-- Pick a random K rational point on the scheme X defined by I
randomMutableMatrix
-- see
randomMutableMatrix(ZZ,ZZ,RR,ZZ)
-- a random mutable matrix of integers
rank
-- compute the rank
rays
-- 1-dimensional cones of a fan or polyhedral object
read
-- read from a file
readDirectory
-- read the contents of a directory
readlink
-- readlink fn
readPackage
-- read the package preamble
realPart
-- real part
realpath
-- convert a filename to one passing through no symbolic links
recursionDepth
-- the current recursion depth
reducedRowEchelonForm
-- compute the reduced row echelon form of a matrix or mutable matrix over a field
reduceHilbert
-- reduce a Hilbert series expression
reductionNumber
-- Reduction number of one ideal with respect to another
reesAlgebra
-- Compute the defining ideal of the Rees Algebra
reesIdeal
-- Compute the defining ideal of the Rees Algebra
regex
-- evaluate a regular expression search
regexQuote
-- escape special characters in regular expressions
registerFinalizer
-- register a string that will be displayed when an object is garbage collected
regSeqInIdeal
-- a regular sequence contained in an ideal
regularity
-- compute the Castelnuovo-Mumford regularity
regularizedBeta
-- regularized beta function
regularizedGamma
-- upper regularized gamma function
relations
-- the defining relations
relativizeFilename
-- relativize a file name
remainder
-- matrix remainder
remainder'
-- matrix quotient and remainder (opposite)
remove
-- remove an entry from a mutable hash table, list, or database
removeDirectory
-- remove a directory
removeFile
-- remove a file
removeLowestDimension
-- remove components of lowest dimension
reorganize
-- reorganize a database file
replace
-- replacement in strings and lists
reshape
-- reshape a matrix
resolution
-- projective resolution
resultant
-- see
resultant(RingElement,RingElement,RingElement)
reverse
-- reverse a list, sequence, or string
right
-- right endpoint of an interval
ring
-- get the associated ring of an object
ringFromFractions
-- find presentation for f.g. ring
roots
-- see
roots(RingElement)
-- compute the roots of a polynomial
rotate
-- see
rotate(ZZ,VisibleList)
-- rotate a list
round
-- round a number
rowAdd
-- add a multiple of one row to another
rowMult
-- multiply a row by a ring element
rowPermute
-- permute some rows
rowRankProfile
-- find the row rank profile of a mutable matrix
rowSwap
-- interchange rows
rsort
-- sort a list or matrix in reverse order
run
-- run an external command
runHooks
-- run the hook functions stored in an object
runLengthEncode
-- see
runLengthEncode(VisibleList)
-- run length encoding
runProgram
-- run an external program
same
-- whether everything in a list is the same
saturate
-- saturation of ideal or submodule
scan
-- apply a function to each element in a list or sequence
scanKeys
-- apply a function to each key in a hash table or database
scanLines
-- apply a function to each line of a file
scanPairs
-- apply a function to the pairs in a hash table
scanValues
-- apply a function to each value in a hash table or database
schedule
-- schedule a task for execution
schreyerOrder
-- create or obtain free modules with Schreyer monomial orders
Schubert
-- see
Schubert(ZZ,ZZ,VisibleList)
-- find the Plücker ideal of a Schubert variety
searchPath
-- see
searchPath(List,String)
-- search a path for a file
sec
-- secant
sech
-- hyperbolic secant
seeParsing
-- display parsing precedence table for Macaulay2 operators
select
-- select from a list, hash table, or string
selectInSubring
-- select columns in a subring
selectVariables
-- see
selectVariables(List,PolynomialRing)
-- make a subring of a polynomial ring generated by selected variables
separate
-- split a string into substrings using a regular expression
separateRegexp
-- see
separate
-- split a string into substrings using a regular expression
sequence
-- make a sequence
serialNumber
-- serial number of a dictionary, task, symbol, mutable hash table, or mutable list,
set
-- make a set
setEcho
-- turn on echoing
setGroupID
-- set the process group identifier
setIOExclusive
-- exclusive I/O for the current thread
setIOSynchronized
-- synchronized I/O for threads
setIOUnSynchronized
-- unsynchronized I/O for threads
setRandomSeed
-- set starting point for random number generator
setup
setupEmacs
sheaf
-- make a coherent sheaf
sheafHom
-- sheaf Hom
sheafMap
(missing documentation)
show
-- display various objects in an external viewer
showHtml
-- see
show
-- display various objects in an external viewer
showTex
-- see
show
-- display various objects in an external viewer
simpleDocFrob
-- a sample documentation node
sin
-- compute the sine
singularLocus
-- singular locus
sinh
-- compute the hyperbolic sine
size
-- the size of an object
size2
-- number of binary digits to the left of the point
sleep
-- sleep for a while
smithNormalForm
-- see
smithNormalForm(Matrix)
-- smith normal form for a matrix over ZZ or a PID
solve
-- solve linear equation(s)
someTerms
-- select some terms of a polynomial
sort
-- sort a list or columns of a matrix
sortColumns
-- permutation giving sort order
source
-- source of a map
span
-- construct smallest interval
Spec
-- make an affine variety
specialFiber
-- Special fiber of a blowup
specialFiberIdeal
-- Special fiber of a blowup
splice
-- remove subsequences from a sequence or list
splitWWW
-- separate an http response into header and body
sqrt
-- square root function
stack
-- join nets or string vertically
stacksProject
-- link to a Stacks Project tag
standardForm
-- convert to standard form
standardPairs
-- find the standard pairs of a monomial ideal
stashValue
-- stash values of functions in their arguments
status
-- status of a resolution computation
store
-- store the value of an atomic integer
sublists
-- process interspersed subsequences of a visible list
submatrix
submatrix'
-- exclude rows and/or columns of a matrix
submatrixByDegrees
-- submatrix consisting of rows and columns in an interval or box of degrees
subquotient
-- make a subquotient module
subsets
-- produce the subsets of a set or list
substitute
-- substituting values for variables
substring
-- extract part of a string
subtable
-- extract a subtable from a table
sum
-- compute the sum
super
-- get the ambient module
support
-- list of variables occurring in a polynomial or matrix
SVD
-- singular value decomposition of a matrix
switch
-- copy a list, switching two elements
sylvesterMatrix
-- see
sylvesterMatrix(RingElement,RingElement,RingElement)
symbolBody
-- symbol bodies
symlinkDirectory
-- see
symlinkDirectory(String,String)
-- make symbolic links for all files in a directory tree
symlinkFile
-- make a symbolic link to a file
symmetricAlgebra
-- the symmetric algebra of a module
symmetricAlgebraIdeal
-- Ideal of the symmetric algebra of an ideal or module
symmetricKernel
-- Compute the Rees ring of the image of a matrix
symmetricPower
-- symmetric power
synonym
-- synonym for members of a class
SYNOPSIS
-- a standardized synopsis for use in documentation nodes
syz
-- the syzygy matrix
syzygyScheme
-- construct a syzygy scheme
table
-- make a table from a binary function
take
-- Take some elements from a list or sequence.
tally
-- tally the elements of a list, sequence, array, or string
tan
-- compute the tangent
tangentCone
-- see
tangentCone(Ideal)
tangentSheaf
-- tangent sheaf of a projective variety
tanh
-- compute the hyperbolic tangent
target
-- target of a map
taskResult
-- see
taskResult(Task)
-- retrieve the value returned by a task
temporaryFileName
-- make a temporary file name
tensor
-- tensor product
tensorAssociativity
-- associativity isomorphisms for tensor products
terms
-- provide a list of terms of a polynomial
TEST
-- add a test for a package
testHunekeQuestion
-- tests a conjecture on integral closures strengthening the Eisenbud-Mazur conjecture
tests
-- locate a package's tests
tex
-- convert to $\TeX$ format
texMath
-- convert to TeX math format
times
-- multiplication
toAbsolutePath
-- the absolute path version of a file name
toCC
-- convert to high-precision complex number
toDividedPowers
-- Translates to divided power monomial basis from ordinary monomial basis
toDual
-- finds the inverse system to an ideal up to a given degree
toExternalString
-- convert to a readable string
toField
-- see
toField(Ring)
-- declare that a ring is a field
toList
-- create a list
toLower
-- convert to lower case
topCoefficients
-- first variable and its coefficient of a polynomial or matrix
topComponents
-- compute top dimensional component of an ideal or module
toRR
-- convert to high-precision real number
toRRi
-- construct an interval
toSequence
-- convert to sequence
toString
-- convert to a string
toUpper
-- convert to upper case
trace
-- see
trace(Matrix)
-- trace of a matrix
transpose
-- transpose a table or a matrix
trim
-- minimize generators and relations
truncate
truncateOutput
tutorial
-- convert documentation from tutorial format
ultimate
-- ultimate value for an iteration
unbag
-- see
unbag(Bag)
uncurry
-- uncurry a function
undocumented
-- declare that something need not be documented
uniform
-- whether all elements of a list are the same class
uninstallAllPackages
uninstallPackage
-- uninstall a package and remove its documentation
unique
-- eliminate duplicates from a list
uniquePermutations
-- produce all unique permutations of a list
unsequence
-- extract the single element from a sequence of length 1
unstack
-- list the rows of a net
urlEncode
-- URL encoding
use
-- install or activate object
userSymbols
-- a list of the user's symbols
utf8
-- encode and decode unicode utf-8-encoded strings
utf8check
-- see
utf8
-- encode and decode unicode utf-8-encoded strings
utf8substring
-- extract part of a utf8 string
value
-- evaluate
values
-- values in a hash table
variety
-- get the variety
vars
-- variables
vector
-- make a vector
versalEmbedding
-- Compute a versal embedding
wait
-- wait for child process
wedgeProduct
-- see
wedgeProduct(ZZ,ZZ,Module)
-- the exterior multiplication map
weightRange
-- the pair of lowest and highest weights of the monomials
isWeylAlgebra
-- see
Weyl algebras
whichGm
-- Largest Gm satisfied by an ideal
width
-- width of a file or net
wikipedia
-- link to a Wikipedia article
wrap
youngest
-- the youngest member of a sequence
zero
-- whether something is zero
zeta
-- Riemann zeta function
For the programmer
The object
Function
is
a
type
, with ancestor class
Thing
.