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AngouriMath.Entity

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Classes within the AngouriMath.Entity namespace

• Absf

  Summary

  A node of abs
  

• Alephf

  Summary

  The aleph number aleph(k), the size of an infinite set: aleph(0) is the
  size of the whole numbers, and aleph(k + 1) the least size after
  aleph(k).
  

  Remarks

  card of an infinite set is one of these, or a power of 2 of one:
  card(ZZ) is aleph(0) and card(RR) is 2^aleph(0), which is an
  aleph but not a named one -- that it is aleph(1) is the continuum hypothesis,
  which ZFC does not decide. A sum or a product of sizes is the larger of them, and the
  sizes compare.
  https://github.com/asc-community/AngouriMath/issues/1409
  

• Andf

  Summary

  Is true iff both operands are true
  

• Application

  Summary

  Application of arguments to the given expression
  

• Arccosecantf

  Summary

  A node of arccosecant
  

• Arccosf

  Summary

  A node of arccosine
  

• Arccotanf

  Summary

  A node of arccotangent
  

• Arcsecantf

  Summary

  A node of arcsecant
  

• Arcsinf

  Summary

  A node of arcsine
  

• Arctanf

  Summary

  A node of arctangent
  

• Argmaxf

  Summary

  The set of points at which an expression takes its largest value over a set:
  argmax(f(t), t in S). See Maximumf.
  

• Argminf

  Summary

  The set of points at which an expression takes its smallest value over a set:
  argmin(f(t), t in S). See Maximumf.
  

• Binomialf

  Summary

  A node of the binomial coefficient binomial(n, k), "n choose k".
  

  Remarks

  A node rather than n!/(k! (n - k)!): that quotient is correct and useless as a
  spelling -- it loses the integrality, it is undefined for a negative n where
  the falling factorial n (n - 1) ... (n - k + 1)/k! is not, and no rule can
  recognise Pascal's identity or Vandermonde's in a ratio of three factorials. The
  falling factorial is the definition for a whole k, whatever n is; a
  whole k below zero gives 0; and for a k that is not whole the
  value is Γ(n + 1)/(Γ(k + 1) Γ(n - k + 1)), which has no value where n is a negative whole number. What cannot be settled is left as this node.
  https://github.com/asc-community/AngouriMath/issues/1409
  https://github.com/asc-community/AngouriMath/issues/809
  

• Boolean

  Summary

  This node represents all possible values a boolean node might be of
  

• CalculusOperator

  Summary

  Describes any calculus operator
  

  Remarks

  Every one of these bindsVar, so this is where a bound name is
  read — see Binding for the one name in the language that needs
  reading rather than taking as it arrives.
  

• Cardf

  Summary

  This node represents the cardinality of a set: card(S) is the number of
  elements of S. Counted for a finite set whose elements are numbers, and for
  an interval whose ends are; left as written for an infinite set, which has a
  cardinality this library has no number for, and for a set whose elements are not
  yet known to be distinct.
  https://github.com/asc-community/AngouriMath/issues/1212
  

• Ceilf

  Summary

  A node of ceil: the least integer not below the argument.
  

  Remarks

  Componentwise on a complex argument, for the same reason as
  Floorf.
  

• Chif

  Summary

  The hyperbolic cosine integral, Chi(z) = gamma + ln z + int_0^z (cosh t - 1)/t dt, with the cut of ln.
  

  Remarks

  https://github.com/asc-community/AngouriMath/issues/1501

• Cif

  Summary

  The cosine integral, Ci(z) = gamma + ln z + int_0^z (cos t - 1)/t dt, with the cut of ln.
  

  Remarks

  https://github.com/asc-community/AngouriMath/issues/1501

• ComparisonSign

  Summary

  =, <, >, >=, <=
  Nodes Equalsf, Lessf, LessOrEqualf, Greaterf, GreaterOrEqualf are assignable to this type.
  

• Congruentf

  Summary

  The statement that a and b are congruent modulo n: written
  a = b (mod n), or a ≡ b (mod n), and meaning n divides a - b. A
  relation between integers with the modulus written once at the end, as mathematics
  writes it — not the remainder operator a mod n, which is a number, and which the
  reference this follows refuses to write in this sense at all. The modulus may be any
  integer: modulo 0 it is equality, and its sign is immaterial, exactly as in
  n divides a - b. A statement about integers: NaN elsewhere.
  https://github.com/asc-community/AngouriMath/issues/1409
  

• Constant

  Summary

  A mathematical constant: a number whose spelling happens to be a legal identifier.
  

  Remarks

  It is a Variable by inheritance, so everything that reads a leaf by name
  keeps working, and a distinct type by record identity, which is the whole point: a
  binder that declares pi and the constant pi are no longer one object, so
  evaluation can tell them apart without being told where it is.
  #984 A binder binds occurrences, not values, and that is the whole of the second
  distinction here. Every constant a writer types is the one object in
  NamedConstants, because the parser and pi hand that
  object back; the base of ln and of exp is
  EulerIntrinsic, a separate object, because it is Euler's number standing
  in an operator's own definition and not a reference to the name e. A binder
  replaces the occurrences it was handed — compared by reference — so
  sum(ln(x), e, 1, 2) is 2 * ln(x) while sum(log(e, x), e, 1, 2),
  where the writer did name e, is log(1, x) + log(2, x).
  The two are equal, and deliberately so: they are the same number, they print
  alike, and any rule that matches one matches the other, so nothing about canonical form,
  equality or substitution changes. Only the identity of the occurrence differs, and only
  while a binder is deciding what it binds — which is at construction, before anything
  evaluates.
  

• ContinuousNode

  Summary

  Any node that might be counted as a number, derived, etc. is continuous
  

• Cosecantf

  Summary

  A node of secant
  

• Cosf

  Summary

  A node of cosine
  

• Cotanf

  Summary

  A node of cotangent
  

• Derivativef

  Summary

  A node of derivative
  

  Remarks

  Negative iterations convert to integrals.
  

• Divf

  Summary

  A node of division
  

• Dividesf

  Summary

  This node represents whether the first argument divides the second: a divides b is the statement that b is a whole multiple of a, which for integers is
  b mod a = 0, and 0 divides b exactly when b is 0. A statement
  about integers: over anything else it is NaN.
  https://github.com/asc-community/AngouriMath/issues/1212
  

• Eif

  Summary

  The exponential integral, Ei(z) = gamma + (ln z - ln(1/z))/2 + sum_(k >= 1) z^k/(k k!),
  which on the real line is the principal value of int_-oo^x e^t/t dt.
  

  Remarks

  https://github.com/asc-community/AngouriMath/issues/1501

• Equalsf

  Summary

  It is true if left and right are equal
  

• Erfcf

  Summary

  The complementary error function, erfc(z) = 1 - erf(z).
  

  Remarks

  https://github.com/asc-community/AngouriMath/issues/1501

• Erff

  Summary

  The error function, erf(z) = 2/sqrt(pi) int_0^z e^(-t^2) dt, entire and odd.
  

  Remarks

  https://github.com/asc-community/AngouriMath/issues/1501

• Erfif

  Summary

  The imaginary error function, erfi(z) = -i erf(i z), which on the real line is
  2/sqrt(pi) int_0^x e^(t^2) dt.
  

  Remarks

  https://github.com/asc-community/AngouriMath/issues/1501

• EvaledValue

  Summary

  A value of Evaled, with the node and the precision it was computed for.

• Existsf

  Summary

  exists x in S : P: some member of S satisfies P. False of the empty
  set. See Quantifier.
  

• ExistsUniquef

  Summary

  exists! x in S : P: exactly one member of S satisfies P, which is
  exists x in S : P and forall y in S : P(y) implies y = x. See Quantifier.
  

• Factorialf

  Summary

  A node of factorial
  

• Floorf

  Summary

  A node of floor: the greatest integer not above the argument.
  

  Remarks

  On a complex argument it is taken componentwise, so that
  floor(3/2 + 5/2i) is 1 + 2i. That is what SymPy and Mathematica
  both do, and it is the only reading under which floor of a real number
  keeps its meaning when the imaginary part happens to be zero.
  

• Forallf

  Summary

  forall x in S : P: every member of S satisfies P. True of the empty
  set. See Quantifier.
  

• Function

  Summary

  Describes any node that is a function (e. g. sin, cos, etc.) or calculus operator (e. g. derivative, integral, limit)
  but not an arithmetic operator or leaf
  

• Gcdf

  Summary

  A node of gcd: the greatest common divisor of its two arguments.
  

  Remarks

  Not integers only. SymPy's gcd is the polynomial gcd with the integer case
  as a special case, and this library already computes one — see
  PolynomialGcd, which the
  PolynomialGcdCancellation rule set uses. What cannot be settled is left as
  this node rather than guessed at.
  

• Greaterf

  Summary

  It is true iff both parts are numeric and real, and left number is greater
  than the right one
  It is false iff both parts are numeric and real, and left number is less or equal
  the right one
  It is NaN/unsimplified otherwise.
  

• GreaterOrEqualf

  Summary

  It is true iff both parts are numeric and real, and left number is greater
  than the right one or equal to it
  It is false iff both parts are numeric and real, and left number is less
  the right one
  It is NaN/unsimplified otherwise.
  

• Impliesf

  Summary

  Is true iff assumption is false or conclusion is true
  

• Integralf

  Summary

  A node of integral
  

• Iversonf

  Summary

  The Iverson bracket iverson(P): 1 where the statement P holds and
  0 where it does not.
  

  Remarks

  It is how a count is written as a sum: sum(iverson(k divides 12), k, 1, 12) is
  the number of divisors of 12. A named function rather than a bracket, since every
  bracket shape the grammar has already means something: a vector, a matrix, an interval.
  A statement that is not decided keeps the node, and so does a number, which is not a
  statement, as a connective keeps one.
  https://github.com/asc-community/AngouriMath/issues/1478
  

• Lambda

  Summary

  A lambda with the only parameter and its body
  

• Lcmf

  Summary

  The least common multiple of two numbers: lcm(a, b), non-negative by the same
  convention as Gcdf, and 0 where either is 0. Over
  rationals lcm(a/b, c/d) is lcm(a, c) / gcd(b, d), which is what SymPy
  gives. Left as this node where it cannot be settled.
  https://github.com/asc-community/AngouriMath/issues/1409
  

• Lessf

  Summary

  It is true iff both parts are numeric and real, and left number is less
  than the right one
  It is false iff both parts are numeric and real, and left number is greater or equal
  the right one
  It is NaN/unsimplified otherwise.
  

• LessOrEqualf

  Summary

  It is true iff both parts are numeric and real, and left number is less
  than the right one or equal to it
  It is false iff both parts are numeric and real, and left number is greater
  the right one
  It is NaN/unsimplified otherwise.
  

• Lif

  Summary

  The logarithmic integral, li(z) = Ei(ln z), which on x > 0 is the principal
  value of int_0^x dt/ln t.
  

  Remarks

  https://github.com/asc-community/AngouriMath/issues/1501

• Limitf

  Summary

  A node of limit
  

• Logf

  Summary

  A node of logarithm
  

• Matrix

  Summary

  Basic matrix implementation: https://en.wikipedia.org/wiki/Matrix_(mathematics)

• Maxf

  Summary

  A node of max: the greater of its two arguments.
  

  Remarks

  See Minf for why this is a node.

• Maximumf

  Summary

  The largest value of an expression as a variable ranges over a set:
  max(f(t), t in S). A binder, like Summationf: the variable is
  bound throughout the expression and the set.
  https://github.com/asc-community/AngouriMath/issues/1212
  

• Minf

  Summary

  A node of min: the lesser of its two arguments.
  

  Remarks

  A node rather than sugar over (a + b - abs(a - b)) / 2. The closed form is
  what this simplifies to where that helps; it is not what it should print
  as, and the round-trip contract would make that permanent. SymPy keeps
  Min as a node for the same reason.
  Only ordered arguments compare, so a complex one is left alone rather than
  guessed at.
  

• Minimumf

  Summary

  The smallest value of an expression as a variable ranges over a set:
  min(f(t), t in S). See Maximumf.
  

• Minusf

  Summary

  A node of difference
  

• Modf

  Summary

  A node of modulus, that is, the floored remainder after division,
  a - b * floor(a / b), which takes the sign of the divisor:
  (-7) mod 3 is 2 and 7 mod (-3) is -2, the convention of
  SymPy, Mathematica and Maxima and the one under which the residues modulo n are 0 to n - 1. Not C#'s %, which truncates; this comment said
  the opposite of what the evaluation and the documentation do.
  

• Mulf

  Summary

  A node of product
  

• Notf

  Summary

  Whatever its argument is, the result will be inverted
  

• Number

  Summary

  Number node.
  This class represents all possible numerical values as a hierarchy,
  NumberComplexRealRationalInteger

• Orf

  Summary

  Is true iff at least one operand is true,
  

• Phif

  Summary

  This node represents the Euler totient function (phi)
  

• Piecewise

  Summary

  This is a node which defined on different subsets differently. When evaluating, it will turn
  into a particular case once all cases' predicates before are false, and this case's predicate is true.
  
  That is, the order counts. An example:
  Piecewise(a provided false, b provided true, c provided true)
  Will be evaluated into b,
  
  Piecewise(a provided false, b provided c, c provided false)
  Will remain as it is (although unreachable cases will be removed)
  

• Powf

  Summary

  A node of exponential (power)
  

• Primef

  Summary

  The n-th prime, prime(n): prime(1) is 2, prime(25) is 97. An enumeration of PP in its order, which a set does not carry, so
  a function of its own -- SymPy's prime(n). Defined on the positive whole numbers.
  https://github.com/asc-community/AngouriMath/issues/1450
  

• Productf

  Summary

  A product over a bound variable: product(f, i, from, to) is
  f(from) · … · f(to). See Summationf, which it mirrors exactly.
  

  Remarks

  #248

• Providedf

  Summary

  That is a node which equals Expression if Predicate is true, otherwise NaN

• Quantifier

  Summary

  A quantified statement: forall x in S : P, exists x in S : P and
  exists! x in S : P. A binder, like a set builder: the name is bound throughout
  the body and the set, and the statement is a function of what else the body mentions.
  The quantification set is mandatory, as Sullivan and Mackey's proofs book insists — a
  statement is quantified over something, and forall x : x^2 >= 0 is true of
  the reals and false of the complex numbers.
  https://github.com/asc-community/AngouriMath/issues/1409
  https://github.com/asc-community/AngouriMath/issues/225
  

• Roundf

  Summary

  A node of round: the nearest integer, with a tie going to the even one.
  

  Remarks

  Half to even, which is what Python, SymPy, Mathematica and IEEE 754 all do — and
  what .NET's Math.Round does by default, though not what most people expect
  of it. So round(1/2) is 0 and round(3/2) is 2, and it
  is notfloor(x + 1/2), which differs at every tie.
  

• Secantf

  Summary

  A node of secant
  

• Set

  Summary

  https://en.wikipedia.org/wiki/Set_(mathematics) A set might be a set operator, conditinal set, finite set, or interval
  

• Shif

  Summary

  The hyperbolic sine integral, Shi(z) = int_0^z sinh(t)/t dt, entire and odd.
  

  Remarks

  https://github.com/asc-community/AngouriMath/issues/1501

• Sif

  Summary

  The sine integral, Si(z) = int_0^z sin(t)/t dt, entire and odd.
  

  Remarks

  https://github.com/asc-community/AngouriMath/issues/1501

• Signumf

  Summary

  A node of signum
  

• Sinf

  Summary

  A node of sine
  

• SortKeyCache

  Summary

  The keys, behind a struct that is equal to every other, for the reason
  LazyPropertyA is: an Entity is a record and compares every
  field, so a bare array here made two equal trees unequal the moment one had been
  sorted -- and the simplifier, which recognises a repeated candidate by equality,
  then never recognised one. Measured: 30 GB and 57 s on SimplifyHard.
  

• Statement

  Summary

  This class describes every node whose value is boolean, that is, true or false.
  

  Example

  Entity t1 = "a and b"; // Statement
  Entity t2 = "a > b > c"; // Statement
  Entity t3 = "a or b implies x > 0"; // Statement
  
  Entity t4 = "a"; // Not Statement
  Entity t5 = "a + 3"; // Not Statement
  Entity t6 = "a provided c and d"; // Not Statement

• Sumf

  Summary

  A node of sum
  

• Summationf

  Summary

  A summation over a bound variable: sum(f, i, from, to) is
  f(from) + … + f(to).
  

  Remarks

  A CalculusOperator rather than a Sumf of many terms,
  because the bounds may be symbolic — sum(i, i, 1, n) has no finite expansion
  and is still a well-formed expression. Where they are concrete integers it expands,
  which is what makes the common case useful without the general case being wrong.
  The index is bound: it is the summation index, not a free variable of
  the expression, so substituting it from outside must not reach inside. That is the
  property ConditionalSet already has and the one
  #878 was about
  getting wrong.
  #248

• SumOverSetf

  Summary

  A sum over a set: sum(f(x), x in S) is the sum of f(x) over the members
  x of S, each counted once. A binder, like Summationf, whose
  range is a set rather than two bounds, and so has no index and no order: the sum over
  the roots of a polynomial is sum(f(w), w in { w : p(w) = 0 }), which needs no
  first root.
  https://github.com/asc-community/AngouriMath/issues/1285
  

• Tanf

  Summary

  A node of tangent
  

• TrigonometricFunction

  Summary

  Describes any function that is related to trigonometry
  

• Valuationf

  Summary

  The p-adic valuation, valuation(n, p): the exponent of the prime
  p in n, so valuation(12, 2) is 2 and valuation(12, 5) is 0; +oo at n = 0, by the convention every text uses. Sullivan and
  Mackey's Ex 5.5.1 writes n = 2^m (2 l + 1) with m the 2-adic valuation, and
  n = product(p^valuation(n, p), p in PP) is what a factorisation says.
  https://github.com/asc-community/AngouriMath/issues/1450
  

• Variable

  Summary

  Variable node. It only has a name.
  Construct a Variable with an implicit conversion from String.
  It has no type, so you can substitute any value under a given variable.
  

• Xorf

  Summary

  Is true iff one operand is true