Source code for diofant.functions.special.tensor_functions

from ...core import Function, Integer, prod
from ...utilities import default_sort_key, has_dups


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# #################### Kronecker Delta, Levi-Civita etc. #################### #
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[docs]def Eijk(*args, **kwargs): """ Represent the Levi-Civita symbol. This is just compatibility wrapper to ``LeviCivita()``. See Also ======== diofant.functions.special.tensor_functions.LeviCivita """ return LeviCivita(*args, **kwargs)
[docs]def eval_levicivita(*args): """Evaluate Levi-Civita symbol.""" from .. import factorial n = len(args) return prod( prod(args[j] - args[i] for j in range(i + 1, n)) / factorial(i) for i in range(n))
# converting factorial(i) to int is slightly faster
[docs]class LeviCivita(Function): """Represent the Levi-Civita symbol. For even permutations of indices it returns 1, for odd permutations -1, and for everything else (a repeated index) it returns 0. Thus it represents an alternating pseudotensor. Examples ======== >>> from diofant.abc import i, j >>> LeviCivita(1, 2, 3) 1 >>> LeviCivita(1, 3, 2) -1 >>> LeviCivita(1, 2, 2) 0 >>> LeviCivita(i, j, k) LeviCivita(i, j, k) >>> LeviCivita(i, j, i) 0 See Also ======== diofant.functions.special.tensor_functions.Eijk """ is_integer = True @classmethod def eval(cls, *args): if all(isinstance(a, (int, Integer)) for a in args): return eval_levicivita(*args) if has_dups(args): return Integer(0) def doit(self, **hints): return eval_levicivita(*self.args)
[docs]class KroneckerDelta(Function): """The discrete, or Kronecker, delta function. A function that takes in two integers `i` and `j`. It returns `0` if `i` and `j` are not equal or it returns `1` if `i` and `j` are equal. Parameters ========== i : Number, Symbol The first index of the delta function. j : Number, Symbol The second index of the delta function. Examples ======== A simple example with integer indices:: >>> KroneckerDelta(1, 2) 0 >>> KroneckerDelta(3, 3) 1 Symbolic indices:: >>> from diofant.abc import i, j >>> KroneckerDelta(i, j) KroneckerDelta(i, j) >>> KroneckerDelta(i, i) 1 >>> KroneckerDelta(i, i + 1) 0 >>> KroneckerDelta(i, i + 1 + k) KroneckerDelta(i, i + k + 1) See Also ======== diofant.functions.special.tensor_functions.KroneckerDelta.eval diofant.functions.special.delta_functions.DiracDelta References ========== * https://en.wikipedia.org/wiki/Kronecker_delta """ is_integer = True
[docs] @classmethod def eval(cls, i, j): """ Evaluates the discrete delta function. Examples ======== >>> from diofant.abc import i, j >>> KroneckerDelta(i, j) KroneckerDelta(i, j) >>> KroneckerDelta(i, i) 1 >>> KroneckerDelta(i, i + 1) 0 >>> KroneckerDelta(i, i + 1 + k) KroneckerDelta(i, i + k + 1) """ diff = i - j if diff.is_zero: return Integer(1) elif diff.is_nonzero: return Integer(0) if i._assumptions.get("below_fermi") and \ j._assumptions.get("above_fermi"): return Integer(0) if j._assumptions.get("below_fermi") and \ i._assumptions.get("above_fermi"): return Integer(0) # to make KroneckerDelta canonical # following lines will check if inputs are in order # if not, will return KroneckerDelta with correct order if i is not min(i, j, key=default_sort_key): return cls(j, i)
def _eval_power(self, expt): if expt.is_positive: return self if expt.is_negative and -expt != 1: return 1/self @property def is_above_fermi(self): """ True if Delta can be non-zero above fermi Examples ======== >>> a = Symbol('a', above_fermi=True) >>> i = Symbol('i', below_fermi=True) >>> p = Symbol('p') >>> q = Symbol('q') >>> KroneckerDelta(p, a).is_above_fermi True >>> KroneckerDelta(p, i).is_above_fermi False >>> KroneckerDelta(p, q).is_above_fermi True See Also ======== diofant.functions.special.tensor_functions.KroneckerDelta.is_below_fermi diofant.functions.special.tensor_functions.KroneckerDelta.is_only_below_fermi diofant.functions.special.tensor_functions.KroneckerDelta.is_only_above_fermi """ if self.args[0]._assumptions.get("below_fermi"): return False if self.args[1]._assumptions.get("below_fermi"): return False return True @property def is_below_fermi(self): """ True if Delta can be non-zero below fermi Examples ======== >>> a = Symbol('a', above_fermi=True) >>> i = Symbol('i', below_fermi=True) >>> p = Symbol('p') >>> q = Symbol('q') >>> KroneckerDelta(p, a).is_below_fermi False >>> KroneckerDelta(p, i).is_below_fermi True >>> KroneckerDelta(p, q).is_below_fermi True See Also ======== diofant.functions.special.tensor_functions.KroneckerDelta.is_above_fermi diofant.functions.special.tensor_functions.KroneckerDelta.is_only_above_fermi diofant.functions.special.tensor_functions.KroneckerDelta.is_only_below_fermi """ if self.args[0]._assumptions.get("above_fermi"): return False if self.args[1]._assumptions.get("above_fermi"): return False return True @property def is_only_above_fermi(self): """ True if Delta is restricted to above fermi Examples ======== >>> a = Symbol('a', above_fermi=True) >>> i = Symbol('i', below_fermi=True) >>> p = Symbol('p') >>> q = Symbol('q') >>> KroneckerDelta(p, a).is_only_above_fermi True >>> KroneckerDelta(p, q).is_only_above_fermi False >>> KroneckerDelta(p, i).is_only_above_fermi False See Also ======== diofant.functions.special.tensor_functions.KroneckerDelta.is_above_fermi diofant.functions.special.tensor_functions.KroneckerDelta.is_below_fermi diofant.functions.special.tensor_functions.KroneckerDelta.is_only_below_fermi """ return (self.args[0]._assumptions.get("above_fermi") or self.args[1]._assumptions.get("above_fermi") or False) @property def is_only_below_fermi(self): """ True if Delta is restricted to below fermi Examples ======== >>> a = Symbol('a', above_fermi=True) >>> i = Symbol('i', below_fermi=True) >>> p = Symbol('p') >>> q = Symbol('q') >>> KroneckerDelta(p, i).is_only_below_fermi True >>> KroneckerDelta(p, q).is_only_below_fermi False >>> KroneckerDelta(p, a).is_only_below_fermi False See Also ======== diofant.functions.special.tensor_functions.KroneckerDelta.is_above_fermi diofant.functions.special.tensor_functions.KroneckerDelta.is_below_fermi diofant.functions.special.tensor_functions.KroneckerDelta.is_only_above_fermi """ return (self.args[0]._assumptions.get("below_fermi") or self.args[1]._assumptions.get("below_fermi") or False) @property def indices_contain_equal_information(self): """ Returns True if indices are either both above or below fermi. Examples ======== >>> a = Symbol('a', above_fermi=True) >>> i = Symbol('i', below_fermi=True) >>> p = Symbol('p') >>> q = Symbol('q') >>> KroneckerDelta(p, q).indices_contain_equal_information True >>> KroneckerDelta(p, q+1).indices_contain_equal_information True >>> KroneckerDelta(i, p).indices_contain_equal_information False """ if (self.args[0]._assumptions.get("below_fermi") and self.args[1]._assumptions.get("below_fermi")): return True if (self.args[0]._assumptions.get("above_fermi") and self.args[1]._assumptions.get("above_fermi")): return True # if both indices are general we are True, else false return self.is_below_fermi and self.is_above_fermi @property def preferred_index(self): """ Returns the index which is preferred to keep in the final expression. The preferred index is the index with more information regarding fermi level. If indices contain same information, 'a' is preferred before 'b'. Examples ======== >>> a = Symbol('a', above_fermi=True) >>> i = Symbol('i', below_fermi=True) >>> j = Symbol('j', below_fermi=True) >>> p = Symbol('p') >>> KroneckerDelta(p, i).preferred_index i >>> KroneckerDelta(p, a).preferred_index a >>> KroneckerDelta(i, j).preferred_index i See Also ======== diofant.functions.special.tensor_functions.KroneckerDelta.killable_index """ if self._get_preferred_index(): return self.args[1] else: return self.args[0] @property def killable_index(self): """ Returns the index which is preferred to substitute in the final expression. The index to substitute is the index with less information regarding fermi level. If indices contain same information, 'a' is preferred before 'b'. Examples ======== >>> a = Symbol('a', above_fermi=True) >>> i = Symbol('i', below_fermi=True) >>> j = Symbol('j', below_fermi=True) >>> p = Symbol('p') >>> KroneckerDelta(p, i).killable_index p >>> KroneckerDelta(p, a).killable_index p >>> KroneckerDelta(i, j).killable_index j See Also ======== diofant.functions.special.tensor_functions.KroneckerDelta.preferred_index """ if self._get_preferred_index(): return self.args[0] else: return self.args[1] def _get_preferred_index(self): """ Returns the index which is preferred to keep in the final expression. The preferred index is the index with more information regarding fermi level. If indices contain same information, index 0 is returned. """ if not self.is_above_fermi: if self.args[0]._assumptions.get("below_fermi"): return 0 else: return 1 elif not self.is_below_fermi: if self.args[0]._assumptions.get("above_fermi"): return 0 else: return 1 else: return 0 @staticmethod def _latex_no_arg(printer): return r'\delta'