282 lines
8.3 KiB
Python
282 lines
8.3 KiB
Python
from typing import Type
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from sympy.vector.basisdependent import (BasisDependent, BasisDependentAdd,
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BasisDependentMul, BasisDependentZero)
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from sympy.core import S, Pow
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from sympy.core.expr import AtomicExpr
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from sympy import ImmutableMatrix as Matrix
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import sympy.vector
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class Dyadic(BasisDependent):
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"""
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Super class for all Dyadic-classes.
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References
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==========
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.. [1] https://en.wikipedia.org/wiki/Dyadic_tensor
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.. [2] Kane, T., Levinson, D. Dynamics Theory and Applications. 1985
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McGraw-Hill
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"""
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_op_priority = 13.0
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_expr_type = None # type: Type[Dyadic]
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_mul_func = None # type: Type[Dyadic]
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_add_func = None # type: Type[Dyadic]
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_zero_func = None # type: Type[Dyadic]
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_base_func = None # type: Type[Dyadic]
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zero = None # type: DyadicZero
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@property
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def components(self):
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"""
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Returns the components of this dyadic in the form of a
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Python dictionary mapping BaseDyadic instances to the
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corresponding measure numbers.
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"""
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# The '_components' attribute is defined according to the
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# subclass of Dyadic the instance belongs to.
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return self._components
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def dot(self, other):
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"""
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Returns the dot product(also called inner product) of this
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Dyadic, with another Dyadic or Vector.
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If 'other' is a Dyadic, this returns a Dyadic. Else, it returns
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a Vector (unless an error is encountered).
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Parameters
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==========
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other : Dyadic/Vector
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The other Dyadic or Vector to take the inner product with
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Examples
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========
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>>> from sympy.vector import CoordSys3D
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>>> N = CoordSys3D('N')
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>>> D1 = N.i.outer(N.j)
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>>> D2 = N.j.outer(N.j)
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>>> D1.dot(D2)
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(N.i|N.j)
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>>> D1.dot(N.j)
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N.i
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"""
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Vector = sympy.vector.Vector
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if isinstance(other, BasisDependentZero):
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return Vector.zero
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elif isinstance(other, Vector):
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outvec = Vector.zero
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for k, v in self.components.items():
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vect_dot = k.args[1].dot(other)
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outvec += vect_dot * v * k.args[0]
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return outvec
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elif isinstance(other, Dyadic):
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outdyad = Dyadic.zero
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for k1, v1 in self.components.items():
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for k2, v2 in other.components.items():
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vect_dot = k1.args[1].dot(k2.args[0])
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outer_product = k1.args[0].outer(k2.args[1])
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outdyad += vect_dot * v1 * v2 * outer_product
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return outdyad
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else:
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raise TypeError("Inner product is not defined for " +
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str(type(other)) + " and Dyadics.")
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def __and__(self, other):
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return self.dot(other)
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__and__.__doc__ = dot.__doc__
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def cross(self, other):
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"""
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Returns the cross product between this Dyadic, and a Vector, as a
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Vector instance.
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Parameters
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==========
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other : Vector
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The Vector that we are crossing this Dyadic with
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Examples
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========
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>>> from sympy.vector import CoordSys3D
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>>> N = CoordSys3D('N')
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>>> d = N.i.outer(N.i)
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>>> d.cross(N.j)
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(N.i|N.k)
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"""
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Vector = sympy.vector.Vector
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if other == Vector.zero:
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return Dyadic.zero
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elif isinstance(other, Vector):
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outdyad = Dyadic.zero
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for k, v in self.components.items():
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cross_product = k.args[1].cross(other)
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outer = k.args[0].outer(cross_product)
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outdyad += v * outer
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return outdyad
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else:
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raise TypeError(str(type(other)) + " not supported for " +
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"cross with dyadics")
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def __xor__(self, other):
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return self.cross(other)
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__xor__.__doc__ = cross.__doc__
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def to_matrix(self, system, second_system=None):
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"""
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Returns the matrix form of the dyadic with respect to one or two
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coordinate systems.
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Parameters
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==========
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system : CoordSys3D
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The coordinate system that the rows and columns of the matrix
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correspond to. If a second system is provided, this
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only corresponds to the rows of the matrix.
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second_system : CoordSys3D, optional, default=None
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The coordinate system that the columns of the matrix correspond
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to.
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Examples
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========
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>>> from sympy.vector import CoordSys3D
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>>> N = CoordSys3D('N')
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>>> v = N.i + 2*N.j
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>>> d = v.outer(N.i)
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>>> d.to_matrix(N)
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Matrix([
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[1, 0, 0],
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[2, 0, 0],
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[0, 0, 0]])
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>>> from sympy import Symbol
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>>> q = Symbol('q')
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>>> P = N.orient_new_axis('P', q, N.k)
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>>> d.to_matrix(N, P)
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Matrix([
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[ cos(q), -sin(q), 0],
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[2*cos(q), -2*sin(q), 0],
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[ 0, 0, 0]])
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"""
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if second_system is None:
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second_system = system
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return Matrix([i.dot(self).dot(j) for i in system for j in
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second_system]).reshape(3, 3)
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def _div_helper(one, other):
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""" Helper for division involving dyadics """
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if isinstance(one, Dyadic) and isinstance(other, Dyadic):
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raise TypeError("Cannot divide two dyadics")
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elif isinstance(one, Dyadic):
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return DyadicMul(one, Pow(other, S.NegativeOne))
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else:
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raise TypeError("Cannot divide by a dyadic")
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class BaseDyadic(Dyadic, AtomicExpr):
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"""
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Class to denote a base dyadic tensor component.
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"""
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def __new__(cls, vector1, vector2):
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Vector = sympy.vector.Vector
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BaseVector = sympy.vector.BaseVector
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VectorZero = sympy.vector.VectorZero
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# Verify arguments
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if not isinstance(vector1, (BaseVector, VectorZero)) or \
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not isinstance(vector2, (BaseVector, VectorZero)):
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raise TypeError("BaseDyadic cannot be composed of non-base " +
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"vectors")
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# Handle special case of zero vector
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elif vector1 == Vector.zero or vector2 == Vector.zero:
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return Dyadic.zero
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# Initialize instance
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obj = super().__new__(cls, vector1, vector2)
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obj._base_instance = obj
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obj._measure_number = 1
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obj._components = {obj: S.One}
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obj._sys = vector1._sys
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obj._pretty_form = ('(' + vector1._pretty_form + '|' +
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vector2._pretty_form + ')')
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obj._latex_form = ('(' + vector1._latex_form + "{|}" +
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vector2._latex_form + ')')
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return obj
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def _sympystr(self, printer):
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return "({}|{})".format(
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printer._print(self.args[0]), printer._print(self.args[1]))
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class DyadicMul(BasisDependentMul, Dyadic):
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""" Products of scalars and BaseDyadics """
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def __new__(cls, *args, **options):
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obj = BasisDependentMul.__new__(cls, *args, **options)
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return obj
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@property
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def base_dyadic(self):
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""" The BaseDyadic involved in the product. """
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return self._base_instance
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@property
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def measure_number(self):
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""" The scalar expression involved in the definition of
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this DyadicMul.
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"""
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return self._measure_number
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class DyadicAdd(BasisDependentAdd, Dyadic):
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""" Class to hold dyadic sums """
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def __new__(cls, *args, **options):
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obj = BasisDependentAdd.__new__(cls, *args, **options)
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return obj
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def _sympystr(self, printer):
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items = list(self.components.items())
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items.sort(key=lambda x: x[0].__str__())
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return " + ".join(printer._print(k * v) for k, v in items)
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class DyadicZero(BasisDependentZero, Dyadic):
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"""
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Class to denote a zero dyadic
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"""
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_op_priority = 13.1
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_pretty_form = '(0|0)'
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_latex_form = r'(\mathbf{\hat{0}}|\mathbf{\hat{0}})'
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def __new__(cls):
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obj = BasisDependentZero.__new__(cls)
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return obj
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Dyadic._expr_type = Dyadic
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Dyadic._mul_func = DyadicMul
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Dyadic._add_func = DyadicAdd
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Dyadic._zero_func = DyadicZero
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Dyadic._base_func = BaseDyadic
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Dyadic.zero = DyadicZero()
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