diff --git a/tfields/bases/udo/christoffelSymbols.py b/tfields/bases/udo/christoffelSymbols.py new file mode 100644 index 0000000000000000000000000000000000000000..66457d8d1920270c0b2b443a010d98b1947fa895 --- /dev/null +++ b/tfields/bases/udo/christoffelSymbols.py @@ -0,0 +1,24 @@ +class ChristoffelSymbol2ndKind(object): + """ + Calculate Christoffel Gamma_ij^k symbols of metric g + """ + + def __init__(self, g, x): + self.g = g + self.x = x + + def ddu(self, i, j, k): + g = self.g + x = self.x + chr = 0 + for m in [0, 1, 2, 3]: + chr += ( + g.uu(k, m) + / 2 + * ( + g.dd(m, i).diff(x[j]) + + g.dd(m, j).diff(x[i]) + - g.dd(i, j).diff(x[m]) + ) + ) + return chr.simplify() diff --git a/tfields/bases/udo/coordinate_system_implementation_generator.py b/tfields/bases/udo/coordinate_system_implementation_generator.py new file mode 100644 index 0000000000000000000000000000000000000000..a891ae47c268220e0f5dee4bf522c87419732cd5 --- /dev/null +++ b/tfields/bases/udo/coordinate_system_implementation_generator.py @@ -0,0 +1,648 @@ +import re + +import numpy as np +import sympy as sym + +from collections import OrderedDict +from metric import Metric + + +class JavaCoordinateSystemCreator(object): + + # TODOs: + # - limits for variables + # - document + # - to java code routine needs to be enhanced to handle all possible cases + + def __init__( + self, java_class_name, Metric, axes, java_name=None, extra_parameters=None + ): + self.java_class_name = java_class_name + self.g = Metric + self.code = "" + + if java_name == "" or java_name is None: + index = java_class_name.index("Coordinates") + if index == -1: + raise Exception("Cannot guess java_name, please supply explicitly") + self.java_name = java_class_name[:index] + else: + self.java_name = java_name + + if axes is None: + raise Exception("Cannot create default axes!") + else: + self.axes = axes + + # needs to be a dictionary with type info + self.extra_parameters = extra_parameters + + self.unable_to_invert = False + self.inv_transformation = None + + def __str__(self): + if self.code == "": + self.__build() + return self.code + + def __build(self): + self.code = "" + self.__build_header() + self.__build_constructor() + self.__build_dimension() + self.__build_isOrthogonal() + self.__build_to_base_point() + self.__build_from_base_transformation() + self.__build_to_base_transformation_vector() + self.__build_from_base_transformation_vector() + self.__build_metric_coefficient() + self.__build_metric_tensor() + # self.__build_jacobian_determinant() + # self.__build_christoffel_symbols_2nd_kind() + # self.__build_christoffel_symbols_1st_kind() + # self.__build_riemann_tensor() + # self.__build_ricci_tensor() + # self.__build_ricci_scalar() + + self.code += "}\n" + + def __build_header(self): + """ + Builds the header of the Java implementation of the eu.hoefel.coordinates.CoordinateSystem interface + and attaches it to the variable representing the Java code. + """ + self.code += "import java.util.NavigableSet;\n" + if self.extra_parameters is not None: + self.code += "import java.util.Objects;\n" + + self.code += "\n" + self.code += "import eu.hoefel.coordinates.axes.Axes;\n" + self.code += "import eu.hoefel.coordinates.axes.Axis;\n" + if self.extra_parameters is not None: + self.code += "import eu.hoefel.coordinates.CoordinateSystems;\n" + + self.code += "\n" + self.code += "@CoordinateSystemSymbols({\"\"}) // TODO: add symbols representing the coordinate system, e.g. 'cartesian' and 'cart' for the cartesian coordinate system\n" + self.code += ( + "public final record " + self.java_class_name + "(NavigableSet<Axes> axes" + ) + if self.extra_parameters is not None: + for name, type in self.extra_parameters.items(): + self.code += ", " + type + " " + name + + self.code += ") implements CoordinateSystem {\n\n" + + self.code += " public static final NavigableSet<Axis> DEFAULT_AXES = Axes.of(" + + axes = [] + for dimension, unit in self.axes.items(): + axes.append("new Axis(" + str(dimension) + ", ") + if "." in unit: # probably smth like SiBaseUnit.METER + axes[len(axes) - 1] += unit + else: + axes[len(axes) - 1] += 'Unit.of("' + unit + '")' + axes[len(axes) - 1] += ")" + self.code += ", ".join(axes) + ");\n\n" + + if self.extra_parameters is None: + self.code += ( + ' private static final Consumer<Object[]> ARG_CHECK = args -> Axes.DEFAULT_ARG_CHECK.accept("' + + self.java_name + + '", args);\n\n' + ) + + def __build_constructor(self): + if self.extra_parameters is None: + self.code += " public " + self.java_class_name + "(Object... args) {\n" + self.code += " this(Axes.fromArgs(DEFAULT_AXES, ARG_CHECK, args));\n" + self.code += " }\n\n" + else: + self.code += " public " + self.java_class_name + " {\n" + self.code += ' Objects.requireNonNull(axes, "Axes may not be null. "\n' + self.code += ( + ' + "Use the DEFAULT_AXES or the constructor that just requires ' + + str([*self.extra_parameters.keys()]) + + ' to get a reasonable default.");\n' + ) + # TODO add limits for parameters + self.code += " }\n\n" + + self.code += " public " + self.java_class_name + "(Object... args) {\n" + self.code += " this(Axes.fromArgs(DEFAULT_AXES, args)" + + doubleIndex = 0 + for name, type in self.extra_parameters.items(): + if type == "double": + self.code += ( + ", CoordinateSystems.doubleFromArgs(" + + str(doubleIndex) + + ", args)" + ) + doubleIndex += 1 + else: + raise Exception( + "Don't know how to handle " + + type + + " for variable '" + + name + + "'!" + ) + self.code += ");\n" + self.code += " }\n\n" + + self.code += " public " + self.java_class_name + "(" + + parms = "" + for name, type in self.extra_parameters.items(): + parms += ", " + type + " " + name + + self.code += parms[2:] + self.code += ") {\n" + self.code += " this(DEFAULT_AXES, " + + parms = "" + for name, type in self.extra_parameters.items(): + parms += ", " + name + + self.code += parms[2:] + self.code += ");\n" + self.code += " }\n\n" + + def __build_to_base_point(self): + self.code += " @Override\n" + self.code += " public double[] toBasePoint(double[] position) {\n" + if self.g.to_base_point is None: + self.code += ' throw new UnsupportedOperationException("Transformation not implemented!")' + else: + self.code += ( + " double[] pointInBase = new double[" + str(self.g.dim()) + "];\n" + ) + for i in range(self.g.dim()): + self.code += ( + " pointInBase[" + + str(i) + + "] = " + + self.symbolicToJava(self.g.to_base_point[i], java_name="position") + + ";\n" + ) + self.code += " return pointInBase;\n" + self.code += " }\n" + self.code += "\n" + + def __build_from_base_transformation(self): + self.code += " @Override\n" + self.code += " public double[] fromBasePoint(double[] position) {\n" + + if not self.unable_to_invert and self.inv_transformation is None: + ( + self.unable_to_invert, + self.inv_transformation, + self.inv_symbols, + ) = self.__invert_forward_transformation() + + if self.g.to_base_point is None or self.unable_to_invert: + self.code += ' throw new UnsupportedOperationException("Transformation not implemented!")\n' + else: + self.code += ( + " double[] pointInCurrentSystem = new double[" + + str(self.g.dim()) + + "];\n" + ) + for i in range(self.g.dim()): + self.code += ( + " pointInCurrentSystem[" + + str(i) + + "] = " + + self.symbolicToJava( + self.inv_transformation[i], + java_name="position", + symbols=self.inv_symbols, + ) + + ";\n" + ) + self.code += " return pointInCurrentSystem;\n" + self.code += " }\n" + self.code += "\n" + + def __invert_forward_transformation(self): + """ + Inverts the given to_base_point transformation, i.e. it finds the transformation for a point given + in base coordinates to the current coordinate system. + + Returns + ------- + unable_to_invert + True if there was any failure in inverting + inv_results + The resulting inverse transformation or None if not successful + inv_symbols + The symbols corresponding to the inverse transformation + """ + unable_to_invert = False + try: + inv_symbols = [i.as_dummy() for i in self.g.symbols] + inv_results = sym.solve( + [t[0] - t[1] for t in zip(inv_symbols, self.g.to_base_point)], + list(self.g.symbols), + dict=True, + )[0] + inv_results = [ + sym.trigsimp(inv_results[s].simplify()) for s in self.g.symbols + ] + except Exception as e: + unable_to_invert = True + inv_results = None + inv_symbols = None + print("inversion failed: " + str(e)) + return unable_to_invert, inv_results, inv_symbols + + def __build_to_base_transformation_vector(self): + self.code += " @Override\n" + self.code += ( + " public double[] toBaseVector(double[] position, double[] vector) {\n" + ) + if self.g.to_base_point is None: + self.code += ' throw new UnsupportedOperationException("Transformation not implemented!")' + else: + self.code += " double[] vectorInBaseSys = new double[vector.length];\n" + + vec = [] + for i in range(self.g.dim()): + for j in range(self.g.dim()): + expr = self.g.to_base_point[j].diff(self.g.symbols[i]).simplify() + if expr == 0: + continue + + if expr == 1: + expr_as_string = "vector[" + str(j) + "]" + else: + expr_as_string = ( + self.symbolicToJava(expr) + "*vector[" + str(j) + "]" + ) + + if len(vec) <= i: + vec.append(expr_as_string) + else: + vec[i] += "+" + expr_as_string + if vec[i] != 0: + self.code += " vectorInBaseSys[" + str(i) + "] = " + vec[i] + ";\n" + self.code += " return vectorInBaseSys;\n" + self.code += " }\n" + self.code += "\n" + + def __build_from_base_transformation_vector(self): + self.code += " @Override\n" + self.code += ( + " public double[] fromBaseVector(double[] position, double[] vector) {\n" + ) + + if not self.unable_to_invert and self.inv_transformation is None: + ( + self.unable_to_invert, + self.inv_transformation, + self.inv_symbols, + ) = self.__invert_forward_transformation() + + if self.g.to_base_point is None or self.unable_to_invert: + self.code += ' throw new UnsupportedOperationException("Transformation not implemented!")' + else: + self.code += " double[] vectorInCurrentSys = new double[vector.length];\n" + + vec = [] + for i in range(self.g.dim()): + for j in range(self.g.dim()): + expr = ( + self.inv_transformation[j].diff(self.inv_symbols[i]).simplify() + ) + if expr == 0: + continue + + if expr == 1: + expr_as_string = "vector[" + str(j) + "]" + else: + expr_as_string = ( + self.symbolicToJava(expr, symbols=self.inv_symbols) + + "*vector[" + + str(j) + + "]" + ) + + if len(vec) <= i: + vec.append(expr_as_string) + else: + vec[i] += "+" + expr_as_string + if vec[i] != 0: + self.code += ( + " vectorInCurrentSys[" + str(i) + "] = " + vec[i] + ";\n" + ) + self.code += " return vectorInCurrentSys;\n" + self.code += " }\n" + self.code += "\n" + + def __build_metric_tensor(self): + self.code += " @Override\n" + self.code += " public double[][] metricTensor(double[] position, TensorTransformation behavior) {\n" + self.code += " int dim = " + str(self.g.dim()) + ";\n" + self.code += " double[][] g = new double[dim][dim];\n" + self.code += "\n" + self.code += " // Note that we skip all elements that are zero anyways\n" + + # check whether the same elements in covariant and covariant entries are nonnull + sameElementsAreNonNull = True + for i in range(self.g.dim()): + for j in range(self.g.dim()): + if (self.g.gdd[i, j] != 0 and self.g.guu[i, j] == 0) or ( + self.g.gdd[i, j] == 0 and self.g.guu[i, j] != 0 + ): + sameElementsAreNonNull = False + else: + # Continue if the inner loop wasn't broken. + continue + # Inner loop was broken, break the outer. + break + + if self.g.gdd != self.g.guu and not sameElementsAreNonNull: + self.code += " return switch (behavior) {\n" + self.code += " case COVARIANT -> {\n" + for i in range(self.g.dim()): + for j in range(self.g.dim()): + if self.g.gdd[i, j] == 0: + continue + self.code += ( + " g[" + + str(i) + + "][" + + str(j) + + "] = metricCoefficient(position, behavior, " + + str(i) + + ", " + + str(j) + + ");\n" + ) + self.code += " return g;\n" + self.code += " }\n" + + self.code += " case CONTRAVARIANT -> {\n" + for i in range(self.g.dim()): + for j in range(self.g.dim()): + if self.g.guu[i, j] == 0: + continue + self.code += ( + " g[" + + str(i) + + "][" + + str(j) + + "] = metricCoefficient(position, behavior, " + + str(i) + + ", " + + str(j) + + ");\n" + ) + self.code += " return g;\n" + self.code += " }\n" + self.code += " };\n" + self.code += " }\n" + else: + for i in range(self.g.dim()): + for j in range(self.g.dim()): + if self.g.gdd[i, j] == 0: + continue + self.code += ( + " g[" + + str(i) + + "][" + + str(j) + + "] = metricCoefficient(position, behavior, " + + str(i) + + ", " + + str(j) + + ");\n" + ) + self.code += " return g;\n" + self.code += " }\n" + self.code += "\n" + + def __build_metric_coefficient(self): + self.code += " @Override\n" + self.code += " public double metricCoefficient(double[] position, TensorTransformation behavior, int i, int j) {\n" + + self.code += " if (i < 0 || j < 0) {\n" + self.code += ' throw new IllegalArgumentException(("Metric coefficient not available for i=%d, j=%d "\n' + self.code += ( + ' + "(too low dimension, only ' + + str(self.g.dim()) + + " dimensions " + + str([*range(self.g.dim())]).replace(" ", "") + + " are supported for " + + self.java_name.lower() + + ' coordinates)")\n' + ) + self.code += " .formatted(i, j));\n" + self.code += ( + " } else if (i > " + + str(self.g.dim() - 1) + + " || j > " + + str(self.g.dim() - 1) + + ") {\n" + ) + self.code += ' throw new IllegalArgumentException(("Metric coefficient not available for i=%d, j=%d "\n' + self.code += ( + ' + "(too high dimension, only ' + + str(self.g.dim()) + + " dimensions " + + str([*range(self.g.dim())]).replace(" ", "") + + " are supported for " + + self.java_name.lower() + + ' coordinates)")\n' + ) + self.code += " .formatted(i, j));\n" + self.code += " }\n\n" + + if self.g.gdd != self.g.guu: + self.code += " return switch (behavior) {\n" + self.code += " case COVARIANT -> {\n" + for i in range(self.g.dim()): + for j in range(self.g.dim()): + if self.g.gdd[i, j] == 0: + continue + self.code += ( + " if (i == " + + str(i) + + " && j == " + + str(j) + + ") return " + + self.symbolicToJava(self.g.gdd[i, j]) + + ";\n" + ) + self.code += " return 0;\n" + self.code += " }\n" + + self.code += " case CONTRAVARIANT -> {\n" + for i in range(self.g.dim()): + for j in range(self.g.dim()): + if self.g.guu[i, j] == 0: + continue + self.code += ( + " if (i == " + + str(i) + + " && j == " + + str(j) + + ") return " + + self.symbolicToJava(self.g.guu[i, j]) + + ";\n" + ) + self.code += " return 0;\n" + self.code += " }\n" + self.code += " };\n" + self.code += " }\n" + else: + for i in range(self.g.dim()): + for j in range(self.g.dim()): + if self.g.gdd[i, j] == 0: + continue + self.code += ( + " if (i == " + + str(i) + + " && j == " + + str(j) + + ") return " + + self.symbolicToJava(self.g.gdd[i, j]) + + ";\n" + ) + self.code += " return 0;\n" + self.code += " }\n" + self.code += "\n" + + def __build_dimension(self): + self.code += " @Override\n" + self.code += " public int dimension() {\n" + self.code += " return " + str(self.g.dim()) + ";\n" + self.code += " }\n" + self.code += "\n" + + def __build_isOrthogonal(self): + self.code += " @Override\n" + self.code += " public boolean isOrthogonal() {\n" + + isOrthogonal = True + for i in range(self.g.dim()): + for j in range(self.g.dim()): + if i == j: + continue + if self.g.gdd[i, j] != 0: + isOrthogonal = False + break + else: + # Continue if the inner loop wasn't broken. + continue + # Inner loop was broken, break the outer. + break + + self.code += " return " + str(isOrthogonal).lower() + ";\n" + self.code += " }\n" + self.code += "\n" + + def symbolicToJava(self, expression, java_name="position", symbols=None): + if symbols is None: + symbols = self.g.symbols + + expression = str(expression) + mapping = {} + for i in range(len(symbols)): + mapping[str(symbols[i])] = java_name + "[" + str(i) + "]" + + # add math symbols + mapping["atan"] = "Math.atan" + mapping["sin"] = "Math.sin" + mapping["cos"] = "Math.cos" + mapping["tan"] = "Math.tan" + mapping["abs"] = "Math.abs" + mapping["Abs"] = "Math.abs" + mapping["sqrt"] = "Math.sqrt" + ## TODO and many more + + for key in OrderedDict( + sorted(mapping.items(), reverse=True, key=lambda t: len(str(t[0]))) + ): + if key == "tan": + # we need to be a bit careful, as it may find Math.atan... + expression = re.sub("(?<!Math.a)tan", mapping[key], expression) + else: + expression = expression.replace(key, mapping[key]) + + # handle exponents + while expression.find("**") != -1: + index = expression.find("**") + numBrackets = 0 + for i in range(index - 1, -1, -1): + if expression[i] == ")": + numBrackets += 1 + if expression[i] == "(": + numBrackets -= 1 + + if numBrackets <= 0 and ( + np.any( + np.array([" ", "*", "+", "-", "/", "%", "("]) == expression[i] + ) + or i == 0 + ): + indexForBracketClose = len(expression) + bracketForExponent = False + for j, c in enumerate(expression[index + len("**") :]): + if j == 0 and c == "(": + bracketForExponent = True + continue + + if not c.isdigit() and c != ".": + indexForBracketClose = ( + len(expression[: index + len("**")]) + j + ) + break + if bracketForExponent: + expression = ( + expression[: index + len("**")] + + expression[index + len("**") + len("(") :] + ) + else: + expression = ( + expression[:indexForBracketClose] + + ")" + + expression[indexForBracketClose:] + ) + expression = ( + expression[:index] + ", " + expression[index + len("**") :] + ) + + if i != 0: + i += 1 + expression = expression[:i] + "Math.pow(" + expression[i:] + break + + return expression + + +if __name__ == "__main__": + # TODO vector trafo + # TODO parameters + + r = sym.symbols("r", real=True, positive=True) + theta = sym.symbols("theta", real=True) + syms = sym.symbols("r theta z", real=True) + + axes = {0: "m", 1: "rad", 2: "rad"} + extra_parms = {"a": "double"} + + m = Metric.fromTransformation( + syms, + to_base_point=[syms[0] * sym.cos(syms[1]), syms[0] * sym.sin(syms[1]), syms[2]], + ) + + code = JavaCoordinateSystemCreator( + "CylindricalCoordinates", m, axes, extra_parameters=extra_parms + ) + print(code) + print("----") + + m = Metric.fromTransformation(syms, to_base_point=[syms[0], syms[1], syms[2]]) + print(JavaCoordinateSystemCreator("CartesianCoordinates", m, axes)) + print("----") diff --git a/tfields/bases/udo/metric.py b/tfields/bases/udo/metric.py new file mode 100644 index 0000000000000000000000000000000000000000..2d83f96dfa464aff7b2ba468e6372c04d478259d --- /dev/null +++ b/tfields/bases/udo/metric.py @@ -0,0 +1,139 @@ +import numpy as np +import sympy as sym + +from sympy.matrices import dotprodsimp + + +class Metric(object): + """ + Represents a metric. + + Parameters + ---------- + symbols : numpy array or array_like of sympy.core.symbol.Symbol + The symbols representing the different dimensions, e.g. `sympy.symbols('r theta z', real=True)`. + Note that it is good to provide as much information on the symbols as possible, e.g. in the + aforementioned example that 'r' is positive + m : Matrix + The (covariant) metric represented by a Matrix + to_base_point : numpy array or array_like, optional + The transformation equation to get from the current coordinate system into the base coordinate system. + Should be expressed using the `symbols` and sympy functions for, e.g., the cosine. + Note that not providing this may limit the scope on subsequent operations with this metric. + + Returns + ------- + Metric + the (analytical) metric as derived from the transformation for a point into the base coordinate system + """ + + def __init__(self, symbols, m: sym.Matrix, to_base_point=None): + self.gdd = m # covariant + print("Inverting metric...") + + # necessary due to a regression in sympy + with dotprodsimp(False): + self.guu = m.inv() # contravariant + + print("Metric successfully inverted") + self.symbols = symbols + self.to_base_point = to_base_point + + @staticmethod + def fromTransformation(symbols, to_base_point): + """ + Creates a new metric from the given to_base_point transformation. + + Parameters + ---------- + symbols : numpy array or array_like of sympy.core.symbol.Symbol + The symbols representing the different dimensions, e.g. `sympy.symbols('r theta z', real=True)` + to_base_point : numpy array or array_like + The transformation equation to get from the current coordinate system into the base coordinate system. + Should be expressed using the `symbols` and sympy functions for, e.g., the cosine. + + Returns + ------- + Metric + the (analytical) metric as derived from the transformation for a point into the base coordinate system + """ + if to_base_point is None: + raise Exception("You need to specify the point-to-base-transformation!") + + print("Reconstructing metric from point-to-base-transformation...") + dim = len(to_base_point) + partial_derivatives = [[0 for i in range(dim)] for i in range(dim)] + + for i in range(dim): + for j in range(dim): + partial_derivatives[i][j] = sym.diff(to_base_point[j], symbols[i]) + + g = [[0 for i in range(dim)] for i in range(dim)] + for i in range(dim): + for j in range(dim): + for k in range(dim): + g[i][j] += partial_derivatives[i][k] * partial_derivatives[j][k] + g[j][i] = g[i][j] = g[i][j].simplify() + + print("Metric successfully reconstructed!") + + return Metric(symbols, sym.Matrix(g), to_base_point) + + def __str__(self): + """ + Creates a representative string of the (covariant) metric. + + Returns + ------- + str + the analytic metric wth covariant components represented by a string + """ + return "g_dd = " + str(self.gdd) + + def dd(self, i, j): + """ + Gets the covariant metric coefficient at the specified position. + + Parameters + ---------- + i : int + The row index in the metric + j : int + The column index in the metric + + Returns + ------- + sympy expression + the (analytical) covariant metric coefficient g_ij + """ + return self.gdd[i, j] + + def uu(self, i, j): + """ + Gets the contravariant metric coefficient at the specified position. + + Parameters + ---------- + i : int + The row index in the metric + j : int + The column index in the metric + + Returns + ------- + sympy expression + the (analytical) contravariant metric coefficient g^ij + """ + return self.guu[i, j] + + def dim(self) -> int: + """ + Gets the dimensionality of the metric. + For example, a metric with size 3x3 will be of dimensionality 3. + + Returns + ------- + int + the dimension of the metric + """ + return int(np.sqrt(len(self.gdd))) diff --git a/tfields/bases/udo/sympyGradient.py b/tfields/bases/udo/sympyGradient.py new file mode 100644 index 0000000000000000000000000000000000000000..037d1f10001c0065862db918a3a80b3e3425a2c0 --- /dev/null +++ b/tfields/bases/udo/sympyGradient.py @@ -0,0 +1,19 @@ +import sys +from sympy import * +import numpy as np +from collections import OrderedDict +import re + +from sympy.matrices import dotprodsimp +from sympyTensors import Metric + +r = symbols("r", real=True, positive=True) +theta = symbols("theta", real=True) +z = symbols("z", real=True) + +from sympy.vector import gradient +from sympy.vector import CoordSys3D + +c = CoordSys3D("c", transformation="cylindrical", variable_names=("r", "theta", "z")) + +gradient(c.r) diff --git a/tfields/bases/udo/sympyTensors.py b/tfields/bases/udo/sympyTensors.py new file mode 100644 index 0000000000000000000000000000000000000000..c48a3bd6421198c1ee3e08e71858ef284a0dc96d --- /dev/null +++ b/tfields/bases/udo/sympyTensors.py @@ -0,0 +1,849 @@ +import sys +from sympy import * +import numpy as np +from collections import OrderedDict +import re + +from sympy.matrices import dotprodsimp + + +class Metric(object): + """ + Turn matrix into upper and lower metric + + m: covariant matrix + """ + + def __init__(self, symbols, m: Matrix, to_base=None): + self.gdd = m # covariant + print("Inverting metric...") + print(type(symbols)) + print(type(symbols[0])) + # necessary due to a regression in sympy + with dotprodsimp(False): + self.guu = m.inv() # contravariant + + print("Metric successfully inverted") + self.symbols = symbols + self.to_base = to_base + + """ + Creates a new metric from the given to_base_point transformation. + """ + + @staticmethod + def fromTransformation(symbols, to_base_point): + if to_base_point is None: + raise Exception( + "You need to specify either the to-base-transformation or the from-base-transformation" + ) + + print("Reconstructing metric from to-base-transformation") + dim = len(to_base_point) + partial_derivatives = [[0 for i in range(dim)] for i in range(dim)] + + for i in range(dim): + for j in range(dim): + partial_derivatives[i][j] = diff(to_base_point[j], symbols[i]) + + g = [[0 for i in range(dim)] for i in range(dim)] + for i in range(dim): + for j in range(dim): + for k in range(dim): + g[i][j] += partial_derivatives[i][k] * partial_derivatives[j][k] + g[j][i] = g[i][j] = g[i][j].simplify() + print("Metric successfully reconstructed") + return Metric(symbols, Matrix(g), to_base_point) + + def __str__(self): + return "g_dd = \n" + str(self.gdd) + + def dd(self, i, j): + return self.gdd[i, j] + + def uu(self, i, j): + return self.guu[i, j] + + def dim(self): + return int(np.sqrt(len(self.gdd))) + + +class Gamma(object): + """ + Calculate Christoffel Gamma_ij^k symbols of metric g + """ + + def __init__(self, g: Metric, x): + self.g = g + self.x = x + + """ + + """ + + def ddu(self, m, i, j): + g = self.g + x = self.x + chr = 0 + for k in range(self.g.dim()): + chr += ( + g.uu(m, k) + / 2 + * ( + g.dd(k, i).diff(x[j]) + + g.dd(k, j).diff(x[i]) + - g.dd(i, j).diff(x[k]) + ) + ) + return chr.simplify() + # return chr + + +class Riemann(object): + """ + Calculate Riemann tensor R^mu_nu,rho,sigma + """ + + def __init__(self, g: Metric, G: Gamma, x): + self.g = g + self.G = G + self.x = x + + def uddd(self, mu, nu, rho, sigma): + G = self.G + x = self.x + R = G.ddu(mu, nu, sigma).diff(x[rho]) - G.ddu(mu, nu, rho).diff(x[sigma]) + for lam in range(self.g.dim()): + R += G.ddu(mu, rho, lam) * G.ddu(lam, nu, sigma) - G.ddu( + mu, sigma, lam + ) * G.ddu(lam, nu, rho) + return R.simplify() + + def dddd(self, mu, nu, rho, sigma): + g = self.g + R = 0 + for lam in range(self.g.dim()): + R += g.dd(mu, lam) * self.uddd(lam, nu, rho, sigma) + return R.simplify() + + +class Ricci(object): + """ + Calculate Ricci tensor from Riemann tensor + """ + + def __init__(self, R: Riemann): + self.R = R + + def dd(self, mu, nu): + R = self.R + Ric = 0 + for rho in range(self.R.g.dim()): + Ric += R.uddd(rho, mu, rho, nu) + return Ric.simplify() + + +class RicciScalar(object): + """ + Calculate Ricci scalar from Ricci tensor + """ + + def __init__(self, Ric, g): + self.Ric = Ric + self.g = g + + def value(self): + Ric = self.Ric + g = self.g + RS = 0 + for mu in range(self.g.dim()): + for nu in range(self.g.dim()): + RS += g.uu(mu, nu) * Ric.dd(mu, nu) + return RS.simplify() + + +class JavaCoordinateSystemCreator(object): + def __init__(self, java_class_name, Metric, java_name=None, extra_parameters=None): + self.java_class_name = java_class_name + self.g = Metric + self.code = "" + + if java_name == "" or java_name is None: + index = java_class_name.index("Coordinates") + if index == -1: + raise Exception("Cannot guess java_name, please supply explicitly") + self.java_name = java_class_name[:index] + else: + self.java_name = java_name + + # TODO handle extra parameters + + def __str__(self): + self.__build() + return self.code + + def __build(self): + self.__build_header() + self.__build_dimension() + self.__build_to_base_transformation() + self.__build_from_base_transformation() + self.__build_to_base_transformation_vector() + self.__build_from_base_transformation_vector() + self.__build_metric_coefficient() + self.__build_metric_tensor() + self.__build_jacobian_determinant() + self.__build_isOrthogonal() + self.__build_christoffel_symbols_2nd_kind() + self.__build_christoffel_symbols_1st_kind() + self.__build_riemann_tensor() + self.__build_ricci_tensor() + self.__build_ricci_scalar() + + self.code += "}\n" + + def __build_header(self): + self.code += "@CoordinateSystemSymbols({\"\"}) // TODO: add symbols representing the coordinate system, e.g. 'cartesian' and 'cart' for the cartesian coordinate system\n" + self.code += ( + "public final record " + + self.java_class_name + + "(NavigableSet<Axes> axes) implements CoordinateSystem {\n\n" + ) + + def __build_to_base_transformation(self): + self.code += " @Override\n" + self.code += " public Function<double[],double[]> toBaseTransformation() {\n" + if self.g.to_base is None: + self.code += ( + ' throw new IllegalStateException("Transformation not implemented!")' + ) + else: + self.code += " return pos -> {\n" + self.code += ( + " double[] pointInBase = new double[" + str(self.g.dim()) + "];\n" + ) + for i in range(self.g.dim()): + self.code += ( + " pointInBase[" + + str(i) + + "] = " + + self.symbolicToJava(self.g.to_base[i], java_name="pos") + + ";\n" + ) + self.code += " return pointInBase;\n" + self.code += " };\n" + self.code += " }\n" + self.code += "\n" + + def __build_from_base_transformation(self): + self.code += " @Override\n" + self.code += " public Function<double[],double[]> fromBaseTransformation() {\n" + + unable_to_invert = False + try: + inv_syms = [i.as_dummy() for i in self.g.symbols] + inv_results = solve( + [t[0] - t[1] for t in zip(inv_syms, self.g.to_base)], + list(self.g.symbols), + dict=True, + )[0] + inv_results = [inv_results[s] for s in self.g.symbols] + except: + unable_to_invert = True + + if self.g.to_base is None or unable_to_invert: + self.code += ( + ' throw new IllegalStateException("Transformation not implemented!")' + ) + else: + self.code += " return pos -> {\n" + self.code += ( + " double[] pointInCurrentSystem = new double[" + + str(self.g.dim()) + + "];\n" + ) + for i in range(self.g.dim()): + self.code += ( + " pointInCurrentSystem[" + + str(i) + + "] = " + + self.symbolicToJava( + trigsimp(inv_results[i].simplify()), + java_name="pos", + symbols=inv_syms, + ) + + ";\n" + ) + self.code += " return pointInCurrentSystem;\n" + self.code += " };\n" + self.code += " }\n" + self.code += "\n" + + def __build_to_base_transformation_vector(self): + self.code += " @Override\n" + self.code += ( + " public double[] toBaseVector(double[] position, double[] vector) {\n" + ) + if self.g.to_base is None: + self.code += ( + ' throw new IllegalStateException("Transformation not implemented!")' + ) + else: + self.code += " double[] vectorInBaseSys = new double[vector.length];\n" + + vec = [] + for i in range(self.g.dim()): + for j in range(self.g.dim()): + expr = self.g.to_base[j].diff(self.g.symbols[i]).simplify() + if expr == 0: + continue + + if expr == 1: + expr_as_string = "vector[" + str(j) + "]" + else: + expr_as_string = ( + self.symbolicToJava(expr) + "*vector[" + str(j) + "]" + ) + + if len(vec) <= i: + vec.append(expr_as_string) + else: + vec[i] += "+" + expr_as_string + if vec[i] != 0: + self.code += " vectorInBaseSys[" + str(i) + "] = " + vec[i] + ";\n" + self.code += " return vectorInBaseSys;\n" + self.code += " }\n" + self.code += "\n" + + def __build_from_base_transformation_vector(self): + self.code += " @Override\n" + self.code += ( + " public double[] fromBaseVector(double[] position, double[] vector) {\n" + ) + + unable_to_invert = False + try: + inv_syms = [i.as_dummy() for i in self.g.symbols] + inv_results = solve( + [t[0] - t[1] for t in zip(inv_syms, self.g.to_base)], + list(self.g.symbols), + dict=True, + )[0] + inv_results = [trigsimp(inv_results[s].simplify()) for s in self.g.symbols] + except: + unable_to_invert = True + + if self.g.to_base is None or unable_to_invert: + self.code += ( + ' throw new IllegalStateException("Transformation not implemented!")' + ) + else: + self.code += " double[] vectorInCurrentSys = new double[vector.length];\n" + + vec = [] + for i in range(self.g.dim()): + for j in range(self.g.dim()): + expr = inv_results[j].diff(inv_syms[i]).simplify() + if expr == 0: + continue + + if expr == 1: + expr_as_string = "vector[" + str(j) + "]" + else: + expr_as_string = ( + self.symbolicToJava(expr, symbols=inv_syms) + + "*vector[" + + str(j) + + "]" + ) + + if len(vec) <= i: + vec.append(expr_as_string) + else: + vec[i] += "+" + expr_as_string + if vec[i] != 0: + self.code += ( + " vectorInCurrentSys[" + str(i) + "] = " + vec[i] + ";\n" + ) + self.code += " return vectorInCurrentSys;\n" + self.code += " }\n" + self.code += "\n" + + def __build_metric_tensor(self): + self.code += " @Override\n" + self.code += " public double[][] metricTensor(double[] position, BasisChangeBehavior behavior) {\n" + self.code += " int dim = " + str(self.g.dim()) + ";\n" + self.code += " double[][] g = new double[dim][dim];\n" + self.code += "\n" + self.code += " // Note that we skip all elements that are zero anyways\n" + + # check whether the same elements in covariant and covariant entries are nonnull + sameElementsAreNonNull = True + for i in range(self.g.dim()): + for j in range(self.g.dim()): + if (self.g.gdd[i, j] != 0 and self.g.guu[i, j] == 0) or ( + self.g.gdd[i, j] == 0 and self.g.guu[i, j] != 0 + ): + sameElementsAreNonNull = False + else: + # Continue if the inner loop wasn't broken. + continue + # Inner loop was broken, break the outer. + break + + if self.g.gdd != self.g.guu and not sameElementsAreNonNull: + self.code += " return switch (behavior) {\n" + self.code += " case COVARIANT -> {\n" + for i in range(self.g.dim()): + for j in range(self.g.dim()): + if self.g.gdd[i, j] == 0: + continue + self.code += ( + " g[" + + str(i) + + "][" + + str(j) + + "] = metricCoefficient(position, behavior, " + + str(i) + + ", " + + str(j) + + ");\n" + ) + self.code += " return g;\n" + self.code += " }\n" + + self.code += " case CONTRAVARIANT -> {\n" + for i in range(self.g.dim()): + for j in range(self.g.dim()): + if self.g.guu[i, j] == 0: + continue + self.code += ( + " g[" + + str(i) + + "][" + + str(j) + + "] = metricCoefficient(position, behavior, " + + str(i) + + ", " + + str(j) + + ");\n" + ) + self.code += " return g;\n" + self.code += " }\n" + self.code += " };\n" + self.code += " }\n" + else: + for i in range(self.g.dim()): + for j in range(self.g.dim()): + if self.g.gdd[i, j] == 0: + continue + self.code += ( + " g[" + + str(i) + + "][" + + str(j) + + "] = metricCoefficient(position, behavior, " + + str(i) + + ", " + + str(j) + + ");\n" + ) + self.code += " return g;\n" + self.code += " }\n" + self.code += "\n" + + def __build_metric_coefficient(self): + self.code += " @Override\n" + self.code += " public double metricCoefficient(double[] position, BasisChangeBehavior behavior, int i, int j) {\n" + + self.code += " if (i < 0 || j < 0) {\n" + self.code += ' throw new IllegalArgumentException(("Metric coefficient not available for i=%d, j=%d "\n' + self.code += ( + ' + "(too low dimension, only ' + + str(self.g.dim()) + + " dimensions " + + str([*range(self.g.dim())]).replace(" ", "") + + " are supported for " + + self.java_name.lower() + + ' coordinates)")\n' + ) + self.code += " .formatted(i, j));\n" + self.code += ( + " } else if (i > " + + str(self.g.dim() - 1) + + " || j > " + + str(self.g.dim() - 1) + + ") {\n" + ) + self.code += ' throw new IllegalArgumentException(("Metric coefficient not available for i=%d, j=%d "\n' + self.code += ( + ' + "(too high dimension, only ' + + str(self.g.dim()) + + " dimensions " + + str([*range(self.g.dim())]).replace(" ", "") + + " are supported for " + + self.java_name.lower() + + ' coordinates)")\n' + ) + self.code += " .formatted(i, j));\n" + self.code += " }\n\n" + + if self.g.gdd != self.g.guu: + self.code += " return switch (behavior) {\n" + self.code += " case COVARIANT -> {\n" + for i in range(self.g.dim()): + for j in range(self.g.dim()): + if self.g.gdd[i, j] == 0: + continue + self.code += ( + " if (i == " + + str(i) + + " && j == " + + str(j) + + ") return " + + self.symbolicToJava(self.g.gdd[i, j]) + + ";\n" + ) + self.code += " return 0;\n" + self.code += " }\n" + + self.code += " case CONTRAVARIANT -> {\n" + for i in range(self.g.dim()): + for j in range(self.g.dim()): + if self.g.guu[i, j] == 0: + continue + self.code += ( + " if (i == " + + str(i) + + " && j == " + + str(j) + + ") return " + + self.symbolicToJava(self.g.guu[i, j]) + + ";\n" + ) + self.code += " return 0;\n" + self.code += " }\n" + self.code += " };\n" + self.code += " }\n" + else: + for i in range(self.g.dim()): + for j in range(self.g.dim()): + if self.g.gdd[i, j] == 0: + continue + self.code += ( + " if (i == " + + str(i) + + " && j == " + + str(j) + + ") return " + + self.symbolicToJava(self.g.gdd[i, j]) + + ";\n" + ) + self.code += " return 0;\n" + self.code += " }\n" + self.code += "\n" + + def __build_dimension(self): + self.code += " @Override\n" + self.code += " public int dimension() {\n" + self.code += " return " + str(self.g.dim()) + ";\n" + self.code += " }\n" + self.code += "\n" + + def __build_isOrthogonal(self): + self.code += " @Override\n" + self.code += " public boolean isOrthogonal() {\n" + + isOrthogonal = True + for i in range(self.g.dim()): + for j in range(self.g.dim()): + if i == j: + continue + if self.g.gdd[i, j] != 0: + isOrthogonal = False + break + else: + # Continue if the inner loop wasn't broken. + continue + # Inner loop was broken, break the outer. + break + + self.code += " return " + str(isOrthogonal).lower() + ";\n" + self.code += " }\n" + self.code += "\n" + + def __build_jacobian_determinant(self): + self.code += " @Override\n" + self.code += " public double jacobianDeterminant(double[] position) {\n" + + det = sqrt(self.g.gdd.det()).simplify() + + self.code += " return " + self.symbolicToJava(det) + ";\n" + self.code += " }\n" + self.code += "\n" + + def __build_christoffel_symbols_2nd_kind(self): + self.code += " @Override\n" + self.code += " public double christoffelSymbol2ndKind(double[] position, int m, int i, int j) {\n" + + gamma = Gamma(self.g, self.g.symbols) + + for m in range(self.g.dim()): + for i in range(self.g.dim()): + for j in range(self.g.dim()): + if gamma.ddu(m, i, j) == 0: + continue + self.code += ( + " if (m == " + + str(m) + + " && i == " + + str(i) + + " && j == " + + str(j) + + ") return " + + self.symbolicToJava(gamma.ddu(m, i, j)) + + ";\n" + ) + self.code += " return 0;\n" + self.code += " }\n" + self.code += "\n" + + def __build_christoffel_symbols_1st_kind(self): + self.code += " @Override\n" + self.code += " public double christoffelSymbol1stKind(double[] position, int i, int j, int k) {\n" + + gamma = Gamma(self.g, self.g.symbols) + + for i in range(self.g.dim()): + for j in range(self.g.dim()): + for k in range(self.g.dim()): + christoffelSymbol1stKind = 0 + for u in range(self.g.dim()): + christoffelSymbol1stKind += self.g.gdd[u, k] * gamma.ddu( + u, i, j + ) + christoffelSymbol1stKind = christoffelSymbol1stKind.simplify() + + if christoffelSymbol1stKind == 0: + continue + self.code += ( + " if (i == " + + str(i) + + " && j == " + + str(j) + + " && k == " + + str(k) + + ") return " + + self.symbolicToJava(christoffelSymbol1stKind) + + ";\n" + ) + self.code += " return 0;\n" + self.code += " }\n" + self.code += "\n" + + def __build_riemann_tensor(self): + self.code += " @Override\n" + self.code += " public double riemannTensor(double[] position, int i, int j, int k, int l) {\n" + + gamma = Gamma(self.g, self.g.symbols) + riem = Riemann(self.g, gamma, self.g.symbols) + + for i in range(self.g.dim()): + for j in range(self.g.dim()): + for k in range(self.g.dim()): + for l in range(self.g.dim()): + if riem.dddd(i, j, k, l) == 0: + continue + self.code += ( + " if (i == " + + str(i) + + " && j == " + + str(j) + + " && k == " + + str(k) + + " && l == " + + str(l) + + ") return " + + self.symbolicToJava(riem.dddd(i, j, k, l)) + + ";\n" + ) + self.code += " return 0;\n" + self.code += " }\n" + self.code += "\n" + + def __build_ricci_tensor(self): + self.code += " @Override\n" + self.code += " public double ricciTensor(double[] position, int i, int j) {\n" + + gamma = Gamma(self.g, self.g.symbols) + riem = Riemann(self.g, gamma, self.g.symbols) + ric = Ricci(riem) + + for i in range(self.g.dim()): + for j in range(self.g.dim()): + if ric.dd(i, j) == 0: + continue + self.code += ( + " if (i == " + + str(i) + + " && j == " + + str(j) + + ") return " + + self.symbolicToJava(ric.dd(i, j)) + + ";\n" + ) + self.code += " return 0;\n" + self.code += " }\n" + self.code += "\n" + + def __build_ricci_scalar(self): + self.code += " @Override\n" + self.code += " public double ricciScalar(double[] position) {\n" + + gamma = Gamma(self.g, self.g.symbols) + riem = Riemann(self.g, gamma, self.g.symbols) + ric = Ricci(riem) + ric_scalar = RicciScalar(ric, self.g) + + self.code += " return " + self.symbolicToJava(ric_scalar.value()) + ";\n" + self.code += " }\n" + + def symbolicToJava(self, expression, java_name="position", symbols=None): + if symbols is None: + symbols = self.g.symbols + + expression = str(expression) + mapping = {} + for i in range(len(symbols)): + mapping[str(symbols[i])] = java_name + "[" + str(i) + "]" + + # add math symbols + mapping["atan"] = "Math.atan" + mapping["sin"] = "Math.sin" + mapping["cos"] = "Math.cos" + mapping["tan"] = "Math.tan" + mapping["abs"] = "Math.abs" + mapping["Abs"] = "Math.abs" + mapping["sqrt"] = "Math.sqrt" + ## TODO and many more + + for key in OrderedDict( + sorted(mapping.items(), reverse=True, key=lambda t: len(str(t[0]))) + ): + if key == "tan": + # we need to be a bit careful, as it may find Math.atan... + expression = re.sub("(?<!Math.a)tan", mapping[key], expression) + else: + expression = expression.replace(key, mapping[key]) + + # handle exponents + while expression.find("**") != -1: + index = expression.find("**") + numBrackets = 0 + for i in range(index - 1, -1, -1): + if expression[i] == ")": + numBrackets += 1 + if expression[i] == "(": + numBrackets -= 1 + + if numBrackets <= 0 and ( + np.any( + np.array([" ", "*", "+", "-", "/", "%", "("]) == expression[i] + ) + or i == 0 + ): + indexForBracketClose = len(expression) + bracketForExponent = False + for j, c in enumerate(expression[index + len("**") :]): + if j == 0 and c == "(": + bracketForExponent = True + continue + + if not c.isdigit() and c != ".": + indexForBracketClose = ( + len(expression[: index + len("**")]) + j + ) + break + if bracketForExponent: + expression = ( + expression[: index + len("**")] + + expression[index + len("**") + len("(") :] + ) + else: + expression = ( + expression[:indexForBracketClose] + + ")" + + expression[indexForBracketClose:] + ) + expression = ( + expression[:index] + ", " + expression[index + len("**") :] + ) + + if i != 0: + i += 1 + expression = expression[:i] + "Math.pow(" + expression[i:] + break + + return expression + + +if __name__ == "__main__": + # TODO vector trafo + # TODO parameters + + r = symbols("r", real=True, positive=True) + theta = symbols("theta", real=True) + syms = symbols("r theta z", real=True) + + print(type(syms[0] * cos(syms[1]) / 2)) + m = Metric.fromTransformation( + syms, to_base_point=[syms[0] * cos(syms[1]), syms[0] * sin(syms[1]), syms[2]] + ) + + print(JavaCoordinateSystemCreator("CylindricalCoordinates", m)) + print("----") + + m = Metric.fromTransformation(syms, to_base_point=[syms[0], syms[1], syms[2]]) + print(JavaCoordinateSystemCreator("CartesianCoordinates", m)) + print("----") + + a = symbols("a", real=True, constant=True) + sigma = symbols("sigma", real=True) + tau = symbols("tau", real=True, positive=True) + theta = symbols("theta", real=True, positive=True) + syms = [sigma, tau, theta] + + to_base_point = [ + a * sinh(syms[1]) * cos(syms[2]) / (cosh(syms[1]) - cos(syms[0])), + a * sinh(syms[1]) * sin(syms[2]) / (cosh(syms[1]) - cos(syms[0])), + a * sign(syms[0]) * abs(sin(syms[0]) / (cosh(syms[1]) - cos(syms[0]))), + ] + + # m = Metric.fromTransformation(syms, to_base_point=to_base_point) + # print(JavaCoordinateSystemCreator("ToroidalCoordinates", m)) + # print("----") + + # jcsc = JavaCoordinateSystemCreator("CylindricalCoordinates", m) + # print(jcsc.symbolicToJava("r**2+z**2")) + + # syms = symbols('r theta z') + ## define the metric for 3d cylindrical coordinates + # m = [[0 for i in range(3)] for i in range(3)] + # m[0][0] = 1 + # m[1][1] = syms[0]**2 + # m[2][2] = 1 + # + # metric = Metric(Matrix(m)) + # + # + # gamma = Gamma(metric, syms) + # riem = Riemann(metric, gamma, syms) + # ric = Ricci(riem) + # ric_scalar = RicciScalar(ric,metric) + # + # codeprint_christoffels(gamma) + # print("") + # codeprint_riem(riem) + # print("") + # codeprint_ricci(ric) + # print("") + # print(ric_scalar.value()) diff --git a/tfields/bases/udo/tensorTransformation.py b/tfields/bases/udo/tensorTransformation.py new file mode 100644 index 0000000000000000000000000000000000000000..a42ee90e5d17a76e51bbf9d0a451dc72f7b4e2c2 --- /dev/null +++ b/tfields/bases/udo/tensorTransformation.py @@ -0,0 +1,68 @@ +import sys +from sympy import * +import numpy as np +from collections import OrderedDict +import re + +from sympy.matrices import dotprodsimp +from sympyTensors import Metric + +r = symbols("r", real=True, positive=True) +theta = symbols("theta", real=True) +z = symbols("z", real=True) + +g = [[1, 0, 0], [0, r ** 2, 0], [0, 0, 1]] +# g = [[1,0,0],[0,1,0],[0,0,1]] + +syms = [r, theta, z] + +metric = Metric(symbols, Matrix(g)) + +# sigma = [[1,2,3],[1.5,3,4.5],[2,4,6]] +sigma = [[2, 3, 4], [3, 4.5, 6], [4, 6, 8]] + +result = [[0, 0, 0], [0, 0, 0], [0, 0, 0]] +for i in range(3): + for j in range(3): + result[i][j] += ( + sigma[0][j] * metric.guu[0, i] + + sigma[1][j] * metric.guu[1, i] + + sigma[2][j] * metric.guu[2, i] + ) +print("sigma^i_j " + str(result)) + +sigma = result +result = [[0, 0, 0], [0, 0, 0], [0, 0, 0]] +for i in range(3): + for j in range(3): + result[i][j] += ( + sigma[i][0] * metric.guu[0, j] + + sigma[i][1] * metric.guu[1, j] + + sigma[i][2] * metric.guu[2, j] + ) +print("sigma^ij " + str(result)) + +sigma = result +result = [[0, 0, 0], [0, 0, 0], [0, 0, 0]] +for i in range(3): + for j in range(3): + result[i][j] += ( + sigma[0][j] * metric.gdd[0, i] + + sigma[1][j] * metric.gdd[1, i] + + sigma[2][j] * metric.gdd[2, i] + ) +print("sigma_i^j " + str(result)) + +sigma = result +result = [[0, 0, 0], [0, 0, 0], [0, 0, 0]] +for i in range(3): + for j in range(3): + result[i][j] += ( + sigma[i][0] * metric.gdd[0, j] + + sigma[i][1] * metric.gdd[1, j] + + sigma[i][2] * metric.gdd[2, j] + ) +print("sigma_ij " + str(result)) + + +print("Done") diff --git a/tfields/bases/udo/testSympy.py b/tfields/bases/udo/testSympy.py new file mode 100644 index 0000000000000000000000000000000000000000..28731b5019fafb7a5e30046c4a5e93935abf5511 --- /dev/null +++ b/tfields/bases/udo/testSympy.py @@ -0,0 +1,10 @@ +import matplotlib.pyplot as plt +import numpy as np +import sympy as sym + +r = np.linspace(-1.6, np.pi, 1000) +theta = np.linspace(-np.pi, np.pi, len(r)) + +plt.plot(r, np.arctan2(theta, r)) +plt.plot(r, -2 * np.arctan((r - np.sqrt(r ** 2 + theta ** 2)) / theta)) +plt.show() diff --git a/theory/1901.05512.pdf b/theory/1901.05512.pdf new file mode 100644 index 0000000000000000000000000000000000000000..99f47d6a002c843adceb4ba4e13a432c110e8339 Binary files /dev/null and b/theory/1901.05512.pdf differ diff --git a/theory/CatalogueOfSpacetimes.pdf b/theory/CatalogueOfSpacetimes.pdf new file mode 100644 index 0000000000000000000000000000000000000000..33e156b2bbb17898b076eaf19b4636115294ea46 --- /dev/null +++ b/theory/CatalogueOfSpacetimes.pdf @@ -0,0 +1,19065 @@ +%PDF-1.4 +%���� +1 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363 0 R +/Next 395 0 R +/First 371 0 R +/Last 391 0 R +/Count 6 +>> endobj +363 0 obj << +/Title 364 0 R +/A 361 0 R +/Parent 139 0 R +/Prev 359 0 R +/Next 367 0 R +>> endobj +359 0 obj << +/Title 360 0 R +/A 357 0 R +/Parent 139 0 R +/Prev 327 0 R +/Next 363 0 R +>> endobj +355 0 obj << +/Title 356 0 R +/A 353 0 R +/Parent 327 0 R +/Prev 351 0 R +>> endobj +351 0 obj << +/Title 352 0 R +/A 349 0 R +/Parent 327 0 R +/Prev 347 0 R +/Next 355 0 R +>> endobj +347 0 obj << +/Title 348 0 R +/A 345 0 R +/Parent 327 0 R +/Prev 343 0 R +/Next 351 0 R +>> endobj +343 0 obj << +/Title 344 0 R +/A 341 0 R +/Parent 327 0 R +/Prev 339 0 R +/Next 347 0 R +>> endobj +339 0 obj << +/Title 340 0 R +/A 337 0 R +/Parent 327 0 R +/Prev 335 0 R +/Next 343 0 R +>> endobj +335 0 obj << +/Title 336 0 R +/A 333 0 R +/Parent 327 0 R +/Prev 331 0 R +/Next 339 0 R +>> endobj +331 0 obj << +/Title 332 0 R +/A 329 0 R +/Parent 327 0 R +/Next 335 0 R +>> endobj +327 0 obj << +/Title 328 0 R +/A 325 0 R +/Parent 139 0 R +/Prev 315 0 R +/Next 359 0 R +/First 331 0 R +/Last 355 0 R +/Count 7 +>> endobj +323 0 obj << +/Title 324 0 R +/A 321 0 R +/Parent 315 0 R +/Prev 319 0 R +>> endobj +319 0 obj << +/Title 320 0 R +/A 317 0 R +/Parent 315 0 R +/Next 323 0 R +>> endobj +315 0 obj << +/Title 316 0 R +/A 313 0 R +/Parent 139 0 R +/Prev 311 0 R +/Next 327 0 R +/First 319 0 R +/Last 323 0 R +/Count 2 +>> endobj +311 0 obj << +/Title 312 0 R +/A 309 0 R +/Parent 139 0 R +/Prev 307 0 R +/Next 315 0 R +>> endobj +307 0 obj << +/Title 308 0 R +/A 305 0 R +/Parent 139 0 R +/Prev 303 0 R +/Next 311 0 R +>> endobj +303 0 obj << +/Title 304 0 R +/A 301 0 R +/Parent 139 0 R +/Prev 299 0 R +/Next 307 0 R +>> endobj +299 0 obj << +/Title 300 0 R +/A 297 0 R +/Parent 139 0 R +/Prev 291 0 R +/Next 303 0 R +>> endobj +295 0 obj << +/Title 296 0 R +/A 293 0 R +/Parent 291 0 R +>> endobj +291 0 obj << +/Title 292 0 R +/A 289 0 R +/Parent 139 0 R +/Prev 287 0 R +/Next 299 0 R +/First 295 0 R +/Last 295 0 R +/Count 1 +>> endobj +287 0 obj << +/Title 288 0 R +/A 285 0 R +/Parent 139 0 R +/Prev 283 0 R +/Next 291 0 R +>> endobj +283 0 obj << +/Title 284 0 R +/A 281 0 R +/Parent 139 0 R +/Prev 279 0 R +/Next 287 0 R +>> endobj +279 0 obj << +/Title 280 0 R +/A 277 0 R +/Parent 139 0 R +/Prev 275 0 R +/Next 283 0 R +>> endobj +275 0 obj << +/Title 276 0 R +/A 273 0 R +/Parent 139 0 R +/Prev 263 0 R +/Next 279 0 R +>> endobj +271 0 obj << +/Title 272 0 R +/A 269 0 R +/Parent 263 0 R +/Prev 267 0 R +>> endobj +267 0 obj << +/Title 268 0 R +/A 265 0 R +/Parent 263 0 R +/Next 271 0 R +>> endobj +263 0 obj << +/Title 264 0 R +/A 261 0 R +/Parent 139 0 R +/Prev 247 0 R +/Next 275 0 R +/First 267 0 R +/Last 271 0 R +/Count 2 +>> endobj +259 0 obj << +/Title 260 0 R +/A 257 0 R +/Parent 247 0 R +/Prev 255 0 R +>> endobj +255 0 obj << +/Title 256 0 R +/A 253 0 R +/Parent 247 0 R +/Prev 251 0 R +/Next 259 0 R +>> endobj +251 0 obj << +/Title 252 0 R +/A 249 0 R +/Parent 247 0 R +/Next 255 0 R +>> endobj +247 0 obj << +/Title 248 0 R +/A 245 0 R +/Parent 139 0 R +/Prev 243 0 R +/Next 263 0 R +/First 251 0 R +/Last 259 0 R +/Count 3 +>> endobj +243 0 obj << +/Title 244 0 R +/A 241 0 R +/Parent 139 0 R +/Prev 239 0 R +/Next 247 0 R +>> endobj +239 0 obj << +/Title 240 0 R +/A 237 0 R +/Parent 139 0 R +/Prev 235 0 R +/Next 243 0 R +>> endobj +235 0 obj << +/Title 236 0 R +/A 233 0 R +/Parent 139 0 R +/Prev 231 0 R +/Next 239 0 R +>> endobj +231 0 obj << +/Title 232 0 R +/A 229 0 R +/Parent 139 0 R +/Prev 219 0 R +/Next 235 0 R +>> endobj +227 0 obj << +/Title 228 0 R +/A 225 0 R +/Parent 219 0 R +/Prev 223 0 R +>> endobj +223 0 obj << +/Title 224 0 R +/A 221 0 R +/Parent 219 0 R +/Next 227 0 R +>> endobj +219 0 obj << +/Title 220 0 R +/A 217 0 R +/Parent 139 0 R +/Prev 215 0 R +/Next 231 0 R +/First 223 0 R +/Last 227 0 R +/Count 2 +>> endobj +215 0 obj << +/Title 216 0 R +/A 213 0 R +/Parent 139 0 R +/Prev 211 0 R +/Next 219 0 R +>> endobj +211 0 obj << +/Title 212 0 R +/A 209 0 R +/Parent 139 0 R +/Prev 207 0 R +/Next 215 0 R +>> endobj +207 0 obj << +/Title 208 0 R +/A 205 0 R +/Parent 139 0 R +/Prev 171 0 R +/Next 211 0 R +>> endobj +203 0 obj << +/Title 204 0 R +/A 201 0 R +/Parent 171 0 R +/Prev 199 0 R +>> endobj +199 0 obj << +/Title 200 0 R +/A 197 0 R +/Parent 171 0 R +/Prev 195 0 R +/Next 203 0 R +>> endobj +195 0 obj << +/Title 196 0 R +/A 193 0 R +/Parent 171 0 R +/Prev 191 0 R +/Next 199 0 R +>> endobj +191 0 obj << +/Title 192 0 R +/A 189 0 R +/Parent 171 0 R +/Prev 187 0 R +/Next 195 0 R +>> endobj +187 0 obj << +/Title 188 0 R +/A 185 0 R +/Parent 171 0 R +/Prev 183 0 R +/Next 191 0 R +>> endobj +183 0 obj << +/Title 184 0 R +/A 181 0 R +/Parent 171 0 R +/Prev 179 0 R +/Next 187 0 R +>> endobj +179 0 obj << +/Title 180 0 R +/A 177 0 R +/Parent 171 0 R +/Prev 175 0 R +/Next 183 0 R +>> endobj +175 0 obj << +/Title 176 0 R +/A 173 0 R +/Parent 171 0 R +/Next 179 0 R +>> endobj +171 0 obj << +/Title 172 0 R +/A 169 0 R +/Parent 139 0 R +/Prev 143 0 R +/Next 207 0 R +/First 175 0 R +/Last 203 0 R +/Count 8 +>> endobj +167 0 obj << +/Title 168 0 R +/A 165 0 R +/Parent 143 0 R +/Prev 163 0 R +>> endobj +163 0 obj << +/Title 164 0 R +/A 161 0 R +/Parent 143 0 R +/Prev 159 0 R +/Next 167 0 R +>> endobj +159 0 obj << +/Title 160 0 R +/A 157 0 R +/Parent 143 0 R +/Prev 155 0 R +/Next 163 0 R +>> endobj +155 0 obj << +/Title 156 0 R +/A 153 0 R +/Parent 143 0 R +/Prev 151 0 R +/Next 159 0 R +>> endobj +151 0 obj << +/Title 152 0 R +/A 149 0 R +/Parent 143 0 R +/Prev 147 0 R +/Next 155 0 R +>> endobj +147 0 obj << +/Title 148 0 R +/A 145 0 R +/Parent 143 0 R +/Next 151 0 R +>> endobj +143 0 obj << +/Title 144 0 R +/A 141 0 R +/Parent 139 0 R +/Next 171 0 R +/First 147 0 R +/Last 167 0 R +/Count 6 +>> endobj +139 0 obj << +/Title 140 0 R +/A 137 0 R +/Parent 2751 0 R +/Prev 3 0 R +/Next 411 0 R +/First 143 0 R +/Last 407 0 R +/Count 67 +>> endobj +135 0 obj << +/Title 136 0 R +/A 133 0 R +/Parent 119 0 R +/Prev 131 0 R +>> endobj +131 0 obj << +/Title 132 0 R +/A 129 0 R +/Parent 119 0 R +/Prev 127 0 R +/Next 135 0 R +>> endobj +127 0 obj << +/Title 128 0 R +/A 125 0 R +/Parent 119 0 R +/Prev 123 0 R +/Next 131 0 R +>> endobj +123 0 obj << +/Title 124 0 R +/A 121 0 R +/Parent 119 0 R +/Next 127 0 R +>> endobj +119 0 obj << +/Title 120 0 R +/A 117 0 R +/Parent 3 0 R +/Prev 107 0 R +/First 123 0 R +/Last 135 0 R +/Count 4 +>> endobj +115 0 obj << +/Title 116 0 R +/A 113 0 R +/Parent 107 0 R +/Prev 111 0 R +>> endobj +111 0 obj << +/Title 112 0 R +/A 109 0 R +/Parent 107 0 R +/Next 115 0 R +>> endobj +107 0 obj << +/Title 108 0 R +/A 105 0 R +/Parent 3 0 R +/Prev 103 0 R +/Next 119 0 R +/First 111 0 R +/Last 115 0 R +/Count 2 +>> endobj +103 0 obj << +/Title 104 0 R +/A 101 0 R +/Parent 3 0 R +/Prev 95 0 R +/Next 107 0 R +>> endobj +99 0 obj << +/Title 100 0 R +/A 97 0 R +/Parent 95 0 R +>> endobj +95 0 obj << +/Title 96 0 R +/A 93 0 R +/Parent 3 0 R +/Prev 71 0 R +/Next 103 0 R +/First 99 0 R +/Last 99 0 R +/Count 1 +>> endobj +91 0 obj << +/Title 92 0 R +/A 89 0 R +/Parent 71 0 R +/Prev 87 0 R +>> endobj +87 0 obj << +/Title 88 0 R +/A 85 0 R +/Parent 71 0 R +/Prev 83 0 R +/Next 91 0 R +>> endobj +83 0 obj << +/Title 84 0 R +/A 81 0 R +/Parent 71 0 R +/Prev 79 0 R +/Next 87 0 R +>> endobj +79 0 obj << +/Title 80 0 R +/A 77 0 R +/Parent 71 0 R +/Prev 75 0 R +/Next 83 0 R +>> endobj +75 0 obj << +/Title 76 0 R +/A 73 0 R +/Parent 71 0 R +/Next 79 0 R +>> endobj +71 0 obj << +/Title 72 0 R +/A 69 0 R +/Parent 3 0 R +/Prev 67 0 R +/Next 95 0 R +/First 75 0 R +/Last 91 0 R +/Count 5 +>> endobj +67 0 obj << +/Title 68 0 R +/A 65 0 R +/Parent 3 0 R +/Prev 55 0 R +/Next 71 0 R +>> endobj +63 0 obj << +/Title 64 0 R +/A 61 0 R +/Parent 55 0 R +/Prev 59 0 R +>> endobj +59 0 obj << +/Title 60 0 R +/A 57 0 R +/Parent 55 0 R +/Next 63 0 R +>> endobj +55 0 obj << +/Title 56 0 R +/A 53 0 R +/Parent 3 0 R +/Prev 51 0 R +/Next 67 0 R +/First 59 0 R +/Last 63 0 R +/Count 2 +>> endobj +51 0 obj << +/Title 52 0 R +/A 49 0 R +/Parent 3 0 R +/Prev 19 0 R +/Next 55 0 R +>> endobj +47 0 obj << +/Title 48 0 R +/A 45 0 R +/Parent 19 0 R +/Prev 43 0 R +>> endobj +43 0 obj << +/Title 44 0 R +/A 41 0 R +/Parent 19 0 R +/Prev 39 0 R +/Next 47 0 R +>> endobj +39 0 obj << +/Title 40 0 R +/A 37 0 R +/Parent 19 0 R +/Prev 35 0 R +/Next 43 0 R +>> endobj +35 0 obj << +/Title 36 0 R +/A 33 0 R +/Parent 19 0 R +/Prev 31 0 R +/Next 39 0 R +>> endobj +31 0 obj << +/Title 32 0 R +/A 29 0 R +/Parent 19 0 R +/Prev 27 0 R +/Next 35 0 R +>> endobj +27 0 obj << +/Title 28 0 R +/A 25 0 R +/Parent 19 0 R +/Prev 23 0 R +/Next 31 0 R +>> endobj +23 0 obj << +/Title 24 0 R +/A 21 0 R +/Parent 19 0 R +/Next 27 0 R +>> endobj +19 0 obj << +/Title 20 0 R +/A 17 0 R +/Parent 3 0 R +/Prev 15 0 R +/Next 51 0 R +/First 23 0 R +/Last 47 0 R +/Count 7 +>> endobj +15 0 obj << +/Title 16 0 R +/A 13 0 R +/Parent 3 0 R +/Prev 11 0 R +/Next 19 0 R +>> endobj +11 0 obj << +/Title 12 0 R +/A 9 0 R +/Parent 3 0 R +/Prev 7 0 R +/Next 15 0 R +>> endobj +7 0 obj << +/Title 8 0 R +/A 5 0 R +/Parent 3 0 R +/Next 11 0 R +>> endobj +3 0 obj << +/Title 4 0 R +/A 1 0 R +/Parent 2751 0 R +/Next 139 0 R +/First 7 0 R +/Last 119 0 R +/Count 33 +>> endobj +2752 0 obj << +/Names [(Doc-Start) 419 0 R (Hfootnote.1) 628 0 R (Hfootnote.2) 846 0 R (chapter*.1) 511 0 R (chapter*.8) 2506 0 R (chapter.1) 2 0 R] +/Limits [(Doc-Start) (chapter.1)] +>> endobj +2753 0 obj << +/Names [(chapter.2) 138 0 R (cite.Kastor1993) 1798 0 R (cite.alcubierre1994) 1219 0 R (cite.aryal1986) 1338 0 R (cite.bardeen1972) 1829 0 R (cite.barriola1989) 1245 0 R] +/Limits [(chapter.2) (cite.barriola1989)] +>> endobj +2754 0 obj << +/Names [(cite.bicak2001) 2285 0 R (cite.bini2000) 700 0 R (cite.bini2002) 2483 0 R (cite.bini2003b) 2484 0 R (cite.bonnor1982) 2198 0 R (cite.boyer1967) 1872 0 R] +/Limits [(cite.bicak2001) (cite.boyer1967)] +>> endobj +2755 0 obj << +/Names [(cite.brill1966) 1874 0 R (cite.buchdahl1985) 1042 0 R (cite.chandrasekhar1989) 1490 0 R (cite.chandrasekhar2006) 654 0 R (cite.clark1999) 1221 0 R (cite.cruz2005) 1923 0 R] +/Limits [(cite.brill1966) (cite.cruz2005)] +>> endobj +2756 0 obj << +/Names [(cite.dhurandhar1983) 1417 0 R (cite.eddington1924) 1072 0 R (cite.ehlers1962) 2150 0 R (cite.eiroa2002) 2262 0 R (cite.ellis1973) 2069 0 R (cite.ernst1976) 1396 0 R] +/Limits [(cite.dhurandhar1983) (cite.ernst1976)] +>> endobj +2757 0 obj << +/Names [(cite.exact2003) 606 0 R (cite.finkelstein1958) 1073 0 R (cite.goedel1949) 1662 0 R (cite.griffiths1997) 1369 0 R (cite.griffiths2009) 607 0 R (cite.hackmann2008) 1922 0 R] +/Limits [(cite.exact2003) (cite.hackmann2008)] +>> endobj +2758 0 obj << +/Names [(cite.halilsoy1988) 1715 0 R (cite.hartleHawking1972) 1491 0 R (cite.hawking1999) 818 0 R (cite.janis1968) 1739 0 R (cite.kajari2004) 1663 0 R (cite.karas1992) 1395 0 R] +/Limits [(cite.halilsoy1988) (cite.karas1992)] +>> endobj +2759 0 obj << +/Names [(cite.kasner1921) 1785 0 R (cite.kerr1963) 1828 0 R (cite.kottler1918) 1920 0 R (cite.kramer1999) 1314 0 R (cite.kruskal1960) 1098 0 R (cite.martel2001) 1143 0 R] +/Limits [(cite.kasner1921) (cite.martel2001)] +>> endobj +2760 0 obj << +/Names [(cite.morris1988) 2051 0 R (cite.mtw) 610 0 R (cite.mueller2004) 2071 0 R (cite.mueller2008grg) 993 0 R (cite.mueller2008prdA) 2072 0 R (cite.mueller2009b) 608 0 R] +/Limits [(cite.morris1988) (cite.mueller2009b)] +>> endobj +2761 0 obj << +/Names [(cite.mueller2009grg) 994 0 R (cite.mueller2010a) 609 0 R (cite.nakahara) 653 0 R (cite.oppenheimer1939b) 2133 0 R (cite.penroseRindler) 722 0 R (cite.perlick2004) 1261 0 R] +/Limits [(cite.mueller2009grg) (cite.perlick2004)] +>> endobj +2762 0 obj << +/Names [(cite.pfenning1997) 1220 0 R (cite.pravda2000) 2197 0 R (cite.rindler) 629 0 R (cite.rindler1998) 1281 0 R (cite.schwarzschild1916a) 991 0 R (cite.schwarzschild1916b) 992 0 R] +/Limits [(cite.pfenning1997) (cite.schwarzschild1916b)] +>> endobj +2763 0 obj << +/Names [(cite.stephani1990) 784 0 R (cite.stephani2003) 1315 0 R (cite.stucklik1999) 1418 0 R (cite.sultana2005) 2451 0 R (cite.tolman) 2373 0 R (cite.vanDenBroeck1999) 1222 0 R] +/Limits [(cite.stephani1990) (cite.vanDenBroeck1999)] +>> endobj +2764 0 obj << +/Names [(cite.visser1995) 2070 0 R (cite.wald) 840 0 R (cite.weyl1919) 1921 0 R (cite.wilkins1972) 1873 0 R (cite.wuensch2013) 1493 0 R (cite.yurtsever1994Chaos) 1492 0 R] +/Limits [(cite.visser1995) (cite.yurtsever1994Chaos)] +>> endobj +2765 0 obj << +/Names [(equation.1.1.1) 593 0 R (equation.1.1.2) 601 0 R (equation.1.10.1) 803 0 R (equation.1.10.2) 804 0 R (equation.1.11.1) 812 0 R (equation.1.11.10) 830 0 R] +/Limits [(equation.1.1.1) (equation.1.11.10)] +>> endobj +2766 0 obj << +/Names [(equation.1.11.11) 831 0 R (equation.1.11.12) 832 0 R (equation.1.11.13) 833 0 R (equation.1.11.14) 834 0 R (equation.1.11.15) 835 0 R (equation.1.11.16) 836 0 R] +/Limits [(equation.1.11.11) (equation.1.11.16)] +>> endobj +2767 0 obj << +/Names [(equation.1.11.17) 837 0 R (equation.1.11.18) 838 0 R (equation.1.11.19) 839 0 R (equation.1.11.2) 813 0 R (equation.1.11.3) 814 0 R (equation.1.11.4) 815 0 R] +/Limits [(equation.1.11.17) (equation.1.11.4)] +>> endobj +2768 0 obj << +/Names [(equation.1.11.5) 816 0 R (equation.1.11.6) 817 0 R (equation.1.11.7) 827 0 R (equation.1.11.8) 828 0 R (equation.1.11.9) 829 0 R (equation.1.2.1) 603 0 R] +/Limits [(equation.1.11.5) (equation.1.2.1)] +>> endobj +2769 0 obj << +/Names [(equation.1.2.2) 616 0 R (equation.1.3.1) 617 0 R (equation.1.3.10) 626 0 R (equation.1.3.11) 637 0 R (equation.1.3.12) 638 0 R (equation.1.3.12a) 639 0 R] +/Limits [(equation.1.2.2) (equation.1.3.12a)] +>> endobj +2770 0 obj << +/Names [(equation.1.3.12b) 640 0 R (equation.1.3.13) 641 0 R (equation.1.3.14) 642 0 R (equation.1.3.15) 643 0 R (equation.1.3.16) 644 0 R (equation.1.3.2) 618 0 R] +/Limits [(equation.1.3.12b) (equation.1.3.2)] +>> endobj +2771 0 obj << +/Names [(equation.1.3.3) 619 0 R (equation.1.3.4) 620 0 R (equation.1.3.5) 621 0 R (equation.1.3.6) 622 0 R (equation.1.3.7) 623 0 R (equation.1.3.8) 624 0 R] +/Limits [(equation.1.3.3) (equation.1.3.8)] +>> endobj +2772 0 obj << +/Names [(equation.1.3.9) 625 0 R (equation.1.4.1) 646 0 R (equation.1.4.10) 666 0 R (equation.1.4.11) 667 0 R (equation.1.4.12) 668 0 R (equation.1.4.13) 669 0 R] +/Limits [(equation.1.3.9) (equation.1.4.13)] +>> endobj +2773 0 obj << +/Names [(equation.1.4.14) 670 0 R (equation.1.4.15) 671 0 R (equation.1.4.16) 672 0 R (equation.1.4.16a) 673 0 R (equation.1.4.16b) 674 0 R (equation.1.4.16c) 675 0 R] +/Limits [(equation.1.4.14) (equation.1.4.16c)] +>> endobj +2774 0 obj << +/Names [(equation.1.4.17) 682 0 R (equation.1.4.18) 683 0 R (equation.1.4.19) 685 0 R (equation.1.4.2) 647 0 R (equation.1.4.20) 686 0 R (equation.1.4.21) 687 0 R] +/Limits [(equation.1.4.17) (equation.1.4.21)] +>> endobj +2775 0 obj << +/Names [(equation.1.4.22) 688 0 R (equation.1.4.23) 689 0 R (equation.1.4.24) 690 0 R (equation.1.4.24a) 691 0 R (equation.1.4.24b) 692 0 R (equation.1.4.25) 710 0 R] +/Limits [(equation.1.4.22) (equation.1.4.25)] +>> endobj +2776 0 obj << +/Names [(equation.1.4.26) 711 0 R (equation.1.4.27) 712 0 R (equation.1.4.28) 713 0 R (equation.1.4.29) 714 0 R (equation.1.4.3) 648 0 R (equation.1.4.4) 649 0 R] +/Limits [(equation.1.4.26) (equation.1.4.4)] +>> endobj +2777 0 obj << +/Names [(equation.1.4.5) 650 0 R (equation.1.4.5a) 651 0 R (equation.1.4.5b) 652 0 R (equation.1.4.6) 662 0 R (equation.1.4.7) 663 0 R (equation.1.4.8) 664 0 R] +/Limits [(equation.1.4.5) (equation.1.4.8)] +>> endobj +2778 0 obj << +/Names [(equation.1.4.9) 665 0 R (equation.1.5.1) 715 0 R (equation.1.5.2) 716 0 R (equation.1.5.3) 717 0 R (equation.1.5.3a) 718 0 R (equation.1.5.3b) 719 0 R] +/Limits [(equation.1.4.9) (equation.1.5.3b)] +>> endobj +2779 0 obj << +/Names [(equation.1.5.3c) 720 0 R (equation.1.5.3d) 721 0 R (equation.1.6.1) 729 0 R (equation.1.6.10) 757 0 R (equation.1.6.10a) 758 0 R (equation.1.6.10b) 759 0 R] +/Limits [(equation.1.5.3c) (equation.1.6.10b)] +>> endobj +2780 0 obj << +/Names [(equation.1.6.2) 730 0 R (equation.1.6.3) 732 0 R (equation.1.6.4) 733 0 R (equation.1.6.4a) 734 0 R (equation.1.6.4b) 735 0 R (equation.1.6.4c) 736 0 R] +/Limits [(equation.1.6.2) (equation.1.6.4c)] +>> endobj +2781 0 obj << +/Names [(equation.1.6.5) 737 0 R (equation.1.6.5a) 738 0 R (equation.1.6.5b) 739 0 R (equation.1.6.5c) 740 0 R (equation.1.6.6) 741 0 R (equation.1.6.7) 752 0 R] +/Limits [(equation.1.6.5) (equation.1.6.7)] +>> endobj +2782 0 obj << +/Names [(equation.1.6.8) 753 0 R (equation.1.6.9) 754 0 R (equation.1.6.9a) 755 0 R (equation.1.6.9b) 756 0 R (equation.1.7.1) 760 0 R (equation.1.7.2) 761 0 R] +/Limits [(equation.1.6.8) (equation.1.7.2)] +>> endobj +2783 0 obj << +/Names [(equation.1.7.3) 762 0 R (equation.1.8.1) 777 0 R (equation.1.8.10) 795 0 R (equation.1.8.11) 796 0 R (equation.1.8.2) 778 0 R (equation.1.8.3) 779 0 R] +/Limits [(equation.1.7.3) (equation.1.8.3)] +>> endobj +2784 0 obj << +/Names [(equation.1.8.4) 780 0 R (equation.1.8.5) 781 0 R (equation.1.8.6) 782 0 R (equation.1.8.7) 783 0 R (equation.1.8.8) 793 0 R (equation.1.8.9) 794 0 R] +/Limits [(equation.1.8.4) (equation.1.8.9)] +>> endobj +2785 0 obj << +/Names [(equation.1.9.12) 797 0 R (equation.1.9.13) 798 0 R (equation.1.9.13a) 799 0 R (equation.1.9.13b) 800 0 R (equation.1.9.13c) 801 0 R (equation.1.9.13d) 802 0 R] +/Limits [(equation.1.9.12) (equation.1.9.13d)] +>> endobj +2786 0 obj << +/Names [(equation.2.1.1) 873 0 R (equation.2.1.10) 889 0 R (equation.2.1.10a) 890 0 R (equation.2.1.10b) 891 0 R (equation.2.1.11) 892 0 R (equation.2.1.11a) 893 0 R] +/Limits [(equation.2.1.1) (equation.2.1.11a)] +>> endobj +2787 0 obj << +/Names [(equation.2.1.11b) 894 0 R (equation.2.1.11c) 895 0 R (equation.2.1.12) 896 0 R (equation.2.1.13) 897 0 R (equation.2.1.14) 898 0 R (equation.2.1.15) 899 0 R] +/Limits [(equation.2.1.11b) (equation.2.1.15)] +>> endobj +2788 0 obj << +/Names [(equation.2.1.16) 900 0 R (equation.2.1.17) 901 0 R (equation.2.1.18) 902 0 R (equation.2.1.18a) 903 0 R (equation.2.1.18b) 904 0 R (equation.2.1.19) 911 0 R] +/Limits [(equation.2.1.16) (equation.2.1.19)] +>> endobj +2789 0 obj << +/Names [(equation.2.1.19a) 912 0 R (equation.2.1.19b) 913 0 R (equation.2.1.19c) 914 0 R (equation.2.1.2) 874 0 R (equation.2.1.20) 915 0 R (equation.2.1.21) 916 0 R] +/Limits [(equation.2.1.19a) (equation.2.1.21)] +>> endobj +2790 0 obj << +/Names [(equation.2.1.22) 917 0 R (equation.2.1.23) 918 0 R (equation.2.1.24) 919 0 R (equation.2.1.25) 920 0 R (equation.2.1.26) 921 0 R (equation.2.1.27) 922 0 R] +/Limits [(equation.2.1.22) (equation.2.1.27)] +>> endobj +2791 0 obj << +/Names [(equation.2.1.28) 923 0 R (equation.2.1.29) 924 0 R (equation.2.1.3) 875 0 R (equation.2.1.30) 929 0 R (equation.2.1.31) 930 0 R (equation.2.1.32) 931 0 R] +/Limits [(equation.2.1.28) (equation.2.1.32)] +>> endobj +2792 0 obj << +/Names [(equation.2.1.33) 932 0 R (equation.2.1.33a) 933 0 R (equation.2.1.33b) 934 0 R (equation.2.1.34) 935 0 R (equation.2.1.35) 936 0 R (equation.2.1.36) 937 0 R] +/Limits [(equation.2.1.33) (equation.2.1.36)] +>> endobj +2793 0 obj << +/Names [(equation.2.1.37) 938 0 R (equation.2.1.38) 939 0 R (equation.2.1.39) 940 0 R (equation.2.1.4) 876 0 R (equation.2.1.40) 941 0 R (equation.2.1.5) 877 0 R] +/Limits [(equation.2.1.37) (equation.2.1.5)] +>> endobj +2794 0 obj << +/Names [(equation.2.1.6) 878 0 R (equation.2.1.7) 879 0 R (equation.2.1.8) 880 0 R (equation.2.1.9) 888 0 R (equation.2.10.1) 1424 0 R (equation.2.10.10) 1459 0 R] +/Limits [(equation.2.1.6) (equation.2.10.10)] +>> endobj +2795 0 obj << +/Names [(equation.2.10.11) 1460 0 R (equation.2.10.11a) 1461 0 R (equation.2.10.11b) 1462 0 R (equation.2.10.12) 1473 0 R (equation.2.10.13) 1474 0 R (equation.2.10.13a) 1475 0 R] +/Limits [(equation.2.10.11) (equation.2.10.13a)] +>> endobj +2796 0 obj << +/Names [(equation.2.10.13b) 1476 0 R (equation.2.10.13c) 1477 0 R (equation.2.10.13d) 1478 0 R (equation.2.10.13e) 1479 0 R (equation.2.10.14) 1480 0 R (equation.2.10.14a) 1481 0 R] +/Limits [(equation.2.10.13b) (equation.2.10.14a)] +>> endobj +2797 0 obj << +/Names [(equation.2.10.14b) 1482 0 R (equation.2.10.15) 1483 0 R (equation.2.10.15a) 1484 0 R (equation.2.10.15b) 1485 0 R (equation.2.10.15c) 1486 0 R (equation.2.10.15d) 1487 0 R] +/Limits [(equation.2.10.14b) (equation.2.10.15d)] +>> endobj +2798 0 obj << +/Names [(equation.2.10.15e) 1488 0 R (equation.2.10.16) 1489 0 R (equation.2.10.2) 1425 0 R (equation.2.10.3) 1426 0 R (equation.2.10.3a) 1427 0 R (equation.2.10.3b) 1428 0 R] +/Limits [(equation.2.10.15e) (equation.2.10.3b)] +>> endobj +2799 0 obj << +/Names [(equation.2.10.3c) 1429 0 R (equation.2.10.3d) 1430 0 R (equation.2.10.3e) 1431 0 R (equation.2.10.4) 1432 0 R (equation.2.10.4a) 1433 0 R (equation.2.10.4b) 1434 0 R] +/Limits [(equation.2.10.3c) (equation.2.10.4b)] +>> endobj +2800 0 obj << +/Names [(equation.2.10.4c) 1435 0 R (equation.2.10.4d) 1436 0 R (equation.2.10.5) 1442 0 R (equation.2.10.5a) 1443 0 R (equation.2.10.5b) 1444 0 R (equation.2.10.5c) 1445 0 R] +/Limits [(equation.2.10.4c) (equation.2.10.5c)] +>> endobj +2801 0 obj << +/Names [(equation.2.10.5d) 1446 0 R (equation.2.10.5e) 1447 0 R (equation.2.10.6) 1448 0 R (equation.2.10.6a) 1449 0 R (equation.2.10.6b) 1450 0 R (equation.2.10.7) 1451 0 R] +/Limits [(equation.2.10.5d) (equation.2.10.7)] +>> endobj +2802 0 obj << +/Names [(equation.2.10.8) 1452 0 R (equation.2.10.8a) 1453 0 R (equation.2.10.8b) 1454 0 R (equation.2.10.8c) 1455 0 R (equation.2.10.8d) 1456 0 R (equation.2.10.8e) 1457 0 R] +/Limits [(equation.2.10.8) (equation.2.10.8e)] +>> endobj +2803 0 obj << +/Names [(equation.2.10.9) 1458 0 R (equation.2.11.1) 1498 0 R (equation.2.11.10) 1521 0 R (equation.2.11.10a) 1522 0 R (equation.2.11.10b) 1523 0 R (equation.2.11.11) 1524 0 R] +/Limits [(equation.2.10.9) (equation.2.11.11)] +>> endobj +2804 0 obj << +/Names [(equation.2.11.12) 1525 0 R (equation.2.11.13) 1526 0 R (equation.2.11.13a) 1527 0 R (equation.2.11.13b) 1528 0 R (equation.2.11.13c) 1529 0 R (equation.2.11.13d) 1530 0 R] +/Limits [(equation.2.11.12) (equation.2.11.13d)] +>> endobj +2805 0 obj << +/Names [(equation.2.11.13e) 1531 0 R (equation.2.11.14) 1532 0 R (equation.2.11.14a) 1533 0 R (equation.2.11.14b) 1534 0 R (equation.2.11.14c) 1535 0 R (equation.2.11.15) 1536 0 R] +/Limits [(equation.2.11.13e) (equation.2.11.15)] +>> endobj +2806 0 obj << +/Names [(equation.2.11.15a) 1537 0 R (equation.2.11.15b) 1538 0 R (equation.2.11.16) 1544 0 R (equation.2.11.17) 1545 0 R (equation.2.11.18) 1546 0 R (equation.2.11.18a) 1547 0 R] +/Limits [(equation.2.11.15a) (equation.2.11.18a)] +>> endobj +2807 0 obj << +/Names [(equation.2.11.18b) 1548 0 R (equation.2.11.19) 1549 0 R (equation.2.11.2) 1499 0 R (equation.2.11.20) 1550 0 R (equation.2.11.20a) 1551 0 R (equation.2.11.20b) 1552 0 R] +/Limits [(equation.2.11.18b) (equation.2.11.20b)] +>> endobj +2808 0 obj << +/Names [(equation.2.11.21) 1553 0 R (equation.2.11.22) 1555 0 R (equation.2.11.23) 1556 0 R (equation.2.11.23a) 1557 0 R (equation.2.11.23b) 1558 0 R (equation.2.11.23c) 1559 0 R] +/Limits [(equation.2.11.21) (equation.2.11.23c)] +>> endobj +2809 0 obj << +/Names [(equation.2.11.23d) 1560 0 R (equation.2.11.24) 1566 0 R (equation.2.11.24a) 1567 0 R (equation.2.11.24b) 1568 0 R (equation.2.11.24c) 1569 0 R (equation.2.11.25) 1570 0 R] +/Limits [(equation.2.11.23d) (equation.2.11.25)] +>> endobj +2810 0 obj << +/Names [(equation.2.11.25a) 1571 0 R (equation.2.11.25b) 1572 0 R (equation.2.11.26) 1573 0 R (equation.2.11.27) 1574 0 R (equation.2.11.28) 1575 0 R (equation.2.11.28a) 1576 0 R] +/Limits [(equation.2.11.25a) (equation.2.11.28a)] +>> endobj +2811 0 obj << +/Names [(equation.2.11.28b) 1577 0 R (equation.2.11.29) 1578 0 R (equation.2.11.3) 1500 0 R (equation.2.11.30) 1579 0 R (equation.2.11.30a) 1580 0 R (equation.2.11.30b) 1581 0 R] +/Limits [(equation.2.11.28b) (equation.2.11.30b)] +>> endobj +2812 0 obj << +/Names [(equation.2.11.31) 1582 0 R (equation.2.11.32) 1584 0 R (equation.2.11.33) 1585 0 R (equation.2.11.33a) 1586 0 R (equation.2.11.33b) 1587 0 R (equation.2.11.33c) 1588 0 R] +/Limits [(equation.2.11.31) (equation.2.11.33c)] +>> endobj +2813 0 obj << +/Names [(equation.2.11.33d) 1589 0 R (equation.2.11.34) 1594 0 R (equation.2.11.34a) 1595 0 R (equation.2.11.34b) 1596 0 R (equation.2.11.34c) 1597 0 R (equation.2.11.35) 1598 0 R] +/Limits [(equation.2.11.33d) (equation.2.11.35)] +>> endobj +2814 0 obj << +/Names [(equation.2.11.35a) 1599 0 R (equation.2.11.35b) 1600 0 R (equation.2.11.36) 1601 0 R (equation.2.11.37) 1602 0 R (equation.2.11.38) 1603 0 R (equation.2.11.38a) 1604 0 R] +/Limits [(equation.2.11.35a) (equation.2.11.38a)] +>> endobj +2815 0 obj << +/Names [(equation.2.11.38b) 1605 0 R (equation.2.11.39) 1606 0 R (equation.2.11.3a) 1501 0 R (equation.2.11.3b) 1502 0 R (equation.2.11.3c) 1503 0 R (equation.2.11.3d) 1504 0 R] +/Limits [(equation.2.11.38b) (equation.2.11.3d)] +>> endobj +2816 0 obj << +/Names [(equation.2.11.3e) 1505 0 R (equation.2.11.4) 1506 0 R (equation.2.11.40) 1607 0 R (equation.2.11.40a) 1608 0 R (equation.2.11.40b) 1609 0 R (equation.2.11.41) 1610 0 R] +/Limits [(equation.2.11.3e) (equation.2.11.41)] +>> endobj +2817 0 obj << +/Names [(equation.2.11.42) 1612 0 R (equation.2.11.43) 1613 0 R (equation.2.11.43a) 1614 0 R (equation.2.11.43b) 1615 0 R (equation.2.11.43c) 1616 0 R (equation.2.11.43d) 1617 0 R] +/Limits [(equation.2.11.42) (equation.2.11.43d)] +>> endobj +2818 0 obj << +/Names [(equation.2.11.44) 1623 0 R (equation.2.11.44a) 1624 0 R (equation.2.11.44b) 1625 0 R (equation.2.11.44c) 1626 0 R (equation.2.11.45) 1627 0 R (equation.2.11.45a) 1628 0 R] +/Limits [(equation.2.11.44) (equation.2.11.45a)] +>> endobj +2819 0 obj << +/Names [(equation.2.11.45b) 1629 0 R (equation.2.11.46) 1630 0 R (equation.2.11.47) 1631 0 R (equation.2.11.48) 1632 0 R (equation.2.11.48a) 1633 0 R (equation.2.11.48b) 1634 0 R] +/Limits [(equation.2.11.45b) (equation.2.11.48b)] +>> endobj +2820 0 obj << +/Names [(equation.2.11.49) 1635 0 R (equation.2.11.4a) 1507 0 R (equation.2.11.4b) 1508 0 R (equation.2.11.4c) 1509 0 R (equation.2.11.5) 1510 0 R (equation.2.11.50) 1636 0 R] +/Limits [(equation.2.11.49) (equation.2.11.50)] +>> endobj +2821 0 obj << +/Names [(equation.2.11.50a) 1637 0 R (equation.2.11.50b) 1638 0 R (equation.2.11.51) 1639 0 R (equation.2.11.5a) 1511 0 R (equation.2.11.5b) 1512 0 R (equation.2.11.6) 1513 0 R] +/Limits [(equation.2.11.50a) (equation.2.11.6)] +>> endobj +2822 0 obj << +/Names [(equation.2.11.7) 1514 0 R (equation.2.11.8) 1519 0 R (equation.2.11.9) 1520 0 R (equation.2.12.1) 1647 0 R (equation.2.12.10) 1673 0 R (equation.2.12.11) 1674 0 R] +/Limits [(equation.2.11.7) (equation.2.12.11)] +>> endobj +2823 0 obj << +/Names [(equation.2.12.12) 1675 0 R (equation.2.12.13) 1676 0 R (equation.2.12.13a) 1677 0 R (equation.2.12.13b) 1678 0 R (equation.2.12.13c) 1679 0 R (equation.2.12.13d) 1680 0 R] +/Limits [(equation.2.12.12) (equation.2.12.13d)] +>> endobj +2824 0 obj << +/Names [(equation.2.12.14) 1681 0 R (equation.2.12.14a) 1682 0 R (equation.2.12.14b) 1683 0 R (equation.2.12.15) 1684 0 R (equation.2.12.16) 1689 0 R (equation.2.12.17) 1690 0 R] +/Limits [(equation.2.12.14) (equation.2.12.17)] +>> endobj +2825 0 obj << +/Names [(equation.2.12.18) 1691 0 R (equation.2.12.18a) 1692 0 R (equation.2.12.18b) 1693 0 R (equation.2.12.19) 1694 0 R (equation.2.12.19a) 1695 0 R (equation.2.12.2) 1648 0 R] +/Limits [(equation.2.12.18) (equation.2.12.2)] +>> endobj +2826 0 obj << +/Names [(equation.2.12.20) 1696 0 R (equation.2.12.20a) 1697 0 R (equation.2.12.20b) 1698 0 R (equation.2.12.21) 1699 0 R (equation.2.12.22) 1700 0 R (equation.2.12.22a) 1701 0 R] +/Limits [(equation.2.12.20) (equation.2.12.22a)] +>> endobj +2827 0 obj << +/Names [(equation.2.12.2a) 1649 0 R (equation.2.12.2b) 1650 0 R (equation.2.12.2c) 1651 0 R (equation.2.12.2d) 1652 0 R (equation.2.12.3) 1653 0 R (equation.2.12.3a) 1654 0 R] +/Limits [(equation.2.12.2a) (equation.2.12.3a)] +>> endobj +2828 0 obj << +/Names [(equation.2.12.3b) 1655 0 R (equation.2.12.4) 1656 0 R (equation.2.12.5) 1657 0 R (equation.2.12.6) 1658 0 R (equation.2.12.7) 1659 0 R (equation.2.12.7a) 1660 0 R] +/Limits [(equation.2.12.3b) (equation.2.12.7a)] +>> endobj +2829 0 obj << +/Names [(equation.2.12.7b) 1661 0 R (equation.2.12.8) 1668 0 R (equation.2.12.8a) 1669 0 R (equation.2.12.9) 1670 0 R (equation.2.12.9a) 1671 0 R (equation.2.12.9b) 1672 0 R] +/Limits [(equation.2.12.7b) (equation.2.12.9b)] +>> endobj +2830 0 obj << +/Names [(equation.2.13.1) 1708 0 R (equation.2.13.2) 1709 0 R (equation.2.13.2a) 1710 0 R (equation.2.13.2b) 1711 0 R (equation.2.13.2c) 1712 0 R (equation.2.13.3) 1713 0 R] +/Limits [(equation.2.13.1) (equation.2.13.3)] +>> endobj +2831 0 obj << +/Names [(equation.2.13.4) 1714 0 R (equation.2.14.1) 1722 0 R (equation.2.14.10) 1746 0 R (equation.2.14.10a) 1747 0 R (equation.2.14.10b) 1748 0 R (equation.2.14.11) 1749 0 R] +/Limits [(equation.2.13.4) (equation.2.14.11)] +>> endobj +2832 0 obj << +/Names [(equation.2.14.12) 1750 0 R (equation.2.14.12a) 1751 0 R (equation.2.14.12b) 1752 0 R (equation.2.14.13) 1753 0 R (equation.2.14.14) 1754 0 R (equation.2.14.15) 1755 0 R] +/Limits [(equation.2.14.12) (equation.2.14.15)] +>> endobj +2833 0 obj << +/Names [(equation.2.14.16) 1756 0 R (equation.2.14.2) 1723 0 R (equation.2.14.2a) 1724 0 R (equation.2.14.2b) 1725 0 R (equation.2.14.2c) 1726 0 R (equation.2.14.3) 1727 0 R] +/Limits [(equation.2.14.16) (equation.2.14.3)] +>> endobj +2834 0 obj << +/Names [(equation.2.14.3a) 1728 0 R (equation.2.14.3b) 1729 0 R (equation.2.14.3c) 1730 0 R (equation.2.14.4) 1731 0 R (equation.2.14.4a) 1732 0 R (equation.2.14.4b) 1733 0 R] +/Limits [(equation.2.14.3a) (equation.2.14.4b)] +>> endobj +2835 0 obj << +/Names [(equation.2.14.4c) 1734 0 R (equation.2.14.5) 1735 0 R (equation.2.14.6) 1736 0 R (equation.2.14.7) 1737 0 R (equation.2.14.8) 1738 0 R (equation.2.14.9) 1745 0 R] +/Limits [(equation.2.14.4c) (equation.2.14.9)] +>> endobj +2836 0 obj << +/Names [(equation.2.15.1) 1763 0 R (equation.2.15.10) 1780 0 R (equation.2.15.11) 1781 0 R (equation.2.15.12) 1782 0 R (equation.2.15.12a) 1783 0 R (equation.2.15.12b) 1784 0 R] +/Limits [(equation.2.15.1) (equation.2.15.12b)] +>> endobj +2837 0 obj << +/Names [(equation.2.15.2) 1764 0 R (equation.2.15.3) 1765 0 R (equation.2.15.4) 1766 0 R (equation.2.15.4a) 1767 0 R (equation.2.15.4b) 1768 0 R (equation.2.15.5) 1769 0 R] +/Limits [(equation.2.15.2) (equation.2.15.5)] +>> endobj +2838 0 obj << +/Names [(equation.2.15.5a) 1770 0 R (equation.2.15.5b) 1771 0 R (equation.2.15.6) 1772 0 R (equation.2.15.6a) 1773 0 R (equation.2.15.6b) 1774 0 R (equation.2.15.7) 1775 0 R] +/Limits [(equation.2.15.5a) (equation.2.15.7)] +>> endobj +2839 0 obj << +/Names [(equation.2.15.7a) 1776 0 R (equation.2.15.7b) 1777 0 R (equation.2.15.8) 1778 0 R (equation.2.15.9) 1779 0 R (equation.2.16.1) 1791 0 R (equation.2.16.2) 1792 0 R] +/Limits [(equation.2.15.7a) (equation.2.16.2)] +>> endobj +2840 0 obj << +/Names [(equation.2.16.2a) 1793 0 R (equation.2.16.2b) 1794 0 R (equation.2.16.2c) 1795 0 R (equation.2.16.3) 1796 0 R (equation.2.16.4) 1797 0 R (equation.2.17.1) 1808 0 R] +/Limits [(equation.2.16.2a) (equation.2.17.1)] +>> endobj +2841 0 obj << +/Names [(equation.2.17.10) 1841 0 R (equation.2.17.10a) 1842 0 R (equation.2.17.10b) 1843 0 R (equation.2.17.10c) 1844 0 R (equation.2.17.10d) 1845 0 R (equation.2.17.10e) 1846 0 R] +/Limits [(equation.2.17.10) (equation.2.17.10e)] +>> endobj +2842 0 obj << +/Names [(equation.2.17.10f) 1847 0 R (equation.2.17.10g) 1848 0 R (equation.2.17.10h) 1849 0 R (equation.2.17.10i) 1850 0 R (equation.2.17.10j) 1851 0 R (equation.2.17.10k) 1852 0 R] +/Limits [(equation.2.17.10f) (equation.2.17.10k)] +>> endobj +2843 0 obj << +/Names [(equation.2.17.10l) 1853 0 R (equation.2.17.11) 1854 0 R (equation.2.17.12) 1864 0 R (equation.2.17.13) 1865 0 R (equation.2.17.13a) 1866 0 R (equation.2.17.13b) 1867 0 R] +/Limits [(equation.2.17.10l) (equation.2.17.13b)] +>> endobj +2844 0 obj << +/Names [(equation.2.17.14) 1868 0 R (equation.2.17.15) 1869 0 R (equation.2.17.16) 1870 0 R (equation.2.17.17) 1871 0 R (equation.2.17.2) 1809 0 R (equation.2.17.3) 1810 0 R] +/Limits [(equation.2.17.14) (equation.2.17.3)] +>> endobj +2845 0 obj << +/Names [(equation.2.17.4) 1812 0 R (equation.2.17.4a) 1813 0 R (equation.2.17.4b) 1814 0 R (equation.2.17.5) 1815 0 R (equation.2.17.6) 1816 0 R (equation.2.17.7) 1834 0 R] +/Limits [(equation.2.17.4) (equation.2.17.7)] +>> endobj +2846 0 obj << +/Names [(equation.2.17.8) 1835 0 R (equation.2.17.8a) 1836 0 R (equation.2.17.8b) 1837 0 R (equation.2.17.9) 1838 0 R (equation.2.17.9a) 1839 0 R (equation.2.17.9b) 1840 0 R] +/Limits [(equation.2.17.8) (equation.2.17.9b)] +>> endobj +2847 0 obj << +/Names [(equation.2.18.1) 1880 0 R (equation.2.18.10) 1907 0 R (equation.2.18.11) 1908 0 R (equation.2.18.12) 1909 0 R (equation.2.18.13) 1910 0 R (equation.2.18.14) 1911 0 R] +/Limits [(equation.2.18.1) (equation.2.18.14)] +>> endobj +2848 0 obj << +/Names [(equation.2.18.14a) 1912 0 R (equation.2.18.14b) 1913 0 R (equation.2.18.15) 1914 0 R (equation.2.18.15a) 1915 0 R (equation.2.18.15b) 1916 0 R (equation.2.18.16) 1917 0 R] +/Limits [(equation.2.18.14a) (equation.2.18.16)] +>> endobj +2849 0 obj << +/Names [(equation.2.18.17) 1918 0 R (equation.2.18.18) 1919 0 R (equation.2.18.2) 1881 0 R (equation.2.18.3) 1882 0 R (equation.2.18.4) 1883 0 R (equation.2.18.5) 1884 0 R] +/Limits [(equation.2.18.17) (equation.2.18.5)] +>> endobj +2850 0 obj << +/Names [(equation.2.18.5a) 1885 0 R (equation.2.18.5b) 1886 0 R (equation.2.18.5c) 1887 0 R (equation.2.18.6) 1888 0 R (equation.2.18.6a) 1889 0 R (equation.2.18.6b) 1890 0 R] +/Limits [(equation.2.18.5a) (equation.2.18.6b)] +>> endobj +2851 0 obj << +/Names [(equation.2.18.6c) 1891 0 R (equation.2.18.7) 1892 0 R (equation.2.18.8) 1893 0 R (equation.2.18.9) 1894 0 R (equation.2.18.9a) 1895 0 R (equation.2.18.9b) 1896 0 R] +/Limits [(equation.2.18.6c) (equation.2.18.9b)] +>> endobj +2852 0 obj << +/Names [(equation.2.19.1) 1437 0 R (equation.2.19.10) 1981 0 R (equation.2.19.11) 1982 0 R (equation.2.19.11a) 1983 0 R (equation.2.19.11b) 1984 0 R (equation.2.19.12) 1985 0 R] +/Limits [(equation.2.19.1) (equation.2.19.12)] +>> endobj +2853 0 obj << +/Names [(equation.2.19.13) 1995 0 R (equation.2.19.13a) 1996 0 R (equation.2.19.13b) 1997 0 R (equation.2.19.13c) 1998 0 R (equation.2.19.13d) 1999 0 R (equation.2.19.13e) 2000 0 R] +/Limits [(equation.2.19.13) (equation.2.19.13e)] +>> endobj +2854 0 obj << +/Names [(equation.2.19.13f) 2001 0 R (equation.2.19.13g) 2002 0 R (equation.2.19.13h) 2003 0 R (equation.2.19.13i) 2004 0 R (equation.2.19.14) 2005 0 R (equation.2.19.14a) 2006 0 R] +/Limits [(equation.2.19.13f) (equation.2.19.14a)] +>> endobj +2855 0 obj << +/Names [(equation.2.19.14b) 2007 0 R (equation.2.19.14c) 2008 0 R (equation.2.19.15) 2009 0 R (equation.2.19.15a) 2010 0 R (equation.2.19.15b) 2011 0 R (equation.2.19.15c) 2012 0 R] +/Limits [(equation.2.19.14b) (equation.2.19.15c)] +>> endobj +2856 0 obj << +/Names [(equation.2.19.15d) 2013 0 R (equation.2.19.15e) 2014 0 R (equation.2.19.15f) 2015 0 R (equation.2.19.15g) 2016 0 R (equation.2.19.15h) 2017 0 R (equation.2.19.15i) 2018 0 R] +/Limits [(equation.2.19.15d) (equation.2.19.15i)] +>> endobj +2857 0 obj << +/Names [(equation.2.19.2) 1931 0 R (equation.2.19.3) 1932 0 R (equation.2.19.3a) 1933 0 R (equation.2.19.3b) 1934 0 R (equation.2.19.3c) 1935 0 R (equation.2.19.3d) 1936 0 R] +/Limits [(equation.2.19.2) (equation.2.19.3d)] +>> endobj +2858 0 obj << +/Names [(equation.2.19.3e) 1937 0 R (equation.2.19.4) 1938 0 R (equation.2.19.4a) 1939 0 R (equation.2.19.4b) 1940 0 R (equation.2.19.4c) 1941 0 R (equation.2.19.4d) 1942 0 R] +/Limits [(equation.2.19.3e) (equation.2.19.4d)] +>> endobj +2859 0 obj << +/Names [(equation.2.19.4e) 1943 0 R (equation.2.19.4f) 1944 0 R (equation.2.19.5) 1950 0 R (equation.2.19.5a) 1951 0 R (equation.2.19.5b) 1952 0 R (equation.2.19.5c) 1953 0 R] +/Limits [(equation.2.19.4e) (equation.2.19.5c)] +>> endobj +2860 0 obj << +/Names [(equation.2.19.5d) 1954 0 R (equation.2.19.5e) 1955 0 R (equation.2.19.5f) 1956 0 R (equation.2.19.5g) 1957 0 R (equation.2.19.5h) 1958 0 R (equation.2.19.5i) 1959 0 R] +/Limits [(equation.2.19.5d) (equation.2.19.5i)] +>> endobj +2861 0 obj << +/Names [(equation.2.19.6) 1960 0 R (equation.2.19.6a) 1961 0 R (equation.2.19.6b) 1962 0 R (equation.2.19.6c) 1963 0 R (equation.2.19.7) 1964 0 R (equation.2.19.8) 1970 0 R] +/Limits [(equation.2.19.6) (equation.2.19.8)] +>> endobj +2862 0 obj << +/Names [(equation.2.19.8a) 1971 0 R (equation.2.19.8b) 1972 0 R (equation.2.19.8c) 1973 0 R (equation.2.19.8d) 1974 0 R (equation.2.19.8e) 1975 0 R (equation.2.19.8f) 1976 0 R] +/Limits [(equation.2.19.8a) (equation.2.19.8f)] +>> endobj +2863 0 obj << +/Names [(equation.2.19.8g) 1977 0 R (equation.2.19.8h) 1978 0 R (equation.2.19.8i) 1979 0 R (equation.2.19.9) 1980 0 R (equation.2.2.1) 946 0 R (equation.2.2.10) 978 0 R] +/Limits [(equation.2.19.8g) (equation.2.2.10)] +>> endobj +2864 0 obj << +/Names [(equation.2.2.11) 979 0 R (equation.2.2.11a) 980 0 R (equation.2.2.11b) 981 0 R (equation.2.2.12) 982 0 R (equation.2.2.12a) 983 0 R (equation.2.2.12b) 984 0 R] +/Limits [(equation.2.2.11) (equation.2.2.12b)] +>> endobj +2865 0 obj << +/Names [(equation.2.2.12c) 985 0 R (equation.2.2.13) 986 0 R (equation.2.2.14) 987 0 R (equation.2.2.15) 988 0 R (equation.2.2.16) 989 0 R (equation.2.2.17) 990 0 R] +/Limits [(equation.2.2.12c) (equation.2.2.17)] +>> endobj +2866 0 obj << +/Names [(equation.2.2.18) 1001 0 R (equation.2.2.19) 1002 0 R (equation.2.2.2) 947 0 R (equation.2.2.20) 1003 0 R (equation.2.2.20a) 1004 0 R (equation.2.2.20b) 1005 0 R] +/Limits [(equation.2.2.18) (equation.2.2.20b)] +>> endobj +2867 0 obj << +/Names [(equation.2.2.21) 1006 0 R (equation.2.2.21a) 1007 0 R (equation.2.2.21b) 1008 0 R (equation.2.2.22) 1010 0 R (equation.2.2.23) 1011 0 R (equation.2.2.24) 1012 0 R] +/Limits [(equation.2.2.21) (equation.2.2.24)] +>> endobj +2868 0 obj << +/Names [(equation.2.2.25) 1013 0 R (equation.2.2.25a) 1014 0 R (equation.2.2.25b) 1015 0 R (equation.2.2.25c) 1016 0 R (equation.2.2.26) 1024 0 R (equation.2.2.26a) 1025 0 R] +/Limits [(equation.2.2.25) (equation.2.2.26a)] +>> endobj +2869 0 obj << +/Names [(equation.2.2.26b) 1026 0 R (equation.2.2.26c) 1027 0 R (equation.2.2.27) 1028 0 R (equation.2.2.28) 1029 0 R (equation.2.2.28a) 1030 0 R (equation.2.2.28b) 1031 0 R] +/Limits [(equation.2.2.26b) (equation.2.2.28b)] +>> endobj +2870 0 obj << +/Names [(equation.2.2.29) 1032 0 R (equation.2.2.29a) 1033 0 R (equation.2.2.29b) 1034 0 R (equation.2.2.2a) 948 0 R (equation.2.2.2b) 949 0 R (equation.2.2.2c) 950 0 R] +/Limits [(equation.2.2.29) (equation.2.2.2c)] +>> endobj +2871 0 obj << +/Names [(equation.2.2.3) 951 0 R (equation.2.2.30) 1035 0 R (equation.2.2.31) 1036 0 R (equation.2.2.31a) 1037 0 R (equation.2.2.31b) 1038 0 R (equation.2.2.32) 1040 0 R] +/Limits [(equation.2.2.3) (equation.2.2.32)] +>> endobj +2872 0 obj << +/Names [(equation.2.2.33) 1041 0 R (equation.2.2.34) 1050 0 R (equation.2.2.34a) 1051 0 R (equation.2.2.34b) 1052 0 R (equation.2.2.34c) 1053 0 R (equation.2.2.34d) 1054 0 R] +/Limits [(equation.2.2.33) (equation.2.2.34d)] +>> endobj +2873 0 obj << +/Names [(equation.2.2.34e) 1055 0 R (equation.2.2.35) 1056 0 R (equation.2.2.36) 1057 0 R (equation.2.2.37) 1058 0 R (equation.2.2.38) 1059 0 R (equation.2.2.38a) 1060 0 R] +/Limits [(equation.2.2.34e) (equation.2.2.38a)] +>> endobj +2874 0 obj << +/Names [(equation.2.2.38b) 1061 0 R (equation.2.2.38c) 1062 0 R (equation.2.2.39) 1063 0 R (equation.2.2.39a) 1064 0 R (equation.2.2.39b) 1065 0 R (equation.2.2.39c) 1066 0 R] +/Limits [(equation.2.2.38b) (equation.2.2.39c)] +>> endobj +2875 0 obj << +/Names [(equation.2.2.39d) 1067 0 R (equation.2.2.39e) 1068 0 R (equation.2.2.3a) 952 0 R (equation.2.2.3b) 953 0 R (equation.2.2.3c) 954 0 R (equation.2.2.3d) 955 0 R] +/Limits [(equation.2.2.39d) (equation.2.2.3d)] +>> endobj +2876 0 obj << +/Names [(equation.2.2.4) 956 0 R (equation.2.2.40) 1069 0 R (equation.2.2.40a) 1070 0 R (equation.2.2.40b) 1071 0 R (equation.2.2.41) 1080 0 R (equation.2.2.42) 1081 0 R] +/Limits [(equation.2.2.4) (equation.2.2.42)] +>> endobj +2877 0 obj << +/Names [(equation.2.2.43) 1082 0 R (equation.2.2.44) 1083 0 R (equation.2.2.45) 1084 0 R (equation.2.2.45a) 1085 0 R (equation.2.2.45b) 1086 0 R (equation.2.2.46) 1087 0 R] +/Limits [(equation.2.2.43) (equation.2.2.46)] +>> endobj +2878 0 obj << +/Names [(equation.2.2.47) 1089 0 R (equation.2.2.48) 1090 0 R (equation.2.2.49) 1091 0 R (equation.2.2.49a) 1092 0 R (equation.2.2.49b) 1093 0 R (equation.2.2.49c) 1094 0 R] +/Limits [(equation.2.2.47) (equation.2.2.49c)] +>> endobj +2879 0 obj << +/Names [(equation.2.2.4a) 957 0 R (equation.2.2.4b) 958 0 R (equation.2.2.5) 959 0 R (equation.2.2.50) 1095 0 R (equation.2.2.50a) 1096 0 R (equation.2.2.50b) 1097 0 R] +/Limits [(equation.2.2.4a) (equation.2.2.50b)] +>> endobj +2880 0 obj << +/Names [(equation.2.2.51) 1103 0 R (equation.2.2.51a) 1104 0 R (equation.2.2.51b) 1105 0 R (equation.2.2.51c) 1106 0 R (equation.2.2.51d) 1107 0 R (equation.2.2.51e) 1108 0 R] +/Limits [(equation.2.2.51) (equation.2.2.51e)] +>> endobj +2881 0 obj << +/Names [(equation.2.2.51f) 1109 0 R (equation.2.2.52) 1110 0 R (equation.2.2.52a) 1111 0 R (equation.2.2.52b) 1112 0 R (equation.2.2.52c) 1113 0 R (equation.2.2.53) 1114 0 R] +/Limits [(equation.2.2.51f) (equation.2.2.53)] +>> endobj +2882 0 obj << +/Names [(equation.2.2.54) 1115 0 R (equation.2.2.55) 1116 0 R (equation.2.2.55a) 1117 0 R (equation.2.2.55b) 1118 0 R (equation.2.2.56) 1119 0 R (equation.2.2.57) 1120 0 R] +/Limits [(equation.2.2.54) (equation.2.2.57)] +>> endobj +2883 0 obj << +/Names [(equation.2.2.58) 1127 0 R (equation.2.2.58a) 1128 0 R (equation.2.2.58b) 1129 0 R (equation.2.2.58c) 1130 0 R (equation.2.2.59) 1131 0 R (equation.2.2.59a) 1132 0 R] +/Limits [(equation.2.2.58) (equation.2.2.59a)] +>> endobj +2884 0 obj << +/Names [(equation.2.2.59b) 1133 0 R (equation.2.2.59c) 1134 0 R (equation.2.2.6) 960 0 R (equation.2.2.60) 1135 0 R (equation.2.2.61) 1136 0 R (equation.2.2.62) 1137 0 R] +/Limits [(equation.2.2.59b) (equation.2.2.62)] +>> endobj +2885 0 obj << +/Names [(equation.2.2.63) 1138 0 R (equation.2.2.63a) 1139 0 R (equation.2.2.63b) 1140 0 R (equation.2.2.64) 1141 0 R (equation.2.2.65) 1142 0 R (equation.2.2.66) 1149 0 R] +/Limits [(equation.2.2.63) (equation.2.2.66)] +>> endobj +2886 0 obj << +/Names [(equation.2.2.67) 1150 0 R (equation.2.2.67a) 1151 0 R (equation.2.2.67b) 1152 0 R (equation.2.2.67c) 1153 0 R (equation.2.2.67d) 1154 0 R (equation.2.2.67e) 1155 0 R] +/Limits [(equation.2.2.67) (equation.2.2.67e)] +>> endobj +2887 0 obj << +/Names [(equation.2.2.68) 1156 0 R (equation.2.2.68a) 1157 0 R (equation.2.2.68b) 1158 0 R (equation.2.2.68c) 1159 0 R (equation.2.2.69) 1160 0 R (equation.2.2.7) 961 0 R] +/Limits [(equation.2.2.68) (equation.2.2.7)] +>> endobj +2888 0 obj << +/Names [(equation.2.2.70) 1161 0 R (equation.2.2.71) 1162 0 R (equation.2.2.72) 1163 0 R (equation.2.2.73) 1164 0 R (equation.2.2.74) 1165 0 R (equation.2.2.74a) 1166 0 R] +/Limits [(equation.2.2.70) (equation.2.2.74a)] +>> endobj +2889 0 obj << +/Names [(equation.2.2.74b) 1167 0 R (equation.2.2.75) 1173 0 R (equation.2.2.76) 1174 0 R (equation.2.2.77) 1175 0 R (equation.2.2.77a) 1176 0 R (equation.2.2.77b) 1177 0 R] +/Limits [(equation.2.2.74b) (equation.2.2.77b)] +>> endobj +2890 0 obj << +/Names [(equation.2.2.77c) 1178 0 R (equation.2.2.77d) 1179 0 R (equation.2.2.77e) 1180 0 R (equation.2.2.78) 1181 0 R (equation.2.2.78a) 1182 0 R (equation.2.2.78b) 1183 0 R] +/Limits [(equation.2.2.77c) (equation.2.2.78b)] +>> endobj +2891 0 obj << +/Names [(equation.2.2.79) 1184 0 R (equation.2.2.8) 962 0 R (equation.2.2.80) 1185 0 R (equation.2.2.80a) 1186 0 R (equation.2.2.80b) 1187 0 R (equation.2.2.81) 1188 0 R] +/Limits [(equation.2.2.79) (equation.2.2.81)] +>> endobj +2892 0 obj << +/Names [(equation.2.2.81a) 1189 0 R (equation.2.2.81b) 1190 0 R (equation.2.2.9) 977 0 R (equation.2.20.1) 2024 0 R (equation.2.20.2) 2025 0 R (equation.2.20.2a) 2026 0 R] +/Limits [(equation.2.2.81a) (equation.2.20.2a)] +>> endobj +2893 0 obj << +/Names [(equation.2.20.3) 2027 0 R (equation.2.20.4) 2028 0 R (equation.2.21.1) 2034 0 R (equation.2.21.10) 2050 0 R (equation.2.21.11) 2061 0 R (equation.2.21.12) 2062 0 R] +/Limits [(equation.2.20.3) (equation.2.21.12)] +>> endobj +2894 0 obj << +/Names [(equation.2.21.13) 2063 0 R (equation.2.21.13a) 2064 0 R (equation.2.21.13b) 2065 0 R (equation.2.21.14) 2066 0 R (equation.2.21.15) 2067 0 R (equation.2.21.16) 2068 0 R] +/Limits [(equation.2.21.13) (equation.2.21.16)] +>> endobj +2895 0 obj << +/Names [(equation.2.21.2) 2035 0 R (equation.2.21.2a) 2036 0 R (equation.2.21.2b) 2037 0 R (equation.2.21.3) 2038 0 R (equation.2.21.3a) 2039 0 R (equation.2.21.3b) 2040 0 R] +/Limits [(equation.2.21.2) (equation.2.21.3b)] +>> endobj +2896 0 obj << +/Names [(equation.2.21.3c) 2041 0 R (equation.2.21.4) 2042 0 R (equation.2.21.5) 2043 0 R (equation.2.21.6) 2044 0 R (equation.2.21.6a) 2045 0 R (equation.2.21.6b) 2046 0 R] +/Limits [(equation.2.21.3c) (equation.2.21.6b)] +>> endobj +2897 0 obj << +/Names [(equation.2.21.7) 2047 0 R (equation.2.21.8) 2048 0 R (equation.2.21.9) 2049 0 R (equation.2.22.1) 2077 0 R (equation.2.22.10) 2107 0 R (equation.2.22.11) 2108 0 R] +/Limits [(equation.2.21.7) (equation.2.22.11)] +>> endobj +2898 0 obj << +/Names [(equation.2.22.11a) 2109 0 R (equation.2.22.11b) 2110 0 R (equation.2.22.11c) 2111 0 R (equation.2.22.11d) 2112 0 R (equation.2.22.12) 2113 0 R (equation.2.22.12a) 2114 0 R] +/Limits [(equation.2.22.11a) (equation.2.22.12a)] +>> endobj +2899 0 obj << +/Names [(equation.2.22.12b) 2115 0 R (equation.2.22.13) 2116 0 R (equation.2.22.14) 2122 0 R (equation.2.22.15) 2123 0 R (equation.2.22.16) 2124 0 R (equation.2.22.16a) 2125 0 R] +/Limits [(equation.2.22.12b) (equation.2.22.16a)] +>> endobj +2900 0 obj << +/Names [(equation.2.22.16b) 2126 0 R (equation.2.22.16c) 2127 0 R (equation.2.22.17) 2128 0 R (equation.2.22.18) 2129 0 R (equation.2.22.18a) 2130 0 R (equation.2.22.18b) 2131 0 R] +/Limits [(equation.2.22.16b) (equation.2.22.18b)] +>> endobj +2901 0 obj << +/Names [(equation.2.22.19) 2132 0 R (equation.2.22.2) 2078 0 R (equation.2.22.2a) 2079 0 R (equation.2.22.2b) 2080 0 R (equation.2.22.2c) 2081 0 R (equation.2.22.2d) 2082 0 R] +/Limits [(equation.2.22.19) (equation.2.22.2d)] +>> endobj +2902 0 obj << +/Names [(equation.2.22.2e) 2083 0 R (equation.2.22.2f) 2084 0 R (equation.2.22.2g) 2085 0 R (equation.2.22.3) 2086 0 R (equation.2.22.3a) 2087 0 R (equation.2.22.3b) 2088 0 R] +/Limits [(equation.2.22.2e) (equation.2.22.3b)] +>> endobj +2903 0 obj << +/Names [(equation.2.22.3c) 2089 0 R (equation.2.22.4) 2090 0 R (equation.2.22.5) 2091 0 R (equation.2.22.5a) 2092 0 R (equation.2.22.5b) 2093 0 R (equation.2.22.6) 2099 0 R] +/Limits [(equation.2.22.3c) (equation.2.22.6)] +>> endobj +2904 0 obj << +/Names [(equation.2.22.6a) 2100 0 R (equation.2.22.6b) 2101 0 R (equation.2.22.7) 2102 0 R (equation.2.22.8) 2103 0 R (equation.2.22.8a) 2104 0 R (equation.2.22.8b) 2105 0 R] +/Limits [(equation.2.22.6a) (equation.2.22.8b)] +>> endobj +2905 0 obj << +/Names [(equation.2.22.9) 2106 0 R (equation.2.23.1) 2141 0 R (equation.2.23.10) 2156 0 R (equation.2.23.11) 2157 0 R (equation.2.23.12) 2158 0 R (equation.2.23.13) 2159 0 R] +/Limits [(equation.2.22.9) (equation.2.23.13)] +>> endobj +2906 0 obj << +/Names [(equation.2.23.14) 2160 0 R (equation.2.23.15) 2161 0 R (equation.2.23.16) 2162 0 R (equation.2.23.16a) 2163 0 R (equation.2.23.17) 2168 0 R (equation.2.23.18) 2169 0 R] +/Limits [(equation.2.23.14) (equation.2.23.18)] +>> endobj +2907 0 obj << +/Names [(equation.2.23.19) 2170 0 R (equation.2.23.2) 2142 0 R (equation.2.23.20) 2171 0 R (equation.2.23.21) 2172 0 R (equation.2.23.22) 2173 0 R (equation.2.23.23) 2174 0 R] +/Limits [(equation.2.23.19) (equation.2.23.23)] +>> endobj +2908 0 obj << +/Names [(equation.2.23.24) 2175 0 R (equation.2.23.24a) 2176 0 R (equation.2.23.24b) 2177 0 R (equation.2.23.25) 2185 0 R (equation.2.23.25a) 2186 0 R (equation.2.23.25b) 2187 0 R] +/Limits [(equation.2.23.24) (equation.2.23.25b)] +>> endobj +2909 0 obj << +/Names [(equation.2.23.26) 2188 0 R (equation.2.23.27) 2189 0 R (equation.2.23.28) 2190 0 R (equation.2.23.28a) 2191 0 R (equation.2.23.28b) 2192 0 R (equation.2.23.29) 2193 0 R] +/Limits [(equation.2.23.26) (equation.2.23.29)] +>> endobj +2910 0 obj << +/Names [(equation.2.23.29a) 2194 0 R (equation.2.23.29b) 2195 0 R (equation.2.23.3) 2143 0 R (equation.2.23.30) 2203 0 R (equation.2.23.30a) 2204 0 R (equation.2.23.30b) 2205 0 R] +/Limits [(equation.2.23.29a) (equation.2.23.30b)] +>> endobj +2911 0 obj << +/Names [(equation.2.23.31) 2206 0 R (equation.2.23.31a) 2207 0 R (equation.2.23.31b) 2208 0 R (equation.2.23.4) 2144 0 R (equation.2.23.5) 2145 0 R (equation.2.23.5a) 2146 0 R] +/Limits [(equation.2.23.31) (equation.2.23.5a)] +>> endobj +2912 0 obj << +/Names [(equation.2.23.6) 2147 0 R (equation.2.23.7) 2148 0 R (equation.2.23.8) 2149 0 R (equation.2.23.9) 2155 0 R (equation.2.24.1) 2214 0 R (equation.2.24.2) 2215 0 R] +/Limits [(equation.2.23.6) (equation.2.24.2)] +>> endobj +2913 0 obj << +/Names [(equation.2.24.3) 2216 0 R (equation.2.24.4) 2217 0 R (equation.2.24.5) 2218 0 R (equation.2.24.6) 2219 0 R (equation.2.24.7) 2220 0 R (equation.2.25.1) 2226 0 R] +/Limits [(equation.2.24.3) (equation.2.25.1)] +>> endobj +2914 0 obj << +/Names [(equation.2.25.10) 2249 0 R (equation.2.25.11) 2250 0 R (equation.2.25.12) 2251 0 R (equation.2.25.13) 2252 0 R (equation.2.25.13a) 2253 0 R (equation.2.25.13b) 2254 0 R] +/Limits [(equation.2.25.10) (equation.2.25.13b)] +>> endobj +2915 0 obj << +/Names [(equation.2.25.14) 2255 0 R (equation.2.25.15) 2256 0 R (equation.2.25.15a) 2257 0 R (equation.2.25.15b) 2258 0 R (equation.2.25.16) 2259 0 R (equation.2.25.17) 2260 0 R] +/Limits [(equation.2.25.14) (equation.2.25.17)] +>> endobj +2916 0 obj << +/Names [(equation.2.25.18) 2261 0 R (equation.2.25.2) 2227 0 R (equation.2.25.3) 1945 0 R (equation.2.25.4) 2228 0 R (equation.2.25.4a) 2229 0 R (equation.2.25.4b) 2230 0 R] +/Limits [(equation.2.25.18) (equation.2.25.4b)] +>> endobj +2917 0 obj << +/Names [(equation.2.25.4c) 2231 0 R (equation.2.25.5) 2232 0 R (equation.2.25.5a) 2233 0 R (equation.2.25.5b) 2234 0 R (equation.2.25.5c) 2235 0 R (equation.2.25.6) 2236 0 R] +/Limits [(equation.2.25.4c) (equation.2.25.6)] +>> endobj +2918 0 obj << +/Names [(equation.2.25.7) 2237 0 R (equation.2.25.8) 2238 0 R (equation.2.25.8a) 2239 0 R (equation.2.25.8b) 2240 0 R (equation.2.25.8c) 2241 0 R (equation.2.25.9) 2248 0 R] +/Limits [(equation.2.25.7) (equation.2.25.9)] +>> endobj +2919 0 obj << +/Names [(equation.2.26.1) 2270 0 R (equation.2.26.10) 2292 0 R (equation.2.26.11) 2293 0 R (equation.2.26.12) 2294 0 R (equation.2.26.12a) 2295 0 R (equation.2.26.12b) 2296 0 R] +/Limits [(equation.2.26.1) (equation.2.26.12b)] +>> endobj +2920 0 obj << +/Names [(equation.2.26.13) 2297 0 R (equation.2.26.13a) 2298 0 R (equation.2.26.13b) 2299 0 R (equation.2.26.14) 2300 0 R (equation.2.26.15) 2301 0 R (equation.2.26.16) 2302 0 R] +/Limits [(equation.2.26.13) (equation.2.26.16)] +>> endobj +2921 0 obj << +/Names [(equation.2.26.17) 2303 0 R (equation.2.26.18) 2311 0 R (equation.2.26.19) 2312 0 R (equation.2.26.19a) 2313 0 R (equation.2.26.19b) 2314 0 R (equation.2.26.19c) 2315 0 R] +/Limits [(equation.2.26.17) (equation.2.26.19c)] +>> endobj +2922 0 obj << +/Names [(equation.2.26.2) 2271 0 R (equation.2.26.20) 2316 0 R (equation.2.26.20a) 2317 0 R (equation.2.26.20b) 2318 0 R (equation.2.26.21) 2319 0 R (equation.2.26.22) 2320 0 R] +/Limits [(equation.2.26.2) (equation.2.26.22)] +>> endobj +2923 0 obj << +/Names [(equation.2.26.23) 2321 0 R (equation.2.26.24) 2322 0 R (equation.2.26.25) 2323 0 R (equation.2.26.26) 2324 0 R (equation.2.26.27) 2325 0 R (equation.2.26.28) 2330 0 R] +/Limits [(equation.2.26.23) (equation.2.26.28)] +>> endobj +2924 0 obj << +/Names [(equation.2.26.29) 2331 0 R (equation.2.26.29a) 2332 0 R (equation.2.26.29b) 2333 0 R (equation.2.26.29c) 2334 0 R (equation.2.26.29d) 2335 0 R (equation.2.26.2a) 2272 0 R] +/Limits [(equation.2.26.29) (equation.2.26.2a)] +>> endobj +2925 0 obj << +/Names [(equation.2.26.2b) 2273 0 R (equation.2.26.2c) 2274 0 R (equation.2.26.2d) 2275 0 R (equation.2.26.3) 2276 0 R (equation.2.26.30) 2336 0 R (equation.2.26.30a) 2337 0 R] +/Limits [(equation.2.26.2b) (equation.2.26.30a)] +>> endobj +2926 0 obj << +/Names [(equation.2.26.30b) 2338 0 R (equation.2.26.30c) 2339 0 R (equation.2.26.31) 2340 0 R (equation.2.26.32) 2341 0 R (equation.2.26.33) 2342 0 R (equation.2.26.34) 2343 0 R] +/Limits [(equation.2.26.30b) (equation.2.26.34)] +>> endobj +2927 0 obj << +/Names [(equation.2.26.34a) 2344 0 R (equation.2.26.34b) 2345 0 R (equation.2.26.35) 2346 0 R (equation.2.26.36) 2347 0 R (equation.2.26.36a) 2348 0 R (equation.2.26.36b) 2349 0 R] +/Limits [(equation.2.26.34a) (equation.2.26.36b)] +>> endobj +2928 0 obj << +/Names [(equation.2.26.37) 2356 0 R (equation.2.26.38) 2357 0 R (equation.2.26.39) 2358 0 R (equation.2.26.39a) 2359 0 R (equation.2.26.39b) 2360 0 R (equation.2.26.39c) 2361 0 R] +/Limits [(equation.2.26.37) (equation.2.26.39c)] +>> endobj +2929 0 obj << +/Names [(equation.2.26.3a) 2277 0 R (equation.2.26.3b) 2278 0 R (equation.2.26.3c) 2279 0 R (equation.2.26.4) 2280 0 R (equation.2.26.40) 2362 0 R (equation.2.26.41) 2363 0 R] +/Limits [(equation.2.26.3a) (equation.2.26.41)] +>> endobj +2930 0 obj << +/Names [(equation.2.26.42) 2364 0 R (equation.2.26.43) 2365 0 R (equation.2.26.44) 2366 0 R (equation.2.26.45) 2367 0 R (equation.2.26.46) 2368 0 R (equation.2.26.47) 2369 0 R] +/Limits [(equation.2.26.42) (equation.2.26.47)] +>> endobj +2931 0 obj << +/Names [(equation.2.26.47a) 2370 0 R (equation.2.26.47b) 2371 0 R (equation.2.26.48) 2372 0 R (equation.2.26.5) 2281 0 R (equation.2.26.6) 2282 0 R (equation.2.26.7) 2283 0 R] +/Limits [(equation.2.26.47a) (equation.2.26.7)] +>> endobj +2932 0 obj << +/Names [(equation.2.26.8) 2284 0 R (equation.2.26.9) 2291 0 R (equation.2.27.1) 2379 0 R (equation.2.27.2) 2380 0 R (equation.2.27.2a) 2381 0 R (equation.2.27.2b) 2382 0 R] +/Limits [(equation.2.26.8) (equation.2.27.2b)] +>> endobj +2933 0 obj << +/Names [(equation.2.27.3) 2383 0 R (equation.2.27.3a) 2384 0 R (equation.2.27.3b) 2385 0 R (equation.2.27.3c) 2386 0 R (equation.2.27.3d) 2387 0 R (equation.2.27.3e) 2388 0 R] +/Limits [(equation.2.27.3) (equation.2.27.3e)] +>> endobj +2934 0 obj << +/Names [(equation.2.27.3f) 2389 0 R (equation.2.27.3g) 2390 0 R (equation.2.27.3h) 2391 0 R (equation.2.27.3i) 2392 0 R (equation.2.27.3j) 2393 0 R (equation.2.27.3k) 2394 0 R] +/Limits [(equation.2.27.3f) (equation.2.27.3k)] +>> endobj +2935 0 obj << +/Names [(equation.2.27.3l) 2395 0 R (equation.2.27.4) 2396 0 R (equation.2.27.5) 2397 0 R (equation.2.28.1) 2403 0 R (equation.2.28.10) 2414 0 R (equation.2.28.10a) 2415 0 R] +/Limits [(equation.2.27.3l) (equation.2.28.10a)] +>> endobj +2936 0 obj << +/Names [(equation.2.28.10b) 2416 0 R (equation.2.28.10c) 2417 0 R (equation.2.28.11) 2423 0 R (equation.2.28.2) 2404 0 R (equation.2.28.3) 2405 0 R (equation.2.28.4) 2406 0 R] +/Limits [(equation.2.28.10b) (equation.2.28.4)] +>> endobj +2937 0 obj << +/Names [(equation.2.28.5) 2407 0 R (equation.2.28.6) 2408 0 R (equation.2.28.7) 2409 0 R (equation.2.28.8) 2410 0 R (equation.2.28.8a) 2411 0 R (equation.2.28.8b) 2412 0 R] +/Limits [(equation.2.28.5) (equation.2.28.8b)] +>> endobj +2938 0 obj << +/Names [(equation.2.28.9) 2413 0 R (equation.2.29.1) 2430 0 R (equation.2.29.2) 2431 0 R (equation.2.29.2a) 2432 0 R (equation.2.29.2b) 2433 0 R (equation.2.29.2c) 2434 0 R] +/Limits [(equation.2.28.9) (equation.2.29.2c)] +>> endobj +2939 0 obj << +/Names [(equation.2.29.2d) 2435 0 R (equation.2.29.2e) 2436 0 R (equation.2.29.2f) 2437 0 R (equation.2.29.3) 2438 0 R (equation.2.29.3a) 2439 0 R (equation.2.29.3b) 2440 0 R] +/Limits [(equation.2.29.2d) (equation.2.29.3b)] +>> endobj +2940 0 obj << +/Names [(equation.2.29.3c) 2441 0 R (equation.2.29.3d) 2442 0 R (equation.2.29.3e) 2443 0 R (equation.2.29.3f) 2444 0 R (equation.2.29.3g) 2445 0 R (equation.2.29.3h) 2446 0 R] +/Limits [(equation.2.29.3c) (equation.2.29.3h)] +>> endobj +2941 0 obj << +/Names [(equation.2.29.4) 2447 0 R (equation.2.29.4a) 2448 0 R (equation.2.29.4b) 2449 0 R (equation.2.29.4c) 2450 0 R (equation.2.29.5) 2457 0 R (equation.2.29.5a) 2458 0 R] +/Limits [(equation.2.29.4) (equation.2.29.5a)] +>> endobj +2942 0 obj << +/Names [(equation.2.29.5b) 2459 0 R (equation.2.29.6) 2460 0 R (equation.2.29.7) 2461 0 R (equation.2.3.1) 1199 0 R (equation.2.3.2) 1200 0 R (equation.2.3.2a) 1201 0 R] +/Limits [(equation.2.29.5b) (equation.2.3.2a)] +>> endobj +2943 0 obj << +/Names [(equation.2.3.2b) 1202 0 R (equation.2.3.2c) 1203 0 R (equation.2.3.3) 1204 0 R (equation.2.3.4) 1205 0 R (equation.2.3.4a) 1206 0 R (equation.2.3.4b) 1207 0 R] +/Limits [(equation.2.3.2b) (equation.2.3.4b)] +>> endobj +2944 0 obj << +/Names [(equation.2.3.4c) 1208 0 R (equation.2.3.4d) 1209 0 R (equation.2.3.4e) 1210 0 R (equation.2.3.4f) 1211 0 R (equation.2.3.5) 1212 0 R (equation.2.3.5a) 1213 0 R] +/Limits [(equation.2.3.4c) (equation.2.3.5a)] +>> endobj +2945 0 obj << +/Names [(equation.2.3.5b) 1214 0 R (equation.2.3.5c) 1215 0 R (equation.2.3.5d) 1216 0 R (equation.2.3.6) 1217 0 R (equation.2.3.7) 1218 0 R (equation.2.30.1) 2469 0 R] +/Limits [(equation.2.3.5b) (equation.2.30.1)] +>> endobj +2946 0 obj << +/Names [(equation.2.30.2) 2470 0 R (equation.2.30.2a) 2471 0 R (equation.2.30.2b) 2472 0 R (equation.2.30.2c) 2473 0 R (equation.2.30.2d) 2474 0 R (equation.2.30.2e) 2475 0 R] +/Limits [(equation.2.30.2) (equation.2.30.2e)] +>> endobj +2947 0 obj << +/Names [(equation.2.30.2f) 2476 0 R (equation.2.30.2g) 2477 0 R (equation.2.30.2h) 2478 0 R (equation.2.30.3) 2479 0 R (equation.2.30.4) 2480 0 R (equation.2.30.5) 2481 0 R] +/Limits [(equation.2.30.2f) (equation.2.30.5)] +>> endobj +2948 0 obj << +/Names [(equation.2.30.6) 2482 0 R (equation.2.4.1) 1228 0 R (equation.2.4.10) 1244 0 R (equation.2.4.11) 1254 0 R (equation.2.4.12) 1255 0 R (equation.2.4.13) 1256 0 R] +/Limits [(equation.2.30.6) (equation.2.4.13)] +>> endobj +2949 0 obj << +/Names [(equation.2.4.13a) 1257 0 R (equation.2.4.13b) 1258 0 R (equation.2.4.14) 1259 0 R (equation.2.4.15) 1260 0 R (equation.2.4.2) 1229 0 R (equation.2.4.2a) 1230 0 R] +/Limits [(equation.2.4.13a) (equation.2.4.2a)] +>> endobj +2950 0 obj << +/Names [(equation.2.4.2b) 1231 0 R (equation.2.4.3) 1232 0 R (equation.2.4.3a) 1233 0 R (equation.2.4.3b) 1234 0 R (equation.2.4.3c) 1235 0 R (equation.2.4.4) 1236 0 R] +/Limits [(equation.2.4.2b) (equation.2.4.4)] +>> endobj +2951 0 obj << +/Names [(equation.2.4.5) 1237 0 R (equation.2.4.6) 1238 0 R (equation.2.4.6a) 1239 0 R (equation.2.4.6b) 1240 0 R (equation.2.4.7) 1241 0 R (equation.2.4.8) 1242 0 R] +/Limits [(equation.2.4.5) (equation.2.4.8)] +>> endobj +2952 0 obj << +/Names [(equation.2.4.9) 1243 0 R (equation.2.5.1) 1267 0 R (equation.2.5.10) 1278 0 R (equation.2.5.11) 1279 0 R (equation.2.5.12) 1280 0 R (equation.2.5.13) 1288 0 R] +/Limits [(equation.2.4.9) (equation.2.5.13)] +>> endobj +2953 0 obj << +/Names [(equation.2.5.14) 1289 0 R (equation.2.5.14a) 1290 0 R (equation.2.5.14b) 1291 0 R (equation.2.5.15) 1292 0 R (equation.2.5.16) 1293 0 R (equation.2.5.17) 1294 0 R] +/Limits [(equation.2.5.14) (equation.2.5.17)] +>> endobj +2954 0 obj << +/Names [(equation.2.5.2) 1268 0 R (equation.2.5.3) 1269 0 R (equation.2.5.4) 1270 0 R (equation.2.5.5) 1271 0 R (equation.2.5.6) 1272 0 R (equation.2.5.7) 1273 0 R] +/Limits [(equation.2.5.2) (equation.2.5.7)] +>> endobj +2955 0 obj << +/Names [(equation.2.5.7a) 1274 0 R (equation.2.5.7b) 1275 0 R (equation.2.5.8) 1276 0 R (equation.2.5.9) 1277 0 R (equation.2.6.1) 1303 0 R (equation.2.6.2) 1304 0 R] +/Limits [(equation.2.5.7a) (equation.2.6.2)] +>> endobj +2956 0 obj << +/Names [(equation.2.6.3) 1305 0 R (equation.2.6.4) 1306 0 R (equation.2.6.4a) 1307 0 R (equation.2.6.4b) 1308 0 R (equation.2.6.4c) 1309 0 R (equation.2.6.5) 1310 0 R] +/Limits [(equation.2.6.3) (equation.2.6.5)] +>> endobj +2957 0 obj << +/Names [(equation.2.6.6) 1311 0 R (equation.2.6.7) 1312 0 R (equation.2.6.8) 1313 0 R (equation.2.7.1) 1321 0 R (equation.2.7.10) 1346 0 R (equation.2.7.10a) 1347 0 R] +/Limits [(equation.2.6.6) (equation.2.7.10a)] +>> endobj +2958 0 obj << +/Names [(equation.2.7.10b) 1348 0 R (equation.2.7.11) 1349 0 R (equation.2.7.12) 1350 0 R (equation.2.7.2) 1322 0 R (equation.2.7.2a) 1323 0 R (equation.2.7.2b) 1324 0 R] +/Limits [(equation.2.7.10b) (equation.2.7.2b)] +>> endobj 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(subsection.1.9.1) 98 0 R] +/Limits [(subsection.1.8.1) (subsection.1.9.1)] +>> endobj +2998 0 obj << +/Names [(subsection.2.1.1) 146 0 R (subsection.2.1.2) 150 0 R (subsection.2.1.3) 154 0 R (subsection.2.1.4) 158 0 R (subsection.2.1.5) 162 0 R (subsection.2.1.6) 166 0 R] +/Limits [(subsection.2.1.1) (subsection.2.1.6)] +>> endobj +2999 0 obj << +/Names [(subsection.2.11.1) 250 0 R (subsection.2.11.2) 254 0 R (subsection.2.11.3) 258 0 R (subsection.2.12.1) 266 0 R (subsection.2.12.2) 270 0 R (subsection.2.17.1) 294 0 R] +/Limits [(subsection.2.11.1) (subsection.2.17.1)] +>> endobj +3000 0 obj << +/Names [(subsection.2.2.1) 174 0 R (subsection.2.2.2) 178 0 R (subsection.2.2.3) 182 0 R (subsection.2.2.4) 186 0 R (subsection.2.2.5) 190 0 R (subsection.2.2.6) 194 0 R] +/Limits [(subsection.2.2.1) (subsection.2.2.6)] +>> endobj +3001 0 obj << +/Names [(subsection.2.2.7) 198 0 R (subsection.2.2.8) 202 0 R (subsection.2.22.1) 318 0 R (subsection.2.22.2) 322 0 R (subsection.2.23.1) 330 0 R (subsection.2.23.2) 334 0 R] +/Limits [(subsection.2.2.7) (subsection.2.23.2)] +>> endobj +3002 0 obj << +/Names [(subsection.2.23.3) 338 0 R (subsection.2.23.4) 342 0 R (subsection.2.23.5) 346 0 R (subsection.2.23.6) 350 0 R (subsection.2.23.7) 354 0 R (subsection.2.26.1) 370 0 R] +/Limits [(subsection.2.23.3) (subsection.2.26.1)] +>> endobj +3003 0 obj << +/Names [(subsection.2.26.2) 374 0 R (subsection.2.26.3) 378 0 R (subsection.2.26.4) 382 0 R (subsection.2.26.5) 386 0 R (subsection.2.26.6) 390 0 R (subsection.2.6.1) 222 0 R] +/Limits [(subsection.2.26.2) (subsection.2.6.1)] +>> endobj +3004 0 obj << +/Names [(subsection.2.6.2) 226 0 R] +/Limits [(subsection.2.6.2) (subsection.2.6.2)] +>> endobj +3005 0 obj << +/Kids [2752 0 R 2753 0 R 2754 0 R 2755 0 R 2756 0 R 2757 0 R] +/Limits [(Doc-Start) (cite.hackmann2008)] +>> endobj +3006 0 obj << +/Kids [2758 0 R 2759 0 R 2760 0 R 2761 0 R 2762 0 R 2763 0 R] +/Limits [(cite.halilsoy1988) (cite.vanDenBroeck1999)] +>> endobj +3007 0 obj << 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2810 0 R 2811 0 R] +/Limits [(equation.2.11.15a) (equation.2.11.30b)] +>> endobj +3015 0 obj << +/Kids [2812 0 R 2813 0 R 2814 0 R 2815 0 R 2816 0 R 2817 0 R] +/Limits [(equation.2.11.31) (equation.2.11.43d)] +>> endobj +3016 0 obj << +/Kids [2818 0 R 2819 0 R 2820 0 R 2821 0 R 2822 0 R 2823 0 R] +/Limits [(equation.2.11.44) (equation.2.12.13d)] +>> endobj +3017 0 obj << +/Kids [2824 0 R 2825 0 R 2826 0 R 2827 0 R 2828 0 R 2829 0 R] +/Limits [(equation.2.12.14) (equation.2.12.9b)] +>> endobj +3018 0 obj << +/Kids [2830 0 R 2831 0 R 2832 0 R 2833 0 R 2834 0 R 2835 0 R] +/Limits [(equation.2.13.1) (equation.2.14.9)] +>> endobj +3019 0 obj << +/Kids [2836 0 R 2837 0 R 2838 0 R 2839 0 R 2840 0 R 2841 0 R] +/Limits [(equation.2.15.1) (equation.2.17.10e)] +>> endobj +3020 0 obj << +/Kids [2842 0 R 2843 0 R 2844 0 R 2845 0 R 2846 0 R 2847 0 R] +/Limits [(equation.2.17.10f) (equation.2.18.14)] +>> endobj +3021 0 obj << +/Kids [2848 0 R 2849 0 R 2850 0 R 2851 0 R 2852 0 R 2853 0 R] +/Limits [(equation.2.18.14a) (equation.2.19.13e)] +>> endobj +3022 0 obj << +/Kids [2854 0 R 2855 0 R 2856 0 R 2857 0 R 2858 0 R 2859 0 R] +/Limits [(equation.2.19.13f) (equation.2.19.5c)] +>> endobj +3023 0 obj << +/Kids [2860 0 R 2861 0 R 2862 0 R 2863 0 R 2864 0 R 2865 0 R] +/Limits [(equation.2.19.5d) (equation.2.2.17)] +>> endobj +3024 0 obj << +/Kids [2866 0 R 2867 0 R 2868 0 R 2869 0 R 2870 0 R 2871 0 R] +/Limits [(equation.2.2.18) (equation.2.2.32)] +>> endobj +3025 0 obj << +/Kids [2872 0 R 2873 0 R 2874 0 R 2875 0 R 2876 0 R 2877 0 R] +/Limits [(equation.2.2.33) (equation.2.2.46)] +>> endobj +3026 0 obj << +/Kids [2878 0 R 2879 0 R 2880 0 R 2881 0 R 2882 0 R 2883 0 R] +/Limits [(equation.2.2.47) (equation.2.2.59a)] +>> endobj +3027 0 obj << +/Kids [2884 0 R 2885 0 R 2886 0 R 2887 0 R 2888 0 R 2889 0 R] +/Limits [(equation.2.2.59b) (equation.2.2.77b)] +>> endobj +3028 0 obj << +/Kids [2890 0 R 2891 0 R 2892 0 R 2893 0 R 2894 0 R 2895 0 R] +/Limits [(equation.2.2.77c) 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