julianandJulian Piribauer
6600379681
--------- Co-authored-by: Julian Piribauer <julian.piribauer@gmail.com> Reviewed-on: #20
345 lines
15 KiB
Python
345 lines
15 KiB
Python
import logging
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from sage.all import sage_eval, PolynomialRing, QQ, SR, ceil, factorial, floor, log, matrix, prod, var
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load("sage/operators.sage")
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# Logger
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logger = logging.getLogger(__name__)
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logging.basicConfig(level=logging.DEBUG)
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class Period:
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"""
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A class representing a period as a formal power series in z-variables and their logs.
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The coefficients are stored in a dictionary of dictionaries, where the first key is the multi-index
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of the logarithmic part and the second key is the multi-index of the z-variables.
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So, for example, the coefficient of
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log(z0)^2 * log(z1) * z0^3 * z1^2 would be stored as coefficients[(2, 1)][(3, 2)]
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"""
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def _initialise_ring(self):
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z_names = ['z%d' % i for i in range(self.no_variables)]
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log_names = ['L%d' % i for i in range(self.no_variables)]
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ring = PolynomialRing(QQ, z_names + log_names)
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z_gens = ring.gens()[:self.no_variables]
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log_gens = ring.gens()[self.no_variables:]
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return ring, z_gens, log_gens # type: (PolynomialRing, list, list)
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def _period_from_string(self, period_string: str) -> dict:
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ring, z_gens, log_gens = self._initialise_ring()
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log_of_z = dict(zip(z_gens, log_gens))
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def log(zi):
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try:
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return log_of_z[zi]
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except (KeyError, TypeError):
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raise ValueError(
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"log(...) may only be applied to one of the z-variables z0, ..., z%d"
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% (self.no_variables - 1)
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)
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parse_locals = dict(ring.gens_dict())
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parse_locals['log'] = log
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try:
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expression = ring(sage_eval(period_string, locals=parse_locals))
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except Exception as e:
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raise ValueError("Invalid period string: %s" % e)
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coefficients = {}
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for coeff, monomial in expression:
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exponents = monomial.exponents()[0]
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z_index = tuple(exponents[:self.no_variables])
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log_index = tuple(exponents[self.no_variables:])
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coefficients.setdefault(log_index, {})[z_index] = coeff
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return coefficients
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def _period_to_string(self) -> str:
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ring, z_gens, log_gens = self._initialise_ring()
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expression = ring(0)
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for log_index, z_dict in self.coefficients.items():
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for z_index, coeff in z_dict.items():
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monomial = coeff
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for i in range(self.no_variables):
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monomial *= (log_gens[i] ** log_index[i]) * (z_gens[i] ** z_index[i])
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expression += monomial
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log_substitutions = {log_gens[i]: log(SR(z_gens[i])) for i in range(self.no_variables)}
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result = SR(expression).subs(log_substitutions)
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# A nonzero indicial ρ_i means the coefficients above are for z_i^k, but the
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# actual solution is z_i^(ρ_i + k); make that explicit in the printed form.
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indicial_prefactor = prod(
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(SR(z_gens[i]) ** self.indicials[i] for i in range(self.no_variables) if self.indicials[i] != 0),
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SR(1),
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)
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if indicial_prefactor != 1:
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result *= indicial_prefactor
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return str(result)
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def _max_z_degree(self) -> int:
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# Highest total z-degree (sum of the z-multi-index) among all coefficients.
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if not self.coefficients:
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return 0
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return max(
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sum(z_index)
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for z_dict in self.coefficients.values()
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for z_index in z_dict
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)
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def _truncate_coefficients(self, order: int) -> dict:
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# Drop every (log_index, z_index) entry whose total z-degree exceeds order,
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# removing it from the dictionary rather than merely zeroing it out.
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truncated = {}
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for log_index, z_dict in self.coefficients.items():
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kept = {z_index: coeff for z_index, coeff in z_dict.items() if sum(z_index) <= order}
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if kept:
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truncated[log_index] = kept
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return truncated
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def _apply_theta(self, coefficients: dict, index: int) -> dict:
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"""
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Applies the logarithmic derivative theta_i = z_i * d/dz_i once to a coefficients dict of the same
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shape as self.coefficients. It uses the product rule
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theta_i(z^a log(z)^k) = a_i * z^a log(z)^k + k_i * z^a log(z)^(k - e_i).
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z_index entries are offsets from self.indicials: a stored z_index of a really means
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z^(a + self.indicials[index]), so theta_i's eigenvalue is a_i + self.indicials[index]
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rather than the bare a_i (self.indicials is all-zero unless explicitly given, in which
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case this reduces to the ordinary power-series rule).
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"""
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result = {}
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for log_index, z_dict in coefficients.items():
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for z_index, coeff in z_dict.items():
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a_i = z_index[index] + self.indicials[index]
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if a_i != 0:
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inner = result.setdefault(log_index, {})
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inner[z_index] = inner.get(z_index, 0) + a_i * coeff
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k_i = log_index[index]
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if k_i != 0:
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lowered_log_index = log_index[:index] + (k_i - 1,) + log_index[index + 1:]
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inner = result.setdefault(lowered_log_index, {})
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inner[z_index] = inner.get(z_index, 0) + k_i * coeff
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return result
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def apply_operator(self, pf_operator: PFOperator) -> "Period":
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"""
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Applies a PFOperator to this period and returns the result as a new Period,
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truncated to self.order. Each monomial of the operator is normalised as
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coeff * z^p * theta^q (see PFOperator's docstring), so theta^q is applied
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to the period first and the result is then multiplied by coeff * z^p.
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"""
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if pf_operator.no_variables != self.no_variables:
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raise ValueError(
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"Variable count mismatch: period has %d variable(s), operator has %d."
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% (self.no_variables, pf_operator.no_variables)
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)
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result_coefficients = {}
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for monomial_coeff, monomial in pf_operator.operator:
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exponents = monomial.exponents()[0]
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z_exponents = exponents[:self.no_variables]
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theta_exponents = exponents[self.no_variables:]
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term = self.coefficients
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for i, power in enumerate(theta_exponents):
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for _ in range(power):
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term = self._apply_theta(term, i)
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for log_index, z_dict in term.items():
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inner = result_coefficients.setdefault(log_index, {})
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for z_index, coeff in z_dict.items():
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shifted_z_index = tuple(z_index[i] + z_exponents[i] for i in range(self.no_variables))
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inner[shifted_z_index] = inner.get(shifted_z_index, 0) + monomial_coeff * coeff
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result_coefficients = {
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log_index: {z_index: c for z_index, c in z_dict.items() if c != 0}
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for log_index, z_dict in result_coefficients.items()
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}
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result_coefficients = {log_index: z_dict for log_index, z_dict in result_coefficients.items() if z_dict}
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return Period(
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no_variables=self.no_variables,
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coefficients=result_coefficients,
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order=self.order,
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indicials=self.indicials,
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)
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def find_annihilating_operators(self, z_degree: int, theta_degree: int) -> list:
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"""
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Finds operators annihilating this period among PFOperator Ansätze of the given
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z- and theta-degree.
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"""
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ansatz = PFOperatorAnsatz(self.no_variables, theta_degree, z_degree)
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unknowns = ansatz.unknowns
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equations = [
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coeff
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for z_dict in self.apply_operator(ansatz).coefficients.values()
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for coeff in z_dict.values()
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]
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coefficient_rows = [[SR(equation).coefficient(b) for b in unknowns] for equation in equations]
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kernel_basis = matrix(QQ, coefficient_rows, ncols=len(unknowns)).right_kernel().basis()
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operators = []
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for basis_vector in kernel_basis:
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solution = {unknowns[j]: basis_vector[j] for j in range(len(unknowns))}
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solved_operator = ansatz.operator.map_coefficients(lambda c: SR(c).subs(solution))
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operators.append(PFOperator(str(solved_operator), no_variables=self.no_variables))
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return operators
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def __init__(
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self,
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no_variables: int = 1,
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coefficients: dict = None,
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period_string: str = None,
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order: int = None,
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indicials: list = None,
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):
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self.no_variables = no_variables
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# indicials[i] is the (rational) Frobenius exponent ρ_i of z_i: a stored
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# z-index of a really represents z^(a + indicials[i]).
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self.indicials = list(indicials) if indicials is not None else [0] * no_variables
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# Use coefficients or period_string to initialize the period
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if period_string is not None:
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if coefficients is not None:
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raise ValueError("Provide either coefficients or period_string, not both.")
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self.coefficients = self._period_from_string(period_string)
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self.period_string = period_string
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logger.debug("Using string for initialisation.")
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elif coefficients is None:
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self.coefficients = {}
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else:
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self.coefficients = coefficients
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self.period_string = self._period_to_string()
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logger.debug("Using coefficients for initialisation.")
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if order is None:
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self.order = self._max_z_degree()
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logger.debug("Order not provided, using maximum z-degree: %d", self.order)
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else:
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self.coefficients = self._truncate_coefficients(order)
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self.order = order
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self.period_string = self._period_to_string()
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logger.debug("Truncated coefficients to order %d.", self.order)
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logger.info("Initialised Period in %d variables at order %d.", self.no_variables, self.order)
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@classmethod
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def from_l_vectors(cls, l_vectors: list, order: int) -> "Period":
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"""
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Computes the holomorphic (fundamental) period directly from the GLSM/Mori charge
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vectors l^(1), ..., l^(m) of a toric Calabi-Yau (m = number of moduli), via the
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hypergeometric-type sum
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varpi_0(z) = sum_{a in Z_{>=0}^m}
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(prod_j Gamma(1 - l_num^(j).a)) / (prod_i Gamma(1 + l_den^(i).a)) * z^a,
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see e.g. Hosono-Klemm-Theisen-Yau. l_vectors is a list of the m charge vectors, each
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given as [numerator part, denominator part]: the components landing in the numerator
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resp. denominator Gamma-functions (for a Calabi-Yau hypersurface/CICY, these are the
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nef-partition weight(s) resp. the toric divisor charges). For m = 1 the outer list may
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be dropped, e.g. the quintic is given by [[-5], [1, 1, 1, 1, 1]], while P^4_{1,1,2,2,6}[12]
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is given by [[[-6], [3, 2, 1, 0, 0, 0]], [[0], [0, 0, -3, 1, 1, 1]]].
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For a Calabi-Yau charge vector the numerator Gamma-functions never hit a pole, since
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their argument only grows as a grows; only the denominator factors can vanish - at a
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non-positive-integer argument - which is exactly what truncates each a_k to a finite
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range once the others are fixed. This makes summing directly over that range far
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cheaper than solving the full Picard-Fuchs system term by term.
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"""
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# A single charge vector [numerator part, denominator part] has integer entries in its parts.
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if l_vectors and l_vectors[0] and not isinstance(l_vectors[0][0], (list, tuple)):
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l_vectors = [l_vectors]
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if any(len(l_vector) != 2 for l_vector in l_vectors):
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raise ValueError("Each charge vector must be given as [numerator part, denominator part].")
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m = len(l_vectors)
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numerator_charges = [l_vector[0] for l_vector in l_vectors]
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denominator_charges = [l_vector[1] for l_vector in l_vectors]
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len0 = len(numerator_charges[0]) if m else 0
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len1 = len(denominator_charges[0]) if m else 0
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# Each denominator Gamma-factor is linear in a; pivot it on the smallest index j with
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# a nonzero coefficient there. Since a is built up from a_{m-1} down to a_0, every
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# other index still appearing in that linear form is then already fixed, turning the
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# factor's pole at non-positive integers into a concrete bound on a_j.
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pivots = [[] for _ in range(m)]
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for i in range(len1):
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charges = [denominator_charges[k][i] for k in range(m)]
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j = next(k for k in range(m) if charges[k] != 0)
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pivots[j].append(charges)
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coefficients = {}
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a = [0] * m
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def recurse(j, num_partial, den_partial, degree):
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if j < 0:
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numerator = prod(factorial(-x) for x in num_partial)
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denominator = prod(factorial(x) for x in den_partial)
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coefficients[tuple(a)] = QQ(numerator) / QQ(denominator)
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return
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lower, upper = 0, order - degree
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for charges in pivots[j]:
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c = charges[j]
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bound = -QQ(sum(charges[k] * a[k] for k in range(j + 1, m))) / QQ(c)
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if c < 0:
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upper = min(upper, floor(bound))
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else:
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lower = max(lower, ceil(bound))
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for a_j in range(lower, upper + 1):
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a[j] = a_j
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recurse(
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j - 1,
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[num_partial[jj] + numerator_charges[j][jj] * a_j for jj in range(len0)],
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[den_partial[ii] + denominator_charges[j][ii] * a_j for ii in range(len1)],
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degree + a_j,
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)
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recurse(m - 1, [0] * len0, [0] * len1, 0)
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logger.info("Computed holomorphic period from l-vectors for %d moduli at order %d.", m, order)
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return cls(no_variables=m, coefficients={tuple([0] * m): coefficients}, order=order)
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class PeriodAnsatz(Period):
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"""
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A class for period Ansätze, characterised by number of variables and their z- and log-multi-degrees.
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"""
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def __init__(self, no_variables: int, z_degree: int, log_degree: int, indicials: list = None):
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self.no_variables = no_variables
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self.z_degree = z_degree
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self.log_degree = log_degree
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log_indices = _multi_indices(self.no_variables, log_degree)
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z_indices = _multi_indices(self.no_variables, z_degree)
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# Every (log_index, z_index) pair allowed by the degree bounds gets its own
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# fresh unknown, to be solved for once a PFOperator is applied to the Ansatz.
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coefficients = {
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log_index: {
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z_index: var("a_" + "_".join(str(x) for x in log_index + z_index))
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for z_index in z_indices
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}
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for log_index in log_indices
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}
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super().__init__(no_variables=no_variables, coefficients=coefficients, order=z_degree, indicials=indicials)
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self.expansion_coefficients = self.coefficients
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logger.info(
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"Initialised PeriodAnsatz with %d unknown coefficient(s).",
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sum(len(z_dict) for z_dict in self.expansion_coefficients.values()),
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) |