field.py 31.2 KB
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from __future__ import division
import numpy as np

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from d2o import distributed_data_object,\
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    STRATEGIES as DISTRIBUTION_STRATEGIES
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from nifty.config import about,\
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                         nifty_configuration as gc
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from nifty.field_types import FieldType
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from nifty.spaces.space import Space
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from nifty.spaces.power_space import PowerSpace
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import nifty.nifty_utilities as utilities
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from nifty.random import Random

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class Field(object):
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    # ---Initialization methods---
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    def __init__(self, domain=None, val=None, dtype=None, field_type=None,
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                 distribution_strategy=None, copy=False):
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        self.domain = self._parse_domain(domain=domain, val=val)
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        self.domain_axes = self._get_axes_tuple(self.domain)
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        self.field_type = self._parse_field_type(field_type, val=val)
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        try:
            start = len(reduce(lambda x, y: x+y, self.domain_axes))
        except TypeError:
            start = 0
        self.field_type_axes = self._get_axes_tuple(self.field_type,
                                                    start=start)
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        self.dtype = self._infer_dtype(dtype=dtype,
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                                       val=val,
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                                       domain=self.domain,
                                       field_type=self.field_type)
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        self.distribution_strategy = self._parse_distribution_strategy(
                                distribution_strategy=distribution_strategy,
                                val=val)
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        self.set_val(new_val=val, copy=copy)

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    def _parse_domain(self, domain, val=None):
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        if domain is None:
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            if isinstance(val, Field):
                domain = val.domain
            else:
                domain = ()
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        elif isinstance(domain, Space):
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            domain = (domain,)
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        elif not isinstance(domain, tuple):
            domain = tuple(domain)

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        for d in domain:
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            if not isinstance(d, Space):
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                raise TypeError(about._errors.cstring(
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                    "ERROR: Given domain contains something that is not a "
                    "nifty.space."))
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        return domain

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    def _parse_field_type(self, field_type, val=None):
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        if field_type is None:
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            if isinstance(val, Field):
                field_type = val.field_type
            else:
                field_type = ()
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        elif isinstance(field_type, FieldType):
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            field_type = (field_type,)
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        elif not isinstance(field_type, tuple):
            field_type = tuple(field_type)
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        for ft in field_type:
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            if not isinstance(ft, FieldType):
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                raise TypeError(about._errors.cstring(
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                    "ERROR: Given object is not a nifty.FieldType."))
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        return field_type

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    def _get_axes_tuple(self, things_with_shape, start=0):
        i = start
        axes_list = []
        for thing in things_with_shape:
            l = []
            for j in range(len(thing.shape)):
                l += [i]
                i += 1
            axes_list += [tuple(l)]
        return tuple(axes_list)
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    def _infer_dtype(self, dtype, val, domain, field_type):
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        if dtype is None:
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            if isinstance(val, Field) or \
               isinstance(val, distributed_data_object):
                dtype = val.dtype
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            dtype_tuple = (np.dtype(gc['default_field_dtype']),)
        else:
            dtype_tuple = (np.dtype(dtype),)
        if domain is not None:
            dtype_tuple += tuple(np.dtype(sp.dtype) for sp in domain)
        if field_type is not None:
            dtype_tuple += tuple(np.dtype(ft.dtype) for ft in field_type)
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        dtype = reduce(lambda x, y: np.result_type(x, y), dtype_tuple)
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        return dtype
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    def _parse_distribution_strategy(self, distribution_strategy, val):
        if distribution_strategy is None:
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            if isinstance(val, distributed_data_object):
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                distribution_strategy = val.distribution_strategy
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            elif isinstance(val, Field):
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                distribution_strategy = val.distribution_strategy
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            else:
                about.warnings.cprint("WARNING: Datamodel set to default!")
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                distribution_strategy = gc['default_distribution_strategy']
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        elif distribution_strategy not in DISTRIBUTION_STRATEGIES['global']:
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            raise ValueError(about._errors.cstring(
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                    "ERROR: distribution_strategy must be a global-type "
                    "strategy."))
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        return distribution_strategy
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    # ---Factory methods---
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    @classmethod
    def from_random(cls, random_type, domain=None, dtype=None, field_type=None,
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                    distribution_strategy=None, **kwargs):
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        # create a initially empty field
        f = cls(domain=domain, dtype=dtype, field_type=field_type,
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                distribution_strategy=distribution_strategy)
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        # now use the processed input in terms of f in order to parse the
        # random arguments
        random_arguments = cls._parse_random_arguments(random_type=random_type,
                                                       f=f,
                                                       **kwargs)

        # extract the distributed_dato_object from f and apply the appropriate
        # random number generator to it
        sample = f.get_val(copy=False)
        generator_function = getattr(Random, random_type)
        sample.apply_generator(
            lambda shape: generator_function(dtype=f.dtype,
                                             shape=shape,
                                             **random_arguments))
        return f

    @staticmethod
    def _parse_random_arguments(random_type, f, **kwargs):

        if random_type == "pm1":
            random_arguments = {}

        elif random_type == "normal":
            mean = kwargs.get('mean', 0)
            std = kwargs.get('std', 1)
            random_arguments = {'mean': mean,
                                'std': std}

        elif random_type == "uniform":
            low = kwargs.get('low', 0)
            high = kwargs.get('high', 1)
            random_arguments = {'low': low,
                                'high': high}

#        elif random_type == 'syn':
#            pass

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        else:
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            raise KeyError(about._errors.cstring(
                "ERROR: unsupported random key '" + str(random_type) + "'."))
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        return random_arguments
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    # ---Powerspectral methods---

    def power_analyze(self, spaces=None, log=False, nbin=None, binbounds=None,
                      real_signal=True):
        # assert that all spaces in `self.domain` are either harmonic or
        # power_space instances
        for sp in self.domain:
            if not sp.harmonic and not isinstance(sp, PowerSpace):
                raise AttributeError(
                    "ERROR: Field has a space in `domain` which is neither "
                    "harmonic nor a PowerSpace.")

        # check if the `spaces` input is valid
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        spaces = utilities.cast_axis_to_tuple(spaces, len(self.domain))
        if spaces is None:
            if len(self.domain) == 1:
                spaces = (0,)
            else:
                raise ValueError(about._errors.cstring(
                    "ERROR: Field has multiple spaces as domain "
                    "but `spaces` is None."))

        if len(spaces) == 0:
            raise ValueError(about._errors.cstring(
                "ERROR: No space for analysis specified."))
        elif len(spaces) > 1:
            raise ValueError(about._errors.cstring(
                "ERROR: Conversion of only one space at a time is allowed."))

        space_index = spaces[0]

        if not self.domain[space_index].harmonic:
            raise ValueError(about._errors.cstring(
                "ERROR: Conversion of only one space at a time is allowed."))

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        # Create the target PowerSpace instance:
        # If the associated signal-space field was real, we extract the
        # hermitian and anti-hermitian parts of `self` and put them
        # into the real and imaginary parts of the power spectrum.
        # If it was complex, all the power is put into a real power spectrum.

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        distribution_strategy = \
            self.val.get_axes_local_distribution_strategy(
                self.domain_axes[space_index])

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        if real_signal:
            power_dtype = np.dtype('complex')
        else:
            power_dtype = np.dtype('float')

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        harmonic_domain = self.domain[space_index]
        power_domain = PowerSpace(harmonic_domain=harmonic_domain,
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                                  distribution_strategy=distribution_strategy,
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                                  log=log, nbin=nbin, binbounds=binbounds,
                                  dtype=power_dtype)
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        # extract pindex and rho from power_domain
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        pindex = power_domain.pindex
        rho = power_domain.rho
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        if real_signal:
            hermitian_part, anti_hermitian_part = \
                harmonic_domain.hermitian_decomposition(
                                            self.val,
                                            axes=self.domain_axes[space_index])

            [hermitian_power, anti_hermitian_power] = \
                [self._calculate_power_spectrum(
                                            x=part,
                                            pindex=pindex,
                                            rho=rho,
                                            axes=self.domain_axes[space_index])
                 for part in [hermitian_part, anti_hermitian_part]]

            power_spectrum = hermitian_power + 1j * anti_hermitian_power
        else:
            power_spectrum = self._calculate_power_spectrum(
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                                            x=self.val,
                                            pindex=pindex,
                                            rho=rho,
                                            axes=self.domain_axes[space_index])

        # create the result field and put power_spectrum into it
        result_domain = list(self.domain)
        result_domain[space_index] = power_domain

        result_field = self.copy_empty(domain=result_domain)
        result_field.set_val(new_val=power_spectrum, copy=False)

        return result_field

    def _calculate_power_spectrum(self, x, pindex, rho, axes=None):
        fieldabs = abs(x)
        fieldabs **= 2

        if axes is not None:
            pindex = self._shape_up_pindex(
                                    pindex=pindex,
                                    target_shape=x.shape,
                                    target_strategy=x.distribution_strategy,
                                    axes=axes)
        power_spectrum = pindex.bincount(weights=fieldabs,
                                         axis=axes)
        if axes is not None:
            new_rho_shape = [1, ] * len(power_spectrum.shape)
            new_rho_shape[axes[0]] = len(rho)
            rho = rho.reshape(new_rho_shape)
        power_spectrum /= rho

        power_spectrum **= 0.5
        return power_spectrum

    def _shape_up_pindex(self, pindex, target_shape, target_strategy, axes):
        if pindex.distribution_strategy not in \
                DISTRIBUTION_STRATEGIES['global']:
            raise ValueError("ERROR: pindex's distribution strategy must be "
                             "global-type")

        if pindex.distribution_strategy in DISTRIBUTION_STRATEGIES['slicing']:
            if ((0 not in axes) or
                    (target_strategy is not pindex.distribution_strategy)):
                raise ValueError(
                    "ERROR: A slicing distributor shall not be reshaped to "
                    "something non-sliced.")

        semiscaled_shape = [1, ] * len(target_shape)
        for i in axes:
            semiscaled_shape[i] = target_shape[i]
        local_data = pindex.get_local_data(copy=False)
        semiscaled_local_data = local_data.reshape(semiscaled_shape)
        result_obj = pindex.copy_empty(global_shape=target_shape,
                                       distribution_strategy=target_strategy)
        result_obj.set_full_data(semiscaled_local_data, copy=False)

        return result_obj

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    def power_synthesize(self, spaces=None, real_signal=True):
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        # assert that all spaces in `self.domain` are either of signal-type or
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        # power_space instances
        for sp in self.domain:
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            if not sp.harmonic and not isinstance(sp, PowerSpace):
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                raise AttributeError(
                    "ERROR: Field has a space in `domain` which is neither "
                    "harmonic nor a PowerSpace.")

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        # check if the `spaces` input is valid
        spaces = utilities.cast_axis_to_tuple(spaces, len(self.domain))
        if spaces is None:
            if len(self.domain) == 1:
                spaces = (0,)
            else:
                raise ValueError(about._errors.cstring(
                    "ERROR: Field has multiple spaces as domain "
                    "but `spaces` is None."))

        if len(spaces) == 0:
            raise ValueError(about._errors.cstring(
                "ERROR: No space for synthesis specified."))
        elif len(spaces) > 1:
            raise ValueError(about._errors.cstring(
                "ERROR: Conversion of only one space at a time is allowed."))

        power_space_index = spaces[0]
        power_domain = self.domain[power_space_index]
        if not isinstance(power_domain, PowerSpace):
            raise ValueError(about._errors.cstring(
                "ERROR: A PowerSpace is needed for field synthetization."))

        # create the result domain
        result_domain = list(self.domain)
        harmonic_domain = power_domain.harmonic_domain
        result_domain[power_space_index] = harmonic_domain

        # create random samples: one or two, depending on whether the
        # power spectrum is real or complex

        if issubclass(power_domain.dtype.type, np.complexfloating):
            result_list = [None, None]
        else:
            result_list = [None]

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        result_list = [self.__class__.from_random(
                             'normal',
                             result_domain,
                             dtype=harmonic_domain.dtype,
                             field_type=self.field_type,
                             distribution_strategy=self.distribution_strategy)
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                       for x in result_list]

        # from now on extract the values from the random fields for further
        # processing without killing the fields.
        # if the signal-space field should be real, hermitianize the field
        # components
        if real_signal:
            result_val_list = [harmonic_domain.hermitian_decomposition(
                                    x.val,
                                    axes=x.domain_axes[power_space_index])[0]
                               for x in result_list]
        else:
            result_val_list = [x.val for x in result_list]

        # weight the random fields with the power spectrum
        # therefore get the pindex from the power space
        pindex = power_domain.pindex
        # take the local data from pindex. This data must be compatible to the
        # local data of the field given the slice of the PowerSpace
        local_distribution_strategy = \
            result_list[0].val.get_axes_local_distribution_strategy(
                result_list[0].domain_axes[power_space_index])

        if pindex.distribution_strategy is not local_distribution_strategy:
            about.warnings.cprint(
                "WARNING: The distribution_stragey of pindex does not fit the "
                "slice_local distribution strategy of the synthesized field.")

        # Now use numpy advanced indexing in order to put the entries of the
        # power spectrum into the appropriate places of the pindex array.
        # Do this for every 'pindex-slice' in parallel using the 'slice(None)'s
        local_pindex = pindex.get_local_data(copy=False)
        local_spec = self.val.get_local_data(copy=False)

        local_blow_up = [slice(None)]*len(self.shape)
        local_blow_up[self.domain_axes[power_space_index][0]] = local_pindex

        # here, the power_spectrum is distributed into the new shape
        local_rescaler = local_spec[local_blow_up]

        # apply the rescaler to the random fields
        result_val_list[0].apply_scalar_function(
                                            lambda x: x * local_rescaler.real,
                                            inplace=True)

        if issubclass(power_domain.dtype.type, np.complexfloating):
            result_val_list[1].apply_scalar_function(
                                            lambda x: x * local_rescaler.imag,
                                            inplace=True)

        # store the result into the fields
        [x.set_val(new_val=y, copy=False) for x, y in
            zip(result_list, result_val_list)]

        if issubclass(power_domain.dtype.type, np.complexfloating):
            result = result_list[0] + 1j*result_list[1]
        else:
            result = result_list[0]

        return result
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    # ---Properties---
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    def set_val(self, new_val=None, copy=False):
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        new_val = self.cast(new_val)
        if copy:
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            new_val = new_val.copy()
        self._val = new_val
        return self._val
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    def get_val(self, copy=False):
        if copy:
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            return self._val.copy()
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        else:
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            return self._val
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    @property
    def val(self):
        return self._val
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    @val.setter
    def val(self, new_val):
        self._val = self.cast(new_val)
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    @property
    def shape(self):
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        shape_tuple = ()
        shape_tuple += tuple(sp.shape for sp in self.domain)
        shape_tuple += tuple(ft.shape for ft in self.field_type)
        try:
            global_shape = reduce(lambda x, y: x + y, shape_tuple)
        except TypeError:
            global_shape = ()
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        return global_shape
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    @property
    def dim(self):
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        dim_tuple = ()
        dim_tuple += tuple(sp.dim for sp in self.domain)
        dim_tuple += tuple(ft.dim for ft in self.field_type)
        try:
            return reduce(lambda x, y: x * y, dim_tuple)
        except TypeError:
            return 0
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    @property
    def dof(self):
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        dof = self.dim
        if issubclass(self.dtype.type, np.complexfloating):
            dof *= 2
        return dof

    @property
    def total_volume(self):
        volume_tuple = tuple(sp.total_volume for sp in self.domain)
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        try:
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            return reduce(lambda x, y: x * y, volume_tuple)
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        except TypeError:
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            return 0
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    # ---Special unary/binary operations---
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    def cast(self, x=None, dtype=None):
        if dtype is None:
            dtype = self.dtype
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        else:
            dtype = np.dtype(dtype)
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        casted_x = self._actual_cast(x, dtype=dtype)
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        for ind, sp in enumerate(self.domain):
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            casted_x = sp.complement_cast(casted_x,
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                                          axes=self.domain_axes[ind])
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        for ind, ft in enumerate(self.field_type):
            casted_x = ft.complement_cast(casted_x,
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                                          axes=self.field_type_axes[ind])
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        return casted_x
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    def _actual_cast(self, x, dtype=None):
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        if isinstance(x, Field):
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            x = x.get_val()

        if dtype is None:
            dtype = self.dtype

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        return_x = distributed_data_object(
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                            global_shape=self.shape,
                            dtype=dtype,
                            distribution_strategy=self.distribution_strategy)
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        return_x.set_full_data(x, copy=False)
        return return_x
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    def copy(self, domain=None, dtype=None, field_type=None,
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             distribution_strategy=None):
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        copied_val = self.get_val(copy=True)
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        new_field = self.copy_empty(
                                domain=domain,
                                dtype=dtype,
                                field_type=field_type,
                                distribution_strategy=distribution_strategy)
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        new_field.set_val(new_val=copied_val, copy=False)
        return new_field
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    def copy_empty(self, domain=None, dtype=None, field_type=None,
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                   distribution_strategy=None):
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        if domain is None:
            domain = self.domain
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        else:
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            domain = self._parse_domain(domain)
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        if dtype is None:
            dtype = self.dtype
        else:
            dtype = np.dtype(dtype)
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        if field_type is None:
            field_type = self.field_type
        else:
            field_type = self._parse_field_type(field_type)
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        if distribution_strategy is None:
            distribution_strategy = self.distribution_strategy
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        fast_copyable = True
        try:
            for i in xrange(len(self.domain)):
                if self.domain[i] is not domain[i]:
                    fast_copyable = False
                    break
            for i in xrange(len(self.field_type)):
                if self.field_type[i] is not field_type[i]:
                    fast_copyable = False
                    break
        except IndexError:
            fast_copyable = False

        if (fast_copyable and dtype == self.dtype and
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                distribution_strategy == self.distribution_strategy):
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            new_field = self._fast_copy_empty()
        else:
            new_field = Field(domain=domain,
                              dtype=dtype,
                              field_type=field_type,
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                              distribution_strategy=distribution_strategy)
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        return new_field
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    def _fast_copy_empty(self):
        # make an empty field
        new_field = EmptyField()
        # repair its class
        new_field.__class__ = self.__class__
        # copy domain, codomain and val
        for key, value in self.__dict__.items():
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            if key != '_val':
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                new_field.__dict__[key] = value
            else:
                new_field.__dict__[key] = self.val.copy_empty()
        return new_field

    def weight(self, power=1, inplace=False, spaces=None):
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        if inplace:
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            new_field = self
        else:
            new_field = self.copy_empty()

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        new_val = self.get_val(copy=False)
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        if spaces is None:
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            spaces = range(len(self.domain))
        else:
            spaces = utilities.cast_axis_to_tuple(spaces, len(self.domain))
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        for ind, sp in enumerate(self.domain):
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            if ind in spaces:
                new_val = sp.weight(new_val,
                                    power=power,
                                    axes=self.domain_axes[ind],
                                    inplace=inplace)
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        new_field.set_val(new_val=new_val, copy=False)
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        return new_field

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    def dot(self, x=None, bare=False):
        if isinstance(x, Field):
            try:
                assert len(x.domain) == len(self.domain)
                for index in xrange(len(self.domain)):
                    assert x.domain[index] == self.domain[index]
                for index in xrange(len(self.field_type)):
                    assert x.field_type[index] == self.field_type[index]
            except AssertionError:
                raise ValueError(about._errors.cstring(
                    "ERROR: domains are incompatible."))
            # extract the data from x and try to dot with this
            x = x.get_val(copy=False)

        # Compute the dot respecting the fact of discrete/continous spaces
        if bare:
            y = self
        else:
            y = self.weight(power=1)

        y = y.get_val(copy=False)

        # Cast the input in order to cure dtype and shape differences
        x = self.cast(x)

        dotted = x.conjugate() * y

        return dotted.sum()

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    def norm(self, q=2):
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        """
            Computes the Lq-norm of the field values.

            Parameters
            ----------
            q : scalar
                Parameter q of the Lq-norm (default: 2).

            Returns
            -------
            norm : scalar
                The Lq-norm of the field values.

        """
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        if q == 2:
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            return (self.dot(x=self)) ** (1 / 2)
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        else:
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            return self.dot(x=self ** (q - 1)) ** (1 / q)
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    def conjugate(self, inplace=False):
        """
            Computes the complex conjugate of the field.

            Returns
            -------
            cc : field
                The complex conjugated field.

        """
        if inplace:
            work_field = self
        else:
            work_field = self.copy_empty()

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        new_val = self.get_val(copy=False)
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        new_val = new_val.conjugate()
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        work_field.set_val(new_val=new_val, copy=False)
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        return work_field

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    # ---General unary/contraction methods---
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    def __pos__(self):
        return self.copy()
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    def __neg__(self):
        return_field = self.copy_empty()
        new_val = -self.get_val(copy=False)
        return_field.set_val(new_val, copy=False)
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        return return_field

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    def __abs__(self):
        return_field = self.copy_empty()
        new_val = abs(self.get_val(copy=False))
        return_field.set_val(new_val, copy=False)
        return return_field
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    def _contraction_helper(self, op, spaces, types):
        # build a list of all axes
        if spaces is None:
            spaces = xrange(len(self.domain))
        else:
            spaces = utilities.cast_axis_to_tuple(spaces, len(self.domain))
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        if types is None:
            types = xrange(len(self.field_type))
        else:
            types = utilities.cast_axis_to_tuple(types, len(self.field_type))
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        axes_list = ()
        axes_list += tuple(self.domain_axes[sp_index] for sp_index in spaces)
        axes_list += tuple(self.field_type_axes[ft_index] for
                           ft_index in types)
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        try:
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            axes_list = reduce(lambda x, y: x+y, axes_list)
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        except TypeError:
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            axes_list = ()
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        # perform the contraction on the d2o
        data = self.get_val(copy=False)
        data = getattr(data, op)(axis=axes_list)
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        # check if the result is scalar or if a result_field must be constr.
        if np.isscalar(data):
            return data
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        else:
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            return_domain = tuple(self.domain[i]
                                  for i in xrange(len(self.domain))
                                  if i not in spaces)
            return_field_type = tuple(self.field_type[i]
                                      for i in xrange(len(self.field_type))
                                      if i not in types)
            return_field = Field(domain=return_domain,
                                 val=data,
                                 field_type=return_field_type,
                                 copy=False)
            return return_field
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    def sum(self, spaces=None, types=None):
        return self._contraction_helper('sum', spaces, types)
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    def prod(self, spaces=None, types=None):
        return self._contraction_helper('prod', spaces, types)
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    def all(self, spaces=None, types=None):
        return self._contraction_helper('all', spaces, types)
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    def any(self, spaces=None, types=None):
        return self._contraction_helper('any', spaces, types)
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    def min(self, spaces=None, types=None):
        return self._contraction_helper('min', spaces, types)
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    def nanmin(self, spaces=None, types=None):
        return self._contraction_helper('nanmin', spaces, types)
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    def max(self, spaces=None, types=None):
        return self._contraction_helper('max', spaces, types)
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    def nanmax(self, spaces=None, types=None):
        return self._contraction_helper('nanmax', spaces, types)
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    def mean(self, spaces=None, types=None):
        return self._contraction_helper('mean', spaces, types)
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    def var(self, spaces=None, types=None):
        return self._contraction_helper('var', spaces, types)
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    def std(self, spaces=None, types=None):
        return self._contraction_helper('std', spaces, types)
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    # ---General binary methods---
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    def _binary_helper(self, other, op, inplace=False):
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        # if other is a field, make sure that the domains match
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        if isinstance(other, Field):
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            try:
                assert len(other.domain) == len(self.domain)
                for index in xrange(len(self.domain)):
                    assert other.domain[index] == self.domain[index]
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                assert len(other.field_type) == len(self.field_type)
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                for index in xrange(len(self.field_type)):
                    assert other.field_type[index] == self.field_type[index]
            except AssertionError:
                raise ValueError(about._errors.cstring(
                    "ERROR: domains are incompatible."))
            other = other.get_val(copy=False)
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        self_val = self.get_val(copy=False)
        return_val = getattr(self_val, op)(other)
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        if inplace:
            working_field = self
        else:
            working_field = self.copy_empty()

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        working_field.set_val(return_val, copy=False)
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        return working_field

    def __add__(self, other):
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        return self._binary_helper(other, op='__add__')
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    def __radd__(self, other):
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        return self._binary_helper(other, op='__radd__')
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    def __iadd__(self, other):
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        return self._binary_helper(other, op='__iadd__', inplace=True)
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    def __sub__(self, other):
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        return self._binary_helper(other, op='__sub__')
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    def __rsub__(self, other):
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        return self._binary_helper(other, op='__rsub__')
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    def __isub__(self, other):
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        return self._binary_helper(other, op='__isub__', inplace=True)
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    def __mul__(self, other):
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        return self._binary_helper(other, op='__mul__')
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    def __rmul__(self, other):
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        return self._binary_helper(other, op='__rmul__')
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    def __imul__(self, other):
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        return self._binary_helper(other, op='__imul__', inplace=True)
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    def __div__(self, other):
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        return self._binary_helper(other, op='__div__')
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    def __rdiv__(self, other):
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        return self._binary_helper(other, op='__rdiv__')
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    def __idiv__(self, other):
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        return self._binary_helper(other, op='__idiv__', inplace=True)
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    def __pow__(self, other):
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        return self._binary_helper(other, op='__pow__')
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    def __rpow__(self, other):
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        return self._binary_helper(other, op='__rpow__')
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    def __ipow__(self, other):
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        return self._binary_helper(other, op='__ipow__', inplace=True)
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    def __lt__(self, other):
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        return self._binary_helper(other, op='__lt__')
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    def __le__(self, other):
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        return self._binary_helper(other, op='__le__')
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    def __ne__(self, other):
        if other is None:
            return True
        else:
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            return self._binary_helper(other, op='__ne__')
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    def __eq__(self, other):
        if other is None:
            return False
        else:
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            return self._binary_helper(other, op='__eq__')
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    def __ge__(self, other):
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        return self._binary_helper(other, op='__ge__')
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    def __gt__(self, other):
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        return self._binary_helper(other, op='__gt__')

    def __repr__(self):
        return "<nifty_core.field>"

    def __str__(self):
        minmax = [self.min(), self.max()]
        mean = self.mean()
        return "nifty_core.field instance\n- domain      = " + \
               repr(self.domain) + \
               "\n- val         = " + repr(self.get_val()) + \
               "\n  - min.,max. = " + str(minmax) + \
               "\n  - mean = " + str(mean)
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class EmptyField(Field):
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    def __init__(self):
        pass