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# iris.coords¶

Definitions of coordinates and other dimensional metadata.

In this module:

class iris.coords.AncillaryVariable(data, standard_name=None, long_name=None, var_name=None, units=None, attributes=None)[source]

Constructs a single ancillary variable.

Args:

• data:

The values of the ancillary variable.

Kwargs:

• standard_name:

CF standard name of the ancillary variable.

• long_name:

Descriptive name of the ancillary variable.

• var_name:

The netCDF variable name for the ancillary variable.

• units

The Unit of the ancillary variable’s values. Can be a string, which will be converted to a Unit object.

• attributes

A dictionary containing other cf and user-defined attributes.

__binary_operator__(other, mode_constant)

Common code which is called by add, sub, mul and div

Mode constant is one of ADD, SUB, MUL, DIV, RDIV

Note

The unit is not changed when doing scalar operations on a metadata object. This means that a metadata object which represents “10 meters” when multiplied by a scalar i.e. “1000” would result in a metadata object of “10000 meters”. An alternative approach could be taken to multiply the unit by 1000 and the resultant metadata object would represent “10 kilometers”.

__getitem__(keys)

Returns a new dimensional metadata whose values are obtained by conventional array indexing.

Note

Indexing of a circular coordinate results in a non-circular coordinate if the overall shape of the coordinate changes after indexing.

convert_units(unit)

Change the units, converting the values of the metadata.

copy(values=None)

Returns a copy of this dimensional metadata object.

Kwargs:

• values

An array of values for the new dimensional metadata object. This may be a different shape to the original values array being copied.

core_data()[source]

The data array at the core of this ancillary variable, which may be a NumPy array or a dask array.

cube_dims(cube)[source]

Return the cube dimensions of this AncillaryVariable.

Equivalent to “cube.ancillary_variable_dims(self)”.

has_bounds()

Return a boolean indicating whether the current dimensional metadata object has a bounds array.

has_lazy_data()[source]

Return a boolean indicating whether the ancillary variable’s data array is a lazy dask array or not.

is_compatible(other, ignore=None)

Return whether the current dimensional metadata object is compatible with another.

lazy_data()[source]

Return a lazy array representing the ancillary variable’s data.

Accessing this method will never cause the data values to be loaded. Similarly, calling methods on, or indexing, the returned Array will not cause the ancillary variable to have loaded data.

If the data have already been loaded for the ancillary variable, the returned Array will be a new lazy array wrapper.

Returns

A lazy array, representing the ancillary variable data array.

name(default=None, token=False)

Returns a string name representing the identity of the metadata.

First it tries standard name, then it tries the long name, then the NetCDF variable name, before falling-back to a default value, which itself defaults to the string ‘unknown’.

Kwargs:

• default:

The fall-back string representing the default name. Defaults to the string ‘unknown’.

• token:

If True, ensures that the name returned satisfies the criteria for the characters required by a valid NetCDF name. If it is not possible to return a valid name, then a ValueError exception is raised. Defaults to False.

Returns

String.

rename(name)

If ‘name’ is a valid standard name it will assign it to standard_name, otherwise it will assign it to long_name.

xml_element(doc)

Create the xml.dom.minidom.Element that describes this _DimensionalMetadata.

Args:

• doc:

The parent xml.dom.minidom.Document.

Returns

The xml.dom.minidom.Element that will describe this _DimensionalMetadata, and the dictionary of attributes that require to be added to this element.

property attributes
property data
property dtype

The NumPy dtype of the current dimensional metadata object, as specified by its values.

property long_name

The CF Metadata long name for the object.

property metadata
property ndim

Return the number of dimensions of the current dimensional metadata object.

property shape

The fundamental shape of the metadata, expressed as a tuple.

property standard_name

The CF Metadata standard name for the object.

property units

The S.I. unit of the object.

property var_name

The NetCDF variable name for the object.

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A CF auxiliary coordinate.

class iris.coords.AuxCoord(*args, **kwargs)[source]

Create a coordinate with mutable points and bounds.

Args:

• points:

The values (or value in the case of a scalar coordinate) for each cell of the coordinate.

Kwargs:

• standard_name:

CF standard name of the coordinate.

• long_name:

Descriptive name of the coordinate.

• var_name:

The netCDF variable name for the coordinate.

• units

The Unit of the coordinate’s values. Can be a string, which will be converted to a Unit object.

• bounds

An array of values describing the bounds of each cell. Given n bounds for each cell, the shape of the bounds array should be points.shape + (n,). For example, a 1D coordinate with 100 points and two bounds per cell would have a bounds array of shape (100, 2) Note if the data is a climatology, climatological should be set.

• attributes

A dictionary containing other CF and user-defined attributes.

• coord_system

A CoordSystem representing the coordinate system of the coordinate, e.g., a GeogCS for a longitude coordinate.

• climatological (bool):

When True: the coordinate is a NetCDF climatological time axis. When True: saving in NetCDF will give the coordinate variable a ‘climatology’ attribute and will create a boundary variable called ‘<coordinate-name>_climatology’ in place of a standard bounds attribute and bounds variable. Will set to True when a climatological time axis is loaded from NetCDF. Always False if no bounds exist.

__binary_operator__(other, mode_constant)

Common code which is called by add, sub, mul and div

Mode constant is one of ADD, SUB, MUL, DIV, RDIV

Note

The unit is not changed when doing scalar operations on a metadata object. This means that a metadata object which represents “10 meters” when multiplied by a scalar i.e. “1000” would result in a metadata object of “10000 meters”. An alternative approach could be taken to multiply the unit by 1000 and the resultant metadata object would represent “10 kilometers”.

__getitem__(keys)

Returns a new dimensional metadata whose values are obtained by conventional array indexing.

Note

Indexing of a circular coordinate results in a non-circular coordinate if the overall shape of the coordinate changes after indexing.

cell(index)

Return the single Cell instance which results from slicing the points/bounds with the given index.

cells()

Returns an iterable of Cell instances for this Coord.

For example:

for cell in self.cells():
...

collapsed(dims_to_collapse=None)

Returns a copy of this coordinate, which has been collapsed along the specified dimensions.

Replaces the points & bounds with a simple bounded region.

contiguous_bounds()

Returns the N+1 bound values for a contiguous bounded 1D coordinate of length N, or the (N+1, M+1) bound values for a contiguous bounded 2D coordinate of shape (N, M).

Only 1D or 2D coordinates are supported.

Note

If the coordinate has bounds, this method assumes they are contiguous.

If the coordinate is 1D and does not have bounds, this method will return bounds positioned halfway between the coordinate’s points.

If the coordinate is 2D and does not have bounds, an error will be raised.

convert_units(unit)

Change the coordinate’s units, converting the values in its points and bounds arrays.

For example, if a coordinate’s units attribute is set to radians then:

coord.convert_units('degrees')

will change the coordinate’s units attribute to degrees and multiply each value in points and bounds by 180.0/$$\pi$$.

copy(points=None, bounds=None)

Returns a copy of this coordinate.

Kwargs:

• points: A points array for the new coordinate.

This may be a different shape to the points of the coordinate being copied.

• bounds: A bounds array for the new coordinate.

Given n bounds for each cell, the shape of the bounds array should be points.shape + (n,). For example, a 1d coordinate with 100 points and two bounds per cell would have a bounds array of shape (100, 2).

Note

If the points argument is specified and bounds are not, the resulting coordinate will have no bounds.

core_bounds()

The points array at the core of this coord, which may be a NumPy array or a dask array.

core_points()

The points array at the core of this coord, which may be a NumPy array or a dask array.

cube_dims(cube)

Return the cube dimensions of this Coord.

Equivalent to “cube.coord_dims(self)”.

classmethod from_coord(coord)

Create a new Coord of this type, from the given coordinate.

guess_bounds(bound_position=0.5)

Add contiguous bounds to a coordinate, calculated from its points.

Puts a cell boundary at the specified fraction between each point and the next, plus extrapolated lowermost and uppermost bound points, so that each point lies within a cell.

With regularly spaced points, the resulting bounds will also be regular, and all points lie at the same position within their cell. With irregular points, the first and last cells are given the same widths as the ones next to them.

Kwargs:

• bound_position:

The desired position of the bounds relative to the position of the points.

Note

An error is raised if the coordinate already has bounds, is not one-dimensional, or is not monotonic.

Note

Unevenly spaced values, such from a wrapped longitude range, can produce unexpected results : In such cases you should assign suitable values directly to the bounds property, instead.

has_bounds()

Return a boolean indicating whether the coord has a bounds array.

has_lazy_bounds()

Return a boolean indicating whether the coord’s bounds array is a lazy dask array or not.

has_lazy_points()

Return a boolean indicating whether the coord’s points array is a lazy dask array or not.

intersect(other, return_indices=False)

Returns a new coordinate from the intersection of two coordinates.

Both coordinates must be compatible as defined by is_compatible().

Kwargs:

• return_indices:

If True, changes the return behaviour to return the intersection indices for the “self” coordinate.

is_compatible(other, ignore=None)

Return whether the coordinate is compatible with another.

Compatibility is determined by comparing iris.coords.Coord.name(), iris.coords.Coord.units, iris.coords.Coord.coord_system and iris.coords.Coord.attributes that are present in both objects.

Args:

• other:

An instance of iris.coords.Coord, iris.common.CoordMetadata or iris.common.DimCoordMetadata.

• ignore:

A single attribute key or iterable of attribute keys to ignore when comparing the coordinates. Default is None. To ignore all attributes, set this to other.attributes.

Returns

Boolean.

is_contiguous(rtol=1e-05, atol=1e-08)

Return True if, and only if, this Coord is bounded with contiguous bounds to within the specified relative and absolute tolerances.

1D coords are contiguous if the upper bound of a cell aligns, within a tolerance, to the lower bound of the next cell along.

2D coords, with 4 bounds, are contiguous if the lower right corner of each cell aligns with the lower left corner of the cell to the right of it, and the upper left corner of each cell aligns with the lower left corner of the cell above it.

Args:

• rtol:

The relative tolerance parameter (default is 1e-05).

• atol:

The absolute tolerance parameter (default is 1e-08).

Returns

Boolean.

is_monotonic()

Return True if, and only if, this Coord is monotonic.

lazy_bounds()

Return a lazy array representing the coord bounds.

Accessing this method will never cause the bounds values to be loaded. Similarly, calling methods on, or indexing, the returned Array will not cause the coord to have loaded bounds.

If the data have already been loaded for the coord, the returned Array will be a new lazy array wrapper.

Returns

A lazy array representing the coord bounds array or None if the coord does not have bounds.

lazy_points()

Return a lazy array representing the coord points.

Accessing this method will never cause the points values to be loaded. Similarly, calling methods on, or indexing, the returned Array will not cause the coord to have loaded points.

If the data have already been loaded for the coord, the returned Array will be a new lazy array wrapper.

Returns

A lazy array, representing the coord points array.

name(default=None, token=False)

Returns a string name representing the identity of the metadata.

First it tries standard name, then it tries the long name, then the NetCDF variable name, before falling-back to a default value, which itself defaults to the string ‘unknown’.

Kwargs:

• default:

The fall-back string representing the default name. Defaults to the string ‘unknown’.

• token:

If True, ensures that the name returned satisfies the criteria for the characters required by a valid NetCDF name. If it is not possible to return a valid name, then a ValueError exception is raised. Defaults to False.

Returns

String.

nearest_neighbour_index(point)

Returns the index of the cell nearest to the given point.

Only works for one-dimensional coordinates.

For example:

>>> cube = iris.load_cube(iris.sample_data_path('ostia_monthly.nc'))
>>> cube.coord('latitude').nearest_neighbour_index(0)
9
>>> cube.coord('longitude').nearest_neighbour_index(10)
12

Note

If the coordinate contains bounds, these will be used to determine the nearest neighbour instead of the point values.

Note

For circular coordinates, the ‘nearest’ point can wrap around to the other end of the values.

rename(name)

If ‘name’ is a valid standard name it will assign it to standard_name, otherwise it will assign it to long_name.

xml_element(doc)

Create the xml.dom.minidom.Element that describes this Coord.

Args:

• doc:

The parent xml.dom.minidom.Document.

Returns

The xml.dom.minidom.Element that will describe this DimCoord, and the dictionary of attributes that require to be added to this element.

property attributes
property bounds

The coordinate bounds values, as a NumPy array, or None if no bound values are defined.

Note

The shape of the bound array should be: points.shape + (n_bounds, ).

property bounds_dtype

The NumPy dtype of the coord’s bounds. Will be None if the coord does not have bounds.

property climatological

A boolean that controls whether the coordinate is a climatological time axis, in which case the bounds represent a climatological period rather than a normal period.

Always reads as False if there are no bounds. On set, the input value is cast to a boolean, exceptions raised if units are not time units or if there are no bounds.

property coord_system

The coordinate-system of the coordinate.

property dtype

The NumPy dtype of the current dimensional metadata object, as specified by its values.

property long_name

The CF Metadata long name for the object.

property metadata
property nbounds

Return the number of bounds that this coordinate has (0 for no bounds).

property ndim

Return the number of dimensions of the current dimensional metadata object.

property points

The coordinate points values as a NumPy array.

property shape

The fundamental shape of the metadata, expressed as a tuple.

property standard_name

The CF Metadata standard name for the object.

property units

The S.I. unit of the object.

property var_name

The NetCDF variable name for the object.

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An immutable representation of a single cell of a coordinate, including the sample point and/or boundary position.

Notes on cell comparison:

Cells are compared in two ways, depending on whether they are compared to another Cell, or to a number/string.

Cell-Cell comparison is defined to produce a strict ordering. If two cells are not exactly equal (i.e. including whether they both define bounds or not) then they will have a consistent relative order.

Cell-number and Cell-string comparison is defined to support Constraint matching. The number/string will equal the Cell if, and only if, it is within the Cell (including on the boundary). The relative comparisons (lt, le, ..) are defined to be consistent with this interpretation. So for a given value n and Cell cell, only one of the following can be true:

n < cell
n == cell
n > cell

Similarly, n <= cell implies either n < cell or n == cell. And n >= cell implies either n > cell or n == cell.

class iris.coords.Cell(point=None, bound=None)[source]

Construct a Cell from point or point-and-bound information.

__common_cmp__(other, operator_method)[source]

Common method called by the rich comparison operators. The method of checking equality depends on the type of the object to be compared.

Cell vs Cell comparison is used to define a strict order. Non-Cell vs Cell comparison is used to define Constraint matching.

__eq__(other)[source]

Compares Cell equality depending on the type of the object to be compared.

static __new__(cls, point=None, bound=None)[source]

Construct a Cell from point or point-and-bound information.

contains_point(point)[source]

For a bounded cell, returns whether the given point lies within the bounds.

Note

The test carried out is equivalent to min(bound) <= point <= max(bound).

count(value, /)

Return number of occurrences of value.

index(value, start=0, stop=9223372036854775807, /)

Return first index of value.

Raises ValueError if the value is not present.

bound

Alias for field number 1

point

Alias for field number 0

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A CF Cell Measure, providing area or volume properties of a cell where these cannot be inferred from the Coordinates and Coordinate Reference System.

class iris.coords.CellMeasure(data, standard_name=None, long_name=None, var_name=None, units=None, attributes=None, measure=None)[source]

Constructs a single cell measure.

Args:

• data:

The values of the measure for each cell. Either a ‘real’ array (numpy.ndarray) or a ‘lazy’ array (dask.array.Array).

Kwargs:

• standard_name:

CF standard name of the coordinate.

• long_name:

Descriptive name of the coordinate.

• var_name:

The netCDF variable name for the coordinate.

• units

The Unit of the coordinate’s values. Can be a string, which will be converted to a Unit object.

• attributes

A dictionary containing other CF and user-defined attributes.

• measure

A string describing the type of measure. Supported values are ‘area’ and ‘volume’. The default is ‘area’.

__binary_operator__(other, mode_constant)

Common code which is called by add, sub, mul and div

Mode constant is one of ADD, SUB, MUL, DIV, RDIV

Note

The unit is not changed when doing scalar operations on a metadata object. This means that a metadata object which represents “10 meters” when multiplied by a scalar i.e. “1000” would result in a metadata object of “10000 meters”. An alternative approach could be taken to multiply the unit by 1000 and the resultant metadata object would represent “10 kilometers”.

__getitem__(keys)

Returns a new dimensional metadata whose values are obtained by conventional array indexing.

Note

Indexing of a circular coordinate results in a non-circular coordinate if the overall shape of the coordinate changes after indexing.

convert_units(unit)

Change the units, converting the values of the metadata.

copy(values=None)

Returns a copy of this dimensional metadata object.

Kwargs:

• values

An array of values for the new dimensional metadata object. This may be a different shape to the original values array being copied.

core_data()

The data array at the core of this ancillary variable, which may be a NumPy array or a dask array.

cube_dims(cube)[source]

Return the cube dimensions of this CellMeasure.

Equivalent to “cube.cell_measure_dims(self)”.

has_bounds()

Return a boolean indicating whether the current dimensional metadata object has a bounds array.

has_lazy_data()

Return a boolean indicating whether the ancillary variable’s data array is a lazy dask array or not.

is_compatible(other, ignore=None)

Return whether the current dimensional metadata object is compatible with another.

lazy_data()

Return a lazy array representing the ancillary variable’s data.

Accessing this method will never cause the data values to be loaded. Similarly, calling methods on, or indexing, the returned Array will not cause the ancillary variable to have loaded data.

If the data have already been loaded for the ancillary variable, the returned Array will be a new lazy array wrapper.

Returns

A lazy array, representing the ancillary variable data array.

name(default=None, token=False)

Returns a string name representing the identity of the metadata.

First it tries standard name, then it tries the long name, then the NetCDF variable name, before falling-back to a default value, which itself defaults to the string ‘unknown’.

Kwargs:

• default:

The fall-back string representing the default name. Defaults to the string ‘unknown’.

• token:

If True, ensures that the name returned satisfies the criteria for the characters required by a valid NetCDF name. If it is not possible to return a valid name, then a ValueError exception is raised. Defaults to False.

Returns

String.

rename(name)

If ‘name’ is a valid standard name it will assign it to standard_name, otherwise it will assign it to long_name.

xml_element(doc)[source]

Create the xml.dom.minidom.Element that describes this CellMeasure.

Args:

• doc:

The parent xml.dom.minidom.Document.

Returns

The xml.dom.minidom.Element that describes this CellMeasure.

property attributes
property data
property dtype

The NumPy dtype of the current dimensional metadata object, as specified by its values.

property long_name

The CF Metadata long name for the object.

property measure

String naming the measure type.

property metadata
property ndim

Return the number of dimensions of the current dimensional metadata object.

property shape

The fundamental shape of the metadata, expressed as a tuple.

property standard_name

The CF Metadata standard name for the object.

property units

The S.I. unit of the object.

property var_name

The NetCDF variable name for the object.

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Represents a sub-cell pre-processing operation.

class iris.coords.CellMethod(method, coords=None, intervals=None, comments=None)[source]

Args:

• method:

The name of the operation.

Kwargs:

• coords:

A single instance or sequence of Coord instances or coordinate names.

• intervals:

A single string, or a sequence strings, describing the intervals within the cell method.

__str__()[source]

Return a custom string representation of CellMethod

xml_element(doc)[source]

Create the xml.dom.minidom.Element that describes this CellMethod.

Args:

• doc:

The parent xml.dom.minidom.Document.

Returns

The xml.dom.minidom.Element that describes this CellMethod.

comments = None

coord_names = None

The tuple of coordinate names over which the operation was applied.

intervals = None

A description of the original intervals over which the operation was applied.

method = None

The name of the operation that was applied. e.g. “mean”, “max”, etc.

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Abstract base class for coordinates.

class iris.coords.Coord(points, standard_name=None, long_name=None, var_name=None, units=None, bounds=None, attributes=None, coord_system=None, climatological=False)[source]

Coordinate abstract base class. As of v3.0.0 you cannot create an instance of Coord.

Args:

• points:

The values (or value in the case of a scalar coordinate) for each cell of the coordinate.

Kwargs:

• standard_name:

CF standard name of the coordinate.

• long_name:

Descriptive name of the coordinate.

• var_name:

The netCDF variable name for the coordinate.

• units

The Unit of the coordinate’s values. Can be a string, which will be converted to a Unit object.

• bounds

An array of values describing the bounds of each cell. Given n bounds for each cell, the shape of the bounds array should be points.shape + (n,). For example, a 1D coordinate with 100 points and two bounds per cell would have a bounds array of shape (100, 2) Note if the data is a climatology, climatological should be set.

• attributes

A dictionary containing other CF and user-defined attributes.

• coord_system

A CoordSystem representing the coordinate system of the coordinate, e.g., a GeogCS for a longitude coordinate.

• climatological (bool):

When True: the coordinate is a NetCDF climatological time axis. When True: saving in NetCDF will give the coordinate variable a ‘climatology’ attribute and will create a boundary variable called ‘<coordinate-name>_climatology’ in place of a standard bounds attribute and bounds variable. Will set to True when a climatological time axis is loaded from NetCDF. Always False if no bounds exist.

__binary_operator__(other, mode_constant)

Common code which is called by add, sub, mul and div

Mode constant is one of ADD, SUB, MUL, DIV, RDIV

Note

The unit is not changed when doing scalar operations on a metadata object. This means that a metadata object which represents “10 meters” when multiplied by a scalar i.e. “1000” would result in a metadata object of “10000 meters”. An alternative approach could be taken to multiply the unit by 1000 and the resultant metadata object would represent “10 kilometers”.

__getitem__(keys)

Returns a new dimensional metadata whose values are obtained by conventional array indexing.

Note

Indexing of a circular coordinate results in a non-circular coordinate if the overall shape of the coordinate changes after indexing.

cell(index)[source]

Return the single Cell instance which results from slicing the points/bounds with the given index.

cells()[source]

Returns an iterable of Cell instances for this Coord.

For example:

for cell in self.cells():
...

collapsed(dims_to_collapse=None)[source]

Returns a copy of this coordinate, which has been collapsed along the specified dimensions.

Replaces the points & bounds with a simple bounded region.

contiguous_bounds()[source]

Returns the N+1 bound values for a contiguous bounded 1D coordinate of length N, or the (N+1, M+1) bound values for a contiguous bounded 2D coordinate of shape (N, M).

Only 1D or 2D coordinates are supported.

Note

If the coordinate has bounds, this method assumes they are contiguous.

If the coordinate is 1D and does not have bounds, this method will return bounds positioned halfway between the coordinate’s points.

If the coordinate is 2D and does not have bounds, an error will be raised.

convert_units(unit)[source]

Change the coordinate’s units, converting the values in its points and bounds arrays.

For example, if a coordinate’s units attribute is set to radians then:

coord.convert_units('degrees')

will change the coordinate’s units attribute to degrees and multiply each value in points and bounds by 180.0/$$\pi$$.

copy(points=None, bounds=None)[source]

Returns a copy of this coordinate.

Kwargs:

• points: A points array for the new coordinate.

This may be a different shape to the points of the coordinate being copied.

• bounds: A bounds array for the new coordinate.

Given n bounds for each cell, the shape of the bounds array should be points.shape + (n,). For example, a 1d coordinate with 100 points and two bounds per cell would have a bounds array of shape (100, 2).

Note

If the points argument is specified and bounds are not, the resulting coordinate will have no bounds.

core_bounds()[source]

The points array at the core of this coord, which may be a NumPy array or a dask array.

core_points()[source]

The points array at the core of this coord, which may be a NumPy array or a dask array.

cube_dims(cube)[source]

Return the cube dimensions of this Coord.

Equivalent to “cube.coord_dims(self)”.

classmethod from_coord(coord)[source]

Create a new Coord of this type, from the given coordinate.

guess_bounds(bound_position=0.5)[source]

Add contiguous bounds to a coordinate, calculated from its points.

Puts a cell boundary at the specified fraction between each point and the next, plus extrapolated lowermost and uppermost bound points, so that each point lies within a cell.

With regularly spaced points, the resulting bounds will also be regular, and all points lie at the same position within their cell. With irregular points, the first and last cells are given the same widths as the ones next to them.

Kwargs:

• bound_position:

The desired position of the bounds relative to the position of the points.

Note

An error is raised if the coordinate already has bounds, is not one-dimensional, or is not monotonic.

Note

Unevenly spaced values, such from a wrapped longitude range, can produce unexpected results : In such cases you should assign suitable values directly to the bounds property, instead.

has_bounds()[source]

Return a boolean indicating whether the coord has a bounds array.

has_lazy_bounds()[source]

Return a boolean indicating whether the coord’s bounds array is a lazy dask array or not.

has_lazy_points()[source]

Return a boolean indicating whether the coord’s points array is a lazy dask array or not.

intersect(other, return_indices=False)[source]

Returns a new coordinate from the intersection of two coordinates.

Both coordinates must be compatible as defined by is_compatible().

Kwargs:

• return_indices:

If True, changes the return behaviour to return the intersection indices for the “self” coordinate.

is_compatible(other, ignore=None)[source]

Return whether the coordinate is compatible with another.

Compatibility is determined by comparing iris.coords.Coord.name(), iris.coords.Coord.units, iris.coords.Coord.coord_system and iris.coords.Coord.attributes that are present in both objects.

Args:

• other:

An instance of iris.coords.Coord, iris.common.CoordMetadata or iris.common.DimCoordMetadata.

• ignore:

A single attribute key or iterable of attribute keys to ignore when comparing the coordinates. Default is None. To ignore all attributes, set this to other.attributes.

Returns

Boolean.

is_contiguous(rtol=1e-05, atol=1e-08)[source]

Return True if, and only if, this Coord is bounded with contiguous bounds to within the specified relative and absolute tolerances.

1D coords are contiguous if the upper bound of a cell aligns, within a tolerance, to the lower bound of the next cell along.

2D coords, with 4 bounds, are contiguous if the lower right corner of each cell aligns with the lower left corner of the cell to the right of it, and the upper left corner of each cell aligns with the lower left corner of the cell above it.

Args:

• rtol:

The relative tolerance parameter (default is 1e-05).

• atol:

The absolute tolerance parameter (default is 1e-08).

Returns

Boolean.

is_monotonic()[source]

Return True if, and only if, this Coord is monotonic.

lazy_bounds()[source]

Return a lazy array representing the coord bounds.

Accessing this method will never cause the bounds values to be loaded. Similarly, calling methods on, or indexing, the returned Array will not cause the coord to have loaded bounds.

If the data have already been loaded for the coord, the returned Array will be a new lazy array wrapper.

Returns

A lazy array representing the coord bounds array or None if the coord does not have bounds.

lazy_points()[source]

Return a lazy array representing the coord points.

Accessing this method will never cause the points values to be loaded. Similarly, calling methods on, or indexing, the returned Array will not cause the coord to have loaded points.

If the data have already been loaded for the coord, the returned Array will be a new lazy array wrapper.

Returns

A lazy array, representing the coord points array.

name(default=None, token=False)

Returns a string name representing the identity of the metadata.

First it tries standard name, then it tries the long name, then the NetCDF variable name, before falling-back to a default value, which itself defaults to the string ‘unknown’.

Kwargs:

• default:

The fall-back string representing the default name. Defaults to the string ‘unknown’.

• token:

If True, ensures that the name returned satisfies the criteria for the characters required by a valid NetCDF name. If it is not possible to return a valid name, then a ValueError exception is raised. Defaults to False.

Returns

String.

nearest_neighbour_index(point)[source]

Returns the index of the cell nearest to the given point.

Only works for one-dimensional coordinates.

For example:

>>> cube = iris.load_cube(iris.sample_data_path('ostia_monthly.nc'))
>>> cube.coord('latitude').nearest_neighbour_index(0)
9
>>> cube.coord('longitude').nearest_neighbour_index(10)
12

Note

If the coordinate contains bounds, these will be used to determine the nearest neighbour instead of the point values.

Note

For circular coordinates, the ‘nearest’ point can wrap around to the other end of the values.

rename(name)

If ‘name’ is a valid standard name it will assign it to standard_name, otherwise it will assign it to long_name.

xml_element(doc)[source]

Create the xml.dom.minidom.Element that describes this Coord.

Args:

• doc:

The parent xml.dom.minidom.Document.

Returns

The xml.dom.minidom.Element that will describe this DimCoord, and the dictionary of attributes that require to be added to this element.

property attributes
property bounds

The coordinate bounds values, as a NumPy array, or None if no bound values are defined.

Note

The shape of the bound array should be: points.shape + (n_bounds, ).

property bounds_dtype

The NumPy dtype of the coord’s bounds. Will be None if the coord does not have bounds.

property climatological

A boolean that controls whether the coordinate is a climatological time axis, in which case the bounds represent a climatological period rather than a normal period.

Always reads as False if there are no bounds. On set, the input value is cast to a boolean, exceptions raised if units are not time units or if there are no bounds.

property coord_system

Relevant coordinate system (if any).

property dtype

The NumPy dtype of the current dimensional metadata object, as specified by its values.

property long_name

The CF Metadata long name for the object.

property metadata
property nbounds

Return the number of bounds that this coordinate has (0 for no bounds).

property ndim

Return the number of dimensions of the current dimensional metadata object.

property points

The coordinate points values as a NumPy array.

property shape

The fundamental shape of the metadata, expressed as a tuple.

property standard_name

The CF Metadata standard name for the object.

property units

The S.I. unit of the object.

property var_name

The NetCDF variable name for the object.

↑ top ↑

Defines a range of values for a coordinate.

class iris.coords.CoordExtent(name_or_coord, minimum, maximum, min_inclusive=True, max_inclusive=True)[source]

Create a CoordExtent for the specified coordinate and range of values.

Args:

• name_or_coord

Either a coordinate name or a coordinate, as defined in iris.cube.Cube.coords().

• minimum

The minimum value of the range to select.

• maximum

The maximum value of the range to select.

Kwargs:

• min_inclusive

If True, coordinate values equal to minimum will be included in the selection. Default is True.

• max_inclusive

If True, coordinate values equal to maximum will be included in the selection. Default is True.

static __new__(cls, name_or_coord, minimum, maximum, min_inclusive=True, max_inclusive=True)[source]

Create a CoordExtent for the specified coordinate and range of values.

Args:

• name_or_coord

Either a coordinate name or a coordinate, as defined in iris.cube.Cube.coords().

• minimum

The minimum value of the range to select.

• maximum

The maximum value of the range to select.

Kwargs:

• min_inclusive

If True, coordinate values equal to minimum will be included in the selection. Default is True.

• max_inclusive

If True, coordinate values equal to maximum will be included in the selection. Default is True.

count(value, /)

Return number of occurrences of value.

index(value, start=0, stop=9223372036854775807, /)

Return first index of value.

Raises ValueError if the value is not present.

max_inclusive

Alias for field number 4

maximum

Alias for field number 2

min_inclusive

Alias for field number 3

minimum

Alias for field number 1

name_or_coord

Alias for field number 0

↑ top ↑

A coordinate that is 1D, and numeric, with values that have a strict monotonic ordering. Missing values are not permitted in a DimCoord.

class iris.coords.DimCoord(points, standard_name=None, long_name=None, var_name=None, units=None, bounds=None, attributes=None, coord_system=None, circular=False, climatological=False)[source]

Create a 1D, numeric, and strictly monotonic coordinate with immutable points and bounds.

Missing values are not permitted.

Args:

• points:

1D numpy array-like of values (or single value in the case of a scalar coordinate) for each cell of the coordinate. The values must be strictly monotonic and masked values are not allowed.

Kwargs:

• standard_name:

CF standard name of the coordinate.

• long_name:

Descriptive name of the coordinate.

• var_name:

The netCDF variable name for the coordinate.

• units:

The Unit of the coordinate’s values. Can be a string, which will be converted to a Unit object.

• bounds:

An array of values describing the bounds of each cell. Given n bounds and m cells, the shape of the bounds array should be (m, n). For each bound, the values must be strictly monotonic along the cells, and the direction of monotonicity must be consistent across the bounds. For example, a DimCoord with 100 points and two bounds per cell would have a bounds array of shape (100, 2), and the slices bounds[:, 0] and bounds[:, 1] would be monotonic in the same direction. Masked values are not allowed. Note if the data is a climatology, climatological should be set.

• attributes:

A dictionary containing other CF and user-defined attributes.

• coord_system:

A CoordSystem representing the coordinate system of the coordinate, e.g., a GeogCS for a longitude coordinate.

• circular (bool):

Whether the coordinate wraps by the modulus i.e., the longitude coordinate wraps around the full great circle.

• climatological (bool):

When True: the coordinate is a NetCDF climatological time axis. When True: saving in NetCDF will give the coordinate variable a ‘climatology’ attribute and will create a boundary variable called ‘<coordinate-name>_climatology’ in place of a standard bounds attribute and bounds variable. Will set to True when a climatological time axis is loaded from NetCDF. Always False if no bounds exist.

__binary_operator__(other, mode_constant)

Common code which is called by add, sub, mul and div

Mode constant is one of ADD, SUB, MUL, DIV, RDIV

Note

The unit is not changed when doing scalar operations on a metadata object. This means that a metadata object which represents “10 meters” when multiplied by a scalar i.e. “1000” would result in a metadata object of “10000 meters”. An alternative approach could be taken to multiply the unit by 1000 and the resultant metadata object would represent “10 kilometers”.

__deepcopy__()Deep copy of coordinate.[source]

Used if copy.deepcopy is called on a coordinate.

cell(index)

Return the single Cell instance which results from slicing the points/bounds with the given index.

cells()

Returns an iterable of Cell instances for this Coord.

For example:

for cell in self.cells():
...

collapsed(dims_to_collapse=None)[source]

Returns a copy of this coordinate, which has been collapsed along the specified dimensions.

Replaces the points & bounds with a simple bounded region.

contiguous_bounds()

Returns the N+1 bound values for a contiguous bounded 1D coordinate of length N, or the (N+1, M+1) bound values for a contiguous bounded 2D coordinate of shape (N, M).

Only 1D or 2D coordinates are supported.

Note

If the coordinate has bounds, this method assumes they are contiguous.

If the coordinate is 1D and does not have bounds, this method will return bounds positioned halfway between the coordinate’s points.

If the coordinate is 2D and does not have bounds, an error will be raised.

convert_units(unit)

Change the coordinate’s units, converting the values in its points and bounds arrays.

For example, if a coordinate’s units attribute is set to radians then:

coord.convert_units('degrees')

will change the coordinate’s units attribute to degrees and multiply each value in points and bounds by 180.0/$$\pi$$.

copy(points=None, bounds=None)[source]

Returns a copy of this coordinate.

Kwargs:

• points: A points array for the new coordinate.

This may be a different shape to the points of the coordinate being copied.

• bounds: A bounds array for the new coordinate.

Given n bounds for each cell, the shape of the bounds array should be points.shape + (n,). For example, a 1d coordinate with 100 points and two bounds per cell would have a bounds array of shape (100, 2).

Note

If the points argument is specified and bounds are not, the resulting coordinate will have no bounds.

core_bounds()

The points array at the core of this coord, which may be a NumPy array or a dask array.

core_points()

The points array at the core of this coord, which may be a NumPy array or a dask array.

cube_dims(cube)

Return the cube dimensions of this Coord.

Equivalent to “cube.coord_dims(self)”.

classmethod from_coord(coord)

Create a new Coord of this type, from the given coordinate.

classmethod from_regular(zeroth, step, count, standard_name=None, long_name=None, var_name=None, units=None, attributes=None, coord_system=None, circular=False, climatological=False, with_bounds=False)[source]

Create a DimCoord with regularly spaced points, and optionally bounds.

The majority of the arguments are defined as for Coord, but those which differ are defined below.

Args:

• zeroth:

The value prior to the first point value.

• step:

The numeric difference between successive point values.

• count:

The number of point values.

Kwargs:

• with_bounds:

If True, the resulting DimCoord will possess bound values which are equally spaced around the points. Otherwise no bounds values will be defined. Defaults to False.

guess_bounds(bound_position=0.5)

Add contiguous bounds to a coordinate, calculated from its points.

Puts a cell boundary at the specified fraction between each point and the next, plus extrapolated lowermost and uppermost bound points, so that each point lies within a cell.

With regularly spaced points, the resulting bounds will also be regular, and all points lie at the same position within their cell. With irregular points, the first and last cells are given the same widths as the ones next to them.

Kwargs:

• bound_position:

The desired position of the bounds relative to the position of the points.

Note

An error is raised if the coordinate already has bounds, is not one-dimensional, or is not monotonic.

Note

Unevenly spaced values, such from a wrapped longitude range, can produce unexpected results : In such cases you should assign suitable values directly to the bounds property, instead.

has_bounds()

Return a boolean indicating whether the coord has a bounds array.

has_lazy_bounds()

Return a boolean indicating whether the coord’s bounds array is a lazy dask array or not.

has_lazy_points()

Return a boolean indicating whether the coord’s points array is a lazy dask array or not.

intersect(other, return_indices=False)

Returns a new coordinate from the intersection of two coordinates.

Both coordinates must be compatible as defined by is_compatible().

Kwargs:

• return_indices:

If True, changes the return behaviour to return the intersection indices for the “self” coordinate.

is_compatible(other, ignore=None)

Return whether the coordinate is compatible with another.

Compatibility is determined by comparing iris.coords.Coord.name(), iris.coords.Coord.units, iris.coords.Coord.coord_system and iris.coords.Coord.attributes that are present in both objects.

Args:

• other:

An instance of iris.coords.Coord, iris.common.CoordMetadata or iris.common.DimCoordMetadata.

• ignore:

A single attribute key or iterable of attribute keys to ignore when comparing the coordinates. Default is None. To ignore all attributes, set this to other.attributes.

Returns

Boolean.

is_contiguous(rtol=1e-05, atol=1e-08)

Return True if, and only if, this Coord is bounded with contiguous bounds to within the specified relative and absolute tolerances.

1D coords are contiguous if the upper bound of a cell aligns, within a tolerance, to the lower bound of the next cell along.

2D coords, with 4 bounds, are contiguous if the lower right corner of each cell aligns with the lower left corner of the cell to the right of it, and the upper left corner of each cell aligns with the lower left corner of the cell above it.

Args:

• rtol:

The relative tolerance parameter (default is 1e-05).

• atol:

The absolute tolerance parameter (default is 1e-08).

Returns

Boolean.

is_monotonic()[source]

Return True if, and only if, this Coord is monotonic.

lazy_bounds()

Return a lazy array representing the coord bounds.

Accessing this method will never cause the bounds values to be loaded. Similarly, calling methods on, or indexing, the returned Array will not cause the coord to have loaded bounds.

If the data have already been loaded for the coord, the returned Array will be a new lazy array wrapper.

Returns

A lazy array representing the coord bounds array or None if the coord does not have bounds.

lazy_points()

Return a lazy array representing the coord points.

Accessing this method will never cause the points values to be loaded. Similarly, calling methods on, or indexing, the returned Array will not cause the coord to have loaded points.

If the data have already been loaded for the coord, the returned Array will be a new lazy array wrapper.

Returns

A lazy array, representing the coord points array.

name(default=None, token=False)

Returns a string name representing the identity of the metadata.

First it tries standard name, then it tries the long name, then the NetCDF variable name, before falling-back to a default value, which itself defaults to the string ‘unknown’.

Kwargs:

• default:

The fall-back string representing the default name. Defaults to the string ‘unknown’.

• token:

If True, ensures that the name returned satisfies the criteria for the characters required by a valid NetCDF name. If it is not possible to return a valid name, then a ValueError exception is raised. Defaults to False.

Returns

String.

nearest_neighbour_index(point)

Returns the index of the cell nearest to the given point.

Only works for one-dimensional coordinates.

For example:

>>> cube = iris.load_cube(iris.sample_data_path('ostia_monthly.nc'))
>>> cube.coord('latitude').nearest_neighbour_index(0)
9
>>> cube.coord('longitude').nearest_neighbour_index(10)
12

Note

If the coordinate contains bounds, these will be used to determine the nearest neighbour instead of the point values.

Note

For circular coordinates, the ‘nearest’ point can wrap around to the other end of the values.

rename(name)

If ‘name’ is a valid standard name it will assign it to standard_name, otherwise it will assign it to long_name.

xml_element(doc)[source]

Create the xml.dom.minidom.Element that describes this DimCoord.

Args:

• doc:

The parent xml.dom.minidom.Document.

Returns

The xml.dom.minidom.Element that describes this DimCoord.

property attributes
property bounds

The coordinate bounds values, as a NumPy array, or None if no bound values are defined.

Note

The shape of the bound array should be: points.shape + (n_bounds, ).

property bounds_dtype

The NumPy dtype of the coord’s bounds. Will be None if the coord does not have bounds.

property circular

Whether the coordinate wraps by coord.units.modulus.

property climatological

A boolean that controls whether the coordinate is a climatological time axis, in which case the bounds represent a climatological period rather than a normal period.

Always reads as False if there are no bounds. On set, the input value is cast to a boolean, exceptions raised if units are not time units or if there are no bounds.

property coord_system

The coordinate-system of the coordinate.

property dtype

The NumPy dtype of the current dimensional metadata object, as specified by its values.

property long_name

The CF Metadata long name for the object.

property metadata
property nbounds

Return the number of bounds that this coordinate has (0 for no bounds).

property ndim

Return the number of dimensions of the current dimensional metadata object.

property points

The coordinate points values as a NumPy array.

property shape

The fundamental shape of the metadata, expressed as a tuple.

property standard_name

The CF Metadata standard name for the object.

property units

The S.I. unit of the object.

property var_name

The NetCDF variable name for the object.


text:
# NMR and MRI - questions!

1. Apr 21, 2007

### Coto

NMR and MRI -- questions!

I'm currently doing an experiment on nuclear magnetic resonance for an undergrad physics lab. Our experiment was based around determining the effects of proton concentration on spin-lattice and spin-spin relaxation times. We used two separate sets of ionic solutions (cupric sulfate solutions and ferric nitrate solutions.)

To analyze the results I'll be looking at T1 and T2 vs. concentration level as well as the supposed 1/T1 1/T2 relaxivity vs concentration. My problem is I can find little to no theory whatsoever that describes to me in detail the importance of relaxivity. I'm aware that relaxivity affects contrast in MRI images and to what extent it can do so, however I'm not sure about what makes a substance good for T2 imaging versus good for T1 imaging?

I guess in short, other than that main question at the end, I'm looking for sources about MRI relaxivity theory. Any help would be greatly appreciated.

EDIT: Sorry about posting this in the main forum. Since I was just looking for sources on MRI theory, I thought it'd be just as fine to post in the general forum.

Last edited: Apr 21, 2007
2. Apr 22, 2007

### Coto

Found the answer to my own question.. may as well post it in case someone else is needing the same help one day.

MRI imaging uses voxels that are governed by an intensity equation based off of T1 and T2 relaxation times. For 2d images, the pixel intensity is related as:

$$I = \kappa \rho(x,y) (1-exp({\frac{-\tau_R}{T_1}})) exp({\frac{-\tau_e}{T2}}})$$

Where $$\tau_R$$ and $$\tau_e$$ are the repetition time and echo pulse time respectively, $$\rho(x,y)$$ is the proton density w.r.t 2 dimensions, and $$\kappa$$ is a proportionality (normalization) constant.

As can be seen shortening T1 will increase pixel intensity, where as shortening T2 will decrease pixel intensity.. this naturally leads to T1 and T2-weighted imaging in MRI application.

Gyah, I hope that helps someone someday. Ah. For sources, try and find any modern book on MRI technology.. NMR books generally don't mention this stuff.

Last edited: Apr 22, 2007

format: plain


text:
Can you make a definition of velocity in QM ?

1. Nov 4, 2011

gurhkn

Can you make a definition of velocity in QM ?

I am trying to find a general definition of velocity in QM. QM is totally different world !

2. Nov 4, 2011

martinbn

Why not the "same" as in classical mechanics. It is the time derivative of position. So the x-component of the velocity is dx/dt, which by Heisenberg's equations is some constant times the comutator [x,H].

3. Nov 5, 2011

juanrga

$$v_{QM} \equiv \frac{dx_{QM}}{dt}$$

4. Nov 5, 2011

jfy4

Sorry to maybe be a little pedantic, but are those definitions satisfying? When we write
$$\frac{d\hat{x}}{dt}=\frac{i}{\hbar}[\hat{x},\hat{H}]$$
we have some idea in our head that something, 'x', is changing by such and such amount in time, 't'. But what is 'x' talking about? It's easy when we talk about a whole set-up and system that we are describing and we say "that system is translating." Are we asking how fast the approximate location of a particle is changing? I think the question can be much deeper, and I'm not sure if the OP had this in mind, but I would love to see a good definition of velocity for QM. For me it seems that the best at hand is actually
$$v^\mu \equiv \frac{p^\mu}{m}$$
which might seem bland and not anything new to the discussion, but when it comes down to it, in QFT calculations and the study of say, the Dirac equation, momentum is the main quantity one looks at (especially in scattering), and so it seems like a good place to at least look for a velocity. But, in general I also would like to see a solid definition, or if what has been given above is satisfactory, I feel like I haven't been given the whole deal about what it means to 'move' in QM.

5. Nov 6, 2011

bbbeard

I basically agree with jfy4: if you're studying a particle of mass m, the velocity is p/m, where p is the momentum. I think this relation is true for both the operator $\hat{p}$ and the eigenvalue p. Note that the commutator of position and velocity operators is not zero, so there is an uncertainty relation for x and v, but since the commutator of momentum and velocity is zero, there is no uncertainty relation of momentum and velocity. In other words, velocity acts like momentum in quantum mechanics.

BBB

6. Nov 6, 2011

juanrga

The definition of velocity in QM was given above.

Your is $v^i \equiv {p^i}/{m}$ is not valid for a low velocity particle in presence of a magnetic field for instance. It is also wrong for a Dirac electron. The application of the quantum mechanical definition gives the well-known result

$$v^i = c \alpha^i$$

for the velocity operator of an electron in QED.

Last edited: Nov 6, 2011
7. Nov 6, 2011

jfy4

This is new news to me! I'm not even sure what $\alpha^\mu$ is!? care to expand a little for me? or where can I see a reference for what you just wrote?

8. Nov 6, 2011

juanrga

It is Dirac alpha matrix. Any reference in QED uses that velocity operator for the interaction of an electron with the vector potential.

9. Nov 6, 2011

jfy4

Okay, thanks, I'll try and read up a little more on that.

Edit: wait! you mean the gamma matrices $\gamma^\mu$! Im completely familiar with those! your referencing $\bar{\Psi}\gamma^\mu \Psi$ as the current 4-vector with $\bar{\Psi}\Psi$ as the charge density correct? In which case we agree completely. I meant that in calculating scattering amplitudes (say the H diagram) the final amplitude $\mathcal{M}$ is expressed in terms of initial momenta and energy, and that we normally associate these with how the electron is moving. How do you feel about that statement?

Last edited: Nov 6, 2011
10. Nov 7, 2011

juanrga

No I said alpha matrix.

The charge density is $q\Psi^\dagger \Psi$

and the associated current $q\Psi^\dagger c\alpha^i \Psi$

this is often rewritten as $cq \bar{\Psi} \gamma^i \Psi$, where $\bar{\Psi} \equiv \Psi^\dagger \beta$ and $\gamma^i \equiv \beta \alpha^i$.

$\beta$ is Dirac beta matrix and satisfies $\beta^2 =1$

The four current is $cq \bar{\Psi} \gamma^\mu \Psi$ if you introduce a $\gamma^0 \equiv \beta$

Last edited: Nov 7, 2011
11. Nov 7, 2011

jfy4

Ok, then never heard of them, but they look like they fit in pretty similarly. Thanks again.

12. Nov 7, 2011

dextercioby

The alpha and beta matrices are the original matrices used by Dirac in 1928 in his famous articles. They are part of the so-called non-covariant version of Dirac's equation. Most books dealing in one way or another with the Dirac equation from a historical perspective use them.

13. Nov 7, 2011

akhmeteli

For example, Dirac's book: Principles of quantum mechanics (in the chapter where Zitterbewegung is discussed). Or http://en.wikipedia.org/wiki/Zitterbewegung

14. Nov 7, 2011

DrDu

m is a positive parameter in the Dirac equation. However the actual mass, i.e. the energy of a particle at rest, is -m for a negative energy solution. Hence this paradox can be avoided if we replace m by $m\beta$. Then instead of the alpha matrices, the gamma matrices appear, as they should.

Now to the problem of how to define velocity in QM: Beginners often think that it is necessary to perform two position measurements at two (infinitesimally) different times to measure velocity and then wonder how that should be possible as the velocity operator doesn't commute with the position operator. However, only the difference has to be measured which can be done without absolute measurement of position. In fact, this is what happens e.g. in the Doppler effect, which is probably the most routinely used method to measure the velocity of qm particles.

15. Nov 8, 2011

jewbinson

Given juanrga's definition of QM velocity is correct, that is v^i = c(alpha)^i, is it true that

<v^i> = <dx/dt> ?

(I'm talking about averages here...).

I'm just interested because the other day I went through the Ehrenfest Theorem which was used to show that d/dt<p_x> = -<dV/dx> (partial for V, but I suck at Latex).

16. Nov 8, 2011

dextercioby

Averages follow from the operator equation, obviously.

Pages 19-->26 of B. Thaller's <The Dirac equation> describe the issues with the velocity and position operators for the Dirac equation.

17. Nov 8, 2011

bbbeard

Thanks for your clarification; I agree that this formulation is correct for the relativistic case. Risking the fallacy of "appeal to authority", I would like to offer this excerpt from Bjorken and Drell's Relativistic Quantum Mechanics, p.37:

Now we come to an important difference in the relativistic theory. In the Schrödinger theory the velocity operator appearing in the current is just p/m and is a constant of motion for free particles. The current is not, however, proportional to the momentum in the Dirac theory, and whereas the Ehrenfest relation (1.27) has shown that dp/dt =0 for free-particle motion, the velocity operator cα is not constant, since [α,H]≠0. Indeed in constructing eigenfunctions of cα we have to include both positive- and negative-energy solutions, since the eigenvalues of cαi are ±c whereas |<cαi>+|<c, according to (3.29).

So, just to reiterate: for the (non-relativistic) Schrödinger equation, it is perfectly fine to use p/m for the velocity operator.

BBB

18. Nov 10, 2011

juanrga

It depends what you mean by the non-relativistic Schrödinger equation. If you mean strictly that the equation does not contain c, then you are right. However, usually people calls non-relativistic also to the Schrödinger equation for a charged particle in an electromagnetic field {*}

$$H = \frac{(p - eA/c)^2}{2m} + e\phi$$

In that case the velocity operator is $\frac{p - eA/c}{m}$.

{*} They mean that the electron is being treated non-relativistically with a quadratic momenta kinetic Hamiltonian

http://quantummechanics.ucsd.edu/ph130a/130_notes/node484.html

Of course rigorously it cannot be non-relativistic because contains c.

Last edited: Nov 10, 2011
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text:
The statement that is not correct for periodic classification of elements is

1. The properties of elements are periodic function of their atomic numbers

2.  Non-metallic elements are less in number than metallic elements

3.  For transition elements, the 3d-orbitals are filled with electrons after 3p-orbitals and before 4s-orbitals

4.  The first ionisation enthalpies of elements generally increase with increase in atomic number as we go along a period

Subtopic:  AUFBAU Principle |
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Comprehension given below is followed by some multiple choice questions. Each question has one correct option. Choose the correct option.

In the modern periodic table, elements are arranged in order of configuration. Depending upon the type of orbitals receiving the last electron, the elements in the periodic table have been divided into four block, viz s, p, d and f.

The modern periodic table consists of 7 periods and 18 groups. Each period begins with the filling of a new energy shell. In accordance with the Aufbau principle, the seven periods (1 to 7) have 2, 8, 8, 18, 18, 32 and 32 elements respectively.

The seventh period is still incomplete. To avoid the periodic table being too long, the two series of f-block elements, called lanthanoids and actinoids are placed at the bottom of the periodic table

(i) The element with atomic number 57 belongs to

(1) s-block

(2) p-block

(3) d-block

(4) f-block

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The electronic configuration of the element which is just above the element with atomic number 43 in the same group is ______.

(1) $1{\mathrm{s}}^{2}2{\mathrm{s}}^{2}2{\mathrm{p}}^{6}3{\mathrm{s}}^{2}3{\mathrm{p}}^{6}3{\mathrm{d}}^{5}4{\mathrm{s}}^{2}$

(2) $1{\mathrm{s}}^{2}2{\mathrm{s}}^{2}2{\mathrm{p}}^{6}3{\mathrm{s}}^{2}3{\mathrm{p}}^{6}3{\mathrm{d}}^{5}4{\mathrm{s}}^{3}4{\mathrm{p}}^{6}$

(3) $1{\mathrm{s}}^{2}2{\mathrm{s}}^{2}2{\mathrm{p}}^{6}3{\mathrm{s}}^{2}3{\mathrm{p}}^{6}3{\mathrm{d}}^{6}4{\mathrm{s}}^{2}$

(4) $1{\mathrm{s}}^{2}2{\mathrm{s}}^{2}2{\mathrm{p}}^{6}3{\mathrm{s}}^{2}3{\mathrm{p}}^{6}3{\mathrm{d}}^{7}4{\mathrm{s}}^{2}$

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# A box contains 10 light bulbs, fewer than half of which are

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A box contains 10 light bulbs, fewer than half of which are defective. Two bulbs are to be drawn simultaneously from the box. If n of the bulbs in box are defective, what is the value of n?

(1) The probability that the two bulbs to be drawn will be defective is 1/15.
(2) The probability that one of the bulbs to be drawn will be defective and the other will not be defective is 7/15.
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28 Aug 2010, 06:09
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udaymathapati wrote:
A box contains 10 light bulbs, fewer than half of which are defective. Two bulbs are to be drawn simultaneously from the box. If n of the bulbs in box are defective, what is the value of n?
(1) The probability that the two bulbs to be drawn will be defective is 1/15.
(2) The probability that one of the bulbs to be drawn will be defective and the other will not be defective is 7/15.

Given: $$bulbs=10$$ and $$defective=n<5$$. Question: $$n=?$$

(1) The probability that the two bulbs to be drawn will be defective is 1/15 --> clearly sufficient, as probability, $$p$$, of drawing 2 defective bulbs out of total 10 bulbs, obviously depends on # of defective bulbs, $$n$$, so we can calculate uniques value of $$n$$ if we are given $$p$$.

To show how it can be done: $$\frac{n}{10}*\frac{n-1}{9}=\frac{1}{15}$$ --> $$n(n-1)=6$$ --> $$n=3$$ or $$n=-2$$ (not a valid solution as $$n$$ represents # of defective bulbs and can not be negative). Sufficient.

(2) The probability that one of the bulbs to be drawn will be defective and the other will not be defective is 7/15 --> also sufficient, but a little bit trickier: if it were 3 defective and 7 good bulbs OR 7 defective and 3 good bulbs, then the probability of drawing one defective and one good bulb would be the same for both cases (symmetric distribution), so info about the probability, 7/15, of drawing one defective and one good bulb would give us 2 values of $$n$$ one less than 5 and another more than 5 (their sum would be 10), but as we are given that $$n<5$$, we can stiil get unique value of $$n$$ which is less than 5.

To show how it can be done: $$2*\frac{n}{10}*\frac{10-n}{9}=\frac{7}{15}$$ --> $$n(10-n)=21$$ --> $$n=3$$ or $$n=7$$ (not a valid solution as $$n<5$$). Sufficient.

Hope it's clear.
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29 Aug 2010, 19:52
Bunuel,

I did not understand the below part in option 2. How you got '2' in the equation. I know n < 5 but why do we need to multiply by '2'

$$2 * n/10 * (n-1)/9 = 1/15$$

Thanks
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29 Aug 2010, 20:00
Seekmba,

There are 2 ways to draw 2 bulbs in which 1 works and 1 is defective.
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29 Aug 2010, 21:07
But that is why we have

$$n/10 * (n-1)/9 = 1/15$$.

I am sorry but I still dont understand why multiply by 2.
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29 Aug 2010, 21:54
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Think of it like this:
Can select 2 bulbs out of 10 in 10C2 ways

1 defective and 1 ok bulb in N C1 x 10-N C1

So probability is 2 x N x 10-N / (9 x 10) = 7/15
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seekmba wrote:
Bunuel,

I did not understand the below part in option 2. How you got '2' in the equation. I know n < 5 but why do we need to multiply by '2'

$$2 * n/10 * (n-1)/9 = 1/15$$

Thanks

The probability that one of the bulbs to be drawn will be defective and the other will not be defective is the sum of the probabilities of 2 events: the first one is defective and the second is not PLUS the first is not defective and the second one is defective = $$\frac{n}{10}*\frac{10-n}{9}+\frac{10-n}{10}*\frac{n}{9}=2*\frac{n}{10}*\frac{10-n}{9}=\frac{7}{15}$$.
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31 Aug 2010, 04:32
Bunuel Rocks.....
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03 Feb 2011, 22:46
Yes, that it was only for two ways. However, we must see the option into the works of Bunuel. That it was the possible way to check and see the probability of the light bulb.
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04 Feb 2011, 00:54
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Defective bulbs: n
Non-defective bulbs: 10-n

and 0<=n<5.

Two bulbs can be drawn from a lot of 10 in
$$C^{10}_{2} = \frac{10!}{2!8!}=\frac{(10*9)}{2}=45$$
ways.

(1) Both defective:

Probability(Pick both defective bulbs) = Number of favorable choices/Number of total choices

Number of favorable choices : If we draw both defective bulbs
i.e. {draw 2 balls out of n defective balls} * {0 balls out of (10-n) balls}
The number of ways that can be done is:

$$C^{n}_{2} * C^{(10-n)}_{0}$$

$$=> \frac{n!}{(n-2)!2!} * 1$$

$$\frac{n(n-1)(n-2)!}{(n-2)!2!}=\frac{n(n-1)}{2}$$

Number of favorable choices: $$\frac{n(n-1)}{2}$$

Number of total choices: 45

Probability(Pick both defective bulbs) = $$\frac{n(n-1)}{(2*45)}$$

Given, Probability(Pick both defective bulbs) = 1/15

$$\frac{n(n-1)}{90} = \frac{1}{15}$$
$$n(n-1) = 6$$
$$n^2-n-6=0$$

n=3, n=-2

Since,
0<=n<5
n = 3

Sufficient.

(2) One defective and one non-defective:

Probability(One defective and one non-defective) = Number of favorable choices/Number of total choices

Number of favorable choices : If we draw exactly one defective and one non-defective bulb
i.e. {draw 1 balls out of n defective balls} * {1 balls out of (10-n) balls}
The number of ways that can be done is:

$$C^{n}_{1} * C^{(10-n)}_{1}$$

$$=> \frac{n!}{(n-1)!1!} * \frac{(10-n)!}{(10-n-1)!1!}$$

$$=> \frac{n(n-1)!}{(n-1)!1!} * \frac{(10-n)(10-n-1)!}{(10-n-1)!1!}$$

$$n(10-n)$$

Number of favorable choices: $$n(10-n)$$

Number of total choices: 45

Probability(One defective and one non-defective) = $$\frac{n(10-n)}{45}$$

Given, Probability(One defective and one non-defective) = 7/15

$$\frac{n(10-n)}{45} = \frac{7}{15}$$
$$n(10-n) = 21$$
$$n^2-10n+21=0$$

n=3 and n=7

Since,
0<=n<5
n = 3

Sufficient.

Ans: "D"
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08 May 2012, 05:37
Hi Bunuel ,
I am not very sure about the concept of symmetric probability , and when to use it ?
Can you please explain with an example.
Thanks
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09 May 2012, 01:33
shikhar wrote:
Hi Bunuel ,
I am not very sure about the concept of symmetric probability , and when to use it ?
Can you please explain with an example.
Thanks

Hope it helps.
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08 Jun 2013, 02:08
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udaymathapati wrote:
A box contains 10 light bulbs, fewer than half of which are defective. Two bulbs are to be drawn simultaneously from the box. If n of the bulbs in box are defective, what is the value of n?

(1) The probability that the two bulbs to be drawn will be defective is 1/15.
(2) The probability that one of the bulbs to be drawn will be defective and the other will not be defective is 7/15.

As an alternative way i suggest to use plug in method.

(1) since fewer than 5 bulbs are defective we are limited to choices 4,3,2 or 1. So max 4 numbers to plug, not many. First lets take 4 defective bulbs. (4/10)*(3/9)=12/90=2/15 a little bigger than 1/15, so the number of defective bulbs must be 3, but lets check it. (3/10)*(2/9)=6/90=1/15. Yes we could find the number of defective bulbs using the statement 1 - Sufficient.

(2) we can use the same logic here as well. The only difference is that here we need to look for one defective and one not defective sequence and multiple it to 2, because the sequence could start with defective as well as with non-defective bulb. Lets take 4 as the number of defective bulbs: (4/10)*(7/9)=24/90=8/30 after multiplying to 2 we have 8/15 not the ration we are looking for. Lets take 3 as the number of defective bulbs: (3/10)*(7/9)=21/90=7/30 after multiplying it to 2 we have 7/15. This is the ratio that we were looking for, so the statement 2 is sufficient.

Since both statements are sufficient on their own the answer is D.
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01 Apr 2014, 06:19
Bunuel wrote:
udaymathapati wrote:
A box contains 10 light bulbs, fewer than half of which are defective. Two bulbs are to be drawn simultaneously from the box. If n of the bulbs in box are defective, what is the value of n?
(1) The probability that the two bulbs to be drawn will be defective is 1/15.
(2) The probability that one of the bulbs to be drawn will be defective and the other will not be defective is 7/15.

Given: $$bulbs=10$$ and $$defective=n<5$$. Question: $$n=?$$

(1) The probability that the two bulbs to be drawn will be defective is 1/15 --> clearly sufficient, as probability, $$p$$, of drawing 2 defective bulbs out of total 10 bulbs, obviously depends on # of defective bulbs, $$n$$, so we can calculate uniques value of $$n$$ if we are given $$p$$.

To show how it can be done: $$\frac{n}{10}*\frac{n-1}{9}=\frac{1}{15}$$ --> $$n(n-1)=6$$ --> $$n=3$$ or $$n=-2$$ (not a valid solution as $$n$$ represents # of defective bulbs and can not be negative). Sufficient.

(2) The probability that one of the bulbs to be drawn will be defective and the other will not be defective is 7/15 --> also sufficient, but a little bit trickier: if it were 3 defective and 7 good bulbs OR 7 defective and 3 good bulbs, then the probability of drawing one defective and one good bulb would be the same for both cases (symmetric distribution), so info about the probability, 7/15, of drawing one defective and one good bulb would give us 2 values of $$n$$ one less than 5 and another more than 5 (their sum would be 10), but as we are given that $$n<5$$, we can stiil get unique value of $$n$$ which is less than 5.

To show how it can be done: $$2*\frac{n}{10}*\frac{10-n}{9}=\frac{7}{15}$$ --> $$n(10-n)=21$$ --> $$n=3$$ or $$n=7$$ (not a valid solution as $$n<5$$). Sufficient.

Hope it's clear.

In the second statement, you didn't subtract the defective bulb that was taken first. Was this because the question says that they are taken simultaneously?

Thanks for clarifying
Cheers
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01 Apr 2014, 08:27
jlgdr wrote:
Bunuel wrote:
udaymathapati wrote:
A box contains 10 light bulbs, fewer than half of which are defective. Two bulbs are to be drawn simultaneously from the box. If n of the bulbs in box are defective, what is the value of n?
(1) The probability that the two bulbs to be drawn will be defective is 1/15.
(2) The probability that one of the bulbs to be drawn will be defective and the other will not be defective is 7/15.

Given: $$bulbs=10$$ and $$defective=n<5$$. Question: $$n=?$$

(1) The probability that the two bulbs to be drawn will be defective is 1/15 --> clearly sufficient, as probability, $$p$$, of drawing 2 defective bulbs out of total 10 bulbs, obviously depends on # of defective bulbs, $$n$$, so we can calculate uniques value of $$n$$ if we are given $$p$$.

To show how it can be done: $$\frac{n}{10}*\frac{n-1}{9}=\frac{1}{15}$$ --> $$n(n-1)=6$$ --> $$n=3$$ or $$n=-2$$ (not a valid solution as $$n$$ represents # of defective bulbs and can not be negative). Sufficient.

(2) The probability that one of the bulbs to be drawn will be defective and the other will not be defective is 7/15 --> also sufficient, but a little bit trickier: if it were 3 defective and 7 good bulbs OR 7 defective and 3 good bulbs, then the probability of drawing one defective and one good bulb would be the same for both cases (symmetric distribution), so info about the probability, 7/15, of drawing one defective and one good bulb would give us 2 values of $$n$$ one less than 5 and another more than 5 (their sum would be 10), but as we are given that $$n<5$$, we can stiil get unique value of $$n$$ which is less than 5.

To show how it can be done: $$2*\frac{n}{10}*\frac{10-n}{9}=\frac{7}{15}$$ --> $$n(10-n)=21$$ --> $$n=3$$ or $$n=7$$ (not a valid solution as $$n<5$$). Sufficient.

Hope it's clear.

In the second statement, you didn't subtract the defective bulb that was taken first. Was this because the question says that they are taken simultaneously?

Thanks for clarifying
Cheers
J

Not sure I understand what you mean...

Anyway, mathematically the probability of picking two balls simultaneously, or picking them one at a time (without replacement) is the same.
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22 Jul 2014, 09:57
D is correct

b=10 , d=n<5 , Select 2

The total possibilities to select 2 out of 10 bulbs: 10C2 = 45
.
Total possibilities to select 2 out of n defective bulbs: nC2 = 1/2 * n(n-1)
.
(1) => (1/2 * n(n-1))/45 =1/5 => n={3;-2} => (1) sufficient
.
Total possibilities to either select 2 out of 10-n bulbs or n defective bulbs: nC2 + (10-n)C2 = 1/2 * n(n-1) + 1/2 * (10-n)(9-n)
.
(2) => the probability to either select 2 out of 10-n bulbs or n defective bulbs is 8/15
we have the equation:
(1/2 * n(n-1) + 1/2 * (10-n)(9-n))/45 = 8/15
=> n = {3;7} => (2) sufficient
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11 Dec 2014, 03:49
Hi Bunuel, I think I have the same question as jlgdr. (In the second statement, you didn't subtract the defective bulb that was taken first. Was this because the question says that they are taken simultaneously?)

In other words, why is the equation 2x(n/10)x((10-n)/9) and not 2x(n/10)x((10-n-1)/9). Is this because they are taken simultaneously?
Or is this because the second term is (9-(n-1))/9 = (9-n+1)/9 =(10-n)/9 as there is one "n" less to deduct after pulling it out in the first draw?

Many thanks

Last edited by Parco on 11 Dec 2014, 05:17, edited 1 time in total.
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11 Dec 2014, 04:49
as per question n ranges from 0 to 4 and is an integer
statement 1
nC2/10C2 = 1/15
this gives us a quadratic equation with solutions as 3 & -2. 3 is valid. Hence sufficient
statement 2
nC1*(10-n)C1/10C2 = 7/15
this also gives us a quadratic equation with 2 +ve solns of 3 & 7. But as per the range of n it cannot be 7. So n=3. Hence sufficient.

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11 Dec 2014, 06:51
Parco wrote:
Hi Bunuel, I think I have the same question as jlgdr. (In the second statement, you didn't subtract the defective bulb that was taken first. Was this because the question says that they are taken simultaneously?)

In other words, why is the equation 2x(n/10)x((10-n)/9) and not 2x(n/10)x((10-n-1)/9). Is this because they are taken simultaneously?
Or is this because the second term is (9-(n-1))/9 = (9-n+1)/9 =(10-n)/9 as there is one "n" less to deduct after pulling it out in the first draw?

Many thanks

After 1 defective bulb is drawn there are total 9 bulbs left out of which n-1 are defective and 9 - (n-1) = 10 - n non-defective. So, the probability of drawing non-defective is (non-defective)/(total) = (10 - n)/9.
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text:
SHOGUN  4.1.0
colwise_sum< Backend, Matrix > Struct Template Reference

Detailed Description

template<enum Backend, class Matrix> struct shogun::linalg::implementation::colwise_sum< Backend, Matrix >

Generic class colwise_sum which provides a static compute method. This class is specialized for different types of matrices and backend, providing a means to deal with various matrices directly without having to convert.

Definition at line 131 of file Sum.h.

Public Types

typedef Matrix::Scalar T

Static Public Member Functions

static SGVector< Tcompute (Matrix m, bool no_diag)
static SGVector< Tcompute (Block< Matrix > b, bool no_diag)
static void compute (SGMatrix< T > m, SGVector< T > result, bool no_diag)
static void compute (Block< SGMatrix< T > > b, SGVector< T > result, bool no_diag)

Member Typedef Documentation

 typedef Matrix::Scalar T

Scalar type

Definition at line 134 of file Sum.h.

Member Function Documentation

 static SGVector compute ( Matrix m, bool no_diag )
static

Method that computes column wise sum of co-efficients of a dense matrix

Parameters
 m the matrix whose colwise sum of co-efficients has to be computed no_diag if true, diagonal entries are excluded from the sum
Returns
the colwise sum of co-efficients computed as $$s_j=\sum_{i}m_{i,j}$$
 static SGVector compute ( Block< Matrix > b, bool no_diag )
static

Method that computes column wise sum of co-efficients of dense matrix blocks

Parameters
 b the matrix-block whose colwise sum of co-efficients has to be computed no_diag if true, diagonal entries are excluded from the sum
Returns
the colwise sum of co-efficients computed as $$s_j=\sum_{i}b_{i,j}$$
 static void compute ( SGMatrix< T > m, SGVector< T > result, bool no_diag )
static

Method that computes the column wise sum of co-efficients of SGMatrix using Eigen3

Parameters
 m the matrix whose colwise sum of co-efficients has to be computed no_diag if true, diagonal entries are excluded from the sum result Pre-allocated vector for the result of the computation
 static void compute ( Block< SGMatrix< T > > b, SGVector< T > result, bool no_diag )
static

Method that computes the column wise sum of co-efficients of SGMatrix blocks using Eigen3

Parameters
 b the matrix-block whose colwise sum of co-efficients has to be computed no_diag if true, diagonal entries are excluded from the sum result Pre-allocated vector for the result of the computation

The documentation for this struct was generated from the following file:

SHOGUN Machine Learning Toolbox - Documentation


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text:
What is the empirical formula of a compound that contains 75.7% arsenic and 24.3% oxygen?

Apr 14, 2016

$A {s}_{2} {O}_{3}$

Explanation:

Empirical formula is the simplest formula for a compound.

Arsenic's symbol is $A s$ while oxygen's symbol is $O$

Firstly, we need to make an assumption that the mass of the compound is 100g so that it is easier for us to convert the percentage of each of the element into its respective mass .

Based on the assumption;

Compound contain 75.7% arsenic;

$\frac{75.7}{100} \times 100 g = 75.7 g$

Compound contain 24.3% oxygen;

$\frac{24.3}{100} \times 100 g = 24.3 g$

Then, we convert each of the element mass into the number of moles by dividing it with its respective molar mass;

To calculate the number of moles of arsenic;

$\frac{75.7 g}{74.9 g \quad m o {l}^{-} 1} = 1.01 m o l$

To calculate the number of moles of oxygen;

$\frac{24.3 g}{16.0 g \quad m o {l}^{-} 1} = 1.52 m o l$

To find the relation between arsenic and oxygen, we divide each elements' number of moles with the smallest number of moles available .

For arsenic; $\frac{1.01 m o l}{1.01 m o l} = 1.00$

For oxygen; $\frac{1.52 m o l}{1.01 m o l} = 1.50$

Convert both values into whole number, which is by multiplying with $2$;

For arsenic; $1.00 \times 2 = 2$

For oxygen; $1.50 \times 2 = 3$

Put the values all together with respective element's symbol and we get the empirical formula of $A {s}_{2} {O}_{3}$, namely arsenic trioxide.


text:
# The lilypond model: existence, uniqueness and absence of percolation

#### by Prof Gunter Last

Institution: Uni Karlsruhe (Germany)
Date: Fri 29th July 2005
Time: 3:15 PM
Location: Theatre 2, Ground Floor, ICT Building (111 Barry St, Carlton)

Abstract: The lilypond system model based on a locally finite subset
$\varphi$ of the Euclidean space R^n is defined as follows. At time 0
every point of $\varphi$ starts growing with unit speed in all directions
to form a system of balls in which any particular ball ceases its growth
at the instant that it collides with another ball. A stochastic version of
this model has been introduced by HÃ¤ggstrÃ¶m and Meester (1996) in the
context of percolation and by Daley, Stoyan and Stoyan (1999) in the
context of hard-sphere models with maximal non-overlapping spheres.

In the first part of the talk we will present recent results obtained
jointly with Matthias Heveling (Karlsruhe) showing that the lilypond model
is uniquely defined for any locally finite subset $\varphi$. Actually we
will show that these results apply in the far more general setting, where
$\varphi$ is a locally finite subset of some pseudo-metric space. We will
also discuss an algorithm approximating the system with at least linearly
decreasing error.

In the second part of the talk we will consider a stochastic lilypond
model based on a stationary point process N. We present analytic
conditions on N implying the absence of percolation in this model.
Examples are Cox, Poisson cluster and Gibbs processes satisfying certain
exponential moment conditions. This part of the talk is based on joint
work with Daryl Daley (Canberra).

For More Information: Emma Lockwood tel: 8344-1617 email: emmal@ms.unimelb.edu.au


text:
# How do you prove that 1 is the supremum of the set: $A=\{\frac{m}{n} | m,n\in \mathbb{N}\}$

it might be a simple question.

I need to prove that $1$ is the supremum of the following set: $$A=\left\{\frac{m}{n}\mathrel{}\middle|\mathrel{} m<n\wedge m,n\in \mathbb{N}\right\}$$ So, actually I need to prove 2 things:

1. $\forall x\in A .x\le 1$
2. $\forall \varepsilon>0 \ \exists x\in A .x>1-\varepsilon$

So, the the first requirement is easy to prove, from the defeinition of $A$. but the second is not so simple for me. I've tried to show that this $x$ exists, but I can't show how it looks like.. I guess I should express it with $\varepsilon$, but I have no success so far.

If there are other ways to prove it, it'll fine too, and I'll be happy to hear about them.

Thanks!

-
Can you find an integer $N$ with $N>\frac 1 {\epsilon}$? (The Axiom of Archimedes, if you have encountered it) –  Mark Bennet May 6 '12 at 15:24

Hint: ${n\over n+1} =1-{1\over n+1}$.
I'm not sure how to use it. I guess I should look at $\frac{1}{n+1}$ as $\varepsilon$, but still I don't understand how to build $x$ from it :( –  rboy May 6 '12 at 15:41
@RB14 Given $\epsilon>0$, select $n$ such that ${1\over n+1}<\epsilon$ (why can you do this?). Then show that $1-{1\over{n+1}} >1-\epsilon$. –  David Mitra May 6 '12 at 15:45
there supposed to be a continuation to that comment, but I accidently pressed enter, and my chrome browser did not allow me to delete that comment. any way, I get it now, I should select $x$ to be $\frac{n}{n+1}$ where $n > \frac{1}{\epsilon}$. that is possible from the reason Mark mentioned in his comment - the axiom of archimedes. thanks guys! –  rboy May 6 '12 at 15:54