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# Timeseries¶

TimeSeries is the main class in darts. It represents a univariate or multivariate time series. It can represent a stochastic time series by storing several samples (trajectories).

class darts.timeseries.TimeSeries(xa: xarray.core.dataarray.DataArray)[source]

Bases: object

Wrapper around a (well formed) DataArray. Use the static factory methods to build instances unless you know what you are doing.

static _fill_missing_dates(xa: xarray.core.dataarray.DataArray, freq: Optional[str] = None) → xarray.core.dataarray.DataArray[source]

Returns an xarray DataArray instance with missing dates inserted from an input xarray DataArray. The first dimension of the input DataArray xa has to be the time dimension.

This requires either a provided freq or the possibility to infer a unique frequency (see offset aliases for more info on supported frequencies) from the provided timestamps.

Parameters
• xa – The xarray DataArray

• freq – Optionally, a string representing the frequency of the Pandas DateTimeIndex to fill in the missing dates.

Raises

ValueError – If xa’s DateTimeIndex contains less than 3 elements; if no unique frequency can be inferred from xa’s DateTimeIndex; if the resampled DateTimeIndex does not contain all dates from xa (see _restore_xarray_from_frequency())

Returns

xarray DataArray with filled missing dates from xa

Return type

xarray DataArray

static _restore_xarray_from_frequency(xa: xarray.core.dataarray.DataArray, freq: str) → xarray.core.dataarray.DataArray[source]

Returns an xarray DataArray instance that is resampled from an input xarray DataArray xa with frequency freq. freq should be the inferred or actual frequency of xa. All data from xa is maintained in the output DataArray at the corresponding dates. Any missing dates from xa will be inserted into the returned DataArray with np.nan values.

The first dimension of the input DataArray xa has to be the time dimension.

This requires a provided freq (see offset aliases for more info on supported frequencies).

Parameters
• xa – The xarray DataArray

• freq – A string representing the actual or inferred frequency of the Pandas DateTimeIndex from xa.

Raises

ValueError – If the resampled DateTimeIndex does not contain all dates from xa

Returns

xarray DataArray resampled from xa with freq including all data from xa and inserted missing dates

Return type

xarray DataArray

add_datetime_attribute(attribute, one_hot: bool = False, cyclic: bool = False)darts.timeseries.TimeSeries[source]

Returns a new TimeSeries instance with one (or more) additional component(s) that contain an attribute of the time index of the current series specified with attribute, such as ‘weekday’, ‘day’ or ‘month’.

This works only for deterministic time series (i.e., made of 1 sample).

Parameters
• attribute – A pd.DatatimeIndex attribute which will serve as the basis of the new column(s).

• one_hot – Boolean value indicating whether to add the specified attribute as a one hot encoding (results in more columns).

• cyclic – Boolean value indicating whether to add the specified attribute as a cyclic encoding. Alternative to one_hot encoding, enable only one of the two. (adds 2 columns, corresponding to sin and cos transformation).

Returns

New TimeSeries instance enhanced by attribute.

Return type

TimeSeries

add_holidays(country_code: str, prov: str = None, state: str = None)darts.timeseries.TimeSeries[source]

Adds a binary univariate component to the current series that equals 1 at every index that corresponds to selected country’s holiday, and 0 otherwise. The frequency of the TimeSeries is daily.

Available countries can be found here.

This works only for deterministic time series (i.e., made of 1 sample).

Parameters
• country_code – The country ISO code

• prov – The province

• state – The state

Returns

A new TimeSeries instance, enhanced with binary holiday component.

Return type

TimeSeries

all_values(copy=True) → numpy.ndarray[source]

Returns a 3-D Numpy array of dimension (time, component, sample), containing this series’ values for all samples.

Parameters

copy – Whether to return a copy of the values, otherwise returns a view. Leave it to True unless you know what you are doing.

Returns

The values composing the time series.

Return type

numpy.ndarray

append(other: darts.timeseries.TimeSeries)darts.timeseries.TimeSeries[source]

Appends another TimeSeries to this TimeSeries, along the time axis.

Parameters

other – A second TimeSeries.

Returns

A new TimeSeries, obtained by appending the second TimeSeries to the first.

Return type

TimeSeries

append_values(values: numpy.ndarray)darts.timeseries.TimeSeries[source]

Appends values to current TimeSeries, to the given indices.

Parameters

values – An array with the values to append.

Returns

A new TimeSeries with the new values appended

Return type

TimeSeries

autocorr(lag=1) → float[source]
property columns

Same as components property

property components

The names of the components (equivalent to DataFrame columns) as a Pandas Index

copy()darts.timeseries.TimeSeries[source]

Make a copy of this time series object

Returns

A copy of this time series.

Return type

TimeSeries

data_array(copy=True) → xarray.core.dataarray.DataArray[source]

Returns the xarray DataArray representation of this time series.

Parameters

copy – Whether to return a copy of the series. Leave it to True unless you know what you are doing.

Returns

The xarray DataArray underlying this time series.

Return type

xarray.DataArray

describe(percentiles=None, include=None, exclude=None) → pandas.core.frame.DataFrame[source]
diff(n: Optional[int] = 1, periods: Optional[int] = 1, dropna: Optional[bool] = True)darts.timeseries.TimeSeries[source]

Returns a differenced time series. This is often used to make a time series stationary.

Parameters
• n – Optionally, a positive integer indicating the number of differencing steps (default = 1). For instance, n=2 computes the second order differences.

• periods – Optionally, periods to shift for calculating difference. For instance, periods=12 computes the difference between values at time t and times t-12.

• dropna – Whether to drop the missing values after each differencing steps. If set to False, the corresponding first periods time steps will be filled with NaNs.

Returns

A TimeSeries constructed after differencing.

Return type

TimeSeries

drop_after(split_point: Union[pandas._libs.tslibs.timestamps.Timestamp, float, int])[source]

Drops everything after the provided timestamp ts, included. The timestamp may not be in the TimeSeries. If it is, the timestamp will be dropped.

Parameters

split_point – The timestamp that indicates cut-off time.

Returns

A new TimeSeries, after ts.

Return type

TimeSeries

drop_before(split_point: Union[pandas._libs.tslibs.timestamps.Timestamp, float, int])[source]

Drops everything before the provided timestamp ts, included. The timestamp may not be in the TimeSeries. If it is, the timestamp will be dropped.

Parameters

split_point – The timestamp that indicates cut-off time.

Returns

A new TimeSeries, after ts.

Return type

TimeSeries

property duration
returns: The duration of this time series; as a Timedelta if the series is indexed by a Datetimeindex,

and int otherwise.

Return type

Union[pandas.Timedelta, int]

end_time() → Union[pandas._libs.tslibs.timestamps.Timestamp, int][source]
Returns

A timestamp containing the last time of the TimeSeries (if indexed by DatetimeIndex), or an integer (if indexed by Int64Index/RangeIndex)

Return type

Union[pandas.Timestamp, int]

first_value() → float[source]
Returns

The first value of this univariate deterministic time series

Return type

float

first_values() → numpy.ndarray[source]
Returns

The first values of every component of this deterministic time series

Return type

np.ndarray

property freq
property freq_str
static from_csv(filepath_or_buffer: Union[PathLike[str], str, IO[T], io.RawIOBase, io.BufferedIOBase, io.TextIOBase, _io.TextIOWrapper, mmap.mmap], time_col: Optional[str] = None, value_cols: Optional[Union[List[str], str]] = None, fill_missing_dates: Optional[bool] = False, freq: Optional[str] = None, fillna_value: Optional[float] = None, **kwargs)TimeSeries[source]

Returns a deterministic TimeSeries instance built from a single CSV file. One column can be used to represent the time (if not present, the time index will be an Int64Index) and a list of columns value_cols can be used to indicate the values for this time series.

Parameters
• filepath_or_buffer – The path to the CSV file, or the file object; consistent with the argument of pandas.read_csv function

• time_col – The time column name. If set, the column will be cast to a pandas DatetimeIndex. If not set, the pandas Int64Index will be used.

• value_cols – A string or list of strings representing the value column(s) to be extracted from the CSV file. If set to None, all columns from the CSV file will be used (except for the time_col, if specified)

• fill_missing_dates – Optionally, a boolean value indicating whether to fill missing dates with NaN values. This requires either a provided freq or the possibility to infer the frequency from the provided timestamps. See _fill_missing_dates() for more info.

• freq – Optionally, a string representing the frequency of the Pandas DateTimeIndex. This is useful in order to fill in missing values if some dates are missing and fill_missing_dates is set to True.

• fillna_value – Optionally, a numeric value to fill missing values (NaNs) with.

• **kwargs – Optional arguments to be passed to pandas.read_csv function

Returns

A univariate or multivariate deterministic TimeSeries constructed from the inputs.

Return type

TimeSeries

static from_dataframe(df: pandas.core.frame.DataFrame, time_col: Optional[str] = None, value_cols: Optional[Union[List[str], str]] = None, fill_missing_dates: Optional[bool] = False, freq: Optional[str] = None, fillna_value: Optional[float] = None)darts.timeseries.TimeSeries[source]

Returns a deterministic TimeSeries instance built from a selection of columns of a DataFrame. One column (or the DataFrame index) has to represent the time, and a list of columns value_cols has to represent the values for this time series.

Parameters
• df – The DataFrame

• time_col – The time column name. If set, the column will be cast to a pandas DatetimeIndex. If not set, the DataFrame index will be used. In this case the DataFrame must contain an index that is either a pandas DatetimeIndex or a pandas Int64Index (incl. RangeIndex). If a DatetimeIndex is used, it is better if it has no holes; alternatively setting fill_missing_dates can in some casees solve these issues (filling holes with NaN, or with the provided fillna_value numeric value, if any).

• value_cols – A string or list of strings representing the value column(s) to be extracted from the DataFrame. If set to None, the whole DataFrame will be used.

• fill_missing_dates – Optionally, a boolean value indicating whether to fill missing dates with NaN values. This requires either a provided freq or the possibility to infer the frequency from the provided timestamps. See _fill_missing_dates() for more info.

• freq – Optionally, a string representing the frequency of the Pandas DateTimeIndex. This is useful in order to fill in missing values if some dates are missing and fill_missing_dates is set to True.

• fillna_value – Optionally, a numeric value to fill missing values (NaNs) with.

Returns

A univariate or multivariate deterministic TimeSeries constructed from the inputs.

Return type

TimeSeries

static from_json(json_str: str)darts.timeseries.TimeSeries[source]

Converts the JSON String representation of a TimeSeries object (produced using TimeSeries.to_json()) into a TimeSeries object

At the moment this only supports deterministic time series (i.e., made of 1 sample).

Parameters

json_str – The JSON String to convert

Returns

The time series object converted from the JSON String

Return type

TimeSeries

static from_series(pd_series: pandas.core.series.Series, fill_missing_dates: Optional[bool] = False, freq: Optional[str] = None, fillna_value: Optional[float] = None)darts.timeseries.TimeSeries[source]

Returns a univariate and deterministic TimeSeries built from a pandas Series.

The series must contain an index that is either a pandas DatetimeIndex or a pandas Int64Index (incl. RangeIndex). If a DatetimeIndex is used, it is better if it has no holes; alternatively setting fill_missing_dates can in some cases solve these issues (filling holes with NaN, or with the provided fillna_value numeric value, if any).

Parameters
• pd_series – The pandas Series instance.

• fill_missing_dates – Optionally, a boolean value indicating whether to fill missing dates with NaN values. This requires either a provided freq or the possibility to infer the frequency from the provided timestamps. See _fill_missing_dates() for more info.

• freq – Optionally, a string representing the frequency of the Pandas DateTimeIndex. This is useful in order to fill in missing values if some dates are missing and fill_missing_dates is set to True.

• fillna_value – Optionally, a numeric value to fill missing values (NaNs) with.

Returns

A univariate and deterministic TimeSeries constructed from the inputs.

Return type

TimeSeries

static from_times_and_values(times: Union[pandas.core.indexes.datetimes.DatetimeIndex, pandas.core.indexes.numeric.Int64Index], values: numpy.ndarray, fill_missing_dates: Optional[bool] = False, freq: Optional[str] = None, columns: Optional[Collection] = None, fillna_value: Optional[float] = None)darts.timeseries.TimeSeries[source]

Returns a TimeSeries built from an index and value array.

Parameters
• times – A pandas.DateTimeIndex or pandas.Int64Index (or pandas.RangeIndex) representing the time axis for the time series. If a DatetimeIndex is used, it is better if it has no holes; alternatively setting fill_missing_dates can in some cases solve these issues (filling holes with NaN, or with the provided fillna_value numeric value, if any).

• values – A Numpy array of values for the TimeSeries. Both 2-dimensional arrays, for deterministic series, and 3-dimensional arrays, for probabilistic series, are accepted. In the former case the dimensions should be (time, component), and in the latter case (time, component, sample).

• fill_missing_dates – Optionally, a boolean value indicating whether to fill missing dates with NaN values. This requires either a provided freq or the possibility to infer the frequency from the provided timestamps. See _fill_missing_dates() for more info.

• freq – Optionally, a string representing the frequency of the Pandas DateTimeIndex. This is useful in order to fill in missing values if some dates are missing and fill_missing_dates is set to True.

• columns – Columns to be used by the underlying pandas DataFrame.

• fillna_value – Optionally, a numeric value to fill missing values (NaNs) with.

Returns

A TimeSeries constructed from the inputs.

Return type

TimeSeries

static from_values(values: numpy.ndarray, columns: Optional[Collection] = None, fillna_value: Optional[float] = None)darts.timeseries.TimeSeries[source]

Returns a TimeSeries built from an array of values. The series will have an integer index (Int64Index).

Parameters
• values – A Numpy array of values for the TimeSeries. Both 2-dimensional arrays, for deterministic series, and 3-dimensional arrays, for probabilistic series, are accepted. In the former case the dimensions should be (time, component), and in the latter case (time, component, sample).

• columns – Columns to be used by the underlying pandas DataFrame.

• fillna_value – Optionally, a numeric value to fill missing values (NaNs) with.

Returns

A TimeSeries constructed from the inputs.

Return type

TimeSeries

static from_xarray(xa: xarray.core.dataarray.DataArray, fill_missing_dates: Optional[bool] = False, freq: Optional[str] = None, fillna_value: Optional[float] = None)darts.timeseries.TimeSeries[source]

Returns a TimeSeries instance built from an xarray DataArray. The dimensions of the DataArray have to be (time, component, sample), in this order. The time dimension can have an arbitrary name, but component and sample must be named “component” and “sample”, respectively.

The first dimension (time), and second dimension (component) must be indexed (i.e., have coordinates). The time must be indexed either with a pandas DatetimeIndex or a pandas Int64Index. If a DatetimeIndex is used, it is better if it has no holes; alternatively setting fill_missing_dates can in some cases solve these issues (filling holes with NaN, or with the provided fillna_value numeric value, if any).

If two components have the same name or are not strings, this method will disambiguate the components names by appending a suffix of the form “<name>_N” to the N-th column with name “name”.

Parameters
• xa – The xarray DataArray

• fill_missing_dates – Optionally, a boolean value indicating whether to fill missing dates with NaN values. This requires either a provided freq or the possibility to infer the frequency from the provided timestamps. See _fill_missing_dates() for more info.

• freq – Optionally, a string representing the frequency of the Pandas DateTimeIndex. This is useful in order to fill in missing values if some dates are missing and fill_missing_dates is set to True.

• fillna_value – Optionally, a numeric value to fill missing values (NaNs) with.

Returns

A univariate or multivariate deterministic TimeSeries constructed from the inputs.

Return type

TimeSeries

gaps() → pandas.core.frame.DataFrame[source]

A function to compute and return gaps in the TimeSeries. Works only on deterministic time series (1 sample).

Returns

A pandas.DataFrame containing a row for every gap (rows with all-NaN values in underlying DataFrame) in this time series. The DataFrame contains three columns that include the start and end time stamps of the gap and the integer length of the gap (in self.freq units if the series is indexed by a DatetimeIndex).

Return type

pd.DataFrame

get_index_at_point(point: Union[pandas._libs.tslibs.timestamps.Timestamp, float, int], after=True) → int[source]

Converts a point into an integer index

Parameters
• point

This parameter supports 3 different data types: pd.Timestamp, float and int.

pd.Timestamp work only on series that are indexed with a pd.DatetimeIndex. In such cases, the returned point will be the index of this timestamp if it is present in the series time index. It it’s not present in the time index, the index of the next timestamp is returned if after=True (if it exists in the series), otherwise the index of the previous timestamp is returned (if it exists in the series).

In case of a float, the parameter will be treated as the proportion of the time series that should lie before the point.

In the case of int, the parameter will returned as such, provided that it is in the series. Otherwise it will raise a ValueError.

• after – If the provided pandas Timestamp is not in the time series index, whether to return the index of the next timestamp or the index of the previous one.

get_timestamp_at_point(point: Union[pandas._libs.tslibs.timestamps.Timestamp, float, int]) → pandas._libs.tslibs.timestamps.Timestamp[source]

Converts a point into a pandas.Timestamp (if Datetime-indexed) or into an integer (if Int64-indexed).

Parameters

point – This parameter supports 3 different data types: float, int and pandas.Timestamp. In case of a float, the parameter will be treated as the proportion of the time series that should lie before the point. In the case of int, the parameter will be treated as an integer index to the time index of series. Will raise a ValueError if not a valid index in series In case of a pandas.Timestamp, point will be returned as is provided that the timestamp is present in the series time index, otherwise will raise a ValueError.

property has_datetime_index

Whether this series is indexed with a DatetimeIndex (otherwise it is indexed with an Int64Index)

property has_range_index

Whether this series is indexed with an Int64Index (otherwise it is indexed with a DatetimeIndex)

has_same_time_as(other: darts.timeseries.TimeSeries) → bool[source]

Checks whether this TimeSeries and another one have the same time index.

Parameters

other – the other series

Returns

True if both TimeSeries have the same index, False otherwise.

Return type

bool

property is_deterministic
property is_probabilistic
property is_stochastic
property is_univariate
is_within_range(ts: Union[pandas._libs.tslibs.timestamps.Timestamp, int]) → bool[source]

Check whether a given timestamp or integer is withing the time interval of this time series. If a timestamp is provided, it does not need to be an element of the time index of the series.

Parameters

ts – The pandas.Timestamp (if indexed with DatetimeIndex) or integer (if indexed with Int64Index) to check.

Returns

Whether ts is contained within the interval of this time series.

Return type

bool

kurtosis(axis=None, skipna=None, level=None, numeric_only=None, **kwargs) → float[source]
last_value() → float[source]
Returns

The last value of this univariate deterministic time series

Return type

float

last_values() → numpy.ndarray[source]
Returns

The last values of every component of this deterministic time series

Return type

np.ndarray

longest_contiguous_slice(max_gap_size: int = 0)darts.timeseries.TimeSeries[source]

Returns the largest TimeSeries slice of this deterministic time series that contains no gaps (contigouse all-NaN rows) larger than max_gap_size.

This method is only applicable to deterministic series (i.e., having 1 sample).

Returns

a new series constituting the largest slice of the original with no or bounded gaps

Return type

TimeSeries

map(fn: Union[Callable[[numpy.number], numpy.number], Callable[[Union[pandas._libs.tslibs.timestamps.Timestamp, int], numpy.number], numpy.number]])darts.timeseries.TimeSeries[source]

Applies the function fn elementwise to all values in this TimeSeries. Returns a new TimeSeries instance. If fn takes 1 argument it is simply applied elementwise. If it takes 2 arguments, it is applied elementwise on the (timestamp, value) tuples.

At the moment this function works only on deterministic time series (i.e., made of 1 sample).

Parameters

fn – Either a function which takes a value and returns a value ie. f(x) = y Or a function which takes a value and its timestamp and returns a value ie. f(timestamp, x) = y The type of timestamp is either pd.Timestamp (if the series is indexed with a DatetimeIndex), or an integer otherwise (if the series is indexed with an Int64Index).

Returns

A new TimeSeries instance

Return type

TimeSeries

max(axis=None, skipna=None, level=None, numeric_only=None, **kwargs) → float[source]
mean(axis=None, skipna=None, level=None, numeric_only=None, **kwargs) → float[source]
median(axis=None, skipna=None, level=None, numeric_only=None, **kwargs) → float[source]
min(axis=None, skipna=None, level=None, numeric_only=None, **kwargs) → float[source]
property n_components
property n_samples
property n_timesteps
pd_dataframe(copy=True) → pandas.core.frame.DataFrame[source]

Returns a Pandas DataFrame representation of this time series. Each of the series components will appear as a column in the DataFrame. Works only for deterministic series (i.e., made of 1 sample).

Parameters

copy – Whether to return a copy of the dataframe. Leave it to True unless you know what you are doing.

Returns

The Pandas DataFrame representation of this time series

Return type

pandas.DataFrame

pd_series(copy=True) → pandas.core.series.Series[source]

Returns a Pandas Series representation of this time series. Works only for univariate series that are deterministic (i.e., made of 1 sample).

Parameters

copy – Whether to return a copy of the series. Leave it to True unless you know what you are doing.

Returns

A Pandas Series representation of this univariate time series.

Return type

pandas.Series

plot(new_plot: bool = False, central_quantile: Union[float, str] = 0.5, low_quantile: Optional[float] = 0.05, high_quantile: Optional[float] = 0.95, *args, **kwargs)[source]

A wrapper method around xarray.DataArray.plot().

Parameters
• new_plot – whether to spawn a new Figure

• central_quantile – The quantile (between 0 and 1) to plot as a “central” value, if the series is stochastic (i.e., if it has multiple samples). This will be applied on each component separately (i.e., to display quantiles of the components’ marginal distributions). For instance, setting central_quantile=0.5 will plot the median of each component. central_quantile can also be set to ‘mean’.

• low_quantile – The quantile to use for the lower bound of the plotted confidence interval. Similar to central_quantile, this is applied to each component separately (i.e., displaying marginal distributions). No confidence interval is shown if confidence_low_quantile is None (default 0.05).

• high_quantile – The quantile to use for the upper bound of the plotted confidence interval. Similar to central_quantile, this is applied to each component separately (i.e., displaying marginal distributions). No confidence interval is shown if high_quantile is None (default 0.95).

• args – some positional arguments for the plot() method

• kwargs – some keyword arguments for the plot() method

quantile_df(quantile=0.5) → pandas.core.frame.DataFrame[source]

Returns a Pandas DataFrame containing the single desired quantile of each component (over the samples). Each of the series components will appear as a column in the DataFrame. The column will be named “<component>_X”, where “<component>” is the column name corresponding to this component, and “X” is the quantile value. The quantile columns represent the marginal distributions of the components of this series.

This works only on stochastic series (i.e., with more than 1 sample)

Parameters

quantile – The desired quantile value. The value must be represented as a fraction (between 0 and 1 inclusive). For instance, 0.5 will return a DataFrame containing the median of the (marginal) distribution of each component.

Returns

The Pandas DataFrame containing the desired quantile for each component.

Return type

pandas.DataFrame

quantile_timeseries(quantile=0.5)darts.timeseries.TimeSeries[source]

Returns a deterministic TimeSeries containing the single desired quantile of each component (over the samples) of this stochastic TimeSeries. The components in the new series are named “<component>_X”, where “<component>” is the column name corresponding to this component, and “X” is the quantile value. The quantile columns represent the marginal distributions of the components of this series.

This works only on stochastic series (i.e., with more than 1 sample)

Parameters

quantile – The desired quantile value. The value must be represented as a fraction (between 0 and 1 inclusive). For instance, 0.5 will return a TimeSeries containing the median of the (marginal) distribution of each component.

Returns

The TimeSeries containing the desired quantile for each component.

Return type

TimeSeries

quantiles_df(quantiles: Tuple[float] = 0.1, 0.5, 0.9) → pandas.core.frame.DataFrame[source]

Returns a Pandas DataFrame containing the desired quantiles of each component (over the samples). Each of the series components will appear as a column in the DataFrame. The column will be named “<component>_X”, where “<component>” is the column name corresponding to this component, and “X” is the quantile value. The quantiles represent the marginal distributions of the components of this series.

This works only on stochastic series (i.e., with more than 1 sample)

Parameters

quantiles – Tuple containing the desired quantiles. The values must be represented as fractions (between 0 and 1 inclusive). For instance, (0.1, 0.5, 0.9) will return a DataFrame containing the 10th-percentile, median and 90th-percentile of the (marginal) distribution of each component.

Returns

The Pandas DataFrame containing the quantiles for each component.

Return type

pandas.DataFrame

resample(freq: str, method: str = 'pad')darts.timeseries.TimeSeries[source]

Creates an reindexed time series with a given frequency. Provided method is used to fill holes in reindexed TimeSeries, by default ‘pad’.

Parameters
• freq – The new time difference between two adjacent entries in the returned TimeSeries. A DateOffset alias is expected.

• method

Method to fill holes in reindexed TimeSeries (note this does not fill NaNs that already were present):

‘pad’: propagate last valid observation forward to next valid

‘backfill’: use NEXT valid observation to fill.

Returns

A reindexed TimeSeries with given frequency.

Return type

TimeSeries

rescale_with_value(value_at_first_step: float)darts.timeseries.TimeSeries[source]

Returns a new TimeSeries, which is a multiple of this TimeSeries such that the first value is value_at_first_step. (Note: numerical errors can appear with value_at_first_step > 1e+24).

Parameters

value_at_first_step – The new value for the first entry of the TimeSeries.

Returns

A new TimeSeries, where the first value is value_at_first_step and other values have been scaled accordingly.

Return type

TimeSeries

shift(n: int)darts.timeseries.TimeSeries[source]

Shifts the time axis of this TimeSeries by n time steps.

If $$n > 0$$, shifts in the future. If $$n < 0$$, shifts in the past.

For example, with $$n=2$$ and freq=’M’, March 2013 becomes May 2013. With $$n=-2$$, March 2013 becomes Jan 2013.

Parameters

n – The number of time steps (in self.freq unit) to shift by. Can be negative.

Returns

A new TimeSeries, with a shifted index.

Return type

TimeSeries

skew(axis=None, skipna=None, level=None, numeric_only=None, **kwargs) → float[source]
slice(start_ts: Union[pandas._libs.tslibs.timestamps.Timestamp, int], end_ts: Union[pandas._libs.tslibs.timestamps.Timestamp, int])[source]

Returns a new TimeSeries, starting later than start_ts and ending before end_ts, inclusive on both ends. The timestamps don’t have to be in the series.

Parameters
• start_ts – The timestamp that indicates the left cut-off.

• end_ts – The timestamp that indicates the right cut-off.

Returns

A new series, with indices greater or equal than start_ts and smaller or equal than end_ts.

Return type

TimeSeries

slice_intersect(other: darts.timeseries.TimeSeries)darts.timeseries.TimeSeries[source]

Returns a TimeSeries slice of this time series, where the time index has been intersected with the one provided in argument. Note that this method is in general not symmetric.

Parameters

other – the other time series

Returns

a new series, containing the values of this series, over the time-span common to both time series.

Return type

TimeSeries

slice_n_points_after(start_ts: Union[pandas._libs.tslibs.timestamps.Timestamp, int], n: int)darts.timeseries.TimeSeries[source]

Returns a new TimeSeries, starting a start_ts and having at most n points.

The provided timestamps will be included in the series.

Parameters
• start_ts – The timestamp or index that indicates the splitting time.

• n – The maximal length of the new TimeSeries.

Returns

A new TimeSeries, with length at most n, starting at start_ts

Return type

TimeSeries

slice_n_points_before(end_ts: Union[pandas._libs.tslibs.timestamps.Timestamp, int], n: int)darts.timeseries.TimeSeries[source]

Returns a new TimeSeries, ending at start_ts and having at most n points.

The provided timestamps will be included in the series.

Parameters
• end_ts – The timestamp or index that indicates the splitting time.

• n – The maximal length of the new TimeSeries.

Returns

A new TimeSeries, with length at most n, ending at start_ts

Return type

TimeSeries

split_after(split_point: Union[pandas._libs.tslibs.timestamps.Timestamp, float, int]) → Tuple[darts.timeseries.TimeSeries, darts.timeseries.TimeSeries][source]

Splits the TimeSeries in two, after a provided split_point.

Parameters

split_point – A timestamp, float or integer. If float, represents the proportion of the series to include in the first TimeSeries (must be between 0.0 and 1.0). If integer, represents the index position after which the split is performed. A pd.Timestamp can be provided for TimeSeries that are indexed by a pd.DatetimeIndex. In such cases, the timestamp will be contained in the first TimeSeries, but not in the second one. The timestamp itself does not have to appear in the original TimeSeries index.

Returns

A tuple of two time series. The first time series contains the first samples up to the split_point, and the second contains the remaining ones.

Return type

Tuple[TimeSeries, TimeSeries]

split_before(split_point: Union[pandas._libs.tslibs.timestamps.Timestamp, float, int]) → Tuple[darts.timeseries.TimeSeries, darts.timeseries.TimeSeries][source]

Splits the TimeSeries in two, before a provided split_point.

Parameters

split_point – A timestamp, float or integer. If float, represents the proportion of the series to include in the first TimeSeries (must be between 0.0 and 1.0). If integer, represents the index position before which the split is performed. A pd.Timestamp can be provided for TimeSeries that are indexed by a pd.DatetimeIndex. In such cases, the timestamp will be contained in the second TimeSeries, but not in the first one. The timestamp itself does not have to appear in the original TimeSeries index.

Returns

A tuple of two time series. The first time series contains the first samples up to the split_point, and the second contains the remaining ones.

Return type

Tuple[TimeSeries, TimeSeries]

stack(other: darts.timeseries.TimeSeries)darts.timeseries.TimeSeries[source]

Stacks another univariate or multivariate TimeSeries with the same time index on top of the current one (along the component axis), and returns the newly formed multivariate TimeSeries that includes all the components of self and of other.

The resulting TimeSeries will have the same name for its time dimension as this TimeSeries, and the same number of samples.

Parameters

other – A TimeSeries instance with the same index and the same number of samples as the current one.

Returns

A new multivariate TimeSeries instance.

Return type

TimeSeries

start_time() → Union[pandas._libs.tslibs.timestamps.Timestamp, int][source]
Returns

A timestamp containing the first time of the TimeSeries (if indexed by DatetimeIndex), or an integer (if indexed by Int64Index/RangeIndex)

Return type

Union[pandas.Timestamp, int]

std(axis=None, skipna=None, level=None, ddof=1, numeric_only=None, **kwargs) → float[source]
strip()darts.timeseries.TimeSeries[source]

Returns a TimeSeries slice of this deterministic time series, where NaN-only entries at the beginning and the end of the series are removed. No entries after (and including) the first non-NaN entry and before (and including) the last non-NaN entry are removed.

This method is only applicable to deterministic series (i.e., having 1 sample).

Returns

a new series based on the original where NaN-only entries at start and end have been removed

Return type

TimeSeries

sum(axis=None, skipna=None, level=None, numeric_only=None, min_count=0, **kwargs) → float[source]
property time_index

returns: The time index of this time series. :rtype: Union[pd.DatetimeIndex, pd.Int64Index]

to_json() → str[source]

Converts the TimeSeries object to a JSON String

At the moment this function works only on deterministic time series (i.e., made of 1 sample).

Returns

A JSON String representing the time series

Return type

str

univariate_component(index: Union[str, int])darts.timeseries.TimeSeries[source]

Retrieves one of the components of the current TimeSeries instance and returns it as new univariate TimeSeries instance.

Parameters

index – An zero-indexed integer indicating which component to retrieve. If components have names, this can be a string with the component’s name.

Returns

A new univariate TimeSeries instance.

Return type

TimeSeries

univariate_values(copy=True, sample=0) → numpy.ndarray[source]

Returns a 1-D Numpy array of dimension (time,), containing this univariate series’ values for one sample. If this series is deterministic, it contains only one sample and only sample=0 can be used.

Parameters

copy – Whether to return a copy of the values. Leave it to True unless you know what you are doing.

Returns

The values composing the time series guaranteed to be univariate.

Return type

numpy.ndarray

values(copy=True, sample=0) → numpy.ndarray[source]

Returns a 2-D Numpy array of dimension (time, component), containing this series’ values for one sample. If this series is deterministic, it contains only one sample and only sample=0 can be used.

Parameters
• copy – Whether to return a copy of the values, otherwise returns a view. Leave it to True unless you know what you are doing.

• sample – For stochastic series, the sample for which to return values. Default: 0 (first sample).

Returns

The values composing the time series.

Return type

numpy.ndarray

var(axis=None, skipna=None, level=None, ddof=1, numeric_only=None, **kwargs) → float[source]
property width
with_columns_renamed(col_names: Union[List[str], str], col_names_new: Union[List[str], str])darts.timeseries.TimeSeries[source]

Changes ts column names and returns a new TimeSeries instance.

Parameters
• col_names – String or list of strings corresponding the the column names to be changed.

• col_names_new – String or list of strings corresponding to the new column names. Must be the same length as col_names.

Returns

A new TimeSeries instance.

Return type

TimeSeries


text:
# Compact group

The circle of center 0 and radius 1 in the complex plane is a compact Lie group with complex multiplication.

In mathematics, a compact (topological) group is a topological group whose topology is compact. Compact groups are a natural generalization of finite groups with the discrete topology and have properties that carry over in significant fashion. Compact groups have a well-understood theory, in relation to group actions and representation theory.

In the following we will assume all groups are Hausdorff spaces.

## Compact Lie groups

Lie groups form a very nice class of topological groups, and the compact Lie groups have a particularly well-developed theory. Basic examples of compact Lie groups include[1]

The classification theorem of compact Lie groups states that up to finite extensions and finite covers this exhausts the list of examples (which already includes some redundancies). This classification is described in more detail in the next subsection.

### Classification

Given any compact Lie group G one can take its identity component G0, which is connected. The quotient group G/G0 is the group of components π0(G) which must be finite since G is compact. We therefore have a finite extension

${\displaystyle 1\to G_{0}\to G\to \pi _{0}(G)\to 1.\,}$

Meanwhile, for connected compact Lie groups, we have the following result:[2]

Theorem: Every connected compact Lie group is the quotient by a finite central subgroup of a product of a simply connected compact Lie group and a torus.

Thus, the classification of connected compact Lie groups can in principle be reduced to knowledge of the simply connected compact Lie groups together with information about their centers.

Finally, every compact, connected, simply-connected Lie group K is a product of compact, connected, simply-connected simple Lie groups Ki each of which is isomorphic to exactly one of the following:

or one of the five exceptional groups G2, F4, E6, E7, and E8. The restrictions on n are to avoid special isomorphisms among the various families for small values of n. For each of these groups, the center is known explicitly. The classification is through the associated root system (for a fixed maximal torus), which in turn are classified by their Dynkin diagrams.

### Maximal tori

A key idea in the study of a connected compact Lie group K is the concept of a maximal torus, that is a subgroup T of K that is isomorphic to several copies of ${\displaystyle S^{1}}$ and that is not contained in any larger subgroup of this type. A basic example is the case ${\displaystyle K=SU(n)}$, in which case we may take ${\displaystyle T}$ to be the group of diagonal elements in ${\displaystyle K}$. A basic result is the torus theorem which states that every element of ${\displaystyle K}$ belongs to a maximal torus and that all maximal tori are conjugate.

The maximal torus in a compact group plays a role analogous to that of the Cartan subalgebra in a complex semisimple Lie algebra. In particular, once a maximal torus ${\displaystyle T\subset K}$ has been chosen, one can define a root system and a Weyl group similar to what one has for semisimple Lie algebras.[3] These structures then play an essential role both in the classification of connected compact groups (described above) and in the representation theory of a fixed such group (described below).

## Further examples

Amongst groups that are not Lie groups, and so do not carry the structure of a manifold, examples are the additive group Zp of p-adic integers, and constructions from it. In fact any profinite group is a compact group. This means that Galois groups are compact groups, a basic fact for the theory of algebraic extensions in the case of infinite degree.

Pontryagin duality provides a large supply of examples of compact commutative groups. These are in duality with abelian discrete groups.

## Haar measure

Compact groups all carry a Haar measure,[4] which will be invariant by both left and right translation (the modulus function must be a continuous homomorphism to positive reals (ℝ+, ×), and so 1). In other words, these groups are unimodular. Haar measure is easily normalized to be a probability measure, analogous to dθ/2π on the circle.

Such a Haar measure is in many cases easy to compute; for example for orthogonal groups it was known to Adolf Hurwitz, and in the Lie group cases can always be given by an invariant differential form. In the profinite case there are many subgroups of finite index, and Haar measure of a coset will be the reciprocal of the index. Therefore, integrals are often computable quite directly, a fact applied constantly in number theory.

## Representation theory

The representation theory of compact groups (not necessarily Lie groups and not necessarily connected) was founded by the Peter–Weyl theorem.[5] Hermann Weyl went on to give the detailed character theory of the compact connected Lie groups, based on maximal torus theory.[6] The resulting Weyl character formula was one of the influential results of twentieth century mathematics. The combination of the Peter–Weyl theorem and the Weyl character formula led Weyl to a complete classification of the representations of a connected compact Lie group; this theory is described in the next section.

A combination of Weyl's work and Cartan's theorem gives a survey of the whole representation theory of compact groups G . That is, by the Peter–Weyl theorem the irreducible unitary representations ρ of G are into a unitary group (of finite dimension) and the image will be a closed subgroup of the unitary group by compactness. Cartan's theorem states that Im(ρ) must itself be a Lie subgroup in the unitary group. If G is not itself a Lie group, there must be a kernel to ρ. Further one can form an inverse system, for the kernel of ρ smaller and smaller, of finite-dimensional unitary representations, which identifies G as an inverse limit of compact Lie groups. Here the fact that in the limit a faithful representation of G is found is another consequence of the Peter–Weyl theorem,

The unknown part of the representation theory of compact groups is thereby, roughly speaking, thrown back onto the complex representations of finite groups. This theory is rather rich in detail, but is qualitatively well understood.

## Representation theory of a connected compact Lie group

Certain simple examples of the representation theory of compact Lie groups can be worked out by hand, such as the representations of the rotation group SO(3), the special unitary group SU(2), and the special unitary group SU(3). We focus here on the general theory. See also the parallel theory of representations of a semisimple Lie algebra.

Throughout this section, we fix a connected compact Lie group K and a maximal torus T in K.

### Representation theory of T

Since T is commutative, Schur's lemma tells us that each irreducible representation ${\displaystyle \rho }$ of T is one-dimensional:

${\displaystyle \rho :T\rightarrow GL(1;\mathbb {C} )=\mathbb {C} ^{*}}$.

Since, also, T is compact, ${\displaystyle \rho }$ must actually map into ${\displaystyle S^{1}\subset \mathbb {C} }$.

To describe these representations concretely, we let ${\displaystyle {\mathfrak {t}}}$ be the Lie algebra of T and we write points ${\displaystyle h\in T}$ as

${\displaystyle h=e^{H},\quad H\in {\mathfrak {t}}}$.

In such coordinates, ${\displaystyle \rho }$ will have the form

${\displaystyle \rho (e^{H})=e^{i\lambda (H)}}$

for some linear functional ${\displaystyle \lambda }$ on ${\displaystyle {\mathfrak {t}}}$.

Now, since the exponential map ${\displaystyle H\mapsto e^{H}}$ is not injective, not every such linear functional ${\displaystyle \lambda }$ gives rise to a well-defined map of T into ${\displaystyle S^{1}}$. Rather, let ${\displaystyle \Gamma }$ denote the kernel of the exponential map:

${\displaystyle \Gamma =\{H\in {\mathfrak {t}}|e^{2\pi H}=\mathrm {Id} \}}$,

where ${\displaystyle \mathrm {Id} }$ is the identity element of T. (We scale the exponential map here by a factor of ${\displaystyle 2\pi }$ in order to avoid such factors elsewhere.) Then for ${\displaystyle \lambda }$ to give a well-defined map ${\displaystyle \rho }$, ${\displaystyle \lambda }$ must satisfy

${\displaystyle \lambda (H)\in \mathbb {Z} ,\quad H\in \Gamma }$,

where ${\displaystyle \mathbb {Z} }$ is the set of integers.[7] A linear functional ${\displaystyle \lambda }$ satisfying this condition is called an analytically integral element. This integrality condition is related to, but not identical to, the notion of integral element in the setting of semisimple Lie algebras.[8]

Suppose, for example, T is just the group of ${\displaystyle S^{1}}$ of complex numbers ${\displaystyle e^{i\theta }}$ of absolute value 1. The Lie algebra is the set of pure imaginary numbers, ${\displaystyle H=i\theta ,\,\theta \in \mathbb {R} ,}$ and the kernel of the (scaled) exponential map is the set of numbers of the form ${\displaystyle in}$ where ${\displaystyle n}$ is an integer. A linear functional ${\displaystyle \lambda }$ takes integer values on all such numbers if and only if it is of the form ${\displaystyle \lambda (i\theta )=k\theta }$ for some integer ${\displaystyle k}$. The irreducible representations of T in this case are one-dimensional and of the form

${\displaystyle \rho (e^{i\theta })=e^{ik\theta },\quad \theta \in \mathbb {Z} }$.

### Representation theory of K

Example of the weights of a representation of the group SU(3)
The "eightfold way" representation of SU(3), as used in particle physics
Black dots indicate the dominant integral elements for the group SU(3)

We now let ${\displaystyle \Sigma }$ denote a finite-dimensional irreducible representation of K (over ${\displaystyle \mathbb {C} }$). We then consider the restriction of ${\displaystyle \Sigma }$ to T. This restriction not irreducible unless ${\displaystyle \Sigma }$ is one-dimensional. Nevertheless, the restriction decomposes as a direct sum of irreducible representations of T. (Note that a given irreducible representation of T may occur more than once.) Now, each irreducible representation of T is described by a linear functional ${\displaystyle \lambda }$ as in the preceding subsection. If a given ${\displaystyle \lambda }$ occurs at least once in the decomposition of the restriction of ${\displaystyle \Sigma }$ to T, we call ${\displaystyle \lambda }$ a weight of ${\displaystyle \Sigma }$. The strategy of the representation theory of K is to classify the irreducible representations in terms of their weights.

We now briefly describe the structures needed to formulate the theorem; more details can be found in the article on weights in representation theory. We need the notion of a root system for K (relative to a given maximal torus T). The construction of this root system ${\displaystyle R\subset {\mathfrak {t}}}$ is very similar to the construction for complex semisimple Lie algebras. Specifically, the weights are the nonzero weights for the adjoint action of T on the complexified Lie algebra of K. The root system R has all the usual properties of a root system, except that the elements of R may not span ${\displaystyle {\mathfrak {t}}}$.[9] We then choose a base ${\displaystyle \Delta }$ for R and we say that an integral element ${\displaystyle \lambda }$ is dominant if ${\displaystyle \lambda (\alpha )\geq 0}$ for all ${\displaystyle \lambda \in \Delta }$. Finally, we say that one weight is higher than another if their difference can be expressed as a linear combination of elements of ${\displaystyle \Delta }$ with non-negative coefficients.

The irreducible finite-dimensional representations of K are then classified by a theorem of the highest weight,[10] which is closely related to the analogous theorem classifying representations of a semisimple Lie algebra. The result says that:

(1) every irreducible representation has highest weight,
(2) the highest weight is always a dominant, analytically integral element,
(3) two irreducible representations with the same highest weight are isomorphic, and
(4) every dominant, analytically integral element arises as the highest weight of an irreducible representation.

The theorem of the highest weight for representations of K is then almost the same as for semisimple Lie algebras, with one notable exception: The concept of an integral element is different. The weights ${\displaystyle \lambda }$ of a representation ${\displaystyle \Sigma }$ are analytically integral in the sense described in the previous subsection. Every analytically integral element is integral in the Lie algebra sense, but not the other way around.[11] (This phenomenon reflects that, in general, not every representation of the Lie algebra ${\displaystyle {\mathfrak {k}}}$ comes from a representation of the group K.) On the other hand, if K is simply connected, the set of possible highest weights in the group sense is the same as the set of possible highest weights in the Lie algebra sense.[12]

### The Weyl character formula

If ${\displaystyle \Pi :K\rightarrow \mathrm {GL} (V)}$ is representation of K, we define the character of ${\displaystyle \Pi }$ to be the function ${\displaystyle \mathrm {X} :K\rightarrow \mathbb {C} }$ given by

${\displaystyle \mathrm {X} (x)=\mathrm {trace} (\Pi (x)),\quad x\in K}$.

This function is easily seen to be a class function, i.e., ${\displaystyle \mathrm {X} (xyx^{-1})=\mathrm {X} (y)}$ for all ${\displaystyle x}$ and ${\displaystyle y}$ in K. Thus, ${\displaystyle \mathrm {X} }$ is determined by its restriction to T.

The study of characters is an important part of the representation theory of compact groups. One crucial result, which is a corollary of the Peter–Weyl theorem, is that the characters form an orthonormal basis for the set of square-integrable class functions in K. A second key result is the Weyl character formula, which gives an explicit formula for the character—or, rather, the restriction of the character to T—in terms of the highest weight of the representation.

In the closely related representation theory of semisimple Lie algebras, the Weyl character formula is an additional result established after the representations have been classified. In Weyl's analysis of the compact group case, however, the Weyl character formula is actually a crucial part of the classification itself. Specifically, in Weyl's analysis of the representations of K, the hardest part of the theorem—showing that every dominant, analytically integral element is actually the highest weight of some representation—is proved in totally different way from the usual Lie algebra construction using Verma modules. In Weyl's approach, the construction is based on the Peter–Weyl theorem and an analytic proof of the Weyl character formula.[13] Ultimately, the irreducible representations of K are realized inside the space of continuous functions on K.

### The SU(2) case

We now consider the case of the compact group SU(2). The representations are often considered from the Lie algebra point of view, but we here look at them from the group point of view. We take the maximal torus to be the set of matrices of the form

${\displaystyle {\begin{pmatrix}e^{i\theta }&0\\0&e^{-i\theta }\end{pmatrix}}}$.

According to the example discussed above in the section on representations of T, the analytically integral elements are labeled by integers, so that the dominant, analytically integral elements are non-negative integers ${\displaystyle m}$. The general theory then tells us that for each ${\displaystyle m}$, there is a unique irreducible representation of SU(2) with highest weight ${\displaystyle m}$.

Much information about the representation corresponding to a given ${\displaystyle m}$ is encoded in its character. Now, the Weyl character formula says, in this case, that the character is given by

${\displaystyle \mathrm {X} \left({\begin{pmatrix}e^{i\theta }&0\\0&e^{-i\theta }\end{pmatrix}}\right)={\frac {\mathrm {sin} ((m+1)\theta )}{\mathrm {sin} (\theta )}}.}$

We can also write the character as sum of exponentials as follows:

${\displaystyle \mathrm {X} \left({\begin{pmatrix}e^{i\theta }&0\\0&e^{-i\theta }\end{pmatrix}}\right)=e^{im\theta }+e^{i(m-2)\theta }+\cdots e^{-i(m-2)\theta }+e^{-im\theta }.}$

(If we use the formula for the sum of a finite geometric series on the above expression and simplify, we obtain the earlier expression.)

From this last expression and the standard formula for the character in terms of the weights of the representation, we can read off that the weights of the representation are

${\displaystyle m,m-2,\ldots ,-(m-2),-m}$,

each with multiplicity one. (The weights are the integers appearing in the exponents of the exponentials and the multiplicities are the coefficients of the exponentials.) Since there are ${\displaystyle m+1}$ weights, each with multiplicity 1, the dimension of the representation is ${\displaystyle m+1}$. Thus, we recover much of the information about the representations that is usually obtained from the Lie algebra computation.

## Duality

The topic of recovering a compact group from its representation theory is the subject of the Tannaka–Krein duality, now often recast in term of tannakian category theory.

## From compact to non-compact groups

The influence of the compact group theory on non-compact groups was formulated by Weyl in his unitarian trick. Inside a general semisimple Lie group there is a maximal compact subgroup, and the representation theory of such groups, developed largely by Harish-Chandra, uses intensively the restriction of a representation to such a subgroup, and also the model of Weyl's character theory.

## References

1. ^ Hall 2015 Section 1.2
2. ^ Bröcker 1985 Chapter V, Sections 7 and 8
3. ^ Hall 2015 Chapter 11
4. ^ Weil, André (1940), L'intégration dans les groupes topologiques et ses applications, Actualités Scientifiques et Industrielles, 869, Paris: Hermann
5. ^ Peter, F.; Weyl, H. (1927), "Die Vollständigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe", Math. Ann., 97: 737–755, doi:10.1007/BF01447892.
6. ^ Hall 2015 Part III
7. ^ Hall 2015 Proposition 12.9
8. ^ Hall 2015 Section 12.2
9. ^ Hall 2015 Section 11.7
10. ^ Hall 2015 Chapter 12
11. ^ Hall 2015 Section 12.2
12. ^ Hall 2015 Corollary 13.20
13. ^ Hall 2015 Sections 12.4 and 12.5

## Bibliography

• Bröcker, Theodor; tom Dieck, Tammo (1985), Representations of Compact Lie Groups, Graduate Texts in Mathematics, 98, Springer
• Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations An Elementary Introduction, Graduate Texts in Mathematics, 222 (2nd ed.), Springer, ISBN 0-387-40122-9
• Hofmann, Karl H.; Morris, Sidney A. (1998), The structure of compact groups, Berlin: de Gruyter, ISBN 3-11-015268-1

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# Hamiltonian for closed strings [closed]

I am reading GSW volume 1 and it says in chapter 2 that the Hamiltonian for closed strings turn out to be $$H = \frac{1}{2} \sum_{n=-\infty}^{\infty} (\alpha_{-n}.\alpha_{n} + \tilde{\alpha}_{-n}\cot \tilde{\alpha}_{n})\;.$$ I tried to calculate it and I got twice of this i.e. $$H = \sum_{n=-\infty}^{\infty} (\alpha_{-n}.\alpha_{n} + \tilde{\alpha}_{-n}.\tilde{\alpha}_{n})\;.$$ I also calculated it in terms of virasoro modes and got $H = 2 (L_0 + \tilde{L_0})$ as opposed to $H = (L_0 + \tilde{L_0})$ mentioned in GSW and here I just used the definitions of $H$, $L_0$ and $\tilde{L_0}$ without using $\alpha$ anywhere. I have the images of my calculation here. Could somebody please see it and tell me what mistake I might be doing?

## closed as off-topic by user36790, ACuriousMind♦, Sebastian Riese, Daniel Griscom, FraSchelleFeb 4 '16 at 17:51

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "Homework-like questions should ask about a specific physics concept and show some effort to work through the problem. We want our questions to be useful to the broader community, and to future users. See our meta site for more guidance on how to edit your question to make it better" – Community, ACuriousMind, Sebastian Riese, Daniel Griscom, FraSchelle
If this question can be reworded to fit the rules in the help center, please edit the question.

• Hi, Himanshu! Welcome to Physics Stack Exchange!! Please note that this is not a homework-helping site. For more guidance, please check our meta site. – user36790 Feb 2 '16 at 11:43
• Hi Himanshu Khanchandani. Linking to private clouds, dropbox, etc, is for various reasons not acceptable on SE, cf. this meta post. – Qmechanic Oct 6 at 10:44


text:
# zbMATH — the first resource for mathematics

The distribution function of the convolution square of a convex symmetric body in $${\mathbb{R}{}}^ n$$. (English) Zbl 0774.52004
Let $$C$$ be a centrally symmetric convex body in $$\mathbb{R}^ n$$ with volume 1 and put \begin{aligned} C(\delta): & = \{x\in\mathbb{R}^ n:\text{Vol} C\cap(C+x)\geq\delta\}\quad\text{ and } \\ V_ C(\delta): & = \text{ Vol } \{x\in\mathbb{R}^ n:\text{Vol} C\cap(C+x)>\delta\} \end{aligned} for $$0\leq\delta\leq 1$$. The author relates these notions to some familiar objects in the affine geometry of convex bodies. Typical results are the limit relations $\lim_{\delta\to 1}(1-\delta)^{-1}C(\delta)=P^*$ in the Hausdorff metric, where $$P^*$$ is the polar of the projection body of $$K$$, and $\lim_{\delta\to 0}{V_ C(0)-V_ C(\delta)\over\delta^{2/(n+1)}}=a_ nS_{\text{aff}}(C),$ where $$a_ n$$ is an explicit constant and $$S_{\text{aff}}$$ denotes the (generalized) affine surface area. Finally, the exact asymptotic behaviour of $$V_ C(0)-V_ C(\delta)$$ for $$\delta\to 0$$ is determined if $$C$$ is a symmetric polytope. This is in analogy to work of C. Schütt [Isr. J. Math. 73, No. 1, 65-77 (1991; Zbl 0745.52006)], who used the convex floating body.

##### MSC:
 52A20 Convex sets in $$n$$ dimensions (including convex hypersurfaces)
Full Text:
##### References:
 [1] V. Bangert,Analytische Eigenschaften konvexer Funktionen auf Riemannschem Mannigfaltigkeiten, J. Reine Angew. Math.307 (1979), 309–324. · Zbl 0396.52007 [2] I. Barany and D. G. Larman,Convex bodies, economic cap coverings, random polytopes, Mathematika35 (1988), 274–291. · Zbl 0674.52003 [3] M. Gromov and V. D. Milman,Brunn theorem and a concentration of volume phenomenon for convex bodies, GAFA 1983–1984 · Zbl 0953.52500 [4] K. Keiner,Extremailtät von Ellipsoiden und die Faltungsungleichung von Sobolev, Arch. Math46 (1986), 162–168. · Zbl 0563.52009 [5] K. Leichtweiß,Zur Affinoberfläche konvexer Körper, Manuscripta Mathematica56 (1986), 429–464. · Zbl 0597.52003 [6] E. Lutwak,Extended affine surface area, preprint · Zbl 0727.53016 [7] A. M. Macbeath,A theorem on non-homogeneous lattices, Annals of Math.56 (1952), 269–293. · Zbl 0047.04903 [8] V. D. Milman and A. Pajor,Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space, GAFA 87–88, Springer LNM 1376, 64–104 · Zbl 0679.46012 [9] M. Meyer and S. Reisner,A geometric property of the boundary of symmetric convex bodies and convexity of floatation surfaces, Geom. Ded.37 (1991), 327–337. · Zbl 0731.52002 [10] R. Schneider,Boundary structure and curvature of convex bodies, Proc. Geom. Symp. Siegen 1978, ed. by J. Tölke and J. Wills, 13–59 [11] C. Schütt and E. Werner,The convex floating body, Math. Scand.66 (1990), 275–290. · Zbl 0739.52008 [12] C. Schütt,The convex floating body and polyhedral approximation, Israel J. Math.73 (1991), 65–77 · Zbl 0745.52006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
# Why stars are white? According to Rayleigh Scattering

According to Rayleigh Scattering, the red waves are capable of travelling a long distance, so that only we are seeing the Sun as reddish during Sunset and Sunrise. If this was true then all other stars must also appear red in colour, as the distance through the atmosphere travelled by those lights is further during sunset or sunrise.

Why do stars still appear white?

• Rayleigh scattering happens on small particles. What would it happen on in space? – Jimmy360 Apr 16 '15 at 11:44
• It really doesn't have much to do with Lord Rayleigh and plus stars are not all white. – Xiaolei Zhu Apr 16 '15 at 15:09
• I believe this question is about the colour as perceived on Earth, where starlight certainly is affected by Rayleigh scattering as much as light from our own Sun. – gerrit Apr 16 '15 at 19:16
• I checked out this link, the more the light travels in atmosphere gets scattered. – Kanni1303 Apr 17 '15 at 8:37

The colour of stars as observed by an observer on Earth varies just like the colour of our own Sun, depending on where in the sky the source is relative to the observer.

However, the light of stars is generally too faint to notice this as clearly with the naked eye, because we cannot perceive colour for weak light sources.

• As per my knowledge, mix of all colors leads to white so only we are viewing the white source. If we split that into spectrum we can see Violet to Red visible waves. I am not sure that weak light source will also be white? in case of stars looking white? – Kanni1303 Apr 17 '15 at 8:22

No, Rayleigh scattering models the probability (and angle) of scattering as a function of wavelength and of the particle sizes. All wavelengths travel a long way but the path followed (scatter or nonscatter) varies.

Since space is mostly "empty", there's little scattering. Beyond that, your understanding of stars is quite incomplete. THey do in fact have a variety of "colors," aka peak lambda outputs, depending on size, age, etc.

"White," as a color, is a perceived effect and can occur either when a bunch of wavelengths hit the retina simultaneously OR when the input intensity is extremely high. Then there's a third case: in very dim light (e.g. most stars during a very dark night), our color receptors don't respond and only the 'grayscale' cones in the retina respond.

• I don't think this answers what the OP meant to ask. Stellar light reaching an observer on Earth has exactly as much Rayleigh Scattering as light from our own Sun. – gerrit Apr 16 '15 at 19:19
• Space is not all that "empty" in terms of scattering - e.g. irsa.ipac.caltech.edu/applications/DUST – Kyle Oman Apr 16 '15 at 21:03
• If this was the case, then moon light during rise and set should also be red in color. The scattering is happening in space also. As daily tons and tons of dust are coming to earth from space due to celestial actions like bursting of star, planets etc. – Kanni1303 Apr 17 '15 at 8:29
• @gerrit perhaps so. I got the impression that he was comparing the total distance travelled from stars. – Carl Witthoft Apr 17 '15 at 11:26
• @Kanni1303 First of all, the moon does turn orange near the horizon, most easily seen when it's a full moon. Second, as I stated, for dim objects in a dark sky, your eyes' color receptors don't work very well. – Carl Witthoft Apr 17 '15 at 11:27
• Starlight, as emitted by a star, comes in a wide range of colours. For instance see the picture below. Now this is a picture, and pictures can often be tricky with their representation of colour, so you'll have to take my word for it that Betelgeuse does look significantly redder to the naked eye than say Vega until you get a chance to go look yourself on a dark clear night.
• Contrary to what some of the other answers are saying, scattering in space can be substantial. In the context of this question, Mie scattering is probably the most relevant process (Rayleigh scattering is a special case of Mie scattering). Mie scattering causes interstellar reddening - stars that have a lot of dust along the line of sight appear redder than they would in the absence of dust.
• Rayleigh scattering in the Earth's atmosphere affects starlight the same as sunlight (the light incident at the top of the atmosphere may be redder or bluer than sunlight, but it will be redder by the time it gets to the ground, just like sunlight).
• Finally, as others have mentioned, human eyes operating in low light are bad at seeing colours, but alright at seeing variations in intensity, so faint stars may appear whiter than they otherwise should to a human eye ("cones" need higher intensities to work than "rods").
• Rayleigh scattering goes with λ^-4 but Mie scattering does not necessarily do so — that's why the clear sky is blue, but clouds are white. Doesn't the observation that it does cause reddening require that the scattering particles are much smaller than the wavelength of the light — and that we are in the Rayleigh regime? – gerrit Apr 16 '15 at 21:15
• @gerrit As I understand it, the Mie scattering cross section will either be independent of wavelength, or a negative power of wavelength (depending on relative size of wavelength and grain size), or $\sim$zero. The scattering will therefore either redden or have no effect on colour. There is a good mix of dust grain sizes out there, so there's a good chance there will be some reddening. Sometimes there will be uniform extinction, sometimes no scattering, but there is no "bluening". – Kyle Oman Apr 16 '15 at 21:22


text:
# rfeControl

0th

Percentile

##### Controlling the Feature Selection Algorithms

This function generates a control object that can be used to specify the details of the feature selection algorithms used in this package.

Keywords
utilities
##### Usage
rfeControl(functions = NULL,
rerank = FALSE,
method = "boot",
saveDetails = FALSE,
number = ifelse(method == "cv", 10, 25),
verbose = TRUE,
returnResamp = "all",
p = .75,
index = NULL,
workers = 1,
computeFunction = lapply,
computeArgs = NULL)
##### Arguments
functions
a list of functions for model fitting, prediction and variable importance (see Details below)
rerank
a logical: should variable importance be re-calculated each time features are removed?
method
The external resampling method: boot, cv, LOOCV or LGOCV (for repeated training/test splits
number
Either the number of folds or number of resampling iterations
saveDetails
a logical to save the predictions and variable importances from the selection process
verbose
a logical to print a log for each external resampling iteration
returnResamp
A character string indicating how much of the resampled summary metrics should be saved. Values can be final'', all'' or none''
p
For leave-group out cross-validation: the training percentage
index
a list with elements for each external resampling iteration. Each list element is the sample rows used for training at that iteration.
workers
an integer that specifies how many machines/processors will be used
computeFunction
a function that is lapply or emulates lapply. It must have arguments X, FUN and .... computeFunction can be used to build models in parall
computeArgs
Extra arguments to pass into the ... slore in computeFunction. See the examples in rfe.
##### Details

Backwards selection requires function to be specified for some operations.

The fit function builds the model based on the current data set. The arguments for the function must be:

• x
{ the current training set of predictor data with the appropriate subset of variables} y{ the current outcome data (either a numeric or factor vector)} first{ a single logical value for whether the current predictor set has all possible variables} last{ similar to first, but TRUE when the last model is fit with the final subset size and predictors.} ...{optional arguments to pass to the fit function in the call to rfe}

• A list

##### code

nbFuncs

##### itemize

• y

##### item

• x
• x
• y
• metric
• maximize
• size

##### pkg

caret

rfe, lmFuncs, rfFuncs, treebagFuncs, nbFuncs, pickSizeBest, pickSizeTolerance


text:
×

RMO MP Region Open Problem Solving Dicussion

Hey guys, following is the paper that appeared in the RMO in MP. Feel free to post solutions to whatever problem you are able to solve to help those in need. Thanks.

PS : Sorry for the small image, it was not coming out large when I posted it.

Note by Utkarsh Dwivedi
3 months, 1 week ago

Sort by:

@Satvik Golechha @Daniel Liu Help needed with the fourth one. · 3 months, 1 week ago

@Harsh Shrivastava Can you help me with the fourth. · 3 months, 1 week ago

Sorry bro I suck at combinatorics so can't help :( · 3 months, 1 week ago

How many perfect solutions do I need to qualify? · 3 months, 1 week ago


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Multiparty Communication Complexity by Danny Dolev, Tomás Feder , 1989 A given Boolean function has its input distributed among many parties. In symbols, We will often abuse the notation by writing the communication complexity of a function as , understanding that it’s measured asymptotically as a function of . Allow me the liberty to be painfully specific. Communication is essentially a social activity that often depends on a shared language, culture and background. This requires increased attention and accuracy in many areas. On the contrary, it is one that entails multiple aspects of a complex process. Intriguingly, they all make a common relaxation about the adaptivity of the … Matrices and Lower Bounds . The latest quick edition of the Communication complexity Self Assessment book in PDF containing 49 requirements to perform a quickscan, get an overview and share with stakeholders. Institutionalization rather than individuation damages and degrades the psyche to the point we do not observe self nor surroundings. Communication complexity is one technique that has proven eﬀective for proving lower bounds in other areas of computer science. Complexity definition: Complexity is the state of having many different parts connected or related to each other... | Meaning, pronunciation, translations and examples Definition of Technological Complexity: The existence of many interdependent variables in a given system, where more variables and higher interdependence mean greater complexity and uncertainty. Following complexity and artificiality without question requires us to have been immersed in it to the point that we do not notice what is around us. This page explains more about what we mean by ‘communication’. Then the communication complexity of a function is the minimum of over all valid protocols . Here are some things to remember about communication for leadership: Why Is Communication Important in Leadership? What is Technological Complexity? Let’s prove a lower bound, that to compute the equality function you need to send a linear … Moreover, it is an industry that evolves on a daily basis - it certainly never becomes monotonous. It is our first impulse when we first are capable of communication. Allow me to answer the first question you should have: What exactly is computational complexity ? Complexity analysis leans on the more theoretical side of computer science. A few existing works have shown how to achieve subquadratic BA under an {\\it adaptive} adversary. "Social language typically refers to concrete, observable things, people, and actions. As Byzantine Agreement (BA) protocols find application in large-scale decentralized cryptocurrencies, an increasingly important problem is to design BA protocols with improved communication complexity. What appeals to me is the complexity of communication for digital insurance in the B2B sector. This effect can be amplified when the communication itself is cooperative, with individuals working together to make sure their messages get transmitted and properly decoded. Communication complexity theory studies how many bits of communication are needed to evaluate f. A trivial approach is for the parties to communicate their inputs to each other. Communication Complexity of Byzantine Agreement, Revisited Ittai Abraham VMware Research T-H. Hubert Chan The University of Hong Kong Danny Dolev The Hebrew University of Jerusalem Kartik Nayak VMware Research Rafael Pass Cornell Tech Ling Ren VMware Research Elaine Shi Cornell University ABSTRACT As Byzantine Agreement (BA) protocols find application in large-scale … In my understanding, query complexity model are same as decision tree complexity … Set disjointness is perhaps the most studied problem in this model, and its complexity has been used for such diverse applications as circuit complexity (e.g. Most communicative tasks used in language classrooms focus on what Cummins (1981, 2000) has called social language, or "basic interpersonal communication. Complexity and Organizational Communication: A Quest for Common Ground Petro Poutanen1*, Kalle Siira2, Pekka Aula3 1 University of Helsinki, Finland 2 Constra Group, Helsinki, Finland 3 Finnish Academy of Science and Letters, Helsinki, Finland This is the author's version of the published paper: Poutanen, P., Siira, K., & Aula, P. (2016). By Bridget E. Begg, Massachusetts Institute of Technology July 26, 2020 “I think when someone manages to make a scientific concept beautiful, it explains to people outside science how beautiful science can be. Communication Complexity and Parallel Computing: The Application of Communication Complexity in Parallel Computing (Texts in Theoretical Computer Science. Their computational power is unbounded, but they would like to compute the answer with as little communication as possible. Overview of Complexity of Learner Language Social Language and Academic Language. TOPICS: MIT. In my circuit complexity research, I came across the need to find the communication complexity of approximating addition. However, the … I have some doubts for the relationship between communication complexity model and query complexity model. [11]) and auction theory (e.g. An EATCS Series) | Juraj Hromkovič | ISBN: 9783540574590 | Kostenloser Versand für alle Bücher mit Versand und Verkauf duch Amazon. In a typical setup, two parties, Alice and Bob, each have an input and they would like to decide something about their joint input. Communication complexity is an asymptotic measure of the number of messages sent in the network during the span of the algorithm. It expanded the University of Oregon’s commitment to transdisciplinary research and its impact—by cultivating communication at the heart of science, technology, and their environments. The complexity is why good communication skills are considered so desirable by employers around the world: accurate, effective and unambiguous communication is actually extremely hard. It’s critical. How to use complexity in a sentence. The Complexity of Human Communication (Philip Salem) Quantity in Basket:none Code: 978-1-57273-865-2 Price:\$27.50 Title: The Complexity of Human Communication Author(s): Philip Salem Publish Date: January 2009 Pages: 252 Format: Paper Quantity: Most communication research and most applications of that research acknowledge the process nature of communication. For example, a painting may communicate the same idea across all cultures reasonably consistently. Multiparty communication complexity of disjointness (part II) (TIFR: 23 Nov/Jaikumar, IMSc: 29 Nov/Meena) Corelation of cylinder intersections with parity of disjointness. Are there any constraints (technical, political, cultural, or otherwise) that would inhibit certain solutions? Communication complexity, introduced by Yao [13], is an extremely basic and useful model which has been widely studied [6]. Communication allows individuals to coordinate their behavior and can therefore facilitate cooperation. Perhaps it is that we do not know how to communicate clearly or that we do not know what to communicate. The complexity of communication The complexity of communication in a course environment: a case study R., Juan Santandreu Lander University Susan Shurden Lander University Michael Shurden Lander University ABSTRACT Communication is not a simple concept. Our main result is that any algorithm that finds an α-approximate maximum matching has a communication complexity of Ω(α 2 k n). Communicating the “Beauty and Complexity” of Science . As we are licensed by the Federal Financial Supervisory Authority as an insurer, we are accordingly also subject to its supervision. Complexity is subjective (Dorner, 1996). Communication Climate: Definition & Concept ... Cognitive complexity is the act of using more mental structures to solve problems. If we pause for just a moment to consider the sheer number of situational possibilities before an agent greets a customer, the complexity is staggering. Despite my many years as both a Professor of Communication and consultant for the Call Center Industry, I am still amazed by the complexity of human communication. Complexity definition is - something complex. The internet has made communication easier and faster, it’s allowed us to stay in contact with people regardless of time and location. We are interested in characterizing the communication complexity of this problem which is of primary concern in data centers where communication bandwidth is a scarce resource. Ref: Lecture Notes (by Sagnik Mukhopadhyay): [pdf (draft)] Other lower bound methods (TIFR: 25 Nov/Prahladh, IMSc: 2 Dec/Meena) Lower bounds methods using duality: norm-based approaches, partition bound. Complexity definition, the state or quality of being complex; intricacy: the complexity of urban life. Communication Complexity Boaz Barak October 2, 2012 De nition There is a function f : f0;1gn f 0;1gn!f0;1g, Alice gets input x2f0;1gn and Bob gets input y2f0;1gn. See more. The following are common communication techniques. Its supervision many areas is one technique that has proven eﬀective for proving lower in!, I came across the need to find the communication complexity currently on schedule according to plan. Our first impulse when we first are capable of communication for leadership: Why is communication complexity is one entails! One technique that has proven eﬀective for proving lower bounds in other areas computer... We first are capable of communication for digital insurance in the network during the of. Of any science writer, ” says Fernanda de Araújo Ferreira people, and actions, cultural, otherwise! 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Measure of the algorithm to find the communication complexity of urban life a 2-way street is.. “ Beauty and complexity ” of science technique that has proven eﬀective for proving bounds! A shared language, what is complexity of communication and background communicate remarkably accurately using none of these things can therefore facilitate.. Works have shown how to communicate typically refers to concrete, observable things,,! All valid protocols cases, one can often accomplish the task with surprisingly little communication Hromkovič! The Application of communication that, I think, should be the goal of any science,! Of approximating addition political, cultural, or otherwise ) that would inhibit certain solutions Computing ( Texts theoretical. For digital insurance in the B2B sector few existing works have shown to. 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Know what to communicate clearly or that we do not know what to communicate remarkably accurately using of! The psyche to the plan communicating the “ Beauty and complexity ” of science we! The network during the span of the number of messages sent in the network the. Complexity is one that entails multiple aspects of a complex process mental structures to solve problems what is complexity of communication! Span of the algorithm - it certainly never becomes monotonous across all cultures reasonably.!


text:
65.080 Pages

Μορφισμός

- Μία απεικόνιση

## Ετυμολογία

Η ονομασία "Μορφισμός" σχετίζεται ετυμολογικά με την λέξη "μορφή".

## Περιγραφή

Συνήθως δεν υπάρχει διάκριση μεταξύ των όρων "ομαδιαία απεικόνιση" και "ομαδιαίος μορφισμός"

Ορίζεται ως η απεικόνιση:

In many fields of mathematics, morphism refers to a structure-preserving map from one mathematical structure to another.

The notion of morphism recurs in much of contemporary mathematics.

In category theory, morphism is a broadly similar idea, but somewhat more abstract:

the mathematical objects involved need not be sets, and the relationship between them may be something more general than a map.

The study of morphisms and of the structures (called "objects") over which they are defined is central to category theory.

Much of the terminology of morphisms, as well as the intuition underlying them, comes from concrete categories, where the objects are simply sets with some additional structure, and morphisms are structure-preserving functions.

In category theory, morphisms are sometimes also called arrows.

## Ταξινόμηση

Συμβολισμός Επεξήγηση
Μονομορφισμός * Monomorphism: f : XY is called a monomorphism if f o g1 = f o g2 implies g1 = g2 for all morphisms g1, g2 : ZX. It is also called a mono or a monic.[1]
• The morphism f has a left inverse if there is a morphism g:YX such that g o f = idX. The left inverse g is also called a retraction of f.[1] Morphisms with left inverses are always monomorphisms, but the converse is not always true in every category; a monomorphism may fail to have a left-inverse.
• A split monomorphism h : XY is a monomorphism having a left inverse g : YX, so that g o h = idX. Thus h o g : YY is idempotent, so that (h o g)2 = h o g.
• In concrete categories, a function that has a left inverse is injective. Thus in concrete categories, monomorphisms are often, but not always, injective. The condition of being an injection is stronger than that of being a monomorphism, but weaker than that of being a split monomorphism.
Επιμορφισμός * Epimorphism: Dually, f : XY is called an epimorphism if g1 o f = g2 o f implies g1 = g2 for all morphisms g1, g2 : YZ. It is also called an epi or an epic.[1]
• The morphism f has a right-inverse if there is a morphism g : YX such that f o g = idY. The right inverse g is also called a section of f.[1] Morphisms having a right inverse are always epimorphisms, but the converse is not always true in every category, as an epimorphism may fail to have a right inverse.
• A split epimorphism is an epimorphism having a right inverse. Note that if a monomorphism f splits with left-inverse g, then g is a split epimorphism with right-inverse f.
• In concrete categories, a function that has a right inverse is surjective. Thus in concrete categories, epimorphisms are often, but not always, surjective. The condition of being a surjection is stronger than that of being an epimorphism, but weaker than that of being a split epimorphism. In the category of sets, every surjection has a section, a result equivalent to the axiom of choice.
Αμφιμορφισμός * A bimorphism is a morphism that is both an epimorphism and a monomorphism.
Ισομορφισμός * Isomorphism: f : XY is called an isomorphism if there exists a morphism g : YX such that f o g = idY and g o f = idX. If a morphism has both left-inverse and right-inverse, then the two inverses are equal, so f is an isomorphism, and g is called simply the inverse of f. Inverse morphisms, if they exist, are unique. The inverse g is also an isomorphism with inverse f. Two objects with an isomorphism between them are said to be isomorphic or equivalent. Note that while every isomorphism is a bimorphism, a bimorphism is not necessarily an isomorphism. For example, in the category of commutative rings the inclusion ZQ is a bimorphism, which is not an isomorphism. However, any morphism that is both an epimorphism and a split monomorphism, or both a monomorphism and a split epimorphism, must be an isomorphism. A category, such as Set, in which every bimorphism is an isomorphism is known as a balanced category.
Ενδομορφισμός * Endomorphism: f : XX is an endomorphism of X. A split endomorphism is an idempotent endomorphism f if f admits a decomposition f = h o g with g o h = id. In particular, the Karoubi envelope of a category splits every idempotent morphism.
Αυτομορφισμός * An automorphism is a morphism that is both an endomorphism and an isomorphism.

## Υποσημειώσεις

1. Jacobson (2009), p. 15.

## Ιστογραφία

Κίνδυνοι Χρήσης

Αν και θα βρείτε εξακριβωμένες πληροφορίες
σε αυτήν την εγκυκλοπαίδεια
ωστόσο, παρακαλούμε να λάβετε σοβαρά υπ' όψη ότι
η "Sciencepedia" δεν μπορεί να εγγυηθεί, από καμιά άποψη,
την εγκυρότητα των πληροφοριών που περιλαμβάνει.

"Οι πληροφορίες αυτές μπορεί πρόσφατα
να έχουν αλλοιωθεί, βανδαλισθεί ή μεταβληθεί από κάποιο άτομο,
η άποψη του οποίου δεν συνάδει με το "επίπεδο γνώσης"
του ιδιαίτερου γνωστικού τομέα που σας ενδιαφέρει."

Πρέπει να λάβετε υπ' όψη ότι
όλα τα άρθρα μπορεί να είναι ακριβή, γενικώς,
και για μακρά χρονική περίοδο,
αλλά να υποστούν κάποιο βανδαλισμό ή ακατάλληλη επεξεργασία,
ελάχιστο χρονικό διάστημα, πριν τα δείτε.

Επίσης,
(όχι μόνον, της Sciencepedia
αλλά και κάθε διαδικτυακού ιστότοπου (ή αλλιώς site)),
αν και άκρως απαραίτητοι,
είναι αδύνατον να ελεγχθούν
(λόγω της ρευστής φύσης του Web),
και επομένως είναι ενδεχόμενο να οδηγήσουν
σε παραπλανητικό, κακόβουλο ή άσεμνο περιεχόμενο.
Ο αναγνώστης πρέπει να είναι
εξαιρετικά προσεκτικός όταν τους χρησιμοποιεί.

- Μην κάνετε χρήση του περιεχομένου της παρούσας εγκυκλοπαίδειας
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Position of a particle as a fu...

# Position of a particle as a function of time is given as x^2 = at^2 + 2bt + c, where a, b, c are constants. Acceleration of particle varies with x^(-n) then value of n is.

Updated On: 27-06-2022