text:
Warning

This is A PREVIEW for NEST 3.0 and NOT an OFFICIAL RELEASE! Some functionality may not be available and information may be incomplete!

# PyNEST API¶

Here is a list of functions for the PyNEST interface.

## Functions related to models¶

Functions for model handling

nest.lib.hl_api_models.ConnectionRules()

Return a typle of all available connection rules, sorted by name.

Returns

Available connection rules

Return type

tuple

nest.lib.hl_api_models.CopyModel(existing, new, params=None)

Create a new model by copying an existing one.

Parameters
• existing (str) – Name of existing model

• new (str) – Name of the copied model

• params (dict, optional) – Default parameters assigned to the copy. Not provided parameters are taken from the existing model.

nest.lib.hl_api_models.GetDefaults(model, keys=None, output='')

Return default parameters of the given model, specified by a string.

Parameters
• model (str) – Name of the model

• keys (str or list, optional) – String or a list of strings naming model properties. GetDefaults then returns a single value or a list of values belonging to the keys given.

• output (str, optional) – Whether the returned data should be in a format (output='json'). Default is ‘’.

Returns

• dict – A dictionary of default parameters.

• type – If keys is a string, the corrsponding default parameter is returned.

• list – If keys is a list of strings, a list of corrsponding default parameters is returned.

• str – If output is json, returns parameters in JSON format.

Raises
nest.lib.hl_api_models.Models(mtype='all', sel=None)

Return a tuple of model names, sorted by name.

All available models are neurons, devices and synapses.

Parameters
• mtype (str, optional) – Use 'mtype='nodes' to only see neuron and device models, or 'type='synapses' to only see synapse models.

• sel (str, optional) – String used to filter the result list and only return models containing it.

Returns

Available model names

Return type

tuple

Raises

ValueError – Description

Notes

• Synapse model names ending with '_hpc' provide minimal memory requirements by using thread-local target neuron IDs and fixing the 'rport' to 0.

• Synapse model names ending with '_lbl' allow to assign an individual integer label ('synapse_label') to created synapses at the cost of increased memory requirements.

nest.lib.hl_api_models.SetDefaults(model, params, val=None)

Set the default parameter values of the given model.

New default values are used for all subsequently created instances of the model.

Parameters
• model (str) – Name of the model

• params (str or dict) – Dictionary of new default parameter values

• val (str, optional) – If given, params has to be the name of a model property.

## Functions related to the creation and retrieval of nodes (neurons, devices)¶

Functions for node handling

nest.lib.hl_api_nodes.Create(model, n=1, params=None, positions=None)

Create one or more nodes.

Generates n new network objects of the supplied model type. If n is not given, a single node is created. Note that if setting parameters of the nodes fail, the nodes will still have been created.

Parameters
• model (str) – Name of the model to create

• n (int, optional) – Number of nodes to create

• params (dict or list, optional) –

Parameters for the new nodes. Can be any of the following:

• A dictionary with either single values or lists of size n. The single values will be applied to all nodes, while the lists will be distributed across the nodes. Both single values and lists can be given at the same time.

• A list with n dictionaries, one dictionary for each node.

Values may be Parameter objects. If omitted, the model’s defaults are used.

• positions (spatial.grid or spatial.free object, optional) – Object describing spatial positions of the nodes. If omitted, the nodes have no spatial attachment.

Returns

Object representing the IDs of created nodes, see NodeCollection for more.

Return type

NodeCollection

Raises
• NESTError – If setting node parameters fail. However, the nodes will still have been created.

• TypeError – If the positions object is of wrong type.

nest.lib.hl_api_nodes.GetLocalNodeCollection(nc)

Get local nodes of a NodeCollection as a new NodeCollection.

This function returns the local nodes of a NodeCollection. If there are no local elements, an empty NodeCollection is returned.

Parameters

nc (NodeCollection) – NodeCollection for which to get local nodes

Returns

Object representing the local nodes of the given NodeCollection

Return type

NodeCollection

nest.lib.hl_api_nodes.GetNodes(properties={}, local_only=False)

Return all nodes with the given properties as NodeCollection.

Parameters
• properties (dict, optional) – Only node IDs of nodes matching the properties given in the dictionary exactly will be returned. Matching properties with float values (e.g. the membrane potential) may fail due to tiny numerical discrepancies and should be avoided. Note that when a params dict is present, thread parallelization is not possible, the function will be run thread serial.

• local_only (bool, optional) – If True, only node IDs of nodes simulated on the local MPI process will be returned. By default, node IDs of nodes in the entire simulation will be returned. This requires MPI communication and may slow down the script.

Returns

NodeCollection of nodes

Return type

NodeCollection

nest.lib.hl_api_nodes.PrintNodes()

Print the node ID ranges and model names of all the nodes in the network.

## Functions related to setting and getting parameters¶

Functions to get information on NEST.

nest.lib.hl_api_info.GetStatus(nodes, keys=None, output='')

Return the parameter dictionaries of nodes or connections.

If keys is given, a list of values is returned instead. keys may also be a list, in which case the returned list contains lists of values.

Parameters
• nodes (NodeCollection or SynapseCollection) – Either a NodeCollection representing nodes, or a SynapseCollection of connection handles as returned by GetConnections().

• keys (str or list, optional) – string or a list of strings naming model properties. GetStatus then returns a single value or a list of values belonging to the keys given.

• output (str, optional) – Whether the returned data should be in a selected format (output='json').

Returns

• list of dicts – All parameters in a dict for each node or connection.

• list of values – If keys is a string, the value of the corresponding parameter for each node or connection is returned.

• list of lists of values – If keys is a list of strings, a list of values of the corresponding parameters for each node or connection is returned.

• str – If output is json, the above formats are converted to JSON format before they are returned.

Raises

TypeError – If nodes or keys are on the wrong form.

Examples

For nodes:

>>>    nest.GetStatus(nodes)
({'archiver_length': 0,
'beta_Ca': 0.001,
...
'global_id': 1,
...
'vp': 0},
...
{'archiver_length': 0,
'beta_Ca': 0.001,
...
'global_id': 3,
...
'vp': 0})

>>>    nest.GetStatus(nodes, 'V_m')
(-70.0, -70.0, -70.0)

>>>    nest.GetStatus(nodes, ['V_m', 'C_m'])
((-70.0, 250.0), (-70.0, 250.0), (-70.0, 250.0))

>>>    nest.GetStatus(nodes, ['V_m', 'C_m'], output='json')
'[[-70.0, 250.0], [-70.0, 250.0], [-70.0, 250.0]]'

For connections:

>>>    nest.GetStatus(conns)
({'delay': 1.0,
...
'source': 1,
...
'weight': 1.0},
...
{'delay': 1.0,
...
'source': 3,
...
'weight': 1.0})

>>>    nest.GetStatus(conns, 'weight')
(1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0)

>>>    nest.GetStatus(conns, ['source', 'delay'])
((1, 1.0),
...
(3, 1.0))

>>>    nest.GetStatus(conns, ['source', 'delay'], output='json')
'[[1, 1.0], [1, 1.0], [1, 1.0], [2, 1.0], [2, 1.0], [2, 1.0],
[3, 1.0], [3, 1.0], [3, 1.0]]'

nest.lib.hl_api_info.SetStatus(nodes, params, val=None)

Set parameters of nodes or connections.

Parameters of nodes or connections, given in nodes, is set as specified by params. If val is given, params has to be a string with the name of an attribute, which is set to val on the nodes/connections. val can be a single value or a list of the same size as nodes.

Parameters
• nodes (NodeCollection or SynapseCollection) – Either a NodeCollection representing nodes, or a SynapseCollection of connection handles as returned by GetConnections().

• params (str or dict or list) – Dictionary of parameters (either lists or single values) or list of dictionaries of parameters of same length as nodes. If val is given, this has to be a string giving the name of a model property.

• val (int, list, optional) – If given, params has to be the name of a model property.

Raises

TypeError – If nodes is not a NodeCollection of nodes, a SynapseCollection of synapses, or if the number of parameters don’t match the number of nodes or synapses.

nest.lib.hl_api_info.authors()

Print the authors of NEST.

nest.lib.hl_api_info.get_argv()

Return argv as seen by NEST.

This is similar to Python sys.argv but might have changed after MPI initialization.

Returns

Argv, as seen by NEST

Return type

tuple

nest.lib.hl_api_info.get_verbosity()

Return verbosity level of NEST’s messages.

• M_ALL=0, display all messages

• M_INFO=10, display information messages and above

• M_DEPRECATED=18, display deprecation warnings and above

• M_WARNING=20, display warning messages and above

• M_ERROR=30, display error messages and above

• M_FATAL=40, display failure messages and above

Returns

The current verbosity level

Return type

int

nest.lib.hl_api_info.help(obj=None, pager=None, return_text=False)

Show the help page for the given object using the given pager.

The default pager is more (See .nestrc).

Parameters
• obj (object, optional) – Object to display help for

• pager (str, optional) – Pager to use

• return_text (bool, optional) – Option for returning the help text

Returns

The help text of the object if return_text is True.

Return type

None or str

nest.lib.hl_api_info.helpdesk()

Open the NEST helpdesk in browser.

Use the system default browser.

nest.lib.hl_api_info.message(level, sender, text)

Print a message using message system of NEST.

Parameters
• level – Level

• sender – Message sender

• text (str) – Text to be sent in the message

nest.lib.hl_api_info.set_verbosity(level)

Change verbosity level for NEST’s messages.

• M_ALL=0, display all messages

• M_INFO=10, display information messages and above

• M_DEPRECATED=18, display deprecation warnings and above

• M_WARNING=20, display warning messages and above

• M_ERROR=30, display error messages and above

• M_FATAL=40, display failure messages and above

Parameters

level (str) – Can be one of ‘M_FATAL’, ‘M_ERROR’, ‘M_WARNING’, ‘M_DEPRECATED’, ‘M_INFO’ or ‘M_ALL’.

nest.lib.hl_api_info.sysinfo()

Print information on the platform on which NEST was compiled.

## Functions related to connections¶

Functions for connection handling

nest.lib.hl_api_connections.Connect(pre, post, conn_spec=None, syn_spec=None, return_synapsecollection=False)

Connect pre nodes to post nodes.

Nodes in pre and post are connected using the specified connectivity (all-to-all by default) and synapse type (static_synapse by default). Details depend on the connectivity rule.

Parameters
• pre (NodeCollection (or array-like object)) – Presynaptic nodes, as object representing the IDs of the nodes

• post (NodeCollection (or array-like object)) – Postsynaptic nodes, as object representing the IDs of the nodes

• conn_spec (str or dict, optional) – Specifies connectivity rule, see below

• syn_spec (str or dict, optional) – Specifies synapse model, see below

• return_synapsecollection (bool) – Specifies whether or not we should return a SynapseCollection of pre and post connections

Raises

kernel.NESTError

Notes

It is possible to connect NumPy arrays of node IDs one-to-one by passing the arrays as pre and post, specifying ‘one_to_one’ for conn_spec. In that case, the arrays may contain non-unique IDs. You may also specify weight, delay, and receptor type for each connection as NumPy arrays in the syn_spec dictionary. This feature is currently not available when MPI is used; trying to connect arrays with more than one MPI process will raise an error.

If pre and post have spatial positions, a mask can be specified as a dictionary. The mask define which nodes are considered as potential targets for each source node. Connections with spatial nodes can also use nest.spatial_distributions as parameters, for instance for the probability p.

Connectivity specification (conn_spec)

Available rules and associated parameters:

- 'all_to_all' (default)
- 'one_to_one'
- 'fixed_indegree', 'indegree'
- 'fixed_outdegree', 'outdegree'
- 'fixed_total_number', 'N'
- 'pairwise_bernoulli', 'p'
- 'symmetric_pairwise_bernoulli', 'p'

See Connection Rules for more details, including example usage.

Synapse specification (syn_spec)

The synapse model and its properties can be given either as a string identifying a specific synapse model (default: static_synapse) or as a dictionary specifying the synapse model and its parameters.

Available keys in the synapse specification dictionary are:

- 'synapse_model'
- 'weight'
- 'delay'
- 'receptor_type'
- any parameters specific to the selected synapse model.

See Synapse Specification for details, including example usage.

All parameters are optional and if not specified, the default values of the synapse model will be used. The key ‘synapse_model’ identifies the synapse model, this can be one of NEST’s built-in synapse models or a user-defined model created via CopyModel().

If synapse_model is not specified the default model static_synapse will be used.

Distributed parameters can be defined through NEST’s different parametertypes. NEST has various random parameters, spatial parameters and distributions (only accesseable for nodes with spatial positions), logical expressions and mathematical expressions, which can be used to define node and connection parameters.

To see all available parameters, see documentation defined in distributions, logic, math, random and spatial modules.

nest.lib.hl_api_connections.Disconnect(pre, post, conn_spec='one_to_one', syn_spec='static_synapse')

Disconnect pre neurons from post neurons.

Neurons in pre and post are disconnected using the specified disconnection rule (one-to-one by default) and synapse type (static_synapse by default). Details depend on the disconnection rule.

Parameters
• pre (NodeCollection) – Presynaptic nodes, given as NodeCollection

• post (NodeCollection) – Postsynaptic nodes, given as NodeCollection

• conn_spec (str or dict) – Disconnection rule, see below

• syn_spec (str or dict) – Synapse specifications, see below

Notes

conn_spec

Apply the same rules as for connectivity specs in the Connect() method

Possible choices of the conn_spec are

- 'one_to_one'
- 'all_to_all'

syn_spec

The synapse model and its properties can be inserted either as a string describing one synapse model (synapse models are listed in the synapsedict) or as a dictionary as described below.

Note that only the synapse type is checked when we disconnect and that if syn_spec is given as a non-empty dictionary, the ‘synapse_model’ parameter must be present.

If no synapse model is specified the default model static_synapse will be used.

Available keys in the synapse dictionary are:

- 'synapse_model'
- 'weight'
- 'delay',
- 'receptor_type'
- parameters specific to the synapse model chosen

‘synapse_model’ determines the synapse type, taken from pre-defined synapse types in NEST or manually specified synapses created via CopyModel().

All other parameters are not currently implemented.

Notes

Disconnect only disconnects explicitly specified nodes.

nest.lib.hl_api_connections.GetConnections(source=None, target=None, synapse_model=None, synapse_label=None)

Return a SynapseCollection representing the connection identifiers.

Any combination of source, target, synapse_model and synapse_label parameters is permitted.

Parameters
• source (NodeCollection, optional) – Source node IDs, only connections from these pre-synaptic neurons are returned

• target (NodeCollection, optional) – Target node IDs, only connections to these postsynaptic neurons are returned

• synapse_model (str, optional) – Only connections with this synapse type are returned

• synapse_label (int, optional) – (non-negative) only connections with this synapse label are returned

Returns

Object representing the source-node_id, target-node_id, target-thread, synapse-id, port of connections, see SynapseCollection for more.

Return type

SynapseCollection

Raises

Notes

Only connections with targets on the MPI process executing the command are returned.

## Functions related to simulation¶

Functions for simulation control

nest.lib.hl_api_simulation.Cleanup()

Cleans up resources after a Run call. Not needed for Simulate.

Closes state for a series of runs, such as flushing and closing files. A Prepare is needed after a Cleanup before any more calls to Run.

nest.lib.hl_api_simulation.DisableStructuralPlasticity()

Disable structural plasticity for the network simulation

nest.lib.hl_api_simulation.EnableStructuralPlasticity()

Enable structural plasticity for the network simulation

nest.lib.hl_api_simulation.GetKernelStatus(keys=None)

Obtain parameters of the simulation kernel.

Parameters

keys (str or list, optional) – Single parameter name or list of parameter names

Returns

• dict – Parameter dictionary, if called without argument

• type – Single parameter value, if called with single parameter name

• list – List of parameter values, if called with list of parameter names

Raises

TypeError – If keys are of the wrong type.

Notes

See SetKernelStatus for documentation on each parameter key.

nest.lib.hl_api_simulation.Install(module_name)

Parameters

module_name (str) – Name of the dynamically linked module

Returns

Return type

handle

Notes

Dynamically linked modules are searched in the NEST library directory (<prefix>/lib/nest) and in LD_LIBRARY_PATH (on Linux) or DYLD_LIBRARY_PATH (on OSX).

Example

nest.Install("mymodule")

nest.lib.hl_api_simulation.Prepare()

Calibrate the system before a Run call. Not needed for Simulate.

Call before the first Run call, or before calling Run after changing the system, calling SetStatus or Cleanup.

nest.lib.hl_api_simulation.ResetKernel()

Reset the simulation kernel.

This will destroy the network as well as all custom models created with CopyModel(). Calling this function is equivalent to restarting NEST.

In particular,

• all network nodes

• all connections

• all user-defined neuron and synapse models

are deleted, and

• time

• random generators

are reset. The only exception is that dynamically loaded modules are not unloaded. This may change in a future version of NEST.

nest.lib.hl_api_simulation.Run(t)

Simulate the network for t milliseconds.

Parameters

t (float) – Time to simulate in ms

Notes

Call between Prepare and Cleanup calls, or within a with RunManager clause.

Simulate(t): t’ = t/m; Prepare(); for _ in range(m): Run(t’); Cleanup()

Prepare must be called before Run to calibrate the system, and Cleanup must be called after Run to close files, cleanup handles, and so on. After Cleanup, Prepare can and must be called before more Run calls. Any calls to SetStatus between Prepare and Cleanup have undefined behaviour.

nest.lib.hl_api_simulation.RunManager()

ContextManager for Run

Calls Prepare before a series of Run calls, and calls Cleanup at end.

E.g.:

with RunManager():
for i in range(10):
Run()

nest.lib.hl_api_simulation.SetKernelStatus(params)

Set parameters for the simulation kernel.

Parameters

params (dict) – Dictionary of parameters to set.

Params dictionary

Some of the keywords in the kernel status dictionary are internally calculated, and cannot be defined by the user. These are flagged as read only in the parameter list. Use GetKernelStatus to access their assigned values.

Time and resolution

Parameters
• resolution (float) – The resolution of the simulation (in ms)

• time (float) – The current simulation time (in ms)

• to_do (int, read only) – The number of steps yet to be simulated

• max_delay (float) – The maximum delay in the network

• min_delay (float) – The minimum delay in the network

• ms_per_tic (float) – The number of milliseconds per tic

• tics_per_ms (float) – The number of tics per millisecond

• tics_per_step (int) – The number of tics per simulation time step

• T_max (float, read only) – The largest representable time value

• T_min (float, read only) – The smallest representable time value

Parallel processing

Parameters
• total_num_virtual_procs (int) – The total number of virtual processes

• num_processes (int, read only) – The number of MPI processes

• off_grid_spiking (bool) – Whether to transmit precise spike times in MPI communication

• grng_seed (int) – Seed for global random number generator used synchronously by all virtual processes to create, e.g., fixed fan-out connections.

• rng_seeds (array) – Seeds for the per-virtual-process random number generators used for most purposes. Array with one integer per virtual process, all must be unique and differ from grng_seed.

MPI buffers

Parameters
• adaptive_spike_buffers (bool) – Whether MPI buffers for communication of spikes resize on the fly

• adaptive_target_buffers (bool) – Whether MPI buffers for communication of connections resize on the fly

• buffer_size_secondary_events (int, read only) – Size of MPI buffers for communicating secondary events (in bytes, per MPI rank, for developers)

• buffer_size_spike_data (int) – Total size of MPI buffer for communication of spikes

• buffer_size_target_data (int) – Total size of MPI buffer for communication of connections

• growth_factor_buffer_spike_data (float) – If MPI buffers for communication of spikes resize on the fly, grow them by this factor each round

• growth_factor_buffer_target_data (float) – If MPI buffers for communication of connections resize on the fly, grow them by this factor each round

• max_buffer_size_spike_data (int) – Maximal size of MPI buffers for communication of spikes.

• max_buffer_size_target_data (int) – Maximal size of MPI buffers for communication of connections

Waveform relaxation method (wfr)

Parameters
• use_wfr (bool) – Whether to use waveform relaxation method

• wfr_comm_interval (float) – Desired waveform relaxation communication interval

• wfr_tol (float) – Convergence tolerance of waveform relaxation method

• wfr_max_iterations (int) – Maximal number of iterations used for waveform relaxation

• wfr_interpolation_order (int) – Interpolation order of polynomial used in wfr iterations

Synapses

Parameters
• max_num_syn_models (int, read only) – Maximal number of synapse models supported

• sort_connections_by_source (bool) – Whether to sort connections by their source; increases construction time of presynaptic data structures, decreases simulation time if the average number of outgoing connections per neuron is smaller than the total number of threads

• structural_plasticity_synapses (dict) – Defines all synapses which are plastic for the structural plasticity algorithm. Each entry in the dictionary is composed of a synapse model, the pre synaptic element and the postsynaptic element

• structural_plasticity_update_interval (int) – Defines the time interval in ms at which the structural plasticity manager will make changes in the structure of the network (creation and deletion of plastic synapses)

Output

Returns

• data_path (str) – A path, where all data is written to (default is the current directory)

• data_prefix (str) – A common prefix for all data files

• overwrite_files (bool) – Whether to overwrite existing data files

• print_time (bool) – Whether to print progress information during the simulation

• network_size (int, read only) – The number of nodes in the network

• num_connections (int, read only, local only) – The number of connections in the network

• local_spike_counter (int, read only) – Number of spikes fired by neurons on a given MPI rank since NEST was started or the last ResetKernel. Only spikes from “normal” neurons (neuron models with proxies) are counted, not spikes generated by devices such as poisson_generator.

Miscellaneous

Other Parameters
• dict_miss_is_error (bool) – Whether missed dictionary entries are treated as errors

• keep_source_table (bool) – Whether to keep source table after connection setup is complete

nest.lib.hl_api_simulation.Simulate(t)

Simulate the network for t milliseconds.

Parameters

t (float) – Time to simulate in ms

## Functions related to parallel computing¶

Functions for parallel computing

nest.lib.hl_api_parallel_computing.NumProcesses()

Return the overall number of MPI processes.

Returns

Number of overall MPI processes

Return type

int

nest.lib.hl_api_parallel_computing.Rank()

Return the MPI rank of the local process.

Returns

MPI rank of the local process

Return type

int

Note

DO NOT USE Rank() TO EXECUTE ANY FUNCTION IMPORTED FROM THE nest MODULE ON A SUBSET OF RANKS IN AN MPI-PARALLEL SIMULATION.

This will lead to unpredictable behavior. Symptoms may be an error message about non-synchronous global random number generators or deadlocks during simulation. In the worst case, the simulation may complete but generate nonsensical results.

nest.lib.hl_api_parallel_computing.SetAcceptableLatency(port_name, latency)

Set the acceptable latency (in ms) for a MUSIC port.

Parameters
• port_name (str) – MUSIC port to set latency for

• latency (float) – Latency in ms

nest.lib.hl_api_parallel_computing.SetMaxBuffered(port_name, size)

Set the maximum buffer size for a MUSIC port.

Parameters
• port_name (str) – MUSIC port to set buffer size for

• size (int) – Buffer size

nest.lib.hl_api_parallel_computing.SyncProcesses()

Synchronize all MPI processes.

## Functions related to NEST types¶

Classes defining the different PyNEST types

class nest.lib.hl_api_types.CollocatedSynapses(*args)

Class for collocated synapse specifications.

Wrapper around a list of specifications, used when calling Connect().

Example

nodes = nest.Create('iaf_psc_alpha', 3)
syn_spec = nest.CollocatedSynapses({'weight': 4., 'delay': 1.5},
{'synapse_model': 'stdp_synapse'},
{'synapse_model': 'stdp_synapse', 'alpha': 3.})
nest.Connect(nodes, nodes, conn_spec='one_to_one', syn_spec=syn_spec)

conns = nest.GetConnections()

print(conns.alpha)
print(len(syn_spec))

nest.lib.hl_api_types.CreateParameter(parametertype, specs)

Create a parameter.

Parameters
• parametertype (string) – Parameter type with or without distance dependency. Can be one of the following: ‘constant’, ‘linear’, ‘exponential’, ‘gaussian’, ‘gaussian2D’, ‘uniform’, ‘normal’, ‘lognormal’, ‘distance’, ‘position’

• specs (dict) – Dictionary specifying the parameters of the provided parametertype, see Parameter types.

Returns

Object representing the parameter

Return type

Parameter

Notes

• Instead of using CreateParameter you can also use the various parametrizations embedded in NEST. See for

instance uniform().

Parameter types

Some available parameter types (parametertype parameter), their function and acceptable keys for their corresponding specification dictionaries

• Constant
'constant' :
{'value' : float} # constant value

• Randomization
# random parameter with uniform distribution in [min,max)
'uniform' :
{'min' : float, # minimum value, default: 0.0
'max' : float} # maximum value, default: 1.0

# random parameter with normal distribution, optionally truncated
# to [min,max)
'normal':
{'mean' : float, # mean value, default: 0.0
'sigma': float, # standard deviation, default: 1.0
'min'  : float, # minimum value, default: -inf
'max'  : float} # maximum value, default: +inf

# random parameter with lognormal distribution,
# optionally truncated to [min,max)
'lognormal' :
{'mu'   : float, # mean value of logarithm, default: 0.0
'sigma': float, # standard deviation of log, default: 1.0
'min'  : float, # minimum value, default: -inf
'max'  : float} # maximum value, default: +inf

class nest.lib.hl_api_types.Mask(datum)

Masks are used when creating connections when nodes have spatial extent. A mask describes the area of the pool population that shall be searched to find nodes to connect to for any given node in the driver population. Masks are created using the CreateMask() command.

Inside(point)

Test if a point is inside a mask.

Parameters

point (tuple/list of float values) – Coordinate of point

Returns

out – True if the point is inside the mask, False otherwise

Return type

bool

class nest.lib.hl_api_types.NodeCollection(data=None)

Class for NodeCollection.

NodeCollection represents the nodes of a network. The class supports iteration, concatenation, indexing, slicing, membership, length, conversion to and from lists, test for membership, and test for equality. By using the membership functions get() and set(), you can get and set desired parameters.

A NodeCollection is created by the Create() function, or by converting a list of nodes to a NodeCollection with nest.NodeCollection(list).

If your nodes have spatial extent, use the member parameter spatial to get the spatial information.

Example

import nest

nest.ResetKernel()

# Create NodeCollection representing nodes
nc = nest.Create('iaf_psc_alpha', 10)

# Convert from list
node_ids_in = [2, 4, 6, 8]
new_nc = nest.NodeCollection(node_ids_in)

# Convert to list
nc_list =  nc.tolist()

# Concatenation
Enrns = nest.Create('aeif_cond_alpha', 600)
Inrns = nest.Create('iaf_psc_alpha', 400)
nrns = Enrns + Inrns

# Slicing and membership
print(new_nc[2])
print(new_nc[1:2])
6 in new_nc

get(*params, **kwargs)

Get parameters from nodes.

Parameters
• params (str or list, optional) –

Parameters to get from the nodes. It must be one of the following:

• A single string.

• A list of strings.

• One or more strings, followed by a string or list of strings. This is for hierarchical addressing.

• output (str, ['pandas','json'], optional) – If the returned data should be in a Pandas DataFrame or in a JSON serializable format.

Returns

• int or float – If there is a single node in the NodeCollection, and a single parameter in params.

• array_like – If there are multiple nodes in the NodeCollection, and a single parameter in params.

• dict – If there are multiple parameters in params. Or, if no parameters are specified, a dictionary containing aggregated parameter-values for all nodes is returned.

• DataFrame – Pandas Data frame if output should be in pandas format.

Raises
• TypeError – If the input params are of the wrong form.

• KeyError – If the specified parameter does not exist for the nodes.

Examples

>>>    nodes.get()
{'archiver_length': (0, 0, 0),
'beta_Ca': (0.001, 0.001, 0.001),
'C_m': (250.0, 250.0, 250.0),
...
'V_th': (-55.0, -55.0, -55.0),
'vp': (0, 0, 0)}

>>>    nodes.get('V_m')
(-70.0, -70.0, -70.0)

>>>    nodes[0].get('V_m')
-70.0

>>>    nodes.get('V_m', 'C_m')
{'V_m': (-70.0, -70.0, -70.0), 'C_m': (250.0, 250.0, 250.0)}

>>>    voltmeter.get('events', 'senders')
array([...], dtype=int64)

index(node_id)

Find the index of a node ID in the NodeCollection.

Parameters

node_id (int) – Global ID to be found.

Raises

ValueError – If the node ID is not in the NodeCollection.

set(params=None, **kwargs)

Set the parameters of nodes to params.

NB! This is almost the same implementation as SetStatus.

If kwargs is given, it has to be names and values of an attribute as keyword argument pairs. The values can be single values or list of the same size as the NodeCollection.

Parameters
• params (str or dict or list) – Dictionary of parameters (either lists or single values) or list of dictionaries of parameters of same length as the NodeCollection.

• kwargs (keyword argument pairs) – Named arguments of parameters of the elements in the NodeCollection.

Raises
• TypeError – If the input params are of the wrong form.

• KeyError – If the specified parameter does not exist for the nodes.

tolist()

Convert NodeCollection to list.

class nest.lib.hl_api_types.Parameter(datum)

Class for parameters

A parameter may be used as a probability kernel when creating connections and nodes or as synaptic parameters (such as weight and delay). Parameters are created using the CreateParameter() command.

GetValue()

Compute value of parameter.

Returns

out – The value of the parameter

Return type

value

Example

import nest

# normal distribution parameter
P = nest.CreateParameter('normal', {'mean': 0.0, 'sigma': 1.0})

# get out value
P.GetValue()

class nest.lib.hl_api_types.SynapseCollection(data)

Class for Connections.

SynapseCollection represents the connections of a network. The class supports indexing, iteration, length and equality. You can get and set connection parameters by using the membership functions get() and set(). By using the membership function sources() you get an iterator over source nodes, while targets() returns an interator over the target nodes of the connections.

A SynapseCollection is created by the GetConnections() function.

get(keys=None, output='')

Return a parameter dictionary of the connections.

If keys is a string, a list of values is returned, unless we have a single connection, in which case the single value is returned. keys may also be a list, in which case a dictionary with a list of values is returned.

Parameters
• keys (str or list, optional) – String or a list of strings naming model properties. get then returns a single value or a dictionary with lists of values belonging to the given keys.

• output (str, ['pandas','json'], optional) – If the returned data should be in a Pandas DataFrame or in a JSON serializable format.

Returns

• dict – All parameters, or, if keys is a list of strings, a dictionary with lists of corresponding parameters

• type – If keys is a string, the corresponding parameter(s) is returned

Raises
• TypeError – If input params are of the wrong form.

• KeyError – If the specified parameter does not exist for the connections.

Examples

>>>    conns.get()
{'delay': [1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0],
...
'weight': [1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0]}

>>>    conns.get('weight')
[1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0]

>>>    conns[0].get('weight')
1.0

>>>    nodes.get(['source', 'weight'])
{'source': [1, 1, 1, 2, 2, 2, 3, 3, 3],
'weight': [1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0]}

set(params=None, **kwargs)

Set the parameters of the connections to params.

NB! This is almost the same implementation as SetStatus

If kwargs is given, it has to be names and values of an attribute as keyword argument pairs. The values can be single values or list of the same size as the SynapseCollection.

Parameters
• params (str or dict or list) – Dictionary of parameters (either lists or single values) or list of dictionaries of parameters of same length as SynapseCollection.

• kwargs (keyword argument pairs) – Named arguments of parameters of the elements in the SynapseCollection.

Raises
• TypeError – If input params are of the wrong form.

• KeyError – If the specified parameter does not exist for the connections.

sources()

Returns iterator containing the source node IDs of the SynapseCollection.

targets()

Returns iterator containing the target node IDs of the SynapseCollection.

nest.lib.hl_api_types.serializable(data)

Make data serializable for JSON.

Parameters

data (any) –

Returns

data_serialized – Data can be encoded to JSON

Return type
nest.lib.hl_api_types.to_json(data, **kwargs)

Serialize data to JSON.

Parameters
• data (any) –

• kwargs (keyword argument pairs) – Named arguments of parameters for json.dumps function.

Returns

data_json – JSON format of the data

Return type

str

## Functions related to helper info¶

These are helper functions to ease the definition of the high-level API of the PyNEST wrapper.

class nest.lib.hl_api_helper.SuppressedDeprecationWarning(no_dep_funcs)

Context manager turning off deprecation warnings for given methods.

Think thoroughly before use. This context should only be used as a way to make sure examples do not display deprecation warnings, that is, used in functions called from examples, and not as a way to make tedious deprecation warnings dissapear.

nest.lib.hl_api_helper.broadcast(item, length, allowed_types, name='item')

Parameters
• item (object) – Object to broadcast

• length (int) – Length to broadcast to

• allowed_types (list) – List of allowed types

• name (str, optional) – Name of item

Returns

The original item broadcasted to sequence form of length

Return type

object

Raises
nest.lib.hl_api_helper.deprecated(alt_func_name, text=None)

Decorator for deprecated functions.

Shows a warning and calls the original function.

Parameters
• alt_func_name (str, optional) – Name of the function to use instead, may be empty string

• text (str, optional) – Text to display instead of standard text

Returns

Decorator function

Return type

function

nest.lib.hl_api_helper.get_help_filepath(hlpobj)

Get file path of help object

Prints message if no help is available for hlpobj.

Parameters

hlpobj (string) – Object to display help for

Returns

Filepath of the help object or None if no help available

Return type

string

nest.lib.hl_api_helper.get_parameters(nc, param)

Get parameters from nodes.

Used by NodeCollections get() function.

Parameters
• nc (NodeCollection) – nodes to get values from

• param (string or list of strings) – string or list of string naming model properties.

Returns

• int, list – param is a string so the value(s) is returned

• dict – param is a list of string so a dictionary is returned

nest.lib.hl_api_helper.get_parameters_hierarchical_addressing(nc, params)

Get parameters from nodes, hierarchical case.

Used by NodeCollections get() function.

Parameters
• nc (NodeCollection) – nodes to get values from

• params (tuple) – first value in the tuple should be a string, second can be a string or a list of string. The first value corresponds to the path into the hierarchical structure while the second value corresponds to the name(s) of the desired properties.

Returns

• int, list – params[-1] is a string so the value(s) is returned

• dict – params[-1] is a list of string so a dictionary is returned

nest.lib.hl_api_helper.get_unistring_type()

Returns string type dependent on python version.

Returns

Depending on Python version

Return type

str or basestring

nest.lib.hl_api_helper.get_wrapped_text(text, width=80)

Formats a given multiline string to wrap at a given width, while preserving newlines (and removing excessive whitespace).

Parameters

text (str) – String to format

Returns

Wrapped string

Return type

str

nest.lib.hl_api_helper.is_coercible_to_sli_array(seq)

Checks whether a given object is coercible to a SLI array

Parameters

seq (object) – Object to check

Returns

True if object is coercible to a SLI array

Return type

bool

nest.lib.hl_api_helper.is_iterable(seq)

Return True if the given object is an iterable, False otherwise.

Parameters

seq (object) – Object to check

Returns

True if object is an iterable

Return type

bool

nest.lib.hl_api_helper.is_literal(obj)

Check whether obj is a “literal”: a unicode string or SLI literal

Parameters

obj (object) – Object to check

Returns

True if obj is a “literal”

Return type

bool

nest.lib.hl_api_helper.is_sequence_of_connections(seq)

Checks whether low-level API accepts seq as a sequence of connections.

Parameters

seq (object) – Object to check

Returns

True if object is an iterable of dictionaries or subscriptables of CONN_LEN

Return type

bool

nest.lib.hl_api_helper.is_sequence_of_node_ids(seq)

Checks whether the argument is a potentially valid sequence of node IDs (non-negative integers).

Parameters

seq (object) – Object to check

Returns

True if object is a potentially valid sequence of node IDs

Return type

bool

nest.lib.hl_api_helper.is_string(obj)

Check whether obj is a unicode string

Parameters

obj (object) – Object to check

Returns

True if obj is a unicode string

Return type

bool

nest.lib.hl_api_helper.load_help(hlpobj)

Returns documentation of the object

Parameters

hlpobj (object) – Object to display help for

Returns

The documentation of the object or None if no help available

Return type

string

nest.lib.hl_api_helper.model_deprecation_warning(model)

Checks whether the model is to be removed in a future version of NEST. If so, a deprecation warning is issued.

Parameters

model (str) – Name of model

nest.lib.hl_api_helper.restructure_data(result, keys)

Restructure list of status dictionaries or list of parameter values to dict with lists or single list or int.

Parameters
• result (list) – list of status dictionaries or list (of lists) of parameter values.

• keys (string or list of strings) – name(s) of properties

Returns

Return type

int, list or dict

nest.lib.hl_api_helper.show_deprecation_warning(func_name, alt_func_name=None, text=None)

Shows a deprecation warning for a function.

Parameters
• func_name (str) – Name of the deprecated function

• alt_func_name (str, optional) – Name of the function to use instead. Needed if text=None

• text (str, optional) – Text to display instead of standard text

nest.lib.hl_api_helper.show_help_with_pager(hlpobj, pager=None)

Output of doc in python with pager or print

Parameters
• hlpobj (object) – Object to display

• pager (str, optional) – pager to use, False if you want to display help using print().

nest.lib.hl_api_helper.uni_str

alias of str

## Functions related to randomization¶

nest.random.hl_api_random.exponential(beta=1.0)

Draws samples from an exponential distribution.

Parameters

beta (float, optional) – Scale parameter the distribution. Default value is 1.0.

Returns

Object yielding values drawn from the distribution.

Return type

Parameter

nest.random.hl_api_random.lognormal(mean=0.0, std=1.0)

Draws samples from a log-normal distribution.

Parameters
• mean (float, optional) – Mean value of the underlying normal distribution. Default value is 0.

• std (float, optional) – Standard deviation of the underlying normal distribution. Default value is 1.0.

Returns

Object yielding values drawn from the distribution.

Return type

Parameter

nest.random.hl_api_random.normal(mean=0.0, std=1.0)

Draws samples from a normal distribution.

Parameters
• mean (float, optional) – Mean of the distribution. Default value is 0.

• std (float, optional) – Standard deviation of the distribution. Default value is 1.0.

Returns

Object yielding values drawn from the distribution.

Return type

Parameter

nest.random.hl_api_random.uniform(min=0.0, max=1.0)

Draws samples from a uniform distribution.

Samples are distributed uniformly in [min, max) (includes min, but excludes max).

Parameters
• min (float, optional) – Lower boundary of the sample interval. Default value is 0.

• max (float, optional) – Upper boundary of the sample interval. Default value is 1.0.

Returns

Object yielding values drawn from the distribution.

Return type

Parameter

## Functions related to spatial distributions¶

nest.spatial_distributions.hl_api_spatial_distributions.exponential(x, beta=1.0)

Applies an exponential distribution on a Parameter.

Parameters
• x (Parameter) – Input Parameter.

• beta (float, optional) – Scale parameter. Default is 1.0.

Returns

Object yielding values drawn from the distribution.

Return type

Parameter

nest.spatial_distributions.hl_api_spatial_distributions.gamma(x, kappa=1.0, theta=1.0)

Applies a gamma distribution on a Parameter.

This function requires SciPy, and will raise an error if SciPy cannot be imported.

Parameters
• x (Parameter) – Input Parameter.

• kappa (float, optional) – Shape parameter. Default is 1.0.

• theta (float, optional) – Scale parameter. Default is 1.0.

Returns

Object yielding values drawn from the distribution.

Return type

Parameter

nest.spatial_distributions.hl_api_spatial_distributions.gaussian(x, mean=0.0, std=1.0)

Applies a gaussian distribution on a Parameter.

Parameters
• x (Parameter) – Input Parameter.

• mean (float, optional) – Mean of the distribution. Default is 0.0.

• std (float, optional) – Standard deviation of the distribution. Default is 1.0.

Returns

Object yielding values drawn from the distribution.

Return type

Parameter

nest.spatial_distributions.hl_api_spatial_distributions.gaussian2D(x, y, mean_x=0.0, mean_y=0.0, std_x=1.0, std_y=1.0, rho=0.0)

Applies a bivariate gaussian distribution on two Parameters, representing values in the x and y direction.

Parameters
• x (Parameter) – Input Parameter for the x-direction.

• y (Parameter) – Input Parameter for the y-direction.

• mean_x (float, optional) – Mean of the distribution in the x-direction. Default is 0.0.

• mean_y (float, optional) – Mean of the distribution in the y-direction. Default is 0.0.

• std_x (float, optional) – Standard deviation of the distribution in the x-direction. Default is 1.0.

• std_y (float, optional) – Standard deviation of the distribution in the y-direction. Default is 1.0.

• rho (float, optional) – Correlation of x and y. Default is 0.0

Returns

Object yielding values drawn from the distribution.

Return type

Parameter

## Functions related to mathematical expressions¶

nest.math.hl_api_math.cos(parameter)

Calculate the cosine of the parameter

Parameters

parameter (Parameter) – Input Parameter.

Returns

Object representing the cosine of the parameter.

Return type

Parameter

nest.math.hl_api_math.exp(parameter)

Calculate the exponential of the parameter

Parameters

parameter (Parameter) – Input Parameter.

Returns

Object representing the exponential of the parameter.

Return type

Parameter

nest.math.hl_api_math.max(parameter, value)

Yields the largest value of the value of a parameter and a given value

Parameters
• parameter (Parameter) – Input Parameter.

• value (float) – Value to compare against.

Returns

Object yielding the largest value.

Return type

Parameter

nest.math.hl_api_math.min(parameter, value)

Yields the smallest value of the value of a parameter and a given value

Parameters
• parameter (Parameter) – Input Parameter.

• value (float) – Value to compare against.

Returns

Object yielding the smallest value.

Return type

Parameter

nest.math.hl_api_math.redraw(parameter, min, max)

Redraws the value of the parameter if it is outside of the given limits

Both min and max values are included in the limit. If the number of redraws exceeds 1000, an error is thrown.

Parameters
• parameter (Parameter) – Input Parameter.

• min (float) – Lower bound of the value.

• max (float) – Upper bound of the value.

Returns

Object redrawing the parameter until it can yield a value within the given limits.

Return type

Parameter

nest.math.hl_api_math.sin(parameter)

Calculate the sine of the parameter

Parameters

parameter (Parameter) – Input Parameter.

Returns

Object representing the sine of the parameter.

Return type

Parameter

nest.logic.hl_api_logic.conditional(condition, param_if_true, param_if_false)

Yields one value or another, based on the condition.

Parameters
• condition (Parameter) – A comparing Parameter, created with the usual comparators.

• param_if_true ([Parameter | float]) – Value or Parameter used to get a value used if the condition evaluates to true.

• param_if_false ([Parameter | float]) – Value or Parameter used to get a value used if the condition evaluates to false.

Returns

Object representing the conditional.

Return type

Parameter


text:
# How to prove that $Ax = e^x$ has two solutions when $e < A < \infty$

How to prove that $Ax = e^x$ has two solutions when $e < A < \infty$? This is easy to visualise graphically, but how can it be shown with algebra?

• I'm thinking you want to use the Intermediate Value Theorem, but that's not really "algebra." – Braindead Nov 24 '12 at 16:17
• Does "algebra" include the material in a basic introduction to calculus? – André Nicolas Nov 24 '12 at 16:19

Consider the function $f(x)=e^x-Ax$. Then $f(0)=1$. We have $f'(x)=e^x-A$, $f''(x)=e^x$. As the second derivative is always positive, our function is convex. The derivative has a single zero at $x=\log A$, so $f$ has a minimum at that point. This means that $$f(x)\geq f(\log A)=A-A\log A.$$ As $\lim_{x\to\infty}f(x)=\infty$, if the minimum is negative, then $f$ will have two roots (and none if the minimum is positive).

Assuming $A>0$, we have $A\log A>A$ precisely when $\log A>1$, i.e. $A>e$.

In conclusion, we have

1. Two points where $Ax=e^x$ when $A>e$;
2. One point where $Ax=e^x$ when $A=e$;
3. No points where $Ax=e^x$ when $A<e$;

This is a proof that uses the intermedate value theorem (I don't know an algebraic proof of this fact):

Let $g(x)=e^x-Ax$. $g(0)=1$ and $g(1)=e-A<0$. Using the intermediate value theorem, we deduce that there is a root $r_1$ of $g(x)$ in the interval $[0,1]$

Since $g$ goes increases without bound after sufficiently large x, thus there exists M>1 such that $g(M)>0$. By the intermediate value theorem, we know that there exists another root $r_2$ of $g$ in the interval $]1,M[$

Here is one with calculus.
If $f(x)=e^x-Ax$ then:

• $f(0)=1>0.$
• $f(1)=e-A<0.$
• $f(e)=e^e-Ae>e^e-e^2>0.$

Now use intermediate value theorem.

Since $f$ is differentiable in $\mathbb{R}$ with $f'(x)=e^x-A$ and $f'(x)<0 \iff x<\ln A , \ \ f'(x)>0 \iff x>\ln A \Rightarrow f(x)=0$ has at most two solutions.

format: plain


text:
# Properties of Equivalent Fractions

The properties of equivalent fractions are discussed here step-by-step.

1. If the numerator and denominator of a fraction is multiplied by the same number, except zero, the value of the fraction remains the same and an equivalent fraction is obtained.

As:

(i) 2/3 = 2 x 2/3 x 2 = 4/6; 2 x 3/3 x 3 = 6/9; 2 x 4/3 x 4 = 8/12;

2 x 5/3 x 5 = 10/15

So, 2/3, 4/6, 6/9, 8/12, 10/15, etc., are equivalent fractions.

(ii) 5/6 = 5 x 3/6 x 3 = 15/18; 5 x 7/6 x 7 = 35/42; 5 x 4/6 x 4 = 20/24;

5 x 9/6 x 9 = 45/54

So, 5/6, 15/18, 35/42, 20/24, 45/54, etc., are equivalent fractions.

2. If the numerator and denominator of a fraction is divided by the same number, except zero, the value of the fraction remains the same and an equivalent fraction is obtained.

(i) 60/90 = 60 ÷ 10/90 ÷ 10 = 6/9; 60 ÷ 2/90 ÷ 2 = 30/45;

60 ÷ 3/90 ÷ 3 = 20/30, 60 ÷ 5/90 ÷ 5 = 12/18

So, 60/90, 6/9, 30/45, 20/30, 2/3 etc., are equivalent fractions.

32/72 = 32 ÷ 2/72 ÷ 2 = 16/36, 32 ÷ 4/72 ÷ 4 = 8/18, 32 ÷ 8/72 ÷ 8 = 4/9

So, 32/72, 16/36, 8/18, 4/9 are equivalent fractions.

3. In case of two equivalent fractions, the product of the numerator of one fraction and denominator of the second fraction is equal to the product of the denominator of the first fraction and numerator of the second fraction.

Accordingly, the two fractions are equivalent if:

numerator of the first fraction × denominator of the second fraction = denominator of the first fraction × numerator of the second fraction

As:

 1/3 = 2/6 5/7 = 15/21 So, 1 x 6 = 3 x 2 = 6 So, 5 x 21 = 7 x 15 = 105

4. A fraction may be reduced to its lowest term. If a factor or factors is / are common to numerator and denominator of a fraction, then the common factor or factors may be removed to get it in its lowest term.

If there is a fraction 12/18 and we have to reduce it to its lowest term,

Since, 12 = 2 x 2 x 3 and 18 = 2 x 3 x 3, thus, 2 x 3 = 6 is a common factor in numerator and denominator of 12/18

So, 12 ÷ 6/18 ÷ 6 = 2/3

By dividing both 12 and 18 by 6, we get the fraction 2/3 as the lowest of 12/18.

These are the properties of equivalent fractions explained along with the examples.

Related Concept


text:
Linear preservers for matrix inequalities and partial orderings.(English)Zbl 0985.15018

Let $$M_{n}(F)$$ denote the algebra of $$n\times n$$ matrices over the field $$F$$. The author calls a linear transformation $$T$$ of $$M_{n}(F)$$ into itself of “standard form” if $$T$$ either has the form $$X\mapsto AXB$$ or the form $$X\mapsto AX^{T}B$$ for some fixed invertible matrices $$A$$ and $$B$$. There have been many papers by a series of authors in which it has been shown that transformations $$T$$ which preserve a wide variety of properties of $$M_{n}(F)$$ are necessarily of standard form [see, for example, C.-K. Li and N.-K. Tsing, Linear Algebra Appl. 162-164, 217-235 (1992; Zbl 0762.15016)].
In the present paper the author considers the partial ordering $$\preceq$$ on $$M_{n}(F)$$ defined by $$A\preceq B$$ when $$rank(B-A)=rank(B)-rank(A).$$ (This is called the rank-subtractivity ordering and is equivalent to the minus ordering). He shows that, provided $$\left|F\right|>n$$, every invertible linear transformation $$T$$ of $$M_{n}(F)$$ which preserves $$\preceq$$ is of standard form, and also proves a number of related results.
Reviewer: J.D.Dixon (Ottawa)

MSC:

 15B48 Positive matrices and their generalizations; cones of matrices 15A04 Linear transformations, semilinear transformations 15A45 Miscellaneous inequalities involving matrices 15A30 Algebraic systems of matrices

Zbl 0762.15016
Full Text:

References:

 [1] Baksalary, J.K., A relationship between star and minus-ordering, Linear algebra appl., 82, 163-167, (1986) · Zbl 0603.15002 [2] Baksalary, J.K.; Hauke, J., A further algebraic version of Cochran’s theorem and matrix partial orderings, Linear algebra appl., 127, 157-169, (1990) · Zbl 0696.15005 [3] Baksalary, J.K.; Pukelsheim, F.; Styan, G.P.H., Some properties of matrix partial orderings, Linear algebra appl., 119, 57-85, (1989) · Zbl 0681.15005 [4] Baksalary, J.K.; S̆emrl, P.; Styan, G.P.H., A note on rank-additivity and range-additivity, Linear algebra appl., 237-238, 489-498, (1996) · Zbl 0856.47001 [5] Beasley, L.R.B., Linear operators which preserve pairs on which the rank is additive, J. Korean soc. ind. appl. math., 2, 2, 27-36, (1998) [6] Beasley, L.R.B., Spaces of matrices of equal rank, Linear algebra appl., 40, 183-187, (1981) [7] A.H. Clifford, G.B. Preston, The algebraic theory of semigroups, I, II, AMS Math. Survey 7, Providence, RI, 1961 · Zbl 0111.03403 [8] Cline, R.E.; Funderlic, R.E., The rank of a difference of matrices and associated generalized inverses, Linear algebra appl., 24, 185-215, (1979) · Zbl 0393.15005 [9] Dieudonné, J., Sur une généralisation du groupe orthogonal àquatre variables, Arch. math., 1, 282-287, (1949) · Zbl 0032.10601 [10] Drazin, M.P., Natural structures on semigroups with involution, Bull. am. math. soc., 84, 1, 139-141, (1978) · Zbl 0395.20044 [11] Dixon, J.D., Rigid embeddings of simple groups in the general linear group, Can. J. math., 29, 384-391, (1979) · Zbl 0332.20016 [12] G. Frobenius, Über die Darstellung der endlichen Gruppen durch lineare Substitutionen, Sitzungsber. Preuss. Akad. Wiss. Berlin (1897) 994-1015 · JFM 28.0130.01 [13] Groß, J., A note on the rank-subtractivity ordering, Linear algebra appl., 289, 151-160, (1999) · Zbl 0941.15011 [14] Guterman, A.; Li, C.-K.; S̆emrl, P., Some general techniques on linear preserver problems, Linear algebra appl., 315, 61-81, (2000) · Zbl 0964.15004 [15] Hartwig, R.E., How to partially order regular elements, Math. japonica, 25, 1, 1-13, (1980) · Zbl 0442.06006 [16] Hartwig, R.E., A note on rank-additivity, Linear multilinear algebra, 10, 59-61, (1981) · Zbl 0456.15002 [17] Hartwig, R.E.; Omladic̆, M.; S̆emrl, P.; Styan, G.P.H., On some characterizations of pairwise star orthogonality using rank and dagger additivity and subtractivity, Linear algebra appl., 237-238, 499-507, (1996) · Zbl 0848.15013 [18] Hartwig, R.E.; Styan, G.P.H., On some characterizations of the star partial ordering and rank-subtractivity, Linear algebra appl., 82, 145-161, (1986) · Zbl 0603.15001 [19] Loewy, R.; Radwan, N., Spaces of symmetric matrices of bounded rank, Linear algebra appl., 162-164, 189-215, (1992) · Zbl 0801.15011 [20] Li, C.-K.; Tsing, N.-K., Linear preserver problems: a brief introduction and some special techniques, Linear algebra appl., 162-164, 217-235, (1992) · Zbl 0762.15016 [21] Mitra, S.K., The minus partial order and the shorted matrix, Linear algebra appl., 83, 1-27, (1986) · Zbl 0605.15004 [22] Marsaglia, G.; Styan, G.P.H., Equalities and inequalities for ranks of matrices, Linear multilinear algebra, 2, 269-292, (1974) [23] Nambooripad, K.S.S., The natural partial order on a regular semigroup, Proc. Edinburgh math. soc., 23, 249-260, (1980) · Zbl 0459.20054 [24] Pierce, S., Linear maps on algebraic groups, Linear algebra appl., 162-164, 237-242, (1992) · Zbl 0745.20042 [25] S. Pierce, et al., A survey of linear preserver problems, Linear Multilinear Algebra 33 (1992) 1-119 [26] S̆emrl, P., On a matrix version of Cochran’s statistical theorem, Linear algebra appl., 237-238, 477-487, (1996) · Zbl 0847.62043 [27] Westwick, R., Spaces of matrices of fixed rank, Linear multilinear algebra, 20, 171-174, (1987) · Zbl 0611.15002
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text:
Condensed Matter Physics, 1999, vol. 2, No. 4(20), p. 721-725, English
DOI:10.5488/CMP.2.4.721

Title: PIEZOELECTRIC PROPERTIES OF Bi_2TeO_5 SINGLE CRYSTALS
Author(s): A.M.Antonenko, K.V.Domoratsky, A.Yu.Kudzin, L.Ya.Sadovskaya (Dnipropetrovsk State University, 20 Kazakova Str., 320625 Dnipropetrovsk, Ukraine)

This paper is devoted to the measurement results of piezoelectric, dielectric and elastic tensor components in the temperature range between 20 $^\circ{\rm C}$ and 400 $^\circ{\rm C}$. It was revealed that excitation resonance and antiresonance signals were possible without preliminary sample polarizing. Besides, the sound velocities measured along the main axes by echo-pulse method at room temperature are presented.

Comments: Figs. 3, Refs. 5, Tabs. 1.

 [ps,pdf] << Contents of Vol.2 No.4(20)


text:
problem for interpretation of resulted ComBat() plot
1
0
Entering edit mode
modarzi ▴ 10
@modarzi-16296
Last seen 9 months ago

HI, I ran ComBat() for batch effect correction in my TCGA RNA-seq. consequently, I have two choices for batch effect correction: Parametric and nonparametric adjustment. So, I have down Parametric adjustment and resulted plot available in this link. Now, I need to know should I apply nonparametric adjustment ComBat() on my data or not? I would like to know more about the interpretation of that plot.

I appreciate it if anybody shares his/her comment with me.

Best

ComBat sva batch effect RNA-seq • 226 views
0
Entering edit mode
@w-evan-johnson-5447
Last seen 15 months ago
United States

The parametric approach should work just fine on your data. You are just looking for gross deviations from these distributions (bi-modal or extreme skewness). Your plots look good to me!

Activity on a post you are following on support.bioconductor.orghttps://support.bioconductor.org/

User modarzihttps://support.bioconductor.org/u/16296/ wrote Question: problem for interpretation of resulted ComBat() plothttps://support.bioconductor.org/p/126946/:

HI, I ran ComBat() for batch effect correction in my TCGA RNA-seq. consequently, I have two choices for batch effect correction: Parametric and nonparametric adjustment. So, I have down Parametric adjustment and resulted plot available in this linkhttps://www.dropbox.com/s/knay47rl9hxecib/Rplot_Combat_parametric%20adjustment_980818.bmp?dl=0. Now, I need to know should I apply nonparametric adjustment ComBat() on my data or not? I would like to know more about the interpretation of that plot.

I appreciate it if anybody shares his/her comment with me.

Best


text:
## Filters: edu-sharing Filter

filter_edusharing
Maintained by Christian Ernst, Steffen Hippeli
The edu-sharing filter displays embedded edu-sharing content.
29 sites
2 fans
Moodle 2.9, 3.0, 3.1, 3.2, 3.3, 3.4, 3.5, 3.6, 3.7

The edu-sharing filter displays embedded edu-sharing content.

### Dependencies

Depends on the edu-sharing activity module. The filter is only needed together the editor plugin.

### Installation

Activate the edu-sharing filter, set it up for moodle and html texts and push it to the first position in the filter overview list

For a full documentation with screenshots of the post installation steps for the edu-sharing plugin pakage visit the documentation pages.

### Awards

• Wed, Apr 27, 2016, 3:54 PM
Prechecker results: 367 errors/47 warnings
http://integration.moodle.org/job/Precheck%20remote%20branch/23128/artifact/work/smurf.html

* Please provide a full description of the plugin. You can describe the plugin's features, purpose and typical usage scenarios there, for example.
* Bug tracker is not specified for your plugin. Providing a place for users of your plugin to report issues encourages participation and provides a way for users to report bugs, make feature requests, or suggest other types of improvements.
* In order to facilitate easier sharing and further development of your open-source plugin, please provide publicly accessible URL of your code repository. The suggested naming convention of the repository is moodle-{plugintype}_{pluginname}.
* Please add a screenshot of your plugin to help folks get an idea of what it looks like when installed.
• Sat, Aug 27, 2016, 12:03 AM

Thanks Steffen for sharing this filter with the Moodle community. I've been looking at this together with reviewing the edusharing activity module. One thing I noticed is that your filter does a lot of things on all pages by default - such as it tries to prevent the browser from caching them, it includes the module's jQuery and other JS modules and it attempts to connect to the edu-sharing server via SOAP. This all seems to happen on every single page even if there is no edu-sharing contents on it.

I am afraid of potential performance impacts due to such behaviour. I believe it would be much cleaner if the filter's filter() method first checked via a quick and check strpos()call that the passed \$text contains a substring like es:resource_id. It would then perform all the required steps only then, otherwise it would quit early.

Such an approach is recommended even for plugins that make much less than your filter does. I am marking this as needing more work now so you can evaluate this and provide a new fixed version. Thanks for your patience with the plugin approval process.

• Thu, Sep 29, 2016, 10:22 PM
Hi David,
finally i had the time to change this issue. Now the scripts will be loaded only if actually there is some edu-sharaing content in the text.
• Thu, Nov 24, 2016, 5:20 AM

Hi Steffen, and sorry for the delay in finishing the review of your plugin. There are couple of issues I spotted that should be addressed in a future version.

• You call reset_text_filters_cache() every time you initialize the filter. As a consequence, your filter effectively switches off all caching for all filters (and even worse, adds some extra processing on top of it). I don't think the filter is supposed to use this function like this.
• Performance wise, the preg_match_all() is generally much slower than strpos() check. I still think it would make sense to use the latter one first to see if a substring like es:resource_id is present at all. And only if so, call preg_match_all() and gather the actual matches.
• Your CSS file styles.css uses too generic selectors like .inner and is likely to affect other parts of the Moodle interface. Please refer to https://docs.moodle.org/dev/Plugin_contribution_checklist#CSS_styles
• As you are using your own low level curl calls to obtain the remote content, the plugin does not respect other related site's settings (such as proxy etc). It might be better (and safer) to stick to using Moodle core API for these things.

Some of these issues are quite serious and normally I would prefer to see them fixed before publishing the plugin here. But I am going to approve this now anyway as the filter is an essential companion of the edusharing set and I believe you will improve it soon. You are cleared to land, welcome to the Plugins directory.


text:
Professional repair and maintenance of both desktops and notebooks.Over 17 years experience.Specializing in virus removal and general computer maintenance. Upgrades

Address 905 N College Ave, Clarksville, AR 72830 (479) 754-0254

# definition of standard error Hagarville, Arkansas

Specifically, the standard error equations use p in place of P, and s in place of σ. The standard error is a measure of variability, not a measure of central tendency. Copyright (c) 2010 Croatian Society of Medical Biochemistry and Laboratory Medicine. The table below shows how to compute the standard error for simple random samples, assuming the population size is at least 20 times larger than the sample size.

estimate – Predicted Y values close to regression line     Figure 2. Chegg Chegg Chegg Chegg Chegg Chegg Chegg BOOKS Rent / Buy books Sell books My books STUDY Textbook solutions Expert Q&A TUTORS TEST PREP ACT prep ACT pricing SAT prep SAT The researchers report that candidate A is expected to receive 52% of the final vote, with a margin of error of 2%. Available at: http://damidmlane.com/hyperstat/A103397.html.

Biochemia Medica 2008;18(1):7-13. Gurland and Tripathi (1971)[6] provide a correction and equation for this effect. Accessed September 10, 2007. 4. Relative standard error See also: Relative standard deviation The relative standard error of a sample mean is the standard error divided by the mean and expressed as a percentage.

Learn the correct uses of these two commonly confused homophones. For example, the sample mean is the usual estimator of a population mean. Scenario 2. Suppose the mean number of bedsores was 0.02 in a sample of 500 subjects, meaning 10 subjects developed bedsores.

On the other hand, if the sample represents a significant fraction (say, 1/20) of the population size, the standard error will be noticeably smaller, when we sample without replacement. However, while the standard deviation provides information on the dispersion of sample values, the standard error provides information on the dispersion of values in the sampling distribution associated with the population Similarly, the sample standard deviation will very rarely be equal to the population standard deviation. In each of these scenarios, a sample of observations is drawn from a large population.

It represents the standard deviation of the mean within a dataset. For illustration, the graph below shows the distribution of the sample means for 20,000 samples, where each sample is of size n=16. Notation The following notation is helpful, when we talk about the standard deviation and the standard error. The following expressions can be used to calculate the upper and lower 95% confidence limits, where x ¯ {\displaystyle {\bar {x}}} is equal to the sample mean, S E {\displaystyle SE}

This estimate may be compared with the formula for the true standard deviation of the sample mean: SD x ¯   = σ n {\displaystyle {\text{SD}}_{\bar {x}}\ ={\frac {\sigma }{\sqrt {n}}}} Population parameter Sample statistic N: Number of observations in the population n: Number of observations in the sample Ni: Number of observations in population i ni: Number of observations in sample This capability holds true for all parametric correlation statistics and their associated standard error statistics. All rights reserved.

Because of random variation in sampling, the proportion or mean calculated using the sample will usually differ from the true proportion or mean in the entire population. The computations derived from the r and the standard error of the estimate can be used to determine how precise an estimate of the population correlation is the sample correlation statistic. In cases where the standard error is large, the data may have some notable irregularities.Standard Deviation and Standard ErrorThe standard deviation is a representation of the spread of each of the The sample mean x ¯ {\displaystyle {\bar {x}}} = 37.25 is greater than the true population mean μ {\displaystyle \mu } = 33.88 years.

When the S.E.est is large, one would expect to see many of the observed values far away from the regression line as in Figures 1 and 2.     Figure 1. To help you learn and understand key math terms and concepts, we’ve identified some of the most important ones and provided detailed definitions for them, written and compiled by Chegg experts. Statistic Standard Deviation Sample mean, x σx = σ / sqrt( n ) Sample proportion, p σp = sqrt [ P(1 - P) / n ] Difference between means, x1 - HyperStat Online.

The standard deviation cannot be computed solely from sample attributes; it requires a knowledge of one or more population parameters. Retrieved 17 July 2014. Blackwell Publishing. 81 (1): 75–81. Hints help you try the next step on your own.

Therefore, it is essential for them to be able to determine the probability that their sample measures are a reliable representation of the full population, so that they can make predictions Using a sample to estimate the standard error In the examples so far, the population standard deviation σ was assumed to be known. The standard error (SE) is the standard deviation of the sampling distribution of a statistic,[1] most commonly of the mean. An R of 0.30 means that the independent variable accounts for only 9% of the variance in the dependent variable.

The mean age was 23.44 years. The confidence interval so constructed provides an estimate of the interval in which the population parameter will fall. As discussed previously, the larger the standard error, the wider the confidence interval about the statistic. People invent new words all the time, but which ones actually make it?

CRC Standard Mathematical Tables and Formulae. Decode the pieces of our favorite... It will be shown that the standard deviation of all possible sample means of size n=16 is equal to the population standard deviation, σ, divided by the square root of the Ecology 76(2): 628 – 639. ^ Klein, RJ. "Healthy People 2010 criteria for data suppression" (PDF).

If one survey has a standard error of $10,000 and the other has a standard error of$5,000, then the relative standard errors are 20% and 10% respectively. Allison PD. Taken together with such measures as effect size, p-value and sample size, the effect size can be a useful tool to the researcher who seeks to understand the accuracy of statistics The standard error indicates the likely accuracy of the sample mean as compared with the population mean.


text:
## Featured Answers

1
Active contributors today

## What is the exact value of #cos(tan^-1(2)+tan^-1(3))# ?

mason m
Featured 2 months ago

#### Answer:

$\cos \left({\tan}^{-} 1 \left(2\right) + {\tan}^{-} 1 \left(3\right)\right) = - \frac{1}{\sqrt{2}}$

#### Explanation:

First use the cosine angle-addition formula:

$\cos \left(a + b\right) = \cos \left(a\right) \cos \left(b\right) - \sin \left(a\right) \sin \left(b\right)$

Then the original expression equals:

$= \cos \left({\tan}^{-} 1 \left(2\right)\right) \cos \left({\tan}^{-} 1 \left(3\right)\right) - \sin \left({\tan}^{-} 1 \left(2\right)\right) \sin \left({\tan}^{-} 1 \left(3\right)\right)$

Note that if $\theta = {\tan}^{-} 1 \left(2\right)$, then $\tan \left(\theta\right)$. That is, a right triangle with angle $\theta$ has $\tan \left(\theta\right) = 2$, which is the triangle with the leg opposite $\theta$ being $2$ and the leg adjacent to $\theta$ being $1$. The Pythagorean theorem tells us that the hypotenuse of this triangle is $\sqrt{5}$.

So, when $\theta = {\tan}^{-} 1 \left(2\right)$, we see that:

$\cos \left({\tan}^{-} 1 \left(2\right)\right) = \cos \left(\theta\right) = \text{adjacent"/"hypotenuse} = \frac{1}{\sqrt{5}}$

$\sin \left({\tan}^{-} 1 \left(2\right)\right) = \sin \left(\theta\right) = \text{opposite"/"hypotenuse} = \frac{2}{\sqrt{5}}$

Now let angle $\phi$ be defined by $\phi = {\tan}^{-} 1 \left(3\right)$, such that $\tan \left(\phi\right) = 3$. This is the right triangle with:

$\left\{\begin{matrix}\text{opposite"=3 \\ "adjacent"=1 \\ "hypotenuse} = \sqrt{10}\end{matrix}\right.$

Then:

$\cos \left({\tan}^{-} 1 \left(3\right)\right) = \cos \left(\phi\right) = \text{adjacent"/"hypotenuse} = \frac{1}{\sqrt{10}}$

$\sin \left({\tan}^{-} 1 \left(3\right)\right) = \sin \left(\phi\right) = \text{opposite"/"hypotenuse} = \frac{3}{\sqrt{10}}$

Plugging these into the expression from earlier we get:

$= \frac{1}{\sqrt{5}} \left(\frac{1}{\sqrt{10}}\right) - \frac{2}{\sqrt{5}} \left(\frac{3}{\sqrt{10}}\right)$

Note that $\sqrt{5} \left(\sqrt{10}\right) = \sqrt{50} = 5 \sqrt{2}$:

$= \frac{1 - 6}{5 \sqrt{2}} = - \frac{1}{\sqrt{2}}$

## Given f(x)=2sin(x-pi/4)? a) State all x-intercepts over the interval (0,4pi) b) State the point(s) in (0,2pi) where the graph of this function reaches a maximum. Give your answer as a point (x,y).

dk_ch
Featured 2 months ago

Given
$y = f \left(x\right) = 2 \sin \left(x - \frac{\pi}{4}\right)$

To get x-intercepts we can put y=0 and solve for $x \in \left[0 , 4 \pi\right]$

So $2 \sin \left(x - \frac{\pi}{4}\right) = 0$

$\implies x - \frac{\pi}{4} = n \pi \text{ where } n \in \mathbb{Z}$

$\implies x = n \pi + \frac{\pi}{4} \text{ where } n \in \mathbb{Z}$

Putting $n = 0$ we get

$x = \frac{\pi}{4}$

Putting $n = 1$ we get

$x = \frac{5 \pi}{4}$

Putting $n = 2$ we get

$x = \frac{9 \pi}{4}$

Putting $n = 3$ we get

$x = \frac{13 \pi}{4}$

(a) So points of x-intercepts over $\left[0 , 4 \pi\right]$ are

#(pi/4,0);((5pi)/4,0);((9pi)/4,0);((13pi)/4,0)#

b) For maximum value of y the value of $\sin \left(x - \frac{\pi}{4}\right) = 1$

So $x - \frac{\pi}{4} = \frac{\pi}{2}$
$\implies x = \frac{\pi}{4} + \frac{\pi}{2} = \frac{3 \pi}{4}$

For $x = \frac{3 \pi}{4} \text{ } y = 2$

The point(in (0,2pi) where the graph of this function reaches
maximum is $\left(\frac{3 \pi}{4} , 2\right)$

## #cosx+cos3x=0#?

HSBC244
Featured 1 month ago

#### Answer:

$x = \frac{\pi}{2} , \frac{3 \pi}{2} , \frac{\pi}{4} , \frac{3 \pi}{4} , \frac{5 \pi}{4} \mathmr{and} \frac{7 \pi}{4}$

#### Explanation:

Note that $\cos 3 x$ can be rewritten as $\cos \left(2 x + x\right)$.

$\cos x + \cos \left(2 x + x\right) = 0$

Now use $\cos \left(A + B\right) = \cos A \cos B - \sin A \sin B$.

$\cos x + \cos 2 x \cos x - \sin 2 x \sin x = 0$

Apply $\cos 2 x = 2 {\cos}^{2} x - 1$ and $\sin 2 x = 2 \sin x \cos x$.

$\cos x + \left(2 {\cos}^{2} x - 1\right) \cos x - 2 \sin x \cos x \left(\sin x\right) = 0$

$\cos x + 2 {\cos}^{3} x - \cos x - 2 {\sin}^{2} x \cos x = 0$

Use ${\sin}^{2} x + {\cos}^{2} x = 1$:

$\cos x + 2 {\cos}^{3} x - \cos x - 2 \left(1 - {\cos}^{2} x\right) \cos x = 0$

$\cos x + 2 {\cos}^{3} x - \cos x - 2 \cos x + 2 {\cos}^{3} x = 0$

$4 {\cos}^{3} x - 2 \cos x = 0$

Factor:

$2 \cos x \left(2 {\cos}^{2} x - 1\right) = 0$

We have

$\cos x = 0$

$x = \frac{\pi}{2} , \frac{3 \pi}{2}$

AND

$\cos x = \pm \frac{1}{\sqrt{2}}$

$x = \frac{\pi}{4} , \frac{3 \pi}{4} , \frac{5 \pi}{4} \mathmr{and} \frac{7 \pi}{4}$

Hopefully this helps!

## How do you graph # y=1+5cos(x-pi)#?

Rina Takahashi
Featured 1 month ago

#### Answer:

Here is the graph of the function (it's not drawn to scale):

I will explain how I graph it below:

#### Explanation:

Your given equation is $y = 1 + 5 \cos \left(x - \pi\right)$ .

I would write the given equation in the standard form according to $y = A \cos B \left(x - C\right) + D$

You get: $y = 5 \cos \left(x - \pi\right) + 1$

This way, it is easier to define your amplitude, phase shift, vertical shift, and period multiplier.

In order to graph, we need to know the following:

1. Amplitude , which is $| A |$
2. Period Multiplier , which is B
3. Phase Shift , which is C
4. Vertical Shift, which is D
5. Period, which is $P e r i o d = \frac{2 \pi}{B}$

Let's get all our required values from our equation: $y = 5 \cos \left(x - \pi\right) + 1$

1. Amplitude = $| 5 |$ = 5
2. Period Multiplier = B = 1
3. Phase Shift = $- \pi$ = $\pi$ units to the right
4. Vertical Shift = $+ 1$ = 1 units up
5. Period = $P e r i o d = \frac{2 \pi}{1}$ = $2 \pi$

It is important to note that B is not apparent in this equation because it is equal to 1. The actual equation should really be $y = 5 \cos 1 \left(x - \pi\right) + 1$ .

Our period multiplier is important for determining our period, and it also affects our phase shift. However, we do not have to worry about it for this equation since it's just equal to 1.

Since we have both an amplitude and a vertical shift, we get new minimum and maximum y values for the graph:

Maximum Y Value = Vertical Shift + Amplitude = $1 + 5 = 6$

Minimum Y Value = Vertical Shift - Amplitude = $1 - 5 = - 4$

Most importantly, this graph is a positive cosine graph because our equation is 5cos.. and not -5cos.. . A positive cosine graph always starts at the maximum value.

Also, our vertical shift is +1, so our new x-axis start has shifted up by 1. We are starting at y = 1 and not y = 0.

As with our phase shift, it is $\pi$ units to the right. How I like to think of the phase shift is that it's just a new start to the graph. You can treat the phase shift as a place to actually begin your graph.

So.. with all this information, we can now actually graph the equation of $y = 5 \cos \left(x - \pi\right) + 1$ .

Here are some visual cues.

My art is pretty bad, sographing the graph on a website should be much more different.

I hope this helps.

## How do you sketch the graph of #y=3 cos (x+π) -3#?

HSBC244
Featured yesterday

I like getting rid of the phase shift (the $x + \pi$ part) using the sum and difference formulas. The one that is applicable here is

$\cos \left(A + B\right) = \cos A \cos B - \sin A \sin B$.

We have:

$y = 3 \left(\cos x \cos \left(\pi\right) - \sin x \sin \pi\right) - 3$

$y = 3 \left(\cos x \left(- 1\right) - 0\right) - 3$

$y = - 3 \cos x - 3$

Now you need a little bit of knowledge on the basic cosine function, $y = \cos x$. Here's the graph:

graph{y = cosx [-10, 10, -5, 5]}

Whenever there is a coefficient $a$ next to the cosine, you have an altered amplitude, which is the distance between the centre (the line $y = 0$) and the top or bottom of the curve.

In the graph of $y = \cos x$, the amplitude is simply $1$. In the graph of $y = - 3 \cos x - 3$, the amplitude will be $3$.

The $-$ is in front of the $3$ to signify a reflection over the x-axis.

Finally, the $- 3$ to the far right of the equation signifies a vertical transformation of $3$ units down. We are left with the following graph:

graph{y = -3cosx - 3 [-10, 10, -5, 5]}

Hopefully this helps!

## Expressing in different form and finding a maximum? See picture below (7 marks)

dk_ch
Featured yesterday

(a)

$\sqrt{15} \sin \left(2 x\right) + \sqrt{5} \cos \left(2 x\right)$

$= \sqrt{20} \left(\frac{\sqrt{15}}{\sqrt{20}} \sin \left(2 x\right) + \frac{\sqrt{5}}{\sqrt{20}} \cos \left(2 x\right)\right)$

[Taking $\frac{\sqrt{15}}{\sqrt{20}} = \cos \alpha \mathmr{and} \frac{\sqrt{5}}{\sqrt{20}} = \sin \alpha$]

$= \sqrt{20} \left(\cos \alpha \sin \left(2 x\right) + \sin \alpha \cos \left(2 x\right)\right)$

$= 2 \sqrt{5} \sin \left(2 x + \alpha\right)$

Where $\alpha = {\tan}^{-} 1 \left(\frac{\sqrt{5}}{\sqrt{15}}\right) = {\tan}^{-} 1 \left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6}$

(b)

(i)
$f \left(x\right) = \frac{2}{5 + \sqrt{15} \sin \left(2 x\right) + \sqrt{\cos} \left(2 x\right)}$

$\implies f \left(x\right) = \frac{2}{5 + 2 \sqrt{5} \sin \left(2 x + \alpha\right)}$

The value of $f \left(x\right)$ will be maximum when $\sin \left(2 x + \alpha\right)$ is minimum i.e. $\sin \left(2 x + \alpha\right) = - 1$

So

$f {\left(x\right)}_{\text{max}} = \frac{2}{5 - 2 \sqrt{5}}$

$\implies f {\left(x\right)}_{\text{max}} = \frac{2 \left(5 + 2 \sqrt{5}\right)}{\left(5 - 2 \sqrt{5}\right) \left(5 + 2 \sqrt{5}\right)}$

$\implies f {\left(x\right)}_{\text{max}} = \frac{10 + 4 \sqrt{5}}{25 - 20} = 2 + \frac{4}{5} \sqrt{5}$

(ii)Here $f \left(x\right)$ is maximum when

$\sin \left(2 x + \alpha\right) = - 1$

$\implies \sin \left(2 x + \frac{\pi}{6}\right) = \sin \left(\frac{3 \pi}{2}\right)$

$\implies 2 x + \frac{\pi}{6} = \frac{3 \pi}{2}$

$\implies 2 x = \frac{3 \pi}{2} - \frac{\pi}{6} = \frac{4 \pi}{3}$

$\implies x = \frac{2 \pi}{3}$

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# Math Help - compound intrest

1. ## compound intrest

$2750 invested at 4.825% annual interest for 6 years compounded daily (n=365). find the amount that will be in the account given the stated conditions i cant figure this one out for the life of me 2. Originally Posted by mattdale321$2750 invested at 4.825% annual interest for 6 years compounded daily
(n=365).

i cant figure this one out for the life of me
$p(t)=P_0(1+\frac{r}{n})^{nt}$

$P(6)=2750 \left(1+\frac{.04825}{365} \right)^{(365)(6)} \approx 3673.27$