text:
Cantera  2.2.1
SimpleTransport Class Reference

Class SimpleTransport implements mixture-averaged transport properties for liquid phases. More...

#include <SimpleTransport.h>

Inheritance diagram for SimpleTransport:
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## Public Member Functions

SimpleTransport (thermo_t *thermo=0, int ndim=1)
Default constructor. More...

SimpleTransport (const SimpleTransport &right)

SimpleTransportoperator= (const SimpleTransport &right)

virtual TransportduplMyselfAsTransport () const
Duplication routine for objects which inherit from Transport. More...

virtual bool initLiquid (LiquidTransportParams &tr)
Initialize the transport object. More...

virtual int model () const
Transport model. More...

virtual doublereal viscosity ()
Returns the mixture viscosity of the solution. More...

virtual void getSpeciesViscosities (doublereal *const visc)
Returns the pure species viscosities. More...

virtual void getBinaryDiffCoeffs (const size_t ld, doublereal *const d)
Returns the binary diffusion coefficients. More...

virtual void getMixDiffCoeffs (doublereal *const d)
Get the Mixture diffusion coefficients. More...

virtual void getThermalDiffCoeffs (doublereal *const dt)
Return the thermal diffusion coefficients. More...

virtual doublereal thermalConductivity ()
Returns the mixture thermal conductivity of the solution. More...

virtual void getMobilities (doublereal *const mobil_e)
Get the Electrical mobilities (m^2/V/s). More...

virtual void getFluidMobilities (doublereal *const mobil_f)
Get the fluid mobilities (s kmol/kg). More...

Specify the value of the gradient of the voltage. More...

Specify the value of the gradient of the temperature. More...

Specify the value of the gradient of the MoleFractions. More...

virtual void getSpeciesVdiff (size_t ndim, const doublereal *grad_T, int ldx, const doublereal *grad_X, int ldf, doublereal *Vdiff)
Get the species diffusive velocities wrt to the averaged velocity, given the gradients in mole fraction and temperature. More...

virtual void getSpeciesVdiffES (size_t ndim, const doublereal *grad_T, int ldx, const doublereal *grad_X, int ldf, const doublereal *grad_Phi, doublereal *Vdiff)
Get the species diffusive velocities wrt to the averaged velocity, given the gradients in mole fraction, temperature and electrostatic potential. More...

virtual void getSpeciesFluxes (size_t ndim, const doublereal *const grad_T, size_t ldx, const doublereal *const grad_X, size_t ldf, doublereal *const fluxes)
Get the species diffusive mass fluxes wrt to the specified solution averaged velocity, given the gradients in mole fraction and temperature. More...

virtual void getSpeciesFluxesExt (size_t ldf, doublereal *fluxes)
Return the species diffusive mass fluxes wrt to the mass averaged velocity,. More...

Public Member Functions inherited from Transport
Transport (thermo_t *thermo=0, size_t ndim=1)
Constructor. More...

Transport (const Transport &right)

Transportoperator= (const Transport &right)

thermo_tthermo ()

void setNDim (const int ndim)
Set the number of dimensions to be expected in flux expressions. More...

size_t nDim () const
Return the number of dimensions in flux expressions. More...

void checkSpeciesIndex (size_t k) const
Check that the specified species index is in range Throws an exception if k is greater than nSpecies() More...

void checkSpeciesArraySize (size_t kk) const
Check that an array size is at least nSpecies() Throws an exception if kk is less than nSpecies(). More...

virtual doublereal getElectricConduct ()
Compute the mixture electrical conductivity (S m-1) at the current conditions of the phase (Siemens m-1) More...

virtual void getElectricCurrent (int ndim, const doublereal *grad_T, int ldx, const doublereal *grad_X, int ldf, const doublereal *grad_V, doublereal *current)
Compute the electric current density in A/m^2. More...

virtual void getSpeciesFluxesES (size_t ndim, const doublereal *grad_T, size_t ldx, const doublereal *grad_X, size_t ldf, const doublereal *grad_Phi, doublereal *fluxes)
Get the species diffusive mass fluxes wrt to the mass averaged velocity, given the gradients in mole fraction, temperature and electrostatic potential. More...

virtual void getMolarFluxes (const doublereal *const state1, const doublereal *const state2, const doublereal delta, doublereal *const cfluxes)
Get the molar fluxes [kmol/m^2/s], given the thermodynamic state at two nearby points. More...

virtual void getMassFluxes (const doublereal *state1, const doublereal *state2, doublereal delta, doublereal *mfluxes)
Get the mass fluxes [kg/m^2/s], given the thermodynamic state at two nearby points. More...

virtual void getMultiDiffCoeffs (const size_t ld, doublereal *const d)
Return the Multicomponent diffusion coefficients. Units: [m^2/s]. More...

virtual void getMixDiffCoeffsMole (doublereal *const d)
Returns a vector of mixture averaged diffusion coefficients. More...

virtual void getMixDiffCoeffsMass (doublereal *const d)
Returns a vector of mixture averaged diffusion coefficients. More...

virtual void setParameters (const int type, const int k, const doublereal *const p)
Set model parameters for derived classes. More...

void setVelocityBasis (VelocityBasis ivb)
Sets the velocity basis. More...

VelocityBasis getVelocityBasis () const
Gets the velocity basis. More...

virtual doublereal bulkViscosity ()
The bulk viscosity in Pa-s. More...

virtual doublereal ionConductivity ()
The ionic conductivity in 1/ohm/m. More...

virtual void getSpeciesIonConductivity (doublereal *const ionCond)
Returns the pure species ionic conductivity. More...

virtual void mobilityRatio (double *mobRat)
Returns the pointer to the mobility ratios of the species in the phase. More...

virtual void getSpeciesMobilityRatio (double **mobRat)
Returns the pure species limit of the mobility ratios. More...

virtual void selfDiffusion (doublereal *const selfDiff)
Returns the self diffusion coefficients of the species in the phase. More...

virtual void getSpeciesSelfDiffusion (double **selfDiff)
Returns the pure species self diffusion in solution of each species. More...

virtual doublereal electricalConductivity ()

virtual void init (thermo_t *thermo, int mode=0, int log_level=0)
Initialize a transport manager. More...

virtual bool initSolid (SolidTransportData &tr)
Called by TransportFactory to set parameters. More...

virtual void setThermo (thermo_t &thermo)
Specifies the ThermoPhase object. More...

## Protected Member Functions

virtual bool update_T ()
Handles the effects of changes in the Temperature, internally within the object. More...

virtual bool update_C ()
Handles the effects of changes in the mixture concentration. More...

void updateViscosity_T ()
Update the temperature-dependent viscosity terms. More...

void updateCond_T ()
Update the temperature-dependent parts of the mixture-averaged thermal conductivity. More...

void updateViscosities_C ()
Update the concentration parts of the viscosities. More...

void updateDiff_T ()
Update the binary diffusion coefficients wrt T. More...

Protected Member Functions inherited from Transport
void finalize ()
Enable the transport object for use. More...

## Private Attributes

int tempDepType_
Temperature dependence type. More...

enum LiquidTranMixingModel compositionDepType_
Composition dependence of the transport properties. More...

Boolean indicating whether to use the hydrodynamic radius formulation. More...

bool doMigration_
Boolean indicating whether electro-migration term should be added. More...

vector_fp m_mw
Local Copy of the molecular weights of the species. More...

std::vector< LTPspecies * > m_coeffVisc_Ns
Pure species viscosities in Arrhenius temperature-dependent form. More...

std::vector< LTPspecies * > m_coeffLambda_Ns
Pure species thermal conductivities in Arrhenius temperature-dependent form. More...

std::vector< LTPspecies * > m_coeffDiff_Ns
Pure species viscosities in Arrhenius temperature-dependent form. More...

Hydrodynamic radius in LTPspecies form. More...

Internal value of the gradient of the mole fraction vector. More...

Internal value of the gradient of the Temperature vector. More...

Internal value of the gradient of the Pressure vector. More...

Internal value of the gradient of the Electric Voltage. More...

vector_fp m_diffSpecies
Vector of Species Diffusivities. More...

vector_fp m_viscSpecies
Species viscosities. More...

vector_fp m_condSpecies
Internal value of the species individual thermal conductivities. More...

int m_iStateMF
State of the mole fraction vector. More...

vector_fp m_molefracs
Local copy of the mole fractions of the species in the phase. More...

vector_fp m_concentrations
Local copy of the concentrations of the species in the phase. More...

doublereal concTot_
Local copy of the total concentration. More...

doublereal meanMolecularWeight_
Mean molecular weight. More...

doublereal dens_
Density. More...

vector_fp m_chargeSpecies
Local copy of the charge of each species. More...

doublereal m_temp
Current Temperature -> locally stored. More...

doublereal m_press
Current value of the pressure. More...

doublereal m_lambda
Saved value of the mixture thermal conductivity. More...

doublereal m_viscmix
Saved value of the mixture viscosity. More...

vector_fp m_spwork
work space More...

vector_fp m_fluxes

bool m_visc_mix_ok
Boolean indicating that the top-level mixture viscosity is current. More...

bool m_visc_temp_ok
Boolean indicating that weight factors wrt viscosity is current. More...

bool m_diff_mix_ok
Boolean indicating that mixture diffusion coeffs are current. More...

bool m_diff_temp_ok
Boolean indicating that binary diffusion coeffs are current. More...

bool m_cond_temp_ok
Flag to indicate that the pure species conductivities are current wrt the temperature. More...

bool m_cond_mix_ok
Boolean indicating that mixture conductivity is current. More...

size_t m_nDim
Number of dimensions. More...

double rhoVc [3]
Temporary variable that stores the rho Vc value. More...

Protected Attributes inherited from Transport
thermo_tm_thermo
pointer to the object representing the phase More...

true if finalize has been called More...

size_t m_nsp
Number of species. More...

size_t m_nDim
Number of dimensions used in flux expressions. More...

int m_velocityBasis
Velocity basis from which diffusion velocities are computed. More...

## Detailed Description

Class SimpleTransport implements mixture-averaged transport properties for liquid phases.

The model is based on that described by Newman, Electrochemical Systems

The velocity of species i may be described by the following equation p. 297 (12.1)

$c_i \nabla \mu_i = R T \sum_j \frac{c_i c_j}{c_T D_{ij}} (\mathbf{v}_j - \mathbf{v}_i)$

This as written is degenerate by 1 dof.

To fix this we must add in the definition of the mass averaged velocity of the solution. We will call the simple bold-faced $$\mathbf{v}$$ symbol the mass-averaged velocity. Then, the relation between $$\mathbf{v}$$ and the individual species velocities is $$\mathbf{v}_i$$

$\rho_i \mathbf{v}_i = \rho_i \mathbf{v} + \mathbf{j}_i$

where $$\mathbf{j}_i$$ are the diffusional fluxes of species i with respect to the mass averaged velocity and

$\sum_i \mathbf{j}_i = 0$

and

$\sum_i \rho_i \mathbf{v}_i = \rho \mathbf{v}$

Using these definitions, we can write

$\mathbf{v}_i = \mathbf{v} + \frac{\mathbf{j}_i}{\rho_i}$

$c_i \nabla \mu_i = R T \sum_j \frac{c_i c_j}{c_T D_{ij}} (\frac{\mathbf{j}_j}{\rho_j} - \frac{\mathbf{j}_i}{\rho_i}) = R T \sum_j \frac{1}{D_{ij}} (\frac{x_i \mathbf{j}_j}{M_j} - \frac{x_j \mathbf{j}_i}{M_i})$

The equations that we actually solve are

$c_i \nabla \mu_i = = R T \sum_j \frac{1}{D_{ij}} (\frac{x_i \mathbf{j}_j}{M_j} - \frac{x_j \mathbf{j}_i}{M_i})$

and we replace the 0th equation with the following:

$\sum_i \mathbf{j}_i = 0$

When there are charged species, we replace the RHS with the gradient of the electrochemical potential to obtain the modified equation

$c_i \nabla \mu_i + c_i F z_i \nabla \Phi = R T \sum_j \frac{1}{D_{ij}} (\frac{x_i \mathbf{j}_j}{M_j} - \frac{x_j \mathbf{j}_i}{M_i})$

With this formulation we may solve for the diffusion velocities, without having to worry about what the mass averaged velocity is.

## Viscosity Calculation

The viscosity calculation may be broken down into two parts. In the first part, the viscosity of the pure species are calculated In the second part, a mixing rule is applied. There are two mixing rules. Solvent-only and mixture-averaged.

For the solvent-only mixing rule, we use the pure species viscosity calculated for the solvent as the viscosity of the entire mixture. For the mixture averaged rule we do a mole fraction based average of the pure species viscosities:

Solvent-only:

$\mu = \mu_0$

Mixture-average:

$\mu = \sum_k {\mu_k X_k}$

## Calculate of the Binary Diffusion Coefficients

The binary diffusion coefficients are obtained from the pure species diffusion coefficients using an additive process

$D_{i,j} = \frac{1}{2} \left( D^0_i(T) + D^0_j(T) \right)$

## Electrical Mobilities

The mobility $$\mu^e_k$$ is calculated from the diffusion coefficient using the Einstein relation.

$\mu^e_k = \frac{F D_k}{R T}$

The diffusion coefficients, $$D_k$$ , is calculated from a call to the mixture diffusion coefficient routine.

## Species Diffusive Fluxes

The diffusive mass flux of species k is computed from the following formula

Usually the specified solution average velocity is the mass averaged velocity. This is changed in some subclasses, however.

$j_k = - c^T M_k D_k \nabla X_k - \rho Y_k V_c$

where V_c is the correction velocity

$\rho V_c = - \sum_j {c^T M_j D_j \nabla X_j}$

In the above equation, $$D_k$$ is the mixture diffusivity for species k calculated for the current conditions, which may depend on T, P, and X_k. $$C^T$$ is the total concentration of the phase.

When this is electrical migration, the formulas above are enhanced to

$j_k = - C^T M_k D_k \nabla X_k + F C^T M_k \frac{D_k}{ R T } X_k z_k \nabla V - \rho Y_k V_c$

where V_c is the correction velocity

$\rho V_c = - \sum_j {c^T M_j D_j \nabla X_j} + \sum_j F C^T M_j \frac{D_j}{ R T } X_j z_j \nabla V$

## Species Diffusional Velocities

Species diffusional velocities are calculated from the species diffusional fluxes, within this object, using the following formula for the diffusional velocity of the kth species, $$V_k^d$$

$j_k = \rho Y_k V_k^d$

TODO

This object has to be made compatible with different types of reference velocities. Right now, elements of the formulas are only compatible with the mass-averaged velocity.

Definition at line 180 of file SimpleTransport.h.

## Constructor & Destructor Documentation

 SimpleTransport ( thermo_t * thermo = 0, int ndim = 1 )

Default constructor.

This requires call to initLiquid(LiquidTransportParams& tr) after filling LiquidTransportParams to complete instantiation. The filling of LiquidTransportParams is currently carried out in the TransportFactory class, but might be moved at some point.

Parameters
 thermo ThermoPhase object holding species information. ndim Number of spatial dimensions.

Definition at line 12 of file SimpleTransport.cpp.

Referenced by SimpleTransport::duplMyselfAsTransport().

## Member Function Documentation

 Transport * duplMyselfAsTransport ( ) const
virtual

Duplication routine for objects which inherit from Transport.

This virtual routine can be used to duplicate objects derived from Transport even if the application only has a pointer to Transport to work with.

These routines are basically wrappers around the derived copy constructor.

Reimplemented from Transport.

Definition at line 135 of file SimpleTransport.cpp.

References SimpleTransport::SimpleTransport().

 bool initLiquid ( LiquidTransportParams & tr )
virtual

Initialize the transport object.

Here we change all of the internal dimensions to be sufficient. We get the object ready to do property evaluations.

Parameters
 tr Transport parameters for all of the species in the phase.

Reimplemented from Transport.

Definition at line 156 of file SimpleTransport.cpp.

 virtual int model ( ) const
inlinevirtual

Transport model.

The transport model is the set of equations used to compute the transport properties. This method returns an integer flag that identifies the transport model implemented. The base class returns 0.

Reimplemented from Transport.

Definition at line 209 of file SimpleTransport.h.

 doublereal viscosity ( )
virtual

Returns the mixture viscosity of the solution.

The viscosity is computed using the general mixture rules specified in the variable compositionDepType_.

Solvent-only:

$\mu = \mu_0$

Mixture-average:

$\mu = \sum_k {\mu_k X_k}$

Here $$\mu_k$$ is the viscosity of pure species k.

units are Pa s or kg/m/s

updateViscosity_T();

Reimplemented from Transport.

Definition at line 268 of file SimpleTransport.cpp.

Referenced by SimpleTransport::updateDiff_T().

 void getSpeciesViscosities ( doublereal *const visc )
virtual

Returns the pure species viscosities.

The pure species viscosities are to be given in an Arrhenius form in accordance with activated-jump-process dominated transport.

units are Pa s or kg/m/s

Parameters
 visc Return the species viscosities as a vector of length m_nsp

Reimplemented from Transport.

Definition at line 297 of file SimpleTransport.cpp.

 void getBinaryDiffCoeffs ( const size_t ld, doublereal *const d )
virtual

Returns the binary diffusion coefficients.

Parameters
 ld d

Reimplemented from Transport.

Definition at line 306 of file SimpleTransport.cpp.

 void getMixDiffCoeffs ( doublereal *const d )
virtual

Get the Mixture diffusion coefficients.

Parameters
 d vector of mixture diffusion coefficients units = m2 s-1. length = number of species

Reimplemented from Transport.

Definition at line 538 of file SimpleTransport.cpp.

 void getThermalDiffCoeffs ( doublereal *const dt )
virtual

Return the thermal diffusion coefficients.

These are all zero for this simple implementation

Parameters
 dt thermal diffusion coefficients

Reimplemented from Transport.

Definition at line 393 of file SimpleTransport.cpp.

References Transport::m_nsp.

 doublereal thermalConductivity ( )
virtual

Returns the mixture thermal conductivity of the solution.

The thermal is computed using the general mixture rules specified in the variable compositionDepType_.

Controlling update boolean = m_condmix_ok

Units are in W/m/K or equivalently kg m / s3 / K

Solvent-only:

$\lambda = \lambda_0$

Mixture-average:

$\lambda = \sum_k {\lambda_k X_k}$

Here $$\lambda_k$$ is the thermal conductivity of pure species k.

updateCond_T();

Reimplemented from Transport.

Definition at line 369 of file SimpleTransport.cpp.

 void getMobilities ( doublereal *const mobil_e )
virtual

Get the Electrical mobilities (m^2/V/s).

This function returns the mobilities. In some formulations this is equal to the normal mobility multiplied by Faraday's constant.

Frequently, but not always, the mobility is calculated from the diffusion coefficient using the Einstein relation

$\mu^e_k = \frac{F D_k}{R T}$

Parameters
 mobil_e Returns the mobilities of the species in array mobil_e. The array must be dimensioned at least as large as the number of species.

Reimplemented from Transport.

Definition at line 325 of file SimpleTransport.cpp.

 void getFluidMobilities ( doublereal *const mobil_f )
virtual

Get the fluid mobilities (s kmol/kg).

This function returns the fluid mobilities. Usually, you have to multiply Faraday's constant into the resulting expression to general a species flux expression.

Frequently, but not always, the mobility is calculated from the diffusion coefficient using the Einstein relation

$\mu^f_k = \frac{D_k}{R T}$

Parameters
 mobil_f Returns the mobilities of the species in array mobil. The array must be dimensioned at least as large as the number of species.

Reimplemented from Transport.

Definition at line 334 of file SimpleTransport.cpp.

virtual

Specify the value of the gradient of the voltage.

Parameters

Definition at line 343 of file SimpleTransport.cpp.

Referenced by SimpleTransport::getSpeciesVdiffES().

virtual

Specify the value of the gradient of the temperature.

Parameters

Definition at line 354 of file SimpleTransport.cpp.

virtual

Specify the value of the gradient of the MoleFractions.

Parameters

Definition at line 361 of file SimpleTransport.cpp.

 void getSpeciesVdiff ( size_t ndim, const doublereal * grad_T, int ldx, const doublereal * grad_X, int ldf, doublereal * Vdiff )
virtual

Get the species diffusive velocities wrt to the averaged velocity, given the gradients in mole fraction and temperature.

The average velocity can be computed on a mole-weighted or mass-weighted basis, or the diffusion velocities may be specified as relative to a specific species (i.e. a solvent) all according to the velocityBasis input parameter.

Units for the returned velocities are m s-1.

Parameters
 ndim Number of dimensions in the flux expressions grad_T Gradient of the temperature (length = ndim) ldx Leading dimension of the grad_X array (usually equal to m_nsp but not always) grad_X Gradients of the mole fraction Flat vector with the m_nsp in the inner loop. length = ldx * ndim ldf Leading dimension of the fluxes array (usually equal to m_nsp but not always) Vdiff Output of the diffusive velocities. Flat vector with the m_nsp in the inner loop. length = ldx * ndim

Reimplemented from Transport.

Definition at line 400 of file SimpleTransport.cpp.

 void getSpeciesVdiffES ( size_t ndim, const doublereal * grad_T, int ldx, const doublereal * grad_X, int ldf, const doublereal * grad_Phi, doublereal * Vdiff )
virtual

Get the species diffusive velocities wrt to the averaged velocity, given the gradients in mole fraction, temperature and electrostatic potential.

The average velocity can be computed on a mole-weighted or mass-weighted basis, or the diffusion velocities may be specified as relative to a specific species (i.e. a solvent) all according to the velocityBasis input parameter.

Units for the returned velocities are m s-1.

Parameters
 ndim Number of dimensions in the flux expressions grad_T Gradient of the temperature (length = ndim) ldx Leading dimension of the grad_X array (usually equal to m_nsp but not always) grad_X Gradients of the mole fraction Flat vector with the m_nsp in the inner loop. length = ldx * ndim ldf Leading dimension of the fluxes array (usually equal to m_nsp but not always) grad_Phi Gradients of the electrostatic potential (length = ndim) Vdiff Output of the species diffusion velocities Flat vector with the m_nsp in the inner loop. length = ldx * ndim

Reimplemented from Transport.

Definition at line 425 of file SimpleTransport.cpp.

 void getSpeciesFluxes ( size_t ndim, const doublereal *const grad_T, size_t ldx, const doublereal *const grad_X, size_t ldf, doublereal *const fluxes )
virtual

Get the species diffusive mass fluxes wrt to the specified solution averaged velocity, given the gradients in mole fraction and temperature.

units = kg/m2/s

The diffusive mass flux of species k is computed from the following formula

Usually the specified solution average velocity is the mass averaged velocity. This is changed in some subclasses, however.

$j_k = - \rho M_k D_k \nabla X_k - Y_k V_c$

where V_c is the correction velocity

$V_c = - \sum_j {\rho M_j D_j \nabla X_j}$

Parameters
 ndim The number of spatial dimensions (1, 2, or 3). grad_T The temperature gradient (ignored in this model). ldx Leading dimension of the grad_X array. grad_X Gradient of the mole fractions(length nsp * num dimensions); ldf Leading dimension of the fluxes array. fluxes Output fluxes of species.

Reimplemented from Transport.

Definition at line 449 of file SimpleTransport.cpp.

 void getSpeciesFluxesExt ( size_t ldf, doublereal * fluxes )
virtual

Return the species diffusive mass fluxes wrt to the mass averaged velocity,.

units = kg/m2/s

Internally, gradients in the in mole fraction, temperature and electrostatic potential contribute to the diffusive flux

The diffusive mass flux of species k is computed from the following formula

$j_k = - \rho M_k D_k \nabla X_k - Y_k V_c$

where V_c is the correction velocity

$V_c = - \sum_j {\rho M_j D_j \nabla X_j}$

Parameters
 ldf stride of the fluxes array. Must be equal to or greater than the number of species. fluxes Vector of calculated fluxes

Definition at line 458 of file SimpleTransport.cpp.

 bool update_T ( void )
protectedvirtual

Handles the effects of changes in the Temperature, internally within the object.

This is called whenever a transport property is requested. The first task is to check whether the temperature has changed since the last call to update_T(). If it hasn't then an immediate return is carried out.

Returns
Returns true if the temperature has changed, and false otherwise

Definition at line 635 of file SimpleTransport.cpp.

 bool update_C ( )
protectedvirtual

Handles the effects of changes in the mixture concentration.

This is called for every interface call to check whether the concentrations have changed. Concentrations change whenever the pressure or the mole fraction has changed. If it has changed, the recalculations should be done.

Note this should be a lightweight function since it's part of all of the interfaces.

Definition at line 551 of file SimpleTransport.cpp.

 void updateViscosity_T ( )
protected

Update the temperature-dependent viscosity terms.

Updates the array of pure species viscosities, and the weighting functions in the viscosity mixture rule.

The flag m_visc_temp_ok is set to true.

Definition at line 622 of file SimpleTransport.cpp.

Referenced by SimpleTransport::getSpeciesViscosities(), and SimpleTransport::viscosity().

 void updateCond_T ( )
protected

Update the temperature-dependent parts of the mixture-averaged thermal conductivity.

Definition at line 587 of file SimpleTransport.cpp.

Referenced by SimpleTransport::thermalConductivity().

 void updateViscosities_C ( )
protected

Update the concentration parts of the viscosities.

Internal routine is run whenever the update_boolean is false. This routine will calculate internal values for the species viscosities.

Definition at line 618 of file SimpleTransport.cpp.

 void updateDiff_T ( )
protected

Update the binary diffusion coefficients wrt T.

These are evaluated from the polynomial fits at unit pressure (1 Pa).

Definition at line 600 of file SimpleTransport.cpp.

Referenced by SimpleTransport::getBinaryDiffCoeffs(), and SimpleTransport::getMixDiffCoeffs().

## Member Data Documentation

 int tempDepType_
private

Temperature dependence type.

The following coefficients are allowed to have simple temperature dependencies: mixture viscosity mixture thermal conductivity diffusitivy

Types of temperature dependencies: 0 - Independent of temperature (only one implemented so far) 1 - extended arrhenius form 2 - polynomial in temperature form

Definition at line 500 of file SimpleTransport.h.

 enum LiquidTranMixingModel compositionDepType_
private

Composition dependence of the transport properties.

The following coefficients are allowed to have simple composition dependencies

mixture viscosity
mixture thermal conductivity

Permissible types of composition dependencies 0 - Solvent values (i.e., species 0) contributes only 1 - linear combination of mole fractions;

Definition at line 513 of file SimpleTransport.h.

private

Boolean indicating whether to use the hydrodynamic radius formulation.

If true, then the diffusion coefficient is calculated from the hydrodynamic radius.

Definition at line 520 of file SimpleTransport.h.

Referenced by SimpleTransport::initLiquid(), and SimpleTransport::updateDiff_T().

 bool doMigration_
private

Boolean indicating whether electro-migration term should be added.

Definition at line 523 of file SimpleTransport.h.

 vector_fp m_mw
private

Local Copy of the molecular weights of the species.

Length is Equal to the number of species in the mechanism.

Definition at line 529 of file SimpleTransport.h.

Referenced by SimpleTransport::initLiquid().

 std::vector m_coeffVisc_Ns
private

Pure species viscosities in Arrhenius temperature-dependent form.

Definition at line 532 of file SimpleTransport.h.

Referenced by SimpleTransport::initLiquid(), and SimpleTransport::updateViscosity_T().

 std::vector m_coeffLambda_Ns
private

Pure species thermal conductivities in Arrhenius temperature-dependent form.

Definition at line 535 of file SimpleTransport.h.

Referenced by SimpleTransport::initLiquid(), and SimpleTransport::updateCond_T().

 std::vector m_coeffDiff_Ns
private

Pure species viscosities in Arrhenius temperature-dependent form.

Definition at line 538 of file SimpleTransport.h.

Referenced by SimpleTransport::initLiquid(), and SimpleTransport::updateDiff_T().

private

Definition at line 541 of file SimpleTransport.h.

Referenced by SimpleTransport::initLiquid(), and SimpleTransport::updateDiff_T().

private

Internal value of the gradient of the mole fraction vector.

Note, this is the only gradient value that can and perhaps should reflect the true state of the mole fractions in the application solution vector. In other words no cropping or massaging of the values to make sure they are above zero should occur. - developing ....

m_nsp is the number of species in the fluid k is the species index n is the dimensional index (x, y, or z). It has a length equal to m_nDim

Definition at line 558 of file SimpleTransport.h.

private

Internal value of the gradient of the Temperature vector.

Generally, if a transport property needs this in its evaluation it will look to this place to get it.

No internal property is precalculated based on gradients. Gradients are assumed to be freshly updated before every property call.

Definition at line 568 of file SimpleTransport.h.

private

Internal value of the gradient of the Pressure vector.

Generally, if a transport property needs this in its evaluation it will look to this place to get it.

No internal property is precalculated based on gradients. Gradients are assumed to be freshly updated before every property call.

Definition at line 578 of file SimpleTransport.h.

Referenced by SimpleTransport::initLiquid().

private

Internal value of the gradient of the Electric Voltage.

Generally, if a transport property needs this in its evaluation it will look to this place to get it.

No internal property is precalculated based on gradients. Gradients are assumed to be freshly updated before every property call.

Definition at line 588 of file SimpleTransport.h.

 vector_fp m_diffSpecies
private

Vector of Species Diffusivities.

Depends on the temperature. We have set the pressure dependence to zero for this liquid phase constituitve model

units m2/s

Definition at line 599 of file SimpleTransport.h.

 vector_fp m_viscSpecies
private

Species viscosities.

Viscosity of the species Length = number of species

Depends on the temperature. We have set the pressure dependence to zero for this model

controlling update boolean -> m_visc_temp_ok

Definition at line 611 of file SimpleTransport.h.

 vector_fp m_condSpecies
private

Internal value of the species individual thermal conductivities.

Then a mixture rule is applied to get the solution conductivities

Depends on the temperature and perhaps pressure, but not the species concentrations

controlling update boolean -> m_cond_temp_ok

Definition at line 622 of file SimpleTransport.h.

 int m_iStateMF
private

State of the mole fraction vector.

Definition at line 625 of file SimpleTransport.h.

Referenced by SimpleTransport::update_C().

 vector_fp m_molefracs
private

Local copy of the mole fractions of the species in the phase.

The mole fractions here are assumed to be bounded by 0.0 and 1.0 and they are assumed to add up to one exactly. This mole fraction vector comes from the ThermoPhase object. Derivative quantities from this are referred to as bounded.

Update info? length = m_nsp

Definition at line 637 of file SimpleTransport.h.

 vector_fp m_concentrations
private

Local copy of the concentrations of the species in the phase.

The concentrations are consistent with the m_molefracs vector which is bounded and sums to one.

Update info? length = m_nsp

Definition at line 647 of file SimpleTransport.h.

Referenced by SimpleTransport::initLiquid(), and SimpleTransport::update_C().

 doublereal concTot_
private

Local copy of the total concentration.

This is consistent with the m_concentrations[] and m_molefracs[] vector.

Definition at line 654 of file SimpleTransport.h.

Referenced by SimpleTransport::update_C().

 doublereal meanMolecularWeight_
private

Mean molecular weight.

Definition at line 657 of file SimpleTransport.h.

Referenced by SimpleTransport::update_C().

 doublereal dens_
private

Density.

Definition at line 660 of file SimpleTransport.h.

Referenced by SimpleTransport::update_C().

 vector_fp m_chargeSpecies
private

Local copy of the charge of each species.

Contains the charge of each species (length m_nsp)

Definition at line 666 of file SimpleTransport.h.

Referenced by SimpleTransport::getSpeciesFluxesExt(), and SimpleTransport::initLiquid().

 doublereal m_temp
private

Current Temperature -> locally stored.

This is used to test whether new temperature computations should be performed.

Definition at line 673 of file SimpleTransport.h.

 doublereal m_press
private

Current value of the pressure.

Definition at line 676 of file SimpleTransport.h.

Referenced by SimpleTransport::update_C().

 doublereal m_lambda
private

Saved value of the mixture thermal conductivity.

Definition at line 679 of file SimpleTransport.h.

Referenced by SimpleTransport::thermalConductivity().

 doublereal m_viscmix
private

Saved value of the mixture viscosity.

Definition at line 682 of file SimpleTransport.h.

Referenced by SimpleTransport::viscosity().

 vector_fp m_spwork
private

work space

Length is equal to m_nsp

Definition at line 688 of file SimpleTransport.h.

 bool m_visc_mix_ok
private

Boolean indicating that the top-level mixture viscosity is current.

This is turned false for every change in T, P, or C.

Definition at line 697 of file SimpleTransport.h.

 bool m_visc_temp_ok
private

Boolean indicating that weight factors wrt viscosity is current.

Definition at line 700 of file SimpleTransport.h.

 bool m_diff_mix_ok
private

Boolean indicating that mixture diffusion coeffs are current.

Definition at line 703 of file SimpleTransport.h.

 bool m_diff_temp_ok
private

Boolean indicating that binary diffusion coeffs are current.

Definition at line 706 of file SimpleTransport.h.

 bool m_cond_temp_ok
private

Flag to indicate that the pure species conductivities are current wrt the temperature.

Definition at line 710 of file SimpleTransport.h.

 bool m_cond_mix_ok
private

Boolean indicating that mixture conductivity is current.

Definition at line 713 of file SimpleTransport.h.

 size_t m_nDim
private

Number of dimensions.

Either 1, 2, or 3

Definition at line 719 of file SimpleTransport.h.

 double rhoVc[3]
private

Temporary variable that stores the rho Vc value.

Definition at line 722 of file SimpleTransport.h.

Referenced by SimpleTransport::getSpeciesFluxesExt().

The documentation for this class was generated from the following files:


text:
Thread: The dual nature of Nabla ? View Single Post
 Sci Advisor HW Helper Thanks P: 26,167 hi abrowaqas! ∇ behaves just like an ordinary vector … the vector itself is ∇f (where f is a scalar)and you can dot-product it, or cross-product it, with a vector A … ∇.A (divA) or ∇xA (curlA)

format: plain


text:
# Knapsack dynamic programming complexity issue for $W=1$, is it $O(n)$?

The 0-1 knapsack problem is given by

\begin{align}&\text{maximize } \sum_{i=1}^n v_ix_i,\tag{P1}\\& \text{subject to } \sum_{i=1}^n w_i x_i \leq W,\\&\text{and } x_i \in \{0,1\}.\end{align}

Running the dynamic programming algorithm for this problem would give an optimal solution in $$O(nW)$$.

If I define $$p_i=w_i/W$$, I get the following knapsack problem:

\begin{align}&\text{maximize } \sum_{i=1}^n v_ix_i,\tag{P2}\\& \text{subject to } \sum_{i=1}^n p_i x_i \leq 1,\\&\text{and } x_i \in \{0,1\}.\end{align}

Running the dynamic programming algorithm for this problem would give an optimal solution in $$O(n)$$. This is not true, isn't it?

The running time is $$O(nW)$$ if all weights are integers. This is because the dynamic programming table is indexed by possible weights of subsets. If all weights are integers, then since we are only interested in subsets whose sum of weights is at most $$W$$, there are only $$W+1$$ weights of subsets that could possibly interest us. This is where the $$W$$ factor in the running time is coming from.
Dividing all weights by $$W$$ accomplishes exactly nothing. You haven't changed the problem, so the required time for solving it is remains exactly the same.


text:
# [Srednicki] Charge Conjugation of Dirac Spinor

1. Jan 15, 2014

### cedricyu803

1. The problem statement, all variables and given/known data
I am reading Srednicki's QFT up to CPT symmetries of Spinors
In eq. 40.42 of
http://web.physics.ucsb.edu/~mark/ms-qft-DRAFT.pdf
I attempted to get the 2nd equation:
$$C^{-1}\bar{\Psi}C=\Psi^{T}C$$

from the first one:
$$C^{-1}\Psi C=\bar{\Psi}^{T}C$$

2. Relevant equations

$$\bar{\Psi}=\Psi^\dagger \beta$$
where numerically $$\beta=\gamma^0$$

$$C^\dagger=C^{-1}=C^T=-C$$

3. The attempt at a solution

h.c. of the first equation:
$$C^{-1}\Psi^\dagger C=(C^{-1}\Psi C)^\dagger=(C\bar{\Psi}^T)^\dagger =(C(\Psi^\dagger \beta)^T)^\dagger=(C\beta\Psi^\ast)^\dagger=\Psi^T\beta C^\dagger=\Psi^T C\beta$$

So

$$C^{-1}\bar{\Psi}C=C^{-1}\Psi^\dagger \beta C=-C^{-1}\Psi^\dagger C \beta=-(\Psi^T C\beta) \beta=-\Psi^{T}C$$

I got an extra minus sign.

However, if I start from takingg transpose of the first equation I got the equation correctly.

What have I done wrong?

Also, for eq. 40.43
A is some general combination of gamma matrices.
Should it not be
$$C^{-1}\bar{\Psi}A\Psi C=\Psi^TA\bar{\Psi}^T$$
?

Why are there C's wedging A??

Thanks a lot

2. Jan 16, 2014

### strangerep

You seem to be using the same "C" on both sides on the equations. But they should be, e.g.,
$$C^{-1}\Psi C ~=~ \bar{\Psi}^{T} {\mathcal C} ~.$$(The "C" on the rhs is in caligraphic font.)

${\mathcal C}$ is a matrix. $C$ is a unitary representation of the charge-conjugation operator on this Hilbert space.

Maybe a review of ch23 would help to clarify this distinction?

3. Jan 16, 2014

### cedricyu803

Oh right. I wasn't aware of the difference between the operator and the matrix.
So the charge conjugation of $$\beta$$ does nothing to it. That's why I got a minus sign.

Thanks very much!

4. Jan 20, 2014

### russelljbarry15

There is a book out by partick labelle. It is a DeMystified book. But this one is really good. Called Supersymmetry. Chapters 2-4 cover Weyl, Majorana and Dirac Spinors. The book does it in a way that it writes out all the components of the matrices even the C one so you can really see what it going on. All the problems have solutions in the back. If you are that far in Srednicki's book you could probably read the three chapters and understand them in about a day. The book is cheap and those sections make it worth the price, since Srednicki will use it more through out the book. I found this book after Srednicki and I wish I had found it first. Then later if you want to do Supersymmetry go ahead. But it cover those topics like no other book I have ever seen, and those three chapters only focus on that subject. I think the level you are at you might want a really good understanding of those topics. Good Luck.

5. Jan 22, 2014

### cedricyu803

Thanks very much for your recommendation.

I will look for this book in a public library.

I am lucky to have been recommended Srednicki's book by my prof as well, after I told him I was having a hard time Peskin & Schroeder. Hopefully I can finish the whole book by the end of this semester.

Cheers


text:
# A strategy in a possibly-infinite game determined by coin toss

A pair of positive integers $x_0$ and $y_0$ are selected and further pairs are generated according to the following rule:

After the values of $x_n$ and $y_n$ have been found, a coin is tossed. If it lands heads, then $x_{n+1} = x_n −1$ and $y_{n+1} = x_n$. Otherwise, if it lands tails, then $x_{n+1} = y_n − 2$ and $y_{n+1} = x_n + 1$.

The process ends when a non-positive value of $x$ or $y$ is generated. Is there a choice of initial values for $x_0$ and $y_0$ and sequence of coin toss outcomes for which the sequence of $x$’s and $y$’s does not terminate?

-
By the "Q2" in the original post, and the lack of motivation I have added the [homework] tag, if this is not actually a homework question feel free to remove it. –  Asaf Karagila Jan 28 '12 at 21:05
No (probability) here. –  Did Jan 30 '12 at 18:35

Call $h:(x,y)\mapsto(x-1,x)$ and $t:(x,y)\mapsto(y-2,x+1)$. For every $x$, let $u_x=(x,x+1)$. Call $U$ the set of points $u_x$ and $x$ the rank of the point $u_x$.
Then $t\circ t:(x,y)\mapsto(x-1,y-1)$ hence if one uses $t$ only, the process ends. As soon as one uses $h$, the result is $h(x,y)=u_{x-1}$ hence one lands in $U$.
Now, $h(u_x)=u_{x-1}$, $t\circ t(u_x)=u_{x-1}$ and $h\circ t(u_x)=u_{x-2}$, hence, from this moment on, one stays in $U$ every one or two steps. The sequence of the ranks is decreasing hence the ranks cannot stay positive forever.
It seems that no path starting from $(x,y)$ with length at least twice the maximum of $x$ and $y$, lies entirely in the positive orthant.


text:
42.30 Gysin homomorphisms and rational equivalence

In this section we use the key formula to show the Gysin homomorphism factor through rational equivalence. We also prove an important commutativity property.

Lemma 42.30.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $X$ be integral and $n = \dim _\delta (X)$. Let $i : D \to X$ be an effective Cartier divisor. Let $\mathcal{N}$ be an invertible $\mathcal{O}_ X$-module and let $t$ be a nonzero meromorphic section of $\mathcal{N}$. Then $i^*\text{div}_\mathcal {N}(t) = c_1(\mathcal{N}|_ D) \cap [D]_{n - 1}$ in $\mathop{\mathrm{CH}}\nolimits _{n - 2}(D)$.

Proof. Write $\text{div}_\mathcal {N}(t) = \sum \text{ord}_{Z_ i, \mathcal{N}}(t)[Z_ i]$ for some integral closed subschemes $Z_ i \subset X$ of $\delta$-dimension $n - 1$. We may assume that the family $\{ Z_ i\}$ is locally finite, that $t \in \Gamma (U, \mathcal{N}|_ U)$ is a generator where $U = X \setminus \bigcup Z_ i$, and that every irreducible component of $D$ is one of the $Z_ i$, see Divisors, Lemmas 31.26.1, 31.26.4, and 31.27.2.

Set $\mathcal{L} = \mathcal{O}_ X(D)$. Denote $s \in \Gamma (X, \mathcal{O}_ X(D)) = \Gamma (X, \mathcal{L})$ the canonical section. We will apply the discussion of Section 42.27 to our current situation. For each $i$ let $\xi _ i \in Z_ i$ be its generic point. Let $B_ i = \mathcal{O}_{X, \xi _ i}$. For each $i$ we pick generators $s_ i \in \mathcal{L}_{\xi _ i}$ and $t_ i \in \mathcal{N}_{\xi _ i}$ over $B_ i$ but we insist that we pick $s_ i = s$ if $Z_ i \not\subset D$. Write $s = f_ i s_ i$ and $t = g_ i t_ i$ with $f_ i, g_ i \in B_ i$. Then $\text{ord}_{Z_ i, \mathcal{N}}(t) = \text{ord}_{B_ i}(g_ i)$. On the other hand, we have $f_ i \in B_ i$ and

$[D]_{n - 1} = \sum \text{ord}_{B_ i}(f_ i)[Z_ i]$

because of our choices of $s_ i$. We claim that

$i^*\text{div}_\mathcal {N}(t) = \sum \text{ord}_{B_ i}(g_ i) \text{div}_{\mathcal{L}|_{Z_ i}}(s_ i|_{Z_ i})$

as cycles. More precisely, the right hand side is a cycle representing the left hand side. Namely, this is clear by our formula for $\text{div}_\mathcal {N}(t)$ and the fact that $\text{div}_{\mathcal{L}|_{Z_ i}}(s_ i|_{Z_ i}) = [Z(s_ i|_{Z_ i})]_{n - 2} = [Z_ i \cap D]_{n - 2}$ when $Z_ i \not\subset D$ because in that case $s_ i|_{Z_ i} = s|_{Z_ i}$ is a regular section, see Lemma 42.24.2. Similarly,

$c_1(\mathcal{N}) \cap [D]_{n - 1} = \sum \text{ord}_{B_ i}(f_ i) \text{div}_{\mathcal{N}|_{Z_ i}}(t_ i|_{Z_ i})$

The key formula (Lemma 42.27.1) gives the equality

$\sum \left( \text{ord}_{B_ i}(f_ i) \text{div}_{\mathcal{N}|_{Z_ i}}(t_ i|_{Z_ i}) - \text{ord}_{B_ i}(g_ i) \text{div}_{\mathcal{L}|_{Z_ i}}(s_ i|_{Z_ i}) \right) = \sum \text{div}_{Z_ i}(\partial _{B_ i}(f_ i, g_ i))$

of cycles. If $Z_ i \not\subset D$, then $f_ i = 1$ and hence $\text{div}_{Z_ i}(\partial _{B_ i}(f_ i, g_ i)) = 0$. Thus we get a rational equivalence between our specific cycles representing $i^*\text{div}_\mathcal {N}(t)$ and $c_1(\mathcal{N}) \cap [D]_{n - 1}$ on $D$. This finishes the proof. $\square$

Lemma 42.30.2. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $(\mathcal{L}, s, i : D \to X)$ be as in Definition 42.29.1. The Gysin homomorphism factors through rational equivalence to give a map $i^* : \mathop{\mathrm{CH}}\nolimits _{k + 1}(X) \to \mathop{\mathrm{CH}}\nolimits _ k(D)$.

Proof. Let $\alpha \in Z_{k + 1}(X)$ and assume that $\alpha \sim _{rat} 0$. This means there exists a locally finite collection of integral closed subschemes $W_ j \subset X$ of $\delta$-dimension $k + 2$ and $f_ j \in R(W_ j)^*$ such that $\alpha = \sum i_{j, *}\text{div}_{W_ j}(f_ j)$. Set $X' = \coprod W_ i$ and consider the diagram

$\xymatrix{ D' \ar[d]_ q \ar[r]_{i'} & X' \ar[d]^ p \\ D \ar[r]^ i & X }$

of Remark 42.29.7. Since $X' \to X$ is proper we see that $i^*p_* = q_*(i')^*$ by Lemma 42.29.8. As we know that $q_*$ factors through rational equivalence (Lemma 42.20.3), it suffices to prove the result for $\alpha ' = \sum \text{div}_{W_ j}(f_ j)$ on $X'$. Clearly this reduces us to the case where $X$ is integral and $\alpha = \text{div}(f)$ for some $f \in R(X)^*$.

Assume $X$ is integral and $\alpha = \text{div}(f)$ for some $f \in R(X)^*$. If $X = D$, then we see that $i^*\alpha$ is equal to $c_1(\mathcal{L}) \cap \alpha$. This is rationally equivalent to zero by Lemma 42.28.2. If $D \not= X$, then we see that $i^*\text{div}_ X(f)$ is equal to $c_1(\mathcal{O}_ D) \cap [D]_{n - 1}$ in $\mathop{\mathrm{CH}}\nolimits _{n - 2}(D)$ by Lemma 42.30.1. Of course capping with $c_1(\mathcal{O}_ D)$ is the zero map (Lemma 42.25.2). $\square$

Lemma 42.30.3. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $(\mathcal{L}, s, i : D \to X)$ be as in Definition 42.29.1. Then $i^*i_* : \mathop{\mathrm{CH}}\nolimits _ k(D) \to \mathop{\mathrm{CH}}\nolimits _{k - 1}(D)$ sends $\alpha$ to $c_1(\mathcal{L}|_ D) \cap \alpha$.

Proof. This is immediate from the definition of $i_*$ on cycles and the definition of $i^*$ given in Definition 42.29.1. $\square$

Lemma 42.30.4. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $(\mathcal{L}, s, i : D \to X)$ be a triple as in Definition 42.29.1. Let $\mathcal{N}$ be an invertible $\mathcal{O}_ X$-module. Then $i^*(c_1(\mathcal{N}) \cap \alpha ) = c_1(i^*\mathcal{N}) \cap i^*\alpha$ in $\mathop{\mathrm{CH}}\nolimits _{k - 2}(D)$ for all $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X)$.

Lemma 42.30.5. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $(\mathcal{L}, s, i : D \to X)$ and $(\mathcal{L}', s', i' : D' \to X)$ be two triples as in Definition 42.29.1. Then the diagram

$\xymatrix{ \mathop{\mathrm{CH}}\nolimits _ k(X) \ar[r]_{i^*} \ar[d]_{(i')^*} & \mathop{\mathrm{CH}}\nolimits _{k - 1}(D) \ar[d]^{j^*} \\ \mathop{\mathrm{CH}}\nolimits _{k - 1}(D') \ar[r]^{(j')^*} & \mathop{\mathrm{CH}}\nolimits _{k - 2}(D \cap D') }$

commutes where each of the maps is a gysin map.

Proof. Denote $j : D \cap D' \to D$ and $j' : D \cap D' \to D'$ the closed immersions corresponding to $(\mathcal{L}|_{D'}, s|_{D'}$ and $(\mathcal{L}'_ D, s|_ D)$. We have to show that $(j')^*i^*\alpha = j^* (i')^*\alpha$ for all $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X)$. Let $W \subset X$ be an integral closed subscheme of dimension $k$. Let us prove the equality in case $\alpha = [W]$. We will deduce it from the key formula.

We let $\sigma$ be a nonzero meromorphic section of $\mathcal{L}|_ W$ which we require to be equal to $s|_ W$ if $W \not\subset D$. We let $\sigma '$ be a nonzero meromorphic section of $\mathcal{L}'|_ W$ which we require to be equal to $s'|_ W$ if $W \not\subset D'$. Write

$\text{div}_{\mathcal{L}|_ W}(\sigma ) = \sum \text{ord}_{Z_ i, \mathcal{L}|_ W}(\sigma )[Z_ i] = \sum n_ i[Z_ i]$

and similarly

$\text{div}_{\mathcal{L}'|_ W}(\sigma ') = \sum \text{ord}_{Z_ i, \mathcal{L}'|_ W}(\sigma ')[Z_ i] = \sum n'_ i[Z_ i]$

as in the discussion in Section 42.27. Then we see that $Z_ i \subset D$ if $n_ i \not= 0$ and $Z'_ i \subset D'$ if $n'_ i \not= 0$. For each $i$, let $\xi _ i \in Z_ i$ be the generic point. As in Section 42.27 we choose for each $i$ an element $\sigma _ i \in \mathcal{L}_{\xi _ i}$, resp. $\sigma '_ i \in \mathcal{L}'_{\xi _ i}$ which generates over $B_ i = \mathcal{O}_{W, \xi _ i}$ and which is equal to the image of $s$, resp. $s'$ if $Z_ i \not\subset D$, resp. $Z_ i \not\subset D'$. Write $\sigma = f_ i \sigma _ i$ and $\sigma ' = f'_ i\sigma '_ i$ so that $n_ i = \text{ord}_{B_ i}(f_ i)$ and $n'_ i = \text{ord}_{B_ i}(f'_ i)$. From our definitions it follows that

$(j')^*i^*[W] = \sum \text{ord}_{B_ i}(f_ i) \text{div}_{\mathcal{L}'|_{Z_ i}}(\sigma '_ i|_{Z_ i})$

as cycles and

$j^*(i')^*[W] = \sum \text{ord}_{B_ i}(f'_ i) \text{div}_{\mathcal{L}|_{Z_ i}}(\sigma _ i|_{Z_ i})$

The key formula (Lemma 42.27.1) now gives the equality

$\sum \left( \text{ord}_{B_ i}(f_ i) \text{div}_{\mathcal{L}'|_{Z_ i}}(\sigma '_ i|_{Z_ i}) - \text{ord}_{B_ i}(f'_ i) \text{div}_{\mathcal{L}|_{Z_ i}}(\sigma _ i|_{Z_ i}) \right) = \sum \text{div}_{Z_ i}(\partial _{B_ i}(f_ i, f'_ i))$

of cycles. Note that $\text{div}_{Z_ i}(\partial _{B_ i}(f_ i, f'_ i)) = 0$ if $Z_ i \not\subset D \cap D'$ because in this case either $f_ i = 1$ or $f'_ i = 1$. Thus we get a rational equivalence between our specific cycles representing $(j')^*i^*[W]$ and $j^*(i')^*[W]$ on $D \cap D' \cap W$. By Remark 42.19.6 the result follows for general $\alpha$. $\square$

Comments (2)

Comment #2983 by Xia on

In line 7 of the proof of the first lemma, the latter L should be N

Comment #3107 by on

THanks, fixed here. Please leave comments on lemmas on the page of the lemma.

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text:
1. ## find the value

$\tan\left(i\log\left[\frac{a-ib}{a+ib}\right]\right)$
Thanks...

2. ## Re: find the value

Start by writing the tangent function in terms of sines and cosines, then write the sines and cosines in terms of the complex exponential. Things should simplify greatly...


text:
# If $\forall x\in X:P(x)$ is true for all $X\neq\emptyset$, is it true when $X=\emptyset$?

If $\forall x\in X:P(x)$ is true for all $X\neq\emptyset$, is it true when $X=\emptyset$?

For the statement to be true, there at least must be an element of $X$ such that $P(x)$ holds, but there is no element in the empty set, so is it true, false or undefined?

• $\forall x \in \varnothing (P(x))$ is true no matter what $P(x)$ is. – amrsa Jan 12 '18 at 20:33
• $\forall x\in X: P(x)$ is short for $\forall x (x\in X\implies P(x))$... – Robert Wolfe Jan 12 '18 at 21:12
• @Robert Thanks. – Garmekain Jan 12 '18 at 21:23

No, in fact $\forall x\in X :P(x)$ is always true for $X=\emptyset$, regardless of whether it's true for $X\ne\emptyset$. Your reasoning is backwards; in fact for it to be false would mean that $P(x)$ was false for some $x\in\emptyset$.
No, the statement $\forall x \in X : P(x)$ does not require that there be an $x \in X$ (at least, under the standard interpretation). Generally, $\forall$ is taken to mean the same as $\neg\exists\neg$ - in other words, $\forall x \in X : P(x)$ should be the same as $\neg\exists x \in X : \neg P(x)$. In order for this new sentence to be false, we would need to have an $x \in X$ with $\neg P(x)$; for $X = \emptyset$, there is no such $x$, so $\neg\exists x \in X : \neg P(x)$ can't be false, so $\forall x \in X : P(x)$ must be true.


text:
In the simplest case, the output value of the layer with input size $$(N, C, H, W)$$, output $$(N, C, H_{out}, W_{out})$$ and kernel_size $$(kH, kW)$$ can be precisely described as:

## Usage

nn_avg_pool2d(
kernel_size,
stride = NULL,
ceil_mode = FALSE,
divisor_override = NULL
)

## Arguments

kernel_size

the size of the window

stride

the stride of the window. Default value is kernel_size

ceil_mode

when TRUE, will use ceil instead of floor to compute the output shape

when TRUE, will include the zero-padding in the averaging calculation

divisor_override

if specified, it will be used as divisor, otherwise kernel_size will be used

## Details

$$out(N_i, C_j, h, w) = \frac{1}{kH * kW} \sum_{m=0}^{kH-1} \sum_{n=0}^{kW-1} input(N_i, C_j, stride[0] \times h + m, stride[1] \times w + n)$$

If padding is non-zero, then the input is implicitly zero-padded on both sides for padding number of points.

The parameters kernel_size, stride, padding can either be:

• a single int -- in which case the same value is used for the height and width dimension

• a tuple of two ints -- in which case, the first int is used for the height dimension, and the second int for the width dimension

## Shape

• Input: $$(N, C, H_{in}, W_{in})$$

• Output: $$(N, C, H_{out}, W_{out})$$, where

$$H_{out} = \left\lfloor\frac{H_{in} + 2 \times \mbox{padding}[0] - \mbox{kernel\_size}[0]}{\mbox{stride}[0]} + 1\right\rfloor$$ $$W_{out} = \left\lfloor\frac{W_{in} + 2 \times \mbox{padding}[1] - \mbox{kernel\_size}[1]}{\mbox{stride}[1]} + 1\right\rfloor$$

## Examples

if (torch_is_installed()) {

# pool of square window of size=3, stride=2
m <- nn_avg_pool2d(3, stride = 2)
# pool of non-square window
m <- nn_avg_pool2d(c(3, 2), stride = c(2, 1))
input <- torch_randn(20, 16, 50, 32)
output <- m(input)
}


text:
# Excel Random Number Poisson Distribution

The function names are easy to remember: the letter r followed by the name of the distribution. For example, the rpois function is the random number generator for the Poisson distribution and it has only the parameter argument lambda. If not provided, the distribution defaults to 0 mean and 1 standard deviation. For my simulation, I want to generate a random number through RAND() and then figure out where on the cumulative Poisson distribution that number falls. , is given, and you are asked to calculate a probability of n events happening in a given time/number of pages/number of miles cycled, then the Poisson Distribution is used. The randomness comes from atmospheric noise, which for many purposes is better than the pseudo-random number algorithms typically used in computer programs. Given the mean number of successes (μ) that occur in a specified region, we can compute the Poisson probability based on the following formula:. One commonly used discrete distribution is that of the Poisson distribution. which is the correct probability for a uniform distribution on [1/4,1]. Excel was asked to generate 50 Poisson random numbers with mean λ = 5. Re: rewrite VBA code : inverse Poisson function The PoissonInv function returns the max number of events that have a cumulative probability = a given probability. The standard deviation computation is provided as a statistical function in the Excel functional operators and the. (a)If Xis equal to the number of calls received in an hour, produce the (partial) probability distri-bution of Xand its (partial) histogram. A Poisson random variable is the number of successes that result from a Poisson experiment. 01), we can use a Poisson with λ = nπ (≤20) to approximate it!. A Pólya–Aeppli distribution is a compound Poisson distribution in which the random numbers are generated by NegativeBinomial(1, 1-p)+1 for a fixed p. In statistics we have a term called a lognormal distribution which is calculated to find out the distribution of a variable whose logarithm is normally distributed, the original formula is a very complex formula to calculate it but in excel we have an inbuilt function to calculate the lognormal distribution which Lognorm. A binomial random number is the number of heads in N tosses of a coin with probability p of a heads on any single toss. The Poisson Distribution This DIGMath spreadsheet lets you investigate two different aspects of the Poisson distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate and independently of the time since the last event. Menascé, Ph. The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since the last event. In Excel, you may calculate the Poisson probabilities using the POISSON. 0425 is the answer. For small values of p, Poisson is a good approximation for the binomial distribution. A random variable, X, represents the number of roller coaster cars to pass through the circuit between 6pm and 6:10pm. The number of traffic fatalities per year in the state of Florida e. Hi, I am writing a sub that where I have to generate numbers from the Poisson distribution with different values of Lambda. Period, in terms of random number generation, is the number of calls that can be made to the RNG before it begins to repeat. Random variates from various distributions. This is because Microsoft Excel comes with an inbuilt Poisson function. Let's fill the numbers as shown in the image below. Random numbers from a Poisson or Binomial distribution. The random numbers generated are not consistent between different REXX interpreters or even the same REXX interpreters executing on different hardware. It can also be used for the number of events in other specified intervals such as distance, area or volume. GENPOISSON(m): Returns a an integer chosen from a Poisson distribution with mean value m. You might need to create randomized samples of normally distributed data for which the mean and the standard deviation of the distribution are known. There is a corresponding version for each of these functions but Poisson. For my simulation, I want to generate a random number through RAND() and then figure out where on the cumulative Poisson distribution that number falls. The Poisson distribution is a discrete probability function that is used to calculate the probability of a number of events occurring in a specified time period. Excel was asked to generate 50 Poisson random numbers with mean λ = 5. 2,False) = 10. The Poisson distribution provides a realistic model for many random phenomena. (Enter a specific random seed to repeat the same results based on that seed. The variance of a distribution of a random variable is an important feature. Examples Check areppim's Poisson Probability Calculator. Description Function to generate random outcomes from a Negative Binomial distribution, with mean mu and variance mu + mu^2/theta. Normal distribution. The rbinom function is the random number generator for the binomial distribution and it takes two arguments: size and prob. The poisson distribution is used to model rates, such as rabbits per acre, defects per unit, or arrivals per hour. The Data Analysis command in Excel also includes a Random Number Generation tool. The Poisson process is a random process which counts the number of random events that have occurred up to some point tin time. It's often related to rare events where the number of trials are indefinitely large and the probability of success P(x) is very small. This is the number of rows in the output table generated by Excel. Let X be be the number of hits in a day 2. It depends on a uniform random number generator function GetUniform such as you can find here. where λ is a positive number known as the Poisson parameter. Finally we just need to count how many time intervals totally contained in a unit time, that is the Poisson number. The probability of a success during a small time interval is proportional to the entire length of the time interval. I will leave that for you to explore on your own. We’ll then present the probability distribution of the binomial random variable, which will be presented as a formula, and explain why the formula makes sense. Unlike previous labs where the homework was done via OHMS, this lab will require you to submit short answers, submit plots (as aesthetic as possible!!), and also some code. 232), (Sharpie, De Veaux,. If x is a poisson distributed random variable, the 1/ x is an exponential random variable. CLARKS Women's Rosalyn Belle Dress Pump,Aldo Men's Slip-On Leather,NEW Business wallet Casual Clutch coins pocket luxury portfolio Double zipper ph. To shift distribution use the loc parameter. Extracted a set of features from irregularly spaced time series which included the exponential distribution parameter used to model the waiting time before the first occurrence of price change and the poisson distribution parameter used to model the number of price changes taking place. distribution, we can generate any random variable with a known distribution. Several important probability distributions arise naturally from the Poisson process—the Poisson distribution, the exponential distribution, and the gamma distribution. Day 2: Functions of Random Variables. The RAND function uses the Mersenne-Twister random number generator (RNG) that was developed by Matsumoto and Nishimura (1998). The Excel function NEGBINOMDIST(number_f, number_s, probability_s) calculates the probability of k = number_f failures before s = number_s successes where p. of the binomial distribution. CS177 Notes on Q-Q Plots Mart Molle, February 2008 I. For example, the number of car accidents in a day or the number of dandelions in a square meter plot of land. Sadly most random number generators only give uniformly distributed random numbers, however using a uniform random number p U ~U(0,1) one can calculate a Poisson distributed random number p Po ~Po(λ). Normal is easy. You can suppress the static values and have @RISK generate new random values for each distribution when Excel does an automatic recalculation or when you press F9 to force a manual recalculation. the Binomial (B ernoulli) distribution when the number, A Binomial random variable represents the number of successes in a series of independent and probabilistically homogenous trials distribution to the Binomial distribution Veaux, Velleman, Bock 2006, p. The number or bad checks drawn on Upright Bank on a day selected at random c. The Poisson distribution is used for sampling plans involving the number of defects or defects per unit rather than the number of defective parts. It is well known that „ = ‚ and ¾ = p ‚ so the coe–cient of variation ¾=„ becomes. However, this page came about because I have often been asked specifically how to create a Normal distribution curve in Excel. Problem with the Compound Poisson Interpretation of the Tweedie Distribution • A constant φwill force an artificial relationship between the claim frequency, λ, or the claim severity, αθ. 0 and later, you can select a random number generator on the Sampling tab of the Simulation Settings dialog. real-statistics. Given the above conditions, then k is a random variable, and the distribution of k is a Poisson Distribution. Chapter 13 The Poisson Distribution Jeanne Antoinette Poisson (1721–1764), Marquise de Pompadour, was a member of the French court and was the ofﬁcial chief mistress of Louis XV from 1745 until her death. Patterned -Characterized by a lower and upper bound, a step, repetition rate for values, and repetition rate for the sequence. All the 329 functions and what they do. Sadly most random number generators only give uniformly distributed random numbers, however using a uniform random number p U ~U(0,1) one can calculate a Poisson distributed random number p Po ~Po(λ). We will see how to calculate the variance of the Poisson distribution with parameter λ. The following types of distributions are available in Analysis Toolpak: Uniform distribution. dpois gives the (log) density, ppois gives the (log) distribution function, qpois gives the quantile function, and rpois generates random deviates. to generate NURNs from a Poisson distribution. A logical value that determines the form of the probability distribution returned. This Poisson distribution calculator can help you find the probability of a specific number of events taking place in a fixed time interval and/or space if these events take place with a known average rate. Poisson Statistics σ= n, (15) and Equation (14) can be approximated by () 1 ()n-n 2 n= exp-2πn 2n P ⎡ ⎤ ⎢ ⎥ ⎢⎣ ⎥⎦. where F-1 (u) is the inverse cumulative distribution function (ICDF) of the same distribution, and x is the resulting random number. The Data Analysis command in Excel also includes a Random Number Generation tool. org are unblocked. A Poisson random variable is the number of successes, denoted x, that results from a Poisson experiment, and it follows a Poisson distribution. This distribution is used for calculating the possibilities for an event with the given average rate of value(λ). A random variable, X, represents the number of roller coaster cars to pass through the circuit between 6pm and 6:10pm. The demand for jackets is expected to follow a Poisson distribution with an average rate of 80. To implement this method in Excel, you can use the RAND function to generate random numbers from the Uniform distribution, and apply the built-in functions to calculate the ICDF. The Poisson Distribution Description. (a)If Xis equal to the number of calls received in an hour, produce the (partial) probability distri-bution of Xand its (partial) histogram. But I've also built a simple Excel template that will help make this process a lot easier. There is more to distribution fitting than just overlaying a distribution on top of the histogram. Poisson method (Excel) 05/24/2019; 2 minutes to read +1; In this article. power (a[, size]) Draws samples in [0, 1] from a power distribution with positive exponent a - 1. In other words, the Poisson distribution is the probability distribution that results from a Poisson experiment. Enter the Statistics Workbench. There is no easy way to calculate Poisson distribution results using a single Excel function, but in this movie, I will show you a straightforward way to get the values you need using a series of. Once we have random numbers between 0 and 1, we can generate random numbers from an exponential distribution using the following formula: (1) This formula takes a random number R between 0 and 1 and gives a random number T from an exponential distribution with a mean of θ. Alvin Carreon if you can assign a number to each individual and pick a randomly 60 persons (based on the random numbers generated by a software Eg-Excel Random Function ) that would be ideal (The. Returns the Poisson distribution. Common random numbers Can have big effect on estimate quality (or computational effort required) Generating from truncated distributions (pp. The formula NORMINV() function in excel basically generates a number that has a probability represented by RAND() and which belongs to a normal distribution curve with a mean represented by mean_value and standard deviation represented by standard_deviation in the above formula. By incorporating Poisson distribution with historical data, football bettors can begin to determine the probable number of goals scored in a game. The random variable x is the. Speci cally, if Y ˘B(n;ˇ) then the distribution of Y as n!1. is called a Poisson random variable and the proba-bility distribution of X is called the Poisson distrib-ution. Patterned -Characterized by a lower and upper bound, a step, repetition rate for values, and repetition rate for the sequence. 6) Look to the right of 7 and under P(Exactly x) to see 0. Any unsold jacket at the end of the season are sold for $45. 0996 and 14. 2010 Poisson distribution When dealing with the number of occurrences of an event over a specified interval of time or space the Poisson distribution is often useful. In Excel 2003, Microsoft obtained an accurate answer in the central region of the distribution in exchange for inaccurate results in the tail. WorksheetFunction. The randomness comes from atmospheric noise, which for many purposes is better than the pseudo-random number algorithms typically used in computer programs. It also requires a function LogFactorial that returns the natural logarithm of the factorial of its argument. X = X 1 + X 2 + + X n: 2 The mean and variance of each X i can easily be calculated as: E(X i) = p;V(X i) = p(1 p):. hypergeometric probability distribution. 116) contains examples. Poisson distribution calculator, formulas, work with steps, real world and practice problems to learn how to find the probability of given number of events that occurred in a fixed interval of time with respect to the known average rate of events occurred. is a discrete probability distribution of a random variable x that satisfies the following conditions. where {N(t),t ≥ 0} is a Poisson process and {Y i,i ≥ 0} is a family of independent and identically distributed random variables which are also indepen-dent of {N(t),t ≥ 0}. As shown in Graph A, below, the fit between the observed distribution and the theoretical Poisson distribution defined by mean=variance=. Returns the Poisson distribution. Hi, I am writing a sub that where I have to generate numbers from the Poisson distribution with different values of Lambda. Discrete Probability Distributions * * Random Variable Random experiment is an experiment with random outcome. In this experiment, the trials are to be random and could have only two outcomes whether it can be success or failure. The exponential distribution can be used to determine the probability that it will take a given number of trials to arrive at the first success in a Poisson distribution; i. Random events • Arrival process – Packets arrive according to a random process – Typically the arrival process is modeled as Poisson • The Poisson process – Arrival rate of λ packets per second – Over a small interval δ, P(exactly one arrival) = λδ + ο(δ) P(0 arrivals) = 1 - λδ + ο(δ) P(more than one arrival) = 0(δ). The Poisson distribution is the discrete probability distribution of the number of events occurring in a given time period, given the average number of times the event occurs over that time period. The average number of events in an interval is designated (lambda). In this worksheet we are going to create the probability distribution function (pdf) and cumulative distribution function (cdf) for a Poisson distribution and then use the pdf and cdf to answer some probability questions. is called a Poisson random variable and the proba-bility distribution of X is called the Poisson distrib-ution. Parameters double lambda. The RAND function uses the Mersenne-Twister random number generator (RNG) that was developed by Matsumoto and Nishimura (1998). 8% of the measurements would lie within of the mean. Every observation is fed into every decision tree. Bernoulli Distribution Fitting. 447-448 and Problem 8. Tahoma Wingdings Times New Roman Arial Symbol Blends Microsoft Equation 3. To summarize, F-1 (U) is a random number with a probability distribution function f(x) if U \in \left(0,1\right). rpoisson(m) generates Poisson(m) random numbers. The Excel function NEGBINOMDIST(number_f, number_s, probability_s) calculates the probability of k = number_f failures before s = number_s successes where p. The Excel Functions covered here are: VLOOKUP, INDEX, MATCH, RANK, AVERAGE, SMALL, LARGE, LOOKUP, ROUND, COUNTIFS, SUMIFS, FIND, DATE, and many more. 116) contains examples. Donald Knuth describes a way to generate such values in §3. where λ is a positive number known as the Poisson parameter. It can also be used for the number of events in other specified intervals such as distance, area or volume. In general, a Poisson random variable represents the number of counts in some interval. Excel: There is no built-in Poisson analog to BINOM. Stata also provides functions that generate random numbers from other distributions. A new random real number is returned every time the worksheet is calculated or F9 key is pressed. All products Azure AS Excel 2016 Excel 2019 Power BI SSAS 2012 SSAS 2014 SSAS 2016 SSAS 2017 SSAS 2019 SSDT Any attribute Context transition Row context Iterator CALCULATE. Menascé, Ph. It also requires a function LogFactorial that returns the natural logarithm of the factorial of its argument. Once we have random numbers between 0 and 1, we can generate random numbers from an exponential distribution using the following formula: (1) This formula takes a random number R between 0 and 1 and gives a random number T from an exponential distribution with a mean of θ. 447-448 and Problem 8. Notation Used in the Hypergeometric Probability Distribution • The population is size N. x is the number of "successes" in question. If the data stray significantly from this line, Poisson capability analysis may not provide reliable results. In this recording, what I want to do is examine that question, show you how those data are related to a Poisson distribution and also give you some demos of techniques of how to generate random numbers that are drawn from a Poisson distribution. This is a part of our spreadcheats series of posts where we aim to solve 30 common work related excel problems, one at a time. In Weibull++, the Chi-Squared distribution has been used for reliability demonstration test design when the failure rate behavior of the product to be tested follows an exponential distribution. Poisson distribution can work if the data set is a discrete distribution, each and every occurrence is independent of the other occurrences happened, describes discrete events over an interval, events in each interval can range from zero to infinity and mean a number of occurrences must be constant throughout the process. >> Math Calculators >> Statistics Calculators Statistics Calculators >> Poisson Distribution Calculator Poisson Distribution Calculator. Sadly most random number generators only give uniformly distributed random numbers, however using a uniform random number p U ~U(0,1) one can calculate a Poisson distributed random number p Po ~Po(λ). The intervals are independent. Statalist archive (ordered by date) (last updated Thu Jan 31 23:05:02 2013). The hallmark of the Poisson distribution is that the mean is equal to the variance. Wherever possible, the simplest form of the distribution is used. This random variable has a Poisson distribution if the time elapsed between two successive occurrences of the event has an exponential distribution and it is independent of previous occurrences. Random number distribution that produces integers according to a Poisson distribution, which is described by the following probability mass function: This distribution produces random integers where each value represents a specific count of independent events occurring within a fixed interval, based on the observed mean rate at which they appear to happen (μ). The most direct way of generating random samples from a Poisson distribution is efficient for some parameters and inefficient for others. For example, you have data for class sections with the number of students in each section. One issue/problem that we will encounter is that the possible values for x for a Poisson are 0, 1, 2, … to infinity. If X1, X2,…,Xn are independent Poisson random variables with parameters λ1, λ2,… λn, then their sum is also a Poisson random variable with parameter λ1 +λ2+…+λn. Returns the Poisson distribution. The Poisson process is a collection {N(t) : t 0} of random variables, where N(t) is the number of events that have occurred up to time t (starting from time 0). To learn the situation that makes a discrete random variable a Poisson random variable. You are encouraged to do this with a spread-sheet (like Excel) if possible. Poisson random variables 5. This is the probability density function for the normal distribution in Excel. So you get the same result as this User-Defined formula with the in-built formula "=BINOM. abscissa = number of counts, ordinate = number of measurements yielding that number of counts). INV(100000, /100000,)". It will produce random numbers in a sequence which are greater than/equal to zero but less than 25. To illustrate the POISSON function, create a blank Excel worksheet, copy the following table, select cell A1 in your blank Excel worksheet, and then paste the entries so that the following table fills cells A1:C29 in your worksheet. Excel VBA Statistics and Mathematics Examples This page contains simple Excel VBA Statistics and Mathematics examples. 71828) k! is the factorial of k. For example, specifying 5,3,2 or [5,3,2] generates a 5-by-3-by-2 array of random numbers from the specified probability distribution. Examples of this are number of falls, asthma attacks, number of cells, and so on. Poisson's Law of Large Numbers synonyms, Poisson's Law of Large Numbers pronunciation, Poisson's Law of Large Numbers translation, English dictionary definition of Poisson's Law of Large Numbers. I got asked the question of how to generate random numbers with a normal distribution. DIST(x, mean, cumulative) Example 5. Also, the probability of having no major accident over 50 years,. n the fundamental statistical result that the average of a sequence of n identically distributed independent random variables. The Random Number Generation tool isn’t really a tool for descriptive statistics. Why doesn't Excel have a POISSON. We’ve gone through the process of creating a random normal distribution of numbers manually. GENPOISSON(m): Returns a an integer chosen from a Poisson distribution with mean value m. We’ll then present the probability distribution of the binomial random variable, which will be presented as a formula, and explain why the formula makes sense. Poisson Probability Calculator. I was hoping someone could help me with this problem. Please post any new questions, ideas, or discussions to our new community (we call it Answers) at https. It is considered a discrete distribution because the individual outcomes are discrete, such as the number of defects or the number of customers. This method can be used for any distribution in theory. Excel Function: Excel provides the following function for the Poisson distribution: POISSON ( x, μ, cum ) where μ = the mean of the distribution and cum takes the values TRUE and FALSE POISSON( x, μ , FALSE) = probability density function value f ( x ) at the value x for the Poisson distribution with mean μ. Generate an array of random numbers from one Poisson distribution. tail = TRUE, log. Most such sources would more precisely be called pseudorandom numbers, since a deterministic algorithm cannot, by definition, ever produce truly random numbers. If you want to your sequences of random numbers to be repeatable, see. A discrete random variable X is said to have a Poisson distribution with parameter λ > 0, if, for k = 0, 1, 2, …, the probability mass function of X is given by: where. How many such events will occur during a fixed time interval? Under the right circumstances, this is a random number with a Poisson distribution. We produce 500 random numbers between -100 and 100, with mean m = 0 and standard deviation s = 30. Excel has very powerful formulas and add-ins for performing almost any kind of statistical analysis. hypergeometric probability distribution. In Weibull++, the Chi-Squared distribution has been used for reliability demonstration test design when the failure rate behavior of the product to be tested follows an exponential distribution. The Poisson Random Variable. 3-2 RANDOM VARIATE GENERATION Table 3. INV(), so Poisson-distributed numbers can't be generated in the same way. For instance, it is well-known for testing the goodness-of-fit. Several important probability distributions arise naturally from the Poisson process—the Poisson distribution, the exponential distribution, and the gamma distribution. poisson ([lam, size]) Draw samples from a Poisson distribution. The number of traffic accidents per week at intersection R has a Poisson distribution with mean 1. The normal distribution is the "bell-shaped" curve; many variables are distributed approximately normally, including many physical traits such as height or weight, and also variables such as IQ. 6 THE POISSON DISTRIBUTION Exercise 3b: Can you guess what the mean and standard deviation for the sum of the throw of 100 dice might be? Exercise 3c: Devise a Monte Carlo simulation on Excel for the sum of the throw of two dice. p = FALSE) rpois(n, lambda) Arguments. Solution Preview. 2) Determine the mean, n, and then plot the expected Poisson distribution for this mean. Thanks to Scott Marryman an Roger Harbord for the useful advices. Any unsold jacket at the end of the season are sold for$45. The random number generator is not user selectable in RISKOptimizer 5. The Poisson probability mass function calculates the probability that there will be exactly x occurrences and is given by the formula:. x or in earlier versions of either product. Binomial is there (although I haven't used it)*. If a random event has a mean number of occurrences l in a given time period, then the number of occurrences within that time period will follow a Poisson distribution. 7 goals per game, so by putting the Poisson Distribution formula tells us that this average equates to Manchester City scoring 0 goals 18. Dist function. The Poisson distribution equation is very useful in finding out a number of events with a given time frame and known rate. You observe that the number of telephone calls that arrive each day on your mobile phone over a period of a year, and note that the average is 3. Under the right circumstances, this is a random number with a Poisson distribution. The variance is also equal to μ. The Poisson probability distribution is often used as a model of the number of arrivals at a facility within a given period of time. The random number generator which is used to draw random samples. POISSON: Returns the value of the Poisson distribution function (or Poisson cumulative distribution function) for a specified value and mean. A measurement is simply a numerical assignment to something, usually a. If one absolutely has to generate Poisson-distributed numbers in Excel, one should look up how to create a VBA (i. The amount of gasoline needed to drive your car 200 miles d. We can generate Poisson random variables in Python using poisson. For example, the number of car accidents in a day or the number of dandelions in a square meter plot of land. 447-448 and Problem 8. A Poisson random variable is the number of successes, denoted x, that results from a Poisson experiment, and it follows a Poisson distribution. The function returns one number. There is more to distribution fitting than just overlaying a distribution on top of the histogram. The law of small numbers. You might need to create randomized samples of normally distributed data for which the mean and the standard deviation of the distribution are known. Therefore, for every number genera ted with a uniform random number gen-erator, compute as where. The binomial is a finite discrete random variable whereas the Poisson distribution has an infinite number of possibilities. Usage rnegbin(n, mu = n, theta = stop("'theta' must be specified")) road 131 Arguments n If a scalar, the number of sample values required. Tutorial: Generating Random Numbers in Excel The field Number of Variables will be the number of columns Excel will generate with this random variable; the field Number of Random Numbers is the number of rows Excel will fill with random numbers. Poisson showed that the probability of observing a number n of such rare random events, in a spatial or temporal interval finished and always identical, that is: with μ the mean value of the number of events per interval that are observed in a number N of identical observations. The Poisson distribution is a discrete distribution that models the number of events based on a constant rate of occurrence. You want to calculate the probability (Poisson Probability) of a given number of occurrences of an event (e. The number of errors per page typed by Lena is assumed to follow a Poisson distribution with a mean of 0. In this worksheet we are going to create the probability distribution function (pdf) and cumulative distribution function (cdf) for a Poisson distribution and then use the pdf and cdf to answer some probability questions. Suppose orders follow a Poisson distribution with a mean of 0. I found a formula-based solution to this that would work for most situations where the mean is not huge. Lab 3: Simulations in R. So while number of jobs that arrive according to Poisson process during a time interval x follow Poisson distribution with parameter λx , the inter-arrival times of this process are distributed exponentially. POISSON (Microsoft Excel - found under Function Category: Statistical) Returns the Poisson distribution. Parameters of the Poisson Distribution The mean is µ. 1) distribution. I got asked the question of how to generate random numbers with a normal distribution. The diagonal line shows where the data would fall if they perfectly followed the Poisson distribution. The mean number of micro- organisms per square in this distribution is. Example Arrivals at a bus-stop follow a Poisson distribution with an average of 4. Recall from the previous. The random numbers generated are not consistent between different REXX interpreters or even the same REXX interpreters executing on different hardware. Central limit theorem. 1 for several values of the parameter ν. Poisson Distribution Example The number of visitors to a webserver per minute follows a Poisson distribution. quiz submissions chapter quiz n1 the ounces of soda consumed by an adult next month are an example of discrete random variable. (i) For part (i), we need the average number of. At some point, we got into List comprehensions – a convenient syntax to generate lists - via an example along these lines (which returns a list of the multiple of 2, from 2 to 40):. Let X 1 denote the number on the. Then in the Random Number Generation dialog box fill in: The 1,000 random draws have sample mean close to 8, sample variance close to 4. In Excel, you may calculate the Poisson probabilities using the POISSON. A = how many variables that are to be randomly generated B = number of random numbers generated per variable C = number corresponding to a distribution 1= Uniform 2= Normal 3= Bernoulli 4= Binomial 5= Poisson 6= Patterned 7= Discrete D = random number seed E = parameter of distribution (mu, lambda, etc. Wherever possible, the simplest form of the distribution is used. The number of times they occur can be viewed as a random variable and this random variable can be approximated by the Poisson distribution. If one absolutely has to generate Poisson-distributed numbers in Excel, one should look up how to create a VBA (i. This article describes how to easily create a random sample of a normal distribution with Excel. , for 0 ≤ x, the probability of X ≤ x is given by Without the option Seed = s , an initial seed is chosen internally. Usage rnegbin(n, mu = n, theta = stop("'theta' must be specified")) road 131 Arguments n If a scalar, the number of sample values required. If visitors come an average of ten seconds and the time period is one minute, then is 6. We've gone through the process of creating a random normal distribution of numbers manually. Replacing 1-u with u, and sum these time intervals will give us: That is what the fifth line of Knuth’s algorithm is doing. The Poisson Distribution was developed by the French mathematician Simeon Denis Poisson in 1837. I'll only add a shameful self-serving advertisement: In my book, Simulating Data with SAS, I provide several examples of using simulation to compute power and sample size (Ch 5 [p. What is the probability that you will not be late for class? My attempted solution: 1. Each function has its own set of parameter arguments. Finally we just need to count how many time intervals totally contained in a unit time, that is the Poisson number. we can use another distribution, known as the t distribution, that has a slightly different shape than the normal distribution. laplace Software - Free Download laplace - Top 4 Download - Top4Download. Uniform distribution is trivial. See law of averages. Examples include: • The number of misspelled words per page in a newspaper. In Poisson distribution mean is denoted by m i. We need to specify the number of samples to be generated. 1, we are able to set up the relationship below,. The sample is size n. In the Number of Variables you can enter the number of columns and in the Number of Random Numbers the number of rows. Uniform distribution is trivial. You can use the RAND() function. The following distributions are available: Uniform Normal Bernouilli Binomial Poisson Patterned 1. μ is the mean number of "successes". If you typed "77" into the box, and typed "77" the next time you run the random number generator, Excel will display that same set of random numbers. , constants. In fact, R can create lots of different types of random numbers ranging from familiar families of distributions to specialized ones. 7% of the area under the normal probability density curve falls within 3 standard deviations from the mean. In this equation, Rand is a random number drawn from a Gaussian distribution with mean=0 and SD=1. See law of averages. Function File: hygernd (t, m, n) Function File: hygernd (t, m, n, r) Function File: hygernd (t, m, n, r, c, …) Function File: hygernd (t, m, n, [sz]). Patterned -Characterized by a lower and upper bound, a step, repetition rate for values, and repetition rate for the sequence.


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# Integer pick with pigeon hole principle

Consider any set of $7$ unique positive integers. Prove that there's alway a way to pick a pair where either the difference or the sum of the pair is a multiple of $10$.

I know this is a trivial problem but it's driving me nuts. For some reason I can't find the right approach.

I've tried to consider all $\binom{7}{2} = 21$ possible pairs, and tried fitting them into $mod\space 10$ integer spaces. By the pigeon hole principle, there will be at least a set of $3$'s such that all three pair have the same congruence mod $10$. But I'm not sure how to proceed from that point, or if I'm on the right path.

If two of these 7 numbers, say $p$ and $q$, leave the same remainder, when divided by 10, then $p-q$ is divisible by 10.

If all 7 leave the 7 different remainders, when divided by 10, then at least 5 of their remainders are non-zero and different from 5, and hence are 5 different members of $$1, 9,\,\,\,\, 2,8,\,\,\,\, 3,7,\,\,\,\,4,6$$ Apparently, since they are five different ones, they occupy at least one of the four pairs above - each of these pairs has sum 10.

Let the nos be a1 , a2 , a3 ,a4 ,a5 ,a6 , a7 and without loss of generality , let a1 be the largest no

Define two groups Group 1 : a1+a2 a1+a3 a1+a4 ... a1+a7

Group 2 a1-a2 a1-a3 a1-a4 ... a1-a7

Now we have 14 nos , dividing them by 10 and comparing their remainders , we find that , by pigeon hole principle , atleast 2 leave the same remainder (as no of numbers = 14 , while no of possible remainders = 10)

Case 1 : Both numbers that have the same remainder lie in group 1 - Without loss of generality , let the nos be a1+a2 and a1+a3 , then the required no is a2-a3 ( subtracting two no that leave equal remainders (when divided by 10 ) results in a no divisible by 10

Case 2 : both nos lie in group 2 - Perform same operation as in case 1

Case 3: one lies in group 1 and one lies in group 2 - Perform same operation

Therefore , there is always a number divisible by 10


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# Browse Dissertations and Theses - Physics by Title

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