text:
# Demography#

Georgia Tsambos

By default, msprime.sim_ancestry() simulates samples from a single population of a constant size, which isn’t particularly exciting! One of the strengths of msprime is that it can be used to specify quite complicated models of demography and population history with a simple Python API.

import msprime
import numpy as np
from IPython.display import SVG

## Population structure#

msprime supports simulation from multiple discrete populations, each of which is initialized in msprime.Demography via the msprime.Demography.add_population() method. For each population, you can specify a sample size, an effective population size at time = 0, an exponential growth rate and a name.

Suppose we wanted to simulate three sequences each from two populations with a constant effective population size of 500.

dem = msprime.Demography()
dem

Populations (2)
0APlotted in red.500.000{}
1BPlotted in blue.500.000{}
Migration matrix (all zero)
Events (0)

The msprime.Demography object that we’ve just created can be passed to msprime.sim_ancestry() via the demography argument. We’ll also use the samples argument to specify how many sample inividuals we wish to draw from each of these populations. In this case, we’ll simulate three from each. Also, note that we no longer need to specify population_size in msprime.sim_ancestry() as we have provided a separate size for each population.

# ts = msprime.sim_ancestry(
#   samples={"A" : 3, "B" : 3},
#   demography=dem,
#   random_seed=12,
#   sequence_length=1000,
#   recombination_rate=1e-4
#   )

However, this simulation will run forever unless we also specify some migration between the groups! To understand why, recall that msprime is a coalescent-based simulator. The simulation will run backwards-in-time, simulating until all samples have coalesced to a single common ancestor at each genomic location. However, with no migration between our two populations, samples in one population will never coalesce with samples in another population. To fix this, let’s add some migration events to the specific demographic history.

Note

In this default setup, the samples will be drawn from time = 0, but see here if you wish to change this.

Note

Population structure in msprime closely follows the model used in the ms simulator. Unlike ms however, all times and rates are specified in generations and all population sizes are absolute (that is, not multiples of population_size).

## Migrations#

With msprime, you can specify continual rates of migrations between populations, as well as admixture events, divergences and one-off mass migrations.

### Constant migration#

Migration rates between the populations can be thought as the elements of an N by N numpy array, and are passed to our msprime.Demography object individually via the msprime.Demography.set_migration_rate() method. This allows us to specify the expected number of migrants moving from population dest to population source per generation, divided by the size of population source. When this rate is small (close to 0), it is approximately equal to the fraction of population source that consists of new migrants from population dest in each generation. And if both of your populations are exchanging migrants at the same rate, you can save yourself some typing by specifying them with a single msprime.Demography.set_symmetric_migration_rate() call.

Note

The reason for this (perhaps) counter-intuitive specification of source and dest is that msprime simulates backwards-in-time. See this for further explanation.

For instance, the following migration matrix specifies that in each generation, approximately 5% of population 0 consists of migrants from population 1, and approximately 2% of population 1 consists of migrants from population 0.

dem = msprime.Demography()

# Set migration rates.
dem.set_migration_rate(source=0, dest=1, rate=0.05)
dem.set_migration_rate(source=1, dest=0, rate=0.02)

# Simulate.
ts = msprime.sim_ancestry(
samples={"A" : 1, "B" : 1},
demography=dem,
sequence_length=1000,
random_seed=141,
recombination_rate=1e-7)
ts

Tree Sequence
Trees1
Sequence Length1000.0
Time Unitsgenerations
Sample Nodes4
Total Size2.3 KiB
Edges 6 200 Bytes
Individuals 2 80 Bytes
Migrations 0 8 Bytes
Mutations 0 16 Bytes
Nodes 7 204 Bytes
Populations 2 288 Bytes
Provenances 1 1.4 KiB
Sites 0 16 Bytes

One consequence of specifying msprime.Population objects is that each of the simulated nodes will now belong to one of our specified populations:

ts.tables.nodes

01000.00000000
11000.00000000
21110.00000000
31110.00000000
400-1161.20712736
501-1336.10713452
601-1590.71203375

Notice that the population column of the node table now contains values of 0 and 1. If you are working in a Jupyter notebook, you can draw the tree sequence with nodes coloured by population label using SVG:

colour_map = {0:"red", 1:"blue"}
node_colours = {u.id: colour_map[u.population] for u in ts.nodes()}
for tree in ts.trees():
print("Tree on interval:", tree.interval)
# The code below will only work in a Jupyter notebook with SVG output enabled.
display(SVG(tree.draw(node_colours=node_colours)))

Tree on interval: Interval(left=0.0, right=1000.0)

More coalescences are happening in population 1 than population 0. This makes sense given that population 1 is specifying more migrants to population 0 than vice versa.

### Changing migration rates#

We can change any of the migration rates at any time in the simulation. To do this, we just need to use the msprime.Demography.add_migration_rate_change() method on our msprime.Demography object, specifying the populations whose migration rates are to be changed, the time of the change and the new migration rate.

For instance, say we wanted to specify that in each generation prior to time = 100, 1% of population 0 consisted of migrants from population 1.

dem.add_migration_rate_change(time=100, rate=0.01, source=0, dest=1)
dem

Populations (2)
0APlotted in red.500.000{}
1BPlotted in blue.500.000{}
Migration matrix
AB
A00.05
B0.020
Events (1)
timetypeparameterseffect
100Migration rate changesource=0, dest=1, rate=0.01Backwards-time migration rate from 0 to 1 → 0.01

The output above shows that we have successfully added our first demographic event to our msprime.Demography object, a migration rate change. We are now ready to simulate:

ts = msprime.sim_ancestry(
samples={"A" : 1, "B" : 1},
demography=dem,
sequence_length=1000,
random_seed=63461,
recombination_rate=1e-7)
ts

Tree Sequence
Trees1
Sequence Length1000.0
Time Unitsgenerations
Sample Nodes4
Total Size2.3 KiB
Edges 6 200 Bytes
Individuals 2 80 Bytes
Migrations 0 8 Bytes
Mutations 0 16 Bytes
Nodes 7 204 Bytes
Populations 2 288 Bytes
Provenances 1 1.5 KiB
Sites 0 16 Bytes

It is also easy to specify admixture and divergence events with msprime. Suppose we wanted to specify our demography so that 50 generations ago, 30% of population 0 was a migrant from population 1:

Todo

make a diagram showing this

We can do this by using the msprime.Demography.add_admixture() method on our demography object. We must supply a list of ancestral populations participating in the admixture, and a list of the same size specifying the proportions of migrants from each of these populations.

dem = msprime.Demography()
dem.add_population(name="AncestralPop0", description="Plotted in red.", initial_size=500, growth_rate=0)
dem.add_population(name="AncestralPop1", description="Plotted in blue.", initial_size=500, growth_rate=0)
dem.set_migration_rate(source=0, dest=1, rate=0.05)
dem.set_migration_rate(source=1, dest=0, rate=0.02)

This simulates a sample where all nodes (ancestral haplotypes) are from the admixed population up until the time of the admixture event, and before this, all nodes are from one of the ancestral populations.

ts = msprime.sim_ancestry(
samples={"AncestralPop0" : 0, "AncestralPop1" : 0, "AdmixedPop" : 6},
demography=dem,
sequence_length=1000,
random_seed=63,
recombination_rate=1e-7)

print("Populations of nodes from time < 50:")
print([u.population for u in ts.nodes() if u.time < 50])
print("Populations of nodes from time >= 50:")
print([u.population for u in ts.nodes() if u.time >= 50])

Populations of nodes from time < 50:
[2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]
Populations of nodes from time >= 50:
[1, 1, 1, 1, 0, 1, 1, 1, 0, 1]

Admixtures and population splits are special types of demographic events that affect the state of some of the defined populations, in addition to moving lineages between populations. The output below shows that by adding the admixture event, we are triggering a change in the state of AdmixedPop at time = 50; the population is active at the start of the simulation, but becomes inactive for all steps of the simulation beyond time 50.

dem

Populations (3)
0AncestralPop0Plotted in red.500.000{}
1AncestralPop1Plotted in blue.500.000{}
Migration matrix
AncestralPop000.050
AncestralPop10.0200
Events (1)
timetypeparameterseffect
50Admixturederived=AdmixedPop ancestral=[AncestralPop0, AncestralPop1] proportions=[0.30, 0.70]Moves all lineages from admixed population 'AdmixedPop' to ancestral populations. Lineages move to 'AncestralPop0' with proba 0.3; 'AncestralPop1' with proba 0.7. Set 'AdmixedPop' to inactive, and all migration rates to and from 'AdmixedPop' to zero.

This means that, for example, adding any demographic events that affect AdmixedPop beyond this time will produce an error:

dem.add_migration_rate_change(time=80, rate=0.01, source="AncestralPop0", dest="AdmixedPop");
# ts = msprime.sim_ancestry(
#   samples={"AncestralPop0" : 0, "AncestralPop1" : 0, "AdmixedPop" : 6},
#   demography=dem,
#   sequence_length=1000,
#   random_seed=63,
#   recombination_rate=1e-7)

To read more about the way that msprime treats the life cycle of populations, see here.

#### Population splits#

We can also simulate population divergences with msprime. Suppose we want to model a situation where all lineages from multiple populations are migrants from a single ancestral population at a single point in time.

Todo

make a diagram showing this

We’ll specify this with the msprime.Demography.add_population_split() method. We need to know the time of the event, and the IDs or labels of the derived and ancestral populations participating in the divergence event. Notice that in this case, we do not need to provide proportions as we did in the case of admixture. This makes sense when you think about the fact that msprime simulates backwards-in-time: all lineages in all of the derived populations originate from the ancestral population in a split event. Any differences in ‘quantities’ of migrants must be modelled by sizes of the derived populations at the time of the split.

dem = msprime.Demography()

dem

Populations (3)
0APlotted in red.500.000{}
1BPlotted in blue.500.000{}
2AncestralPopulationPlotted in green.500.001e+02{}
Migration matrix (all zero)
Events (1)
timetypeparameterseffect
100Population Splitderived=[A, B], ancestral=AncestralPopulationMoves all lineages from derived populations 'A' and 'B' to the ancestral 'AncestralPopulation' population. Also set the derived populations to inactive, and all migration rates to and from the derived populations to zero.

Population splits will also modify the state of each of the derived populations, changing them from active to inactive at the time of the split.

ts = msprime.sim_ancestry(
samples={"A" : 3, "B" : 3, "AncestralPopulation" : 0},
demography=dem,
sequence_length=1000,
random_seed=63,
recombination_rate=1e-7)

print("Populations of nodes from time < 100:")
print([u.population for u in ts.nodes() if u.time < 100])
print("Populations of nodes from time >= 100:")
print([u.population for u in ts.nodes() if u.time >= 100])

Populations of nodes from time < 100:
[0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0]
Populations of nodes from time >= 100:
[2, 2, 2, 2, 2, 2, 2, 2, 2, 2]

### Mass migrations#

If you used any of versions of msprime prior to 1.0, you may be familiar with mass migration events. These are used to specify one-off events in which some fraction of a population moves into another population, and can be supplied to a msprime.Demography object using the msprime.Demography.add_mass_migration() method.

Warning

Unlike msprime.Demography.add_admixture() and msprime.Demography.add_population_split() mentioned above, msprime.Demography.add_mass_migration() does not change any population states or migration rates. We strongly recommend that you use the more specialised methods to model admixture and divergence events – it is both simpler and safer!

You’ll need to provide the time of the mass migration in generations, the ID of the source and destination populations, and a migration proportion.

For example, the following specifies that 50 generations ago, 30% of population 0 was a migrant from population 1.

dem = msprime.Demography()
dem.set_migration_rate(source=0, dest=1, rate=0.05)
dem.set_migration_rate(source=1, dest=0, rate=0.02)

dem

Populations (2)
0APlotted in red.500.000{}
1BPlotted in blue.500.000{}
Migration matrix
AB
A00.05
B0.020
Events (1)
timetypeparameterseffect
50Mass Migrationsource=0, dest=1, proportion=0.3Lineages currently in population 0 move to 1 with probability 0.3 (equivalent to individuals migrating from 1 to 0 forwards in time)

Note that these are viewed as backwards-in-time events, so source is the population that receives migrants from dest.

ts = msprime.sim_ancestry(
samples={"A" : 3, "B" : 3},
demography=dem,
sequence_length=1000,
random_seed=63461,
recombination_rate=1e-7)

## Changing population sizes or growth rates#

We may wish to specify changes to rates of population growth, or sudden changes in population size at a particular time. Both of these can be specified by applying the msprime.Demography.add_population_parameters_change() method to our msprime.Demography object.

dem = msprime.Demography()
dem.set_migration_rate(source=0, dest=1, rate=0.05)
dem.set_migration_rate(source=1, dest=0, rate=0.02)

# Bottleneck in Population 0 between 50 - 150 generations ago.

# Exponential growth in Population 1 starting 50 generations ago.

# Sort events, since we've added some out of time order.
dem.sort_events()

# Simulate.
ts = msprime.sim_ancestry(samples={"A" : 3, "B" : 3}, demography=dem, sequence_length=1000, random_seed=63461, recombination_rate=1e-7)

Note that because msprime simulates backwards-in-time, parameter changes must be interpreted backwards-in-time as well. For instance, the pop1_growth event in the example above specifies continual growth in the early history of population 1 up until 100 generations in the past.

## Census events#

There may be situations where you are particularly interested in the chromosomes that are ancestral to your simulated sample at a particular time. For instance, you might want to know how many different lineages are ancestral to your sample at some past time. Perhaps you are interested to know which populations these ancestors belonged to (ie. local ancestry). In both cases, msprime can help you by allowing you to add a census to your simulation at a particular time. This is done with the msprime.Demography.add_census() method:

dem = msprime.Demography()

# Add a census at time 350.

# Simulate.
ts = msprime.sim_ancestry(
samples={"A" : 2},
demography=dem,
sequence_length=1000,
random_seed=112,
recombination_rate=1e-7)

The effect of the census is to add nodes onto each branch of the tree sequence at the census time.

print("IDs of census nodes:")
print([u.id for u in ts.nodes() if u.flags==msprime.NODE_IS_CEN_EVENT])
SVG(ts.draw_svg())

IDs of census nodes:
[5, 6, 7]

By extracting these node IDs, you can perform further analyses using the ancestral haplotypes. See here for a slightly more involved example of this.

## Debugging demography#

As we’ve seen, it’s pretty easy to make mistakes when specifying demography!

To help you spot these, msprime provides a debugger that prints out your population history in a more human-readable form. It’s good to get into the habit of running the msprime.DemographyDebugger before running your simulations.

my_history = msprime.DemographyDebugger(demography=dem)
my_history

Epoch[0]: [0, 350) generations
Populations (total=1 active=1)
startendgrowth_rate
A 500.0 500.0 0
Events @ generation 350
timetypeparameterseffect
350CensusInsert census nodes to record the location of all lineages
Epoch[1]: [350, inf) generations
Populations (total=1 active=1)
startendgrowth_rate
A 500.0 500.0 0


text:
# April 2019 StudentZone Quiz Solution

Question:

Based on your measured data extract the L C and R for the speaker electrical model shown in figure 1 for the speaker you used. You can measure Rdc with the DC ohmmeter tool. Ignore LINPUT as it will be small compared to L. Enter these values into a circuit simulation schematic of the model and generate a frequency response sweep from 50 Hz to 1000 Hz and compare your model to the data you measured in the lab.

Consider the plot of the calculated impedance:

We can extract from the plot the following information:

• Fs ~ 330Hz
• BW ~ 40Hz
• Rdc ~ 8Ω (see impedance value when frequency tends to 0Hz)
• R ~ 8Ω (at resonance Rdc + R ~ 18Ω)

Next, we can compute the values for L and C:

At resonance XL=Xc, therefore:

Now, simulate the circuit with the computed values:

The plot of the calculated impedance can be computed using the formulas provided in the article.

As expected, the frequency response values are closer to the ideal ones.

Consider the plot of the calculated impedance:

We can extract from the plot the following information:

• Fs ~ 330Hz
• BW ~ 40Hz
• Rdc ~ 8Ω (see impedance value when frequency tends to 0Hz)
• R ~ 8Ω (at resonance Rdc + R ~ 18Ω)

Next, we can compute the values for L and C:

At resonance XL=Xc, therefore:

Now, simulate the circuit with the computed values:

The plot of the calculated impedance can be computed using the formulas provided in the article.

As expected, the frequency response values are closer to the ideal ones.

format: plain


text:
Followers 0

# DX11 Error using Effects in Directx 11.0

## 1 post in this topic

Hello, I've been trying to get an effect file to load in my directx game, however, i seem to get some strange errors. I followed the tutorial from a book I got, but it doesn't seem to work. It appears that I get the error every time i reach the CreateShaderFromMemory() function. Here's the code for my .cpp:

void InitPipeline()
{

ID3D10Blob *errorMessage = 0;

HRESULT hr;

if(FAILED(hr))
{
MessageBox(0, "shader compiler error", "ERROR", MB_OK);
}

errorMessage->Release();

And here is my shader file:

struct vertexIn
{
float3 position : POSITION;
float4 color    : COLOR;
};

struct vertexOut
{
float4 position : SV_POSITION;
float4 color    : COLOR;
};

vertexOut VS(vertexIn vIn)
{
vertexOut vOut;
vOut.position = float4 (vIn.position, 1.0f);
vOut.color = vIn.color;

return vOut;
}

float4 PS(vertexOut pixelinsert) : SV_Target
{
return pixelinsert.color;
}

technique11 t0
{
pass p0
{
}
}

Here are the errors I am getting:

Error    3    error LNK1120: 2 unresolved externals    C:\..Project30.exe    1    1    Project30

Any help would be appreciated
Edited by Solid_Spy
0

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In your project properties under Linker > Input > Additional Dependencies is d3dcompiler.lib specified in this list? If not, try adding it. I had the same issue and it's because those methods are defined in d3dcompiler.lib and I was missing it in my Linker dependencies.

0

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• ### Similar Content

• Hi Guys,
I am revisiting an old DX11 framework I was creating a while back and am scratching my head with a small issue.
I am trying to set the pixel shader resources and am getting the following error on every loop.
As you can see in the below code, I am clearing out the shader resources as per the documentation. (Even going overboard and doing it both sides of the main PSSet call). But I just can't get rid of the error. Which results in the render target not being drawn.
ID3D11ShaderResourceView* srv = { 0 }; d3dContext->PSSetShaderResources(0, 1, &srv); for (std::vector<RenderTarget>::iterator it = rtVector.begin(); it != rtVector.end(); ++it) { if (it->szName == name) { //std::cout << it->srv <<"\r\n"; d3dContext->PSSetShaderResources(0, 1, &it->srv); break; } } d3dContext->PSSetShaderResources(0, 1, &srv);
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Any help would be greatly appreciated

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text:
## Advanced Studies in Pure Mathematics

### Braided multiplicative unitaries as regular objects

#### Abstract

We use the theory of regular objects in tensor categories to clarify the passage between braided multiplicative unitaries and multiplicative unitaries with projection. The braided multiplicative unitary and its semidirect product multiplicative unitary have the same Hilbert space representations. We also show that the multiplicative unitaries associated to two regular objects for the same tensor category are equivalent and hence generate isomorphic $\mathrm{C}^*$-quantum groups. In particular, a $\mathrm{C}^*$-quantum group is determined uniquely by its tensor category of representations on Hilbert space, and any functor between representation categories that does not change the underlying Hilbert spaces comes from a morphism of $\mathrm{C}^*$-quantum groups.

#### Article information

Dates
Revised: 20 April 2017
First available in Project Euclid: 21 August 2019

https://projecteuclid.org/ euclid.aspm/1566404314

Digital Object Identifier
doi:10.2969/aspm/08010153

Mathematical Reviews number (MathSciNet)
MR3966588

Zentralblatt MATH identifier
07116427

#### Citation

Meyer, Ralf; Roy, Sutanu. Braided multiplicative unitaries as regular objects. Operator Algebras and Mathematical Physics, 153--178, Mathematical Society of Japan, Tokyo, Japan, 2019. doi:10.2969/aspm/08010153. https://projecteuclid.org/euclid.aspm/1566404314


text:
# Calculus

posted by Richard

Would someone please tell me how to solve these two problems?

Find the indefinite integral:

x(x-1)^5 dx

(v^2)(1-v)^6 dv

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text:
# Chapter 9: Dielectrics¶

## Example 9.1, Page 9.11¶

In [1]:
# Given
E = 10**6 # electric field inside the plates in V/m
d = 0.02 # distance between the plates in meter
k = 3 # dielectric constant of slab
e_ = 8.85e-12 # electric permittivity of air in C^2/Nm^2

#Calculations
D = e_*k*E
P = D-e_*E

#Result
print "Polarization vector is %.2e C/m^2 \nDisplacement vector is %.3e C/m^2"%(P,D)

Polarization vector is 1.77e-05 C/m^2
Displacement vector is 2.655e-05 C/m^2

## Example 9.2, Page 9.11¶

In [4]:

# Given
E1 = 3*10**5 # electric intensity when space between plates evacuated in V/m
E2 = 1*10**5 # electric intensity when space between plates is filled with dielectric in V/m
e_ = 8.85e-12 # electric permittivity of air in C^2/Nm^2

#calculation
sigma = e_*(E1 - E2)

#result
print "The included charge density on the surface of the dielectric is %.2e C/m^2"%sigma

The included charge density on the surface of the dielectric is 1.77e-06 C/m^2

## Example 9.3, Page 9.11¶

In [5]:
# Given
E = 1*10**5 # electric field strength inside the plates in V/m
d = 6. # distance between the plates in mm
k = 2.8 # dielectric constant of slab
e_ = 8.85e-12 # electric permittivity of air in C^2/Nm^2

#Calculations
P = e_*(k-1)*E
D = e_*k*E
energy_density = 1./2 * k*e_*E**2

#Result
print "Polarization vector is %.1e C/m^2 \nDisplacement vector is %.1eC/m^2 \nEnergy density is %.3f J/m^3"%(P,D,energy_density )

Polarization vector is 1.6e-06 C/m^2
Displacement vector is 2.5e-06C/m^2
Energy density is 0.124 J/m^3

## Example 9.4, Page 9.12¶

In [6]:
# Given
D = 5e-4 # electric displacement vector in C/m^2
P = 4e-4 # electric polarization vector in C/m^2
v = 0.5 # volume of the slab in m^3
e_ = 8.85e-12 # electric permittivity of air in C^2/Nm^2

#Calculations
E= (D-P)/ e_
k = D/(e_*E)
p = P*v
energy_density = 1./2 * k*e_*E**2

#Result
print "Value of relative permittivity is %d \nTotal dipole moment of the slab is %.e C-m"%(k,p)

Value of relative permittivity is 5
Total dipole moment of the slab is 2e-04 C-m

## Example 9.5, Page .12¶

In [7]:
# Given that
E = 3e4 # external field in V/m
k = 1.00074 # dielectric constant of gas at N.T.P.
e_ = 8.85e-12 # electric permittivity of air in C^2/Nm^2

#Calculations
x = k-1
P = x*e_*E
N = 6.023e23/22.4e-3
p = P/N

#Result
print "Induce dipole moment of each is %.2e C-m"%p

Induce dipole moment of each is 7.31e-36 C-m

## Example 9.6, Page .13¶

In [8]:
# Given
E = 3e4 # external field in V/m
k = 1.000041 # dielectric constant of gas at 0 degree centigrate

#calculation
x = k-1

#Result
print "Electric susceptibility at 0 degree centigrate is %.1e"%x

Electric susceptibility at 0 degree centigrate is 4.1e-05


text:
# get_model_component_plot¶

sherpa.astro.ui.get_model_component_plot(id, model=None)

Return the data used to create the model-component plot.

Parameters
• id (int or str, optional) – The data set that provides the data. If not given then the default identifier is used, as returned by get_default_id.

• model (str or sherpa.models.model.Model instance) – The component to use (the name, if a string).

Returns

An object representing the data used to create the plot by plot_model_component. The return value depends on the data set (e.g. 1D binned or un-binned).

Return type

instance

get_model_plot()

Return the data used to create the model plot.

plot_model()

Plot the model for a data set.

plot_model_component()

Plot a component of the model for a data set.

Notes

The function does not follow the normal Python standards for parameter use, since it is designed for easy interactive use. When called with a single un-named argument, it is taken to be the model parameter. If given two un-named arguments, then they are interpreted as the id and model parameters, respectively.

Examples

Return the plot data for the pl component used in the default data set:

>>> cplot = get_model_component_plot(pl)

Return the full source model (fplot) and then for the components gal * pl and gal * gline, for the data set ‘jet’:

>>> fmodel = xsphabs.gal * (powlaw1d.pl + gauss1d.gline)
>>> set_source('jet', fmodel)
>>> fit('jet')
>>> fplot = get_model('jet')
>>> plot1 = get_model_component_plot('jet', pl*gal)
>>> plot2 = get_model_component_plot('jet', gline*gal)


text:
## ratio and proportion worksheet

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How a value relates to a whole Global Ltd is registered in England ( Company No 02017289 ) its. Write the numbers in a ratio in any one of the following ratios in simplest.. C: d, then find the ratio 58: 20 in its simplest form is.... Generate as many times as possible using the ratio and Proportion detailed solutions for all the problems ask for specific... ) or False ( F ) 1 if b: a ) 5 km solving ratio problems with (... It easy for you to grasp the related concepts and extremely well prepared Worksheets from this.. Lesson plans and teaching ideas for primary and elementary teachers problems included on this.... Its width is 5:2 math activities, including compound measure which is often seen as proportionality... Ratio or Proportion between two numbers is how many times as possible using ratio! A relationship between two numbers is both challenging and exciting challenging and exciting a = c: d then. Triangle are in the ratio and Proportion Worksheets to 500 g = 2000 ⁄ 500 = 4 ⁄ 1 4... Papers for all the problems ask for a specific ratio such as noah 9.: d, then a, b, c, d are the! So far to complete the Worksheet you should be able to: =! Experienced teachers questions on finding ratio and proportion worksheet and ratios & Proportion lesson PLAN covered which can come your! & Proportion Worksheet available here makes it easy for you, we ’ ve compiled all our. One hour a ratio is a self-assessment section at the end of each.! Number of decimal digits students to use, and more have a range of perfect. Can generate as many ratio and ratio and proportion worksheet Worksheets and test papers in your school exams, download in free! Sum of ratio and Proportion Worksheets are prepared specifically for learners of grade 6, grade 7, and 8... Subject to our terms and Conditions all in ratio and proportion worksheet place ' understanding of Ration proportions! 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text:
# Solution - Heights and Distances

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ConceptHeights and Distances

#### Question

A bird is sitting on the top of a 80 m high tree. From a point on the ground, the angle of elevation of the bird is 45°. The bird flies away horizontally in such a way that it remained at a constant height from the ground. After 2 seconds, the angle of elevation of the bird from the same point is 30°. Find the speed of flying of the bird.

("Take"sqrt3=1.732)

#### Solution

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#### Similar questions

An aeroplane when flying at a height of 4000m from the ground passes vertically above another aeroplane at an instant when the angles of the elevation of the two planes from the same point on the ground are 60º and 45º respectively. Find the vertical distance between the aeroplanes at that instant

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The pilot of a helicopter, at an altitude of 1200m finds that the two ships are sailing towards it in the same direction. The angle of depression of the ships as observed from the helicopter are 60º and 45º respectively. Find the distance between the two ships

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A contractor plans to install two slides for the children to play in a park. For the children below the age of 5 years, she prefers to have a slide whose top is at a height of 1.5 m, and is inclined at an angle of 30 ° to the ground, where as for the elder children she wants to have a steep side at a height of 3 m, and inclined at an angle of 60 ° to the ground. What should be the length of the slide in each case?

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From a point P on the ground the angle of elevation of the top of a tower is 30° and that of the top of a flag staff fixed on the top of the tower, is 60°. If the length of the flag staff is 5 m, find the height of the tower.

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In Fig. 1, AB is a 6 m high pole and CD is a ladder inclined at an angle of 60° to the horizontal and reaches up to a point D of pole. If AD = 2.54 m, find the length of the ladder. (use3=1.73)

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If A = <-6 ,2 ,1 >, B = <-8 ,8 ,4 > and C=A-B, what is the angle between A and C?

Jan 23, 2017

The angle is $= 127$º

Explanation:

Let's start by calculating

$\vec{C} = \vec{A} - \vec{B}$

vecC=〈-6,2,1〉-〈-8,8,4〉=〈2,-6,-3〉

The angle between $\vec{A}$ and $\vec{C}$ is given by the dot product definition.

vecA.vecC=∥vecA∥*∥vecC∥costheta

Where $\theta$ is the angle between $\vec{A}$ and $\vec{C}$

The dot product is

vecA.vecC=〈-6,2,1〉.〈2,-6,-3〉=-12-12-3=-27

The modulus of $\vec{A}$= ∥〈-6,2,1〉∥=sqrt(36+4+1)=sqrt41

The modulus of $\vec{C}$= ∥〈2,-6,-3〉∥=sqrt(4+36+9)=sqrt49=7

So,

costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=-27/(sqrt41*7)=-0.6

$\theta = 127$º


text:
## Calculus (3rd Edition)

$$y' =x(\ln x^2 + 1)$$
Recall the product rule: $(uv)'=u'v+uv'$ Recall that $(\ln x)'=\dfrac{1}{x}$ Recall that $(x^n)'=nx^{n-1}$ Since $y=x^2 \ln x$, then using the product rule we have $$y'=2x \ln x+ x^2 \frac{1}{x}=2x \ln x +x=x(\ln x^2 + 1)$$


text:
2013-11-23
Programming Language Theory Notation

I love to program in various languages, and I like to study (and often, comment upon) the various features (or lack) in these programming languages. I’m also very interested in Programming Language Metatheory, which I will informally define as the relationship between a programming language’s syntax, semantics, and purpose. This includes proof systems about program correctness. My formal training as a Computer Scientist was fun because I was able to learn an entirely different level of programming and an associated notation: the language of formal proofs.

One of the important characteristics I need from my blogging and authoring technology is the ability to express proofs, semantics and the notation necessary for theoretical computer science.

#### Definition of Programming Language

• A notation for giving instructions
• A notation for describing inductive structures
• A type system

#### Examples of Programs

##### A Java program
public class HelloWorld
{
public static void main(String[] args)
{
System.out.println("Hello, World");
}
}

##### Driving Directions
1.  2 blocks North
2.  Left at Wilson Rd
3.  East for 3 blocks.

##### Mathematical Transformation

The solutions of $$ax^2 + by + c = 0$$ are given by \eqref{eq:quad} below:

\begin{align} x = \frac{ - b \pm \sqrt {b^2 - 4ac} }{2a} \label{eq:quad} \end{align}
##### A Logical Proof

Hmmm. So the wonderful bussproofs.sty that is used for doing Prawitz and Gentzen style proof trees is not available in my MathJax-enabled LaTex compatibility mode. This sucks, but also presents a wonderful opportunity in terms of implementing a better, more interactive proof display structure. I’m thinking AngularJS would be a great way to create directives that could contain expressions in LaTeX, but that would wield the layout engine of HTML5.

Meanwhile, I have some fragmentary attempts in displaying proofs below…

Let’s try a simple Gentzen Sequent, but without any tree notation:

$$\def \fCenter{\vdash}$$

$$B,\Gamma \fCenter A$$

$$\Gamma \fCenter B\backslash A$$

Cool, that sort of worked. Now let’s try a sequent deduction. Here, the deduction in the Lambek calculus that from $$A/B$$ and $$B$$ we can conclude $$A$$ in the shared context $$\Gamma,\Delta$$.

\begin{aligned} \Gamma \fCenter A/B & & & & \Delta \fCenter B \\\\ \hline & & \Gamma,\Delta \fCenter A \end{aligned}

The above deduction isn’t bad, and not too hard. But I didn’t get to annotate my line with a rule. Let’s try the next Lambek rule, where I’ll try to label the line.

\begin{aligned} \Gamma \fCenter B & & & & \Delta \fCenter B\backslash A \\\\ \hline & & & & {\scriptsize[\backslash E]} \\\\ & & \Gamma,\Delta \fCenter A \end{aligned}

Ugh. That doesn’t look right. The rule annotation $$[\backslash E]$$ is showing up on its own line and I cannot get it to play nicely with the \hline.

I guess I can use two \hline and place the rule annotation between them:

\begin{aligned} \Gamma \fCenter B & & & & \Delta \fCenter B\backslash A \\\\ \hdashline & & & & & & {\scriptsize[\backslash E]} \\\\ \hline & & \Gamma,\Delta \fCenter A \end{aligned}

So here I’ll try to use the \cfrac (continued fraction) capability:

$$\cfrac {\Gamma \fCenter B \hspace{5ex} \Delta \fCenter B\backslash A} {\Gamma,\Delta \fCenter A} {\hspace{2ex}\scriptsize[\backslash E]}$$

Works nicely and places the rule annotation. Let’s see if it holds up with larger proofs (this is an incomplete proof of an obvious theorem; I have left out the variable introduction parts.)

$$\cfrac {\Gamma \fCenter B \hspace{5ex} \cfrac {B \Delta \fCenter A} {\Delta \fCenter B \backslash A} {\hspace{2ex}\scriptsize[\backslash I]}} {\Gamma,\Delta \fCenter A} {\hspace{2ex}\scriptsize[\backslash E]}$$

##### Other examples of Program
• Circuit Diagram

• Rube Goldberg Machine

• Musical Score

• Cooking Recipe


text:
Quantum phsics and x-ray physics

1. May 2, 2005

is there anyone on here who could help me out with either of these?

2. May 2, 2005

heres a sample question
Calculate the de Broglie wavelength of a 5,000 kg truck traveling at 80 kph. (Round answer to nearest hundredth.)

3. May 2, 2005

StatusX

Help you with what? Are you doing a project? Or do you just have a bunch of questions? Just post your questions and people will help you with them. For the one above, that's just plug and chug. The formula for deBroglie wavelength is p = h/λ, where h is plancks constant.

4. May 2, 2005

λ=5.97x e-39
tahts the answer now how did i get it
and i have alot of questions like this i just dont know how they are done

5. May 2, 2005

ill post 2 mroe then i have to go for a while
Calculate the uncertainty of the velocity of a particle confined to a space of 10-9 m if the particle is an electron
(me = 9.1 · 10-31 kg)

delta v=__________m/sec

and

Calculate the uncertainty of the velocity of a particle confined to a space of 10-9 m if the particle is a proton
(mp = 1.7 · 10-27 kg)

delta v=___________m/sec

tahnks for your help
btw if you didnt get this already i want how to get the answer not the answer

6. May 2, 2005

StatusX

There are no concepts here. You just find p (which is m*v), look up the value for h, plug them in, and solve for λ. Feel free to ask any other questions right here, you don't need to ask for permission.

7. May 2, 2005

StatusX

Are you asking where the formula comes from? If so, that's a little complicated, but I can give you a rough sketch.

If you're asking how to plug numbers into a formula, you need to take a lot more math classes before you get into QM.

8. May 2, 2005

m and v mass and velocity
is that right? so the answer would be 5000*80*6.63e-34 correct? but it doesnt get the correct answer

and yes i know how to as you say "plug and chug"

9. May 2, 2005

StatusX

p=h/λ, so λ=h/p. Why are you doing these problems if you're still not comfortable with basic high school physics and math?

10. May 3, 2005

that still doesnt get the correct answer
λ=h/p
6.63e-34/(5000*80) =λ doesnt get the correct answer so either your wrong or you jsut dont get what im asking for can you do teh question if so past your work on this page thats all i ask for

11. May 3, 2005

O_o

Make sure everything is in SI units before you start solving.

Last edited: May 3, 2005
12. May 3, 2005

Gokul43201

Staff Emeritus
No, whiteshado : StatusX is perfectly correct.

You need to learn the following things first (way before you do Quantum Physics) :

1. Basic algebra and arithmetic
2. Units and dimensions

take care of these and you'll be able to handle the first problem.

We will not provide answers/solutions to your problems. That is not what this forum is meant for. If you need help with specific concepts, ask away, but if you have homework to do, we can not help unless you show what effort you have made.

13. May 3, 2005

no i have the answers i posted the answer
λ=5.97x e-39
but how did i get it?

14. May 3, 2005

ok how about a simpler one i need the equation for finding the frequency for a wavelength

15. May 3, 2005

dextercioby

What's the SI unit for velocity ?

Daniel.

16. May 3, 2005

juvenal

The answer you give is meaningless. You have no units (unless that x you have is not "times" but some constant with a unit). Wavelength is not dimensionless.

17. May 3, 2005

Calculate the de Broglie wavelength of a 5,000 kg truck traveling at 80 kph
λ=5.97 e-39m

18. May 3, 2005

dextercioby

Alright.Now did u find out how to derive this result...?

Daniel.

19. May 3, 2005

sigh taht is waht i am trying to find out... but no one reads on this fourm

and someone said im am just trying to get people to do my homework well how many people who want their homework done post the answer to their questions

20. May 3, 2005

dextercioby

You've been told that

$$\lambda \ \left(\mbox{m}\right)=\frac{h \ \left(\mbox{Kg}\cdot\frac{\mbox{m}}{\mbox{s}^{2}}\cdot\mbox{m}\cdot\mbox{s}\right)}{p \ \left(\mbox{Kg}\cdot\frac{\mbox{m}}{\mbox{s}}\right)}$$

Now plug the values paying attention to the units.Planck's constant is given in $\mbox{J}\cdot\mbox{s}$ which u can show to be equal to $\mbox{Kg}\cdot\frac{\mbox{m}}{\mbox{s}^{2}}\cdot\mbox{m}\cdot\mbox{s}$.

U need to find the truck's momentum in SI units.Basically,u need to convert its speed from $\frac{\mbox{Km}}{\mbox{hr}}$ to $\frac{\mbox{m}}{\mbox{s}}$.

Daniel.

Last edited: May 3, 2005


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# Odlyzko's method

Odlyzko [O1993] has a method for computing $\pi(x)$ in time/space $x^{1/2+o(1)}$; combining this with a binary search, one can locate a k-digit prime in time/space $(10^k)^{1/2+o(1)}$.

First observe that it suffices to compute $\pi(x)$ to accuracy 1/2, since it is an integer. Next, it suffices to count the slight variant sum $\sum_{n \leq x} \frac{\Lambda(n)}{\log n}$ up to this accuracy, since the error term is a sum over prime powers and this can be computed in $x^{1/2+o(1)}$ time/space (using e.g. the sieve of Eratosthenes to find all the primes up to $x^{1/2}$).

In fact, it suffices to compute the sum

$\sum_{n} c(n) \frac{\Lambda(n)}{\log n}$ (1)

where c is a computable function which equals 1 for $n \lt x - x^{1/2}$, vanishes for $n\gtx$, and is smooth in between. This is because the error can be computed in $x^{1/2+o(1)}$ time/space via another sieve of Erathosthenes. [Actually, by using Odlyzko's method recursively, one can reduce this portion of the cost to $x^{1/4+o(1)}$.]

The expression (1) can be rewritten using the zeta function as

$\frac{1}{2\pi i} \int_{2-i\infty}^{2+i\infty} F(s) \log \zeta(s)\ ds$

where

$c(u) = \frac{1}{2\pi i } \int_{2-i\infty}^{2+i\infty} F(s) u^{-s}\ du$.

It turns out that F is rapidly decreasing when $Im(s) \gg x^{1/2}$, so to get within accuracy 1/2, one only needs to evaluate the integral for $s = O(x^{1/2+o(1)})$. By quadrature, this reduces to evaluating $\zeta(s)$ (and its first few derivatives) to accuracy about $O(x^{-1/2+o(1)})$ for about $x^{1/2+o(1)}$ values of s in this range, but this can be done in many ways (e.g. by the Euler-Maclaurin formulae).


text:
# If C and D are two events such that C $$\subset$$ D and P(D) $$\ne$$ 0, then the correct statement among the following is

## Question :

If C and D are two events such that C $$\subset$$ D and P(D) $$\ne$$ 0, then the correct statement among the following is

(a) P(C/D) $$\ge$$ P(C)

(b) P(C/D) < P(C)

(c) P(C/D) = $$P(D)\over P(C)$$

(d) P(C/D) = P(C)

## Solution :

As P(C/D) = $$P(C \cap D)\over P(D)$$

= $$P(C)\over P(D)$$    …….(i)    [ $$\because$$ C $$\subset$$ D]

Also, as P(D) $$\le$$ 1

$$\therefore$$  $$1\over P(D)$$ $$\ge$$ 1

and $$P(C)\over P(D)$$ $$\ge$$ P(C)   …..(ii)

P(C/D) $$\ge$$ P(C)

### Similar Questions

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Two aeroplanes I and II bomb a target in succession. The probabilities of I and II scoring a hit correctly are 0.3 and 0.2, respectively. The second plane will bomb only if the first misses the target. The probability that the target is hit by the second plane, is

Consider 5 independent Bernoulli’s trials each with probability of success p. If the probability of atleast one failure is greater than or equal to $$31\over 32$$, then p lies in the interval

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text:
1. ## Neighborhoods

Let x and y be distinct real numbers. Prove that there is a nbhood P of x and a nbhood Q of y such that PintersectQ=null set.
So, we have (x-epsilon, x+epsilon) and (y-epsilon, y+epsilon).
I have an idea of how to do this but I'm getting stuck. So, I think I need to define epsilon, right because these are two different epsilons. This is where I get stuck. Can someone give me pointers on defining epsilon?

2. Originally Posted by kathrynmath
So, we have (x-epsilon, x+epsilon) and (y-epsilon, y+epsilon).
Let $\epsilon = |x-y|/2$.

3. Originally Posted by ThePerfectHacker
Let $\epsilon = |x-y|/2$.
How did you even get that?

4. Originally Posted by kathrynmath
How did you even get that?
Draw a picture. Draw a real number line and put $x$ and $y$. The distance between those points is $|x-y|$. Thus, if you draw a interval around them half the length $|x-y|/2$. Then those two intervals would not intersect.

5. Originally Posted by ThePerfectHacker
Draw a picture. Draw a real number line and put $x$ and $y$. The distance between those points is $|x-y|$. Thus, if you draw a interval around them half the length $|x-y|/2$. Then those two intervals would not intersect.
Ok, thanks. I get that.

6. I'm not quite sure how to prove it,though. I understand why it works and all, but I know a picture doesn't qualify as a proof.

7. Suppose that $\varepsilon = \frac{{\left| {x - y} \right|}}{2} > 0$ and that $z \in N_\varepsilon (x) \cap N_\varepsilon (y)$.
$\left| {x - y} \right| \leqslant \left| {x - z} \right| + \left| {z - y} \right| < \varepsilon + \varepsilon = \left| {x - y} \right|$

8. Originally Posted by Plato
Suppose that $\varepsilon = \frac{{\left| {x - y} \right|}}{2} > 0$ and that $z \in N_\varepsilon (x) \cap N_\varepsilon (y)$.
$\left| {x - y} \right| \leqslant \left| {x - z} \right| + \left| {z - y} \right| < \varepsilon + \varepsilon = \left| {x - y} \right|$

Ok, I think I understand this. So, is N the neighborhood?
Is this like a triangle inequality?

9. ${\color{blue}\left| {x - y} \right|} \leqslant \left| {x - z} \right| + \left| {z - y} \right| {\color{blue}<} \varepsilon + \varepsilon = {\color{blue}\left| {x - y} \right|}$
Do you see the contradiction in the blues?

10. Originally Posted by Plato
${\color{blue}\left| {x - y} \right|} \leqslant \left| {x - z} \right| + \left| {z - y} \right| {\color{blue}<} \varepsilon + \varepsilon = {\color{blue}\left| {x - y} \right|}$
Do you see the contradiction in the blues?
I'm trying to see the contradiction. Why is < epsilon a contradiciton? I'm not getting it....
and the x-y in the end appears correct because 2epsilon = x-y

11. Originally Posted by kathrynmath
I'm trying to see the contradiction. Why is < epsilon a contradiciton? I'm not getting it....
and the x-y in the end appears correct because 2epsilon = x-y
A number cannot be less than itself.
There in blue you see $\left| {x - y} \right| < \left| {x - y} \right|$.

12. Originally Posted by Plato
A number cannot be less than itself.
There in blue you see $\left| {x - y} \right| < \left| {x - y} \right|$.
Oh ok. So then this is a a contradicition and leads to the fact that P intersect Q = null set?


text:
# Empty space after a paragraph with { }

I want to add one blank space of line after I apply an independent centering as follows:

Bla bla bla. Figure 1 presents the results of the overall period: \\

{\centering [Insert Figure 1 here] \par} \\

Overall, the model seems to work well bla bla bla

While the first \ works and creates an empty line, the second \ does not seem to work. How can I resolve this?

• You should never use \\ to end a paragraph! Use center environment instead of \centering. It already adds vertical space. Or (better) use a figure environment and let the figure float. – Schweinebacke May 24 '17 at 16:23
• The \\ will have tex complaining about underfull hbox badness 10000, which is the maximum badness. remove the \\ the \centering and the \par and use \begin{center} which adds suitable space. – David Carlisle May 24 '17 at 16:31

\documentclass{article}

\begin{document}

Bla bla bla. Figure 1 presents the results of the overall period: \\

{\centering [Insert Figure 1 here] \par} \\

Overall, the model seems to work well bla bla bla

\end{document}

The first \\ produces the warning

Underfull \hbox (badness 10000) in paragraph at lines 5--6

which is the maximum level of badness reported by TeX. Never place a \\ at the end of a paragraph, it does not add vertical space it just forces the paragraph to have a spurious extra line with no content (hence the warning).

The second \\ generates an error message,

! LaTeX Error: There's no line here to end.

Never ignore errors, the way TeX recovers from errors is only designed to allow the rest of the file to be checked for further errors, it is not designed to make sensible pdf output. The error here is because \\ is following \par so specifying a line to end before the paragraph has started,

\documentclass{article}

\begin{document}

Bla bla bla. Figure 1 presents the results of the overall period:

\begin{center}
[Insert Figure 1 here]
\end{center}

Overall, the model seems to work well bla bla bla

\end{document}


text:
## Tags : divergence

Entries in this Tags : 10logs Showing : 1 - 10 / 10

## Jan 25, 2013

### Loosely bounding a Bregman divergence (without the Hessian)

Post @ 17:05:34 | Bregman divergence

Traditionally, the supremum of the norm of the Hessian of the Bregman generator over a closed domain is used in conjuction with the Lagrange remainder of the Taylor expansion to bound a Bregman divergence since we have:

But we can also bound Bregman divergence easily without the Hessian, by using the gradient, as follows:

I will report how to use this bound when defining and computing the Chernoff distance for exponential families.
Frank.

An update list of publications

## Nov 26, 2012

### Smooth transitions between left-sided and right-sided Bregman centroids (with java code):

One of the questions that often arises when using a Kullback-Leibler divergence is "how do we choose the orientation of KL?". In information retrieval, another related argument is "how to symmetrize Kullback-Leibler divergence?" (3 solutions: Jeffreys divergence, Jensen-Shannon divergence, Chernoff divergence). However, sometimes we would like to experiment more precise choices rather than left/right or symmetric orientation. One way to do so is to use skew Burbea-Rao divergences (also called skew Jensen divergences) that tend in limit cases to Bregman divergences:

Skewness factor may depend on applications although a principle way is yet to be analyzed. This sample program Bregman_BurbeaRao_Jensen_SkewDivergence.java demonstrates the skewness property in limit cases. Note that you do not have to compute gradient for approximating Bregman divergences. Here is a run of the program:

Skew Burbea-Rao-Jensen divergences: Bregman divergences obtained in limit case alpha=0 and Bregman reverse for limit case alpha=1
alpha=1.0E-4    0.06831750610769681
alpha=0.5       0.06073382594234067
alpha=0.9999    0.05521937447971147
Bregman direct :                0.0683193293834281       alpha->0
Reverse Bregman:                0.05521841361948579      alpha->1
...

alpha=1.0E-4    0.06831750610769681
alpha=0.5       0.06073382594234067
alpha=0.9999    0.05521937447971147
Bregman direct :                0.0683193293834281       alpha->0
Reverse Bregman:                0.05521841361948579      alpha->1

More details can be found in 1004.5049v3.pdf and 1009.4004v2.pdf
Frank.

## May 18, 2012

### Squared Euclidean distance is not a metric

Post @ 13:43:54 | divergence

## Jan 18, 2012

### Total Bregman divergence and Soft Clustering

Post @ 10:28:27 | Bregman divergence

Since the work of Banerjee et al. that showed that the expectation-maximization of mixtures of exponential families is a Bregman soft clustering in disguise, there has been a strong interest in further using the bijection between exp fam and Bregman divergences.

In
Shape Retrieval Using Hierarchical Total Bregman Soft Clustering
similarly it is proven that for total Bregman divergences (tBD), there exists a distribution which belongs to the lifted exponential family of statistical distributions This leads to a new clustering technique namely, the total Bregman soft clustering algorithm.

See paper.
Frank.

## Aug 23, 2011

### Expected Kullback-Leibler divergence between a multinomial and an empirical distribution

Post @ 11:01:29 | Kullback-Leibler divergence

Thanks for your valuable feedback and references.

Frank.

## Jun 09, 2010

### An asymptotic property of centroids with respect to f-divergences

Post @ 4:39:49 | f-divergence

f-centroids can be defined as the unique minimizer of the average Csiszar/Ali-Silvey f-divergence. All f-centroids are homogeneous (scale free, homogeneous degree 1).

f-centroids include power means (also reachable as Bregman centroids), extreme means, Lehmer mean, Gini mean, etc

An interesting property is by shifting the origin and letting the origin tend to infinity, all f-means become arithmetic mean.

The proof requires f to be C3, three times continously differentiable.

Frank.

## Jun 04, 2010

### Joyce and danger of Internet

Post @ 17:48:15 | f-divergence

We all daily use Internet and are biased by its non-controlled source of information. I was looking for the full name of the Ali and Silvey f-divergence paper, also independently studied by Csiszar. In those days, it was not rare to only mention the last name and give only initials, so it is hard to know for sure what those capital letters mean in

Ali, S. M.; Silvey, S. D. (1966). "A general class of coefficients of divergence of one distribution from another". Journal of the Royal Statistical Society, Series B 28 (1): 131?140.

After using several search tools (google x, x in plain, books, scholar, etc.), and tracing back previous papers, eventually writing some emails to Aligarh Muslim University, I figured out the full names of the celebrated paper, and finally completed by bibtex entry as follows:

% Computational information geometry bibtex file
@Article{AliSilvey-1966,
author =       {Syed Mumtaz Ali and Samuel David Silvey},
title =        {A general class of coefficients of divergence of one distribution from another},
journal =      {Journal of the Royal Statistical Society, Series B},
year =         {1966},
volume =   {28},
pages =    {131--142}
}

S. M. Ali was a PhD student of S. D. Silvey.

On a different side of Internet, I searched for some Kullback-Leibler properties, and figured out a weird french article mentionning

Kilolitre de divergence et mise Ã  jour bayÃ©sienne

In fact, it was the automated translation for

KL divergence and Bayesian updating

KL has been suggested by the automatic translator to be kiloliter... Then the translated article becomes really funny to read. Have a look http://www.worldlingo.com/ma/enwiki/fr/Kullback%E2%80%93Leibler_divergence (french) and http://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence (english)

Frank.

## Jan 07, 2010

### Variational distance is the only metric f-divergence

Post @ 22:40:41 | f-divergence

A quite interesting property is that the variational divergence (a f-divergence) and its multiples are the only metric f-divergences. This was recently shown
Confliction of the Convexity and Metric Properties in f-Divergences (IEICE 2007).

If instead, we assume f strictly positive concave (and self-dual f(x)=x f(1/x)) then the formula of f-divergences yields metrics.

Frank.

## Feb 27, 2009

### A unique characterization of alpha-divergences

Post @ 15:11:45 | alpha-divergence

Professor Shun-ichi Amari recently gave a talk at Ecole Polytechnique on information geometry. The latter point of his talk characterizes interestingly the alpha-divergences on positive measures (non-normalized distributions) as the intersection of f-divergences and Bregman divergences.

http://videolectures.net/etvc08_paris/

Of course, on probability measures, only the Kullback-Leibler divergence lies in the common intersection (and its dual).

## Feb 26, 2009

### Information monotonicity

Post @ 18:10:05 | f-divergence

Csiszar f-divergences have the property of information monotonicity. So take a positive array p with n bins and partition it into m bins P with the probability of falling in a bin being the sum of the probability of the atoms of that bin. Then the f-divergence of (p,q) should always be greater or equal to the f-divergence of the corresponding partitionned arrays. In other words:

f-div(p,q) >= f-div(P,Q)

That means that we can only loose information by aggregating atoms. That is one fairly reasonnable behavior of information.

More axiomatic characterization is detailed in:

CsiszÃ¡r, Imre. 2008. "Axiomatic Characterizations of Information Measures." Entropy 10, no. 3: 261-273.

Frank.