text:
## anonymous 4 years ago How exactly do I solve this equation? x2 - 12x = -36

1. anonymous

$x^2-12x+36=0$ now solve the quadratic

2. anonymous

gives x=6 can yuo see why

3. precal

you can use the quadratic formula or you can factor it you can also, look at its graph and see where it crosses the x axis

4. anonymous

here yuo can factor the eqn, Do you know how to do this?

5. anonymous

I don't understand how you got x = 6...

6. precal

|dw:1338677027789:dw|

7. precal

|dw:1338677062779:dw|

8. anonymous

I'm sorry... I'm so confused. /:

9. anonymous

Oh! Wait I think I get it!!

10. anonymous

Took me a little bit :P Thank you both a ton!!

11. anonymous

if you factorise the equation you get (x-6)(x-6) so when (x-6)=0 x=6 Do you see

12. anonymous

Here is how I practice factorising write the factorised end point i.e. (x+3)x+4) and expand the brackets to give$x^2+4x+3x+12 \rightarrow x^2+7x+12$ now |I take $x^2+7x+12$ and work backwards because I know it will definitely factorise

13. precal

|dw:1338678477379:dw|

14. precal

|dw:1338678551861:dw|

15. anonymous

wow that's the hard way! but it does always work. Try to factorise it before going through all this stress

16. precal

just presenting multiple ways to solve it. Quadratic formula works 100% of the time, factoring does not

17. anonymous

that's what I said :P


text:
# Sleeved Triaxial Test of a Bonded Material

Problem Statement

Note

To view this project, use the menu command Help ► Examples…. Choose “Example Applications/SleevedTriaxialTest” and select “SleevedTriaxialTest.f3prj” to load. The main data files used are shown at the end of this example. The remaining data files can be found in the project.

This example demonstrates the use of shell-based structural elements to perform a set of sleeved triaxial tests on a bonded-particle model (BPM). A model of this nature could be investigated using PFC alone. However, in that case, a fair amount of FISH would be required for the sleeved wall to mimic the behavior of an elastic membrane. Here, a sleeve is created from a shell and a BPM is synthesized inside the sleeve as seen in Figure 1. As described in detail in the Wall-Zone Coupling Scheme section, PFC walls are slaved to shell-based structural element. Contacts exist between PFC balls and wall facets; the contact forces and moments at each ball-facet contact are used to determine an equivalent force system to apply to the corresponding shell nodes. In this model, the specimen is created with no initial stress for simplicity and the platens are displaced with a constant velocity. Using this example as a basis, one could easily define the failure envelope of a BPM.

Figure 1: Geometry of the sleeved triaxial test. The BPM is created inside the sleeve.

Coupled PFC-FLAC3D Model

Prior to creating a coupled PFC-FLAC3D model, one must load the PFC modules. These are the DLLs that contain the code responsible for instantiating all PFC related model components. In addition, a set of modules exist to couple PFC walls with FLAC3D zones and shell-based structural elements. By default these modules are not loaded when the FLAC3D application is opened. One may either load these modules via the Tools ► Load PFC menu item or with the following commands:

program load module 'contact'

Once a project has been saved with any additional modules loaded, these modules will be loaded whenever the project is opened once again. Note that the additional modules remain loaded until the FLAC3D application is closed.

PFC balls are created within the cylindrical sleeve composed of FLAC3D shells. In order to create the same initial cloud of balls each time the data files are run, the model random command is used to initialize the random seed to a specified value. Changing the random seed will result in a different realization of balls, though statistically, the realizations will be the same. Unlike zones and structural elements, all PFC model components (i.e., balls, clumps, walls and contacts) can only exist within a user-specified model domain (see the model domain command). This restriction allows for contact detection and spatial searching to be simplified and optimized. As PFC always operates in large-strain mode, one must explicitly activate large-strain mode for zones and structural elements (see the model large-strain command). PFC, by default, solves the dynamic equations of motion using the real inertia properties of balls and clumps. In this model, we desire to operate with timestep scaling in PFC (see the model mechanical command), an approach which is similar to the default FLAC3D behavior.

With these preliminary steps complete, we dive into creating the initial model.

;define relevant parameters for the cylinder
[height = 3.0]
[segments = 6]
[halfLen = height/2.0]
[freeRegion = height/2.0*0.8]

;create the cylinder with the geometry logic
;first create arcs making a circle of edges
geometry edge create by-arc origin (0,0,[-halfLen]) ...
segments [segments]
geometry edge create by-arc origin (0,0,[-halfLen]) ...
segments [segments]
geometry edge create by-arc origin (0,0,[-halfLen]) ...
segments [segments]
geometry edge create by-arc origin (0,0,[-halfLen]) ...
segments [segments]
;extrude the edges to make a cylinder
geometry generate from-edges extrude (0,0,[height]) segments [segments*2]

The cylindrical sleeve is created in the geometry logic to be imported as a shell. Arcs are created as geometry edges and extruded vertically to create the cylindrical sleeve. Instead of using the ‘@’ operator, one can use the bracket notation (i.e., ‘[]’) to define FISH variables and to use these values in commands.

With the geometry created, we turn our attention to dealing with the structural elements.

;import the structural elements as a shell from the geometry logic
;and assign groups and properties
structure shell import from-geometry 'Default' element-type dkt-cst
structure node group 'middle' range position-z [-freeRegion] [freeRegion]
structure node group 'top' range position-z [freeRegion] [freeRegion+1]
structure node group 'bot' range position-z [-freeRegion-1] [-freeRegion]
structure shell group 'middle' range position-z [-freeRegion] [freeRegion]
structure shell property isotropic (1e6, 0.0) thick 0.25 density 930.0
;initialize the structural element related data structures
model cycle 0
;In order to use the STRUCTURE APPLY command
;set the local system of each structural
;element to be pointing to the center of the triaxial cell
fish define setLocalSystem
loop foreach local s struct.node.list()
local p = struct.node.pos(s)
local nid = struct.node.id.component(s)
local mvec = vector(0,0,comp.z(p))
zdir = math.unit(p-mvec)
ydir = vector(0,0,1)
command
structure node system-local z [zdir] y [ydir] ...
range component-id [nid]
endcommand
endloop
command
structure node fix system-local
endcommand
end
[setLocalSystem]
;make sure that local damping is active for the structural elements
structure damp local
;fix the structural element nodes for specimen creation
structure node fix velocity rotation

The structure shell import command is used to create the shell from the created geometry. Both element and node groups are assigned to simplify boundary condition assignment. The code snippet also shows a FISH function that is used to set the local axis system of each node as pointing toward the center of the cylinder so that the structure shell apply command can be used to apply a pressure on shell elements in the middle of the model. The z- direction points directly to the center of the cylinder and the y- directions points up. The velocity and rotation of the nodes are fixed so that balls can be equilibrated inside the sleeve.

Now it is time to create the walls for both the sleeve and the platens.

;create a wall representation of the structural element
wall-structure create
;parameter to set the platen width relative to the cylinder radius
;create the platens
;set the wall resolution strategy to full and set the cutoff angle for
;proximity contacts
wall resolution full
wall attribute cutoff-angle 20

The sleeve is created with the wall-structure create command. In this case wall facets are created for each shell face and the wall facet vertex velocities and positions are slaved to the corresponding shell nodes. The top and bottom platens are created with the wall generate command and the wall resolution mode is set to full with the wall resolution command. In general, contacts exist between PFC balls and wall facets. If the resolution mode is set the none, all contacts between facets and balls will be delineated and contribute to the force-displacement response of the balls. When the wall facet sizes are near to the ball sizes, though, this may produce many more contacts than if the wall was idealized as a perfect surface without facets. The full scheme is used to mitigate this situation, and the cutoff-angle is used to control the “smoothness” of the wall. The wall resolution is discussed in detail in the Faceted Walls in PFC topic.

After the walls are created, the BPM is created within the sleeve.

;set the ball modulus and generate a cloud of balls with arbitrary overlap
[ballemod = 1.0e8]
porosity 0.3 radius 0.05 0.1 ...
range cylinder end-1 (0,0,[-halfLen]) end-2 (0,0,[height]) ...

;set the ball attributes
ball attribute density 2600 damp 0.8
;set the default contact behavior -
;the deformability method sets properties of the
;linear portion of the contact model
contact cmat default model linearpbond method deformability ...
emod [ballemod] kratio 1.0
;allow the balls to rearrange,
;nulling the linear and angular velocities every 100 cycles
model cycle 1000 calm 100

;provide some friction to kill additional energy
;and apply this to all of the contacts
contact cmat default model linearpbond method deformability ...
emod [ballemod] kratio 1.0 property fric 0.3
contact cmat apply
;solve to a small average ratio
model solve ratio-average 1e-8

;bond the ball-ball contacts
contact method bond gap 1.0e-2 range contact type 'ball-ball'
;change the existing contact properties
;so that the linear force is incremental and
;supply strength
contact property lin_mode 1 lin_force 0 0 0 pb_ten 2e5 pb_coh 2e6
;set the stiffness of the parallel-bond portion of the contact model
contact method pb_deformability emod [ballemod] kratio 1.0 ...
range contact type 'ball-ball'

;this set of operations removes all ball velocities
;and all contact forces in the model
;so that the specimen is completely stress free and bonded
model calm
ball fix velocity spin
model cycle 2
ball free velocity spin

;free the nodes in the middle section of the sleeve
;while keeping the top and bottom edges fixed
structure node free velocity rotation range group 'middle'

PFC balls and clumps can either be created without initial overlap or with arbitrary initial overlap. In this case, the ball distribute command is used to create a cloud of balls with arbitrary overlaps to a specified volume fraction within a parallelepiped volume. Ball attributes are then assigned and the contact cmat command is used to specify the contact model behavior. With these steps taken the balls can be allowed to rearrange. The calm keyword in the model cycle command is used to periodically remove all kinetic energy from the system. Subsequently the contact cmat command is given to modify the Contact Model Assignment Table and this change is applied to all contacts. Some friction is supplied to remove additional energy from the system of balls is cycled to a small average ratio. The contact method bond command is given to create the BPM by installing parallel bonds at all ball-ball contacts with gap less than or equal to the specified value. A contact method is a set of operations that operate to perform a task on contacts, possibly modifying multiple contact properties. After the bonds have been installed, the parallel bond strength is specified and the normal force calculation of the linear part of the linear parallel bond contact model is set to incremental model by specifying lin_mode = 1. This allows one to produce a BPM quickly without internal stresses. In addition, the stiffnesses of the parallel bond component are set. By removing all velocities and fixing the ball velocity and spin, all contact forces are nulled after a few cycles. As a result, the BPM is in equilibrium with no internal stresses. Finally, the shell nodes in the middle of the sleeve are freed in preparation for the triaxial tests. The initial contacts of the BPM with no stess are shown in the figure below.

Figure 2: Initial contact forces of the BPM without stress.

With the stress-free BPM created, some additional work is required to build the environment to perform the triaxial tests. The first is a FISH function to calculate the stress and strain based on the platens.

;define the stress FISH function to measure the stress and strain
fish define stress
local topForce = math.abs(comp.z(wall.force.contact(platenTop)))
local botForce = math.abs(comp.z(wall.force.contact(platenBottom)))
currentStress = 0.5*(topForce+botForce)/area
stress = currentStress
strain = (height - (comp.z(wall.pos(platenTop)) - ...
comp.z(wall.pos(platenBottom))))/height * 100
if failureStress <= currentStress
failureStress = currentStress
failureStrain = strain
endif
end

Here the forces on the platens in the z- direction are used to calculate the average stress on the platens. The strain is also calculated based on the specimen height. The FISH parameters failureStress and failureStrain are used to record the peak stress and strain. This information is used in the FISH function below as a means to terminate cycling.

;define the halt FISH function to stop cycling
;as the platens displace and the material fails
fish define halt
halt = 0
if currentStress < failureStress * 0.85
halt = 1
endif
end

Once the current stress level is less than 85% of the maximum stress, cycling will cease. A FISH function can be used to terminate cycling with the model solve fish-halt command. Once the returned value of the function halt is 1 cycling will terminate.

The following FISH function is used to increase the confining stress on the specimen incrementally:

;define a FISH function to increase the isotropic confining pressure
;in increments so that the bonded material does not break during loading
fish define rampUp(beginIn,ending,increment)
command
ball attribute displacement (0,0,0)
structure node initialize displacement (0,0,0)
endcommand
begin = beginIn
loop while (math.abs(begin) < math.abs(ending))
begin = begin + increment
command
;apply the confining stress
structure shell apply [begin] range group 'middle'
;apply the same confining stress on the platens
wall servo force (0,0,[begin*area]) activate true ...
range name 'platenTop'
wall servo force (0,0,[-begin*area]) activate true ...
range name 'platenBottom'
model cycle 200
model calm
endcommand
endloop
command
;once the stress state has been installed cycle and turn off the servo
;mechanism on the walls
model cycle 1000
wall servo activate false
wall attribute velocity (0,0,0) range name 'platenTop'
wall attribute velocity (0,0,0) range name 'platenBottom'
endcommand
end

The core of this function is to incrementally increase the confining stress on both the platens and sleeve. Since we have specified the local system for the shells, it is simple to use the structure shell apply command to provide confining stress. Also, the built-in wall servo mechanism is employed, with the wall servo command, to provide stress on the platens. Now it is time to squeeze some synthetic materials.

Results and Discussion

First the unconfined compression (UCS) test is performed by removing the sleeve and fixing the velocities of the platens. The results of this tests are shown in the figure below. The broken bonds are also shown to demonstrate the pervasive damage of the BPM at failure. Note that the response is quite linear to failure and that the test automatically stopped at 85% of the peak stress. This peak UCS tests of this BPM was 2.20e+06 Pa at 1.13% strain.

Figure 3: Stress-strain response of the BPM to a UCS test along with the broken contact bonds.

To undertake the first triaxial test, the original BPM is loaded and the stress is ramped up from to 1e4 Pa in 1e3 Pa increments. The contacts are show in Figure 4. Notice that the stress is isotropic as the contacts retain nearly their original configuration. One is able to tell this as the contacts are plotted in the force vector direction as opposed to the direction connecting the balls in question.

Figure 4: Contact forces oriented in the direction of the force after the stress installation.

The test results with 1e4 Pa confinement are shown in Figure 5. The specimen failed at 1.15% strain with a peak stress of 2.16e+06 Pa. Figure 6 and Figure 7 show the stress-strain responses and broken contact bonds for 5e4 Pa and 1e5 Pa, respectively. The table below presents the peak stress at which the BPM failed for each level of confinement. Notice that, as with real materials, the peak strength increases with confinement. In addition, the stress-strain curves remain linear and similar in slope as with increasing confinement. The slight drop in stress with low confinement can be attributed to using the walls to measure the stress and the relatively small number of balls.

Table 1: Peak strength of BPM as a function of confinement stress
Confinement (Pa) Peak Strength (Pa) Strain (%)
0 2.20e+06 1.13
1e4 2.16e+06 1.15
5e4 2.60e+06 1.37
1e5 2.96e+06 1.54

Figure 5: Stress-strain response of the BPM to the triaxial test with 1e4 confinement along with the broken contact bonds.

Figure 6: Stress-strain response of the BPM to the triaxial test with 5e4 confinement along with the broken contact bonds.

Figure 7: Stress-strain response of the BPM to the triaxial test with 1e5 confinement along with the broken contact bonds.

Taking this example as a starting point, one can effectively delineated the failure envelope of a BPM under realistic triaxial conditions.

Data Files

SleevedTriaxialTest.dat

;-------------------------------------------------------------------
;                Sleeved Triaxial Test of a Bonded
;                Material
;-------------------------------------------------------------------
model new
model title 'Sleeved Triaxial Test of a Bonded Material'
;make repeatable by setting the random number seed
model random 10001
;set the model domain for PFC balls and walls
model domain extent -2 2 condition destroy
;largestrain mode must always be on for coupled simulations
model large-strain on
;apply timestep scaling so the PFC timestep will be 1
model mechanical timestep scale

;define relevant parameters for the cylinder
[height = 3.0]
[segments = 6]
[halfLen = height/2.0]
[freeRegion = height/2.0*0.8]

;create the cylinder with the geometry logic
;first create arcs making a circle of edges
geometry edge create by-arc origin (0,0,[-halfLen]) ...
segments [segments]
geometry edge create by-arc origin (0,0,[-halfLen]) ...
segments [segments]
geometry edge create by-arc origin (0,0,[-halfLen]) ...
segments [segments]
geometry edge create by-arc origin (0,0,[-halfLen]) ...
segments [segments]
;extrude the edges to make a cylinder
geometry generate from-edges extrude (0,0,[height]) segments [segments*2]

;import the structural elements as a shell from the geometry logic
;and assign groups and properties
structure shell import from-geometry 'Default' element-type dkt-cst
structure node group 'middle' range position-z [-freeRegion] [freeRegion]
structure node group 'top' range position-z [freeRegion] [freeRegion+1]
structure node group 'bot' range position-z [-freeRegion-1] [-freeRegion]
structure shell group 'middle' range position-z [-freeRegion] [freeRegion]
structure shell property isotropic (1e6, 0.0) thick 0.25 density 930.0
;initialize the structural element related data structures
model cycle 0
;In order to use the STRUCTURE APPLY command
;set the local system of each structural
;element to be pointing to the center of the triaxial cell
fish define setLocalSystem
loop foreach local s struct.node.list()
local p = struct.node.pos(s)
local nid = struct.node.id.component(s)
local mvec = vector(0,0,comp.z(p))
zdir = math.unit(p-mvec)
ydir = vector(0,0,1)
command
structure node system-local z [zdir] y [ydir] ...
range component-id [nid]
endcommand
endloop
command
structure node fix system-local
endcommand
end
[setLocalSystem]
;make sure that local damping is active for the structural elements
structure damp local
;fix the structural element nodes for specimen creation
structure node fix velocity rotation

;create a wall representation of the structural element
wall-structure create
;parameter to set the platen width relative to the cylinder radius
;create the platens
;set the wall resolution strategy to full and set the cutoff angle for
;proximity contacts
wall resolution full
wall attribute cutoff-angle 20

;set the ball modulus and generate a cloud of balls with arbitrary overlap
[ballemod = 1.0e8]
porosity 0.3 radius 0.05 0.1 ...
range cylinder end-1 (0,0,[-halfLen]) end-2 (0,0,[height]) ...

;set the ball attributes
ball attribute density 2600 damp 0.8
;set the default contact behavior -
;the deformability method sets properties of the
;linear portion of the contact model
contact cmat default model linearpbond method deformability ...
emod [ballemod] kratio 1.0
;allow the balls to rearrange,
;nulling the linear and angular velocities every 100 cycles
model cycle 1000 calm 100

;provide some friction to kill additional energy
;and apply this to all of the contacts
contact cmat default model linearpbond method deformability ...
emod [ballemod] kratio 1.0 property fric 0.3
contact cmat apply
;solve to a small average ratio
model solve ratio-average 1e-8

;bond the ball-ball contacts
contact method bond gap 1.0e-2 range contact type 'ball-ball'
;change the existing contact properties
;so that the linear force is incremental and
;supply strength
contact property lin_mode 1 lin_force 0 0 0 pb_ten 2e5 pb_coh 2e6
;set the stiffness of the parallel-bond portion of the contact model
contact method pb_deformability emod [ballemod] kratio 1.0 ...
range contact type 'ball-ball'

;this set of operations removes all ball velocities
;and all contact forces in the model
;so that the specimen is completely stress free and bonded
model calm
ball fix velocity spin
model cycle 2
ball free velocity spin

;free the nodes in the middle section of the sleeve
;while keeping the top and bottom edges fixed
structure node free velocity rotation range group 'middle'

;function for calculating stress and strain as the platens are displaced
;these values will be recorded as a history
;first find the top and bottom platens
[platenTop = wall.find('platenTop')]
[platenBottom = wall.find('platenBottom')]
;define some variables for the calculation
[failureStress = 0]
[currentStress = 0]
[failureStrain = 0]
;define the stress FISH function to measure the stress and strain
fish define stress
local topForce = math.abs(comp.z(wall.force.contact(platenTop)))
local botForce = math.abs(comp.z(wall.force.contact(platenBottom)))
currentStress = 0.5*(topForce+botForce)/area
stress = currentStress
strain = (height - (comp.z(wall.pos(platenTop)) - ...
comp.z(wall.pos(platenBottom))))/height * 100
if failureStress <= currentStress
failureStress = currentStress
failureStrain = strain
endif
end

;define the halt FISH function to stop cycling
;as the platens displace and the material fails
fish define halt
halt = 0
if currentStress < failureStress * 0.85
halt = 1
endif
end

;define a FISH function to increase the isotropic confining pressure
;in increments so that the bonded material does not break during loading
fish define rampUp(beginIn,ending,increment)
command
ball attribute displacement (0,0,0)
structure node initialize displacement (0,0,0)
endcommand
begin = beginIn
loop while (math.abs(begin) < math.abs(ending))
begin = begin + increment
command
;apply the confining stress
structure shell apply [begin] range group 'middle'
;apply the same confining stress on the platens
wall servo force (0,0,[begin*area]) activate true ...
range name 'platenTop'
wall servo force (0,0,[-begin*area]) activate true ...
range name 'platenBottom'
model cycle 200
model calm
endcommand
endloop
command
;once the stress state has been installed cycle and turn off the servo
;mechanism on the walls
model cycle 1000
wall servo activate false
wall attribute velocity (0,0,0) range name 'platenTop'
wall attribute velocity (0,0,0) range name 'platenBottom'
endcommand
end
;set the platen velocity
[platenVel = 0.000003]

;save the model for future use, including these FISH utility function, before
;any confinement has been applied
model save 'beforeApplication'

;-------------------------------------------------------------------
;test 1: perform a UCS test on the specimen
structure shell delete
wall attribute velocity-z [-platenVel] range name 'platenTop'
wall attribute velocity-z [platenVel] range name 'platenBottom'
ball attribute displacement (0,0,0)
fish history stress
fish history strain
model solve fish-halt halt
model save 'ucs'
[io.out(string(failureStress) + 'Pa ')]
[io.out('at' + string(failureStrain) + '% strain')]

;-------------------------------------------------------------------
;test 2: perform a triaxial test with isotropic confining stress 1e4
model restore 'beforeApplication'
[rampUp(0,-1e4,-1e3)]
model save 'to1e4'
wall attribute velocity-z [-platenVel] range name 'platenTop'
wall attribute velocity-z [platenVel] range name 'platenBottom'
ball attribute displacement (0,0,0)
structure node initialize displacement (0,0,0)
fish history stress
fish history strain
model solve fish-halt halt
model save 'triaxial1e4'
[io.out(string(failureStress) + 'Pa ')]
[io.out('at' + string(failureStrain) + '% strain')]

;-------------------------------------------------------------------
;test 3: perform a triaxial test with isotropic confining stress 5e4
model restore 'to1e4'
[rampUp(-1e4,-5e4,-1e3)]
model save 'to5e4'
wall attribute velocity-z [-platenVel] range name 'platenTop'
wall attribute velocity-z [platenVel] range name 'platenBottom'
ball attribute displacement (0,0,0)
structure node initialize displacement (0,0,0)
fish history stress
fish history strain
model solve fish-halt halt
model save 'triaxial5e4'
[io.out(string(failureStress) + 'Pa ')]
[io.out('at' + string(failureStrain) + '% strain')]

;-------------------------------------------------------------------
;test 4: perform a triaxial test with isotropic confining stress 1e5
model restore 'to5e4'
[rampUp(-5e4,-1e5,-1e3)]
model save 'to1e5'
wall attribute velocity-z [-platenVel] range name 'platenTop'
wall attribute velocity-z [platenVel] range name 'platenBottom'
ball attribute displacement (0,0,0)
structure node initialize displacement (0,0,0)
fish history stress
fish history strain
model solve fish-halt halt
model save 'triaxial1e5'
[io.out(string(failureStress) + 'Pa ')]
[io.out('at' + string(failureStrain) + '% strain')]

format: plain


text:
Chapter 5, Problem 21AT

### Contemporary Mathematics for Busin...

8th Edition
Robert Brechner + 1 other
ISBN: 9781305585447

Chapter
Section

### Contemporary Mathematics for Busin...

8th Edition
Robert Brechner + 1 other
ISBN: 9781305585447
Textbook Problem

# Set up and solve equations for each of the following business situations.Discount Electronics ordered three dozen cell phones from the manufacturer. If the total order amounted to $1,980, what was the cost of each phone? To determine To calculate: The cost of one phone if the discount electronics ordered 3 dozen cell phones in the amount of$1,980 from the manufacturer.

Explanation

Given Information:

The discount electronics ordered 3 dozen cell phones in the amount of \$1,980 from the manufacturer.

Formula used:

To solve the equation and prove the solution, follow the steps as mentioned below.

Step 1: Understand the problem and write the facts of the equation.

Step2: List the known and the unknows from the problem.

Step3: Write the equation with the help of following table of key word,

 Equal Sign Addition Subtraction Multiplication Division Parentheses Is And Less of Divide Times the quantity of Are Added to Less than Multiply Divided by Was Totals Smaller than Times Divided into Equals The sum of Minus Product of Quotient of Gives Plus Difference Multiplied by Ratio of Giving More than Decreased by Twice Leaves Larger than Reduced by Double Results in Increased by Take away Triple Produces Greater than Loss of At yields exceeds Fewer than @

Step4: Solve the equation by rearrange the unknown values to the left side and known values to the right side of the equal sign.

Step5: Prove the solution by substituting the value obtained from step 4 in the equation formed in step 4 then check the left and right of the equation if, they are equal then the answer is correct

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text:
# A calibration scheme for quantitative concentration measurements using simultaneous PIV and PLIF

A calibration scheme for quantitative concentration measurements using simultaneous PIV and PLIF A new approach for calibration of planar laser-induced fluorescence (PLIF) measurements is presented. The calibration scheme is based on the fact that there is a constant concentration flux through each cross-section of a fluorescent plume in a given flow field and makes use of simultaneous measurements of particle image velocimetry (PIV) and PLIF. The following are the advantages of the current technique: (1) it is experimentally less demanding and (2) it does not require in situ calibration for generating the calibration curves. The technique can be implemented in many experimental setups (both in water and gaseous flows) provided the geometry of the time-averaged scalar field is known. Using the calibration scheme, an analysis is carried out on the measurements of concentration fields in grid turbulence to validate the proposed technique. To demonstrate the feasibility of the scheme, the distributed second-order moments (μ 2), and concentration and velocity correlations ( $$\left\langle {u^{\prime}c^{\prime}} \right\rangle$$ and $$\left\langle {v^{\prime}c^{\prime}} \right\rangle$$ ) are computed. Good agreement is found with previous studies. In addition, a quantitative appraisal of a simple closure approximation of the moment-based transport equation is also presented using simultaneous PIV and PLIF. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Experiments in Fluids Springer Journals

# A calibration scheme for quantitative concentration measurements using simultaneous PIV and PLIF

, Volume 52 (1) – Oct 28, 2011
13 pages

/lp/springer_journal/a-calibration-scheme-for-quantitative-concentration-measurements-using-OmhN3r6BTQ
Publisher
Springer-Verlag
Subject
Engineering; Engineering Fluid Dynamics; Engineering Thermodynamics, Heat and Mass Transfer; Fluid- and Aerodynamics
ISSN
0723-4864
eISSN
1432-1114
D.O.I.
10.1007/s00348-011-1219-7
Publisher site
See Article on Publisher Site

### Abstract

A new approach for calibration of planar laser-induced fluorescence (PLIF) measurements is presented. The calibration scheme is based on the fact that there is a constant concentration flux through each cross-section of a fluorescent plume in a given flow field and makes use of simultaneous measurements of particle image velocimetry (PIV) and PLIF. The following are the advantages of the current technique: (1) it is experimentally less demanding and (2) it does not require in situ calibration for generating the calibration curves. The technique can be implemented in many experimental setups (both in water and gaseous flows) provided the geometry of the time-averaged scalar field is known. Using the calibration scheme, an analysis is carried out on the measurements of concentration fields in grid turbulence to validate the proposed technique. To demonstrate the feasibility of the scheme, the distributed second-order moments (μ 2), and concentration and velocity correlations ( $$\left\langle {u^{\prime}c^{\prime}} \right\rangle$$ and $$\left\langle {v^{\prime}c^{\prime}} \right\rangle$$ ) are computed. Good agreement is found with previous studies. In addition, a quantitative appraisal of a simple closure approximation of the moment-based transport equation is also presented using simultaneous PIV and PLIF.

### Journal

Experiments in FluidsSpringer Journals

Published: Oct 28, 2011

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Question

# A ray of light travelling in a transparent medium falls on a surface separating the medium from air, at an angle of incidence of ${{45}^{o}}$. The ray undergoes total internal reflection. If $n$is the refractive index of the medium with respect to air, select the possible values of $n$ from the followinga)$1.3$b)$1.4$c)$1.5$d)$1.2$

Hint: The angle of incidence is given in the question, so, substitute the value in the total internal reflection formula. The value of the refractive index of air is one. For total internal refraction to occur, the angle of incidence must exceed the critical angle.

Formula used:
$i>{{\sin }^{-1}}(\dfrac{1}{\mu })$

For total internal reflection to occur, the angle of incidence must be greater than the critical angle.
Let the refractive index be assumed as $\mu$and the angle of incidence as $i$,
The critical angle is calculated as,
${{\sin }^{-1}}(\dfrac{1}{\mu })$
Therefore, the formula turns into,
\begin{align} & i > {{\sin }^{-1}}(\dfrac{1}{\mu }) \\ & \sin (i) > \dfrac{1}{\mu } \\ & \sin ({{45}^{0}}) > \dfrac{1}{\mu } \\ & \mu > \sqrt{2} \\ & \mu > 1.44 \\ & \\ \end{align}

So, the correct answer is “Option C”.


text:
# Find the value of 'n' in a Taylor series.

I have the following problem: I want to find the value of $n$ (degree of the Taylor series) such that the taylor polynomial $Pn(x)$ approximates the function $f(x)$ with an error no greater than $10^{-t}$ in the interval $[a,b]$. So, i have this: $$\frac{\lvert f^{(n+1)}(\xi)\lvert}{(n+1)!} * \lvert(x-\bar{x})^{(n+1)}\lvert\le 10^{-t}$$

My problem is: How can i solve for n?, What can i do with the $n$ at $\lvert f^{(n+1)}(\xi)\lvert$ where $(n+1)$ is the derivative? The idea is to use the formula to calculate $n$ with any specific $f(x)$. Thank you.

• Generally, just plug in a few values. Without more information on the specific problem, this is the best I advice I can give you. – Simply Beautiful Art Aug 25 '17 at 1:09
• The best I think we can do is to first try to get an exact form or an approximation for $f^{(n)}(\xi)$. You can get some suggestions here and here. – Prasun Biswas Aug 25 '17 at 1:57

I tried to do something similar some months ago and this is what I got so far. Maybe it helps you. And no, I did not manage to solve this problem completely.

First thing, the function has to be analytic on that domain, so that the Taylor series converges to the function.

I will start with an observation related to the way you formulated the question. I will take $\bar x=0$ for convenience in writing the relations.

Let $T_n(x)$ be the Taylor polynomial of degree $n$ and $\epsilon$ the maximum value of the difference between $T_n(x)$ and $f(x)$ on a neighborhood around $\bar x = 0$. In other words, if you fix $\epsilon$, then $|T_n(x)-f(x)|<\epsilon$ only for $|x|<x_{\epsilon}$. So you can take values for $x$ only inside the interval $(-x_{\epsilon}, x_{\epsilon})$ so that $|T_n(x)-f(x)|<\epsilon$. There are 2 aspects here:

1. If you take bigger values of $\epsilon$ but keep the degree of the Taylor polynomial fixed, the allowed interval $(-x_{\epsilon}, x_{\epsilon})$ for $x$ will get bigger.
2. If you keep the same value for $\epsilon$ but take Taylor polynomials of greater degree, the interval $(-x_{\epsilon}, x_{\epsilon})$ will also get bigger.

Those result should be easy to check if you want to. What is important for your question is that you can approximate the function on an interval $(-x_{\epsilon}, x_{\epsilon})$, and that interval gets bigger as you take a Taylor polynomial of greater degree. So by incrementing the value of $n$ you 'cover' a bigger domain of what you considered as $[a,b]$. Now where is the problem with this. Based on observation 2. if you consider your Taylor polynomial centered at $\bar x = a$ and want to cover the interval $[a,b]$, you will actually need to cover the interval $[a-(b-a), b]$ which is double of what you initially needed. A different situation is if you start at the center of $[a,b]$ that is $\bar x = a+\frac{b-a}{2}$. In this scenario you will have to cover only $[a,b]$. So what should be clear from those 2 cases is that in the first one you will need a greater degree for the Taylor polynomial than in the second scenario, so your problem also depends on the variable $\bar x$. Thus, you have an equation with 3 variables ($\bar x$, $n$ and $\xi$), and trying to solve this won't get you very close to what you probably wanted. That is unless you impose some additional conditions.

Now, about the evaluation part, assuming that $\bar x = a+\frac{b-a}{2}$. As Prasun Biswas linked in his comment, in order to evaluate the reminder you'll have to be able to say something about $f^{n}(x)$. What I tried was to write my function as:

$$f(x) = \sum_{j=1}^{N} a_j sin(\omega _j x)$$

or using any other function that had a general formula for the $n$-th derivative. As an example I also considered writing the function as a discrete version of a Weierstrass transform.

In this form, you should be able to take each term $a_j sin(\omega _j x)$, plug it into the reminder formula, and see the contribution of each of them on the sum. At this point the $j$ term of the reminder should be something like this:

$$|a_j sin(\omega _j \xi)\frac{(x-\bar x)^{n+1}}{(n+1)!}\omega_j^{n+1}|$$

and your analysis should be focused on:

$$|\frac{(x-\bar x)^{n+1}}{(n+1)!}\omega_j^{n+1}|$$

I hope this helps you somehow.


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• 1.
Umeå University, Faculty of Science and Technology, Department of Physics.
Umeå University, Faculty of Science and Technology, Department of Physics.
Flow-flame interaction in a closed chamber2008In: Physics of fluids, ISSN 1070-6631, E-ISSN 1089-7666, Vol. 20, no 5, p. 055107-055121Article in journal (Refereed)

Numerous studies of flame interaction with a single vortex andrecent simulations of burning in vortex arrays in open tubesdemonstrated the same tendency for the turbulent burning rate$\propto U_{rms}\lambda^{2/3}$, where  $U_{rms}$ is theroot-mean-square velocity and $\lambda$ is the vortex size. Here itis demonstrated that this tendency is not universal for turbulentburning. Flame interaction with vortex arrays is investigated forthe geometry of a closed burning chamber using direct numericalsimulations of the complete set of gas-dynamic combustion equations.Various initial conditions in the chamber are considered, includinggas at rest and several systems of vortices of different intensitiesand sizes. It is found that the burning rate in a closed chamber(inverse burning time) depends strongly on the vortex intensity; atsufficiently high intensities it increases with $U_{rms}$approximately linearly in agreement with the above tendency. On thecontrary, dependence of the burning rate on the vortex size isnon-monotonic and qualitatively different from the law$\lambda^{2/3}$. It is shown that there is an optimal vortex size ina closed chamber, which provides the fastest total burning rate. Inthe present work the optimal size is 6 times smaller than thechamber height.

• 2.
Princeton University.
Princeton University. Umeå University, Faculty of Science and Technology, Department of Physics. CTH.
Analysis of flame acceleration induced by wall friction in open tubes2010In: Physics of fluids, ISSN 1070-6631, E-ISSN 1089-7666, Vol. 22, p. 053606-Article in journal (Refereed)

Spontaneous flame acceleration leading to explosion triggering in open tubes/channels due to wall friction was analytically and computationally studied. It was first demonstrated that the acceleration is affected when the thermal expansion across the flame exceeds a critical value depending on the combustion configuration. For the axisymmetric flame propagation in cylindrical tubes with both ends open, a theory of the initial (exponential) stage of flame acceleration in the quasi-isobaric limit was developed and substantiated by extensive numerical simulation of the hydrodynamics and combustion with an Arrhenius reaction. The dynamics of the flame shape, velocity, and acceleration rate, as well as the velocity profile ahead and behind the flame, have been determined.

• 3.
Department of Mechanical and Aerospace Engineering, West Virginia University, Morgantown, West Virginia 26506, USA.
Department of Mechanical and Aerospace Engineering, West Virginia University, Morgantown, West Virginia 26506, USA. Umeå University, Faculty of Science and Technology, Department of Applied Physics and Electronics. Center for Combustion Energy, Key Laboratory for Thermal Science and Power Engineering of Ministry of Education of China, Department of Energy and Power Engineering, Tsinghua University, Beijing, China. Department of Mechanical and Aerospace Engineering, West Virginia University, Morgantown, West Virginia 26506, USA.
Impact of the Lewis number on finger flame acceleration at the early stage of burning in channels and tubes2019In: Physics of fluids, ISSN 1070-6631, E-ISSN 1089-7666, Vol. 31, no 8, article id 083606Article in journal (Refereed)

For premixed combustion in channels and tubes with one end open, when a flame is ignited at the centerline at the closed end of the pipe and it propagates toward the open one, significant flame acceleration occurs at an early stage of the combustion process due to formation of a finger-shaped flame front. This scenario is tagged “finger flame acceleration” (FFA), involving an initially hemispherical flame kernel, which subsequently acquires a finger shape with increasing surface area of the flame front. Previous analytical and computational studies of FFA employed a conventional assumption of equidiffusivity when the thermal-to-mass-diffusivity ratio (the Lewis number) is unity (Le = 1). However, combustion is oftentimes nonequidiffusive (Le ≠ 1) in practice such that there has been a need to identify the role of Le in FFA. This demand is addressed in the present work. Specifically, the dynamics and morphology of the Le ≠ 1 flames in two-dimensional (2D) channels and cylindrical tubes are scrutinized by means of the computational simulations of the fully compressible reacting flow equations, and the role of Le is identified. Specifically, the Le > 1 flames accelerate slower as compared with the equidiffusive ones. In contrast, the Le < 1 flames acquire stronger distortion of the front, experience the diffusional-thermal combustion instability, and thereby accelerate much faster than the Le = 1 flames. In addition, combustion in a cylindrical configuration shows stronger FFA than that under the same burning conditions in a 2D planar geometry.

• 4.
FFA, the Aeronautical Research Institute of Sweden, Bromma.
FFA, the Aeronautical Research Institute of Sweden, Bromma. FFA, the Aeronautical Research Institute of Sweden, Bromma.
Optimal disturbances and bypass transition in boundary layers1999In: Physics of fluids, ISSN 1070-6631, E-ISSN 1089-7666, Vol. 11, no 1, p. 134-150Article in journal (Refereed)
• 5.
Department of Mechanical and Aerospace Engineering, West Virginia University, Morgantown, West Virginia 26506-6106, USA.
Umeå University, Faculty of Science and Technology, Department of Physics. Department of Mechanical and Aerospace Engineering, West Virginia University, Morgantown, West Virginia 26506-6106, USA. Umeå University, Faculty of Science and Technology, Department of Applied Physics and Electronics. Center for Combustion Energy, Tsinghua University, Beijing, China. Department of Mechanical and Aerospace Engineering, West Virginia University, Morgantown, West Virginia 26506-6106, USA.
Critical role of blockage ratio for flame acceleration in channels with tightly spaced obstacles2016In: Physics of fluids, ISSN 1070-6631, E-ISSN 1089-7666, Vol. 28, no 9, article id 093602Article in journal (Refereed)

A conceptually laminar mechanism of extremely fast flame acceleration in obstructed channels, identified by Bychkov et al. ["Physical mechanism of ultrafast flame acceleration," Phys. Rev. Lett. 101, 164501 (2008)], is further studied by means of analytical endeavors and computational simulations of compressible hydrodynamic and combustion equations. Specifically, it is shown how the obstacles length, distance between the obstacles, channel width, and thermal boundary conditions at the walls modify flamepropagation through a comb-shaped array of parallel thin obstacles. Adiabatic and isothermal (cold and preheated) side walls are considered, obtaining minor difference between these cases, which opposes the unobstructed channel case, where adiabatic and isothermal walls provide qualitatively different regimes offlame propagation. Variations of the obstructed channel width also provide a minor influence on flamepropagation, justifying a scale-invariant nature of this acceleration mechanism. In contrast, the spacing between obstacles has a significant role, although it is weaker than that of the blockage ratio (defined as the fraction of the channel blocked by obstacles), which is the key parameter of the problem. Evolution of the burning velocity and the dependence of the flame acceleration rate on the blockage ratio are quantified. The critical blockage ratio, providing the limitations for the acceleration mechanism in channels with comb-shaped obstacles array, is found analytically and numerically, with good agreement between both approaches. Additionally, this comb-shaped obstacles-driven acceleration is compared to finger flameacceleration and to that produced by wall friction.

• 6.
Princeton University.
Umeå University, Faculty of Science and Technology, Department of Physics. Princeton University. CTH. Princeton University.
Quasi-steady stages in the process of premixed flame acceleration in narrow channels2013In: Physics of fluids, ISSN 1070-6631, E-ISSN 1089-7666, Vol. 25, no 9, p. 096101-Article in journal (Refereed)
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Fast Steering Mirror Example

 Fast Steering Mirror Example All Examples All Modeling Articles In order to prevent spam, users must register before they can edit or create articles.

## 1 Simple Mirror Model

Modeling of even simple systems, such as a Fast Steering Mirror (FSM), can appear complex to those who have not done much modeling. This section will include a simple model of the system. Being a simple model the following items will be ignored:

• Noise
• Conversion factors (Amps to torque for driving circuits, Volts to radians for position sensors, etc.)

The simple example will focus on the creation of 2nd order system models for each piece of the system. Those pieces are:

1. mirror
2. driver circuit
3. position
4. controller

At the end of the simple example the complexities of a true to hardware model will be addressed.

### 1.1 Mirror

The chosen mirror is by Axsys Technologies Imaging Systems and it is described in this whitepaper (used to be free in a couple of places but I can only find it as a paid item anymore).

Figure 1: FSM Block Diagram

Keep Figure 1 in mind as the example progresses. $LaTeX: G(s)$ is the plant, in this case the physical dynamics of the mirror and its flexures. $LaTeX: A(s)$ is the mirror's driver circuit. $LaTeX: H(s)$ is the feedback sensor, in this example a position sensor built by Kaman. $LaTeX: W(s)$ is a prefilter that could be designed but is unnecessary in this example so $LaTeX: W(s)=1$. The reference command $LaTeX: r(s)$ is typically a step command provided by another sensor somewhere else in the parent system. This example will follow along the Axsys whitepaper's laser communications example. For laser communications the outside sensor providing the reference command would typically be an optical feedback sensor which returned the position of the communication target relative to the platform this mirror resided on.

$LaTeX: K(s)$ is what needs to be designed.

#### 1.1.1 Mirror Parameters

The whitepaper suggests that the mirror can be accurately described by a 2nd order system. Therefore the governing transfer function is

 $LaTeX: H(s)=K\frac{\omega_n^2}{s^2+2\zeta\omega_n s+\omega_n^2} \mbox{ for } 0 \le \zeta \le 1$ Generic 2nd Order System

where
$LaTeX: K$ is the system gain
$LaTeX: \omega_n$ is the system's natural frequency
$LaTeX: \zeta$ is the system's damping ratio.
Figure 2: Single-Axis Mirror Plant Transfer Function

For the Fast Sterring Mirror described in the whitepaper:

$LaTeX: K=1$
$LaTeX: \omega_n \approx 23$ Hz
$LaTeX: \zeta \approx 0.3875$

Since the mirror is our plant and the plant transfer function is denoted by $LaTeX: G(s)$ in Figure 1 replace $LaTeX: H(s)$ with $LaTeX: G(s)$.

Figure 2 shows the frequency response of the Axsys FSM. The whitepaper provides only 1 value for $LaTeX: \omega_n$. So only a single axis is modeled. The 2nd axis could be modeled identically, at least in early design stages.

#### 1.1.2 Driver Circuit

Figure 9 of the whitepaper includes a circuit which converts voltage commands from the controller to amps used by the mirror motor.

Figure 3: 10 kHz Driver Circuit Transfer Function

The driver circuit is not described in detail. However, the proposed closed loop bandwidth is between 500 and 1000 Hz. This requires that the driver circuit have a bandwidth of at least 10x the maximum desired closed loop bandwidth. This means the driver circuit has a minimum bandwidth of 10 kHz. The driver circuit will be modeled with a 2nd order system with the following parameters:

$LaTeX: K=1$
$LaTeX: \omega_n=10$ kHz
$LaTeX: \zeta=\frac{1}{\sqrt{2}}$

In Figure 1 the driver circuit is denoted by $LaTeX: A(s)$. With the bandwidth of the driver circuit at only 10 kHz it will effect the phase margin of any system with a closed loop bandwidth of near 1 kHz. If the phase margin was not going to be effected we could model the driver circuit as a unity gain.

#### 1.1.3 Sensor Parameters

Figure 10 of the whitepaper includes a position sensor. The position sensor is not described but a common sensor for FSMs is a differential impedence transducer.

Figure 4: 20 kHz Position Sensor Transfer Function

A position sensor for mirror feedback is required when the system is meant to follow step commands. The Kaman DIT 5200 is described in this datasheet. The datasheet provides the following parameters

$LaTeX: K=1$
$LaTeX: \omega_n=20$ kHz
$LaTeX: \zeta=\frac{1}{\sqrt{2}}$

The sensor is traditionally denoted with $LaTeX: H(s)$. With the bandwidth of the sensor at only 20 kHz it will also effect the phase margin of any system with a closed loop bandwidth of near 1 kHz.

### 1.2 Controller

The Figure 9 of whitepaper also inlcudes reference to a PID controller with a parallel Integral Control and Lead Lag Controller network. The authors present a circuit drawing of their controller. They show a bandwidth of 560 Hz with closed loop peaking of 8 dB.

#### 1.2.1 Plant and Sensor System

Combining the Mirror, Driver Circuit, and Sensor transfer functions we get an overall transfer function.

Figure 5: Combined system to be controlled

Below are controller examples.

#### 1.2.2 Axsys Whitepaper Author's Controller

Figure 6: Frequency response of Axsys FSM controller

The author's of the Axsys FSM whitepaper presented what they called a PID controller. However, the form of the controller is a Lead/Lag network in parallel with an integrator.

Figure 7: Simulink diagram for Axsys FSM controller
Figure 8: Frequncy response of closed loop response

Below are the system metrics of for the closed loop system

A phase margin of 24 degrees is not very good. The whitepaper reports a closed loop bandwidth of 560 Hz and a phase margin of 35 degrees. A phase margin of 35 degrees is just barely good enough. The 8 dB of closed loop peaking is related to the low phase margin. The poor phase margin in both closed loop systems leads to excessive closed loop peaking. The closed loop peaking is related the system overshoot as well. So the lower the phase margin, the higher the closed loop peaking, the more the system rings in reponse to a step input. Figure 7 shows the Simulink model built to test this controller. Figure 9 shows the system's reponse.

Figure 9: System reponse to a step input with the Axsys controller

#### 1.2.3 PI with Lead/Lag Network

With a little tuning the controller can be made significantly better.

Figure 10: Frequncy response of retuned controller

Notice that the phase margin is larger and the peak closed loop magnitude is smaller. This controller's better response is largely due to more phase gain from the lead/lag network. The phase gain was achieved by a larger pole/zero separation.

Today analog lead/lag circuits can achieve this much pole/zero separation and as such this much phase gain. However, some years ago this controller was not achievable with an analog controller. It is easily accomplished with a digital controller.

Figure 11 shows the step response of the retuned controller. Note the smaller overshoot and shorter settling time.

Figure 11: Step response of system with retuned controller

Designing and tuning controllers is a broad topic beyond the scope of this example.

## 2 True to Hardware Mirror Model

The most important difference between a simple model used in early design phases and a mature model used for accurate system performance is an accurate accounting of the noise. However, there are also important factors such as conversion factors and quantization error (for digital systems).

### 2.1 Noise

There are several types of noise to be considered

1. bias
2. white noise
3. shaped noise
4. quantization error

For this example we are assuming all noises are either white noise or quantization error. In Simulink white noise is easily modeled using the Band-Limited White Noise block. Simulink also includes a block specific to quantization errors called the quantizer.

#### 2.1.1 Driver Circuit

Depending on the circuit components and the length of cabling used to connect the circuit to the rest of the system, electrical circuits can pick up noise from radio stations, cell phones, the electrical power system of the building and any number of other sources. Electrical noise is usually modeled very well by white noise. See white noise for more information.

#### 2.1.2 Position Sensor

Position sensor noise is complicated by sensor misalignment, linearity, proximity, and temperature. However, systems design will take care of these items. In other words, buy a sensor with small (by comparison to required error) misalignment, good linearity across the entire operating range, etc. If a sensor with all these characteristics is not available then really good calibration is required so the negative impacts can be minimized.

Assuming a good sensor can be procured, the noise source that eliminated or minimized is the electrical noise of the sensor. This is usually white noise.

### 2.2 Conversion Factors

Conversion factors are typically gains. This means they are easily handled once a plant transfer function is measured. However, sometimes it is preferrable to include these conversion factors in the model so that it is as accurate as possible.

#### 2.2.1 Mirror Gain

The Axsys whitepaper states that the conversion factors are

• 0.365 Amps/Volt
• the second factor $LaTeX: \frac{3.3}{IS^2}$ radians/amp is provided but this author does not know what $LaTeX: IS^2$ stands for, it assumed to be related to inertia

In reality the mirror transfer function will not have a $LaTeX: K=1$ but instead will have a non-unity DC gain. The gain will be a function of the mirror's inertia.

The mirror gain does not necessarily need to be modelled accurately. The change to the controller is only the inverse of the mirror's gain (K). An accurately modeled mirror gain is only necessary for the following reasons:

1. driver circuit saturation is a concern
2. hardware is to be tested

In the case of driver circuit saturation the circuit needs to be capable of generating enough amps (torque) to achieve the necessary response. If it cannot do this without saturating then something in the loop needs redesign.

In the case of hardware being built models of the mirror are not likely to be quite accurate enough. Therefore the mirror plant transfer function needs to be measured. The DC gain from this measurement can then be used to update the model. The controller gain will need to be updated to reflect this new gain in order to maintain the same open loop crossover and closed loop bandwidth.

#### 2.2.2 Driver Circuit - Amps to Torque

The driver circuit conversion factor was not specified in the whitepaper. There is an Amps/Volt factor and radians/Amp factor. In reality the driver circuit will produce a current. The voice coils will produce a force based on this current. The force acts via a lever arm to produce a torque about the mirror's center of rotation.

Ultimately the conversion factor will be a combination of all the factors to create a Torque/Amp or Torque/Volt.

#### 2.2.3 Position Sensors - Volts to Angular Position (radians)

The Axsys whitepaper specifies a position sensor conversion factor of

If the controller works well when all the 2nd order systems have a gain of 1 then it will work fine with only a magnitude adjustment. The adjustment should be equivalent to the inverse of all the conversion factors. The easy way to adjust the magnitude is to form the system

 $LaTeX: OL=A(s)G(s)H(s)K(s)$ (see Figure 1)

where
$LaTeX: A(s)$ is the driver circuit transfer function
$LaTeX: G(s)$ is the mirror plant transfer function
$LaTeX: H(s)$ is the position sensor transfer function
$LaTeX: K(s)$ is the controller transfer function

Then adjust the magnitude in order to get the same open loop crossover frequency that the system had before the conversion factors.

### 2.3 Digital Considerations

The Axsys whitepaper also shows that the controller is digital. There is a 12 bit DAC which turns the digital controller command into an analog voltage. The analog output from the position sensor is passed through a 12 bit ADC to turn the sensor output into a digital value for the controller. The analog/digital conversion introduce a courseness to the signals which introduces yet another error source.

Modelling an ADC in the frequency domain is not possible. In order to model an ADC in the time domain (Simulink) use the quantizer block.

#### 2.3.2 DAC

Modelling a DAC is a simple matter of using a gain equal to the Least Significant Bit (LSB) which will be the DAC range divided by 2^(number of effective bits).

## 3 Final Mirror Model

Figure 12: Full System Model with Noise and Quantization included

The Kaman 5200 DITs specificaton state that the 15N sensor head has 14 nanoinches of error RMS. If we assume that the mirror is a square mirror, 4 inches on a side then the position error is 7 nanoradians $LaTeX: \left( \tan^-1 \left(\frac{1.4E-8}{2} \right)\right)$.

The Axsys whitepaper states that there is a driver circuit but it does not specify the RMS noise for that circuit. It also specifies a mechnical range of +/-3 degrees (52.4 mrad). The table below show the results of the simulation in Figure 12 as various parameters are changed.

Simulation Results
Sample Rate (kHz) Bits Mechanical Range (deg)Residual RMS (rad)
101231.60E-5
10120.32.88E-5
101432.70E-5
201231.28E-5
20120.38.05E-6
201438.00E-6
301236.23E-6
30120.34.07E-6
10001234.28E-6
1000120.33.29E-6


text:
# SN110 Nodes

#### MHSTech

##### Active Member
I'm looking into getting some Strand SN110 Nodes. Does anybody know what the list price is on these? And what's everybody's thought on them?

#### gafftaper

##### Senior Team
Senior Team
Fight Leukemia
I just asked my Strand Dealer that same question. $700 OUCH! Hopefully the street price is down around$500... but I don't know that one yet. I was thinking about 5 to minimize my DMX runs... no thanks... I can get by with 2.

MHSTech

#### MHSTech

##### Active Member
Hmmm...I don't like those figures at all..

#### gafftaper

##### Senior Team
Senior Team
Fight Leukemia
Yeah my feeling exactly. And they've kind of got you if you want to use anything with DMX you're stuck.

#### Footer

##### Senior Team
Senior Team
I have two sitting on my desk right now, we had a new strand install, they sent us two without clamps, called the dealer and sent them back to strand to trade out with clamps and a jack, they sent us the ones with clamps and sent back the ones that didnt have clamps, never charged us for the extra two....

#### MHSTech

##### Active Member
you better let me take those off your hands...

#### Grog12

##### CBMod
CB Mods
you better let me take those off your hands...
Raise your hand if you saw this one coming... *slowly raises hand*

Senior Team


text:
This is a testing site only. See the live Public Lab site here »

# [Testing] New inline maps

by sagarpreet | 06 Oct 05:58

[map:layers::23:77:mapknitter] gets you:

[map:layers::23:77:mapknitter] gets you:

[map:layers::23:77:mapknitter] gets you:

`

Okay, the popup works if there is only 1 inline map. There is a bug if there are multiple inline maps. Sorry did not tested this case.

Sounds like we need some more unique variable names, and maybe element id names?

Is this a question? Click here to post it to the Questions page.


text:
# Extending smooth irreducible representations

Hi, Let $G_1, G_2$ be topological groups with $G_1 \subset G_2$ is closed. Let $\rho:G_1 \to Aut(V)$ be a smooth irreducible representation. Can anyone tell me if there is a criterion/example/idea about when $\rho$ cannot be extended to $G_2$, i.e., there does not exist smooth representation $\rho':G_2 \to Aut(V)$ such that $\rho'|_{G_1}=\rho$.

-
I haven't thought this all the way through, but many examples of non-normal abelian $G_1$ inside non-abelian $G_2$ give obstructions. For example, let $G_1$ be strictly upper-triangular matrices in $G_2=GL_2(\mathbb R)$. All irreducible representations of $G_1$ are $1$-dimensional, but the only $1$-dimensional representations of $G_2$ factor through the determinant, so are trivial on $G_1$. This depends on what you mean by "extending/restricting" and the result being "equal to" the original. For example, Frobenius reciprocity says that induction and restriction are adjoint functors. – B R Mar 30 '12 at 18:43
Take $\mathbb Z$ (discrete topology) as closed subgroup of $\mathbb R$, suppose $V$ is of dimension one over $F$, when $F$ is not "big enough", there are only trivial character of $Z$ to $F^{*}$ which could be extended to $\mathbb R$, e.g $F=\mathbb Q_p$. – user20421 Mar 30 '12 at 19:07


text:
# Proofpoint: How to make email miserable to use

Apologies to the non-technical readers. Some parts of this rant are a bit technical. But most of it should make sense to most readers.

This past fall, Grinnell College decided to install the Proofpoint spam protection service on our email system. Now, I certainly understand the need for spam protection. There is way too much spam. But the goal of a spam protection service should be to make your life easier, not harder. And Proofpoint clearly makes my life harder. Why? Let’s look at some reasons.

First, and most importantly, Proofpoint protects URLs. That is, it takes any URL that appears in an email message you receive and makes it direct to one of their servers, where they check to see whether or not it’s valid and then redirects to the appropriate location. For example, suppose Amazon sends me a message like the following.

Dear Amazon.com Customer,

See this week's new music!

http://www.amazon.com/gp/browse.html/ref=pe_supersubjectlink/?ie=UTF8&node=307026011

Instead of that message, I get something more like the following.

Dear https://urldefense.proofpoint.com/v2/url?u=http-3A__Amazon.com&d=DQIFAg&c=HUrdOLg_tCr0UMeDjWLBOM9lLDRpsndbROGxEKQRFzk&r=6rcUljFJZnpk5uomPd3v3WCzboqh0RuwO-BZyxMfi0U&m=fo458hhJrF87dIYyHwDAWZkegyOy6sGJVAGrntX1mP0&s=2r8EJkvOhZj1zr2Emwwgjav6t4vvg-O42jL_dHQUDkk&e= Customer,

See this week's new music!

https://urldefense.proofpoint.com/v2/url?u=http-3A__www.amazon.com_gp_browse.html_ref-3Dpe-5Fsupersubjectlink_-3Fie-3DUTF8-26node-3D307026011&d=DQICaQ&c=HUrdOLg_tCr0UMeDjWLBOM9lLDRpsndbROGxEKQRFzk&r=6rcUljFJZnpk5uomPd3v3WCzboqh0RuwO-BZyxMfi0U&m=WQlZQItScdT7feAdRWA96BPThpNgqjCFcXEDdMoYBTk&s=vYDegbEPK2TILjuGa8Nb7qxaFDk7cwzl1lOkD52e_KA&e=

That’s right. Not only did it change the URL in the message to something that I can’t read, it also changed Amazon.com to something I can’t read. Who cares if people can read their email? At least it’s safe.

Proofpoint is also aggressive about changing things that might by site names to protected URLs. For example, if someone writes to me about Web services and says

Have you tried haml.rb on a Node.js server?

The message I get says

Have you tried https://urldefense.proofpoint.com/v2/url?u=http-3A__haml.rb&d=DQICAg&c=HUrdOLg_tCr0UMeDjWLBOM9lLDRpsndbROGxEKQRFzk&r=6rcUljFJZnpk5uomPd3v3WCzboqh0RuwO-BZyxMfi0U&m=DmoxnEjN2hRl9XiigsgXIg_9HROvObWYCLCX-aYhZCo&s=a-NQAgTh9ladcMIlklxm1FnXIFambZJaXoF_x4z6s2E&e= on a https://urldefense.proofpoint.com/v2/url?u=http-3A__Node.js&d=DQICAg&c=HUrdOLg_tCr0UMeDjWLBOM9lLDRpsndbROGxEKQRFzk&r=6rcUljFJZnpk5uomPd3v3WCzboqh0RuwO-BZyxMfi0U&m=DmoxnEjN2hRl9XiigsgXIg_9HROvObWYCLCX-aYhZCo&s=3gVxYxxmljpf7wphNFOjxu0Ll06JH1yLOEZ6BAm2VtA&e= server?

I dare you to read that!

They also seem to think that any four numbers with dots between them should be protected. Suppose a German phisher writes me the following message.

Our client has left you 500.000.000.000 Euros!

Instead, I get this much safer request.

Our client has left you https://urldefense.proofpoint.com/v2/url?u=http-3A__500.000.000.000&d=DQICAg&c=HUrdOLg_tCr0UMeDjWLBOM9lLDRpsndbROGxEKQRFzk&r=6rcUljFJZnpk5uomPd3v3WCzboqh0RuwO-BZyxMfi0U&m=vppZ9BGkRx0OQXQ5bvasdGB8rHyHdO-cRar5WvbqGdQ&s=WL0NaP3yYZv23KsMRUmmF0izq90Glwd5GxpeY6E6JVI&e= Euros!

Of course, when I click on the link, it leads nowhere (because 500.000.000.000 is not a valid address).

It wouldn’t be so bad if they provided a way to retrieve the original message. But that’s not part of their strategy. I’ve hacked something together (ah, the joys of knowing programming, particularly regular expressions), but as far as I can tell, they protect the terms amazon.com and http://amazon.com identically. That means that there is no way to go back to the original message. (They also add some extra spaces and other things.)

It also means that I can’t readily forward email to colleagues outside of Grinnell. What computer scientist would trust someone who provides links that clearly go to the wrong site? (So yes, almost every time that I need to forward a message, I have to run it through my script or manually change the links. Our ITS department seems to think this is acceptable.)

I also think that URL rewriting leads to bad habits. Sensible people read URLs. Proofpoint defended URLs are unreadable, so it’s impossible to read them, and you just click them and hope for the best.

And, as Grinnell talks about accessibility, it’s clear that these defended URLs are not accessible. Just think about the joy of having to listen to one of those URLs!

The obtrusive and incompetent URL rewriting alone is enough to make me despise Proofpoint, but it gets worse!

They clearly monitor the links we click. Our ITS department tells us to trust them that they don’t use the information except for appropriate purposes, but they clearly keep the information for awhile, because it sometimes takes a few hours for them to determine that a link you click is dangerous. Are we sure that they don’t do anything else with our data? I’m told, Trust us. It’s difficult.

They break some of the links. I haven’t been able to come up with a particular characteristic for those links, but every once in a while I get something that should have been a valid link and is instead invalid.

Proofpoint also delays messages as it thinks about URLs. I’ve clicked reset my password links on credit cards and it’s taken so long for the message to get to me that the company times out the password reset opportunity.

And they store some of our email on their servers. These are messages that are quarantined; not quite spam, but things that they don’t think we’d like to read. I think keeping email on a non-Grinnell server is a FERPA violation. (I’m told that our lawyers don’t think so. But we had to make Microsoft an officer of the college in order to let them store our messages on their servers without violating FERPA.)

To make matters worse, the software was broken for the first six months or so that we had it, so things ended up in the quarantine, but Proofpoint did not notify you that they had quarantined anything. Who knows how much mail got lost? I know that I missed a few student questions that came from their Gmail accounts.

And, as you’d expect, Proofpoint customer service is a black hole. The known issues go in. Nothing changes. When I ask about updates, I’m told that they know about the problems I reported, and that they have provided us with no estimate as to when those problems will be fixed.

It’s lovely to see how software makes our lives better.

I do understand that too many people on campus fall victim to phishing links and that we need to increase protection. However, we should be choosing software that is sensible and that provides mechanisms for people to read their email in its original form.

My favorite email to get urldefended is the regular Risks in Computing Systems digest, which is filled with URLs and whose subject matter includes the danger of using systems that can track your actions.

While using Proofpoint frustrates me, at least I don’t have a broken \$20,000 computer that ITS refuses to ship back to the manufacturer while it’s still under warranty [1].

[1] Followup from a few months after I wrote the essay. Yes, ITS shipped it back. But I still think it took the faculty member six months to convince them to do so, and we even had SNAFUs when the computer came back to campus.

Version 1.0.1 of 2016-10-02.


text:
So far, we have covered HMM-GMM acoustic modeling and some practical issues related to context and less frequent phonemes.

Introduction to Neural Network acoustic modeling

There is an alternative way to build an acoustic model, and it consists in using neural networks to:

• take an input acoustic frame
• and output a score for each phone

Let’s consider a single-layer neural network, that takes as an input an acoustic frame $X_t$ and outputs phonetic scores $f(t)$ (one score for each phone).

It can be expressed as:

Where $W$ is the weight matrix made of weights $w_{ij}$ that reflect the weifht between input $i$ and output $j$, and $b$ is the bias term.

How do we learn the parameters $W$ and $b$? We target the minimization of the error function $E$, the Mean Square Error (MSE) between the output and the target:

Where $r(t)$ are the target outputs.

The error minimization is typically done using gradient descent, and we must compute the terms:

$\frac{d E}{d W}$ and $\frac{d E}{d b}$

In SGD, we:

• intialize weights and biases with small random numbers
• randomise the order of training data example
• then for each epoch:
• take a minibact
• compute network outputs
• backpropagate and update the weights

The network that predicts phonetic scores is a classifier, so we need to take a softmax to force output values to act as probabilities:

Where $f(x_j) = \sum_{d=1}^D w_{jd} x_d + b_j$

However, the MSE is not the wisest choice when working with probabilities. We can directly maximize the log probability of observing the correct label using the Cross-Entropy (CE) error function:

Using CE, the gradients of the outputs weights simplify to:

There are several extensions possible to this very very simple model:

• use more context by taking multiple frames into account
• add more hidden layers to obtains DNNs
• use activation functions such as ReLU

This looks interesting, and overall simpler than the HMM-GMM. But there is a major limitation, since we cannot do speech recognition with this approach. There are several phone recognition tasks:

• frame classification: classify each frame of data
• phone classification: classify each segment of data (what Neural Networks can do)
• phone recongition: segment the data and label the segment

Using only DNN, we lack the notion of data segmentation that HMMs are good at doing. But can’t we mix HMMs and DNNs ?

HMM-DNN acoustic modeling

In an HMM-GMM, replacing the GMM by a DNN to estimate output pdfs build a so-called HMM-DNN architecture. In a HMM-DNN, we consider one-state per phone, and train a NN as a phone-state classifier.

It can be shown that the outputs corresponding to class $j$ given an input $x_t$ are an estimate of the posterior probability $P(q_t = j \mid x_t)$, $q_t$ being a state, because we have softmax outputs and use a CE loss function.

And using Bayes Rule, we can relate the posterior $P(q_t = j \mid x_t)$ to the likelihood $P(x_t \mid q_t = j)$:

If we want HMM-DNNs to output probabilities, we should scale the likelihoods:

This means we can obtain scaled likelihoods by dividing each network output by the prior, i.e. the relative frequency of class $j$ in training data.

There are several approaches to continuous speech recognition with HMM-DNN:

• 1 state per phone (each NN can output typically 60 classes if 60 phone classes)
• 3 state context-independent (CI) models, where each phone has 3 states, modeled by 3 NNs
• State-clustered context-dependent (CD) models, 1 NN output per tied state

The architecture of a HMM-DNN is presented below:

NN are more flexible, learn richer representations and handle correlated features. In terms of speech features, experiments indicate that mel-scaled filter bank features (FBANK) work better than MFCCs, and results in better clustering when applying t-SNE on the hidden layers. Indeed, in FBANK the useful information is distributed over all the features, whereas in MFCC it is concentrated in the first few.

Modelling phonetic context with DNNs

Modling the phonetic context with DNNs was considered hard until 2011, but a simple solution emerged: use the state-tying process from a GMM system, and train a HMM-DNN on it.

More precisely, the context-dependent hybrid HMM/DNN approach is:

• Train a GMM/GMM system on your data
• Use a phonetic decision tree to determine the HMM tied states for infrequent states
• Perform Viterbi alignment using the trained HMM/GMM and the training data
• Train a neural network to map the input speech features to a label representing a context-dependent tied HMM-state, instead of contenxt-independent phones
• this increases the number of possible labels, and each frame is labelled with Viterbi aligned tied states
• we then train the NN using gradient descent

Concretely, we model the acoustic context by including neighbour frames in the input layer.

We use richer Neural Network models for acoustic context:

• Recurrent Neural Networks (RNNs)
• Time-delay Neural Networks (TDNNs)

Time-delay Neural Networks

In TDNNs, higher hidden layers take input from a larger acoustic context, and lower hidden layers from narrower contexts. TDNNs can be seen as a 1D convolutional network.

A TDNN with a context (-2, 2) has 5 times more weights than a regular DNN. For this reason, sub-sampled TDNNs are explored. Due to the large overlaps between input contexts at adjacent time steps, which are likely to be correlated, we take a sub-sample window of hidden unit activations. It reduces computation time and model weights.

Recurrent Neural Networks

In its unfolded representation, the RNN for a sequence of T inputs is a T-layer network with shared weights. It can keep information through time.

LSTMs avoid the vanishing gradient problem of RNNs. Bidirectional RNNs consider both the right and the left context, in a forward layer and a backward layer. Deep RNNs have several hidden layers, and deep bidirectional LSTM combine the advantages of all these methods:

Here is an example of a Bidirectional LSTM Acoustic Model training on the Switchboard dataset:

• LSTM has 4-6 bidirectional layers with 1024 cells/layer (512 each direction)
• 256 unit linear bottleneck layer
• 32k context-dependent state outputs
• Input features:
• 40-dimension linearly transformed MFCCs (plus ivector)
• 64-dimension log mel filter bank features (plus first and second derivatives)
• concatenation of MFCC and FBANK features
• Training: 14 passes frame-level cross-entropy training, 1 pass sequence training (2 weeks on a K80 GPU)

LSTMs + feature fusion currently reach close to state-of-the-art.


text:
# Hartree atomic units

(Redirected from Atomic units)

The Hartree atomic units are a system of natural units of measurement which is especially convenient for atomic physics calculations. They are named after the physicist Douglas Hartree.[1] In this system the numerical values of the following four fundamental physical constants are all unity by definition:

• Electron mass ${\displaystyle m_{\text{e}}}$, also known as the atomic unit of mass[2][3]
• Elementary charge ${\displaystyle e}$, also known as the atomic unit of charge[4]
• Reduced Planck constant ${\displaystyle \hbar }$, also known as the atomic unit of action[5]
• Inverse Coulomb constant ${\displaystyle 4\pi \epsilon _{0}}$, also known as the atomic unit of permittivity[6]

In Hartree atomic units, the speed of light is approximately 137 atomic units of velocity. Atomic units are often abbreviated "a.u." or "au", not to be confused with the same abbreviation used also for astronomical units, arbitrary units, and absorbance units in other contexts.

## Use and notation

Atomic units, like SI units, have a unit of mass, a unit of length, and so on. However, the use and notation is somewhat different from SI.

Suppose a particle with a mass of m has 3.4 times the mass of electron. The value of m can be written in three ways:

• "${\displaystyle m=3.4~m_{\text{e}}}$". This is the clearest notation (but least common), where the atomic unit is included explicitly as a symbol.[7]
• "${\displaystyle m=3.4~{\text{a.u.}}}$" ("a.u." means "expressed in atomic units"). This notation is ambiguous: Here, it means that the mass m is 3.4 times the atomic unit of mass. But if a length L were 3.4 times the atomic unit of length, the equation would look the same, "${\displaystyle L=3.4~{\text{a.u.}}}$" The dimension needs to be inferred from context.[7]
• "${\displaystyle m=3.4}$". This notation is similar to the previous one, and has the same dimensional ambiguity. It comes from formally setting the atomic units to 1, in this case ${\displaystyle m_{\text{e}}=1}$, so ${\displaystyle 3.4~m_{\text{e}}=3.4}$.[8][9]

## Defining constants

Each unit in this system can be expressed as a product of powers of four physical constants without a multiplying constant. This makes it a coherent system of units, as well as making the numerical values of the defining constants in atomic units equal to unity.

Defining constants
Name Symbol/Definition Value in SI units
electron rest mass ${\displaystyle m_{\mathrm {e} }}$ 9.1093837015(28)×10−31 kg[10]
elementary charge ${\displaystyle e}$ 1.602176634×10−19 C[11]
reduced Planck constant ${\displaystyle \hbar =h/(2\pi )}$ 1.054571817...×10−34 J⋅s[12]
inverse Coulomb constant ${\displaystyle 4\pi \epsilon _{0}=1/k_{\text{e}}}$ 1.11265005545(17)×10−10 F⋅m−1[13]

These defining constants are used to define four units that take the role of base units, in the sense that other units are generally expressed symbolically in terms of these:

Nominal base units
Dimension Symbol Definition
electric charge ${\displaystyle e}$ ${\displaystyle e}$
mass ${\displaystyle m_{\text{e}}}$ ${\displaystyle m_{\text{e}}}$
action ${\displaystyle \hbar }$ ${\displaystyle \hbar }$
length ${\displaystyle a_{0}}$ ${\displaystyle 4\pi \epsilon _{0}\hbar ^{2}/(m_{\text{e}}e^{2})}$

A further unit is derived from these that is used for simplifying expression of all other units in terms of these five units:

Auxiliary unit symbol
Dimension Symbol Definition
energy ${\displaystyle E_{\text{h}}}$ ${\displaystyle \hbar ^{2}/(m_{\text{e}}a_{0}^{2})}$

## Units

Below are listed units that can be derived in the system. A few are given names, as indicated in the table.

Derived atomic units
Atomic unit of Name Expression Value in SI units Other equivalents
1st hyperpolarizability ${\displaystyle e^{3}a_{0}^{3}/E_{\text{h}}^{2}}$ 3.2063613061(15)×10−53 C3⋅m3⋅J−2[14]
2nd hyperpolarizability ${\displaystyle e^{4}a_{0}^{4}/E_{\text{h}}^{3}}$ 6.2353799905(38)×10−65 C4⋅m4⋅J−3[15]
action ${\displaystyle \hbar }$ 1.054571817...×10−34 J⋅s[16]
charge ${\displaystyle e}$ 1.602176634×10−19 C[17]
charge density ${\displaystyle e/a_{0}^{3}}$ 1.08120238457(49)×1012 C·m−3[18]
current ${\displaystyle eE_{\text{h}}/\hbar }$ 6.623618237510(13)×10−3 A[19]
electric dipole moment ${\displaystyle ea_{0}}$ 8.4783536255(13)×10−30 C·m[20] 2.541746473 D
electric field ${\displaystyle E_{\text{h}}/(ea_{0})}$ 5.14220674763(78)×1011 V·m−1[21] 5.14220674763(78) GV·cm−1, 51.4220674763(78) V·Å−1
electric field gradient ${\displaystyle E_{\text{h}}/(ea_{0}^{2})}$ 9.7173624292(29)×1021 V·m−2[22]
electric polarizability ${\displaystyle e^{2}a_{0}^{2}/E_{\text{h}}}$ 1.64877727436(50)×10−41 C2⋅m2⋅J−1[23]
electric potential ${\displaystyle E_{\text{h}}/e}$ 27.211386245988(53) V[24]
electric quadrupole moment ${\displaystyle ea_{0}^{2}}$ 4.4865515246(14)×10−40 C·m2[25]
energy hartree ${\displaystyle E_{\text{h}}}$ 4.3597447222071(85)×10−18 J[26] ${\displaystyle 2R_{\infty }hc}$, ${\displaystyle \alpha ^{2}m_{\text{e}}c^{2}}$, 27.211386245988(53) eV
force ${\displaystyle E_{\text{h}}/a_{0}}$ 8.2387234983(12)×10−8 N[27] 82.387 nN, 51.421 eV·Å−1
length bohr ${\displaystyle a_{0}}$ 5.29177210903(80)×10−11 m[28] ${\displaystyle \hbar /(m_{\text{e}}c\alpha )}$, 0.529177210903(80) Å
magnetic dipole moment ${\displaystyle e\hbar /m_{\text{e}}}$ 1.85480201566(56)×10−23 J⋅T−1[29] ${\displaystyle 2\mu _{\text{B}}}$
magnetic flux density ${\displaystyle \hbar /(ea_{0}^{2})}$ 2.35051756758(71)×105 T[30] 2.35051756758(71)×109 G
magnetizability ${\displaystyle e^{2}a_{0}^{2}/m_{\text{e}}}$ 7.8910366008(48)×10−29 J⋅T−2[31]
mass ${\displaystyle m_{\mathrm {e} }}$ 9.1093837015(28)×10−31 kg[32]
momentum ${\displaystyle \hbar /a_{0}}$ 1.99285191410(30)×10−24 kg·m·s−1[33]
permittivity ${\displaystyle e^{2}/(a_{0}E_{\text{h}})}$ 1.11265005545(17)×10−10 F⋅m−1[34] ${\displaystyle 4\pi \epsilon _{0}}$
pressure ${\displaystyle E_{\text{h}}/{a_{0}}^{3}}$ 2.9421015697(13)×1013 Pa
temperature ${\displaystyle E_{\text{h}}/k_{\text{B}}}$ 315775.02480407(60) K[35][citation needed]
time ${\displaystyle \hbar /E_{\text{h}}}$ 2.4188843265857(47)×10−17 s[36]
velocity ${\displaystyle a_{0}E_{\text{h}}/\hbar }$ 2.18769126364(33)×106 m·s−1[37] ${\displaystyle \alpha c}$

Here,

${\displaystyle c}$ is the speed of light
${\displaystyle \epsilon _{0}}$ is the vacuum permittivity
${\displaystyle R_{\infty }}$ is the Rydberg constant
${\displaystyle h}$ is the Planck constant
${\displaystyle \alpha }$ is the fine-structure constant
${\displaystyle \mu _{\text{B}}}$ is the Bohr magneton
${\displaystyle k_{\text{B}}}$ is the Boltzmann constant

## Physical constants

Dimensionless physical constants retain their values in any system of units. Of note is the fine-structure constant ${\displaystyle \alpha ={\frac {e^{2}}{(4\pi \epsilon _{0})\hbar c}}\approx 1/137}$, which appears as a consequence of the choice of units. For example, the numeric value of the speed of light, expressed in atomic units, has a value related to the fine structure constant.

Some physical constants expressed in atomic units
Name Symbol/Definition Value in atomic units
speed of light ${\displaystyle c}$ ${\displaystyle (1/\alpha ){\text{ a.u.}}\approx 137{\text{ a.u.}}}$
classical electron radius ${\displaystyle r_{\mathrm {e} }={\frac {1}{4\pi \epsilon _{0}}}{\frac {e^{2}}{m_{\mathrm {e} }c^{2}}}}$ ${\displaystyle \alpha ^{2}{\text{ a.u.}}\approx 5.32\times 10^{-5}{\text{ a.u.}}}$
reduced Compton wavelength
of the electron
ƛe ${\displaystyle ={\frac {\hbar }{m_{\text{e}}c}}}$ ${\displaystyle \alpha {\text{ a.u.}}\approx 0.007297{\text{ a.u.}}}$
proton mass ${\displaystyle m_{\mathrm {p} }}$ ${\displaystyle m_{\mathrm {p} }\approx 1836{\text{ a.u.}}}$

## SI and Gaussian-CGS variants, and magnetism-related units

There are two common variants of atomic units, one where they are used in conjunction with SI units for electromagnetism, and one where they are used with Gaussian-CGS units.[38] Although most of the units listed above are the same either way (including the unit for electric field), the units related to magnetism are not. In the SI system, the atomic unit for magnetic field is

1 a.u. = ${\displaystyle {\frac {\hbar }{ea_{0}^{2}}}\approx }$ 2.35×105 T = 2.35×109 G,

and in the Gaussian-cgs unit system, the atomic unit for magnetic field is

1 a.u. = ${\displaystyle {\frac {e}{a_{0}^{2}c}}\approx }$ 1.72×103 T = 1.72×107 G.

(These differ by a factor of ${\displaystyle \alpha }$.)

Other magnetism-related quantities are also different in the two systems. An important example is the Bohr magneton: In SI-based atomic units,[39]

${\displaystyle \mu _{\text{B}}={\frac {e\hbar }{2m_{\text{e}}}}=1/2}$ a.u.

and in Gaussian-based atomic units,[40]

${\displaystyle \mu _{\text{B}}={\frac {e\hbar }{2m_{\text{e}}c}}=\alpha /2\approx 3.6\times 10^{-3}}$ a.u.

## Bohr model in atomic units

Atomic units are chosen to reflect the properties of electrons in atoms. This is particularly clear from the classical Bohr model of the hydrogen atom in its ground state. The ground state electron orbiting the hydrogen nucleus has (in the classical Bohr model):

• Mass = 1 a.u. of mass
• Orbital radius = 1 a.u. of length
• Orbital velocity = 1 a.u. of velocity
• Orbital period = 2π a.u. of time
• Orbital angular velocity = 1 radian per a.u. of time
• Orbital angular momentum = 1 a.u. of momentum
• Ionization energy = 1/2 a.u. of energy
• Electric field (due to nucleus) = 1 a.u. of electric field
• Electrical attractive force (due to nucleus) = 1 a.u. of force

## Non-relativistic quantum mechanics in atomic units

The Schrödinger equation for an electron in SI units is

${\displaystyle -{\frac {\hbar ^{2}}{2m_{\text{e}}}}\nabla ^{2}\psi (\mathbf {r} ,t)+V(\mathbf {r} )\psi (\mathbf {r} ,t)=i\hbar {\frac {\partial \psi }{\partial t}}(\mathbf {r} ,t)}$.

The same equation in atomic units is

${\displaystyle -{\frac {1}{2}}\nabla ^{2}\psi (\mathbf {r} ,t)+V(\mathbf {r} )\psi (\mathbf {r} ,t)=i{\frac {\partial \psi }{\partial t}}(\mathbf {r} ,t)}$.

For the special case of the electron around a hydrogen atom, the Hamiltonian in SI units is:

${\displaystyle {\hat {H}}=-{{{\hbar ^{2}} \over {2m_{\text{e}}}}\nabla ^{2}}-{1 \over {4\pi \epsilon _{0}}}{{e^{2}} \over {r}}}$,

while atomic units transform the preceding equation into

${\displaystyle {\hat {H}}=-{{{1} \over {2}}\nabla ^{2}}-{{1} \over {r}}}$.

## Comparison with Planck units

Both Planck units and atomic units are derived from certain fundamental properties of the physical world, and are free of anthropocentric considerations. It should be kept in mind that atomic units were designed for atomic-scale calculations in the present-day universe, while Planck units are more suitable for quantum gravity and early-universe cosmology. Both atomic units and Planck units normalize the reduced Planck constant. Beyond this, Planck units normalize to 1 the two fundamental constants of general relativity and cosmology: the gravitational constant ${\displaystyle G}$ and the speed of light in a vacuum, ${\displaystyle c}$. Atomic units, by contrast, normalize to 1 the mass and charge of the electron, and, as a result, the speed of light in atomic units is a large value, ${\displaystyle 1/\alpha \approx 137}$. The orbital velocity of an electron around a small atom is of the order of 1 in atomic units, so the discrepancy between the velocity units in the two systems reflects the fact that electrons orbit small atoms much more slowly than the speed of light (around 2 orders of magnitude slower).

There are much larger discrepancies in some other units. For example, the unit of mass in atomic units is the mass of an electron, while the unit of mass in Planck units is the Planck mass, a mass so large that if a single particle had that much mass it might collapse into a black hole. Indeed, the Planck unit of mass is 22 orders of magnitude larger than the atomic unit of mass. Similarly, there are many orders of magnitude separating the Planck units of energy and length from the corresponding atomic units.

## Notes and references

• Shull, H.; Hall, G. G. (1959). "Atomic Units". Nature. 184 (4698): 1559. Bibcode:1959Natur.184.1559S. doi:10.1038/1841559a0.
1. ^ Hartree, D. R. (1928). "The Wave Mechanics of an Atom with a Non-Coulomb Central Field. Part I. Theory and Methods". Mathematical Proceedings of the Cambridge Philosophical Society. 24 (1). Cambridge University Press. pp. 89–110. Bibcode:1928PCPS...24...89H. doi:10.1017/S0305004100011919.
2. ^ "2018 CODATA Value: atomic unit of mass". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-15.
3. ^ Not to be confused with the unified atomic mass unit.
4. ^ "2018 CODATA Value: atomic unit of charge". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-15.
5. ^ "2018 CODATA Value: atomic unit of action". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-15.
6. ^ "2018 CODATA Value: atomic unit of permittivity". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
7. ^ a b Pilar, Frank L. (2001). Elementary Quantum Chemistry. Dover Publications. p. 155. ISBN 978-0-486-41464-5.
8. ^ Bishop, David M. (1993). Group Theory and Chemistry. Dover Publications. p. 217. ISBN 978-0-486-67355-4.
9. ^ Drake, Gordon W. F. (2006). Springer Handbook of Atomic, Molecular, and Optical Physics (2nd ed.). Springer. p. 5. ISBN 978-0-387-20802-2.
10. ^ "2018 CODATA Value: electron mass". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
11. ^ "2018 CODATA Value: elementary charge". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
12. ^ "2018 CODATA Value: reduced Planck constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-28.
13. ^ "2018 CODATA Value: atomic unit of permittivity". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
14. ^ "2018 CODATA Value: atomic unit of 1st hyperpolarizability". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
15. ^ "2018 CODATA Value: atomic unit of 2nd hyperpolarizability". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
16. ^ "2018 CODATA Value: atomic unit of action". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
17. ^ "2018 CODATA Value: atomic unit of charge". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
18. ^ "2018 CODATA Value: atomic unit of charge density". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
19. ^ "2018 CODATA Value: atomic unit of current". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
20. ^ "2018 CODATA Value: atomic unit of electric dipole moment". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
21. ^ "2018 CODATA Value: atomic unit of electric field". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
22. ^ "2018 CODATA Value: atomic unit of electric field gradient". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
23. ^ "2018 CODATA Value: atomic unit of electric polarizability". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
24. ^ "2018 CODATA Value: atomic unit of electric potential". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
25. ^ "2018 CODATA Value: atomic unit of electric quadrupole moment". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
26. ^ "2018 CODATA Value: atomic unit of energy". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
27. ^ "2018 CODATA Value: atomic unit of force". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
28. ^ "2018 CODATA Value: atomic unit of length". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
29. ^ "2018 CODATA Value: atomic unit of magnetic dipole moment". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
30. ^ "2018 CODATA Value: atomic unit of magnetic flux density". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
31. ^ "2018 CODATA Value: atomic unit of magnetizability". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
32. ^ "2018 CODATA Value: atomic unit of mass". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
33. ^ "2018 CODATA Value: atomic unit of momentum". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
34. ^ "2018 CODATA Value: atomic unit of permittivity". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
35. ^ This value has been derived from the CODATA 2018 values of Eh and kB.
36. ^ "2018 CODATA Value: atomic unit of time". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
37. ^ "2018 CODATA Value: atomic unit of velocity". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-08-31.
38. ^ "A note on Units" (PDF). Physics 7550 — Atomic and Molecular Spectra. University of Colorado lecture notes.
39. ^ Chis, Vasile. "Atomic Units; Molecular Hamiltonian; Born-Oppenheimer Approximation" (PDF). Molecular Structure and Properties Calculations. Babes-Bolyai University lecture notes.
40. ^ Budker, Dmitry; Kimball, Derek F.; DeMille, David P. (2004). Atomic Physics: An Exploration through Problems and Solutions. Oxford University Press. p. 380. ISBN 978-0-19-850950-9.


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# Partition into c and 1-c

Let $$c\in(0,1/2]$$ be a constant. Given a set of positive integers with sum $$S$$, is there a partition into two subsets such that both subsets have sum at least $$cS$$?

If $$c=1/2$$, this is the famous partition problem which is NP-complete. What is known about the complexity for other values of $$c$$?

• It is NP-hard for every fixed constant $c$ with $0<c<1/2$. – Gamow Mar 24 at 17:42
• @Gamow Thanks. Could you please give a reference? – user52378 Mar 24 at 18:10
• @Gamow This is incorrect, according to Tom van der Zanden's answer on CS SE. – David Richerby Jun 19 at 15:07