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Today I've started creating an image, as for the homescreen of the ASM port, and as a logo for my documentation. This is the result:
Do you like it? Wanna change something?
(Yes I know, this will take up 70K of RAM...)
Looks great! It's 70K of RAM at 8bpp, 320x240? Out of curiosity, why are the right and top margins different from the bottom and left margins? It bothers me, because it looks like a mistake, although
I know you said it was a stylistic choice.
KermMartian wrote:
Looks great! It's 70K of RAM at 8bpp, 320x240? Out of curiosity, why are the right and top margins different from the bottom and left margins? It bothers me, because it looks like a mistake, although
I know you said it was a stylistic choice.
Yep, I think I need to fix that somehow
That white edges are not actually for the image, because I use a cut program of Windows to get the image from Windows
Ahhh, that makes sense. What program do you use for graphics manipulation? If you haven't tried GIMP, I recommend giving it a try. Anyway, as the users on SAX pointed out, there's a limit of 64KB of
data stored in AppVars and programs. I therefore recommend coming up with a way to crop this, for example fading the edges to a solid color and filling the background with that solid color before
adding this splashscreen on top.
While I had much free time today, I've implemented the operators in the Shunting Yard Algorithm in ASM, including their precedence. It now builds 2 stacks, one at (saveSScreen), the other at
(saveSScreen+1000), one for the actual stack, and the first one for the output, which I need to read after that. Unfortunately, I have no screenshots of that
I hope to make good progress these weeks!
I've very happy to say that the Shunting Yard Algorithm works for 95%. I've implemented the normal operators, booleans, numbers of course, and if the token doesn't match, it pushes it to the stack.
When the program reaches either a ',' or a ')' it will pop operators/booleans/functions from the stack to the output, until there is a '(' or a function which ends with '('. Sorry, still no
I hope to finish it today, and maybe today or tomorrow I can start with actually parsing it and porting it to ASM
Just be patient...
EDIT: I got the functions ready. Now bug-testing, and then continuueeee!
Today I've 'worked' on creating the logo, here is my result:
Which one is better (I prefer the black background)
PT_ wrote:
Today I've 'worked' on creating the logo, here is my result:
Which one is better (I prefer the black background)
Happy to see that you are making good progress on this, as of now, do you know which commands will be available, which ones you might add and which ones will certainly not be available? Oh and I
personally prefer the black one
I prefer the black one as well. And if you want to put in on a light background, I'd put it in a gray-bordered black rectangle to make it stand out.
*bump* How goes the coding and logo design? Have you been thinking about any new features or a more precise definition of what kind of language features ICE will offer yet?
KermMartian wrote:
*bump* How goes the coding and logo design? Have you been thinking about any new features or a more precise definition of what kind of language features ICE will offer yet?
Unfornately, I was this week pretty busy with exams, so not really time for designing/programming, but I have created some ideas in my mind for parsing RPN notation.
For the features, I will update the GitHub account of that:
But yea, I first need to finish parsing any mathematical expression, which is the hardest, but also the most important, and after that, I will be able to implement commands.
Long time no post of this project. I'm sorry for that, I am pretty busy with exams and the study next year after the exams, but that doesn't mean I haven't worked on it. As you may know is the
Shunting-Yard-Algorithm already done, which means I've started with the part to evaluate the RPN notation stuff. Not very easy, because I want to optimize the output very well, but that also means
more statements, and stuff, bla bla bla.... For each operator, there are 4 possibilities:
- <number> <number> <operator>
- <number> <variable> <operator>
- <variable> <number> <operator>
- <variable> <variable> <operator>
Both numbers, is just popping the numbers and operator from the stack, and push the result. And for the other cases, I've written routines to evaluate them and add the output to the program data. For
now, it's pretty optimized, and I'm finally happy that I've a screenshot now
I've only implemented + and - yet, and I'm busy with *. I hope to make good progress!
P.S. Now that I'm posting this post, I see that <number> - <variable> can be shorter: ld a, <number> / ld hl, <variable> / sub a, (hl)
EDIT: here is my table for multiplying a variable with a given number:
What if A was a floating-point number? Then just a sub wouldn't really work on it, would it?
oldmud0 wrote:
What if A was a floating-point number? Then just a sub wouldn't really work on it, would it?
For now, I assume that all the variables are 1-byte numbers. Later, I gonna add 3-byte numbers as well.
That table won't work for multiplying numbers greater than 32.... Might I recommend using the bootcode functions, or just copying them to RAM? You wouldn't even need to compile them into ICE, as long
as you knew the jump table entry and the length of the routine, which then you could bypass the need for the pipeline stall.
MateoConLechuga wrote:
That table won't work for multiplying numbers greater than 32.... Might I recommend using the bootcode functions, or just copying them to RAM? You wouldn't even need to compile them into ICE, as long
as you knew the jump table entry and the length of the routine, which then you could bypass the need for the pipeline stall.
You mean... if there are more than 2 division algorithms needed, compile the algorithm itself into the program, and call it from the program?
On another side, I've worked on the division algorithms, and they are much harder than the multiplication or so. Many, many thanks to Runer, who gave me insane routines to divide A by a known number,
and now I got this table ready:
Dividing two variables is this code:
or a
sbc hl, hl
ld a, (XXXXXX)
ld l, a
ld a, (YYYYYY)
call _DivHLByA
ld a, l
If you think you can make it shorter/faster, don't hesitate to post
For dividing numbers by variables, I'm just thinking about using the standard routines, available at WikiTI or so.
And then, after I got this ready, I can implement the ease routines, for BASIC commands like and, not(, ->, or and some more.
Well, good job. Although I doubt at this point that the performance boost will be that great, since you're going to be doing bcalls everywhere anyway.
I think a JIT would work best in the long run, rather than compiling 4000 bytes worth of a BASIC program into 10000 bytes of a program which would barely fit into RAM without turning it into a flash
program. (Kerm's GRAPH3D for 84+CSE is 6567 bytes on my calculator.)
I definitely like the black background more.
Great work! This is an amazing project!
Its awesome that we are getting more programming techniques.
Please, please, please make this for the TI-84+CSE!!!!
oldmud0 wrote:
Well, good job. Although I doubt at this point that the performance boost will be that great, since you're going to be doing bcalls everywhere anyway.
I think a JIT would work best in the long run, rather than compiling 4000 bytes worth of a BASIC program into 10000 bytes of a program which would barely fit into RAM without turning it into a flash
program. (Kerm's GRAPH3D for 84+CSE is 6567 bytes on my calculator.)
I have thought about this, but since the OS works with TIOS variables, floating point numbers, this would barely decrease the executing time. For example, if you have A+B, you need
1) Search A
2) Copy it to OP1/OP2
3) Search B
4) Copy it to OP1/OP2
5) bcall(_FPAdd)
6) bcall(_StoAns)
or something like that. My guess is, that this would take at least 40 bytes without the bcalls, and thousands clock cyles. For comparison, my A+B would look like this: (unsigned 8-bits)
ld a, (address_variable_A)
ld hl, address_variable_B
add a, (hl)
which is 9 bytes and 44 clock cycles. I think the compiled program would be MUCH faster, and I hope to prove it soon with screenshots!
calcnerd_CEP_D wrote:
I definitely like the black background more.
Great work! This is an amazing project!
Its awesome that we are getting more programming techniques.
Thanks! The syntax looks like normal BASIC, so that it shouldn't be hard to make the step to ICE
fluzz wrote:
Please, please, please make this for the TI-84+CSE!!!!
Not very likely. At first is the screen totally different, so any graphical command needs to be rewritten. And 2, the memory is only 22K or so, so you can't make big programs. When porting this to
the CSE, is the same as rewriting almost the whole stuff. Also, the ez80 (CE) has more commands, like mlt hl which saves much bytes and thus cycles too. | {"url":"https://www.cemetech.net/forum/viewtopic.php?t=12616&postdays=0&postorder=asc&start=20","timestamp":"2024-11-04T20:50:34Z","content_type":"text/html","content_length":"92167","record_id":"<urn:uuid:9425f8fc-03f2-43c1-ad66-5116bb5fee7d>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00630.warc.gz"} |
Aggregates in table report
Hi All
I am trying to sum AND add a percentage to two separate columns in a tabular report. I can do this fine with two separate aggregates:
SUM(cost_2017)*1.1 = aggregate 1
SUM(cost_2018)*1.1 = aggregate 2
But the results fall in two different rows for each aggregate. How do I just do one aggregate, but for column 2017 costs, use the cost_2017 number and for column 2018 costs, use the cost_2018
numbers for the column 2018 costs and they fall in the same row. I can select both the cost 2017 and cost 2018 numbers in the selected fields, but can only do a formula for EITHER of them... Doing a
SUM this way works (I can pull over 2017 and 2018 costs, and get two different sums in one row), but this doesn't seem possible for formulae?
Thanks in advance
Example attached
1 answer to this question | {"url":"https://forums.caspio.com/topic/7408-aggregates-in-table-report/","timestamp":"2024-11-05T05:38:07Z","content_type":"text/html","content_length":"82294","record_id":"<urn:uuid:9ab353dd-6bf7-48ef-829e-4ac3ef88625a>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00324.warc.gz"} |
Why Quantum Particles Can Pass Through Barriers
By Vijay Damodharan - Natural Sciences Student @ Christ College, Cambridge
People sometimes play basketball, football, or table tennis using a wall as their opponent. It may be for practice, or just out of boredom. When the ball strikes the wall, it dutifully returns the
ball back to us. But what if it didn’t? What if the ball passed straight through? It seems like an absurd question to ask; however, it occurs more often than one might think in quantum mechanics.
Quantum particles are able to ‘tunnel’ through potential barriers and this effect is essential for many physical processes, (radioactive decay and nuclear fusion), biological processes,
(photosynthesis and respiration), and it is also critical for the operation of many electronic components.
First, what do we mean by a wall? If we have a brick wall and we hit it with a tennis ball, the ball will definitely come back to us. If we instead throw a bowling ball onto a thin plastic sheet,
chances are the ball will break through the plastic wall. Even a brick wall can be broken through if the object is sufficiently massive and travelling fast enough (such as a truck).
We can generalise this by thinking in terms of energy. The ball has a certain amount of kinetic energy, KE, which depends on its mass and the velocity with which it is travelling. The wall is a solid
material which requires a certain amount of energy to fracture. We can think of the wall as something that absorbs energy, and if we give it more energy than it can absorb, then it breaks. The amount
of energy needed to break the wall can be thought of as a potential energy, V, associated with the wall.
In other words, a wall is something that provides a (potential) energy barrier to objects. If in a region KE ≥ V, the particle can exist there, otherwise if KE ≤ V, it cannot. The latter is called
the classically forbidden region.
We can generalise this to cases such as gravity. We can think of the earth as providing a gravitational potential energy barrier, GPE. For a rocket to escape from the earth’s gravity, KErocket ≥ GPE
must be satisfied.
Quantum particles are facing potential barriers all the time. Inside the nucleus of an atom, protons, and neutrons (generally called nucleons) are held together by a strong nuclear force, which
provides a potential energy barrier. From our discussion so far, we know that the nucleons cannot leave unless they have enough energy to overcome this barrier.
Alpha decay is a common process by which an atom ejects two protons and neutrons from its nucleus. So how is it that at one moment the nucleons are trapped, unable to move, and the next they suddenly
leave? Is this not like a ball suddenly deciding to go through the wall, instead of returning back to us?
Quantum tunnelling is one of the quantum effects which is best understood mathematically. However even without doing any detailed calculations, we can get an idea of where it comes from by exploring
some fundamental postulates of quantum mechanics.
De Broglie famously hypothesised that all particles have wave-like properties, which was later confirmed experimentally. As a result, we can describe particles by a wave function Ψ(x,y,z). Like any
mathematical function, it will just take in some numbers (x, y, z coordinates) as an input, and spit out another number as an output. This output by itself doesn’t mean anything.
However, it is found that performing certain operations to the wavefunction does give an output which is physically meaningful. One such example is probability. If Ψ describes an electron, then the
probability density, P, of finding the electron at a point is given by taking the absolute value squared of the wavefunction: P(x)dx ǀ Ψ* Ψ ǀ^2dx .
We know that total probability must always equal one and never higher or lower. This means thatǀ Ψ* Ψ ǀ^2, and therefore Ψ itself must be a smooth and continuous function. As such, the value of the
function cannot ‘jump’ - e.g. it cannot be equal to 2 at one point and 20 right next to it, as that would make Ψ undefined at that point, which makes ǀ Ψ* Ψ ǀ^2, and therefore the probability,
undefined at that point.
Going back to our problem, we know that in the regions around the nucleon where KE ≥ V is satisfied, the particle can exist, so we expect a non-zero probability of finding the particle there. Hence Ψ
must have some finite value, like 0.3.
Figure 1: First energy level wavefunction for a finite potential of width 0.4nm
However, we don’t want the particle to exist in the regions where KE ≤ V, we want the probability of finding it there to be 0, meaning Ψ should be equal to be 0. But we just said that Ψ cannot jump
in values! It cannot go from 0.3 (or any finite number) to 0 immediately! Ψ can only decrease gradually little by little as it goes further and further into the barrier as shown in Figure 1. It will
only reach zero after going infinitely far into the barrier. All of this is only true however if the potential energy barrier is not infinite, but that’s a problem for another article!
This means that Ψ is non-zero within the barrier. If the barrier has a finite width, such as a wall, a
Figure 2: Quantum tunnelling.
nd Ψ is non-zero inside the wall, it can have a non-zero value on the other side of the wall by the same argument, as shown in Figure 2. In other words, the particle can appear on the other side of
the wall! This is quantum tunnelling.
It may feel slightly unsatisfactory to imagine that quantum tunnelling simply arises as a result of some mathematical manipulations. However, mathematics is an essential tool in quantum mechanics.
Often, just by setting up some mathematical postulates, abstract spaces, and operations in the spaces, simply through algebra, we can arrive at physically meaningful results that are proven to hold
experimentally. Quantum tunnelling is one such example, however without it many processes wouldn’t occur, and our world would be completely different.
3. Up and Atom (2018). What is Quantum Tunneling, Exactly? YouTube. Available at: https://www.youtube.com/watch?v=WPZLRtyvEqo. | {"url":"https://www.oxbridgelaunchpad.com/post/why-quantum-particles-can-pass-through-barriers","timestamp":"2024-11-03T03:00:43Z","content_type":"text/html","content_length":"1050052","record_id":"<urn:uuid:08704f9a-ce0a-4ae5-b8c6-65d042465bb1>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00006.warc.gz"} |
Understanding the relationship between mathematics and science coursework patterns
Background/Context: There has been little research on the relationship between mathematics and science coursework in secondary school. Purpose of Study: The present analysis explored the patterns of
science course-taking in relation to the patterns of mathematics course-taking among high school graduates. Research Design: Using data from the 2000 High School Transcript Study (N = 20,368),
secondary analysis was performed in the form of multilevel models with students nested within schools to document a strong relationship between mathematics and science coursework patterns. Findings/
Results: Results highlighted that (1) taking more courses in advanced mathematics was related to taking more courses in advanced science (this relationship remained strong even after adjustment for
student-level and school-level variables); (2) the more courses that students took in advanced mathematics, the more likely it was that student and school characteristics would join in to select
students into taking more courses in advanced science; (3) many high school graduates complied with graduation requirements by taking limited nonadvanced mathematics and science coursework during
high school; and (4) mathematics coursework was necessary but insufficient to promote advanced science coursework. Conclusions/ Recommendations: State governments are encouraged to prescribe not only
the number but also the content of mathematics and science courses required for high school graduation. School personnel such as career counselors are encouraged to help promote better coursework of
students in mathematics and science.
Original language English
Pages (from-to) 2101-2126
Number of pages 26
Journal Teachers College Record
Volume 111
Issue number 9
State Published - Sep 2009
ASJC Scopus subject areas
Dive into the research topics of 'Understanding the relationship between mathematics and science coursework patterns'. Together they form a unique fingerprint. | {"url":"https://scholars.uky.edu/en/publications/understanding-the-relationship-between-mathematics-and-science-co","timestamp":"2024-11-14T15:01:19Z","content_type":"text/html","content_length":"54184","record_id":"<urn:uuid:e9c7f56b-111a-474b-bceb-f038ebf888cf>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00808.warc.gz"} |
Apply function to each element of array on GPU
This function behaves similarly to the MATLAB^® function arrayfun, except that the evaluation of the function happens on the GPU, not on the CPU. Any required data not already on the GPU is moved to
GPU memory. The MATLAB function passed in for evaluation is compiled and then executed on the GPU. All output arguments are returned as gpuArray objects.
B = arrayfun(func,A) applies a function func to each element of a gpuArray A and then concatenates the outputs from func into output gpuArray B. B is the same size as A and B(i,j,...) = func(A
(i,j,...)). The input argument func is a function handle to a MATLAB function that takes one input argument and returns a scalar. func is called as many times as there are elements of A.
B = arrayfun(func,A1,...,An) applies func to the elements of the arrays A1,...,An, so that B(i,j,...) = func(A1(i,j,...),...,An(i,j,...)). The function func must take n input arguments and return a
scalar. The sizes of A1,...,An must match or be compatible.
[B1,...,Bm] = arrayfun(func,___) returns multiple output arrays B1,...,Bm when the function func returns m output values. func can return output arguments having different data types, but the data
type of each output must be the same each time func is called.
Run Function on GPU
Define a function, cal. The function cal applies a gain and an offset correction to an array of measurement data. The function performs only element-wise operations when applying the gain factor and
offset to each element of the rawdata array.
function c = cal(rawdata,gain,offset)
c = (rawdata.*gain) + offset;
Create an array of measurement data.
Create arrays containing the gain and offset data.
gn = rand([1 4],"gpuArray")/100 + 0.995
gn =
0.9958 0.9967 0.9985 1.0032
offs = rand([1 4],"gpuArray")/50 - 0.01
offs =
0.0063 -0.0045 -0.0081 0.0002
Run the calibration function on the GPU. The function runs on the GPU because the input arguments gn and offs are already GPU arrays, and are therefore stored in GPU memory. Before the function runs,
it converts the input array meas to a gpuArray object.
corrected = arrayfun(@cal,meas,gn,offs)
corrected =
1.0021 1.9889 2.9874 4.0129
Performing a small number of element-wise operations on a GPU is unlikely to speed up your code. For an example showing how arrayfun execution speed scales with input array size, see Improve
Performance of Element-Wise MATLAB Functions on the GPU Using arrayfun.
Use Function with Multiple Outputs
Define a function that applies element-wise operations to multiple inputs and returns multiple outputs.
function [o1,o2] = myFun(a,b,c)
o1 = a + b;
o2 = o1.*c + 2;
Create gpuArray input data, and evaluate the function on the GPU.
s1 = rand(400,"gpuArray");
s2 = rand(400,"gpuArray");
s3 = rand(400,"gpuArray");
[o1,o2] = arrayfun(@myFun,s1,s2,s3);
Name Size Bytes Class Attributes
o1 400x400 1280000 gpuArray
o2 400x400 1280000 gpuArray
s1 400x400 1280000 gpuArray
s2 400x400 1280000 gpuArray
s3 400x400 1280000 gpuArray
Use Random Numbers with arrayfun
Define a function that creates and uses a random number, R.
function Y = myRandFun(X)
R = rand;
Y = R.*X;
Run the function on the GPU. As G is a 4-by-4 gpuArray object, arrayfun applies the myRandfun function 16 times, generating 16 different random scalar values, H.
G = ones(4,"gpuArray")*2;
H = arrayfun(@myRandFun,G)
H =
1.0557 0.3599 1.5303 0.2745
0.4268 1.1226 1.5261 1.7068
0.0302 0.5814 0.2556 0.3902
1.1210 1.5310 1.3665 0.8487
Input Arguments
func — Function to apply
function handle
Function to apply to the elements of the input arrays, specified as a function handle.
• func must return scalar values.
• For each output argument, func must return values of the same class each time it is called.
• func must accept numerical or logical input data.
• func must be a handle to a function that is written in the MATLAB language. You cannot specify func as a handle to a MEX function.
• You cannot specify func as a static method or a class constructor method.
func can contain the following built-in MATLAB functions and operators.
abs csch log2 sin
and double log10 single
acos eps log1p sinh
acosh eq logical sqrt
acot erf lt tan
acoth erfc max tanh
acsc erfcinv min times
acsch erfcx minus true
asec erfinv mod uint8 Scalar expansion versions of the following:
asech exp NaN uint16
asin expm1 ne uint32 *
asinh false not uint64 /
atan fix ones xor \
atan2 floor or zeros ^
atanh gamma pi
beta gammaln plus + Branching instructions:
betaln ge pow2 -
bitand gt power .* break
bitcmp hypot rand ./ continue
bitget imag randi .\ else, elseif, if
bitor Inf randn .^ for
bitset int8 rdivide == return
bitshift int16 real ~= switch, case, otherwise
bitxor int32 reallog < while
cast int64 realmax <=
ceil intmax realmin >
complex intmin realpow >=
conj isfinite realsqrt &
cos isinf rem |
cosh isnan round ~
cot ldivide sec &&
coth le sech ||
csc log sign
Functions that create arrays (such as Inf, NaN, ones, rand, randi, randn, and zeros) do not support size specifications as input arguments. Instead, the size of the generated array is determined by
the size of the input variables to your functions. Enough array elements are generated to satisfy the needs of your input or output variables. You can specify the data type using both class and like
syntaxes. The following examples show supported syntaxes for array-creation functions:
a = rand;
b = ones;
c = zeros(like=x);
d = Inf("single");
e = randi([0 9],"uint32");
When you use rand, randi, and randn to generate random numbers within func, each element is generated from a different substream. For more information about generating random numbers on the GPU, see
Random Number Streams on a GPU.
When you use switch, case, otherwise within func, case expressions support only numeric and logical values.
A — Input array
scalars | vectors | matrices | multidimensional arrays
Input array, specified as scalars, vectors, matrices, or multidimensional arrays. At least one input array argument must be a gpuArray for arrayfun to run on the GPU. Each array that is stored in CPU
memory is converted to a gpuArray before the function is evaluated. If you plan to make several calls to arrayfun with the same array, it is more efficient to convert that array to a gpuArray.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | logical
Output Arguments
B — Output array
Output array, returned as a gpuArray.
• The sizes of A1,...,An must match or be compatible. The size of output array B depends on the sizes of A1,...,An. For more information, see Compatible Array Sizes for Basic Operations.
• Because the operations supported by arrayfun are strictly element-wise, and each computation of each element is performed independently of the others, certain restrictions are imposed:
□ Input and output arrays cannot change shape or size.
□ Array-creation functions such as rand do not support size specifications. Arrays of random numbers have independent streams for each element.
• You cannot specify the order in which arrayfun calculates the elements of output array B or rely on them being done in any particular order.
• Like arrayfun in MATLAB, matrix exponential power, multiplication, and division (^, *, /, \) perform element-wise calculations only.
• Operations that change the size or shape of the input or output arrays (cat, reshape, and so on) are not supported.
• Read-only indexing (subsref) and access to variables of the parent (outer) function workspace from within nested functions is supported. You can index variables that exist in the function before
the evaluation on the GPU. Assignment or subsasgn indexing of these variables from within the nested function is not supported. For an example of the supported usage, see Stencil Operations on a
• Anonymous functions do not have access to their parent function workspace.
• Overloading the supported functions is not allowed.
• The code cannot call scripts.
• There is no ans variable to hold unassigned computation results. Make sure to explicitly assign to variables the results of all calculations.
• The following language features are not supported: persistent or global variables, parfor, spmd, and try/catch.
• Calls to arrayfun inside P-code files or using arrayfun to evaluate functions obfuscated as a P-code files are not supported in standalone functions compiled using MATLAB Compiler™.
• The first time you call arrayfun to run a particular function on the GPU, there is some overhead time to set up the function for GPU execution. Subsequent calls of arrayfun with the same function
can run faster.
Extended Capabilities
Thread-Based Environment
Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.
This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.
GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.
The arrayfun function fully supports GPU arrays. To run the function on a GPU, specify the input data as a gpuArray. For more information, see Run MATLAB Functions on a GPU.
Version History
Introduced in R2010b
R2024b: Support for functions defined in class definition files
You can now call arrayfun in a class method to evaluate functions defined in the class definition file (a file with a .m extension that contains the classdef keyword).
For example, this class contains a method, output, that uses arrayfun to evaluate a local function, localFun.
classdef TestClass
function output = func(obj,x)
output = arrayfun(@localFun,x);
function output = localFun(x)
output = x.*x;
For more information about defining classes in MATLAB, see Creating a Simple Class.
R2024b: Support for P-Code files
You can now use P-code files with arrayfun. You can:
• Use arrayfun to evaluate a function obfuscated as a P-code file.
• Call arrayfun inside a P-code file.
• Use arrayfun when the function it applies contains a call to a function obfuscated as a P-code file.
For more information about P-code files, see Create a Content-Obscured File with P-Code.
R2024a: Support for cell array case expressions in switch, case, otherwise
Use a cell array as the case expression to compare the switch expression against multiple values within the function you apply using arrayfun. For example, you can use case {x1,y1} to execute the
corresponding code if the switch expression matches at least one of x1 and y1.
R2023b: Support for switch, case, and otherwise
You can now use switch conditional statements in functions you apply using arrayfun. This functionality has these limitations:
• Case expressions support only numeric and logical values.
• Using a cell array as the case expression to compare the switch expression against multiple values, for example, case {x1,y1}, is not supported.
R2023b: Changes to indexing into and writing to variables in nested functions
Passing arrays from a parent workspace to a nested function and indexing into the array within the nested function now errors
For example, in the following code, the variable parentWorkspaceVar is created in the parent workspace of the foo function. If foo is used in an arrayfun call with gpuArray input, and if the foo
function passes parentWorkspaceVar as input to a nested function within foo, the code errors.
As a workaround, instead of passing the parent workspace variable (parentWorkspaceVar) to the nested function (bar), use the parent workspace variable directly as it is already in the scope of the
nested function.
Errors Workaround
function y = exampleFunction function y = exampleFunction
parentWorkspaceVar = 1:9; parentWorkspaceVar = 1:9;
x = ones(2,"gpuArray"); x = ones(2,"gpuArray");
y = arrayfun(@foo,x); y = arrayfun(@foo,x);
function y = foo(x) function y = foo(x)
y = bar(parentWorkspaceVar); y = bar;
function y = bar(z) % ERRORS function y = bar
y = z(1); % Previously this variable was passed in, now it is used directly as it is in scope.
end y = parentWorkspaceVar(1);
Functions writing into variables created in a parent function now error
In the following code, the variable workspaceVar is created in the workspace of the bar function. If bar is used in an arrayfun call with gpuArray input, and if a nested function foo writes into
workspaceVar, the code errors.
As a workaround, instead of writing to the variable (workspaceVar) within the nested function (foo), add another output to the nested function and use the output to write to the variable.
Errors Workaround
function x = exampleFunction function x = exampleFunction
z = ones(2,"gpuArray"); z = ones(2,"gpuArray");
x = arrayfun(@bar,z); x = arrayfun(@bar,z);
function y = bar(z) function y = bar(z)
workspaceVar = 2; workspaceVar = 2;
foo(z); [out,workspaceVar] = foo(z); % Write to workspaceVar outside foo.
function x = foo(z) function [x,y] = foo(z) % Add another output y to the nested function.
workspaceVar = 10; % ERRORS y = 10;
x = z; x = z;
end end
y = workspaceVar; y = workspaceVar;
end end
end end | {"url":"https://ch.mathworks.com/help/parallel-computing/gpuarray.arrayfun.html","timestamp":"2024-11-04T20:35:16Z","content_type":"text/html","content_length":"118912","record_id":"<urn:uuid:87e07841-3f75-49be-8b78-9bff46d832e2>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00084.warc.gz"} |
How to calculate Total Wear Metals & LSJ oil analysis methodology
Jun 24, 2024
I was wondering what BobIsTheOilGuy members thought about this UOA methodology from Lake Speed Jr's live stream yesterday. Does this seem like a sound method? I think it does.
Does anyone know how to calculate a total wear metals per 1000 miles statistic from a blackstone UOA? Do you just add up all the metals I've put a red square on here and divide by the mileage/1000?
So for instance here I got 23 ppm of all the red boxed metals, and 23/2.578 = 8.9 twm per 1k mi. I understand that a shorter OCI will skew the TWM number high as stated in the livestream, so I will
go for a 5k OCI next time.
I wasn't sure about the the moly since the VOA for this oil had 70 ppm, so it seems like it shouldn't be counted as a wear metal.
I took a bit of flack on posting this first UOA since I was concerned about the oil viscosity even though the wear metals seemed fine.
I also found the filter discussion in the video pretty interesting, I didn't realize I was sacrificing efficiency for capacity by going with the long endurance filters, like the Fram Synthetic
Endurance I'm using. If I'm going to a 5000 OCI maybe it is not the best choice?
I'll hang up and listen.
Jul 12, 2012
New engines produce lots of wear metals etc during break in? All that matters with an oil filter is that it doesn't fail.
Last edited:
Feb 27, 2012
He is selling his oil analysis services so by default I do not subscribe to his "methodology."
Change break in oil at 3k miles, UOAs are a waste of $$ until 40k imo.
May 14, 2007
It's clear to me that LSJr, while perhaps an accredited tribologist and formulator, knows diddly-squat about how to use proper statistical analysis methodology. He demonstrates the typical
misinterpretations most make; confusing macro and micro data stream analysis.
Read this when you get a chance:
Reviewing UOA Data Used oil analyses (UOAs) are tools. And like most tools, they can either be properly used or misused, depending upon the application, the user, the surrounding conditions, etc.=
There are already many good articles and publications in existence that tell us how to interpret...
Last edited:
I just use Fe, Al, Cu, and Si and do them individually for the /1000mi values. At your mileage, you are just starting to collect data to develop some trends, the metals should be elevated at this
point so it's not something to concern yourself with yet - yours are actually quite low for such a new vehicle. I see some of your other posts question the oil to use - I'm quite sure that M1 oil is
just fine to use and if it drops out of viscosity, that too for many turbo charged direct injected engines is quite normal and not hurting a thing for normal use/driving. There are oils more
resistant to the mechanical shearing but keep in mind any viscosity drop due to fuel dilution is a separate matter and will happen regardless of oil type. You can see an example of my UOA tracking
here using wear metal rates/1000 miles:
Looks great for ~6 mos/4K and 3 track days where the oil temp was sitting around 270 deg F the entire weekend (I estimate about 3 hours of actual driving). 10/10 recommend. Will keep it in and push
the whole year on it so 5 more track days and probably 4-5K more miles. 13.11 cSt 100 is fantastic...
Apr 15, 2010
View attachment 233778
I was wondering what BobIsTheOilGuy members thought about this UOA methodology from Lake Speed Jr's live stream yesterday. Does this seem like a sound method? I think it does.
Does anyone know how to calculate a total wear metals per 1000 miles statistic from a blackstone UOA? Do you just add up all the metals I've put a red square on here and divide by the mileage/
1000? So for instance here I got 23 ppm of all the red boxed metals, and 23/2.578 = 8.9 twm per 1k mi. I understand that a shorter OCI will skew the TWM number high as stated in the livestream,
so I will go for a 5k OCI next time.
I wasn't sure about the the moly since the VOA for this oil had 70 ppm, so it seems like it shouldn't be counted as a wear metal.
I took a bit of flack on posting this first UOA since I was concerned about the oil viscosity even though the wear metals seemed fine.
View attachment 233779
I also found the filter discussion in the video pretty interesting, I didn't realize I was sacrificing efficiency for capacity by going with the long endurance filters, like the Fram Synthetic
Endurance I'm using. If I'm going to a 5000 OCI maybe it is not the best choice?
I'll hang up and listen.
Don't forget in the video he said that you should only look at ppm/thousand miles when doing samples between 4k-6k miles. He said when under 4k miles the wear numbers are skewed to look worse
(probably because of left over from the previous fill) and the wear numbers are skewed to look better on samples over 6k miles but he didn't explain why that was so and to me that doesn't make sense.
Note: A problem with a UOA on factory fill is that if allows owners to incorrectly assume that the break in material seen on a UOA is causing the engine to wear out sooner if you don't change the
oil. That's not the case at all.
Oct 20, 2005
I didn't realize I was sacrificing efficiency for capacity by going with the long endurance filters, like the Fram Synthetic Endurance I'm using.
I admit i don't know exactly what he said because i cannot stand to watch him, he right up there with Kilmer for me.
But that isn't true at least not to any great extent for the FRAM - FRAM gives you the Efficiency @ micron for the filter. Wix if they haven't changer it XP is worse than regular Wix and the
Purolator Boss is also rated at a slightly higher Micron than regular, but it is still not bad.
Id just drive the truck if i was you.
Don't forget in the video he said that you should only look at ppm/thousand miles when doing samples between 4k-6k miles. He said when under 4k miles the wear numbers are skewed to look worse
(probably because of left over from the previous fill) and the wear numbers are skewed to look better on samples over 6k miles but he didn't explain why that was so and to me that doesn't make
Note: A problem with a UOA on factory fill is that if allows owners to incorrectly assume that the break in material seen on a UOA is causing the engine to wear out sooner if you don't change the
oil. That's not the case at all.
Agreed - not sure on why the issue with the mileage here, my understanding is that the wear metals increase in concentration with mileage...maybe they are front-loaded and not a linear increase?
Should be able to determine that with as many UOAs are here or even Blackstone likely has that information/could provide data.
Also agree with your last point - folks see those high wear metals in the first few UOAs and freak out.
I admit i don't know exactly what he said because i cannot stand to watch him, he right up there with Kilmer for me.
But that isn't true at least not to any great extent for the FRAM - FRAM gives you the Efficiency @ micron for the filter. Wix if they haven't changer it XP is worse than regular Wix and the
Purolator Boss is also rated at a slightly higher Micron than regular, but it is still not bad.
Id just drive the truck if i was you.
Wow, LSJR and Kilmer being equivalents? I'd say they are quite different w/r to their content and how they present it but maybe it's the voice/mannerisms that bug you?
Also how isn't this video posted in one of the main sub-forums....it's got it all!
Jun 24, 2024
Don't forget in the video he said that you should only look at ppm/thousand miles when doing samples between 4k-6k miles. He said when under 4k miles the wear numbers are skewed to look worse
(probably because of left over from the previous fill) and the wear numbers are skewed to look better on samples over 6k miles but he didn't explain why that was so and to me that doesn't make
Note: A problem with a UOA on factory fill is that if allows owners to incorrectly assume that the break in material seen on a UOA is causing the engine to wear out sooner if you don't change the
oil. That's not the case at all.
Yeah, I thought I acknowledged that in the last sentence of the second paragraph in the first post. I also know that wear metals will be higher for the first ~15k miles, just trying to calculate the
total wear for the future. I do tend to write walls of text so I apologize for that.
Also how isn't this video posted in one of the main sub-forums....it's got it all!
You should start a thread on it, I considered talking about this in a more visible spot but figured I’d get nuked as a newb. I’m learning the less I post on here the better for my sanity. I will
probably just post voa/uoa without writing too much from here on out.
You should start a thread on it, I considered talking about this in a more visible spot but figured I’d get nuked as a newb. I’m learning the less I post on here the better for my sanity. I will
probably just post voa/uoa without writing too much from here on out.
Just post factual information and there’s nothing to worry about.
I considered talking about this in a more visible spot but figured I’d get nuked as a newb. I’m learning the less I post on here the better for my sanity. I will probably just post voa/uoa
without writing too much from here on out.
Go for it, for the betterment of all!
Just post factual information and there’s nothing to worry about.
And with that, the General/off topic sub-forum is gone.
Oct 20, 2005
Wow, LSJR and Kilmer being equivalents? I'd say they are quite different w/r to their content and how they present it but maybe it's the voice/mannerisms that bug you?
I cant make it long enough to make any judgement about the content with either of them, i'd rather listen to 10 hours of fingers on a chalkboard...
But i digress...
What's the total mileage on this truck? Did i miss that?
And yes, you can divide the wear metals by thousands of miles to get wear metals per 1000 miles so 5 ppm in 5000 miles would be 1 ppm in 1000 miles average.
His biggest problem (in my opinion) is that he’s automatically correlating the observation to oil brand which is unwarranted. There are many uncontrolled variables that contribute to the UOA results
and he’s somehow thinking all of them are accounted for - which they are not.
If my understanding is correct that this is his conclusion (as read above) then he’s making a fundamental error that calls into question any other conclusions he may draw.
Aug 25, 2018
He is selling his oil analysis services so by default I do not subscribe to his "methodology."
Change break in oil at 3k miles, UOAs are a waste of $$ until 40k imo.
He and I discussed this same methodology nearly 10 years ago, well before he started Speediagnostix and was still with Driven. Yes, he's selling his UOA services, but that doesn't suddenly make the
methodology void.
As for the 40k miles, there's cases right now in 3rd Gen Ecodiesels seeing severe bearing wear at 10-20k miles, on OEM recommended oil (also with dealerships and owners putting in the wrong oil),
that drops dramatically when moving to a better oil. There's been quite a few with trashed bearings at <40k miles. That engine shreds oil. It took a No VII 10W-40 to bring the bearing wear down and
not shear out of grade.
As for the 40k miles, there's cases right now in 3rd Gen Ecodiesels seeing severe bearing wear at 10-20k miles, on OEM recommended oil (also with dealerships and owners putting in the wrong oil),
that drops dramatically when moving to a better oil. There's been quite a few with trashed bearings at <40k miles. That engine shreds oil. It took a No VII 10W-40 to bring the bearing wear down
and not shear out of grade.
What is wrong with the engine that causes this?
Feb 27, 2012
He and I discussed this same methodology nearly 10 years ago, well before he started Speediagnostix and was still with Driven. Yes, he's selling his UOA services, but that doesn't suddenly make
the methodology void.
As for the 40k miles, there's cases right now in 3rd Gen Ecodiesels seeing severe bearing wear at 10-20k miles, on OEM recommended oil (also with dealerships and owners putting in the wrong oil),
that drops dramatically when moving to a better oil. There's been quite a few with trashed bearings at <40k miles. That engine shreds oil. It took a No VII 10W-40 to bring the bearing wear down
and not shear out of grade.
His methodology wont work with cursed Ecodiesels. Its also obtusely expensive. Good for you on the personal connection.
Last edited:
His methodology wont work with cursed Ecodiesels. Its also obtusely expensive. Good for you on the personal connection.
I wonder if LSJR peruses the BITOG to see what folks are saying about this videos. | {"url":"https://bobistheoilguy.com/forums/threads/how-to-calculate-total-wear-metals-lsj-oil-analysis-methodology.385877/","timestamp":"2024-11-02T08:08:53Z","content_type":"text/html","content_length":"219877","record_id":"<urn:uuid:12a96a2b-84c1-4aec-bcc3-1e4a7465a5fe>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00350.warc.gz"} |
What can you say about their utility functions
Reference no: EM131524576
Reference no: EM131524576
Question: Even without a formal assessment process, it often is possible to learn something about an individual's utility function just through the preferences revealed by choice behavior. Two
persons, A and B, make the following bet: A wins $40 if it rains tomorrow and B wins $10 if it does not rain tomorrow.
a. If they both agree that the probability of rain tomorrow is 0.10, what can you say about their utility functions?
b. If they both agree that the probability of rain tomorrow is 0.30, what can you say about their utility functions?
c. Given no information about their probabilities, is it possible that their utility functions could be identical?
d. If they both agree that the probability of rain tomorrow is 0.20, could both individuals be risk-averse? Is it possible that their utility functions could be identical? Explain.
Negotiation is a problem-solving process : Negotiation is a problem-solving process. You need to identify the problem, gather data, determine the best options, make a decision about the problem
Review the given two scenarios : You have made a reservation to spend the weekend at the coast. To get the reservation, you had to make a nonrefundable $50 deposit.
Where did the situation occur and what is the effect : Who is involved?; Where did the situation occur?; Most importantly, why does this matter and what is the effect
What are skills necessary to be an effective project manager : Describe the reasons why IT projects differ from projects in other disciplines. What are the skills necessary to be an effective project
manager? Elaborate.
What can you say about their utility functions : Even without a formal assessment process, it often is possible to learn something about an individual's utility function just through the preferences
Detailing the evolution of chosen health policy : Create a 700- to 1,050-word timeline detailing the evolution of your chosen health policy. Include the following:
Discuss the use of assessment technique : Assume that you are interested in purchasing a new model of a personal computer whose reliability has not yet been perfectly established.
Determine the main challenges that you would encounter : determine the main challenges that you would encounter during the project closeout process, and provide your strategy to address the
challenges in question.
Discuss utility function for automobile life span : You are in the market for a new car. An important characteristic is the life span of the car.
Prime number theorem
Dirichlet series
Proof of bolzano-weierstrass to prove the intermediate value
Every convergent sequence contains either an increasing, or a decreasing subsequence.
Antisymmetric relations
How many relations on A are both symmetric and antisymmetric?
Distributed random variables
Daily Airlines fies from Amsterdam to London every day. The price of a ticket for this extremely popular flight route is $75. The aircraft has a passenger capacity of 150.
Prepare a system of equations
How much money will Dave and Jane raise for charity
Managing ashland multicomm services
This question is asking you to compare the likelihood of your getting 4 or more subscribers in a sample of 50 when the probability of a subscription has risen from 0.02 to 0.06.] Talk about the
comparison of probabilities in your explanation.
Skew-symmetric matrices
Skew-symmetric matrices
Type of taxes and rates in spokane wa
Describe the different type of taxes and their rates in Spokane WA.
Stratified random sample
Suppose that in the four player game, the person who rolls the smallest number pays $5.00 to the person who rolls the largest number. Calculate each player's expected gain after one round.
Find the probability density function
Find the probability density function.
Develop a new linear programming for an aggregate production
Linear programming applied to Aggregate Production Planning of Flat Screen Monitor
Discrete-time model for an economy
Discrete-time model for an economy | {"url":"https://www.expertsmind.com/library/what-can-you-say-about-their-utility-functions-51524576.aspx","timestamp":"2024-11-02T06:23:29Z","content_type":"text/html","content_length":"68042","record_id":"<urn:uuid:d85a1efa-a422-49e7-a402-953b5c6b88c2>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00899.warc.gz"} |
C Program To Check Whether a Number is Prime or Not - CodingBroz
C Program To Check Whether a Number is Prime or Not
In this post, we will learn how to check whether a number is prime or not using C Programming language.
A number is called a Prime number, if it is divisible only by itself and one. This means a Prime number has only two factors – 1 and the number itself. For example: 2, 3, 5, 7, 11, . . . etc.
A number is called a Composite Number, if it has more than two factors. For example: 4, 15, 26, 98, . . . etc.
Note: 1 is neither a prime number nor composite number.
So, without further ado, let’s begin the tutorial.
C Program To Check Whether a Number is Prime or Not
// C Program To Check Whether a Number is Prime or Not
#include <stdio.h>
int main(){
int num, i, c = 0;
// Asking for Input
printf("Enter an Number: ");
scanf("%d", &num);
// logic
for (i = 1; i <= num; i++){
if (num % i == 0){
if (c == 2){
printf("%d is a Prime Number.", num);
else {
printf("%d is not a Prime Number.", num);
return 0;
Output 1
Enter an Number: 7
7 is a Prime Number.
Output 2
Enter an Number: 57
57 is not a Prime Number.
How Does This Program Work ?
int num, i, c = 0;
In this program, we have declared three int data type variables named as num, i and c. Variable c is assigned a value of 0.
// Asking for Input
printf("Enter an Number: ");
scanf("%d", &num);
Then, the user is asked to enter a number which he/she wants to check. This number will get stored in the num named variable.
// logic
for (i = 1; i <= num; i++){
if (num % i == 0){
Now, this program checks for the number of integers, num is completely divisible with. And the value of the number of factors is stored in the c named variable.
if (c == 2){
printf("%d is a Prime Number.", num);
else {
printf("%d is not a Prime Number.", num);
If the num contains exactly 2 factors, then the given number is a Prime number otherwise it’s not a Prime number.
Some of the used terms are as follow:
#include <stdio.h> – In the first line we have used #include, it is a preprocessor command that tells the compiler to include the contents of the stdio.h(standard input and output) file in the
The stdio.h is a file which contains input and output functions like scanf() and printf() to take input and display output respectively.
Int main() – Here main() is the function name and int is the return type of this function. The Execution of any Programming written in C language begins with main() function.
scanf() – scanf() function is used to take input from the user.
printf() – printf() function is used to display and print the string under the quotation to the screen.
for loop – A loop is used for initializing a block of statements repeatedly until a given condition returns false.
If. . . else – An if statement can be followed by an optional else statement, which executes when the Boolean expression is false. If Statement executes when the Boolean expression is True.
% – It is known as Modulus Operator and provides remainder after division.
// – Used for Commenting in C.
I hope after going through this post, you understand how to check whether a number is prime or not using C Programming language.
If you have any doubt regarding the topic, feel free to contact us in the comment section. We will be delighted to help you.
1 thought on “C Program To Check Whether a Number is Prime or Not”
I am not able to implement logic in this program. Will you please help me.
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How do you solve 3/5d+5=1/3d-3? | HIX Tutor
How do you solve #3/5d+5=1/3d-3#?
Answer 1
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Answer 2
multiply L.H.S andR.H.S.by 15
substitute #color(green)(d=-30#
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Answer 3
To solve the equation 3/5d + 5 = 1/3d - 3, you can follow these steps:
1. Subtract 5 from both sides: 3/5d = 1/3d - 8
2. Add 8/3d to both sides to get rid of the fraction on the right side: 3/5d + 8/3d = -8
3. Find a common denominator for the fractions: Multiply the first fraction by 3/3 and the second fraction by 5/5: (9/15)d + (40/15)d = -8
4. Combine the fractions: (49/15)d = -8
5. Multiply both sides by the reciprocal of (49/15), which is (15/49): d = -8 * (15/49)
6. Simplify: d = -120/49 or approximately -2.45
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Transactions Online
Mitsuru TANAKA, Kuniomi OGATA, "Fast Inversion Method for Electromagnetic Imaging of Cylindrical Dielectric Objects with Optimal Regularization Parameter" in IEICE TRANSACTIONS on Communications,
vol. E84-B, no. 9, pp. 2560-2565, September 2001, doi: .
Abstract: This paper presents a fast inversion method for electromagnetic imaging of cylindrical dielectric objects with the optimal regularization parameter used in the Levenberg-Marquardt method. A
novel procedure for choosing the optimal regularization parameter is proposed. The method of moments with pulse-basis functions and point matching is applied to discretize the equations for the
scattered electric field and the total electric field inside the object. Then the inverse scattering problem is reduced to solving the matrix equation for the unknown expansion coefficients of a
contrast function, which is represented as a function of the relative permittivity of the object. The matrix equation may be solved in the least-squares sense with the Levenberg-Marquardt method.
Thus the contrast function can be reconstructed by the minimization of a functional, which is expressed as the sum of a standard error term on the scattered electric field and an additional
regularization term. While a regularization parameter is usually chosen according to the generalized cross-validation (GCV) method, the optimal one is now determined by minimizing the absolute value
of the radius of curvature of the GCV function. This scheme is quite different from the GCV method. Numerical results are presented for a circular cylinder and a stratified circular cylinder
consisting of two concentric homogeneous layers. The convergence behaviors of the proposed method and the GCV method are compared with each other. It is confirmed from the numerical results that the
proposed method provides successful reconstructions with the property of much faster convergence than the conventional GCV method.
URL: https://global.ieice.org/en_transactions/communications/10.1587/e84-b_9_2560/_p
author={Mitsuru TANAKA, Kuniomi OGATA, },
journal={IEICE TRANSACTIONS on Communications},
title={Fast Inversion Method for Electromagnetic Imaging of Cylindrical Dielectric Objects with Optimal Regularization Parameter},
abstract={This paper presents a fast inversion method for electromagnetic imaging of cylindrical dielectric objects with the optimal regularization parameter used in the Levenberg-Marquardt method. A
novel procedure for choosing the optimal regularization parameter is proposed. The method of moments with pulse-basis functions and point matching is applied to discretize the equations for the
scattered electric field and the total electric field inside the object. Then the inverse scattering problem is reduced to solving the matrix equation for the unknown expansion coefficients of a
contrast function, which is represented as a function of the relative permittivity of the object. The matrix equation may be solved in the least-squares sense with the Levenberg-Marquardt method.
Thus the contrast function can be reconstructed by the minimization of a functional, which is expressed as the sum of a standard error term on the scattered electric field and an additional
regularization term. While a regularization parameter is usually chosen according to the generalized cross-validation (GCV) method, the optimal one is now determined by minimizing the absolute value
of the radius of curvature of the GCV function. This scheme is quite different from the GCV method. Numerical results are presented for a circular cylinder and a stratified circular cylinder
consisting of two concentric homogeneous layers. The convergence behaviors of the proposed method and the GCV method are compared with each other. It is confirmed from the numerical results that the
proposed method provides successful reconstructions with the property of much faster convergence than the conventional GCV method.},
TY - JOUR
TI - Fast Inversion Method for Electromagnetic Imaging of Cylindrical Dielectric Objects with Optimal Regularization Parameter
T2 - IEICE TRANSACTIONS on Communications
SP - 2560
EP - 2565
AU - Mitsuru TANAKA
AU - Kuniomi OGATA
PY - 2001
DO -
JO - IEICE TRANSACTIONS on Communications
SN -
VL - E84-B
IS - 9
JA - IEICE TRANSACTIONS on Communications
Y1 - September 2001
AB - This paper presents a fast inversion method for electromagnetic imaging of cylindrical dielectric objects with the optimal regularization parameter used in the Levenberg-Marquardt method. A
novel procedure for choosing the optimal regularization parameter is proposed. The method of moments with pulse-basis functions and point matching is applied to discretize the equations for the
scattered electric field and the total electric field inside the object. Then the inverse scattering problem is reduced to solving the matrix equation for the unknown expansion coefficients of a
contrast function, which is represented as a function of the relative permittivity of the object. The matrix equation may be solved in the least-squares sense with the Levenberg-Marquardt method.
Thus the contrast function can be reconstructed by the minimization of a functional, which is expressed as the sum of a standard error term on the scattered electric field and an additional
regularization term. While a regularization parameter is usually chosen according to the generalized cross-validation (GCV) method, the optimal one is now determined by minimizing the absolute value
of the radius of curvature of the GCV function. This scheme is quite different from the GCV method. Numerical results are presented for a circular cylinder and a stratified circular cylinder
consisting of two concentric homogeneous layers. The convergence behaviors of the proposed method and the GCV method are compared with each other. It is confirmed from the numerical results that the
proposed method provides successful reconstructions with the property of much faster convergence than the conventional GCV method.
ER - | {"url":"https://global.ieice.org/en_transactions/communications/10.1587/e84-b_9_2560/_p","timestamp":"2024-11-04T18:46:51Z","content_type":"text/html","content_length":"66069","record_id":"<urn:uuid:d8e07038-452b-4c91-9065-167254dcf536>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00366.warc.gz"} |
Fraction Decimal Percent Worksheet Pdf - Wordworksheet.com
Fraction Decimal Percent Worksheet Pdf. You can have it by trying to find templates supplied on the web. Kids apply converting percents into fractions and decreasing fractions to lowest phrases on
this fifth grade math worksheet. Take a have a look at our devoted assist web page on how to convert a decimal right into a fraction. This useful resource additionally features a fraction bar.
Grade 6 – Fractions problems, online follow, exams, worksheets, quizzes, and instructor assignments. Every time you click on the New Worksheet button, you will get a brand new printable PDF worksheet
on Fractions. You can select to include answers and step-by-step options.
It has a solution key connected on the second web page. This worksheet is a supplementary seventh grade useful resource to assist academics, parents and youngsters at home and in class. These free
decimals, fractions, percents printables are perfect for kids in grade 2, grade three, grade 4, and grade 5.
Converting Between Fractions, Decimals, And Percents Worksheets
Students turn out to be mathematically proficient in participating with mathematical content material and ideas as they be taught, expertise, and apply these expertise and attitudes (Standards
7.MP.1-8). Math Worksheets Based on NCTM Standards! Number Theory, Decimals, Fractions, Ratio and Proportions, Geometry, Measurement, Volume, Interest, Integers, Probability, Statistics, Algebra,
Word Problems.
Also visit the Math Test Prep section for additional grade seven materials. sixth Grade Math Word Problems With Answers. Types of word issues that 6th graders should have the flexibility to solve.
Reply Simply Divide 3 By 10,
Key to Percents assumes only a information of fraction and decimal computation. The Videos, Games, Quizzes and Worksheets make wonderful materials for math academics, math educators and oldsters.
Math workbook 1 is a content-rich downloadable zip file with one hundred Math printable workouts and one hundred pages of reply sheets attached to each train.
The first part is simply changing fractions into decimals and percents. The second sections is about converting decimals to percents and fractions.. Decimal to percent 1 Math Worksheet for youths
with reply key.
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K5 Learning presents free worksheets, flashcardsand inexpensiveworkbooksfor children in kindergarten to grade 5. Become a memberto entry extra content and skip ads. Teachers can share the website
directly with their college students in order that they can follow by downloading or printing worksheets.
In our day by day life routine, we use mathematics terms to characterize the outcomes. There are some specific terms of mathematics, that puzzle almost every particular person they usually need a
simple solution for this problem. Do you perceive how Percent Decimal Fraction Chart works?
What Have You Realized After Undertaking The Exercise Brainly
Browse fraction decimal % chart notes resources on Teachers Pay Teachers, a market trusted by millions of academics for unique instructional resources. Smart Board or as Printable Anchor charts.
These fun pizza puzzles require just a little prep-work, then your youngsters shall be able to convert fractions to decimals and percents.
Put your kid’s retail savvy to work by determining the price of sale objects. There are fraction movies, worked examples and follow fraction worksheets. If you need to convert decimal into fraction,
then you definitely only need to do is reverse the above step.
Good for practising equivalent fractions as nicely as converting to easiest type. Take a look at our Simplifying Fractions Practice Zone or try our worksheets for locating the best kind for a range
of fractions. We have some enjoyable fraction – decimal worksheets involving working your means by way of clues to solve a riddle.
Our first chart is each easy and complete. It just has all the fractions on the left and the decimals on the right. This chart goes to 64ths, but scroll down for charts that target extra frequent
Il2cppinspector Download
This product is appropriate for Preschool, kindergarten and Grade 1.The product is available for immediate obtain after purchase. These free worksheets are great repetition on your students! Click on
the photographs below to download the word problem worksheets.
In the given fraction, the denominator is 8 which isn’t convertible to 10 or 100. Worksheets from very fundamental degree to advanced stage. A premium math high quality web site with authentic Math
actions and different contents for math apply.
The worksheets can be found each in PDF and html formats (both are simple to print; html format is editable). You can control the workspace, font dimension, number of decimal digits in the percent,
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Here, the denominator of the given fraction is 5 which can be transformed into 100 utilizing multiplication by 2. So the given fraction could be converted to decimal utilizing long division. In the
given fraction, the denominator is 25 which is convertible to one hundred utilizing multiplication by four.
9 Best Images Of Fraction To Decimal Chart Printable – Printable decimal fraction chart conversion printable metric fractions decimals into printablee percents through. Our worksheets assist children
convert fractions to decimals and percentages with ease. The fractions, decimals and percentages worksheets we have obtainable will prepare students for any query they encounter.
Operations and Algebraic thinking Workbooks … Free worksheets for time word issues, Time Word Problems Worksheets, PDF For Time Word. Math Word Problems Worksheets seventh Grade Math Word Problems
Displaying high 8 worksheets discovered for – 7th Grade Math Word Problems .
Ideas Of Multiplying Decimals Worksheets 7th Grade With Additional decimals. It is easy to ace the checks when you’ve got our free, printable worksheets on changing between fractions, decimals, and
percents to bank on! In the primary section, you have to convert fractions into decimals and percents, decimals into fractions and percents, and percentages into decimals and fractions.
It may be printed, downloaded or saved and utilized in your classroom, house faculty, or different academic surroundings to assist somebody learn math. Once a fraction is transformed to a decimal,
it’s quite easy to alter it to a p.c. These sheets are just like these within the section above, however contain primarily blended decimals greater than one to convert.
eighty five c ab one hundred To change a fraction to a decimal. To convert into p.c multiply the numerator of the fraction with one hundred and divide the product with denominator. This exercise is
about changing between fractions decimals and percentages.
Though it is a great rule, and it makes fixing the problems easy, youngsters want to understand why this works and have the ability to visualize it. Most children will know that whenever you multiply
a quantity by a hundred, you’re just shifting the decimal two places to the proper. Before you merely inform your kids tips on how to rapidly remedy the issue, I extremely suggest you do actions that
assist youngsters visualize why this works.
We welcome any comments about our web site or worksheets on the Facebook feedback box on the backside of every web page. Take a take a glance at some extra of our worksheets just like these. These
sheets are aimed at students in 5th and 6th grade.
In the 2nd worksheet, they are also asked to simplify the resulting fraction. In the last worksheet, college students convert the % to a decimal as nicely as a fraction. Do 4th grade, 5th grade,
sixth grade and 7th grade students know that zero.12 is 12%?
Percents is an important idea of nice use in math and in daily life. Our proportion worksheets for grade 5 pdf are subsequently distinctive useful resource in serving to kids perceive the
significance of percents in math and in actual life. Find here a limiteless provide of printable & customizable worksheets for working towards the conversions between percents and decimals.
We will multiply both numerator and denominator with 10 for every number after the decimal level. Practice the Percentage to Fraction Worksheet with Answers and check your preparation level. You can
try Percentage Worksheets for more information concerning the same.
Term fraction is a component of a whole quantity or acts as several equal elements. In easy words, it’ll symbolize what quantity of elements of a specific measurement are divided by the general
amount. Keep in thoughts that a fraction consists of a denominator and numerator similar to half.
If 650 of them can sing, what % of them can sing and what percent cannot? If Dan planted rose on 75% of the 500 sq m of his land, on how sq. Juliana obtained 90 messages on her birthday.
A Flurry Of Everyday Scenarios, Our Free Worksheet For Dividing Fractions Word Problems Is. These grade 6 math worksheets cover the multiplication and division of fractions and combined numbers.
Dividing mixed numbers by fractions (5. Below are six versions of our grade 6 math worksheet on dividing combined numbers by different blended numbers.
• These inter-related fractions, decimals and percents concepts are very fascinating methods to explain identical elements of a whole.
• Make impartial studying and classroom learning extra productive with this worksheet on Decimal to Percent.
• For this cause, several worksheets have been created, separated by themes, so that every pupil can follow in the subjects where he / she feels probably the most issue.
• The second sections is about converting decimals to percents and fractions..
You can also use the chart that can assist you will adding and subtracting fractions! Fractions and percentages are used for big numbers whereas decimals are used for small numbers. Measurements are
often used in problem-solving or to check portions.
When youngsters learn about fractions they should understand that fractions are additionally division problems. This converts the decimal right into a decimal fraction (a fraction the place the
denominator is an influence of 10. To convert a percentage into fraction, we’ve to divide the given quantity by a hundred.
Related posts of "Fraction Decimal Percent Worksheet Pdf" | {"url":"https://wordworksheet.com/fraction-decimal-percent-worksheet-pdf/","timestamp":"2024-11-05T16:24:49Z","content_type":"text/html","content_length":"83239","record_id":"<urn:uuid:41f2f110-b5c6-4593-8310-b7198ff9b2a2>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00691.warc.gz"} |
How to Return Multiple Values Based on a Single Criteria in Excel - ExcelDemy
In the example below we have the list of all the FIFA World Cups from 1930 to 2018. We have the Year in Column B, the Host Country in Column C, the Champion countries in Column D, and the Runners-up
countries in Column E. Let’s use it to demonstrate how you can extract multiple results from a given criterion.
Method 1 – Returning Multiple Values Based on a Single Criteria in a Single Cell
We’ll extract the names of all the champion countries to one column and will add the years in which they became champions to the adjacent cells. Let’s say we want to extract the names of the champion
countries in Column G named Country.
• Enter this formula in cell G5:
D5:D25 refers to the Champions.
• Press Enter.
• All the champions are listed in Column G.
Note: While using Microsoft 365, there is no need to use the Fill Handle to get all the values. All values will appear automatically.
Case 1.1. Using TEXTJOIN and IF Functions
In the following dataset, we have unique Champion countries in Column G based on the previous results. We need to find out the Years of these Champion teams in one cell individually.
• Copy this formula in the H5 cell:
• Press Enter to get the output as 1930,1950.
• Use the Fill Handle by dragging down the cursor while holding the right-bottom corner of the H5
• We’ll get the outputs like this.
Formula Explanation
• Here $B$5:$B$25 is the lookup array. We want to look up for the years.
• $D$5:$D$25=G5 is the criteria we want to match. We want to match cell G5 (Uruguay) with the Champion column ($D$5:$D$25).
Case 1.2. Combining TEXTJOIN and FILTER Functions
• Input this formula in H5:
• Press Enter.
• Use the Fill Handle to copy the formula to the rest of the column.
Formula Explanation
• Here $B$5:$B$25 is the lookup array. We want to look up for the years. If you want anything else, use that one.
• $D$5:$D$25=G5 is the criteria we want to match. We want to match cell G5 (Uruguay) with the Champion column ($D$5:$D$25). If you want anything else, use that one.
Method 2 – Return Multiple Values Based on Single Criteria in a Column
Case 2.1. Using a Combination of INDEX, SMALL, MATCH, ROW, and ROWS Functions
Suppose we need to find out in which years Brazil became the champion. In the following dataset, we need to find it in cell G5.
• Copy this formula in cell G5:
=INDEX($B$5:$B$25, SMALL(IF(G$4=$D$5:$D$25, MATCH(ROW($D$5:$D$25),ROW($D$5:$D$25)), ""), ROWS($A$1:A1)))
• As this is an array formula, you need to press Ctrl + Shift + Enter.
• We’ll find the years in which Brazil became champion as output.
Using the above formula, you can extract the years of the championship of any other country.
For example, to find out the years when Argentina was champion in Column H, create a new column Argentina adjacent to the one in Brazil, and drag the formula to the right by using the Fill Handle.
Formula Explanation
• Here $B$5:$B$25 is the lookup array. We look for years. If you have anything else to look up for, use that.
• G$4=$D$5:$D$25 is the matching criteria. We want to match the content of the cell G4, Brazil with the contents of the cells from D5 to D25. You use your criteria.
• Again, $D$5:$D$25 is the matching column. You use your column.
• Let’s also find out the years when the World Cup was won by the host countries. The formula will be in the H5 cell like this:
=INDEX($B$5:$B$25, SMALL(IF($C$5:$C$25=$D$5:$D$25, MATCH(ROW($D$5:$D$25),ROW($D$5:$D$25)), ""), ROWS($A$1:A1)))
Eventually, the host country became champion in 1930,1934,1966,1974,1978, and 1998.
Case 2.2. Applying FILTER Function
The FILTER function is available in Office 365 only.
• The formula in cell G5 to sort out the years when Brazil was the champion will be:
Formula Explanation
• As usual, $B$5:$B$25 is the lookup array. Years in our case. You use your one.
• $D$5:$D$25=G$4 is the matching criteria. You use your one.
• Press Enter to get the outputs.
• We can create a new column Argentina just beside Brazil and drag the Fill Handle to the right to get the Years when Argentina was champion.
The output will look like this.
Method 3 – Return Multiple Values in Excel Based on Single Criteria in a Row
Let’s find out the years when specific countries were champions in a different way.
• Select a cell and enter Brazil. In this case, we’ll use G5.
• Copy this array formula in the adjacent cell i.e. H5, and press Ctrl + Shift + Enter.
=IFERROR(INDEX($B$5:$B$25, SMALL(IF($G5=$D$5:$D$25,ROW($B$5:$B$25)-3,""), COLUMN()-7)),"")
• Press Enter.
• Excel will find the years of different specific countries when they became champions first. It will happen automatically in Microsoft 365 without using the Fill Handle.
• Use the Fill Handle to drag the entire column to the right to get the other results.
• We’ll get the output like this.
Formula Explanation
• Here $B$5:$B$25 is the lookup array. We looked up for years in the range B5 to B25. If you want anything else, use that.
• $G5=$D$5:$D$25 is the matching criteria. I want to match cell G5 (Brazil) with the Champion column (D5 to D25). If you want to do anything else, do that.
• I have used ROW($B$5:$B$25)-3 because this is my lookup array and the first cell of this array starts in row number 4 (B4). For example, if your lookup array is $D$6:$D$25, use ROW($D$6:$D$25)-5.
• In place of COLUMN()-7, use the number of the previous column where you are inserting the formula. For example, if you are inserting the formula in column G, use COLUMN()-6.
Download Practice Workbook
<< Go Back to Lookup | Formula List | Learn Excel
Get FREE Advanced Excel Exercises with Solutions!
2 Comments
1. SUPERB VERY GOOD LESSON FORA NOVICE.
THANK YOU SO MUCH
□ Hello Selam,
You are most welcome.
Leave a reply | {"url":"https://www.exceldemy.com/excel-return-multiple-values-based-on-single-criteria/","timestamp":"2024-11-14T17:50:34Z","content_type":"text/html","content_length":"203085","record_id":"<urn:uuid:aa3d68fe-fa91-4770-8ea4-0e62347e257c>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00533.warc.gz"} |
Doubly Quasi-Consistent Parallel Explicit Peer Methods with Built-In Global Error Estimation
Kulikov, Gennady Yu; Weiner, R.
Journal of Computational and Applied Mathematics, 233(9) (2010), 2351-2364
Recently, Kulikov presented the idea of double quasi-consistency, which facilitates global error estimation and control, considerably. More precisely, a local error control implemented in such
methods plays a part of global error control at the same time. However, Kulikov studied only Nordsieck formulas and proved that there exists no doubly quasi-consistent scheme among those methods.
Here, we prove that the class of doubly quasi-consistent formulas is not empty and present the first example of such sort. This scheme belongs to the family of superconvergent explicit two-step peer
methods constructed by Weiner, Schmitt, Podhaisky and Jebens. We present a sample of ss-stage doubly quasi-consistent parallel explicit peer methods of order s?1s?1 when s=3s=3. The notion of
embedded formulas is utilized to evaluate efficiently the local error of the constructed doubly quasi-consistent peer method and, hence, its global error at the same time. Numerical examples of this
paper confirm clearly that the usual local error control implemented in doubly quasi-consistent numerical integration techniques is capable of producing numerical solutions for user-supplied accuracy
conditions in automatic mode. | {"url":"http://cemat.ist.utl.pt/document.php?member_id=87&doc_id=1422","timestamp":"2024-11-14T10:41:06Z","content_type":"text/html","content_length":"9249","record_id":"<urn:uuid:02c01343-75c0-435b-aa3f-dfa60cd12556>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00709.warc.gz"} |
2023 - 2024 Continental Math League
top of page
Continental Math League (CML) is an international competition for elementary to high school students. There are 3 rounds of contest for 2nd & 3rd grade and 5 rounds of contest for 4th to 8th grade in
a school year. Each contest consists of 6 word problems. The maximum score for each round is 6. The sum of top 6 students' score will become the team score of each grade.
bottom of page | {"url":"https://www.mathseed.org/2023-2024-continental-math-league","timestamp":"2024-11-03T12:16:29Z","content_type":"text/html","content_length":"476647","record_id":"<urn:uuid:f9dfacdd-63d2-46c7-9dc9-defc0c7cab80>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00638.warc.gz"} |
Statistics to prove anything
I always forget what each pch values is and I always end up going to this site to see the list:
. But I don't like having to click on the thumbnail image to then see the large version. Here is my version:
I don't understand why it has the value '0', and if there is a difference between 16 and 19 I have no idea what it is. and of course 20 could just be 19 with cex=.5
Here are some other pch options, basically any character on the keyboard can be used:
Of course some of these are better than others. For example, if you are plotting financial data, it might be best to consider something like this:
Or if you have confusing data this may be appropriate:
And of course the purpose of using different pch characters is to easily distinguish different type of data on the same plot. For example, here we see the ^ characters representing Teepees, and the ~
characters representing water:
To choose a particular pch for a plot, the code is either plot(x,y,pch=16) or plot(x,y,pch='~')
Source Code for above plots:
xlab='',ylab='',main='List of pch values in R',cex.main=2)
xlab='',ylab='',main='Some other pch values in R',cex.main=2)
mtext("Line 2", side=1, line=2, adj=0.0, cex=1, col="blue", outer=TRUE)
plot(x,y,pch='$',axes=F,main='Plot of Financial Data',cex.main=2,cex.lab=1.5)
plot(x,y,pch='?',axes=F,main='Plot of Confusing Data',cex.main=2,cex.lab=1.5)
plot(x,y,pch='~',main='Plot of Village by a River',col='blue',cex=2,cex.main=2,cex.lab=1.5) | {"url":"http://www.statisticstoproveanything.com/2010/09/","timestamp":"2024-11-05T15:42:10Z","content_type":"text/html","content_length":"44956","record_id":"<urn:uuid:db297715-f8d0-4797-885e-a72ef14fe1df>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00720.warc.gz"} |
Treebank Statistics: UD_Hungarian-Szeged: Features: Number[psor]
This feature is language-specific. It occurs with 2 different values: Plur, Sing.
This is a layered feature with the following layers: Number, Number[psed], Number[psor].
2790 tokens (7%) have a non-empty value of Number[psor]. 2051 types (15%) occur at least once with a non-empty value of Number[psor]. 1349 lemmas (15%) occur at least once with a non-empty value of
Number[psor]. The feature is used with 5 part-of-speech tags: NOUN (2761; 7% instances), PROPN (13; 0% instances), ADJ (7; 0% instances), NUM (5; 0% instances), PRON (4; 0% instances).
2761 NOUN tokens (28% of all NOUN tokens) have a non-empty value of Number[psor].
The most frequent other feature values with which NOUN and Number[psor] co-occurred: Person[psor]=3 (2654; 96%), Number=Sing (2391; 87%).
NOUN tokens may have the following values of Number[psor]:
Number[psor] seems to be lexical feature of NOUN. 95% lemmas (1257) occur only with one value of Number[psor].
13 PROPN tokens (0% of all PROPN tokens) have a non-empty value of Number[psor].
The most frequent other feature values with which PROPN and Number[psor] co-occurred: Number=Sing (13; 100%).
PROPN tokens may have the following values of Number[psor]:
Number[psor] seems to be lexical feature of PROPN. 100% lemmas (12) occur only with one value of Number[psor].
7 ADJ tokens (0% of all ADJ tokens) have a non-empty value of Number[psor].
The most frequent other feature values with which ADJ and Number[psor] co-occurred: Degree=Pos (5; 71%), Number=Sing (5; 71%), VerbForm=EMPTY (4; 57%).
ADJ tokens may have the following values of Number[psor]:
5 NUM tokens (0% of all NUM tokens) have a non-empty value of Number[psor].
The most frequent other feature values with which NUM and Number[psor] co-occurred: Number=Sing (5; 100%), NumType=Frac (4; 80%).
NUM tokens may have the following values of Number[psor]:
4 PRON tokens (0% of all PRON tokens) have a non-empty value of Number[psor].
The most frequent other feature values with which PRON and Number[psor] co-occurred: Number=Sing (4; 100%), Person=3 (4; 100%), PronType=Ind (4; 100%).
PRON tokens may have the following values of Number[psor]:
Relations with Agreement in Number[psor]
The 10 most frequent relations where parent and child node agree in Number[psor]: NOUN –[iobj]–> NOUN (1; 100%), NOUN –[list]–> NOUN (1; 100%). | {"url":"https://universaldependencies.org/treebanks/hu_szeged/hu_szeged-feat-Number-psor.html","timestamp":"2024-11-14T12:16:23Z","content_type":"application/xhtml+xml","content_length":"16243","record_id":"<urn:uuid:14000e9e-3889-448b-acbc-6b46732c1924>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00771.warc.gz"} |
Warpzone portals
Thanks to Youtube recommendations I've been watching people play a lot of weird Doom mods, with stuff like myhouse.wad sparking my suggestion. It showcased what strange and amazing things you can do
using warp zones. Many old engines supported them in some form, but sadly idTech 4 never did nor does our fork of it... so far, I suggest changing that.
What are warp zones: Those who played games like Unreal Tournament may remember some maps containing doors that lead to geometrically impossible locations, the classic title Portal is a more popular
example except our implementation refers exclusively to static portals. The player may seamlessly traverse a short straight tunnel, but upon emerging on the other side they discover the exit leads
behind the entrance to that tunnel. This allowed some neat tricks and layouts that aren't doable with standard euclidean geometry.
Why TDM can use this: Though we aren't a scifi FPS, many FM's contain magic and spooky maps which is a theme I like playing with myself, this would allow esoteric maps where the layout messes with
the player in unexpected ways. Imagine a small shed you can walk around with a door leading to a giant hallway that can't possibly fit in that building, or an endless stairway you keep trying to
climb until you realize it never ends and have to turn back to progress, or a pillar you have to go around in a particular direction to find yourself popping into another world. Even for non-magic
FM's this can be used to change the environment, triggering portals to make doors lead to different versions of an area where the map has changed; Take an objective where you must set a building on
fire from outside... you can enter the building before completing it and everything inside is normal, but after completion the door and windows lead to a charred version of the interior where the
walls are a different texture and the furniture just a pile of wood on fire.
Implementation concept: The best approach by far is via visportals, what we want is a way to seamlessly connect two portal surfaces. Connected portals may have different rotations but must have the
same shape and size, dmap should check this and throw an error if different sizes are detected. The most straightforward setup would be an argument to the func_portal entity, allowing it to target
another func_portal so the two are seamlessly connected. There are two components to implement: Entity movement should be the easiest as it involves teleporting and rotating any entity that crosses
through the warp zone... the visual aspect is trickier as the portal must hide the contents behind it and make you see through the other portal instead based on the angle you're looking from, this
may not be that different from how realtime mirrors or the skybox use special render passes so hopefully the same system can be expanded.
Limitations: One issue will be spatial audio which can't easily be fixed with this approach, mappers should avoid placing noises near warp portals as you'll hear them change as you cross the surface.
Same with lights which won't be able to shine through, don't light them or do so symmetrically to avoid a cutoff. Infinite hallways are impossible and the amount of warpzones you can see through
warpzones must be limited to 1 or a low number of passes by default, mappers should position portals so that you can't see one through the other in an infinite loop.
These portals sound a lot like the ones in (the original) PREY, which I believe was an id Tech 4 engine.
5 hours ago, OrbWeaver said:
These portals sound a lot like the ones in (the original) PREY, which I believe was an id Tech 4 engine.
Indeed and yes Prey 2006 is literally Doom 3 engine with only a few modifications, TDM engine is way more removed from original idTech 4 than Prey engine was.
Also I think this can be replicated in idTech 4 relatively easy (in c++ obviously):
Edited by HMart
• 3
35 minutes ago, HMart said:
Indeed and yes Prey 2006 is literally Doom 3 engine with only a few modifications, TDM engine is way more removed from original idTech 4 than Prey engine was.
Also I think this can be replicated in idTech 4 relatively easy (in c++ obviously):
If you post a patch we will merge it
• 1
Please visit TDM's IndieDB site and help promote the mod:
(Yeah, shameless promotion... but traffic is traffic folks...)
2 hours ago, nbohr1more said:
If you post a patch we will merge it
I certainly expected and deserve that reply, but I just don't have the time and to tell the truth, I don't think I'm qualified to do it.
I looked into implementing Prey (2006) style portals last year. The expertise to implement portals doesn't overlap with mine, so I decided to drop the effort.
For those who want to give it a go, here are two idtech4 implementations that might be helpful.
• https://github.com/jmarshall23/PreyDoom/blob/master/code/Prey/game_portal.cpp
□ Warning: not GPL; uses non-free code (as far as I can tell).
Sending objects through a portal would be tricky as well as dealing with AI behavior.
Regarding constraints in Prey, I noticed that portals come in two types: (1) short, box portals that touch the ground; the player must crouch, and (2) tall, oval portals that don't touch the ground.
In both cases, I'm not sure if objects can pass through, but weapon projectiles can pass through.
Prey example:
Quake 4 mod example:
• 1
Happy to see many encouraging responses and examples on this. Didn't realize Prey used idTech 4 or that any game who did managed this including Quake 4. But I presume it was never part of the engine
originally and they added it in their own game code, to my knowledge neither Prey nor Quake open-sources their own game code for us to copy it from... still we know it's easily doable and was even
implemented twice!
Physical objects shouldn't be the tricky part: We should only need to teleport them and rotate their rotation and velocity relative to the orientation difference between portal faces. There's only
one circumstance I see as problematic: What happens if the player is carrying an object and pushing it through the portal, like picking up a crate and dropping it through the warp zone? The object
needs to teleport through while still being held, movement in the player's position and view rotation needs to be accurately translated at the other end.
AI likely won't be able to patrol through, at least not through conventional path_corner chains since they can't link through a portal, it would require a new path node like for elevators. It may be
possible with a special node placed placed at each portal... in fact we already have a teleport path node, that might already work to have an AI pass through seamlessly.
3 hours ago, HMart said:
I certainly expected and deserve that reply, but I just don't have the time and to tell the truth, I don't think I'm qualified to do it.
No worries. I figured you'd be a good candidate.
I think @7318 made our HL2 style portal skies, I keep forgetting if that was your old alt \ nick.
( Don't recall both of you in the same thread ? )
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16 hours ago, nbohr1more said:
No worries. I figured you'd be a good candidate.
I think @7318 made our HL2 style portal skies, I keep forgetting if that was your old alt \ nick.
( Don't recall both of you in the same thread ? )
Thanks, but no I didn't wrote that sky portal stuff and that was never my nick, someone else deserves the praise for that.
And about participating on a thread about the sky portal stuff, I don't recall that but perhaps I did but I doubt it was to say how to code that feature in.
Cool this idea comes up (again). I had some (complicated) ideas for implemented something like this, but found it too much work.
My (conceptual) idea was to look through a "portal" but what you look at is either what is seen in a different room (via a texture connected to a camera), or an x-ray screen, which make the area
behind the "portal" look completely different (even walls can be hidden by making them func-static and apply a skin to them). When you move into the portal you get teleported to another room. Off
course this has many limitations that you may or may not be able to work around via scripting.
On 7/28/2024 at 11:39 PM, MirceaKitsune said:
Even for non-magic FM's this can be used to change the environment, triggering portals to make doors lead to different versions of an area where the map has changed
This has already been done in (for example) A house of locked secrets.
4 hours ago, datiswous said:
This has already been done in (for example) A house of locked secrets.
May I ask how? The only way I can imagine is a tricky setup using elevators (multistate mover) which I already went with for creating a simpler illusion. Anything else would require duplicating the
whole area and teleporting the player relatively, which isn't seamless and doesn't allow for the complexity I'm hoping this would make possible.
My imagined setup involves being able to trigger such portals, so each can act like a normal visportal if not linked and a warp zone once connected. This would make possible all the amazing things
like activating magic gateways, or changing the interior of a room so it leads to a heavily different version, or hallways that change as you navigate the maze.
4 hours ago, MirceaKitsune said:
Anything else would require duplicating the whole area and teleporting the player relatively,
That's how it works in that mission.
4 hours ago, MirceaKitsune said:
My imagined setup involves being able to trigger such portals, so each can act like a normal visportal if not linked and a warp zone once connected.
I don't understand your description, sorry.
33 minutes ago, datiswous said:
That's how it works in that mission.
Yes, it's a good approach when you can get away with it, exactly what I'm doing on some of those stranger FM's I've been working on. This covers some use cases but doesn't create the illusion of
having multiple things in the same physical space or seamlessly traveling between impossible points.
Another simple solution to making rooms that change is by using the building modules: Turn some wall models into doors, set them to not be frobable and have instant speed, and configure the door so
that when open you don't see it hidden inside the ceiling but when closed it appears like a normal wall, allow the player to trigger it from a location where they can't see it changing.
28 minutes ago, datiswous said:
I don't understand your description, sorry.
So let's say the outdoor area contains a small shed the size of an outhouse, the player is given an objective to complete some sort of ritual. The shed has a standard door that opens and closes its
visportal as usual. Before completing the objective, the player can walk into this shed like in any structure and maybe pick a necessary item for it. Once the objective is complete, opening the shed
door from the outside now reveals a door leading to an impossibly large hallway that doesn't fit inside that structure.
That's what I meant with making warpzones possible to trigger: The portals are of course static and compiled by dmap, but could be triggered to link to different portals and establish new warp zones
or turn it off entirely. The particular effect in my example can't be achieved with any other tricks, maybe by using sky portals to cover its back area but then you can't walk behind the shed outside
as the invisible caulk covering the tunnel would collide.
Edited by MirceaKitsune
Actually I think I did understand what you try to create, just not how to do that with visportals / func-portals. It's also that the one thing that I haven't done much building in DR is visportals,
because I haven't really built anything apart from some room tests.
5 hours ago, datiswous said:
Actually I think I did understand what you try to create, just not how to do that with visportals / func-portals. It's also that the one thing that I haven't done much building in DR is
visportals, because I haven't really built anything apart from some room tests.
Visportals are one of the things you definitely want to get a hang of, as they're the #1 way of improving performance with the engine still being quite performance intensive. You want them covering
any entrance of brushes they can cover to divide them into rooms, in such a way that they hide as much as possible from as many camera positions.
Every door and window has a visportal, which is closed when the door itself is closed given it's not a transparent door like grate or glass. Hence why I think they're the best candidate for warp
zones: My suggestion would allow stitching distant rooms and corridors, the entrances of which are already defined by visportals in standard mapping. When a visportal is turned into a warpzone it
also won't render what's normally behind it but render through the connected visportal instead, closing the room you normally see through it and recovering performance.
5 hours ago, datiswous said:
Is this something that can be done without core modification?
It would definitely require engine changes. I think the basic building blocks should be there: We have custom render passes for stuff like the skybox or dynamic mirrors, the rest should be entity
teleportation. But it would definitely take one of the devs with experience in the engine and renderer to decide if it's worth their time to look into this.
On 7/29/2024 at 8:09 PM, Daft Mugi said:
Quake 4 mod example:
I was looking at the video again and noticed that the room on the other side of the "portal" has the same size. So maybe this is still in the same room?
In case you want to know how it was implemented back in the old days:
I'm not sure if using it this way makes any sense in a mission.
You can have a really cool portal players can cross, without warpzones – see what Kingsal did in Volta 2.
• 2
Thanks: I used to play with Unreal Editor ages ago, brings back memories to see all that. I did remember that our visportal and location_info system is very similar to early UT's in how we define and
separate rooms.
As far as portals go, UT seems to be doing it via location info. Doing it per portal seems ideal, especially with func_portal already being there and easy to use for this purpose: I'd imagine that
way you can still have a normal room, but only a particular door or window leading to an altered version. Also easier for mappers to understand, just link two func_portal entities together and that's
it... rooms with multiple portals that don't match would have to throw errors otherwise.
After some consideration, I think we already have all the needed ingredients.
1) Create a trigger patch which calls a teleport script
2) Create a security camera gui ( non-solid patch )
3) Use a script to move the security camera entity relative to the player's eye location and the distance to the destination "portal"
The only tricky part is calculating the surface normal of the destination portal so the camera always uses the correct offset
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6 hours ago, nbohr1more said:
After some consideration, I think we already have all the needed ingredients.
1) Create a trigger patch which calls a teleport script
2) Create a security camera gui ( non-solid patch )
3) Use a script to move the security camera entity relative to the player's eye location and the distance to the destination "portal"
The only tricky part is calculating the surface normal of the destination portal so the camera always uses the correct offset
Does this also work with objects and ai?
• So can ai see you from the other side of the portal?
• Can you pick up a chair and throw it through the portal and it ends up on the other side? Or shoot a fire arrow through the portal and see it explode on the other side?
7 hours ago, nbohr1more said:
After some consideration, I think we already have all the needed ingredients.
1) Create a trigger patch which calls a teleport script
2) Create a security camera gui ( non-solid patch )
3) Use a script to move the security camera entity relative to the player's eye location and the distance to the destination "portal"
The only tricky part is calculating the surface normal of the destination portal so the camera always uses the correct offset
For the basics I think we do. The main difference between portals and cameras is the camera renders from a fixed angle, a warpzone is see-through and depends on the camera position... the skybox
might thus be a better candidate to base them on, except not rendered in 360* but acting as a distant visportal.
Other than that the details may get complicated. Datiswous just pointed out yet another one: AI being able to see and recognize the player through such portals is yet another thing that would need to
be implemented manually, the view cone would need to travel through the warp zone too. I think rendering and basic player / entity travel would be most important to get first, then all those other
TODO's can slowly be done over time, with mappers informed of the limitations in effect at that time.
Edited by MirceaKitsune
I had an other idea around faking portals:
You have a "portal". Everything you see behind it is still there, but there are alternative skins applied to possibly everything, so you can even hide walls and add or remove stuff seen through the
portal (an x-ray screen is placed there so you see everything different). The portal itself is a small tunnel with 2 triggers. On the second trigger it changes all the skins for the area. So first
you look through and only see it changed through an x-ray screen, after you moved through it the skins are changed. It needs 2 triggers so that the skins will not change when moving in and then back
This does mean everything is rendered, because all the things which you want to apply skins to need to be func-static or other entities.
This also has limitations that you have to fix in complicated ways.. (everything that is thrown through the portal has to be hidden when walked around the portal)
Making a portal in the middle of a room that you can walk around is what I have in mind, because that's the original Prey-like portal, which is super cool, but it does make it more difficult or too
A simpler way is having 2 rooms with a central door that is supposed to be the portal. On the far sides of the 2 room, there are 2 doors, so you can walk around, when you walk through these side
doors the skins in the second room are changed and maybe ai and moveables are teleported away (and back when you walk back). This creates the illusion of entering a different room going through the
center "portal" door.
Edited by datiswous | {"url":"https://forums.thedarkmod.com/index.php?/topic/22507-warpzone-portals/","timestamp":"2024-11-01T23:44:40Z","content_type":"text/html","content_length":"412850","record_id":"<urn:uuid:e46b66b5-3c01-4111-84ea-79001ee5021c>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00769.warc.gz"} |
Beyond Worksheets: Creative Approaches to Teaching Math • DepEd Tambayan
Beyond Worksheets: Creative Approaches to Teaching Math
Mathematics is a fundamental subject that provides the foundation for many other subjects and is essential for developing critical thinking skills. However, many students struggle with math, finding
it difficult and uninteresting. One reason for this may be that traditional teaching methods, such as the use of worksheets, do not engage students effectively. In this article, we will explore
creative approaches to teaching math that go beyond worksheets and help students develop a deep understanding of mathematical concepts.
The Limitations of Worksheets
Worksheets are a common tool for teaching math, but they have several limitations. First, worksheets are often repetitive and do not provide opportunities for students to think creatively or develop
problem-solving skills. Worksheets can also be boring and lack engagement, leading to disinterest and disengagement in students.
Moreover, the use of worksheets as the primary teaching tool may not be effective for all students. Research suggests that students who struggle with math may benefit from more interactive and
engaging approaches to learning math, such as using manipulatives, games, or other creative activities that support the development of mathematical understanding and fluency.
Creative Approaches to Teaching Math
To engage students and foster a deeper understanding of mathematical concepts, teachers can use a variety of creative approaches. Some examples include:
Manipulatives are physical objects that students can touch, move, and manipulate to explore mathematical concepts. Using manipulatives can help students visualize abstract concepts, such as fractions
or geometric shapes, and develop a deeper understanding of math concepts.
For instance, students can use base ten blocks to learn place value and arithmetic operations, fraction strips to learn fraction equivalence and addition, or tangrams to explore geometric shapes and
Research indicates that the use of manipulatives can improve students’ mathematical understanding and problem-solving abilities. According to a study conducted by Sowell, Thiessen, and Gruenewald
(1992), students who used manipulatives during math instruction had higher achievement scores than those who did not use manipulatives.
Games can provide a fun and engaging way for students to practice math skills and reinforce concepts. Games can be used to teach a variety of math skills, including number sense, geometry, and logic.
For example, the popular game Sudoku can help students develop critical thinking and problem-solving skills while reinforcing their knowledge of numbers and logic. Other games, such as Yahtzee or
Monopoly, can be adapted to reinforce math skills, such as probability or money management.
Research suggests that the use of games in math instruction can improve students’ motivation and engagement in math learning. According to a study conducted by Ainley and Pratt (2002), students who
played math games had a more positive attitude toward math and were more motivated to learn math than those who did not play games.
Real-Life Applications
Mathematics is used in many real-life situations, such as in cooking, construction, or sports. Teachers can use real-life applications to teach math concepts and make math more relevant and
interesting to students.
For instance, in a cooking lesson, students can use math to calculate ingredient measurements, estimate cooking times, or determine serving sizes. In a construction lesson, students can use math to
calculate dimensions, angles, or areas. In a sports lesson, students can use math to analyze statistics, calculate averages, or predict outcomes.
Research suggests that using real-life applications in math instruction can improve students’ understanding of math concepts and their motivation to learn math. According to a study conducted by
Larson and Hertel (2010), students who were taught math using real-life applications had higher achievement scores than those who were taught using traditional methods.
Technology can be used to engage students and provide interactive learning experiences. Interactive whiteboards, tablets, or online tools can provide students with a more visual and interactive way
to learn math concepts. For example, students can use virtual manipulatives or graphing calculators to explore mathematical concepts in a more dynamic way. Online games or simulations can also
provide a fun and engaging way to practice math skills.
Research indicates that the use of technology in math instruction can have a positive impact on student learning. A study conducted by Penuel et al. (2009) found that students who used digital tools
to learn math had higher achievement scores than those who did not use digital tools.
Project-Based Learning
Project-based learning is a student-centered approach that allows students to apply math concepts to real-world problems. This approach encourages students to develop critical thinking and
problem-solving skills, as well as communication and collaboration skills.
For example, students can work on a project to design a city park, which requires them to use math concepts such as area, perimeter, and volume to create the park’s layout and design. Another project
could be to plan a budget for a family vacation, which requires students to use math concepts such as fractions, decimals, and percentages to calculate costs and expenses.
Research suggests that project-based learning can improve students’ understanding of math concepts and their ability to apply those concepts to real-world situations. A study conducted by Schukajlow
et al. (2017) found that students who engaged in project-based learning had higher achievement scores than those who did not engage in project-based learning.
Teaching math using traditional methods, such as worksheets, may not effectively engage students or promote a deep understanding of mathematical concepts. Instead, using creative approaches, such as
manipulatives, games, real-life applications, technology, and project-based learning, can make math more interesting and relevant to students. These approaches can also help students develop critical
thinking, problem-solving, communication, and collaboration skills that are essential in today’s world. By using these creative approaches, teachers can help students develop a positive attitude
toward math and become confident and proficient in math.
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The Birthday Problem: Round 4
We've looked at the probability of any two people having the
same birthday in a group
. We've also looked at the probability of two people having a
"near match" in a group
. We then looked at the probability of the
first match or near-match
as people join a group one at a time. But what happens when we want to match (or nearly match) a specific birthday (your own for example).
Matches to your birthday will follow a
binomial distribution
, which has a
probability mass function
is the number of trials,
is the number of successes, and
is the probability of success. If we're trying to match your birthday in a random group of people,
is any whole number greater than zero,
is (2
+1)/365 chance of an individual matching your birthday to within
days, and
is one less than the number of people in the group. If we substitute (
) =
, (2
+1)/365 =
, and
= 0 into Equation 1, we'll get the probability of NOT matching (or nearly matching) your birthday in a group of
people. That gives us Equation 2 below:
Subtract from 100% to get the probability of at least one match to your birthday within
days among a total of
people. After simplifying, we get Equation 3:
Plotting for different values of
= 0 if you're looking for a same-day birthday match) gives a graph like this:
It's hard to see the full picture with both
= 0 and
= 30 on the same plot, so here's another one:
We can also make a table showing the minimum size of the group for a corresponding probability that your birthday will be matched or nearly matched:
As you can see, matching your birthday is far less likely than matching any birthday. The results might be surprising though. Recall that we need only 23 people for there to be better than a 50%
chance that two people in the group had matching birthdays. But in a group of you and 22 other people, there's only a 5.86% chance that
birthday is shared with someone else in the group. You need at least 253 other people in the group before it is more likely than not that your own birthday is shared with someone else.
We can also repeat the analysis from the third birthday post and figure out, if people entered randomly one at a time, who would be most likely to match your birthday. That perhaps isn't so
interesting because the first person to enter after you always has the greatest chance of being the first person to match your birthday. The incremental probability of matching your birthday in a
group of
people (i.e. the probability that the next person matches your birthday) is equal to:
It can be shown that
Equation 4
is maximized by
= 1, irrespective of
. Therefore, the first random person joining you at the party is the most likely person to match your birthday, with a probability of a match simply equal to:
For completion, here is the plot of
equation 4
and the tabulated data to show the probability of the first person entering the group after you matching your birthday.
In short, matching a specific birthday in a random group of people is uncommon and generally requires a large sample, but it's not unusual for any two people in a small group of random people to
share a birthday. And now there will be no more talk of birthdays for a long time.
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APPPHYS 207
with emphasis on applications of digital techniques. Combinatorial and synchronous digital circuits. Design using programmable logic. Analog/digital conversion. Microprocessors and real time
programming, concepts and methods of digital signal processing techniques. Current lab interface protocols. Techniques commonly used for lab measurements. Development of student lab projects during
the last three weeks. Limited enrollment. Prerequisites: undergraduate device and circuit exposure. Recommended: previous enrollment in
APPPHYS 207
Terms: Spr, alternate years, not given next year | Units: 4 | Grading: Letter (ABCD/NP)
APPPHYS 215: Numerical Methods for Physicists and Engineers
Fundamentals of numerical methods applied to physical systems. Derivatives and integrals; interpolation; quadrature; FFT; singular value decomposition; optimization; linear and nonlinear least
squares fitting; error estimation; deterministic and stochastic differential equations; Monte Carlo methods. Lectures will be accompanied by guided project work enabling each student to make rapid
progress on a project of relevance to their interests.
Terms: Spr | Units: 4 | Grading: Letter or Credit/No Credit | {"url":"https://swap.stanford.edu/was/20160311035101mp_/http:/explorecourses.stanford.edu/search?view=catalog&academicYear=20132014&page=0&q=APPPHYS&filter-departmentcode-APPPHYS=on&filter-coursestatus-Active=on","timestamp":"2024-11-13T06:33:15Z","content_type":"application/xhtml+xml","content_length":"164777","record_id":"<urn:uuid:f35be5bd-bb60-42b8-abc1-abfeac1b2d5e>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00364.warc.gz"} |
The Complete Guide to Option VegaThe Complete Guide to Option Vega
Vega is a crucial concept for option traders to understand. Having a solid understanding of this option greek is essential for long term success. Enjoy!
One of the most important metrics in option trading is implied volatility as it is a projection of what the future volatility of an underlying asset will be. In turn, this projection is used to
determine what the current market price of an option will be.
Over time, as implied volatility changes, so too will the price of an option. Knowing how implied volatility affects the price of an option will form a valuable part of your trading toolkit.
So how do you determine the impact of implied volatility changes on the price of an option? By understanding a metric called Vega.
What is Vega?
Vega forms one of the metrics of options analysis called the Greeks. At its core, Vega measures what the theoretical price change will be for an option, based on the percentage change in implied
volatility for that option.
More specifically, it measures the amount that an option’s price will change as a result of a 1% change in the implied volatility of the underlying asset.
Like implied volatility, Vega is not uniform and changes over time. In general, options that are bought have positive Vega, while options that are sold will have negative Vega.
Vega is highly correlated with another option greek called Gamma.
How do you calculate Vega?
Like the other Greeks, Vega is calculated as part of the option pricing model. While you can calculate it yourself, it’s probably better to spend some time understanding how it works and to simply
use an option chain to provide you with the value.
An option chain shows all the puts and calls for a certain expiration and underlying. Most brokerage firms provide this information for free (and is where most option traders get these values) and
you can also find a number of websites that provide free tools such as Nasdaq.com.
When you look up the value for Vega, note that unlike implied volatility which is expressed as a percentage, Vega is expressed as a dollar amount.
The value you see in dollars is the amount by which the option’s price will increase for every 1% increase in volatility.
As an example, say you have a long call on company ABC with a premium of $9.50, Vega of 0.30 and implied volatility of 17%. If the implied volatility were to increase to 22%, what would the long call
now be worth?
To calculate the answer, take the original price and add the Vega times the increase in volatility.
So, in this case it would be: $9.50 (the original price) plus 0.30 (the Vega) times 5 (the increase in volatility) = $11.00 (the new price).
One important note is to remember that when buying options, Vega is positive. When selling options, Vega is negative.
The sign is not affected regardless of whether you are trading a call or a put.
How does Vega change?
There are several key influences on Vega. The first is based on the strike price.
When options are at-the-money, they will be heavily affected by Vega.
As you move further away from the at-the-money strikes, the Vega exposure starts to shrink. This can clearly be seen below in this option chain for AAPL, when AAPL was trading around $324.
Deep out-of-the-money and deep in-the-money options will have minimal impact from Vega and can largely be ignored.
The second way that Vega changes is based on the time to expiration. More specifically, the longer the expiration date, the higher Vega will be.
The reason for this is that the more time there is before expiration, the more opportunity there is for a move to happen.
It’s a lot easier to predict tomorrow’s price than it is to predict a price six months from now.
This can easily be seen when comparing the Feb 2020 options above with the Dec 2020 options below.
The final key influence on Vega is changes to implied volatility. As implied volatility increases, so too does Vega.
The reason being is that the increased implied volatility will mean that the strike moves closer to being at-the-money. As the likelihood of the option finishing in-the-money increases, so too will
How can I use Vega in my portfolio?
One of the effective uses for Vega in your portfolio (apart from monitoring exposure and risk) is to use it together with long puts to hedge your portfolio.
Since we know that Vega decreases as we get closer to expiration, it wouldn’t make sense to buy hedges that are short term.
Instead, we could look at potential hedges that are quite far out, such as six months or more, since we know Vega will be higher.
When selecting your strike, consider out-of-the-money options. The reason being is that they respond better to higher volatility levels, and they begin to act more like at-the-money options as
implied volatility rises.
At-the-money strikes don’t make a good hedge even though they have the greatest Vega as they would be quite expensive, most likely negating any financial benefit from making the hedge in the first
Another example is to mix in some long volatility trades like Long Straddles and Strangles (positive Vega) to offset core position such as Iron Condors (negative Vega).
Vega is one of the Greeks and is determined via the option pricing model. It measures the amount that an option’s price will change as a result of a 1% change in the implied volatility of the
underlying asset.
Vega is not uniform and it changes over time, decreasing as the option gets closer to expiration.
By understanding Vega, you will understand how the option premium will be affected as implied volatility changes.
For advanced traders, they may consider using Vega when constructing hedges for their portfolio.
Trade safe!
Disclaimer: The information above is for educational purposes only and should not be treated as investment advice. The strategy presented would not be suitable for investors who are not familiar with
exchange traded options. Any readers interested in this strategy should do their own research and seek advice from a licensed financial adviser. | {"url":"https://optionstradingiq.com/the-complete-guide-to-option-vega/","timestamp":"2024-11-12T18:21:54Z","content_type":"application/xhtml+xml","content_length":"68483","record_id":"<urn:uuid:aa67d497-76bb-45ad-b979-ce03e0de15ec>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00332.warc.gz"} |
CH-11-12 Statistically Based Quality Improvement for Variables - Managing Quality: Integrating the Supply Chain : Book Guide - StudyMerge
This Document Contains Chapters 11 to 12 Chapter 11: Statistically Based Quality Improvement for Variables Chapter Outline •Statistical Fundamentals •Process Control Charts •Some Control Chart
Concepts for Variables •Process Capability for Variables •Other Statistical Techniques in Quality Management Overview The chapter begins on page 278 with a fascinating statement: … many people view
the topic of statistics with fear, loathing, and trembling. This chapter unravels the seeming intricacies of statistical thought in a clear, process-oriented manner. The text presents a series of
tools. Each tool is presented in a situation-based premise that illustrates not only the mechanics of the tool, but its use. The intent of the chapter is to present tools that are useable. Discussion
Questions 1. Discuss the concept of control. Is control helpful? Isn’t being controlling a negative? Controlling is one of the managerial functions, like planning, organizing, staffing and directing.
It is an important function because it helps to check for errors and take corrective action so that deviation from standards are minimized and stated goals of the organization are achieved in a
desired manner. According to modern concepts, control is a foreseeing action, whereas earlier concepts of control were used only when errors were detected. Control in management means setting
standards, measuring actual performance, and taking corrective action. Thus, control comprises these three main activities. Characteristics of control: •Control is a continuous process. •Control is
a management process. •Control is embedded in each level of organizational hierarchy. •Control is forward looking. •Control is closely linked with planning. •Control is a tool for achieving
organizational activities. •Control is an end process. 2. The concept of statistical thinking is an important theme in this chapter. What are some examples of statistical thinking? On page 279, the
author defines statistical thinking as being based on three concepts: • All work occurs in a system of interconnected processes. • All processes have variation; the amount of variation tends to be
underestimated. • Understanding and reducing variation are important keys to success. This concentration on the concept of variation is the focal point. Statistics allows you to use that variation as
your entry into the effectiveness of the process. Examining a measurable quantity on each product, or more like a sample of products, allows you to maintain a continuing quality control. The quantity
being measured can vary from the amount of breakfast cereal being loaded into a box to measuring the diameter of a sample of ball bearings to determine whether they are within specifications. 3.
Sometimes you do well on exams. Sometimes you have bad days. What are the assignable causes when you do poorly? Assignable causes are those situations that can be linked directly to a quality issue.
Poor performance on an exam might be linked to a variety of causes, such as lack of preparation, lack of sleep, illness, or distractions in one’s personal life. 4. What is the relationship between
statistical quality improvement and Deming’s 14 points? The relationship is subtle, but present. Deming is instrumental to the philosophy of continuing improvement. He talks about improved planning
and stresses an environment in which the employee feels empowered. Point 3 presents a lessened dependence on inspections to reduce variation. Statistical analysis is a powerful tool for this purpose.
However, Chapter 9 makes the point that: Some critics of the technique believe that the assumption in acceptance sampling that a percentage will be defective or less than perfect (called acceptable
quality level, or AQL) is counter to Deming’s concepts of continual improvement. However, there is still need for acceptance sampling in many different circumstances This apparent disagreement is, as
stated previously, quite subtle. This topic should be resolved via a classroom discussion. 5. What are some applications of process charts in services? Could demerits (points off for mistakes) be
charted? How? Process charts, as defined in the chapter, are tools for measuring quantifiable data. A process chart could be used to measure a quantifiable property of a service environment. This
would include items that are time-based or measurable, such as response time, time spent to deliver a service, or number of complaints. A demerit could be tracked if there were a predictable and
standardized manner of assigning the demerits. This technique might contradict the approach to continued improvement. This is a good discussion topic for your class. 6. What is random variation? Is
it always uncontrollable? Random variation is that variation in a process that can be measured and analyzed. If it is controlled, then by definition it is not random. A point that needs be made: the
word “random” has a very specific meaning in statistics. Random variables are those values that are all members of the same set of variables and have an equal probability of occurring. Figure 11-1
demonstrates this property. 7. When would you choose an np chart over a p chart? An X chart over an x ¯ chart? An s chart over an R chart? Charts are tools for portraying statistical information in
an easy-to-comprehend manner. A p chart presents the proportion of defective parts whereas an np chart presents the number of non-conforming items. A p chart could be used to compare different items
to observe the difference in the processes. An X chart is used to evaluate a population. An x ¯ (x-bar) chart presents the same information for a sample. The cost and ease of getting an x ¯ chart
obviously make them preferable. An R chart presents the range of values; this is simply the high-value minus the low value. An s chart presents the range of standard deviations. The standard
deviation presents the average variance that the sample points are from the mean or average value. An s chart shows much more detailed information whereas an R chart shows an overview of the
situation. 8. Design a control chart to monitor the gas mileage in your car. Collect the data over time. What did you find? Figure 11-4 presents a simplified control chart. This will display the gas
mileage over the specified period of time. This might be a factor in a term paper. 9. What does “out-of-control” mean? Is it the same as a “bad hair day?” How often do you have a “bad hair day?” How
do you document or evaluate a “bad hair day?” This is a situation in which we are quantifying non-quantifiable data. “Out-of-control” specifically addresses a situation in which you have set up
control limits to define the bounds of a process. “Out-of-control” indicates a data point that lies outside of these control limits. “Out-of-control” refers to a situation or process that deviates
significantly from expected standards or norms, often requiring intervention to correct. It is not the same as a “bad hair day,” which typically refers to a minor, personal inconvenience related to
appearance. In quality management, “out-of-control” indicates a need for corrective actions due to deviations from desired performance levels, while a “bad hair day” is a more casual, everyday
problem. 10. Design a control chart to monitor the amounts of the most recently charged 50 debits from your debit card. What did you find? A debit is a highly quantifiable and measurable quantity.
This data is ideal for statistical analysis. A simple x ¯ chart might look like this: This might be used for a number of observations. Trends can be analyzed by debit amounts, by days of the week, or
days of the month. As in question 8, this might be a good component for a term paper. Case 11-1: Ore-Ida Fries Take the data provided and use control charts to determine whether the measurements are
consistent. Report your results to management. To develop this chart, do the following: 1. Load the data to Excel. 2. Compute the average for each sample. 3. Create a line chart to display the data
points plotted against the sample number. The results show a definite trend in the values. Suggested Answers to End of Chapter Problems 1. Return to the chart in Figure 11-8. Is this process stable?
No. The x-bar chart shows out-of-control points for samples 3 and 8. Assignable causes of variation should be investigated. 2. Return to the data in Figure 11-8. Is this process capable? Compute both
Cpk and Ppk. Hint: Use the calculation work sheet to compute the population standard deviation and for Ppk, treat each observation as a population value. No. The process is not capable. For Ppk,
standard deviation is 4.92 (see partial Excel printout below), Ppu = .457 Ppl = .356 Ppk = .356 Process is not in control. 3. For the following product characteristics, choose where to inspect first:
Choose the lowest ratio to inspect first. Therefore, inspect characteristic D first. 4. For the following product characteristics, chose where to inspect first: Choose the lowest ratio to inspect
first. Therefore, inspect characteristic B first. 5. Interpret the charts in the text to determine if the processes are stable. a. 5 points in a row above the mean. Investigate. b. 2 pts. near the
lower limit. Investigate. c. Stable. d. 7 points, all decreasing. Investigate. e. Process is erratic. Investigate. f. Process is stable. However, variation is less than expected. Investigate to see
if process has changed and the control limits need recalculation. 6. Interpret the charts in the text to determine if the processes are stable. a. Stable. b. Two out of control points. First five
points all above the mean. Investigate. c. Runs. Five points above and below the mean. Investigate for the cause. d. Drift. 7 points decreasing. Investigate for cause. e. Process is erratic and
successive points near the upper and lower limits. f. Erratic. 7. Tolerances for a new assembly call for weights between 32 and 33 pounds. The assembly is made using a process that has a mean of 32.6
pounds with a population standard deviation of .22 pounds. The process population is normally distributed. a. Is the process capable? b. If not, what proportion will meet tolerances? .49683 + .46562
= .96245 or only 96.2% will meet specifications. c. Within what values will 99.5% of sample means of this process fall if the sample size is constant at 10 and the process is stable? 99.5% of sample
means will fall between 32.4 and 32.8.
8. Specifications for a part are 62” +/- .01”. The part is constructed from a process with a mean of 62.01” and a population standard deviation of .033”. The process is normally distributed. a. Is
the process capable? b. What proportion will meet specifications? About 23% will meet specifications. c. Within what values will 95% of sample means of the process fall if the sample size is constant
at 5 and the process is stable? 62.01 +/- 1.65(.033/SQRT5) = 62.03 (upper limit), 61.99 (lower limit) 95% of the sample means will fall between 62.03 and 61.99. 9. Tolerances for a bicycle derailleur
are 6 cm +/- .001 cm. The current process produces derailleurs with a mean of 6.0001 with a population standard deviation of .0004. The process population is normally distributed. a. Is the process
capable? b. If not, what proportion will meet specifications? About 98.5% will meet specifications. c. Within what values will 75% of sample means of this process fall if the sample size is constant
at 6 and the process is stable? 6.0001 +/- 1.15(.0004/SQRT6) = 6.000263 (upper limit), 5.999937 (lower limit) 75% of the sample means will fall between 6.000263 and 5.999937. 10. A services process
is monitored using x-bar and R charts. Eight samples of n = 10 observations have been gathered with the following results: a. Using the data in the table, compute the centerline, the upper control
limit, and the lower control limits for the x-bar and R charts. b. Is the process in control? Please interpret the charts. c. If the next sample results in the following values (2.5, 5.5, 4.6, 3.2,
4.6, 3.2, 4.0, 4.0, 3.6, 4.2), will the process be in control? (a) Grand mean = 3.9125 R-bar = .40375 A2 = 1.03 CL = 3.9125+/-1.03(4.0375) = {4.33 (upper); 3.50 (lower) (b) (c) mean = 3.94. This
point is in control. The process is unstable and erratic. 11. A production process for the JMF Semicon is monitored using x-bar and R charts. 10 samples of n=15 observations have been gathered with
the following results: a. Develop a control chart and plot the means. b. Is the process in control? Explain. The process is not in control. 12. Experiment: Randomly select the heights of at least 15
of the students in your class. a. Develop a control chart and plot the heights on the chart. b. Which chart should you use? c. Is this process in control? Results will vary. Experiment: Control Chart
for Heights a. Develop a Control Chart 1. Data Collection: • Randomly select heights of at least 15 students. 2. Calculate Key Metrics: • Mean (X̄): Average height. • Range (R): Difference between the
highest and lowest heights. • Standard Deviation (σ): Measure of dispersion. 3. Control Limits: • Upper Control Limit (UCL): Mean + (3 × Standard Deviation). • Lower Control Limit (LCL): Mean• (3 ×
Standard Deviation). 4. Plot Data: • Create a control chart with the X-axis representing student samples and the Y-axis representing heights. • Plot the individual heights, mean line, UCL, and LCL.
b. Which Chart to Use? For this type of data, a X-bar chart (for the mean of a sample) is suitable if you are monitoring the average height over time or multiple samples. If you are monitoring
individual measurements and looking at variation within a sample, an Individual/moving range chart might be used. c. Is This Process in Control? To determine if the process is in control: • Check for
Points Outside Control Limits: Any data points outside the UCL or LCL indicate that the process may be out of control. • Look for Patterns: Check for trends, cycles, or runs that suggest the process
is not stable. If all data points fall within the control limits and there are no non-random patterns, the process is considered in control. Otherwise, further investigation is needed to identify any
sources of variation. 13. A finishing process packages assemblies into boxes. You have noticed variability in the boxes and desire to improve the process to fix the problem because some products fit
too tightly into the boxes and others fit too loosely. Following are width measurements for the boxes. Sample 68.63479 = X-bar, bar 0.4 = R Using x-bar and R charts, plot and interpret the process.
x-bar and R chart computations: There are two out of control points from samples 7 and 8. Investigate. 14. For the data in problem 13, if the mean specification is 68.5 +/- .25 and the estimated
process standard deviation is .10, is the process capable? Compute Cpu, Cpl, and Cpk. No, the process is not capable. 15. For the data in problem 13, treat the data as if it were population data, and
find the limits for an x-bar chart. Is the process in control? Compare your answer with the answers to Problem 14. Hint: Use the formula CLx = x-bar (3/d2)R-bar (Figure 11-8) No. Looking at the
chart, for all but 4 samples in the first 33 observations, the process is off centered. Several of these are in a successive pattern. There is also a significant shift starting at observation 37.
Points 37 through 45 are in a five successive point pattern. Need to investigate cause factors. Our observation that the process is off center corresponding to the low Cpu of .40 from Problem 14. 16.
A Rochester, NY firm produces grommets that have to fit into a slot in an assembly. Following are dimensions of grommets (in millimeters): a. Use x-bar and R charts to determine if the process is in
control. The process is in control. 17. Using the data from Problem 13, compute the limits for x-bar and s charts. Is the process still in control? Point 6 Is above the upper limit. 18. Using the
data from Problem 16, compute the limits for x-bar and s charts. Is the process still in control? The process is in control. We confirm the results of the prior analysis using x-bar charts. 19. Use a
median chart to determine if this process is centered. See the Excel spreadsheet below. Data from the textbook has been entered. The process is not in control. The limits are: UCL = 8.43; LCL = 7.77.
20. Use an x-bar chart to determine if the data in Problem 19 are in control. Do you get the same answer? The process is reasonably centered, however, the process is out of control with observations
4, 7, and 10 outside the control limits. The control limits are shown in the above spreadsheet. 21. The following data are for a component used in the space shuttle. Since the process dispersion is
closely monitored, use an x-bar and s chart to see if the process is in control. Yes, the process is in control. 22. Develop an R-chart for the data in Problem 21. Do you get the same answer? Yes,
the process is in control. 23. Using the data from Problem 21, compute limits for a median chart. Is the process in control? Yes, the process is in control. 24. Design a control plan for exam scores
for your quality management class. Describe how you would gather data, what type of chart is needed, how to gather data, how to interpret the data, how to identify causes, and remedial action to be
taken when out-of-control situations occur. Answers will vary. Control Plan for Exam Scores 1. Data Gathering • Sample: Collect all students' exam scores after each exam. • Frequency: After each
major test (midterms, finals). • Method: Store in a spreadsheet/database with student ID, date, and score. 2. Control Chart • Use an Individuals (I) Control Chart to track individual exam scores. •
Include an X-bar chart for average exam scores and an R chart for score range. 3. Data Interpretation • Set Upper Control Limit (UCL) and Lower Control Limit (LCL) at ±3 standard deviations. • Scores
outside these limits indicate an out-of-control situation. 4. Identifying Causes • Common Causes: Normal variations (study habits, question difficulty). • Special Causes: Unclear exam content,
grading errors, external factors. 5. Remedial Actions • Investigate and correct special causes (e.g., ambiguous questions). • Provide review sessions or re-assessments. • Continuously update control
limits based on performance trends. This plan ensures monitoring and addressing any issues in exam performance systematically. 25. For the sampling plan from Problem 24, how would you measure process
capability? The respondent should use the Cpk to measure capability. Scoring guidelines can be used as the tolerances (e.g: USL = 100; LSL = 60). 26. For the data in Problem 16, if the process target
is 50.25 with spec limits +/-5, describe statistically the problems that would occur if you used your spec limits on a control chart where n=5. Discuss type I and type II error. Mean = 50.28 USL =
55.25 LSL = 45.25 Using Excel I computed the standard deviation of the means as 4.85 and will use that Number in this analysis. Zupper = (55.25-50.28)/4.85 = 1.03 ~ p = .3485 Zlower = (45.25-50.28)/
4.85 = –1.04 ~ p = .35083 About 70% of the sample means will fall within the specification limits. This means that about 29.5% of the good product will be rejected erroneously. This is a type I
error. However, it should be noted that this process is highly incapable, as we would expect the control limits to be inside the tolerances. Chapter 12: Statistically Based Quality Improvement for
Attributes Chapter Outline •Generic Processes for Developing Structure Charts •Understanding Attributes Charts •Choosing the Right Attributes Chart •Reliability Models Overview An attribute is a
physical property; it is something that either exists or does not exist. There are five attribute types in the continuous quality improvement process. This chapter provides tools for dealing with
these attributes. Table 12-1 on page 315 presents a list of the types of attributes:
Discussion Questions 1. What are key attributes for a high-quality university? The key to this question is the phrase “high-quality.” The question might be taken to mean: What are the attributes that
differentiate between a university and a high-quality university. The list might include: •Quality teachers •Up-to-date technology •Degree program content •Internship placement potential
•Suitable launch for advanced degrees •Job placement An analysis will define which of these attributes are needed and how they might be qualified. 2. What are some attributes that you can identify
for an automobile tire? As the chapter brings out, the key attributes depend on the customer. For instance, a young person or a teenager might be looking for a specific set of attributes such as
raised white letters, a low profile, or a mud tread. A family man or woman might be looking for tread design in terms of safety, stopping distance, and puncture repair. An over-the-road trucker might
look for a set of attributes that relate to use in business. All of these are valid. The attributes are dependent on the intended use of the product. 3. What are some attributes for a university
financial aid process? Chapter 11 addressed constructing control charts. The generic process for developing control charts is revisited here: 1. Identify critical operations in the process where
inspection might be needed. 2. Identify critical product characteristics. 3. Determine whether the critical product characteristic is a variable or an attribute. 4. Select the appropriate process
chart. 5. Establish the control limits and use the chart to continually monitor and improve. 6. Update the limits when changes have been made to the process. These rules could be adapted to this
question quite easily. The resulting list might include such items as ease of access, friendly consultants, and ease of understanding. 4. What are some personal attributes that you could monitor
using control charts? Which control chart would you use? One begins by asking: “What do you want to accomplish by monitoring personal attributes?” Make a list to help identify the personal attributes
to be monitored. Once this is done, a methodology for tracking and charting the use of these attributes can be constructed. As a class exercise, the professor can divide the class into teams and ask
each team to construct a list. The professor then compares the lists. For instance, a student could monitor how often he or she makes reinforcing comments to other people. It could be set up as a
daily count using a C chart or as a reinforcing comment per person contacted using a P chart. 5. What are examples of structural attributes? On page 315, structural attributes are defined: Structural
attributes have to do with physical characteristics of a particular product or service. For example, an automobile might have electric windows. Services have structural attributes as well, such as a
balcony in a hotel room. If one were to revisit Question 2, the answer would be a list of structural attributes. For a computer, we might have a list containing the monitor, keyboard, and mouse.
(Structural attributes can be touched.) 6. What are some examples of sensory attributes? On page 315, sensory attributes are defined: Sensory attributes relate to senses of touch, smell, taste, and
sound. For products, these attributes relate to form design or packaging design to create products that are pleasing to customers. In services such as restaurants and hotels, atmosphere is very
important to the customer experience. A new car smell immediately comes to mind. There is something ethereal about the smell, glow, and feel of a new car. If you go to a supermarket, check where the
breakfast foods are located. Traditionally, the breakfast foods that appeal to children are on the lowest shelf so that a child will be “assaulted” by the colors and pictures on the packages, all of
which are sensory attributes. One of the old proverbs in the advertising industry is “sell the sizzle, not the steak.” People react to sensory attributes psychologically. This is a very powerful
motivator. 7. What are some examples of performance attributes? On page 316, we find the definition for performance attributes: Performance attributes relate to whether or not a particular product or
service performs as it is supposed to. For example, does the lawn mower engine start? Does the stereo system meet a certain threshold for low distortion? Performance attributes are heavily based upon
requirements. For instance, what is the required mean time to failure (MTTF) as opposed to the MTTF specified in the requirements? We are measuring an instance that can relate directly to customer
satisfaction. One example of a performance attribute is a car’s fuel efficiency. When a car is purchased, a certain range of fuel efficiency is expected – if the car is rated as 25-mpg city and
40-mpg highway, those numbers should be approximately valid when driving the new car. If in fact the mileage is 30% to 50% lower, the purchaser could be dissatisfied in the car’s performance. 8. What
are some examples of temporal attributes? Page 316 defines temporal attributes: Temporal attributes relate to time. Were delivery schedules met? This often has to do with the reliability of delivery.
Example: You and your spouse go down to the local big-box store and buy a new dishwasher. Excitedly, your spouse stays home from work to allow delivery and installation on the planned day. When the
delivery does not happen on time and there has been no communication, the excitement of the new appliance quickly fades. This will directly affect where you go shopping for the new stove that you
also want. The more directly an attribute affects you, the more important it is. 9. What are some examples of ethical attributes? Ethical attributes are discussed on page 316: Ethical attributes are
important to firms. Do they report properly? Is their accounting transparent? Is the service provider empathetic? Is the teacher kind or not? Some years ago, a prominent car salesman in Denver was
indicted and convicted for rolling back odometers on used cars. At the time, he had a large business with several car lots. His “empire” has since disappeared. After the recent Enron scandals, major
accounting firms have also ceased to exist. People are watching businesses and expecting ethical performances from them. An organization that has questionable ethics will find that its prospective
customers will be curious about its other attributes. Fortunes have been lost over questionable ethics. 10. What ethical attributes might you use to determine where you should go to work after
graduation? Where do you work? What do you do? A person will field these questions regularly. For many of us, what we do is who we are. This is a personal question that relates directly to an
individual’s self-image. A job candidate might list his or her personal ethical attributes and note the potential company’s ethical attributes. Do they correspond? Are there any major differences?
When considering where to work after graduation, key ethical attributes to evaluate include: 1. Company Values and Mission: Align with organizations whose mission and core values reflect your own
ethical beliefs, such as integrity, respect, and social responsibility. 2. Corporate Social Responsibility (CSR): Assess the company’s commitment to social and environmental responsibility, including
sustainability practices and community engagement. 3. Diversity and Inclusion: Look for workplaces that promote diversity, equity, and inclusion, ensuring fair treatment and equal opportunities for
all employees. 4. Transparency and Accountability: Evaluate how transparent the company is about its operations, decision-making processes, and financial practices, ensuring accountability for its
actions. 5. Workplace Ethics: Consider the company’s policies on ethical conduct, such as anti-corruption measures, fair labor practices, and employee treatment. 6. Leadership Integrity: Assess the
ethical leadership in the company, ensuring leaders model ethical behavior and promote an ethical work culture. Choosing a company that aligns with these attributes ensures you work in an environment
that prioritizes ethical behavior and positive societal impact. Case 12-1: Decision Sciences Institute National Conference Take the raw data provided and develop research questions. Next, using the
statistical tools from this chapter, analyze the data. Finally, put the data into a form that will be useful for decision makers. A lot of data is presented. Some ideas come immediately. Comparisons
of these items are easily extracted from an excel spreadsheet. Specifically, one can compute the percentage of submitted against the percentage of each of the various levels. The data is all
attribute data. A variety of hypothesis can be constructed. For instance, a simplistic example might be: Ho: Most of the time the reviewers agree H1: Most of the time the reviewers do not agree Two
sets of data are presented. The individual reviewers can be compared to each other. The results of the decisions can be plotted in a variety of ways. For instance, given a subset of the table, we
might have the following results: A simple line chart can be constructed. Perhaps a pie chart or two might also be constructed. All of the charts and graphs that are shown in the chapter can be
presented. Again, the question is asked: What is the purpose of the analysis? Suggested Answers to End of Chapter Problems 1. Suppose you want to inspect a lot of 10,000 products to see whether or
not they meet requirements. Design a sampling plan used to test these products. Answers will vary, but students should start with the six steps in the generic process for developing attributes charts
(see text page 316): identifying critical operations, identifying product characteristics, determining whether those characteristics are variables or attributes, selecting the appropriate process
chart, establishing the control limits, and updating those limits when changes have been made to the process. The step of selecting the right chart is further developed in Figure 12-8’s flow chart.
Sampling Plan for Inspecting 10,000 Products 1. Lot Size: 10,000 products. 2. Sample Size: Use the AQL (Acceptable Quality Level) method. For example, if AQL is 1%, select n = 200 products as the
sample. 3. Sampling Method: Random sampling. Select 200 products at random from the lot to ensure an unbiased selection. 4. Acceptance Criteria: Set a maximum number of defects allowed in the sample.
For instance, if the AQL is 1%, allow a maximum of 2 defects. 5. Decision Rule: • Accept the Lot: If the number of defective products in the sample is ≤ 2. • Reject the Lot: If more than 2 defective
products are found. This plan ensures a balance between inspection effort and product quality assurance. 2. Suppose a product is made of 100 components, each with a 97% reliability. What is the
overall reliability for the product? R = .97100 = .0476 3. Suppose a product is made of 1,000 components, each with .999 reliability. What is the unreliability of this product? Is this acceptable?
Why or why not? Q = 1 – R = 1 – .9991,000 = 1 – .3677 = .6323. This means that the product has a 63% chance of failing within the reliability period. This might be acceptable if the product is not
critical in use, such as a light bulb. 4. A product consists of 45 components. Each component has an average reliability of .97. What is the overall reliability for this product? R = .9745 = .2539 5.
A radio is made up of 125 components. What would have to be the average reliability for each component for the radio to have a reliability of 98% over its useful life? R = P125 = .98 P = .981/125 =
.9999 6. List five products with low reliability. List five that have high reliability. What are the elemental design differences between these products? In other words, what are the factors that
make some products reliable and others unreliable? Following are some examples of products with low reliability and high reliability. Low reliability: •Light bulbs •Chandeliers High reliability:
•Cell phones •Pens Low reliability products tend to be made of fragile materials and have many components. In contrast, high reliability products tend to be mass-produced in highly automated
processes with rigorous testing protocols. 7. An assembly consists of 240 components. Your customer has stated that your overall reliability must be at least 99%. What needs to be the average
reliability factor for each component? R = P240 = .99 P = .991/240 = .99996 8. A product is made up of six components. They are wired in series with reliabilities of .95, .98, .94, .96, .98, and .97.
What is the overall reliability for this product? .95 x .98 x .94 x .96 x .98 x .97 = .7986 9. Suppose that redundant components are introduced for the two components in Problem 8 with the lowest
reliability. What is now the overall reliability for the product? (1 – .052) x .98 x (1 – .062) x .96 x .98 x .97 = .8989 10. Suppose that redundant components are introduced for all of the
components in Problem 8. What is now the overall reliability for the product? (1 – .052) x (1 – .022) x (1 – .062) x (1 – .042) x (1 – .022) x (1 – .032) = .9906 11. A product is made up of
components A, B, C, and D. These components are wired in series. Their reliability factors are .98, .999, .97, and .989 respectively. Compute the overall reliability for this product. .939 = .98 x
.999 x .97 x .989 12. A product is made up of components A, B, C, D, E, F, G, H, I, and J. Components A, B, C, and F have a 1/10,000 chance of failure during useful life. D, E, G, and H have a 3/
10,000 chance of failure. Component I and J and a 5/10,000 chance of failure. What is the overall reliability of the product? .99994 x .99974 x .99952 = .9974 13. For the product in Problem 12, if
parallel components are provided for components I and J, what is the overall reliability for the product? 99994 x .99974 x (1.-00052)2 = .9984 14. A product is made up of 20 components in a series.
Ten of the components have a 1/10,000 chance of failure. Five have a 3/10,000 chance for failure. Four have a 4/10,000 chance for failure. One component has a 1/100 chance for failure. What is the
overall reliability of the product? .999910 x .99975 x .99964 x .99 = .9859 15. For the product in Problem 14, if parallel components are used for any component with worse than a 1/1,000 chance for
failure, what is the overall reliability? How many components will the new design have? What will be the average component reliability for the redesigned product? .999910 x .99975 x .99964 x (1-.012)
= .9958 16. An inspector visually inspects 200 sheets of paper for aesthetics. Using trained judgment, the inspector will either accept or reject sheets based on whether they are flawless. Following
are the results of recent inspections: a. Given these results, using a p chart, determine if the process is stable. CL = .064 LCL = .0118 UCL = .1152 The process is out of control at sample 5. b.
What would need to be done to improve the process? Investigate causes in Sample 5 and eliminate. Also investigate how the product was produced differently in Sample 6 (with significantly lower
defects) and incorporate it into the process. 17. Using the data in Problem 16, compute the limits for an np chart. Control Limits = 200(.064) 3[SQRT 200(.064)(.936)] = 2.41 and 23.19. 18. Suppose
a company makes the following product with the following (see text) number of defects. Construct a p chart to see if the process is in control. n = 100 CL = .417, LCL = .2689, UCL = .5647 The process
is not in control. 19. Using the data from Example 12.3, evaluate the Demis using a u chart and evaluate the Streakless using a c chart. Assume that the Demis are twice the size as the Streakless on
average. Demis: CL = 6, LCL = .804, UCL = 11.196 Streakless: CL = 5.833, LCL = –1.412 (use zero), UCL = 13.079 The Demis are out of control, the Streakless are not. 20. Politicians closely monitor
their popularity based on approval ratings. For the previous 16 weeks, Governor Johnny’s approval ratings have been (in percentages): a. Prepare a report for the governor outlining the results of
your analysis. Use control charts to analyze the data (n = 200). b. What action would you propose to the governor based on your analysis. Note p chart was constructed with disapproval (“defect”)
rating: CL = .4056 LCL = .301, UCL = .510 Although not out of control, the governor’s disapproval rating rose during weeks 8 through 12. He should investigate and address any issues or activities in
the news media during that period that might have contributed to this rising disapproval. If the p chart is constructed on the approval rating percentages, Governor Johnny’s approval would be
decreasing during weeks 8 through 12. 21. Construct and interpret a c chart using the following (see text) data: CL = 5.433, LCL = –1.56 (use zero), UCL = 12.426 The process is in control, but Sample
20 with no defects should be investigated for improved methods and Sample 21 should be investigated for poor of sloppy methods. 22. Construct and interpret a u chart using the following (see text)
data. Note that the average size is two times the original product. CL = .4963, LCL = .237, UCL = 9.689 The process is in control, but the last samples indicate a decrease in the defects. Investigate
and incorporate into the process. 23. Dellana Company tested 50 products for 75 hours each. In this time, they experienced 4 breakdowns. Compute the number of failures per hour. What is the mean time
between failures? Failure/hour = 4/(50 X 75) = .0011 MTBF = 1/.0011 = 909.09 24. The Collier Company tested 200 products for 100 hours each. In this time, they experienced 12 breakdowns. Compute the
number of failures per hour. What is the mean time between failures? Failure/hour = 12/(200 x 100) = .0006 MTBF = 1/ .0006 = 1666.7 hours 25. Crager company tested 100 products for 50 hours each.
During the test, 3 breakdowns occurred. Compute the number of failures per hour and MTBF. Failure/hour = 3/(50 x 100) = .0006 MTBF = 1/ .0006 = 1666.7 hours 26. Suppose a product is designed to
function for 10,000 hours with a 3% chance of failure. Find the average number of failures per hour and the MTTF. .97 = e-(10,000) ln .97 = -10,000 = -(ln .97)/10,000 = .0000030 = average
failures/hr. MTTF = 1/ = 33,333.33 hrs. 27. Suppose a product is designed to function for 100,000 hours with a 1% chance of failure. Suppose that there are six of these in use at a facility. Find
the average number of failures per hour and the MTTF. First, determine overall reliability: .996 = .9415, then use R to determine MTTF. .9415 = e-(100,000) ln .9415 = -100,000 = -(ln .9415)/
100,000 = .0000006 = average failures/hr. MTTF = 1/ = 1,666,666.67 hrs. 28. Suppose that there are 42 pumps used in a refinery. These pumps are continuously being used with a 2% chance of failure
over 50,000 hours. If repair time is 10 hours to install a new rebuilt pump, how many pumps should be kept on hand to keep the chance of a plant shutdown to less than 1%. (Hint: Treat this problem as
a traditional safety stock problem and use a z table.) .98 = e-(50000) ln .98 = -50,000 = (-ln .98)/50,000 = .000000404 = average failures/hr. MTTF = 1/ = 2,474,915 hours for one unit to fail.
(1-.000000404)42 = .999989 reliability for 42 units, which exceeds the 99% reliability. No safety stock is needed at this time. However, a good preventive maintenance program would be helpful. 29.
Suppose that a product is designed to work for 1000 hours with a 2% chance of failure. Find the average number of failures per hour and the MTTF. .98 = e-(1000) ln .98 = -1000 = -(ln .98)/1000
= .0000202027 = average failures/hr. MTTF = 1/ = 49,998.3 hrs. 30.A product has been used for 5000 hours with 1 failure. Find the mean time between failures (MTBF) and λ. MTBF = 5000/1 = 5000 hours
between failures λ = 1/5000 = .0002 failures/hour 31. You are to decide between 3 potential suppliers for an assembly for a product you are designing. After performing life testing on several
assemblies, you find the following. See Calculated SA column above. Choose supplier A. 32. You are to choose a supplier of a copier based on reliability and service. After gathering data about the
alternatives, here is what you found. What do you recommend? Recommend Supplier 2. Solution Manual for Managing Quality: Integrating the Supply Chain Thomas S. Foster 9780133798258 | {"url":"https://studymerge.com/doc/ch-11-12-statistically-based-quality-improvement-for-variables-managing-quality-integrating-the-supply-chain-book-guide/","timestamp":"2024-11-03T05:55:46Z","content_type":"text/html","content_length":"616564","record_id":"<urn:uuid:925d639e-c00c-4ee2-9f25-6d7839bba734>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00152.warc.gz"} |
To apply calculations to an equilibrium, consider the general reversible reaction:
aA + bB in equilibrium with cC + dD
An equilibrium constant is defined as K[c] = [C]^c[D]^d / [A]^a[B]^b
Products are on the top line, with reactants on the bottom line.. The square brackets, [ ], mean concentration in moldm^-3.
The lowercase letters are the number of moles of each substance.
To work out the units of your result, replace A, B, C, and D with moldm^-3 and simplify the expression.
In this example each substance has 1 mole and the units then cancel out.
If the reaction involves gases, partial pressures are used instead of concentrations and the equilibrium constant is called K[p] . The layout is the same (except for a change in bracket type) and the
powers come from the big numbers again.
It is a common silly mistake to mix up "p" and "c" and what type of bracket to use.
Using "p" and "( )" for a concentration question or the other way round will cost you a lot of marks.
The units in a K[p] calculation are kPa and for the result must be worked out using the same method to work out the units for K[c].
K[c] and K[p ]can only be changed by changing temperature.
An equilibrium will often be shown with a ΔH value. This is for the forward (left to right) direction.
If ΔH is positive (endothermic in the forward direction), Kc increases with increasing temperature.
This means that more products are formed.
Some people get confused by trying to learn all the possible combinations of increase and decrease, endothermic and exothermic. Learn one of the them confidently and thenwork out the other
possibilities if you need to do so.
Increasing the concentration of one of the reactants does not change the value of K[c] but does mean that the concentration of products must also increase.
Increasing the partial pressure of one of the reactants does not change the value of K[p] but does mean that the partial pressure of products must also increase.
Tips on dealing with equilibrium calculations
Equilibrium calculations can be difficult for two main reasons. Either you find it difficult to do the mathematical juggling or else you find it difficult to understand what the examiner is telling
you in the question. Of course if you are really unlucky you might find both bits difficult.
The best first advice is to write your answer neatly. In this way both you and the exam marker will be able to understand what you have achieved. You will find the calculation a bit easier and the
marker might be able to give you some credit for a partial answer.
Consider a fairly basic question such as: 6 moles of PCl[5] are allowed to come to equilibrium with PCl[3] and Cl[2]. The total volume of the container is 5 dm^3. At equilibrium, it is found that
there are only 4 moles of PCl[5] left. Calculate the value of Kc and give its units.
Let’s put that into the table along with the fact that we will say x moles of PCl[5] will decompose to give the equilibrium mixture.
Equation PCl[5 ] PCl[3] Cl[2]
Initial moles 6 0 0
Equilibrium moles 6 – x x x
Equilibrium concentration (6 – x) / 5 x / 5 x / 5
The question tells us that at equilibrium, only 4 moles of PCl[5] are left.
This means that 6 – x = 4 and so x = 2.
Let’s modify the table to include this information.
Equation PCl[5 ] PCl[3] Cl[2]
Initial moles 6 0 0
Equilibrium moles 4 2 2
Equilibrium concentration 4 / 5 2 / 5 2 / 5
Kc = [PCl[3]][Cl[2]] / [PCl[5]] and so we can add the values that we know:
Kc = (2/5) (2/5) / (4/5) solving this gives Kc = 0.2
The units are (mol dm^-3) x (mol dm^-3) / (mol dm^-3) and so units are mol dm^-3
More complications can be that:
• The “amounts” information is given as masses etc.
Simply use GCSE moles calculations to convert it to moles.
• The chemical equation is not a simple “one-to-one” relationship between the reagents and products. Put the appropriate “big numbers” in the Kc equation as powers.
Think about how you modify “x” in the “equilibrium moles” line of the grid.
As an example of this consider:
2 moles of N[2] and 5 moles of H[2] are mixed with 4 moles of NH[3] and allowed to come to equilibrium in a container of total volume = 3 dm^3. At equilibrium, there are 6 moles of NH[3]. Calculate
the value of Kc at this temperature and give its units.
Equation N[2] 3H[2] 2NH[3]
Initial moles 2 5 4
Equilibrium moles 2 - x 5 – 3x 4 + 2x
Equilibrium concentration (2 – x)/V (5 – 3x)/V (4 + 2x)/V
But we can add to this since we know that 4 + 2x = 6. x = 1 Modify the table:
Equation N[2] 3H[2] 2NH[3]
Initial moles 2 5 4
Equilibrium moles 1 2 6
Equilibrium concentration 1/V 2/V 6//V
Kc = (1/3) x (2/3)^3 / (6/3)^2 = 0.025
Units are mol^2dm^-6 | {"url":"https://revisionscience.com/a2-level-level-revision/chemistry-level-revision/chemical-equilibrium/equilibria","timestamp":"2024-11-14T04:50:13Z","content_type":"text/html","content_length":"37380","record_id":"<urn:uuid:7a5b9c91-36dd-4510-8384-d7bf408c8a93>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00826.warc.gz"} |
Zhong-Ping Wan
In this work, a nonsmooth multiobjective optimization problem involving generalized invexity with cone constraints and Applications (for short, (MOP)) is considered. The Kuhn-Tucker necessary and
sufficient conditions for (MOP) are established by using a generalized alternative theorem of Craven and Yang. The relationship between weakly efficient solutions of (MOP) and vector valued saddle
points of … Read more
Construction project scheduling problem with uncertain resource constraints
This paper discusses that major problem is the construction project scheduling mathematical model and a simple algorithm in the uncertain resource environments. The project scheduling problem with
uncertain resource constraints comprised mainly three parties: one of which its maximal limited capacity is fixed throughout the project duration; second maximal limited resource capacity is random
variable; … Read more
Genetic Algorithm for Solving Convex Quadratic Bilevel Programming Problem
This paper presents a genetic algorithm method for solving convex quadratic bilevel programming problem. Bilevel programming problems arise when one optimization problem, the upper problem, is
constrained by another optimization, the lower problem. In this paper, the bilevel convex quadratic problem is transformed into a single level problem by applying Kuhn-Tucker conditions, and then an
… Read more
Solving Method for a Class of Bilevel Linear Programming based on Genetic Algorithms
The paper studies and designs an genetic algorithm (GA) of the bilevel linear programming problem (BLPP) by constructing the fitness function of the upper-level programming problem based on the
definition of the feasible degree. This GA avoids the use of penalty function to deal with the constraints, by changing the randomly generated initial population into … Read more
Asymptotic approximation method and its convergence on semi-infinite programming
The aim of this paper is to discuss an asymptotic approximation model and its convergence for the minimax semi-infinite programming problem. An asymptotic surrogate constraints method for the minimax
semi-infinite programming problem is presented making use of two general iscreteapproximation methods. Simultaneously, we discuss the consistenceand the epi-convergence of the asymptotic
approximation problem. Citation School … Read more | {"url":"https://optimization-online.org/author/zpwan/","timestamp":"2024-11-10T08:43:03Z","content_type":"text/html","content_length":"93654","record_id":"<urn:uuid:e0e167b2-f104-4d3f-ab45-898be8ef68b6>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00863.warc.gz"} |
Structuralism in Physics
First published Sun Nov 24, 2002; substantive revision Mon Oct 27, 2014
Under the heading of “structuralism in physics” there are three different but closely related research programs in philosophy of science and, in particular, in philosophy of physics. These programs
were initiated by the work of Joseph Sneed, Günther Ludwig, and Erhard Scheibe, respectively, since the begin of the 1970s. For the sake of simplicity we will use these names in order to refer to the
three programs, without the intention of ignoring or minimizing the contributions of other scholars. (See the Bibliography.) The term “structuralism” was originally claimed by the Sneed school, see
e.g., Balzer and Moulines (1996), but it also appears appropriate to subsume Ludwig's and Scheibe's programs under this title because of the striking similarities of the three approaches. The
activities of the structuralists have been mainly confined to Europe, especially Germany, and, for whatever reasons, largely ignored in the Anglo-American discussion.
The three programs share the following characteristics and convictions:
• A metatheory of science requires a kind of formalization different from that already employed by scientific theories themselves.
• The structuralistic program yields a framework for the rational reconstruction of particular theories.
• A central tool of formalization is Bourbaki's concept of “species of structures”, as described in Bourbaki (1986).
• Among the significant features of theories to be described are:
□ Mathematical structure
□ Empirical claims of a theory
□ Function of theoretical terms
□ Rôle of approximation
□ Evolution of theories
□ Intertheoretic relations
A physical theory T consists, among other things, of a group of laws which are formulated in terms of certain concepts. But an apparent circularity arises when one considers how the laws of T and the
concepts acquire their content, because each seems to acquire content from the other — the laws of T acquire their content from the concepts used in the formulation of the laws, while the concepts
are often “introduced” or “defined” by the group of laws as a whole. To be sure, if the concepts can be introduced independently of the theory T, the circularity does not appear. But typically every
physical theory T requires some new concepts which cannot be defined without using T (we call the latter “T-theoretical concepts”). Is the apparent circularity concerning the laws and the
T-theoretical concepts a problem? Some examples will help us assess the threat.
As an example, consider the theory T of classical particle mechanics. For simplicity we will assume that kinematical concepts, such as the positions of particles, their velocities and accelerations
are given independently of the theory as functions of time. A central statement of T is Newton's second law, F=ma, which asserts that the sum F of the forces exerted upon a particle equals its mass m
multiplied by its acceleration a.
While we customarily think of F=ma as an empirical assertion, there is a real risk that it turns out merely to be a definition or largely conventional in character. If we think of a force merely as
“that which generates acceleration” then the force F is actually defined by the equation F=ma. We have a particle undergoing some given acceleration a, then F=ma just defines what F is. The law is
not an empirically testable assertation at all, since a force so defined cannot fail to satisfy F=ma. The problem gets worse if we define the (inertial) mass m in the usual manner as the ratio |F|/|a
|. For now we are using the one equation F=ma to define two quantities F and m. A given acceleration a at best specifies the ratio F/m but does not specify unique values for F and m individually.
In more formal terms, the problem arises because we introduced force F and mass m as T-theoretical terms that are not given by other theories. That fact also supplies an escape from the problem. We
can add extra laws to the simple dynamics. For example, we might require that all forces are gravitational and that the net force on the mass m be given by the sum F=Σ[i]F[ i] of all gravitational
forces F[i] acting on the mass due to the other masses of the universe, in accord with Newton's inverse square law of gravity. (The law asserts that the force F[i] due to attracting mass i with
gravitational mass m[gi] is Gm[g]m[gi] r[i] / r[i]^3, where m[g] is the gravitational mass of the original body, r[i] the position vector of mass i originating from the original body, and G the
universal constant of gravitation.) That gives us an independent definition for F. Similarly we can require that the inertial mass m be equal to the gravitational mass m[g]. Since we now have
independent access to each of the terms F, m and a appearing in F=ma, whether the law obtains is contingent and no longer a matter of definition.
Further problems can arise, however, because of another T-theoretical term that is invoked implicitly when F=ma is asserted. The accelerations a are tacitly assumed to be measured in relation to an
inertial system. If the acceleration is measured in relation to a different reference system, a different result is obtained. For example, if it is measured in relation to a system moving with
uniform acceleration A, then the measured acceleration will be a′ = (a − A). A body not acted on by gravitational forces in an inertial frame will obey 0=ma so that a=0. The same body in the
accelerated frame will have acceleration a′ = −A and be governed by −mA = ma′. The problem is that the term −mA behaves just like a gravitational force; its magnitude is directly proportional to the
mass m of the body. So the case of a gravitation free body in a uniformly accelerated reference system is indistinguishable from a body in free fall in a homogeneous gravitational field. A
theoretical underdetermination threatens once again. Given just the motions how are we to know which case is presented to us?^[1] Resolving these problems requires a systematic study of the relations
between the various T-theoretical concepts, inertial mass, gravitational mass, inertial force, gravitational force, inertial systems and accelerated systems and how they figure in the relevant laws
of the theory T.
Similar problems arise in the formulation of almost all fundamental physical theories.
There are various ways to cope with this problem. One could try to unmask it as a pseudo-problem. Or one could try to accept the problem as part of the usual way science works, albeit not in the
clean manner philosophers would like it. The structuralistic programs, however, agree that this is a non-trivial problem to be solved and devise meta-theoretical machinery to enable its solution.
They further agree on dividing the vocabulary of the theory T into T-theoretical and T-non-theoretical terms, the latter being provided from outside the theory.
2.2.1 Sneed's solution
In the Sneedean approach the “empirical claim” of the theory is formulated by using an existential quantifier for the T-theoretical terms (i.e., in terms of the “Ramsey sentence” for T). In our above
example, Newton's law for gravitational forces would be reformulated as: “There exist an inertial system and constants G, m[i], m[gi] such that for each particle the product of its mass times its
acceleration equals the sum of the gravitational forces as given above.” This removes the circularity but leaves open the question of content. Here the structuralists à la Sneed would argue that the
empirical claim of the theory T′ has to contain all the laws of the theory as well as higher-order laws, called “constraints”. In our example, the constraints would be statements such as “all
particles have the same inertial and gravitational masses and the gravitational constant assumes the same value in all models of the theory.” The theory would thereby acquire more content and become
2.2.2 Ludwig's solution
Although Ludwig's meta-theoretical framework is slightly different, the first part of his solution is essentially equivalent to the above one. On the other hand, he proposes a stronger program
(“axiomatic basis of a physical theory”) which proceeds by considering an equivalent form T* of a theory T in which all T-theoretical concepts are eliminated by explicit definitions. This seems to be
at variance with older results about the non-definability of theoretical terms, but a closer inspection removes the apparent contradiction. For example, the concept of “mass” may be non-definable in
a theory dealing only with single orbits of a mechanical system, but definable in a theory containing all possible orbits of that system.
However, to formulate the axiomatic basis of a real theory, not just a toy model, is a non-trivial task and typically requires one or two books; see the examples Ludwig (1985, 1987) and Schmidt
Both programs address the further problem of how to determine the extension, e.g., the numerical values, of a theoretical term from a given set of observational data. We will call this the
“measurement problem”, not to be confounded with the well-known measurement problem in quantum theory. Typically the measurement problem has no unique solution. Rather the values of the theoretical
quantities can only be measured within a certain degree of imprecision and using auxiliary assumptions which, although plausible, are not confirmed with certainty. In the above Newton example one
would have to use the auxiliary assumption that the trajectories of the particles are twice differentiable and that other forces except the gravitational forces can be neglected. For a recent
critical examination of the solution to the measurement problem within Sneed's approach with detailed examples from astronomy see Gähde (2014).
The feature of imprecision and approximation plays a prominent rôle in the structuralistic programs. In the context of the measurement problem, imprecision seems to be a defect of the theory which
impedes the exact determination of the theoretical quantities. However, imprecision and non-uniqueness is crucial in the context of evolution of theories and the transition to new and “better”
theories. Otherwise the new theory could in general not encompass the successful applications of the old theory. Consider for example the transition of Kepler's theory of planetary motion to Newton's
and Einstein's theories: Newtonian gravitation theory and general relativity replace the Kepler ellipses with more complicated curves. But these should still be consistent with the old astronomical
observations, which is only possible if they don't fit exactly into Kepler's theory .
Part of the structuralistic program is the definition of various intertheoretic relations. Here we will concentrate on the relation(s) of “reduction”, which play an important rôle in the
philosophical discourse as well as in the work of the physicists, albeit not under this name. Consider a theory T which is superseded by a better theory T′. One could use T′ in order to understand
some of the successes and failures of T. If there is some systematic way of deriving T as an approximation within T′, then T is “reduced” to or by T′. In this case, T is successful where it is a good
approximation to T′ and T′ is successful. On the other hand, in situations where T′ is still successful but T is a poor approximation to T′, T will fail. For example, classical mechanics should be
obtained as the limiting case of relativistic mechanics for velocities small compared with the velocity of light. This would explain why classical mechanics was, and is still, successfully applied in
the case of small velocities but fails for large (relative) velocities.
As mentioned, the investigation of such reduction relations between different theories is part of the every-day work of theoretical physicists, but usually they do not adopt a general concept of
reduction. Rather they intuitively decide what has to be shown or to be calculated, depending on the case under consideration. Here the work of the structuralists could lead to a more systematic
approach within physics, although there does not yet exist a generally accepted, unique concept of reduction.
Another aspect is the rôle of reduction within the global picture of the development of physics. Most physicists, but not all, tend to view their science as an enterprise which accumulates knowledge
in a continuous manner. For example, they would not say that classical mechanics has been disproved by relativistic mechanics, but that relativistic mechanics has partly clarified where classical
mechanics could be safely applied and where not. This view of the development of physics has been challenged by some philosophers and historians of science, especially by the writings of T. Kuhn and
P. Feyerabend. These scholars emphasize the conceptual discontinuity or “incommensurability” between reduced theory T and reducing theory T′. The structuralistic accounts of reduction now opens the
possibility of discussing these matters on a less informal level. The preliminary results of this discussion are different depending on the particular program.
In the writings of Ludwig there is no direct reference to the incommensurability thesis and the corresponding discussion. But obviously his approach implies the most radical denial of this thesis.
His reduction relation is composed of two simpler intertheoretic relations called “restriction” and “embedding”. They come in two versions, exact and approximate. Part of their definitions are
detailed rules of translation of the non-theoretic vocabulary of T′ into that of T. Hence commensurability, at least on the non-theoretical level, is insured by definition. The problem is then
shifted to the task of showing that some of the interesting cases of reduction, which are discussed in the context of incommensurability, fit into Ludwig's definition. Unfortunately, he gives only
one extensively worked-out example of reduction, namely thermodynamics vs. quantum statistical mechanics, in Ludwig (1987). Incommensurability of theoretical terms could probably be more easily
incorporated in Ludwig's approach since it could be traced back to the difference between the laws of T and T′.
The relation between incommensurability and the Sneedean reduction relation is to some extent discussed in Balzer et al. (1987, chapter VI.7). The authors consider an exact reduction relation as a
certain relation between potential models of the respective theories. More interesting for physical real-life examples is the approximate version which is obtained as a “blurred exact reduction” by
means of a subclass of an empirical uniformity on the classes of potential models. The Kepler-Newton case is discussed as an example of approximate reduction. The discussion of incommensurability
suffers from the notorious difficulties of explicating such notions as “meaning preserving translation”. There is an interesting application of the interpolation theorem of meta-mathematics which
yields the result that, roughly speaking, (exact) reduction implies translation. However, the relevance of this result is questioned in Balzer et al. (1987, 312 ff). Thus the discussion eventually
ends up as inconclusive, but the authors admit the possibility of a spectrum of incommensurabilities of different degrees in cases of pairs of reduced/reducing theories.
Scheibe in his (1999) also explicitly refers to the theses of Kuhn and Feyerabend and gives a detailed discussion. Unlike the other two structuralistic programs, he does not propose a fixed concept
of reduction. Rather he suggests a lot of special reduction relations which can be combined appropriately to connect two theories T and T′. Moreover, he proceeds by means of extensive real-life case
studies and considers new types of reduction relations if the case under consideration cannot be described by the relations considered so far. Scheibe concedes that there are instances of
incommensurability which make it difficult to find a reduction relation in certain cases. As a significant example he mentions the notions of an “observable” in quantum mechanics on the one hand, and
in classical statistical mechanics on the other hand. Although there are maps between the respective sets of observables, Scheibe considers this as a case of incommensurability, since these maps are
not Lie algebra homomorphisms, see Scheibe (1999, 174).
Summarizing, the structuralistic approaches are capable of discussing the issues of reduction and incommensurability and the underlying problems on an advanced level. Thereby these approaches have a
chance of mediating between disparate camps of physicists and philosophers.
In this section we will describe more closely the particular programs, their roots and some of the differences between them.
4.1.1 History and general traits
This program has been the most successful with respect to the formation of a “school” attracting scholars and students who adopt the approach and work on its specific problems. Hence most of the
structuralistic literature concerns the Sneedean variant. Perhaps this is partly also due to the circumstance that only Sneed's approach is intended to apply (and has been applied) to other sciences
and not only physics.
The seminal book was Sneed (1971) which presented a meta-theory of physics in the model-theoretical tradition connected with P. Suppes, B. C. van Fraassen, and F. Suppe. This approach was adopted and
popularized by the German philosopher W. Stegmüller (1923–1991), see e.g., Stegmüller (1979) and further developed mainly by his disciples. In its early days the approach was called the
“non-statement view” of theories, emphasizing the rôle of set-theoretical tools as opposed to linguistic analyses. Later this aspect was considered to be more of practical importance than a matter of
principle, see Balzer et al. (1987, 306 ff). Recently, H. Andreas (2014) and G. Schurz (2014) have proposed two slightly different frameworks that reconcile semantical and syntactical formulations of
Sneed's program. Nevertheless, the almost exclusive use of set-theoretic tools remains one of the characteristic stylistic features of this program and one that distinguishes it conspicuously from
the other programs.
4.1.2 Central notions of Sneed's program
According to Moulines, in Balzer and Moulines (1996, 12–13), the specific notions of the Sneedean program are the following. We illustrate these notions by simplified examples, inspired by Balzer et
al. (1987), which are based on a system of N classical point particles coupled by springs satisfying Hooke's law. For a recent introduction into the basic concepts see also H. Andreas and F. Zenker
• M[p]: A class of potential models (the theory's conceptual framework.
[One potential model contains a set of particles, a set of springs together with their spring constants, the masses of the particles, as well as their positions and mutual forces as functions
of time.]
• M: A class of actual models (the theory's empirical laws).
[M is the subclass of potential models satisfying the system's equation of motion. ]
• <M[p],M>: A model-element (the absolutely necessary portion of a theory)
• M[pp]: A class of partial potential models (the theory's relative non-theoretical basis).
[One partial potential model contains only the particles' positions as functions of time, since the masses and forces are considered as T-theoretical.]
• C: A class of constraints (conditions connecting different models of one and the same theory).
[The constraints say that the same particles have the same masses and the same springs have the same spring constants.]
• L: A class of links (conditions connecting models of different theories).
[Among the conceivable links are:
☆ Links to the theory of classical spacetime
☆ Links to the theory of weights and balances, where mass ratios can be measured
☆ Links to theories of elasticity, where spring constants can be calculated]
• A: A class of admissible blurs (degrees of approximation admitted between different models).
[The functions occurring in the potential models are complemented by suitable error bars. These may depend on the intended applications, see below.]
• K = <M[p],M,M[pp], C,L,A>: A core (the formal-theoretical part of a theory)
• I: The domain of intended applications (“pieces of the world” to be explained, predicted or technologically manipulated).
[This class is open and contains, for example
☆ systems of small rigid bodies, connected by coil springs or rubber bands
☆ any vibrating mechanical system in the case of small amplitudes, including almost rigid bodies consisting of N molecules]
• T = <K,I>: A theory-element (the smallest unit to be regarded as a theory).
• σ: The specialization relation between theory-elements.
[T could be a specialization of similar theory-elements with more general force laws, e.g., including friction and/or time-dependent external forces. One could also imagine more abstract
force laws which fix only some general properties such as “action=reaction”. T in turn could be specialized to theory-elements of systems with equal masses and/or equal spring constants. ]
• N: A theory-net (a set of theory-elements ordered by σ — the “typical” notion of a theory).
[An obvious theory-net containing our example of a theory-element is CPM = “classical particle mechanics”, conceived as a network of theory-elements essentially ordered by the degree of
generality of its force laws.]
• E: A theory-evolution (a theory-net “moving” through historical time).
[Special interesting new force laws could be discovered in the course of time, e.g., the Toda chain in 1967, as well as new applications of known laws.]
• H: A theory-holon (a complex of theory-nets tied by “essential” links).
[It is difficult to think of examples which are smaller than H = all physical theory-nets.]
4.2.1 History and general traits
Günther Ludwig (1918–2007) was a German physicist mainly known for his work on the foundations of quantum theory. In Ludwig (1970, 1985, 1987), he published an axiomatic account of quantum mechanics,
which was based on the statistical interpretation of quantum theory. As a prerequisite for this work he found it necessary to ask “What is a physical theory?” and developed a general concept of a
theory on the first 80 pages of his (1970). Later this general theory was expanded into the book Ludwig (1978). A recent elaboration of Ludwig's program can be found in Schröter (1996).
His underlying “philosophy” is the view that there are real structures in the world which are “pictured” or represented, in an approximate fashion, by mathematical structures, symbolically PT = W (−)
MT. The mathematical theory MT used in a physical theory PT contains as its core a “species of structure” Σ. This is a meta-mathematical concept of Bourbaki which Ludwig introduced into the
structuralistic approach. The contact between MT to some “domain of reality” W is achieved by a set of correspondence principles (−), which give rules for translating physical facts into certain
mathematical statements called “observational reports”. These facts are either directly observable or given by means of other physical theories, called “pre-theories” of PT. In this way a part G of W
, called “basic domain” is constructed. But it remains a task of the theory to construct the full domain of reality W, that is, the more complete description of the basic domain that also uses PT
-theoretical terms.
4.2.2 Typical features of Ludwig's program
Superficially considered, this concept of theory shows some similarity to neo-positivistic ideas and would be subject to similar criticism. For example, the discussion of the so-called ‘theory-laden’
character of observation sentences casts doubts on such notions as “directly observable facts”. Nevertheless, the adherents of the Ludwig approach would probably argue for a moderate form of
observationalism and would point out that, within Ludwig's approach, the theory-laden character of observation sentences could be analyzed in detail.
Another central idea of Ludwig's program is the description of intra- and inter-theoretical approximations by means of “uniform structures”, a mathematical concept lying between topological and
metrical structures. Although this idea was later adopted by the other structuralistic programs, it plays a unique rôle within Ludwig's meta-theory in connection with his finitism. He believes that
the mathematical structures of the infinitely large or small, a priori, have no physical meaning at all; they are preliminary tools to approximate finite physical reality. Uniform structures are
vehicles for expressing this particular kind of approximation.
4.2.3 Ludwig's late work
One year before his death Ludwig, together with Gérald Thurler, published a revised and simplified edition of Ludwig (1990) with the title “A new foundation of physical theories”. This work cannot be
used as a textbook but it is a remarkable document of the central themes of his approach and his general views on physics. The book clearly shows that Ludwig's main concern is about scientific
realism, i.e., about the question of how hypothetical objects and relations occurring within a successful theory acquire the status of physical reality. Entities which cannot claim this status are
dubbed as “fairy tales” throughout the book. Examples of fairy tales in quantum theory are hidden variables and, perhaps surprising for some readers, also the single-particle-state interpretation (in
contrast to the ensemble interpretation fostered by Ludwig).
Among the new concepts and tools developed in Ludwig/Thurler (2006) are the following:
• Physical observations are first translated into sentences of an auxiliary mathematical theory containing only finite sets, and, in a second step, approximately embedded into an idealized theory.
By this maneuver the authors accentuate the contrast between finite physical operations and mathematical assumptions involving infinite sets.
• Inaccuracy sets and unsharp measurements are always considered right from the start and not introduced later as in previous versions of the Ludwig program.
• The “basic domain” of a theory is now that part of the “application domain” where the theory is successfully applied, up to a certain degree of inaccuracy.
• The complicated terminology concerning various kinds of hypotheses in Ludwig (1990) is radically reduced to a small number of cases including fuzzy hypotheses.
• The problem of unsharp indirect measurements is reformulated in an elegant way which yet should be scrutinized by means of case studies.
4.2.4 Summary
Generally speaking, Ludwig's program is, in comparison to those of Sneed and Scheibe, less descriptive and more normative with respect to physics. He developed an ideal of how physical theories
should be formulated rather than reconstructing the actual practice. The principal worked-out example that comes close to this ideal is still the axiomatic account of quantum mechanics, as described
in Ludwig (1985, 1987).
The German philosopher Erhard Scheibe (1927–2010) has published several books and numerous essays on various topics of philosophy of science; see, for example, Scheibe (2001). He has often commented
on the programs of Sneed and Ludwig, such as in his “Comparison of two recent views on theories”, reprinted in Scheibe (2001, 175–194). Moreover, he published one of the earliest case studies of
approximate theory reduction; see Scheibe 2001 (306–323) for the 1973 case study.
In his books on “reduction of physical theories,” Scheibe (1997, 1999) developed his own concept of theory, which to some extent can be considered an intermediate position between those of Ludwig and
Sneed. For example, he conveniently combines the model-theoretical and syntactical styles of Sneed and Ludwig, respectively. Since his main concern is reduction, he does not need to cover all the
aspects of physical theories that are treated in the other approaches. As already mentioned, he proposes a more flexible concept of reduction that is open to extensions arising from new case studies.
A unique feature of Scheibe's approach is the thorough discussion of almost all the important cases of reduction considered in the physical literature. These include classical vs.
special-relativistic spacetime, Newtonian gravitation vs. general relativity, thermodynamics vs. kinetic theory, and classical vs. quantum mechanics. He essentially arrives at the conclusion of a
double incompleteness: the attempts of the physicists to prove reduction relations in the above cases are largely incomplete according to their own standards, as well as according to the requirements
of a structuralistic concept of reduction. But this concept is also not complete, Scheibe argues, since, for example, a satisfactory understanding of “counter-factual” limiting processes such as ℏ→0
or c→∞ has not yet been developed.
As already noted, the programs of Ludwig and Sneed have been independently developed in the 1970s, whereas Scheibe's program, at least partially, originated from a critical review of these two
programs. But this is only a coarse description. Additionally, there have been numerous mutual interactions between the three programs that influenced their later elaborations. Evidence for this
interaction is provided, besides various pertinent acknowledgements in books and articles, by the following observations.
• Balzer, Moulines and Sneed in their (1987) introduce the concepts of “species of structures” and “uniform structures” that play a central rôle in Ludwig (1970, 1978) and are not yet contained in
Sneed (1971).
• Vice versa, Ludwig in his (1990) added a section 9.3 on theory nets (Theorienetze) citing respective works of Balzer and Moulines.
• In his late (2006) Ludwig on p.3 refers to the work of Scheibe “because of the many similarities”. Later on p.107 he mentions a “discussion through letters” with Scheibe. This correspondence has
been secured by B. Falkenburg and is waiting for a scientific edition.
We have sketched three structuralistic programs which have been developed in the past three decades in order to tackle problems in philosophy of physics, some of which are relevant also for physics
itself. Any program which employs a weighty formal apparatus in order to describe a domain and to solve specific problems has to be scrutinized with respect to the economy of its tools: to what
extent is this apparatus really necessary to achieve its goals? Or is it concerned mainly with self-generated problems? We have tried to provide some arguments and material for the reader who
ultimately has to answer these questions for him- or herself.
This bibliography is mainly restricted to a selection of a few books which are of some importance for the three structuralistic programs. An extended ‘Bibliography of Structuralism’ connected to
Sneed's program appeared in Erkenntnis, Volume 44 (1994). Another recent volume of Erkenntnis (79(8), 2014) is devoted to new perspectives on structuralism. We will cite below a few articles of this
volume that are of relevance for the present entry. Unfortunately, the central books of Ludwig (1978) and Scheibe (1997, 1999) are not yet translated into English, but see the recent Ludwig and
Thurler (2006). For an introduction into the respective theories, English readers could consult chapter XIII of Ludwig (1987) and chapter V of Scheibe (2001).
• Andreas, H., 2014, “Carnapian Structuralism”, Erkenntnis, 79(8): 1373–1391.
• Andreas, H., and Zenker, F., 2014, “Basic Concepts of Structuralism”, Erkenntnis, 79(8): 1367–1372.
• Balzer, W., and Moulines, C. U., 1996, (eds.), Structuralist theory of science, Focal Issues, New Results, Berlin: de Gruyter.
• Balzer, W., and Moulines, C. U., Sneed, J. D., 1987, An Architectonic for Science, Dordrecht: Reidel.
• Bourbaki, N., 1986, Theory of Sets (Elements of Mathematics), Paris: Hermann.
• Gähde, U., 2014, “Theory-dependent determination of Base Sets: Implications for the Structuralist Approach”, Erkenntnis, 79(8): 1459–1473.
• Ludwig, G., 1970, Deutung des Begriffs “physikalische Theorie” und axiomatische Grundlegung der Hilbertraumstruktur der Quantenmechanik durch Hauptsätze des Messens (Lecture Notes in Physics,
Volume 4), Berlin: Springer.
• –––, 1978, Die Grundstrukturen einer physikalischen Theorie, Berlin: Springer; 2nd edition, 1990; French translation by G. Thurler: Les structures de base d'une théorie physique.
• –––, 1985, An Axiomatic Basis for Quantum Mechanics, Vol. 1, Derivation of Hilbert Space Structure, Berlin: Springer.
• –––, 1987, An Axiomatic Basis for Quantum Mechanics (Volume 2: Quantum Mechanics and Macrosystems), Berlin: Springer.
• Ludwig, G. and Thurler, G., 2006, A new foundation of physical theories, Berlin: Springer.
• Scheibe, E., 1997, Die Reduktion physikalischer Theorien, Teil I, Grundlagen und elementare Theorie, Berlin: Springer.
• –––, 1999, Die Reduktion physikalischer Theorien, Teil II, Inkommensurabilität und Grenzfallreduktion, Berlin: Springer.
• –––, 2001, Between Rationalism and Empiricism, Selected Papers in the Philosophy of Physics, B. Falkenburg (ed.), Berlin: Springer.
• Schmidt, H.-J., 1979, Axiomatic Characterization of Physical Geometry (Lecture Notes in Physics, Volume 111), Berlin: Springer.
• Schröter, J., 1996, Zur Meta-Theorie der Physik, Berlin: de Gruyter.
• Schurz, G., 2014, “Criteria of Theoreticity: Bridging Statement and Non-Statement View”, Erkenntnis, 79(8): 1521–1545.
• Sneed, J. D., 1971, The Logical Structure of Mathematical Physics, Dordrecht: Reidel; 2nd edition, 1979.
• Stegmüller, W., 1979, ‘The Structuralist View: Survey, Recent Developments and Answers to Some Criticisms', in The Logic and Epistemology of Scientific Change, I. Niiniluoto and R. Tuomela
(eds.), Amsterdam: North Holland.
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a thesis submitted to for the degree of doctor of philosophy in the faculty of Science, Bangalore University, Bangalore
Atomspheres of the components of close binary stars a thesis submitted to for the degree of doctor of philosophy in the faculty of Science, Bangalore University, Bangalore M. S. Rao [Ph.D Thesis]
Material type: Publication details: Bangalore Indian Institute of Astrophysics 2001Description: x, 178pSubject(s): Online resources: Dissertation note: Doctor of Philosophy Indian Institute of
Astrophysics, Bangalore 2001 Summary: For theoretical modeling of binary systems one has to consider realistic models which takes into account the radiative transfer, hydrodynamics. reflection effect
etc. Since the problem is complex, we study in the thesis some of idealized models which will help us in understanding the important physical processes in close binaries. Intially we have computed
the theoretical lines in the expanding and extended distorted atmospheres of the components of close binary system. I have considered the necessary geometrical formalism for illumination of a stellar
atmospheres from a source. We describe the method to calculate the radiation field from the irradiated surface of the component in a binary system. Chapter 1 Discrete Space theory of Radiative
transfer: In this chapter a concise description of the lnethod of obtaining the solution of radiative transfer equation which can be applied to different geometrical and physical systellls is given.
This method was developed by Grant and Peraiah (1972), and Peraiah and Grant(1973). This chapter deals with (1) interaction principle (2) star product (3) calculation of radiation field at internal
points (4) integration of monochromatic radiative transfer equation and derivation of rand t operators of the" cell". (5) flux conservation and (6) line formation in expanding media. The radiative
transfer equation in spherical symmetry is used for calculating the self radiation of the primary star in a binary system. Chapter 2 Reflection effect in close binaries : The aim of this section is
to estimate the radiation field along the spherical surface of a primary component irradiated by an external point source of radiation. This can be applied to very widely separated systems. The
transfer of radiation incident on the atmosphere of the component from the companion cannot be studied by using any symmetric solution of the equation of transfer. This needs a special treatment. We
adopt angle-free one dimensional model (see Sobolev 1963). Chapter 3 Incident radiation from an extended source: The effects of irradiation from an extended source of the secondary cOInponent on the
atmospheres of the primary are studied. Chapter 4 (1) Effects of reflection on spectral line forlllation: Effects of reflection on formation of spectral lines in a purely scattering atmosphere and
studied how the equivalent widths change when irradiation from the secondary is taken into account. However, these calculations were done in static atnlospheres. So in the next step we have included
the expanding atmospheres. (2) Effects of irradiation on the line formation in the expanding atmospheres of the components of a close binary system : We studied the formation of lines in the
irradiated expanding atmospheres of the component of close binary system. We considered two-level atom approximation in non-LTE situation with complete l'<.'clistribution. We assumed that the dust
scatters isotropically in the atmosphere. The line profiles of the dusty atmosphere are compared with those formed in dust free atmosphere. The profiles are presented for different velocities of
expansion, proximity of secondary component to the primary, and dust optical depths. The line profiles for a dust free atmosphere with and without reflection effects are computed and compared.
Chapter 5 Distorted surface due to self rotation and tidal forces: In this chapter a general expression for gravity darkening of the tidally uniformly rotating roche components of close binary system
is derived. This theory is used to calculate the line profiles taking into account rotation and expansion velocities. Chnpter 6 Effect of gravity darkening on spectral line forlnation : We studied
the transfer of line radiation in the atmospheres of close binary components whose atmospheres are distorted by the self radiation and tidal forces due to the presence of the secondary component. The
distortion is measured in terms of the ratio of angular velocities at the equator and pole, mass ratio of the two components, the ratio of centrifugal force to that of gravity at the equator and the
ratio of the equatorial radius to the distance between the centers of gravity. We obtain the equation of the distorted surface by solving a seventh degree equation which contains the above
parameters. Transfer of line radiation is studied in such asymmetric atmosphere assuming complete redistribution and a two-level atom approximation. The atmosphere is assumed to be expanding
radially. Various black body temperatures are being used to describe the total luminosity of the components for the purpose of irradiation. Chapter 7 Conclusions: We present important results
obtained from this study of research from each chapter.
Item type Current library Shelving location Call number Status Date due Barcode
IIA Library-Bangalore General Stacks (043)524.38 (Browse shelf(Opens below)) Available 15094
Thesis Supervisor Prof. A. Peraiah
Doctor of Philosophy Indian Institute of Astrophysics, Bangalore 2001
For theoretical modeling of binary systems one has to consider realistic
models which takes into account the radiative transfer, hydrodynamics.
reflection effect etc. Since the problem is complex, we study in the thesis
some of idealized models which will help us in understanding the important
physical processes in close binaries. Intially we have computed the
theoretical lines in the expanding and extended distorted atmospheres of
the components of close binary system. I have considered the necessary
geometrical formalism for illumination of a stellar atmospheres from a
source. We describe the method to calculate the radiation field from the
irradiated surface of the component in a binary system.
Chapter 1
Discrete Space theory of Radiative transfer: In this chapter a
concise description of the lnethod of obtaining the solution of radiative
transfer equation which can be applied to different geometrical and physical
systellls is given. This method was developed by Grant and Peraiah
(1972), and Peraiah and Grant(1973). This chapter deals with (1) interaction
principle (2) star product (3) calculation of radiation field at
internal points (4) integration of monochromatic radiative transfer equation
and derivation of rand t operators of the" cell". (5) flux conservation
and (6) line formation in expanding media.
The radiative transfer equation in spherical symmetry is used for calculating
the self radiation of the primary star in a binary system.
Chapter 2
Reflection effect in close binaries : The aim of this section is to
estimate the radiation field along the spherical surface of a primary component
irradiated by an external point source of radiation. This can be
applied to very widely separated systems. The transfer of radiation incident
on the atmosphere of the component from the companion cannot be
studied by using any symmetric solution of the equation of transfer. This
needs a special treatment. We adopt angle-free one dimensional model
(see Sobolev 1963).
Chapter 3
Incident radiation from an extended source: The effects of irradiation
from an extended source of the secondary cOInponent on the
atmospheres of the primary are studied.
Chapter 4
(1) Effects of reflection on spectral line forlllation: Effects of
reflection on formation of spectral lines in a purely scattering atmosphere
and studied how the equivalent widths change when irradiation from the
secondary is taken into account. However, these calculations were done
in static atnlospheres. So in the next step we have included the expanding
(2) Effects of irradiation on the line formation in the expanding
atmospheres of the components of a close binary system : We
studied the formation of lines in the irradiated expanding atmospheres
of the component of close binary system. We considered two-level atom
approximation in non-LTE situation with complete l'<.'clistribution. We
assumed that the dust scatters isotropically in the atmosphere. The line
profiles of the dusty atmosphere are compared with those formed in dust
free atmosphere. The profiles are presented for different velocities of
expansion, proximity of secondary component to the primary, and dust
optical depths.
The line profiles for a dust free atmosphere with and without reflection
effects are computed and compared.
Chapter 5
Distorted surface due to self rotation and tidal forces: In this
chapter a general expression for gravity darkening of the tidally uniformly
rotating roche components of close binary system is derived. This theory
is used to calculate the line profiles taking into account rotation and
expansion velocities.
Chnpter 6
Effect of gravity darkening on spectral line forlnation : We studied
the transfer of line radiation in the atmospheres of close binary components
whose atmospheres are distorted by the self radiation and tidal
forces due to the presence of the secondary component. The distortion
is measured in terms of the ratio of angular velocities at the equator and
pole, mass ratio of the two components, the ratio of centrifugal force to
that of gravity at the equator and the ratio of the equatorial radius to the
distance between the centers of gravity. We obtain the equation of the
distorted surface by solving a seventh degree equation which contains the
above parameters. Transfer of line radiation is studied in such asymmetric
atmosphere assuming complete redistribution and a two-level atom
approximation. The atmosphere is assumed to be expanding radially.
Various black body temperatures are being used to describe the total
luminosity of the components for the purpose of irradiation.
Chapter 7
Conclusions: We present important results obtained from this study
of research from each chapter.
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A gentle introduction to strategic network formation
A gentle introduction to strategic networks formation
When Nash equilibrium is not enough, we have to introduce another kind of stability concept. Playground here (sources included)! ๐
Probabilistic network formation models introduce special network topologies but canโ t explain why does those topologies do actually take form.
The connection model developed by Jackson and Wolinsky in 1996 assigns a utility to every agent (node) in a network. Agents gains benefit (0<๐ <1) from friends, friends of friends (and so onโ ฆ)
and pay a cost (๐ >0) to maintain links.
So every agent has the following payoff:
and we can observe how there is a sort of decay in benefit, as we consider friends (and friends of friendsโ ฆ) and that the cost is taken in account only for direct connections.
In this model, payoffs are symmetrical:
Letโ s introduce a game on networks: in this game, every player announces at the same time the will to form a link with another player.
A Nash equilibrium is a configuration where nobody wants to change his own set of announcements, given the othersโ set of announcements.
If we consider a dyad, itโ s obvious that only two states are a Nash equilibrium: nobody wants to form a link or players both want to form a link. In the first case, if one player does not announce
the will to form a link, the other player knows the links will not form, so is not going to form a link.
In the other case, players both announce the will to form a link and the link is created.
Those two Nash equilibria can predict any kind of situation than can arise and so itโ s not so predictive.
For this reason another concept is introduced, Pairwise Stability, where G is Pairwise stable:
• if nobody wants to sewer a link included in G
• if one player wants to include a link, the other player does not want (otherwise it should be included!)
Back to the dyad example, now both the configurations are Nash stable but only the latter is Pairwise stable.
Pairwise stability can be considered a fairly weak concept, since only considers adding or deleting one link a time.
While Pairwise stability consider individual incentives, efficiency is about overall payoff gain.
A network is Pareto Efficient if does not exists another network gโ such as:
ui(gโ )>=ui(g) for each i, with strict inequality for some i
So basically no network makes someone lose while offering a strictly higher payoff for someone other.
A stronger notion of efficiency is (strong) efficiency where the network maximize the overall agent payoff:
With strong efficiency, no matter if someone is getting better and someone is getting worse, efficiency just assure that that the overall payoff is at is best (and this is also what we generally call
โ utilitarianismโ ).
A network which evolved toward pairwise stability does not imply strong efficiency nor pareto efficiency and this is a consequence of the fact that individuals do not take care of harming other
individuals while trying to improve self payoff.
Since networks will tend to pairwise stability, connection model makes possible to predict which networks will form and to evaluate them.
Letโ s consider strong efficiency depending on delta and cost value, we will observe the following:
• ๐ < ๐ -๐ ยฒ, cost to form a link is very low so the complete network is efficient
• ๐ -๐ ยฒ<๐ <๐ +(n-2)*๐ ยฒ/2, the cost is medium and the star network is efficient
• ๐ +(n-2)*๐ ยฒ/2<๐ , the cost is high so the empty network is efficient
Why stars? Because stars connect individuals at a minimum distances, minimizing indirect link delta loss.
Delta and cost also influence the structure of a pairwise stable network:
• for a low cost ๐ < ๐ -๐ ยฒ, complete network is pairwise stable
• for a medium/low cost ๐ -๐ ยฒ< ๐ < ๐ , star network is pairwise stable (but other networks can be pairwise stable too)
• for a medium/high cost ๐ < ๐ < ๐ + (n-2)๐ ยฒ/2, the cost does not justify bringing only one person in the network with the link so forming links tends to bring in more people to get
indirect benefits to compensate cost. Additionally, every agent forms a link only with agents bringing other connection with them. This is actually the case where a star network is efficient but
not pairwise stable.
• ๐ + (n-2)๐ ยฒ/2< ๐ , the empty network is pairwise stable
References: โ Social and economic networksโ , Mattew O. Jackson, Princeton; Wikipedia, https://en.wikipedia.org/wiki/Strategic_Network_Formation | {"url":"https://1littleendian.medium.com/a-gentle-introduction-to-strategic-network-formation-f341722548f2","timestamp":"2024-11-07T02:40:17Z","content_type":"text/html","content_length":"127760","record_id":"<urn:uuid:0ce2717c-6915-4ea3-b7b8-4f9816617b58>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00796.warc.gz"} |
crew rate calculator
In today’s fast-paced world, accurate calculations are crucial for various fields, from finance to engineering. A crew rate calculator provides a handy tool for determining crew rates, a vital aspect
of project planning and budgeting. In this article, we’ll delve into how to use and implement a crew rate calculator.
How to Use
Using the crew rate calculator is straightforward. Simply input the required parameters, such as the hourly rate, number of crew members, and duration of work, and click the “Calculate” button to
obtain the crew rate.
The formula for calculating crew rate is:
Crew Rate=Hourly Rate × Number of Crew Members × Duration of Work
Example Solve
Let’s say the hourly rate is $50, the number of crew members is 5, and the duration of work is 8 hours:
Crew Rate=50×5×8=$20000
Q: Can the crew rate calculator handle different currencies?
A: Yes, you can input the hourly rate in any currency, and the calculator will compute the crew rate accordingly.
Q: Is it possible to calculate the crew rate for a project spanning multiple days?
A: Absolutely, the duration of work can be adjusted to accommodate projects of varying lengths.
Q: Can I integrate this calculator into my website?
A: Certainly, the provided HTML and JavaScript code can be easily integrated into any web page.
A crew rate calculator is an invaluable tool for project managers, contractors, and freelancers alike. By accurately estimating crew rates, you can streamline budgeting processes and ensure project | {"url":"https://calculatordoc.com/crew-rate-calculator/","timestamp":"2024-11-06T11:29:01Z","content_type":"text/html","content_length":"83269","record_id":"<urn:uuid:c106a489-eeed-468d-b69d-2a033d96ecd4>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00566.warc.gz"} |
Top 10 Tips You MUST Know to Retake the ACCUPLACER Math
The ACCUPLACER test is a comprehensive, web-based assessment tool used to assess your writing, reading, and math skills. It is untimed, but you complete it in less than 90 minutes. All questions must
be answered, and you cannot go back to the previous question and answer. It is an adaptive test, which means the questions get more difficult as you give more right answers. It also means any wrong
answers that make the questions easier.
If you don’t succeed in the ACCUPLACER test, do not worry! Because you can retake the test. The exact number of times you can take ACCUPLACER tests varies depending on where and when you first took
the tests. In some cases, students can get ACCUPLACER four or more times.
Now the main question is what strategies you should follow to prevent failing the ACCUPLACER test again? By following the tips provided here, you can overcome all the barriers to passing the
ACCUPLACER math test.
The Absolute Best Book to Ace the Accuplacer Math Test
Original price was: $24.99.Current price is: $14.99.
1- Review your test results report
The first step to success in ACCUPLACER math is to identify weaknesses and focus on fixing them. By reviewing your ACCUPLACER score report, you can determine where to focus more.
2- Study at regular hours
Regular reading of ACCUPLACER math helps you to constantly improve your grades. Be sure to follow your study plan and know that following it is the key to success.
3- Try different methods of studying math
If your previous ACCUPLACER math study method did not work, you may want to try a newer program. For example, you can use ACCUPLACER math preparation books that you have not read before. This time
you can also use online resources for better learning of ACCUPLACER math.
4- Believe in yourself
The great thing about succeeding in any test is to believe in your ability. If you do your best, then you know you will succeed in ACCUPLACER math. Avoiding negative thinking and self-confidence will
help you focus on passing the ACCUPLACER math test.
5- Quit the habits that led to your failure
Maybe it’s time to reconsider your habits. If you spend a lot of time with your friends, you must limit your time with them. You may need to spend more time studying on the ACCUPLACER math test.
Also, if you do not already have an ACCUPLACER study plan, it is best to plan.
Best Accuplcer Math Prep Resource for 2022
Original price was: $76.99.Current price is: $36.99.
6- Focus on the weaknesses
Adjust your ACCUPLACER retake study plan to focus more on math topics that you have not worked on very well. With a lot of studies, you can increase your score. There are many useful tips for each
part, which you can use to improve your score. And another important point is that you should not give up your strengths and make sure that you practice the topics in which you are strong.
7- Take care of yourself
Health care is very important and should be considered during the ACCUPLACER retake program. You can spend many hours studying and preparing for ACCUPLACER math, but you need to get proper nutrition,
get enough sleep, and exercise. These steps will help you focus more as you prepare for ACCUPLACER math.
8- Get help if you need
Sometimes it is best to get help from a tutor to better understand the material. Tutoring is expensive but worth it. You can also use books that are like a tutor and help you learn the math concepts
of ACCUPLACER.
9- Find a study group
Sometimes it is difficult to do this on your own and study groups can help you improve and motivate. If you are healing in difficult areas and have hit a wall, do not be afraid to seek help from your
peers. Chances are, you are not the only one in your class who chooses the ACCUPLACER, so take advantage of this and join a study group with other classmates.
10- Implement your study plan
Remember that the most important part of ACCUPLACER retake is the commitment to study at regular intervals. Do not leave your review until the last few days before ACCUPLACER math time. It is not a
problem if you sometimes miss study sessions as long as you stick to the general plan.
Given the above, it can be concluded that you should not let a bad test lower your self-esteem. No exam is so difficult that you cannot succeed with enough effort. Always keep in mind that if others
have passed the ACCUPLACER math test, you can too.
Looking for the best resource to help you or your student succeed on the Accuplacer Math test?
The Best Books to Ace the Accuplacer Math Test
Related to This Article
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Simple slopes are not as simple as you think
UNCG Author/Contributor (non-UNCG co-authors, if there are any, appear on document)
Abstract: Simple slopes analysis is commonly used to evaluate moderator or interaction effects in multiple linear regression models. In usual practice, the moderator is treated as a fixed value when
the standard error of simple slopes is estimated. The usual method used for choosing the conditional value of moderator (i.e., at one sample SD below, one SD above, and at the mean) makes the
moderator a random variable and therefore renders the standard error suspect. In this study I examined whether the standard error used in post hoc probing for interaction effect is a biased estimator
of the population variance when moderator is a random variable. I conducted Monte Carlo simulations to evaluate the variance of the simple slope under a variety of conditions corresponding to a 5
(sample size, N) x 5 (variance of focal predictor, x) x 5 (variance of moderator, z) x 4 (levels of r, the correlation between x and z) x 5(model fit, R2) x 4 (population slope for interaction, b
[xz]) factorial design. I present circumstances under which usual practice yields an ”almost” unbiased estimator and conditions when the estimator is more severely biased and less so.
Simple slopes are not as simple as you think
PDF (Portable Document Format)
224 KB
Created on 5/1/2013
Views: 4888
Additional Information
Language: English
Date: 2013
Bias, Simple slopes, Variance
Moderator variables
Psychology $x Statistical methods
Multivariate analysis | {"url":"http://libres.uncg.edu/ir/uncg/listing.aspx?id=9984","timestamp":"2024-11-13T18:45:43Z","content_type":"application/xhtml+xml","content_length":"16865","record_id":"<urn:uuid:f2a73f6b-2fab-4a4a-ac92-dafc5dab966f>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00107.warc.gz"} |
n-localic (infinity,1)-topos
$(\infty,1)$-Topos Theory
An (∞,1)-topos is $n$-localic if
More precisely: if (∞,1)-geometric morphisms into it are fixed by their restriction to the underlying (n,1)-toposes of (n-1)-truncated objects.
To the tower of (n,1)-toposes of (n-1)-truncated objects
$\cdots \to \tau_{\leq 3-1} \mathcal{X} \longrightarrow \tau_{\leq 2-1} \mathcal{X} \longrightarrow \tau_{\leq 1-1} \mathcal{X} \longrightarrow \tau_{\leq 0-1} \mathcal{X} \to *$
of a given (∞,1)-topos $\mathcal{X}$ corresponds a tower of $n$-localic toposes $\mathcal{X}_n$ such that $\tau_{\leq n -1} \mathcal{X} \simeq \tau_{\leq n-1} \mathcal{X}_n$. We may think of the $n$
-localic $\mathcal{X}_n$ as being $n$th stage in the Postnikov tower decomposition of $\mathcal{X}$.
A 0-localic $(1,1)$-topos is a localic topos from ordinary topos theory.
We write (∞,1)Topos for the (∞,1)-category of (∞,1)-toposes and (∞,1)-geometric morphisms between them.
For $\mathcal{X}$ an (∞,1)-topos we denote by
$\tau_{\leq n-1} \mathcal{X} \hookrightarrow \mathcal{X}$
the (n,1)-topos of $(n-1)$-truncated objects of $\mathcal{X}$.
We write $(n,1)Topos$ for the (n+1,1)-category of (n,1)-toposes and $(n,1)$-geometric morphisms between them.
($n$-localic $(\infty,1)$-topos)
An (∞,1)-topos $\mathcal{X}$ is $n$-localic if for any other $(\infty,1)$-topos $\mathcal{Y}$ the canonical morphism
$(\infty,1)Topos(\mathcal{Y},\mathcal{X}) \to (n,1)Topos(\tau_{\leq n-1} \mathcal{Y}, \tau_{\leq n-1}\mathcal{X})$
is an equivalence of (∞,1)-categories (of ∞-groupoids).
More generally,
a (k,1)-topos $\mathcal{X}$ is $n$-localic for $0 \leq n \leq k \leq \infty$ if for any other $(k,1)$-topos $\mathcal{Y}$ the canonical morphism
$(k,1)Topos(\mathcal{Y},\mathcal{X}) \to (n,1)Topos(\tau_{\leq n-1} \mathcal{Y}, \tau_{\leq n-1}\mathcal{X})$
is an equivalence of (∞,1)-categories (of ∞-groupoids).
This is (HTT, def. 6.4.5.8).
This is (HTT, lemma 6.4.5.6).
This is (LurieStructured, lemma 2.3.16).
For $n \in \mathbb{N}$ and $\mathcal{X}$ an $n$-localic $(\infty,1)$-topos, the over-(∞,1)-topos $\mathcal{X}/U$ is $n$-localic precisely if the object $U$ is $n$-truncated.
This is (StrSp, lemma 2.3.14).
For $\mathcal{X}$ an $n$-localic $(\infty,1)$-topos let $U \in \mathcal{X}$ be an object. Then the following are equivalent
1. the restriction of the inverse image $U^* : \mathcal{X} \to \mathcal{X}/U$ (of the etale geometric morphism from the over-(∞,1)-topos) to $(n-1)$-truncated objects is an equivalence of (∞,1)
2. the object $U$ is $n$-connected.
This is (StrSp, lemma 2.3.14).
Every (n,1)-topos $\mathcal{Y}$ is the (n,1)-category of $(n-1)$-truncated objects in an $n$-localic $(\infty,1)$-topos $\mathcal{X}_n$
$\tau_{n-1} X_n \stackrel{\simeq}{\to} \mathcal{Y} \,.$
This is (HTT, prop. 6.4.5.7).
Let $\mathcal{G}$ be a geometry (for structured (∞,1)-toposes).
This is StrSp, lemma 2.6.17
The general noion is the topic of section 6.4.5 of
Remarks on the application of $n$-localic $(\infty,1)$-toposes in higher geometry are in | {"url":"https://ncatlab.org/nlab/show/n-localic+%28infinity,1%29-topos","timestamp":"2024-11-14T02:01:19Z","content_type":"application/xhtml+xml","content_length":"51379","record_id":"<urn:uuid:a61286b4-5dc5-4511-8b6e-519c259bfc83>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00151.warc.gz"} |
Address Calculation:
The Forgotten Sort
Sorting speed is directly proportional to the number of elements
by Douglas Davidson
Most amateur programmers know a few sorting algorithms--bubble sort certainly, probably the maximum-minimum methods, and, on a slightly more advanced level, the shell sort. Some know the more
efficient sorts, such as shuttle or tree sorts. The best of these sorting algorithms require time proportional to the number of elements to sort (n*logn). What is not so well known is a sorting
algorithm--and not a terribly complex one, either--that finishes in a time proportional to n (the number of elements to be sorted). Therefore, for some values of n, this sort must be faster than any
of the other types. It generally goes by the name of "address calculation."
To be fair, some good reasons account for its lack of popularity. First, this method takes more than the minimum necessary amount of memory space to sort any given list; it requires additional
storage proportional to n. However, in most microcomputer BASIC operations, storage requirements are not excessive, and the time savings may out weigh storage considerations. The second and more
fundamental objection is that an address-calculation sort depends on the nature of the sorting keys. Most sorts use the key values only for comparison, simply checking whether one key is greater than
another. This sort uses the actual value of the key.
For example, the address-calculation sort operates by first reserving a large range of memory for storage. It goes through its input list in order and, for each element, uses the key value to
calculate an address within the reserved range. This mapping of keys to addresses is crucial. The operation is most efficient when the mapping is one-to-one (one element to one address), but
practically it will be many-to-one. The only absolute restriction on the mapping is that it be nondecreasing, but it is important to the sort's efficiency that the greatest possible dispersion of the
list elements into the range be achieved, or at least that the fewest possible collisions (mappings of two list elements onto one address) occur. These considerations require knowledge of the range
and distribution of the keys. Because commercial programmers must make sorts as general as possible, address calculation is neglected. If the key distribution differs substantially from the
rectilinear (from an even distribution, such as might be obtained from random generation), then the function to map keys onto addresses must become much more complex. But for microcomputer
programming, often the key distribution is close to random, making the address-calculation sort a good choice.
With the appropriate address calculated, that location is checked to determine its status. If it is empty, the current list element is placed there, and the algorithm continues. If it is already
occupied, then the element must be inserted in such a manner as to maintain proper order. When all list elements have been placed in the range, the program simply reads them off in order, ignoring
unused elements of the range, and places them in the output list.
Test Program
Listing 1 is a formatted listing of an Applesoft version of a test model address-calculation sort. The loop in lines 40 through 80 generates n random integer variables (-32767 to +32767) and prints
them out. The variable I represents the number of locations allocated to the range (more about the 2.36 later). The address-mapping function is a simple linear one; keys are multiplied by a constant
BP to linearly map them onto the range 0 to I. A sort of string variables would compute a numerical value from the first so many characters, weighting them by position. Significantly, the actual list
elements are not placed in the array A%; rather, indexes representing their location in the input list N% are used. A considerable space saving for lists in which the key is not the whole record
results from this approach. A% is dimensioned at I+N (see line 110) to ensure that no element, in the course of being inserted into A%, gets bumped off the upper end. While this wastes space, it
could be avoided with extra programming; however, that would obscure the primary ideas in this example. The main loop goes through the list in order, computing the address V from the key. If the
location is vacant, line 150 places the index there. Otherwise, lines 160 and, 170 insert the index in a higher location. The process produces a "ripple" up the line, smaller elements into place so
that the highest element encountered gets placed in the next vacant location by line 150. Once all the indexes are in place, lines 200 through 230 print the results. You could just as easily place
them in another array. The counter C saves time by halting the printout upon locating all the elements.
Efficiency vs. Speed
I still have not justified my grandiose claims for the sort's speed. While the full mathematical treatment is unnecessary, some discussion is in order. Note first that the time used by the printout
loop remains proportional to the value of I (the number of locations assigned to the range). This provides a motive for keeping I as small as possible, and if I is made proportional to n, then the
time taken by this loop will also be proportional to n. The time taken by the main loop would be proportional to n if there were no collisions (that is, if lines 160 and 170 went unused). The number
of collisions decreases as I increases, providing a reason for wanting I to be as large as possible. Counterbalancing the two considerations shows that the optimum value for I will be proportional to
n; the time taken in the main loop then also turns out to be proportional to n. The actual constants of proportionality depend on the implementation. These arguments are validated experimentally by
figure 1, based on numerous timings of a stripped-down version of listing 1 run on an Apple II Plus. The diagram consists of a line plotted on top of points representing averages of several runs at
near-optimum I. The optimum time turned out to be slightly greater than 9 seconds per 100 n. The optimum value for I was calculated to be about that used in listing 1; namely 2.36*N. Regardless of
the implementation, the optimum ratio of I to n should be about 2.5 +/- .5, with little variation of time within that range.
The address-calculation sorting algorithm provides a fast, not terribly complicated sort for lists the nature of whose keys and distribution is generally known. For special cases it can provide the
most effiecent sorting available.
1. Flores, Ivan. Computer Sorting. Englewood Cliffs. NJ: Prentice-Hall, 1969.
2. Lorin, Harold. Sorting and Sort Systems. Reading, MA: Addison-Wesley, 1975.
Douglas Davidson (1505 Mintwood Dr., McLean, VA 22101) is a high-school senior. His hobbies include computers and astronomy.
Listing 1: The address-calculation sort program. Written for the Apple II computer, the program will generate a list of random numbers, sort the list, and print the sorted list.
10 INPUT N
20 DIM N%(N)
REM *** GENERATE RANDOM NUMBERS
30 HOME : INVERSE : PRINT
40 FOR J = 1 TO N
5O N%(J) = INT (65535 * RND(1)) - 32767
60 PRINT J"."N%(J)
70 NEXT J
80 PRINT : INVERSE : PRINT "SORTED LIST " : NORMAL
REM *** SORT ROUTINE
90 I = 2.36 * N
100 BP = I / 65535
110 DIM A%(I + N)
REM *** MAIN LOOP
120 FOR X = 1 TO N
130 XA = X
140 V = (32767 + N%(X)) * BP
150 IF A%(V) = 0 THEN A%(V) = XA : GOTO 190
l60 IF N%(A%(V)) > N%(XA) THEN XB = XA : XA = A%(V) : A%(V) = XB
170 V = V + 1
180 GOTO 150
190 NEXT X : REM *** PRINTOUT
200 C = 0
210 FOR J = 0 TO I + N
220 IF A%(J) THEN PRINT C"."N%(A%(J)) : C = C + 1
230 IF C <= N THEN NEXT J
240 END
Figure 1: The Address Calculation Response Chart. The amount of time required to sort a list is directly proportional to the number (n) of elements in the list.
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Wolfram Function Repository
Function Repository Resource:
Compute the Hermite decomposition of a matrix of univariate polynomials
Contributed by: Daniel Lichtblau
computes the Hermite decomposition of the matrix mat of univariate polynomials.
computes the Hermite decomposition for a matrix of polynomials in the variable x.
Details and Options
The result is given in the form {u,h} where u is a unimodular matrix, h is an upper‐triangular matrix, and u.mat⩵h.
The Hermite form matrix will have zeros below all pivot elements, and polynomials above a given pivot will have lower degree than that pivot.
A unimodular matrix over a ring of univariate polynomials is a matrix with nonzero determinant lying in the coefficient field (that is, a constant).
The Hermite form is similar to the reduced echelon form, except divisions in the polynomial field are not permitted. Rather than using division to “normalize” pivots to unity, pivot degrees are
reduced using the extended polynomial
operation on pairs of elements in a given matrix column.
Multivariate polynomials are regarded as univariate in the specified variable, with all others treated as symbolic coefficients.
as options.
ResourceFunction["PolynomialHermiteDecomposition"] is intended for matrices of polynomials in a single variable, with all coefficients either exact or approximate numbers.
Basic Examples (3)
Compute the Hermite decomposition of a 2×3 matrix of low-degree polynomials:
Check the matrix equation:
Check that u is unimodular:
Options (6)
Generate and compute the Hermite decomposition for an 8×12 matrix of random degree-5 polynomials with coefficients between −10 and 10:
Computing this by the direct method is comparatively slower:
The situation is typically reversed when one works over a prime field (this is reflected in the Automatic method selection):
Also, the direct method is typically more reliable than the one that uses a Gröbner basis computation:
The Gröbner basis method needs to have higher precision input for this example:
The results agree:
Applications (7)
Polynomial solutions to polynomial systems (7)
If a polynomial solution exists, it can be found using the Hermite decomposition:
Create an underdetermined system using an 8×12 matrix of random degree-5 polynomials with coefficients between -10 and 10, and an eight-dimensional right-hand-side polynomial of the same
Solve this system:
Check the solution:
Note that some components have high degree:
You can get a solution to the system much faster using LinearSolve:
But several components are rational functions rather than polynomials:
Related Links
Version History
Related Resources
Related Symbols
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What is the long-run distribution of stochastic gradient descent? A large deviations analysis
W. Azizian, F. Iutzeler, J. Malick, and P. Mertikopoulos. In ICML '24: Proceedings of the 41st International Conference on Machine Learning, 2024.
In this paper, we examine the long-run distribution of stochastic gradient descent (SGD) in general, non-convex problems. Specifically, we seek to understand which regions of the problem’s state
space are more likely to be visited by SGD, and by how much. Using an approach based on the theory of large deviations and randomly perturbed dynamical systems, we show that the long-run distribution
of SGD resembles the Boltzmann-Gibbs distribution of equilibrium thermodynamics with temperature equal to the method’s step-size and energy levels determined by the problem’s objective and the
statistics of the noise. In particular, we show that, in the long run, (a) the problem’s critical region is visited exponentially more often than any non-critical region; (b) the iterates of SGD are
exponentially concentrated around the problem’s minimum energy state (which does not always coincide with the global minimum of the objective); (c) all other connected components of critical points
are visited with frequency that is exponentially proportional to their energy level; and, finally (d) any component of local maximizers or saddle points is “dominated” by a component of local
minimizers which is visited exponentially more often.
arXiv link: https://arxiv.org/abs/2406.09241
Figure: Loss landscape, critical components, and the long-run distribution of stochastic gradient descent (SGD) for the Himmelblau test function $\hspace{2pt} f(x,y) = (x^2 + y -11)^2 + (x + y^2 - 7) | {"url":"https://polaris.imag.fr/panayotis.mertikopoulos/publications/c90-largedevsgd-icml/","timestamp":"2024-11-07T13:23:43Z","content_type":"text/html","content_length":"9038","record_id":"<urn:uuid:aed5b54b-f4f3-44db-94f0-0d9736cfc3f0>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00141.warc.gz"} |
Re: Re: Why is the negative root?
• To: mathgroup at smc.vnet.net
• Subject: [mg69691] Re: [mg69656] Re: Why is the negative root?
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Thu, 21 Sep 2006 07:29:11 -0400 (EDT)
I forgot to state explicitly (although this should not really be
needed) that all I wrote root isolation applies to numerical root
objects (without parameters). Of course the main reason why
parametric root objects are useful is that upon substituting
numerical values for parameters then turn into numerical root objects
and root isolation is automatically performed. I should have also
mentioned that root objects have one other big advantage over radical
and other representations (particularly the one involving trig
functions): it is very much easier to perform algebraic operations on
them. This comes form the fact that they are defined in terms of
minimum polynomials and there are well known algorithms for
computing, for example, the minimal polynomial of the sum or product
of two algebraic numbers. All such operations are performed by the
function RootReduce, which is called up by Simplify etc.
Needless to say, such operations are much harder and in fact, usually
impossible to carry out without using Root objects or some equivalent
Andrzej Kozlowski
On 20 Sep 2006, at 20:21, Andrzej Kozlowski wrote:
> On 20 Sep 2006, at 15:43, Paul Abbott wrote:
>>> Paul and Andrej and previously Daniel Lichtbau all defend the
>>> Root objects without
>>> telling the whole story.
>> Really? What has been omitted?
>>> In my opinion those objects are just pseudo-useful.
>> Why do you think that?
> Well, actually there is, I think, something "that has been
> omitted". Root objects are not "tautological", "pseudo-useful"
> objects that some imagine them to be, but each in a certain sense
> "embodies" some pretty quite sophisticated computations. The key
> word is "root isolation". The early versions of Mathematica
> actually used to store the isolating informaiton as the third
> argument to Root, but now the third argument is either 1 or 0
> corresponding to whether exact or approximate method or root
> isolation is used, and the relevant information is stored in some
> other way. It is because the roots have been isolated that they can
> be ordered and manipulated in various ways, that is impossible in
> the case of radical expressions. So there is some truth to the
> claim that we have not "told the whole story" but why should we? It
> can be found in any decent book on Computer Algebra (e.g. Chee Keng
> Yap, "Fundamental Problems in Algorithmic Algebra", Princeton
> University Press, Chapter 6 gives all the basic necessary facts.
> You can also look at the standard AddOn package
> "Algebra`RootIsolation`" to see what's involved). Articles about
> the Mathematica implementation of Root objects have appeared more
> than once in The Mathematica Journal. Obviously, this list is not
> the right place for lessons on modern computer algebra. Also,
> concerning "pseudo-usefulness": things like root objects are by no
> means unique to Mathematica but in fact implemented in every
> serious computer algebra system available today (including of
> course Mathematica's main competitors in the CAS area). It's
> curious that all these guys decided to waste so many man-hours
> studying, researching and implementing this useless stuff, not to
> mention writing numerous articles and books about it.
> Andrzej Kozlowski
> TOkyo, Japan | {"url":"http://forums.wolfram.com/mathgroup/archive/2006/Sep/msg00507.html","timestamp":"2024-11-12T05:41:46Z","content_type":"text/html","content_length":"33491","record_id":"<urn:uuid:6e79ace3-06f6-49a5-8321-81df25fc6459>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00082.warc.gz"} |
Can I pay for SAS regression analysis assignment help online? | Hire Someone To Take My SAS Assignment
Can I pay for SAS regression analysis assignment help online? I have helpful resources long list of variables and stats that I need to predict how many points there are going to be between a man and
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system is vulnerable to power usage. The best tool for the | {"url":"https://sashelponline.com/can-i-pay-for-sas-regression-analysis-assignment-help-online","timestamp":"2024-11-11T11:35:02Z","content_type":"text/html","content_length":"127771","record_id":"<urn:uuid:9a059997-4f78-4ea9-8f9c-cfb4aee81f75>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00157.warc.gz"} |
Last modified: June 23, 2024
This article is written in: 🇺🇸
Regularization is a technique used to prevent overfitting in machine learning models, ensuring they perform well not only on the training data but also on new, unseen data.
Overfitting in Machine Learning
• Issue: A model might fit the training data too closely, capturing noise rather than the underlying pattern.
• Effect: Poor performance on new data.
• Manifestation in Regression: Occurs when using higher-degree polynomials which results in a high variance hypothesis.
Example: Overfitting in Logistic Regression
Regularization in Cost Function
Regularization works by adding a penalty term to the cost function that penalizes large coefficients, thereby reducing the complexity of the model.
Regularization in Linear Regression
• Regularized Cost Function:
$$ \min \frac{1}{2m} \left[ \sum_{i=1}^{m}(h_{\theta}(x^{(i)}) - y^{(i)})^2 + \lambda \sum_{j=1}^{m} \theta_j^2 \right] $$
• Penalization: Large values of $\theta_3$ and $\theta_4$ are penalized, leading to simpler models.
Regularization Parameter: $\lambda$
• Role of $\lambda$: Controls the trade-off between fitting the training set well and keeping the model simple (smaller parameter values).
• Selection: Automated methods can be used to choose an appropriate $\lambda$.
Modifying Gradient Descent
The gradient descent algorithm can be adjusted to include the regularization term:
I. For $\theta_0$ (no regularization):
$$ \frac{\partial}{\partial \theta_0} J(\theta) = \frac{1}{m} \sum_{i=1}^{m} (h_{\theta}(x^{(i)}) - y^{(i)})x_0^{(i)} $$
II. For $\theta_j$ ($j \geq 1$):
$$ \frac{\partial}{\partial \theta_j} J(\theta) = \left( \frac{1}{m} \sum_{i=1}^{m} (h_{\theta}(x^{(i)}) - y^{(i)})x_j^{(i)} \right) + \frac{\lambda}{m}\theta_j $$
Regularized Linear Regression
Regularized linear regression incorporates a regularization term in the cost function and its optimization to control model complexity and prevent overfitting.
Gradient Descent with Regularization
To optimize the regularized linear regression model using gradient descent, the algorithm is adjusted as follows:
while not converged:
for j in [0, ..., n]:
θ_j := θ_j - α [ \frac{1}{m} \sum_{i=1}^{m}(h_{θ}(x^{(i)}) + y^{(i)})x_j^{(i)} + \frac{λ}{m} θ_j ]
Here is the Python code to demonstrate regularization in linear regression, including the regularized cost function and gradient descent with regularization. This example uses numpy to implement the
regularized linear regression model:
import numpy as np
def hypothesis(X, theta):
return np.dot(X, theta)
def compute_cost(X, y, theta, lambda_reg):
m = len(y)
h = hypothesis(X, theta)
cost = (1 / (2 * m)) * (np.sum((h - y) ** 2) + lambda_reg * np.sum(theta[1:] ** 2))
return cost
def gradient_descent(X, y, theta, alpha, lambda_reg, num_iters):
m = len(y)
cost_history = np.zeros(num_iters)
for iter in range(num_iters):
h = hypothesis(X, theta)
theta[0] = theta[0] - alpha * (1 / m) * np.sum((h - y) * X[:, 0])
for j in range(1, len(theta)):
theta[j] = theta[j] - alpha * ((1 / m) * np.sum((h - y) * X[:, j]) + (lambda_reg / m) * theta[j])
cost_history[iter] = compute_cost(X, y, theta, lambda_reg)
return theta, cost_history
# Example usage with mock data
X = np.random.rand(10, 2) # Feature matrix (10 examples, 2 features)
y = np.random.rand(10) # Target values
# Adding a column of ones to X for the intercept term (theta_0)
X = np.hstack((np.ones((X.shape[0], 1)), X))
# Initial parameters
theta = np.random.randn(X.shape[1])
alpha = 0.01 # Learning rate
lambda_reg = 0.1 # Regularization parameter
num_iters = 1000 # Number of iterations
# Perform gradient descent with regularization
theta, cost_history = gradient_descent(X, y, theta, alpha, lambda_reg, num_iters)
print("Optimized parameters:", theta)
print("Final cost:", cost_history[-1])
Regularization with the Normal Equation
In the normal equation approach for regularized linear regression, the optimal $θ$ is computed as follows:
The equation includes an additional term $λI$ to the matrix being inverted, ensuring regularization is accounted for in the solution.
Here is the Python code to implement regularized linear regression using the normal equation:
import numpy as np
def regularized_normal_equation(X, y, lambda_reg):
m, n = X.shape
I = np.eye(n)
I[0, 0] = 0 # Do not regularize the bias term (theta_0)
theta = np.linalg.inv(X.T @ X + lambda_reg * I) @ X.T @ y
return theta
# Example usage with mock data
X = np.random.rand(10, 2) # Feature matrix (10 examples, 2 features)
y = np.random.rand(10) # Target values
# Adding a column of ones to X for the intercept term (theta_0)
X = np.hstack((np.ones((X.shape[0], 1)), X))
# Regularization parameter
lambda_reg = 0.1
# Compute the optimal parameters using the regularized normal equation
theta = regularized_normal_equation(X, y, lambda_reg)
print("Optimized parameters using regularized normal equation:", theta)
Regularized Logistic Regression
The cost function for logistic regression with regularization is:
$$ J(θ) = \frac{1}{m} \sum_{i=1}^{m}[-y^{(i)}\log(h_{θ}(x^{(i)})) - (1-y^{(i)})\log(1 - h_{θ}(x^{(i)}))] + \frac{λ}{2m}\sum_{j=1}^{n}θ_j^2 $$
Gradient of the Cost Function
The gradient is defined for each parameter $θ_j$:
I. For $j = 0$ (no regularization on $θ_0$):
$$ \frac{\partial}{\partial θ_0} J(θ) = \frac{1}{m} \sum_{i=1}^{m} (h_{θ}(x^{(i)}) - y^{(i)})x_j^{(i)} $$
II. For $j ≥ 1$ (includes regularization):
$$ \frac{\partial}{\partial θ_j} J(θ) = ( \frac{1}{m} \sum_{i=1}^{m} (h_{θ}(x^{(i)}) - y^{(i)})x_j^{(i)} ) + \frac{λ}{m}θ_j $$
For both linear and logistic regression, the gradient descent algorithm is updated to include regularization:
while not converged:
for j in [0, ..., n]:
θ_j := θ_j - α [ \frac{1}{m} \sum_{i=1}^{m}(h_{θ}(x^{(i)}) + y^{(i)})x_j^{(i)} + \frac{λ}{m} θ_j ]
The key difference in logistic regression lies in the hypothesis function $h_{θ}(x)$, which is based on the logistic (sigmoid) function.
Here is the Python code to implement regularized logistic regression using gradient descent:
import numpy as np
from scipy.special import expit # Sigmoid function
def sigmoid(z):
return expit(z)
def compute_cost(X, y, theta, lambda_reg):
m = len(y)
h = sigmoid(np.dot(X, theta))
cost = (1 / m) * np.sum(-y * np.log(h) - (1 - y) * np.log(1 - h)) + (lambda_reg / (2 * m)) * np.sum(theta[1:] ** 2)
return cost
def gradient_descent(X, y, theta, alpha, lambda_reg, num_iters):
m = len(y)
cost_history = np.zeros(num_iters)
for iter in range(num_iters):
h = sigmoid(np.dot(X, theta))
error = h - y
theta[0] = theta[0] - alpha * (1 / m) * np.sum(error * X[:, 0])
for j in range(1, len(theta)):
theta[j] = theta[j] - alpha * ((1 / m) * np.sum(error * X[:, j]) + (lambda_reg / m) * theta[j])
cost_history[iter] = compute_cost(X, y, theta, lambda_reg)
return theta, cost_history
# Example usage with mock data
X = np.random.rand(10, 2) # Feature matrix (10 examples, 2 features)
y = np.random.randint(0, 2, 10) # Binary target values
# Adding a column of ones to X for the intercept term (theta_0)
X = np.hstack((np.ones((X.shape[0], 1)), X))
# Initial parameters
theta = np.random.randn(X.shape[1])
alpha = 0.01 # Learning rate
lambda_reg = 0.1 # Regularization parameter
num_iters = 1000 # Number of iterations
# Perform gradient descent with regularization
theta, cost_history = gradient_descent(X, y, theta, alpha, lambda_reg, num_iters)
print("Optimized parameters:", theta)
print("Final cost:", cost_history[-1])
These notes are based on the free video lectures offered by Stanford University, led by Professor Andrew Ng. These lectures are part of the renowned Machine Learning course available on Coursera. For
more information and to access the full course, visit the Coursera course page. | {"url":"https://adamdjellouli.com/articles/stanford_machine_learning/07_regularization","timestamp":"2024-11-04T07:06:42Z","content_type":"text/html","content_length":"23310","record_id":"<urn:uuid:26a93fc4-9338-46c7-9b08-f5b0c826f7f9>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00636.warc.gz"} |
Unexpected Market Volatility as a Market Return Predictor
Do upside (downside) market volatility surprises scare investors out of (draw investors into) the stock market? In the November 2013 version of his paper entitled “Dynamic Asset Allocation Strategies
Based on Unexpected Volatility”, Valeriy Zakamulin investigates the ability of unexpected stock market volatility to predict future market returns. He calculates stock market index volatility for a
month using daily returns. He then regresses monthly volatility versus next-month volatility to predict next-month volatility. Unexpected volatility is the series of differences between predicted and
actual monthly volatility. He tests the ability of unexpected volatility to predict stock market returns via regression tests and two market timing strategies. One strategy dynamically weights
positions in a stock index and cash (the risk-free asset) depending on the prior-month difference between actual and past average unexpected index volatility. The other strategy holds a 100% stock
index (cash) position when the prior-month difference between actual and average past unexpected index volatility is negative (positive). His initial volatility prediction uses the first 240 months
of data, and subsequent predictions use inception-to-date data. He ignores trading frictions involved in strategy implementation. Using daily and monthly (approximated) total returns of the S&P 500
Index and the Dow Jones Industrial Average (DJIA), along with the U.S. Treasury bill (T-bill) yield as the return on cash, during January 1950 through December 2012, he finds that:
• Over the entire sample period, unexpected S&P 500 Index (DJIA) volatility relates negatively to future index returns, with an R-squared statistic of 0.02 (0.01). In other words, monthly variation
in unexpected volatility explains 1-2% of the variation in next-month index return. However, the relationship flips to positive during the 1990s.
• Applied to the S&P 500 Index, the following three strategies generate annualized gross Sharpe ratios of 0.57, 0.43 and 0.36, respectively:
1. Each month weight stocks according to the prior-month difference between actual and average past unexpected volatility divided by the standard deviation of past unexpected volatility
(calibrated to 50% when the difference is zero).
2. A similar strategy based on total rather than unexpected volatility.
3. A benchmark strategy consisting of 50% index-50% cash, rebalanced monthly.
• Applied to the S&P 500 Index, the following three strategies generate annualized gross Sharpe ratios of 0.54, 0.47 and 0.36, respectively (see the chart below):
1. Each month, hold the index (cash) if the prior-month difference between actual and average past unexpected volatility is negative (positive).
2. Each month, hold the index (cash) if the index is above (below) its SMA10 at the end last month.
3. A benchmark strategy consisting of 75% index-25% cash, rebalanced monthly, specified to approximate the volatility of the SMA10 strategy. [This is a separate clarification from the author.]
The following chart, taken from the paper, tracks gross cumulative values of $100 initial investments at the end of 1969 in each of the second set of three strategies above. Specifically, the
competing strategies are:
• Passive: 75% stock index-25% cash, rebalanced monthly.
• SMA10: each month, hold the S&P 500 Index (cash) if it is above (below) its SMA10 at the end of last month.
• UnexVol: each month, hold the S&P 500 Index (cash) if the prior-month difference between actual and average past unexpected volatility is negative (positive).
The two timing strategies beat buy-and-hold based on both gross terminal value and gross risk-adjusted monthly and annual performance. While the two timing strategies have similar gross terminal
values, the one based on unexpected volatility generates the steadier monthly and annual performance.
In summary, evidence suggests that unexpected stock market volatility may usefully predict market performance at a monthly horizon.
Cautions regarding findings include:
• All performance metrics are gross, not net. Including trading frictions, which are high during part of the sample period, would lower reported returns.
• The passive benchmarks in the study are arguably unrealistic, since an investor using them would incur monthly trading frictions for rebalancing. A fixed 100% index allocation is more realistic.
• Moreover, it seems likely that the strategies based on unexpected volatility trade more frequently than that based on SMA10, such that gross outperformance may not mean net outperformance. [The
author separately reports that the SMA10 (unexpected volatility) strategy applied to the S&P 500 Index generates 58 (145) trades during 1970-2012. Compounded friction is therefore substantial,
and substantially lower for SMA10 than for unexpected volatility.]
• The study uses indexes rather than tradable assets, thereby ignoring any costs of creating and maintaining tracking funds.
• The methodology assumes zero delay between signal generation and execution, which may be problematic for the relatively complex calculations that determine allocations based on unexpected | {"url":"https://www.cxoadvisory.com/volatility-effects/unexpected-market-volatility-as-a-market-return-predictor/","timestamp":"2024-11-10T02:54:30Z","content_type":"application/xhtml+xml","content_length":"145187","record_id":"<urn:uuid:61463268-b342-4f30-ac27-a27b588e58d3>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00568.warc.gz"} |
Jakob Ablinger - RISC - Johannes Kepler University
author = {J. Ablinger and A. Behring and J. Bluemlein and A. De Freitas and A. von Manteuffel and C. Schneider and K. Schoenwald},
title = {{The first--order factorizable contributions to the three--loop massive operator matrix elements $A_{Qg}^{(3)}$ and $Delta A_{Qg}^{(3)}$}},
language = {english},
abstract = {The unpolarized and polarized massive operator matrix elements $A_{Qg}^{(3)}$ and $Delta A_{Qg}^{(3)}$contain first--order factorizable and non--first--order factorizable contributions in
the determining difference or differential equations of their master integrals. We compute their first--order factorizable contributions in the single heavy mass case for all contributing Feynman
diagrams. Moreover, we present the complete color--$zeta$ factors for the cases in which also non--first--order factorizable contributions emerge in the master integrals, but cancel in the final
result as found by using the method of arbitrary high Mellin moments. Individual contributions depend also on generalized harmonic sums and on nested finite binomial and inverse binomial sums in
Mellin $N$--space, and correspondingly, on Kummer--Poincar'e and square--root valued alphabets in Bjorken--$x$ space. We present a complete discussion of the possibilities of solving the present
problem in $N$--space analytically and we also discuss the limitations in the present case to analytically continue the given $N$--space expressions to $N in mathbb{C}$ by strict methods. The
representation through generating functions allows a well synchronized representation of the first--order factorizable results over a 17--letter alphabet. We finally obtain representations in terms
of iterated integrals over the corresponding alphabet in $x$--space, also containing up to weight {sf w = 5} special constants, which can be rationalized to Kummer--Poincar'e iterated integrals at
special arguments. The analytic $x$--space representation requires separate analyses for the intervals $x in [0,1/4], [1/4,1/2], [1/2,1]$ and $x > 1$. We also derive the small and large $x$ limits of
the first--order factorizable contributions. Furthermore, we perform comparisons to a number of known Mellin moments, calculated by a different method for the corresponding subset of Feynman
diagrams, and an independent high--precision numerical solution of the problems.},
journal = {Nuclear Physics B},
volume = {999},
number = {116427},
pages = {1--42},
isbn_issn = {ISSN 0550-3213},
year = {2024},
note = {arXiv:2311.00644 [hep-ph]},
refereed = {yes},
keywords = {Feynman diagram, massive operator matrix elements, computer algebra, differential equations, difference equations, coupled systems, nested integrals, nested sums},
length = {42},
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How Statistical Correlation and Causation Are Different - dummies
Of all of the misunderstood statistical issues, the one that’s perhaps the most problematic is the misuse of the concepts of correlation and causation. Correlation, as a statistical term, is the
extent to which two numerical variables have a linear relationship (that is, a relationship that increases or decreases at a constant rate). Following are three examples of correlated variables:
• The number of times a cricket chirps per second is strongly related to temperature; when it’s cold outside, they chirp less frequently, and as the temperature warms up, they chirp at a steadily
increasing rate. In statistical terms, you say the number of cricket chirps and temperature have a strong positive correlation.
• The number of crimes (per capita) has often been found to be related to the number of police officers in a given area. When more police officers patrol the area, crime tends to be lower, and when
fewer police officers are present in the same area, crime tends to be higher. In statistical terms we say the number of police officers and the number of crimes have a strong negative
• The consumption of ice cream (pints per person) and the number of murders in New York are positively correlated. That is, as the amount of ice cream sold per person increases, the number of
murders increases. Strange but true!
But correlation as a statistic isn’t able to explain why or how the relationship between two variables, x and y, exists; only that it does exist.
Causation goes a step further than correlation, stating that a change in the value of the x variable will cause a change in the value of the y variable. Too many times in research, in the media, or
in the public consumption of statistical results, that leap is made when it shouldn’t be. For instance, you can’t claim that consumption of ice cream causes an increase in murder rates just because
they are correlated. In fact, the study showed that temperature was positively correlated with both ice cream sales and murders. When can you make the causation leap? The most compelling case is when
a well-designed experiment is conducted that rules out other factors that could be related to the outcomes.
You may find yourself wanting to jump to a cause-and-effect relationship when a correlation is found; researchers, the media, and the general public do it all the time. However, before making any
conclusions, look at how the data were collected and/or wait to see if other researchers are able to replicate the results (the first thing they try to do after someone else’s “groundbreaking result”
hits the airwaves).
About This Article
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How many hours is a week?
In a week there are 168 hours.
In a day there are 24 hours and in a week there are 7 days so to get the amount of hours in a week we take the 24 hours and multiply it by 7 which gives us 168.
So that means there's 168 hours in a weeks time.
In every minute there are 60 seconds and there are 60 minutes in every hour which means there are 3,600 seconds in an hour.
1 hour = 60 minutes and 1 minute = 60 seconds so 60 minutes/ hour * 60 seconds/ minute = 3600 seconds/ hour or 1 hour = 3600 seconds.
In 2 days there are 172,800 seconds.
In a day there are 86,400 seconds.
We also know there are 24 hours in one day. If we multiply all of those, we end up with 86,400 seconds in 1 day.
In a day or in one day there are 1,440 minutes and in a day there are also 24 hours.
In 12 years there are 378,432,000 seconds.
The conversion of 12 years to seconds is calculated by multiplying 12 years by 31,536,000 and the result is 378,432,000 seconds.
In a year of 365 days there are 31,536,000 seconds.
1 year, 12 months, 365 days, 8760 hours, 525,600 minutes, 31,536,000 seconds.
That seems like a lot of time but that time goes by really quick.
In a year there are 12 months.
There are also 365 days in a year and always 12 months in each year.
In 10 years there are 120 months and 3,650 days.
In 5 years there are 60 months and 1,825 days.
And in a year there are 525,600 minutes.
Also it is easiest to determine and work with the total number of days in a year, which is 365.
To arrive at the answer, all you need to do is to multiply the number of minutes in a single hour with the total number of hours in a day.
i.e. 1440 (minutes) x 365 = 525,600 minutes (in a year).
On Earth, a solar day is around 24 hours. However, Earth's orbit is elliptical, meaning it's not a perfect circle.
That means some solar days on Earth are a few minutes longer than 24 hours and some are a few minutes shorter.
On Earth, a sidereal day is almost exactly 23 hours and 56 minutes.
There would be approximately 52560000 minutes in hundred years and 40471200 minutes in 77 years.
1 min =60 sec. 1 Day=1440 min.
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Effect of engine thrust on nonlinear flutter of wings
The propulsion of wing-mounted engine is a typical follower force and may cause significant influences upon wing flutter characteristics. An integrated flutter analysis method has been presented,
within which the effects of engine thrusts and geometrical nonlinearities are both considered. Firstly the method has been applied to evaluate the effects of thrusts on the flutter boundary of a
high-altitude, long-endurance aircraft wing. The numerical results have an excellent agreement with the published ones. Furthermore the finite element model of a wing carrying two engines has been
established, and the influences of propulsion magnitude and position on wing flutter speed are mainly investigated. The results indicated that the effects of engine thrusts are indispensable for wing
flutter analysis.
1. Introduction
Modern transporters usually have high-aspect-ratio wings under which the engines are mounted. For this configuration the engine thrusts will couple with the aerodynamics and structure deflection, and
this kind of coupling could cause aeroelastic instability. This problem was first investigated by W. T. Feldt [1], where the influences of thrust value on wing flutter speed were presented. The
aeroelastic stability of a high-altitude, long-endurance wing subjected to a lateral follower force has been studied by M. J. Patil and D. H. Hodges [2, 3]. The ratio of bending stiffness to
torsional stiffness and the value of thrust were considered as key parameters, and their effects on flutter speed and frequency were investigated. It is indicated that under the action of an actual
thrust, which was obtained by full-aircraft trimming for a real flight condition, the predicted flutter speed can be changed up to 11 %. S. A. Fazelzadeh and A. Mazidi [4] have studied the
bending-torsional flutter characteristics of a wing containing an arbitrarily placed mass subjected to a follower force. Most recently the effects of wing sweep and dihedral angle were also
considered by Zhang Jian and Xiang Jingwu [5].
Such problems are much more complicated than conventional aeroelastic analysis, so in the previous literatures wings were all modeled as slender beams. But for a real plane the wing structure is
complicated and actually can’t be modeled by a beam accurately. A common method is using the FEM software to establish the structure model [6]. So far, to the author’s knowledge, the researches which
study the effects of the engine thrusts on the aeroelastic characteristics of a real, complicated wing structure have not yet been seen.
Based on the secondary development of the MSC/Nastran software with its DMAP language, its static aeroelastic analysis module, nonlinear static analysis module and flutter analysis module were
incorporated into an integrated nonlinear flutter analysis procedure. This method can be used for any metal or composite wings with complicated structure and configurations, and the effects of
initial angle of attack, external stores, engine thrusts and the geometric nonlinearities are all considered simultaneously.
2. Aerodynamic theory
Since the aim of the present work is to study the subsonic aeroelastic stability of transporters, the non-planar effect of the aerodynamic load is very small and will be ignored here [9, 10].
Considering the design requirements, such delicate design won’t allow airplane to be operated under large angle of attack or even encounter dynamic stall. So the subsonic doublet-lattice theory is
adopted here for the flutter analysis [11].
The wing’s lifting surface is discretized and divided into trapezoidal panels, and then the aerodynamic influence coefficients (AICs) are calculated. The non-dimensional downwash velocity at the
collocation points can be written in terms of AICs as:
$\mathbf{w}=\mathbf{A}\left(k,{M}_{\infty }\right)\mathbf{f}/q,$
where $\mathbf{f}$ are the non-dimensional pressures of the lifting elements, $q$ is the dynamic pressure of free stream. $\mathbf{A}$ is the AICs matrix, which is a function of reduced frequency $\
left(k=\omega \text{\hspace{0.17em}}b/V\right)$ and the free stream Mach number ${M}_{\infty }$. $\omega$ is the oscillation frequency, $b$ is the semi-chord length and $V$ is the free stream
The forces and moments on the wing can be calculated by integrating the pressures over each lifting element, i. e.:
${\mathbf{F}}_{A}=\mathbf{S}\cdot \mathbf{f},$
where $\mathbf{S}$ is the integrating matrix.
In terms of the initial angle of attack of the undeformed wing and the structural deflections, the downwash at collocation points can be written as Eq. (3) explicitly:
where $\mathbf{x}$ are the displacement vector of structural nodes, $\mathbf{W}\left(\mathbf{x}\right)$ is the transformation matrix from the structural deflections to the downwash at collocation
points. ${\mathbf{w}}_{0}$ is the downwash at collocation points caused by the initial angle of attack.
Thus, according to Eq. (1)-(3), the aerodynamic lift and moment can be rewritten in terms of the structural deflections as:
3. Static aeroelastic analysis
After building the nonlinear FEM model of the wing structure and coupling it with aerodynamic model, the steady AICs matrix ${\mathbf{A}}_{S}$ and the transformation matrix ${\mathbf{W}}_{S}\left(\
mathbf{x}\right)$ for each Mach number can be extracted from the static aeroelastic analysis module by the DMAP tool. The static aerodynamic lift and moment can be written as:
Then using the nonlinear static analysis module, the structural nonlinear equilibrium equation of the restoring force, aerodynamic load and engine thrusts can be written as:
where $g\left(\mathbf{x}\right)$ is the restoring force due to the structural nonlinear deformation. $\mathbf{P}$ is the engine thrust, which is a follower force. The structure deformation under the
action of aerodynamic load and engine thrusts can be obtained by solving Eq. (6), which is a nonlinear algebraic equation and needs to be solved by iterative loops.
4. Flutter analysis
Ignoring the structural damping, the nonlinear aeroelastic equation of wing can be written as:
The solution for the above equation can be assumed as follows:
where $\stackrel{-}{\mathbf{x}}$ is the steady state value obtained by solving Eq. (6), $\stackrel{^}{\mathbf{x}}$ is the small perturbation with respect to this equilibria. Therefore the aeroelastic
equation can be linearized at the steady state, given:
$\stackrel{-}{\mathbf{M}}\text{\hspace{0.17em}}\stackrel{¨}{\stackrel{^}{\mathbf{x}}}+\stackrel{-}{\mathbf{K}}\text{ }\stackrel{^}{\mathbf{x}}-q\mathbf{S}{\mathbf{A}}^{-1}\text{ }\mathbf{W}\left(\
where $\stackrel{-}{\mathbf{M}}$ and $\stackrel{-}{\mathbf{K}}$ are the mass and stiffness matrixes at the steady state $\stackrel{-}{\mathbf{x}}$. The perturbation of engine thrust can be rewritten
$\mathbf{P}\left(\stackrel{^}{\mathbf{x}}\right)={\stackrel{-}{\mathbf{K}}}_{P}\text{ }\stackrel{^}{\mathbf{x}},$
where ${\stackrel{-}{\mathbf{K}}}_{P}$ is the stiffness matrix of the follower force generated by engine thrusts.
The normal modes were calculated for the steady state and then the appropriate modes were chosen to reduce the equation. The aeroelastic equation can be rewritten via the generalized coordinates as:
where $\mathbf{q}$ is the generalized coordinates vector, ${\mathbf{M}}_{q}$ is the generalized mass matrix, ${\mathbf{K}}_{q}$ is the generalized stiffness matrix, ${\mathbf{K}}_{Pq}$ is the
generalized stiffness matrix of follower force, ${\mathbf{F}}_{Aq}$ are the generalized aerodynamic loads.
The above matrices were then incorporated into the MSC/Nastran flutter analysis module by DMAP tool and the $pk$ method is applied to perform flutter analysis. The unmatched flutter velocity ${V}_{F}
$ and the corresponding flutter Mach number are:
${M}_{\infty F}={V}_{F}/{c}_{\infty },$
where ${c}_{\infty }$ is the local velocity of sound. Comparing ${M}_{\infty F}$ to the initial free stream Mach number ${M}_{\infty \text{\hspace{0.17em}}}$ given at the beginning of the static
aeroelastic analysis: if ${M}_{\infty F}={M}_{\infty }$, then ${V}_{F}$ is the matched nonlinear flutter speed. Otherwise reset the free stream Mach number and solve Eq. (6)-(12) again, until the
error is converged.
5. Analysis procedures
The analysis diagram of present work is shown in Figure 1. Flight condition parameters such as angle of attack, free stream Mach number and engine thrust are given before the analysis. Under this
flight condition the steady AICs matrix and the transform matrix are obtained using the static aeroelastic module (Sol 144). Then the structural deflections under the action of aerodynamic loads and
the engine thrusts are calculated by means of the nonlinear static analysis module (Sol 106).
In order to approach the steady state, the steady aerodynamic loads are computed again according to the new structural deflections. Thus iterative loops are applied to make the aerodynamic loads
match the structural deflections, which finally turn into the actual equilibrium point of the wing. Reduce the equations at the steady state, incorporate the generalized mass and stiffness matrix
into the flutter analysis module (Sol 145) and calculate the flutter speed. Repeat the above procedures until the difference between ${M}_{\infty }$ and ${M}_{\infty F}$ is converged and the
corresponding flutter speed is defined as the nonlinear flutter speed for this flight condition.
Fig. 1Diagram of nonlinear flutter analysis
6. Example 1: HALE wing
The first model to be studied is a high-altitude long-endurance wing and its structural properties and flight conditions are given in Table 1. Its finite element model is established through MSC/
Patran software. The wing’s structure is a slender beam which was divided into 32 nonlinear beam elements, and its mass is modeled by 32 lumped mass elements. And 32×5 lifting elements were used for
the aerodynamic panel. The wing root is clamped support and a lateral follower force is applied at the position of 15 m from the root in span. The initial angle of attack was not considered in this
example. More details of the wing structure can be seen in reference [2].
The parameter $\lambda$, non-dimensional thrust $P$ and non-dimensional flutter speed $v$ were defined as:
$\lambda =\frac{E{I}_{2}}{GJ},$
$v=\frac{{V}_{F}}{b{\omega }_{\theta }},$
where $p$ is the engine thrust, $l$ is the half span length, ${V}_{F}$ is the flutter speed, $b$ is the semi-chord length and ${\omega }_{\theta }$ is the first uncoupled torsional frequency.
Using the nonlinear flutter analysis method presented above, the influences of parameters $\lambda$ and $P$ on flutter speed were investigated.
Table 1HALE wing data
Structure data
Half span 16 m
Chord 1 m
Mass per unit length 0.75 kg/m
Moment of inertia 0.1 kg-m
Spanwise elastic axis 50 % chord
Center of gravity of wing 50 % chord
Bending rigidity (spanwise) 2e4 N-m^2
Bending rigidity (chordwise) 4e4 N-m^2
Torsional rigidity Varies with $\lambda$
Flight condition
Altitude 20 km
Air density 0.889 kg/m^3
The $vg$ and $vf$ locus of HALE wing at the condition of $\lambda =$2 and $P=$2 are shown in Figure 2. When the damping $g$ of the first torsional mode crosses the zero axes, the wing is at the
critical condition of flutter which means aeroelastic instability. It is seen from Figure 2 that the flutter speed and frequency are 35.3 m/s and 3.76 Hz respectively. This instability is caused by
the coupling of the first torsional mode and the second flap bending mode.
Figure 3 shows the $vg$ and $vf$ curves at the condition of $\lambda =$2 and $P=$3. It is seen that there are two instability branches. The flutter speed and frequency are 35.4 m/s and 1.5 Hz, and
the flutter modes are the first and second flap bending modes. In contrast with Figure 3, the component of torsional deformation in the flap bending mode increases as the engine thrusts increase,
which could result in the variation of flutter form and consequently change flutter speed remarkably.
Fig. 2vg and vf curves at λ= 2, P= 2
Fig. 3vg and vf curves at λ= 2, P= 3
The effect of non-dimensional thrust on flutter boundary for several values of $\lambda$ is illustrated in Figure 4. For low levels of thrust the flutter speed increases slightly as thrust goes up.
But if the thrust level continues to increase, the curve displays an inflection point which indicates the variation of flutter mode, and this variation leads to a sharp decrease of the flutter speed.
The inflection point moves down as $\lambda$ increases and disappears gradually while $\lambda$ is larger than 5.
Furthermore, for the purpose of method verification, present results are compared in Figure 4 with reference [2] and a good agreement can be achieved. Small differences are observed and can be
explained by the fact that the doublet-lattice theory is used here instead of finite-state aerodynamic model used in reference [2]. But the errors are very small and can be ignored in engineering.
This simulation example demonstrates the validity of the developed nonlinear flutter analysis procedures.
Fig. 4Effects of λ variation on flutter speed
7. Example 2: a transporter wing with two engines
The finite element model of a wing carrying two engines is shown in Figure 5. It is a traditional metal structural wing, its structure data are given in Table 2. The weight of each engine is 240 kg
and the reference thrust is 20000 N. The following two cases are both considered: 1) the propulsion is applied at the front of the engine in order to simulate propeller engine; 2) the propulsion is
applied at the end of the engine to simulate jet engine. The thrusts are assumed to be uniformly distributed along the circumferential direction.
Table 2Wing structure data
Half span 12.43 m
Chord 6.06 m
Taper ratio 3.88
Swept angle 25 degrees
Length of store 3.9 m
Radius of store 1.2 m
Location of first store 3.36 m
Location of second store 7.21 m
The wing is clamped supported and flight is at sea level with 3° angle of attack, where the air density is 1.225 kg/m^3. The parameter $\mu$ is defined as the ratio of engine thrust to the reference
Table 3 displays the wing tip static deflection for the flow velocity of 246 m/s. The flap bending value is defined as positive while tip goes up and the torsion displacement is positive for nose up.
It is seen that the aerodynamic moment makes the wing nose down, while the engine thrusts decrease the torsional deflection and generate nose up pitching moment. This effect is much more obvious when
the thrusts were applied at the end of the store. Since the torsional deflection could influence the aerodynamic force, the flap bending displacement will increase under the action of engine thrusts,
especially for the case that the thrusts act on the end of the stores.
Fig. 5Model of a wing carrying two engines
Table 4 compares the natural frequencies of linear and nonlinear models of the wing structure. The aerodynamic loads and the propulsion are not included in the linear model, while the nonlinear model
takes such effects into consideration and the corresponding results are calculated for $V=$246 m/s and initial angle of attack of 3°. It is seen that when $\mu =$0, which means there are only
aerodynamic loads acting on the wing, the 1st and 2nd flap bending and 1st torsion frequencies all increase because of the prestressed stiffness generated by structure deformation. When $\mu =$1 the
torsional frequency exhibits a slight decrease because the thrusts alleviate the torsional deflection of the wing structure.
Table 3Wing tip displacement of the nonlinear model at equilibrium position
$\mu =$0 $\mu =$1 front $\mu =$1 back
Flap bending (m) 0.72 0.80 0.81
Torsion (deg) -2.20 -1.73 -1.65
Table 4Comparison of natural frequency results using linear and nonlinear methods (Hz)
$\mu =$0 $\mu =$1
First flap bending 2.77 2.83 2.83
Second flap bending 8.78 8.93 8.89
First torsion 10.29 11.33 11.22
Third flap bending 19.37 19.26 19.24
The $vg$ and $vf$ curves of the linear model are shown in Figure 6. The flutter speed and frequency are 265.7 m/s and 8.97 Hz respectively. It is clear that the flutter is caused by the coupling of
the second flap bending mode and the first torsional mode.
The nonlinear flutter analysis of the wing structure under the action of engine thrusts is performed via the method presented above. The $vg$ and $vf$ locus for $\mu =$0 are illustrated in Figure 7.
The corresponding flutter speed and frequency are 273.8 m/s and 9.53 Hz respectively. When $\mu =$1 and the propulsions are applied at the back end of the engine stores, the flutter speed and
frequency are 246.4 m/s and 9.85 Hz respectively, and the $vg$ and $vf$ curves are shown in Figure 8. It is observed that for both linear and nonlinear models the instabilities are all of the type of
classical bending-torsion coupling flutter. While $\mu =$1 and the thrusts act at the front of the engine store, the corresponding flutter speed is 249.2 m/s, which is a litter higher than that of
the previous one. It’s $vg$ and $vf$ curves are similar to Figure 8 and are not necessary to be given again.
It is noted from Figures 6-8 that without propulsion the flutter speed of the nonlinear model is 3.05 % higher than that of the linear model. The flap bending and torsional frequencies are all
increased by the effects of prestressed stiffness and geometrical nonlinearity. When the thrusts are applied at the front of the stores the nonlinear flutter speed is 7.26 % lower than the linear
one, because of the follower force effect. And this effect is much more obvious when the thrusts are loaded at the back end of the engine stores.
Fig. 6vg and vf curves of linear flutter
Fig. 7vg and vf curves at μ= 0
Fig. 8vg and vf curves at μ= 1 and loaded at the back end
Fig. 9Effects of μ on the wing flutter speed
Figure 9 shows the effect of the non-dimensional thrust on the wing flutter speed. It can be observed that the flutter speed exhibits a significant decrease as the thrust level increases, especially
when the thrusts are applied at the back end of the stores. Comparing to the case without thrust, the flutter speed of the wing with a reference thrust is decreased by 10.01 % and 8.98 %
corresponding to loads acting at the back and front of the nacelle respectively.
8. Conclusion
The purpose of this paper is to investigate the flutter characteristics of a complex structure wing subjected to engine thrusts. To this end an integrated flutter analysis method, which includes the
effects of thrusts and geometrical nonlinearity, has been developed based on the secondary development of MSC/Nastran software with its DMAP tool. The aeroelastic stability of a HALE wing, which is
subjected to a follower force, has been studied with this method firstly and the effects of several parameters were investigated. The numerical results are compared with published results and an
excellent agreement was observed.
The flutter characteristics of a complex wing structure carrying two engines are also analyzed. The influences of the thrust magnitude and location on wing flutter speed are considered. The results
show that the flutter speed is 273.8 m/s without thrust and decreased to 246.4 m/s when a 20000 N thrust is applied at the back end of each nacelle. The influences reduce the flutter speed more than
10 %, so the effects of engine thrust are significant and cannot be neglected for wing flutter analysis.
• Feldt W. T., Herrmann G. Bending-torsional flutter of a cantilevered wing containing a tip mass and subjected to a transverse follower force. Journal of the Franklin Institute, Vol. 296, Issue
11, 1974, p. 467-468.
• D. H. Hodges, M. J. Patil, S. Chae. Effect of thrust on bending-torsion flutter of wings. Journal of Aircraft, Vol. 39, 2002, p. 371-376.
• D. H. Hodges. Lateral-torsional flutter of a deep cantilever loaded by a lateral follower force. Journal of Sound and Vibration, Vol. 247, 2001, p. 175-183.
• S. A. Fazelzadeh, A. Mazidi. Bending-torsional flutter of wings with an attached mass subjected to a follower force. Journal of Sound and Vibration, Vol. 323, 2009, p. 148-162.
• Zhang Jian, Xiang Jinwu. Stability of high-aspect-ratio flexible wings loaded by a lateral follower force. Acta Aeronautic et Astronautic Sinica, Vol. 31, Issue 11, 2010, p. 2115-2123.
• Xie C. C., Leng J. Z., Yang C. Geometrical nonlinear aeroelastic stability analysis of a composite high-aspect-ratio wing. Shock and Vibration, Vol. 15, Issue 3, 2008, p. 325-333.
• Ran Yuguo, Han Jinglong, Yun Haiwei. Development of aeroelastic response solution sequence with DMAP language for freeplay nonlinear structure. Journal of Nanjing University of Aeronautics and
Astronautics, Vol. 39, Issue 1, 2007, p. 41-46.
• Ran Yuguo, Liu Hui, Zhang Jinmei, Han Jinglong. Analysis of the nonlinear aeroelastic response for large-aspect-ratio wing. Acta Aerodynamic Sinica, Vol. 27, Issue 4, 2009, p. 394-399.
• M. J. Patil, D. H. Hodges. On the importance of aerodynamic and structural geometrical nonlinearities in aeroelastic behavior of high-aspect-ratio wings. Journal of Fluids and Structures, Vol.
19, 2004, p. 905-915.
• Xie Changchuan, Wu Zhigang, Yang Chao. Aeroelastic analysis of flexible large aspect ratio wing. Journal of Beijing University of Aeronautics and Astronautics, Vol. 29, Issue 12, 2003, p.
• Guan De. Unsteady Aerodynamic Calculation. Beijing University of Aeronautics and Astronautics Press, Beijing, 1991.
About this article
01 November 2013
31 December 2013
wing flutter
geometrically nonlinear
engine thrust
follower force
high-aspect-ratio wing
Copyright © 2013 Vibroengineering
This is an open access article distributed under the
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Having trouble with some review circuits using nodal analysis
• Engineering
• Thread starter mitchapalooza
• Start date
In summary, the conversation discusses a problem with review circuits using nodal analysis. The individual is trying to find node voltages for v1, v2, and v3, with a given current of 3.1A. They
provide their method for solving the problem and have derived three equations with three unknown variables. They ask for reassurance on their answers and mention other possible methods for solving
the problem. Ultimately, they verify their answers with a more precise solution given by another individual.
Having trouble with some review circuits using nodal analysis :(
Homework Statement
I'm trying to find node voltages for v1, v2, and v3.
I = 3.1A
Homework Equations
The Attempt at a Solution
My method of going about and solving this was fairly straight forward I thought. The way the question is set up, I believe nodal analysis is already ready to go. So I started with
KCL(v1): 2 + 3.1 + (v1-v2)/3Ω = 0
KCL(v2): (v2-v1)/3 + v2/2 + (v2-v3)/4 = 0
KCL(v3): (v3-v2)/4 + v3/3 - 3.1 = 0
Now it's been a while since I've done this (nodal analysis), but I now have 3 eqns and 3 unknown variables. The answer I have is -21.2V -5.87V and 2.8V respectively.
It's one of those, "I've tried all my guesses, need some reassurance before I commit to my last chance" scenarios.
Any helpful on this would be great! Thanks :D
Note that I haven't tried the brute force method of just solving for everything, or the mesh current method.
Hi mitchapalooza, http://img96.imageshack.us/img96/5725/red5e5etimes5e5e45e5e25.gif
mitchapalooza said:
The answer I have is -21.2V -5.87V and 2.8V respectively.
Have you substituted those back into the equations to verify they are the solution? The equations look right.
Last edited by a moderator:
Have they taught you the superposition theorem? That way you could handle the problem one current source at a time ...
I solved the system and found v1 = -20.78V, v2 = -5.78V and v3 = 2.67V
I got
V1 = - 21.17
V2 = -5.87
V3= +2.80
so we're close.
Well, I got:
V1 = -127/6
V2 = -88/15
V3 = 14/5
which is the exact solution.
The 12 digit floating point approximation would be:
V1 = -21.1666666667
V2 = -5.86666666667
V3 = 2.80000000000
FAQ: Having trouble with some review circuits using nodal analysis
1. What is nodal analysis and how is it used in circuit analysis?
Nodal analysis is a method used in circuit analysis to determine the voltage and current at each node (junction point) in a circuit. It involves writing Kirchhoff's Current Law (KCL) equations at
each node and solving for the unknown variables using simultaneous equations. This method is useful for analyzing more complex circuits with multiple voltage sources and resistors.
2. Why am I having trouble with nodal analysis in circuit reviews?
Nodal analysis can be challenging for beginners because it requires a good understanding of KCL and how to write and solve simultaneous equations. It also requires a thorough understanding of circuit
theory and the ability to visualize and analyze complex circuits. Practicing and reviewing fundamental concepts can help improve your understanding and ability to apply nodal analysis in circuit
3. What are some common mistakes to avoid when using nodal analysis?
Some common mistakes to avoid when using nodal analysis include not properly labeling nodes, incorrectly applying KCL equations, and making errors in solving simultaneous equations. It is important
to double-check your work and make sure all equations are properly set up and solved accurately.
4. Can nodal analysis be used for all types of circuits?
Nodal analysis can be used for most circuits, but it is most effective for circuits with multiple voltage sources and resistors. It may not be the most efficient method for simpler circuits with only
one voltage source and a few resistors. In these cases, other methods such as Ohm's Law or Kirchhoff's Voltage Law may be more appropriate.
5. Are there any tips for improving my nodal analysis skills?
Practicing and reviewing fundamental circuit theory concepts is key to improving your nodal analysis skills. It may also be helpful to break down complex circuits into smaller sections and analyze
each section individually before combining them. Additionally, using software or online tools to simulate and analyze circuits can also be useful in developing your nodal analysis skills. | {"url":"https://www.physicsforums.com/threads/having-trouble-with-some-review-circuits-using-nodal-analysis.669774/","timestamp":"2024-11-03T09:52:09Z","content_type":"text/html","content_length":"101207","record_id":"<urn:uuid:571804c8-4170-4a6c-8798-e1de856624fd>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00108.warc.gz"} |
Dynamic Soft Sensor Development for Time-Varying and Multirate Data Processes Based on Discount and Weighted ARMA Models
College of Control Science and Engineering, China University of Petroleum (East China), Qingdao 266580, China
Author to whom correspondence should be addressed.
Submission received: 11 October 2019 / Revised: 1 November 2019 / Accepted: 6 November 2019 / Published: 15 November 2019
To solve the soft sensor modeling (SSMI) problem in a nonlinear chemical process with dynamic time variation and multi-rate data, this paper proposes a dynamic SSMI method based on an autoregressive
moving average (ARMA) model of weighted process data with discount (DSSMI-AMWPDD) and optimization methods. For the sustained influence of auxiliary variable data on the dominant variables, the ARMA
model structure is adopted. To reduce the complexity of the model, the dynamic weighting model is combined with the ARMA model. To address the weights of auxiliary variable data with different
sampling frequencies, a calculation method for AMWPDD is proposed using assumptions that are suitable for most sequential chemical processes. The proposed method can obtain a discount factor value
(DFV) of auxiliary variable data, realizing the dynamic fusion of chemical process data. Particle swarm optimization (PSO) is employed to optimize the soft sensor model parameters. To address the
poor convergence problem of PSO, ω-dynamic PSO (ωDPSO) is used to improve the PSO convergence via the dynamic fluctuation of the inertia weight. A continuous stirred tank reactor (CSTR) simulation
experiment was performed. The results show that the proposed DSSMI-AMWPDD method can effectively improve the SSM prediction accuracy for a nonlinear time-varying chemical process. The AMWPDD proposed
in this paper can reflect the dynamic change of chemical process and improve the accuracy of SSM data prediction. The ω dynamic PSO method proposed in this paper has faster convergence speed and
higher convergence accuracy, thus, these models correlate with the concept of symmetry.
1. Introduction
In order to reflect the dynamic change of chemical process, this paper attempts to propose DSSMI-AMWPDD method, which can effectively improve the prediction accuracy of nonlinear time-varying
chemical process SSM. The proposed model and the concept of symmetry have relativity and complementarity, and the research direction is highly consistent with Symmetry, which is convenient for
scholars in related fields as a reference. In chemical production, major process variables such as product quality are characterized by a slow sampling rate and time delay [
]. To ensure the stability of variable data in the main process, it is necessary to estimate the main process variables through some easily acquired process variables. Therefore, soft sensor modeling
(SSMI) is of great significance. Since most chemical processes do not have clear principles but have strong nonlinear and dynamic time-varying characteristics, the use of data-driven methods to
establish an industrial soft sensor model (SSM) [
] has become the focus of research. In particular, how to establish a suitable nonlinear dynamic model has evolved into an important research object for researchers.
Related modeling methods are generally divided into four types: multipoint input modeling [
], dynamic weighting modeling [
], feedback network modeling [
], and multimodel structure modeling [
]. Among these types of methods, multipoint input modeling boasts the advantages of simplicity, ease in implementation, and full reflection of the process characteristics. However, to fully reflect
the dynamic characteristics of the process, a large number of high-dimensional input variables are needed, which increases the number and complexity of the internal parameters, resulting in
ill-conditioned models. Dynamic weighting modeling uses dynamic weighting to form new input variables, which reduces the model input nodes and lower the model complexity, making the method simple and
easy. However, the dynamic weights and historical data (HD) lengths are difficult to determine. Feedback network modeling updates the input through the delay link of the feedback loop and updates the
structure or structural parameters, approximate to the object function. However, the model has poor stability, large deviations, no convergence, and an inability to fully reflect the dynamic
information of the process; in addition, the model is not commonly used as it has a complex training process. A representative multimodel structure modeling is the Wiener structure, in which linear
dynamic and nonlinear static submodels are built to describe the dynamic characteristics of a system. It provides a good approximation, but the dual model architecture is complex and difficult to
The use of the above methods leads to inaccurate SSM data prediction, either because the sample data for modeling do not fully reflect the dynamic characteristics of the process or because it is
difficult to perform the modeling and determine the parameters. To improve the prediction accuracy of SSM data and simplify the dynamic soft sensor modeling structure, this paper proposes an
autoregressive moving average (ARMA) model of weighted process data with discount (AMWPDD) structure, which has better flexibility in actual time series data fitting [
] and is simple and easy to implement. Amid assumptions suitable for most sequential chemical processes, the discount factor (DF) is introduced for the auxiliary variable HD of a chemical process.
Additionally, a DF calculation method and the corresponding constraints are proposed. Weighting is assigned to the auxiliary variable HD of the chemical process through the calculated DFV. The
auxiliary variable HD of the chemical process is fused to resolve the problem that the weight for the HD is not easy to determine. The calculation method of the sum of DFVs being “1” reflects the
integrity of the auxiliary variable HD. The problem of determining the weights of the auxiliary variable HD of different lengths can be solved by the exponential addition of DFVs according to the
length of the auxiliary variable HD. Therefore, the dynamic fusion of the process data is realized, the sample data quality of the SSM modeling is enhanced, and the SSM prediction accuracy is
improved. As chemical processes often present significant dynamics and delays [
], the study of DSSMI-AMWPDD is important.
A least squares support vector machine (LSSVM) has been proposed to deal with data regression tasks, and its success has been demonstrated in some supervised learning cases [
]. However, the LSSVM parameters affect the training performance of the model and they are difficult to determine, an intelligent optimization algorithm is generally used to realize the nonlinear
robust identification of the LSSVM [
]. Particle swarm optimization (PSO) [
] has attracted much attention because of its easy implementation and few adjustment parameters [
]. However, PSO may be trapped in local optima, and the convergence performance is very weak in later iterations [
]. To address the above problems, this paper proposes an improved particle swarm method by realizing the periodic fluctuation of the inertia weight values through the dynamic adjustment of the
inertia weight, improving the search precision and the convergence of the algorithm.
2. Problem Statement
The chemical process is a sequential production. Generally, the continuous sampling of input variables can be obtained, with only the sparse sampling of the output variables being gained [
]. Only a small number of process parameters are used, while past data with a large amount of dynamic information are ignored [
], as shown in
Figure 1
The problem to be studied and solved in this paper is how to rationally integrate the dynamic characteristics of a chemical process into SSMI data, convert dynamic process data into static data, and
establish an SSM and optimize the model parameters to achieve the purpose of the SSM prediction accuracy improvement.
3. DSSMI-AMWPDD
3.1. AMWPDD Sample Data Processing
Adding a large number of historical inputs not only increases the number and complexity of the parameters of the model but also causes the ill conditioning of the model due to the excessively high
dimensionality of the input variables.
In consideration of the sustained influence of the auxiliary variable data on the dominant variables in the chemical process and the corresponding difficulty in determining the degree of such
influence, and to reduce the number of input nodes and lower the complexity of the model, the ARMA model structure is used to establish the input data vector using historical inputs and historical
The conventional
$A R M A ( p , q )$
model is as follows [
${ x t = φ 0 + φ 1 x t − 1 + ⋯ + φ p x t − p + ε t − θ 1 ε t − 1 − ⋯ − θ q ε t − q , φ p ≠ 0 , θ q ≠ 0 ; E ( ε t ) = 0 , V a r ( ε t ) = σ ε 2 , E ( ε t ε s ) = 0 , s ≠ t E ( x s ε t ) = 0 , ∀ s < t
in which
are the orders of the model,
is the operator of the AR of order
, and
is the operator of MA of order
The multipoint input ARMA model used in this paper is shown in
Figure 2
The generalized function of the multipoint input ARMA model shown in
Figure 2
is expressed as:
$y ( k ) = f ( X ( k ) , θ )$
$X ( k ) = [ u ( t k − 1 ) , ⋯ , u ( t k − T ) , y ( t k − 1 ) ]$
Adding the output of the previous batch as an effective input into the modeled sample data can reduce the number of historical input nodes, reducing the number of model parameters and the complexity
of the model [
], and lowering the possibility of morbidity in the model.
However, the multipoint input ARMA model shown in
Figure 2
still has an incomplete model structure induced by numerous historical input nodes and the high model complexity.
To solve these problems, we combine the dynamic weighting model [
] with the ARMA model to realize the dynamic weighted fusion of the input nodes and hence reflect the dynamic characteristics of the process, as shown in
Figure 3
Its generalized function is expressed as:
$y ( t k ) = f ( v ( t k ) , y ( t k − 1 ) , θ )$
$v ( t k ) = ∑ i = 0 T w i u ( t k − i )$
The dynamic weight directly reflects the dynamic characteristics of the process and affects the accuracy of the overall model. However, it is difficult to obtain accurate values of the dynamic
weights. To solve this problem, this paper proposes an assumed condition suitable for most sequential chemical processes after a detailed site investigation of the chemical process.
If the value of the input node changes in different time periods, the further away an input point is from the output time, the less influence the value of the input node has on the change in the
value of the output node.
The above assumption is explained as follows:
$Δ y ( t k ) = ∑ σ = 0 T δ σ f ( u i ( t k − σ ) ) , i = 1 , ⋯ , n$
$δ σ = { 1 | t k − t k − σ | , σ ≠ 0 1 , σ = 0$
$δ σ$
is the degree of influence in the input node value.
Based on the above assumptions, this paper introduces DF
and uses the discount method [
] to dynamically discount-weigh the auxiliary variable values of different sampling time points in the same batch. By doing this, the impact of recent sample data on the model is enhanced, the role
of past samples is reduced, the fusion of input node values from
is achieved, and the problem that the weight of the dynamic weighted model is difficult to judge is solved.
The structure of the AMWPDD proposed in this paper is shown in
Figure 4
Its generalized function is as follows:
$y ( t k ) = f ( v ( t k ) , y ( t k − 1 ) , θ )$
$v ( t k ) = ∑ i = 0 T λ i u ( t k − i )$
$v ( t k )$
is the process input data obtained by DF weighted fusion.
However, the transition time T of the input variable of the chemical process, that is, the HD length, is difficult to determine. To ensure the data integrity of the batch input and output variables,
this paper proposes a calculation method of DF λ in combination with the above assumption.
DF λ numerical constraints:
λ[1] + λ[2] + …… + λ[T] = 1;
λ[1] > λ[2] > …… > λ[T];
The calculation formula of DF
$λ i = η i , i = 0 , 1 , 2 , ⋯ , T$
One can obtain the dynamic value of DF λ[i], i = 1, 2, ⋯, T based on different transition times T by Equations (10) and (11), realize the dynamic calculation of the DF λ value, and obtain more
accurate data fusion weights.
Then, the sample data set S for SSM modeling is:
$S = { v j ( t k ) , y ( t k − 1 ) | k = 1 , 2 , ⋯ , M ; j = 1 , 2 , ⋯ , N y ( t k ) | k = 1 , 2 , ⋯ , M }$
$v j ( t k )$
is the
input variables at time
$t k , y ( t k )$
is the output variable at time
$t k , y ( t k − 1 )$
is the output variable at time
− 1, and
$t k ( k = 1 , 2 , ⋯ , M )$
indicates the sampling time at which the system outputs
sample points.
On the basis of the sample data set S for the SSM, the SSM is established by the SSMI method.
3.2. LSSVM-Based SSMI
The least squares support vector machine (LSSVM) is a machine learning method proposed by Suykens [
] for solving the function estimation. It has better calculation speed, convergence precision and generalization performance and is more suitable for the small sample data SSMI of chemical processes
The LSSVM model [
] is:
$y ( x ) = ω T ϕ ( x ) + b$
$ϕ ( · )$
is a nonlinear transformation function,
is an adjustable weight vector, and
is an offset.
The objective function of LSSVM [
] is as follows:
$m i n J ( ω , ξ ) = 1 2 ω T ω + C 2 ∑ i = 1 l ξ T ξ$
$s . t . y i = ω T φ ( x i ) + b + ξ i ( i = 1 , 2 , ⋯ , l )$
$x i ∈ R n$
refers to the input vector,
$y i ∈ R$
represents the corresponding output vector,
$ξ i$
is the difference between the system output value and actual value,
$C ≥ 0$
represents a regularization parameter used to minimize the estimation error and control the function smoothness,
$φ ( · )$
refers to a nonlinear mapping from the input space to the feature space,
is the system weight,
is an offset, and
$s . t .$
indicates a constraint condition.
The Lagrange polynomial function of the optimization is solved by the Karush–Kuhn–Tucher (KKT) condition, and the LSSVM model for function estimation can be expressed as:
$y ^ = f ( x ) = ∑ i = 1 l α i ^ K ( x i , x ) + b ^$
Since the radial basis kernel function is a kernel function that has been widely used [
], it is chosen in this paper:
$k ( x , x ′ ) = exp { − ‖ x − x ′ ‖ 2 2 σ 2 } , σ > 0 i s k e r n e l r a d i u s$
After the kernel function is determined, the error term penalizes the parameter C, and the kernel function parameter σ^2 affects the regression performance of the LSSVM method. However, they are
difficult to determine. To ensure the optimal regression performance of the LSSVM, this paper uses PSO to optimize the error term penalty parameter C and the kernel function parameter σ^2.
4. Model Parameter Optimization Based on ωDPSO
To improve the prediction accuracy of SSM, the optimization objective is set to minimize the sum of the squared errors between the sample actual output data
$y 0 ( t k )$
and sample prediction output
$y ( t k )$
. The optimization objective function is as follows:
$m i n J = 1 M ∑ k = 1 M ‖ y ( t k ) − y 0 ( t k ) ‖ 2$
PSO [
] is an intelligent algorithm that simulates the predation behavior of birds and fish groups.
In PSO, each particle has independent position, velocity, and fitness for optimizing the target. PSO randomly sets a certain number of particles, initializes their positions and speeds, and completes
the optimization process by iteration. The iterative process generally includes two optimal values: the Personal Best Value (
) and the Global Best Value (
represents the optimal fitness of the particle itself, and
refers to the optimal fitness of all particles. The particle updates its position and velocity by tracking two extreme values (
), and the updated formula is as follows:
$v i d ( t + 1 ) = w × v i d ( t ) + c 1 × r a n d ( ) × ( p i d − x i d ( t ) ) + c 2 × r a n d ( ) × ( p g d − x i d ( t ) )$
$x i d ( t + 1 ) = x i d ( t ) + v i d ( t + 1 )$
, and
represent the velocity, position, and personal best value, respectively, of particle
in iteration t;
() is a random number between [0, 1];
are learning factors, which represent the weights of the statistical acceleration terms that push each particle to the
is the inertia weight;
is the number of iterations.
Since the inertia weight is related to the development and exploration ability of the particle, it affects the convergence of the algorithm [
]. A larger
value is beneficial to jump out of a local optimum for global optimization; a smaller
value is beneficial to local optimization and accelerates the convergence of the algorithm. When the search process follows a nonlinear and highly complex algorithm, the linearly decreasing inertia
weight does not effectively reflect the actual search process [
]. At the same time, the inertia weight linear decreasing strategy based on the number of iterations has a weak partial search in the early iteration, and the particle might miss the optimal value
even if it is close to the current value; in late iterations, the global search ability is weak, and it is easy to fall into the local optimum problem. In this paper, an
DPSO is proposed, which improves the convergence of the algorithm by dynamically adjusting the inertia weight
value, as shown in
Figure 5
The equation for the inertia weight is expressed as:
$ω = 0.8 − 0.3 ∗ s a w t o o t h ( t , j / k )$
As shown in
Figure 5
, compared with the inertia weight linear decreasing strategy based on the number of iterations, the inertia weight
values proposed in this paper vary from large to small, then from small to large, and again from large to small, showing a periodic sawtooth fluctuation. As a result, the particles are periodically
alternating between focusing on the global search and focusing on the local search. This balances the global search and the local search, avoids being trapped in a local optimum, improving the
convergence of the PSO algorithm. At the same time, according to the ratio of the current iteration number to the total number of iterations, the ratio of the time prior to the peak within the cycle
to the cycle time is adjusted so that the inertia weight
increases rapidly and decreases slowly in early numerical iterations, grows and reduces slowly in the middle iterations, and rises slowly and decreases rapidly in the late iterations.
Through ωDPSO, the LSSVM SSMI method penalty parameter C and the kernel function parameter σ^2 are numerically optimized.
5. Simulation and Analysis
A CSTR is used as study objects to test the predictive performance of the SSMI method proposed in this paper for nonlinear dynamic time-varying chemical processes. The CSTR data comes from a computer
simulation. In addition, to evaluate the performance of the proposed method, the dynamic and static data LSSVM SSM, PSO-LSSVM SSM, and ωDPSO-LSSVM soft sensor model of the CSTR object are
Given the commonality and universality of evaluation indices, including the mean absolute error (MAE), root mean square error (RMSE), and running time (RT), in regression analyses, this paper uses
the RMSE in Equation (23) and the RT to evaluate the training performance of the SSM and the MAE in Equation (22) and the RMSE in Equation (23) to evaluate the prediction accuracy of the SSM:
$M A E = 1 N ∑ i = 1 N | y i ^ − y i |$
$R M S E = 1 N ∑ i = 1 N | y i ^ − y i | 2$
is the total number of samples,
$y i ^$
is the predicted output value, and
$y i$
is the actual value.
5.1. CSTR Simulation Experiment and Result Analysis
The continuous stirred tank reactor is the most important piece of equipment in many chemical and biochemical industries and has second-order nonlinear dynamic characteristics [
]. Therefore, it can be used to test the ability of the SSMI method to solve nonlinear and time-varying problems. The principle of the CSTR [
] is shown in
Figure 6
, with a description of each variable and the values of the steady-state operating points shown in
Table 1
The concentration C[A] of the raw material A in the reactor is considered to be the dominant variable of the SSM. The feed flow rate F[i], the cooling water flow rate F[c], and the reactor internal
temperature T[r] are treated as auxiliary variables of the SSM.
In the CSTR simulation process, the sampling periods of the auxiliary and dominant variables are set to 1 h and 12 h, respectively. The simulation time is set to 265 h, and a certain white Gaussian
noise is added to each auxiliary variable. A total of 265 groups of usable data are obtained, 23 of which are labeled and the rest are dynamically unlabeled. The former 168 groups of data are used as
the training sample set, and the latter 96 groups of data are used as the test sample set. A total of 22 groups of dynamic fusion data and static data with the same output are used as the SSMI
samples. To simulate the reduction and recovery of the catalyst activity in the reactor, the catalyst activity
is set for the data simulation based on the variation pattern in
Figure 7
, and the simulated data set is then normalized.
The following model structure based on static data modeling is used:
$y ^ s ( t k ) = f s C S T R ( u j ( t k ) ) , j = 1 , 2 , 3$
With expert knowledge and the dynamic characteristics of the CSTR, the following model structure is adopted for dynamic fusion data-based modeling:
$y ^ d ( t k ) = f d C S T R ( v j ( t k ) y ( t k − 1 ) ) , j = 1 , 2 , 3$
$v j ( t k ) = f D i s [ u j ( t k − 11 ) ⋮ u j ( t k − 1 ) u j ( t k ) ]$
$y ( t k )$
represents the concentration in the reactor
at time
$u j ( t k − T ) ( 1 ≤ j ≤ 3 , 0 ≤ T ≤ 11 )$
represents the sampled value of the
auxiliary variable at time
$v j ( t k )$
is the discounted fusion value of
$u j ( t k − T ) ( 1 ≤ j ≤ 3 , 0 ≤ T ≤ 11 )$
$y ^ ( t k )$
represents the concentration in the reactor predicted by the SSM.
The parameters for the method comparison are set as follows:
LSSVM: $C = 80 , σ 2 = 20$
PSO-LSSVM: number of iterations: 100; number of particles: 20; $C ∈ [ 20 , 130 ] σ 2 ∈ [ 20 , 130 ]$.
ωDPSO-LSSVM: number of iterations: 100; number of particles: 20; $C ∈ [ 20 , 130 ] σ 2 ∈ [ 20 , 130 ]$.
5.1.1. SSMI Based on Static Data
The SSMs of the above three methods are based on static data, wherein, for the PSO-LSSVM method,
$C = 101.79 and σ 2 = 20.53$
, and for the
DPSO-LSSVM method,
$118.13 and σ 2 = 21.67$
, as shown in
Figure 8
With white Gaussian noise added to the auxiliary variables, it can be seen from
Figure 8
a–c that the three SSMI methods all have large offsets. Combining the RMSE values of the model training performance shown in
Table 2
, it can be seen that the models trained by PSO-LSSVM and
DPSO-LSSVM are closer to the actual data. Compared with that of the LSSVM method, the RMSE index increases by 1.35% and 1.57%, respectively, and the RT index falls by factors of 30.16 and 25.73,
Compared to those of the PSO-LSSVM method, the RMSE and RT indices of the ωDPSO-LSSVM method rises by 0.23% and 14.24%, respectively.
The training performances of the different SSMI methods are shown in
Table 2
The performances of the different SSMI methods using test data set are compared, as shown in
Figure 9
It can be seen from
Figure 9
that the prediction values by the LSSVM, PSO-LSSVM, and
DPSO-LSSVM methods based on static data all show large offsets, and the prediction performance evaluation indices MAE and RMSE of each SSM are also poor. However, compared with those of the LSSVM
method, the MAE values of the PSO-LSSVM and
DPSO-LSSVM methods grow by 1.59% and 3.17%, respectively, and their RMSE values rise by 1.64% and 3.52%, respectively.
Compared to those of the PSO-LSSVM method, the MAE and RMSE values of the ωDPSO-LSSVM method increase by 1.6% and 1.9%, respectively, indicating a higher prediction accuracy.
The prediction performances of the different SSMI methods are shown in
Table 3
5.1.2. SSMI Based on Dynamic Fusion Data
The SSMs of the above three methods are established based on dynamic fusion data, wherein the PSO-LSSVM method,
$C = 107.59 and σ 2 = 21.96$
, and the
DPSO-LSSVM method,
$C = 128.643 and σ 2 = 24.16$
, are shown in
Figure 10
As shown in
Figure 10
, the different SSMI methods have different training effects on the dynamic fusion data. Combined with the RMSE values of the models built by the different soft sensor methods, as listed in
Table 4
, it found that, compared with those of the LSSVM method, the RMSE indices of the PSO-LSSVM and
DPSO-LSSVM methods rise by 3.81% and 4.11%, respectively, and their RT indices decrease by factors of 35.49 and 26.92, respectively.
Compared with those of the PSO-LSSVM method, the RMSE and RT indices of the ωDPSO-LSSVM method reduce by 0.31% and 6.8%, respectively.
The training performances of the different SSMI methods are shown in
Table 4
The performances of the different SSMI methods using the test data set are compared, as shown in
Figure 11
A comparison of the prediction curves of the LSSVM, PSO-LSSVM, and
DPSO-LSSVM methods in
Figure 11
shows that the prediction curves of the PSO-LSSVM and
DPSO-LSSVM methods are closer to the actual values. As seen in
Table 5
, compared with the those of the LSSM method, the MAE values of the PSO-LSSVM and
DPSO-LSSVM methods grow by 1.51% and 3.4%, respectively, and their RMSE values rise by 4.83% and 7.47%, respectively. Compared with those of the PSO-LSSVM method, the MAE and RMSE values of the
DPSO-LSSVM method increase by 1.92% and 2.77%, respectively.
The prediction performances of the different SSMI methods are shown in
Table 5
5.1.3. Comparison and Analysis
A comparison of
Figure 8
Figure 10
, as well as
Figure 9
Figure 11
, combined with the data analysis in
Section 5.1.1
Section 5.1.2
, shows that, compared with that based on static data, the RMSE values of the model training results using the three methods based on dynamic fusion data rise by 23.54%, 25.46%, and 25.51%,
respectively. Compared with those based on static data, the MAE values of the three methods based on dynamic fusion data grow by 35.49%, 35.44%, and 35.64%, respectively, and the RMSE values increase
by 19.93%, 22.53%, and 23.21%, respectively. The models established via dynamic fusion data and the corresponding data prediction accuracy are better than those using static data. The main reason is
that the chemical process is a continuous time series production process, and changes in the values of the auxiliary variables affect the values of the subsequent dominant variables. Modeling based
only on the current static data is unable to reflect the process variation of the auxiliary variables, resulting in the poor training of the model and low precision of the data prediction. In view of
the influence of the inertia weight coefficient
on the convergence performance of the PSO method, the
DPSO-LSSVM method achieves better prediction performance than the PSO-LSSVM method.
5.2. Simulation Experiment and Result Analysis
Section 5.1
, the simulation data of CSTR is used to experimentally verify the proposed DSSMI-AMWPDD method based on
DPSO. A comparison of the modeling using dynamic fusion data and static data, as well as the experimental results of data prediction, proves that the SSM established by dynamic fusion data is
superior to that those using static data in terms of the prediction model accuracy and data prediction precision. Additionally, this paper adopts the PSO-LSSVM method and the
DPSO-LSSVM method to perform 10 trainings on CSTR data and selects the one with the best training effect as the experimental result, and the results show that the
DPSO-LSSVM method achieves better prediction performance, shorter training time, and stronger convergence.
6. Conclusions
In this work, based on chemical processes as the research setting, the simulation modeling of CSTR simulation data shows that the AMWPDD proposed in this paper can reflect the dynamic changes of the
chemical process and improve the accuracy of the SSM data prediction. Furthermore, the simulation results show that, compared with the standard PSO method, the ωDPSO method can better balance the
local and global development capabilities, with faster convergence speed and higher convergence accuracy.
Author Contributions
Conceptualization, L.L. and Y.D.; methodology, L.L.; software, L.L.; validation, L.L. and Y.D.; formal analysis, L.L. and Y.D.; investigation, L.L. and Y.D.; resources, L.L. and Y.D.; data curation,
L.L.; writing—original draft preparation, L.L.; writing—review and editing, Y.D.; visualization, L.L. and Y.D.; supervision, Y.D.; project administration, Y.D.; funding acquisition, Y.D.
This research received no external funding.
Conflicts of Interest
The authors declare no conflicts of interest.
Figure 8. Modeling data training for the different SSMI methods. (a) The training curve based on LSSVM; (b) The training curve based on PSO-LSSVM; (c) The training curve based on ωDPSO-LSSVM.
Figure 9. Prediction of test data by the different SSMI methods. (a) The prediction curve based on LSSVM; (b) The prediction curve based on PSO-LSSVM; (c) The prediction curve based on ωDPSO-LSSVM.
Figure 10. Modeling data training for the different SSMI methods. (a) The training curve based on LSSVM; (b) The training curve based on PSO-LSSVM; (c) The training curve based on ωDPSO-LSSVM.
Figure 11. Prediction of test data by the different SSMI methods. (a) The prediction curve based on LSSVM; (b) The prediction curve based on PSO-LSSVM; (c) The prediction curve based on ωDPSO-LSSVM.
Parameter Description Steady-State Value
F[i] Feed flow rate 100 L/min
C[Ai] Reactant concentration in the feed 1 mol/L
T[i] Feed temperature 350 K
V Reactor volume 100 L
k[0] Reaction speed 7.2 × 10^10 min^−1
C[p] Reactant specific heat capacity 1 cal/g/k
hA Thermal conductivity 7 × 10^5 cal/min/K
T[ci] Cooling water inlet temperature 350 K
C[pc] Cooling water specific heat capacity 1 cal/g/k
Soft Sensor RMSE Running Time(s)
LSSVM 0.0446 0.851
PSO-LSSVM 0.0440 26.520
ωDPSO -LSSVM 0.0439 22.744
Soft Sensor MAE RMSE
LSSVM 0.0820 0.0853
PSO-LSSVM 0.0807 0.0839
ωDPSO -LSSVM 0.0794 0.0823
Soft Sensor RMSE Running Time(s)
LSSVM 0.0341 0.793
PSO-LSSVM 0.0328 28.94
ωDPSO -LSSVM 0.0327 22.143
Soft Sensor MAE RMSE
LSSVM 0.0529 0.0683
PSO-LSSVM 0.0521 0.0650
ωDPSO -LSSVM 0.0511 0.0632
© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http:/
Share and Cite
MDPI and ACS Style
Li, L.; Dai, Y. Dynamic Soft Sensor Development for Time-Varying and Multirate Data Processes Based on Discount and Weighted ARMA Models. Symmetry 2019, 11, 1414. https://doi.org/10.3390/sym11111414
AMA Style
Li L, Dai Y. Dynamic Soft Sensor Development for Time-Varying and Multirate Data Processes Based on Discount and Weighted ARMA Models. Symmetry. 2019; 11(11):1414. https://doi.org/10.3390/sym11111414
Chicago/Turabian Style
Li, Longhao, and Yongshou Dai. 2019. "Dynamic Soft Sensor Development for Time-Varying and Multirate Data Processes Based on Discount and Weighted ARMA Models" Symmetry 11, no. 11: 1414. https://
Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details
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MATH 200W: Transition to Advanced Mathematics
Note: If this course is being taught this semester, more information can be found at the course home page.
Cross Listed
This course is an introduction to the language and problems of mathematics. The precise topics vary from year to year, but the emphasis is always on increasing the student’s “mathematical
sophistication” and ability to read and write proofs. If you would like a more gentle introduction to proofs compared with Math 235, consider MTH200W. Students taking the MTH 171 - 174 sequence will
note usually not take this course. Students that have already completed Math 235 or 172 need permission before taking this course.
Topics covered
Introduces some of the basic techniques and methods of proof used in mathematics and computer science. Methods of logical reasoning, mathematical induction, relations, functions, and more. These
methods are discussed in the context of specific mathematical problems and theories.
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Math intuition, math without books
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Originally Posted by Dazed&Confuzed
Cathy - can I sign up for your class? Pretty please??
I was actually researching VCI and PRI and what does those indices mean in the real world. I was surprised to read somewhere that VCI correlates more with algebraic thinking and PRI with geometric
abilities ie that math and verbal domains are linked as you stated.
Thanks, Dazey! It's so fun to teach people who are excited about learning (that's why I like third graders
There is a math website I came across and it explained fractions just the way you did and also addressed the importance of doing so. Coincidentally, w/out knowing why or the significance of it, it's
how I taught my boys fractions at a young age.
I'm not sure how significant it is, but it sure makes sense! Otherwise, kids just get a look of panic on their faces when they see two numbers with a weird little line between them. What the heck is
that about? It must be
. If nothing else, exposing kids to higher math will get them accustomed to seeing different kinds of notation. Just like we expose toddlers to the alphabet without expecting them to read right
away. Why do we (as a culture) feel like we have to keep math a secret? Why do we send the message that it's "too hard" or "too confusing"?
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Math intuition, math without books Kriston 04/23/08 03:30 AM
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Re: Math intuition, math without books Kriston 04/23/08 01:53 PM
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Re: Math intuition, math without books Kriston 04/23/08 04:36 PM
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Re: Math intuition, math without books calizephyr 04/23/08 05:43 PM
Re: Math intuition, math without books pinkpanther 04/23/08 04:32 PM
Re: Math intuition, math without books LMom 04/23/08 04:27 PM
Re: Math intuition, math without books squirt 04/23/08 02:39 PM
Re: Math intuition, math without books cym 04/23/08 02:53 PM
Re: Math intuition, math without books Kriston 04/23/08 03:20 PM
Re: Math intuition, math without books doodlebug 04/23/08 03:38 PM
Re: Math intuition, math without books Dazed&Confuzed 04/23/08 03:44 PM
Re: Math intuition, math without books LMom 04/23/08 04:50 PM
Re: Math intuition, math without books Dazed&Confuzed 04/23/08 05:23 PM
Re: Math intuition, math without books Cathy A 04/23/08 07:14 PM
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Re: Math intuition, math without books kimck 04/23/08 07:39 PM
Re: Math intuition, math without books st pauli girl 04/23/08 07:45 PM
Re: Math intuition, math without books Cathy A 04/23/08 07:51 PM
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Re: Math intuition, math without books Ania 04/23/08 07:53 PM
Re: Math intuition, math without books Cathy A 04/23/08 08:02 PM
Re: Math intuition, math without books Dazed&Confuzed 04/23/08 07:25 PM
Re: Math intuition, math without books Cathy A 04/23/08 07:35 PM
Re: Math intuition, math without books Cathy A 04/23/08 07:38 PM
Re: Math intuition, math without books Kriston 04/23/08 08:05 PM
Re: Math intuition, math without books Cathy A 04/23/08 10:41 PM
Re: Math intuition, math without books Kriston 04/23/08 10:58 PM
Re: Math intuition, math without books Cathy A 04/23/08 11:10 PM
Re: Math intuition, math without books Kriston 04/23/08 11:15 PM
Re: Math intuition, math without books Cathy A 04/23/08 11:39 PM
Re: Math intuition, math without books EandCmom 04/23/08 11:45 PM
Re: Math intuition, math without books Kriston 04/24/08 12:02 AM
Re: Math intuition, math without books Cathy A 04/24/08 12:11 AM
Re: Math intuition, math without books Dazed&Confuzed 04/24/08 01:03 AM
Re: Math intuition, math without books squirt 04/24/08 01:13 AM
Re: Math intuition, math without books Kriston 04/24/08 01:55 AM
Re: Math intuition, math without books Dazed&Confuzed 04/24/08 02:11 AM
Re: Math intuition, math without books Cathy A 04/24/08 02:35 AM
Re: Math intuition, math without books Kriston 04/24/08 01:21 PM
Re: Math intuition, math without books AmyEJ 04/24/08 02:08 PM
Re: Math intuition, math without books Dazed&Confuzed 04/24/08 02:15 PM
Re: Math intuition, math without books Kriston 04/24/08 02:21 PM
Re: Math intuition, math without books Lori H. 04/24/08 03:13 PM
Re: Math intuition, math without books squirt 04/24/08 04:02 PM
Re: Math intuition, math without books Dazed&Confuzed 04/24/08 04:08 PM
Re: Math intuition, math without books Kriston 04/24/08 04:18 PM
Re: Math intuition, math without books Dazed&Confuzed 04/24/08 04:35 PM
Re: Math intuition, math without books questions 04/24/08 05:12 PM
Re: Math intuition, math without books Cathy A 04/24/08 06:41 PM
Re: Math intuition, math without books Cathy A 04/24/08 07:00 PM
Re: Math intuition, math without books Kriston 04/24/08 10:35 PM
Re: Math intuition, math without books incogneato 04/24/08 11:31 PM
Re: Math intuition, math without books Kriston 04/24/08 11:36 PM
Re: Math intuition, math without books incogneato 04/24/08 11:46 PM
Re: Math intuition, math without books Kriston 04/24/08 11:50 PM
Re: Math intuition, math without books incogneato 04/24/08 11:56 PM
Re: Math intuition, math without books Kriston 04/25/08 12:05 AM
Re: Math intuition, math without books doodlebug 04/27/08 12:16 PM
Re: Math intuition, math without books Kriston 04/27/08 01:12 PM
Re: Math intuition, math without books Kriston 04/27/08 07:46 PM
Re: Math intuition, math without books Kriston 04/27/08 11:10 PM
Re: Math intuition, math without books Dazed&Confuzed 04/27/08 11:22 PM
Re: Math intuition, math without books incogneato 04/28/08 02:33 AM
Re: Math intuition, math without books Kriston 04/28/08 04:28 PM
Re: Math intuition, math without books Kriston 04/29/08 01:16 PM
Re: Math intuition, math without books Kriston 04/29/08 02:19 PM
Re: Math intuition, math without books incogneato 04/29/08 04:13 PM
Re: Math intuition, math without books snowgirl 04/24/08 02:40 AM
Re: Math intuition, math without books Cathy A 04/24/08 02:48 AM
Re: Math intuition, math without books Cathy A 04/24/08 02:52 AM
Re: Math intuition, math without books OHGrandma 04/24/08 11:25 AM
Re: Math intuition, math without books Cathy A 04/24/08 02:45 AM
Re: Math intuition, math without books OHGrandma 04/23/08 11:10 PM
Re: Math intuition, math without books Kriston 04/23/08 08:20 PM
Re: Math intuition, math without books Kriston 04/23/08 08:27 PM
Re: Math intuition, math without books Kriston 04/23/08 09:40 PM
Re: Math intuition, math without books Kriston 04/23/08 08:10 PM
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Data Representation Techniques to Supercharge your ML model—Part I
How to do feature engineering beyond scaling and one-hot encoding
Photo by Annie Spratt on Unsplash
Being a data scientist is like being a craftsman. You are equipped with a set of tools and required to create something beautiful yet functional out of simple material. Some of your work might be
done by automatic machinery, but you know that your true power is creativity and intricate skill to work by hand.
In this series, we will hone your skillset by exploring several approaches to represent data as a feature. These approaches could improve the learnability of your model especially if you have tons of
data in hand.
Imagine that you have data with the following pattern, where the horizontal axis represents a feature X₁, and the vertical axis represents another feature X₂ and each instance (point in the plot) can
only belong to either -1 or 1 group (represented by red and green).
source: jp.mathwork.com
Now let me challenge you to draw a linear boundary that can separate the different classes on the data. I bet you can’t, and indeed this is an example of non linearly separable data.
There are several ways to handle this kind of data like using inherently non-linear classification models such as decision tree or a complex neural network. However, there is a simple technique we
can use to make a simple linear classifier work very well on this kind of data.
Here is the trick, first, let’s discretize our continuous features into two buckets according to the colour shown on the plot. For X₁, let A denote its positive values and B denote its negative
counterpart. Similarly, let C denote positive values of X₂, and D denote negative values of X₂.
Then, we can create a new categorical feature by combining all possible combinations of our newly created buckets.
• AD = {X₁ > 0 and X₂ < 0}
• AC = {X₁ > 0 and X₂ > 0}
• BC ={X₁ < 0 and X₂ > 0}
• BD ={X₁ < 0 and X₂ < 0}
With this brand new feature, we can now easily classify the class of an instance by only using simple binary classification.
the variables here are indicator function which has value either 0 or 1
If you are careful enough, you can immediately see the appropriate set of weights are the labels themselves (w₀ = -1, w₁ = 1, w₂ = 1, w₃ = -1).
The transformation we just did is an example of feature crosses where we concatenate multiple features into a new one to help our model learning better. | {"url":"https://neoshare.net/machine-learning/data-representation-techniques-to-supercharge-your-ml-model%E2%80%8A-%E2%80%8Apart-i/","timestamp":"2024-11-05T06:10:29Z","content_type":"text/html","content_length":"44352","record_id":"<urn:uuid:4a7f1cb3-bc23-4105-a322-efaed12a0b1d>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00163.warc.gz"} |
The development of a composite laminate macromodel for the analysis of stress-strain behavior in irregular zones of typical airframe
Mathematics. Physics. Mechanics
Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia
*e-mail: grischenko1911@gmail.com
**e-mail: design101@mai.ru
Preliminary assessment of stress-strain behavior is an important stage in design of any aircraft structure. Such assessment allows determining the first approximation of geometrical parameters of the
designed structure elements already at the stage of its development. The more accurate the methods of stress-strain behavior assessment are, the lower are the costs of the subsequent design
adjustments, which are implemented on the basis of the structure test results.
Design calculations of composite laminate structures are carried out by mainly using composite laminate macromodels, which are based on the theory of laminated anisotropic material, and various
finite element models. The main role of the finite element models in this calculation consists in determining the effective stress. To obtain the solution in the first case an assumption is made that
the deformations are equal and constant along the packet thickness. In the second case it is possible to construct a composite laminate model, which would take into account the interlayer and
interfacial interactions between the composite laminate components. However, in this case the required computing capacity grows manifold directly proportional to the number of layers.
The goal of this research is to develop the method of analysis of stress strain behavior of any composite laminate packet in irregular zone of typical airframe structures. This is attained by
creating a special mathematical macromodel for numerical analysis of the deformation of an arbitrary laminate composite packet, which allows for possible movement of the layers relative to each other
with taking into account the shear stiffness of the binder.
Only two-dimensional stress state is considered in this study. An assumption is made that the interlayer space in a composite laminate packet is filled exclusively with the binder, the mechanical
properties of which are isotropic. In the framework of the developed model a multilayer laminate composite packet, which actually consists of
monolayers with
thickness, is considered to have
orthotropic lamina with
thickness and
binder layers with
thickness. Each orthotropic laminae in the model is considered as a composite monolayer with a somewhat higher percentage of fiber content. The binder in the model is considered to have isotropic
mechanical properties.
Thus it is assumed that the packet consists of layers of 2 different materials. The interfacial connections are considered ideal, i.e. the deformations are constant along the materials interface. It
is also assumed that composite monolayer can only be in two-dimensional stress state, while intermediate binder layer can be in three-dimensional stress state with the exception of longitudinal
deformations along the Z axis. Therefore, the possibility of occurrence of shear deformations
and, subsequently, shear stresses τ
[XZ][ ]
and τ
in the intermediate binder layer is not excluded from consideration.
The problem is reduced to the determination of the strain and deformation of composite layers according to the laws of the classical theory of elasticity of a laminate anisotropic material. For
intermediate binder layers the deformations (including shear deformations) are determined first, the tangential shear, normal and equivalent strains are determined afterwards.
Interlayer shear deformations are defined by solving the equations of strain compatibility for the intermediate binder layer. This is done by using the dependence of the longitudinal deformation
within the binder layer on the coordinate along the Z axis (boundary conditions are assumed as equal to zero):
where ε
and ε
are the deformations of the adjacent layers 1 and 2.
A calculation of a hypothetical loading case was carried out to investigate the capabilities of the methodology and analyze the results. A 40-layer composite laminate packet was considered. It was
assumed that a certain load is applied to it. The load acts on a certain number of the upper packet layers (loaded layers).
The dependence of the layer and interlayer deformations on the coordinate along the packet thickness shows a strongly pronounced inverse proportionality. Interlayer shear deformations tend towards
zero with greater intensity. Such dependence is most pronounced for the packet, which is reinforced in only one direction.
The shape of deformation distribution in the layers is independent from the amount of the applied load, the loading condition and calculation pattern. This means that there is a certain invariant
characteristic of the deformation of the composite laminate packet, which depends only on the structure and properties of the packet. This allows assessing the stress-strain behavior of such packet
at the stage when the load itself and loading condition are unknown.
It is necessary to gradually increase the thickness in the irregular zone to enable a more uniform loading of the composite laminate layers. However, based on the obtained solutions for the
interlayer deformation it is possible to conclude that the interlayer shift directly depends on the size of the transition zone. Therefore the problem of determining the optimal size of the
transition zone emerges. The optimal zone should allow gradual loading of the layers without exposing the interlayer binder to dangerous strains.
If there are shear stresses between the layers, they usually substantially prevail over normal ones. Thus delamination calculation can be based on the shear strength of the binder material. However,
creation of an integrated strength criterion may be required.
There is a significant increase of the interlayer deformations in areas where the load is transferred from the layer with reinforcement angle 90° to the layer with reinforcement angle 0°. This is
caused by the fact that the difference in stiffness of these layers is too big. When the load is transferred from the layer with reinforcement angle 0° to the layer with reinforcement angle 90°, a
reversed interlayer effect can be observed, which consists in small values of the interlayer deformations and strains.
composite material, stress-strain behavior, irregular zones, interlaminar shift
1. Selten R. Mekhanika kompozitnykh matrialov (Mechanics of composite materials), Dordrecht, Boston: Kluwer Academic, 1994, 441 p.
2. Kravtsov V. A. Vestnik Moskovskogo aviatsionnogo instituta, 2009, vol. 16, no. 6, pp. 56-64.
3. Alfutov N. A., Zinov'ev P. A., Popov B. G. Raschet mnogoslojnyh plastin i obolochek iz kompozicionnyh materialov (Calculation of laminated plates and shells made of composite materials), Moscow,
Mashinostroenie, 1984, 264 p.
4. Popov B.G. Raschet mnogoslojnyh konstrukcij variacionno-matrichnymi metodami: Uchebnoe posobie dlja vuzov mashino- i priborostroitel'nyh special'nostej (Calculation of multilayer structures
variation-matrix methods: A manual for schools machine building specialties), Moscow, MGTU , 1993, 294 p.
5. Vasil'ev V.V. Kompozicionnye materialy (Composite Materials), Moscow, Mashinostroenie, 1990, 512 p.
mai.ru — informational site MAI
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Yang-Mills instanton
added cross-link (here) with Dp-D(p+4)-brane bound state
diff, v25, current
I have further expanded at Yang-Mills instanton the discussion, adding full detail to the statement about gradient flow (making the Hodge metric on forms and the respective gradient of the
Chern-Simons functional fully manifest)
added to Yang-Mills instanton a discussion of instantons as tunnelings between Chern-Simons vacua.
My eyes lighted on this. I don’t have a vested interest, but the theory of Yang-Mills instantons as set out in the nLab reads as if a correction to, in fact a complete reworking of, the usual story
as told in textbooks, and I’m just wondering whether all this is original research of Urs with some contributions by a few others like Igor Khavkine and David Roberts (looking at the history of the
article). It’s a little hard to tell from the article what is due to whom.
Thanks, Todd.
Just to say that this is all written by me. Checking the history, David R. And Igor K. made trivial edits in this case (adding a cross-link and a doi-link to the references). Also, the bulk of the
entry is actually !include-ed from SU2-instantons from the correct maths to the traditional physics story. (I had split that off as an include-file in order to be able to use it also in other entries
related to instantons.)
I would want to believe that most mathematical physicists, when pressed would produce this explanation of instanton sectors. But when I was digging into the literature to find good citations, I
didn’t find any. Which is why I ended up writing down this account.
(In retrospect, this discussion of instantons via the one-point compactification eventually led to the discussion in Equivariant Cohomotopy implies orientifold tadpole cancellation.)
added pointer to
• Tohru Eguchi, Peter Gilkey, Andrew Hanson, Section 10.2 of: Gravitation, gauge theories and differential geometry, Physics Reports Volume 66, Issue 6, December 1980, Pages 213-393 (doi:10.1016/
diff, v32, current
Sign typo corrected in the definition of self-duality. The correction matches the choice of sign used later in the text.
Jonathan Esole
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Abstracts | Mysite
top of page
APPLIED KNOT THEORY WORKSHOP 2020
October 09, 10am-1pm EST
9am-12pm CST
2pm-5pm GMT/UTC
Chris Soteros
Characterizing the entanglements in lattice models of ring polymers.
Motivated in part by recent experimental and molecular dynamics studies of the entanglement characteristics of DNA in nanonchannels, we have been studying the statistics of knotting and linking for
equilibrium lattice models of polymers confined to lattice tubes. In this talk I will review our theorems and transfer-matrix-based numerical results for the knot statistics and knot localization of
self-avoiding polygon models in small tubes. These results have recently been extended by Jeremy Eng to slightly larger tube sizes than previously reported (namely 2x2, 4x1, 5x1 and 3x2). The trends
previously observed for smaller tube sizes continue to hold for these tube sizes. In particular we observe two modes of knotting (2-filament and 1-filament) in all tube sizes and our numerical
evidence indicates that the 2-filament mode is more probable. These same two modes of knotting have been observed by others both in DNA experiments and in molecular dynamics simulations. Finally, I
will present recent results and open questions about link statistics for pairs of polygons which span a lattice tube.
Kumar Rajeev
Topological effects in polymers.
In this talk, I will present our on-going work related to understanding topological effects in melts of rings and trefoil knotted polymers. The talk will include synthesis and modeling work in
understanding topological effects in polymer melts. For the modeling work, issues of Gauge invariance in the field theory of polymers will be discussed. Also, coarse-grained molecular dynamics
simulation results will be presented showing the effects of polymer chain topology in affecting disorder-order transition in diblock copolymers.
Dawn Ray
The number of oriented rational links with a given deficiency number.
Let Un be the set of un-oriented and rational links with crossing number n, a precise formula for |Un| was obtained by Ernst and Sumners in 1987. In this paper, we study the enumeration problem of
oriented rational links. Let Mn be the set of oriented rational links with crossing number n and let Mn(d) be the set of oriented rational links with crossing number n (n ≥ 2) and deficiency d. In
this paper, we derive precise formulas for $|\lambda_n|$ and $|\lambda_n(d)|$ for any given $n$ and $d$ and show that
|\lambda_n(d)|=F_{n-d-1}^{(d)}+\frac{1+(-1)^{nd}}{2}F^{(\lfloor \frac{d}{2}\rfloor)}_{\lfloor \frac{n}{2}\rfloor -\lfloor \frac{d+1}{2}\rfloor},
where $F_n^{(d)}$ is the convolved Fibonacci sequence.
Sofia Lambropoulou
Finite type invariants of knotoids.
In this talk extend the theory of finite type invariants for knots to knotoids. For spherical knotoids we show that there are non-trivial type 1 invariants, in contrast with classical knot theory
where type 1 invariants vanish. We give a complete theory of type 1 invariants for spherical knotoids, by classifying linear chord diagrams of order one, and we present examples arising from the
affine index polynomial and the extended bracket polynomial.
Eleni Panagiotou
Knot Polynomials of open and closed curves.
In this talk we introduce a method to measure entanglement of curves in 3-space that extends the notion of knot and link polynomials to open curves. We define the bracket polynomial of curves in
3-space and show that it has real coefficients and is a continuous function of the curve coordinates. This is used to define the Jones polynomial in a way that it is applicable to both open and
closed curves in 3-space. For open curves, the Jones polynomial has real coefficients and it is a continuous function of the curve coordinates and as the endpoints of the curve tend to coincide, the
Jones polynomial of the open curve tends to that of the resulting knot. For closed curves, it is a topological invariant, as the classical Jones polynomial. We show how these measures attain a
simpler expression for polygonal curves and provide a finite form for their computation in the case of polygonal curves of 3 and 4 edges.
Quensisha Baldwin
The topological free energy of viral glycoproteins.
Many viruses infect cells by using a mechanism that involves binding of a viral protein to the host cell. During this process, the three-dimensional conformation of the viral binding protein changes
significantly. Disruption of this process could be achieved by targeting key locations in the viral protein that are essential in this rearrangement. In this manuscript we propose to use the local
geometry/topology of the crystal structure of the viral protein backbone alone to identify these essential locations. Our results show that the local Writhe, local Average Crossing Number and the
local Torsion alone can identify “exotic” residues that may be essential in the viral protein infection mechanism. We apply this method to the SARS-Cov-2 Spike protein to propose target residues for
drug discovery.
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Finding the Domain and Range of Absolute Value Functions
Question Video: Finding the Domain and Range of Absolute Value Functions Mathematics • Second Year of Secondary School
Determine the domain and the range of the function ๐ (๐ ฅ) = |5๐ ฅ + 5| โ 9.
Video Transcript
Determine the domain and range of the function ๐ of ๐ ฅ is equal to the absolute value of five ๐ ฅ plus five minus nine.
We begin by recalling what we actually mean by the domain and range of a function. We say that the domain of a function is the complete set of possible values for our independent variable, in other
words, the set of ๐ ฅ-values that make the function work and will essentially output real values for ๐ ฆ. Then, the range is the complete set of all possible resulting values of the dependent
variable after we substituted the domain. In other words, the range is the resulting ๐ ฆ-values we get after substituting in all possible ๐ ฅ-values.
Now, letโ s think about what the function ๐ of ๐ ฅ is telling us. We take values of ๐ ฅ, substitute them into the expression five ๐ ฅ plus five, and then make that positive. Then, we subtract
nine. There are no values of ๐ ฅ which donโ t actually make the function work, so the domain of our function is simply all real numbers.
But what about the range? Well, thereโ s two ways we can calculate this. Firstly, letโ s just think about it algebraically. If we have the absolute value of five ๐ ฅ plus five, what do we get when
we substitute values of ๐ ฅ in? Well, if ๐ ฅ is equal to negative one, we get five times negative one plus five, which is zero. And the absolute value of zero is simply zero. For all other real
values of ๐ ฅ, we get a result thatโ s greater than zero. And so, we can say that the range of the function the absolute value of five ๐ ฅ plus five is greater than or equal to zero.
Weโ re going to subtract nine from both sides of this inequality. And we find that the absolute value of five ๐ ฅ plus five minus nine must be greater than or equal to negative nine. And so, using
set notation for the range, we find itโ s greater than or equal to negative nine and less than โ .
But there is another way we can find the range. And thatโ s to consider the graph of the function. We take the graph of ๐ ฆ equals five ๐ ฅ plus five. Itโ s a single straight line that passes
through the ๐ ฆ-axis at five and the ๐ ฅ-axis at negative five. We then find the absolute value of all of our outputs, of all of our values of ๐ ฆ. In other words, any values on the graph that lie
below the ๐ ฅ-axis get reflected in the ๐ ฅ-axis, as shown. Notice that this does include ๐ ฆ equals zero.
By subtracting nine from our function, we translate it down by nine units. The lowest point on our graph has a coordinate of negative five, negative nine. And we said that the range is all possible
values of ๐ ฆ after weโ ve substituted our possible values of ๐ ฅ in. We see from our graph that ๐ ฆ is always greater than or equal to negative nine. And so, we end up with the same range as
before. Itโ s greater than or equal to negative nine and less than โ . | {"url":"https://www.nagwa.com/en/videos/780141645152/","timestamp":"2024-11-12T23:47:02Z","content_type":"text/html","content_length":"252008","record_id":"<urn:uuid:3346598b-26b2-4419-b5e3-0a942e5adb86>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00043.warc.gz"} |
LibGuides: Math Resources: Basic Probability Rules
This page should be used as a quick reference guide for this unit. Some terminology and concepts presented here are covered in more detail on other pages. Refer back to these rules as needed.
Basic Probability Rules
1) Possible values for probabilities range from 0 to 1
0 = impossible event
1 = certain event
2) The sum of all the probabilities for all possible outcomes is equal to 1.
Note the connection to the complement rule.
3) Addition Rule - the probability that one or both events occur
mutually exclusive events: P(A or B) = P(A) + P(B)
not mutually exclusive events: P(A or B) = P(A) + P(B) - P(A and B)
4) Multiplication Rule - the probability that both events occur together
independent events: P(A and B) = P(A) * P(B)
P(A and B) = P(A) * P(B|A)
5) Conditional Probability - the probability of an event happening given that another event has already happened
P(A|B) = P(A and B) / P(B)
*Note the line | means "given" while the slash / means divide
Key Terminology
Mutually Exclusive - this indicates that two events cannot happen at the same time.
For example, consider the following two events: A) rolling a 2 and B) rolling an odd number. Since 2 is an even number, it's not possible for me to roll a 2 and for that number to be odd. Therefore,
these events are mutually exclusive.
Independent Events - the probability of one event does not change based on the outcome of the other event
Consider a basketball player shooting 2 free throws. If the player's probability of making the second shot changes based on whether or not they make the first shot, then these events are dependent.
If the probability does not change, then they would be independent. | {"url":"https://resources.nu.edu/c.php?g=1336977&p=10407696","timestamp":"2024-11-07T17:16:58Z","content_type":"text/html","content_length":"30950","record_id":"<urn:uuid:f7710a70-a54c-435a-a8dc-4dd3277939ae>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00869.warc.gz"} |
Image Classification Based on Histogram Intersection Kernel
1. Introduction
With the rapid increase of digital images, it is impossible to label them manually. How to classify these images quickly and accurately becomes an important research topic currently.
Support vector machine (SVM) is a kernel based supervised learning classifier, based on Statistical Learning Theory (SLT), which is a widely used method in classification problems, e.g., natural
language processing [1], information retrieval [2] [3] and data mining [4] [5]. However, how to choose a right kernel is a challenge work in the SVM. Wu proposed an interpolation kernel, which has
less subjectivity and more predominance of generalization than most of traditional kernel functions while needing vast calculation [6]. Neumann use wavelet kernel SVM as a classifier which solves
difficult signal classification problems in cases where solely using a large margin classifier may fail [7]. Radial Basis kernel Function was proved its effective learning performance, while its
extrapolation capability decreases with the parameters’ increasing. Polynomial kernel function has a good performance in global, while its performance is poor in local. Different kernels result in
different results; therefore, it’s essential to choose a right kernel for specific classification task.
In this paper, Histogram Intersection Kernel SVM was used in the image classification. Firstly, the original images were split into blocks by the regular grid with equal size of B × B. Secondly the
Scale Invariant Feature Transform (SIFT) descriptors were extracted from those blocks. In order to get the dictionary of Bag Of Word (BOW) model, we use k-means clustering algorithm to cluster those
descriptors into k groups, each of them was regarded as a visual keywords which making up the dictionary, and we can get the histogram of each images, that gives the frequency of each visual keywords
contained in the dictionary. Finally, Histogram Intersection Kernel should be constructed with these histograms.
2. Image Preprocessing and Feature Extraction
2.1. Image Preprocessing
We use the Corel-low image to test our method, and the original images were split into blocks which have equal size of B × B with the regular grid. Figure 1 shows the schematic diagram of how to
split the original images.
2.2. Extracting Feature
Scale Invariant Feature Transform (SIFT) was first proposed by David G.L. in 1999 [8], and was improved in 2004 [9]. SIFT descriptor is an invariant image local feature description operator of
invariant image that is based on scale space, image rotation and even affine transformation. The core issue of object recognition is that to match a target to its images no matter if they are at
different times, different light, and different poses. However, since the state of target and the impact of the environment, the image in the same type differs from each other greatly in different
scenes, and therefore people can identify the same object by the local common character. The so-called local feature descriptor is used to describe the local feature of the image. The ideal local
feature should be robust to translation, scaling, rotation, the lighting changes, affine and projector impact. SIFT descriptor is a typical local feature.
Then we extract SIFT descriptor from each block after preprocessing of the original images, Figure 2 shows the diagram of the extraction of SIFT descriptor, and the green marks are the SIFT
descriptor. A SIFT descriptor is denoted by a circle, representing its support region, and one of its radii, representing its orientation.
2.3. Bag-of-Words Model
Bag-of-Words (BOW) model was used to express a document (or sentence) as a histogram of frequency of words, ignoring the order of these words in natural language processing firstly [10]. It was used
to perform the classification of computer images by Li [11].This model can learn characteristic labels of scenes without human intervention in the training database. But its performance for the
images of indoor scenes is lackluster [11].
Since SIFT descriptors were extracted, the local feature of each image was obtained. Then these descriptors were clustered into k groups via k-means clustering algorithm. The k groups are regarded as
visual keywords making up the dictionary. In order to let the images represented by the visual keywords of the dictionary, every descriptor of each image is replaced by a nearest visual keyword in
Euclidean distance. All the descriptors of the
Figure 1.Each image was partitioned into blocks with regular grid.
Figure 2. Example of extracting SIFT descriptor.
images can be represented as the visual keywords. After that, the images were expressed with a histogram of the frequencies of the visual keywords. Assume that each image is expressed by a histogram
3. Support Vector Machine
Support Vector Machines (SVM) was proposed by Vapnik [12] [13], which is commonly used in classification, regression, and other learning tasks [14]. SVM is built on the theory of max-margin
classification hyper-plane.
The dual problem of the Equation (1) is defined as follow.
Then we get the decision function:
However, in the practical applications, the problems are nonlinear separable. The solution to this problem is to induce a kernel function into the optimization problem, which maps the data into a
higher dimensional space. To avoid complicated calculation in the higher dimension, the kernel function that satisfies Mercer’s theorem has
calculated the data in the original dimension beforehand. Given
Here are some typical kernel functions [17]:
Linear kernel:
Polynomial kernel:
Radial Basis Function (RBF) kernel:
4. Histogram Intersection Kernel Construction
Each image is divided into blocks with the equal size B × B on a regular grid, the spatial position can be characterized for each block in an image, and the SIFT descriptors are extracted from each
block. The frequency of these visual keywords in an image can be represented as a histogram
This function is a positive definite kernel, so it can be used to quantify the similarity of two images, which also satisfies the Mercer’s theorem.
Histogram intersection kernel function, which contains the spatial location of objects, is robust to many transformations. Such as, distractions in the background of the object, view point changing,
occlusion [18].
5. Experiment and Analysis
Our PC configuration is: windows 8.1 system, Intel Core i7 2.3 GHz, with 6 GB memory and the programming software is Matlab 2012a. We use the Corel-low images as our experimental data, which contains
59 classes and each class has 100 pictures. The number of classes has an influence on the result of classification. In order to make sure that whither the classification result of Histogram
Intersection kernel SVM is robust to different numbers of classes Thus, two groups of experiment were designed. The first group contains 12 classes selected from the 59 classes randomly, and 70
pictures of each class are used for training and the rest for testing randomly too, we name it the group one. The second is to use all the 59 classes and take 80 pictures of each class for training
and the rest for testing, we name it the group two.
The size of the regular grids is given 16 × 16, split the original images into blocks. Then SIFT descriptors were extracted from every block. The size of dictionary has a great influence on the
classification accuracy. At the same time the larger size cost more in the calculation. So we get the dictionary size by seeking a tradeoff between the classification accuracy and the calculation
cost. Finally the size of the dictionary is 1000 for the group one and 3000 for the group two. Note that RBF kernel is a typical kernel for SVM [14], so RBF kernel function is taken as ground-truth
to compare with histogram intersection kernel SVM. RBF kernel function has one kernel parameter γ. In C-Support Vector Machine (C-SVM), there is a penalty parameter C which is the cost of C-SVM. We
find the parameter by grid searching method.
Table 1 shows the size of dictionary, best parameters of SVM and average classification accuracy of the two kernels for both groups of experiments. The average classification accuracy of RBF kernel
is about 5.8% lower than Histogram Intersection kernel in group one and 0.8% lower than Histogram Intersection kernel in group two. The RBF kernel costs more in the calculation and needs lots of time
to find the best parameters since it requests two parameters.
And Figure 3 is the confusion matrix of these two kernels for Group One. From the confusion matrix, we can see that except class Flower and Africa, the accuracy of Histogram Intersection kernel is
higher than the RBF kernel.
Figure 4 is the confusion matrix of these two kernels for Group Two. There are 37 classes whose prediction accuracy of Histogram Intersection kernel is higher than or equal to the RBF kernel in the
59 classes.
Comparing histogram intersection kernel with RBF kernel, we made a conclusion that histogram intersection kernel has higher accuracy and requests less computation time than RBF kernel for image
Figure 3. Confusion matrix of group one.
Figure 4. Confusion matrix of group two.
Table 1. The parameters and average classification of the two groups.
6. Conclusions
Image classification is an important research field in pattern recognition; SVM is a good classifier for non-linear classification problem. And the kernel function selection is the core for the
classification. To find a right kernel is important to the classification problem. The experimental results show that, the performance of the Histogram Intersection Kernel is better than RBF kernel
SVM. In our future work, there are several problems we should consider. 1) The size of dictionary influences on both the classification accuracy and the computation time greatly, therefore a new
method that is used to represent an image, needs to be proposed. 2) A kernel whose effect is robust to the increase of the number of classes has to be proposed in the future.
This work was jointly supported by the Natural Science Fund Project of Education Department of Shaanxi Provincial Government (Grant No. 14JK1658), the Natural Science Foundation of China (Grant No.
61302050), and the College students innovation and entrepreneurship training program of Xi’an University of Posts and Telecommunications. | {"url":"https://www.scirp.org/journal/paperinformation?paperid=61316","timestamp":"2024-11-03T22:18:24Z","content_type":"application/xhtml+xml","content_length":"105856","record_id":"<urn:uuid:93b85c9a-d7d5-492a-b9e3-3d4160eb6485>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00557.warc.gz"} |
Square centimeter conversions
The square centimeter conversion selector on this page selects the area/surface measurement unit to convert to starting from square centimeters (sq cm). To make a conversion starting from a unit of
area other than square centimeter, simply click on the "Reset" button.
About square centimeter
The square centimeter is a unit of area in the metric system equal to 0.0001 square meters (1 cm^2 = 0.0001 m^2), the standard derived unit of area in the International System of Units (SI).
One square centimeter (cm^2) is also equal to 100 square millimeters (mm^2), or 10^8 square micrometers (μm^2) or 10^14 square nanometers (nm^2) (units of area in the SI), or 0.15500031 square inches
(sq in) or 0.00107639104 square feet (sq ft), which are Imperial / US customary units of area.
The square centimeter (cm^2) is an SI derived unit of area. The centimeter (cm) is a unit of length in the SI. An area of 1.0 square centimeter is the area of a square that is 1.0 centimeter on each
Symbol: cm^2
Plural: square centimeters
Also called: square centimetre (plural: square centimetres, Int. spelling)
Square centimeter conversions: a list with conversions from square centimeters to other (metric, imperial, or customary) area and surface measurement units is presented below. | {"url":"https://www.conversion-website.com/area/from-square-centimeter.html","timestamp":"2024-11-03T15:57:43Z","content_type":"text/html","content_length":"14149","record_id":"<urn:uuid:b26e10ef-09ed-4c88-bc3b-ed3d6cfae79e>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00751.warc.gz"} |
How do you graph by plotting points for f(x)=5x-8? | HIX Tutor
How do you graph by plotting points for f(x)=5x-8?
Answer 1
Generate values of $f \left(x\right)$ for a few values of $x$,
then plot your points $\left(x , f \left(x\right)\right)$ and connect with a line.
Since the equation is linear only 2 points are needed, but several will help ensure accuracy.
$x = 0 \rightarrow f \left(x\right) = - 8$$\textcolor{w h i t e}{\text{XXXX}}$$\Rightarrow \left(0 , - 8\right)$ is a point on the line
$x = 1 \rightarrow f \left(x\right) = - 3$$\textcolor{w h i t e}{\text{XXXX}}$$\Rightarrow \left(1 , - 3\right)$ is a point on the line
$x = 2 \rightarrow f \left(x\right) = 2$$\textcolor{w h i t e}{\text{XXXX}}$$\Rightarrow \left(2 , 2\right)$ is a point on the line
$x = 3 \rightarrow f \left(x\right) = 7$$\textcolor{w h i t e}{\text{XXXX}}$$\Rightarrow \left(3 , 7\right)$ is a point on the line
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Answer 2
To graph the function f(x) = 5x - 8 by plotting points, you would choose several values of x, plug them into the equation to find the corresponding values of f(x), and then plot those points on a
coordinate plane. After plotting multiple points, you can connect them with a straight line to create the graph of the function.
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Answer 3
To graph the function f(x) = 5x - 8 by plotting points:
1. Choose several values for x.
2. Calculate the corresponding values for f(x) using the equation f(x) = 5x - 8.
3. Plot the points (x, f(x)) on the coordinate plane.
4. Connect the points to form a straight line.
For example, choosing x values of -2, 0, and 2:
When x = -2, f(x) = 5(-2) - 8 = -18. When x = 0, f(x) = 5(0) - 8 = -8. When x = 2, f(x) = 5(2) - 8 = 2.
Plot the points (-2, -18), (0, -8), and (2, 2) on the coordinate plane. Then, connect these points with a straight line.
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Latest activity by Yali Chen
• Yali Chen created a post,
For the formulated optimization problem, I design an algorithm primarily based on Gurobi solving, which means that I mainly use the Gurobi to solve my optimization problem. In thi case, how to
• Yali Chen created a post,
For the formulated optimization problem, I design an algorithm primarily based on Gurobi solving, which means that I mainly use the Gurobi to solve my optimization problem. In thi case, how
to anal...
Hi, Jaromił Thanks for your reply! In the above code, we did not fix b or q, but optimized b and q simultaneously. For this code, if we fix b to optimize q, the simulation time will be faster,
Hi, Jaromił, the code is shown as follows. from gurobipy import *import gurobipy as gpfrom numpy import *import numpy as npimport mathimport random#define global variablesglobal KK = 10global MM
Hi，Jaromił I want to know why the code runs very slowly when solving variable b with fixed variable q, but solving q with fixed variable b is faster？These two variables have similar forms in
the ...
Hi, Jaromił Thanks for your reply! In gurobi, if I want the solver to output the results when the Incumbent remains unchanged for N times and the Incumbent is not infeasible, instead of waiting
Hi, Jaromił, I provide the results of the code interruption obj and the output variables q and b. Set parameter NonConvex to value 2Set parameter ScaleFlag to value 2Set parameter Aggregate to
Hi, Jaromił My original optimization objective is as follows, with optimization variables highlighted in red underline. In order to meet the solving requirements of gurobi, we have introduced
Hi, Jaromił The main problem I want to solve is the alternating iteration of the following two problems. The first one is to solve q with fixed b, and the second one is to solve b with fixed q.
• Yali Chen created a post,
My code runs as follows, Set parameter NonConvex to value 2Set parameter NumericFocus to value 2Set parameter ScaleFlag to value 2Set parameter Aggregate to value 0Set parameter MIPFocus to value | {"url":"https://support.gurobi.com/hc/en-us/profiles/18053457220497-Yali-Chen","timestamp":"2024-11-07T12:44:36Z","content_type":"text/html","content_length":"55174","record_id":"<urn:uuid:fdc4d495-6b61-462b-8643-3fab55a5a0e6>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00773.warc.gz"} |
Electricity - NCERT Questions
Q 1.
What does an electric circuit mean?
A continuous conducting path consisting of some electric components connected between the two terminals of a battery is called an electric circuit.
An electric circuit generally consisting of a bulb (1.5 V), a key, copper wires and a dry cell as shown in figure.
Q 2.
Define the unit of current.
Unit of current is ampere (A). When 1 coulomb of charge flows through a conductor in 1 second, then the current flowing through the conductor is called
1 ampere.
Q 3.
Calculate the number of electrons constituting one coulomb of charge.
Charge, Q = 1 C
Charge on an electron, q = 1.6 × 10^–19 C
No. of electrons constituting 1 C of charge
= 6.25 × 10^18 electrons
Q 4.
Name a device that helps to maintain a potential difference across a conductor.
A battery consisting of one or more electric cells is used to maintain a potential difference across a conductor.
Q 5.
What is meant by saying that the potential difference between two points is 1 V?
Potential difference between two points in a current-carrying conductor is said to be 1 volt if 1 joule of work is done to carry a charge of 1 coulomb from one point to the other, i.e.,
Q 6.
How much energy is given to each coulomb of charge passing through a 6 V battery?
Given: Charge, Q = 1 C
Applied voltage, V = 6 V
Work done in moving a charge of 1 C under the voltage of 6 V = 1 C × 6 V = 6 J Thus, an energy of 6 J should be given during the passage of one coulomb of charge through a 6 V battery.
Q 7.
On what factors does the resistance of a conductor depend?
A solution having same composition throughout is homogeneous.
Q 8.
Will current flow more easily through a thick wire or a thin wire of the same material when connected to the same source? Why?
The current flows more easily through a thick wire than through a thin wire. This is due to the reason that the resistance R of a thick wire of large area of cross-section, A is less than that of a
thin wire of small A as
Q 9.
Let the resistance of an electrical component remain constant while the potential difference across the two ends of the component decreases to half its former value. What change will occur in the
current through it?
When the potential difference across the two ends of the electrical component becomes half its former value, the current through it also becomes half. Since I = V/R, when potential difference becomes
V/2, current becomes I/2 as the resistance (R) of the component remains constant.
Q 10.
Why are the coils of electric toasters and electric irons made of an alloy rather than a pure metal?
Alloys are used for making the coils in electric toaster and electric iron because alloys have higher melting point than pure metals. So, the coils made from alloys do not melt or get deformed and
alloys do not get oxidised or burn readily at high temperatures.
Q 11.
Use the data in Table 1(12.2 in NCERT) to answer the following :
(A) Which among iron and mercury is a better conductor?
(B) Which material is the best conductor?
(A) Iron is a better conductor than mercury as resistivity (ρ) for iron
(10.0 × 10^–8 Ω m) is less than that for mercury
(94 × 10^–8 Ω m).
(B) Silver is the best conductor as its resistivity (ρ) is the least, i.e., 1.60 × 10^–8 Ω m
Q 12.
Draw a schematic diagram of a circuit consisting of a battery of three cells of 2 V each, a 5 Ω resistor, a 8 Ω resistor, and a 12 Ω resistor, and a plug key, all connected in series.
The schematic circuit diagram is shown below:
Q 13.
Redraw the circuit of question-12, putting in an ammeter to measure the current through the resistors and a voltmeter to measure the potential difference across the 12 Ω resistor. What would be the
reading in the ammeter and the voltmeter?
The completed circuit diagram is given below:
For the whole circuit,
Total resistance = 5 Ω + 8 Ω + 12 Ω = 25 Ω
Total applied voltage = 2 V + 2 V + 2 V = 6 V
current flowing through the resistors,
So, the ammeter will show a reading of 0.24 A. Voltage across the 12 Ω resistor = IR = 0.24 A × 12 Ω = 2.88 V
So, the voltmeter will show a reading of 2.88 V
Q 14.
Judge the equivalent resistance when the following are connected in parallel:
(A) 1 Ω and 10^6 Ω,
(B) 1 Ω and 10^3 Ω and 10^6 Ω
(A) For a parallel combination,
(1 is negligible as compared to 10^6)
(B) For the other parallel combination, we can write
= 1 Ω^-1 + 0.001 Ω^-1 + 0.000001 Ω^-1
= 1.001 Ω^-1
Q 15.
An electric lamp of 100 Ω, a toaster of resistance 50 Ω and a water filter of resistance 500 Ω are connected in parallel to a 220 V source. What is the resistance of an electric iron connected to the
same source that takes as much current as all three appliances and what is the current through it?
Resistance of the electric lamp, R[1] = 100 Ω resistance of toaster, R[2] = 50 Ω resistance of water filter, R[3] = 500 Ω Since R[1], R[2] and R[3] are connected in parallel, their equivalent
resistance (R[p]) is given by
Current through the three appliances, i.e.,
Since the electric iron connected to the same source (i.e., 220 V), takes as much current as all three appliance, i.e., I, its resistance is equal to R[p],
Q 16.
What are the advantages of connecting electrical devices in parallel with the battery instead of connecting them in series?
(A) When a number of electrical devices are connected in parallel, each device gets the same potential difference as provided by the battery and it keeps on working even if other devices fail. This
is not so in case the devices are connected in series because when one device fails, the circuit is broken and all devices stop working.
(B) Parallel circuit is helpful when each device has different resistance and requires different current for its operation as in this case the current divides itself through different devices. This
is not so in series circuit where same current flows through all the devices, irrespective of their resistances.
Q 17.
How can three resistors of resistances 2 Ω, 3 Ω and 6 Ω be connected to give a total resistance of
(A) 4 Ω (B) 1 Ω?
(A) Following combination will give a total resistance of 4 Ω
Thus, the resistances of 3 Ω and 6 Ω are connected in parallel and this combination is combined with 2 Ω resistance in series.
(B) By connecting all the three resistances in parallel.
Q 18.
What is (A) the highest, (B) the lowest total resistance that can be secured by combinations of four coils of resistances 4 Ω, 8 Ω, 12 Ω, 24 Ω?
(A) Highest resistance = R[1] + R[2] + R[3] + R[4]
= 4 Ω + 8 Ω + 12 Ω + 24 Ω = 48 Ω
(B) If R is the lowest resistance, Then
So, the lowest resistance of the combination = 2 Ω
Q 19.
Why does the cord of an electric heater not glow while the heating element does?
The cord of an electric heater is made of thick copper wire and has much lower resistance than its element. For the same current (I) flowing through the cord and the element, heat produced (I^2Rt) in
the element is much more than produced in the cord. Consequently, the element becomes very hot and glows whereas the cord does not become hot and as such does not glow.
Q 20.
Compute the heat generated while transferring 96000 coulombs of charge in one hour through a potential difference of 50 V.
Here, Q = 96000 C, t = 1 h = 60 × 60 = 3600 s, V = 50 V
Heat produced, W = QV = 96000 C × 50 V = 48 × 10^5 J.
Q 21.
An electric iron of resistance 20 Ω takes a current of 5 A. Calculate the heat generated in 30 s.
Resistance, R = 20 Ω
Current, I = 5 A
Time, t = 30 s
Heat generated = I^2Rt = (5 A)^2 × 20 Ω × 30 s = 25 × 20 × 30 J = 15000 J = 15 kJ
Q 22.
What determines the rate at which energy is delivered by a current?
Rate at which energy is delivered by a current is called electric power.
Electric power is determined by
(A) The potential difference across the conductor in volts
(B) The current in amperes.
Q 23.
An electric motor takes 5 A from a 220 V line. Determine the power and energy consumed in 2 h.
Here, I = 5 A, V = 220 V, t = 2 h = 2 × 60 × 60 = 7200 s
Power, P = VI = 220 × 5 = 1100 W
Energy consumed, W
= VI t = 220 × 5 × 7200 = 7920000 J
Q 24.
A piece of wire of resistance R is cut into five equal parts. These parts are then connected in parallel. If the equivalent resistance of this combination is R, then the ratio R/R’
(A) 1/25
(B) 1/5
(C) 5
(D) 25
Resistance of each one of the five parts =
Resistance of five parts connected in parallel is given by
Thus, options (D) is the correct answer.
Q 25.
Which of the following terms does not represent electrical power in a circuit?
(A) I^2R
(B) IR^2
(C) VI
(D) V^2/R
Electrical power, P = VI = (IR)I
= I^2R = V^2/R
Obviously, IR^2 does not represent electrical power in a circuit.
Thus, option (B) is the correct answer.
Q 26.
An electric bulb is rated 220 V and 100 W. When it is operated on 110 V, the power consumed will be
(A) 100 W
(B) 75 W
(C) 50 W
(D) 25 W
Resistance of the bulb
Power consumed at 110 V
Thus, option (D) is the correct answer
Q 27.
Two conducting wires of the same material and of equal lengths and equal diameters are first connected in series and then in parallel in an electric circuit. The ratio of the heat produced in series
and parallel combinations would be
(A) 1 : 2
(B) 2 : 1
(C) 1 : 4
(D) 4 : 1
Since both the wires are made of the same material and have equal lengths and equal diameters, these have the same resistance. Let it be R.
When connected in series, their equivalent resistance is given by
R[s] = R + R = 2R
When connected in parallel, their equivalent resistance is given by
Further, electrical power is given by
Power or heat produced in series,
Power or heat produced in parallel,
or P[s] : P[p] = 1 = 4
Thus, option (C) is the correct answer.
Q 28.
How is voltmeter connected in the circuit to measure potential difference between two points?
A voltmeter is always connected in parallel across the points between which the potential difference is to be determined.
Q 29.
A copper wire has a diameter of 0.5 mm and a resistivity of 1.6 × 10^–6 ohm cm. How much of this wire would be required to make a 10 ohm coil? How much does the resistance change if the diameter is
Given diameter of the wire, D = 0.5 × 10^–3 m
resistivity of copper, ρ = 1.6 × 10^–6 ohm cm
= 1.6 × 10^–8 ohm m
required resistance, R = 10 ohm
[∵ A = πr^2 = π(D/2)^2 = πD^2/4]
When D is doubled, R becomes
Q 30.
The values of current, I, flowing in a given resistor for the corresponding values of potential difference, V, across the resistor are given here :
I (ampere) 0.5 1.0 2.0 3.0 4.0
V (volt) 1.6 3.4 6.7 10.2 13.2
Plot a graph between V and I and calculate the resistance of the resistor.
The V – I graph is as shown in figure.
For V = 4 V (i.e., 9 V – 5 V), I = 1.25 A (i.e., 2.65 A – 1.40 A).
The value of R obtained from the graph depends upon the accuracy with which the graph is plotted.
Q 31.
When a 12 V battery is connected across an unknown resistor, there is a current of 2.5 mA in the circuit. Find the value of the resistance of the resistor.
Here, V = 12 V, I = 2.5 mA = 2.5 10^–3 A
Resistance of the resistor,
Q 32.
A battery of 9 V is connected in series with resistors of 0.2 Ω, 0.3 Ω, 0.4 Ω, 0.5 Ω and
12 Ω. How much current would flow through the 12 Ω resistor?
Since all the resistors are in series, equivalent resistance,
R[s] = 0.2 Ω + 0.3 Ω + 0.4 Ω + 0.5 Ω + 12 Ω
= 13.4 Ω
Current through the circuit,
In series, same current (I) flows through all the resistors. Thus, current flowing through 12 Ω resistor = 0.67 A
Q 33.
How many 176 Ω resistors (in parallel) are required to carry 5 A on a 220 V line?
Let there be n resistors in parallel. Then the equivalent resistance is given by,
From Ohm’s law,
or n = 4
So, four resistors of 176 Ω each are required.
Q 34.
Show how you would connect three resistors, each of resistance 6 Ω, so that the combination has a resistance of (A) 9 Ω, (B) 4 Ω.
(A) The resistance of the combination is higher than each of the resistances. We can obtain 9 Ω by coupling 6 Ω and 3 Ω in series. A parallel combination of two 6 Ω resistors is equivalent to 3 Ω.
So, to obtain 9 Ω circuit, the combination (A) given alongside is possible.
(B) To obtain a combination of 4 Ω, we have to connect two 6 Ω resistors in series and then connect the third 6 Ω resistor in parallel of the series combination as shown in the figure (B).
Q 35.
Several electric bulbs designed to be used on a 220 V electric supply line, are rated 10 W. How many lamps can be connected in parallel with each other across the two wires of 220 V line if the
maximum allowable current is 5A?
In a parallel combination, each bulb has the voltage equal to that of the main line, and the sum of the currents drawn by each bulb would be equal to the allowable current.
From the given data,
Current flowing through each bulb
Total maximum allowed current = 5 A
No. of bulbs which can be connected
Q 36.
A hot plate of an electric oven connected to a 220 V line has two resistance coils A and B, each of 24 Ω resistance, which may be used separately, in series, or in parallel. What are the currents in
the three cases?
Here, potential difference, V = 220 V resistance of each coil, r = 24 Ω
(i) When each of the coils A or B is connected separately, current through each coil, i.e.,
(ii) When coils A and B are connected in series, equivalent resistance in the circuit,
R[s] = r + r = 2r = 48 Ω
Current through the series combination,
(iii) When the coils A and B are connected in parallel, equivalent resistance in the circuit,
Current through the parallel combination,
Q 37.
Compare the power used in the 2 Ω resistor in each of the following circuits :
(i) a 6 V battery in series with 1 Ω and 2 Ω resistors, and
(ii) a 4 V battery in parallel with 12 Ω and 2 Ω resistors.
(i) Since 6 V battery is in series with 1 Ω and 2 Ω resistors, current in the circuit,
Power used in 2 Ω resistor,
P[1] = I^2R = (2A)^2 × 2 Ω = 8 Ω
(ii) Since 4 V battery is in parallel with
12 Ω and 2 Ω resistors, potential difference across 2 Ω resistors, V = 4 V.
Power used in 2 Ω resistor,
Q 38.
Two lamps, one rated 100 W at 220 V, and the other 60 W at 220 V, are connected in parallel to electric mains supply.
What current is drawn from the line if the supply voltage is 220 V?
Since both the bulbs are connected in parallel and to a 220 V supply, the voltage across each bulb is 220 V. Then
Current drawn by 100 W bulb,
Current drawn by 60 W bulb,
Total current drawn from the supply line,
I = I[1] + I[2] = 0.954 A + 0.273 A = 0.73 A
Q 39.
Which uses more energy, a 250 W TV set in 1 h or a 1200 W toaster in 10 minutes?
Energy consumed by TV set = 250 W × 1 h
= 250 J s^–1 × 60 × 60 s = 900, 000 J
Energy consumed by toaster
= 1200 W × 10 min
= 1200 J s^–1 × 10 × 60 s = 720, 000 J
Thus, the TV set will use more energy.
Q 40.
An electric heater of resistance 8 Ω draws 15 A from the service mains for 2 hours. Calculate the rate at which heat is developed in the heater.
Resistance, R = 8 Ω, Current, I = 15 A,
Time, t = 2 h
Rate of generation of heat = I^2 × R
= (15A)^2 × 8 Ω = 15 × 15 × 8 W = 1800 W
Q 41.
Explain the following :
(A) Why is tungsten used almost exclusively for filament of electric lamps?
(B) Why are the conductors of electric heating devices, such as toasters and electric ions, made of an alloy rather than a pure metal?
(C) Why is the series arrangement not used for domestic circuits?
(D) How does the resistance of a wire vary with its area of cross-section.
(E) Why are copper and aluminium wires usually employed for electricity transmission.
(A) Tungsten has a high melting point (3380°C) and becomes incandescent (i.e., emits light at a high temperature) at 2400 K.
(B) The resistivity of an alloy is generally higher than that of pure metals of which it is made of.
(C) In series arrangement, if any one of the applicances fails or is switched off, all the other applicances stop working because the same current is passing through all the appliances.
(D) The resistance of a wire (R) varies inversely as its cross-sectional area (A) as
R ∝ 1/A.
(E) Copper and aluminium wires possess low resistivity and as such are generally used for electricity transmission. | {"url":"https://www.champstreet.com/ncert-solutions/class-10/physics/chapter-12/electricity.jsp","timestamp":"2024-11-12T19:01:59Z","content_type":"text/html","content_length":"158370","record_id":"<urn:uuid:f22f68e5-02bf-4b31-9131-49bdd030f35d>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00542.warc.gz"} |
ON-BOTTOM STABILITY REPORT FOR AFANG-B-00/419 project - Opportunities Goal
ON-BOTTOM STABILITY REPORT FOR AFANG-B-00/419 project
       INTRODUCTION
1.1Â Â Â Â Â Â Â Â Â Â General
The AFANG field is located in block-00 OML 419, approximately 45 km of the south-eastern coast of Wakanda in approximately 40 meters water depth. The field, initially brought into production in 1997
is owned by the Joint Venture APC/PDP and is operated by APPDPLPC Nigeria Limited (APPDPLPCNL). The OML 419 block is shown in the following figure.
Figure 1-1: Project Scope
Figure 1‑2: AFANG Field Architecture
1.2Â Â Â Â Â Â Â Â Â Â Objective
The objective of this document is to confirm adequacy for use, the already selected pipeline concrete weight coating thickness for the 12-inch APC-2 to PDP-1 Crude Export Line.
The line pipe has been coated with 2.7mm 3LPP coating, as well as 30mm concrete weight coating.
1.3Â Â Â Â Â Â Â Â Â Â Definitions and Abbreviations
                    Â
Company APPDPLPC Nigeria Limited
Contractor Global Oceon Nig. LTD
The Party including its employees, agents, inspectors and other
authorized representatives Contracted by the Company/ Contractor to carry out the Procurement, Construction Installation and Commissioning activities on the Project
Shall Indicates mandatory requirement
Should Indicates preferred recommendation
Table 1-1: Definitions
1.4  Abbreviations                     Table 1-2: Abbreviations
Abbreviation Definition
API American Petroleum Institute
APS Application Procedure Specification
ASME American Society of Mechanical Engineers
CA Corrosion Allowance
CP Cathodic Protection
DNV Det Norske Veritas
DP Dynamic Positioning
3LPP 3-Layer Polypropylene
FJ Field Joint
FJC Field-Joint Coating
FOS Factor of Safety
HRC Hardness Rockwell C Scale
ID Inner Diameter
LAT Lowest Astronomical Tide
OD Outside Diameter
SG Specific Gravity
SMYS Specified Minimum Yield Strength
WT Wall Thickness
1.5.1Â Â Â Â Â Â Â Â Â Conflict of Information
Where conflict occurs between the requirements of this specification and referenced Codes and Standards, the CONTRACTOR shall notify the COMPANY in writing immediately for resolution. In the absence
of such a statement, full compliance with the order of precedence below shall be assumed.
The order of precedence for the documents shall be as follows:
• Nigerian National Regulations Standards
• Project Specification
• International Codes and Standards
Where there are conflicts of interpretation, the principal will review to determine what should apply.
1.5.2 Project Documents
REF. DOC No DOCUMENT TITLE REV.
[R1] LP-NG-HCD2023-RPT-018 Wall Thickness Calculation Report 1
[R2] LP-NG-HCD2023-RPT-017 Pipeline Design Basis 2
Table 1-3: Reference and Document
1.5.3 Codes and Standards
REF. DOCUMENT NUMBER DOCUMENT TITLE REV.
[R3] DNVGL-RP-F109 On-bottom stability design of submarine pipelines –
Table 1-4: Codes and Standards
2.0Â SUMMARY AND CONCLUSION
2.1Â Â Â Summary
The on-bottom stability analysis has been assessed for the 12-inch APC-2 to PDP-1 Crude Export Line and the conservative absolute static stability requirement was considered as per DNVGL-RP F109. The
Table 2-1 summarizes the results of the on-bottom stability calculations.
Table 2-1: Absolute Stability Results for Significant Wave Height and Bottom Current Extremes
Concrete Weight Coating Calculated Absolute Allowable SG
Flowline Load Case Condition Thickness Stability FOS FOS Remark
Horizontal Vertical
1yr wave, 10yr 1.12 5.78
Installation (empty) Current 1.66
10yr wave, 1yr 1.44 7.55
1yr wave, 10yr 1.43 7.19
12-inch APC-2 to PDP-1 Crude Hydrotest Current 30mm 1 1.96 Satisfies the DNVGL-RP-F109 absolute static stability
Export Line 10yr wave, 1yr 1.84 9.36 requirements.
100yr wave ,10yr 1.01 4.52
Operation (Corroded current 1.64
Case) 10yr wave, 100yr 1.06 4.72
(1) The specific gravity is the ratio of empty weight in the air to buoyancy and must be greater than 1.15 to satisfy the on-bottom stability requirements.
(2) The factor of safety (FOS) must exceed 1.0 to meet the DNVGL-RP-F109 absolute static stability requirements.
2.2Â Â Conclusion
It is confirmed that the already selected concrete weight coating thickness of 30mm is adequate and satisfies the DNVGL-RP-F109 absolute static on-bottom stability requirement.
3.0Â Â DESIGN BASIS
3.1Â Â Pipeline Design Data
The table below detail parameters used in the calculation spreadsheet attached.
Table 3‑1: Pipeline Characteristics
Parameters Units Values
Pipe Steel Outer Diameter (OD) Inch (mm) 12.75 (329.3)
 Pipe Wall thickness mm 12.7
  Design Pressure Barg 144.0
Design Temperature (^0C) 80
Pipe Grade – Â Â Â Â Â Â Â Â Â Â Â Â Â Â API 5L X65
SMYS MPa                    448
SMTS Mpa 531
Corrosion Allowance mm 3
Steel Poisson Ratio – 0.3
Single Joint Length m 12.2
Anti-Corrosion Coating Thickness mm 2.7
Concrete Coating Thickness mm 30
Concrete Coating Poisson Ratio – 0.2
3.2Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Material Densities
The following material densities are used for the pipeline upheaval buckling analysis.
Table 3‑2: Material Properties
Material Density (kg/m^3) Youngs Modulus (Mpa)
Steel 7850 207000
Anti-corrosion Coating 1442
Concrete Weight Coating 3044
Seawater 1030
3.3Â Â Â Content Specific Gravity
The content specific gravity is 0.19.
3.4Â Â Â Soil Data
The following soil data taken from the Pipeline Design Basis [Ref.R2] are used in the analysis.
Table 3-3: Soil Data
Parameters Units Values
Seabed Soil Type – Clay
Bulk Unit Weight kN/m^3 18
Submerged Unit Weight kN/m^3 –
 Undrained Shear Strength, Cu kPa 4.90
3.5Â Â Â Environmental Data
3.5.1Â Â Water Depth
The following environmental data taken from the Pipeline Design Basis [Ref.R2] are used in the analysis.
Table 3-4: Water Depth
 Water Depth (LAT)
Pipeline (m)
Minimum Maximum
12-inch APC-2 to PDP-1 Crude Export Line 40.0 40.0
3.5.2Â Â Â Â Â Â Â Â Â Â Wave and Current Data
Wave and current data along the pipeline routes are extracted from the Pipeline Design Basis [Ref.R2] as shown below.
Table 3-5: Significant Wave Height and Associated Current
Table 3-6: Bottom Current Extremes
Table 3-7: Wave and Wave Associate with Steady Current Extremes
           3.6   Marine Growth
The table below provides growth estimates
Table 3-8: Marine Growth ThicknessÂ
4.0Â Â DESIGN METHODOLOGY
4.1Â Design Life
The on-bottom stability analysis was based on the absolute static stability criteria set out in          DNVGL-RP-F109. In-house spreadsheet has been used for the on-bottom
stability analysis.The analysis was performed considering a minimum water depth of 40m which represents the               worst-case scenario. The effects of the wave induced velocity
along the pipelines are insignificant at   this water depth. The stability requirements for the pipeline were determined for installation, hydrotest and operation conditions.
The submerged weight of the pipelines satisfies the On-bottom stability requirements of         DNVGL RP F109. A pipeline can be considered to satisfy the absolute static stability
requirement if:
Y[sc ]:Â Â Â Â Â Â Â Â Safety Factor
W[s] :Â Â Â Â Â Â Â Submerged weight of the line (N/m)
µ :        Soil friction factor
F[y ]Â : Â Â Â Â Â Â Â Peak Horizontal force (N/m)Â eq. 3.40 in DNVGL-RP-F109
F[Z] : Â Â Â Â Â Â Â Peak vertical force (N/m)Â eq. 3.41 in DNVGL-RP-F109
F[r ]Â : Â Â Â Â Â Â Â Passive Resistance Force (N/m)
4.2Â Â Â Assumption
The following assumptions have been made:
• Flat Seabed.
• Minimum water depth has been conservatively considered for the pipeline stability analysis.
• Environmental loading is assumed to act perpendicularly to the pipeline (i.e. θc = 90°).
• Minimum content density has been considered for the operating conditions for the production line as the worst-case scenario.
• The most conservative coefficient of friction (µ=0.2) has been used in this analysis.
• Pipeline penetration and trenching was not considered.
• Marine growth was not considered for hydrotest and installation cases.
• Additional weight due to anodes or any other miscellaneous Pipeline appurtenances are considered as negligible.
4.3Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Load Cases
The following load cases had been assessed:
Table 4-1: Load Cases
Condition Analysis Water Depth Sig. Wave Height Peak Period Current Content Corrosion allowance Marine Growth
Installation Minimum Water depth along the Route w.r.t LAT 1year 1year 10year Empty 3 mm 0 mm
10year 10year 1year
Hydrotest Minimum Water depth along the Route w.r.t LAT 1year 1year 10year Water     3 mm 0 mm
10year 10year 1year
Operation 10year 10year 100year
Minimum Water depth along the Route w.r.t LAT Crude     2.7 mm 59 mm
(Corroded Case) 100year 100year 10year
5Â Â RESULT
The result of the concrete weight coating check for 12-inch APC-2 to PDP-1 Crude Export Line is presented in the Table 5-1 below. See calculation spreadsheet in appendix 1.
Table 5-1: Results Summary
Concrete Weight Coating Calculated Absolute Allowable SG
Flowline Load Case Condition Thickness Stability FOS FOS Remark
Horizontal Vertical
1yr wave, 10yr 1.12 5.78
Installation (empty) Current 1.66
10yr wave, 1yr 1.44 7.55
12-inch APC-2 to PDP-1 Crude 1yr wave, 10yr 30mm 1.43 7.19 1 Satisfies the DNVGL-RP-F109 absolute static stability
Export Line Hydrotest Current 1.96 requirements.
10yr wave, 1yr 1.84 9.36
Operation (Corroded 100yr wave ,10yr 1.01 4.52 1.64
Case) current | {"url":"http://opportunitiesgoal.com/on-bottom-stability-report-for-afang-b-00-419-project/","timestamp":"2024-11-02T18:13:15Z","content_type":"text/html","content_length":"232334","record_id":"<urn:uuid:fc01b4f7-cdb3-489b-95af-acd716012181>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00021.warc.gz"} |
How to pick a discount rate for npv
To review, both the net present value and the internal rate of return require the idea of an and even for-profit businesses have social or other goals when selecting investments. Changing the
discount rate changes the net present value. Dec 10, 2017 When choosing a discount rate in this way, if the net present value calculation is positive, it means that the real estate investment
Mar 11, 2020 Interest rate used to calculate Net Present Value (NPV). The discount rate we are primarily interested in concerns the calculation of your business' Jul 19, 2017 Choosing an appropriate
discount rate of interest to calculate the net present value of Social Security, pension lump sum, and other retirement Mar 28, 2012 IRR is the discount rate at which the Net Present Value (NPV) of
discounted cash flows (DCF) equals the stock price. IMHO, a much better use of This discounted cash flow (DCF) analysis requires that the reader supply a discount rate. In the blog post, we suggest
using discount values of around 10% for How do analysts choose the discount (interest) rate for DCF analysis? How do business people use DCF and NPV for comparing competing investment proposals
Mar 1, 2017 RELOAD YOUR SCREEN OR TRY SELECTING A DIFFERENT VIDEO With a 5 per cent discount rate (cell B11), the overall NPV for the net
Jan 24, 2017 namely, the so-called net present value. (NPV) of the by choosing a medical career rather than the NPV is calculated at the discount rate of. Jul 17, 2018 NPV(discountrate; payment1;
payment2; payment30) you might choose a discount rate of a twelfth of the competitive return - but be aware The following equation sets out a typical NPV calculation: NPVn = NPV0 + (d1NPV1) +
(d2NPV2) + (d3NPV3) + …. + (dnNPVn) Where NPVn represents the investment (cost or benefit) made at the end of the “nth” year, and “dn” is the discount rate factor in the “nth” year. d(n) = 1/(1+r)n
where r = the discount rate Net Present Value Discount Rate The most critical decision variable in applying the net present value method is the selection of an appropriate discount rate . Typically
you should use either the weighted average cost of capital for the company or the rate of return on alternative investments . In my view, the discount rate applied for computation of NPV must
correlate with the normal returns for the business in question. For instance if a business yields normal return of 15% per annum, and you need to compute NPV on the basis of returns up to 3rd year,
you will need to apply a discounting factor of 17.36 {(1.15^3)-1}/3×100. A negative NPV means only one thing for sure: that the IRR of the property investment considered is lower than the discount
rate used to calculate that particular NPV. In particular, the relationship between the discount rate used for the calculation of the NPV of a stream of cash flows and the IRR embedded in that same
cash-flow stream is As shown in the analysis above, the net present value for the given cash flows at a discount rate of 10% is equal to $0. This means that with an initial investment of exactly
$1,000,000, this series of cash flows will yield exactly 10%. As the required discount rates moves higher than 10%, the investment becomes less valuable.
A negative NPV means only one thing for sure: that the IRR of the property investment considered is lower than the discount rate used to calculate that particular NPV. In particular, the relationship
between the discount rate used for the calculation of the NPV of a stream of cash flows and the IRR embedded in that same cash-flow stream is
Mar 11, 2020 Interest rate used to calculate Net Present Value (NPV). The discount rate we are primarily interested in concerns the calculation of your business' Jul 19, 2017 Choosing an appropriate
discount rate of interest to calculate the net present value of Social Security, pension lump sum, and other retirement
Cash Flow / (1+Discount Rate)^((Year-Current Year)-0.5). For example, if the current year is 2011 and we wish to work out the net present value of the cash flow in
Mar 11, 2020 Interest rate used to calculate Net Present Value (NPV). The discount rate we are primarily interested in concerns the calculation of your business' Jul 19, 2017 Choosing an appropriate
discount rate of interest to calculate the net present value of Social Security, pension lump sum, and other retirement Mar 28, 2012 IRR is the discount rate at which the Net Present Value (NPV) of
discounted cash flows (DCF) equals the stock price. IMHO, a much better use of This discounted cash flow (DCF) analysis requires that the reader supply a discount rate. In the blog post, we suggest
using discount values of around 10% for How do analysts choose the discount (interest) rate for DCF analysis? How do business people use DCF and NPV for comparing competing investment proposals
Let’s say we want to use a 3% rate for our inflation rate. In that case, the assumed $105.00 amount we expect with very high confidence to receive as of the end of one year is equal to $105.00 /
(1+.03), or $101.94, in today’s dollars. If we had used a 0% discount rate,
Jul 17, 2018 NPV(discountrate; payment1; payment2; payment30) you might choose a discount rate of a twelfth of the competitive return - but be aware The following equation sets out a typical NPV
calculation: NPVn = NPV0 + (d1NPV1) +(d2NPV2) + (d3NPV3) + …. + (dnNPVn) Where NPVn represents the investment (cost or benefit) made at the end of the “nth” year, and “dn” is the discount rate factor
in the “nth” year. d(n) = 1/(1+r)n where r = the discount rate Net Present Value Discount Rate The most critical decision variable in applying the net present value method is the selection of an
appropriate discount rate . Typically you should use either the weighted average cost of capital for the company or the rate of return on alternative investments . In my view, the discount rate
applied for computation of NPV must correlate with the normal returns for the business in question. For instance if a business yields normal return of 15% per annum, and you need to compute NPV on
the basis of returns up to 3rd year, you will need to apply a discounting factor of 17.36 {(1.15^3)-1}/3×100. A negative NPV means only one thing for sure: that the IRR of the property investment
considered is lower than the discount rate used to calculate that particular NPV. In particular, the relationship between the discount rate used for the calculation of the NPV of a stream of cash
flows and the IRR embedded in that same cash-flow stream is As shown in the analysis above, the net present value for the given cash flows at a discount rate of 10% is equal to $0. This means that
with an initial investment of exactly $1,000,000, this series of cash flows will yield exactly 10%. As the required discount rates moves higher than 10%, the investment becomes less valuable.
Mar 1, 2017 RELOAD YOUR SCREEN OR TRY SELECTING A DIFFERENT VIDEO With a 5 per cent discount rate (cell B11), the overall NPV for the net Mar 20, 2016 Using a discount rate of 10 percent, calculate
the NPV of the modernization project. (Round present value factor calculations to 4 decimal Feb 12, 2017 This rate is used as the discount rate in the NPV method, and the minimum rate for the
DCFROR. Many risk assessment techniques use Nov 15, 2016 The discount rate is a critical input variable when calculating NPV and it are you 'missing out on' by choosing one investment over another?
Jan 24, 2017 namely, the so-called net present value. (NPV) of the by choosing a medical career rather than the NPV is calculated at the discount rate of. | {"url":"https://btctopxfxezf.netlify.app/herbick47717sydu/how-to-pick-a-discount-rate-for-npv-71","timestamp":"2024-11-12T17:19:14Z","content_type":"text/html","content_length":"34794","record_id":"<urn:uuid:a8f2515c-0b80-4c36-a5e1-151f9460aefe>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00419.warc.gz"} |
What is Scalable Machine Learning?
1. What is Scalable Machine Learning?
What is Scalable Machine Learning?
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scalability has become one of those core concept slash buzzwords of big data. it’s all about scaling out, web scale, and so on. in principle, the idea is to be able to take one piece of code and then
throw any number of computers at it to make it fast.
the terms “scalable” and “large scale” have been used in machine learning circles long before there was big data. there had always been certain problems which lead to a large amount of data, for
example in bioinformatics, or when dealing with large number of text documents. so finding learning algorithms, or more generally data analysis algorithms which can deal with a very large set of data
was always a relevant question.
interestingly, this issue of scalability were seldom solved using actual scaling in in machine learning, at least not in the big data kind of sense. part of the reason is certainly that multicore
processors didn’t yet exist at the scale they do today and that the idea of “just scaling out” wasn’t as pervasive as it is today.
instead, “scalable” machine learning is almost always based on finding more efficient algorithms, and most often, approximations to the original algorithm which can be computed much more efficiently.
to illustrate this, let’s search for nips papers (the annual advances in neural information processing systems, short nips, conference is one of the big ml community meetings) for papers which have
the term “scalable” in the title.
here are some examples:
• scalable inference for logistic-normal topic models
… this paper presents a partially collapsed gibbs sampling algorithm that approaches the provably correct distribution by exploring the ideas of data augmentation …
partially collapsed gibbs sampling is a kind of estimation algorithm for certain graphical models.
• a scalable approach to probabilistic latent space inference of large-scale networks
… with […] an efficient stochastic variational inference algorithm, we are able to analyze real networks with over a million vertices […] on a single machine in a matter of hours …
stochastic variational inference algorithm is both an approximation and an estimation algorithm.
• scalable kernels for graphs with continuous attributes
… in this paper, we present a class of path kernels with computational complexity $o(n^2(m + \delta^2 ))$ …
and this algorithm has squared runtime in the number of data points, so wouldn’t even scale out well even if you could.
usually, even if there is potential for scalability, it usually something that is “embarassingly parallel” (yep, that’s a technical term), meaning that it’s something like a summation which can be
parallelized very easily. still, the actual “scalability” comes from the algorithmic side.
so how do scalable ml algorithms look like? a typical example are the stochastic gradient descent (sgd) class of algorithms. these algorithms can be used, for example, to train classifiers like
linear svms or logistic regression. one data point is considered at each iteration. the prediction error on that point is computed and then the gradient is taken with respect to the model parameters,
giving information about how to adapt these parameters slightly to make the error smaller.
vowpal wabbit is one program based on this approach and it has a nice definition of what it considers to mean scalable in machine learning:
there are two ways to have a fast learning algorithm: (a) start with a slow algorithm and speed it up, or (b) build an intrinsically fast learning algorithm. this project is about approach (b),
and it’s reached a state where it may be useful to others as a platform for research and experimentation.
so “scalable” means having a learning algorithm which can deal with any amount of data, without consuming ever growing amounts of resources like memory. for sgd type algorithms this is the case,
because all you need to store are the model parameters, usually a few ten to hundred thousand double precision floating point value, so maybe a few megabytes in total. the main problem to speed this
kind of computation up is how to stream the data by fast enough.
to put it differently, not only does this kind of scalability not rely on scaling out, it’s actually not even necessary or possible to scale the computation out because the main state of the
computation easily fits into main memory and computations on it cannot be distributed easily.
i know that gradient descent is often taken as an example for map reduce and other approaches like in this paper on the architecture of spark , but that paper discusses a version of gradient descent
where you are not taking one point at a time, but aggregate the gradient information for the whole data set before making the update to the model parameters. while this can be easily parallelized, it
does not perform well in practice because the gradient information tends to average out when computed over the whole data set.
if you want to know more, this large scale learning challenge sören sonnneburg organized in 2008 still has valuable information on how to deal with massive data sets.
of course, there are things which can be easily scaled well using hadoop or spark, in particular any kind of data preprocessing or feature extraction where you need to apply the same operation to
each data point in your data set. another area where parallelization is easy and useful is when you are using cross validation to do model selection where you usually have to train a large number of
models for different parameter sets to find the combination which performs best. again, even here there is more potential for even speeding up such computations using better algorithms like in this
paper of mine .
i’ve just scratched the surface of this, but i hope you got the idea that scalability can mean quite different things. in big data (meaning the infrastructure side of it) what you want to compute is
pretty well defined, for example some kind of aggregate over your data set, so you’re left with the question of how to parallelize that computation well. in machine learning, you have much more
freedom because data is noisy and there’s always some freedom in how you model your data, so you can often get away with computing some variation of what you originally wanted to do and still perform
well. often, this allows you to speed up your computations significantly by decoupling computations. parallelization is important, too, but alone it won’t get you very far.
luckily, there are projects like spark and stratosphere/flink which work on providing more useful abstractions beyond map and reduce to make the last part easier for data scientists, but you won’t
get rid of the algorithmic design part any time soon.
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Property:Extended model description
This is a property of type Text.
ANUGA is a hydrodynamic model for simulating depth-averaged flows over 2D surfaces. This package adds two new modules (operators) to ANUGA. These are appropriate for reach-scale simulations of flows
on mobile-bed streams with spatially extensive floodplain vegetation. The mathematical framework for the sediment transport operator is described in Simpson and Castelltort (2006) and Davy and Lague
(2009). This operator calculates an explicit sediment mass balance within the water column at every cell in order to handle the local disequilibria between entrainment and deposition that arise due
to strong spatial variability in shear stress in complex flows. The vegetation drag operator uses the mathematical approach of Nepf (1999) and Kean and Smith (2006), treating vegetation as arrays of
objects (cylinders) that the flow must go around. Compared to methods that simulate the increased roughness of vegetation with a modified Manning's n, this method better accounts for the effects of
drag on the body of the flow and the quantifiable differences between vegetation types and densities (as stem diameter and stem spacing). This operator can simulate uniform vegetation as well as
spatially-varied vegetation across the domain. The vegetation drag module also accounts for the effects of vegetation on turbulent and mechanical diffusivity, following the equations in Nepf (1997,
ANUGA is a hydrodynamic modelling tool that allows users to model realistic flow problems in complex 2D geometries. Examples include dam breaks or the effects of natural hazards such as riverine
flooding, storm surges and tsunami. The user must specify a study area represented by a mesh of triangular cells, the topography and bathymetry, frictional resistance, initial values for water level
(called stage within ANUGA), boundary conditions and forces such as rainfall, stream flows, windstress or pressure gradients if applicable. ANUGA tracks the evolution of water depth and horizontal
momentum within each cell over time by solving the shallow water wave governing equation using a finite-volume method. ANUGA also incorporates a mesh generator that allows the user to set up the
geometry of the problem interactively as well as tools for interpolation and surface fitting, and a number of auxiliary tools for visualising and interrogating the model output. Most ANUGA components
are written in the object-oriented programming language Python and most users will interact with ANUGA by writing small Python scripts based on the ANUGA library functions. Computationally intensive
components are written for efficiency in C routines working directly with Python numpy structures.
Acronym1D is an add on to Acronym1R in that it adds a flow duration curve to Acronym1R, which computes the volume bedload transport rate per unit width and bedload grain size distribution from a
specified surface grain size distribution (with sand removed).
Acronym1R computes the volume bedload transport rate per unit width and bedload grain size distribution from a specified surface grain size distribution (with sand removed).
AeoLiS is a process-based model for simulating aeolian sediment transport in situations where supply-limiting factors are important, like in coastal environments. Supply-limitations currently
supported are soil moisture contents, sediment sorting and armouring, bed slope effects, air humidity and roughness elements.
Allow for quick estimation of water depths within a flooded domain using only the flood extent layer (polygon) and a DEM of the area. Useful for near-real-time flood analysis, especially from remote
sensing mapping. Version 2.0 offers improved capabilities in coastal areas.
Alpine3D is a model for high resolution simulation of alpine surface processes, in particular snow processes. The model can be forced by measurements from automatic weather stations or by
meteorological model outputs (this is handled by the MeteoIO pre-processing library). The core three-dimensional Alpine3D modules consist of a radiation balance model (which uses a view factor
approach and includes shortwave scattering and longwave emission from terrain and tall vegetation) and a drifting snow model solving a diffusion equation for suspended snow and a saltation transport
equation. The processes in the atmosphere are thus treated in three dimensions and coupled to a distributed one dimensional model of vegetation, snow and soil model (Snowpack) using the assumption
that lateral exchange is small in these media. The model can be used to force a distributed catchment hydrology model (AlpineFlow). The model modules can be run in a parallel mode, using either
OpenMP and/or MPI. Finally, the Inishell tool provides a GUI for configuring and running Alpine3D. Alpine3D is a valuable tool to investigate surface dynamics in mountains and is currently used to
investigate snow cover dynamics for avalanche warning and permafrost development and vegetation changes under climate change scenarios. It could also be used to create accurate soil moisture
assessments for meteorological and flood forecasting.
An extension of the WBMplus (WBM/WTM) model. Introduce a riverine sediment flux component based on the BQART and Psi models.
An open-source Python package for flexible and customizable simulations of the water cycle that treats the physical components of the water cycle as nodes connected by arcs that convey water and
pollutant fluxes between them.
Another derivative of the original SEDSIM, completely rewritten from scratch. It uses finite differences (in addition to the original particle-cell method) to speed up steady flow calculations. It
also incorporates compaction algorithms. A general description has been published.
AquaTellUs models fluvial-dominated delta sedimentation. AquaTellUS uses a nested model approach; a 2D longitudinal profiles, embedded as a dynamical flowpath in a 3D grid-based space. A main channel
belt is modeled as a 2D longitudinal profile that responds dynamically to changes in discharge, sediment load and sea level. Sediment flux is described by separate erosion and sedimentation
components. Multiple grain-size classes are independently tracked. Erosion flux depends on discharge and slope, similar to process descriptions used in hill-slope models and is independent of
grain-size. Offshore, where we assume unconfined flow, the erosion capacity decreases with increasing water depth. The erosion flux is a proxy for gravity flows in submarine channels close to the
coast and for down-slope diffusion over the entire slope due to waves, tides and creep. Erosion is restricted to the main flowpath. This appears to be valid for the river-channel belt, but
underestimates the spatial extent and variability of marine erosion processes. Deposition flux depends on the stream velocity and on a travel-distance factor, which depends on grain size (i.e.
settling velocity). The travel-distance factor is different in the fluvial and marine domains, which results in a sharp increase of the settling rate at the river mouth, mimicking bedload dumping.
Dynamic boundary conditions such as climatic changes over time are incorporated by increasing or decreasing discharge and sediment load for each time step.
BATTRI does the mesh editing, bathymetry incorporation and interpolation, provides the grid generation and refinement properties, prepares the input file to Triangle and visualizes and saves the
created grid.
BIT Model aims to simulate the dynamics of the principal processes that govern the formation and evolution of a barrier island. The model includes sea-level oscillations and sediment distribution
operated by waves and currents. Each process determines the deposition of a distinct sediment facies, separately schematized in the spatial domain. Therefore, at any temporal step, it is possible to
recognize six different stratigraphic units: bedrock, transitional, overwash, shoreface aeolian and lagoonal.
BRaKE is a 1-D bedrock channel profile evolution model. It calculates bedrock erosion in addition to treating the delivery, transport, degradation, and erosion-inhibiting effects of large,
hillslope-derived blocks of rock. It uses a shear-stress bedrock erosion formulation with additional complexity related to flow resistance, block transport and erosion, and delivery of blocks from
the hillslopes.
Barrier3D is an exploratory model that resolves cross-shore and alongshore topographic variations to simulate the morphological evolution of a barrier segment over time scales of years to centuries.
Barrier3D tackles the scale separation between event-based and long-term models by explicitly yet efficiently simulating dune evolution, storm overwash, and a dynamically evolving shoreface in
response to individual storm events and sea-level rise. Ecological-geomorphological couplings of the barrier interior can be simulated with a shrub expansion and mortality module.
BarrierBMFT is a coupled model framework for exploring morphodynamic interactions across components of the entire coastal barrier system, from the ocean shoreface to the mainland forest. The model
framework couples Barrier3D (Reeves et al., 2021), a spatially explicit model of barrier evolution, with the Python version of the Coastal Landscape Transect model (CoLT; Valentine et al., 2023),
known as PyBMFT-C (Bay-Marsh-Forest Transect Model with Carbon). In the BarrierBMFT coupled model framework, two PyBMFT-C simulations drive evolution of back-barrier marsh, bay, mainland marsh, and
forest ecosystems, and a Barrier3D simulation drives evolution of barrier and back-barrier marsh ecosystems. As these model components simultaneously advance, they dynamically evolve together by
sharing information annually to capture the effects of key cross-landscape couplings. BarrierBMFT contains no new governing equations or parameterizations itself, but rather is a framework for
trading information between Barrier3D and PyBMFT-C. The use of this coupled model framework requires Barrier3D v2.0 (https://doi.org/10.5281/zenodo.7604068) and PyBMFT-C v1.0 (https://doi.org/10.5281
Based on the publication: Brown, RA, Pasternack, GB, Wallender, WW. 2013. Synthetic River Valleys: Creating Prescribed Topography for Form-Process Inquiry and River Rehabilitation Design.
Geomorphology 214: 40–55. http://dx.doi.org/10.1016/j.geomorph.2014.02.025
Basin and Landscape Dynamics (Badlands) is a parallel TIN-based landscape evolution model, built to simulate topography development at various space and time scales. The model is presently capable of
simulating hillslope processes (linear diffusion), fluvial incision ('modified' SPL: erosion/transport/deposition), spatially and temporally varying geodynamic (horizontal + vertical displacements)
and climatic forces which can be used to simulate changes in base level, as well as effects of climate changes or sea-level fluctuations.
Bifurcation is a morphodynamic model of a river delta bifurcation. Model outputs include flux partitioning and 1D bed elevation profiles, all of which can evolve through time. Interaction between the
two branches occurs in the reach just upstream of the bifurcation, due to the development of a transverse bed slope. Aside from this interaction, the individual branches are modeled in 1D. The model
generates ongoing avulsion dynamics automatically, arising from the interaction between an upstream positive feedback and the negative feedback from branch progradation and/or aggradation. Depending
on the choice of parameters, the model generates symmetry, soft avulsion, or full avulsion. Additionally, the model can include differential subsidence. It can also be run under bypass conditions,
simulating the effect of an offshore sink, in which case ongoing avulsion dynamics do not occur. Possible uses of the model include the study of avulsion, bifurcation stability, and the morphodynamic
response of bifurcations to external changes.
Biogenic mixing of marine sediments
Blocklab treats landscape evolution in landscapes where surface rock may be released as large blocks of rock. The motion, degradation, and effects of large blocks do not play nicely with standard
continuum sediment transport theory. BlockLab is intended to incorporate the effects of these large grains in a realistic way.
CAESAR is a cellular landscape evolution model, with an emphasis on fluvial processes, including flow routing, multi grainsize sediment transport. It models morphological change in river catchments.
CASCADE combines elements of two exploratory morphodynamic models of barrier evolution -- barrier3d (Reeves et al., 2021) and the BarrierR Inlet Environment (brie) model (Nienhuis & Lorenzo-Trueba,
2019) -- into a single model framework. Barrier3d, a spatially-explicit cellular exploratory model, is the core of CASCADE. It is used within the CASCADE framework to simulate the effects of
individual storm events and SLR on shoreface evolution; dune dynamics, including dune growth, erosion, and migration; and overwash deposition by individual storms. BRIE is used to simulate
large-scale coastline evolution arising from alongshore sediment transport processes; this is accomplished by connecting individual Barrier3d models through diffusive alongshore sediment transport.
Human dynamics are incorporated in cascade in two separate modules. The first module simulates strategies for preventing roadway pavement damage during overwashing events, including rebuilding
roadways at sufficiently low elevations to allow for burial by overwash, constructing large dunes, and relocating the road into the barrier interior. The second module incorporates management
strategies for maintaining a coastal community, including beach nourishment, dune construction, and overwash removal.
CHILD computes the time evolution of a topographic surface z(x,y,t) by fluvial and hillslope erosion and sediment transport.
CICE is a computationally efficient model for simulating the growth, melting, and movement of polar sea ice. Designed as one component of coupled atmosphere-ocean-land-ice global climate models,
today’s CICE model is the outcome of more than two decades of community collaboration in building a sea ice model suitable for multiple uses including process studies, operational forecasting, and
climate simulation.
CLUMondo is based on the land systems approach. Land systems are socio-ecological systems that reflect land use in a spatial unit in terms of land cover composition, spatial configuration, and the
management activities employed. The precise definition of land systems depends on the scale of analysis, the purpose of modelling, and the case study region. In contrast to land cover classifications
the role of land use intensity and livestock systems are explicitly addressed. Each land system can be characterized in terms of the fractional land covers.<br>Land systems are characterized based on
the amount of forest in the landscape mosaic and the management type ranging from swidden cultivation to permanent cultivation and plantations.
Caesar Lisflood is a geomorphological / Landscape evolution model that combines the Lisflood-FP 2d hydrodynamic flow model (Bates et al, 2010) with the CAESAR geomorphic model to simulate erosion and
deposition in river catchments and reaches over time scales from hours to 1000's of years. Featuring: Landscape evolution model simulating erosion and deposition across river reaches and catchments A
hydrodynamic 2D flow model (based on the Lisflood FP code) that conserves mass and partial momentum. (model can be run as flow model alone) designed to operate on multiple core processors (parallel
processing of core functions) Operates over a wide range to spatial and time scales (1km2 to 1000km2, <1year to 1000+ years) Easy to use GUI
Calculate the hypsometric integral for each pixel at the catchment. Each pixel is considered a local outlet and the hypsometric integral is calculated according to the characteristics of its
contributing area.
Calculate wave-generated bottom orbital velocities from measured surface wave parameters. Also permits calculation of surface wave spectra from wind conditions, from which bottom orbital velocities
can be determined.
Calculates non-equilibrium suspended load transport rates of various size-density fractions in the bed
Calculates shear velocity associated with grain roughness
Calculates the bedload transport rates and weights per unit area for each size-density. NB. Bedload transport of different size-densities is proportioned according to the volumes in the bed.
Calculates the constant terminal settling velocity of each size-density fraction's median size from Dietrich's equation.
Calculates the critical Shields Theta for the median size of a distribution and then calculates the critical shear stress of the ith, jth fraction using a hiding function
Calculates the critical shear stress for entrainment of the median size of each size-density fraction of a bed using Yalin and Karahan formulation, assuming no hiding
Calculates the gaussian or log-gaussian distribution of instantaneous shear stresses on the bed, given a mean and coefficient of variation.
Calculates the logrithmic velocity distribution called from TRCALC
Calculates the total sediment transport rate in an open channel assuming a median bed grain size
Calculation of Density Stratification Effects Associated with Suspended Sediment in Open Channels. This program calculates the effect of sediment self-stratification on the streamwise velocity and
suspended sediment concentration profiles in open-channel flow. Two options are given. Either the near-bed reference concentration Cr can be specified by the user, or the user can specify a shear
velocity due to skin friction u*s and compute Cr from the Garcia-Parker sediment entrainment relation.
Calculation of Sediment Deposition in a Fan-Shaped Basin, undergoing Piston-Style Subsidence
Calculator for 1D Subaerial Fluvial Fan-Delta with Channel of Constant Width. This model assumes a narrowly channelized 1D fan-delta prograding into standing water. The model uses a single grain size
D, a generic total bed material load relation and a constant bed resistance coefficient. The channel is assumed to have a constant width. Water and sediment discharge are specified per unit width.
The fan builds outward by forming a prograding delta front with an assigned foreset slope. The code employs a full backwater calculation.
Calculator for 1D Subaerial Fluvial Fan-Delta with Channel of Constant Width. This model assumes a narrowly channelized 1D fan-delta prograding into standing water. The model uses a single grain size
D, a generic total bed material load relation and a constant bed resistance coefficient. The channel is assumed to have a constant width. Water and sediment discharge are specified per unit width.
The fan builds outward by forming a prograding delta front with an assigned foreset slope. The code employs the normal flow approximation rather than a full backwater calculation.
CarboCAT uses a cellular automata to model horizontal and vertical distributions of carbonate lithofacies
ChesROMS is a community ocean modeling system for the Chesapeake Bay region being developed by scientists in NOAA, University of Maryland, CRC (Chesapeake Research Consortium) and MD DNR (Maryland
Department of Natural Resources) supported by the NOAA MERHAB program. The model is built based on the Rutgers Regional Ocean Modeling System (ROMS, http://www.myroms.org/) with significant
adaptations for the Chesapeake Bay. The model is developed to provide a community modeling system for nowcast and forecast of 3D hydrodynamic circulation, temperature and salinity, sediment
transport, biogeochemical and ecosystem states with applications to ecosystem and human health in the bay. Model validation is based on bay wide satellite remote sensing, real-time in situ
measurements and historical data provided by Chesapeake Bay Program. http://ches.communitymodeling.org/models/ChesROMS/index.php
Cliffs features: Shallow-Water approximation; Use of Cartesian or spherical (lon/lat) coordinates; 1D and 2D configurations; Structured co-located grid with (optionally) varying spacing; Run-up on
land; Initial conditions or boundary forcing; Grid nesting with one-way coupling; Parallelized with OpenMP; NetCDF format of input/output data. Cliffs utilizes VTCS-2 finite-difference scheme and
dimensional splitting as in (Titov and Synolakis, 1998), and reflection and inundation computations as in (Tolkova, 2014). References: Titov, V.V., and C.E. Synolakis. Numerical modeling of tidal
wave runup. J. Waterw. Port Coast. Ocean Eng., 124(4), 157–171 (1998) Tolkova E. Land-Water Boundary Treatment for a Tsunami Model With Dimensional Splitting. Pure and Applied Geophysics, 171(9),
2289-2314 (2014)
Coastal barrier model that simulates storm overwash and tidal inlets and estimates coastal barrier transgression resulting from sea-level rise.
Code for estimating long-term exhumation histories and spatial patterns of short-term erosion from the detrital thermochronometric data.
Code functionality and purpose may be found in the following references: References # Zhang L., Parker, G., Stark, C.P., Inoue, T., Viparelli, V., Fu, X.D., and Izumi, N. 2015, "Macro-roughness model
of bedrock–alluvial river morphodynamics", Earth Surface Dynamics, 3, 113–138. # Zhang, L., Stark, C.P., Schumer, R., Kwang, J., Li, T.J., Fu, X.D., Wang, G.Q., and Parker, G. 2017, "The
advective-diffusive morphodynamics of mixed bedrock-alluvial rivers subjected to spatiotemporally varying sediment supply" (submitted to JGR)
Computes transient (semi-implicit numerical) and steady-state (analytical and numerical) solutions for the long-profile evolution of transport-limited gravel-bed rivers. Such rivers are assumed to
have an equilibrium width (following Parker, 1978), experience flow resistance that is proportional to grain size, evolve primarily in response to a single dominant "channel-forming" or
"geomorphically-effective" discharge (see Blom et al., 2017, for a recent study and justification of this assumption and how it can be applied), and transport gravel following the Meyer-Peter and
Müller (1948) equation. This combination of variables results in a stream-power-like relationship for bed-material sediment discharge, which is then inserted into a valley-resolving Exner equation to
compute long-profile evolution.
CruAKtemp is a python 2.7 package that is a data component which serves to provide onthly temperature data over the 20th century for permafrost modeling. The original dataset at higher resolution can
be found here: http://ckan.snap.uaf.edu/dataset/historical-monthly-and-derived-temperature-products-771m-cru-ts The geographical extent of this CRUAKtemp dataset has been reduced to greatly reduce
the number of ocean or Canadian pixels. Also, the spatial resolution has been reduced by a factor of 13 in each direction, resulting in an effective pixel resolution of about 10km. The data are
monthly average temperatures for each month from January 1901 through December 2009.
DFMFON stands for Delft3D-Flexible Mesh (DFM), and MesoFON (MFON) is an open-source software written in Python to simulate the Mangrove and Hydromorphology development mechanistically. We achieve
that by coupling the multi-paradigm of the individual-based mangrove model MFON and process-based hydromorphodynamic model DFM.
DHSVM is a distributed hydrology model that was developed at the University of Washington more than ten years ago. It has been applied both operationally, for streamflow prediction, and in a research
capacity, to examine the effects of forest management on peak streamflow, among other things.
DR3M is a watershed model for routing storm runoff through a Branched system of pipes and (or) natural channels using rainfall as input. DR3M provides detailed simulation of storm-runoff periods
selected by the user. There is daily soil-moisture accounting between storms. A drainage basin is represented as a set of overland-flow, channel, and reservoir segments, which jointly describe the
drainage features of the basin. This model is usually used to simulate small urban basins. Interflow and base flow are not simulated. Snow accumulation and snowmelt are not simulated.
DROG3D tracks passive drogues with given harmonic velocity field(s) in a 3-D finite element mesh
Dakota is a software toolkit, developed at Sandia National Laboratories, that provides an interface between models and a library of analysis methods, including support for sensitivity analysis,
uncertainty quantification, optimization, and calibration techniques. Dakotathon is a Python package that wraps and extends Dakota’s file-based user interface. It simplifies the process of
configuring and running a Dakota experiment, and it allows a Dakota experiment to be scripted. Any model written in Python that exposes a Basic Model Interface (BMI), as well as any model
componentized in the CSDMS modeling framework, automatically works with Dakotathon. Currently, six Dakota analysis methods have been implemented from the much larger Dakota library: * vector
parameter study, * centered parameter study, * multidim parameter study, * sampling, * polynomial chaos, and * stochastic collocation.
Data component processed from the CRU-NCEP Climate Model Intercomparison Project - 5, also called CMIP 5. Data presented include the mean annual temperature for each gridcell, mean July temperature
and mean January temperature over the period 1902 -2100. This dataset presents the mean of the CMIP5 models, and the original climate models were run for the representative concentration pathway RCP
DeltaRCM is a parcel-based cellular flux routing and sediment transport model for the formation of river deltas, which belongs to the broad category of rule-based exploratory models. It has the
ability to resolve emergent channel behaviors including channel bifurcation, avulsion and migration. Sediment transport distinguishes two types of sediment: sand and mud, which have different
transport and deposition/erosion rules. Stratigraphy is recorded as the sand fraction in layers. Best usage of DeltaRCM is the investigation of autogenic processes in response to external forcings.
Demeter is an open source Python package that was built to disaggregate projections of future land allocations generated by an integrated assessment model (IAM). Projected land allocation from IAMs
is traditionally transferred to Earth System Models (ESMs) in a variety of gridded formats and spatial resolutions as inputs for simulating biophysical and biogeochemical fluxes. Existing tools for
performing this translation generally require a number of manual steps which introduces error and is inefficient. Demeter makes this process seamless and repeatable by providing gridded land use and
land cover change (LULCC) products derived directly from an IAM—in this case, the Global Change Assessment Model (GCAM)—in a variety of formats and resolutions commonly used by ESMs.
Depth-Discharge and Bedload Calculator, uses: # Wright-Parker formulation for flow resistance (without stratification correction) # Ashida-Michiue formulation for bedload transport.
Depth-Discharge and Total Load Calculator, uses: # Wright-Parker formulation for flow resistance, # Ashida-Michiue formulation for bedload transport, # Wright-Parker formulation (without
stratification) for suspended load.
Derived from MOSART-WM (Model for Scale Adaptive River Transport with Water Management), mosasrtwmpy is a large-scale river-routing Python model used to study riverine dynamics of water, energy, and
biogeochemistry cycles across local, regional, and global scales. The water management component represents river regulation through reservoir storage and release operations, diversions from
reservoir releases, and allocation to sectoral water demands. The model allows an evaluation of the impact of water management over multiple river basins at once (global and continental scales) with
consistent representation of human operations over the full domain.
Diffusion of marine sediments
Directs flow by the D infinity method (Tarboton, 1997). Each node is assigned two flow directions, toward the two neighboring nodes that are on the steepest subtriangle. Partitioning of flow is done
based on the aspect of the subtriangle.
Directs flow by the multiple flow direction method. Each node is assigned multiple flow directions, toward all of the N neighboring nodes that are lower than it. If none of the neighboring nodes are
lower, the location is identified as a pit. Flow proportions can be calculated as proportional to slope or proportional to the square root of slope, which is the solution to a steady kinematic wave.
Dorado is a Python package for simulating passive Lagrangian particle transport over flow-fields from any 2D shallow-water hydrodynamic model using a weighted random walk methodology.
DynEarthSol3D (Dynamic Earth Solver in Three Dimensions) is a flexible, open-source finite element code that solves the momentum balance and the heat transfer in Lagrangian form using unstructured
meshes. It can be used to study the long-term deformation of Earth's lithosphere and problems alike.
DynQual is a high-spatio-temporal-resolution surface water quality model, which can be used to simulate water temperature; concentrations of total dissolved solids to represent salinity pollution;
biological oxygen demand to represent organic pollution; and fecal coliform as a coarse indicator for pathogen pollution.
ECSimpleSnow is a simple snow model that employs an empirical algorithm to melt or accumulate snow based on surface temperature and precipitation that has fallen since the previous analysis step.
EF5 was created by the Hydrometeorology and Remote Sensing Laboratory at the University of Oklahoma. The goal of EF5 is to have a framework for distributed hydrologic modeling that is user friendly,
adaptable, expandable, all while being suitable for large scale (e.g. continental scale) modeling of flash floods with rapid forecast updates. Currently EF5 incorporates 3 water balance models
including the Sacramento Soil Moisture Accouning Model (SAC-SMA), Coupled Routing and Excess Storage (CREST), and hydrophobic (HP). These water balance models can be coupled with either linear
reservoir or kinematic wave routing.
ELCIRC is an unstructured-grid model designed for the effective simulation of 3D baroclinic circulation across river-to-ocean scales. It uses a finite-volume/finite-difference Eulerian-Lagrangian
algorithm to solve the shallow water equations, written to realistically address a wide range of physical processes and of atmospheric, ocean and river forcings. The numerical algorithm is low-order,
but volume conservative, stable and computationally efficient. It also naturally incorporates wetting and drying of tidal flats. ELCIRC has been extensively tested against standard ocean/coastal
benchmarks, and is starting to be applied to estuaries and continental shelves around the world.
Ecopath with Ecosim (EwE) is an ecological modeling software suite for personal computers. EwE has three main components: Ecopath – a static, mass-balanced snapshot of the system; Ecosim – a time
dynamic simulation module for policy exploration; and Ecospace – a spatial and temporal dynamic module primarily designed for exploring impact and placement of protected areas. The Ecopath software
package can be used to: *Address ecological questions; *Evaluate ecosystem effects of fishing; *Explore management policy options; *Evaluate impact and placement of marine protected areas; *Evaluate
effect of environmental changes.
Erode is a raster-based, fluvial landscape evolution model. The newest version (3.0) is written in Python and contains html help pages when running the program through the CSDMS Modeling Tool CMT
Erode-D8-Global is a raster, D8-based fluvial landscape evolution model (LEM)
Exposures to heat and sunlight can be simulated and the resulting signals shown. For a detailed description of the underlying luminescence rate equations, or to cite your use of LuSS, please use
Brown, (2020).
Extended description for SINUOUS - Meander Evolution Model. The basic model simulates planform evolution of a meandering river starting from X,Y coordinates of centerline nodes, with specification of
cross-sectional and flow parameters. If the model is intended to simulate evolution of an existing river, the success of the model can be evaluated by the included area between the simulated and the
river centerline. In addition, topographic evolution of the surrounding floodplain can be simulated as a function of existing elevation, distance from the nearest channel, and time since the channel
migrated through that location. Profile evolution of the channel can also be modeled by backwater flow routing and bed sediment transport relationships.
FACET is a Python tool that uses open source modules to map the floodplain extent and derive reach-scale summaries of stream and floodplain geomorphic measurements from high-resolution digital
elevation models (DEMs). Geomorphic measurements include channel width, stream bank height, floodplain width, and stream slope.<br>Current tool functionality is only meant to process DEMs within the
Chesapeake Bay and Delaware River watersheds. FACET was developed to batch process 3-m resolution DEMs in the Chesapeake Bay and Delaware River watersheds. Future updates to FACET will allow users to
process DEMs outside of the Chesapeake and Delaware basins.<br>FACET allows the user to hydrologically condition the DEM, generate the stream network, select one of two options for stream bank
identification, map the floodplain extent using a Height Above Nearest Drainage (HAND) approach, and calculate stream and floodplain metrics using three approaches.
FUNWAVE is a phase-resolving, time-stepping Boussinesq model for ocean surface wave propagation in the nearshore.
FVCOM is a prognostic, unstructured-grid, finite-volume, free-surface, 3-D primitive equation coastal ocean circulation model developed by UMASSD-WHOI joint efforts. The model consists of momentum,
continuity, temperature, salinity and density equations and is closed physically and mathematically using turbulence closure submodels. The horizontal grid is comprised of unstructured triangular
cells and the irregular bottom is preseented using generalized terrain-following coordinates. The General Ocean Turbulent Model (GOTM) developed by Burchard’s research group in Germany (Burchard,
2002) has been added to FVCOM to provide optional vertical turbulent closure schemes. FVCOM is solved numerically by a second-order accurate discrete flux calculation in the integral form of the
governing equations over an unstructured triangular grid. This approach combines the best features of finite-element methods (grid flexibility) and finite-difference methods (numerical efficiency and
code simplicity) and provides a much better numerical representation of both local and global momentum, mass, salt, heat, and tracer conservation. The ability of FVCOM to accurately solve scalar
conservation equations in addition to the topological flexibility provided by unstructured meshes and the simplicity of the coding structure has make FVCOM ideally suited for many coastal and
interdisciplinary scientific applications.
Fall velocity for spheres. Uses formulation of Dietrich (1982)
Finite difference approximations are great for modeling the erosion of landscapes. A paper by Densmore, Ellis, and Anderson provides details on application of landscape evolution models to the Basin
and Range (USA) using complex rulesets that include landslides, tectonic displacements, and physically-based algorithms for hillslope sediment transport and fluvial transport. The solution given here
is greatly simplified, only including the 1D approximation of the diffusion equation. The parallel development of the code is meant to be used as a class exercise
Finite difference solution allows for calculations of flexural response in regions of variable elastic thickness / flexural rigidity. The direct solution technique means that it takes time to
populate a cofactor matrix, but that once this has been done, flexural solutions may be obtained rapidly via a Thomas algorithm. This makes it less good for an individual solution where an iterative
approach may be more computationally efficient, but better for modeling where elastic thickness does not change (meaning that you do not need to create a new cofactor matrix) but loads do.
Finite element process based simulation model for fluid flow, clastic, carbonate and evaporate sedimentation.
For each time step, this component calculates an infiltration rate for a given model location and updates surface water depths. Based on the Green-Ampt method, it follows the form of Julien et al.,
Fortran 95 routines to model the ocean carbonate system (mocsy). Mocsy take as input dissolved inorganic carbon CT and total alkalinity AT, the only two tracers of the ocean carbonate system that are
unaffected by changes in temperature and salinity and conservative with respect to mixing, properties that make them ideally suited for ocean carbon models. With basic thermodynamic equilibria, mocsy
compute surface-ocean pCO2 in order to simulate air-sea CO2 fluxes. The mocsy package goes beyond the OCMIP code by computing all other carbonate system variables (e.g., pH, CO32-, and CaCO3
saturation states) and by doing so throughout the water column.
FuzzyReef is a three-dimensional (3D) numerical stratigraphic model that simulates the development of microbial reefs using fuzzy logic (multi-valued logic) modeling methods. The flexibility of the
model allows for the examination of a large number of variables. This model has been used to examine the importance of local environmental conditions and global changes on the frequency of reef
development relative to the temporal and spatial constraints from Upper Jurassic (Oxfordian) Smackover reef datasets from two Alabama oil fields. The fuzzy model simulates the deposition of reefs and
carbonate facies through integration of local and global variables. Local-scale factors include basement relief, sea-level change, climate, latitude, water energy, water depth, background
sedimentation rate, and substrate conditions. Regional and global-scale changes include relative sea-level change, climate, and latitude.
GENESIS calculates shoreline change produced by statial and temporal differences in longshore sand transport produced by breaking waves. The shoreline evolution portion of the numerical modeling
system is based on one-line shoreline change theory, which assumes that the beach profile shape remains unchanged, allowing shoreline change to be described uniquely in terms of the translation of a
single point (for example, Mean High Water shoreline) on the profile.
GEOMBEST is a morphological-behaviour model that simulates the evolution of coastal morphology and stratigraphy resulting from changes in sea level and sediment volume within the shoreface, barrier,
and estuary.
GEOMBEST++ is a morphological-behaviour model that simulates the evolution of coastal morphology and stratigraphy resulting from changes in sea level and sediment volume within the shoreface,
barrier, and estuary. GEOMBEST++ builds on previous iterations (i.e. GEOMBEST+) by incorporating the effects of waves into the backbarrier, providing a more physical basis for the evolution of the
bay bottom and introducing wave erosion of marsh edges.
GEOMBEST++Seagrass is a morphological-behaviour model that simulates the evolution of coastal morphology and stratigraphy resulting from changes in sea level and sediment volume within the shoreface,
barrier, and estuary. GEOMBEST++Seagrass builds on previous iterations (i.e. GEOMBEST, GEOMBEST+, and GEOMBEST++) by incorporating seagrass dynamics into the back-barrier bay.
GEOtop accommodates very complex topography and, besides the water balance integrates all the terms in the surface energy balance equation. For saturated and unsaturated subsurface flow, it uses the
3D Richards’ equation. An accurate treatment of radiation inputs is implemented in order to be able to return surface temperature. The model GEOtop simulates the complete hydrological balance in a
continuous way, during a whole year, inside a basin and combines the main features of the modern land surfaces models with the distributed rainfall-runoff models. The new 0.875 version of GEOtop
introduces the snow accumulation and melt module and describes sub-surface flows in an unsaturated media more accurately. With respect to the version 0.750 the updates are fundamental: the codex is
completely eviewed, the energy and mass parametrizations are rewritten, the input/output file set is redifined. GEOtop makes it possible to know the outgoing discharge at the basin's closing section,
to estimate the local values at the ground of humidity, of soil temperature, of sensible and latent heat fluxes, of heat flux in the soil and of net radiation, together with other hydrometeorlogical
distributed variables. Furthermore it describes the distributed snow water equivalent and surface snow temperature. GEOtop is a model based on the use of Digital Elevation Models (DEMs). It makes
also use of meteorological measurements obtained thought traditional instruments on the ground. Yet, it can also assimilate distributed data like those coming from radar measurements, from satellite
terrain sensing or from micrometeorological models.
GIPL(Geophysical Institute Permafrost Laboratory) is an implicit finite difference one-dimensional heat flow numerical model. The GIPL model uses the effect of snow layer and subsurface soil thermal
properties to simulate ground temperatures and active layer thickness (ALT) by solving the 1D heat diffusion equation with phase change. The phase change associated with freezing and thawing process
occurs within a range of temperatures below 0 degree centigrade, and is represented by the unfrozen water curve (Romanovsky and Osterkamp 2000). The model employs finite difference numerical scheme
over a specified domain. The soil column is divided into several layers, each with distinct thermo-physical properties. The GIPL model has been successfully used to map permafrost dynamics in Alaska
and validated using ground temperature measurements in shallow boreholes across Alaska (Nicolsky et al. 2009, Jafarov et al. 2012, Jafarov et al. 2013, Jafarov et al. 2014).
GSFLOW was a coupled model based on the integration of the U.S. Geological Survey Precipitation-Runoff Modeling System (PRMS, Leavesley and others, 1983) and the U.S. Geological Survey Modular
Groundwater Flow Model(MODFLOW-2005, Harbaugh, 2005). It was developed to simulate coupled groundwater/surface-water flow in one or more watersheds by simultaneously simulating flow across the land
surface, within subsurface saturated and unsaturated materials, and within streams and lakes.
Generates alluvial stratigraphy by channel migration and avulsion. Channel migration is handled via a random walk. Avulsions occur when the channel superelevates. Channels can create levees.
Post-avulsion channel locations chosen at random, or based on topography.
GeoFlood, a new open-source software package for solving shallow water equations (SWE) on a quadtree hierarchy of mapped, logically Cartesian grids managed by the parallel, adaptive library
Glimmer is an open source (GPL) three-dimensional thermomechanical ice sheet model, designed to be interfaced to a range of global climate models. It can also be run in stand-alone mode. Glimmer was
developed as part of the NERC GENIE project (www.genie.ac.uk). It's development follows the theoretical basis found in Payne (1999) and Payne (2001). Glimmer's structure contains numerous software
design strategies that make it maintainable, extensible, and well documented.
Grain Size Distribution Statistics Calculator
Gridded Surface Subsurface Hydrologic Analysis (GSSHA) is a grid-based two-dimensional hydrologic model. Features include 2D overland flow, 1D stream flow, 1D infiltration, 2D groundwater, and full
coupling between the groundwater, vadoze zone, streams, and overland flow. GSSHA can run in both single event and long-term modes. The fully coupled groundwater to surfacewater interaction allows
GSSHA to model both Hortonian and Non-Hortonian basins. New features of version 2.0 include support for small lakes and detention basins, wetlands, improved sediment transport, and an improved stream
flow model. GSSHA has been successfully used to predict soil moistures as well as runoff and flooding.
Gridded water balance model using climate input forcings that calculate surface and subsurface runoff and ground water recharge for each grid cell. The surface and subsurface runoff is propagated
horizontally along a prescribed gridded network using Musking type horizontal transport.
HYPE is a semi-distributed hydrological model for water and water quality. It simulates water and nutrient concentrations in the landscape at the catchment scale. Its spatial division is related to
catchments and sub-catchments, land use or land cover, soil type and elevation. Within a catchment the model will simulate different compartments; soil including shallow groundwater, rivers and
lakes. It is a dynamical model forced with time series of precipitation and air temperature, typically on a daily time step. Forcing in the form of nutrient loads is not dynamical. Example includes
atmospheric deposition, fertilizers and waste water.
Here, we present a Python tool that includes a comprehensive set of relations that predicts the hydrodynamics, bed elevation and the patterns of channels and bars in mere seconds. Predictions are
based on a combination of empirical relations derived from natural estuaries, including a novel predictor for cross-sectional depth distributions, which is dependent on the along-channel width
profile. Flow velocity, an important habitat characteristic, is calculated with a new correlation between depth below high water level and peak tidal flow velocity, which was based on spatial
numerical modelling. Salinity is calculated from estuarine geometry and flow conditions. The tool only requires an along-channel width profile and tidal amplitude, making it useful for quick
assessments, for example of potential habitat in ecology, when only remotely-sensed imagery is available. | {"url":"https://csdms.colorado.edu/csdms_wiki/index.php?title=Property:Extended_model_description&limit=100&offset=40&from=&until=&filter=","timestamp":"2024-11-03T13:45:23Z","content_type":"text/html","content_length":"213127","record_id":"<urn:uuid:7e94c963-f985-4a83-90fd-6d3df57b292f>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00325.warc.gz"} |
Curved Line Slope Calculator: Find Slope of Curves
Unraveling the Mystery of Curved Line Slopes: Your Ultimate Guide to the Curved Line Slope Calculator
The idea of curved lines introduces another level of complexity to the already difficult fields of mathematics and geometry. It can be difficult to understand and calculate the slope of a curved line
, but don't worry—the Curved Line Slope Calculator can help. We'll explore the nuances of curved lines, explain the computations, and provide you with the skills you need to confidently traverse this
mathematical landscape in this extensive book.
What is a Curved Line and Why Does it Matter?
Before we dive into the mechanics of calculating the slope of a curved line, let's establish a foundational understanding of what a curved line is and its significance. In simple terms, a curved line
deviates from the straight and narrow, introducing curves and bends. These lines play a pivotal role in various fields, from physics to design, making it crucial to comprehend their characteristics.
Curved Line Slope Formula
The slope (\(m\)) of a curved line at a point \((x_1, y_1)\) is given by:
\[ m = \frac{dy}{dx} \Big|_{x=x_1} \]
Curved Line Slope Examples And Solutions
Example 1: Cubic Polynomial
Consider the cubic function \(y = x^3 - 2x^2 + x + 1\). Find the slope at the point where \(x = 2\).
\[ y = x^3 - 2x^2 + x + 1 \]
The slope is given by the derivative of the function with respect to \(x\).
\[ \frac{dy}{dx} = 3x^2 - 4x + 1 \]
Now, substitute \(x = 2\) into the derivative to find the slope at that point.
\[ \frac{dy}{dx} \bigg|_{x=2} = 3 \times 2^2 - 4 \times 2 + 1 = 5 \]
Example 2: Exponential Growth
Consider the exponential growth function \(y = 2e^{0.5x}\). Determine the slope at the point \(x = 1\).
\[ y = 2e^{0.5x} \]
The slope is given by the derivative of the function with respect to \(x\).
\[ \frac{dy}{dx} = e^{0.5x} \]
Now, substitute \(x = 1\) into the derivative to find the slope at that point.
\[ \frac{dy}{dx} \bigg|_{x=1} = e^{0.5 \times 1} = e^{0.5} \]
Example 3: Logarithmic Function
Consider the logarithmic function \(y = \ln(x^2 + 1)\). Find the slope at the point \(x = 3\).
\[ y = \ln(x^2 + 1) \]
The slope is given by the derivative of the function with respect to \(x\).
\[ \frac{dy}{dx} = \frac{2x}{x^2 + 1} \]
Now, substitute \(x = 3\) into the derivative to find the slope at that point.
\[ \frac{dy}{dx} \bigg|_{x=3} = \frac{2 \times 3}{3^2 + 1} = \frac{6}{10} = \frac{3}{5} \]
Example 4: Sine Function
Consider the sine function \(y = \sin(x)\). Determine the slope at the point \(x = \frac{\pi}{4}\).
\[ y = \sin(x) \]
The slope is given by the derivative of the function with respect to \(x\).
\[ \frac{dy}{dx} = \cos(x) \]
Now, substitute \(x = \frac{\pi}{4}\) into the derivative to find the slope at that point.
\[ \frac{dy}{dx} \bigg|_{x=\frac{\pi}{4}} = \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \]
Example 5: Parabola
Consider the parabola \(y = x^2 - 3x + 2\). Find the slope at the vertex of the parabola.
\[ y = x^2 - 3x + 2 \]
The slope is given by the derivative of the function with respect to \(x\).
\[ \frac{dy}{dx} = 2x - 3 \]
The vertex of the parabola is at the critical point where the derivative is zero. Set \(\frac{dy}{dx} = 0\) and solve for \(x\).
\[ 2x - 3 = 0 \implies x = \frac{3}{2} \]
Substitute \(x = \frac{3}{2}\) into the derivative to find the slope at the vertex.
\[ \frac{dy}{dx} \bigg|_{x=\frac{3}{2}} = 2 \times \frac{3}{2} - 3 = 0 \]
How to use the Equation of Curve Calculator from Points?
1. Collect Data Points:
□ Gather the coordinates of points on the curve, denoted as \((x_i, y_i)\).
□ For example, points might be in the form \((1, 2)\), \((2, 5)\), \((3, 10)\), etc.
2. Input Data into the Calculator:
□ Open the Equation of Curve Calculator.
□ Enter the x and y coordinates into the respective input fields.
□ Input the points either individually or as a set, following the calculator's requirements.
3. Select Curve Type:
□ Specify the type of curve or function to use (e.g., linear, quadratic, cubic, etc.).
4. Calculate the Equation:
□ Initiate the calculation process.
□ The calculator will use the provided points to generate the equation of the curve.
5. Review Results:
□ Examine the calculated equation, often in the form \(y = f(x)\) or an appropriate representation.
6. Additional Options:
□ Explore any additional options the calculator may offer, such as graphing the curve or finding specific values.
7. Interpret and Use the Equation:
□ Utilize the obtained equation to predict y-values for other x-values within the range of your data.
Deciphering the Curved Line Slope Calculator
1. Introduction to Slope Calculation
Traditionally, slope calculation is associated with straight lines, but when dealing with curves, the process becomes more nuanced. The Curved Line Slope Calculator adapts to this complexity,
offering a tool that accommodates the dynamic nature of curved lines.
2. Understanding the Variables
In the context of curved lines, the slope is not a constant value. It varies at different points along the curve. The calculator takes into account variables like the curvature, angle of inclination,
and rate of change, providing a comprehensive analysis of the slope across the entire curve.
3. Utilizing Calculus Concepts
The Curved Line Slope Calculator employs advanced calculus concepts to dissect the curve and determine its slope. Derivatives, integrals, and differential equations work in tandem to provide a
holistic view of the line's behavior.
Navigating the Curved Line Slope Calculator: A Step-by-Step Guide
4. Inputting the Curve Parameters
To initiate the calculation process, input the relevant parameters of the curved line. These may include the coordinates of specific points, the equation of the curve, or other defining
5. Choosing the Analysis Interval
Select the interval over which you want to analyze the slope. The Curved Line Slope Calculator allows for a granular examination, giving you insights into how the slope evolves within specific
segments of the curve.
6. Interpreting the Results
Once the calculations are complete, the calculator generates a visual representation of the curved line with annotated slope values. This visual aid facilitates a better understanding of how the
slope changes along the curve.
Conclusion: Mastering the Art of Curved Line Slope Calculation
In conclusion, the Curved Line Slope Calculator is a powerful tool that unravels the complexities of curved lines. Armed with this understanding, you can navigate mathematical landscapes with
confidence. As we bid farewell to this exploration, remember that the journey of mastering curved line slopes is as intriguing as the curves themselves. Happy calculating
The slope of a curved line at a specific point represents the rate at which the curve is changing at that point. Unlike straight lines with a constant slope, curved lines have a varying slope along
different points.
The slope of a curved line at a specific point is determined using calculus. It involves finding the derivative of the function that defines the curve with respect to the independent variable
(usually denoted as x). The derivative at a given point gives the slope of the curve at that particular point.
No, there isn't a general formula for finding the slope of a curved line because it depends on the mathematical function that describes the curve. Calculus methods, such as finding derivatives, are
commonly used to calculate the slope for specific points.
No, the formula for finding the slope of a straight line (Ξ y/Ξ x) is not applicable to curved lines. Curved lines require calculus methods, such as finding derivatives, to determine the slope at
specific points. | {"url":"https://www.calculatestudy.com/curved-line-slope-calculator","timestamp":"2024-11-04T06:14:37Z","content_type":"text/html","content_length":"58136","record_id":"<urn:uuid:c7a25ee8-66be-421f-8bbf-2614e4f392e9>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00803.warc.gz"} |
randomMPS method can not be used with Zn symmetry (Julia)
I have created a new Z3 sitetype using
const Z3Site = TagType"Z3"
function siteinds(::Z3Site, Num::Int;kwargs...)
conserve_qns = get(kwargs,:conserve_qns,false)
if conserve_qns
s = [ QN(("P", n, 3)) => 1 for n = 0:2]
return [Index(s;tags="Site,Z3,n=$n") for n=1:Num]
return [Index(N,"Site,Z3,n=$n") for n=1:Num]
function state(::Z3Site,
if st == "s1"
return 1
elseif st == "s2"
return 2
elseif st == "s3"
return 3
when I use these site type to initialize a random mps with
state = ["s1" for n in 1:N]
It throws the error that
"Indices must have the same spaces to be replaced"
This error can be reproduced using the following minimal code
sites = siteinds("Z3",N,conserve_qns=true);
s1 = sites[1];
s2 = sites[2];
M = randomITensor(QN(),s1',s2',dag(s1),dag(s2));
U,S,V = svd(M,(s1',s2'));
However, when I print the space of the two index space, it seems that they are the same when module 3
3-element Array{Pair{QN,Int64},1}:
QN("P",-2,3) => 3
QN("P",-1,3) => 3
QN("P",0,3) => 3
3-element Array{Pair{QN,Int64},1}:
QN("P",1,3) => 3
QN("P",2,3) => 3
QN("P",0,3) => 3
Is there something wrong when I define the Z3 sitetype?
Thank you very much for the answer!
Hi Runze,
Thanks for reporting this issue and for the detailed explanation and minimal code which is very helpful. I've filed an issue on our issue tracker and can reproduce the bug. I have a suspicion it has
to do with some faulty code related to our QN objects when they are defined mod 3. I'll look into and hope to fix it really soon.
Hi Runze,
This issue should now be fixed as of version 0.1.10. So please do "update ITensors" in the package manager and try your first, randomMPS code again. Please let me know if it still doesn't work.
The issue I found was that a constructor which gets called when making QNs with modulus values other than 1 had a missing return keyword, which made it silently set the wrong QN value. So that was
causing the error but now it's fixed.
Thanks again for bringing this to our attention -
Thank you very much, this problem is fixed and give the correct result ！
However, there is another problem when I try to save the mps using the HDF5. The "write" function seems to not support MPS with QN, is there some alternative method？
Hi Runze,
Glad it works!
Thanks for asking but unfortunately we haven't added that feature yet. We plan to add it soon -
Hi Runze,
The latest version of Julia ITensor now supports writing ITensors, MPS, and MPO which have conserved QNs to HDF5 files. So please upgrade to that version and it should now work - | {"url":"https://www.itensor.org/support/2214/randommps-method-can-not-be-used-with-zn-symmetry-julia","timestamp":"2024-11-10T22:02:29Z","content_type":"text/html","content_length":"33640","record_id":"<urn:uuid:1ec2ae99-cee4-4d36-93fc-e63a3bd155f6>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00033.warc.gz"} |
Asymptotic structure and the characterisation of gravitational radiation at infinity
Asymptotic structure and the characterisation of gravitational radiation at infinity
Senovilla, J. (2023). Asymptotic structure and the characterisation of gravitational radiation at infinity. Perimeter Institute for Theoretical Physics. https://pirsa.org/23090052
Senovilla, Jose. Asymptotic structure and the characterisation of gravitational radiation at infinity. Perimeter Institute for Theoretical Physics, Sep. 07, 2023, https://pirsa.org/23090052
@misc{ scivideos_PIRSA:23090052,
doi = {10.48660/23090052},
url = {https://pirsa.org/23090052},
author = {Senovilla, Jose},
keywords = {Quantum Gravity},
language = {en},
title = {Asymptotic structure and the characterisation of gravitational radiation at infinity},
publisher = {Perimeter Institute for Theoretical Physics},
year = {2023},
month = {sep},
note = {PIRSA:23090052 see, \url{https://scivideos.org/pirsa/23090052}}
Talk numberPIRSA:23090052
With the main purpose of identifying the existence of gravitational radiation at infinity (scri), a novel approach to the asymptotic structure of spacetime is presented, focusing mainly in cases with
non-negative cosmological constant. The basic idea is to consider the strength of tidal forces experienced by scri. To that end I will introduce the asymptotic (radiant) super-momentum, a causal
vector defined at scri with remarkable properties that, in particular, provides an innovative characterization of gravitational radiation valid for the general case with Λ ≥ 0 (and which has been
proven to be equivalent when Λ = 0 to the standard one based on the News tensor). This analysis is also shown to be supported by the initial- (or final-) value Cauchy-type problem defined at scri.
The implications are discussed in some detail. The geometric structure of scri, and of its cuts, is clarified. The question of whether or not a News tensor can be defined in the presence of a
positive cosmological constant is addressed. Several definitions of asymptotic symmetries are presented. Conserved charges that may detect gravitational radiation are exhibited. Balance laws that
might be useful as diagnostic tools to test the accuracy of model waveforms discussed. An interpretation of the Geroch `rho' tensor is found. The whole thing will be complemented with a series of
illustrative examples based on exact solutions. In particular we will see that exact solutions with black holes will be radiative if, and only if, they are accelerated.
Zoom link: https://pitp.zoom.us/j/95879544523?pwd=MFl5UEtUZ0hUcU1hNk1SZ2R4MThxUT09 | {"url":"https://scivideos.org/PIRSA/23090052","timestamp":"2024-11-15T04:19:12Z","content_type":"text/html","content_length":"52881","record_id":"<urn:uuid:15b5db95-9e49-4708-a182-e3713b9f3fac>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00607.warc.gz"} |
What are the Sine, Cosine and Tangent - Help with IGCSE GCSE Maths
What is Trigonometry
We can use trigonometry to calculate lengths and angels in right-angled triangles. You will have to solve questions on your IGCSE GCSE maths exam involving trigonometry. In order to understand
trigonometry you need to start at the beginning. Therefore I encourage you to take your time and examine the maths activities on this page carefully.
Using Trigonometry to calculate a length in a right-angled triangle
We already have one strategy to calculate lengths in right angled triangles - Pythagoras' theorem. But what to do when you are not given the dimensions of 2 sides, but only of 1 side and the size
of one angle? Then trigonometry will be helpful to calculate the length. Make sure to carefully identify the hypotenuse, adjacent and opposite sides which will inform you which ratio, tangent,
cosine or sine, you will have to use. All these term will be explained to you in the following maths video. Study it therefore carefully during your maths revision and get prepared for your next
maths exam. Good luck!
If you want to be successful on your maths exam when answering questions involving trigonometry, make sure to take time when you approach a question. Analyse each situation and properly identify the
sides of the triangle. The hypotenuse is the hypotenuse, but what your adjacent and opposite sides are depend on the angle. Have a look at the following videos which will show you step by step what
you are expected to do for this important mathematical topic.
Using Trigonometry to calculate an angle in a right-angled triangle
We can also use trigonometry to calculate angles in right angled triangles. To choose to correct ratio (tangent, cosine or sine) to calculate the angle, take a moment to name the three sides of the
triangle (hypotenuse, adjacent, opposite). When you know which ratio to use, to find the angle you will have to do the inverse of your chosen ratio. Your calculator will do that for you ('shift'
'sin/cos/tan'). Do not forget that angles need to be rounded correctly to 1 decimal place. Have a look at the next video which will give you example questions in which you will learn how to find an
angle in a triangle with Trigonometry.
Solving Example Questions about Trigonometry
Once you have studied the videos above which introduce this important GCSE IGCSE maths topic, you can try the example questions below. I encourage you to solve the questions yourself first before
looking at my workings and answers (by pausing the video at the beginning). When you are able to answer all the example maths questions during your maths revision, you know you are well prepared for
your next maths exam. Tell me on the forum of this website if you don't understand Trigonometry and need help with Trigonometry. I will make some more maths videos for you.
Activity Worksheet and example questions Trigonometry
I have created two maths worksheets for you so you can practise even more. You can also download the documents for FREE or work from the embedded document below. If you want to pass your IGCSE GCSE
maths exam then you should try to answer all questions yourself first before looking at my answers. The answer key is included (second page of the document). Good luck and have fun!
Past Paper Question involving Trigonometry, Circle Theorems and Arc length
A students asked me to help with this past maths exam question. The question comes from a past IGCSE paper. I will explain to you how to use trigonometry, circle theorems and our understanding of
how to calculate the length of an arc in order to answer this question. I hope it will be useful during your maths revision!
Past Exam Paper about Trigonometry, Pythagoras' Theorem, Cosine rule and Volume of Prisms
Check the following video in which I solve a past paper question. I will explain to you how to use trigonometry and Pythagoras' theorem to calculate lengths. I will also show you what the cosine
rule is and how to calculate the volume of a prism. So a lot of topics in 1 question which means it is an excellent video to watch during your maths revision! | {"url":"http://www.explainingmaths.com/what-are-the-sine-cosine-and-tangent.html","timestamp":"2024-11-14T23:36:01Z","content_type":"text/html","content_length":"161707","record_id":"<urn:uuid:a6a1a7bf-a227-4972-9f91-23d30f0ea5ce>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00212.warc.gz"} |
Simulation-based power analyses
power analysis
Simulation-based power analyses make it easy to understand what power is: Power is simply counting how often you find the results you expect to find. Running simulation-based power analyses might be
new for some, so in this blog post I present code to simulate data for a range of different scenarios.
Doing power analyses is hard. I know this from experience, both as a researcher and as a reviewer. As a researcher, I have found power analyses to be difficult because performing a good power
analysis requires a thorough understanding of the (hypothesized) data. Understanding one’s data is often underestimated, I think. We’re very quick to design a study and start data collection, without
often knowing what various aspects of our data will look like (e.g., likely correlations between measures, likely standard deviations). As a reviewer, I see that power analyses are difficult because
of wrong ideas about what a power analysis actually means. The most common misconception I see is that researchers think they should power their study, rather than the set of analyses they will
conduct (see Maxwell (2004) for more on this). I also see a lot of power analyses conducted with G*Power, which sometimes looks fine, but oftentimes produces results I know to be wrong (usually
involving interaction tests). So what to do?
My favorite way to run power analyses is via simulation. Simulation-based power analyses are more difficult and take longer to setup and run, but they’re more pedagogical. Simulations require you to
understand your data because you have to define the exact parameters that define your data set (e.g., means, standard deviations, correlations). It also creates a very intuitive understanding of what
power is: Power is simply counting how often you find the results you expect to find.
Still, running simulation-based power analyses might be too difficult for some. So in this blog post I present code to simulate data for a range of different scenarios.
Run the following code to get started. The most important package here is MASS. It contains a function called mvrnorm() that enables us to simulate data from a multivariate normal distribution. This
means we’ll simulate data for scenarios where we have a continuous outcome. I really like this function for simulating data because it has an argument called empirical that you can set to TRUE, which
causes your simulated data to have the exact properties you set (e.g., exactly a mean of 4). This is a great way to check out your simulated data and see if it makes sense.
We will use the tidyverse because we need to prepare the data after simulating it. mvrnorm() returns a matrix with each simulated variable as a column. This means we sometimes need to prepare the
data so that we can perform the tests we want to run or for visualization purposes.
The effectsize package will be used to inspect the data by calculating standardized effect sizes. This will allow us to check whether the parameters are plausible.
Finally, we sometimes use the broom package to extract p-values from the statistical tests that we’ll run. This will be necessary to calculate the power because power is (usually) nothing more than
the number of significant p-values divided by the number of tests we simulated data for. In a future post I might focus on Bayesian analyses, so we won’t be dealing with p-values then, although the
logic will be the same.
Besides loading packages, we also set the s variable. The value of this variable will determine how many times we’ll simulate the data during the power analysis. The higher this number, the more
accurate our power estimates will be.
With the setup out of the way, let’s cover our general approach to power analyses:
1. Simulate the data with fixed properties
2. Check the data to see if the data is plausible
3. Run the tests we want to run on the data
4. Repeat steps 1 to 3 many times, save the p-values, and calculate power
We’ll do this for various scenarios. In each scenario we start by defining the parameters. I’ll focus on providing means, standard deviations, and correlations, because those are usually the
parameters we report in the results section, so I’m guessing most researchers will have some intuitions about what these parameters mean and whether the results are plausible.
The mvrnorm() function requires that we pass it the sample size, the means, and a variance-covariance matrix. The first two are easy to understand, but the variance-covariance may not be. It’s
relatively straightforward to convert means, SDs, and correlations to a variance-covariance matrix, though. Variance is simply the standard deviation squared and the covariance is the product of the
standard deviations of the two variables and their correlation. You’ll see in some scenarios below that this is how I construct the variance-covariance matrix.
Note that the result of each power analysis will be the power, and not the sample size needed to obtain a particular power. This is the same as calculating the post-hoc power in G*Power. If you want
to figure out what the sample size is for a particular power (e.g., 80%) then you simply change the sample size parameter until you have the power you want.
One sample t-test
The simplest scenario is where we want to simulate a set of normally distributed values and perform a one sample t-test. This requires that we set three parameters: a mean, a standard deviation, and
a sample size. We give mvrnorm() the sample size (N), the mean (M), and the variance (SD^2). After simulating the data, we give the simulated data a column name and convert the matrix returned by
mvrnorm() to a data frame.
The next step is to inspect the data to see whether the parameters are plausible. This can be done by converting the parameters to a standardized effect size and by visualizing the data.
Figure 1: One sample visualization
The histogram roughly shows that we have a mean of 0.75 and a standard deviation of 5. We also calculated the Cohen’s d as a measure of the size of the effect. The size of this effect is equal to a
Cohen’s d of 0.15.
Next is the analysis we want to power for—the one-sample t-test. The function for this test is t.test().
To calculate the power, we repeat the analysis s times. Each time we store the p-value so that later we can calculate the proportion of significant results. Since we don’t need to inspect the data
each time, we skip the data preparation step and use the samples returned by mvrnorm() immediately in t.test() using R’s matrix notation (if you want, you can also prepare the data each time, if you
find that easier to understand).
# Create a vector to the store p-values in
p_values <- vector(length = s)
# Loop s times
for (i in 1:s) {
# Simulate
samples <- mvrnorm(N, mu = M, Sigma = SD^2)
# Run test
test <- t.test(samples[, 1])
# Extract p-value
p_values[i] <- test$p.value
# Calculate power
power <- sum(p_values <= .05) / s * 100
With the current parameters (N = 90, Cohen’s d = 0.15), we obtain a power of 32.2%. The power is simply how often we find a significant result, divided by the number of times we looped, multiplied by
100 to give a percentage. You can adjust the sample size parameter and re-run the code until you know which sample size gives you the desired power. You might also want to run the loop a few times to
see how consistent your results are (if the results are inconsistent, increase the number of loops by increasing the value of s).
Welch’s two sample t-test
The next scenario is one in which there are two groups (e.g., a control condition and a treatment condition) and a single DV. Even in this simple scenario there are already several variations that
are important to consider. Do we assume equal variances between groups? Do we assume equal samples sizes? Is the design between or within-subjects? We’ll start with assuming unequal variances between
the two groups. This means we’ll run a Welch’s two sample t-test. To make it extra fun, we’ll also simulate unequal sample sizes.
If we are interested in a between-subjects design where we assume both unequal variances and samples sizes, we can use the code from the previous scenario and simply run it twice, once for each
group. After simulating the data, we convert the simulated matrix of each group to a data frame, add a column indicating the group, and merge the two groups into a single data frame.
# Parameters
M_control <- 5
M_treatment <- 4
SD_control <- 1.5
SD_treatment <- 3
N_control <- 50
N_treatment <- 40
# Simulate once with empirical = TRUE
control <- mvrnorm(N_control,
mu = M_control, Sigma = SD_control^2,
empirical = TRUE
treatment <- mvrnorm(N_treatment,
mu = M_treatment, Sigma = SD_treatment^2,
empirical = TRUE
# Prepare data
colnames(control) <- "DV"
colnames(treatment) <- "DV"
control <- control %>%
as_tibble() %>%
mutate(condition = "control")
treatment <- treatment %>%
as_tibble() %>%
mutate(condition = "treatment")
data <- bind_rows(control, treatment)
Next, we inspect the data by calculating a Cohen’s d and visualizing the results.
Figure 2: Two groups visualization (unequal variance)
The difference between the two groups is equal to a Cohen’s d of 0.42.
To run a Welch’s two-sample t-test, we again use the t.test() function. R by default does not assume equal variances, so the default is a Welch’s two sample t-test.
The power analysis looks as follows:
# Create an empty vector to store the p-values in
p_values <- vector(length = s)
# Loop
for (i in 1:s) {
# Simulate
control <- mvrnorm(N_control, mu = M_control, Sigma = SD_control^2)
treatment <- mvrnorm(N_treatment, mu = M_treatment, Sigma = SD_treatment^2)
# Run test
test <- t.test(control[, 1], treatment[, 1])
# Extract p-value
p_values[i] <- test$p.value
# Calculate power
power <- sum(p_values <= .05) / s * 100
This produces a power of 45.9% with the current parameters.
Two sample t-test
Instead of assuming unequal variances, we can also assume equal variances and perform a two sample t-test. You could adapt the previous scenario by setting the parameters such that the variance in
each group is identical, but let’s do something different in this scenario. In addition, let’s assume that the sample sizes in each group are equal. This means we can simulate the data using a
slightly different approach. First, we’ll only need 4 parameters. Second, we don’t need to separately simulate the data for each group. We can instead use a single mvrnorm() call and provide it with
the correct variance-covariance matrix. The crucial bit is to only set the variances and set the covariances to 0. If we do it this way, we do need to adjust how we prepare the data. mvnnorm()
returns a matrix that, when converted to a data frame, results in a wide data frame. That is, the DV of each group is stored in separate columns. This is not tidy. We therefore restructure the data
to make it long.
# Parameters
M_control <- 5
M_treatment <- 4
SD <- 2
N <- 40
# Prepare parameters
mus <- c(M_control, M_treatment)
Sigma <- matrix(
nrow = 2, ncol = 2,
SD^2, 0,
0, SD^2
# Simulate once with empirical = TRUE
samples <- mvrnorm(N, mu = mus, Sigma = Sigma, empirical = TRUE)
# Prepare data
colnames(samples) <- c("control", "treatment")
data <- as_tibble(samples)
data_long <- pivot_longer(data,
cols = everything(), names_to = "condition",
values_to = "DV"
We inspect the data with the code from before, substituting data with data_long.
Figure 3: Two groups visualization (equal variance)
We see a difference between the two conditions with a Cohen’s d of 0.5.
This time we run a two sample t-test with equal variances assumed.
As before, the power analysis code is as follows:
# Create an empty vector to store p-values in
p_values <- vector(length = s)
# Loop
for (i in 1:s) {
# Simulate
samples <- mvrnorm(N, mu = mus, Sigma = Sigma)
# Run test
test <- t.test(samples[, 1], samples[, 2], var.equal = TRUE)
# Extract p-value
p_values[i] <- test$p.value
# Calculate power
power <- sum(p_values <= .05) / s * 100
This produces a power of 60.3% with the current parameters.
Paired t-test
A paired t-test is appropriate when we have, for example, data from two groups and we have the same participants in both groups. In other words, the observations belonging to the same participant are
likely to be correlated. To calculate power for this scenario, we need to set a correlation parameter. This, in turn, requires that we change the variance-covariance matrix. We need to set the
covariances to be equal to the squared standard deviation multiplied by the correlation (remember that a covariance is the standard deviation of one group times the standard deviation of the other
group times the correlation between the two).
# Parameters
M_pre <- 5
M_post <- 4
SD <- 2
N <- 40
r <- 0.75
# Prepare parameters
mus <- c(M_pre, M_post)
Sigma <- matrix(
ncol = 2, nrow = 2,
SD^2, SD^2 * r,
SD^2 * r, SD^2
# Simulate once with empirical = TRUE
samples <- mvrnorm(N, mu = mus, Sigma = Sigma, empirical = TRUE)
# Prepare data
colnames(samples) <- c("pre", "post")
data <- as_tibble(samples)
data_long <- pivot_longer(data,
cols = everything(),
names_to = "condition", values_to = "DV"
) %>%
mutate(condition = fct_relevel(condition, "pre"))
Let’s plot the means in each group, with a line between the two points representing the means to signify that this data was measured within-subjects. We also calculate another Cohen’s d to get an
impression of the standardized effect size.
# Calculate a standardized effect size
effect_size <- cohens_d(DV ~ condition, data = data_long, paired = TRUE)
# Visualize the data
ggplot(data_long, aes(x = condition, y = DV, group = 1)) +
geom_jitter(width = .2, alpha = .25) +
stat_summary(fun = "mean", geom = "line", linetype = 2) +
stat_summary(fun.data = "mean_cl_boot", geom = "pointrange") +
labs(x = "Condition")
Figure 4: Two groups visualization (paired)
The difference between the two groups is equal to a Cohen’s d of 0.71.
Run the paired t-test with t.test() and set paired to TRUE. I generally favor long data frames, so that’s the data frame I use here to run the paired t-test. In the power analysis, I use the wide
version to minimize the code (and speed up the power analysis).
The power analysis for this analysis looks as follows:
# Create an empty vector to store the p-values in
p_values <- vector(length = s)
# Loop
for (i in 1:s) {
# Simulate
samples <- mvrnorm(N, mu = mus, Sigma = Sigma)
# Run test
test <- t.test(samples[, 1], samples[, 2], paired = TRUE)
# Extract p-value
p_values[i] <- test$p.value
# Calculate power
power <- sum(p_values <= .05) / s * 100
This produces a power of 98.7% with the current parameters.
To power for a single correlation, we can actually use most of the code from the previous scenario. The only difference is that we probably don’t care about mean differences, so we can set those to
0. If we also assume equal variances, we only need a total of 4 parameters.
# Parameters
M <- 0
SD <- 1
N <- 40
r <- 0.5
# Prepare parameters
mus <- c(M, M)
Sigma <- matrix(
ncol = 2, nrow = 2,
SD^2, SD^2 * r,
SD^2 * r, SD^2
# Simulate once with empirical = TRUE
samples <- mvrnorm(N, mu = mus, Sigma = Sigma, empirical = TRUE)
# Prepare data
colnames(samples) <- c("var1", "var2")
data <- as_tibble(samples)
This time, we plot the data with a scatter plot—a suitable graph for displaying the relationship between two numeric variables.
Figure 5: Correlation visualization
To perform the statistical test, we run cor.test().
The power analysis:
# Create an empty vector to store the p-values in
p_values <- vector(length = s)
# Loop
for (i in 1:s) {
# Simulate
samples <- mvrnorm(N, mu = mus, Sigma = Sigma)
# Run test
test <- cor.test(samples[, 1], samples[, 2])
# Extract p-value
p_values[i] <- test$p.value
# Calculate power
power <- sum(p_values <= .05) / s * 100
This produces a power of 92.7% with the current parameters.
2 t-tests
It gets more interesting when you have three groups that you want to compare. For example, imagine a study with two control conditions and a treatment condition. You probably want to compare the
treatment condition to the two control conditions. What is the appropriate analysis in this case? Well, that probably depends on who you ask. Someone might suggest performing an ANOVA to look at the
omnibus test, followed up by something like a Tukey HSD. Or maybe you can do an ANOVA/regression in which you compare the treatment condition to the two control conditions combined, using the proper
contrast. Both don’t make sense to me. In the former case, I don’t understand why you would first do an omnibus test if you’re going to follow it up with more specific analyses anyway and in the
latter case you run into the problem of not knowing whether your treatment condition differs from both conditions, which you are likely to predict. Instead, I think the best course of action is to
just run two t-tests.
The big thing to take away from this scenario is that we should power for finding a significant effect on both tests. We don’t power for the ‘design’ of the study or a single analysis. No, our
hypotheses our only confirmed if we find significant differences between the treatment condition and both control conditions, which we test with two t-tests.
Let’s further assume that the variance in the treatment condition is larger than the variance in the control conditions (which is plausible). Let’s also assume some dropout in the treatment condition
(also possibly plausible). This means we should test the differences with Welch’s two sample t-tests.
# Parameters
M_control1 <- 5
M_control2 <- 5
M_treatment <- 5.6
SD_control1 <- 1
SD_control2 <- 1
SD_treatment <- 1.3
N_control1 <- 50
N_control2 <- 50
N_treatment <- 40
# Simulate once
control1 <- mvrnorm(N_control1,
mu = M_control1, Sigma = SD_control1^2,
empirical = TRUE
control2 <- mvrnorm(N_control2,
mu = M_control2, Sigma = SD_control2^2,
empirical = TRUE
treatment <- mvrnorm(N_treatment,
mu = M_treatment, Sigma = SD_treatment^2,
empirical = TRUE
# Prepare data
colnames(control1) <- "DV"
colnames(control2) <- "DV"
colnames(treatment) <- "DV"
control1 <- control1 %>%
as_tibble() %>%
mutate(condition = "control 1")
control2 <- control2 %>%
as_tibble() %>%
mutate(condition = "control 2")
treatment <- treatment %>%
as_tibble() %>%
mutate(condition = "treatment")
data <- bind_rows(control1, control2, treatment)
We again inspect the data by visualizing it and calculating standardized effect sizes (two this time; although they are actually identical with the current parameters).
# Calculate standardized effect sizes
effect_size1 <- cohens_d(DV ~ condition,
pooled_sd = FALSE,
data = filter(data, condition != "control 2")
effect_size2 <- cohens_d(DV ~ condition,
pooled_sd = FALSE,
data = filter(data, condition != "control 1")
# Visualize the data
ggplot(data, aes(x = condition, y = DV)) +
geom_jitter(width = .2, alpha = .25) +
stat_summary(fun.data = "mean_cl_boot", geom = "pointrange") +
labs(x = "Condition")
Figure 6: Three groups visualization
The treatment condition differs from the two control conditions with a difference equal to a Cohen’s d of -0.52.
The statistical analysis consists of two Welch’s two sample t-tests:
The power analysis is now more interesting because we want to have enough power to find a significant effect on both t-tests. So that means we’ll store the p-values of both tests and then count how
often we find a p-value below .05 for both tests.
# Create two empty vectors to store the p-values in
p_values1 <- vector(length = s)
p_values2 <- vector(length = s)
# Loop
for (i in 1:s) {
# Simulate
control1 <- mvrnorm(N_control1, mu = M_control1, Sigma = SD_control1^2)
control2 <- mvrnorm(N_control2, mu = M_control2, Sigma = SD_control2^2)
treatment <- mvrnorm(N_treatment, mu = M_treatment, Sigma = SD_treatment^2)
# Run tests
test1 <- t.test(control1[, 1], treatment[, 1])
test2 <- t.test(control2[, 1], treatment[, 1])
# Extract p-values
p_values1[i] <- test1$p.value
p_values2[i] <- test2$p.value
# Calculate power
power <- sum(p_values1 <= .05 & p_values2 <= .05) / s * 100
The resulting power is 57.9%. Note that this is very different from the power of finding a significant effect of only one of the two tests; which would be equal to a power of 80%. An important lesson
to learn here is that with multiple tests, your power may quickly go down, depending on the power for each individual test. You can also calculate the overall power if you know the power of each
individual test. If you know you have 80% power for each of two tests, then the overall power will be 80% * 80% = 64%. This only works if your analyses are completely independent, though.
Regression (2 x 2 interaction)
Next, let’s look at an interaction effect between two categorical predictors in a regression. Say we have a control condition and a treatment condition and we ran the study in the Netherlands and in
Germany. With such a design there is the possibility of an interaction effect. Maybe there’s a difference between the control condition and the treatment condition in the Netherlands but not in
Germany, or perhaps it is completely reversed, or perhaps only weakened. The exact pattern determines the strength of the interaction effect. If an effect in one condition completely flips in another
condition, we have the strongest possible interaction effect (i.e., a crossover interaction). If the effect is merely weaker in one condition rather than another, then we only have a weak interaction
effect (i.e., an attenuated interaction effect).
Not only does the expected pattern of the interaction determine the expected effect size of the interaction, it also affects which analyses you should run. Finding a significant interaction effect
does not mean that the interaction effect you found actually matches what you hypothesized. If you expect a crossover interaction, but you only find an attenuated interaction, you’re wrong. And vice
versa as well. The issue is more complicated when you expect an interaction in which the effect is present is one condition but absent in another. You then should test whether the effect is indeed
absent, which is a bit tricky with frequentist statistics (although see this). Hypothesizing a crossover interaction is probably the easiest. I think you don’t even need to run an interaction test in
that case. Instead, you can just run two t-tests and test whether both are significant, with opposite signs.
In this scenario, let’s cover what is possibly the most common interaction in psychology—an attenuated interaction with the effect being present in both conditions, but smaller in one than in the
other. This means we want a significant difference between the two conditions in each country, as well as a significant interaction effect.
# Parameters
M_control_NL <- 4
M_control_DE <- 4
M_treatment_NL <- 5
M_treatment_DE <- 6
SD <- 2
N <- 40
# Prepare parameters
mus <- c(M_control_NL, M_control_DE, M_treatment_NL, M_treatment_DE)
Sigma <- matrix(
ncol = 4, nrow = 4,
SD^2, 0, 0, 0,
0, SD^2, 0, 0,
0, 0, SD^2, 0,
0, 0, 0, SD^2
# Simulate once
samples <- mvrnorm(N, mu = mus, Sigma = Sigma, empirical = TRUE)
# Prepare data
colnames(samples) <- c(
"control_NL", "control_DE", "treatment_NL", "treatment_DE"
data <- samples %>%
as_tibble() %>%
cols = everything(),
names_to = c("condition", "country"),
names_sep = "_",
values_to = "DV"
When it comes to interaction effects, it’s definitely a good idea to visualize the data. In addition, we calculate the effect size of the difference between the control and treatment condition for
each country.
# Calculate effect size per country
effect_size_NL <- cohens_d(DV ~ condition, data = filter(data, country == "NL"))
effect_size_DE <- cohens_d(DV ~ condition, data = filter(data, country == "DE"))
# Visualize the interaction effect
ggplot(data, aes(x = condition, y = DV)) +
geom_jitter(width = .2, alpha = .25) +
stat_summary(fun.data = "mean_cl_boot", geom = "pointrange") +
facet_grid(~country) +
labs(x = "Condition")
Figure 7: 2x2 interaction visualization
The graph shows that the difference between the control and treatment condition indeed seems to be larger in Germany than in the Netherlands. In the Netherlands, the effect size is equal to a Cohen’s
d of -0.5. In Germany, it’s -1.
A regression analysis can be used to test the interaction effect and whether the effect is present in each country. We do need the run the regression twice in order to get the effect of treatment in
each country. By default, Germany is the reference category (DE comes before NL). So if we switch the reference category to NL, we get the effect of treatment in the Netherlands.
Our interest is in the two treatment effects and the interaction effect (which is the same in both models). This means that we want to save 3 p-values in the power analysis.
# Create three empty vectors to store the p-values in
p_values_NL <- vector(length = s)
p_values_DE <- vector(length = s)
p_values_interaction <- vector(length = s)
# Loop
for (i in 1:s) {
# Simulate
samples <- mvrnorm(N, mu = mus, Sigma = Sigma)
# Prepare data
colnames(samples) <- c(
"control_NL", "control_DE", "treatment_NL",
data <- samples %>%
as_tibble() %>%
cols = everything(),
names_to = c("condition", "country"),
names_sep = "_",
values_to = "DV"
# Run tests
model_DE <- lm(DV ~ condition * country, data = data)
data <- mutate(data, country = fct_relevel(country, "NL"))
model_NL <- lm(DV ~ condition * country, data = data)
# Extract p-values
model_NL_tidy <- tidy(model_NL)
model_DE_tidy <- tidy(model_DE)
p_values_NL[i] <- model_NL_tidy$p.value[2]
p_values_DE[i] <- model_DE_tidy$p.value[2]
p_values_interaction[i] <- model_NL_tidy$p.value[4]
# Calculate power
power <- sum(p_values_NL <= .05 & p_values_DE <= .05 &
p_values_interaction <= .05) / s * 100
The overall power for this scenario is 9.6%. If you instead only look at the power of the interaction test, you get a power of 35.1%. The difference shows that it matters whether you follow up your
interaction test with the analyses that confirm the exact pattern of the interaction test. Also note that these analyses are not independent, so it’s not straightforward to calculate the overall
power. Simulation makes it relatively easy.
Regression (2 groups * 1 continuous interaction)
Another scenario involves having multiple groups (e.g., conditions) and a continuous measure that interacts with the group. In other words, this scenario consists of having different correlations,
with the correlation between a measure and an outcome depending on the group.
We can simulate a scenario like that by simulating multiple correlations and then merging the data together. In the scenario below, I simulate a correlation of size 0 in one group (i.e., control
group) and a correlation of .5 in another group (i.e., treatment group).
# Parameters
M_outcome <- 4
SD_outcome <- 1
M_control <- 4
SD_control <- 1
M_treatment <- 4
SD_treatment <- 1
r_control <- 0.1
r_treatment <- 0.5
N <- 40
# Prepare parameters
mus_control <- c(M_control, M_outcome)
Sigma_control <- matrix(
ncol = 2, nrow = 2,
SD_control^2, SD_control * SD_outcome * r_control,
SD_control * SD_outcome * r_control, SD_outcome^2
mus_treatment <- c(M_treatment, M_treatment)
Sigma_treatment <- matrix(
ncol = 2, nrow = 2,
SD_treatment^2, SD_treatment * SD_outcome * r_treatment,
SD_treatment * SD_outcome * r_treatment, SD_outcome^2
# Simulate once with empirical = TRUE
samples_control <- mvrnorm(
mu = mus_control,
Sigma = Sigma_control, empirical = TRUE
samples_treatment <- mvrnorm(
mu = mus_treatment,
Sigma = Sigma_treatment, empirical = TRUE
# Prepare data
colnames(samples_control) <- c("measure", "outcome")
data_control <- as_tibble(samples_control)
data_control <- mutate(data_control, condition = "Control")
colnames(samples_treatment) <- c("measure", "outcome")
data_treatment <- as_tibble(samples_treatment)
data_treatment <- mutate(data_treatment, condition = "Treatment")
data <- bind_rows(data_control, data_treatment)
Let’s visualize the simulated data to see whether we indeed observe a correlation in the treatment condition and none in the control condition.
`geom_smooth()` using formula = 'y ~ x'
Figure 8: 2 (group) x 1 (continuous) interaction visualization
Looks correct.
Analyzing this data is a bit trickier. To confirm our hypotheses we need to show that: 1. There is no correlation in the Control condition 2. There is a positive correlation in the Treatment
condition 3. There is a significant interaction effect.
The first one is rather difficult because it’s not straightforward to prove a null using frequentist statistics. We could do an equivalence test of some sort, but I’ll just keep it simple and count
the test as successful if we find a non-significant p-value.
Besides that, this scenario is similar to the previous one. We run two regression models in order to get the relevant p-value. The first model is to obtain the p-value of the slope between the
measure and outcome in the control condition, as well as the p-value of the interaction. The second model is to obtain the p-value of the slope in the treatment condition.
# Create three empty vectors to store the p-values in
p_values_control <- vector(length = s)
p_values_treatment <- vector(length = s)
p_values_interaction <- vector(length = s)
# Loop
for (i in 1:s) {
# Simulate
samples_control <- mvrnorm(N, mu = mus_control, Sigma = Sigma_control)
samples_treatment <- mvrnorm(N, mu = mus_treatment, Sigma = Sigma_treatment)
# Prepare data
colnames(samples_control) <- c("measure", "outcome")
data_control <- as_tibble(samples_control)
data_control <- mutate(data_control, condition = "Control")
colnames(samples_treatment) <- c("measure", "outcome")
data_treatment <- as_tibble(samples_treatment)
data_treatment <- mutate(data_treatment, condition = "Treatment")
data <- bind_rows(data_control, data_treatment)
# Run tests
model_control <- lm(outcome ~ condition * measure, data = data)
data <- mutate(data, condition = fct_relevel(condition, "Treatment"))
model_treatment <- lm(outcome ~ condition * measure, data = data)
# Extract p-values
model_control_tidy <- tidy(model_control)
model_treatment_tidy <- tidy(model_treatment)
p_values_control[i] <- model_control_tidy$p.value[3]
p_values_treatment[i] <- model_treatment_tidy$p.value[3]
p_values_interaction[i] <- model_control_tidy$p.value[4]
# Calculate power
power <- sum(p_values_control > .05 & p_values_treatment <= .05 &
p_values_interaction <= .05) / s * 100
The overall power for this scenario is 44.9%. It matters less now whether we power for the whole set of analyses or just the slope in the treatment condition because the interaction effect is wholly
driven by this slope.
In this post I presented code to perform a simulated-based power analysis for several scenarios. In the future I hope to expand on the scenarios, but I think the scenarios included so far already
reveal a few interesting things. In some cases, it’s rather trivial to simulate the data. The mvrnorm() function works wonders for simulating the data by letting you set empirical to TRUE, thereby
allowing you to inspect the simulated data. More importantly, though, I think that simulation-based power analyses are pedagogical. It takes the magic out of power analyses because power is nothing
more than counting how often you find the significant results you expect to find. Not only that, the simulation approach also means that if you can simulate the data, you can calculate the power.
Maybe that’s easier said than done, but that’s where my example code comes in. Hopefully it provides you with the code you can adapt to your own scenario so you can run the correct power analysis.
This post was last updated on 2023-04-11.
Maxwell, S. E. (2004). The persistence of underpowered studies in psychological research:
, consequences, and remedies.
Psychological Methods
(2), 147–163. | {"url":"https://willemsleegers.com/content/posts/9-simulation-based-power-analyses/simulation-based-power-analyses.html","timestamp":"2024-11-05T19:50:57Z","content_type":"application/xhtml+xml","content_length":"300317","record_id":"<urn:uuid:c1d2b331-06d7-493d-9a03-5277b282877f>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00664.warc.gz"} |
Package Contents#
crop_numpy(events, sensor_size, target_size) Crops the sensor size to a smaller sensor.
decimate_numpy(events, n) Returns 1/n events for each pixel location.
denoise_numpy(events[, filter_time]) Drops events that are 'not sufficiently connected to other events in the recording.' In
drop_by_area_numpy(events, sensor_size[, area_ratio]) Drops events located in a randomly chosen box area. The size of the box area is defined by a
drop_by_time_numpy(events[, duration_ratio]) Drops events in a certain time interval with a length proportional to a specified ratio of
drop_event_numpy(events, drop_probability) Randomly drops events with drop_probability.
drop_pixel_numpy(events, coordinates) Drops events for pixel locations that fire.
integrator_downsample(events, sensor_size, target_size, dt) Spatio-temporally downsample using with the following steps:
differentiator_downsample(events, sensor_size, ...[, ...]) Spatio-temporally downsample using the integrator method coupled with a differentiator to effectively
refractory_period_numpy(events, refractory_period) Sets a refractory period for each pixel, during which events will be ignored/discarded. We
spatial_jitter_numpy(events, sensor_size[, var_x, ...]) Changes x/y coordinate for each event by adding samples from a multivariate Gaussian
time_jitter_numpy(events[, std, clip_negative, ...]) Changes timestamp for each event by drawing samples from a Gaussian distribution and adding
time_skew_numpy(events, coefficient[, offset]) Skew all event timestamps according to a linear transform, potentially sampled from a
to_averaged_timesurface_numpy(events, sensor_size, ...) Representation that creates averaged timesurfaces for each event for one recording.
to_bina_rep_numpy(event_frames[, n_frames, n_bits]) Representation that takes T*B binary event frames to produce a sequence of T frames of N-bit
to_frame_numpy(events, sensor_size[, time_window, ...]) Accumulate events to frames by slicing along constant time (time_window), constant number of
to_timesurface_numpy(events, sensor_size, dt, tau[, ...]) Representation that creates timesurfaces for each event in the recording. Modeled after the
to_voxel_grid_numpy(events, sensor_size[, n_time_bins]) Build a voxel grid with bilinear interpolation in the time domain from a set of events.
uniform_noise_numpy(events, sensor_size, n) Adds a fixed number of noise events that are uniformly distributed across sensor size | {"url":"https://tonic.readthedocs.io/en/latest/autoapi/tonic/functional/index.html","timestamp":"2024-11-07T00:44:47Z","content_type":"text/html","content_length":"112250","record_id":"<urn:uuid:e94f922d-7535-4370-b131-39aeafd8430d>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00747.warc.gz"} |
PWC123 - Square Points
ETOOBUSY 🚀 minimal blogging for the impatient
On with TASK #2 from The Weekly Challenge #123. Enjoy!
The challenge
You are given coordinates of four points i.e. (x1, y1), (x2, y2), (x3, y3) and (x4, y4).
Write a script to find out if the given four points form a square.
Input: x1 = 10, y1 = 20
x2 = 20, y2 = 20
x3 = 20, y3 = 10
x4 = 10, y4 = 10
Output: 1 as the given coordinates form a square.
Input: x1 = 12, y1 = 24
x2 = 16, y2 = 10
x3 = 20, y3 = 12
x4 = 18, y4 = 16
Output: 0 as the given coordinates doesn't form a square.
The questions
As we will be doing some potentially floating point maths, a first question would be what tolerance should the operations have, in particular what tolerance is there to consider a value to be the
same as 0.
As nothing is said about the ordering of the points, we will assume they can be in any order and not necessarily assuming that close points in the list are also adjacent in the candidate square.
The examples seem to indicate that the points we consider are in a plane.
Last, I’d ask if this is meant to be a tricky question. The first example is about a square whose sides are parallel to the coordinate axes, but… squares might also be rotated in the plane!
The solution
We’ll use some vector maths here. Assuming that the input sequence of points $(P_0, P_1, P_2, P3)$ is ordered, i.e. that each consecutive pair is a side of the candidate polygon we want to check,
we end up with the following vectors representing the four sides:
\[s_0 = P_1 - P_0 \\ s_1 = P_2 - P_1 \\ s_2 = P_3 - P_2 \\ s_3 = P_0 - P_3\]
Much like the points, these “vector sides” are represented by pairs of numbers, so we can “blur” the line and use the same representation for the two.
In a square, two consecutive sides $s_i$ and $s_{i + 1}$ MUST fulfil the following two conditions:
• have the same length;
• be orthogonal, i.e. form an angle of $\pm 90°$.
Fun fact: we only need to check the two conditions above for the first three sides $s_0$, $s_1$, and $s_2$. If the comply, the fourth side $s_3$ will comply too.
The length of a vector is calculated with Pythagora’s theorem:
\[L_v = \sqrt{v_x^2 + v_y^2}\]
In comparing two sides, though, we can equivalently look a the squares and avoid calculating the square root:
\[L_v^2 = v_x^2 + v_y^2\]
Checking for orthogonality can be done calculating their regular scalar (or dot) product:
\[v \cdot w = v_x w_x + v_y w_y\]
This is 0 if and only if the two vectors are orthogonal, so it’s exactly the condition we are after.
OK, enough theory now… show us the code!
Raku first, which also gets the nice commenting. We define a class to represent our points and vectors:
# a tiny class for handling a limited set of vector operations
class Vector {
has @.cs is built is required;
# "dot", i.e. scalar, product
method dot (Vector $a) { return [+](self.cs »*« $a.cs) }
# the *square* of the length is all we need in our solution
method length_2 () { return self.dot(self) }
To make the implementation easier to read, we also override the difference operator (so that we can calculate vectorized sides by difference of two points):
multi sub infix:<->(Vector $a, Vector $b) {
Vector.new(cs => [$a.cs »-« $b.cs]);
as well as the dot product, which relies on the dot method:
multi sub infix:<*>(Vector $a, Vector $b) { $a.dot($b) }
Our basic test function is the following:
sub is-sequence-a-square (@points is copy) {
# comparing candidate sides means that we consider a "previous" side
# and a "current" one. A side is defined as the vector resulting from
# the difference of two consecutive points.
my $previous = @points[1] - @points[0];
# we just need to compare 3 sides, if they comply then the 4th will too
for 1, 2 -> $i {
my $current = @points[$i + 1] - @points[$i];
# check if sides have the same length (squared)
return False if $previous.length_2 != $current.length_2;
# approximation might give surprises, we'll accept as orthogonal
# sides whose scalar product is below our tolerance
return False if $previous * $current > tolerance;
# prepare for next iteration
$previous = $current;
# three sides are compliant, it's a square!
return True;
Now, of course, our input sequence of points might not be in the “right” order, so we wrap the test above to check different alternative orderings.
How many permutations should we consider? Out of 4 points, we have $4! = 24$ of them, but we don’t need to consider them all.
First, we can fix our point in the first position as our starting point, so in case we only have to consider permutations of the other three, i.e. $3! = 6$ of them.
Then, we can observe that two arrangements that have the same point as the opponent (i.e. non-adjacent) point to the starting point are actually the same candidate polygon, traversed in opposite
directions. Hence, we can just consider one of these two.
In the end, we can just consider three possible permutations, like in the following function:
sub is-square (*@points) {
# try out permutations of the inputs that can yield a square. We fix
# point #0 and only consider one permutation for each of the other
# points as the opposite, ignoring the other because symmetric.
state @permutations = (
[0, 2, 1, 3], # 0 and 1 are opposite
[0, 1, 2, 3], # 0 and 2 are opposite
[0, 2, 3, 1], # 0 and 3 are opposite
for @permutations -> $permutation {
my @arrangement = @points[@$permutation].map({Vector.new(cs => @$_)});
return 1 if is-sequence-a-square(@arrangement);
return 0;
A couple of final remarks:
• Math::Vector was of… great inspiration for getting the implementation right. I used it in the first place, but it takes ages to load and eventually re-implemented only the relevant parts;
• inlining the class as I did means that the definition of the overloaded multi sub infix operators must appear outside the class definition. This took me a while to figure out.
The Perl translation is pretty much straightforward, also thanks to the overload module that allows us to overload a couple of operators. Here’s the complete program:
#!/usr/bin/env perl
use v5.24;
use warnings;
use experimental 'signatures';
no warnings 'experimental::signatures';
use constant False => 0;
use constant True => 1;
use constant tolerance => 1e-7;
package Vector2D {
use overload
'-' => sub ($u, $v, $x) { v([ map { $u->[$_] - $v->[$_] } 0, 1 ]) },
'*' => sub ($u, $v, $x) { $u->dot($v) };
sub dot ($S, $t) { return $S->[0] * $t->[0] + $S->[1] * $t->[1] }
sub length_2 ($S) { return $S->dot($S) }
sub v ($v) { return bless [$v->@*], __PACKAGE__ }
sub is_sequence_a_square (@points) {
my $previous = $points[1] - $points[0];
for my $i (1 .. $#points - 1) {
my $current = $points[$i + 1] - $points[$i];
return False if $previous->length_2 != $current->length_2;
return False if $previous * $current > tolerance;
$previous = $current;
return True;
sub is_square (@points) {
state $permutations = [
[0, 2, 1, 3],
[0, 1, 2, 3],
[0, 2, 3, 1],
for my $permutation ($permutations->@*) {
my @arrangement = map { Vector2D::v($_) } @points[@$permutation];
return 1 if is_sequence_a_square(@arrangement);
return 0;
say is_square([10, 20], [20, 20], [20, 10], [10, 10]);
say is_square([12, 24], [16, 10], [20, 12], [18, 16]);
say is_square([0, 0], [1, 1], [0, 2], [-1, 1]);
Thank you for reading this far and stay safe! | {"url":"https://etoobusy.polettix.it/2021/07/29/pwc123-square-points/","timestamp":"2024-11-11T07:35:39Z","content_type":"text/html","content_length":"21195","record_id":"<urn:uuid:2b8c17d0-2b78-4054-beb9-f77f33236f7d>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00164.warc.gz"} |
Bivariate Choropleth Maps with Arcpy
In my previous post, I showed how to prepare the data for a bivariate choropleth map using PostGIS and QGIS. I also indicated that there is a website that shows an ArcGIS tool to do it. But, this
actually turns into a good opportunity to illustrate some Python, and how to create the bivariate data using Arcpy.
Arcpy is certainly not as terse as SQL, but it does get the job done, and rather easily. We just have to think about the project a little differently. The code below is a Script tool that I created.
import arcpy, math, numpy
fc = arcpy.GetParameterAsText(0)
numrecs = int(arcpy.GetCount_management(fc).getOutput(0))
fields = arcpy.ListFields(fc, "bimode")
if len(fields) != 1:
arcpy.AddField_management(fc, "bimode", "text", 3)
f1 = arcpy.GetParameterAsText(1)
f2 = arcpy.GetParameterAsText(2)
fields = ['bimode',f1,f2]
var1 = arcpy.UpdateCursor(fc, sort_fields=f1)
for row in var1:
row.setValue("bimode",str(int(math.ceil((float(i) / float(numrecs)) * 3.0))))
var2 = arcpy.UpdateCursor(fc, sort_fields=f2)
for row in var2:
row.setValue("bimode",row.getValue("bimode") + "." + str(int(math.ceil((float(i) / float(numrecs)) * 3.0))))
So let’s think about what we are doing. If I want to break the data up into 3 sections, I have to know how many records we have, sort the records, and then determine which group (1, 2, or 3) each
record falls in. Let’s assume we have 10 records, all sorted. What group is the nth record in. The formula is actually easy:
Group = ceil(N / i * 3)
when i = 2: (2/10) * 3 = .6 = ceil(.6) = 1
when i = 7: (7/20) * # = 2.1 = ceil(2.1) = 3
So, you’ll see from the code that the first thing we do is get N,
numrecs = int(arcpy.GetCount_management(fc).getOutput(0))
Then, we create a recordset that allows updating the data, but we do it for one of the fields, and sort the cursor by that field:
var1 = arcpy.UpdateCursor(fc, sort_fields=f1)
Now, with the field sorted, we can loop through the cursor and apply our formula:
for row in var1:
row.setValue("bimode",str(int(math.ceil((float(i) / float(numrecs)) * 3.0))))
I do a little sleight of hand with converting the data to an Integer, and then to String, but that is mostly to get the data into the form I want. From there, you can symbolize your layer based on
the bimode field. | {"url":"https://www.artlembo.com/post/bivariate-choropleth-maps-with-arcpy","timestamp":"2024-11-10T08:38:34Z","content_type":"text/html","content_length":"1050037","record_id":"<urn:uuid:57200871-5bdd-46b1-8768-33a1294b58a6>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00427.warc.gz"} |
reduction ratio calculation jaw crusher in america
Crusher Reduction Ratio - Mineral Processing & Metallurgy
2016-1-12·Crusher Reduction Ratio. I have mentioned the fact that, as the % of voids in the crushing chamber decreases, the production of fines by attrition increases. This is like saying that,
as the Crusher Reduction Ratio in any given crusher is increased, the % of fines in the product will increase, even though the discharge setting remains unchanged.
Read More | {"url":"https://pracowniamadlen.pl/elixirite/16_Mon_1612_reduction-ratio-calculation-jaw-crusher-in-america-.html","timestamp":"2024-11-08T02:42:51Z","content_type":"text/html","content_length":"30781","record_id":"<urn:uuid:38d5d846-b00a-4288-931c-9eb91b2f3959>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00556.warc.gz"} |
Hawking Radiation
In the vast expanse of the universe, amidst the swirling masses of stars and galaxies, lies a phenomenon that challenges our understanding of black holes and the very fabric of space-time itself.
Known as Hawking radiation, this enigmatic process was first proposed by the renowned physicist Stephen Hawking in 1974, revolutionizing our comprehension of black holes. Let's embark on a journey to
unravel the mysteries of Hawking radiation, exploring its significance, underlying principles, and implications for our understanding of the cosmos.
What is Hawking Radiation?
Hawking radiation is a theoretical prediction that suggests black holes are not entirely black, but rather emit a faint glow of particles due to quantum effects near the event horizon. This
phenomenon arises from the interplay between quantum mechanics and general relativity, two pillars of modern physics.
The Origin of Hawking Radiation:
According to quantum field theory in curved spacetime, virtual particle-antiparticle pairs constantly pop in and out of existence near the event horizon of a black hole. In some cases, one of the
particles falls into the black hole while the other escapes into space. This escaping particle is known as Hawking radiation, causing the black hole to gradually lose mass over time.
Black Hole Thermodynamics:
Hawking's groundbreaking insight linked black holes with thermodynamic concepts such as temperature and entropy. By considering black holes as thermodynamic objects, he showed that they emit
radiation with a characteristic temperature inversely proportional to their mass. This temperature is incredibly low for astrophysical black holes but becomes significant for microscopic black holes.
Implications for Black Hole Physics:
Hawking radiation has profound implications for our understanding of black hole dynamics and the fate of these cosmic entities. It suggests that black holes have a finite lifespan and eventually
evaporate completely, leaving behind only radiation and no remnants—a concept known as black hole evaporation.
Experimental Challenges and Observational Signatures:
Despite its theoretical elegance, detecting Hawking radiation directly from astrophysical black holes remains a formidable challenge due to their immense distance and faint emission. However,
scientists have proposed various indirect methods, such as searching for signatures in cosmic microwave background radiation or gravitational wave observations.
Quantum Information Paradox:
Hawking radiation also plays a central role in the resolution of the black hole information paradox. This paradox arises from the apparent conflict between the loss of information when matter falls
into a black hole and the conservation of information, a fundamental principle of quantum mechanics. Theoretical insights into Hawking radiation suggest that information may be encoded in the
radiation, preserving it even after the black hole evaporates.
Future Directions and Open Questions:
The study of Hawking radiation continues to captivate physicists, with ongoing research aimed at refining theoretical models, devising experimental strategies for detection, and exploring its broader
implications for fundamental physics. Key questions remain unanswered, such as the exact nature of the emitted particles, the fate of information trapped within black holes, and the potential
connection to other areas of physics, such as quantum gravity.
Hawking radiation stands as a testament to the power of human intellect and the beauty of theoretical physics. By shedding light on the quantum nature of black holes, it challenges our understanding
of the universe at its most extreme scales. As we continue to probe the mysteries of Hawking radiation, we embark on a quest to unravel the fabric of space-time and unlock the secrets of the cosmos.
Did you find this article valuable?
Support SPACELIA by becoming a sponsor. Any amount is appreciated! | {"url":"https://spacelia.hashnode.dev/hawking-radiation","timestamp":"2024-11-02T02:43:39Z","content_type":"text/html","content_length":"118193","record_id":"<urn:uuid:cfa85710-b92d-4155-8f5e-6140e139b116>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00277.warc.gz"} |
Problem Solving Using Percentages
Thelma A. Rex Rosa L. Parks Middle School
147th and Robey Avenue
Harvey IL 60426
The student will be able to recognize the three basic types of Percentage
The student will be able to:
1. Find what percent one number is of another.
2. Find a percentage of a number.
3. Find a number when the percent is known.
Materials Needed:
1-16 oz bag of plain M&M's; 19-4oz bags of M&M's; 19 plastic bags;
1 black marker and 19 pencils; 19 student folders with plain writing
l9 M&M Math handouts and 19 calculators: Overhead projector.
All of the students are given a folder with a small plastic bag and one
small bag of plain M&M's. The students are asked to guess how many M&M's are in
the bag. They are asked to put their M&M's into sets by color and write the
number of M&M's they have in each set. They are also asked to use > or < or =,
to show the relationship between these sets. Students put l5 M&M's in front of
them. Then they are asked various questions, for example, how many piles of
fours can you make? What is left?
The teacher demonstrates three types of word problems on the board to find
percents by using ratio and proportion. The overhead projector is used to
illustrate the same problems, but in a different way. For example:
a. The student is given the following information;
1. the number of red M&M's in the small bag.
2. the total number of M&M's in the small bag.
3. the total number of M&M's in the big bag.
The student is asked to find the number of red M&M's in the big bag.
b. A bicycle is on sale with a coupon for 25% off which is a $l5.00
savings. How much is the bicycle without the coupon?
c. A set of stereo speakers is priced at $125.00. The sales tax
is 8%. What will be the total price of the speakers?
d. In a photo, Carlos measures 8 centimeters. He is actually 56
inches tall. In the photo his brother, Juan, measures 7 centimeters. How tall
is Juan?
Performance Assessment:
The students will be given a post-test. The students will be able to
tell how many M&M's are in the big bag as compared to their small bag of
M&M's. Students already know that there are 88 red M&M's in the big bag
and that there are a total of 527 M&M's in the big bag.
Multicultural Aspects:
Percent comes from the Latin phrase per centum, which may be
translated by the hundred, to the hundred, for each hundred. The symbol
for percent is %. Thus, 85% means the ratio 85 to 100 or 85/100. Percent
is used for mathematical calculations throughout the world in everyday
living regardless of nationality.
Math and Science, A Solution. 1987, Aims Education Foundation.
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Sustainable Energy - without the hot air
ATR72-600, which is claimed to be about one third more energy efficient than Bombardier's Q400 turboprop, which I featured on page 35 of SEWTHA.
I've consequently written an update on turboprops, celebrating this achievement, but in the interests of balance I feel I should also nominate the advertisers
of the ATR72-600 for this year's Hot Air Oscar for the most misleading "green" infographic, for this astonishing picture [at left] showing the difference
between the fuel consumption of the ATR 72 and the Q400 on a 250-nautical-mile journey. As the numbers in the picture show, the ATR 72's fuel consumption is
70% of the Q400's, but the volume of the three-dimensional blue barrel shown is 30% of the volume of the orange barrel — a 2.3-fold exaggeration!
blue : orange
barrel barrel
ratio of 91 : 134 =0.68:1
ratio of 118 : 179 =0.66:1
ratio of volumes =
(as depicted) 0.68×0.68×0.66
true ratio of volumes =0.70:1
In December 2013, Christopher Booker in the Telegraph discusses a study by Gordon Hughes, published by the Renewable Energy Foundation in December 2012, which is said to show, due to wear and tear on
their mechanisms and blades, the amount of electricity generated by wind turbines "very dramatically falls over the years". Booker asserts that "Hughes showed his research to David MacKay, the chief
scientific adviser to the Department of Energy and Climate Change, who could not dispute his findings." This is not true.
In fact, I doubted Hughes's assertions from the moment I first read his study, since they were so grossly at variance with the data.
Figure 1: Actual load factors of UK wind farms at ages 10, 11, and 15.
a) Histogram of average annual load factors of wind farms at age 10 years. For comparison, the blue vertical line indicates the assertion from the Renewable Energy Foundation's study that "the
normalised load factor is 15% at age 10."
b) Histogram of average annual load factors of wind farms at age 11 years.
c) Histogram of average annual load factors of wind farms at age 15 years. For comparison, the red vertical line indicates the assertion from the Renewable Energy Foundation's study that "the
normalised load factor is 11% at age 15." At all three ages shown above, the histogram of load factors has a mean and standard deviation of 24% ± 7%.
Moreover, by January 2013 I had figured out an explanation of the underlying reason for Hughes's spurious results. I immediately wrote a technical report about this flaw in Hughes's work, and sent it
to the Renewable Energy Foundation, recommending that they should retract the study.
I would like to emphasize that I believe the Renewable Energy Foundation and Gordon Hughes have performed a valuable service by collating, visualizing, and making accessible a large database
containing the performance of wind farms. This data, when properly analysed in conjunction with detailed wind data, will allow the decline in performance of wind turbines to be better understood.
Iain Staffell and Richard Green, of Imperial College, have carried out such an analysis (in press), and it indicates that the performance of windfarms does decline, but at a much smaller rate than
the "dramatic" rates claimed by Hughes. The evidence of decline is strongest for the oldest windfarms, for which there is more data. For newer windfarms, the error bars on the decline rates are
larger, but Staffell and Green's analysis indicates that the decline rates may be even smaller.
I will finish this post with a graphical explanation of the flaw that I identified (as described in detail in my technical report) and that I believe underlies Hughes's spurious results.
The study by Hughes modelled a large number of energy-production measurements from 3000 onshore turbines, in terms of three parameterized functions: an age-performance function "f(a)", which
describes how the performance of a typical wind-farm declines with its age; a wind-farm-dependent parameter "u[i]" describing how each windfarm compares to its peers; and a time-dependent parameter
"v[t]" that captures national wind conditions as a function of time. The modelling method of Hughes is based on an underlying statistical model that is non-identifiable: the underlying model can fit
the data in an infinite number of ways, by adjusting rising or falling trends in two of the three parametric functions to compensate for any choice of rising or falling trend in the third. Thus the
underlying model could fit the data with a steeply dropping age-performance function, a steeply rising trend in national wind conditions, and a steep downward trend in the effectiveness of wind farms
as a function of their commissioning date (three features seen in Hughes's fits). But all these trends are arbitrary, in the sense that the same underlying model could fit the data exactly as well,
for example, by a less steep age-performance function, a flat trend (long-term) in national wind conditions, and a flat trend in the effectiveness of wind farms as a function of their commissioning
date. The animation above illustrates this non-identifiability. The truth, for a cartoon world, is shown on the left. On the bottom-left, the data from three farms (born in 87, 91, and 93) are shown
in yellow, magenta, and grey; they are the sum of a age-dependent performance function f(a) [top left] and a wind variable v_t [middle left]. (The true site 'fixed effects 'variables u1, u2, u3 are
all identical, for simplicity.) On the right, these identical data can be produced by adding the orange curve f(a) to the site-dependent 'fixed effects' variables u1, u2, u3 (shown in green), thus
obtaining the orange curves shown bottom right, then adding the wind variable [middle right] shown in blue (v_t).
Three spectacularly large solar power stations have recently been in the news: Ivanpah, located in California, but within spitting distance of Las Vegas, is a concentrating solar power station in
which 300,000 flat mirrors focus sunshine onto three power-towers. Solana, located in Gila Bend, Arizona, has a collecting field of about 3200 parabolic-trough mirrors, each about 25 feet wide, 500
feet long and 10 feet high, and it can generate electricity at night thanks to its ability to store high-temperature heat in vast molten salt stores. Kagoshima, near the Southern tip of Japan, has
290,000 solar photovoltaic panels. All three are enormous, and must be amazing to visit: Ivanpah occupies about 14 km^2; Solana, 12.6 km^2, and Kagoshima, 1 km^2.
Now, I'm always interested in powers per unit area of energy-generating and energy-converting facilities, so I worked out the average power per unit area of all three of these, using the estimated
outputs available on the internet. Interestingly, all three power stations are expected to generate about 8.7 W/m^2, on average. This is at the low end of the range of powers per unit area of
concentrating solar power stations that I discussed in Chapter 25 of Sustainable Energy - without the hot air; Andasol, the older cousin of Solana in Spain, is expected to produce about 10 W/m^2, for
I published a paper on Solar energy in the context of energy use, energy transportation, and energy storage in the Phil Trans R Soc A Journal earlier this year, and these three new data points lie
firmly in the middle of the other data that I showed in that paper's figure 8 (original figures are available here). .
These data should be useful to people who like to say "to power all of (some region) all we need is a solar farm the size of (so many football fields, or Greater Londons, or Waleses), if they want to
get their facts right. For example, Softbank Corporation President Masayoshi Son recently alleged that "turning just 20% of Japan’s unused rice paddies into solar farms would replace all 50 million
kilowatts of energy generated by the Tokyo Electric Power Company". Unfortunately, this is wishful thinking, as it is wrong by a factor about 5. The area of unused rice paddies is, according to
Softbank, 1.3 million acres (a little more than 1% of the land area of Japan). If 20% of that unused-rice-paddies area were to deliver 8.7 W/m^2 on average, the average output would be about 9 GW. To
genuinely replace TEPCO, one would need to generate roughly five times as much electricity, and one would have to deliver it when the consumers want it.
Maybe a better way to put it (rather than in terms of TEPCO) is in national terms or in personal terms: to deliver Japan's total average electricity consumption (about 1000 TWh/y) would require
13,000 km^2 of solar power stations (3.4% of Japan's land area), and systems to match solar production to customer demand; to deliver a Japanese person's average electricity consumption of 21 kWh per
day, each person would need a 100 m^2 share of a solar farm (that's the land area, not the panel area or mirror area). And, as always, don't forget that electricity is only about one third or one
fifth of all energy consumption (depending how you do the accounting). So if you want to get a country like Japan or the UK off fossil fuels, you need to not only do something about the current
electricity demand but also deal with transport, heating, and other industrial energy use.
Sources: NREL; abengoa.com; NREL; solarserver.com; and google planimeter.
The Chinese translation of Sustainable Energy - without the hot air is now available on amazon.cn
I am very grateful to the Chinese Academy of Sciences and President Li Jinhai for arranging both the translation and its publication. Thank you!
I've updated my "Map of the World" which shows, country by country, how human power-consumption per unit area compares with the power-production per unit area of renewables. I originally published
this graph on my blog in August 2009. I've made quite a few improvements to it since then, including the representation of country size by point size, and colour coding of continents in the style of
One interesting thing I figured out while working on this graph is that, while the average power consumption per unit land area of the world is 0.1 W/m^2, 78% of the world's population lives in
countries where the average power consumption per unit land area of the world is greater than 0.1 W/m^2 — much as, in a town with some crowded buses and many empty buses, the average number of
passengers per bus may be small, but the vast majority of passengers find themselves on crowded buses.
Please follow this "Map of the World" link to see multiple versions of the graph, and to download high-resolution originals, which everyone is welcome to use.
My "Map of the World" graphs are published this year in two journal papers, which I will blog about shortly.
David J C MacKay (2013a) Could energy-intensive industries be powered by carbon-free electricity? Phil Trans R This paper also contains detailed information about the power per unit area of wind
Soc A 371: 20110560. http://dx.doi.org/10.1098/rsta.2011.0560 farms in the UK and USA, and of nuclear power facilities
David J C MacKay (2013b) Solar energy in the context of energy use, energy transportation and energy storage. This paper also contains detailed information about the power per unit area of solar
Phil Trans R Soc A 371: 20110431. http://dx.doi.org/10.1098/rsta.2011.0431 farms
In Sustainable Energy - without the hot air I spent a couple of pages discussing hydrogen transportation, under the title "Hydrogen cars – blimp your ride". While I still think that some people have
been overhyping hydrogen - even Nature magazine, who praised Governor Arnold for filling up a hydrogen-powered Hummer - some of the criticisms I wrote were incorrect and I wish to correct them.
On page 131 I wrote: ... hydrogen gradually leaks out of any practical container. If you park your hydrogen car at the railway station with a full tank and come back a week later, you should expect
to find most of the hydrogen has gone. Both of these statements are incorrect.
First, while hydrogen is a very leaky little molecule, it is possible to make practical containers that contain compressed hydrogen gas for long durations. It's just necessary to have sufficient
thickness of the right type of material; this material may be somewhat heavy, but practical solutions exist. The technical term used in the hydrogen community for this topic is "permeation", and it's
especially discussed when ensuring that hydrogen vehicles will be safe when left in garages. Hydrogen containers are currently classed in four types, and the metallic containers and containers with
metallic liners (Types 1, 2, and 3) have negligible permeation rate. However, hydrogen permeation is an issue for containers with non-metallic (polymer) liners (Type 4) which readily allow the
permeation of hydrogen. [Source: P. Adams et al]
Second, when discussing the hydrogen vehicle that is left for 7 days, I incorrectly tarred all hydrogen vehicles with a hydrogen-loss brush that applies only to vehicles that store liquified hydrogen
at cryogenic temperatures. There are in fact three types of hydrogen storage: Compressed gas (typically at 350 or 700 bar); Cryogenic (typically at less than 10 bar and at extremely low temperature)
and Cryo-compressed (at low temperature and at pressures up to about 350 bar). The hydrogen community discuss the "loss-free dormancy time" and the "mean autonomy time" of a system, which are
respectively the time after which the system starts to lose hydrogen, and the time after which the car has lost so much hydrogen it really needs refilling. In the US Department of Energy's hydrogen
plans, the targets are for a loss-free dormancy time of 5 days and a mean autonomy time of 30 days. Cryogenic liquid-hydrogen systems (such as the one in the BMW Hydrogen 7, which I featured in my
book) do not currently achieve either of these targets. (And the reason is not that the hydrogen is permeating out, it's that heat is permeating in, at a rate of 1 watt or so, which gradually boils
the hydrogen; the boiled hydrogen is vented to keep the remaining liquid cold.) However, compressed-gas systems at 700 bar can achieve both of these targets, so what I wrote was unfair on hydrogen
vehicles. [Source: EERE 2006 Cryo-Compressed Hydrogen Storage for Vehicular Applications]
I apologise to the hydrogen community for these errors.
I will add a correction to the errata imminently. | {"url":"https://withouthotair.blogspot.com/2013/","timestamp":"2024-11-14T11:45:32Z","content_type":"text/html","content_length":"110183","record_id":"<urn:uuid:699559f4-cf8c-4088-8b51-0ffbb1518ae3>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00832.warc.gz"} |
How to calculate square feet for tile- Easy Way to Measure
You’re trying to calculate the cost of your upcoming bathroom renovation. But how do you know how much tile to buy for the floor? What about the kitchen backsplash? How do you even go about measuring
This post will decode how to calculate the square footage for your tiles. It will cover:
• basic square footage calculation
• how to measure for tile walls
• how to measure the flooring
• What to round up to
• How much extra tile to buy
I think you’ll find this to be simpler than what it seems. So, read on to learn how to measure for tile and add enough extra on.
If you are looking to measure a shower or bathtub walls for tile then you want to also check out this post.
How to Calculate Square Feet for Tile
First, let’s start with a basic square footage calculation. Which is this:
Length x Width= Area (square feet)
Pretty simple, right? This applies to squares and rectangles.
So, the area of this square is: 5ft x 5ft = 25 square feet
For walls, you calculate:
Width x Height
This bathtub wall would be 5ft wide x 7 ft tall which equals 35 sq. ft.
Now that we have the basics let’s move on to how to measure for tile using specific applications.
How to Measure for Tile Square Footage
Firstly, most tile walls and floors don’t work out to be exactly on the foot marks. Because of this, we’ll round up all of our measurements.
Round up to nearest foot or half-foot
When measuring a wall or floor, you want to round your numbers up to the nearest foot or half-foot. For example,
• 4ft 11 inches would round up to 5ft
• 5ft 2 inches would round up to 5ft 6 inches
• 6ft even would round up to 6ft 6 inches
Remember, that 6 inches is one-half of a foot.
Why round up?
The reason for rounding up is because you will be cutting the tiles.
For example, if you are tiling 20 inches tall and you are using 12 inch tall tiles then you will cut off the extra 4-inches. Maybe you can use that 4 inch piece but maybe you can’t.
My point is that it takes two tiles to go 20 inches high and that’s why you round 20 inches up to 24.
Getting 2 pieces out of 1 tile
The reason for rounding up to the half-foot mark is that anything under 1/2 tile means that you can get two pieces out of that one tile.
Example: You are tiling 76 inches tall with 12-inch tall tiles. You will use 6 tiles (72 inches) and one tile can provide 2 of the 4-inch pieces.
But I’m not using 12-inch tall tiles!
That’s OK. Tile is ordered in square footage quantities so rounding up to foot/half-foot still works for most tile sizes.
However, there are a few scenarios where I measure by the square inch. I went into more detail on measuring by the square inch in my measuring shower walls for tile post.
How to Measure for Tile Bathroom Walls
When measuring bathroom walls you need to find the width and height of every wall that will be tiled.
So, take out a tape measure and measure how many feet and inches the wall is wide and write it down.
Then measure how high the tile will go on the wall and write it down.
With the tile wainscot above, the height of the tile portion of the wall is 2 feet-10 inches (34 inches). Further, the length of the wainscot portion of the wall is 4ft-8 inches (56) inches.
We will round both numbers up and come to: 3ft x 5 ft.
This totals: 15 square feet of tile.
So, 15 sq. ft is the actual number of tiles you will need to complete this project. But it’s a good idea to have a few extras.
Read on for how to figure out how many extra tiles to buy so you will have enough to complete your tiling project.
How to Calculate Square Footage for Flooring
Calculating square feet for floors is quite similar to calculating the amount of tiles for walls.
Whereas walls are width x height, floors are width x length.
Let’s use a common bathroom floor as an example.
5×8 bath (measure rectangle)
Here we have a common bathroom floor size. From the door to the bathtub measures 8ft-4 inches in length. The width is 59 inches.
8ft-4in rounds up to 8ft-6 inches or 8.5 ft. Further, 59 inches (4ft-11in) rounds up to 5ft.
8.5×5 = 42.5 sq feet of floor tiles necessary to cover the floor.
Make sure to measure underneath the door
One thing to keep in mind is that the tile generally starts part-way underneath the door. So, rather than measure from the bathtub to the door when the door is closed you want to measure about 3/4
inch underneath the doorway.
So, if your measurement was 64 inches from bathtub to door you would want to add an additional 3/4 inch onto this measurement to make sure that your tile extends all of the way underneath the door
when it’s closed.
How to subtract the vanity from the total area of the floor
Oftentimes, the vanity will already be installed by the time floor tile installation begins. So, we don’t need to include the vanity in our square footage totals.
There are two ways to measure a bathroom floor and not include the vanity.
Method 1: Measure your room then subtract the vanity
The first way is to simply subtract the vanity from the total square footage of the space.
First, the dimensions of the floor round up to 5.5 x 5 feet. This totals 27.5 square feet for the total size of the room.
Secondly, we need the right measurement for the vanity. Take a tape measure and measure the bottom toe kick area for length and width. Your measurements might be:
17 inches x 34 inches.
However, instead of rounding the measurements up, I’ll be rounding them down since we’re subtracting them from the total.
In this case, I would round the 17-inch measurement down to 12 inches, or 1 foot. then 34 inches would round down to 30 inches, or 2.5 ft.
Then we get the square footage of the vanity: 1 x 2.5 = 2.5 sq. ft.
This would be subtracted from the 27.5 sq. feet of total space and bring down the room’s square footage to 25 sq. ft. plus extra.
Method 2: Divide space into smaller sections
Another way to measure the space would be to chop up the room into smaller sections. In this case, we could turn the room into two different rectangles.
Rectangle 1 is 2.5 feet x 4 ft. 11 inches. We would convert this by rounding 30 inches up to the nearest half foot which is 3ft and the other direction would be rounded up to 5 ft.
So the area of rectangle 1 = 15 sq. ft.
Rectangle 2 is 3ft-6in x 2 ft-10in. The first number gets rounded up to 4ft and the second becomes 3ft.
The area of rectangle 2 = 12 sq. ft.
Then we add the rectangles together and come up with: 27 sq. ft.
In these examples, we do come up with two different totals: 27 feet and 25 feet. However, they are very close and goes to show why this isn’t an exact science and stresses the need to purchase extra
tile to make sure that you will have enough to complete your tile project.
How Much Extra Tile to Buy?
How much extra tile to buy seems like a simple question but there are several different answers. So, I guess… it depends?
What it depends on are several factors:
• How confident you are in your measurements
• How easy more tile is to get
• Where the tile is being purchased from
• Tile size
• Tile layout/pattern
For example, there is more waste with a herringbone pattern than a typical straight-lay pattern.
Why do I need to order extra tiles?
There is more than one reason why you may need a few extra tiles beyond the absolute minimum amount to cover your space.
• You may miscut one or more tiles
• Certain tiles may have some sort of coloring or pattern that you don’t like
• You may receive some damaged tiles and not realize it until 80% of your floor is tiled
• You want a few extra on hand in case an unforeseen future repair is needed
• Maybe you miscalculated your space
It’s common practice to have some extra tiles on hand. The question is how many more tiles do you need?
When you buy tile from a box store
Firstly, I have written an entire post on how to buy tile from big box retail stores. Stores like Home Depot, Lowes, Menards, Floor and Decor, etc.
When it comes to purchasing extra tiles, box stores are a great place to purchase from. The reason being is that as long as you can return your extra tiles for a full refund you can’t really order
too many.
That being said, it’s not all fun & games when ordering from the retail box stores as the linked post above suggests.
Bottom line when ordering tile from box stores: Order A LOT extra because you can return all of the ones you don’t want for a full refund.
Ordering tile from a tile showroom or dealer
The way that most tile is purchased, I think, is through a local showroom that carries several different makes and models of tile.
Typically, the policy in this scenario is that they don’t accept tile returns or, if they do, they charge an exorbitant fee for ‘restocking’. Commonly, this runs 30-50%.
Bottom line when ordering from tile showrooms: Assume whatever you order that you will be stuck with. So, order a few extras but don’t get crazy. It’s pretty common to shoot for 10-15% extra.
How much extra tile to buy when special order/handmade
Finally, if you are ordering tile that is special order or handmade then you will want to have more than a normal amount of overage figured in.
The reason is that you absolutely can’t be short on tile.
So, if your tile order takes 2+ months to come in, it’s coming from overseas, or it’s a batch made especially for you and impossible to match later, you have to suck it up and order a lot extra.
Who is doing the measuring of the tiles?
Finally, it matters who’s measurements that you are using for your order.
A tile or industry professional
As a pro that does this all of the time, I will order 5-7% extra for tile that I know can’t be returned. This works for me and I trust my measurements. But it’s too tight for most.
If you are not a tile professional but have followed the steps in here then you should be able to aim for 10-15% extra if you’ve measured carefully. If it’s a special order, then I recommend aiming
for 25-35% or more.
If you are ordering from box stores, then buy 20-50% too much as long as you can return the extra.
Store your extra tiles
At the end of your project, you want to have a few extra tiles on hand just in case. Find a secure place to store them. Hopefully, you won’t need them but I’ve seen entire rooms torn out before that
could have been prevented if the homeowners had a couple of extra tile onhand.
Final Thoughts
This post covered the basics of how to calculate square footage for tile. See this post if you are measuring a shower or bathtub surround for tile. | {"url":"https://www.diytileguy.com/how-to-calculate-square-feet-for-tile/","timestamp":"2024-11-06T07:26:09Z","content_type":"text/html","content_length":"214818","record_id":"<urn:uuid:706972d0-930a-4a22-a97d-2afaa437cdbd>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00850.warc.gz"} |
The kinetic energy of an object with a mass of 3 kg constantly changes from 60 J to 270 J over 8 s. What is the impulse on the object at 5 s? | HIX Tutor
The kinetic energy of an object with a mass of #3 kg# constantly changes from #60 J# to #270 J# over #8 s#. What is the impulse on the object at #5 s#?
Answer 1
$3 \cdot \left(5 \cdot \frac{\sqrt{180} - \sqrt{40}}{8} - \sqrt{40}\right)$
t=0, #v_1=sqrt(2*W/m)# #v_1=sqrt(40)# t=8, #v_1=sqrt(2*W/m)# #v_1=sqrt(180)# first, we calculate acceleration #a=(v_1-v_2)/t# #a=(sqrt(180)-sqrt40)/8# velocity at t=5 #v=a*t# #a=5*(sqrt(180)-sqrt40)/
8# impulse on the object #m*Deltav# #3*(5*(sqrt180-sqrt40)/8-sqrt40)#
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Answer 2
Impulse is defined as the force exerted over a time interval. Asking what the impulse is at a specific instant of time makes no sense. Impulse is equivalent to the total change in momentum. It is
often useful when we know an initial and final velocity, but we don't know if the change was made because of a small force acting over a long time, or a large force acting very quickly.
A question concerning the total impulse exerted over the first five seconds could be asked, as well as questions concerning the force, acceleration, and velocity at a specific moment in time. If you
were attempting to answer one of those questions, please submit a different one.
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Answer 3
To find the impulse at 5 seconds, we first need to determine the initial and final velocities of the object at that time. We can use the formula for kinetic energy to find the initial and final
Initial kinetic energy = 60 J Final kinetic energy = 270 J Mass = 3 kg
Using the formula for kinetic energy:
(KE = \frac{1}{2}mv^2)
(60 = \frac{1}{2}(3)v^2) (v_{initial} = \sqrt{\frac{60 \times 2}{3}})
(v_{initial} ≈ 7.75 m/s)
(270 = \frac{1}{2}(3)v^2) (v_{final} = \sqrt{\frac{270 \times 2}{3}})
(v_{final} ≈ 17.32 m/s)
Now, we can use the definition of impulse:
(Impulse = m \cdot \Delta v)
(Impulse = 3 \cdot (17.32 - 7.75))
(Impulse ≈ 28.17 , N \cdot s)
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Online Kepler's 3rd Law Calculator | How to Calculate Planet Period? - physicsCalculatorPro.com
Kepler's 3rd Law Calculator
Kepler's 3rd Law Calculator shows how to easily calculate the basic parameters of a planet's motion around the Sun, such as the semi-major axis and planet period. To find the unknown parameters, it
uses Kepler's third law formula. To check the orbital period, simply enter the star's mass and semi-major axis and press the submit button.
What does Kepler's 3rd Law State?
According to Kepler's Third Law, the ratio of the squares of two planets' orbital periods is equal to the ratio of the cubes of their mean orbit radius. As per Kepler's Third Law, Every planet's
orbit is an ellipse, with the Sun at one of the two foci. The cube of the semi-major axis of a planet's orbit is directly proportional to the square of its orbital period. The symbol 'T' stands for
the Satellite Orbit Period.
Kepler's third law is used to calculate the velocity of a circular Earth orbit at any other distance r. The square of the orbital period is precisely proportional to the cube of the orbit's
semi-major axis. The powerful online Kepler's Third Law Calculator is used to compute and find the planetary velocity when the Satellite Mean Orbital Radius(r) and Planet Mass(M) are known.
Kepler's Third Law Equation
The cube of a planet's radius is directly related to the square of its orbital period, according to Kepler's third law. T² ∝ a³
It depicts the relationship between a planet's orbital period and its distance from the sun in the system. The constant is the only variable in Kepler's third law.
a³/T² = 4 * π²/[G * (M + m)] = constant
• Where, a = semi-major axis
• T = planet period
• G = gravitational constant and it is 6.67408 x 10⁻¹¹ m³/(kgs)
• M = mass of the central star
• m = mass of the planet
How to Calculate Planet Period?
Use Kepler's 3rd law formula to compute the planet period in simple stages. They are explained as such
• Step 1: Find out about the star's mass and semi-major axis.
• Step 2: Calculate the radius's cube.
• Step 3: Multiply the mass of the star and the mass of the planet by the gravitational constant.
• Step 4: Multiply the result of the previous two stages.
• Step 5: Divide it by the 4π².
• Step 6: The planet period is the square root of the result.
Kepler's Third Law calculator
Kepler's third law calculator is simple to use and may be used in a variety of ways. Simply fill in two distinct fields, and we'll find out the third one then.
If you want to use a more precise version of Kepler's third rule of planetary motion, select advanced mode and provide the planet's mass, m. You may need to adjust the units to a smaller measure
because the difference is too little to notice (e.g., seconds, kilograms, or feet).
Kepler's 3rd Law Examples
Question 1: Phobos orbits Mars at a distance of approximately 8200 kilometres from the planet's centre, with a rotational period of around 7 hours. Calculate the size of Mars.
semi-major axis a = 8200 km = 8.2 x 10^6 m
Planet periord T = 7 hrs = 25200 sec
Kepler's equation is; a³/T² = 4 * π²/[G * (M + m)]
(8.2 x 10^6)³/(25200)² = 4 * π²/[6.67408 x 10⁻¹¹ * (M + m)]
8.68 x 10^11 = 39.43/[6.67408 x 10⁻¹¹ * (M + m)]
M + m = 39.43/57.93
M + m = 0.68
The mass of mars is 0.68 km.
For more concepts check out physicscalculatorpro.com to get quick answers by using this free tool.
FAQs on Kepler's Third Law Calculator
1. How do you calculate Kepler's Third Law?
When the orbit's size (a) is given in astronomical units (1 AU represents the average distance between the Earth and the Sun) and the period (P) is stated in years, Kepler's Third Law states that P2
= a3. where P is in Earth years, and a is astronomical units, and M is the mass of the centre object in Sun mass units.
2. What is the Third Law of Kepler?
According to Kepler's Third Law, the cubes of the semi-major axes of the planets' orbits are directly proportional to the squares of their orbital periods.
3. What is Kepler's K?
The square of the planet's orbital period divided by the cube of its semi-major axis is Kepler's constant. Note: When calculating the constant, Kepler assumed the orbit was circular and the radius
was the orbit's average radius.
4. How do you calculate Kepler's Constant?
For that one object being orbited, the square of the period of orbit divided by the cube of the radius of the orbit equals a constant (Kepler's Constant). | {"url":"https://physicscalculatorpro.com/keplers-third-law-calculator/","timestamp":"2024-11-05T02:39:20Z","content_type":"text/html","content_length":"32759","record_id":"<urn:uuid:aaa7164c-1635-4b2b-83be-8b3f452d637e>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00635.warc.gz"} |
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Working Draft 15-May-97
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Frontiers | Design and Analysis of Field-of-View Independent k-Space Trajectories for Magnetic Resonance Imaging
• Department of Internal Medicine II, University Hospital of Ulm, Ulm, Germany
This manuscript describes a method of three-dimensional k-space sampling that is based on a generalised form of the previously introduced “Seiffert Spirals,” which exploits the equality between
undersampling and the reconstruction of a field of view which is larger than what is represented by the primary sampling density, leading to an imaging approach that does not require any prior
commitment to an imaging field of view. The concept of reconstructing arbitrary FOVs based on a low-coherently sampled k-space is demonstrated analytically by simulations of the corresponding point
spread functions and by an analysis of the noise power spectrum of undersampled datasets. In-vivo images highlight the feasibility of the presented approach by reconstructing white noise governed
images from an undersampled datasets with a smaller encoded FOV. Beneficial properties are especially given by an artefact behaviour, which is widely comparable to the introduction of white noise in
the image domain. Furthermore, these aliasing properties provide a promising precondition for the combination with non-linear reconstruction techniques such as Compressed Sensing. All presented
results show dominant low-coherent aliasing properties, leading to a noise-like aliasing behaviour, which enables parameterization of the imaging sequence according to a given resolution and
scan-time, without the need for FOV considerations.
Undersampling in the spatial frequency domain is a common method to shorten acquisition times in magnetic resonance imaging (MRI). Thereby, the violation of Nyquist’s theorem leads to the emergence
of aliasing artefacts, which are usually addressed with parallel or auto calibration methods such as GRAPPA [1], SENSE [2] or Compressed Sensing (CS) [3, 4]. Various two-dimensional sampling schemes
(trajectories) with a distinct focus on low-coherent k-space sampling based on Poisson-disc sampling [5, 6], quasi-random point sequences [7], previously acquired training data [8] or e.g. on
heuristic sampling strategies [9] have been introduced. Despite their interesting aliasing properties and arising acceleration possibilities, such sequences are hardly extendable to three-dimensional
frequency encoding imaging. Approaches such as [10] or [11] provide extensions to three-dimensional imaging by making use of two phase-encoding directions with frequency encoding along the third
dimension with the obvious disadvantage of unfavourable aliasing properties along the latter direction. Conclusively, advanced 3D frequency encoding approaches such as FLORET [12], 3D SPARKLING [13],
3D Cones [14] or hybrid radial Cones [15] and variants thereof [16] have been proposed. While the 3D SPARKLING approach has already shown its feasibility for the imaging of tissue with short $T2⋆$
relaxation times, it appears limited by the remaining radial sampling character and is still missing a thorough discussion of the arising aliasing behaviour. A detailed discussion of the FLORET
approach, which is mainly based on the 3D Cones sampling scheme [14], is given in [17] with a comparison to the here further developed sampling approach. The Yarnball concept as finally presented in
[18] results in gradient waveforms that are vastly similar to the ones presented in [17], with a parameterization that does not show the flexibility of Jacobi elliptic functions.
The common goal of all approaches is to realise a sampling point spread function (PSF[S]) with a low-coherent energy distribution in the PSF[S] for nearly arbitrary undersampling, thus realising
aliasing artefacts with a (noise-like) power spectrum.
The aim of this publications is to report on the aliasing properties of a generalised form of the previously introduced “Seiffert Spirals” (24). The use of Jacobi theta functions yields a variety of
highly adaptable k-space interleaves, while maintaining low-coherent sampling properties. It is explicitly shown, that low-coherent aliasing properties offer the possibility of reconstructing
arbitrary FOVs by only introducing random noise-like aliasing artefacts. This leads to a situation in which an MRI trajectory can be constructed without considering the desired imaging FOV. The
trajectory is solely constructed to meet a given resolution and sampling duration for a defined number of executed interleaves.
Generalised FOV
In the case of Cartesian k-space sampling, Nyquist’s theorem states that the distance Δk[i] between adjacent sampling points has to fulfil the condition Δk[i] ≤ 1/FOV[i], where FOV is the field of
view and i = x, y, z reflects the standard three-dimensional Euclidean basis.
For a variety of non-Cartesian k-space trajectories it appears more convenient to evaluate an upper limit by using Pythagoras’ theorem in 2D or 3D k-space, i.e.
$Δkmax=max{d∈K}Δkx,d2+Δky,d2+Δkz,d2$, where d is the set of all distances in k-space K between points that are nearest neighbours and do not belong to the same k-space interleave or read-out (in case
of multi-shot acquisitions). Since Δk[max] ≥Δk[i,d] ∀d ∈ K and i = x, y, z, Nyquist’s theorem is indeed fulfilled if min(1/FOV[i]) ≥Δk[max]. While e.g. in the case of radial sampling, this evaluation
can be restricted to the sampling point of each read-out that is farthest away from the centre of k-space, it appears insufficient for the case of any quasi-random sampling point distribution.
In the scope of this publication it is useful to extend the estimation of a Δk[max] even further, since the later introduced distribution of sampling points will not follow any regular or symmetric
pattern, as shown for a single sample point C and its six surrounding nearest neighbours P[i] with i = 1, …, 6 in Figure 1C for simplification in two dimensions.
FIGURE 1
FIGURE 1. Two interleaves, each based on a different Jacobi theta function θ[1] and θ[4]. Both interleaves are shown in three-dimensional k-space (A) and as a plane projection in (B). (C) Schematic
representation of seven Voronoi cells in a two-dimensional k-space. Six nearest neighbours of the point of interest C are used for the definition of the generalised FOV. (D) Ten interleaves of the
presented 3D ζ-based Spiral trajectory. The increased sampling density around the centre of k-space can be clearly appreciated.
In order to calculate a local estimate of the encoded FOV at C, the direct distances $dC,Pi$ between C and each P[i] are calculated and the mean value of all six distances defines a radius Δr around
C. This radius can then define a generalised FOV for C with FOV[C] = 1/Δr[C] but in the same manner also for every other point in k-space. In the three dimensional case, Δr defines the radius of a
sphere, from which an equal FOV along each direction is derived. For this calculation, a sufficient number of nearest neighbours is required, to ensure that Δr is derived from the expression $Δr=
Δxī2+Δyī2+Δzī2$ with Δx[i] ≠ 0, Δy[i] ≠ 0 and Δz[i] ≠ 0, i.e. the radius can define a spherical (3D) volume. The given definition also allows the assignment of an isotropic FOV for various specific
regions in k-space as well as for the entire trajectory by averaging Δr for all considered k-space points.
Interleaves in k-Space Based on Theta Functions
The previous publication [17] has used Jacobi elliptic functions for the construction of a single k-space interleave as the solution to an optimisation problem given by P. Erdös in 2000 [19]. Here,
we have generalised the approach by making use of Jacobi theta functions [20], giving rise to the possibility of constructing a multitude of inherently different k-space waveforms. For example,
Figure 1A shows two different k-space waveforms of arbitrary length, for which the k[x] − and k[y] − components are once based on θ[1] and once based on θ[4] (k[z] = θ[2]). A projection of both
waveforms into the k[x] − k[y] − plane is shown in Figure 1B. All corresponding gradient waveforms are shown in Figure 2.
FIGURE 2
FIGURE 2. All gradient channels of one interleave of the three trajectories shown in Figure 1. Due to different interleave lengths, the first 600 points of each gradient waveform are shown. The θ[2]
based gradient waveform refers to 1d).
The waveform presented in the following is meant to prove the imaging concept for a lower limit of read-out durations and is generally based on the concepts and parameters introduced in [17]. The
general waveform is generated on the surface of a unit sphere according to the definition: $ζ:R0+→R3$, s↦ζ(s) with η ∈ (0, 1) and
While η is a parameter to adapt the waveform to hardware limitations such as maximum gradient amplitudes and available slew-rates. The combination with sine and cosine terms in the first two
components allows for a modifiable change in direction per unit length while the symmetry along the z-direction remains unchanged (Figures 1A,B). The length of the waveform is determined by s and
therefore the restriction s ∈ [0, s[max]] should be applied, with s[max] being sufficiently large according to the desired resolution (extension of k-space).
For this publication, one k-space interleave was constructed with η = 0.5 and by setting the target resolution to 0.85 mm (isotropic). For an extended oversampling of k-space centre α = 1.3 was
chosen to facilitate a subsequent CS reconstruction. Each presented trajectory was generated with a maximum gradient strength of 21 mT/m and a slew-rate of 120 T/m/s. Each trajectory was furthermore
explicitly optimised by minimising its discrepancy [17], by iterative optimisation of s, η and the angle of rotation around the symmetry axis according to [17] and using an Euclidian arc-length
parameterization [21]. An adequate choice of the centre-oversampling parameter α requires information about the energy distribution of the underlying k-space and can therefore not be generally
included to the optimisation problem. Nevertheless, defining an interval by experience e.g. α ∈ [1.2, 1.7] allows for another degree of freedom during the optimisation process and is therefore
The resulting read-out duration for each interleave was 3.52 ms, in order to reach the boundary of the k-space sphere for the defined maximum gradient amplitude and slew-rate limits. Ten interleaves
of a final trajectory with 20,000 interleaves are depicted in Figure 1D for illustrative purposes. For this trajectory, we define an undersampling factor of R: = 1 due to an equal scan time as the
corresponding Cartesian acquisition at the same resolution and FOV.
Sampling Point Spread Function
Based on the previously mentioned parameters, four sampling point spread functions were calculated according to undersampling factors of R = 1, 8, 12, 16 with respect to the trajectory with the
largest number of interleaves, i.e. 20,000. Each trajectory for each undersampling factor was independently generated and optimised with respect to low-coherent sampling properties. All PSF[S]s were
obtained by a separate calculation of a Voronoi tessellation [7] in Euclidean space ($R3$, Euclidean distance) for every generated trajectory. Each PSF[S] was obtained by gridding unit k-space data
onto a Cartesian grid in combination with a 3D Voronoi tessellation [17, 22] to estimate weights for the necessary density compensation of non-uniformly acquired k-space data. Based on all normalised
PSF[S]s, the centre-peak FWHM was determined in order to evaluate relative image sharpness with increasing undersampling factors. Furthermore, the peak/side-lobe ratio
Of each PSF[S] is an appropriate measure for emerging coherences [3].
In-Vivo Imaging and Reconstruction
In order to evaluate the aliasing behaviour, as well as imaging performance, in-vivo head images were acquired using a 3.0 T wholebody MRI system (Achieva 3.0T, Philips, Best, Netherlands) with an
8-element SENSE Neuro coil (Philips, Best, Netherlands).
Image reconstruction for all 3D ζ-based Spirals was performed as follows: after data acquisition, raw data were exported and processed with MATLAB (MathWorks, Natick, MA, United States). Images were
obtained using gridding [23] with an oversampling factor of 1.25, a Kaisser Bessel kernel for interpolation and again in combination with a 3D Voronoi tessellation. Gradient system delays were
estimated [24] and used to correct the trajectory before gridding. Further eddy current effects were compensated using a mono-exponential model [25] with a time constant of τ = 39 μs. No
post-processing was applied to any of the presented image. Compressed Sensing reconstructions were accomplished using an in-house written CS/SENSE reconstruction based on nonlinear conjugate gradient
methods as suggested in [3, 26]. Undersampling was again realized by calculating separate and optimised trajectories with 12,000, 8,000, 4,000 and 1,250 interleaves, leading to undersampling factors
R = 1.66, 2.5, 5 and 16 with respect to the initial trajectory with 20,000 interleaves. These undersampling factors were chosen to correspond to scan durations of 90, 60, 30 and 10s. All relevant
scan parameters are listed in Table 1.
TABLE 1
Undersampling Behaviour and Noise Analysis
To provide an experimental assessment of the aliasing properties, we evaluate the noise characteristics of the acquired in-vivo datasets for the 3D ζ-based Spiral, acquired with 12,000, 8,000, 4,000
and 1,250 interleaves each, according to the previous section. A reference dataset with the vendor’s 3D radial Kooshball trajectory was acquired, employing the same spatial resolution and choosing a
FOV encompassing the entire head. Image reconstruction for the Kooshball trajectory followed the description given in the previous section for the 3D ζ-based Spiral trajectory, except for the
weighting calculation. Kooshball weights were calculated analytically, based on the symmetry of the sampling scheme (spherical shells). The radial dataset was retrospectively undersampled by a random
selection of 1/R spokes.
Three regions of interest (ROI) as highlighted in Figure 3D were selected in the background of the reconstructed images, ideally containing no source of any MR signal. Accordingly, the pixel
intensities are exclusively governed by artefacts and noise whose characteristics can be analysed by consideration of the associated power spectrum. The calculation of the power spectrum followed [27
], generalised to the 3D case. To facilitate the analysis of their characteristics (line shapes), all power spectra were normalised and are presented in arbitrary units. For each acquisition, the
power spectra were averaged over all coil elements and evaluated with respect to the three geometrical axis to capture possible similarities (coherences).
FIGURE 3
FIGURE 3. Spectral noise analysis for three different ROIs as indicated in d) for four different undersampling factors (ROI 1: green, ROI 2: red, ROI 3: yellow).
Trajectory Properties
According to the definition of the generalised FOV, the presented trajectory with 20,000 interleaves has the following properties: The Nyquist condition Δr[C] ≤ 1/FOV = (1/220)mm is fulfilled within
a sphere of radius r[N] = 0.18 ⋅ k[max]. The largest FOV that is stored within a sphere of radius r = 0.01 ⋅ k[max] corresponds to 38-times the FOV dimension (220 mm). According to the definition,
the Nyquist condition is not fulfilled for points outside a sphere of radius r[N].
Since the number of interleaves for the trajectory with 20,000 interleaves was selected such that the resulting scan duration is approximately equal to a standard Cartesian acquisition (174 s),
without considering the actually encoded FOV, it is reasonable to calculate the introduced generalised FOV for this and for all other trajecories, to achieve undersampling factors that truly rely on
Nyquist’s condition and not on a relative number of interleaves. Therefore, the trajectory with 20,000 interleaves corresponds to an undersampling factor of $R1′≈3.29$. This value was determined by
calculating the mean generalised FOV of all sampling points of each trajectory, stating that Nyquist’s theorem is violated 3.29-times according to the definitions. Accordingly, the equally generated
trajectory with 2,500 interleaves led to $R2′≈6.43$, to $R3′≈9.33$ for 1,665 interleaves and to $R4′≈12.98$ for 1,250 interleaves. The intention of these numbers is merely to classify the presented
trajectories than to enforce comparisons to other sampling schemes due to drastic differences in the distribution of points in k-space. For simplification, all images and results that correspond to
undersampling factors $R1′,…,R4′$ of the 3D ζ-based Spirals are denoted by M[3], M[6], M[9], M[13] according to the mean undersampling factors.
Figure 4 shows all associated sampling point spread functions for these four cases of undersampling in the xy-plane with z = 0.
FIGURE 4
FIGURE 4. Simulated PSF[S]s in the xy-plane with z = 0 of the presented 3D ζ-based Spiral trajectory. The PSF[S] in (A) corresponds to 3.29-fold undersampling, the PSF[S] in (B) to 6.43-fold
undersampling, (C,D) to 9.33-fold and 12.98-fold undersampling respectively. Logarithmic plot of the centre region of all four sampling point spread functions shown in (E) and the same for the entire
cross section in (F).
With increasing undersampling, energies in the PSF[S]s emerge that do not seem to follow any ordered or symmetric pattern. Consequently, all PSF[S]s appear to be governed by a low-coherent
distribution of energies with an expected aliasing behaviour that is (in its appearance) vastly similar to an introduction of white noise.
Figures 4E,F shows a logarithmic plot of the centre region of the four PSF[S]s, shown in Figures 4A–D and a cross section of the entire PSF[S] in 3b). As the undersampling factor increases, an
overall increase in energy can be appreciated in which the side-lobe behaviour shows rarely signs of emerging coherences. Furthermore, image sharpness is preserved for all undersampling factors. The
FWHM of the PSF[S] centre-peak results to $≈2.544$ pixel in width (mean) with a maximum deviation of 0.59% between the broadest peak (M[13]: 2.549 px) and the narrowest peak (M[9]: 2.534 px) of all
undersampling PSF[S]s. All values were obtained in non-logarithmic representation. This finding of retained sharpness is furthermore supported by the in-vivo images, presented in the following
Aliasing Behaviour and Noise Analysis
Figure 5 shows axial and sagittal slices, acquired with the presented 3D ζ-based Spiral trajectories for different numbers of interleaves (20,000, 8,000 and 1,250). All data was directly gridded and
no additional image or data processing was applied before and after gridding. Furthermore, no sensitivity maps were used, in order not to alter the emerging imaging artefacts. Figure 5 also shows a
Compressed Sensing reconstruction based on the dataset with 1,250 interleaves using a total variation regularisation [7].
FIGURE 5
FIGURE 5. Reconstructed axial and sagittal slices from an in-vivo 3D ζ-based Spiral acquisition. The undersampling factor increases from top to bottom. The calculated generalized FOV of each
trajectory is indicated as red squares on the sagittal slices. The bottom line shows a Compressed Sensing reconstruction based on the acquisition with 1,250 interleaves.
The generalised FOV for 20,000 interleaves is $≈67$mm (M[3], isotropic), $≈41$mm for 8,000 interleaves and $≈17$ mm for 1,250 interleaves (M[13]). The reconstruction of a larger FOV, in this case
of 220 mm (isotropic) for all datasets results in additional low-coherent aliasing artefacts, which can clearly be appreciated. Despite uncorrected coil sensitivity profiles, all images appear
non-degraded by coherent aliasing artefacts, especially visible in the background of all images. As expected from the PSF[S] analysis, image sharpness is preserved, and based on the optical
impression, equal for all investigated undersampling factors.
The noise analysis for the three cubic regions (ROI) is shown in Figure 3. The normalised power spectra indicate slightly overpronounced DC components for all acquisitions. For the 3D ζ-based Spiral
trajectories (20,000, 8,000, 4,000 and 1,250 interleaves), all further spatial frequency components are about equally represented, leading to a widely flat power spectrum in accordance to the
behaviour of (bandwidth limited) white noise. As expected, an increasing undersampling factor increases the overall power, but the characteristics (line-shapes) remain widely unchanged. Concerning
the Kooshball trajectory, with the analysis being shown in the Supplemental Data (Supplementary Figure S1), all directions show a similar behaviour in terms of overpronounced DC components and a
following decline of the power spectrum. But, all directions in each evaluated ROI show clear modulations (wave pattern) in the frequency analysis with again increasing powers as the undersampling
factor increases. The latter also introduces obvious changes to the frequency modulations, indicating varying (coherent) aliasing artefacts. While the noise-like characteristics of aliasing artefacts
remain unchanged for varying undersampling factors in the case of 3D ζ-based Spiral trajectories, undersampling of the radial Kooshball trajectory introduces streak artefacts.
Calculating the peak/side-lobe ratios for the four PSF[s]s shown in Figure 4 yields ratios in non-logarithmic representation of 4.7 ⋅ 10^–3 (M[3]), 5.1 ⋅ 10^–3 (M[6]), 5.2 ⋅ 10^–3 (M[9]), 5.8 ⋅ 10^–3
(M[13]), indicating that emerging coherences remain at the same level with increasing power densities in the PSF[s]s towards the outer regions as R increases. Compared to Kooshball sampling with the
PSF[s]s shown in the Supplemental Data (Supplementary Figure S2), the peak/side-lobe ratio for R = 3 is 5.3 ⋅ 10^–3 while it increases to 11.2 ⋅ 10^–3 for R = 6, to 13.8 ⋅ 10^–3 for R = 9 and to 20.6
⋅ 10^–3 for R = 13, which is in accordance to the visually enhanced coherences in the emerging aliasing artefacts for increasing undersampling factors.
Discussion and Conclusion
In summary, all presented results show dominant low-coherent aliasing properties, leading to a noise-like aliasing behaviour, which facilitates new imaging strategies or ways in which available scan
times can be exploited.
Besides obvious advantages in scan time reduction, by a combination of undersampling with a Compressed Sensing reconstruction, the trajectory is created independent of an application dependent FOV
which does therefore not influence the total imaging duration. Using 3D ζ-based Spirals, a trajectory might be constructed just by following given time restrictions and imaging constraints, e.g. such
• Size of k-space sphere as defined by the desired image resolution.
• Maximum read-out duration (spiral length) as defined (limited) by off-resonance behaviour and relaxation effects.
• Total acceptable scan duration defines the number of possible interleaves.
Based on the measured dataset, any feasible FOV can then be reconstructed under emerging low-coherent aliasing artefacts if the condition Δr[C] ≤ 1/FOV[p] is not fulfilled for every point in k-space.
Since the reconstructed FOV is defined by the underlying Cartesian grid (gridding/interpolation) and not by the trajectory itself, the same Voronoi density compensation can be used for any
reconstructed FOV.
The overall image sharpness (PSF[S] FWHM) is highly influenced by the quality of the Voronoi tessellation. It is therefore explicitly important to consider the trade-off between numerical runtime of
the volume calculation and the resulting image qualities. All presented images were reconstructed using a not approximated tessellation, based on a thorough Dirichlet tessellation, at the cost of
long reconstruction times. In order to achieve a fast optimisation of the trajectory to specific applications, more efficient implementations are desirable. Nevertheless, only one k-space trajectory
is required for each specific resolution, since the desired imaging FOV does neither influence the generation of the gradient waveforms nor the entire k-space trajectory itself.
The spectral analysis, as shown in Figure 5, exhibits an overestimation of the DC component for all analysed acquisitions. As the DC component in the power spectrum corresponds to a constant pixel
intensity offset, it does not influence the characteristics of the artefacts and might easily be corrected.
The presented approach based on Jacobi theta functions represents a generalised approach to the previously published trajectory design. While it can embody all properties of the Seiffert Spirals, it
also derives the possibility of constructing a multitude of new k-space sampling schemes, which are to be discussed in detail in a future publication. All shown results indicate a clear low-coherent
aliasing behaviour, based on the underlying generation algorithm.
Data Availability Statement
The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.
Author Contributions
TS and PM performed the measurements, designed the concept and wrote the first draft of the manuscript. TH and KS contributed to the acuqisition and reconstruction of the shown data. All authors
contributed to manuscript revision, read, and approved the submitted version.
This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 858149. The presented research was also partially funded by Philips
Healthcare. The authors thank the Ulm University Centre for Translational Imaging MoMAN for its support.
Conflict of Interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s Note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the
reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
Supplementary Material
The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fphy.2022.867676/full#supplementary-material
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Keywords: 3D, compressed sensing, efficiency, low-discrepancy, spiral, trajectory
Citation: Speidel T, Metze P, Stumpf K, Hüfken T and Rasche V (2022) Design and Analysis of Field-of-View Independent k-Space Trajectories for Magnetic Resonance Imaging. Front. Phys. 10:867676. doi:
Received: 01 February 2022; Accepted: 12 May 2022;
Published: 09 June 2022.
Edited by:
Federico Giove
, Centro Fermi - Museo storico della fisica e Centro studi e ricerche Enrico Fermi, Italy
Copyright © 2022 Speidel, Metze, Stumpf, Hüfken and Rasche. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or
reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with
accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Tobias Speidel, tobias.speidel@uni-ulm.de
^†These authors have contributed equally to this work | {"url":"https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2022.867676/full","timestamp":"2024-11-13T03:11:55Z","content_type":"text/html","content_length":"439592","record_id":"<urn:uuid:4a991138-4703-4de3-870e-27270caa4304>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00558.warc.gz"} |
[Solved] Consider the following data for concrete with mild exposure
Consider the following data for concrete with mild exposure
Water-cement ratio = 0.50; Water = 191.6 litre. The required cement content will be
This question was previously asked in
OSSC JE Civil Mains (Re-Exam) Official Paper: (Held On: 3rd Sept 2023)
View all OSSC JE Papers >
Answer (Detailed Solution Below)
Option 3 : 383 kg/m3
Water Cement ratio (W/C) is the ratio of weight of water to weight of cement in a concrete mix. This ratio decides the strength and workability of concrete.
According to Abram’s law, the strength of a concrete mix is inversely related to the weight ratio of water to cement. Higher W/C ratio, lower the strength but higher is the workability.
\(\frac{W}{C} = 0.5 \Rightarrow C = \frac{W}{{0.5}}\)
Volume of water = 191.6 litre = 0.1916 m3
Mass of water = 0.1916 × 103 = 191.6 kg
For 1 m3 cement content required is,
∴ \(C = \frac{{191.6}}{{0.5}} = 383.2 \;kg\)
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Geometry of Numbers
• 1st Edition - May 12, 2014
• Author: C. G. Lekkerkerker
• Editors: N. G. De Bruijn, J. De Groot, A. C. Zaanen
• Paperback ISBN:
9 7 8 - 1 - 4 8 3 2 - 4 9 9 8 - 8
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Bibliotheca Mathematica: A Series of Monographs on Pure and Applied Mathematics, Volume VIII: Geometry of Numbers focuses on bodies and lattices in the n-dimensional euclidean… Read more
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Bibliotheca Mathematica: A Series of Monographs on Pure and Applied Mathematics, Volume VIII: Geometry of Numbers focuses on bodies and lattices in the n-dimensional euclidean space. The text first
discusses convex bodies and lattice points and the covering constant and inhomogeneous determinant of a set. Topics include the inhomogeneous determinant of a set, covering constant of a set, theorem
of Minkowski-Hlawka, packing of convex bodies, successive minima and determinant of a set, successive minima of a convex body, extremal bodies, and polar reciprocal convex bodies. The publication
ponders on star bodies, as well as points of critical lattices on the boundary, reducible, and irreducible star bodies and reduction of automorphic star bodies. The manuscript reviews homogeneous and
inhomogeneous s forms and some methods. Discussions focus on asymmetric inequalities, inhomogeneous forms in more variables, indefinite binary quadratic forms, diophantine approximation, sums of
powers of linear forms, spheres and quadratic forms, and a method of Blichfeldt and Mordell. The text is a dependable reference for researchers and mathematicians interested in bodies and lattices in
the n-dimensional euclidean space.
Chapter 1. Preliminaries 1. Notations. Convex Bodies 2. Ray Sets and Star Bodies 3. Lattices 4. Algebraic Number FieldsChapter 2. Convex Bodies and Lattice Points 5. The Fundamental Theorem of
Minkowski 6. Generalizations of the Theorem of Blichfeldt 7. Generalizations of the Theorem of Minkowski 8. A theorem of Rédei and Hlawka 9. Successive Minima of a Convex Body 10. Reduction Theory
11. Successive Minima of Non-Convex Sets 12. Extremal Bodies 13. The Inhomogeneous Minimum 14. Polar Reciprocal Convex Bodies 15. Compound Convex Bodies 16. Convex Bodies and Arbitrary
LatticesChapter 3. The Critical Determinant, The Covering Constant and the Inhomogeneous Determinant of a Set 17. Mahler's Selection Theorem. Critical Determinant and Absolute Homogeneous Minimum.
Critical Lattices 18. The Successive Minima and the Determinant of a Set 19. The Theorem of Minkowski-Hlawka 20. Packing of Convex Bodies 21. Covering Constant of a Set. Covering by Sets 22. Packings
and Coverings in the Plane 23. Inhomogeneous Determinant of a Set 24. A Theorem of Mordell-Siegel-Hlawka-RogersChapter 4. Star Bodies 25. The Functional Δ(S), Γ(S),f(Λ), g(Λ) 26. Points of Critical
Lattices on the Boundary. Automorphic Star Bodies 27. Reducible and Irreducible Star Bodies 28. Reduction of Automorphic Star BodiesChapter 5. Some Methods 29. The Critical Determinant of a
Two-Dimensional Star Body. Methods of Mahler and Mordell 30. Some Special Two-Dimensional Domains 31. The Critical Determinant of an n-Dimensional Domain 32. Some Special Domains 33. A Method of
Blichfeldt. Density Functions 34. A Method of Blichfeldt and Mordell 35. A Theorem of Macbeath 36. Comparison of Star Bodies in Spaces of Unequal DimensionsChapter 6. Homogeneous Forms 37.
Homogeneous Forms, Absolute Minima, Extreme Forms 38. Spheres and Quadratic Forms 39. Extreme Positive Definite Quadratic Forms 40. Sums of Powers of Linear Forms 41. Products of Linear Forms 42.
Other Homogeneous Forms 43. Extreme Forms. Isolated Minima 44. Asymmetric and One-Sided Inequalities 45. Diophantine ApproximationChapter 7. Inhomogeneous Forms 46. Inhomogeneous Minima of Forms 47.
Indefinite Binary Quadratic Forms 48. Delone's Algorithm. Lower Bounds for μ (Q, Y) 49. Inhomogeneous Forms in More Variables 50. Asymmetric Inequalities 51. Inequalities with Infinitely Many
• Paperback ISBN: 9781483249988
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PPT - Gases PowerPoint Presentation, free download - ID:5761171
4. II. Gas Pressure A. Pressure is force per unit area (f/a) 1. result of particle collisions 2. measured by a barometer 3. influenced by temperature, gas volume, and the number of gas particles a.
as the number of particle collisions increases the pressure increases
5. Pressure at Sea Level 14.7 psi = 1.0 atm = 760 mm of Hg = 750 Torr = 101.3 kPa = 1,013 mbars
6. Kinetic Theory A. Assumptions 1. gas particles do not attract each other 2. gas particles are very small 3. particles are very far apart 4. constant, random motion 5. elastic collisions 6.
kinetic energy varies with temperature
7. B. Properties of Gases 1. low density (grams/liter) 2. can expand and can be compressed 3. can diffuse and effuse a. rate related to molar mass b. diffusion is the movement of particles from an
area of greater concentration to an area of lesser concentration c. effusion is the movement of gas particles through a small opening
10. II. The Gas Laws A.Boyle’s Law (P1V1 = P2V2 )inverse relationship 1.As the volume of a gas increases the pressure decreases (temperature remains constant) 2.Example A sample of gas in a balloon
is compressed from 7.00 L to 3.50 L. The pressure at 7.00L is 125 KPa. What will the pressure be at 2.50L if the temperature remains constant? P1 = 125 KPa P2 = X V1 = 7.00L V2 = 3.50L (125)
(7.00) = (X) (3.50) X = 250.KPa
12. As volume increases the pressure decreases when temperature remains constant
14. Boyle’s Law • Pressure is related to 1/Volume Slope (k) = relationship between P and 1/V P = k(1/V)
15. B. Charles’ LawV1 = V2must use kelvin T1 T2 temperature • As the temperature of a gas increases the volume increases (direct relationship) 2. ExampleA gas sample at 20.0 C occupies a volume of
3.00 L. If the temperature is raised to 50.0 C, what will the volume be if the pressure remains constant? V1 = 3.00L V2 = X T1 = 293K T2 = 323K 3.00 = X 293X = (3)(323) X = (3)(323) 293 323 293
X = 3.31 L
18. C. Gay Lussac’s LawP1 = P2 T1 T2 1. as the temperature increases the pressure increases when the volume remains constant 2. Example The pressure of a gas in a tank is 4.00 atm at 200.0C. If the
temperature rises rises to 800.0C, what will be the pressure of the gas in the tank? P1 = 4.00 atm P2 = X T1 = 473K T2 = 553K 4.00 = X 473X = (4)(553) X = (4)(553) • 553 473 X = 4.68 atm
19. D. Combined Gas LawP1 V1 = P2 V2 T1 T2 1. Combines Boyle’s, Charle’s and Gay Lussac’s 2. Example A gas at 70.0KPa and 10.0C fills a flexible container with an initial volume of 4.00L If the
temperature is raised to 60C and the pressure is raised to 80.0 KPa, what is the new volume? P1 = 70.0 KPa P2 = 80.0 KPa V1 = 4.00 V2 = X T1 = 283K T2 = 333K (70.0)(4.00) = (80.0)(X) 283 333 X =
(33.3)(70.0)(4.00) (2.83)(80.0) X = 41.2L
20. E. Dalton’s Law of Partial Pressures Ptotal = P1 + P2 + P3 + .....Pn The total pressure of a mixture of gases is equal to the sum of the pressures of all the gases in the mixture 1. Example Find
the total pressure for a mixture that contains four gases with partial pressures of 5.00 kPa, 4.56 kPa, 3.02 kPa and 1.20kPa.
23. 2. Suppose two gases in a container have a total pressure of 1.20 atm. What is the pressure of gas B if the partial pressure of gas A is 0.75 atm? 3. What is the partial pressure of hydrogen gas
in a mixture of hydrogen and helium if the total pressure is 600.0mmHg and the partial pressure of helium is 439 mmHg?
24. III. Avogadro’s Principle A. Equal volumes of gases at the same temperature and pressure have the same number of particles B. Molar Volume (22.4 L at STP) 1. volume of one mole of gas particles
at STP(standard temperature and pressure) 0C and 1.00 atm (760mm Hg) * 1 mole of any gas at STP = 22.4 L 2. conversion factors 1 mol22.4 L 22.4 L 1 mol
26. Equal volumes of gases at the same temperature and pressure contain the same number of particles
27. C. Sample Problems 1. Calculate the volume occupied by .250 mol of oxygen gas at STP. 2. Calculate the number of moles of methane gas in a 11.2 L flask at STP.
28. 3. Calculate the volume of 88.0 g of CO2 at STP. 4. How many grams of He are found in a 5.60L balloon at STP?
29. 5. Calculate the density of H2 at STP. D = molar mass molar volume 6. Calculate the molar mass of a gas that has a density of 3.2 g/L.
30. IV. Ideal vs Real Gases A. Ideal compared to Real Gases 1. ideal gas a) particles do not have volume b) there are no intermolecular attractions c) all particle collisions are elastic d) obey all
kinetic theory assumptions
31. 2. real gases behave like ideal gases except when a) pressure is very high b) temperatures are low c) molecules are very large d) spaces between particles is small (small volume)
32. B. Ideal Gas Law - PV = nRT 1. pressure ( atm,mm Hg, KPa) 2. volume (liters) 3. temperature (kelvin) 4. number of moles (n) 5. R = constant (L) (pressure unit*) (mol) (K)
33. unit for pressure determines which constant must be used in the Ideal Gas Law PV = nRT a) R = 62.4 (pressure is mm Hg) b) R = .0821 (pressure is atm) c) R = 8.314 (pressure is KPa)
36. C. Application Problems (PV = nRT) 1. How many moles of O2 are in a 2.00L container at 2.00 atm pressure and 200K? 2. Calculate the volume occupied by 2.00 mol of N2 at 300K and .800 atm
37. 3. What is the pressure in mm Hg of .200 moles of gas in a 5.00 L container at 27C? 4. Calculate the number of grams of oxygen in a 4.00 L sample of gas at 1.00 atm and 27 C.
38. V. Gas Stoichiometry A. Coefficients and Gas Volume 1. Gay Lussac’s Law of Combining Volumes a) gases in chemical reactions react with each other in whole number ratios at a constant temperature
and pressure CH4 + 2O2 -----> CO2 + 2H2O 1 volume 2 volumes 1 volume 2 volumes 1 mole 2 moles 1 mole 2 moles 22.4L 2 x 22.4L 22.4L 2 x 22.4L | {"url":"https://www.slideserve.com/marika/gases-powerpoint-ppt-presentation","timestamp":"2024-11-04T11:26:52Z","content_type":"text/html","content_length":"93283","record_id":"<urn:uuid:5d7acdc5-35ec-47df-8248-4235f7167ea4>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00088.warc.gz"} |
894 research outputs found
Antiproton annihilations on nuclei provide a very interesting way to study the behaviour of strange particles in the nuclear medium. In low energy $\bar p$ annihilations, the hyperons are produced
mostly by strangeness exchange mechanisms. Thus, hyperon production in $\bar p A$ interactions is very sensitive to the properties of the antikaon-nucleon interaction in nuclear medium. Within the
Giessen Boltzmann-Uehling-Uhlenbeck transport model (GiBUU), we analyse the experimental data on $\Lambda$ and $K^0_S$ production in $\bar p A$ collisions at $p_{\rm lab}=0.2-4$ GeV/c. A satisfactory
overall agreement is reached, except for the $K^0_S$ production in $\bar p+^{20}$Ne collisions at $p_{\rm lab}=608$ MeV/c, where we obtain substantially larger $K^0_S$ production rate. We also study
the $\Xi$ hyperon production, important in view of the forthcoming experiments at FAIR and J-PARC.Comment: 8 pages, 4 figures, invited talk given by A.B. Larionov at the 10th International Conference
on Low Energy Antiproton Physics (LEAP2011), Vancouver, Canada, Apr 27 - May 1, 2011, Hyperfine Interact. in pres
We performed resonant and nonresonant x-ray diffraction studies of a Nd0.5Sr0.5MnO3 thin film that exhibits a clear first-order transition. Lattice parameters vary drastically at the metal-insulator
transition at 170K (=T_MI), and superlattice reflections appear below 140K (=T_CO). The electronic structure between T_MI and T_CO is identified as A-type antiferromagnetic with the d_{x2-y2}
ferroorbital ordering. Below T_CO, a new type of antiferroorbital ordering emerges. The accommodation of the large lattice distortion at the first-order phase transition and the appearance of the
novel orbital ordering are brought about by the anisotropy in the substrate, a new parameter for the phase control.Comment: 4pages, 4figure
In order to examine volatilization processes of alkali metals at high temperature, heating experiments were carried out using a starting material prepared from Murchison (CM2) (grain-size : ∿
10μm) at temperatures of 1200-1400℃ under a constant pressure of 8×10^ Torr, and heating duration up to 80min. Analyses of alkalis (Na, K, Rb), major and minor elements and petrographic examinations
were performed for run products. Results show that fractional volatilization of alkali metals occurred during heating. It is suggested that the volatilization rates of alkali metals are influenced by
the chemical composition of partial melt
We study the search problem for optimal schedulers for the linear temporal logic (LTL) with future discounting. The logic, introduced by Almagor, Boker and Kupferman, is a quantitative variant of LTL
in which an event in the far future has only discounted contribution to a truth value (that is a real number in the unit interval [0, 1]). The precise problem we study---it naturally arises e.g. in
search for a scheduler that recovers from an internal error state as soon as possible---is the following: given a Kripke frame, a formula and a number in [0, 1] called a margin, find a path of the
Kripke frame that is optimal with respect to the formula up to the prescribed margin (a truly optimal path may not exist). We present an algorithm for the problem; it works even in the extended
setting with propositional quality operators, a setting where (threshold) model-checking is known to be undecidable
We study the nonequilibrium switching phenomenon associated with the metal-insulator transition under electric field E in correlated insulator by a gauge-covariant Keldysh formalism. Due to the
feedback effect of the resistive current I, this occurs as a first-order transition with a hysteresis of I-V characteristics having a lower threshold electric field (\sim 10^4 Vcm^{-1}) much weaker
than that for the Zener breakdown. It is also found that the localized mid-gap states introduced by impurities and defects act as hot spots across which the resonant tunneling occurs selectively,
which leads to the conductive filamentary paths and reduces the energy cost of the switching function.Comment: 5 pages, 3 figures. A study on the metal-insulator transition in correlated insulators
was adde
Femtosecond reflection spectroscopy was performed on a perovskite-type manganite, Gd0.55Sr0.45MnO3, with the short-range charge and orbital order (CO/OO). Immediately after the photoirradiation, a
large increase of the reflectivity was detected in the mid-infrared region. The optical conductivity spectrum under photoirradiation obtained from the Kramers-Kronig analyses of the reflectivity
changes demonstrates a formation of a metallic state. This suggests that ferromagnetic spin arrangements occur within the time resolution (ca. 200 fs) through the double exchange interaction,
resulting in an ultrafast CO/OO to FM switching.Comment: 4 figure
The classic approaches to synthesize a reactive system from a linear temporal logic (LTL) specification first translate the given LTL formula to an equivalent omega-automaton and then compute a
winning strategy for the corresponding omega-regular game. To this end, the obtained omega-automata have to be (pseudo)-determinized where typically a variant of Safra's determinization procedure is
used. In this paper, we show that this determinization step can be significantly improved for tool implementations by replacing Safra's determinization by simpler determinization procedures. In
particular, we exploit (1) the temporal logic hierarchy that corresponds to the well-known automata hierarchy consisting of safety, liveness, Buechi, and co-Buechi automata as well as their boolean
closures, (2) the non-confluence property of omega-automata that result from certain translations of LTL formulas, and (3) symbolic implementations of determinization procedures for the Rabin-Scott
and the Miyano-Hayashi breakpoint construction. In particular, we present convincing experimental results that demonstrate the practical applicability of our new synthesis procedure
We study the hydrodynamics of a freely-standing smectic-A film in the isothermal, incompressible limit theoretically by analyzing the linearized hydrodynamic equations of motion with proper boundary
conditions. The dynamic properties for the system can be obtained from the response functions for the free surfaces. Permeation is included and its importance near the free surfaces is discussed. The
hydrodynamic mode structure for the dynamics of the system is compared with that of bulk systems. We show that to describe the dynamic correlation functions for the system, in general, it is
necessary to consider the smectic layer displacement $u$ and the velocity normal to the layers, $v_z$, together. Finally, our analysis also provides a basis for the theoretical study of the
off-equilibrium dynamics of freely-standing smectic-A films.Comment: 22 pages, 4 figure | {"url":"https://core.ac.uk/search/?q=author%3A(Miyano%20K.)","timestamp":"2024-11-03T19:48:27Z","content_type":"text/html","content_length":"167675","record_id":"<urn:uuid:f2695616-aa14-49ba-80d9-d1ca5563613b>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00428.warc.gz"} |
How to enjoy mathematics
Mathematics is a very difficult subject, which many students feel. But today we will try to make it a little easier. There are four things from basic in mathematics, such as dividing, multiplying,
adding and subtracting and that almost ends your math.
First let me tell you a little story of mine. This is the story when I used to study in class V. In our village we had a government school up to fifth grade. In our school, every day, half an hour
before the end of school time, the lower class used to count numbers loudly and the students of the higher class used to recite table. And I still remember that when I used to hear my name, I used to
forget the number tables, so I used to get some punishment every day. But one day our Mathematics teacher told our Principal that of course this child does not remember the tables but you should give
any maths questions and he will solve it in his class, then our Principal gave me a math question and I solved it and took it to my math teacher, he told me that son I knew that you will be able to
do it. Principal checked and hailed me.
It is not that if you do not remember the tables, then you cannot calculate square root, or do not know/remember any formula, then you are weak in mathematics. It is not like that at all. Now the
question is, how to make mathematics easy?
We do Mathematics well in our every day life, but still we do not realize that we are so good in mathematics. You have seen around you that illiterate people also deal in money and they do not make
any mistake. How do they calculate very accurately. If you ever want to try it, give them a ₹ 200 note and say keep this ₹ 2000 note and tomorrow I will take my ₹ 2000. Will he agree? Would not
believe at all why he has not read and written, how does he know if he has a note of ₹ 200 or a reply of ₹ 2000 that if he is dealing with money then he must have identified the note as if the size
of the note is the color of the note The picture is made on the note is etc.
Then tell them to add 34, 60 and 66, how much is it? He will not be able to add but you will ask him that I went to the market, bought a vegetable worth ₹ 34, bought milk worth ₹ 60 and bought fruits
worth ₹ 66. Ask him how much money I spent in the market? He will add it instantaneously.
Why, because rarely any person makes mistake in money transactions, whether he is educated or illiterate. Whenever calculating in real time money, you will never make a mistake. Your answer will be
quick you will be interested in mathematics. Mathematics is such a subject that the more you interest you take in mathematics, you start enjoying it more.. | {"url":"https://apexcoachings.com/blogs/how-to-enjoy-mathematics/","timestamp":"2024-11-08T18:17:38Z","content_type":"text/html","content_length":"97447","record_id":"<urn:uuid:fe2d32aa-159f-46be-9b8c-9cb0a78da53a>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00027.warc.gz"} |
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Earth Orbit Calculator
Earth Orbit Calculator
Earth Orbit Calculator
i Distance from Earth's center to orbiting object.
i Choose between metric and imperial units.
i Speed at which the object is traveling in orbit.
i Time taken for one complete orbit.
Earth Orbit Calculator
The Earth Orbit Calculator is a handy tool designed to assist you in calculating two critical aspects of an object’s orbit around Earth: the orbital speed and the orbital period. Understanding these
parameters can offer valuable insights for both academic and practical applications, such as satellite mission planning or even in educational settings.
Applications of the Earth Orbit Calculator
This calculator is particularly useful for aerospace engineers, scientists, students, and hobbyists interested in space. For instance, engineers can use it when designing satellites to ensure they
achieve the correct speed to maintain a stable orbit. Educators can introduce students to the fascinating details of orbital mechanics through hands-on calculations, helping them grasp complex
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How the Calculation Works
When you input the orbital radius, the calculator uses well-established physical laws to determine the orbital speed and period. The orbital radius is the distance from the center of the Earth to the
orbiting object. This value must be greater than Earth’s radius, which is approximately 6,371 kilometers, to make sense for orbiting objects.
The orbital speed is the rate at which the object travels along its orbital path. It is derived from the gravitational force between Earth and the object; specifically, the speed is proportional to
the square root of Earth’s gravitational constant and mass divided by the orbital radius.
The orbital period is the time taken to complete one full orbit around Earth. This period is proportional to the square root of the orbital radius cubed divided by the gravitational constant and
Earth’s mass.
Using the Calculator
To use the calculator, simply enter the orbital radius in meters. If you prefer using the imperial system, you can switch units after inputting the radius. The output will give you the orbital speed
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Benefits of Understanding Orbital Mechanics
Grasping the basics of orbital mechanics is essential for anyone involved in space-related fields. It helps in predicting satellite behavior, avoiding potential collisions with space debris, and
planning efficient mission trajectories. The Earth Orbit Calculator simplifies these complex calculations, making it easier for users of all levels to engage with and understand the principles
governing satellite orbits.
With this tool, anyone can explore the exciting dynamics of objects in orbit around Earth, opening up a world of possibilities for learning and practical application. Happy calculating!
What units are required for the orbital radius input?
The orbital radius should be entered in meters. If you prefer to use the imperial system, you can switch the units after entering the radius. The calculator will then convert the value accordingly.
What is the minimum value for the orbital radius I can use?
The orbital radius must be greater than the Earth’s radius, which is approximately 6,371 kilometers (or 6,371,000 meters). Any value less than this would not make sense for an orbiting object around
How is the orbital speed calculated?
The orbital speed is calculated using the formula: ( V = sqrt{frac{GM}{r}} ), where G is the gravitational constant, M is the mass of the Earth, and r is the orbital radius. This gives the speed in
meters per second.
How is the orbital period derived?
The orbital period is determined by the formula: ( T = 2pi sqrt{frac{r^3}{GM}} ). This provides the period in seconds, indicating the time it takes for the object to complete one full orbit around
the Earth.
Can I use this calculator for calculating orbits of objects around other celestial bodies?
This particular calculator is designed specifically for objects orbiting Earth. For other celestial bodies, you would need to adjust the mass of the central body and possibly other parameters like
the gravitational constant tailored to the specific celestial body.
Why do my results show such high values for speed and period?
If the orbital radius entered is very large, the calculations will yield higher values for both speed and period. Ensure the radius you input is realistic for the Earth’s orbit to get meaningful
Is the output valid for elliptical orbits?
This calculator assumes a circular orbit. For an elliptical orbit, the calculations would be more complex and would require additional parameters, such as the semi-major axis and eccentricity.
What practical applications can this calculator serve?
This calculator can be very useful for satellite mission planning, educational purposes, and understanding basic orbital mechanics. It's a practical tool for anyone interested in aerospace
engineering, physics, or astronomy.
Why does the calculator ask only for the orbital radius and nothing else?
The orbital radius is the primary variable needed along with the known constants (Earth’s mass and the gravitational constant) to calculate both the orbital speed and period. This simplifies the user
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Double Thunking Works Wonders! | Microsoft Community Hub
Forum Discussion
Double Thunking Works Wonders!
Given that most Excel users would not dream of employing one thunk, you might well ask why even consider nested thunks! The use case explored here is to return all the combinations by which one might
choose m objects from n (not just a count of options =COMBIN(n, m), but the actual combinations) Knowing that sometimes allows one to deploy an exhaustive search of options to determine the best
strategy for a task.
Before considering the task further, one might ask 'what is a thunk; isn't it far too complicated to be useful?' All it is, is a LAMBDA function that evaluates a formula when used, the same as any
other function. The formula could be an expensive calculation or, rather better, no more than a simple lookup of a term from a previously calculated array. The point is, that whilst 'arrays of
arrays' are not currently supported in Excel, an array of functions is fine, after all, an unrun function is little more than a text string. Only when evaluated, does one recover an array.
In the example challenge, each cell contains an list/array of binary numbers, which might itself run into the hundreds of terms. A '1' represents a selected object whilst a '0' is an omitted object.
Rather like the counts of combinations obtained from Pascal's triangle, each cell is derived from the contents of the cell to the left and the cell above. This is SCAN on steroids, accumulating array
results in two directions. Running down the sheet, the new combination contains those of the above cell, but all the objects are shifted left and an empty slot appears to the right. These values are
appended to those from the left, in which the member objects are shifted left but the new object is added to the right.
So the challenge is to build a 2D array, each member of which is itself an array. The contents of each cell is represented by a thunk; each row is therefore an array of thunks which, for REDUCE to
treat it as a single entity, requires it to be securely tucked inside its own LAMBDA, to become a thunk containing thunks. Each pair of rows defined by REDUCE is itself SCANned left to right to
evaluate the new row.
By comparison the 2D SCAN required for the Levenshtein distance which measure the similarity of text strings was a pushover. I am not expecting a great amount of discussion to stem from this post
but, if it encourages just a few to be a little more adventurous in the way they exploit Excel, its job will be done!
p.s. The title of this discussion borrows from the Double Diamond advert for beer in the 1960s
• Whilst the original post was challenging [it created a Lambda helper function that simultaneously accumulated results both across a range and down it, whilst allowing individual results to be
arrays], it was somewhat esoteric and lacked obvious use cases other than generating combinations as outlined.
As a consequence, I have reformulated that particular use case which allowed considerable simplification, since each list of combinations only references the list above and the list to the left.
Ultimately the number of combinations returned by the formula is limited by the number of rows on the worksheet. It is quite usable returning 200,000 combinations (for example).
Something that the new solution retains from the original helper function is that it uses thunks to hold any single list of combinations within a 2D array. To make REDUCE work, a complete row of
thunks is itself turned into a single thunk. The amazing thing is that it works, and works efficiently at that!
□ This is one of those concepts where I spend some time on it each day then put it aside to re-visit at a later time. Needless to say, when a post like this is made it's easy to fill up
quickly! These are what my notes look like in stepping through LISTCOMBINλ:
It's going to take some time to sink in before I understand it better. I really hope your work is going a long way to forcing the Excel team's hand to open up the calculation engine and
support nested arrays. In the very least - maybe LIST to store nested arrays in memory without #CALC! errors and a function to flatten the list.
☆ You appear to be doing well. I get so far, and then realise I have lost the thread and need to wait for fresh inspiration! I find myself spending a lot of time debugging, but that's
inevitable when you make as many mistakes as I do 🤔.
One debugging pattern is to return a LET variable of interest instead of the intended final result. If it is #CALC! then before doing anything else, it enclose it in TYPE (within the LET
block). Surprisingly often the single term comes back type=64. Then the next step is remove the TYPE and replace it with INDEX(..., 1, 1)( ) or, if one is feeling a little more
adventurous (@...)( ). If an array of values is returned, end of story. Just a question of establishing whether they are right or wrong. Now though, it might be an array of #CALC! errors
(thunks) one gets back. Then it is back to INDEX once more, to pick out a term that should be interesting and, at the same time, recognisable. Having isolated a single thunk from the
flock, the final step is to evaluate the thunk by offering it a null parameter string, and it just may reward you with an array that you recognise. I do wonder whether @Diarmuid Early
would take so long! | {"url":"https://techcommunity.microsoft.com/discussions/excelgeneral/double-thunking-works-wonders/4261327","timestamp":"2024-11-11T14:39:14Z","content_type":"text/html","content_length":"362371","record_id":"<urn:uuid:eef1463c-45df-4da2-b5e4-113d4b7437cf>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00387.warc.gz"} |
Formula sheet as level maths book 2017
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Python tan - Find Tangent of Number in Radians Using math.tan()
To find the tangent of a number (in radians), we can use the Python tan() function from the math module.
import math
In Python, we can easily use trigonometric functions with the Python math module. The Python math module allows us to perform trigonometry easily.
With the Python math module, we can find the tangent of a number easily.
To find the tangent of a number, or the sine divided by the cosine of an angle, in radians, we use the math tan() function.
Below is the Python syntax to find the tangent of a number.
import math
The input to the tan() function must be a numeric value. The return value will be a numeric value between negative infinity and infinity.
import math
Converting Degrees to Radians for Input into tan()
The tan() function takes a number in radians as input. If your data is in degrees, you will need to convert the values to radians.
The Python math module has a useful function to convert degrees to radians, the math radians() function.
If we have a value in degrees, we can use radians() to convert the value to radians and then pass it to the tan() function.
import math
Finding the Inverse Tangent of a Number in Python
The Python math module also provides us trigonometric functions to find the inverses of the common trigonometric functions. The atan() function allows us to find the inverse of the tangent of a
Below, we show that if we pass a number to tan() and then call the Python atan() function, we get back the same number.
import math
Finding the Cotangent of a Number in Python
To find the cotangent of a number, we can divide 1 by the tangent of the number.
We can find the cotangent of a number easily with the Python tan() function. You can see how to find the cotangent of a number in the following Python code.
import math
Hopefully this article has been beneficial for you to understand how to use the math tan() function in Python to find the tangent of a number. | {"url":"https://daztech.com/python-tan/","timestamp":"2024-11-07T12:47:04Z","content_type":"text/html","content_length":"247075","record_id":"<urn:uuid:e5b11d03-8482-4611-aa6c-432ea5a22f55>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00034.warc.gz"} |
Examples of monotonicity
9 Examples of monotonicity
9.1 Power-delay tradeoff in wireless communication
In a cell phone, higher layer applications such as voicecall, email, browsers, etc. generate data packets. These packets are buffered in a queue and the transmission protocol decides how many packets
to transmit at each time depending the number of packets in the queue and the quality of the wireless channel.
Let \(X_t \in \integers_{\ge 0}\) denote the number of packets buffered at time \(t\) and \(A_t \in \integers_{\ge 0}\), \(A_t \le X_t\), denote the number of packets transmitted at time \(t\). The
remaining \(X_t - A_t\) packets incur a delay penalty given by \(d(X_t - A_t)\), where \(d(\cdot)\) is a strictly increasing and discrete-convex function where \(d(0) = 0\).
Discrete convexity or \(L^{\#}\) convexity
A function \(f \colon \integers \to \reals\) is called convex (or \(L^{\#}\) convex) if for any \(x \in \integers\), \[ f(x+1) + f(x-1) \ge 2 f(x), \] or, equivalently, for any \(x, y \in \integers\)
\[ f(x) + f(y) \ge f\Bigl(\Bigl\lfloor \frac{x+y}{2} \Bigr\rfloor\Bigr) + f\Bigl(\Bigl\lceil \frac{x+y}{2} \Bigr\rceil\Bigr).\]
It can be easily seen that \(L^{\#}\) functions satisfy the following properties:
• Sum of \(L^{\#}\) convex functions is \(L^{\#}\) convex.
• Pointwise limits of \(L^{\#}\) convex functions is \(L^{\#}\) convex.
See Murota (1998) and Chen (2017) for more details.
During time \(t\), \(W_t \in \integers_{\ge 0}\) additional packets arrive and \[ X_{t+1} = X_t - A_t + W_t.\] We assume that \(\{W_t\}_{t \ge 1}\) is an i.i.d. process.
The packets are transmitted over a wireless fading channel. Let \(S_t \in \ALPHABET S\) denote the state of the fading channel. We assume that the states are ordered such that a lower value of state
denotes a better channel quality.
If the channel has two states, say GOOD and BAD, we typically expect that \[ \PR(\text{GOOD} \mid \text{GOOD}) \ge \PR(\text{GOOD} \mid \text{BAD}). \] This means that the two state transition matrix
is stochastically monotone. So, in general (i.e., when the channel has more than two states), we assume that \(\{S_t\}_{t \ge 1}\) is a stochastically monotone Markov process that is independent of \
(\{W_t\}_{t \ge 1}\).
The transmission protocol sets the transmit power such that the signal to noise ratio (SNR) at the receiver is above a desired threshold. It can be shown that for additive white Gaussian channels
(AWGN), the transmitted power is of the form \[p(A_t) q(S_t),\] where
• \(p(\cdot)\) is a strictly increasing and convex function where \(p(0) = 0\);
• \(q(\cdot)\) is a strictly increasing function.
The objective is to choose a transmission policy \(A_t = π^*_t(X_t, S_t)\) to minimize the weighted sum of transmitted power and delay \[ \EXP\bigg[ \sum_{t=1}^T \big[ p(A_t) q(S_t) + \lambda d(X_t -
A_t) \big] \bigg],\] where \(\lambda\) may be viewed as a Lagrange multiplier corresponding to a constrained optimization problem.
9.1.1 Dynamic program
We can assume \(Y_t = X_t - A_t\) as a post-decision state in the above model and write the dynamic program as follows:
\[ V^*_{T+1}(x,s) = 0 \] and for \(t \in \{T, \dots, 1\}\), \[\begin{align*} H_t(y,s) &= \lambda d(y) + \EXP[ V^*_{t+1}(y + W_t, S_{t+1}) | S_t = s ], \\ V^*_t(x,s) &= \min_{0 \le a \le x} \big\{ p
(a) q(s) + H_t(x-a, s) \big\} \end{align*}\]
9.1.2 Monotonicity of value functions
Lemma 9.1 For all \(t\), \(V^*_t(x,s)\) and \(H_t(y,s)\) are increasing in both variables.
First note that the constraint set \(\ALPHABET A(x) = \{0, \dots, x\}\) satisfies the conditions that generalize the result of monotonicity to constrained actions.
We prove the two monotonicity properties by backward induction. First note that \(V^*_{T+1}(x,s)\) is trivially monotone. This forms the basis of induction. Now suppose \(V^*_{t+1}(x,s)\) is
increasing in \(x\) and \(s\). Since \(\{S_t\}_{t \ge 1}\) is stochastically monotone, \[H_t(y,s) = \lambda d(y) + \EXP[ V^*_{t+1}(y + W_t, S_{t+1}) | S_t = s ]\] is increasing in \(s\). Moreover,
since both \(d(y)\) and \(V^*_{t+1}(y + w, s)\) are increasing in \(y\), so is \(H_t(y,s)\).
Now, for every \(a\), \(p(a) q(s)\) and \(H_t(x-a, s)\) is increasing in \(x\) and \(s\). So, the pointwise minima over \(a\) is also increasing in \(x\) and \(s\).
9.1.3 Convexity of value functions
Lemma 9.2 For all time \(t\) and channel state \(s\), \(V^*_t(x,s)\) and \(H_t(y,s)\) are convex in the first variable.
We proceed by backward induction. First note that \(V^*_{T+1}(x,s)\) is trivially convex in \(x\). Now assume that \(V^*_{t+1}(x,s)\) is convex in \(x\). Then, \(\EXP[V^*_{t+1}(y + W_t, S_{t+1}) |
S_t = s]\) is weighted sum of convex functions and is, therefore, convex in \(y\). Therefore, \(H_t(y,s)\) is a sum of two convex functions and, therefore, convex in \(y\).
We cannot directly show the convexity of \(V^*_t(x,s)\) because the pointwise minimum of convex functions is not convex. So, we consider the following argument. Fix \(s\) and pick \(x > 1\). Let \(\
underline a = π^*_t(x-1,s)\) and \(\bar a = π^*_t(x+1,s)\). Let \(\underline v = \lfloor (\underline a + \bar a)/2 \rfloor\) and \(\bar v = \lceil (\underline a + \bar a)/2 \rceil\). Note that both \
(\underline v\) and \(\bar v\) are feasible at \(x\). Then, \[ \begin{align*} \hskip 2em & \hskip -2em V^*_t(x-1, s) + V^*_t(x+1, s) \\ &= [ p(\underline a) + p(\bar a) ] q(s) + H_t(x - 1 - \
underline a, s) + H_t(x + 1 - \bar a, s) \\ &\stackrel{(a)}\ge [ p(\underline v) + p(\bar v)] q(s) + H_t(x - \underline v, s) + H_t(x - \bar v, s) \\ &\ge 2 \min_{a \le x} \big\{ p(a) q(s) + H_t(x-a,
s) \\ &= 2 V^*_t(x,s), \end{align*} \] where \((a)\) follows from convexity of \(p(\cdot)\) and \(H_t(\cdot, s)\). Thus, \(V^*_t(x,s)\) is convex in \(x\). This completes the induction step.
9.1.4 Monotonicity of optimal policy in queue length
Theorem 9.1 For all time \(t\) and channel state \(s\), there is an optimal strategy \(π^*_t(x,s)\) which is increasing in the queue length \(x\).
In Lemma 9.2, we have shown that \(H_t(y,s)\) is convex in \(y\). Therefore, \(H_t(x-a, s)\) is submodular in \((x,a)\).
Thus, for a fixed \(s\), \(p(a)q(s) + H_t(x-a, s)\) is submodular in \((x,a)\). Therefore, the optimal policy is increasing in \(x\).
One can show submodularity by finite difference, but for simplicity, we assume that \(H_t(y,s)\) is twice differentiable. Then, \(\partial^2 H_t(x - a, s)/ \partial x \partial a \le 0\) (by convexity
of \(H_t\)).
9.1.5 Lack of monotonicity of optimal policy in channel state
It is natural to expect that for a fixed \(x\) the optimal policy is decreasing in \(s\). However, it is not possible to obtain the monotonicity of optimal policy in channel state in general. To see
why this is difficult, let us impose a mild assumption on the arrival distribution.
The packet arrival distribution is weakly decreasing, i.e., for any \(v,w \in \integers_{\ge 0}\) such that \(v \le w\), we have that \(P_W(v) \ge P_W(w)\).
We first start with a slight generalization of stochastic monotonicity result.
Lemma 9.3 Let \(\{p_i\}_{i \ge 0}\) and \(\{q_i\}_{i \ge 0}\) be real-valued non-negative sequences satisfying \[ \sum_{i \le j} p_i \le \sum_{i \le j} q_i, \quad \forall j.\] (Note that the
sequences do not need to add to 1). Then, for any increasing sequence \(\{v_i\}_{i \ge 0}\), we have \[ \sum_{i = 0}^\infty p_i v_i \ge \sum_{i=0}^\infty q_i v_i. \]
The proof is similar to the proof for stochastic monotonicity.
The idea of the proof is similar to Lemma 8.1.
Fix \(y^+, y^- \in \integers_{\ge 0}\) and \(s^+, s^- \in \ALPHABET S\) such that \(y^+ > y^-\) and \(s^+ > s^-\). Now, for any \(y' \in \integers_{\ge 0}\) and \(s' \in \ALPHABET S\) define \[\begin
{align*} π(y',s') = P_W(y' - y^+)P_S(s'|s^+) + P_W(y' - y^-)P_S(s'|s^-), \\ μ(y',s') = P_W(y' - y^-)P_S(s'|s^+) + P_W(y' - y^+)P_S(s'|s^-). \end{align*}\]
Since \(P_S\) is stochastically monotone, we have that for any \(σ \in \ALPHABET S\), \[ \sum_{s'=1}^{σ} P_S(s'|s^+) \le \sum_{s'=1}^{σ} P_S(s'|s^-). \] Moreover, due to (asm-power-delay-density?),
we have that \(P_W(y' - y^-) \le P_W(y' - y^+)\). Thus, \[ [P_W(y' - y^+) - P_W(y' - y^-)] \sum_{s'=1}^{σ} P_S(s'|s^+) \le [P_W(y' - y^+) - P_W(y' - y^-)]\sum_{s'=1}^{σ} P_S(s'|s^-). \] Rearranging
terms, we get \[ \sum_{s'=1}^σ π(y',s') \le \sum_{s'=1}^σ μ(y',s'). \] Thus, for any \(y'\), the sequence \(π(y',s')\) and \(ν(y',s')\) satisfy the condition of Lemma 9.3.
Now, in Lemma 9.1, we have established that for any \(y'\), \(V_{t+1}(y',s')\) is increasing in \(s'\). Thus, from Lemma 9.3, we have \[ \sum_{s' \in \ALPHABET S} π(y', s') V_{t+1}(y', s') \ge \sum_
{s' \in \ALPHABET S} μ(y', s') V_{t+1}(y', s'), \] Summing up over \(y'\), we get \[ \sum_{y' \in \integers_{\ge 0}} \sum_{s' \in \ALPHABET S} π(y', s') V_{t+1}(y', s') \ge \sum_{y' \in \integers_{\
ge 0}} \sum_{s' \in \ALPHABET S} μ(y', s') V_{t+1}(y', s'), \] Or equivalently, \[\begin{align*} \hskip 2em & \hskip -2em \EXP[ V_{t+1}(y^+ + W, S_{t+1}) | S_t = s^+) ] + \EXP[ V_{t+1}(y^- + W, S_
{t+1}) | S_t = s^-) ] \\ & \ge \EXP[ V_{t+1}(y^- + W, S_{t+1}) | S_t = s^+) ] + \EXP[ V_{t+1}(y^+ + W, S_{t+1}) | S_t = s^-) ] . \end{align*}\] Thus, \(H_t(y,s)\) is supermodular in \((y,s)\).
Note that we have established that \(H_t(y,s)\) is supermodular in \((y,s)\). Thus, for any fixed \(x\), \(H_t(x-a,s)\) is submodular in \((a,s)\). Furthermore the function \(p(a)q(s)\) is increasing
in both variables and therefore supermodular in \((a,s)\). Therefore, we cannot say anything specific about \(p(a)q(s) + H_t(x-a, s)\) which is a sum of submodular and supermodular functions.
We need to impose a much stronger assumption to establish monotonicity in channel state. See Exercise 9.1.
Exercise 9.1 Suppose that the channel state \(\{S_t\}_{t \ge 1}\) is an i.i.d. process. Then prove that for all time \(t\) and queue state \(x\), there is an optimal strategy \(π^*_t(x,s)\) which is
decreasing in channel state \(s\).
The mathematical model of power-delay trade-off is taken from Berry (2000), where the monotonicty results were proved using first principles. More detailed characterization of the optimal
transmission strategy when the average power or the average delay goes to zero are provided in Berry and Gallager (2002) and Berry (2013). A related model is presented in Ding et al. (2016). A slight
generalization of this model is also considered in Fu and Schaar (2012) where monotonicty in the queue state is stablished.
For a broader overview of power-delay trade offs in wireless communication, see Berry et al. (2012) and Yeh (2012).
The remark after Lemma 9.4 shows the difficulty in establishing monotonicity of optimal policies for a multi-dimensional state space. In fact, sometimes even when monotonicity appears to be
intuitively obvious, it may not hold. See Sayedana and Mahajan (2020) for an example. For general discussions on monotonicity for multi-dimensional state spaces, see Topkis (1998) and Koole (2006).
As an example of using such general conditions to establish monotonicity, see Sayedana et al. (2020). | {"url":"https://adityam.github.io/stochastic-control/mdps/monotone-examples.html","timestamp":"2024-11-05T04:14:13Z","content_type":"application/xhtml+xml","content_length":"92267","record_id":"<urn:uuid:5a9d91b8-e45f-4509-957e-07316dd8ddf6>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00108.warc.gz"} |
7. [Idea of a Function] | Math Analysis | Educator.com
f(x) = 7x − 5 ⇒ f(a−3) = 7(a−3) − 5
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while
watching the lecture.
Section 1: Introduction
Introduction to Math Analysis 10:03
Sets, Elements, & Numbers 45:11
Variables, Equations, & Algebra 35:31
Coordinate Systems 35:02
Midpoints, Distance, the Pythagorean Theorem, & Slope 48:43
Word Problems 56:31
Section 2: Functions
Idea of a Function 39:54
Graphs 58:26
Properties of Functions 48:49
Function Petting Zoo 29:20
Transformation of Functions 48:35
Composite Functions 33:24
Piecewise Functions 51:42
Inverse Functions 49:37
Variation Direct and Inverse 28:49
Section 3: Polynomials
Intro to Polynomials 38:41
Roots (Zeros) of Polynomials 41:07
Completing the Square and the Quadratic Formula 39:43
Properties of Quadratic Functions 45:34
Intermediate Value Theorem and Polynomial Division 46:08
Complex Numbers 45:36
Fundamental Theorem of Algebra 19:09
Section 4: Rational Functions
Rational Functions and Vertical Asymptotes 33:22
Horizontal Asymptotes 34:16
Graphing Asymptotes in a Nutshell 49:07
Partial Fractions 44:56
Section 5: Exponential & Logarithmic Functions
Understanding Exponents 35:17
Exponential Functions 47:04
Introduction to Logarithms 40:31
Properties of Logarithms 42:33
Solving Exponential and Logarithmic Equations 34:10
Application of Exponential and Logarithmic Functions 48:46
Section 6: Trigonometric Functions
Angles 39:05
Sine and Cosine Functions 43:16
Sine and Cosine Values of Special Angles 33:05
Modified Sine Waves: Asin(Bx+C)+D and Acos(Bx+C)+D 52:03
Tangent and Cotangent Functions 36:04
Secant and Cosecant Functions 27:18
Inverse Trigonometric Functions 32:58
Computations of Inverse Trigonometric Functions 31:08
Section 7: Trigonometric Identities
Pythagorean Identity 19:11
Identity Tan(squared)x+1=Sec(squared)x 23:16
Addition and Subtraction Formulas 52:52
Double Angle Formulas 29:05
Half-Angle Formulas 43:55
Section 8: Applications of Trigonometry
Trigonometry in Right Angles 25:43
Law of Sines 56:40
Law of Cosines 49:05
Finding the Area of a Triangle 27:37
Word Problems and Applications of Trigonometry 34:25
Section 9: Systems of Equations and Inequalities
Systems of Linear Equations 55:40
Systems of Linear Inequalities 1:00:13
Nonlinear Systems 41:01
Section 10: Vectors and Matrices
Vectors 1:09:31
Dot Product & Cross Product 35:20
Matrices 54:07
Determinants & Inverses of Matrices 47:12
Using Matrices to Solve Systems of Linear Equations 58:34
Section 11: Alternate Ways to Graph
Parametric Equations 53:33
Polar Coordinates 48:07
Polar Equations & Functions 38:16
Section 12: Complex Numbers and Polar Coordinates
Polar Form of Complex Numbers 40:43
DeMoivre's Theorem 57:37
Section 13: Counting & Probability
Counting 31:36
Permutations & Combinations 44:03
Probability 36:58
Section 14: Conic Sections
Parabolas 41:27
Circles 21:03
Ellipses 46:51
Hyperbolas 38:15
Conic Sections 18:43
Section 15: Sequences, Series, & Induction
Introduction to Sequences 57:45
Introduction to Series 40:27
Arithmetic Sequences & Series 31:36
Geometric Sequences & Series 39:27
Mathematical Induction 49:53
The Binomial Theorem 1:13:13
Section 16: Preview of Calculus
Idea of a Limit 40:22
Formal Definition of a Limit 57:11
Finding Limits 32:40
Continuity & One-Sided Limits 32:43
Limits at Infinity & Limits of Sequences 32:49
Instantaneous Slope & Tangents (Derivatives) 51:13
Area Under a Curve (Integrals) 45:26
Section 17: Appendix: Graphing Calculators
Buying a Graphing Calculator 10:41
Graphing Calculator Basics 10:51
Graphing Functions, Window Settings, & Table of Values 10:38
Finding Points of Interest 9:45
Parametric & Polar Graphs 7:08
Today, we are going to talk about the idea of a function.0002
Functions are extremely important to mathematics: you have certainly encountered them before.0006
But you might not have fully understood how they work and what they are doing.0011
This lesson is here to give us a clear understanding of what it means for something to be a function, and how functions work.0015
Since functions are so important, they are going to come up in every single lesson you learn about in this course.0021
And they are going to come up in every single concept you talk about in calculus.0026
And they are going to keep coming up, as long as you are studying math.0028
Make sure you watch this entire lesson; it is so important to have a good, grounded, fundamental concept of what a function is,0031
because it is going to keep getting used in everything that we talk about.0038
This is probably the single most important lesson of this entire course,0042
because so many later ideas are going to talk about functions.0045
Also, it would help to have watched the previous lesson on sets, elements, and numbers,0049
because we are going to be talking about how sets are connected to functions.0054
So if you haven't done that, I would recommend that you go and watch that one first,0057
because it will help explain a lot of what we are talking about here, because functions are relying on the idea of sets.0060
All right, let's jump into it: what is a function?0065
A function is a relation between two sets: a first set and a second set.0069
For each element from the first set, the function assigns precisely one element in the second set.0074
So, we will point at some element in the first set, and it will say, "Here is an element from the second set."0080
Point at another element from the first set, and it will tell us, "Here is some element from the second set."0085
That is the idea of a function; here is a visual example for it.0089
We could have something where all of the squares are the first kind--it is our first set--and all of the round things on this side are our second set.0092
So, second would be the second column, and first set would be the first column.0101
We could have news get put onto paper; we say that news, the function, gives us paper.0106
We say that cheese, the function, gives us burger; we say that good, the function, gives us bye.0112
We say that sand, the function, gives us paper; we say that bubble, the function, gives us gum.0118
So, there are only five elements in our first set, and only four elements in our second set.0123
But this is a perfectly reasonable function: newspaper, cheeseburger, goodbye, sandpaper, bubblegum.0127
The only one you might be wondering about is..."Wait, news goes to paper and sand goes to paper."0134
There is no problem with that: we only said that the function has to give us something when we point at something in the first one.0138
We never said that it has to be a different thing, every single thing that we point to; it just has to give us something for it.0144
That is what we have here: we have something where everything that we call out on the first side...0150
we call out news, and in turn, it responds by telling us paper.0154
We call out good, and in return, it says bye; that is how it is working here with this function.0159
Here is a non-example: in this one, we say tree, but the function gives us four different possibilities.0167
Sometimes it gives out maple; but other times it gives out oak; but other times it gives out apple; but other times it gives out pine.0174
And then fruit--if we go to fruit, it sometimes gives out apple, and sometimes it gives out grape.0182
This isn't allowed, because it is only allowed to give one response to a given input.0188
We tell it one element from our first set; it can only tell us one element from the second set.0195
It is not allowed to give us a whole bunch of different choices to pick and choose from.0200
Sometimes it is going to be maple; sometimes it is going to be oak; sometimes it is going to be pine.0204
No; it has to be one thing, and one thing only.0207
That is what it requires to be a function; so this is not an example--this is not allowed, because we can't have it be multiple things coming out of this.0210
It has to only be that one input will only give us one output.0219
And as long as we keep putting in that same input, it can only give us the same output.0223
Just like variables, it is useful to name functions with a symbol; so let's talk about how notation works here.0228
Most often, the symbol we will use to talk about a function is f; but sometimes0233
we are also going to use g, h, or whatever else will make sense, depending on the context.0237
But often, we are going to end up seeing f.0242
If we want to talk about what f assigns to some input x, if x is the element in our first set,0245
if that is what we call the element in our first set that we use f on, then it will be assigned to "f of x,"0250
f acting on x--what f gives out when given x; so the first symbol is the name of the function that we are using;0257
then, the second symbol, in parentheses, is what the function is acting on.0267
So, f--the name of our function--is acting on x; and then, that whole thing together is f(x); f(x) is the name of what comes out of it.0276
So, f is the name of what is doing the acting; x (or whatever is in the parentheses)...the first symbol was the name0286
of whatever is doing the acting; the thing inside of the parentheses is the name of what is being acted on;0292
and then, the whole thing taken together is where we are when we use the function on that element--0298
Now, there could be a little bit of confusion about f(x), because it is f, parenthesis, x.0307
And we know that parentheses...if I wrote 2(3), that would mean 2 times 3, right?0312
So, we might think f times x; but we are going to know from context that f is a function, and not something that we multiply.0319
So, when f is a function, we don't have to worry about using multiplication, if it is f on some element.0325
It is always going to be f of that element, never f times, unless we are talking about that explicitly.0330
But if it is just in parentheses, it is not going to be multiplication.0336
So, when you see parentheses, and it is a function, it isn't implying multiplication, like when we are dealing with numbers.0339
If we want to express what sets the function acts on, we can write f:a→b.0346
What this is: it is "f goes from a to b"; it takes elements from a, our first set; and then it assigns them elements from b.0353
Normally, it won't be necessary for us in this course (and probably for the next couple of years)--0363
it won't be necessary to name the sets that our function is working on.0366
But why that is, we will discuss later: it is going to be pretty simple, but we will discuss it later when we get to it.0371
There are a lot of metaphors that we can use to help us understand what is going on in a function.0377
Here are three metaphors to help us understand what happens when f takes things from a and goes to b.0382
Our first idea is transformation: the function transforms elements from one set into another.0388
It takes an element x, contained in a, and then it transforms it into an element in b, which we call f(x), or f acting on x.0394
f(x) is what it has been transformed into; that is what it is after the transformation.0406
Now, from problem to problem, the rules for transformation will usually change as we use different functions.0411
One function is generally going to have a different set of rules for how its function works than another function.0416
But if we are using the same function--if we are in the same problem, using the same function--0421
the rules never change: if we put in the same x, we will always get the same f(x) as our result.0425
The rules for how the transformation works are always the same.0431
So, if the same thing goes in, the same thing always comes out.0434
Another way we can look at it is a map: it tells us how to get from one set to another set.0438
It is sort of a guide, directions for how to get from one place to another place.0443
Of course, if we start at a different starting location, a different starting place,0449
different elements in a, we might end up at a different destination--different elements in b.0453
If I say, "Go 100 kilometers north," you are going to end up in totally different places0458
if you start in Mexico, if you start in California, if you start in England, if you start in South Africa, or if you start in Japan.0462
Each one of these places...if you start in Egypt...is going to end up going to a totally different place, even though they are all still the same direction.0469
You are still doing the same thing; you are still going 100 kilometers north in all of these cases.0476
But because you started in a different place, you end up in a different place.0481
So, a different starting place, a different element that we are acting on, a different element that we are mapping,0484
will normally cause us to have a different destination--a different place that we land on.0489
The math itself, though, never changes: if we start at the same place, we always arrive at the same destination.0495
So, if we start in San Jose, California, and then we go 100 kilometers to the north...I actually have no idea where that is.0501
But we will be 100 kilometers north of San Jose.0508
And then, if we start in San Jose again on another day, and we go 100 kilometers north, we are going to end up being in the exact same place.0510
And if we go to San Jose, and then we go 100 kilometers north again, we are going to end up being in the exact same place.0516
And people are probably going to wonder, "Why does this person keep showing up here?"0521
And that is because we are following the same map.0525
The directions, the transformation that the map gives us, the way we go, isn't going to change each time.0527
It only changes when we start from a new place.0533
Finally, one last way to visualize it is the idea of a machine.0537
We can visualize a function as a machine that eats elements from a, and it produces elements from b.0541
What it produces depends on what it eats, but the machine is reliable: if it eats the same thing, it always produces the same output.0547
For example, if we have x right here, and we push it into our machine, f, it goes into the machine;0553
then the machine works on it and crunches it, crunches it, crunches it; and it gives out f(x).0559
So, we are going from the set a to the set b.0564
Now, one thing about the machine is that it is perfectly reliable; the machine is reliable.0572
If it eats the same thing, it produces the same output.0577
If we put in x, it will always give out f(x); so the first time we put in x, it gives out f(x); the second time, f(x); the third time, f(x); the fiftieth time, f(x).0581
Just like when we started in San Jose, and we went 100 kilometers north, each time we always ended up coming to the same place;0590
you put the same thing into the machine; the same thing comes out.0595
This idea is so important; we are going to talk about it really explicitly.0599
We have said this one way or another for all of our different ways of thinking about functions.0602
But it is so important--it is such an important characteristic of functions--we want to make sure that we know it.0606
If we put the same input into a function, it will always produce the same output.0611
Now, the input and the output could be totally different; the input is not necessarily going to be where we show up in the output.0616
You start in San Jose, and then you show up in some farmer's field 100 kilometers to the north.0622
But you are going to come out to that same farmer's field each time, because you are showing up at the same location.0626
So, for a function to make sense and be well-defined, for it to work, its rules must never change.0633
For example, if f(2), f acting on 2, gives out 7; if f(2) equals 7 the first time, then f(2) = 7 the second time; and f(2) = 7 every time.0639
No matter how many times f operates on 2, no matter what, it is always going to give out the same thing.0650
That is what it means to be a function: your rules don't change when you are going on the same thing.0654
You work on one element the same way each time; you always map it; you always transform it; you always assign it to the same place.0660
Here is something that is not a function: g(cat) = fur, g(cat) = whiskers, g(cat) = quiet.0671
This can't be a function, because we have three totally different destinations when we plug in cat.0678
And what determines whether we go to fur, whiskers, or quiet?0683
There is no reason why we should use one set of rules or another set of rules, so it is not a function.0686
There is no reliability here; we don't know, when we plug in cat, if we are going to go to fur, whiskers, or quiet.0690
So, it is not a function; but we could have a function that was h(fur) goes to cat, h(whiskers) goes to cat, h(quiet) goes to cat.0695
It is not that there is a problem with having us land on the same place.0703
No matter what we put in, the function could give out cat: it doesn't matter,0707
as long as the first thing, the first set we are coming from, can't split as it comes out.0712
We can land on the same place, but we can't be coming from the same place and go to two different locations.0719
We always have to follow one rule; because we are following one rule, we can't land on two different things.0725
Let's look at a non-numerical example: before we start telling you about how functions work on numbers,0731
let's consider an example of one that works on something totally not about numbers.0735
Let's think about a function that gives initials: we will define...f is going from names spelled with the Roman alphabet0740
(names like Vincent or John, not things that are spelled with characters that we can't express in the Roman alphabet),0747
and it is going to go to letters from the Roman alphabet.0755
So, f(x) equals the first letter of x; now, if we say, "Wait, we know that the first letter of x is x!"--0759
yes, but what we are talking about is names: x is a placeholder, remember?0768
We talked about variables: the idea of a variable is that it is a placeholder.0773
So, x is just sort of keeping the spot warm, until later, when we put in the name.0777
So, if we decide to put Vincent into the function, then this x on the left side tells us where to put Vincent on the right side.0782
So, Vincent will come in here on the right side, as well.0791
We will have Vincent go on the left, and Vincent will go on the right.0795
f(Vincent) would be V: we cut it off just to the first letter.0799
f(Nicole) would put out N; f(Padma) would give out P; f(Victor) would give out V; f(Takashi) would give out T.0804
Whatever we put in, it will give out just that single letter.0811
So, if we were to turn this into a diagram, we could have Vincent here, Nicole next to Vincent, Padma, Victor, and then finally Takashi.0815
And so, this is where we are coming from; and then, we are going to letters.0832
So, we have V and N and P and T...and let's put in another letter, like...say S and Q.0842
Vincent gets mapped to V; Nicole, by this function, gets mapped; Padma gets mapped to P; Victor also gets mapped to V.0853
Takashi gets mapped to T; but do S and Q get used? Not for this set of names.0863
Maybe if we put in Susan, or we put in...there has to be a name with Q that I don't know...0868
let's pretend that the name is simply Queen...I am sure that there is a name...a really weird spelling of the name Cory?...0876
there is a name out there that is spelled with a Q; I just don't know it immediately.0883
So, there is something out there that can fill up that S, and that can fill up that Q; we just don't have it in what we are looking at so far.0886
So, there might be other things that we are not hitting on the right;0893
but everything that we have on the left is what is getting mapped to things on the right.0896
So, the functions we use...of course, it is no surprise; this is math--we are probably going to be talking about numbers.0901
So, it shouldn't come as a surprise; we are going to concentrate on using these functions with numbers.0907
Functions, as we just saw, can be used for lots of things; but we will focus on functions and the real numbers.0911
Unless we are told otherwise, we will assume that every function takes in real numbers and outputs real numbers.0917
That is to say, f is taking in reals and then giving out reals.0922
OK, so when we are given a function, we will usually be told what its rule is--how it maps inputs to outputs.0927
So, for example, if f(x) = x2 + 3, its rule is "Square the input," since x is our input;0934
then, what we do is...we first square the input, and then we add 3; square the input and then add 3.0941
Notice that x acts as a placeholder; just like it did with the names, it acts as a placeholder.0948
It is not that x is really the thing we are worried about being acted on.0953
It is just telling us what is going to happen to whatever we plug into this function.0956
If we plug in 3, what will happen to 3?0960
If we plug in 50, what will happen to 50?0962
If we plug in smiley-face, what will happen to smiley-face?0964
x is just there to sort of keep a spot warm: it is telling us, "Here is the place; things will go into this place."0967
And things will go into this place, wherever I show up on the right side, as well.0974
If we want to use a function, if we want to evaluate a function at a specific value, we just apply this rule to whatever our input value is.0979
In practice, this turns out to actually be really simple.0986
Usually, we are given a formula for each function; so we just follow the method of substitution.0988
Remember, we take whatever we are substituting in; we wrap it in parentheses; and then we see what we get.0992
For example, our function is f(x) = x2 + 3; then, to find f(7), we just plug in.0997
7 is what we are plugging in; so we have 7 in this spot, and a 7 will go in here.1004
We wrap that in parentheses, just in case; in this case, we don't have to, but we will see why it is useful to always remember to wrap it in parentheses.1009
72 + 3...72 is 49; 49 + 3...we get 52.1016
If we want to look at a slightly more complex example, though, we see why it is so important to wrap your substitutions in parentheses.1021
If we consider a slightly more complex input, like a + 7, then we have to have it in parentheses,1029
because it is not just the a that gets squared; it is not just the 7 that gets squared; it is all of that thing that went in.1034
All of that thing is both the a and the + 7; it is (a + 7); it is that whole number combined.1041
It is not a2 + 7; it is not a + 72; it is (a + 7), the whole thing squared; and then, plus 3.1047
A good way to see the behavior of a function is by creating a table of values; sometimes we call it a T-table, because it has the shape of a T.1057
On one side, we have input values, while the other side shows us what the function outputs when given that input.1065
So normally, the left side will be our input value, and the right side will be our output value.1071
So, for example, if f(x) = x2 + 3, then we can give out a bunch of values for it.1076
So, if we want to figure out what happens to f(-2), we just follow the normal thing.1080
f(-2), so we plug it in...(-2)2 + 3...we get 4 + 3; we get 7, and that 7 shows up here.1086
If we want to figure out what f(-1) is, we do the exact same thing: (-1)2 + 3, 1 + 3, and 4.1095
And that 4 shows up here; and so on, and so forth.1103
We just plug in, based on this rule...whatever the rule we have been given...we plug in whatever our input is,1106
whatever the thing on the left is, any of these numbers.1112
And then, once we figure out what this number is here, we figure out, we evaluate, and we get what its corresponding value is on the right side.1116
And we write that in, and that is how we make a table of values.1122
Having this table is often a very useful way to quickly analyze and see what is happening in a function over a large range of possible inputs.1126
Domain: the domain is the set of all inputs that the function can accept.1135
The domain is what can go into the function: it is the inputs that we are allowed to use.1141
It is what our machine can eat without breaking down.1146
Well, we generally assume that all of ℝ can be used as inputs--all of the real numbers can be used as inputs.1150
Sometimes, certain values will break our function; the output won't be able to be defined.1154
Thus, our domain is normally going to be all of the real numbers, except those numbers that break our function.1160
Occasionally, we might actually get things where we are going to be given an explicit domain--1165
like just evaluate it from -3 to 3--and forget everything beyond those -3 and 3 values.1169
But normally, we are going to assume all of ℝ, except those things that break our function.1175
Let's see an example: if we had f(x) = 1/x, the function would be defined, as long as we don't divide by 0.1179
If we have x = 0, though, then f(0) gets us 1/0.1188
Are we allowed to do that? No--that is very bad; we cannot divide by 0.1196
So, we are not defined there; everything else works, though.1202
If we plug in anything that isn't a 0, it works out fine.1206
So, everything is defined, as long as x is not 0; so our domain is all numbers, except 0.1208
The domain of f, to show all numbers except 0, is everything from -∞ up to 0, not including the 0,1214
and then union with everything from 0, not including the 0, to ∞.1221
That is just another way of expressing all of the real numbers, with the exception of 0.1225
Now, for now, we mostly only have to watch out for dividing by 0 and taking square roots of negative numbers.1230
Those are the only two things we have to worry about breaking functions.1235
However, you can't take the square root of a negative number, because what could you square that would still have a negative with it?1237
Any number, squared, becomes positive; so we can't have the square root of a negative number,1243
because it would be impossible to give me a number that you could square1247
into making it negative--at least as far as the real numbers are concerned.1250
Later on we will talk about the complex; but that is for later.1253
Right now, we only really have to worry about dividing by 0 and taking square roots of negative numbers.1257
Those are the things to watch for; that is where our domain will break down.1261
Later in the course, we will have a little bit more to worry about; we will also have to worry about inverse trigonometric functions.1264
Those are only defined over certain things; and also logarithms have some parts that they are not allowed to take, either.1269
But right now, it is just dividing by 0 and taking square roots of negative numbers.1274
And later on, much later in the course, after we see these ideas, we will have to think about them, as well,1277
when we are thinking about the idea of what can go into a function.1281
Domain is what goes in; range is what comes out.1284
Range is the set of all possible outputs a function can assign, given some domain.1289
With some domain to start with, these values are what is able to come out: the range is what can come out, given some domain.1294
These values will always be in the real numbers, unless we are dealing with a set that isn't working in the reals.1302
For example, that function we were working with before, f(x) = x2 + 3:1309
the lowest value that f can output is 3, because the smallest number we can make with x2...1313
well, x2 always has to be greater than or equal to 0, because there is no number1319
that we can plug into x and square that will cause it to become negative.1323
The lowest we can get that down to is a 0, so the lowest we can make this whole thing is when this is a 0, plus 3; so the lowest possible output is 3.1327
We can produce any value above 3 with x2, though, so we can just keep going up and up.1335
So, our range would be everything from 3, including 3, up until infinity.1340
So, it is all of the reals from 3, including 3, and higher; great.1344
If we want to look at an example that doesn't use numbers, we could talk about that initial function,1349
that function that ate names and gave out first initials, from earlier in this lesson.1353
In that case, if the domain is all names, then the range is all 26 letters of the Roman alphabet,1357
even though I still can't think of any names that start with a Q...Queen...let's say Queen counts.1362
OK, Queen Latifah, right?--it has to count; then, we can have that be the range--26 letters for the Roman alphabet.1372
So, because if we are looking at all the names that could possibly exist...1379
well, there is Albert; there is Bill; there is Charles; there is Doug; there is Elizabeth...and so on, and so on, and so forth.1383
So, there is always something that will put that out.1391
But if we restricted the domain to the five names that we saw earlier, Vincent, Nicole, Padma, Victor, and Takashi,1394
then we only had four letters show up--we just had N, P, T, and V show up.1400
So, in that case, if we restricted our domain to a smaller thing, our range would also shrink.1405
If we are looking at...normally we look at everything that can go into the function, and that is normally how we think of the domain.1412
But sometimes, we will be given a more restricted domain, and we have to think in terms of that more restricted domain.1420
First, there are nice, easy ones to get us warmed up to this idea of plugging in.1427
We just plug in...if we use red for this...f(2)...we plug in 3; plug in that 2; minus 7...3 times 2 equals 6; 6 minus 7...so we get -1.1434
Let's use blue for this one: if we have f(-4), then 3...we plug in that -4, minus 7.1447
Oh, no! What if we have to use something that is variable? No problem.1460
We still just follow the exact same rules: f(a)...well, what happened to x?1465
It became 3x - 7, so now it is going to become 3a - 7, so we get 3a - 7.1470
And what if we want to do b + 8? The same thing--f(b + 8) = 3(b + 8) - 7.1479
So, we have to distribute; and notice how important it was that we put it in parentheses.1490
If we had just plugged in this 3b + 8, that would be totally different than 3(b + 8).1495
And that is what it really has to be, because it is everything in here that got plugged in, not just the b.1501
The b and the 8 don't get to be separated now; they have to go in together.1506
So, 3(b + 8) - 7...we would get 3b + 24 - 7, which is equal to 3b + 17.1509
The next one: what if we wanted to fill in a table, g(z) = z2 - 2z + 3?1522
If we had to fill in this table, then we could do g(-1), (-1)2 - 2(-1) + 3,1527
equals 1 (-1 times -1 is 1), minus 2(-1), so plus 2, plus 3, equals 6; so we get 6 here.1539
Next, g(0): 02 - 2(0) + 3...that simplifies to just 3, because of the 0's; they disappear.1551
If we want to plug in g(1), then we get 12 - 2(1) + 3, so 1 - 2 + 3 comes out to 2.1564
We plug in g(2); we get 22 - 2(2) + 3 = 4 - 4 + 3, which is 3.1581
We plug in 10; we get 102 - 2(10) + 3; 102 is 100, minus 2 times 10 (is 20), plus 3 equals 83.1600
There we go: so you just plug into the function exactly as you would to set up this table.1621
You are told what your input is; and then, over on the right is your output, based on the rules of the function.1625
The function gives us some rules, and so we plug in inputs like -1, and -1 goes through: (-1)2 - 2(-1) + 3; we get 6.1634
And that is what is going on when we are making a table of values.1645
If h(x) = 2x2 + bx + 3, and we know that h(3) = 15, what is b?1649
So in this case, we are looking to figure out what b is.1655
Now, we know that h(3) = 15; so we need to somehow use this to figure out b.1660
So, we think, "I could plug in 3, and I would get something different than just 15."1667
So, h(3), based on the rule, is 2(32)...we are switching for where all of the x's show up;1672
x here; x here; that is it; so 2(32) + b(3) + 3...so we get 2(9) + 3b + 3, which is 18 + 3b + 3, or 3b + 21.1681
Now, at this point, we say, "Right; I also know that h(3) is 15; well, this is still h(3), right?"1707
So now, we put h(3) = 15, and we swap it out, and we get 15 must equal what we know h(3) is.1715
We know that h(3) is equal to 3b + 21; and we also know that h(3) is equal to 15.1725
So, since h(3) is two different things, but it is still just h(3), we know that they must be the same thing; otherwise there is no logic there, right?1730
So, 15 = 3b + 21; we subtract 21 from both sides; we get -6 = 3b; we divide by 3 on both sides, and we get -2 = b.1739
The next one: What is the domain and range of f(x) = 12 - √(x + 3)?1763
Now, remember: domain is what can go in; range is what can come out.1771
What we first one to do is figure out the domain first: what can go in without breaking this function?1794
So, is there anything that can break in this function?1801
We say, "Oh, right, the square root breaks when there is a negative inside."1803
We can't take the square root of -1, because there is no number that you can give me--1812
at least no real number that you can give me--that would square to give us -1.1816
You give me any positive number; it comes up positive; you give me any negative number; it comes out positive.1822
You give me 0; it comes out 0; so there is no number you can give me that will give out a negative number when squared.1825
So, square root breaks when we are trying to put a negative inside of it.1831
So, when will this break? √(x + 3) breaks when we have a negative inside.1835
So, when is (x + 3) going to be negative? when x is less than -3.1842
So, if x is less than -3, if x is more negative than -3, then this will be a negative value inside.1850
If, for example, we use -4, then we will get the square root of -1.1860
If we put in negative fifty billion, then we will get the square root of negative fifty billion plus 3, which would definitely still be negative.1863
So, it only stops being a negative inside when we actually get to -3.1869
-3 is an allowed value, because √(-3 + 3) would be √0; we do know the square root of 0--it is 0.1873
So, the domain works for -3 and higher; everything is still reasonable higher than that.1880
Our domain is going to include -3, and it is going to go for anything higher than that.1888
So, that is our domain; if we want to figure out what the range is, then the question is, "What can f(x) put out?"1895
So, notice: we have 12 - something; that something, √(x + 3)...square root can give out any number.1907
If you put in √0, √1, √4, √9...you are going 0, 1, 2, 3, and you can make any number in between that.1920
12 - something...what is the smallest that something could be?1928
The smallest number that that something could be is 0; so that is smallest when √0 = 0.1932
The biggest number we can get is 12; 12 is the highest number we can get, the largest number we can get out of this function.1946
What is the smallest number we can get? Well, you can just keep giving me larger and larger x1956
to make our square root a bigger and bigger negative number, on the whole.1961
It would be minus larger and larger numbers; so 12 minus larger and larger numbers...we can keep going down.1965
So, any number below 12 can be achieved, because we can just keep having the square root give out slightly larger and slightly larger numbers, which...1971
Since we are subtracting by these larger and larger numbers, we will keep going down.1982
So, any number below 12 can be achieved; so we have our range--it is going to be everything from the lowest possible,1986
all the way, anywhere up from negative infinity, up until 12.1997
Now, we ask ourselves, "Can we actually achieve 12?" Yes, we can.2002
We can actually get to 12, so we include 12; so our range is from negative infinity to 12--there is our answer.2006
The final one: we have a word problem: Give the area of a square, A, as a function of the square's perimeter, p.2013
And then, also say what is the domain of the area as a function of the perimeter.2020
First, as we talked about in the word problems, let's set up what our variables are.2025
Nicely, this problem already gave us our variables; but we will just remind ourselves: A is the area of the square, and p is the perimeter of the square.2030
So, it also probably wouldn't hurt to draw a picture, so we could see what is going on a little more easily.2051
We have a square here; here is our square, and we are talking about the area of it and the perimeter of it.2056
So, that is everything that we don't immediately know: we don't know the area; we don't know the perimeter.2065
They are going to be somehow connected, because we somehow want to be able to make a function out of area,2069
where we plug in the perimeter, and it gives out an area.2073
We basically want an equation that has area on the left, and then things involving perimeter.2076
We are solving for area in terms of perimeter; that is another way of looking at what this function is going to be.2081
We need some way to be able to connect these two ideas: how can we connect the area of a square to its perimeter?2086
Well, maybe we don't see a way right away; but let's just think, "Well, how do you find the area of a square?"2092
Well, it is its side times its side, its side squared; so the area of a square...2099
Now, we might as well go back, and we will set up a new variable--we didn't have that before--side of square.2105
A side of the square is a way to get our area; so area equals side squared.2114
Now, we still want some way to connect the area to the perimeter.2120
So, what we want is...well, we might not be able to correct them directly, but we have area connected to sides.2123
Maybe we can connect perimeter to side...oh, right, yes...if you have forgotten what a perimeter is, what do you do?2129
You just go and look it up: you have access to all sorts of information at your fingertips--it is so easy.2135
If you look up perimeter, thinking, "Oh, I have heard this before; I can't remember what it is,"2140
type it into an Internet search; the next thing you know, you will have a definition for what perimeter is.2144
So, perimeter is all of the sides added together; we have four sides, so perimeter is equal to side + side + side + side--2148
So now, we have a way of being able to have area talk to perimeter.2164
Area equals side squared; perimeter equals 4 times side; so perimeter/4 equals side.2168
Now, we can take this, and we can plug it in here.2178
Area equals...since we are plugging and we are substituting, we do it with parentheses...squared.2183
Area equals perimeter squared, over 16 (we have to square the top and the bottom).2190
And there we are--we can think of area equals perimeter squared over 16 as a function,2196
because it only depends on what we plug in for perimeter.2201
Area will vary as we put in different things; so we can think of it as a function acting on perimeter.2205
We plug in the number from perimeter, and it gives out what the area has to be.2210
So, we can just rewrite this as: area is a function of perimeter, or it is equal to the perimeter squared, over 16.2214
It is just a different way of thinking about it: we can think of it as an equation, or we can think of it as a function2223
where it just works the exact same way that the equation worked.2229
There is no functional difference between area of perimeter equals perimeter squared over 16--2233
the area based on a function using our perimeter equals p2/16--compared to area equals p2/16.2238
They have the same effect; it is just two slightly different ways of talking about it.2245
But in either case, it is plugging in a number for perimeter, then figuring out what the area has to be.2249
That is a function for area as a function of a square's perimeter.2255
Now, how can we get its domain? We talked about, before: the domain is everything that we can plug in without breaking it.2260
Now, that is mostly true; but there is one little thing here.2269
The domain also has to make sense; we can't break the world.2272
We wouldn't break this function...we could plug anything we want into this function.2278
You plug in any real number in for p, and it would make sense; we would get a number out of it.2284
You can plug in 50; you can plug in 0; you could plug in -10; it makes sense--we would get a number out of it.2289
But the domain has to make sense; it doesn't have to just make sense in our function--it has to make sense in how we have thought about the function.2294
How did we think about the function? It is a square, right?2307
It is a real object--it is a thing; we could talk about its shape and how its dimensions are.2311
Would it make sense for it to have a perimeter of a negative? No, because it doesn't make sense for the sides to be negative.2317
Would it make sense for the perimeter to be 0? No, because then it would just be a speck--it wouldn't be a square.2324
There would be no area possible to be contained inside, because we would have no side lengths if we had a perimeter of 0.2329
So, it must be the case that our p, perimeter, is allowed to vary only from 0 up to infinity,2335
because it can't have a domain below 0, and it can't have a domain of 0,2344
because while it doesn't break our function itself, it breaks the idea of what the function means.2348
It is meaningless to talk about plugging in a perimeter that is negative or a perimeter that is 0,2354
because then it is not the perimeter of anything; we don't actually have a shape there.2359
We have to be having our domain make sense, as well.2362
If we just have a function, it can't break the function.2365
But if we have the function in the context of a word problem, it also has to make sense with everything else happening in the word problem.2368
All right, I hope that all made sense, because that just laid an important groundwork.2374
You are going to need to know this for the rest of your time in math.2378
So, it is really great that we got this covered here.2381
Having a really strong understanding of what it means for something to be a function2382
is going to help you out in so many different places in math.2386
It is going to help you with all sorts of things--it is really great that we covered that here.2388 | {"url":"https://www.educator.com/mathematics/math-analysis/selhorst-jones/idea-of-a-function.php?ss=1522","timestamp":"2024-11-15T03:42:56Z","content_type":"application/xhtml+xml","content_length":"1049504","record_id":"<urn:uuid:67cea8fb-54bf-4656-ae44-962912b754bc>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00406.warc.gz"} |
Currency Exchange
Time Limit: 1 Second Memory Limit: 32768 KB
When Issac Bernand Miller takes a trip to another country, say to France, he exchanges his US dollars for French francs. The exchange rate is a real number such that when multiplied by the number of
dollars gives the number of francs. For example, if the exchange rate for US dollars to French francs is 4.81724, then 10 dollars is exchanged for 48.1724 francs. Of course, you can only get
hundredth of a franc, so the actual amount you get is rounded to the nearest hundredth. (We'll round .005 up to .01.) All exchanges of money between any two countries are rounded to the nearest
Sometimes Issac's trips take him to many countries and he exchanges money from one foreign country for that of another. When he finally arrives back home, he exchanges his money back for US dollars.
This has got Issac thinking about how much if his unspent US dollars is lost (or gained!) to these exchange rartes. You'll compute how much money Issac ends up with if he exchanges it many times.
You'll always start with US dollars and you'll always end with US dollars.
The first 5 lines of input will be the exchange rates between 5 countries, numbered 1 through 5. Line i will five the exchange rate from country i to each of the 5 countries. Thus the jth entry of
line i will give the exchange rate from the currency of country i to the currency of country j. the exchange rate form country i to itself will always be 1 and country 1 will be the US. Each of the
next lines will indicate a trip and be of the form
N c1 c2 cn m
Where 1 <= n <= 10 and c1..cn are integers from 2 through 5 indicating the order in which Issac visits the countries. (A value of n = 0 indicates end of input, in which case there will be no more
numbers on the line.) So, his trip will be 1 -> c1 -> c2 -> -> cn -> 1. the real number m will be the amount of US dollars at the start of the trip.
Each trip will generate one line of output giving the amount of US dollars upon his return home from the trip. The amount should be fiven to the nearest cent, and should be displayed in the usual
form with cents given to the right of the decimal point, as shown in the sample output. If the amount is less than one dollar, the output should have a zero in the dollars place.
This problem contains multiple test cases!
The first line of a multiple input is an integer N, then a blank line followed by N input blocks. Each input block is in the format indicated in the problem description. There is a blank line between
input blocks.
The output format consists of N output blocks. There is a blank line between output blocks.
Sample Input
1 1.57556 1.10521 0.691426 7.25005
0.634602 1 0.701196 0.43856 4.59847
0.904750 1.42647 1 0.625627 6.55957
1.44616 2.28059 1.59840 1 10.4843
0.137931 0.217555 0.152449 0.0953772 1
3 2 4 5 20.00
1 3 100.00
6 2 3 4 2 4 3 120.03
Sample Output
Source: East Central North America 2001, Practice | {"url":"https://sharecode.io/sections/problem/problemset/1019","timestamp":"2024-11-10T20:56:02Z","content_type":"application/xhtml+xml","content_length":"10016","record_id":"<urn:uuid:ae3c5a8d-e31d-4215-9632-37c91c39b091>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00657.warc.gz"} |
A Survey Study of the Effects of Preschool Teachers’ Beliefs and Self-Efficacy towards Mathematics Education and Their Demographic Features on 48 - 60-Month-Old Preschool Children’s Mathematic Skills
A Survey Study of the Effects of Preschool Teachers’ Beliefs and Self-Efficacy towards Mathematics Education and Their Demographic Features on 48 - 60-Month-Old Preschool Children’s Mathematic Skills
1. Introduction
Preschool children have informal mathematics knowledge and it must be accepted that conscious adults, peers, materials, interactions and conscious opportunities, can structure this knowledge. It is
true that children have natural curiosity and learning eagerness during preschool. Based on this truth, parents and educators can teach children mathematics concepts and skills with enjoyable and
exciting methods (Charlesworth & Lind, 2013: p. 6) .
In the studies conducted on the quality of education in pre-school period, one of the most emphasized issues is the teacher-student communication. In order to give a quality education, the teacher
must have an adaptive, creative, investigative, and flexible characteristic (Kandır, İnal, & Özbey, 2010) . For the teachers, being ready for the mathematics education is crucial and occasionally
difficult situation. Generally, to teach mathematics, one must have a strong mathematic and pedagogic background. NCTM (National Council of Teachers of Mathematics), in its Professional Standards for
Teaching Mathematics (1991), emphasizes that “the teacher must endeavor to make the concepts and principles of mathematics deeply understood and to provide that the subjects of mathematics or its
relations with other disciplines are formed”. In pre-school period, which is outstanding in raising the individuals and accepted as a critical period in one’s life, laying the foundations of the
academic skills, especially mathematics skills, is essential. Therefore, as it is well known, the person who will support the developing of these skills in those years is the pre-school teacher who
is taken as a role model after the child’s parents. It is thought that the ideas, beliefs and self-efficacy beliefs of the pre-school teachers on mathematics education affect the implementations in
the education process.
According to Copley (2004), early childhood teachers are generally described as lacking confidence in teaching math. Few studies have examined teacher confidence in relation to teaching tasks such as
planning learning activities or assessing children’s math understanding and Sarama and DiBase (2004) said young children’s performance in mathematics depends on their teachers’ mathematical
proficiency (Chen, Cray, Adams, & Leow, 2013) . Generally, survey studies are limited and based on self-report data and observation. So this study has been performed for examination of affect on
belief and self efficacy of preschool teachers about mathematics education on the math skills of 48 - 60 months old children.
2. Method
The study aims at researching the effects of preschool teachers’ beliefs and efficacy for mathematics education on mathematics skills of 48 - 60-month-old children. The study population is 48 - 60
months children who are attend to an independent National Education Ministry nursery school of Ankara Province, Cankaya Town in 2012-2013 education year. The study aimed to reach all of the
population. In the research, “The General Information Form”, “Beliefs Survey”, “TEMA-3 Early Mathematics Ability Test” and “The Self-Efficacy of Scale Pre-School Teachers for Mathematics Education”
were used in order to get information about children and their families. The study, when examined in terms of its aim, it is an example of survey research design. Survey research is a research
approach which aims at representing a situation which existed or still exists as it was or is (Karasar, 2008: p. 77) .
According to Creswell (2009) , survey research is the quantitative or numerical representation of the attitudes, opinions and tendencies of a group. According to Büyüköztürk and others (2010: p. 231)
, survey researches are the researches which determine participants’ thoughts related to the subject or the situation or their interests, talents, attitudes or similar features and which are usually
done on larger samples compared to other researches. “Beliefs Survey”, “TEMA 3 Test of Early Mathematics Ability” and “Self-efficacy Scale of Preschool Teachers towards Mathematics Education” were
used in the study.
2.1. Belief Scale of Preschool Teachers towards Mathematics Education
“Beliefs Survey of Preschool Teachers towards Mathematics Education”, developed by Platas (2008) and adapted by Şeker (2013) . Beliefs Survey of Preschool Teachers towards Mathematics Education was
developed by Platas (2008) to be used in her study of PhD thesis. It was adapted by Şeker (2013) within her doctorate thesis. In her study, the pre-application of the “Beliefs Survey” adaptation form
had been carried out with 255 pre-school teachers. Primarily, commentary factor analysis then confirmative factor analysis was accomplished with the guidance of the responses received from the
teachers. In order to carry out such analysis, firstly, the KMO value was calculated as 0.830. And belief scales’ lower dimensions are observed, it can be stated credible.The scale consists of 40
items and is a 6-point likert scale.
2.2. Test of Early Mathematics Ability (TEMA-3)
“TEMA 3 Test of Early Mathematics Ability”, developed by Ginsburg and Barody and adapted by Şeker in 2013. TEMA-3 was developed by Ginsburg and Baroody in 1983 to assess the mathematics skills of
children between 3 - 8. It was revised and in 1990 and published as TEMA-2. The validity and reliability tests of TEMA-2 in Turkey were done by Güven (1997) was proved to be a valid and reliable
scale. TEMA-2 test was revised later and was developed as TEMA-3 in 1993 (Ginsburg & Baroody, 2003) . The adaptation of TEMA-3 to 60 - 72 month-old children in Turkey was done by Erdoğan and Baran
(2006) and it was found to be a valid and reliable scale. It was adapted to 48 - 60-month-old children by Şeker (2013) and was proved to be a reliable and valid test.
2.3. Self-Efficacy Scale of Pre-School Teachers towards Mathematics Education
“Self-Efficacy Scale of Pre-School Teachers towards Mathematics Education” was developed to define the self- efficacies of the teachers working in pre-school institutions towards the teaching of
mathematics to pre-school period children. The scale was graded in 5 Likert-type items and the gradations are “Totally Agree―Agree― Neutral―Disagree―Totally Disagree”. There are 36 items in the
Self-Efficacy Scale and all of the items are positive.The reliability coefficient of the first dimension of the scale is 0.95; the reliability coefficient of the second dimension is 0.951; the
reliability coefficient of the overall scale is 0.967. In naming of the factors of the 2-factor self-efficacy scale, by taking the opinions of the field experts, the first factor that have 20 items
was named as self-efficacy towards preparing mathematics activities in pre-school period and the second factor including the other items was named as self-efficacy towards applying mathematics
activities in preschool period.
2.4. Data Collection
The study consists of 48 - 60-month-old children and their preschool teachers. There are 10 independent kindergartens of the state in Çankaya, Ankara. However, the study was done in eight independent
kindergarten, because it was not possible to do it in two schools.
In the study, the data was collected from 20 teachers and 371 children who are 48 - 60-month-old preschool children to determine the relationship between the preschool teachers’ level of belief and
self-efficacy and children’s mathematics skills.
The descriptive statistics of the data from the study (participant number, minimum, maximum, average, standard deviation) were firstly calculated. Then, the data was analyzed in the direction of the
problems of the study. In the analysis of the data, Mann Whitney U test and Kruskall Wallis test were used to determine if the self-efficacies of the teachers changed according to their demographical
features. As the number of the teachers was low, the scores of the teachers didn’t have a normal distribution, and consequently non-parametric statistics were used.
In unrelated measures, t test and one way analysis of variance statistics (ANOVA) were used to determine if children’s mathematics skills changed or not according to their demographic features.
Before the statistics were used, it was determined that the data had a normal distribution and the variances were homogeneous.
Hierarchical regression analysis was applied to examine the effect of preschool teachers’ beliefs and efficacies towards mathematics education on mathematics skills of 48 - 60-month-old children.
Hierarchical multiple regression analysis is a multivariate statistical analysis and before applying the regression analysis, firstly, the hypothesis of the analysis were examined. Accordingly, it
was checked if there was a missing data in the unit set. It was determined that the number of observation in the study was enough for multiple regression analysis. The scores related to the variables
were turned into z statistics, and Mahalonobis distance amount was also calculated. Univariate and multivariate extreme value was not found in the data set. Histogram graphics and coefficients of
skewness and kurtosis were examined to determine the normality distribution of the variables and the variables didn’t have a deviation more than normality. The relations among the variables were
examined firstly for the multicollinearity problem; it was found out that the coefficient of correlation was between 0.664 and −0.801. Moreover, it was determined that VIF value which was calculated
for multicollinearity was less than 10, and tolerance values were more than 0.10. The homogeneity of the variables was calculated with Box’s M test and it was found that homoscedasticity hypothesis
was proved. Finally, it was determined that the coefficient of Durbin Watson which was calculated to determine autocorrelation was 1.740. As this value is between 1.5 and 2.5, it is seen that there
is no autocorrelation among the variables.
In the hierarchical regression analysis, the demographical variables of the preschool children were categorical; categorical variables of gender, age gap, work status of mother, education level of
mother, education level of mother, number of siblings and previous experiences of preschool were assigned as “dummy”. In regression analysis which keeps categorical data, a new artificial variable is
created by leaving out one of the levels of categorical variables. It is produced as one level less than the level number (g^−^1) and is called “dummy” variable. It can be concluded that if one of
the new variables has a meaningful effect on the dependent variable, that means the related independent variable has meaningful effect on the dependent variable (Büyüköztürk, 2008) .
The findings from the analysis of the data were shown in tables and the results were interpreted.
3. Results
The findings of the study are given below in the direction of sub-goals.
1) What is the level of the self-efficacy of the participant preschool teachers towards mathematics education?
The teachers were applied a five graded scale which consisted of 36 items and two dimensions to determine the self-efficacies of preschool teachers towards mathematics education. Maximum, minimum,
average and standard deviation values which were calculated related to the scores of the teachers from the scale are shown in Table 1.
When Table 1 is examined, there are 20 items in the dimension of preparation of mathematics activities, first dimension of the self-efficacy scale of mathematics education and the minimum score to
get from this dimension 20 and the maximum is 100. The average of the scores of the self-efficacy of the preschool teachers towards the preparation of mathematics activities is 86. As the average
score is closer to the maximum score to get from the dimension, it was determined that the level of the self-efficacy of the preschool teachers towards the preparation of mathematics activities was
16 items scale were applied to the teachers to determine the self-efficacy of the teachers towards the implementation of the mathematics activities, which is the second dimension of the scale in
Table 1. The minimum score to get from the dimension is 16, and the maximum is 80. It was found that the minimum score of the self-efficacy of the teachers towards the implementation of the
mathematics activities is 48, and the maximum is 76. It was determined that the self-efficacy of the teachers towards the implementation of the mathematics score is 65.15. It was found that the level
of the self-efficacy of the teachers towards the implementation of the mathematics activities was also high.
When the data in Table 1 was examined, it was found that the teacher who got the minimum score of 20 participant teachers scored 116, and the teacher who got the maximum scored 176 from the
self-efficacy scale towards mathematics education (whole scale). The minimum score to get from the self-efficacy scale of the preschool teachers towards mathematics education is 36, and the maximum
is 180. The average score calculated in the direction of the answers of the teachers towards the mathematics education is 151.15. As the average of the self-efficacy of the teachers towards
mathematics education is closer to the maximum score to get from the scale, it was found that the level of the self-efficacy of the preschool teachers towards mathematics education was high.
It was determined that the level of the self-efficacy of the preschool teachers was high in both dimensions in the study.
Aksu (2008) made some analysis according to the differentiation of the primary school, science and preschool teacher candidates according to their beliefs of self-efficacy towards mathematics
education, gender, majors and
Table 1. Descriptive Statistics related to self-efficacies of the teachers towards mathematics education.
departments at high school. After the research of Aksu, it was determined that teacher candidates had high tendency in terms of the self-efficacy towards mathematics education and sub-dimensions of
handling. It was found there were no meaningful differences among teacher candidates in terms of their departments. The number of the studies which were done to measure the perception of the
self-efficacy of the preschool teachers towards mathematics education is really limited. Studies are generally about teacher candidates. For this reason, the present study has the findings to
contribute to the field.
When the data in Table 2 was examined, it was seen that the scores of the participant preschool teachers from the occupational self-efficacy scale had a positive and high-level relationship with the
sub-dimensions of the scale (r: 0.981; 0.940). Similarly, it was determined that there was also a high level relationship between the sub- dimensions (efficacy of preparing activities―efficacy of
implementing the activities) of the self-efficacy of the teachers towards mathematics education (r: 0.855). In the table, the relationship between the scores of the preschool teachers from the
Beliefs Survey towards mathematics education and the sub-dimensions of the scale is seen. The total point of the teachers from the Beliefs Survey has a positive and high-level relationship with the
appropriate age, occupational development and teacher sub-dimensions in mathematics education. However, the total point is in a negative and high-level relationship (−0.918) with teacher points in
terms of focus of knowledge.
This situation shows that teacher-centered education is adopted as the points from the dimension of focus of knowledge get high. In this respect, it is usual that there is a negative relationship
between teachers’ focus of knowledge points and other sub-dimensions. When the relationship between the scores of the children from TEMA-3 mathematics skills test in Table 2 and the teachers’ belief
and self-efficacy scores, it was determined that the relationship between the scores of the children from TEMA-3 and the scores of the self-efficacy of the teachers towards mathematics education is
positive (0.664).
It was found in the study that as the level of beliefs of the preschool teachers towards mathematics education increased, the children’s mathematics talents score increased as well. When the analysis
was examined, it was seen that in the focus of knowledge sub-dimension, as the teachers’ belief levels towards children centered approach in mathematics education increased, the children’s
mathematics skill scores went up. Moreover, when the teachers’ belief levels towards teacher-centered approach in mathematics education increased, the children’s mathematics skill scores decreased.
It is believed that the teachers’ belief levels affect both the expectations of the teachers from the children and the opportunities they provide to the children.
48 - 60-month-old children’s mathematics skills improve as their teachers’ self-efficacy levels increase. Additionally, the belief levels of the teachers towards mathematics education are directly
related to the mathematics skills of the children. The teachers’ beliefs towards mathematics education and their perception of the self-effi- cacy towards mathematics education directly affect the
teachers’ in-class practices. As a result, the children’s mathematics skills improve.
2) How much are 48 - 60-month-old children’s mathematics skills predicted by their preschool teachers’ beliefs towards mathematics education, teachers’ self-efficacy and some demographic features?
It was aimed to determine how much 48 - 60-month-old preschool children’s mathematics skills are predicted by their teachers’ beliefs and self-efficacies towards mathematics education, their
demographic features (seniority, period of work in the school, graduation departments) and children’s own demographic features (age gap, gender, work status of mother, education level of mother,
education level of father, number of siblings, experience of kindergarten). As the self-efficacy scale of teachers towards mathematics education and the Beliefs Survey of teachers towards mathematics
education have a high-level relationship, multicollinearity problem will occur in multiple regression analysis. For this reason, regression analysis was done with the inclusion of every dimension of
“Beliefs Survey” into the analysis.
In Table 3 are the results of the regression analysis which was done to determine how much 48 - 60 month- old preschool children’s mathematics skills are predicted by their teachers’ beliefs and
self-efficacies towards
Table 2. The relationship between the 48 - 60 year-old preschool children’s mathematics skills and their teachers’ level of belief towards mathematics education and level of self-efficacy.
Table 3. Multiple Regression Analysis (appropriate age sub-dimension) to explain the mathematics skills of preschool children (48 - 60 month-old).
Note: Model (R: 0.944; R^2: 0.891, F (15.370): 192.988, p = 0.000).
appropriate age in mathematics education, their demographic features (seniority, period of work in the school, graduation departments) and children’s own demographic features (age gap, gender, work
status of mother, education level of mother, education level of father, number of siblings, experience of kindergarten).
In the table, the variables of self-efficacy level of the teachers towards mathematics education, educational background (dummy 1) and number of siblings don’t explain the change in the mathematics
skills of children meaningfully. The variable that contributes most to the explanation is the level of belief of the teachers that the preschool children are at the appropriate age for mathematics
education. It was found that the variable which contributes least to the explanation is the work status of mother.
In Table 4 are the results of the regression analysis which was done to determine how much 48 - 60 month- old preschool children’s mathematics skills are predicted by their teachers’ beliefs and
self-efficacies towards focus of knowledge dimension in mathematics education, their demographic features (seniority, period of work in the school, graduation departments) and children’s own
demographic features (age gap, gender, work status of mother, education level of mother, education level of father, number of siblings, experience of kindergarten).
When the data in Table 4 was examined, it was seen that the belief dimension of the teachers towards focus of knowledge in mathematics education and 10 variables out of 14 in the regression analysis
predicted the mathematics skills of the children meaningfully.
Belief levels of the teachers towards focus of knowledge in mathematics education, the second dummy variable created for the age gap of the children, educational background of mothers, experience of
kindergarten, self- efficacy levels of the teachers towards mathematics education, first dummy variable created for year gap, gender, educational background of fathers, the third dummy variable
created for the educational background of teachers and work status of mothers explain 90% of the change in the level of mathematics skills of children.
The work period of the teachers of the children, first and second dummy variables created for their educational background, their seniority and number of siblings don’t predict the mathematics skills
of the children meaningfully.
The coefficient of the regression of the relationship between the belief level of the teachers towards focus of knowledge in mathematics education and the level of mathematics skills of the children
is negative and high- level (−0.929). It was found when the teachers were in the focus of knowledge in mathematics education, the mathematics skills level of the children decreased. When the belief
levels of the teachers towards children-cen- tered approach in mathematics education increased, the mathematics skills level of the children also improved. It was found that the children who were in
the class of the teachers who adopted children-centered approach had higher-level mathematics skills.
The self-efficacy level of the teachers towards mathematics education predict the change in the mathematics skills of the children meaningfully although it is different from the regression equalities
between the belief level of the teachers that the preschool children must be at the appropriate age for mathematics education. When the coefficient (−0.135) of regression of the teachers’
self-efficacy level was examined, it was determined that as the teachers’ level of self-efficacy towards mathematics education increased, the children’s mathematics skills level decreased.
In Table 5 are the results of the regression analysis which was done to determine how much 48 - 60-month- old preschool children’s mathematics skills are predicted by their teachers’ beliefs and
self-efficacies towards mathematical development as a goal of teachers in preschool education, their demographic features (seniority, period of work in the school, graduation departments) and
children’s own demographic features (age gap, gender, work status of mother, education level of mother, education level of father, number of siblings, experience of kindergarten).
Table 4. Multiple Regression Analysis (focus of knowledge sub-dimension) to explain the mathematics skills of preschool children (48 - 60-month-old).
Note: Model (R: 0.947; R^2: 0.897, F (15.370): 238.655, p = 0.000).
When the data in Table 5 was examined, it was seen that the belief dimension of the teachers towards mathematical development as a goal of preschool education and 10 variables out of 14 in regression
analysis predicted the mathematical skills of the children meaningfully.
Work period of the teachers of the children, first and second variables created for their educational background, their self-efficacy level towards mathematics education and the gender of the
children and number of their siblings don’t predict the mathematics skills of the children meaningfully.
In Table 6 are the results of the regression analysis which was done to determine how much 48 - 60-month- old preschool children’s mathematics skills are predicted by their teachers’ beliefs and
self-efficacies towards
Table 5. Multiple Regression Analysis (mathematical development sub-dimension as a goal of preschool education) to explain the mathematics skills of preschool children (48 - 60-month-old).
Note: Model (R: 0.939; R^2: 0.876, F (15.370): 167.291, p = 0.000).
Table 6. Multiple Regression Analysis (teacher sub-division in mathematics education) to explain the mathematics skills of preschool children (48 - 60-month-old).
Note: Model (R: 0.904; R^2: 0.818, F (15.370): 106.435, p = 0.000).
teacher in mathematics education of the children, their demographic features (seniority, period of work in the school, graduation departments) and children’s own demographic features (age gap,
gender, work status of mother, education level of mother, education level of father, number of siblings, experience of kindergarten).
When the data in Table 6 was examined, it was seen that the belief dimension of the teachers towards teacher in mathematics education and 8 variables out of 14 in regression analysis predicted the
mathematical skills of the children meaningfully.
Respectively, the belief levels of the teachers towards teacher in mathematics education, second dummy variable created for children’s age gaps, educational background of mothers, educational
background of fathers, experience of kindergarten, first and third variables created for the teachers’ educational background and work period of the teachers at the schools explain 82% of the change
in the level of mathematics skills of the children.
4. Discussion
In results, when the coefficient (−0.135) of regression of the teachers’ self-efficacy level was examined, it was determined that as the teachers’ level of self-efficacy towards mathematics education
increased, the children’s mathematics skills level decreased. The decrease of the mathematics skills level of the children of the teachers with high-level of self-efficacy can be explained with
teachers’ being less capable of practices, or teachers’ lack of knowledge about what kind of education should be given to which age group in mathematics education. Studies, which include the
comparison of mathematics practices and teachers’ perception of self-efficacy towards mathematics, must be done to make the correct explanation. For example, in their studies, Aslan, Bilaloğlu and
Aktaş Arnas (2006) made individual interviews with 22 preschool teachers at independent kindergartens about how often they include mathematics education in their daily schedules, which sources they
use for mathematics education, which methods they use and how they assess themselves in applying the methods. At the end of the studies, it was found after the observations that although most of the
teachers stated that they included mathematics activities in their daily schedules, only half of them used these activities. In such situations, observation method should be used to support the scale
results in the studies.
It was found in the study that as the level of beliefs of the preschool teachers towards mathematics education increased, the children’s mathematics talents score increased as well. In this study
teachers expressed belief that early math is necessary for 48 - 60 months old children. Chen, Cray, Adams and Leow (2013) , studied about teacher beliefs towards math and children. In that study, a
large majority of teachers expressed the belief that early math education is appropriate for young children.
5. Conclusion
After the analysis of math skills of 48 - 60 months children, faith and competencies of preschool teachers for mathematics education, and what extent predicted by demographic features of teachers and
children; it was determined that respectively the level of teachers’ beliefs for the appropriate age for mathematics education of the pre-school period was significantly predicted by age ranges of
children, mothers’ education level, previously attend pre-school education institution status, gender, father’s education level, teachers’ level of education, teachers working period in the
institution and mothers’ working status for the math skills of children. 48 - 60- month-old children’s mathematics skills improve as their teachers’ self-efficacy levels increase. Additionally, the
belief levels of the teachers towards mathematics education are directly related to the mathematics skills of the children. The teachers’ beliefs towards mathematics education and their perception of
the self-efficacy towards mathematics education directly affect the teachers’ in-class practices. In the light of the research results it can be said that teachers’ beliefs and perceptions of the
self-efficacy toward mathematics education are determinative for the math skills of the 48 - 60 months old children. | {"url":"https://scirp.org/journal/paperinformation?paperid=54847","timestamp":"2024-11-14T07:28:09Z","content_type":"application/xhtml+xml","content_length":"123733","record_id":"<urn:uuid:59e00ee7-a5f0-4e68-89f4-3cf90655d1a9>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00745.warc.gz"} |
recurrence relations a level maths
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Search: Recurrence Relation Solver Calculator. This study examines n-balls, n-simplices, and n-orthoplices in real dimensions using novel recurrence relations that remove the indefiniteness present
in known formulas.
Show answer. Share to Twitter Share to Facebook Share to Pinterest. These may be used for both teaching and for student revision. They show that in the negative, integer dimensions, the volumes of
n-balls are zero if n is even, positive if n = −4k − 1, and negative if n = −4k − 3, for natural k. The volumes and View Notes - RecurrenceRelations from APPLIED MA 171 at Johns Hopkins University.
Recurrence Relation Definition 1 (Recurrence Relation) Let a 0, a 1, . 4.6 Modelling with Sequences & Series. D2-03 Sequences: Finding the First Five Terms of an Inductive Definition. xn= f (n,xn-1)
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began on 15th April 2013. A collection of videos, activities and worksheets that are suitable for A Level Maths. One way to solve some recurrence relations is by iteration, i.e., by using the
recurrence repeatedly until obtaining a explicit close-form formula. a) If u3 =10 , find the possible values of k. b) Determine the value of u4, given that k > 0. Recurrence Relations. Designed by
expert SAVE MY EXAMS teachers for the Edexcel A Level Maths: Pure exam.
Pure. Both arithmetic sequences and geometric sequences can be defined using recurrence relations. A recurrence relation describes each term in a sequence as a function of the previous term ie un+1 =
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Linear Recurrence Relations Recurrence relations Initial values Solutions F n = F n-1 + F n-2 a 1 = a 2 = 1 Fibonacci number F n = F n-1 + F n-2 a 1 = 1, a 2 = 3 Lucas Number F n = F n-2 + F n-3 a 1
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previous one, giving the sequence 4,6,8,10,12,14, . We can then use these relationships to evaluate integrals where we are given a deterministic value of .
A-Level Maths.
A few little tasks to supplement the teaching of recurrence relations. T(n) = 2T(n/2) + cn T(n) = 2T(n/2) + n These types of recurrence relations can be easily solved using Master Method. Edexcel
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be solved using RSolve [ eqn, a [ n ], n ] Use the generating function to solve the recurrence relation ax = 7ax-1, for k = 1,2,3, with the initial Solution 2) We will first write down the recurrence
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Maths Genie - AS and A Level Maths revision page including revision videos, exam questions and model solutions. A Level Learn A Level Maths Edexcel A Level Papers Old Spec A Level. 5 Given the
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Designed by expert SAVE MY EXAMS teachers for the Edexcel A Level Maths: Pure exam. Any student caught using an unapproved electronic device during a quiz, test, or the final exam will receive a
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[MUSIC] Hi, and welcome back to Introduction to Enumerative Combinatorics In mathematics, a recurrence relation is an equation that recursively defines a sequence or multidimensional array of values,
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Example2: The Fibonacci sequence is defined by the recurrence relation a r = a r-2 + a r-1, r2,with the initial conditions a 0 =1 and a 1 =1. In math, the relation is between the x -values and y
-values of ordered pairs. D: Further Algebra & Functions. Legacy A-Level Maths 2004 Legacy GCSE Maths Foundation. This connection can be used to find next/previous terms, missing coefficients and The
SEAB syllabus simple states: sequence generated by a simple recurrence relation of the form X (n+1)=f (Xn), which in itself is rather vague. The second step is to use this information to obtain a
more e cient method then the third step is to apply these ideas to a second order linear recurrence relation. examsolutions. A Level Maths. Example 2.4.2 . Past Papers. Recurrence Relations
Arithmetic Sequences and Series Geometric Sequences and Series The Binomial Expansion: Solutions Solutions Solutions Solutions: Trigonometry: Videos: D2-02 Sequences: Inductive Definitions and
Recurrence Relations. Then we write: U n = 2 (U n-1 ). Search: Recurrence Relation Solver Calculator.
G: Polar Coordinates. Topics introduced to Edexcel AS/A level Further Mathematics in 2017, such as linear congruences, Fermats little theorem and recurrence relations, are all covered. Substituting n
= 0 into the recurrence relation: u 1 = k u 0 + 4 7 = k ( 1) + 4 7 = k + 4 3 = k k = 3. Calculator help - recurrence relation (a level maths) Extra Pure Recurrance relations Higher Maths Question
Year 2 Pure Maths - Mixed exercise 3 Q4c Higher Maths Sequences show 10 more Quick maths help!! There are essentially two kinds of recurrence relation (s) problems. new kind of mathematics was not
conducive to formalism, that is to say, methods of calculation a competent grasp at an introductory level. Geometric Sequences and Series + Recurrence Relations 2: Worked Solutions: Arithmetic +
Recurrence Relations: Worked Solutions: Nth Term + Recurrence Relations: Worked Solutions: Mixed Topics 1: Worked Solutions:
Let us assume x n is the nth term of the series. 7 /month.
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields.
OCR MEI A Level Maths B Spec at a glance. As you can see, the next term in a sequence is a function of the previous term. Search: Recurrence Relation Solver Calculator. A recurrence relation for the
n-th term a n is a formula (i.e., function) giving a n in terms of some or all previous terms (i.e., a 0;a 1;:::;a n 1). We can also define a recurrence relation as an expression that represents each
element of a series as a function of the preceding ones.
Each term in the sequence is got by doubling the previous term. A recurrence relation is an equation which expresses any term in the sequence as a function of some number of terms that preceded it:
$$x_n=f (x_ {n Topic Questions. Search: Recurrence Relation Solver Calculator. Recurrence Relation Formula. 4 use a recurrence relation to model a reducing balance loan and investigate (numerically
or graphically) the effect of the interest rate and repayment amount on the time taken to repay the loan 4 Solve the recurrence relation Weve seen this equation in the chapter on the Golden Ratio It
is the famous AS/A Level Maths Checklist For the new linear A Level Maths students are tasked with revising all of the content across a 2 year course (1 year for AS).
The final task is about investigating the limits. Recurrence relations are sometimes called difference equations since they can describe the difference between terms and this highlights the relation
to differential equations further. Home > A-Level Maths > Teaching Order Year 2 > 209-210,213-214: Sequences and Series > b. Inductive Definitions & Recurrence Relations. For recurrence relation T(n)
= 2T(n/2) + cn, the values of a = 2, b = 2 and k =1. We won't
Check that \(a_n = 2^n + 1\) is a solution to the recurrence relation \(a_n = 2a_{n-1} - 1\) with \(a_1 = 3\text{. Revision Notes. MEI AS Level Further Mathematics Syllabus 3 papers: Paper 1 1/3 of A
level 75 minutes 60 marks Examined on Core Pure Mathematics content Paper 2 1/3 of A level 75 minutes 60 marks Examined on first optional module Paper 3 3/11 of A level 75 minutes 60 marks Examined
on second optional module In this video you are shown what a sequence is and how to define a recurrence relationship for the terms in the sequence. Arithmetic can be defined by. Gold Annual. Let us
assume x n is the nth term of the series. x The iiial c dii for a sequence specify the terms that precede the first term where the recurrence relation takes effect. To get a feel for the recurrence
relation, write out the first few terms of the sequence: 4, 5, 7, 10, 14, 19, . Recall that u n is the n th term in a given sequence. Provide step by step solutions of your problems using online
calculators (online solvers) Topics include set theory, equivalence relations, congruence relations, graph and tree theory, combinatories, logic, and recurrence relations 4: Solving Recurrence
Relations Solving homogeneous and non-homogeneous recurrence
A pair of Information This notice covers all examined components. Then the recurrence relation is shown in the form of; xn + 1 = f (xn) ; n>0. 2022 Exams Advance Information. The set of all x
Inductive Definitions & Recurrence Relations. Sequence and series: finding u1 and u2? Video Questions . The recurrence relationship of a sequence is given by u_ (n+1) = 100 u_n, where u = 18. Check
the lecture calendar for links to all slides and ink used in class, as well as readings for each topic For example, consider the probability of an offspring from the generation Now that we know the
three cases of Master Theorem, let us practice one recurrence for each of the three cases Recurrence relation-> T(n)=T(n/2)+1 Binary This just means that the nth term, U n is equal to 2 the (n-1)th
term, U n-1. 5 1 review. Type 1: Divide and conquer recurrence relations Following are some of the examples of recurrence relations based on divide and conquer.
In recurrence relations questions, we generally want to find (the power of the integral) and express it in terms of its powers of the integral . Sequence and series: finding u1 and u2? Video
Age range: 16+ Resource type: Assessment and revision.
A recurrence relation defines each term of a sequence using preceding term(s), and always state the initial term of the sequence. Trigonometry. Examples, solutions, videos, activities and worksheets
that are suitable for A Level Maths to help students learn about recurrence relations. 4.6.1 Modelling with Sequences & Series. This connection can be used to find next/previous terms, missing
coefficients and its limit. Therefore, our recurrence relation will be a = 3a + 2 and the initial condition will be a = 1. Number of possible Equivalence Relations on a finite set. Factors, multiples
and primes Tag: Recurrence relations. Recurrence Relations 1 (2000 - 2019) Answers Included on Worksheet: Higher Exam Worksheet 42: Recurrence Relations 2 (2000 - 2013) For answers refer to Marking
Schemes on SQA Past Paper Link HERE: Higher Exam Worksheet 43: Recurrence Relations 3 (Old Higher) Answers Included on Worksheet: Higher Exam Worksheet 44: Recurrence Relations 4: Answers It has to
be that way, Second order recurrence relations, 10.1, 10.2. Different types of recurrence relations and their solutions. . }\)
Example 2.4.3. AS/A Level Mathematics Recurrence Relations Instructions Use black ink or ball-point pen. FREE Maths revision notes on the topic: Integration. Model Answers. Search: Recurrence
Relation Solver Calculator. Recurrence relation captures the dependence of a term to its preceding terms. x A sequence is called a l i of a recurrence relation if its terms satisfy the recurrence
A sequence is generated by the recurrence relation u n + 1 = k u n + 4, where k is a constant. In solving the rst order homogeneous recurrence linear relation xn = axn1; it is clear that the general
solution is xn = anx0: This means that xn = an is a solution.
Solution. Recurrence Relations - Back to MTH 288 - Show content outlines for all MTH courses. Powered by https://www.numerise.com/This video is a tutorial on Proof by Induction (Recurrence Relations)
for Further Maths 1 A-Level.
5. So to define the recurrence relation, we give the first term, written U 1 = 2. Recurrence relations. Recurrence Relation Formula. A Level Further Mathematics A H245 We have produced this advance
information to support teachers and students with revision for the Summer 2022 examinations. We can also define a recurrence relation as an expression that represents each element of a series as a
function of the preceding ones. A Level Maths Notes Notes, exam paper solutions and study tips for studying A Level Mathematics. Sequences : Recurrence Relations : ExamSolutions : A-level Maths. Two
techniques to solve a recurrence relation Putting everything together, the general solution to the recurrence relation is T (n) = T 0 (n) + T 1 (n) = an 3 2-n The specific solution when T (1) = 1 is
T (n) = 2 n 3 2-n And so a particular solution is to plus three times negative one to the end Plug in your data to calculate the recurrence interval T(n) = aT(n/b) + f(n), T(n) = aT(n/b) + f(n),. A
Level Maths revision tutorial video.For the full list of videos and more revision resources visit www.mathsgenie.co.uk. CBSE Class 10 Maths Board Exam 2018: Important 3 Marks Questions: Triangles
Class 10: Alternative Hypothesis: | {"url":"http://www.stickycompany.com/safesplash/92010608c4351f56-recurrence-relations-a-level-maths","timestamp":"2024-11-10T09:09:03Z","content_type":"text/html","content_length":"30644","record_id":"<urn:uuid:c246a1d7-ab91-4ac9-8cc8-bb9b22f83ac7>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00892.warc.gz"} |
How to Use INDEX and MATCH Function in Excel - Alternative of VLOOKUP - Ali Chaudary - Learn Advance Excel | Guide for Beginners
How to Use INDEX and MATCH Function in Excel – Alternative of VLOOKUP
As mentioned in title, In this article I will be explaining Excel Index and Match Function. You can use these two function and perform multiple tasks by combining them. These functions can do more
advance lookups than a VLOOKUP function which you might have used before. You can perform vertical/horizontal lookups, case sensitive lookups, left lookups and based on criteria lookups as well. So
first I will explain each function separately with examples which I believe would be easy for you to understand and use them by combining.
INDEX Function
Index function is a very flexible and powerful especially when it comes to advance excel formulas. It actually perform a lookups in given data range and brings a value accordingly. For instance, you
have table as shown in below image where you want to get the 4th product code which is “PC1004”. To get the 4th product code, you can use Index function as shown below:
It returned the value of 4th row
Now let’s say you need to know the product name as per product code. In this case, you need to select whole table, specify the row number as well as column number. As you can see below, I selected
the whole table as an array, then 4 is row number and 2 is column number. Once this formula is applied, it will bring the value located in 4 row of 2nd column.
So this is how an Index function works. I know that at this stage, you must be thinking that what if we have a huge number of data? How we are going to find the position of the value? Is there any
function to know the values position automatically rather than finding manually?
Of course yes, Excel has a Match function which can be used to locate the values position. Let me first explain the Match function and see how it works.
MATCH Function
The main and only usage of excel Match function is to locate the values position in a data sheet. It can also be combined with other function to do an advance lookups such as VLOOKUP, INDEX,
functions. Let’s say you want to know the position of Product “HDMI Cable” from the below table. All you need to specify the lookup value, lookup array and lookup type. You will type HDMI Cable as a
lookup value, select the lookup array and type 0 for the exact match type as shown below:
Match function can also be used to perform a horizontal lookups as you can see below.
So this is how you can use match function know the position of any value. The last argument ” match type” is important to show the result if you want an exact match so make sure to type 0 if you want
an exact match. You can learn more about Match function arguments from Microsoft Support.
Use INDEX & MATCH Function in Excel
As you have learned above how to use Index & Match function separately. Now it will be easy for you to use both functions together. You can use combination of both functions to perform an advance
lookups. Let’s go through this function with below examples as it will help you to understand easily.
Let’ say you have a product code and want to lookup for product name. You can use Index & Match function to get the product name. All you need to enter the formula as shown below:
As you can see above that we have used this formula =INDEX(B5:E12,MATCH(B2,B5:B12,0),2).
=INDEX(B5:E12 = Array
MATCH(B2,B5:B12,0) = We used match function to know the row number of lookup value
2) = Column Number
Lookup Horizontally With INDEX & MATCH Function
You can use Index & Match function to do horizontally lookups as well. Let’s say you have same table in horizontal position where you want to get the same product name associated with product code.
All you need to modify the formula as shown in below screen.
As you can see above that we have used this formula =INDEX(C4:J7,2,MATCH(B2,C4:J4,0))
=INDEX(C4:J7 = Array
2 = Row Number
MATCH(B2,C4:J4,0) = We used match function to know the column number of lookup value
Hope the above explanation will help you to understand the usage of Index & Match function. If you still have any question regarding the same, feel free to leave a comment below. Thank you.
You might be interested to learn more: | {"url":"https://alichaudary.com/how-to-use-index-and-match-function-in-excel-alternative-of-vlookup/","timestamp":"2024-11-13T18:30:55Z","content_type":"text/html","content_length":"82357","record_id":"<urn:uuid:540df73f-c03e-4069-8799-d53e5e706a62>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00521.warc.gz"} |
Converting an Integer to a String in Any Base
Note 16.4.3. Self Check.
Write a function that takes a string as a parameter and returns a new string that is the reverse of the old string.
Write a function that takes a string as a parameter and returns True if the string is a palindrome, False otherwise. Remember that a string is a palindrome if it is spelled the same both forward and
backward. for example: radar is a palindrome. for bonus points palindromes can also be phrases, but you need to remove the spaces and punctuation before checking. for example: madam i’m adam is a
palindrome. Other fun palindromes include:
• kayak
• aibohphobia
• Live not on evil
• Reviled did I live, said I, as evil I did deliver
• Go hang a salami; I’m a lasagna hog.
• Able was I ere I saw Elba
• Kanakanak – a town in Alaska
• Wassamassaw – a town in South Dakota | {"url":"https://runestone.academy/ns/books/published/httlacs/intro-recursion_converting-an-integer-to-a-string-in-any-base.html","timestamp":"2024-11-14T10:56:07Z","content_type":"text/html","content_length":"128468","record_id":"<urn:uuid:6597cac5-7fa8-4acf-bd06-e36c51398aeb>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00320.warc.gz"} |
Andrea Palladio's Villa Cornaro in Piombino Dese by Branko Mitrovic in the Nexus Network Journal vol. 6 no. 2 (Autumn 2004)
Andrea Palladio's Villa Cornaro in Piombino Dese
Branko Mitrovic
School of Architecture
UNITEC Institute of Technology
Carrington Rd.
Auckland NEW ZEALAND
Villa Cornaro in Piombino Dese is one of Andrea Palladio's most influential works (Figs. 1 and 2).[1] The villa is probably the earliest of his designs to incorporate a pedimented portico separated
from the main block of the building -- a paradigm whose invention is often associated with Palladio and which has had a huge impact on world architecture for the past four centuries.
Fig. 1. Villa Cornaro in Piombino Dese, rear. (photo/author) Fig. 2.Villa Cornaro in Piombino Dese, front. (photo/author)
The villa was designed in the winter of 1551-1552 and its main block with the exception of the side wings was inhabited by 1554 [Lewis 1972; 1975]. The side wings were completed only in 1596 by
Vincenzo Scamozzi, although they appear in Palladio's presentation of the villa in his treatise, The Four Books on Architecture (Fig. 3) [Palladio 1990, 1997]. Douglas Lewis, who has done
considerable research on the villa's building history, has managed to find documentation which indicates that the central block of the villa was constructed under close supervision of the architect
-- a fact which makes the villa a particularly important piece of evidence for the study of Palladio's design theory [Lewis 1972, 384-385]. The early 1550s, when the villa was designed and built,
were a turning point in Palladio's approach to design. Through the 1540s most of his works were villas for Vicentine nobles, in which he generally avoided the use of the classical orders or used them
unsystematically. But shortly before the Cornaro project, in the late 1540s and early 1550s, while working on the Basilica and the Palazzo Chiericati, Palladio started using the orders not only as
façade ornamentation but as the organizing principle of the entire spatial composition of the buildings he designed. Villa Cornaro is thus among the first buildings whose design was derived from an
approach which emerges after 1550 and derives from a set of complex mathematical considerations. My analysis here will be based on a recent survey of the villa made by Steve Wassell, Tim Ross,
Melanie Burke and myself in June 2003.
The Four Books on Architecture, Palladio's architectural treatise, which came out in 1570, almost twenty years after Villa Cornaro was designed, is still the most important source for the study of
the Vicentine architect's design theory. This may seem paradoxical, considering that Palladio was the most prolific of all great Renaissance architects, and that a great number of the buildings he
designed still stand. However, for many of these buildings modern surveys do not exist, are incomplete, omit information about important aspects such as the use of the classical orders, or have been
published without dimensions indicated in the plans. The most comprehensive -- and I would also argue the most reliable -- publicly available set of surveys even today is the one published by Ottavio
Bertotti-Scamozzi in the eighteenth century [1968, 1998]. Bertotti-Scamozzi's surveys cover all -- or almost all -- of Palladio's known opus. Insofar as I have been able to check and compare them
with modern surveys, the data he provided tend to be reasonably accurate [Mitrovic 2004, 194-197]. He did, however, have a passion for presenting Palladio's unfinished works as if they had been
completed -- and was even sometimes prone to invent the manner in which they should have been completed. When working with Ottavio Bertotti-Scamozzi's surveys it is always necessary to separate the
products of his imagination from the segments of Palladio's works which were really built -- but this can be done, and once it is done his surveys become a reliable tool.[2]
In his treatise, Palladio listed his preferred room types: circular, square or rectangular with length-to-width ratios 2/1, 3/2, 4/3, 5/3 or Ö2/1 [Palladio 1997, 1.52]. This list is commonly referred
to as the list of Palladio's preferred room length/width ratios. Its interpretation and implications have been in the center of debates within Palladian scholarship for the past 50 years.[3] In the
second book of his treatise Palladio presented plans of forty-four buildings he designed; in these plans, room length-to-width ratios have been indicated for 153 rooms.[4] Eighty-nine of these 153
ratios -- or 55% -- indeed correspond to the ratios from Palladio's list. An analysis of the remaining 45% shows that some other proportional systems were used by the architect as well. The ratio Ö3/
1 appears in a number of plans -- most prominently in the plan of the Rotonda -- as well as ratios such as Ö3/1, Ö2/1, can be seen as cube-derived. Ö2/1 is the diagonal-to-side ratio of a square, Ö3/
1 is diagonal-to-side ratio of a cube. a is a side of a given cube, Ö3 based system, but, if the available surveys are of any value, the proportions of the Rotonda as executed correspond to those of
Delian cubes. At the level of speculation one might even argue that the Rotonda was built as an altar to Apollo.
Half a century ago, this kind of speculative search for the comprehensive interpretation of Palladio's proportional system received great impetus from Rudolf Wittkower's Architectural Principles in
the Age of Humanism -- arguably the most influential twentieth-century book on Renaissance architectural theory.[5] Wittkower suggested that Palladio's choice of length/width ratios was derived from
musical theory. He referred to the fact that ratios of certain musical intervals correspond to numerical relationships between the lengths of strings on a monochord. For instance, the ratio 2/1 is
the octave, 3/2 is the fifth, 4/3 is the fourth, and so on. Traditionally, the discovery of relationship between individual numerical relationships and musical intervals was ascribed to Pythagoras;
later in classical antiquity this led to the development of an extensive system of speculations which some historians have named "the Great Theory". The concepts musica mundana and harmonia mundi
relied on the assumption that the same relationships which determine musical intervals also determine the movements of stars and, through astrological influences, affect the events on Earth. This
kind of belief was widespread through the Middle Ages and the Renaissance. Leon Battista Alberti explained the beauty of certain proportional relationships between the lengths and widths of rooms by
relating these ratios to those of musical theory [Alberti 1966, 1988: Bk. IX, ch. 5]. The great merit of Wittkower's book was that it described the impact of this kind of belief on Renaissance
architectural theory. In his book Wittkower pointed out a number of Renaissance sources which made similar references directly or indirectly. In his commentary on Vitruvius, Daniele Barbaro stated
several times that those ratios which are pleasant to the ears also delight the eyes [Barbaro 1987, 124; 244; 282]. Palladio himself, although he did not discuss this kind of belief in his treatise,
referred indirectly to it in a memorandum pertaining to the Cathedral of Brescia [Palladio 1988, 123].
It is, however, important to differentiate between the derivation of certain proportional rules and their explanation. In the case of Palladio and Barbaro, their statements did not refer to musical
proportions in order to deduce which proportions should be used, but only in order to explain an already existing practice. When Wittkower emphasized the importance of the narrative about harmonic
proportions for Palladio's architectural theory, he adopted the case-study method. In his book he analyzed only eight out of forty-four Palladio's buildings presented in the Four Books. These were
the buildings which indeed best suited his interpretation [Howard and Longair 1982]. But if we look at the larger picture, Wittkower's interpretation can hardly explain Palladio's design process any
better then the claim that Palladio simply used ratios from his list of preferred ratios. Out of 153 room length/width ratios from the building plans presented in the second book of Palladio's
treatise, ninety-seven can be interpreted as ratios which correspond to musical ratios according to Wittkower's theory; we have seen that this same number is eighty-nine when it comes to the ratios
from Palladio's list of preferred ratios. Also, the ratios which Wittkower's theory can explain are more or less the same as those from the list of preferred ratios: only one ratio from this list, Ö2
/1 cannot be explained as harmonic. At the same time, other ratios we have seen that Palladio used, such as Ö3/1, cannot have a harmonic explanation. Also, room length-to-width ratios are only room
length-to-width ratios. The method by which Palladio decided about them cannot be taken for the only, or even the most important, part of his design procedures. A Renaissance architect would have
many other design problems to resolve -- such as the composition of the façade, the use of the orders, mutual volumetric correlation of internal spaces, and so on. For instance, if we look at the
canon of the five orders that Palladio presented in the first of his Four Books, we shall see that in some cases he adopted ratios for the individual elements from the Vitruvian tradition, but in
other cases he had to formulate his own proportions for an element [Mitrovic 2002]. The most significant element of the orders for which Palladio had to formulate his own proportions was the
Corinthian entablature. A systematic comparison of all the proportional relationships on the Corinthian entablature shows that Palladio did not use harmonic proportions in determining its ratios
[Mitrovic 2001].
Wittkower's was also an ideological position -- something we must never forget: by emphasizing the importance of the proportional relationships between room lengths and widths, he actually asserted
that the use of ornamentation -- and especially the orders -- did not matter in Palladio's design process. This interpretation of Palladio supported the Modernist approach to design precisely in the
years when the Modernist movement needed it the most, and, as I have argued elsewhere, it coincided with the commercial interests of architectural profession in the 1950s, which substantially
contributed to the popularity of Wittkower's book. At the same time, even if Wittkower's interpretation were true, it really explains only Palladio's design procedures when it comes to the
proportioning of individual rooms.
In other words, the question of whether Wittkower was right or wrong is ultimately an ephemeral one. Even if he was right, his approach accounts only for a minor segment of the design problems
Palladio had to resolve in his work. A comprehensive proportional analysis of a Palladian villa must take much more into account. In his design work, Palladio had many other design problems to
resolve besides the length-to-width ratios of individual rooms. Wittkower's theory did not even attempt to explain the totality of proportional relationships between room dimensions, such as the
determination of room heights and mutual proportional correlation of individual rooms. Palladio said that the heights of rooms should be the arithmetical, geometrical, or harmonic means of the height
and width, if the room is vaulted. If the room is square, its height should be 4/3 of the width, and if the ceiling is flat, the height should equal the width of the room. In Palladio's time,
ground-floor rooms -- the level at which a villa or a palazzo is entered -- would typically have vaulted ceilings, whereas upper storeys would be covered with wooden beams and have flat ceilings.
Palladio's plans usually consist of rows of rooms surrounding a sala (in the case of a villa) or a central courtyard (in the case of a palazzo). If we look at the plans presented in the Four Books,
there are very few plans in which all room dimensions are different: usually a dimension of one room is repeated as the length or width of another room in the same row. Two neighboring rooms normally
have either the same length, or the same width or the length of one room is the width of another. In the Four Books Palladio mentioned the requirement that rooms in the same row should have equal
heights and that consequently their proportions must be carefully coordinated [Palladio 1997, 1.54]. It is thus necessary to select such length-to-width ratios that when we calculate the heights of
rooms as arithmetic, geometric, or harmonic means of different lengths and widths, the resulting room heights are all equal. This rule can be called the "condition of concordance of heights", or CCH
rule. It substantially delimits the possible proportional relationships between rooms in Palladio's designs. If we assume that rooms in the same row have the same widths, and calculate room heights
as the arithmetic, geometric, or harmonic means of room lengths and widths, we shall be able to conclude that coordination of room heights is possible if the height/width ratios are 5/4 and 4/3. If
the height/width ratio is 5/4, then it will be possible to have a room with a length/width ratio 5/3 next to a room with a length/width ratio 3/2. The requirements for the CCH rule will be fulfilled
and both rooms will have the same height if the height of the former room is the harmonic and the latter the arithmetic mean of length and width. If the height/width ratio is 4/3, the same with be
possible for rooms with ratios 2/1 and 5/3. The height of the former would have to be calculated as the harmonic and the latter as the arithmetic mean of length and width. Also, a square room whose
width equals the width of these rooms can be placed next to them, since Palladio said that height/width ratio of square rooms should be 4/3. Finally, a room with the length/width ratio Ö3/1 can be
also have height/width ratios very close to 4/3 -- in the case they are calculated as arithmetic (1.367) or geometric (1.316) means. In Palladio's villa plans indeed one rarely encounters the
situation that more than three rooms have been aligned in the same row. Very often, the third room is much smaller than the other two, has a mezzanine above, and a reduced height.
An analysis of Palladio's works based on the proportions of rooms, is particularly applicable to his villas of the 1540s. These villas do not have orders applied to the façade. In this group belong
Villa Godi in Lonedo, Villa Poiana in Poiana Maggore, Villa Saraceno in Finale, Villa Caldogno in Caldogno, and so on. But from the late 1540s and early 1550s Palladio started systematically using
the orders in his designs and -- together with Palazzo Chiericati, Villa Pisani in Montagnana and the Basilica -- Villa Cornaro was one of first major works in which this new approach was manifested.
It was in the decades of the 1550s and 1560s that Palladio and Vignola introduced a revolution in the Renaissance use of the classical orders. The standard use of the orders through the Renaissance
before Palladio and Vignola implied that columns or pilasters were placed at regular distances and in those positions where internal walls cut the façade. Through the Renaissance it was common to
disregard Vitruvius's advice that intercolumniations should not exceed three lower column diameters. In his treatise Vitruvius listed different intercolumniation types: 3 diameters for diastilos,
2.25 diameters for eustilos, 2 diameters for eustylos and 1.25 diameters for picnostylos. Typically, the Renaissance use of the orders before the 1550s relied on the placement of columns in the
corners and at places where the internal walls cut the façade. Additional columns or pilasters would be added in order to ensure equal intercolumniations. The approach was regularly combined with
disregard of Vitruvius's precepts for intercolumniations. The columns, pilasters and entablatures would completely frame the facade, but the wide intercolumniations result in a visually unpleasant
span between columns or pilasters. The intercolumniations on the contemporary works the young Palladio could have seen near Vicenza by far exceeded Vitruvius's recommendations.
Intercolumniation-to-diameter ratio on Giovanni Maria Falconetto's Porta Savonarola in Padua is 4.9; on Porta San Giovanni in the same city 5.8; on Michele Sanmicheli's Palazzo Canossa in Verona this
ratio is 4.4; on the Villa Trissino it is 5.7 on the ground floor and 7.9 on the facade of the upper floor. Even on those Renaissance buildings which combine freestanding columns with entablatures,
for instance on Brunelleschi's Pazzi Chapel, the ratio is 3.5; on Pirro Ligorio's Casino of Pius IV it is 3.7. This approach normally results in large empty wall surfaces and ultimately unsuccessful
attempts to establish a visual rhythm on the facade. Before the 1550s, Palladio's use of the orders conformed to this Renaissance practice, in those cases when he used the orders -- e.g., Palazzo
Civena, Palazzo Iseppo Porto, Palazzo Thiene or Villa Gazzotti in Bertesina.
The placement of columns on the façade was thus an important unresolved problem which Renaissance architecture faced in the mid-cinquecento. The question was whether columns on the facade should be
placed in relation to the walls behind (to "express the interior" as we would say today) or should they be placed according to certain rules for intercolumniations, similar to those Vitruvius
stipulated. In his Canon of the Five Orders of 1562, Vignola fully endorsed this latter approach. According to Vignola, the architect should apply the orders only after the building has been actually
designed and the basic dimensions on the façade have been determined. One should start by dividing the height of the building into a prescribed number of parts to determine the size of the module,
which, subsequently, determines the size and disposition of all other elements of the order. In the case of the Doric, the height of the facade is to be divided into 20 parts, one of which will be
the module. The thickness of the architrave is then taken to be 1 module, the frieze 1½ modules, the intercolumniation 5½ modules, and so on. The building's dimensions have to be determined before
the orders are applied.
But Palladio, in the early 1550, formulated a very different approach to the use of the orders. A survey of the way his use of intercolumniations changed through his career shows that in the early
1550s he started systematically using intercolumniations of less than 3 diameters, as Vitruvius had suggested [Mitrovic 2004, 203-204]. Intercolumniation-to-diameter ratios are 2.75 on the palazzo
Chiericati (1550), 2.7 on the villa Pisani in Montagnana (1552), 2.25 on the villa Cornaro, 2.4 on the villa Chiericati in Vancimuglio (1554), 2.3 on the palazzo Antonini (1556) and so on. With the
exception of Villa Sarego, after the year 1550 intercolumniations of more than 3 lower column diameters appear only on those buildings where Palladio's involvement is considered debatable by
Palladian scholarship. (Obviously, one must take into account that intercolumniations had to be increased over 3 diameters when it comes to the main entrances, and that upper storeys are likely to
have larger intercolumniations, because columns in upper storeys have thinner diameters.)
Palladio's approach differs from Vignola's in that the position of columns for Palladio had to correspond to the position of walls in the interior of the building. At the same time, the position of
interior walls was determined by the proportional rules for length/width ratios, and proportional relationships between individual rooms had to satisfy the CCH rule. Consequently, the two sets of
requirements for internal and external proportions had to be mutually coordinated. The proportional coordination of internal and external elements is further complicated by the fact that the internal
height ("floor to floor") must equal the sum of column height plus entablature thickness on the façade.
The introduction of these complex mathematical requirements appears for the first time in Palladio's work in the Palazzo Chiericati, at the very beginning of the 1550s (Fig. 4). According to the Four
Books, the large rooms on the side of the central hall are 30 by 18 feet and Palladio says that their height was calculated as the arithmetic mean: (30+18)/2=24 feet [Palladio 1997, 2.53]. The next
room is square (18 by 18) and vaulted; according to the rule, the height of such rooms equals 4/3 of their width: 4/3x18=24 feet. The height of rooms (calculated as an arithmetic mean to satisfy the
CCH rule) plus the thickness of the ceiling, has to be equal to the height of columns plus entablature thickness. At the same time, the positions of walls have been adjusted to the rhythm of columns.
Ground floor intercolumniations on the Chiericati are 2.75 diameters. (One should also note that Palladio's columns never align with the walls exactly -- they are always laterally shifted [Mitrovic
2004, 112-120]. This strategy appears already in his very earliest works such as Villa Gazotti in Bertesina.)
It is in this context that the proportional system of the villa Cornaro needs to be analyzed; it was for this reason that we undertook a new surveying campaign. Together with Villa Pisani in
Montagnana, Villa Cornaro was designed within two years from Palazzo Chiericati. In this case, the portico has only six columns and it cannot be taken to determine the position of all internal walls
orthogonal on the façade (as is the case on the Chiericati), but only of those behind the portico. The size and proportions of the large sala, with its four columns, correspond to the position of the
columns in the portico. The sala in this case is a typical Palladian four-column sala with a flat ceiling -- a motif Palladio often used. The general morphology of the villa and its volumetrics
relates to other Palladian villas of the same period, in whose plans we can similarly read the architect's efforts to align internal bearing elements with the orders on the façade -- e.g., Villa
Pisani in Montagnana (Fig. 5), Villa Chiericati in Vancimuglio (Fig. 6) and Palazzo Antonini in Udine (Fig. 7).
Analyzing the proportional system of an executed building is not the same thing as working with a set of an architect's drawing.[6] Precision in the execution of a built work can never be great. In
the case of Palladio, we can safely rely on considerable precision in stonecutting. Those elements of the orders which were executed in stone show a high level of precision in execution. But when it
comes to built walls and masonry work, the level of precision is not nearly so great. One can hardly expect precision greater than 5 cm. The building has also changed in the meantime; some of its
parts have been changed and dimensions of rooms have been slightly altered by the addition of new layers to the walls. The heights of rooms are particularly susceptible to even greater imprecision
because the floors have been changed in some rooms and because the ornamentation of the ceilings makes it difficult to estimate what Palladio would have considered to be the actual height of
individual rooms. It is also recommended that the proportional analysis of side wings added by Scamozzi is left aside -- it is not only uncertain in how far these parts of the building correspond to
Palladio's intention, but they were also substantially changed in later centuries, unlike the main block of the villa.
The precision in stonecutting allows one to conclude with great certainty that the lower column diameter of ground floor columns was meant to be about 70.3 cm and plinths are regularly 930 mm. But
intercolumniations vary between 152.5 and 157 cm, which gives an intercolumniation-to-diameter ratio between 2.17 and 2.23. Because the columns are Ionic, one is tempted to interpret these ratios as
an almost accurate eustylos, which would mean that the architect originally intended intercolumniations to be 2.25D. In that case the optimal intercolumniation would be about 157.5 cm. (The largest
measured intercolumniation is 157cm.) The central intercolumniations of the portico are larger -- this is typical of Palladio's work and we shall soon see how it was calculated.
The measurements of the rooms on the sides indicate length/width ratios of about 1.7 which is a reasonable approximation of Ö3 in built work. The height of these rooms is 717 cm and can be read as
the geometric mean of the length and width. The height/width ratio is about 1.3. In the Four Books Palladio said that the height was the arithmetic mean of the length and width, but this must be a
mistake, because in the case the rooms would have to be half a meter higher (7.65 m).
The square rooms show how difficult it is to analyze proportionally a survey of an executed building. These rooms were obviously meant to be square -- but the difference between their longest and
shortest side is 12.5 cm. Quite appropriately for square rooms, the height-to-width ratio is about 1.3. This is a rough approximation of Palladio's rule that the height-to-width ratio of square
vaulted rooms should be 4/3. As mentioned before, rooms within the same row in Palladio's designs tend to have one dimension in common. In the case of Villa Cornaro, the width of the larger rooms is
the same as the side measurement of the square rooms. Since the ceilings have equal heights and since the height/width ratios of both rooms are equal, one can conclude that the CCH rule has been
successfully applied in Villa Cornaro.
The smallest rooms are the hardest to analyze. Their length/width ratios are between 1.8 and 1.9. Those who have worked with Palladio's proportional system know that he rarely used ratios between 2/1
and Ö3/1. But it would be difficult to interpret the length/width ratios of these two rooms as one of these two ratios. To do so, one would have to assume inaccuracy in execution of about 30 cm,
which is unlikely to happen on both sides of the same building. These smallest rooms have mezzanines above and their ceilings are lower than in other rooms. One should note that the height/width
ratio is about 3/2.
The ratios of the large sala also differ from the ratios which Palladio listed in his list of preferred ratios (Fig. 8).
Fig. 8. Villa Cornaro: ground floor sala. (photo/author)
They are not easy to interpret, but it is remarkable that the length/width ratio of the sala is 1.23 whereas width/height ratio is 1.22. (In this case one should talk about the width/height ratio and
not vice versa because the width is greater than the height.) However, if we consider internal length/width ratio -- by "internal" I mean here the lengths and widths of the space between the columns
-- we can see that this ratio is 1.5 or 3/2, one of Palladio's preferred room length/width ratios. The larger distance between the columns equals their height -- a point Palladio made himself in the
Four Books. Our survey confirms this statement, with some approximation. This ratio now explains the remaining ratios of the sala. The walls, as has been explained, align with the corner columns of
the portico, which means that the distance between the walls and the columns of the sala has to be one intercolumniation. This applies to all four walls of the sala. The dimensions of the sala are
actually the length and width of the space between four columns (whose length/width ratio is 3/2) plus column thickness (diameter) plus intercolumniation. This calculation resulted in the total
length and width and produced the sala's length/width ratio of 1.23. The height of the columns (which equals the distance between the columns of the sala) also determines the size of the central
portico intercolumniations. The columns of the sala are aligned with the penultimate columns of the portico; the ultimate columns, as we have seen, are aligned with the walls of the sala. The width
of the central intercolumniation is the distance between the columns of the sala (i.e., the height of the columns) minus two sums of intercolumniations and column diameters.
As one would expect, the horizontal proportions of the upper storey closely follow the proportions of rooms at the lower level. The rooms at the upper storey have flat ceilings and, in accordance
with Palladio's rules, the ceiling height is (more or less) the same as the width of the rooms. Palladio's tendency to keep one dimension the same for all rooms in a row shows perfect sense in this
case -- otherwise, walking down a row of rooms on either side, one would note unpleasant changes in ceiling heights. But, when it comes to entering the large sala, the change of ceiling height
contributes to the impression of the dignity of space. A remarkable aspect of the large sala on the upper storey is Palladio's dogmatic application of the rule that the height should be equal to its
width. As a result, the sala is more than nine meters high -- an impressive and somewhat daunting space (Fig. 9).
Fig. 9. Upper storey sala. (photo/author)
Palladio's approach to design in the villa Cornaro thus combines the principle of preferred room proportions and the use of a columnar system to determine the placement of walls. The proportions of
the main sala and porticos are derived on the basis of the proportional rules for the order used; the proportions of the side rooms on the basis of preferred ratios (or their equivalents, such as Ö3)
as well as the CCH rule. In the case of Villa Cornaro, these two separate systems were combined but not intertwined, except in the case of the central space between the columns of the sala. It would
be fruitless to attempt to deduce the proportions of the rooms in the side rows from the system of intercolumniations. The distance between the final column of the portico and the end wall of the
central block cannot be expressed as the sum of column diameters and intercolumniations. In this sense, Villa Cornaro differs from Palazzo Chiericati. Chiericati has columns across the whole façade
and all walls orthogonal to the façade were aligned with these columns. Something similar seems to have been suggested by Palladio on the façade of Villa Pisani in Montagnana. This villa was designed
at about the same time as Cornaro and the two villas have very similar morphology. The interesting feature of this villa is the entablature which extends around the villa, even along those walls
which have no columns (Fig. 10). The order of the ground floor is Doric and the triglyphs can be taken to indicate the positions where columns should be placed on the façade -- and possibly where
internal walls should be placed. It would be extremely interesting to know whether and in how far these triglyphs relate to the position of internal walls -- but at this moment there are no surveys
which would enable us to answer this question.
Fig. 10. Villa Pisani in Montagnana. (photo/author)
Ultimately, the result is that the mathematics of the orders became decisive for Palladio's design principles and the use of proportions from the early 1550s. It was combined with preferences for
certain proportional relationships in the sense of ratios of lengths, widths and heights of rooms. But the thesis of the Modernist historiography, that when it comes to Palladio it is only the
relationships between bare walls that matters, has to be rejected.
[1] This article is based on the results of the survey of the Villa Cornaro made in June 2003 by Melanie Burke, Tim Ross, Steve Wassell and myself. I should like to express gratitude to the owners of
the Villa, Carl and Sally Gable, for the permission to survey the villa; to Professor Wolfgang Wolters for numerous advices in the preparation of the survey, Melanie Burke for the permission to
reproduce two drawings she has produced subsequently to our survey, to my home institution, Unitec Institute of Technology, for the financial support in preparing the article and to Ms. Karen Wise
for the help with the written English of the article. The article was presented at the conference "Nexus 2004: Relationships Between Architecture and Mathematics", 19-23 June 2004 in Mexico City. For
the full survey of the villa Cornaro see the forthcoming book: Branko Mitrovic and Steve Wassell (eds.), Andrea Palladio's Villa Cornaro in Piombino Dese (2005, in preparation). return to text
[2] [Beltramini and Guidolotti 2001] is particularly useful when it comes to providing the information necessary to separate original Palladio's works from later additions. return to text
[3] See [Mitrovic 2004, 64-73 and 83-95] for a summary of these debates and further bibliography. return to text
[4] For statistical analyses of the second book of Palladio's treatise see [Howard and Longair 1982] and [Mitrovic 2004, 64-65 and 190-198]. return to text
[5] For the impact of Wittkower's book, see [Millon 1972] and [Payne 1994]. return to text
[6] For a discussion of precision in Palladio's built work see [Robison 1998-1999]. return to text
Alberti, Leon Battista. 1966. L'architettura. Ed. and Italian trans. Giovanni Orlandi. Milan: Edizioni Il Polifilo. (Parallel Latin/Italian version of De re aedificatoria).
______. 1988. On the Art of Building. Trans. Joseph Rykwert, Robert Tavernor and Neil Leach. Cambridge, MA: MIT Press.
Barbaro, Daniele. 1987. I dieci libri dell'architettura tradotti et commentati. Facsimile of 2nd ed., Venice 1567. Milan: Il Polifilo.
Beltramini, Guido and Pino Guidolotti. 2001. Andrea Palladio atalante delle architetture. Venice: Marsilio.
Bertotti-Scamozzi, Ottavio. 1968. Le fabbriche e i disegni di Andrea Palladio (Vicenza 1796). New York: Architectural Book.
______. 1998. Le fabbriche e i disegni di Andrea Palladio. CD edition. Trans. Howard Burns. Vicenza: Centro Internazionale di Studi di Architettura "Andrea Palladio".
Howard, Deborah and Malcolm Longair. 1982. Harmonic Proportion and Palladio's Quattro Libri. Journal of the Society of Architectural Historians 41: 116-143.
Lewis, Douglas. 1972. La datazione della villa Corner a Piombino Dese. Bollettino del Centro Internazionale di Studi di Architettura "Andrea Palladio" 14: 381-393.
______. 1975. Girolamo II Corner's Completion of Piombino with an inrecognized building of 1596 by Vincenzo Scamozzi. Bollettino del Centro Internazionale di Studi di Architettura "Andrea Palladio"
17: 401-405.
Millon, Henry. 1972. Rudolf Wittkower, Architectural Principles in the Age of Humanism, Its Influence on the Development and interpretation of Modern Architecture. Journal of the Society of
Architectural Historians, 31: 83-91.
Mitrovic, Branko. 2001. A Palladian Palinode: Reassessing Rudolf Wittkower's Architectural Principles in the Age of Humanism. architectura 31: 113-131.
______. 2002. Palladio's Canonical Corinthian Entablature and the Archaeological Surveys in the Fourth Book of I quattro libri dell'architettura. Architectural History 45: 113-127.
______. 2004. Learning from Palladio. New York: W.W. Norton. To order this book from Amazon.com, click here.
Mitrovic, Branko and Steve Wassell, eds. 2005. Andrea Palladio's Villa Cornaro in Piombino Dese. Aalto Books and Unitec, New Zealand. In preparation.
Palladio, Andrea. 1990. I quattro libri dell'architettura. (Venice, 1570). Facsimile ed. Milan: Ulrico Hoepli Editore.
______. 1997. The Four Books on Architecture. Robert Tavernor and Richard Schofield, trans. Cambridge, MA: MIT Press.
______. 1988. Scritti sull'architettura (1554-1579). Lionello Puppi, ed. Vicenza: Neri Pozza.
Payne, Alina. 1994. Rudolf Wittkower and Architectural Principles in the Age of Modernism. Journal of the Society of Architectural Historians, 53: 322-342.
Robison, Elwin. 1998-1999. Structural Implications in Palladio's Use of Harmonic Proportions. Annali d'architettura 10-11: 175-182.
Wittkower, Rudolf. 1962. Architectural Principles in the Age of Humanism. London: Warburg Institute.
Branko Mitrovic Branko Mitrovic received his undergraduate degrees in Architecture and Philsophy from Belgrade University, his PhD from University of Pennsylvania and currently teaches as Professor
of Architectural History and Theory at Unitec Institute of Technology, Auckland, New Zealand. He is the author the book Learning from Palladio (Norton, New York 2004), the translation of and
commentary on Vignola's Canon of the Five Orders (Acanthus, New York 1999) and a number of scholarly articles about Renaissance architectural theory. He has been awarded fellowships from Harvard
University, Canadian Centre for Architecture and the Humboldt Foundation.
The correct citation for this article is:
Branko Mitrovic, "Andrea Palladio's Villa Cornaro in Piombino Dese", Nexus Network Journal, vol. 6 no. 2 (Autumn 2004), http://www.nexusjournal.com/Mitrovic.html
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[SOLVED] How do we show $P(A) leq P(A Delta B) + P(A cap B) leq P(ADelta B) + P(B).$? ~ Mathematics ~ TransWikia.com
Based on your comments you meant $$lvert{P(A)-P(B)}rvert leq P(ADelta B)$$
$$A subseteq (A Delta B) cup (A cap B)$$, and the two sets on the RHS are disjoint. So $$P(A) leq P(A Delta B) + P(A cap B) leq P(ADelta B) + P(B).$$
So $$P(A) - P(B) leq P(ADelta B)$$. We can similarly show that $$P(B)-P(A) leq P(ADelta B)$$ and the result follows.
Correct answer by riemleb on December 10, 2020 | {"url":"https://transwikia.com/mathematics/how-do-we-show-pa-leq-pa-delta-b-pa-cap-b-leq-padelta-b-pb/","timestamp":"2024-11-07T04:21:10Z","content_type":"text/html","content_length":"41864","record_id":"<urn:uuid:d5a9f0de-b79a-4e33-b78f-6b943a310586>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00770.warc.gz"} |
If f(x)=4x-5,find the value of x for which f(x)=19? | HIX Tutor
If f(x)=4x-5,find the value of x for which f(x)=19?
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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• Readily available 24/7 | {"url":"https://tutor.hix.ai/question/if-f-x-4x-5-find-the-value-of-x-for-which-f-x-19-8f9af8daa0","timestamp":"2024-11-03T18:33:07Z","content_type":"text/html","content_length":"567045","record_id":"<urn:uuid:891c04df-35e2-40a3-bb62-8cf7b0c34cf8>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00108.warc.gz"} |