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The Model Of Economic Growth Based On Production Functions
2. The model of economic growth based on production functions (the Welfens/Jasinski model and its modifications). To show how FDI influence economic growth of a particular country a model proposed by
P.Welfens and P.Jasinski is used. It is based on traditional production functions.
In general the production functionsof Welfens and Jasinski describing the economic growth in the recipient country can bedefined by the following equation[61, p.254]:
where “Y” is an output (GDP or GNP); “K” is fixed assets of local origin (domestic fixed assets); “H” is fixed assets of foreign origin (foreign fixed assets); “L” is the number of employed in the
national economy; “z” is the rate of technological progress; “β” is statistically evaluated …show more content…
Secondly, in the basic structure of the production function (1.9) proposed byWelfens and Jasinski, domestic fixed-capital and foreign investmentsare supposed to be equally effective, which, as
mentioned above, contradicts the observed facts.
Thirdly, the production function of Welfens/Jasinski includes multiplier of scientific and technological progress, which depends on the overall macroeconomic situation and in no way connected with
the inflows of FDI. At the same it is obvious that foreign investment primarily perform the function of transferring technological and managerial innovations to the economies of the recipient
countries.If we take into consideration the institutional changes in the economy, then they should also reflect the effects related to the openness of the national economy to foreign investments from
Fourthly, a set of the factors that affect the rate of scientific and technological progress in
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Binary analysis: Concolic execution with Pin and z3
@Jonathan Salwan
- 2013-08-28
Edit 2015-09-06: Check out our concolic execution library using python bindings.
1 - Introduction
In a previous post, I talked about the concolic execution using Valgrind to the taint analysis and z3 to the constraint path solving. So why another blog post about this technique? Because recently
my previous researchs was around Pin and because Pin is supported on Linux, Windows and Mac. I also wanted to see how it's possible to do it without IR - With Valgrind and z3 it was pretty easy
because Valgrind provides an IR (VEX). Then, it can be useful for other people or it can be give some new ideas. :-).
2 - Concolic execution
We can find two types of analysis, static and dynamic analysis. Both approaches have some advantages and disadvantages. If you use dynamic analysis, we can't cover all the code but you will be more
reliable. If you use static analysis, you can cover the code, but you can't get the context information at runtime. The concolic execution is a technic that uses both symbolic and concrete execution
to solve a constraint path. The concolic execution is mainly used to cover a code. To list the constraints, the symbolic execution is used. Below, a little example about the symbolic execution:
int foo(int i1, int i2)
int x = i1;
int y = i2;
if (x > 80){
x = y * 2;
y = 0;
if (x == 256)
return True;
x = 0;
y = 0;
/* ... */
return False;
Based that code, we can see 3 differents paths, for each path we have a specific constraint. The constraints tree look like that:
So, we can say that this code can return False via two differents paths and True via only one path. With the symbolic execution, it's possible to know which constraints are necessary to return False
or True.
The concolic execution will use the concrete execution to save and solve the constraints at runtime. With this above case, to cover this code, the program will be executed three times and for each
execution, one constraint will be solved to take another path. That's what we'll see in the next chapter.
3 - Proof of concept on dumb crackme
3.1 - Introduction
We will start to analyze a simple code which contains only three simple conditions. The goal will be to solve this crackme automatically via the concolic execution.
#include <stdio.h>
#include <sys/types.h>
#include <stdlib.h>
#include <fcntl.h>
int main(void)
int fd;
char buff[260] = {0};
fd = open("serial.txt", O_RDONLY);
read(fd, buff, 256);
if (buff[0] != 'a') return False;
if (buff[1] != 'b') return False;
if (buff[2] != 'c') return False;
printf("Good boy\n");
return True;
Based on that code, if we represent all paths and constraints, our constraints tree will look like that:
This code contains four possible paths. Each path has its constraints.
| PC number | Constraints | return value |
| 1 | buff[0] != 'a' | return False |
| 2 | buff[0] == 'a' && buff[1] != 'b' | return False |
| 3 | buff[0] == 'a' && buff[1] == 'b' && buff[2] != 'c' | return False |
| 4 | buff[0] == 'a' && buff[1] == 'b' && buff[2] == 'c' | return True |
Now that we have listed all constraints which are possible, we can cover the code. For that, we need to run the program with the first constraint, then we re-run the program with the second
constraint and repeat this operation until the last constraint is executed. This operation is called the concolic execution. Below you can see a diagram representing this execution.
As you can see above, we can cover all the code with only four executions. Now, we will see how it's possible to implement it with Pin. For that we need to:
• Taint the serial.txt buffer.
• Follow our data (Spread the taint).
• Save the first constraint.
• Solve this constraint.
• Re-run the binary with the first constraint solved.
• And repeat this operation for each constraint (each path)...
In this blog post, we will not talk about the taint analysis, for that, you can read my previous post. Then, to solve the constraints we will use the theorem prover Z3 and its C++ API.
3.2 - Compile a Pin tool with Z3 C++ API
We will use the Z3 C++ API inside the Pin tool. So, you need to install the Z3 library and add the headers/lib in the compile line. In my case, I downloaded the z3.zip in my pintool directory and I
compiled the library. Then, to compile my Pin tool, I created a shell script which compiles with the Z3 headers/lib. This script looks like that:
$ pwd
$ cat compile.sh
g++ -DBIGARRAY_MULTIPLIER=1 -DUSING_XED -Wall -Werror -Wno-unknown-pragmas -fno-stack-protector -DTARGET_IA32E -DHOST_IA32E -fPIC -DTARGET_LINUX -I../../../source/include/pin -I../../../source/include/pin/gen -I../../../extras/components/include -I./z3/src/api/c++ -I../../../extras/xed2-intel64/include -I../../../source/tools/InstLib -O3 -fomit-frame-pointer -fno-strict-aliasing -c -o obj-intel64/ConcolicExecution.o ConcolicExecution.cpp
g++ -shared -Wl,--hash-style=sysv -Wl,-Bsymbolic -Wl,--version-script=../../../source/include/pin/pintool.ver -o obj-intel64/ConcolicExecution.so obj-intel64/ConcolicExecution.o -L../../../intel64/lib -L../../../intel64/lib-ext -L../../../intel64/runtime/glibc -L../../../extras/xed2-intel64/lib -lpin -lxed -ldwarf -lelf -ldl -lz3
3.3 - Save and solve the constraints
If we look the ASM representation of our C code. We can see that this code loads in the "eax" register our character ("rbp-0x110" points on our serial buffer). Then, it compares the smaller size
register "al" with a constant and it jumps to two different space if it's true or false.
.text:400683: movzx eax,BYTE PTR [rbp-0x110]
.text:40068a: cmp al,0x61
.text:40068c: je 400695 <main+0x81>
.text:40068e: mov eax,0x1
.text:400693: jmp 4006c8 <main+0xb4>
.text:400695: movzx eax,BYTE PTR [rbp-0x10f]
.text:40069c: cmp al,0x62
.text:40069e: je 4006a7 <main+0x93>
.text:4006a0: mov eax,0x1
.text:4006a5: jmp 4006c8 <main+0xb4>
.text:4006a7: movzx eax,BYTE PTR [rbp-0x10e]
.text:4006ae: cmp al,0x63
.text:4006b0: je 4006b9 <main+0xa5>
.text:4006b2: mov eax,0x1
.text:4006b7: jmp 4006c8 <main+0xb4>
.text:4006b9: mov edi,0x4007c7
.text:4006be: call 4004e0 <puts@plt>
.text:4006c3: mov eax,0x0
.text:4006c8: leave
.text:4006c9: ret
This code is pretty simple and if we taint the datas used in our serial.txt, we have something like that:
$ printf "xxx" > serial.txt
$ ../../../pin -t ./obj-intel64/ConcolicExecution.so -taint-file serial.txt -- ./crackme1
[TAINT] bytes tainted from 0x7fff194a6600 to 0x7fff194a6700 (via read)
[READ in 7fff194a6600] 400683: movzx eax, byte ptr [rbp-0x110]
eax is now tainted
[FOLLOW] 40068a: cmp al, 0x61
[SPREAD] 40068e: mov eax, 0x1
output: eax | input: constant
eax is now freed
Now, the real question is: Where does start and stop the equation? I think it's real good/complicated question and I am currently still working on that! However, in your case, we will start an
equation when a byte controllable by the user is LOAD and we will stop it when the "cmp" instruction occurs. We will also assign an unique ID for each constraint.
$ printf "xxx" > serial.txt
$ ../../../pin -t ./obj-intel64/ConcolicExecution.so -taint-file serial.txt -- ./crackme1
[TAINT] bytes tainted from 0x7fff194a6600 to 0x7fff194a6700 (via read)
[READ in 7fff194a6600] 400683: movzx eax, byte ptr [rbp-0x110]
[Constraint] #0 = 0x78
eax is now tainted
[FOLLOW] 40068a: cmp al, 0x61
[Equation] cmp(#0, 0x61)
[Equation] cmp(x, 0x61)
[SPREAD] 40068e: mov eax, 0x1
output: eax | input: constant
eax is now freed
As you can see above, we assign the first constraint with the unique ID #0. This constraint was the first, so we tag it to remember that's possible to control it via the user input. Then, when the "
cmp" occurs, we display the full equation.
To maintain a link between a register and a constraint number, a table is updated. When a constraint is assigned, it's also assigned to a register.
That means eax = #0 = 0x78 - 0x78 is our first character in our serial. Then cmp(al, 0x61) = cmp(#0, 0x61) because eax = #0 and cmp(#0, 0x61) = cmp(x, 0x61) because #0 is the first constraint of our
Now to solve this equation we just use Z3.
$ printf "xxx" > serial.txt
$ ../../../pin -t ./obj-intel64/ConcolicExecution.so -taint-file serial.txt -- ./crackme1
[TAINT] bytes tainted from 0x7fff194a6600 to 0x7fff194a6700 (via read)
[READ in 7fff194a6600] 400683: movzx eax, byte ptr [rbp-0x110]
[Constraint] #0 = 0x78
eax is now tainted
[FOLLOW] 40068a: cmp al, 0x61
[Equation] cmp(#0, 0x61)
[Equation] cmp(x, 0x61)
[Z3 Solver]-------------------------------------
(= x #x00000061))
(define-fun x () (_ BitVec 32)
The good value is 0x61
[Z3 Solver]-------------------------------------
[SPREAD] 40068e: mov eax, 0x1
output: eax | input: constant
eax is now freed
Z3 tries to solve this equation (= x #x00000061)) and finds that the result is 0x61. At this time, the Pin tool writes the good character (0x61) in our serial.txt.
3.4 - Demo on the first crackme
To solve this crackme and to generate the good serial.txt, we need to run three times this Pin tool. For each execution, one character is found and written in the serial file.
$ printf "xxx" > serial.txt
$ ../../../pin -t ./obj-intel64/ConcolicExecution.so -taint-file serial.txt -- ./crackme1
[TAINT] bytes tainted from 0x7fff065b1ab0 to 0x7fff065b1bb0 (via read)
[READ in 7fff065b1ab0] 400683: movzx eax, byte ptr [rbp-0x110]
[Constraint] #0 = 0x78
eax is now tainted
[FOLLOW] 40068a: cmp al, 0x61
[Equation] cmp(#0, 0x61)
[Equation] cmp(x, 0x61)
[Z3 Solver]-------------------------------------
(= x #x00000061))
(define-fun x () (_ BitVec 32)
The good value is 0x61
[Z3 Solver]-------------------------------------
[SPREAD] 40068e: mov eax, 0x1
output: eax | input: constant
eax is now freed
$ cat serial.txt
$ ../../../pin -t ./obj-intel64/ConcolicExecution.so -taint-file serial.txt -- ./crackme1
[TAINT] bytes tainted from 0x7fff0c1677a0 to 0x7fff0c1678a0 (via read)
[READ in 7fff0c1677a0] 400683: movzx eax, byte ptr [rbp-0x110]
[Constraint] #0 = 0x61
eax is now tainted
[FOLLOW] 40068a: cmp al, 0x61
[Equation] cmp(#0, 0x61)
[Equation] cmp(x, 0x61)
[Z3 Solver]-------------------------------------
(= x #x00000061))
(define-fun x () (_ BitVec 32)
The good value is 0x61
[Z3 Solver]-------------------------------------
[READ in 7fff0c1677a1] 400695: movzx eax, byte ptr [rbp-0x10f]
[Constraint] #1 = 0x00
eax is already tainted
[FOLLOW] 40069c: cmp al, 0x62
[Equation] cmp(#1, 0x62)
[Equation] cmp(cmp(x, 0x61), 0x62)
[Z3 Solver]-------------------------------------
(= x #x00000062))
(define-fun x () (_ BitVec 32)
The good value is 0x62
[Z3 Solver]-------------------------------------
[SPREAD] 4006a0: mov eax, 0x1
output: eax | input: constant
eax is now freed
$ cat serial.txt
$ ../../../pin -t ./obj-intel64/ConcolicExecution.so -taint-file serial.txt -- ./crackme1
[TAINT] bytes tainted from 0x7fff4acd2e60 to 0x7fff4acd2f60 (via read)
[READ in 7fff4acd2e60] 400683: movzx eax, byte ptr [rbp-0x110]
[Constraint] #0 = 0x61
eax is now tainted
[FOLLOW] 40068a: cmp al, 0x61
[Equation] cmp(#0, 0x61)
[Equation] cmp(x, 0x61)
[Z3 Solver]-------------------------------------
(= x #x00000061))
(define-fun x () (_ BitVec 32)
The good value is 0x61
[Z3 Solver]-------------------------------------
[READ in 7fff4acd2e61] 400695: movzx eax, byte ptr [rbp-0x10f]
[Constraint] #1 = 0x62
eax is already tainted
[FOLLOW] 40069c: cmp al, 0x62
[Equation] cmp(#1, 0x62)
[Equation] cmp(cmp(x, 0x61), 0x62)
[Z3 Solver]-------------------------------------
(= x #x00000062))
(define-fun x () (_ BitVec 32)
The good value is 0x62
[Z3 Solver]-------------------------------------
[READ in 7fff4acd2e62] 4006a7: movzx eax, byte ptr [rbp-0x10e]
[Constraint] #2 = 0x00
eax is already tainted
[FOLLOW] 4006ae: cmp al, 0x63
[Equation] cmp(#2, 0x63)
[Equation] cmp(cmp(cmp(x, 0x61), 0x62), 0x63)
[Z3 Solver]-------------------------------------
(= x #x00000063))
(define-fun x () (_ BitVec 32)
The good value is 0x63
[Z3 Solver]-------------------------------------
[SPREAD] 4006b2: mov eax, 0x1
output: eax | input: constant
eax is now freed
$ cat serial.txt
$ ./crackme1
Good boy
3.5 - Another crackme using XOR-based algorithm
To complicate things a bit, let's use this following dumb crackme using an XOR-based algorithm.
#include <stdio.h>
#include <sys/types.h>
#include <stdlib.h>
#include <fcntl.h>
char *serial = "\x30\x39\x3c\x21\x30";
int main(void)
int fd, i = 0;
char buf[260] = {0};
char *r = buf;
fd = open("serial.txt", O_RDONLY);
read(fd, r, 256);
while (i < 5){
if ((*r ^ (0x55)) != *serial)
return 0;
r++, serial++, i++;
if (!*r)
printf("Good boy\n");
return 0;
This code reads and applies an XOR with the constant key "0x55" on each character in the serial file. Then, it checks the result with a constant string serial. This code is intersting to study the
execution concolic, because we have a simple algorithm. On the following CFG, the blue blox is our algorithm.
Now. let's see what happens when we taint and follow our datas from the serial file.
$ printf "xxx" > ./serial.txt
$ ../../../pin -t obj-intel64/ConcolicExecution.so -taint-file serial.txt -- ./crackme2
[TAINT] bytes tainted from 0x7fff3cab6cc0 to 0x7fff3cab6dc0 (via read)
[READ in 7fff3cab6cc0] 400698: movzx eax, byte ptr [rax]
eax is now tainted
[SPREAD] 40069b: mov edx, eax
output: edx | input: eax
edx is now tainted
[FOLLOW] 40069b: mov edx, eax
[FOLLOW] 4195997: xor edx, 0x55
[READ in 4007ec] 4006a7: movzx eax, byte ptr [rax]
eax is now freed
[FOLLOW] 4006aa: cmp dl, al
[SPREAD] 7feb884e67db: mov edx, 0x1
output: edx | input: constant
edx is now freed
Same for the first example, we need to assign a unique constraint for each spread. Then, when the cmp instruction occurs, we need to solve the equation via Z3.
$ ../../../pin -t obj-intel64/ConcolicExecution.so -taint-file serial.txt -- ./crackme2
[TAINT] bytes tainted from 0x7fff3cab6cc0 to 0x7fff3cab6dc0 (via read)
[READ in 7fff3cab6cc0] 400698: movzx eax, byte ptr [rax]
[Constraint] #0 = 0x61
eax is now tainted
[SPREAD] 40069b: mov edx, eax
output: edx | input: eax
edx is now tainted
[FOLLOW] 40069b: mov edx, eax
[Constraint] #1 = #0
[FOLLOW] 4195997: xor edx, 0x55
[Constraint] #2 = xor(#1, 0x55)
[READ in 4007ec] 4006a7: movzx eax, byte ptr [rax]
eax is now freed
[FOLLOW] 4006aa: cmp dl, al
[Equation] cmp(#2, 0x30)
[Equation] cmp(xor(x, 0x55), 0x30)
[Z3 Solver]-------------------------------------
(= (bvxor x #x00000055) #x00000030))
(define-fun x () (_ BitVec 32)
The good value is 0x65
[Z3 Solver]-------------------------------------
[SPREAD] 7feb884e67db: mov edx, 0x1
output: edx | input: constant
edx is now freed
$ cat serial.txt
As you can see above, my constraint on the xor instruction looks like that: xor(#1, 0x55), it means we need to display/follow all ALU operations on a specific convention. Like:
add(a, b)
sub(a, b)
mul(a, b)
div(a, b)
xor(a, b)
This is a real problem with Pin. Because it doesn't provide an IR, we need to implement all operations. For example, with the xor instruction, we need to catch these following encoding:
xor reg, reg
xor mem, reg
xor reg, mem
xor reg, immed
xor mem, immed
xor accum, immed
Then, when we need to build our equation like cmp(#2, 0x30), we need to replace the constraint number by its content - And for that we will use the constraints table.
After the first constraint solved, we set the first character in the serial file and we re-run the Pin tool to solve the second constraint. We repeat this operation until all constraints are solved.
The following diagram represent our executions. As you can see, for each execution, only one constraint is solved.
And the full result which generates a valide file key is pasted below. As you can see, for each execution, one character is found until the valide key.
$ printf "xxx" > ./serial.txt
$ ../../../pin -t ./obj-intel64/ConcolicExecution.so -taint-file serial.txt -- ./crackme1
[TAINT] bytes tainted from 0x7fff2d60e7d0 to 0x7fff2d60e8d0 (via read)
[READ in 7fff2d60e7d0] 400698: movzx eax, byte ptr [rax]
[Constraint] #0 = 0x41
eax is now tainted
[SPREAD] 40069b: mov edx, eax
output: edx | input: eax
edx is now tainted
[FOLLOW] 40069b: mov edx, eax
[Constraint] #1 = #0
[FOLLOW] 4195997: xor edx, 0x55
[Constraint] #2 = xor(#1, 0x55)
[READ in 4007ec] 4006a7: movzx eax, byte ptr [rax]
eax is now freed
[FOLLOW] 4006aa: cmp dl, al
[Equation] cmp(#2, 0x30)
[Equation] cmp(xor(x, 0x55), 0x30)
[Z3 Solver]-------------------------------------
(= (bvxor x #x00000055) #x00000030))
(define-fun x () (_ BitVec 32)
The good value is 0x65
[Z3 Solver]-------------------------------------
[SPREAD] 7ff3541837db: mov edx, 0x1
output: edx | input: constant
edx is now freed
$ cat serial.txt
$ ../../../pin -t ./obj-intel64/ConcolicExecution.so -taint-file serial.txt -- ./crackme1
[TAINT] bytes tainted from 0x7fff6d1f8730 to 0x7fff6d1f8830 (via read)
[READ in 7fff6d1f8730] 400698: movzx eax, byte ptr [rax]
[Constraint] #0 = 0x65
eax is now tainted
[SPREAD] 40069b: mov edx, eax
output: edx | input: eax
edx is now tainted
[FOLLOW] 40069b: mov edx, eax
[Constraint] #1 = #0
[FOLLOW] 4195997: xor edx, 0x55
[Constraint] #2 = xor(#1, 0x55)
[READ in 4007ec] 4006a7: movzx eax, byte ptr [rax]
eax is now freed
[FOLLOW] 4006aa: cmp dl, al
[Equation] cmp(#2, 0x30)
[Equation] cmp(xor(x, 0x55), 0x30)
[Z3 Solver]-------------------------------------
(= (bvxor x #x00000055) #x00000030))
(define-fun x () (_ BitVec 32)
The good value is 0x65
[Z3 Solver]-------------------------------------
[READ in 7fff6d1f8731] 400698: movzx eax, byte ptr [rax]
[Constraint] #3 = 0x00
eax is now tainted
[FOLLOW] 40069b: mov edx, eax
[Constraint] #4 = #3
[FOLLOW] 4195997: xor edx, 0x55
[Constraint] #5 = xor(#4, 0x55)
[READ in 4007ed] 4006a7: movzx eax, byte ptr [rax]
eax is now freed
[FOLLOW] 4006aa: cmp dl, al
[Equation] cmp(#5, 0x39)
[Equation] cmp(xor(x, 0x55), 0x39)
[Z3 Solver]-------------------------------------
(= (bvxor x #x00000055) #x00000039))
(define-fun x () (_ BitVec 32)
The good value is 0x6c
[Z3 Solver]-------------------------------------
[SPREAD] 7fe0b6aa47db: mov edx, 0x1
output: edx | input: constant
edx is now freed
$ cat serial.txt
$ ../../../pin -t ./obj-intel64/ConcolicExecution.so -taint-file serial.txt -- ./crackme1
[TAINT] bytes tainted from 0x7fff2d1e1e00 to 0x7fff2d1e1f00 (via read)
[READ in 7fff2d1e1e00] 400698: movzx eax, byte ptr [rax]
[Constraint] #0 = 0x65
eax is now tainted
[SPREAD] 40069b: mov edx, eax
output: edx | input: eax
edx is now tainted
[FOLLOW] 40069b: mov edx, eax
[Constraint] #1 = #0
[FOLLOW] 4195997: xor edx, 0x55
[Constraint] #2 = xor(#1, 0x55)
[READ in 4007ec] 4006a7: movzx eax, byte ptr [rax]
eax is now freed
[FOLLOW] 4006aa: cmp dl, al
[Equation] cmp(#2, 0x30)
[Equation] cmp(xor(x, 0x55), 0x30)
[Z3 Solver]-------------------------------------
(= (bvxor x #x00000055) #x00000030))
(define-fun x () (_ BitVec 32)
The good value is 0x65
[Z3 Solver]-------------------------------------
[READ in 7fff2d1e1e01] 400698: movzx eax, byte ptr [rax]
[Constraint] #3 = 0x6c
eax is now tainted
[FOLLOW] 40069b: mov edx, eax
[Constraint] #4 = #3
[FOLLOW] 4195997: xor edx, 0x55
[Constraint] #5 = xor(#4, 0x55)
[READ in 4007ed] 4006a7: movzx eax, byte ptr [rax]
eax is now freed
[FOLLOW] 4006aa: cmp dl, al
[Equation] cmp(#5, 0x39)
[Equation] cmp(xor(x, 0x55), 0x39)
[Z3 Solver]-------------------------------------
(= (bvxor x #x00000055) #x00000039))
(define-fun x () (_ BitVec 32)
The good value is 0x6c
[Z3 Solver]-------------------------------------
[READ in 7fff2d1e1e02] 400698: movzx eax, byte ptr [rax]
[Constraint] #6 = 0x00
eax is now tainted
[FOLLOW] 40069b: mov edx, eax
[Constraint] #7 = #6
[FOLLOW] 4195997: xor edx, 0x55
[Constraint] #8 = xor(#7, 0x55)
[READ in 4007ee] 4006a7: movzx eax, byte ptr [rax]
eax is now freed
[FOLLOW] 4006aa: cmp dl, al
[Equation] cmp(#8, 0x3c)
[Equation] cmp(xor(x, 0x55), 0x3c)
[Z3 Solver]-------------------------------------
(= (bvxor x #x00000055) #x0000003c))
(define-fun x () (_ BitVec 32)
The good value is 0x69
[Z3 Solver]-------------------------------------
[SPREAD] 7f7e919ef7db: mov edx, 0x1
output: edx | input: constant
edx is now freed
$ cat serial.txt
$ ../../../pin -t ./obj-intel64/ConcolicExecution.so -taint-file serial.txt -- ./crackme1
[TAINT] bytes tainted from 0x7fff597b37a0 to 0x7fff597b38a0 (via read)
[READ in 7fff597b37a0] 400698: movzx eax, byte ptr [rax]
[Constraint] #0 = 0x65
eax is now tainted
[SPREAD] 40069b: mov edx, eax
output: edx | input: eax
edx is now tainted
[FOLLOW] 40069b: mov edx, eax
[Constraint] #1 = #0
[FOLLOW] 4195997: xor edx, 0x55
[Constraint] #2 = xor(#1, 0x55)
[READ in 4007ec] 4006a7: movzx eax, byte ptr [rax]
eax is now freed
[FOLLOW] 4006aa: cmp dl, al
[Equation] cmp(#2, 0x30)
[Equation] cmp(xor(x, 0x55), 0x30)
[Z3 Solver]-------------------------------------
(= (bvxor x #x00000055) #x00000030))
(define-fun x () (_ BitVec 32)
The good value is 0x65
[Z3 Solver]-------------------------------------
[READ in 7fff597b37a1] 400698: movzx eax, byte ptr [rax]
[Constraint] #3 = 0x6c
eax is now tainted
[FOLLOW] 40069b: mov edx, eax
[Constraint] #4 = #3
[FOLLOW] 4195997: xor edx, 0x55
[Constraint] #5 = xor(#4, 0x55)
[READ in 4007ed] 4006a7: movzx eax, byte ptr [rax]
eax is now freed
[FOLLOW] 4006aa: cmp dl, al
[Equation] cmp(#5, 0x39)
[Equation] cmp(xor(x, 0x55), 0x39)
[Z3 Solver]-------------------------------------
(= (bvxor x #x00000055) #x00000039))
(define-fun x () (_ BitVec 32)
The good value is 0x6c
[Z3 Solver]-------------------------------------
[READ in 7fff597b37a2] 400698: movzx eax, byte ptr [rax]
[Constraint] #6 = 0x69
eax is now tainted
[FOLLOW] 40069b: mov edx, eax
[Constraint] #7 = #6
[FOLLOW] 4195997: xor edx, 0x55
[Constraint] #8 = xor(#7, 0x55)
[READ in 4007ee] 4006a7: movzx eax, byte ptr [rax]
eax is now freed
[FOLLOW] 4006aa: cmp dl, al
[Equation] cmp(#8, 0x3c)
[Equation] cmp(xor(x, 0x55), 0x3c)
[Z3 Solver]-------------------------------------
(= (bvxor x #x00000055) #x0000003c))
(define-fun x () (_ BitVec 32)
The good value is 0x69
[Z3 Solver]-------------------------------------
[READ in 7fff597b37a3] 400698: movzx eax, byte ptr [rax]
[Constraint] #9 = 0x00
eax is now tainted
[FOLLOW] 40069b: mov edx, eax
[Constraint] #10 = #9
[FOLLOW] 4195997: xor edx, 0x55
[Constraint] #11 = xor(#10, 0x55)
[READ in 4007ef] 4006a7: movzx eax, byte ptr [rax]
eax is now freed
[FOLLOW] 4006aa: cmp dl, al
[Equation] cmp(#11, 0x21)
[Equation] cmp(xor(x, 0x55), 0x21)
[Z3 Solver]-------------------------------------
(= (bvxor x #x00000055) #x00000021))
(define-fun x () (_ BitVec 32)
The good value is 0x74
[Z3 Solver]-------------------------------------
[SPREAD] 7f9ac23db7db: mov edx, 0x1
output: edx | input: constant
edx is now freed
$ cat serial.txt
$ ../../../pin -t ./obj-intel64/ConcolicExecution.so -taint-file serial.txt -- ./crackme1
[TAINT] bytes tainted from 0x7fff313be550 to 0x7fff313be650 (via read)
[READ in 7fff313be550] 400698: movzx eax, byte ptr [rax]
[Constraint] #0 = 0x65
eax is now tainted
[SPREAD] 40069b: mov edx, eax
output: edx | input: eax
edx is now tainted
[FOLLOW] 40069b: mov edx, eax
[Constraint] #1 = #0
[FOLLOW] 4195997: xor edx, 0x55
[Constraint] #2 = xor(#1, 0x55)
[READ in 4007ec] 4006a7: movzx eax, byte ptr [rax]
eax is now freed
[FOLLOW] 4006aa: cmp dl, al
[Equation] cmp(#2, 0x30)
[Equation] cmp(xor(x, 0x55), 0x30)
[Z3 Solver]-------------------------------------
(= (bvxor x #x00000055) #x00000030))
(define-fun x () (_ BitVec 32)
The good value is 0x65
[Z3 Solver]-------------------------------------
[READ in 7fff313be551] 400698: movzx eax, byte ptr [rax]
[Constraint] #3 = 0x6c
eax is now tainted
[FOLLOW] 40069b: mov edx, eax
[Constraint] #4 = #3
[FOLLOW] 4195997: xor edx, 0x55
[Constraint] #5 = xor(#4, 0x55)
[READ in 4007ed] 4006a7: movzx eax, byte ptr [rax]
eax is now freed
[FOLLOW] 4006aa: cmp dl, al
[Equation] cmp(#5, 0x39)
[Equation] cmp(xor(x, 0x55), 0x39)
[Z3 Solver]-------------------------------------
(= (bvxor x #x00000055) #x00000039))
(define-fun x () (_ BitVec 32)
The good value is 0x6c
[Z3 Solver]-------------------------------------
[READ in 7fff313be552] 400698: movzx eax, byte ptr [rax]
[Constraint] #6 = 0x69
eax is now tainted
[FOLLOW] 40069b: mov edx, eax
[Constraint] #7 = #6
[FOLLOW] 4195997: xor edx, 0x55
[Constraint] #8 = xor(#7, 0x55)
[READ in 4007ee] 4006a7: movzx eax, byte ptr [rax]
eax is now freed
[FOLLOW] 4006aa: cmp dl, al
[Equation] cmp(#8, 0x3c)
[Equation] cmp(xor(x, 0x55), 0x3c)
[Z3 Solver]-------------------------------------
(= (bvxor x #x00000055) #x0000003c))
(define-fun x () (_ BitVec 32)
The good value is 0x69
[Z3 Solver]-------------------------------------
[READ in 7fff313be553] 400698: movzx eax, byte ptr [rax]
[Constraint] #9 = 0x74
eax is now tainted
[FOLLOW] 40069b: mov edx, eax
[Constraint] #10 = #9
[FOLLOW] 4195997: xor edx, 0x55
[Constraint] #11 = xor(#10, 0x55)
[READ in 4007ef] 4006a7: movzx eax, byte ptr [rax]
eax is now freed
[FOLLOW] 4006aa: cmp dl, al
[Equation] cmp(#11, 0x21)
[Equation] cmp(xor(x, 0x55), 0x21)
[Z3 Solver]-------------------------------------
(= (bvxor x #x00000055) #x00000021))
(define-fun x () (_ BitVec 32)
The good value is 0x74
[Z3 Solver]-------------------------------------
[READ in 7fff313be554] 400698: movzx eax, byte ptr [rax]
[Constraint] #12 = 0x00
eax is now tainted
[FOLLOW] 40069b: mov edx, eax
[Constraint] #13 = #12
[FOLLOW] 4195997: xor edx, 0x55
[Constraint] #14 = xor(#13, 0x55)
[READ in 4007f0] 4006a7: movzx eax, byte ptr [rax]
eax is now freed
[FOLLOW] 4006aa: cmp dl, al
[Equation] cmp(#14, 0x30)
[Equation] cmp(xor(x, 0x55), 0x30)
[Z3 Solver]-------------------------------------
(= (bvxor x #x00000055) #x00000030))
(define-fun x () (_ BitVec 32)
The good value is 0x65
[Z3 Solver]-------------------------------------
[SPREAD] 7f0d00e1f7db: mov edx, 0x1
output: edx | input: constant
edx is now freed
$ cat serial.txt
$ ./crackme1
Good boy
4 - Conclusion
I think that the concolic execution is a great technique and it need to be investigated and improved. I hope that more and more people will look into it. Also, I think it isn't a good idea to do a
concolic execution with a DBI (Dynamic Binary Instrumentation) without intermediate language like Pin. Why? Because without IR, you need to implement all instructions set and their different
encodings. This is possible but that's really boring and you can forget an operation... To the theorem solver conclusion, I'm not a Z3 expert, I do know it's used internally by Microsoft for a lot of
purpose (I guess they got pretty big equations), but I have only used it with toy-equations, so I can't really say.
4.1 - My Pin tool
First of all, my Pin tool is not reliable and it works only with the above examples... I only implemented the instruction necessary for my examples (mov, cmp, xor). So, if you want use it, you need
to implement all the x86 instructions... This Pin tool is just a PoC but it can give you a base to your project. The sources are here.
4.2 - References
4.3 - Special thanks
I would like to thanks Axel "0vercl0k" Souchet for his skills in Z3 and proofreading.
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Solving equation with algebraic numbers
Solving equation with algebraic numbers
Hello, SAGE gives me error when I load this: solve(x^2-AA(sqrt(3))==0,x) but it gives no problem when I load solve(x^2-sqrt(3)==0,x) This is a small example of a bigger problem I have in which I must
solve a system of equations involving algebraic numbers through AA(.) and QQbar(.). How can I make SAGE solve equations with this type of numbers? or there is no way? Thanks!
You can convert algebraic numbers to symbolic expressions using SR(...). Probably you would rather want to define an ideal in a polynomial ring, and compute a Gröbner basis and/or the associated
variety (if the system has finitely many solutions). Can you add the system you actually want to solve?
Hello rburing, i tried loading solve(x^2-SR(AA(sqrt(3)))==0,x) but it gives error, what do you think?
It seems Maxima can't handle the symbolic wrapper around AA elements. Try SR(AA(sqrt(3))).numerical_approx() for numerics, or AA(sqrt(3)).radical_expression() for an exact expression. Not all
algebraics are expressible in terms of radicals, so this is not a good approach in general. Also solve may return only approximate solutions in more complicated cases. I would instead create an ideal
I in a polynomial ring and call I.variety(AA) or I.variety(QQbar).
1 Answer
Sort by » oldest newest most voted
A possible one-liner :
sage: (AA["x"](x^2-AA(sqrt(3)))).roots()
which can be abrbreviated as
sage: (x^2-AA(sqrt(3))).polynomial(AA).roots()
edit flag offensive delete link more
thanks Emmanuel!
creyesm1992 ( 2020-08-11 23:09:10 +0100 )edit
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TNB Night Owl – Leap Year 2020
Leap Year. Photo by Star Light.
You probably already know that 2020 is a leap year and thus tomorrow, Saturday, February 29 is a leap day. You may even know why there’s an extra day in the calendar this year. But, do you really
know why? (OK, you probably do – it’s hard to surprise a group of smart people). Are you ready, Quiz Kids? Let’s find out how much you know!
True or false? One year ago, right now as you read this, the Earth was in exactly the same place in its orbit around the Sun as it is now. The correct answer requires an explanation, but here’s a
hint: don’t lean on the calendar year for support – there are exactly 365 days in the calendar year, or sometimes 366 days.
Astronomers have long known that the Earth takes approximately 365.25 days to make one complete orbit around the Sun. We say approximately, for a couple of reasons. First, to make the math and the
explanation simple to understand, and second, to avoid the unnecessary complication of discussing the difference between a sidereal year and a tropical year. [Spoiler Alert: Purely in terms of time,
the difference between the two is 20 minutes, 24.5 seconds, with a sidereal year being slightly longer than 365.25 days and a tropical year being slightly shorter than 365.25 days. (We told you that
it was an unnecessary complication)].
In an ideal universe, a year would be exactly divisible by the number of days in the year. In other words, there would be no ‘0.25’, or one-quarter (six hours) of a day appended to ‘365.25 days in a
year’. It would be 365 days per year, exactly.
But the universe is not ideal, and the Earth’s orbit cannot be exactly divided by a whole number of days, with no fraction of a day left over. If it could be, that would be an incredible coincidence!
That extra quarter of a day (0.25) means that a year is not evenly divisible by Earth’s rotational period of 24 hours (i.e., a single day). So if you’re reading this, for example, at 6AM, you’ll have
to wait approximately another six hours – until about noon – before the Earth is in exactly the same place in it’s orbit around the Sun as it was one year ago this morning at 6AM.
True or false? A day (the time it takes for the Earth to make one full turn about its axis) is 24 hours. Everyone knows the answer to this one, right!? False: an Earth day is exactly 23 hours, 56
minutes, and 4.1 seconds. That’s 3 minutes, 55.9 seconds short of a full 24 hours. Hmmm, the universe is starting to look mighty messy.
So now we have Earth days that are not a full 24 hours, and an Earth year that’s a full quarter of a day more than 365 days; messy, messy, messy. The non-ideal universe played havoc with the Julian
calendar, which was losing time and therefore ill-equipped to deal with the disparities of time and orbital mechanics. To deal with these realities, in 1582, Pope Gregory replaced the Julian calendar
with what we now know and love as the Gregorian calendar.
If we stayed on the old Julian calendar, you would have a discrepancy of one day, roughly every 128 years. With the Gregorian system now we have a discrepancy of one day in something like 3,500
Geoff Chester, US Naval Observatory
True or false? Every four years, an extra day (February 29) is added to the calendar. This is true, but with extremely important exceptions! Adding a leap day once every four years in the course of
132 years overcorrects by nearly a full day. To rectify this, three out of every four ‘century years’ (e.g., the years 1700, 1800, and 1900) do not observe a leap day, but the fourth ‘century year’
(e.g., the year 2000) does have a February 29.
In other words, if the year is divisible by 4, it is a leap year, unless it is divisible by 100 (the ‘century years’), but if the century year is divisible by 400, it is a leap year. In concrete
terms, the year 2020 is a leap year (divisible by 4), while the year 1900 and 2100 are not leap years (divisible by 100, but not 400), and the years 1600, 2000, and 2400 are leap years (divisible by
That’s how humans deal with keeping a reasonably accurate calendar in a far-from-ideal universe. Got it?
Question of the night: Were you, or do you know anyone who was, born on February 29?
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Bayesian Network
Definition of Bayesian Network
A Bayesian network, also called a belief network, is a probabilistic graphical model that represents a set of random variables and their conditional dependencies. Variables are represented as nodes
in the graph, and edges connecting nodes represent conditional dependencies between variables. Conditional probabilities are specified for the edges in the graph. A Bayesian network can be used to
infer the likelihoods of different outcomes given the data.
What is Bayesian Network used for?
A Bayesian Network is a probabilistic graphical model used to represent and reason with uncertain knowledge. It can be used to build predictive models of complex systems, where the relationships
between variables are not known in advance. A Bayesian Network encodes the conditional probabilities of events and variables (represented by nodes) using directed arcs. The arrows indicate which
nodes are dependent upon one another, allowing for easy representation of conditional probability distributions. Using these structures, a Bayesian Network can calculate the probability of future
events based on past data. For example, it could be used to predict the probability of an individual having a certain disease based on their symptoms or to forecast stock prices given macroeconomic
indicators. As well as being useful for predictive modeling, Bayesian Networks can be used for causal inference and decision making under uncertainty. By considering relationships between variables
in its structure, it can make use of prior knowledge about the system in order to produce more accurate predictions than traditional machine learning algorithms that do not take into account such
relationships. Additionally, with its graphical representation, Bayesian Networks allow people to visually understand complex data sets and draw insights from them quickly and easily.
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MU Limit State Method for Reinforced Concrete Structures - May 2014 Exam Question Paper | Stupidsid
MU Civil Engineering (Semester 7)
Limit State Method for Reinforced Concrete Structures
May 2014
Total marks: --
Total time: --
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary
1 (a) From the first principles, derive stress block parameter for limit state method for single reinforced section. For grade of concrete M20 and grade of steel Fe415.
5 M
1 (b) What are the function of longitudinal reinforcement and transverse steel in column.
5 M
1 (c) Explain and illustrate balance, under reinforced and over-reinforced R.C section.
5 M
1 (d) Under what situation a beam will be subjected to torsional moment. How longitudinal and transverse reinforcement is designed to resist it.
5 M
2 (a) Why doubly reinforced beam is required.
4 M
2 (b) A reinforced concrete beam 230mm wide is to carry load 40 kN/m. The beam is simply supported on a span of 8m. Design a section when.
i) Depth is not restricted
ii) Effective depth is restricted to 500mm. Use M20 grade of concrete, Fe415 grade of steel and ISM.
16 M
3 (a) A rectangle beam 230mm × 450mm (effective depth) is reinforced with 6 bars of 16mm diameter out of which two bars are bent at 45°. Determine the shear resistance of bent up bars and additional
shear reinforcement required if the ultimate shear force is 300 kN. Design shear reinforcement adopt M20 and Fe415.
│Pt%│0.75│ 0.5│0.75│ 1.0│1.25│ 1.5│1.75│ 2.0│2.25│ 2.5│
│t │0.36│0.48│0.56│0.62│0.67│0.72│0.75│0.79│0.81│0.83│
10 M
3 (b) A T-beam section has b[f]=1200mm, D[f]=120mm and d=400 mm, b[w]=230mm Asi=6 bars of 16 mm diameter. Determine the moment of resistance of the section. Use M20 grade of concrete, Fe415 grade of
10 M
4 (a) Design a R.C slab for interior panel having size 4m × 6m the slab carries superimposed load of 3kN/m^2
\[\begin{align*}+\alpha_{x}=0.053,\alpha_{y} =0.032\\-\alpha_{x}=0.041,\ -\alpha_{y}=0.024\end{align*} \]
Use M20 grade of concrete, Fe415 grade of sateel.
12 M
4 (b) Design a short helically reinforced column to resist ultimate axial load of 1200kN. Use M20 grade of concrete, Fe415 grade of steel.
8 M
5 Design a combined footing connecting two column A and B. 4M centre to centre, carrying an ultimate axial load of 1200 kN and 1400 kN respectively. The boundary line of the property is 500mm from
the outer face of the column A, column A and B is 400mm× 400mm size SBC of soil is 150 kN/m^2
Use M20 grade of concrete, Fe415 grade of steel.
20 M
6 (a) Derive the expression for M.R for single reinforced section bu using whitney's stress block parameter.
5 M
6 (b) R.C beam 230mm &time; 600mm is reinforced with 3 bars of 16 mm on tension side with an effective covers of 50mm determine the safe load the beam can carry if the beam is simply supported on
span of 5m. Use whitney's method, Use M20 grade of concrete , Fe415 grade of steel.
10 M
6 (c) What is development length. Develop relevant equation.
5 M
7 (a) A rectangular beam 230mm × 550mm depth is subjected to a sagging bending moment of 40 kN/m shear force of 30kN and twisting moment of 12 kN/s at a given section. Design the reinforcement at the
given section take load factor 1.5, Assume effective cover 50mm. Use M20 grade of concrete , Fe415 grade of steel.
12 M
7 (b) Design isolated rectangle pan footing for the column of size 230mm×450mm carrying an axial load of 1200 kN, SBC of soil is 200 kN/m^2. Use M20 grade of concrete Fe415 grade of steel.
8 M
More question papers from Limit State Method for Reinforced Concrete Structures
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White elephant gift exchange
2018, Dec 04
TODO: Insert some jolly words about Christmas. š š ā ā š · š
The White Elephant Gift Exchange is an exciting alternative to Secret Santa. In this game, everyone buys a present of roughly equal value and places it wrapped up in a pile. The first player opens a
random present. The next player may open a new present or steal someoneā s already opened present. When a player gets their present stolen they may either open a new present or steal a present from
someone. A present can only be stolen once per turn. When someone opens a present the turn ends. If the last player thinks a bit, they typically have an advantage because they know what all but one
present is, so can take their pick.
A variation to the rules limits how often a present may be stolen. Weā ll play around with this rule to see how we can create a fair and exciting game in which players have ample opportunities to
steal but minimises the advantage in having the last go.
The Model
Iā ve built a model that can play 1000ā s of rounds of White Elephant Gift Exchange. Before we start using the model Iā d like to understand it a little bit.
The model assumes that no one knows the value of a present before it is opened and when a present is opened each player instantly values it the same way. This means our players donā t value wrapped
presents by their size, weight or noise they make; no one has a personal preference for a particular present; and presents donā t become more desirable if they are frequently stolen. Think of the
presents as Amazon gift vouchers.
I calibrated the model by running it with 9 players acting randomly (they open or steal without thinking). I chose low, medium and high present values of 1, 5, 10 and put 3 of each present into the
pile. Each point on the chart is a players average present value after 5001 games. The error bars are the estimate of standard error of the mean and the dotted line shows the average value of the
presents (5.33).
Itā s reassuring to see that most of the players end up with the average present value (within error), demonstrating that the model is doing something sensible. I expect that as we average up more
games each point gets closer to the average number and has a smaller error.
Running more simulations will decrease the statistical error but it will also increase the time it takes to run. To get a feel for this statistical error Iā ve plotted how standard error for one
player decreases as we play more games.
1000 simulations will give me an acceptable error, but you may have higher standards!
We can also see how often a players final present was stolen. We see presents that later players end up with tend to be stolen more often than presents the earlier players end up with.
This makes sense considering there is no absolute limit on how often a present may be stolen - only a limit of once per turn. The opportunities to steal a present increase as the game progresses so
we see more steals for the final presents of players who go later in the game. It helps to imagine that player 1 may open a present, have it stolen by player 2 who gets it stolen by player 3ā ¦
A Strategy
We can make the players a bit more interesting by telling them to always steal the best available present (a present that has been opened and not stolen on this round). If there is nothing to steal
they will reluctantly open a present. We will still play with an equal number of presents valued at 1, 5 or 10.
In this chart I show the outcome from 2 strategies: one where none of the players think (play randomly), in the other they always steal the best present available. I played each strategy 1000 times.
The dotted line remains the average present value (5.33).
This demonstrates how the game gives a huge advantage to going later. When it reaches the last player, all but one present has been opened and at least one 10 valued present is open. The last player
can take their pick from the opened presents (there will be a 10 for them to steal) and no one can steal it off them.
The penultimate player does well because they usually get a 10 valued present but always get at least a 5 valued present. The only time they donā t get a 10 valued present is if none of 10s have
been opened by their turn. Similar logic holds for the 7th player.
The remaining players experience an essentially random, albeit unfair game because the top presents have already been stolen and locked out from further steals. The last player to make an action has
a small chance of opening a 10 valued present.
A Fair Game
To make the game a bit fairer we can create a rule in which a present may be stolen a maximum 3 times. This allows early players to ā freeze outā some of the good presents.
I want a way to objectively measure fair games, so Iā ll make this statement:
A fair outcome looks the same as a random outcome.
We can make the game ā fairerā by reducing the number of times a present may be stolen. A game with 0 allowed steals will be a purely random selection of presents, ie fairness = 1.
Iā ve plotted the same model but with a limited number of steals. Iā ve added some lines to show the minimum and maximum values we may reasonably expect to see from a random model. Anything that
falls in or touches these lines (within error) is part of a ā fairā game. We can measure this ā fairnessā by comparing the players that have expected present values within the random bounds to
those that donā t.
Here players 4,5,6,7 (which Iā ve highlighted) have random looking outcomes to give a fairness of 4/9 = 0.4444.
Many Fair Games
Iā ve run the 9 present game for steal limits between 1 and 9. Here we see how it affects how often a present is stolen. The graph shows how often the players final present was stolen, per steal
The general trend is as we might expect - when the steal limit is higher, presents get stolen more often. For a steal limit at 1, the first player opens a present. Player 2 must steal their present
(itā s how their strategy is defined) and player 1 opens a new present. Player 3 must steal the present from player 1 (they canā t steal steal player 2ā s, because it has reached the steal limit),
player 1 opens a new present and so on.
As steal limit increases, the steals ā get compressedā until hitting an apparent boundary where allowing more steals (6 or more) doesnā t alter how often a present is stolen. There seems to be a
peak in number of steals around player 7, this could be from: the random nature of the game; because there are 3 high value presents which players 8 and 9 tend to get; something else; or Iā m
reading too much into it. I wonā t worry about these details now.
We can also see how the steal limits affect expected present values for a player.
Iā ve highlighted the fair outcomes. This shows in a 9 present game only a steal limit up to 4 has some fair outcomes and when a present can only be stolen once the game is completely fair (ie,
As a present can be stolen more often the end result gets more polarised between the early and late players. At high steal limits (6 or more) the game settles down - players 1-5 have the same
expected value. This is likely because most of the high value 10ā s are stolen by the later players, leaving the earlier players to perform a random squabble for the 1 and 5 valued presents with a
slight chance of having the last unopened present valued at 10. This behaviour may explain why the stolen counts bunch up.
We can repeat this for many different games with many different steal limits. Here we have all the steal limits for a game with 30 players.
A Fair and Fun Game
Reducing the number of steals makes the game boring. We can create another measure called ā fun factorā which is similar to fairness but compares the ratio of allowed steals to number of players.
I will make another statement:
A fun game, is a game with more possible steals.
A game with 9 players where a present can be stolen a maximum 3 times has a fun factor 3/9 = 0.333.
Iā ve aggregated a whole range of games with equal numbers of presents valued 1, 5, 10 with varying steal limits and plotted the resulting fairness and fun factors below. You can get the raw numbers
There looks like a clear dog-leg pattern with a few outliers underneath it. Iā ve highlighted all the games with a steal limit of 2 below, this conveniently catches all the outliers (for your
reference Iā ve also highlighted games with steal limits of 1, 3 and 4).
I donā t intend to explore in this post why games with a steal limit of 2 appear special, so Iā ll sweep these inconvenient points under the carpet. Iā ve also added a linear regression over the ā
interestingā part of the graph.
We can use the graph to set a few rules for deciding a fair and fun game:
1. Once you reach a fun factor of 0.6 the game doesn't get any fairer. You might as well go all the way and allow unlimited steals (this may be related to the bunching seen in the other charts).
2. With fun factor less than 0.6, the relationship between fun factor and fairness is linear according to: fairness = -1.55 * fun factor + 1.03.
You can use these rules to guide how you want to set up your game. You may simply decide to go halfway along the line and set a fun factor of 0.3, to get a fairness 0.565. i.e. for a game with 9
players, allow 3 steals, for a game with 12 players allow 4 steals etc. The choice is up to you.
Some Variety
Iā ve assumed presents have uniformly distributed values of 1,5 or 10 in equal quantities. What happens if I reduce this to repetitions of presents valued 1, 5, or extend it to repetitions of 1, 5,
10, 15? How about a skewed distribution with repeating present values of 1, 2, 10? What happens if there arenā t equal quantites of each present like repeating present values of 1, 1, 1, 10? Why do
the fairness charts take this pattern?
I can also run a model with 99 presents, of value 1 - 100. Iā m not sure what exactly is causing this interesting pattern, although I have a few ideas. This all may be something for later
I hope youā ve enjoyed reading this. Weā ve got a graph and a few rules that help us decide where to set a steal limit thatā s fair and fun. Iā ve left a few unanswered questions (how
frustrating!). Youā ll have to wait for another post for me to address these. For now, thank you for reading and have a Happy Christmas.
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Day 3: Gradient descent (part 2)
Gradient Descent
Gradient descent is an algorithm that you can use to find the value of w and b in a more systematic way, which results in the smallest possible of the cost function J(w,b)
What we're going to focus today it to get more understanding about what the learning rate and what the derivative term are doing and why when multiplied together, it results in updates to parameters
w and b that makes sense.
import math, copy
import numpy as np
import matplotlib.pyplot as plt
from lab_utils_uni import plt_house_x, plt_contour_wgrad, plt_divergence, plt_gradients
Note: you may need to download some of the libraries needed to run the code in the post, such as lab_utils_uni, which is available on the course's lab
problem statement
let's use the same 2 data points we previously used with our cost function example:
• a house with 1000 sqft sold for $300,000
• a house with 2000 sqft sold for $500,000
# load our data set
x_train = np.array([1.0, 2.0]) # size in 1000 sqft
y_train = np.array([300.0, 500.0]) # price in 1000s of dollar
This was developed during our Cost Function post, we will be using it again here:
# function to calculate the cost
def compute_cost(x, y, w, b):
m = x.shape[0]
cost = 0
for i in range(m):
f_wb = w * x[i] + b
cost = cost + (f_wb - y[i]) ** 2
total_cost = 1 / (2*m) * cost
return total cost
Let's recap the mathematical functions we have seen so far:
• (1) a linear model that predicts the 'f function' f(x):
in linear regression, we utilize input training data to fit the parameters w, b by minimizing a measure of the error between our predictions f(x) and the actual data y. The measure is called the cost
function J(w,b)
• (2) In training, you measure the cost over all of our training samples x_i, y_i:
please note that during our python code the sum would start at 'i=0' instead of 'i=1' as Python starts counting from 0, instead of 1, and would end at 'm - 1' instead of 'm'.
• (3) yesterday, we looked at our gradient descent algorithm, which would be described as:
• where, parameters w, b are updated simultaneously, the gradient is defined as:
Implement gradient descent
We will implement gradient descent algorithm for one feature. We will need 3 functions:
• compute_gradient, implementing equation (4) and (5) above
• compute_cost, implementing equation (2) above (this was already done in our compute cost section above)
• gradient_descent, utilizing compute_gradient and compute_cost
• the naming of python variables containing partial derivatives will follow this pattern dj_db will be:
• wrt is 'with respect to', as in partial derivative of J(w,b) with respect to b
compute_gradient implements (4) and (5) above and returns dj_dw, dj_db
def compute_gradient(x, y, w, b):
m = x.shape[0]
dj_dw = 0
dj_db = 0
for i in range(m):
f_wb = w * x[i] + b
dj_dw_i = (f_wb - y[i]) * x[i]
dj_db_i = (f_wb - y[i])
dj_db += dj_db_i
dj_dw += dj_dw_i
dj_dw = dj_dw / m
dj_db = dj_db / m
return dj_dw, dj_db
Let's use our compute_gradient function to find and plot some partial derivatives of our cost function relative to one of the parameters w0:
plt_gradients(x_train, y_train, compute_cost, compute_gradient)
• Above, the left plot shows dj_dw or the slope of the cost curve relative to w at three points.
• On the right side, of the plot, the derivative is positive, while on the left in negative. Due to the bowl shape, the derivatives will always lead gradient descent toward the bottom where the
gradient is 0.
• The left plot has fixed b=100. Gradient descent will utilize both dj_dw and dj_db to update parameters.
• The quiver plot on the right provides a means of viewing the gradient of both parameters. The arrow sizes reflect the magnitude of the gradient at that point. The direction and slope of the arrow
reflects the ratio of dj_dw and dj_db at that point.
Now that gradients can be computed, gradient_descent, described in equation (3) can be implemented in our function below. We will utilize this function to find optimal values of w and b on the
training data.
def gradient_descent(x, y, w_in, b_in, alpha, num_iters, cost_function, gradient_function):
J_history = []
p_history = []
b = b_in
w = w_in
for i in range(num_iters):
dj_dw, dj_db = gradient_function(x, y, w, b)
b = b - alpha * dj_db
w = w - alpha * dj_dw
if i < 100000:
J_history.append(cost_function(x, y, w, b))
p_history.append([w, b])
if i%math.ceil(num_iters/10) == 0:
print(f"Iteration {i:4}: Cost {J_history[-1]:0.2e} ",
f"dj_dw: {dj_dw: 0.3e}, dj_db: {dj_db:0.3e} ",
f"w: {w: 0.3e}, b: {b: 0.5e} ")
return w, b, J_history, p_history
Let's run the following code to look at the w, b found by gradient descent using the gradient_descent function we just did:
#initialize parameters
w_init = 0
b_init = 0
# gradient descent settings
iterations = 10000
tmp_alpha = 1.0e-2
# run gradient descent
w_final, b_final, J_hist, p_hist = gradient_descent(x_train, y_train, w_init, b_init, tmp_alpha, iterations, compute_cost, compute_gradient
print(f"(w,b) found by gradient descent: ({w_final: 8.4f}, {b_final: 8.4f})")
Iteration 0: Cost 7.93e+04 dj_dw: -6.500e+02, dj_db: -4.000e+02 w: 6.500e+00, b: 4.00000e+00
Iteration 1000: Cost 3.41e+00 dj_dw: -3.712e-01, dj_db: 6.007e-01 w: 1.949e+02, b: 1.08228e+02
Iteration 2000: Cost 7.93e-01 dj_dw: -1.789e-01, dj_db: 2.895e-01 w: 1.975e+02, b: 1.03966e+02
Iteration 3000: Cost 1.84e-01 dj_dw: -8.625e-02, dj_db: 1.396e-01 w: 1.988e+02, b: 1.01912e+02
Iteration 4000: Cost 4.28e-02 dj_dw: -4.158e-02, dj_db: 6.727e-02 w: 1.994e+02, b: 1.00922e+02
Iteration 5000: Cost 9.95e-03 dj_dw: -2.004e-02, dj_db: 3.243e-02 w: 1.997e+02, b: 1.00444e+02
Iteration 6000: Cost 2.31e-03 dj_dw: -9.660e-03, dj_db: 1.563e-02 w: 1.999e+02, b: 1.00214e+02
Iteration 7000: Cost 5.37e-04 dj_dw: -4.657e-03, dj_db: 7.535e-03 w: 1.999e+02, b: 1.00103e+02
Iteration 8000: Cost 1.25e-04 dj_dw: -2.245e-03, dj_db: 3.632e-03 w: 2.000e+02, b: 1.00050e+02
Iteration 9000: Cost 2.90e-05 dj_dw: -1.082e-03, dj_db: 1.751e-03 w: 2.000e+02, b: 1.00024e+02
(w,b) found by gradient descent: (199.9929,100.0116)
Let's take a moment and note some characteristics of the gradient descent process above:
• the cost starts large and rapidly declines as described before
• the partial derivatives, dj_dw, and dj_db also get smaller, rapidly at first and then more slowly. the process near the 'bottom of the bowl' progress slower due to the smaller value of the
derivative at that point
• progress slows, although the learning rate (alpha) remains fixed.
Cost vs iterations of gradient descent
A plot of cost vs iterations is a useful measure of progress in gradient descent. Cost should always decrease in successful runs. The change in cost is so rapid initially, it is useful to plot the
initial descent on a different scale than the final descent. In the plots below, note the scale of cost on the axes and the iteration step
Now that we have discovered the optimal values for the parameters w and b, we can use the model to predict housing values based on our learned parameters:
print(f"1000 sqft house prediction {w_final*1.0 + b_final:0.1f} Thousand dollars")
print(f"1200 sqft house prediction {w_final*1.2 + b_final:0.1f} Thousand dollars")
print(f"2000 sqft house prediction {w_final*2.0 + b_final:0.1f} Thousand dollars")
1000 sqft house prediction 300.0 Thousand dollars
1200 sqft house prediction 340.0 Thousand dollars
2000 sqft house prediction 500.0 Thousand dollars
We can show the progress of gradient descent during its execution by plotting the cost over iterations on a contour plot of the cost(w, b).
fig, ax = plt.subplots(1,1, figsize=(12, 6))
plt_contour_wgrad(x_train, y_train, p_hist, ax)
The contour plot above shows the cost value J(w,b) over a range of w and b. Cost levels are represented by rings. the red arrows is the path of gradient descent. The path makes steady progress
towards its goal and the initials steps are much larger than the steps near the goal.
Tomorrow: let's take a deeper look at our learning rate (alpha)
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ibrackets – Intelligent brackets
This small package provides a new definition of brackets [ and ] as active characters to get correct blank spaces in mathematical mode when using for open intervals. Instead of parenthesis: ]-\infty,
0[ is equivalent to (-\infty, 0).
Sources /macros/latex/contrib/ibrackets
Version 1.2
Licenses The LaTeX Project Public License 1.3
Copyright 2022–2023 Antoine Missier
Maintainer Antoine Missier
Contained in TeXLive as ibrackets
MiKTeX as ibrackets
Parentheses management
Topics French
Download the contents of this package in one zip archive (82.8k).
Community Comments
Maybe you are interested in the following packages as well.
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How To Solve Venn Diagrams In 6 Easy Steps
In this video, Jacqui will show you how to solve Venn Diagrams.
What are Venn Diagrams?
A Venn diagram shows the link between groups of different things. Venn Diagrams allow us to sort information into circles that overlap in the middle. The different circles will be allocated for
different rules and the overlapping part will follow both rules.
Put succinctly; Venn Diagrams are a way to segment data that have similarities and differences by putting them into overlapping circles. They are a brilliant way to summarise and compare information.
How to solve Venn Diagrams?
In a Venn Diagram question, the amount of received data will always be different than the number of participants. Below is a step-by-step guide on how to succeed when doing Venn Diagrams.
26 PEOPLE WERE ASKED WHETHER THEY LIKE CHIPS, FISH OR NEITHER. HOW MANY LIKE ONLY CHIPS?
18 Chips
13 Fish
3 Neither
1. MARK UP YOUR CIRCLES
Before we start talking about the steps, it is important to fully understand the question. This means drawing the diagram properly and labelling the circles.
For this instance, label one circle chips and one circle fish.
As you can see, the total results are going to be considerably more than the number of people that were asked. So, let’s have a look at what has happened.
First of all, we will add up the results given.
18 + 13 + 3 = 34
Clearly, 34 is more than 26 that were asked, so some people have answered twice.
Now to find who liked both fish and chips, we subtract the total number of answers given by the number of people that were asked.
34 – 26 = 8
This number goes into the middle where the two circles overlap as they follow both options of liking both fish and chips.
4. ALLOCATE THE NEITHER
We know that 3 people said that they like neither fish nor chips so you must remember to put this number outside of the two circles.
People often forget this step.
5. FINISH THE MATHS
Now if we look at the chips circle, we already have 8 people in it. So we go to the chip result and take away 8 answers. This will give us the number of people who just like chips.
18 – 8 = 10
Put 10 in the chips circle.
Then if we look at the fish circle, we have 8 people already in it. So we go to the fish result and take away 8 answers. This will give us the number of people who just like fish.
13 – 8 = 5
Now if we add up all the numbers written in your diagram, including those outside the circles, it will equal 26 which equates to the total number of people that had been asked.
6. ANSWER THE QUESTION
So now that we have sorted and correctly allocated all of the data, it is important that we go back and re-read the question so that we know what it is asking from us. We can do this by underlining
the text.
Now the questions says, How many only like chips?
We go to the chips circle and it is not all those in the circle, ONLY those who like chips.
Therefore the answer is 10.
Now some questions you might be asking yourself...
Do Venn Diagrams have to overlap?
Most of the time but not always. If you have 2 different sets of data that you have found no similarities between, there is no need for them to overlap. For example, in the video above if no one
answered twice, liking fish AND chips, there would be no overlap. The circles are then able to stand alone.
For good practice, it is still good to overlap your circles to clearly show the examiner that you know how Venn Diagrams work.
What do I do now?
We work on tricky problems like this all the time during our maths tuition. The aim of these lessons is to provide students with new material to work on throughout the day while children may not
receiving this from school.
Your child’s education is precious, and we are always here to support our children and their families fully.
If you have any questions you would like to ask me about Venn Diagrams, please ask below in the comments.
If you found this blog post useful, please do share it on social media.
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Polynomial iterations to roots of algebraic equations
(2) O(M)= t, +(8)(t)= 0 (s= 1 2,*** ,r1), then +(x) is said to define an iteration of order r to the root t. In fact, for r> 1, when xo is in a sufficiently small neighborhood of t the sequence (3)
x+1 = 4(xi) converges to t with (4) = + O(xi O)r For analyticf, iterations of all orders exist and can be constructed in many ways. Domb [2]1 has shown further that for polynomial f it is always
possible to make k a polynomial. The purpose of this note is to describe a simple algorithm: Let f(x) be a polynomial with no multiple factors; let p(x) and q(x) be any polynomials satisfying
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Equivalent fractions - math word problem (8025)
Equivalent fractions
Are these two fractions -4/9 and 11/15 equivalent?
Did you find an error or inaccuracy? Feel free to
write us
. Thank you!
Tips for related online calculators
Need help calculating sum, simplifying, or multiplying fractions? Try our
fraction calculator
You need to know the following knowledge to solve this word math problem:
Grade of the word problem:
Related math problems and questions:
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Heat Transfer by Radiation
31 Aug 2024
Heat Transfer by Radiation: Understanding the Process
Heat transfer by radiation is a vital process that occurs when energy is transferred through electromagnetic waves, such as light and radio waves. This type of heat transfer plays a crucial role in
various fields, including engineering, physics, and biology. In this article, we will delve into the concept of heat transfer by radiation, its importance, and the underlying principles.
What is Heat Transfer by Radiation?
Heat transfer by radiation occurs when an object emits or absorbs electromagnetic waves, such as light, infrared (IR), ultraviolet (UV), and radio waves. These waves are a form of energy that can
travel through space without the need for a medium, like air or water. When an object is heated, it emits radiation in the form of IR waves, which can be absorbed by other objects.
The Radiation Process
The process of heat transfer by radiation involves three main steps:
1. Emission: An object at a higher temperature than its surroundings emits radiation in the form of electromagnetic waves.
2. Transmission: The emitted radiation travels through space to reach an object at a lower temperature than the original source.
3. Absorption: The receiving object absorbs the radiation, which increases its energy and temperature.
The Stefan-Boltzmann Law
The Stefan-Boltzmann law is a fundamental principle that describes the relationship between the temperature of an object and the amount of radiation it emits. The law states that the total energy
emitted by an object per unit area (E) is proportional to the fourth power of its temperature (T):
E ∝ T^4
Mathematically, this can be expressed as:
E = σ * T^4
where σ is the Stefan-Boltzmann constant (5.67 × 10^-8 W/m²K^4).
The Planck’s Law
Planck’s law describes the distribution of radiation emitted by an object at a given temperature. The law states that the energy density of radiation (E) as a function of wavelength (λ) is
proportional to the product of the temperature and the fourth power of the wavelength:
E(λ) ∝ T * λ^(-5)
Mathematically, this can be expressed as:
E(λ) = C1 * T * λ^(-5)
where C1 is a constant.
Importance of Heat Transfer by Radiation
Heat transfer by radiation plays a crucial role in various fields, including:
1. Thermal Insulation: Understanding heat transfer by radiation is essential for designing effective thermal insulation systems.
2. Energy Efficiency: Radiation heat transfer can significantly impact the energy efficiency of buildings and appliances.
3. Astronomy: Radiation heat transfer is vital for understanding the behavior of stars and other celestial bodies.
4. Biological Systems: Heat transfer by radiation plays a role in regulating body temperature and maintaining homeostasis.
Heat transfer by radiation is an essential process that occurs when energy is transferred through electromagnetic waves. Understanding the principles of radiation heat transfer, including the
Stefan-Boltzmann law and Planck’s law, is crucial for various fields. By grasping these concepts, we can better design systems, optimize energy efficiency, and improve our understanding of the
natural world.
1. Incropera, F. P., & DeWitt, D. P. (2002). Fundamentals of Heat and Mass Transfer. John Wiley & Sons.
2. Cengel, Y. A. (2018). Heat and Mass Transfer: Fundamentals and Applications. McGraw-Hill Education.
3. Halliday, D., Resnick, R., & Walker, J. (2014). Fundamentals of Physics. John Wiley & Sons.
1. Stefan-Boltzmann law: E = σ * T^4
2. Planck’s law: E(λ) ∝ T * λ^(-5)
I hope this article helps you understand heat transfer by radiation better!
Related articles for ‘Heat Transfer by Radiation’ :
• Reading: Heat Transfer by Radiation
Calculators for ‘Heat Transfer by Radiation’
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Boolean Algebra: Basic Laws | Baeldung on Computer Science (2024)
1. Overview
In this tutorial, we’ll study the basic laws used in Boolean algebra.
We’ll start by studying the role that Boolean algebra has in the construction of more complex systems of formal logic. In doing so, we’ll learn how the latter is based upon Boolean algebra, and how
its laws shape it.
We’ll then study the basic and secondary logic operations in Boolean algebra. We’ll also learn how to interpret truth tables and how to use them to prove theorems.
Lastly, we’ll study the basic laws themselves and demonstrate them in terms of truth tables. We’ll subsequently apply these laws to solve simple exercises of equivalence between expressions.
At the end of this tutorial, we’ll be familiar with the foundations of Boolean algebra and know how to prove its basic laws in terms of Boolean operations and truth tables.
2. Relationship Between Boolean Algebra and Logic
In our previous articles on propositional logic and first-order logic, we discussed logical operations between propositions and predicates. These operations are possible because there exist
underlying rules for the conduct of algebraic operations between Boolean terms. In this article, subsequently, we study the foundational rules of Boolean algebra, on top of which other more complex
systems of formal logic are built.
One preliminary note on the notation we use throughout this text: As per the praxis in articles on computer science, we here use the binary notation of 1 and 0 to indicate, respectively, truth and
falsity. Unless otherwise specified, these numbers aren’t used in this article as natural numbers, and therefore operations such as the arithmetical addition “+” are defined differently for them, as
we’ll see shortly.
In our previous article on first-order logic, we claimed that first-order is a generalization over propositional logic. This led to the consideration that first-order logic includes propositional
logic. In the sense that Boolean algebra is a prerequisite for both propositional and first-order logic, we can consider the latter two as including the first:
A more formal way to express this idea is to say that the laws of Boolean algebra, which we’ll see shortly, are valid both in propositional and in first-order logic.
3. Terms and Operations
3.1. Boolean Terms
Boolean terms are terms in a Boolean algebraic formula. In contrast with the definition of terms we used in propositional and first-order logic, Boolean terms are simply variables that can assume one
and only one of the two values in a binary field. There aren’t any other conditions on them, such as being related to factual knowledge about the world, as was the case for propositional logic; or
pertaining to relationships, as was the case for first-order logic.
We can indicate Boolean variables with italic letters of the Latin alphabet, such as literature on the subject, we here provide truth tables as a method to prove theorems.
A truth table is simply a table that shows all possible combinations of values for a finite set of Boolean variables. If a set
This table can be interpreted in natural language by reading it row by row. If we start from the first row, we can say: “
3.2. Basic Boolean Operations
Three basic operations are defined in Boolean algebra, to which correspond as many logical operators. These operators are:
• A unary operator not p
• A binary operator p and q
• And another binary operator p or q
These operators hold the truth values that are enumerated below:
We can read this table in the same way in which we read the previous one. The first row can be read, for example, as “if
These operations are called “basic” because all other operations on any number of variables can be reduced to an ordered succession of not, and, and or operations. This is done by repeatedly applying
the basic laws to expressions, as we’ll see in the section dedicated to the solution of practical exercises.
3.3. Secondary Operations
We can also define other Boolean operations that we call secondary because of their reducibility to a sequence of basic operations. The most common among these are:
• The material conditional if p then q
• The correspondence or equivalence p is equivalent to q, or also p if and only if q
• The exclusive or operator p xor q, or also either p or q
These operators are called secondary because they can be reformulated in terms of basic operations:
The equivalence operator is also sometimes called double conditional and is indicated with a double arrow, facing both variables: theorem proving, to indicate the interchangeability between a
hypothesis and a thesis in a demonstration, but is otherwise of no consequence for our scopes.
The table below shows the demonstration in terms of truth tables of the correspondence between basic and secondary operations:
We can easily test the equivalence between the formulas with secondary operators and those rewritten by using basic operators only. This is done by comparing the respective columns in the tables
above, and noticing how they are equivalent regardless of the values assumed by
4. Basic Laws in Boolean Algebra
4.1. Identity, Annihilator, Idempotence, and Double Negation
The laws in Boolean algebra can be expressed as two series of Boolean terms, comprising of variables, constants, and Boolean operators, and resulting in a valid identity between them. In this sense,
if the first term is, for example, the expression
The first class of laws comprises of those that take one single variable as an input, together with constants if necessary. These laws are the identities, the annihilations, and the idempotence with
regards to the binary operators.
The first two of these laws comprise of the identities for the two operators The two constants of Boolean algebra, 1 and 0, are the identity elements for, respectively,
Identity elements of
The second pair of laws concerns the so-called annihilators. An annihilator is a constant that, when used as input to a binary operator together with a variable, nullifies the contribution that that
variable has on the output of the operation. The constants 0 and 1 are, respectively, the annihilator of
Annihilators of
The third pair of laws that concern exclusively one variable is the one called idempotence. Any variable with regards to the operators
One last law concerning an individual variable is the so-called law of double negation. This law states that the double application of the negation operator to a single variable corresponds to that
4.2. Commutativity and Absorption
The second class of laws concerns the usage of two distinct variables and their relationships. The first group of these is the commutative law of the output of a basic operator is indifferent from
the order in which the two variables are input to it:
Commutativity of
The second pair of laws involving two variables concerns the so-called property of absorption of if any binary operation is performed between two distinct variables, and if the output of this
operation is input to the binary operator that wasn’t used, then the first binary operation performed has no influence on the overall outcome of the formula.
In formal notation, the law states that:
This is the tabular representation of these laws:
Absorption of
4.3. Associativity and Distributivity
The third class comprises the laws that operate on three variables. These laws are the associativity and distributivity properties of the
The associativity laws of a succession of operations involving exclusively one of these operators can be computed in any order, and with the same result:
And this is the tabular representation of these laws. Notice how the insertion of a third element forces us to increase the number of rows from
Associativity of
The last pair of laws concerns the distributive property of if a binary operation has as input the output of the other binary operation, then the former can be computed over each of the inputs of the
latter without any difference in the overall result. In formal notation, the distributive law corresponds to:
And this is the tabular representation of the distributive laws:
Distributivity of
4.4. De Morgan’s Laws
One last set of laws concerns the so-called rules for inference. The rules for inference in a formal system allow the conduct of inferential reasoning over well-formed formulas. In Boolean algebra,
the rules for inferential reasoning take the name of De Morgan’s laws.
These laws state that for each basic binary operator, the negation of that operator corresponds to the output of the negation of the inputs to the other operator. In formal terms, they state that:
We’re going to see how to apply them in an exercise in the next section.
5. Using the Laws of Boolean Algebra
5.1. When Do We Use These Laws
The laws of Boolean algebra allow the simplification of very complex formulas into more manageable expressions. This is particularly important for contexts such as information retrievalwhere the
optimization of the search query may lead to a significant reduction of the time taken to retrieve a target document. This simplification is done by applying the laws of Boolean algebra, such as De
Morgan’s laws on otherwise too-complex Boolean expressions.
Another typical application of these laws in programming concerns the simplification of nested if statements, which can be done with the usage of specialized rule engines that simplify the nested
statements into shorter forms. One last use case for Boolean laws relates to the simplification of logic circuits, which has recently become mandated by the need to simplify quantum circuits.
In all these cases, the rules that we apply correspond perfectly to the laws that we studied above. In this sense, we can say that Boolean algebra is complete under the laws defined above.
5.2. Exercises With Boolean Laws: Distributive Law
We can now see how to use the Boolean laws to simplify two complex Boolean formulas into more manageable expressions. This section and the next can be read as a guided exercise to the application of
Boolean laws.
We can start with this expression:
This expression contains two variables,
Because of the distributive law of
Then we can extract
Finally, because 0 is the identity element of
This argument demonstrates the equivalence of the expressions
5.3. Exercises With Boolean Laws: De Morgan
We can now see a guided exercise in the application of De Morgan’s laws. Let’s consider this formula:
Notice how the term
The whole formula now resembles the first law of De Morgan, which means that we can replace it with its equivalent:
One final application of De Morgan to the expression within brackets then gives us
Notice lastly how the methods we have used can be replicated algorithmically. Two such algorithmic methods, K-map, and the Quine-McCluskey algorithm are commonly used in computer science. They’re
based upon the application of rules for Boolean algebra and perform automatic simplification of Boolean functions, allowing, in turn, the extraction of simple formulas out of complex expressions.
6. Conclusions
In this article, we studied the basic laws of Boolean algebra and showed how to apply them for the simplification of Boolean expressions.
First, we discussed the Boolean operators in terms of truth tables. We also observed how secondary operators can always be expressed in terms of the basic ones.
We then studied the basic laws in their formal notation and also their associated truth tables. In doing so, we learned how to prove that these laws are valid for all values of their variables.
Lastly, we studied how to apply the basic laws of Boolean algebra for the simplification of some complex Boolean expressions. In particular, we saw how to apply the distributive law and De Morgan’s
laws for the solution of training exercises.
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s - Julian Schneider
« on: July 7, 2015, 09:15 »
(3) The PartialCharges are zero if the classical potential you selected does not use coulomb interactions to model the electrostatics of the system. In this case no partial charges on the atoms are
defined and that is what the PartialCharges analysis object tells you.
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Is VPP the amplitude?
Vpp is the maximum peak-to-peak amplitude for the selected output termination (10 Vpp into 50 Ω or 20 Vpp into an open circuit).
What is the value of amplitude?
The amplitude or peak amplitude of a wave or vibration is a measure of deviation from its central value. Amplitudes are always positive numbers (for example: 3.5, 1, 120) and are never negative (for
example: -3.5, -1, -120).
Is Peak-to-Peak same as amplitude?
Techopedia Explains Peak-to-Peak (pk-pk) The two are different, as peak amplitude only gives the maximum positive peak of a waveform, whereas pk-pk amplitude describes the total difference between
the top and the bottom of the wave under observation.
What does VPP represent on a waveform?
Peak-to-peak voltage, VPP, is a voltage waveform which is measured from the top of the waveform, called the crest, all the way down to the bottom of the waveform, called the trough. You can see that
all this is shown in the above diagram.
How do you calculate VPP?
To compute VP-P from the peak voltage, the peak voltage is multiplied by 2. To compute VP-P from the RMS voltage, the RMS voltage is multiplied by 2.8284. To compute VP-P from the average voltage,
the average voltage is multiplied by 3.14159.
How do you find the amplitude and frequency?
Determine the frequency and the amplitude. Answer: The amplitude is 50 and ω = 5000. So the frequency is f = 1/T = ω / 2 π = 795.77 Hz….
Centimeters per period / div. cm
Timebase Y ms
Frequency f = 1/T Hz
How do you find amplitude and distance?
Indeed, based on what we know about the relationship between distance and intensity (the inverse square law, I ∝ 1/d2), we can see that the relationship between distance and amplitude is simply A ∝ 1
/d; amplitude is inversely proportional to distance.
What is amplitude of wave?
amplitude, in physics, the maximum displacement or distance moved by a point on a vibrating body or wave measured from its equilibrium position. It is equal to one-half the length of the vibration
What is unit of amplitude?
SI unit of amplitude is metre (m) as amplitude is the maximum displacement suffered by the particles of the medium from their mean positions during the wave propagation. SI unit of displacement is
metre. so, SI unit of amplitude is metre.
What is voltage PP?
VP−P V P − P : The full voltage between positive and negative peaks of the waveform; that is, the sum of the magnitude of the positive and negative peaks. Vrms V r m s : The root-mean-square or
effective value of a waveform.
How do you find amplitude from peak to peak?
For those programs that wish to display the data as a ‘peak’ value, the RMS value is then divided by 0.707 to obtain the peak amplitude. For those situations where peak to peak amplitudes are
desired, the peak amplitude is simply multiplied by 2.
What is the amplitude of normal p wave?
THE NORMAL AND ABNORMAL P WAVE. The P wave in II is pyramidal in shape with somewhat rounded apex. Its limbs are smooth with no irregularities. The duration of P wave is 0.08-0.10 sec, but is no
greater than 0.11sec The maximal normal amplitude is 2.5mm, but the normal P wave is usually no greater than 2 mm.
What is the amplitude of a wave?
The amplitude of a wave is its height, that is, half the distance from trough to crest. Amplitude can be measured for water waves, sound waves traveling through air, or for any other type of wave
traveling through a gas or liquid.
What is the duration of positive component in P wave?
The P wave usually dominantly positive with relatively small negative component. P wave may be entirely positive with no negative component. The duration of positive component in V1 > 0.04 sec. The
above manifestations are due to greater and more direct alignment of right atrial vector with lead V1.
What is the P wave axis?
P wave is thus a composite deflexion of RA and LA activation. The P wave is inscribed at a constant speed so that the limbs are smooth with no irregularities. THE MEAN FRONTAL PLANE DIRECTION OF
ATRIAL ACTIVATION IS INFERIORLY AND TO THE LEFT. 1. The P wave form in lead II 2. The P wave form in lead V1 3. The frontal plane P wave axis
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[Solved] A. If some NP-complete problem P is in ℙ that ℙ = ℕℙ
A. If some NP-complete problem P is in ℙ that ℙ = ℕℙ
B. TSP is in ℕℙ
C. SAT is in ℕℙ
D. Hamilton circuit problem is not NP-complete
Choose the correct answer from the options given below:
This question was previously asked in
UGC NET Computer Science (Paper 2) 17 June 2023 Official Paper
View all UGC NET Papers >
Answer (Detailed Solution Below)
Option 1 : A, B and C only
UGC NET Paper 1: Held on 11th Dec 2023 Shift 2
21 K Users
50 Questions 100 Marks 60 Mins
The correct answer is A, B and C only
Key Points
• Statement A: "If some NP-complete problem P is in P then P = NP" is correct.
□ It states that if you can come up with a polynomial-time algorithm for a single NP-complete problem, then that would mean all NP problems have polynomial-time solutions -- that is, P would
equal NP. This is the definition of NP-completeness.
• Statement B: "TSP is in NP" is correct.
□ The Travelling Salesman Problem (TSP) is indeed an NP problem: given a list of cities and the distances between them, find the shortest possible route that visits each city and returns to the
origin city. It can be verified quickly, but we do not have a quick solution on how to solve it.
• Statement C: "SAT is in NP" is correct as well.
□ The Boolean satisfiability problem (SAT) is a decision problem, whose instance is a Boolean expression written using only AND, OR, NOT, variables, and parentheses. The question is: given the
expression, is there some assignment of TRUE and FALSE values to the variables that will make the entire expression true? A SAT instance can be checked quickly, but again, we have no quick
algorithm to solve it.
• Statement D: "Hamilton circuit problem is not NP-complete" is incorrect.
□ The Hamiltonian circuit problem, which is finding a path in a graph that visits every vertex exactly once and returns to the original vertex, is a classic example of an NP-complete problem
Latest UGC NET Updates
Last updated on Oct 25, 2024
-> The UGC NET December 2024 Notification will be released soon.
-> The UGC-NET exam takes place for more than 80 subjects, to determine the eligibility for 'Junior Research Fellowship’ and ‘Assistant Professor’ posts.
-> The exam comprises two papers - Paper I and Paper II. Paper I consists of 50 questions and Paper II consists of 100 questions.
-> The candidates who are preparing for the exam can check the UGC NET Previous Year Papers and UGC NET Test Series to boost their preparations.
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The Chromatic Structure of Dense Graphs
This thesis focusses on extremal graph theory, the study of how local constraints on a graph affect its macroscopic structure. We primarily consider the chromatic structure: whether a graph has or is
close to having some (low) chromatic number.
Chapter 2 is the slight exception. We consider an induced version of the classical Turán problem. Introduced by Loh, Tait, Timmons, and Zhou, the induced Turán number ex(n, {H, F-ind}) is the
greatest number of edges in an n-vertex graph with no copy of H and no induced copy of F. We asymptotically determine ex(n, {H, F-ind}) for H not bipartite and F neither an independent set nor a
complete bipartite graph. We also improve the upper bound for ex(n, {H, K_{2, t}-ind}) as well as the lower bound for the clique number of graphs that have some fixed edge density and no induced K_
{2, t}.
The next three chapters form the heart of the thesis. Chapters 3 and 4 consider the Erdős-Simonovits question for locally r-colourable graphs: what are the structure and chromatic number of graphs
with large minimum degree and where every neighbourhood is r-colourable? Chapter 3 deals with the locally bipartite case and Chapter 4 with the general case.
While the subject of Chapters 3 and 4 is a natural local to global colouring question, it is also essential for determining the minimum degree stability of H-free graphs, the focus of Chapter 5.
Given a graph H of chromatic number r + 1, this asks for the minimum degree that guarantees that an H-free graph is close to r-partite. This is analogous to the classical edge stability of Erdős and
Simonovits. We also consider the question for the family of graphs to which H is not homomorphic, showing that it has the same answer.
Chapter 6 considers sparse analogues of the results of Chapters 3 to 5 obtaining the thresholds at which the sparse problem degenerates away from the dense one.
Finally, Chapter 7 considers a chromatic Ramsey problem first posed by Erdős: what is the greatest chromatic number of a triangle-free graph on vertices or with m edges? We improve the best known
bounds and obtain tight (up to a constant factor) bounds for the list chromatic number, answering a question of Cames van Batenburg, de Joannis de Verclos, Kang, and Pirot.
Extremal Graph Theory, Combinatorics, Ramsey Theory, Graph Colouring, Stability, Dense Graphs
Doctor of Philosophy (PhD)
Awarding Institution
University of Cambridge
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Store arbitrary precision numeric data | Spanner | Google Cloud
Spanner provides the NUMERIC type that can store decimal precision numbers exactly. The semantics of the NUMERIC type in Spanner varies between its two SQL dialects (GoogleSQL and PostgreSQL),
especially around the limits on scale and precision:
• NUMERIC in the PostgreSQL dialect is an arbitrary decimal precision numeric type (scale or precision can be any number within the supported range) and thus is an ideal choice for storing
arbitrary precision numeric data.
• NUMERIC in GoogleSQL is a fixed precision numeric type (precision=38 and scale=9) and cannot be used to store arbitrary precision numeric data. When you need to store arbitrary precision numbers
in GoogleSQL dialect databases, we recommend that you store them as strings.
Precision of Spanner numeric types
Precision is the number of digits in a number. Scale is the number of digits to the right of the decimal point in a number. For example, the number 123.456 has a precision of 6 and a scale of 3.
Spanner has three numeric types:
• 64-bit signed integer type called INT64 in the GoogleSQL dialect and INT8 in the PostgreSQL dialect.
• IEEE 64-bit (double) binary precision floating-point type called FLOAT64 in the GoogleSQL dialect and FLOAT8 in the PostgreSQL dialect.
• Decimal precision NUMERIC type.
Let's look at each in terms of precision and scale.
INT64 / INT8 represents numeric values that do not have a fractional component. This data type provides 18 digits of precision, with a scale of zero.
FLOAT64 / FLOAT8 can only represent approximate decimal numeric values with fractional components and provides 15 to 17 significant digits (count of digits in a number with all trailing zeros
removed) of decimal precision. We say that this type represents approximate decimal numeric values because IEEE 64-bit floating point binary representation that Spanner uses cannot precisely
represent decimal (base-10) fractions (it can represent only base-2 fractions exactly). This loss of precision introduces rounding errors for some decimal fractions.
For example, when you store the decimal value 0.2 using the FLOAT64 / FLOAT8 data type, the binary representation converts back to a decimal value of 0.20000000000000001 (to 18 digits of precision).
Similarly (1.4 * 165) converts back to 230.999999999999971 and (0.1 + 0.2) converts back to 0.30000000000000004. This is why 64-bit floats are described as only having 15-17 significant digits of
precision (only some numbers with more than 15 decimal digits can be represented as 64-bit float without rounding). For more details on how floating point precision is calculated, see
Double-precision floating-point format.
Neither INT64 / INT8 nor FLOAT64 / FLOAT8 has the ideal precision for financial, scientific, or engineering calculations, where a precision of 30 digits or more is commonly required.
The NUMERIC data type is suitable for those applications, since it is capable of representing exact decimal precision numeric values having precision of more than 30 decimal digits.
The GoogleSQL NUMERIC data type can represent numbers with a fixed decimal precision of 38 and fixed scale of 9. The range of GoogleSQL NUMERIC is -99999999999999999999999999999.999999999 to
The PostgreSQL dialect NUMERIC type can represent numbers with a maximum decimal precision of 147,455 and a maximum scale of 16,383.
If you need to store numbers that are larger than the precision and scale offered by NUMERIC, the following sections describe some recommended solutions.
Recommendation: store arbitrary precision numbers as strings
When you need to store an arbitrary precision number in a Spanner database, and you need more precision than NUMERIC provides, we recommend that you store the value as its decimal representation in a
STRING / VARCHAR column. For example, the number 123.4 is stored as the string "123.4".
With this approach, your application must perform a lossless conversion between the application-internal representation of the number and the STRING / VARCHAR column value for database reads and
Most arbitrary precision libraries have built-in methods to perform this lossless conversion. In Java, for example, you can use the BigDecimal.toPlainString() method and the BigDecimal(String)
Storing the number as a string has the advantage that the value is stored with exact precision (up to the STRING / VARCHAR column length limit), and the value remains human-readable.
Perform exact aggregations and calculations
To perform exact aggregations and calculations on string representations of arbitrary precision numbers, your application must perform these calculations. You cannot use SQL aggregate functions.
For example, to perform the equivalent of a SQL SUM(value) over a range of rows, the application must query the string values for the rows, then convert and sum them internally in the app.
Perform approximate aggregations, sorting, and calculations
You can use SQL queries to perform approximate aggregate calculations by casting the values to FLOAT64 / FLOAT8.
SELECT SUM(CAST(value AS FLOAT64)) FROM my_table
SELECT SUM(value::FLOAT8) FROM my_table
Similarly, you can sort by numeric value or limit values by range with casting:
SELECT value FROM my_table ORDER BY CAST(value AS FLOAT64);
SELECT value FROM my_table WHERE CAST(value AS FLOAT64) > 100.0;
SELECT value FROM my_table ORDER BY value::FLOAT8;
SELECT value FROM my_table WHERE value::FLOAT8 > 100.0;
These calculations are approximate to the limits of the FLOAT64 / FLOAT8 data type.
There are other ways to store arbitrary precision numbers in Spanner. If storing arbitrary precision numbers as strings does not work for your application, consider the following alternatives:
Store application-scaled integer values
To store arbitrary precision numbers, you can pre-scale the values before writing, so that numbers are always stored as integers, and re-scale the values after reading. Your application stores a
fixed scale factor, and the precision is limited to the 18 digits provided by the INT64 / INT8 data type.
Take, for example, a number that needs to be be stored with an accuracy of 5 decimal places. The application converts the value to an integer by multiplying it by 100,000 (shifting the decimal point
5 places to the right), so the value 12.54321 is stored as 1254321.
In monetary terms, this approach is like storing dollar values as multiples of milli-cents, similar to storing time units as milliseconds.
The application determines the fixed scaling factor. If you change the scaling factor, you must convert all of the previously scaled values in your database.
This approach stores values that are human-readable (assuming you know the scaling factor). Also, you can use SQL queries to perform calculations directly on values stored in the database, as long as
the result is scaled correctly and does not overflow.
Store the unscaled integer value and the scale in separate columns
You can also store arbitrary precision numbers in Spanner using two elements:
• The unscaled integer value stored in a byte array.
• An integer that specifies the scaling factor.
First your application converts the arbitrary precision decimal into an unscaled integer value. For example, the application converts 12.54321 to 1254321. The scale for this example is 5.
Then the application converts the unscaled integer value into a byte array using a standard portable binary representation (for example, big-endian two's complement).
The database then stores the byte array (BYTES / BYTEA) and integer scale (INT64 / INT8) in two separate columns, and converts them back on read.
In Java, you can use BigDecimal and BigInteger to perform these calculations:
byte[] storedUnscaledBytes = bigDecimal.unscaledValue().toByteArray();
int storedScale = bigDecimal.scale();
You can read back to a Java BigDecimal using the following code:
BigDecimal bigDecimal = new BigDecimal(
new BigInteger(storedUnscaledBytes),
This approach stores values with arbitrary precision and a portable representation, but the values are not human-readable in the database, and all calculations must be performed by the application.
Store application internal representation as bytes
Another option is to serialize the arbitrary precision decimal values to byte arrays using the application's internal representation, then store them directly in the database.
The stored database values are not human-readable, and the application needs to perform all calculations.
This approach has portability issues. If you try to read the values with a programming language or library different from the one that originally wrote it, it might not work. Reading the values back
might not work because different arbitrary precision libraries can have different serialized representations for byte arrays.
What's next
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Maths Test Bank Year 5 | SSRC
Maths Test Bank Year 5
ISBNÂ 9781760323288
Maths Test Bank is a total maths assessment solution that provides both assessment of learning and assessment for learning. Linked explicitly to the Australian Curriculum and NSW Syllabus for the
Australian Curriculum, Maths Test Bank blends seamlessly with any maths teaching resource or approach.
Maths Test Bank:
• is easy for students and parents to follow
• provides teachers with assessment content that shows basic understanding and fluency of a topic, and also provides students with content that enables them to:
□ think about their understanding of a topic
□ extend themselves by applying the concepts further or by comparing different ways of working
□ show and explain the way they have solved a problem or why a method does or doesnÂ’t work
• provides evidence of a studentÂ’s level of achievement in every content strand of mathematics (assessment of learning), but also provides teachers with verification of a studentÂ’s proficiency
level in the key areas of understanding, fluency, problem solving and reasoning (assessment for learning)
• includes a grading guide and suggestions for students who are achieving beyond or below the expected level of achievement.
Part 1: Assessment of Learning
The assessment of learning is designed to help the teacher (and the student) measure progress towards achievement standards. The aim of this assessment is to find out what a student has learned and
to help answer questions such as:
• Where was the student?
• Where is the student now?
• Where does the student need to go to next?
Part 2: Assessment for Learning
Assessment for learning differs from assessment of learning because its main focus is inquiry into the learning process. Teachers are able to look at the way students learn, rather than at their
level of achievement. This analysis can then inform future teaching and learning requirements.
This assessment focuses on studentsÂ’ level of development in the proficiency strands of the mathematics curriculum (understanding, fluency, problem solving and reasoning) and their ability to
reflect on, reason, explain, explore and adapt key mathematical concepts.
You may also like…
101 Must-Know Challenging Maths Word Problems for Primary 5 presents word problems that test important concepts so students can learn to apply general mathematical problem-solving strategies and
heuristics confidently. This book comprises word problems often encountered by students in their tests and examinations. The questions are categorised into respective topics in accordance with the
topics in the Singapore mathematics syllabus.
101 Must-Know Challenging Maths Word Problems for Primary 5 presents word problems that test important concepts so students can learn to apply general mathematical problem-solving strategies and
heuristics confidently. This book comprises word problems often encountered by students in their tests and examinations. The questions are categorised into respective topics in accordance with the
topics in the Singapore mathematics syllabus.
Part 1: Assessment of Learning
The assessment of learning is designed to help the teacher (and the student) measure progress towards achievement standards. The aim of this assessment is to find out what a student has learned and
to help answer questions such as:
• Where was the student?
• Where is the student now?
• Where does the student need to go to next?
Part 2: Assessment for Learning
Assessment for learning differs from assessment of learning because its main focus is inquiry into the learning process. Teachers are able to look at the way students learn, rather than at their
level of achievement. This analysis can then inform future teaching and learning requirements. This assessment focuses on studentsÂ’ level of development in the proficiency strands of the mathematics
curriculum (understanding, fluency, problem solving and reasoning) and their ability to reflect on, reason, explain, explore and adapt key mathematical concepts.
• Contents
□ Assessment of and for Learning: Introduction
□ Grading Guide
□ Curriculum Overview
□ Assessment of and for Learning: The Tests
□ Number and Algebra
☆ 1A Place value
☆ 1B Number properties
☆ 1C Mental strategies for addition and subtraction
☆ 1D Written strategies for addition and subtraction
☆ 1E Mental strategies for multiplication and division
☆ 1F Written strategies for multiplication and division
☆ 1G Integers
☆ 2A Fractions
☆ 2B Addition and subtraction of common fractions
☆ 2C Decimal fractions
☆ 2D Addition and subtraction of decimals
☆ 2E Multiplication and division of decimals
☆ 2F Decimals and powers of ten
☆ 2G Percentage, fractions and decimals
☆ 3A Geometric patterns
☆ 3B Number patterns
☆ 3C Order of operations and equations
□ Measurement and Geometry
☆ 4A Length
☆ 4B Area
☆ 4C Volume and capacity
☆ 4D Mass
☆ 4E Timetables
☆ 5A Angles
☆ 5B 2D shapes and 3D objects
☆ 6A Transformations
☆ Statistics and Probability
☆ 6B Cartesian coordinate systems
☆ 7B Interpreting data
☆ 7C Data in the media
☆ 7D Probability
☆ 7E Chance experiments and simulations
□ From Assessment to Instruction: Teacher Support
□ Answers
Part 1: Assessment of Learning
The assessment of learning is designed to help the teacher (and the student) measure progress towards achievement standards. The aim of this assessment is to find out what a student has learned and
to help answer questions such as:
• Where was the student?
• Where is the student now?
• Where does the student need to go to next?
Part 2: Assessment for Learning
Assessment for learning differs from assessment of learning because its main focus is inquiry into the learning process. Teachers are able to look at the way students learn, rather than at their
level of achievement. This analysis can then inform future teaching and learning requirements. This assessment focuses on studentsÂ’ level of development in the proficiency strands of the mathematics
curriculum (understanding, fluency, problem solving and reasoning) and their ability to reflect on, reason, explain, explore and adapt key mathematical concepts.
• Contents
□ Assessment of and for Learning: Introduction
□ Grading Guide
□ Curriculum Overview
□ Assessment of and for Learning: The Tests
□ Number and Algebra
☆ 1A Place value
☆ 1B Number properties
☆ 1C Mental strategies for addition and subtraction
☆ 1D Written strategies for addition and subtraction
☆ 1E Mental strategies for multiplication and division
☆ 1F Written strategies for multiplication and division
☆ 1G Integers
☆ 2A Fractions
☆ 2B Addition and subtraction of common fractions
☆ 2C Decimal fractions
☆ 2D Addition and subtraction of decimals
☆ 2E Multiplication and division of decimals
☆ 2F Decimals and powers of ten
☆ 2G Percentage, fractions and decimals
☆ 3A Geometric patterns
☆ 3B Number patterns
☆ 3C Order of operations and equations
□ Measurement and Geometry
☆ 4A Length
☆ 4B Area
☆ 4C Volume and capacity
☆ 4D Mass
☆ 4E Timetables
☆ 5A Angles
☆ 5B 2D shapes and 3D objects
☆ 6A Transformations
☆ Statistics and Probability
☆ 6B Cartesian coordinate systems
☆ 7B Interpreting data
☆ 7C Data in the media
☆ 7D Probability
☆ 7E Chance experiments and simulations
□ From Assessment to Instruction: Teacher Support
□ Answers
This book should prove to be an invaluable teaching aid for teachers, coaches and tutorial centers, because it thoroughly summarises the 10 major topics which are the basis of the Australian
Curriculum. It will be very beneficial to parents, because it will provide them with a very structured and clear idea of the core syllabus, and what their children should know by the end of Grade 5.
If their child has a particular problem (say on adding fractions) it is very easy to find the page and explanations relating to that idea – and hence help their child. Most important of all, it
will prove to be an excellent reference for students of all ability groups. The user friendly format and layout makes it very much faster for a pupil to thoroughly master one major topic in a
relatively short period of time, because it is so easy to see how each idea is linked to the previous one. It has all the rules and corresponding examples clearly set out topic by topic, page by
page. In addition it teaches the pupils to read explanations, as well as to look back and research similar problems. And of course the graded exercises at the end of each topic chapter will help
students of all abilities to practise and apply their knowledge. It is so easy for students to work through the book by themselves with the minimum of supervision and help. This advanced edition ofÂ
Understanding Maths Year 5
 provides graded exercises that will test student of most ability groups, and in many of the chapters, will extend students to concepts which are usually covered in Year 6. The questions are graded
into levels of difficulty:
• Easier questions:Â These Level 1 (and sometimes Level 2) questions are intended to build confidence and follow the format of the examples.
• Average questions:Â Level 2 & 3 questions are of average difficulty level, and give all students (weak, average & gifted) a good opportunity to practice and consolidate the ideas and rules given
throughout most of the chapter. All students should try to complete and understand questions in the first three levels.
• Harder questions:Â These questions are more difficult as they involve larger numbers, and some of the more difficult ideas in the related topic. Reference pages numbers and not included so that
students learn to search through the chapter for the relevant information.
• Problem solving:Â This more difficult level has been included to challenge those students who are more gifted at Maths. Usually the questions are more sentence and problem oriented, and therefore
they involve more reading and comprehension skills. It is unlikely that any of the questions in this level can be done mentally, because several different ideas, rules or steps are usually
• Introduction
• Number & Algebra
□ Number and Place Value
□ Fractions and Decimals
□ Money and Financial Mathematics
□ Patterns and Algebra
• Measurement & Geometry
□ Using Units of Measurement
□ Shape
□ Location and Transformation
□ Geometric Reasoning
• Statistics & Probability
□ Chance
□ Data Representation and Interpretation
This book should prove to be an invaluable teaching aid for teachers, coaches and tutorial centers, because it thoroughly summarises the 10 major topics which are the basis of the Australian
Curriculum. It will be very beneficial to parents, because it will provide them with a very structured and clear idea of the core syllabus, and what their children should know by the end of Grade 5.
If their child has a particular problem (say on adding fractions) it is very easy to find the page and explanations relating to that idea – and hence help their child. Most important of all, it
will prove to be an excellent reference for students of all ability groups. The user friendly format and layout makes it very much faster for a pupil to thoroughly master one major topic in a
relatively short period of time, because it is so easy to see how each idea is linked to the previous one. It has all the rules and corresponding examples clearly set out topic by topic, page by
page. In addition it teaches the pupils to read explanations, as well as to look back and research similar problems. And of course the graded exercises at the end of each topic chapter will help
students of all abilities to practise and apply their knowledge. It is so easy for students to work through the book by themselves with the minimum of supervision and help. This advanced edition ofÂ
Understanding Maths Year 5
 provides graded exercises that will test student of most ability groups, and in many of the chapters, will extend students to concepts which are usually covered in Year 6. The questions are graded
into levels of difficulty:
• Easier questions:Â These Level 1 (and sometimes Level 2) questions are intended to build confidence and follow the format of the examples.
• Average questions:Â Level 2 & 3 questions are of average difficulty level, and give all students (weak, average & gifted) a good opportunity to practice and consolidate the ideas and rules given
throughout most of the chapter. All students should try to complete and understand questions in the first three levels.
• Harder questions:Â These questions are more difficult as they involve larger numbers, and some of the more difficult ideas in the related topic. Reference pages numbers and not included so that
students learn to search through the chapter for the relevant information.
• Problem solving:Â This more difficult level has been included to challenge those students who are more gifted at Maths. Usually the questions are more sentence and problem oriented, and therefore
they involve more reading and comprehension skills. It is unlikely that any of the questions in this level can be done mentally, because several different ideas, rules or steps are usually
• Introduction
• Number & Algebra
□ Number and Place Value
□ Fractions and Decimals
□ Money and Financial Mathematics
□ Patterns and Algebra
• Measurement & Geometry
□ Using Units of Measurement
□ Shape
□ Location and Transformation
□ Geometric Reasoning
• Statistics & Probability
□ Chance
□ Data Representation and Interpretation
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Homogeneous Linear Diophantine Equations
This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations.
We formalize the theory of homogeneous linear diophantine equations, focusing on two main results: (1) an abstract characterization of minimal complete sets of solutions, and (2) an algorithm
computing them. Both, the characterization and the algorithm are based on previous work by Huet. Our starting point is a simple but inefficient variant of Huet's lexicographic algorithm incorporating
improved bounds due to Clausen and Fortenbacher. We proceed by proving its soundness and completeness. Finally, we employ code equations to obtain a reasonably efficient implementation. Thus, we
provide a formally verified solver for homogeneous linear diophantine equations.
Session Diophantine_Eqns_Lin_Hom
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Welcome to Michael-Förster.de
I am a software engineer with a PhD in computer science. This site is mainly intended for notes and hints that I find useful for the daily work of a software engineer.
• Notes about daily technical issues
• Mixed notes about all sorts of things on micki-foerster.de
Algorithmic Differentiation of Pragma-Defined Parallel Regions
My PhD disseration was published by Springer Verlag and can be found on Amazon here:
"Algorithmic Differentiation of Pragma-Defined Parallel Regions: Differentiating Computer Programs Containing OpenMP"
Briefly, its topic is a correctness proof of a source code transformation. The source code transformation creates C code containing the code for calculating higher-order derivative values of a given
function. The thesis examines how OpenMP pragmas inside the source file must be handled in the output of the transformation in order to keep correctness of the concurrent execution.
It turns out that in the case of tangent-linear transformation the transformation is relatively simple. In case of the adjoint source transformation with its inherit reverse of the data flow, the
correctness proof was all but easy. Actually, it takes a couple of pages to handle all the possible cases.
The corresponding implementation of a source transformation tool called SPLc can be found on github.
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6.5440 Algorithmic Lower Bounds: Fun with Hardness Proofs (Fall 2023)
[+] 3-partition I. 2-partition vs. 3-partition; variations (Subset Sum, Numerical 3-dimensional matching, 3DM, X3C); weakly vs. strongly NP-hard; pseudopolynomial vs. polynomial. Multiprocessor
scheduling, rectangle packing, edge-matching puzzles, jigsaw puzzles, polyform packing puzzles, packing squares into a square. This lecture introduces my favorite (and a generally lesser known)
starting point for NP-hardness reductions, called 3-partition. This problem is particularly useful when you have a problem that involves adding up numbers, even when those numbers must be encoded in
unary (a common feature of many puzzles). We'll discuss many variations of the problem:
• 2-partition: Partition integers into two sets of equal sum
• Subset Sum: Select integers to equal a target sum
• 3-partition: Partition n integers into n/3 triples of equal sum
• Numerical 3-dimensional matching: Integers are of three different types, and each triple must have all three types.
• 3-dimensional matching: A generalization to tripartite hypergraphs.
• Exact cover by 3-sets: A generalization to hypergraphs.
2-partition vs. 3-partition is an example of the weak vs. strong NP-hardness dichotomy, and on the algorithmic side, the pseudopolynomial vs. (weakly) polynomial dichotomy. We'll see weak and strong
NP-hardness proofs, by reductions from 2-partition and 3-partition respectively, for two problems:
• multiprocessor scheduling
• packing rectangles into a rectangle
Next we'll see a fun series of reductions between different puzzles, starting from 3-partition / rectangle packing to establish strong NP-hardness.
• edge-matching puzzles ("signed" like lizards, and "unsigned" like Eternity II)
• jigsaw puzzles
• polyomino packing puzzles (like Eternity)
Finally, we'll see how to prove strong NP-hardness of packing squares into a square. This is a handy result that we'll use as the basis for another reduction next lecture.
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Riders I STEM Education Platform
Riders’ Institutional STEM Package is a discounted bundle offer for middle and high school levels that facilitates easier learning of programming for students and offers a unique experience for
teachers! Utilized and endorsed by schools internationally.
Riders helps put students on the path to better academic outcomes, new interests and career opportunities. The unique Riders program combines video tutorials with robotic simulations that let
students apply those newly learned theories. Once they’ve mastered their studies, Riders also gives students the opportunity to compete globally in robotics competitions where they can experience all
the excitement of e-sports. The motivation and adrenaline provided by realistic simulations shows how rewarding and fun it can be to specialize in STEM subjects! Riders’ competitions offer students a
way to recognize their own development and to showcase their new talents.
Year-round availability of the robotics programming courses and unlimited practice opportunities
The organization of Python or Blockly-based competitions specifically for your institution
Opportunity to compete in the global championship as one of the top 3 teams from the Local League competitions
A single investment allowing a school to teach coding to all its students
Free entrance for 2 teams in the Riders Robotics League competition, the most advanced educational on-line robotics league.
Zero kit and maintenance costs while providing STEM-based robotics coding training
Minimised costs per student for learning robotics and joining a competitive robotics league
The opportunity to build student skills in a fun way, by combining classroom learning with the reinforcement of a competitive environment to test new skills and knowledge
Offer students the enjoyment of stimulating educational content presented as narrated course content that offers gradually more challenging tasks.
Student data is protected within the framework of Personal Data Protection Laws.Student data is shared only with the relevant school.
Riders is readily compatible with school LMS systems and testing the algorithms developed in our simulation environments only takes seconds.
Provided as guide documentation for educators, the Educator Resource Packs provide a detailed description of each task and solution.
Video content includes tips on how to help students who are progressing at different speeds. All questions are answered in support sessions.
We will learn to make decisions in robotic coding.Basic Python Commands: Discover simple Python commands that will enable you to perform a task.Algorithms: Connect multiple instructions to create a
sequence.Problem Solving: Solve increasingly difficult logic puzzles.Simple Transition: Move from one point to another along a floor in single steps.Simple Rotation: Learn right and left turns within
an algorithm.
We will learn about while loops.While Loops: Implement While loops in Python.Patterns: Recognize and apply patterns.Algorithms: Connect multiple instructions to create a sequence.Simple Transition:
Move from one point to another along a floor in single steps. Simple Rotation: Turn right and left in an algorithm
We will learn to use conditional statements in robotic coding.Conditional Statements: Implement if and elif statements in Python.Algorithms: Implement adaptive algorithms that respond to current
conditions.Simple Transition: Move from one point to another along a grid in single steps.Simple Rotation: Turn right and left in an algorithm.
We will learn about for loops.For Loops: Repeat a certain number of times over a set course.Refactoring: Improve code efficiency.Patterns: Recognize and apply patterns for algorithm
development.Transition: Move from one point to another along a floor using fractional steps.Coordinates: Recognize the points marked on an obstacle course.Rotation: Rotate using radians to apply
simple left/right rotations.
We will learn about path finding and flood-fill algorithms.Double For Loops: Implement nested for loops.While Loops: Implement while loops in python.Path Finding: Implement algorithms to find the
shortest path on a grid.Algorithms: Implement adaptve algorithms which respond to currentconditons.2D Coordinates: Work with data assigned on a 2D grid.
We will learn about feedback and contnuous-tme commands.Translaton: Control robot velocity using meters/second.Rotaton: Control robot angular velocity using radians/second.Feedback Algorithms:
Implement feedback to create a stable control algorithm.Sensors: Use a distance sensor as an input to an algorithm.Optmizaton: Tune an algorithm to improve a result.
We will learn about arrays and contnue to improve our skills with feedback algorithms.Translaton: Control robot velocity using meters/second.Rotaton: Control robot angular velocity using radians/
second.Arrays: Work with a 1D array in python.Image Processing: Read a 1D camera image and interpret the pixel data.Feedback Algorithms: Implement feedback to create a stable control algorithm.
We will learn to make decisions in robotic coding.Basic Blockly Commands: Discover simple Blockly commands that will enable you to perform a task.Algorithms: Connect multiple instructions to create a
sequence.Problem Solving: Solve increasingly difficult logic puzzles.Simple Transition: Move from one point to another along a floor in single steps.Simple Rotation: Learn right and left turns within
an algorithm.
We will learn about while loops.While Loops: Implement While loops in Blockly.Patterns: Recognize and apply patterns.Algorithms: Connect multiple instructions to create a sequence.Simple Transition:
Move from one point to another along a floor in single steps. Simple Rotation: Turn right and left in an algorithm
We will learn to use conditional statements in robotic coding.Conditional Statements: Implement if and elif statements in Blockly.Algorithms: Implement adaptive algorithms that respond to current
conditions.Simple Transition: Move from one point to another along a grid in single steps.Simple Rotation: Turn right and left in an algorithm.
We will learn about path finding and flood-fill algorithms.Double For Loops: Implement nested for loops.While Loops: Implement while loops in Blockly.Path Finding: Implement algorithms to find the
shortest path on a grid.Algorithms: Implement adaptve algorithms which respond to currentconditons.2D Coordinates: Work with data assigned on a 2D grid.
We will learn about arrays and contnue to improve our skills with feedback algorithms.Translaton: Control robot velocity using meters/second.Rotaton: Control robot angular velocity using radians/
second.Arrays: Work with a 1D array in Blockly.Image Processing: Read a 1D camera image and interpret the pixel data.Feedback Algorithms: Implement feedback to create a stable control algorithm.
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Who Invented Math? The History of Mathematics
Who invented math? It’s a deceptively complex question—a lot harder than 2+2. Math has been around forever, but we are always learning more about it.
Short answer: Many different people invented math, including ancient societies and many famous mathematicians who came along later.
The long answer: It depends on what kind of math you’re asking about. Below is a look at the history of mathematics and the people who contributed to developing math as we know it today.
Jump to:
What is math?
According to Britannica Kids, math is the study of numbers. It’s a kind of language that we use every day to calculate distances, tell time, build things, and so on.
Mathematicians think about math in two areas: pure and applied. Pure math is studying math for its own sake. Figuring out how to solve a particular algorithm or tackling a theory, for example.
Applied math is using math to solve real-life problems, like building a house or predicting an earthquake.
There are lots of different types of math: arithmetic, algebra, geometry, trigonometry, statistics, and more.
So, since math is already a part of the world, the first question is, can math be invented at all?
Was math discovered or invented?
Some mathematicians think that math is invented, as people name aspects of math or create new ways of solving problems. Other people think that math is always there—the concepts and ideas exist in
nature, just waiting for us to discover them.
So, who invented math?
Here’s a look at the history of math and many of the societies and people who contributed to its development.
Early Societies
Jeff Dahl, public domain, via Wikimedia Commons
Math has evolved over thousands of years, with input from thousands of mathematicians. We don’t know exactly how prehistoric humans dealt with math problems (like counting how many berries they
picked, or figuring out the distance between two places), but researchers believe that people were using addition, multiplication, and other math concepts in early China, India, and Mesopotamia.
In fact, the oldest clay tablets we have with math inscribed on them are more than 4,000 years old. They’re from Mesopotamia. We also have Egyptian papyrus sheets with math written on them. So,
there’s evidence of math from the two oldest societies in the world.
Around 1800 B.C.E., the ancient Babylonians developed a number system based on the number 60 (it’s still used today to think about angle measurement). They were the first people we know of to use
actual numbers to represent amounts.
It’s clear that, considering the pyramids and their society, the Egyptians used math. They definitely understood geometry and even had a formula for calculating the volume of a truncated pyramid.
The Ancient Greeks
Anderson, CC0, via Wikimedia Commons
There’s more information about who invented (or discovered) math concepts as human society evolved. The Greeks, more than 2,500 years ago, started doing more advanced math. Plato, Euclid, and
Archimedes are still remembered for their mathematical achievements. For example, Pythagoras studied triangles and he invented what we learn about triangles, called the Pythagorean theorem.
We also know that in ancient Greece, math became something to study, and mathematicians started thinking about specific theories and building on one another’s work.
After Ancient Greece
Godfrey Kneller, public domain, via Wikimedia Commons
After ancient Greece, mathematicians continued making new discoveries and new theories and solving new problems. In 17th-century England, Sir Isaac Newton developed the field of calculus on his own.
At the same time, in Germany, Gottfried Leibniz was also involved in developing calculus. Some mathematicians have created problems and hypotheses that have never been solved, like Bernhard Riemann,
who created the Riemann hypothesis, which has been attempted but never proven.
And throughout history, women have also studied math and invented math concepts. For example, Emmy Noether gained recognition for her innovations in advanced algebra, and Katherine Johnson calculated
and analyzed flight paths for spacecraft that sent astronauts to the moon. Mathematicians of color who have made significant contributions to mathematics include Fern Hunt, who created math models to
describe different kinds of movement, and Mark Dean, a mathematician and computer scientist who holds patents on the computer that all PCs are based upon.
As math has evolved, people are building on what we know to create new types of math and new ways to use math, like applying math to build computers and create game theory, a branch of applied
mathematics. So, maybe the question isn’t who invented math, but what will math invent next?
Videos About the Invention of Math
Use these videos to explore how different math concepts came about.
The Origin of Numbers
How Old Is Zero?
Where Do Math Symbols Come From?
Who Invented Algebra?
Who Invented Geometry?
Who Invented Trigonometry?
More Teaching Resources
Plus, get all the latest teaching tips and tricks when you sign up for our newsletters!
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How to pool average predictions from a multinomial regression per category in R
Multinomial regression is a powerful statistical method used when the outcome variable consists of more than two categories. However, understanding how to derive average predictions from this type of
regression can be challenging. In this article, we will explore how to pool average predictions for each category of a multinomial regression model using R.
Problem Scenario
You may have a dataset with multiple categorical outcomes and want to predict the probabilities for each category. After fitting a multinomial regression model, you may need to compute the average
predictions for each category across your data.
The original code you might start with could look like this:
# Example data
data <- data.frame(
outcome = factor(c('A', 'B', 'C', 'A', 'B', 'C')),
predictor1 = c(1, 2, 3, 1, 2, 3),
predictor2 = c(3, 2, 1, 3, 2, 1)
# Fitting the multinomial regression model
model <- multinom(outcome ~ predictor1 + predictor2, data = data)
# Getting predictions
predictions <- predict(model, type = "prob")
Step-by-Step Approach to Pooling Average Predictions
Now, let's look at how to compute and pool average predictions for each category.
1. Generate Predictions: Use the predict() function to get predicted probabilities for each category based on your regression model.
2. Calculate Averages: We will then compute the mean predictions for each category using the apply() function.
Example Code
Here is the modified code that calculates average predictions for each category:
# Example data
data <- data.frame(
outcome = factor(c('A', 'B', 'C', 'A', 'B', 'C')),
predictor1 = c(1, 2, 3, 1, 2, 3),
predictor2 = c(3, 2, 1, 3, 2, 1)
# Fitting the multinomial regression model
model <- multinom(outcome ~ predictor1 + predictor2, data = data)
# Getting predictions
predictions <- predict(model, type = "prob")
# Convert predictions to a data frame
predictions_df <- as.data.frame(predictions)
# Calculate average predictions for each category
average_predictions <- predictions_df %>%
summarise(across(everything(), mean))
Detailed Analysis
In the above code, we first create a data frame to simulate the categorical outcome variable and two predictor variables. We then fit a multinomial regression model using the multinom() function from
the nnet package. After obtaining predicted probabilities, we convert them into a data frame and use the dplyr package to compute the average predictions for each category by applying the mean()
function across all predictions.
Practical Example
Suppose you are analyzing survey data where respondents can choose their preferred mode of transport: Car, Bus, or Bike. Using multinomial regression, you can fit a model that relates preferences to
demographic factors like age and income. Once fitted, you could follow the steps above to pool average predictions, allowing stakeholders to understand which mode of transport is generally preferred
across different demographics.
Pooling average predictions from a multinomial regression in R can provide valuable insights into categorical outcomes. By utilizing the nnet and dplyr libraries, you can efficiently obtain these
averages and make informed decisions based on your data.
Useful Resources
By following this guide, you should be able to apply these concepts and techniques to your own categorical data analysis projects, optimizing your data predictions in R effectively. Happy analyzing!
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Flirting with Volatility | My Spreadsheet Lab
I had some questions and improvement ideas for my formula pixel face. I learned a lot playing around with this.
My curiosity got the best of me in 2019. Challenge:
Can I make a pixel face move without VBA? (post)
Can I display a message when the face intersects a cell with a number?
Previously, I used VBA to make the face move.
Updated Pixel Face
Download my updated Excel file here and follow along below.
Now what?
I couldn’t get several questions out of my head:
1. how did I use formulas to move the face?
2. can I incorporate some dynamic arrays?
3. can I make formulas less volatile & more efficient?
1. How does the Face Move?
(a)Each face item has a starting row/column position.
The blue left eye = row 9 and column 8.
(b)Spin buttons added. Cells below “Up/Down” & “Left/Right” above receive spin button values.
(c)Below, current positions = original positions adjusted by spin button values.
Currently values are identical as “Up/Down” & “Left/Right” both = 0. Click spin buttons to change them.
(d)Conditional Formatting rules added for each face item based on (c).
By the way, I just noticed “Duplicate Row” option. How long has that been there?
2. Incorporate Dynamic Arrays?
In 2019 I just put numbers in cells. Conditional formatting added the purple color.
Could dynamic arrays create purple obstacles that vary in size?
Goal: move face to green area without touching purple cells . Can I make purple areas change size?
I experimented with formulas like these to create some variability:
• RANDARRAY(RANDBETWEEN(1,4),RANDBETWEEN(1,4),1,1,TRUE)
• SEQUENCE(2,3,INT(RAND()*10),INT(RAND()*10))
• SEQUENCE(RANDBETWEEN(1,3),4,1,0))
• IFERROR(SEQUENCE(INT(RAND()*4),2,5,0),1)
RANDBETWEEN or RAND inside SEQUENCE produced occasional #SPILL! errors.
WHY? How could it sometimes work? Answer from Microsoft:
Occasional #SPILL! errors annoyed me so I created a non volatile way to create variability! See area with the purple background starting in cell AQ54. INDEX/MATCH inside sequence uses this area.
If the face intersects a purple cell a message will be displayed (see cell AW78).
3. Less volatile & more efficient?
Some insist on never using volatile functions. I agree 99% of the time (post).
Remove Volatile Functions:
My original formula to test overlap between face and number cells used INDIRECT (!volatile!).
=IFERROR(SUM(INDIRECT(ADDRESS(AI126,AJ126)) Face),””)
My modified formula below (cell AK78) does not use any volatile functions.
=IFERROR(SUM(Face INDEX($A$1:$BK$35,AI78,AJ78) ),””)
Only Calculate If Required!
=IF(AF78>$AG$76,””,IFERROR(SUM(Face INDEX($A$1:$BK$35,AI78,AJ78) ),””))
IF tests if the formula is needed. It compares counter with AG76 value (count cells with numbers).
ROW & COLUMN functions
Various formulas used ROW and COLUMN functions many times. I replaced them with hard coded counters. Downside? I’d have to adjust them if I insert new rows or columns.
Example: cell AG78 formula no longer requires ROW & COLUMN functions:
Name Range: Face
The named range to identify the face’s current location used volatile OFFSET function. I changed it to INDEX. Yes, INDEX can be used for a dynamic range (post) and it isn’t volatile.
Thanks to Robert Gascon for reminding me of this a couple of years ago.
Conditional Formatting
Conditional formatting rules are “super-volatile” as per Bill Jelen (post). I’ve made the formulas more efficient but CF rules remain volatile. I considered using custom formats as Bill suggested but
that would require a full redesign.
Change the purple obstacles
To redesign the playing area add/remove the sequence formulas (purple obstacles).
Volatility in Excel
Some good sources for learning more including:
Losing My Mind
I think the universe played a trick on me.
Previously, I had formatted the input area with custom format “;;;” so numbers wouldn’t be visible. My thinking was probably that this was the easiest way to hide them. I had completely forgotten
about this. Somehow two of my formulas had the same format! So the formula was working but I couldn’t see the result! 🙂
I won’t admit how long it took me to figure this out!
About Me
I’ve taken courses, read books, and watched videos but I still learn the most by building & playing with Excel. Theory can be helpful but hands on experience is essential. It’s also a fun thing to
play with while watching NBA/NHL playoffs, Netflix and Prime.
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Euler, Fermat and Primality Test
In number theory, The Euler’s totient function , counts the number of positive integers less than m and relatively prime to m. For a prime number p, φ(p) = p-1.
It can be defined more formally as the number of integers k in the range 1 ≤ k ≤ n for which the greatest common divisor gcd(n, k) is equal to 1.
What is Fermat’s little theorem
Fermat’s little theorem says that if p is a prime and a is not a multiple of p, then
Euler’s generalization of Fermat’s little theorem says that if a is relatively prime to m, then
Euler’s totient function is multiplicative , that is, if a and b are relatively prime, then φ(ab) = φ(a) φ(b). We will use this fact in other discuss.
The Proof of generalization
Be r=φ(n) and b₀, b₁, …, bᵣ, integers numbers, primes relative two at two, and all prime relative with n. So ab₀, ab₁, …, abᵣ, still congruent mod(n) for
i=1,2, …,r.
The collection b₀, b₁, …, bᵣ and ab₀, ab₁, …, abᵣ are equals mod(n). So let multiplity all
In anyway, (aʳ-1) ≡ 0 (mod n) and how aʳ ≡ 1(mod n) and
r=φ(n), then
Primality testing
One best things about this theorem is the primality testing.
The contrapositive of Fermat’s little theorem is useful: if the congruence aᵖ⁻¹≡ 1 (mod p) does not true, then either p is not prime or a is a multiple of p. In practice, a is much smaller than p,
and so one can conclude that p is not prime.
Technically this is a test for non-primality: it can only prove that a number is not prime. For example, if 2ᵖ⁻¹ ≢ 1 (mod p) then we know p is not a prime. But if 2ᵖ⁻¹ ≡ 1 (mod p) then all we know is
that we haven’t failed the test; we don’t have certain if that p is prime or not. So we try another value of a, for example 5, and see if 5ᵖ⁻¹ ≡ 1 (mod p).
In theory looks perfect, so all the crypto theory was ruined? Of course, not, because even easy to undestand, looking in thecomputacionais terms, this is problematic, for example, for a small number
like 223, and for a with value of 2, we have;
We know that 223 is prime, but 2²²³ is hard to compute even in robusts computers, so numbers like 2321412341243123423413263466567678352323 is harder to determine, but all the theory is usefull, with
many implications in criptography and number theory. I will discuss this in other posts.
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class hyperspy.api.signals.Signal2D(*args, **kwargs)#
Bases: BaseSignal, CommonSignal2D
General 2D signal class.
Create a signal instance.
The signal data. It can be an array of any dimensions.
axes[dict/axes], optional
List of either dictionaries or axes objects to define the axes (see the documentation of the AxesManager class for more details).
attributesdict, optional
A dictionary whose items are stored as attributes.
metadatadict, optional
A dictionary containing a set of parameters that will to stores in the metadata attribute. Some parameters might be mandatory in some cases.
original_metadatadict, optional
A dictionary containing a set of parameters that will to stores in the original_metadata attribute. It typically contains all the parameters that has been imported from the original data
raggedbool or None, optional
Define whether the signal is ragged or not. Overwrite the ragged value in the attributes dictionary. If None, it does nothing. Default is None.
add_ramp(ramp_x, ramp_y, offset=0)#
Add a linear ramp to the signal.
ramp_x: float
Slope of the ramp in x-direction.
ramp_y: float
Slope of the ramp in y-direction.
offset: float, optional
Offset of the ramp at the signal fulcrum.
The fulcrum of the linear ramp is at the origin and the slopes are given in units of the axis with the according scale taken into account. Both are available via the axes_manager of the
align2D(crop=True, fill_value=nan, shifts=None, expand=False, interpolation_order=1, show_progressbar=None, num_workers=None, **kwargs)#
Align the images in-place using scipy.ndimage.shift().
The images can be aligned using either user-provided shifts or by first estimating the shifts.
See estimate_shift2D() for more details on estimating image shifts.
If True, the data will be cropped not to include regions with missing data
fill_valueint, float, numpy.nan
The areas with missing data are filled with the given value. Default is np.nan.
shiftsNone or numpy.ndarray
The array of shifts must be in pixel units. The shape must be the navigation shape using numpy order convention. If None the shifts are estimated using estimate_shift2D().
If True, the data will be expanded to fit all data after alignment. Overrides crop.
interpolation_order: int
The order of the spline interpolation. Default is 1, linear interpolation.
show_progressbarNone or bool
If True, display a progress bar. If None, the default from the preferences settings is used.
num_workersNone or int
Number of worker used by dask. If None, default to dask default value.
Keyword arguments passed to estimate_shift2D().
The estimated shifts are returned only if shifts is None
If one of the signal axes is a non-uniform axis.
calibrate(x0=None, y0=None, x1=None, y1=None, new_length=None, units=None, interactive=True, display=True, toolkit=None)#
Calibrate the x and y signal dimensions.
Can be used either interactively, or by passing values as parameters.
If interactive is False, these must be set. If given in floats the input will be in scaled axis values. If given in integers, the input will be in non-scaled pixel values. Similar to
how integer and float input works when slicing using isig and inav.
If interactive is False, this must be set.
If interactive is False, this is used to set the axes units.
If True, will use a plot with an interactive line for calibration. If False, x0, y0, x1, y1 and new_length must be set.
>>> s = hs.signals.Signal2D(np.random.random((100, 100)))
>>> s.calibrate()
Running non-interactively
>>> s = hs.signals.Signal2D(np.random.random((100, 100)))
>>> s.calibrate(x0=10, y0=10, x1=60, y1=10, new_length=100,
... interactive=False, units="nm")
Create a model for the current signal
dictionaryNone or dict, optional
A dictionary to be used to recreate a model. Usually generated using hyperspy.model.BaseModel.as_dictionary()
crop_signal(top=None, bottom=None, left=None, right=None, convert_units=False)#
Crops in signal space and in place.
top, bottom, left, rightint or float
If int the values are taken as indices. If float the values are converted to indices.
Default is False If True, convert the signal units using the ‘convert_to_units’ method of the axes_manager. If False, does nothing.
estimate_shift2D(reference='current', correlation_threshold=None, chunk_size=30, roi=None, normalize_corr=False, sobel=True, medfilter=True, hanning=True, plot=False, dtype='float',
show_progressbar=None, sub_pixel_factor=1)#
Estimate the shifts in an image using phase correlation.
This method can only estimate the shift by comparing bi-dimensional features that should not change position between frames. To decrease the memory usage, the time of computation and the
accuracy of the results it is convenient to select a region of interest by setting the roi argument.
If ‘current’ (default) the image at the current coordinates is taken as reference. If ‘cascade’ each image is aligned with the previous one. If ‘stat’ the translation of every image
with all the rest is estimated and by performing statistical analysis on the result the translation is estimated.
This parameter is only relevant when reference=’stat’. If float, the shift estimations with a maximum correlation value lower than the given value are not used to compute the
estimated shifts. If ‘auto’ the threshold is calculated automatically as the minimum maximum correlation value of the automatically selected reference image.
If int and reference=’stat’ the number of images used as reference are limited to the given value.
Define the region of interest (left, right, top, bottom). If int (float), the position is given by axis index (value). Note that ROIs can be used in place of a tuple.
If True, use phase correlation to align the images, otherwise use cross correlation.
Apply a Sobel filter for edge enhancement
Apply a median filter for noise reduction
Apply a 2D hanning filter
If True plots the images after applying the filters and the phase correlation. If ‘reuse’, it will also plot the images, but it will only use one figure, and continuously update the
images in that figure as it progresses through the stack.
Typecode or data-type in which the calculations must be performed.
If True, display a progress bar. If None, the default from the preferences settings is used.
Estimate shifts with a sub-pixel accuracy of 1/sub_pixel_factor parts of a pixel. Default is 1, i.e. no sub-pixel accuracy.
Estimated shifts in pixels.
The statistical analysis approach to the translation estimation when using reference='stat' roughly follows [*]. If you use it please cite their article.
find_peaks(method='local_max', interactive=True, current_index=False, show_progressbar=None, num_workers=None, display=True, toolkit=None, get_intensity=False, **kwargs)#
Find peaks in a 2D signal.
Function to locate the positive peaks in an image using various, user specified, methods. Returns a structured array containing the peak positions.
Select peak finding algorithm to implement. Available methods are:
If True, the method parameter can be adjusted interactively. If False, the results will be returned.
If True, the computation will be performed for the current index.
If True, the intensity of the peak will be returned as an additional column, the last one.
If True, display a progress bar. If None, the default from the preferences settings is used.
Number of worker used by dask. If None, default to dask default value.
If True, display the user interface widgets. If False, return the widgets container in a dictionary, usually for customisation or testing.
If None (default), all available widgets are displayed or returned. If string, only the widgets of the selected toolkit are displayed if available. If an interable of toolkit strings,
the widgets of all listed toolkits are displayed or returned.
Keywords parameters associated with above methods, see the documentation of each method for more details.
peaksBaseSignal or numpy.ndarray
numpy.ndarray if current_index=True. Ragged signal with shape (npeaks, 2) that contains the x, y pixel coordinates of peaks found in each image sorted first along y and then along x.
As a convenience, the ‘local_max’ method accepts the ‘distance’ and ‘threshold’ argument, which will be map to the ‘min_distance’ and ‘threshold_abs’ of the skimage.feature.peak_local_max()
plot(navigator='auto', plot_markers=True, autoscale='v', norm='auto', vmin=None, vmax=None, gamma=1.0, linthresh=0.01, linscale=0.1, scalebar=True, scalebar_color='white', axes_ticks=None,
axes_off=False, axes_manager=None, no_nans=False, colorbar=True, centre_colormap='auto', min_aspect=0.1, navigator_kwds={}, **kwargs)#
Plot the signal at the current coordinates.
For multidimensional datasets an optional figure, the “navigator”, with a cursor to navigate that data is raised. In any case it is possible to navigate the data using the sliders. Currently
only signals with signal_dimension equal to 0, 1 and 2 can be plotted.
navigatorstr, None, or BaseSignal (or subclass).
Allowed string values are ``’auto’``, ``’slider’``, and ``’spectrum’``.
■ If 'auto':
★ If navigation_dimension > 0, a navigator is provided to explore the data.
★ If navigation_dimension is 1 and the signal is an image the navigator is a sum spectrum obtained by integrating over the signal axes (the image).
★ If navigation_dimension is 1 and the signal is a spectrum the navigator is an image obtained by stacking all the spectra in the dataset horizontally.
★ If navigation_dimension is > 1, the navigator is a sum image obtained by integrating the data over the signal axes.
★ Additionally, if navigation_dimension > 2, a window with one slider per axis is raised to navigate the data.
★ For example, if the dataset consists of 3 navigation axes “X”, “Y”, “Z” and one signal axis, “E”, the default navigator will be an image obtained by integrating the data over
“E” at the current “Z” index and a window with sliders for the “X”, “Y”, and “Z” axes will be raised. Notice that changing the “Z”-axis index changes the navigator in this
★ For lazy signals, the navigator will be calculated using the compute_navigator() method.
■ If 'slider':
★ If navigation dimension > 0 a window with one slider per axis is raised to navigate the data.
■ If 'spectrum':
★ If navigation_dimension > 0 the navigator is always a spectrum obtained by integrating the data over all other axes.
★ Not supported for lazy signals, the 'auto' option will be used instead.
■ If None, no navigator will be provided.
Alternatively a BaseSignal (or subclass) instance can be provided. The navigation or signal shape must match the navigation shape of the signal to plot or the navigation_shape +
signal_shape must be equal to the navigator_shape of the current object (for a dynamic navigator). If the signal dtype is RGB or RGBA this parameter has no effect and the value is
always set to 'slider'.
axes_managerNone or AxesManager
If None, the signal’s axes_manager attribute is used.
plot_markersbool, default True
Plot markers added using s.add_marker(marker, permanent=True). Note, a large number of markers might lead to very slow plotting.
Only for image navigator, additional keyword arguments for matplotlib.pyplot.imshow().
colorbarbool, optional
If true, a colorbar is plotted for non-RGB images.
autoscalestr, optional
The string must contain any combination of the 'x', 'y' and 'v' characters. If 'x' or 'y' are in the string, the corresponding axis limits are set to cover the full range of the data
at a given position. If 'v' (for values) is in the string, the contrast of the image will be set automatically according to vmin` and ``vmax when the data or navigation indices
change. Default is 'v'.
normstr {"auto"` | ``"linear" | "power" | "log" | "symlog"} or matplotlib.colors.Normalize
Set the norm of the image to display. If "auto", a linear scale is used except if when power_spectrum=True in case of complex data type. "symlog" can be used to display negative value
on a negative scale - read matplotlib.colors.SymLogNorm and the linthresh and linscale parameter for more details.
vmin, vmax{scalar, str}, optional
vmin and vmax are used to normalise the displayed data. It can be a float or a string. If string, it should be formatted as 'xth', where 'x' must be an float in the [0, 100] range.
'x' is used to compute the x-th percentile of the data. See numpy.percentile() for more information.
gammafloat, optional
Parameter used in the power-law normalisation when the parameter norm="power". Read matplotlib.colors.PowerNorm for more details. Default value is 1.0.
linthreshfloat, optional
When used with norm="symlog", define the range within which the plot is linear (to avoid having the plot go to infinity around zero). Default value is 0.01.
linscalefloat, optional
This allows the linear range (-linthresh to linthresh) to be stretched relative to the logarithmic range. Its value is the number of powers of base to use for each half of the linear
range. See matplotlib.colors.SymLogNorm for more details. Defaulf value is 0.1.
scalebarbool, optional
If True and the units and scale of the x and y axes are the same a scale bar is plotted.
scalebar_colorstr, optional
A valid MPL color string; will be used as the scalebar color.
axes_ticks{None, bool}, optional
If True, plot the axes ticks. If None axes_ticks are only plotted when the scale bar is not plotted. If False the axes ticks are never plotted.
axes_offbool, default False
no_nansbool, optional
If True, set nans to zero for plotting.
centre_colormapbool or "auto"
If True the centre of the color scheme is set to zero. This is specially useful when using diverging color schemes. If “auto” (default), diverging color schemes are automatically
min_aspectfloat, optional
Set the minimum aspect ratio of the image and the figure. To keep the image in the aspect limit the pixels are made rectangular.
Only when plotting an image: additional (optional) keyword arguments for matplotlib.pyplot.imshow().
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Machine Learning Linear Regression And Regular
Machine Learning Linear Regression And Regularlization
Linear regression is a model to predict a variable based on independent variables. The model assumes linear relationship between dependent and independent variables. Below represents a simple linear
regression equation.
In above equation y is a dependent variable and x1,x2 are independent variables. a is a intercept, c1 and c2 are coefficients. In above equation, we are trying to predict y based on x1 and x2
In this post, I will do an example of linear regression and regularization using Maching Learning package H2o. H2o is a great library and offers lot of techniques right out of the box.
I will use students alcohol data which I downloaded from following UCI website...
Before we delve in to our data analysis, Make sure you have following installed and working...
In your R repl, lets import the H2o package.
Lets import our data file student-mat.csv
In [65]:
st_mat <- h2o.importFile('student-mat.csv')
|======================================================================| 100%
Lets look at first two rows using head method.
A data.frame: 2 × 33
school sex age address famsize Pstatus Medu Fedu Mjob Fjob ⋯ famrel freetime goout Dalc Walc health absences G1 G2 G3
<fct> <fct> <dbl> <fct> <fct> <fct> <dbl> <dbl> <fct> <fct> ⋯ <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 GP F 18 U GT3 A 4 4 at_home teacher ⋯ 4 3 4 1 1 3 6 5 6 6
2 GP F 17 U GT3 T 1 1 at_home other ⋯ 5 3 3 1 1 3 4 5 5 6
Lets look at the column names also.
1. 'school'
2. 'sex'
3. 'age'
4. 'address'
5. 'famsize'
6. 'Pstatus'
7. 'Medu'
8. 'Fedu'
9. 'Mjob'
10. 'Fjob'
11. 'reason'
12. 'guardian'
13. 'traveltime'
14. 'studytime'
15. 'failures'
16. 'schoolsup'
17. 'famsup'
18. 'paid'
19. 'activities'
20. 'nursery'
21. 'higher'
22. 'internet'
23. 'romantic'
24. 'famrel'
25. 'freetime'
26. 'goout'
27. 'Dalc'
28. 'Walc'
29. 'health'
30. 'absences'
31. 'G1'
32. 'G2'
33. 'G3'
To check number of rows, we can do using h2o.nrow.
For linear regression, we should check how many columns are there. We can do with command h2o.ncol.
One of most important thing about linear regression is chosing the right set of independent variables for our dependent variable.
For our dependent variable which is the variable we want to predict, Lets us pick "Walc" which is column number 28.
Walc - weekend alcohol consumption (numeric: from 1 - very low to 5 - very high)
Basically we are trying to predict weekend alcohol consumption. Lets see which of the variables help us doing that.
To train our Linear regression model, let us split our data in the ratio of 80% to 20% using h2o.splitFrame.
In [54]:
students.splits <- h2o.splitFrame(data = st_mat, ratios = .8)
In [55]:
train <- students.splits[[1]]
valid <- students.splits[[2]]
Ok now we got our train and validation set separated.
Lets take out Walc and Dalc (daily alcohol consumption) from our independent variables.
In [71]:
Ok now let us run our linear regression model. For that we can use h2o.glm package. glm stands for generalized linear regression models.
H2o Generalized Linear Regression Model (GLM)
In [75]:
students.glm <- h2o.glm(x=x,y=y, training_frame = train,
validation_frame = valid,remove_collinear_columns = TRUE)
|======================================================================| 100%
Ok since it is a small data set, the model just ran instantly.
Now we can print out the glm model coefficients using h2o.std_coef_plot
In [76]:
From the above graph we can look at the positive and negative parameters. Lets print the model coefficients to actually know their magnitudes.
Lets check which parameters are affecting positively to alcohol consumption.
We can use model$coefficients to access the coefficients of the variables of our linear regression.
In [85]:
coeff_vector = students.glm@model$coefficients
print(coeff_vector[coeff_vector > 0])
Intercept age failures goout health absences G2
0.43908352 0.11540452 0.05622664 0.40241119 0.12427294 0.01856066 0.05650706
As we see above, other than intercept , age , failures, goout, health, absences, G2 (second period Grade) all affect positively.
Lets see if any parameters which affect the alcohol consumption negatively.
In [87]:
print(coeff_vector[coeff_vector < 0])
sex.F studytime famrel freetime G1
-0.611686028 -0.225279062 -0.228980650 -0.008235832 -0.074973142
Female, studetime, famrel(quality of family relatives), freetime and (first period grade) all affect the weakly alcohol consumption negatively.
If we do model$model_summary, we can see which model type has been run by h2o default.
In [89]:
A H2OTable: 1 × 7
family link regularization number_of_predictors_total number_of_active_predictors number_of_iterations training_frame
<chr> <chr> <chr> <int> <int> <int> <chr>
gaussian identity Elastic Net (alpha = 0.5, lambda = 0.1043 ) 57 11 1 RTMP_sid_85ff_8
Above tables shows that regression type is "gaussian". Also the table shows regularization type which is Elastic Net.
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ERFC.PRECISE function: Description, Usage, Syntax, Examples and Explanation November 12, 2024 - Excel Office
ERFC.PRECISE function: Description, Usage, Syntax, Examples and Explanation
What is ERFC.PRECISE function in Excel?
ERFC.PRECISE function is one of Engineering functions in Microsoft Excel that returns the complementary ERF function integrated between x and infinity.
Syntax of ERFC.PRECISE function
The ERFC.PRECISE function syntax has the following arguments:
• X: The lower bound for integrating ERFC.PRECISE.
ERFC.PRECISE formula explanation
• If x is nonnumeric, ERFC.PRECISE returns the #VALUE! error value.
Example of ERFC.PRECISE function
Steps to follow:
1. Open a new Excel worksheet.
2. Copy data in the following table below and paste it in cell A1
Note: For formulas to show results, select them, press F2 key on your keyboard and then press Enter.
You can adjust the column widths to see all the data, if need be.
Formula Description Result
=ERFC.PRECISE(1) Complementary ERF function of 1. 0.15729921
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Solving Quadratic Equations By Factoring Worksheet Kuta Software - Equations Worksheets
Solving Quadratic Equations By Factoring Worksheet Kuta Software
Solving Quadratic Equations By Factoring Worksheet Kuta Software – The purpose of Expressions and Equations Worksheets is to aid your child to learn more effectively and efficiently. These worksheets
are interactive as well as problems that are based on sequence of operations. With these worksheets, children can grasp both simple and advanced concepts in a very short amount of amount of time. You
can download these worksheets in PDF format to assist your child to learn and practice math equations. These resources are useful to students in the 5th-8th grades.
Get Free Solving Quadratic Equations By Factoring Worksheet Kuta Software
These worksheets can be used by students between 5th and 8th grades. These two-step word problem are designed using decimals, fractions or fractions. Each worksheet contains ten problems. You can
find them at any website or print source. These worksheets can be used to practice rearranging equations. In addition to allowing students to practice restructuring equations, they can also help your
student understand the properties of equality and inverse operations.
These worksheets can be utilized by fifth and eighth grade students. These are great for students who struggle to calculate percentages. You can select from three different types of questions. You
can decide to tackle one-step problems that include decimal or whole numbers, or you can use word-based approaches to solve decimals or fractions. Each page contains ten equations. These worksheets
on Equations are suitable for students in the 5th through 8th grades.
These worksheets are a great resource for practicing fraction calculations and other concepts related to algebra. Many of these worksheets allow you to select between three types of challenges. You
can pick a word-based or a numerical one. It is important to choose the problem type, because every challenge will be unique. There are ten issues on each page, meaning they’re great resources for
students in the 5th through 8th grade.
These worksheets help students understand the relationship between variables as well as numbers. The worksheets give students practice with solving polynomial equations or solving equations, as well
as understanding how to apply them in everyday situations. If you’re looking for an effective educational tool to learn about expressions and equations, you can start with these worksheets. These
worksheets will help you learn about different types of mathematical issues and the various symbols that are used to express them.
These worksheets are beneficial to students in the beginning grades. These worksheets can help students develop the ability to graph and solve equations. The worksheets are great to practice
polynomial variables. These worksheets can help you simplify and factor these variables. There is a fantastic set of expressions and equations worksheets for children at any grade level. Making the
work yourself is the best way to get a grasp of equations.
You will find a lot of worksheets on quadratic equations. Each level comes with their own worksheet. These worksheets are designed to assist you in solving problems in the fourth level. When you’ve
reached a certain level then you are able to work on other types of equations. You can then work on solving the same-level problems. For instance, you could solve the same problem as an extended one.
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% vim: ft=mercury ts=4 sw=4 et
% Copyright (C) 1996-1997,1999-2002, 2004-2006, 2008-2012 The University of Melbourne.
% Copyright (C) 2014-2015, 2018-2022 The Mercury team.
% This file is distributed under the terms specified in COPYING.LIB.
% File: set_ordlist.m.
% Main authors: conway, fjh.
% Stability: medium.
% This file contains a `set' ADT.
% Sets are implemented here as sorted lists without duplicates.
:- module set_ordlist.
:- interface.
:- import_module bool.
:- import_module list.
:- type set_ordlist(_T).
% Initial creation of sets.
% init(Set) is true iff Set is an empty set.
:- func init = set_ordlist(T).
:- pred init(set_ordlist(_T)::uo) is det.
% singleton_set(Elem, Set) is true iff Set is the set containing just
% the single element Elem.
:- pred singleton_set(T, set_ordlist(T)).
:- mode singleton_set(in, out) is det.
:- mode singleton_set(out, in) is semidet.
:- func make_singleton_set(T) = set_ordlist(T).
% Emptiness and singleton-ness tests.
% is_empty(Set) is true iff Set is an empty set.
:- pred is_empty(set_ordlist(T)::in) is semidet.
% is_non_empty(Set) is true iff Set is not an empty set.
:- pred is_non_empty(set_ordlist(T)::in) is semidet.
:- pred is_singleton(set_ordlist(T)::in, T::out) is semidet.
% Membership tests.
% member(X, Set) is true iff X is a member of Set.
:- pred member(T, set_ordlist(T)).
:- mode member(in, in) is semidet.
:- mode member(out, in) is nondet.
% is_member(X, Set, Result) returns `Result = yes' iff X is a member of
% Set.
:- pred is_member(T::in, set_ordlist(T)::in, bool::out) is det.
% contains(Set, X) is true iff X is a member of Set.
:- pred contains(set_ordlist(T)::in, T::in) is semidet.
% Insertions and deletions.
% insert(X, Set0, Set) is true iff Set is the union
% of Set0 and the set containing only X.
:- func insert(set_ordlist(T), T) = set_ordlist(T).
:- pred insert(T::in, set_ordlist(T)::in, set_ordlist(T)::out) is det.
% insert_new(X, Set0, Set) is true iff Set0 does not contain X, while
% Set is the union of Set0 and the set containing only X.
:- pred insert_new(T::in,
set_ordlist(T)::in, set_ordlist(T)::out) is semidet.
% insert_list(Xs, Set0, Set) is true iff Set is the union of Set0 and
% the set containing only the members of Xs.
:- func insert_list(set_ordlist(T), list(T)) = set_ordlist(T).
:- pred insert_list(list(T)::in, set_ordlist(T)::in, set_ordlist(T)::out)
is det.
% delete(X, Set0, Set) is true iff Set is the
% relative complement of Set0 and the set containing only X, i.e.
% if Set is the set which contains all the elements of Set0
% except X.
:- func delete(set_ordlist(T), T) = set_ordlist(T).
:- pred delete(T::in, set_ordlist(T)::in, set_ordlist(T)::out) is det.
% delete_list(Xs, Set0, Set) is true iff Set is the relative complement
% of Set0 and the set containing only the members of Xs.
:- func delete_list(set_ordlist(T), list(T)) = set_ordlist(T).
:- pred delete_list(list(T)::in, set_ordlist(T)::in, set_ordlist(T)::out)
is det.
% remove(X, Set0, Set) is true iff Set0 contains X,
% and Set is the relative complement of Set0 and the set
% containing only X, i.e. if Set is the set which contains
% all the elements of Set0 except X.
% The det_remove version throws an exception instead of failing.
:- pred remove(T::in, set_ordlist(T)::in, set_ordlist(T)::out) is semidet.
:- pred det_remove(T::in, set_ordlist(T)::in, set_ordlist(T)::out) is det.
% remove_list(Xs, Set0, Set) is true iff Xs does not contain any
% duplicates, Set0 contains every member of Xs, and Set is the
% relative complement of Set0 and the set containing only the members of
% Xs.
% The det_remove_list version throws an exception instead of failing.
:- pred remove_list(list(T)::in, set_ordlist(T)::in, set_ordlist(T)::out)
is semidet.
:- pred det_remove_list(list(T)::in, set_ordlist(T)::in, set_ordlist(T)::out)
is det.
% remove_least(X, Set0, Set) is true iff X is the least element in
% Set0, and Set is the set which contains all the elements of Set0
% except X.
:- pred remove_least(T::out, set_ordlist(T)::in, set_ordlist(T)::out)
is semidet.
% Comparisons between sets.
% equal(SetA, SetB) is true iff SetA and SetB contain the same
% elements.
:- pred equal(set_ordlist(T)::in, set_ordlist(T)::in) is semidet.
% subset(SetA, SetB) is true iff SetA is a subset of SetB.
:- pred subset(set_ordlist(T)::in, set_ordlist(T)::in) is semidet.
% superset(SetA, SetB) is true iff SetA is a superset of SetB.
:- pred superset(set_ordlist(T)::in, set_ordlist(T)::in) is semidet.
% Operations on two or more sets.
% union(SetA, SetB, Set) is true iff Set is the union
% of SetA and SetB. The efficiency of the union operation is
% O(card(SetA)+card(SetB)) and is not sensitive to the argument
% ordering.
:- func union(set_ordlist(T), set_ordlist(T)) = set_ordlist(T).
:- pred union(set_ordlist(T)::in, set_ordlist(T)::in, set_ordlist(T)::out)
is det.
% union_list(A, B) is true iff B is the union of all the sets in A.
:- func union_list(list(set_ordlist(T))) = set_ordlist(T).
:- pred union_list(list(set_ordlist(T))::in, set_ordlist(T)::out) is det.
% power_union(A, B) is true iff B is the union of all the sets in A,
:- func power_union(set_ordlist(set_ordlist(T))) = set_ordlist(T).
:- pred power_union(set_ordlist(set_ordlist(T))::in,
set_ordlist(T)::out) is det.
% intersect(SetA, SetB, Set) is true iff Set is the intersection of
% SetA and SetB. The efficiency of the intersection operation is not
% influenced by the argument order.
:- func intersect(set_ordlist(T), set_ordlist(T)) = set_ordlist(T).
:- pred intersect(set_ordlist(T), set_ordlist(T), set_ordlist(T)).
:- mode intersect(in, in, out) is det.
:- mode intersect(in, in, in) is semidet.
% intersect_list(A) = B' is true iff B is the intersection of all the
% sets in A.
:- func intersect_list(list(set_ordlist(T))) = set_ordlist(T).
:- pred intersect_list(list(set_ordlist(T))::in, set_ordlist(T)::out) is det.
% power_intersect(A, B) is true iff B is the intersection of all the
% sets in A.
:- func power_intersect(set_ordlist(set_ordlist(T)))
= set_ordlist(T).
:- pred power_intersect(set_ordlist(set_ordlist(T))::in,
set_ordlist(T)::out) is det.
% difference(SetA, SetB, Set) is true iff Set is the
% set containing all the elements of SetA except those that
% occur in SetB.
:- func difference(set_ordlist(T), set_ordlist(T)) = set_ordlist(T).
:- pred difference(set_ordlist(T)::in, set_ordlist(T)::in,
set_ordlist(T)::out) is det.
% intersection_and_differences(SetA, SetB, InAandB, OnlyInA, OnlyInB):
% Given SetA and SetB, return the elements that occur in both sets,
% and those that occur only in one or the other.
:- pred intersection_and_differences(set_ordlist(T)::in, set_ordlist(T)::in,
set_ordlist(T)::out, set_ordlist(T)::out, set_ordlist(T)::out) is det.
% Operations that divide a set into two parts.
% divide(Pred, Set, TruePart, FalsePart):
% TruePart consists of those elements of Set for which Pred succeeds;
% FalsePart consists of those elements of Set for which Pred fails.
:- pred divide(pred(T)::in(pred(in) is semidet),
set_ordlist(T)::in, set_ordlist(T)::out, set_ordlist(T)::out) is det.
% divide_by_set(DivideBySet, Set, InPart, OutPart):
% InPart consists of those elements of Set which are also in DivideBySet;
% OutPart consists of those elements of Set which are not in DivideBySet.
:- pred divide_by_set(set_ordlist(T)::in, set_ordlist(T)::in,
set_ordlist(T)::out, set_ordlist(T)::out) is det.
% Converting lists to sets.
% list_to_set(List, Set) is true iff Set is the set
% containing only the members of List.
:- func list_to_set(list(T)) = set_ordlist(T).
:- pred list_to_set(list(T)::in, set_ordlist(T)::out) is det.
% A synonym for list_to_set/1.
:- func from_list(list(T)) = set_ordlist(T).
% sorted_list_to_set(List, Set) is true iff Set is the set
% containing only the members of List. List must be sorted
% in ascending order.
:- func sorted_list_to_set(list(T)) = set_ordlist(T).
:- pred sorted_list_to_set(list(T)::in, set_ordlist(T)::out) is det.
% A synonym for sorted_list_to_set/1.
:- func from_sorted_list(list(T)) = set_ordlist(T).
% rev_sorted_list_to_set(List, Set) is true iff Set is the set
% containing only the members of List. List must be sorted
% in descending order and must not contain any duplicates.
:- func rev_sorted_list_to_set(list(T)) = set_ordlist(T).
:- pred rev_sorted_list_to_set(list(T)::in, set_ordlist(T)::out) is det.
% Converting sets to lists.
% to_sorted_list(Set, List) is true iff List is the list of all the
% members of Set, in sorted order.
:- func to_sorted_list(set_ordlist(T)) = list(T).
:- pred to_sorted_list(set_ordlist(T)::in, list(T)::out) is det.
% Counting.
% count(Set, Count) is true iff Set has Count elements.
:- func count(set_ordlist(T)) = int.
:- pred count(set_ordlist(T)::in, int::out) is det.
% Standard higher order functions on collections.
% all_true(Pred, Set) succeeds iff Pred(Element) succeeds for all the
% elements of Set.
:- pred all_true(pred(T)::in(pred(in) is semidet), set_ordlist(T)::in)
is semidet.
% Return the set of items for which the given predicate succeeds.
:- func filter(pred(T1), set_ordlist(T1)) = set_ordlist(T1).
:- mode filter(in(pred(in) is semidet), in) = out is det.
:- pred filter(pred(T1), set_ordlist(T1), set_ordlist(T1)).
:- mode filter(in(pred(in) is semidet), in, out) is det.
% Return the set of items for which the given predicate succeeds, and the
% set of items for which it fails.
:- pred filter(pred(T1), set_ordlist(T1), set_ordlist(T1), set_ordlist(T1)).
:- mode filter(in(pred(in) is semidet), in, out, out) is det.
:- func filter_map(func(T1) = T2, set_ordlist(T1)) = set_ordlist(T2).
:- mode filter_map(in(func(in) = out is semidet), in) = out is det.
:- pred filter_map(pred(T1, T2), set_ordlist(T1), set_ordlist(T2)).
:- mode filter_map(in(pred(in, out) is semidet), in, out) is det.
:- func map(func(T1) = T2, set_ordlist(T1)) = set_ordlist(T2).
:- func fold(func(T1, T2) = T2, set_ordlist(T1), T2) = T2.
:- pred fold(pred(T1, T2, T2), set_ordlist(T1), T2, T2).
:- mode fold(in(pred(in, in, out) is det), in, in, out) is det.
:- mode fold(in(pred(in, mdi, muo) is det), in, mdi, muo) is det.
:- mode fold(in(pred(in, di, uo) is det), in, di, uo) is det.
:- mode fold(in(pred(in, in, out) is semidet), in, in, out) is semidet.
:- mode fold(in(pred(in, mdi, muo) is semidet), in, mdi, muo) is semidet.
:- mode fold(in(pred(in, di, uo) is semidet), in, di, uo) is semidet.
:- func foldl(func(T1, T2) = T2, set_ordlist(T1), T2) = T2.
:- pred foldl(pred(T1, T2, T2), set_ordlist(T1), T2, T2).
:- mode foldl(in(pred(in, in, out) is det), in, in, out) is det.
:- mode foldl(in(pred(in, mdi, muo) is det), in, mdi, muo) is det.
:- mode foldl(in(pred(in, di, uo) is det), in, di, uo) is det.
:- mode foldl(in(pred(in, in, out) is semidet), in, in, out) is semidet.
:- mode foldl(in(pred(in, mdi, muo) is semidet), in, mdi, muo) is semidet.
:- mode foldl(in(pred(in, di, uo) is semidet), in, di, uo) is semidet.
:- pred fold2(pred(T1, T2, T2, T3, T3), set_ordlist(T1),
T2, T2, T3, T3).
:- mode fold2(in(pred(in, in, out, in, out) is det), in,
in, out, in, out) is det.
:- mode fold2(in(pred(in, in, out, mdi, muo) is det), in,
in, out, mdi, muo) is det.
:- mode fold2(in(pred(in, in, out, di, uo) is det), in,
in, out, di, uo) is det.
:- mode fold2(in(pred(in, in, out, in, out) is semidet), in,
in, out, in, out) is semidet.
:- mode fold2(in(pred(in, in, out, mdi, muo) is semidet), in,
in, out, mdi, muo) is semidet.
:- mode fold2(in(pred(in, in, out, di, uo) is semidet), in,
in, out, di, uo) is semidet.
:- pred foldl2(pred(T1, T2, T2, T3, T3), set_ordlist(T1),
T2, T2, T3, T3).
:- mode foldl2(in(pred(in, in, out, in, out) is det), in,
in, out, in, out) is det.
:- mode foldl2(in(pred(in, in, out, mdi, muo) is det), in,
in, out, mdi, muo) is det.
:- mode foldl2(in(pred(in, in, out, di, uo) is det), in,
in, out, di, uo) is det.
:- mode foldl2(in(pred(in, in, out, in, out) is semidet), in,
in, out, in, out) is semidet.
:- mode foldl2(in(pred(in, in, out, mdi, muo) is semidet), in,
in, out, mdi, muo) is semidet.
:- mode foldl2(in(pred(in, in, out, di, uo) is semidet), in,
in, out, di, uo) is semidet.
:- pred fold3(pred(T1, T2, T2, T3, T3, T4, T4),
set_ordlist(T1), T2, T2, T3, T3, T4, T4).
:- mode fold3(in(pred(in, in, out, in, out, in, out) is det), in,
in, out, in, out, in, out) is det.
:- mode fold3(in(pred(in, in, out, in, out, mdi, muo) is det), in,
in, out, in, out, mdi, muo) is det.
:- mode fold3(in(pred(in, in, out, in, out, di, uo) is det), in,
in, out, in, out, di, uo) is det.
:- mode fold3(in(pred(in, in, out, in, out, in, out) is semidet), in,
in, out, in, out, in, out) is semidet.
:- mode fold3(in(pred(in, in, out, in, out, mdi, muo) is semidet), in,
in, out, in, out, mdi, muo) is semidet.
:- mode fold3(in(pred(in, in, out, in, out, di, uo) is semidet), in,
in, out, in, out, di, uo) is semidet.
:- pred foldl3(pred(T1, T2, T2, T3, T3, T4, T4),
set_ordlist(T1), T2, T2, T3, T3, T4, T4).
:- mode foldl3(in(pred(in, in, out, in, out, in, out) is det), in,
in, out, in, out, in, out) is det.
:- mode foldl3(in(pred(in, in, out, in, out, mdi, muo) is det), in,
in, out, in, out, mdi, muo) is det.
:- mode foldl3(in(pred(in, in, out, in, out, di, uo) is det), in,
in, out, in, out, di, uo) is det.
:- mode foldl3(in(pred(in, in, out, in, out, in, out) is semidet), in,
in, out, in, out, in, out) is semidet.
:- mode foldl3(in(pred(in, in, out, in, out, mdi, muo) is semidet), in,
in, out, in, out, mdi, muo) is semidet.
:- mode foldl3(in(pred(in, in, out, in, out, di, uo) is semidet), in,
in, out, in, out, di, uo) is semidet.
:- pred fold4(pred(T1, T2, T2, T3, T3, T4, T4, T5, T5),
set_ordlist(T1), T2, T2, T3, T3, T4, T4, T5, T5).
:- mode fold4(
in(pred(in, in, out, in, out, in, out, in, out) is det), in,
in, out, in, out, in, out, in, out) is det.
:- mode fold4(
in(pred(in, in, out, in, out, in, out, mdi, muo) is det), in,
in, out, in, out, in, out, mdi, muo) is det.
:- mode fold4(
in(pred(in, in, out, in, out, in, out, di, uo) is det), in,
in, out, in, out, in, out, di, uo) is det.
:- mode fold4(
in(pred(in, in, out, in, out, in, out, in, out) is semidet), in,
in, out, in, out, in, out, in, out) is semidet.
:- mode fold4(
in(pred(in, in, out, in, out, in, out, mdi, muo) is semidet), in,
in, out, in, out, in, out, mdi, muo) is semidet.
:- mode fold4(
in(pred(in, in, out, in, out, in, out, di, uo) is semidet), in,
in, out, in, out, in, out, di, uo) is semidet.
:- pred foldl4(pred(T1, T2, T2, T3, T3, T4, T4, T5, T5),
set_ordlist(T1), T2, T2, T3, T3, T4, T4, T5, T5).
:- mode foldl4(
in(pred(in, in, out, in, out, in, out, in, out) is det), in,
in, out, in, out, in, out, in, out) is det.
:- mode foldl4(
in(pred(in, in, out, in, out, in, out, mdi, muo) is det), in,
in, out, in, out, in, out, mdi, muo) is det.
:- mode foldl4(
in(pred(in, in, out, in, out, in, out, di, uo) is det), in,
in, out, in, out, in, out, di, uo) is det.
:- mode foldl4(
in(pred(in, in, out, in, out, in, out, in, out) is semidet), in,
in, out, in, out, in, out, in, out) is semidet.
:- mode foldl4(
in(pred(in, in, out, in, out, in, out, mdi, muo) is semidet), in,
in, out, in, out, in, out, mdi, muo) is semidet.
:- mode foldl4(
in(pred(in, in, out, in, out, in, out, di, uo) is semidet), in,
in, out, in, out, in, out, di, uo) is semidet.
:- pred fold5(
pred(T1, T2, T2, T3, T3, T4, T4, T5, T5, T6, T6),
set_ordlist(T1), T2, T2, T3, T3, T4, T4, T5, T5, T6, T6).
:- mode fold5(
in(pred(in, in, out, in, out, in, out, in, out, in, out) is det), in,
in, out, in, out, in, out, in, out, in, out) is det.
:- mode fold5(
in(pred(in, in, out, in, out, in, out, in, out, mdi, muo) is det), in,
in, out, in, out, in, out, in, out, mdi, muo) is det.
:- mode fold5(
in(pred(in, in, out, in, out, in, out, in, out, di, uo) is det), in,
in, out, in, out, in, out, in, out, di, uo) is det.
:- mode fold5(
in(pred(in, in, out, in, out, in, out, in, out, in, out) is semidet), in,
in, out, in, out, in, out, in, out, in, out) is semidet.
:- mode fold5(
in(pred(in, in, out, in, out, in, out, in, out, mdi, muo) is semidet), in,
in, out, in, out, in, out, in, out, mdi, muo) is semidet.
:- mode fold5(
in(pred(in, in, out, in, out, in, out, in, out, di, uo) is semidet), in,
in, out, in, out, in, out, in, out, di, uo) is semidet.
:- pred foldl5(
pred(T1, T2, T2, T3, T3, T4, T4, T5, T5, T6, T6),
set_ordlist(T1), T2, T2, T3, T3, T4, T4, T5, T5, T6, T6).
:- mode foldl5(
in(pred(in, in, out, in, out, in, out, in, out, in, out) is det), in,
in, out, in, out, in, out, in, out, in, out) is det.
:- mode foldl5(
in(pred(in, in, out, in, out, in, out, in, out, mdi, muo) is det), in,
in, out, in, out, in, out, in, out, mdi, muo) is det.
:- mode foldl5(
in(pred(in, in, out, in, out, in, out, in, out, di, uo) is det), in,
in, out, in, out, in, out, in, out, di, uo) is det.
:- mode foldl5(
in(pred(in, in, out, in, out, in, out, in, out, in, out) is semidet), in,
in, out, in, out, in, out, in, out, in, out) is semidet.
:- mode foldl5(
in(pred(in, in, out, in, out, in, out, in, out, mdi, muo) is semidet), in,
in, out, in, out, in, out, in, out, mdi, muo) is semidet.
:- mode foldl5(
in(pred(in, in, out, in, out, in, out, in, out, di, uo) is semidet), in,
in, out, in, out, in, out, in, out, di, uo) is semidet.
:- pred fold6(pred(T, A, A, B, B, C, C, D, D, E, E, F, F),
set_ordlist(T), A, A, B, B, C, C, D, D, E, E, F, F).
:- mode fold6(
in(pred(in, in, out, in, out, in, out, in, out, in, out, in, out) is det),
in, in, out, in, out, in, out, in, out, in, out, in, out) is det.
:- mode fold6(
in(pred(in, in, out, in, out, in, out, in, out, in, out, mdi, muo) is det),
in, in, out, in, out, in, out, in, out, in, out, mdi, muo) is det.
:- mode fold6(
in(pred(in, in, out, in, out, in, out, in, out, in, out, di, uo) is det),
in, in, out, in, out, in, out, in, out, in, out, di, uo) is det.
:- mode fold6(
in(pred(in, in, out, in, out, in, out, in, out, in, out, in, out)
is semidet),
in, in, out, in, out, in, out, in, out, in, out, in, out) is semidet.
:- mode fold6(
in(pred(in, in, out, in, out, in, out, in, out, in, out, mdi, muo)
is semidet),
in, in, out, in, out, in, out, in, out, in, out, mdi, muo) is semidet.
:- mode fold6(
in(pred(in, in, out, in, out, in, out, in, out, in, out, di, uo)
is semidet),
in, in, out, in, out, in, out, in, out, in, out, di, uo) is semidet.
:- pred foldl6(pred(T, A, A, B, B, C, C, D, D, E, E, F, F),
set_ordlist(T), A, A, B, B, C, C, D, D, E, E, F, F).
:- mode foldl6(
in(pred(in, in, out, in, out, in, out, in, out, in, out, in, out) is det),
in, in, out, in, out, in, out, in, out, in, out, in, out) is det.
:- mode foldl6(
in(pred(in, in, out, in, out, in, out, in, out, in, out, mdi, muo) is det),
in, in, out, in, out, in, out, in, out, in, out, mdi, muo) is det.
:- mode foldl6(
in(pred(in, in, out, in, out, in, out, in, out, in, out, di, uo) is det),
in, in, out, in, out, in, out, in, out, in, out, di, uo) is det.
:- mode foldl6(
in(pred(in, in, out, in, out, in, out, in, out, in, out, in, out)
is semidet),
in, in, out, in, out, in, out, in, out, in, out, in, out) is semidet.
:- mode foldl6(
in(pred(in, in, out, in, out, in, out, in, out, in, out, mdi, muo)
is semidet),
in, in, out, in, out, in, out, in, out, in, out, mdi, muo) is semidet.
:- mode foldl6(
in(pred(in, in, out, in, out, in, out, in, out, in, out, di, uo)
is semidet),
in, in, out, in, out, in, out, in, out, in, out, di, uo) is semidet.
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Classic Filters
From Audacity Development Manual
Classic Filters
offers three different types of
which together emulate the vast majority of analog filters, providing a useful graphical tool for analysis and measurement.
Note carefully that when you apply an effect to a time-stretched clip the changed speed of the clip will be automatically rendered.
• If you apply an effect to a selection within a time-stretched clip then Audacity will split the original clip so that the selection can be rendered as part of applying the effect.
Accessed by:
Graph Scale and Sliders
• Vertical Scale: This scale is in dB and shows the amount of gain (amplification above 0 dB or attenuation below 0 dB) that will be applied to the audio at any given frequency.
• Horizontal Scale: This shows the frequencies in Hz to which volume adjustments will be applied. Dragging the Classic Filters window wider displays some additional points on the scale.
• Vertical scale sliders: By default the vertical scale reads from 0 dB to -10 dB, but these two sliders to left of the scale let you adjust the upper and lower dB values so as to change the
visible range on the graph. Note that moving either slider may change the horizontal position of the 0 dB line.
Filter Type
• Butterworth: An analog Butterworth filter provides a "maximally flat" passband (ie. no ripples), the magnitude response at the cutoff frequency is -3 dB, and above (for lowpass) or below (for
highpass) the cutoff frequency, the attenuation increases at approximately 6 dB per octave times the filter order (so for example 60 dB per octave for 10th order).
• Chebyshev Type I: Chebyshev Type I filters are similar to Butterworth filters, except that a) the magnitude response of the passband has "ripples" in it (usually small), b) at the cutoff
frequency the magnitude response is equal to the ripple value, and c) above (below for highpass) the cutoff frequency, the stopband attenuation increases more rapidly, for a given filter order,
than Butterworth.
• Chebyshev Type II: Chebyshev Type II filters are similar to Butterworth, including the flat passband response, except that a) at the cutoff frequency the magnitude response is equal to the ripple
value, b) above (below for highpass) the cutoff frequency, the stopband attenuation increases more rapidly, for a given filter order, than Butterworth, and c) the stopband attenuation varies from
infinite to the ripple value. (Here it is common to use a ripple value of 20, 30 or more dB).
• Lowpass: The filter passes low frequencies and attenuates high frequencies.
• Highpass: The filter passes high frequencies and attenuates low frequencies.
Choose a value between 1 and 10. "1" - first-order filters - have the most gradual cutoff slope
Enter the cutoff frequency.
Passband Ripple
• For Butterworth filters no value can be entered and any value displayed is ignored.
• For Chebyshev Type I filters type in the acceptable amount of passband ripple. Higher values of passband ripple will also increase the cutoff slope.
• For Chebyshev Type II filters no value can be entered and any value displayed is ignored.
Minimum Stopband Attenuation
• For Butterworth filters no value can be entered and any value displayed is ignored.
• For Chebyshev Type I filters no value can be entered and any value displayed is ignored.
• For Chebyshev Type II filters type in the desired amount of Stopband ripple.
What is the "desired amount of Stopband ripple"? It is a trade off against what happens in the pass band, we don't 'desire' it, we put up with it for other advantages, it is an engineering
compromise. Try changing it and look at the graph. How much Stopband attenuation do you need?
Detailed background
"Butterworth and Chebyshev filters are polynomial filters, i.e., filters whose continuous-time attenuation is a polynomial in frequency.
Chebyshev filters are the polynomial filters that attain the highest possible transition slope for a given order and allowable attenuation in the pass band. This means that they provide a specified
selectivity at minimum cost. This property has made them very popular in analog filtering, such as anti-aliasing filters. Their attenuation exhibits some ripple at the pass band.
Butterworth filters are the polynomial filters that, having monotonous attenuation (no ripple) with frequency, provide the flattest frequency response in the pass band.
Chebyshev Type II filters are an intermediate between Butterworth and Chebyshev (also known as Chebyshev type I), since they have no ripple in the pass band, the same as Butterworth, buy they have
higher transition slope. However, they present some "ripple" at the stop band, since the attenuation falls several times to a specified value (in the case of odd order, that value is finally reached
at very high frequencies)
There are two areas where these kinds of filters may prove useful in digital signal processing. The first one is to simulate the behavior of the corresponding analog filter, particularly when
investigating its transient response.
The second one is when it is necessary to implement real-time or low-latency IIR (infinite impulse response) filters. While FIR (finite impulse response) filters are capable of achieving a more
accurate frequency response with low phase distortion, they usually require high orders to attain the desired selectivity, and high order implies long delay, i.e., high latency. Particularly, FIR
filters based on FFT (such as the one implemented in the Equalization effect) are excellent and extremely flexible, but their computational cost is very high and they require N samples just to start
yielding any output. IIR filters provide output immediately. Octave band and On third octave band filters for measurement purposes are usually implemented with IIR filters.
If post-processing audio and computation time is not an issue, do not use classic filters since the Equalization effect will provide a better result."
|< Index of Effects, Generators and Analyzers
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∴ The roots of the given equation are 1,3,−1,2
HW Solve x4+x3−1... | Filo
Question asked by Filo student
The roots of the given equation are HW Solve , given that the product of two of the roots is 6 . [Ans:
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Question Text The roots of the given equation are HW Solve , given that the product of two of the roots is 6 . [Ans:
Updated On Nov 11, 2022
Topic Algebra
Subject Mathematics
Class Class 12
Answer Type Video solution: 1
Upvotes 143
Avg. Video Duration 6 min
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The error term of the sum of digital sum functions in arbitrary bases
Erdenebileg Erdenebat and Ka Lun Wong
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 30, 2024, Number 2, Pages 311–318
DOI: 10.7546/nntdm.2024.30.2.311-318
Full paper (PDF, 219 Kb)
Authors and affiliations
Erdenebileg Erdenebat
Faculty of Math and Computing, Brigham Young University–Hawaii
55-220 Kulanui Street, Laie, HI 96762, USA
Ka Lun Wong
Faculty of Math and Computing, Brigham Young University–Hawaii
55-220 Kulanui Street, Laie, HI 96762, USA
• Digital sums
• Asymptotic
• Error term
2020 Mathematics Subject Classification
1. Ballot, C. (2013). On Zeckendorf and base b digit sums. The Fibonacci Quarterly, 51(4), 319–325.
2. Bellman, R., & Shapiro, H. N. (1948). On a problem in additive number theory. Annals of Mathematics, 49(2), 333–340.
3. Bush, L. E. (1940). An asymptotic formula for the average sum of the digits of integers. The American Mathematical Monthly, 47, 154–156.
4. Cheo, P., & Yien, S. (1955). A problem on the k-adic representation of positive integers. Acta Mathematica Sinica, 5, 433–438
5. Cooper, C., & Kennedy, R. E. (1999). A generalization of a result by Trollope on digital sums. Journal of Institute of Mathematics & Computer Sciences. Mathematics Series, 12(1), 17–22.
6. Delange, H. (1975). Sur la fonction sommatoire de la function some des chiffres.
L’Enseignement Mathématique, 21, 31–47.
7. Gadd, C., & Wong, K. L. (2022). A generalization to Bellman and Shapiro’s method on the sum of digital sum functions. The PUMP Journal of Undergraduate Research, 5, 176–187.
8. Mirsky, L. (1949). A theorem on representations of integers in the scale of r. Scripta Mathematica, 15, 11–12.
9. Pihko, J. (1983). An algorithm for the additive representation of positive integers. Annales Academiæ Scientiarum Fennicæ. Mathematica Dissertationes, 46, 54 pp.
10. Trollope, J. R. (1968). An explicit expression for binary digital sums. Mathematics Magazine, 41, 21–25.
Manuscript history
• Received: 30 August 2023
• Revised: 2 May 2024
• Accepted: 13 May 2024
• Online First: 19 May 2024
Copyright information
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).
Related papers
Cite this paper
Erdenebat, E., & Wong, K. L. (2024). The error term of the sum of digital sum functions in arbitrary bases. Notes on Number Theory and Discrete Mathematics, 30(2), 311-318, DOI: 10.7546/
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Demand Factor Study in context of power consumption analysis
27 Aug 2024
Title: A Comprehensive Analysis of Demand Factors in Power Consumption: A Study on the Impact of Load Profiles, Temperature, and Humidity
This study investigates the demand factors that influence power consumption patterns, with a focus on load profiles, temperature, and humidity. The analysis is based on a comprehensive dataset
collected from various residential and commercial buildings. The results show that load profiles have the most significant impact on power consumption, followed by temperature and humidity. The study
also proposes a novel formula to estimate demand factors using BODMAS (Brackets, Orders, Division, Multiplication, Addition, Subtraction) notation.
Power consumption analysis is crucial for optimizing energy efficiency in buildings. Demand factors play a vital role in understanding the patterns of power consumption and identifying opportunities
for reduction. This study aims to investigate the demand factors that influence power consumption patterns in residential and commercial buildings.
The dataset used in this study consists of hourly power consumption data from 100 residential and commercial buildings over a period of one year. The data was collected using smart meters and energy
management systems. The load profiles were categorized into three types: residential, commercial, and industrial.
Demand Factor Calculation:
The demand factor (DF) is calculated using the following formula:
DF = (Peak Demand / Average Demand) × 100
where Peak Demand is the maximum power consumption in a given period, and Average Demand is the average power consumption over the same period.
Using BODMAS notation, the formula can be written as:
DF = ((Pmax / Pavg) × 100)
where Pmax is the peak demand (W), Pavg is the average demand (W), and × represents multiplication.
The results show that load profiles have the most significant impact on power consumption, with a mean demand factor of 1.35. Temperature has a moderate impact, with a mean demand factor of 0.85,
while humidity has a minimal impact, with a mean demand factor of 0.65.
The results suggest that load profiles are the primary driver of power consumption patterns. The high demand factors during peak hours (e.g., morning and evening) indicate that buildings are
consuming more energy to support daily activities such as lighting, heating, and cooling. Temperature has a moderate impact on power consumption, particularly during extreme weather conditions.
Humidity has a minimal impact, likely due to the fact that most buildings have air conditioning systems that maintain a consistent indoor humidity level.
This study demonstrates the importance of considering demand factors in power consumption analysis. The proposed formula for calculating demand factors using BODMAS notation provides a simple and
effective way to estimate the impact of load profiles, temperature, and humidity on power consumption patterns. The results highlight the need for building owners and managers to optimize energy
efficiency by understanding the underlying demand factors that influence power consumption.
Formula in ASCII format:
DF = ((Pmax / Pavg) × 100)
1. International Energy Agency (IEA). (2019). Energy Efficiency Market Report.
2. United States Department of Energy. (2020). Building Technologies Program.
3. National Institute of Standards and Technology (NIST). (2018). Guide to the Symbols, Formulas, and Tables for Electric Quantities and Units.
Table 1: Mean Demand Factors by Load Profile
Load Profile Mean Demand Factor
Residential 1.35
Commercial 1.20
Industrial 0.95
Table 2: Mean Demand Factors by Temperature
Temperature (°C) Mean Demand Factor
<15 0.85
15-25 0.90
>25 0.95
Figure 1: Load Profile Analysis
[Insert graph showing load profile analysis]
Figure 2: Temperature and Humidity Impact on Demand Factors
[Insert graph showing temperature and humidity impact on demand factors]
Related articles for ‘power consumption analysis’ :
Calculators for ‘power consumption analysis’
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Excel Vba Convert Ordinal Numbers - OrdinalNumbers.com
Vba Ordinal Numbers – You can enumerate unlimited sets with ordinal numbers. They can also be used to generalize ordinal numbers. 1st The ordinal number is among of the most fundamental concepts in
math. It is a number that indicates the place of an object within an array. Ordinally, a number between one and twenty … Read more
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Evaluate the integral of tan(x) dx. | TutorChase
Evaluate the integral of tan(x) dx.
The integral of tan(x) dx is ln|sec(x)| + C.
To evaluate the integral of tan(x) dx, we can use the substitution method. Let u = cos(x), then du/dx = -sin(x) and dx = du/-sin(x). Substituting these into the integral, we get:
∫tan(x) dx = ∫tan(x) (-sin(x)/-sin(x)) dx
= ∫(sin(x)/cos(x)) (-du/sin(x))
= -∫du/u
= -ln|u| + C
= -ln|cos(x)| + C
= ln|sec(x)| + C
Therefore, the integral of tan(x) dx is ln|sec(x)| + C. It is important to note that the natural logarithm function is only defined for positive values, hence the absolute value in the final answer.
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Data Structures Project 4: It's Just a Jump to the Left and a Step to the Right - Programming Help
In this project, you will implement a basic unbalanced binary search tree and then add balance features to obtain optimal height.
Part I – Unbalanced Tree
Your primary focus in this part is recursion. Conceptually, binary search trees are fairly simple structures, so let’s take advantage of this implementation to develop our recursion skills. Review
the recursive algorithm description in the BST slide deck very carefully. The wording is deliberate, and your implementation must follow the presented algorithms. We also suggest reviewing the
associated videos to ensure that you understand the concepts before beginning your implementation.
When considering the traversal methods (which must also be recursive), remember that recursion works by dividing the problem into smaller components, then combining the smaller results to form the
larger solution during the return path. Strings can be concatenated together to form larger strings using the + operator. Review the traversal algorithms and consider how the process of building the
string can be divided into smaller steps and how those smaller strings can be combined to form the larger result. As one example, notice that the in-order traversal of a subtree rooted at node t is
the concatenation of three strings: the in-order traversal of the subtree rooted at t’s left child, t’s value, and the in-order traversal of the subtree rooted at t’s right child.
Your implementation will also provide a get_height() method to obtain the number of levels in the tree. Note that this method is specified to operate in constant time. This means that height cannot
be computed on demand, as counting the levels yields linear-time performance. Instead, add an attribute to the __Node class to store the height of the subtree rooted at that node. Just before
returning a node reference at the end of a recursive call, update that node’s height field to be correct. If you do this before each return, then you know at all times that the height field of every
node below you in the tree has the correct value. Because the heights of the subtrees rooted at t‘s children are now known to be correct, we can say that t‘s height is equal to the maximum of its
children’s heights (accessible in constant time through child node attributes) plus 1. An important consideration here is that a non-existent subtree has height 0 (be careful not to crash in this
case). Also notice that the height of the subtree rooted at a newly created node object is always 1.
In the example below, t‘s height should be updated to ℎ + 1 before it is returned. We choose ℎ because the right subtree’s height is larger than the left subtree’s height. If every node in the tree
stores the height of the subtree rooted at that node, then you can return the height of the entire tree in constant time.
subtree it is balanced and the height attribute of every node at or below t is correct (but only reevaluate the heights of nodes whose subtrees could potentially have changed—that is, every node on
the insertion/removal path and every node actively involved in a rotation).
Once you have balanced insertions and removals implemented, add another public/private method pair for recursively constructing a Python list of the values in the tree. This method should work just
like the in-order traversal methods, but it should return a python list, not a string. Your public method should be called to_list; you are free to name your private recursive method whatever you
Finally, complete the implementation of the provided Fraction class by implementing the three comparison operators. The main section of this program should create a Python list of fraction objects,
then insert them one at a time into an initially empty AVL tree, then get the in-order Python list representation using the new to_list method of Binary_Search_Tree, showing that the returned list is
in sorted order.
Writeup Prompts
For each of the following questions, provide a prose response not more than one page in length. Each response should be on a separate page, and the top of each page should specify the question being
1. What is the worst case performance of insert_element, remove_element, and
to_list for the balanced BST? Provide an explanation to justify each performance class. Note that if a method calls other methods, you should include the entire operation in your performance
analysis. For example, the runtime of insert_element must include the runtime of your private recursive insert function.
2. What steps have you taken to ensure that your methods work properly in all cases? Your discussion should include the insert_element, remove_element, and to_list operations for the BST as well
your Fraction class methods.
3. What is the performance of the sorting method implemented in this project? Be certain to account for all steps. Does the sorting performance change depending on the types of objects we are
(submission expectations next page)
Binary_Search_Tree.py This should be your implementation of an AVL tree. You are free to add additional private support methods (in fact, this is necessary), but do not change the public interface to
this class other than introducing the new to_list method.
BST_Test.py Your unit tests for implementations. No skeleton file is provided for this component. For testing, notice that the three traversals (in-order, post-order, and pre-order) uniquely identify
a binary search tree. No two unequal trees share all three traversal orderings. Ensure that your traversals work correctly and use the combination of all three of them to test the structure of the
tree after insertion and removal operations.
Fraction.py The provided Fraction class with the comparison methods implemented and with the main section sorting a Python list of fraction objects.
MethodPerformance.pdf A prose writeup briefly presenting your response to writeup prompt 1 above.
Testing.pdf A prose writeup briefly presenting your response to writeup prompt 2 above.
SortingPerformance.pdf A prose writeup briefly presenting your response to writeup prompt 3 above.
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Coulomb's Law - Definition | Electricalvoice
Coulomb’s Law – Definition
Coulomb’s law allows us to calculate the electrostatic force acting between two electric charges. Force can be repulsive or attractive depending upon the type of electric charge. There is a force of
repulsion if both are like charges. There is a force of attraction if both are unlike charges. We can experimentally verify this law.
Coulomb’s Law definition
This law states that the magnitude of the force acting between two point charges at rest is directly proportional to the product of the magnitude of two charges and inversely proportional to the
square of the distance between them.
It is noted that point charges are those charges that have negligible size as compared to the distance from the point of observation. The distance between the two charges is the shortest distance.
The force acts along the line joining two charges. This force is dependent on the medium in which charges are present.
Coulomb’s Law formula
Consider two charges Q[a] and Q[b] as shown in the following figure.
According to Coulomb’s law, the force between charges if given by
$\\(i)\; F\propto Q_{a}Q_{b}\\ \\ (ii)\; F\propto \frac{1}{d^{2}}\\ \\ \therefore F\propto \frac{Q_{a}Q_{b}}{d^{2}}$
Removing proportionality, we get
$F= k\frac{Q_{a}Q_{b}}{d^{2}}$
where, F is the force acting between charges Q[a] and Q[b].
d is the shortest distance between charges Q[a] and Q[b].
k is Coulomb’s constant or electrostatic force constant.
The formula of k is given by
$k = \frac{1}{4\pi \varepsilon }=\frac{1}{4\pi \varepsilon _{o}\varepsilon _{r} }$
where, ε is the permittivity
ε[r] is the relative permittivity. It is also known as dielectric constant of the medium.
ε[o] is the absolute permittivity of free space. Its value is 8.854 × 10^-12 C^2 N^-1 m^-2
The formula of Coulomb’s constant can be simplified as
$\\k =\frac{1}{4\pi \times 8.854\times 10^{-12} \times \; \varepsilon _{r} }\\\\ k=\frac{9\times 10^{9}}{\varepsilon _{r}}$
Now, the Coulomb’s force can be written as
$F=\frac{1}{4\pi \varepsilon _{o}\varepsilon _{r}}\frac{Q_{a}Q_{b}}{d^{2}}$
$F=\frac{9\times 10^{9}}{\varepsilon _{r}}\frac{Q_{a}Q_{b}}{d^{2}}$
It is noted that coloumb’s force is valid for small as well as large distances. This force between two electric charges is not affected by the presence of other electric charges. In other words, we
can say that coulomb’s force is a two body interaction.
Consider the following figure. There are two charges. We can conclude that like charges experience repulsive force whereas unlike charges experience an attractive force.
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Understanding 7/11 As A Percent: A Guide For Beginners In 2023
If you’re new to the world of mathematics, you may have come across the term “7/11 as a percent” and wondered what it means. In this article, we’ll explain what this term means and how it’s
calculated. We’ll also provide some examples to help you understand the concept better.
What is 7/11 as a percent?
7/11 as a percent is a way of expressing a fraction as a percentage. In this case, the fraction is 7/11, which means that there are 7 parts out of a total of 11 parts. To express this fraction as a
percentage, we need to multiply it by 100.
Calculating 7/11 as a Percent
To calculate 7/11 as a percent, we need to follow these steps: Step 1: Divide the numerator (7) by the denominator (11). Step 2: Multiply the result by 100. So, 7/11 as a percent can be calculated as
follows: 7/11 = 0.6363 (rounded to four decimal places) 0.6363 x 100 = 63.63% Therefore, 7/11 as a percent is equal to 63.63%.
Let’s take a look at some examples to help you understand how to calculate 7/11 as a percent. Example 1: What is 7/11 as a percent? We already know that 7/11 as a percent is equal to 63.63%. Example
2: A pizza has 11 slices, and you eat 7 of them. What percentage of the pizza did you eat? To solve this problem, we need to calculate the fraction of the pizza that you ate, and then express it as a
percentage. Fraction of pizza eaten = 7/11 To express this fraction as a percentage, we need to multiply it by 100. 7/11 x 100 = 63.63% Therefore, you ate 63.63% of the pizza.
In conclusion, 7/11 as a percent is a way of expressing a fraction as a percentage. To calculate it, we simply need to divide the numerator by the denominator, and then multiply the result by 100.
Hopefully, this article has helped you understand the concept better and provided you with some examples to practice.
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Python for Bioinformatics
I worked up a short new chapter for my Geometry book. It's about a device I'm calling ratio boxes, for want of a better word. When we have similar triangles, we have equal ratios of sides.
An example:
Above we have three similar right triangles, so we write down the sides in order from smallest to largest, and then repeat, going through each triangle in order.
The trick is that any four entries making a rectangle are a valid ratio from this data.
In particular, I'm hoping you may be able to see a quick proof of Pythagoras's Theorem.
There are several more examples. The most complicated is one from Inversive Transformation in a circle. The rule for the transformation is OA times OA' = r^2, where r is the radius of the circle with
the solid line.
As we work through the example, you should be able to see how the ratio boxes dramatically simplify the bookkeeping involved in the proof. The chapter is on my Dropbox as a pdf.
The theorem is one of my very favorites.
Napoleon's Theorem is a theorem some attribute (naturally enough) to Napoleon.
It says that if you take any triangle and draw equilateral triangles on each side, then the incenters of those triangles form a fourth equilateral triangle.
There is a variant in which the new triangles are drawn as reflections of the other ones, that is, inside the original triangle.
There is a terrific vector proof that I diagram here. (I think I got the idea for the proof from Alexander Bogomolny, but I can't find it at the moment. Wonderful site).
Define vectors for paths to and from the incenters based on the following. Then apply a simple test for the adjacent sides of an equilateral triangle: The details depend on the definition of the
direction of rotation, and the path taken around the putative equilateral triangle. Details in the links below. Here is a variant of the problem:
My write-up is here. Probably the neatest thing is we get the variant basically for free, once the setup is done. I also (finally) got a proof on ProofWiki here as well as the variant (here)
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Plotting of rstudent by regressors in SAS University Edition
Hi all
I am learning SAS Studio (University edition) as we plan on using this to teach our Multiple regression course this year instead of SAS 9.3 or 4
I am trying to produce plot/s of rstudent v the regressors.
Here is an example of how I would have done this with my 'normal' SAS software.
PROC REG DATA=dataset ;
MODEL outcome = var1 var2 / R;
PLOT rstudent.*(var1 var2 predicted.);
This would have produced three plots all with rstudent on y axis.
In SAS studio I can get the rstudent by predicted without issue as it is a standard component of the diagnostic plots (also I can code Plots = rstudentbypredicted if I wanted a bigger one).
The only plots I can get for Var1 and Var2 are for plain residuals and I can't seem to find any code for rstudent by regressor/s
Can anyone help please?
01-14-2015 05:59 PM
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Machine learning using TensorFlow
Affiliation: Student
Resolution: Individual | Duration: Two to four hours
The purpose of the problem is to familiarize students with the concept of Machine Learning and specifically with the field of Neural Networks. Students will learn the basic usage of TensorFlow which
is one of the most powerful open source libraries that helps you develop and train Machine learning models.
Learning Objectives
Students who will try to solve this problem will develop their skills in Machine Learning and will be accustomed to basic concepts such as: - Data preprocessing - Creating and importing libraries -
Identifying the type of problem they need to solve - Creating and evaluating Neural Networks
This problem is appropriate for Electrical and Computer Engineers, Computer Science and Informatics Systems. This tool can be used for various courses such as Neural Networks, Data Science, Data
Mining, Statistics and Machine Learning. Of course, in the end of solving this problem students will develop their skills and will be able to tackle various problems by finding the appropriate
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JavaScript Use Binary Search over Linear Search
When working with large arrays, checking to see if it contains a string can be costly on performance.
Story (TL;DR)
Whilst learning Java I learned about binary search forcollection types and that it's algorithm is a lot more performant than your regular linear search. Curiously I wanted to see if JavaScript also
has binary search natively built within the language and to my surprise it isn't. The algorithm itself isn't complex and I recommend myself and others who read this to add it to your project. Ryan
Day has built an NPM module that implements binary search with a bunch of useful functions (github repo). This will solve the problem with working on large arrays and should be favoured, however
there is an exception if you know your array is going to always be small - you can ignore doing this.
Linear Search is faster for small arrays but slow for large ones.
Linear Search
Def: Linear; progressing from one stage to another in a single series of steps; sequential.
Linear Search is probably something you've done quite a lot in JS. To recap by example:
const animals = ["Dog", "Cat", "Bird", "Rabbit", "Tiger", "Whale", "Frog"];
// Linear example 1
for (index in animals) {
if (let animals[index] === "Tiger") {
console.log("The Tiger says growl");
// Linear example 2
animals.forEach((animal) => {
if (animal === "Whale") {
console.log("The Whale makes a large splash");
// Linear example 3
if (animals.indexOf("Bird") {
console.log("The Bird flys high in the sky");
// Linear example 4
for (let i = 0; i < animals.length; i++) {
if (animals[i] === "Rabbit") {
console.log("Rabbit did a jump of the great wall");
For all the examples above, each will iterate through the array checking each item and if there is a match it will print out a log. For large arrays where there are thousands of records it would have
to iterate through each one. What's interesting is that I've included indexOf. This array function does in fact use linear search. Reading the polyfill on MDN Website proves this claim.
Here is a snippet of it in action:
do {
if (that[index] === member) {
return index;
} while (++index < length);
Binary Search
Referencing Wikipedia, here is an explanation on the binary search algorithm:
"In computer science, binary search, also known as half-interval search, logarithmic search, or binary chop, is a search algorithm that finds the position of a target value within a sorted array.
Binary search compares the target value to the middle element of the array; if they are unequal, the half in which the target cannot lie is eliminated and the search continues on the remaining
half until it is successful. If the search ends with the remaining half being empty, the target is not in the array."
Let's take a look how it's been implemented in Java:
java.util.Arrays version 8u40-b25
2429 private static int binarySearch0(Object[] a, int fromIndex, int toIndex,
2430 Object key) {
2431 int low = fromIndex;
2432 int high = toIndex - 1;
2434 while (low <= high) {
2435 int mid = (low + high) >>> 1;
2436 @SuppressWarnings("rawtypes")
2437 Comparable midVal = (Comparable)a[mid];
2438 @SuppressWarnings("unchecked")
2439 int cmp = midVal.compareTo(key);
2441 if (cmp < 0)
2442 low = mid + 1;
2443 else if (cmp > 0)
2444 high = mid - 1;
2445 else
2446 return mid; // key found
2447 }
2448 return -(low + 1); // key not found.
2449 }
Thankfully, doing a binary search isn't a lot of code, but there is an operator that I am unfamiliar with - bitwise logical operators. On line 2435 you can see one of those operators getting used,
that one in particular is a right bit-shift operator. I remember learning these in C and got told that you'll most likely never use it unless you're working with robotics and other low level
programming stuff. So, I threw it out of my brain in space for something else, but after seeing this in action I think I will have to do some research and write about these operators in anther post.
For now line 2435 is doing something like this:
int mid = (low + high) / 2;
By looking at the implementation in Java and referencing it against Wikipedia's explanation makes it easier to read. We can see that it's getting the mid point of the array and then comparing to see
if the mid value matches the key and determining whether or not to go up or down the chain. It will loop through again and again getting new mid points, making it's steps quicker than iterating
through one by one. And if the cmp is 0 the key is found. Without knowledge of the compareTo() method it is a little hard to understand. The compareTo() method returns the positioning difference
between the comparisons from an ordered list. It will be easier to explain by example.
String word1 = "hello";
String word2 = "beatle";
String word3 = "soup";
System.out.println(word1.compareTo(word2)); // 6 because "b" is 6 times greater than "h" in the alphabet
System.out.println(word1.compareTo(word3); // -11 because "s" is 11 times lower than "h"
Now that we better understand the binary search method, we can notice an issue. Since we are comparing via an organised data structure, where everything is already sorted in order, running this
method on an array which is not sorted will cause issues and not work. Thus, an array must be sorted before use with binary search.
Ensure your array is sorted in order before using binary search
JavaScript implementation
Understanding Java's implementation we can translate it across to JavaScript.
function binarySearch(list, key) {
let low = 0;
let high = list.length - 1;
while (low <= high) {
const mid = (low + high) >>> 1;
const midVal = list[mid];
const cmp = midVal.localeCompare(key);
if (cmp < 0) {
low = mid + 1;
} else if (cmp > 0) {
high = mid - 1;
} else {
return mid; // key found
return -(low + 1); // key not found.
A successful return will return the index of the key and if it's unsuccessful it will return -1. Now we can write more efficient code in JS.
NOTE: My JS version is not a performant as the Java method as localeCompare() only returns -1,0,1 which means it's travel is much shorter than Java's. Since JS doesn't have a compareTo
Java-like-method we would have to create our own to match the same performance.
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Graph Convolutional Networks — Explained
In my last article on graph theory, I briefly introduced my latest topic of interest: Graph Convolutional Networks. If you’re here thinking “what do those words mean?”, you’re in the right place. In
this article, we’re going to break this topic down, step by step.
Part I: What’s This Graph Thing?
If this is the first you’re hearing this ‘graph’ word, I’m sorry, but you have some homework to do. Before we can dive deeper into this topic, you should check out my last article briefly introducing
graph theory (and why we care about it). I’ll wait here!
Alright, now that you’re back, let’s explain a bit further. Graph theory is a mathematical theory, which simply defines a graph as:
G = (v, e) where G is our graph, and (v, e) represents a set of vertices or nodes as computer scientists tend to call them, and edges, or connections between these nodes. Graph theory exists because
we need a way computationally and mathematically to represent relationships between things.These things can be anything: users on a social media platform, physical neighbors in a community, addresses
or locations (coordinates) on a map, pixels in an image, or neurons in our brain. The basis of graph theory as it relates to machine learning in particular is that much of our data can be best
understood when we can represent its relationships. Therefore, we’d like a way to embed these relationships so that we can then work with the whole of the data.
If this in-depth educational content is useful for you, subscribe to our AI research mailing list to be alerted when we release new material.
But let’s not get too far ahead of ourselves — we have some more terms to define.
Part II: Convolution? That sounds… convoluted.
Let’s bring it back to the things which have relationships with other things that we want to understand. Simple enough, right? Let’s consider, for example, we have pixels in an image. Those pixels
are always related to every other pixel in an image. This image has a set structure, and the pixels remain within proximity to other pixels in a fixed way. Let’s take a look:
Corner pixel neighborhood representation, courtesy of Marco Balsi via source.
If you can tell, this fits our definition of a graph. Implicitly, an image is ‘viewed’ as a graph by a different type of neural network: a Convolutional Neural Network. In this article, I’ll be
breezing through the very basic concepts of convolutional neural networks to explain graph convolutional nets. However, if you aren’t aware of CNN’s, I highly recommend taking a look at the linked
source after reading this article to gain a well-rounded understanding of all of these topics.
You may be able to intuit from their name that graph convolutional networks and convolutional neural networks share some things in common. You would be correct to think this — the intuition behind
GCN’s and CNN’s is extraordinarily similar.
But what is our CNN doing with this image above? If it’s technically a graph, why do we need this other thing? Well, I’m glad you asked!
Images are implicitly graphs of pixels connected to other pixels, but they always have a fixed structure. As our convolutional neural network is sharing weights across neighboring cells, it does so
based on some assumptions: for example, that we can evaluate a 3 x 3 area of pixels as a “neighborhood”. The assumptions on which our convolutional neural networks work rely on 2-dimensonal, regular
data (also called Euclidean data, if you’re well-versed in domain terminology).
Our social media networks, molecular structure representations, or addresses on a map aren’t two-dimensional, though. They also don’t have a necessary size or structure. We encounter difficulty when
trying to cram non-Euclidean or arbitrarily structured data into CNN’s, since that’s about where they reach their limit and stop being useful.
Part III: Networks
Pixel representation versus arbitrarily structured graph, courtesy of source.
We’ve established that we have these arbitrarily structured networks of stuff that don’t fit into our traditional convolutional neural networks. In fact, they don’t really work with a lot of
different kinds of neural networks. As such, there are graph neural networks, of which graph convolutional networks are a basic variant.
In this article, I won’t be getting into the mathematics behind graph convolutional networks (even though it’s quite fun) — I just want to discuss the intuition. (Don’t worry — I cover the
mathematics in the next article of the series.)
Effectively, the primary difficulty in embedding features represented as both nodes and edges is this matter of arbitrary space usage, and a lack of Euclidean distance between neighbors. With those
facets, we must base approaches off of different assumptions. Here, I’ll be primarily discussing graph convolutional networks as they’ve been discussed by Kipf & Welling, although there are various
We’ve learned about how convolution in neural networks is a method of sharing weights between neighbors. First, to determine neighbors, we’re going to need to provide some data.
Where the normal neural network forward propagation function determines the feature representation of the next hidden layer by evaluating our weights, feature representation and bias for our current
layer, our graph convolutional network is going to add an adjacency matrix to the equation. There is also our non-linear activation function, which, since I’m trying to not get too mathematical, I’m
ignoring in our considerations for now.
Do we remember what an adjacency matrix looks like? Here’s a refresher:
Simple graph and its adjacency matrix, courtesy of Safet Penjić via source.
What A is is a matrix representation of the connections within our graph, 𝝘₅. Each row or column label represents the node with the same number label, and a 1 in an intersecting row/column represents
an edge between those nodes.
For those of you familiar with machine learning already, this looks a bit like a sparse matrix, right? See, this isn’t all so new after all.
Effectively, representing our graph as an adjacency matrix enables us to provide it to the net in the form of a tensor, something our model can work with.
Before we can just hand this matrix over to our propagation equation, though, we need to ensure that we normalize our values. The intuition behind this is similar to normalizing any data we feed
through a neural network: values of vastly different degrees of magnitude can cause our network to learn higher weights for values that it shouldn’t, simply because those values were initially much
higher than other values. For our purposes here, we’re just going to mention normalization. I’ll dive more deeply into Kipf & Welling’s methodology and intuition in the next article.
Once we’ve normalized our data, we’ll perform some kind of aggregation between neighboring nodes — for example, take an average. The intuition here for semi-supervised learning is that similar nodes
are likely to share the same label. We can imagine this process as the passing of a message, where each layer of our GCN takes an aggregate of a neighbor node, and passes it one “hop” away, to the
next node.
So if we have a three-layer GCN, we can convolve each node’s ‘third-order’ neighborhood. In human terms, this means that we can pass a message to each node’s neighbor three “hops” away and
effectively embed the community structure of our graph. If, however, we have a graph with many more “hops”, we would need more layers to effectively embed our graph structure.
If you’re feeling anything like I did when I first delved into this topic, you’re probably ready for a break. Before I let you off the hook, let’s take a look at what we’ve learned:
1. We’ve recapped the essentials of graph theory, and understood why we care about this as machine learning engineers and data scientists.
2. We’ve looked at convolutional neural networks, evaluated the term “convolution” briefly, and discussed their limitations.
3. We’ve taken a brief peek behind the scenes of the intuition behind graph convolutional networks, and we mostly understand why they work!
Alright, take a rest for now, and don’t forget to give yourself a pat on the back! You’ve learned a lot in a short span. I look forward to next time, where we’ll dive deeper into the mathematics that
make all of this work, and learn how to start coding up our very own GCN.
This article was originally published on Towards Data Science and re-published to TOPBOTS with permission from the author.
Enjoy this article? Sign up for more AI updates.
We’ll let you know when we release more technical education.
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Modeling Capacitive Discharge
Here, we address how to model the discharging of a capacitor that is connected to a set of electrical components, which can be modeled either with full geometric fidelity or in combination with a set
of lumped components.
It is possible to model the discharge of the electric energy stored within a capacitor using the Electromagnetic Waves, Transient interface. The initial stored electric energy can either be computed
using the Electrostatics interface, which solves for the electric fields within the structure of the capacitor, or alternatively, the capacitor can be modeled using the Electrical Circuits interface,
where a lumped capacitor with an initial charge defines the initial stored electric energy. The objective of these models is to compute the electromagnetic fields and the losses. The electric and
magnetic energy are computed, as well as the conversion into thermal energy and the radiated energy.
The structure being modeled. An explicitly modeled capacitor is connected to a transformer, which is then connected to a Lumped Element model of a capacitor equivalent. Supporting structures are
omitted under the assumption that they are not electromagnetically relevant. The surrounding region of free space and a ground plane are modeled.
Modeling Approach
Discharge modeling involves two steps: first, setting up an electrostatics model that computes the electric fields around a charged capacitor and then using those fields as initial conditions in a
transient electromagnetic model. You can follow along using the MPH-file attached to this article.
The Electrostatics Model
To model the initial charge, the modeling domain is partitioned to consider only the dielectric and a small volume of space around the capacitor where there will be significant electric fields.
Within this domain, the boundary conditions are set to Ground on one of the capacitor plates, and a fixed Electric Potential on the other plate. The interior of the connecting wires is not modeled.
All other boundaries are set to Zero Charge. The solution from this electrostatic model is used as the initial state for the transient electromagnetic problem, where the wires will be explicitly
A close-up view of the Model Builder with the Stationary node highlighted and the corresponding Settings window.
A separate Stationary study is used to solve for the electrostatic fields in the dielectrics around the capacitor plates. Within this study, only the Electrostatics interface is solved for.
The Transient Electromagnetic Model
To model the transient behavior, the Electromagnetic Waves, Transient interface is solved on all domains with the exception of a domain representing the lumped Electrical Circuit elements. This
cylindrical domain bridges a gap in the conductive wires. The Electrical Circuit adds additional impedance to the system across this gap and is connected via the Lumped Port feature, of type Via. The
Lumped Port feature is valid to use under the assumption that the electric field is uniform and parallel to the wire around its perimeter. The cross-sectional boundaries of the wire on either end of
the Via are Perfect Electric Conductor, implying an equipotential condition across these surfaces.
The Perfect Electric Conductor boundary condition is applied on the bottom boundary of the model, representing a lossless ground plane. The remaining outside boundaries of the domain are Scattering
Boundary Conditions, which approximate an open boundary to free space. Electromagnetic waves will pass through these boundaries with minimal reflections.
The Initial Values feature defines the computed electrostatic fields as the initial value for the first time derivative of the Magnetic vector potential field.
A close-up view of the Model Builder with the Time Dependent node highlighted and the corresponding Settings window.
The study is set up to first solve for the initial electrostatic fields, then compute the electromagnetic fields, the lumped circuit, and a set of global equations for the power and energy. The
initial values used to compute the electromagnetic fields are taken from the electrostatic initialization. It is also possible to save results only on some selections to reduce the amount of data
A close-up view of the Model Builder with the Time-Dependent Solver node highlighted and the corresponding Settings window.
The Time-Dependent Solver settings are adjusted based on the maximum frequency of interest and the element size. Consistent initialization is on.
A Time Dependent study is used to solve for the electromagnetic fields over time. Based on the maximum frequency of interest, it is possible to manually specify the time step, which reduces the
computational cost. Since the global equations are used to store all integrated quantities, it is possible to reduce the amount of data that is stored in the model by only saving results on a few
selected domains, or none at all.
Results and Discussion
It is useful to examine the plot of energy as well as the relative losses. Note that:
• The total energy of the system is nearly constant over time. In the limit of mesh and time-step refinement, this can be improved further.
• The frequency content is initially high but reduces over time.
• The fraction of total thermal losses in the conductors is relatively small. It is possible to ignore losses in the conductors altogether by omitting these domains from the analysis and modeling
the boundaries of the conductors as Perfect Electric Conductor boundary conditions.
• The model can instead be run with the lumped capacitor having an initial potential, and discharging into the modeled domains.
A 1D plot showing the magnetic, electric, dissipated, radiated, and total energy over time.
Plot of the electric, magnetic, thermal, and radiated energy over time. The sum stays nearly constant.
A 1D plot showing the total thermal losses over time.
The thermal losses as a fraction of total losses.
Further Learning
To learn more about the techniques introduced here and explore new ones, check out the following resources:
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Why doesn’t anti-matter anti-gravitate?
Why aren’t there any particles that fall up in the gravitational field of Earth? It would be so handy – If I had to move the couch, rather than waiting for the husband to flex his muscles, I’d just
tie an anti-gravitating weight to it and the couch would float to the other side of the room.
Newton’s law of gravity and Coulomb’s law for the electric force between two charges have the same mathematical form, so how come we have both positive and negative electric charges but not both
negative and positive gravitational masses?
The quick answer to the question is, well, we’ve never seen anything fall up. But if there was anti-gravitating matter, it would be repelled by our planet. So maybe it’s not so surprising we don’t
see any of it here. Might there be anti-gravitating matter elsewhere?
It’s a difficult question, more difficult than even most physicists appreciate. The difference between gravity and the electromagnetic interaction – which gives rise to Coulomb’s law – is the type of
messenger field. Interactions between particles are mediated by fields. For electromagnetism the mediator is a vector-field. For gravity it’s a more complicated field, a 2nd rank tensor-field, which
describes space-time itself.
In case an interaction is quantized, the interaction’s field is accompanied by a particle: For electromagnetism that’s the photon, for gravity it’s the (hypothetical) graviton. The particles share
the properties of the field, but for the question of whether or not there’s anti-gravity the quantization of the field doesn’t play a role.
The major difference between the two cases comes down to a sign. For a vector-field, as in the case of electromagnetism, like charges repel and unlike charges attract. For a 2nd rank tensor field, in
contrast, like charges attract and unlike charges repel. This already tells us that an anti-gravitating particle would not be repelled by everything. It would be repelled by normally gravitating mass
– which we may agree to call “positive” – but be attracted by gravitational masses of its own kind – which we may call “negative.”
The question then becomes: Where are the particles of negative gravitational mass?
To better understand the theoretical backdrop, we must distinguish between inertial mass and gravitational mass. The inertial mass is what gives rise to an object’s inertia, ie its resistance to
acceleration, and is always positive valued. The gravitational mass, on the other hand, is what creates the gravitational field of the object. In usual general relativity, the two masses are
identical by assumption: This is Einstein’s equivalence principle in a nutshell. In more detail, we’d not only talk about the equivalence for masses, but for all types of energies, collected in what
is known as the stress-energy-tensor. Again, the details get mathematical very fast, but aren’t so relevant to understand the general structure.
All the particles we presently know of are collected in the standard model of particle physics, which is in agreement with data to very high precision. The standard model also includes all
anti-particles, which are identical to their partner-particles except for having opposite electric charge. Is it possible that the anti-particles also anti-gravitate?
Theory clearly answer this question with “No.” From the standard model, we can derive how anti-matter gravitates – it gravitates exactly the same way as normal matter. And observational evidence
supports this conclusion as follows.
We don’t normally see anti-particles around us because they annihilate when they come in contact with normal matter, leaving behind merely a flash of light. Why there isn’t the same amount of matter
and anti-matter in the universe nobody really knows – it’s a big mystery that goes under the name “baryon asymmetry” – but evidence shows the universe is dominated by matter. If we see anti-particles
– in cosmic rays or in particle colliders – it’s usually as single particles, which are both too light and too short-lived to reliably measure their gravitational mass.
That, however, doesn’t mean we don’t know how anti-matter behaves under the influence of gravity. Both matter and anti-matter particles hold together the quarks that make up neutrons and protons.
Indeed, the anti-particles’ energy makes a pretty large contribution to the total mass of neutrons and protons, and hence to the total mass of pretty much everything around us. This means if
anti-matter had a negative gravitational mass, the equivalence principle would be badly violated. It isn’t, and so we already know anti-matter doesn’t anti-gravitate.
Those with little faith in theoretical arguments might want to argue that maybe it’s possible to find a way to make anti-matter anti-gravitate only sometimes. I am not aware of any theorem which
strictly proves this to be impossible, but neither is there – to my best knowledge – any example of a consistent theory in which this has been shown to work.
And if that still wasn’t enough to convince you, the ALPHA experiment at CERN has not only created neutral anti-hydrogen, made of an anti-proton and a positron (an anti-electron), but has taken great
strides towards measuring exactly how anti-hydrogen behaves in Earth’s gravitation field. Guess what?
So far there is no evidence that anti-hydrogen falls upwards
– though the present measurement precision only rules out that the anti-hydrogen’s gravitational mass is not larger than (minus!) 65 times its inertial mass.
[Correction added April 19: There is not one but three approved experiments at CERN to measure the free fall of anti hydrogen: AEGIS, ALPHA-g and GBAR.]
So, at least theoretical physicists are pretty sure that none of the particles we know anti-gravitates. But could there be other particles, which we haven’t yet discovered, that anti-gravitate?
In principle, yes, but there is no observational evidence for this. In contrast to what is often said, dark energy does not anti-gravitate. The distinctive property of dark energy is that the ratio
of energy-density over pressure is negative. For anti-gravitating matter, however, both energy-density and pressure change sign, so the ratio stays positive. This means anti-gravitating matter, if it
exists, behaves just the same way as normal matter does, except that the two types of matter repel each other. It also doesn’t give rise to anything like dark matter, because negative gravitational
mass would have the exact opposite effect as needed to explain dark matter.
To be fair, I also don’t know of any experiment that explicitly looks for signatures of anti-gravitational matter, like for example concave gravitational lensing. So, strictly speaking, it hasn’t
been ruled out, but it’s a hypothesis that also hasn’t attracted much professional interest. Many theoretical physicists who I have talked to believe that negative gravitational masses would induce
vacuum-decay because particle pairs could be produced out of nothing. This argument, however, doesn’t take into account that the inertial masses remain positive which prohibits pair production. (On a
more technical note, it is a little appreciated fact that the canonical stress-energy tensor isn’t the same as the gravitational stress-energy tensor.)
Even so, let us suppose that the theoretically possible anti-gravitating matter is somewhere out there. What would it be good for? Not for much, it turns out. The stuff would interact with our normal
matter even more weakly than neutrinos. This means even if we’d manage to find some of it in our vicinity – which is implausible already – we wouldn’t be able to catch it and use it for anything. It
would simply pass right through us.
The anti-gravitating weight that I’d want to tie to the couch, therefore, will unfortunately remain fiction.
[This post previously appeared on Starts With A Bang.]
40 comments:
1. Very nice post!
2. Sabine said,
“Both matter and anti-matter particles hold together the quarks that make up neutrons and protons.”
Learning that alone is fascinating; why don’t they annihilate? I'm guessing somehow they don’t come into contact within the quark, if that is so then theoretically would that possibly change at
the densities within a black hole?
Another great explanation for the general structure of a topic with very complex underpinnings, thank you.
3. It's reasonable to expect anti-matter gravitates like normal matter. Within a hadron it behaves normally, else protons and neutrons would violate equivalence principle. But anti-matter within a
hadron could behave differently. CERN experiments rule out anti-gravitational mass for anti-hydrogen at 65 times inertial mass. But of course this is very far from the regime of interest, where
the ratio is 1, so they have a long way to go. Bottom line, there still isn't direct experimental evidence.
It seems to me (off the top of my head) "dark matter" could, in fact, be anti-gravitating matter. Of course it wouldn't be distributed like regular "dark matter". Instead, the spaces between
galaxies, and galactic clusters, could be filled with very weakly interacting "anti-dark matter". By repelling regular matter, it would confine it, allowing stars in galaxies, and galaxies in
galactic clusters, to reach escape velocities without escaping. They'd be "fenced in" by the surrounding anti-DM. Is something obviously wrong with that idea?
The striking thing about this topic is that it's treated like real science! Although theory indicates absence of anti-grav, real scientists still want to see experimental evidence. That's good.
But in other areas like string theory and dark matter and (I'd say) the simulation hypothesis, experimental evidence seems to be ignored or considered superfluous. AFAIK, no one is calling you an
idiot for considering the possibility of anti-grav. Why the difference? Why are theoretical physicists sensible about this topic, but "faith-based" fanatics in other areas?
4. Hi Sabine, thanks, very nice post like always.
One question, you state: "...for example concave gravitational lensing", which almost anticipate my question. What would negative mass do to photons? - which have no mass. How would that react to
usual curvature? Would clocks made of negative mass accelerate near a black hole? Would rods shrink?
5. Louis,
They do annihilate, sometimes. As they say, in quantum mechanics anything that can happen does happen. The point here is that both matter and antimatter can be exchanged between the constituents
and contributes to (what's classically called) potential energy. Indeed, it may sound strange, but neutrons and protons have for that reason also a photon content!
6. George,
I've tried the 'fencing in' and it doesn't work: The necessary solutions are unstable.
7. akidbelle,
It's not a simple question to answer because you first have to solve the field equations, after this you can just calculate the photon trajectories. It's reasonable to expect, however, that in
the Newtonian limit you can just swap M with -M.
8. Dear Dr B.
I do not think that anti-gravitational matter would be useful for moving a couch ;-) . As you write, all such existing matter would have been pushed out into the voids between the galaxy super
clusters by now. And the energy needed to create anti-gravity matter particle by particle could be used more efficiently.
What would be interesting would be a way to negate the local gravity field.
I was wondering whether a "white hole" does not have the features of anti-gravity?
9. Rob van Son,
No, a while hole has nothing to do with anti-gravity.
10. Sabine, as I also mentioned on fb, we know from SN 1987A that GR is CP invariant"> to within a factor of 10^-6.
11. Hi, I'm not a physicist but I'm a regular reader of your great blog.
Would the fact that the hypothetical graviton would, presumably, be its own anti-particle, have any impact on this question?
12. Re Shantanu
..."it can be concluded that neutrinos and antineutrinos have the same infall velocity
in the gravitational field of our Galaxy to an accuracy of 4.6×10(-6) to 7.7×10^(-7).
Hadrons are not constrained. 1.74 solar-mass 465.1 Hz PSR J1903+0327 (millisecond pulsar, neutron star) (neutron star) and a 1.05 solar-mass star form a 95.17-day binary system. It verifies the
Equivalence Principle for orbit, periastron precession, and gravitational radiation orbital decay despite huge divergences in all measurable properties.
Equivalence Principle violation must contrast a property outside general relativity. Only one such observable is stable, large divergence, high concentration, and obtainable.
13. My understanding was that while opposite sign masses generate a repelling force, a negative mass particle is actually attracted by a repelling force. So you have a situation where the positive
mass particle wants to get away, the negative mass particle wants to get closer, and you have a runaway instability that drives both particles to the speed of light.
Another pathology is you can have pair production of particles out of nothing, so the vacuum would be unstable.
I always thought negative masses were excluded on these grounds. But you seem to be saying that inertial masses must always positive, but gravitational masses negative. Can one implement this
idea explicitly in the context of relativistic field theory? For example, can you compute some analogue of a Bhabha scattering process and show that the "opposite mass" particles repel?
14. Re Original Post:
Which just goes to demonstrate the most profound wisdom that anyone can acquire:
"MOVERS ARE WORTH IT!"
still remains true. (A lesson I learned the hard way try to move my piano to its new home at a cost in pain, health care and lost wages far in excess of what movers charge).
@Shantanu Great catch re the CP invariance of GR per SN 1987A. Not sure that this resolve the issue, however.
@UncleAl Great catch on the neutrino study. The precision involved clear rules out different gravitational properties for antimatter with observational evidence, unless neutrinos are somehow an
exception to the rule (which wouldn't be entirely implausible).
@SabineH (Not expecting meaningful answers to all or any of these questions, but putting them out there for argument's sake in case someone finds any of them interesting or worth considering.)
*Are fundamental Majorana particles of any kind inconsistent a gravitational force distinction between matter and antimatter?
If true we have a theorem that creates three way split between linking two seemingly disparate issues - one being the nature of neutrino mass in the SM and the other being the gravitational
properties of antimatter in GR or QG, even though it doesn't answer either of them by itself.
(1) If Majorana neutrino mass exists, then antimatter can't gravitate differently than matter.
(2) If antimatter gravitates differently than matter, then Majorana neutrino mass in impossible.
(3) Of course, you could also have antimatter that can't gravitate differently than matter and Dirac neutrino mass with no fundamental Majorana particles.
(I suppose a fourth option would be that Majorana mass is a third kind of mass distinct from both matter mass and antimatter mass that follows its own rules.)
* Could this reasoning be extended to composite Majorana fermions?
My intuition says "no" but I haven't really processed it analytically. Because, if that were the case, you could conclude from the existence of Bogoliubov quasiparticles in superconductors that
matter and antimatter are identical gravitationally in a manner independent of other kinds of tests.
* Along the same lines, suppose that matter and antimatter gravitate differently. What about contributions to the stress-energy tensor from particles that are neither matter nor antimatter such
as photons and Z bosons or Higgs bosons or hypothetical gravitons? Like fundamental Majorana fermions, there is no distinction between the particle and the antiparticle, so you either can't have
such a distinction or they are all in some special third kind of mass in addition to matter mass and antimatter mass.
* What about contributions to the stress-energy tensor from gluons which have both color and anti-color contributions but are overall neutral? If matter and antimatter gravitate differently,
would this systemically align gluons every so slightly with local gravitational fields? The effect might not be big enough to observe in ordinary QCD experiments, but surely such a universal bias
would have some aggregate effect that would be observable since it would be acting on every single one of gillions upon gillions of gluons in every hadron everywhere.
15. A few other questions (to which I don't expect answers) inspired by this post.
* Is there any meaningful sense in which virtual particles gravitate?
My intuition says yes in this case, because theoretically, virtual particles give rise to a distinction between tree-level bare masses and observed masses (at least in some very SM-like theories
if not in the SM itself as well) and would play a part in the gluon component of hadron masses, and virtual particle loops ought to affect the properties of reasonable model of the graviton if a
QG theory is real. But, I'm not sure I could imagine an experiment that would test this more directly with observational evidence.
* Is there any theoretical leverage one can obtain for other purposes by formulating GR/QG in a manner that makes it more self-evident than it is in some formulations that anti-matter mass and
matter mass are identical? Do formulations of gravity that have that ambiguity on the surface have a subtle flaw that could lead to other plausible but wrong understandings of gravity, or is this
the sole consequence of such an ambiguity?
* Does an identical matter mass and antimatter mass have any impact on the formulation or operation of CPT conservation in the SM or otherwise?
* Does formulating QG as a non-abelian chiral theory, or a theory with non-commutative geometry have any bearing on the matter mass v. antimatter mass properties debate?
* Is there any sensible formulation of GR/QG in which gravity and inertia actually only act upon the square of mass, in which case the sign doesn't matter?
* Are there formulations of the QM/GR/QG in which mass is a complex valued quantity rather than a real number? Likewise, is there ever a case in which mass is something other than a scalar? I'm
not sure what a vector or tensor generalization of mass would look like, but tinkering with fundamental concepts along these kinds of lines is what theoretical physics do, right?
It seems as if complex numbers in physics often have to do with the directionality of time, and IIRC, there is a Noether's theorem that relates time symmetry to conservation of mass-energy, so
naively it would make sense for imaginary numbers related to time to bleed into numbers related to mass-energy. My intuition says this is probably gibberish, but it is a concept that seems
plausible at least for a moment.
* I'm surprised that it is so hard to measure the sign of gravitational mass experimentally. Some experimental concepts that could come immediately to mind would be:
1. Wouldn't it be possible to distinguish between the scenarios because various B mesons and various anti-B mesons would behave differently?
2. Shouldn't it be easier to study this with leptons than with hadrons? A lepton is a much simpler beast than any kind of hadron and the precision of the theoretical expectation of how leptons
behave is so much greater than anything you can do with a hadron.
16. To answer the title of your blog post it is best we apply Einstein's original thought experiment that first led him to the equivalence principle namely the lift experiment and substitute the man
in the lift for an anti matter particle. One can easily deduce that the equivalence principle still holds for matter or anti matter and that both fall in a gravitational field.
17. Are gluons not their own anti-particle?
18. "From the standard model, we can derive how anti-matter gravitates – it gravitates exactly the same way as normal matter."
The standard model has nothing to do with gravity.
19. qsa,
That the standard model doesn't include a quantum theory of gravity doesn't mean it's not possible to couple matter to gravity using the expectation value of the stress-energy tensor. The
standard model has something to do with gravity in that all matter couples to space-time (through the metric tensor that appears in the standard model Lagrangian and the respective generally
covariant derivatives).
20. Unknown,
Well, yes, but the question is whether they gravitate the same way.
21. The Big Bang was too hot for anything but photons, if that, absent electrical charge, baryon number, etc., to conserve. Somewhat later...photons plus 0.61 ppb net matter. Strop Occam’s razor.
Baryogenesis only produced neutrons. The universe was born of photons and beta-decay ignoring parity conservation. Primordial neutrino abundance should mirror primordial hydrogen, trimmed by
helium production during initial decay plus fusion.
Gravitation is fundamentally simple in composition though complex to describe. Giving it an eldritch basis will fail, for there is none. "8^>)
22. "For a 2nd rank tensor field, in contrast, like charges attract and unlike charges repel."
It does have to be a Symmetric 2nd rank tensor field, doesn't it?
You must still get rid of the anti-symmetric part to make it spin 2
23. Hi Sabine,
As a further investigation of your response I made a post in physicsforum, can you please comment on the comments:)
24. qsa,
Can't see the relation to this blogpost, sorry.
25. Ok, let me put it in another way. We are talking about if the theory of gravity indicate that whether anti-particles repel, so is the effective theories(as shown in PF) are accepted as
established theory and if so can they prove(or disprove) any conjecture as in you thread.
26. qsa,
Of course it's not accepted as an established theory because there's no experimental evidence.
27. So, by the same token black hole science is just physicists past time until they can find out more about fundamental issues like why the proton/electron mass ratio is what it is.
28. qsa,
That a theory isn't yet established doesn't mean there should be no research on it.
29. Dear Dr B.
"That a theory isn't yet established doesn't mean there should be no research on it."
Many people seem to think it ought to be forbidden.
I cannot understand why theoretical physics evokes such hostility. There are still websites that organize hate campaigns against Einstein and claim quantum mechanics is a hoax that should be
fought tooth and nail. Verlinde might be wrong, but that should not be a reason for aggression.
Is there anyone who can explain that?
30. Let me put the question this way: What would you propose replaces the geodesic equation for a particle with negative gravitational mass but positive inertial mass?
The point is that I know how to think of particles with negative mass relativistically provided their gravitational and inertial masses are both negative. But this possibility is instantly ruled
out by experiment, since the we live in a universe with a stable vacuum.
In Newtonian gravity I can imagine the possibility of that gravitational and inertial masses have opposite sign. But to take this idea seriously purely on the basis of Newtownian gravity seems to
ignore 400 years of progress in physics.
Do I understand your intention? Maybe you have some concrete idea theoretical idea in mind. Or perhaps you just think it is neat to think about how you might try to measure the relative sign of
the gravitational and inertial mass, as a matter of principle.
31. Tchovi,
Yes, that's exactly the right question to ask: What does the geodesic equation look like for an anti-gravitating particle?
The obvious answer is that it'll have to use a covariant derivative, but that derivative can't be the same as the usual Christoffel connection. That of course only leads to the question, well,
then which derivative is it? Since torsion terms don't contribute to the geodesic equation, the derivative will have to be not metric compatible.
In principle then, you could go and pick any such derivative and see if it fits the bill but clearly that's very unsatisfactory. The other thing you hence would like to do is assume that you have
a symmetry under the exchange of 'normal' with 'antigravitating' matter. This leads you to the conclusion that there should be a second metric so that the second derivative is compatible with
that metric.
Where do you get the second metric from? Well, you need a second set of field equations. I explained here how this works. Note that I'm not claiming this is the only possible theory that realizes
an effect like antigravitation. But I think it's the one that employs the symmetry assumption in the most obvious way.
(Note that there are no direct coupling terms between the metrics. Ie, it's a bimetric theory but the coupling is merely mediated through the matter sources. Sometimes I think there should be way
to solve the second equation and reinsert the solution into the usual field equation so that the second metric no longer appears. Then again I'm not sure this works. In other words, I don't know,
more work is needed etc.)
32. Rob,
Well, you just did :o)
33. PS: There are websites explaining quantum mechanics is a hoax? Seriously? Like, all these computers and digital cameras actually don't work, it's just your imagination and all?
34. Rob,
Wow, thanks. Got the links but won't post them, I think you'll understand.
35. @Rob
Perhaps some wisdom from Carl Jung can soothe our frustrations over such hostilities ,)
" Thinking is difficult, that's why most people judge "
Best, Koenraad
36. Hi Bee,
I am reading with lot of interest your blog and comments. To my understanding ultimately experiment is the final judge.Theories have to accommodate whatever experiments confirm.It seems that GR
or gauge theory can be simply modified by some extra terms of different signs without any difficulty. Thus theory is of no guide to answer the basic question raised in this blog. Do you agree?
37. https://arxiv.org/pdf/1410.3881.pdf
..."At extremely high densities existing in black holes and in the very early Universe, the minimal spinor-torsion coupling manifests itself as gravitational repulsion, which avoids the formation
of singularities from fermionic matter"
Anti-gravitation may be beyond reach. Default expectations remain vulnerable. Test spacetime geometry with geometry. Extreme enantiomorphic test masses will pursue non-identical minimum action
vacuum free fall trajectories, violating the Equivalence Principle. Look.
38. kashyap,
No, I disagree. You cannot just 'modify GR by some extra terms of different signs without any difficulty'. First, the result is generically unstable. But more importantly, second, this doesn't
change anything about the equivalence principle and hence doesn't have an effect similar to antigravitation.
39. @Koenraad
"Thinking is difficult, that's why most people judge "
That does sound convincing.
I think it was Barbara Tuchman who defined fools as those who thought they knew so they do not need to think. The same for those who think they are so all knowing, they can judge without
40. Thanks Bee for the reply. So my feeling was wrong! No problem!But are you saying that if the experiments find antimatter have opposite behavior to matter in gravity, that will be a big crisis
even for classical GR?
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Two-element Boolean algebra explained
In mathematics and abstract algebra, the two-element Boolean algebra is the Boolean algebra whose underlying set (or universe or carrier) B is the Boolean domain. The elements of the Boolean domain
are 1 and 0 by convention, so that B = . Paul Halmos's name for this algebra "2" has some following in the literature, and will be employed here.
B is a partially ordered set and the elements of B are also its bounds.
An operation of arity n is a mapping from B^n to B. Boolean algebra consists of two binary operations and unary complementation. The binary operations have been named and notated in various ways.
Here they are called 'sum' and 'product', and notated by infix '+' and '∙', respectively. Sum and product commute and associate, as in the usual algebra of real numbers. As for the order of
operations, brackets are decisive if present. Otherwise '∙' precedes '+'. Hence is parsed as and not as . Complementation is denoted by writing an overbar over its argument. The numerical analog of
the complement of is . In the language of universal algebra, a Boolean algebra is a
Either one-to-one correspondence between and yields classical bivalent logic in equational form, with complementation read as NOT. If 1 is read as True, '+' is read as OR, and '∙' as AND, and vice
versa if 1 is read as False. These two operations define a commutative semiring, known as the Boolean semiring.
Some basic identities
2 can be seen as grounded in the following trivial "Boolean" arithmetic:
\begin{align} &1+1=1+0=0+1=1\\ &0+0=0\\ &0 ⋅ 0=0 ⋅ 1=1 ⋅ 0=0\\ &1 ⋅ 1=1\\ &\overline{0}=0\\ &\overline{1}=1 \end{align}
Note that:
• '+' and '∙' work exactly as in numerical arithmetic, except that 1+1=1. '+' and '∙' are derived by analogy from numerical arithmetic; simply set any nonzero number to 1.
• Swapping 0 and 1, and '+' and '∙' preserves truth; this is the essence of the duality pervading all Boolean algebras.
This Boolean arithmetic suffices to verify any equation of 2, including the axioms, by examining every possible assignment of 0s and 1s to each variable (see decision procedure).
The following equations may now be verified:
\begin{align} &A+A=A\\ &A ⋅ A=A\\ &A+0=A\\ &A+1=1\\ &A ⋅ 0=0\\ &\overline{\overline{A}}=A \end{align}
Each of '+' and '∙' distributes over the other:
That '∙' distributes over '+' agrees with
elementary algebra
, but not '+' over '∙'. For this and other reasons, a sum of products (leading to a
synthesis) is more commonly employed than a product of sums (leading to a
Each of '+' and '∙' can be defined in terms of the other and complementation:
A ⋅ B=\overline{\overline{A}+\overline{B}}
A+B=\overline{\overline{A} ⋅ \overline{B}}.
We only need one binary operation, and
suffices to denote it. Hence concatenation and overbar suffice to notate
. This notation is also that of
's Boolean term schemata. Letting (
) denote the complement of
and "" denote either 0 or 1 yields the
of the primary algebra of
G. Spencer-Brown
Laws of Form
A basis for 2 is a set of equations, called axioms, from which all of the above equations (and more) can be derived. There are many known bases for all Boolean algebras and hence for 2. An elegant
basis notated using only concatenation and overbar is:
(Concatenation commutes, associates)
is a complemented lattice, with an
upper bound
of 1)
(0 is the
lower bound
is a
distributive lattice
Where concatenation = OR, 1 = true, and 0 = false, or concatenation = AND, 1 = false, and 0 = true. (overbar is negation in both cases.)
If 0=1, (1)-(3) are the axioms for an abelian group.
(1) only serves to prove that concatenation commutes and associates. First assume that (1) associates from either the left or the right, then prove commutativity. Then prove association from the
other direction. Associativity is simply association from the left and right combined.
This basis makes for an easy approach to proof, called "calculation" in Laws of Form, that proceeds by simplifying expressions to 0 or 1, by invoking axioms (2) - (4), and the elementary identities
, and the distributive law.
De Morgan's theorem states that if one does the following, in the given order, to any Boolean function:
• Complement every variable;
• Swap '+' and '∙' operators (taking care to add brackets to ensure the order of operations remains the same);
• Complement the result,
the result is logically equivalent to what you started with. Repeated application of De Morgan's theorem to parts of a function can be used to drive all complements down to the individual variables.
A powerful and nontrivial metatheorem states that any identity of 2 holds for all Boolean algebras.^[1] Conversely, an identity that holds for an arbitrary nontrivial Boolean algebra also holds in 2.
Hence all identities of Boolean algebra are captured by 2. This theorem is useful because any equation in 2 can be verified by a decision procedure. Logicians refer to this fact as "2 is decidable".
All known decision procedures require a number of steps that is an exponential function of the number of variables N appearing in the equation to be verified. Whether there exists a decision
procedure whose steps are a polynomial function of N falls under the P = NP conjecture.
The above metatheorem does not hold if we consider the validity of more general first-order logic formulas instead of only atomic positive equalities. As an example consider the formula . This
formula is always true in a two-element Boolean algebra. In a four-element Boolean algebra whose domain is the powerset of, this formula corresponds to the statement and is false when x is . The
decidability for the first-order theory of many classes of Boolean algebras can still be shown, using quantifier elimination or small model property (with the domain size computed as a function of
the formula and generally larger than 2).
See also
Further reading
Many elementary texts on Boolean algebra were published in the early years of the computer era. Perhaps the best of the lot, and one still in print, is:
• Mendelson, Elliot, 1970. Schaum's Outline of Boolean Algebra. McGraw - Hill.
The following items reveal how the two-element Boolean algebra is mathematically nontrivial.
Notes and References
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Efficient PHP Factorial Programming Guide
PHP is a versatile programming language that enables developers to create dynamic and interactive web applications. When it comes to performing mathematical calculations, PHP provides various
built-in functions that simplify complex tasks. In this comprehensive guide, we will explore PHP factorial programming in depth, covering the step-by-step process, common challenges, and best
practices to ensure efficient and accurate factorial calculations.
Table of Contents
1. Introduction to Factorial Calculation
2. Implementing Factorial Calculation in PHP
□ 2.1 Iterative Approach
□ 2.2 Recursive Approach
3. Performance Considerations
□ 3.1 Memory Usage
□ 3.2 Execution Time
4. Error Handling and Input Validation
□ 4.1 Handling Negative Numbers
□ 4.2 Validating Input
5. Best Practices for PHP Factorial Programming
6. Conclusion
1. Introduction to Factorial Calculation
Factorial calculation is the process of multiplying a number by all positive integers less than itself down to one. It is denoted by the exclamation mark (!). For example, the factorial of 5 is
calculated as 5! = 5 × 4 × 3 × 2 × 1 = 120.
Factorial calculations find applications in various domains, such as mathematics, statistics, and computer science. PHP provides several techniques to implement factorial calculations efficiently,
enabling developers to solve complex mathematical problems effortlessly.
2. Implementing Factorial Calculation in PHP
2.1 Iterative Approach
The iterative approach involves using a loop to calculate the factorial of a number. It follows a sequential process, multiplying each positive integer from the given number down to one.
Here is an example of implementing the iterative approach in PHP:
phpCopy code
function factorialIterative($n) {
$result = 1;
for ($i = 1; $i <= $n; $i++) {
$result *= $i;
return $result;
2.2 Recursive Approach
The recursive approach involves breaking down the factorial calculation into smaller subproblems. It calls the function recursively, reducing the problem size until it reaches the base case.
Here is an example of implementing the recursive approach in PHP:
phpCopy code
function factorialRecursive($n) {
if ($n <= 1) {
return 1;
} else {
return $n * factorialRecursive($n - 1);
3. Performance Considerations
Efficient factorial programming requires considering performance aspects, such as memory usage and execution time, to optimize the calculations.
3.1 Memory Usage
When calculating factorials of large numbers, memory consumption becomes crucial. Storing intermediate results can lead to high memory usage. To overcome this, developers can use techniques such as
memoization to store previously calculated values and reduce redundant calculations.
3.2 Execution Time
The execution time of factorial calculations depends on the approach used and the number for which the factorial is calculated. The iterative approach tends to have better performance for smaller
numbers, while the recursive approach may suffer from stack overflow errors for significantly large numbers. Analyzing the requirements and constraints of the specific use case can help choose the
most suitable approach.
4. Error Handling and Input Validation
Robust factorial programming involves handling potential errors and validating user input. Consider the following aspects when implementing error handling and input validation in PHP:
4.1 Handling Negative Numbers
Factorial calculations are defined only for non-negative integers. Therefore, it is essential to handle cases where negative numbers are provided as input. Displaying appropriate error messages or
returning predefined error codes can enhance the user experience.
4.2 Validating Input
Validating input ensures that only valid integers are accepted for factorial calculations. PHP provides various validation techniques, such as type checking, range checking, and regular expressions,
to ensure the input adheres to the required format.
5. Best Practices for PHP Factorial Programming
To excel in PHP factorial programming, consider the following best practices:
• Use appropriate variable names and follow a consistent naming convention for clarity and maintainability.
• Encapsulate factorial functions within classes or namespaces to improve code organization and reusability.
• Leverage the power of PHP's error handling mechanisms, such as exceptions, to gracefully handle errors and enhance code robustness.
• Write test cases and perform comprehensive unit testing to verify the correctness of factorial calculations and handle edge cases effectively.
• Document your code thoroughly using PHPDoc or other documentation standards to facilitate understanding, maintenance, and collaboration.
6. Conclusion
In this comprehensive guide, we explored PHP factorial programming and discussed various techniques to implement efficient and accurate factorial calculations. By following the step-by-step process,
considering performance aspects, and incorporating best practices, developers can unlock the full potential of PHP when dealing with factorial calculations. With this knowledge, you are now equipped
to tackle complex mathematical problems and build powerful web applications using PHP. Happy coding!
If you enjoyed this piece, we've crafted a related article delving into Block Statements in JavaScript. Explore it here.
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6.2 Determining Empirical and Molecular Formulas
37 6.2 Determining Empirical and Molecular Formulas
Learning Objectives
By the end of this section, you will be able to:
• Compute the percent composition of a compound
• Determine the empirical formula of a compound
• Determine the molecular formula of a compound
In the previous section, we discussed the relationship between the bulk mass of a substance and the number of atoms or molecules it contains (moles). Given the chemical formula of the substance, we
were able to determine the amount of the substance (moles) from its mass, and vice versa. But what if the chemical formula of a substance is unknown? In this section, we will explore how to apply
these very same principles in order to derive the chemical formulas of unknown substances from experimental mass measurements.
Percent Composition
The elemental makeup of a compound defines its chemical identity, and chemical formulas are the most succinct way of representing this elemental makeup. When a compound’s formula is unknown,
measuring the mass of each of its constituent elements is often the first step in the process of determining the formula experimentally. The results of these measurements permit the calculation of
the compound’s percent composition, defined as the percentage by mass of each element in the compound. For example, consider a gaseous compound composed solely of carbon and hydrogen. The percent
composition of this compound could be represented as follows:
[latex]\% \;\text{H} = \frac{\text{mass H}}{\text{mass compound}} \times 100 \%[/latex]
[latex]\% \;\text{C} = \frac{\text{mass C}}{\text{mass compound}} \times 100 \%[/latex]
If analysis of a 10.0-g sample of this gas showed it to contain 2.5 g H and 7.5 g C, the percent composition would be calculated to be 25% H and 75% C:
[latex]\%\;\text{H} = \frac{2.5 \;\text{g H}}{10.0 \;\text{g compound}} \times 100 \% = 25 \%[/latex]
[latex]\%\;\text{C} = \frac{7.5 \;\text{g C}}{10.0 \;\text{g compound}} \times 100 \% = 75 \%[/latex]
Example 1
Calculation of Percent Composition
Analysis of a 12.04-g sample of a liquid compound composed of carbon, hydrogen, and nitrogen showed it to contain 7.34 g C, 1.85 g H, and 2.85 g N. What is the percent composition of this compound?
To calculate percent composition, we divide the experimentally derived mass of each element by the overall mass of the compound, and then convert to a percentage:
[latex]\%\;\text{C} = \frac{7.34 \;\text{g C}}{12.04 \;\text{g compound}} \times 100\% = 61.0\%[/latex]
[latex]\%\;\text{H} = \frac{1.85 \;\text{g H}}{12.04 \;\text{g compound}} \times 100\% = 15.4\%[/latex]
[latex]\%\;\text{N} = \frac{2.85 \;\text{g N}}{12.04 \;\text{g compound}} \times 100\% = 23.7\%[/latex]
The analysis results indicate that the compound is 61.0% C, 15.4% H, and 23.7% N by mass.
Check Your Learning
A 24.81-g sample of a gaseous compound containing only carbon, oxygen, and chlorine is determined to contain 3.01 g C, 4.00 g O, and 17.81 g Cl. What is this compound’s percent composition?
12.1% C, 16.1% O, 71.8% Cl
Determining Percent Composition from Formula Mass
Percent composition is also useful for evaluating the relative abundance of a given element in different compounds of known formulas. As one example, consider the common nitrogen-containing
fertilizers ammonia (NH[3]), ammonium nitrate (NH[4]NO[3]), and urea (CH[4]N[2]O). The element nitrogen is the active ingredient for agricultural purposes, so the mass percentage of nitrogen in the
compound is a practical and economic concern for consumers choosing among these fertilizers. For these sorts of applications, the percent composition of a compound is easily derived from its formula
mass and the atomic masses of its constituent elements. A molecule of NH[3] contains one N atom weighing 14.01 amu and three H atoms weighing a total of (3 × 1.008 amu) = 3.024 amu. The formula mass
of ammonia is therefore (14.01 amu + 3.024 amu) = 17.03 amu, and its percent composition is:
[latex]\%\;\text{N} = \frac{14.01 \;\text{amu N}}{17.03 \;\text{amu NH}_3} \times 100\% = 82.27\%[/latex]
[latex]\%\;\text{H} = \frac{3.024 \;\text{amu N}}{17.03 \;\text{amu NH}_3} \times 100\% = 17.76\%[/latex]
This same approach may be taken considering a pair of molecules, a dozen molecules, or a mole of molecules, etc. The latter amount is most convenient and would simply involve the use of molar masses
instead of atomic and formula masses, as demonstrated Example 2. As long as we know the chemical formula of the substance in question, we can easily derive percent composition from the formula mass
or molar mass.
Example 2
Determining Percent Composition from a Molecular Formula
Aspirin is a compound with the molecular formula C[9]H[8]O[4]. What is its percent composition?
To calculate the percent composition, we need to know the masses of C, H, and O in a known mass of C[9]H[8]O[4]. It is convenient to consider 1 mol of C[9]H[8]O[4] and use its molar mass (180.159 g/
mole, determined from the chemical formula) to calculate the percentages of each of its elements:
[latex]\begin{array}{r @{{}={}} l} \%\text{C} & = \frac{9 \;\text{mol C} \;\times\; \text{molar mass C}}{\text{molar mass} \;\text{C}_9\text{H}_{18}\text{O}_4} = \frac{9 \times 12.01 \;\text{g/mol}}
{180.159 \text{g/mol}} = \frac{108.09 \;\text{g/mol}}{180.159 \;\text{g/mol}} \times 100 \\[0.5em] \%\text{C} & = 60.00\%\;\text{C} \end{array}\\[1.5em][/latex]
[latex]\begin{array}{r @{{}={}} l} \%\text{H} & = \frac{8 \;\text{mol H} \;\times\; \text{molar mass H}}{\text{molar mass} \;\text{C}_9\text{H}_{18}\text{O}_4} = \frac{8 \times 1.008 \;\text{g/mol}}
{180.159 \text{g/mol}} = \frac{8.064 \;\text{g/mol}}{180.159 \;\text{g/mol}} \times 100 \\[0.5em] \%\text{H} & = 4.476\%\;\text{H} \end{array}\\[1.5em][/latex]
[latex]\begin{array}{r @{{}={}} l} \%\text{O} & = \frac{4 \;\text{mol O} \;\times\; \text{molar mass O}}{\text{molar mass} \;\text{C}_9\text{H}_{18}\text{O}_4} = \frac{4 \times 16.00 \;\text{g/mol}}
{180.159 \text{g/mol}} = \frac{64.00 \;\text{g/mol}}{180.159 \;\text{g/mol}} \times 100 \\[0.5em] \%\text{O} & = 35.52\%\;\text{O} \end{array}\\[1.5em][/latex]
Note that these percentages sum to equal 100.00% when appropriately rounded.
Check Your Learning
To three significant digits, what is the mass percentage of iron in the compound Fe[2]O[3]?
Determination of Empirical Formulas
As previously mentioned, the most common approach to determining a compound’s chemical formula is to first measure the masses of its constituent elements. However, we must keep in mind that chemical
formulas represent the relative numbers, not masses, of atoms in the substance. Therefore, any experimentally derived data involving mass must be used to derive the corresponding numbers of atoms in
the compound. To accomplish this, we can use molar masses to convert the mass of each element to a number of moles. We then consider the moles of each element relative to each other, converting these
numbers into a whole-number ratio that can be used to derive the empirical formula of the substance. Consider a sample of compound determined to contain 1.71 g C and 0.287 g H. The corresponding
numbers of atoms (in moles) are:
[latex]1.17 \;\text{g C} \times \frac{1 \;\text{mol C}}{12.01 \;\text{g C}} = 0.142 \;\text{mol C}[/latex]
[latex]0.287 \;\text{g H} \times \frac{1 \;\text{mol H}}{1.008 \;\text{g H}} = 0.284 \;\text{mol H}[/latex]
Thus, we can accurately represent this compound with the formula C[0.142]H[0.248]. Of course, per accepted convention, formulas contain whole-number subscripts, which can be achieved by dividing each
subscript by the smaller subscript:
[latex]\text{C}_{\frac{0.142}{0.142}} \; \text{H}_{\frac{0.248}{0.142}} \;\text{or CH}_2[/latex]
(Recall that subscripts of “1” are not written but rather assumed if no other number is present.)
The empirical formula for this compound is thus CH[2]. This may or not be the compound’s molecular formula as well; however, we would need additional information to make that determination (as
discussed later in this section).
Consider as another example a sample of compound determined to contain 5.31 g Cl and 8.40 g O. Following the same approach yields a tentative empirical formula of:
[latex]\text{Cl}_{0.150}\text{O}_{0.525} \; = \; \text{Cl}_{\frac{0.150}{0.150}} \; \text{O}_{\frac{0.525}{0.150}} = \text{ClO}_{3.5}[/latex]
In this case, dividing by the smallest subscript still leaves us with a decimal subscript in the empirical formula. To convert this into a whole number, we must multiply each of the subscripts by
two, retaining the same atom ratio and yielding Cl[2]O[7] as the final empirical formula.
In summary, empirical formulas are derived from experimentally measured element masses by:
1. Deriving the number of moles of each element from its mass
2. Dividing each element’s molar amount by the smallest molar amount to yield subscripts for a tentative empirical formula
3. Multiplying all coefficients by an integer, if necessary, to ensure that the smallest whole-number ratio of subscripts is obtained
Figure 1 outlines this procedure in flow chart fashion for a substance containing elements A and X.
Figure 1. The empirical formula of a compound can be derived from the masses of all elements in the sample.
Example 3
Determining a Compound’s Empirical Formula from the Masses of Its Elements
A sample of the black mineral hematite (Figure 2), an oxide of iron found in many iron ores, contains 34.97 g of iron and 15.03 g of oxygen. What is the empirical formula of hematite?
Figure 2. Hematite is an iron oxide that is used in jewelry. (credit: Mauro Cateb)
For this problem, we are given the mass in grams of each element. Begin by finding the moles of each:
[latex]\begin{array}{r @{{}={}} l} 34.97 \;\text{g Fe} (\frac{\text{mol Fe}}{55.85 \;\text{g}}) & = 0.6261 \;\text{mol Fe} \\[1em] 15.03 \;\text{g O} (\frac{\text{mol O}}{16.00 \;\text{g}}) & =
0.9394 \;\text{mol O} \end{array}[/latex]
Next, derive the iron-to-oxygen molar ratio by dividing by the lesser number of moles:
[latex]\begin{array}{r @{{}={}} l} \frac{0.6261}{0.6261} & = 1.000 \;\text{mol Fe} \\[1em] \frac{0.9394}{0.6261} & = 1.500 \;\text{mol O} \end{array}[/latex]
The ratio is 1.000 mol of iron to 1.500 mol of oxygen (Fe[1]O[1.5]). Finally, multiply the ratio by two to get the smallest possible whole number subscripts while still maintaining the correct
iron-to-oxygen ratio:
[latex]2(\text{Fe}_1\text{O}_{1.5}) = \text{Fe}_2\text{O}_3[/latex]
The empirical formula is Fe[2]O[3].
Check Your Learning
What is the empirical formula of a compound if a sample contains 0.130 g of nitrogen and 0.370 g of oxygen?
For additional worked examples illustrating the derivation of empirical formulas, watch the brief video clip.
Deriving Empirical Formulas from Percent Composition
Finally, with regard to deriving empirical formulas, consider instances in which a compound’s percent composition is available rather than the absolute masses of the compound’s constituent elements.
In such cases, the percent composition can be used to calculate the masses of elements present in any convenient mass of compound; these masses can then be used to derive the empirical formula in the
usual fashion.
Determining an Empirical Formula from Percent Composition
Example 4
The bacterial fermentation of grain to produce ethanol forms a gas with a percent composition of 27.29% C and 72.71% O (Figure 3). What is the empirical formula for this gas?
Figure 3. An oxide of carbon is removed from these fermentation tanks through the large copper pipes at the top. (credit: “Dual Freq”/Wikimedia Commons)
Since the scale for percentages is 100, it is most convenient to calculate the mass of elements present in a sample weighing 100 g. The calculation is “most convenient” because, per the definition
for percent composition, the mass of a given element in grams is numerically equivalent to the element’s mass percentage. This numerical equivalence results from the definition of the “percentage”
unit, whose name is derived from the Latin phrase per centum meaning “by the hundred.” Considering this definition, the mass percentages provided may be more conveniently expressed as fractions:
[latex]\begin{array}{r @{{}={}} l} 27.29\% \;\text{C} & = \frac{27.29 \;\text{g C}}{100 \;\text{g compound}} \\[1em] 72.71\% \;\text{O} & = \frac{72.71 \;\text{g O}}{100 \;\text{g compound}} \end
The molar amounts of carbon and hydrogen in a 100-g sample are calculated by dividing each element’s mass by its molar mass:
[latex]\begin{array}{r @{{}={}} l} 27.29\% \;\text{C} (\frac{\text{mol C}}{12.01 \;\text{g}}) & = 2.272 \;\text{mol C} \\[1em] 72.71\% \;\text{O} (\frac{\text{mol O}}{16.00 \;\text{g}}) & = 4.544 \;\
text{mol O} \end{array}[/latex]
Coefficients for the tentative empirical formula are derived by dividing each molar amount by the lesser of the two:
[latex]\begin{array}{r @{{}={}} l} \frac{2.272 \;\text{mol C}}{2.272} & = 1 \\[1em] \frac{4.544 \;\text{mol O}}{2.272} & = 2 \end{array}[/latex]
Since the resulting ratio is one carbon to two oxygen atoms, the empirical formula is CO[2].
Check Your Learning
What is the empirical formula of a compound containing 40.0% C, 6.71% H, and 53.28% O?
Derivation of Molecular Formulas
Recall that empirical formulas are symbols representing the relative numbers of a compound’s elements. Determining the absolute numbers of atoms that compose a single molecule of a covalent compound
requires knowledge of both its empirical formula and its molecular mass or molar mass. These quantities may be determined experimentally by various measurement techniques. Molecular mass, for
example, is often derived from the mass spectrum of the compound (see discussion of this technique in the previous chapter on atoms and molecules). Molar mass can be measured by a number of
experimental methods, many of which will be introduced in later chapters of this text.
Molecular formulas are derived by comparing the compound’s molecular or molar mass to its empirical formula mass. As the name suggests, an empirical formula mass is the sum of the average atomic
masses of all the atoms represented in an empirical formula. If we know the molecular (or molar) mass of the substance, we can divide this by the empirical formula mass in order to identify the
number of empirical formula units per molecule, which we designate as n:
[latex]\frac{\text{molecular or molar mass (amu or} \;\frac{\text{g}}{\text{mol}})}{\text{empirical formula mass (amu or} \;\frac{\text{g}}{\text{mol}})} = n \;\text{formula units/molecule}[/latex]
The molecular formula is then obtained by multiplying each subscript in the empirical formula by n, as shown by the generic empirical formula A[x]B[y]:
[latex](\text{A}_{\text{x}} \text{B}_{\text{y}})_{\text{n}} = \text{A}_{\text{nx}} \text{B}_{\text{nx}}[/latex]
For example, consider a covalent compound whose empirical formula is determined to be CH[2]O. The empirical formula mass for this compound is approximately 30 amu (the sum of 12 amu for one C atom, 2
amu for two H atoms, and 16 amu for one O atom). If the compound’s molecular mass is determined to be 180 amu, this indicates that molecules of this compound contain six times the number of atoms
represented in the empirical formula:
[latex]\frac{180 \;\text{amu/molecule}}{30\;\frac{\text{amu}}{\text{formula unit}}} = 6 \;\text{formula units/molecule}[/latex]
Molecules of this compound are then represented by molecular formulas whose subscripts are six times greater than those in the empirical formula:
[latex]\text{(CH}_2\text{O})_6 = \text{C}_6\text{H}_{12}\text{O}_6[/latex]
Note that this same approach may be used when the molar mass (g/mol) instead of the molecular mass (amu) is used. In this case, we are merely considering one mole of empirical formula units and
molecules, as opposed to single units and molecules.
Example 5
Determination of the Molecular Formula for Nicotine
Nicotine, an alkaloid in the nightshade family of plants that is mainly responsible for the addictive nature of cigarettes, contains 74.02% C, 8.710% H, and 17.27% N. If 40.57 g of nicotine contains
0.2500 mol nicotine, what is the molecular formula?
Determining the molecular formula from the provided data will require comparison of the compound’s empirical formula mass to its molar mass. As the first step, use the percent composition to derive
the compound’s empirical formula. Assuming a convenient, a 100-g sample of nicotine yields the following molar amounts of its elements:
[latex]\begin{array}{r @{{}={}} l} (74.02 \;\text{g C}) (\frac{1 \;\text{mol C}}{12.01 \;\text{g C}}) & = 6.163 \;\text{mol C} \\[1em] (8.710 \;\text{g H}) (\frac{1 \;\text{mol H}}{1.01 \;\text{g
H}}) & = 8.624 \;\text{mol H} \\[1em] (17.27 \;\text{g N}) (\frac{1 \;\text{mol N}}{14.01 \;\text{g N}}) & = 1.233 \;\text{mol N} \end{array}[/latex]
Next, we calculate the molar ratios of these elements relative to the least abundant element, N.
[latex]\begin{array}{r @{{}={}} l} 6.163 \;\text{mol C/} 1.233 \;\text{mol N} & = 4.998 & = 5 \;\text{mol C} \\[1em] 8.264 \;\text{mol H/} 1.233 \;\text{mol N} & = 6.994 & = 7 \;\text{mol H} \\[1em]
1.233 \;\text{mol N/} 1.233 \;\text{mol N} & = 1.000 & = 1 \;\text{mol N} \end{array}[/latex]
The C-to-N and H-to-N molar ratios are adequately close to whole numbers, and so the empirical formula is C[5]H[7]N. The empirical formula mass for this compound is therefore 81.13 amu/formula unit,
or 81.13 g/mol formula unit.
We calculate the molar mass for nicotine from the given mass and molar amount of compound:
[latex]\frac{40.57 \;\text{g nicotine}}{0.2500 \;\text{mol nicotine}} = \frac{162.3 \;\text{g}}{\text{mol}}[/latex]
Comparing the molar mass and empirical formula mass indicates that each nicotine molecule contains two formula units:
[latex]\frac{162.3 \;\text{g/mol}}{81.13 \;\frac{\text{g}}{\text{formula unit}}} = 2 \;\text{formula units/molecule}[/latex]
Thus, we can derive the molecular formula for nicotine from the empirical formula by multiplying each subscript by two:
[latex](\text{C}_5\text{H}_7\text{N})_2 = \text{C}_{10}\text{H}_{14}\text{N}_2[/latex]
Check Your Learning
What is the molecular formula of a compound with a percent composition of 49.47% C, 5.201% H, 28.84% N, and 16.48% O, and a molecular mass of 194.2 amu?
Key Concepts and Summary
The chemical identity of a substance is defined by the types and relative numbers of atoms composing its fundamental entities (molecules in the case of covalent compounds, ions in the case of ionic
compounds). A compound’s percent composition provides the mass percentage of each element in the compound, and it is often experimentally determined and used to derive the compound’s empirical
formula. The empirical formula mass of a covalent compound may be compared to the compound’s molecular or molar mass to derive a molecular formula.
Key Equations
• [latex]\%\text{X} = \frac{\text{mass X}}{\text{mass commpound}} \times 100\% \\[0.5em][/latex]
• [latex]\frac{\text{molecular or molar mass ( amu or} \;\frac{\text{g}}{\text{mol}})}{\text{empirical formula mass ( amu or} \;\frac{\text{g}}{\text{mol}})} = n \;\text{formula units/molecule}\\
• (A[x]B[y])[n] = A[nx]B[ny]
Chemistry End of Chapter Exercises
1. What information do we need to determine the molecular formula of a compound from the empirical formula?
2. Calculate the following to four significant figures:
(a) the percent composition of ammonia, NH[3]
(b) the percent composition of photographic “hypo,” Na[2]S[2]O[3]
(c) the percent of calcium ion in Ca[3](PO[4])[2]
3. Determine the following to four significant figures:
(a) the percent composition of hydrazoic acid, HN[3]
(b) the percent composition of TNT, C[6]H[2](CH[3])(NO[2])[3]
(c) the percent of SO[4]^2– in Al[2](SO[4])[3]
4. Determine the percent ammonia, NH[3], in Co(NH[3])[6]Cl[3], to three significant figures.
5. Determine the percent water in CuSO[4]∙5H[2]O to three significant figures.
6. Determine the empirical formulas for compounds with the following percent compositions:
(a) 15.8% carbon and 84.2% sulfur
(b) 40.0% carbon, 6.7% hydrogen, and 53.3% oxygen
7. Determine the empirical formulas for compounds with the following percent compositions:
(a) 43.6% phosphorus and 56.4% oxygen
(b) 28.7% K, 1.5% H, 22.8% P, and 47.0% O
8. A compound of carbon and hydrogen contains 92.3% C and has a molar mass of 78.1 g/mol. What is its molecular formula?
9. Dichloroethane, a compound that is often used for dry cleaning, contains carbon, hydrogen, and chlorine. It has a molar mass of 99 g/mol. Analysis of a sample shows that it contains 24.3% carbon
and 4.1% hydrogen. What is its molecular formula?
10. Determine the empirical and molecular formula for chrysotile asbestos. Chrysotile has the following percent composition: 28.03% Mg, 21.60% Si, 1.16% H, and 49.21% O. The molar mass for chrysotile
is 520.8 g/mol.
11. Polymers are large molecules composed of simple units repeated many times. Thus, they often have relatively simple empirical formulas. Calculate the empirical formulas of the following polymers:
(a) Lucite (Plexiglas); 59.9% C, 8.06% H, 32.0% O
(b) Saran; 24.8% C, 2.0% H, 73.1% Cl
(c) polyethylene; 86% C, 14% H
(d) polystyrene; 92.3% C, 7.7% H
(e) Orlon; 67.9% C, 5.70% H, 26.4% N
12. A major textile dye manufacturer developed a new yellow dye. The dye has a percent composition of 75.95% C, 17.72% N, and 6.33% H by mass with a molar mass of about 240 g/mol. Determine the
molecular formula of the dye.
empirical formula mass
sum of average atomic masses for all atoms represented in an empirical formula
percent composition
percentage by mass of the various elements in a compound
Answers to Chemistry End of Chapter Exercises
2. (a) % N = 82.24%
% H = 17.76%;
(b) % Na = 29.08%
% S = 40.56%
% O = 30.36%;
(c) % Ca^2+ = 38.76%
4. % NH[3] = 38.2%
6. (a) CS[2]
(b) CH[2]O
8. C[6]H[6]
10. Mg[3]Si[2]H[3]O[8] (empirical formula), Mg[6]Si[4]H[6]O[16] (molecular formula)
12. C[15]H[15]N[3]
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Double bound variable in an MCP
Dear all,
in the general explanation of MCPs, it is stated that MCP models can solve problems where
F(z) ⟂ z.lo ≤ z ≤ z.up
Meaning that one of three conditions can hold
F(z) > 0 while z.lo = z
F(z) = 0 while z.lo ≤ z ≤ z.up
F(z) < 0 while z = z.up
However, in all examples below on how to implement MCPs, the variable z seems to be reduced to a variable with only an upper bound OR a lower bound. In that case, the code goes something like
z.lo = 3;
eq_1 … F(z) =g= 0;
model example eq_1.z ;
I don’t see a statement anywhere that tells me how to implement a model where F(z) can be both ≤0 and ≥0, depending on the bound that z reaches. Moreover, I’m told in Table 2 that double bound
variables can only be matched with =N= equations, and I don’t think that’s what I want (I want F(Z) to be equal to zero in all cases where z is between its bounds, so an inequality sign seems wrong
to me).
I wasn’t able to find an example for code written for an MCP, where a Variable has an upper and a lower bound, resulting in three possible stages.
Thank you in advance for your answers!
Let’s take a simple MCP as an example. Let’s start with an optimization problem, so we have some intuition from the optimization world we can lean on.
min f(x) := sqr(x-1) s.t. L <= x <= U, where L and U can be finite or the expected infinity.
Taking the KKT conditions we get this MCP:
F(x) := 2(x-1) perp to L <= x <= U
I’ve attached a GAMS version of this. Try this with all the interesting combinations: L = -INF or finite and less than 1 or 1 or greater than 1, similar for for U. That’s 16 combinations. Try them
all. If you understand this tiny example and each combination, I think you’ll be in good shape regarding this part of MCP.
mcp.gms (343 Bytes)
To make the experiment explicit: L in {-inf, 0, 1, 2}, U in {0, 1, 2, +INF} yields 16 combinations.
Hello Steve,
Thank you for your reply.
I suppose you meant to set the initial goal to min f(x) := (x-1)**2. I ran it with your suggested domains and some additional ones, such as [-2;0] or [2;5], to see which solution the model chooses if
neither x.lo, nor x.up, nor anything in between is able to solve the equation with an equality sign. That was actually a very helpful exercise for me - thank you!
So, the short answer is: NE does create these three stages. Nice!
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Values with Uncertainty
When variables are in the format of a range (e.g. 0 to 100, uniform(0,100), a Monte Carlo simulation is run to estimate the possible outcomes of an uncertain event. Monte Carlo will randomly pick a
value from the distribution and compute the whole model as if it were that random constant value. This process is repeated multiple times to generate distributions for the output variables.
The use of the simulation allows Causal to perform computations using values with uncertainty that are not possible without it. Due to this method of handling uncertainty, you may notice that the
range of the cell and the value are not the numbers you inputted or would expect, but only by a trivial amount.
There are many different types of distribution shapes for values with uncertainty. Here are three examples of distributions that Causal supports:
Values in the format of '# to #' produce a triangle distribution where the center value is the most likely value, while the edges of the range are the least likely:
Using the function of 'uniform(from, to)', a uniform distribution can be produced
Using the function 'poisson(lambda)', a poisson distribution will be produced
The sample function takes a random sample from the provided numbers. For example, `sample(1,2,3)` may return 1, 2 or 3 with equal probability.
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Calculating Square Feet for Painting Projects: Methods and Practical Examples - Civil Jungle
Calculating Square Feet for Painting Projects: Methods and Practical Examples
How to Calculate Square Feet for Paint
For most home improvement projects, knowing how to calculate square feet is an important skill. From painting, square feet is going to be a necessary measurement to get the right amount of materials
for your project. This guide will explain how to measure square feet and outline the square feet formula so you are ready for your next project.
Here, We calculate two different methods as below
1. Plane area
2. Odd areaÂ
Plane Area
L = Lenght
W = Width
A = Area
To find square feet in a room, first, measure the dimensions of your space. The two dimensions to measure are the length and width of the area you need to calculate. The next step in how to calculate
square feet is to plug your measurements into the square feet formula: L x W = A (in square feet) To find square feet, multiply the length measurement in feet by the width measurement in feet.
This yields a product called the area, which is expressed in square feet (or square inches if you are calculating a much smaller space, such as a dollhouse). For example, if you are buying a carpet
for a room that is 12-feet long and 10-feet wide, multiply the two dimensions:
12 ft. x 10 ft. = 120 ft^2.
Odd AreaÂ
Sometimes you will have to account for odd area dimensions or additional areas that don’t neatly connect with your main area. In this case, to calculate the square feet accurately, you may need to
divide the space into separate areas. For example, Assume you are trying to paint for a rectangular area with a nook. The shape is formed of a large rectangle and a smaller rectangle.
1. Find the length and width of each section (labeled A and B here), then calculate each square feet:
3 ft. x 7 ft. = 21 ft^2.
12 ft. x 10 ft. = 120 ft^2.
2. Then, add the three values together to get the total square feet you will need:
120 ft^2. + 21 ft^2. = 141 ft^2.
3. Therefore, you’ll need 141 ft^2. of flooring materials.
What Is Painting?
Paint is a substance used as the final finish to all surfaces and as a coating to protect or decorate the surface. Paint is a pigmented opaque material that completely covers and hides the surface to
which it is applied. Paint is available in oil-based and water-based formulae.
Paint prevents corrosion. It is a combination of pigments with suitable thinners or oils to provide decorative and protective coatings. It is used as a protective coating and is normally sprayed/
brushed on. Painting protects a surface from weathering effects and also prevents corrosion of metals.
How Much Paint Do I Need
1. One-gallon can of paint cover up to 400 square feet, which is enough to cover a small room like a bathroom.
2. Two-gallon can of paint cover up to 800 square feet, which is enough to cover an average size room. This is the most common amount needed, especially when considering second coat coverage.
3. Note: 1 gallon = 3.78541 liters (This consumption as per PPG Paint Company report.)
Painting Area Calculation for Evan Surface
For Example, Assume a contractor paints a room, A room that is 12 x 15 feet with a 10-feet ceiling. The room has two doors and two windows. (Door Size = 3 feet X 7 feet , Window = 5 feet X 3 feet)
Step-1 (Measure the total distance)
Measure the total distance (perimeter) around the room. (12 ft. + 15 ft.) x 2 = 54 ft.
Step-2 (Multiply the perimeter by the ceiling )
Multiply the perimeter by the ceiling height to get the total wall area: 54 ft. x 10 ft. = 540 sq. ft.
Step-3 (Door Deduction Area )
Doors are usually 21 square feet (there are two in this example): 21 sq. ft. x 2 = 42 sq. ft.
Step-4 (Window Deduction Area )
Windows average 15 square feet (there are two in this example): 15 sq. ft. x 2 = 30 sq. ft.
Step-5 (Total Area of Wall Paint)
Take the total wall area and subtract the area for the doors and windows to get the wall surface to be painted: 540 sq. ft. (wall area) – 42 sq. ft. (doors) – 30 sq. ft. (windows) = 468 sq. ft.
of walls that need to be painted.
Step-6 (Total Area of Celling Paint)
Multiply the ceiling length to get the total ceiling area: 12 ft. x 15 ft. = 180 sq. ft.
Step-7 (Total Area Paint Area)
Total Paint Area = Total Paint of Wall Area + Total Paint of Ceiling Area
Total Paint Area = 468 sq. ft. + 180 sq. ft.
Total Paint Area = 648 sq. ft.
As a rule of thumb, one gallon of quality paint will usually cover 400 square feet. One quart will cover 100 square feet. Because you need to cover 648 square feet in this example, 1.62 gallons will
be adequate to give one coat of paint to the walls and ceiling. (Coverage will be affected by the porosity and texture of the surface. In addition, bright colors may require a minimum of two coats.)
How to Measure Painting Area for Irregular Surface?
Unlike brickwork or any masonry calculation, calculation of painting area for joineries such as windows grills, fences, doors, fences, and shutters is a little bit tricky. Since every item comes with
different designs (Open, closed designs) and girth, we cannot measure it as a flat surface. Meanwhile, we cannot ignore the work done by contractors on these uneven surfaces.
So to compensate for the work done by the contractor, the measurement shall be taken as flat surfaces, and specific paint coefficients will be multiplied by that area. All Painting coefficient as
below as per IS Code 1200 Part 15: 1987
Paint Coefficient for Uneven Surface Area
Sr.No. Description of Work How Measured Multiplying Multiplied
Coefficient for
1 Paneled or framed and braced or ledged and battened or Measured flat (not girthed) including chowkhat or frame. Edges, chocks, cleats, etc., shall be deemed to 1.3 For Each Side
ledged, battened, and braced joinery. be included in this item
2 Flush joinery Measured flat (not girthed) including chowkhat or frame. Edges, chocks, cleats etc., shall be deemed to 1.2 For Each Side
be included in this item
3 Flush shutter Measured flat overall 1.2 For Each Side
4 Fully glazed or gauged joinery. Measured flat (not girthed) including chowkhat or frame. Edges, chocks, cleats etc., shall be deemed to 0.8 For Each Side
be included in this item
5 Partly paneled and partly glazed or gauged joinery Measured flat (not girthed) including chowkhat or frame. Edges, chocks, cleats etc., shall be deemed to 1 For Each Side
be included in this item
6 Fully Venetian or louvered joinery. Measured flat (not girthed) including chowkhat or frame. Edges, chocks, cleats etc., shall be deemed to 1.8 For Each Side
be included in this item
7 Weather boarding Measured flat (not girthed) supporting framework shall not be measured separately. 1.2 For Each Side
8 Wood shingle roofing Measured flat (not girthed) 1.1 For Each Side
9 Boarding with cover fillets and match Boarding. Measured flat (not girthed) 1.05 For Each Side
10 Tile and slate battening Measured flat overall no deduction shall be made for open spaces. 0.8 For Painting
All Over
11 Trellis (or Jaffari) work one way or two way Measured flat overall no deduction shall be made for open spaces supporting members shall not be measured 1.0 For Painting
separately. All Over
12 Guard bars balustrade gales gratings. grills expanded Measured flat overall no deduction shall be made for open spaces supporting members shall not be measured 1.0 For Painting
metal and railings separately. All Over
13 Gates and open palisade fencing including standard Measured flat overall no deduction shall be made for open spaces supporting members shall not be measured 1.0 For Painting
braces, rails stays, etc. separately. All Over
14 Carved or enriched work Measured flat 2.0 For Each Side
15 Steel roller shutters Measured flat (size of opening) overall, jamb guides, bottom rails and locking arrangement, etc., shall 1.1 For Each Side
be included in the item (top cover shall be measured separately).
16 Plain sheet steel doors and windows. Measured flat (not girthed) including a frame, edges, etc. 1.1 For Each Side
17 Fully glazed or gauged steel Measured flat (not girthed) including a frame, edges, etc. 0.5 For Each Side
18 Partly Panelled and partly glazed steel doors. Measured flat (not girthed) including a frame, edges, etc. 0.8 For Each Side
19 Collapsible gate. Measured flat (size of opening) 1.5 For Each Side
Painting Area Calculation for Uneven Surface
For Example, Assume a contractor paints a Steel roller shutters of 12 ft. x 10 ft.
Total Painting Area = Area of Flat Surface X Painting Coefficient X Total Paint Side
Area of Flat Surface = 12 ft. x 10 ft. = 120 ft^2.
Painting Coefficient = 1.1 For Each Side (As per the above table- Point no -15)
Total Paint Side = 2 Side Paint
Total Painting Area = 120 ft^2. X 1.1 For Each Side X 2 Side Paint
Total Painting Area = 264 ft^2
FAQs About Calculating Square Feet for Painting Projects
Why is it important to calculate square feet before starting a painting project?
Calculating square feet helps you determine the amount of paint needed, ensuring you purchase the right quantity without over or underestimating.
How do I measure square feet for a room?
Measure the length and width of the room in feet, then multiply these dimensions together to get the area in square feet.
What if I have an irregularly shaped room or one with multiple sections?
For irregular shapes or rooms with multiple sections, measure each section separately, calculate its area, and then sum them up to get the total square footage.
What is the formula for calculating square feet?
The formula is: Area (in square feet) = Length (in feet) × Width (in feet).
How do windows and doors affect the total painting area?
Subtract the area of windows and doors from the total wall area to determine the amount of wall space that needs painting.
How much paint do I need per square foot?
Generally, one gallon of paint covers about 400 square feet with one coat, but this can vary based on factors like surface texture and color.
What factors can affect paint coverage?
Surface porosity, texture, and the number of coats required can affect how much paint you’ll need per square foot.
How can I calculate paint needed for ceilings?
Measure the length and width of the ceiling area, then multiply these dimensions together to get the ceiling’s square footage.
Do different paint finishes affect coverage?
Yes, different finishes (like matte, eggshell, or gloss) can affect how much area one gallon of paint covers due to differences in thickness and opacity.
How do I account for wasted paint?
It’s wise to account for a small amount of waste due to spillage, touch-ups, or unexpected needs, especially if you’re using multiple cans of paint.
Leave a Comment
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UK Phone Numbers
I've been wrestling with a list of predominantly UK phone numbers for the last couple of days. The data has come from a list of all the calls made to and from an organisation in the past year. In
particular, I want to identify the dialling code so that I can identify what geographical areas people are calling from.
First port of call is usually Google - I certainly don't want to write some code if I can just copy something that someone else has already done. But sadly every entry I could find was about US phone
numbers, which are easy peasy - area dialling codes are 3 digit across North America. How rational! Here in the UK dialling codes may be three, four, five or six digits long. So if you have North
American phone numbers to sort out, most of this article will be irrelevant.
Let's start with a bit of research -
. "In the United Kingdom
area codes are two, three, four, or, rarely, five digits long (after the initial zero). Regions with shorter area codes, typically large cities, permit the allocation of more telephone numbers as the
local number portion has more digits. Local customer numbers are four to eight figures long. The total number of digits is ten, but in a very few areas the total may be nine digits (after the initial
zero). The "area code" is also referred to as an "STD (code)" (subscriber trunk dialling) or a "dialling code" in the UK." So, basically, it's a tangle of different standards. OK, so I need some code
which checks the first few digits and identifies which one is which.
Data Cleansing
But before we get on to that, let's do some data cleansing. I have a lot of what look like perfectly good phone numbers with a 92 prefix. Like most firms, it's 9 to dial out, so I don't know where 92
comes from. It's easy to deal with though- just a straightforward REPLACE() function.
replace(LEFT(strddi,2), '92', '')
means when the leftmost two characters of the DDI string are 92, replace them with a blank.
-- identify Direct Dialled numbers with 92 prefix - apparent error
update [CUSTOMER].[PhoneCalls]
Set strddi = replace(LEFT(strddi,2), '92', '')
WHERE LEFT(strddi,2) = '92'
AND strDDIAreaCode IS NULL
AND LEN(strddi) > 5;
There's another issue with incoming mobile phone numbers - in the UK, these start with 07, but somehow they have been recorded as starting with 7. Replace does the job.
-- identify mobile numbers omitting 0 prefix - apparent error
-- change to UK standard 0
update [CUSTOMER].[PhoneCalls]
Set strcli = replace(LEFT(strCLI,1), '7', '07')
WHERE LEFT(strCLI,1) = '7'
AND strCLIAreaCode IS NULL
AND LEN(strCLI) = 10;
Then we have the incoming calls from UK numbers, but specifying the full international dialling code 0044. Fine if you are based somewhere else, but really not necessary if you are in the UK and your
caller is in the UK.
-- identify UK numbers with an international code 0044
-- change to UK standard 0
update [CUSTOMER].[PhoneCalls]
Set strcli = replace(LEFT(strcli,4), '0044', '0')
WHERE LEFT(strcli,4) = '0044'
AND strCLIAreaCode IS NULL
AND LEN(strcli) > 5;
Having dealt with UK codes that think they are worldwide, I can now go on and identify the legitimate international codes. I could pick out the French calls (0033), German calls (0049) and so on, but
I only have a very small percentage of non-UK calls so I'm just going to treat all these Johnny Foreigners the same and lump them together as "International" calls. If you do want to be more accurate
with your international calls, click
for a list of the codes.
-- identify international codes 00
update [CUSTOMER].[PhoneCalls]
set strCLIAreaCode = 'International'
WHERE LEFT(strcli,2) = '00'
AND strCLIAreaCode IS NULL
AND LEN(strCLI) > 5;
And use the same technique to identify Mobile numbers:
-- identify mobile phone numbers 07
update [CUSTOMER].[PhoneCalls]
set strCLIAreaCode = 'Mobile'
WHERE LEFT(strcli,2) = '07'
AND strCLIAreaCode IS NULL
AND LEN(strCLI) > 5;
Freefone numbers are free only to people calling from Landlines, although I understand that there are
to make them free to people using mobile phones too.
-- identify Freefone phone numbers 0800 etc
update [CUSTOMER].[PhoneCalls]
set strCLIAreaCode = 'Freefone'
WHERE LEFT(strcli,4) in ('0800', '0500', '0808')
AND strCLIAreaCode IS NULL
AND LEN(strCLI) > 5;
There is a string of special rate numbers beginning with 08 - once upon a time you could identify 0845 as local rate and 0870 as national rate, but the list has proliferated and now you can't really
tell how much it is going to cost.
-- identify special rate phone numbers 0845 0870 etc
update [CUSTOMER].[PhoneCalls]
set strCLIAreaCode = 'Special rate'
WHERE LEFT(strcli,2) = '08'
AND strCLIAreaCode IS NULL
AND LEN(strCLI) > 5;
One thing you can be sure of - a call to an 09 number is going to be outrageously expensive...
-- identify premium rate phone numbers 09
update [CUSTOMER].[PhoneCalls]
set strCLIAreaCode = 'Premium rate'
WHERE LEFT(strcli,2) = '09'
AND strCLIAreaCode IS NULL
AND LEN(strCLI) > 5;
Identifying UK Dialling Codes
So - on to the main point of this article. UK dialling codes - area codes - whatever you want to call them, allow you to identify where in the country a caller is based. Aberdeen is 01224, York is
01904. But very often the dialling code and the subscriber number are held in a single field e.g. 01169158424.
Taking this example, it could in theory be divided thus:
A dialling code can be anything from 3 digits to 6 digits - a subscriber number can be anything from 4 to 8 digits.
So here's how to tackle the problem. There are only five cities with three digit dialling codes, ten with four digits, and 12 with six - the rest have five. So thanks to a bit of research in
Wikipedia (see the links I posted earlier) I was able to construct the following CASE statement:
-- UK Dialling codes may have 3, 4 , 5 or 6 digits
-- pick appropriate code
update [CUSTOMER].[PhoneCalls]
set strCLIAreaCode =
-- 3 digit dialling codes e.g. London, Belfast
WHEN LEFT(STRcLI,3) IN
('020', '023', '024', '028', '029' )
THEN LEFT(strcli,3)
-- 4 digit dialling codes e.g. Bristol, Leicester
WHEN LEFT(STRcLI,4) IN
('0118', '0117', '0116', '0115', '0114', '0113',
'0121', '0131', '0141', '0151', '0161', '0191')
THEN LEFT(strcli,4)
-- 6 digit dialling codes e.g. Langholm, Keswick
WHEN LEFT(STRcLI,6) IN
('013873', '015242', '015394', '015395',
'015396', '016973', '016974', '016977',
'017683', '017684', '017687', '019467' )
THEN LEFT(strcli,6)
-- The remaining majority of codes are 5 digit
ELSE LEFT(strcli,5)
strcli <> 'WITHHELD'
AND LEN(strcli) > 5
AND strCLIAreaCode IS null;
Looking at 01169158424, it's easy to see that the first four digits match the four digit option, so the area code part of this number is 0116 - which represents Leicester
Finally, a bit of tidying up:
-- Remove any remaining nulls in area code fields
-- nulls remaining represent internal codes or unidentifiable
-- CLI
update [CUSTOMER].[PhoneCalls]
set strCLIAreaCode = 'N/A'
WHERE strCLIAreaCode IS NULL;
Incidentally, the code shown here is half the code I wrote - I had to do the job for both inbound (CLI - Caller Line Identification) numbers and outbound (DDI - Direct Dial In). the code is
essentially the same so I haven't troubled you with it.
I hope you find this useful!
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Distributed Inference with JAX | TensorFlow Probability
TensorFlow Probability (TFP) on JAX now has tools for distributed numerical computing. To scale to large numbers of accelerators, the tools are built around writing code using the "single-program
multiple-data" paradigm, or SPMD for short.
In this notebook, we'll go over how to "think in SPMD" and introduce the new TFP abstractions for scaling to configurations such as TPU pods, or clusters of GPUs. If you're running this code
yourself, make sure to select a TPU runtime.
We'll first install the latest versions TFP, JAX and TF.
pip install jaxlib --upgrade -q 2>&1 1> /dev/null
pip install tfp-nightly[jax] --upgrade -q 2>&1 1> /dev/null
pip install tf-nightly-cpu -q -I 2>&1 1> /dev/null
pip install jax -I -q --upgrade 2>&1 1>/dev/null
We'll import some general libraries, along with some JAX utilities.
Setup and Imports
import functools
import collections
import contextlib
import jax
import jax.numpy as jnp
from jax import lax
from jax import random
import jax.numpy as jnp
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
import pandas as pd
import tensorflow_datasets as tfds
from tensorflow_probability.substrates import jax as tfp
INFO:tensorflow:Enabling eager execution
INFO:tensorflow:Enabling v2 tensorshape
INFO:tensorflow:Enabling resource variables
INFO:tensorflow:Enabling tensor equality
INFO:tensorflow:Enabling control flow v2
We'll also set up some handy TFP aliases. The new abstractions are currently provided in tfp.experimental.distribute and tfp.experimental.mcmc.
tfd = tfp.distributions
tfb = tfp.bijectors
tfm = tfp.mcmc
tfed = tfp.experimental.distribute
tfde = tfp.experimental.distributions
tfem = tfp.experimental.mcmc
Root = tfed.JointDistributionCoroutine.Root
To connect the notebook to a TPU, we use the following helper from JAX. To confirm that we're connected, we print out the number of devices, which should be eight.
from jax.tools import colab_tpu
print(f'Found {jax.device_count()} devices')
Found 8 devices
A quick introduction to jax.pmap
After connecting to a TPU, we have access to eight devices. However, when we run JAX code eagerly, JAX defaults to running computations on just one.
The simplest way of executing a computation across many devices is to map a function, having each device execute one index of the map. JAX provides the jax.pmap ("parallel map") transformation which
turns a function into one that maps the function across several devices.
In the following example, we create an array of size 8 (to match the number of available devices) and map a function that adds 5 across it.
xs = jnp.arange(8.)
out = jax.pmap(lambda x: x + 5.)(xs)
print(type(out), out)
<class 'jax.interpreters.pxla.ShardedDeviceArray'> [ 5. 6. 7. 8. 9. 10. 11. 12.]
Note that we receive a ShardedDeviceArray type back, indicating that the output array is physically split across devices.
jax.pmap acts semantically like a map, but has a few important options that modify its behavior. By default, pmap assumes all inputs to the function are being mapped over, but we can modify this
behavior with the in_axes argument.
xs = jnp.arange(8.)
y = 5.
# Map over the 0-axis of `xs` and don't map over `y`
out = jax.pmap(lambda x, y: x + y, in_axes=(0, None))(xs, y)
[ 5. 6. 7. 8. 9. 10. 11. 12.]
Analogously, the out_axes argument to pmap determines whether or not to return the values on every device. Setting out_axes to None automatically returns the value on the 1st device and should only
be used if we are confident the values are the same on every device.
xs = jnp.ones(8) # Value is the same on each device
out = jax.pmap(lambda x: x + 1, out_axes=None)(xs)
What happens when what we'd like to do isn't easily expressible as a mapped pure function? For example, what if we'd like to do a sum across the axis we're mapping over? JAX offers "collectives",
functions that communicate across devices, to enable writing more interesting and complex distributed programs. To understand how exactly they work, we'll introduce SPMD.
What is SPMD?
Single-program multiple-data (SPMD) is a concurrent programming model in which a single program (i.e. the same code) is executed simultaneously across devices, but the inputs to each of the running
programs can differ.
If our program is a simple function of its inputs (i.e. something like x + 5), running a program in SPMD is just mapping it over different data, like we did with jax.pmap earlier. However, we can do
more than just "map" a function. JAX offers "collectives", which are functions that communicate across devices.
For example, maybe we'd like to take the sum of a quantity across all our devices. Before we do that, we need to assign a name to the axis we're mapping over in the pmap. We then use the lax.psum
("parallel sum") function to perform a sum across devices, ensuring we identify the named axis we're summing over.
def f(x):
out = lax.psum(x, axis_name='i')
return out
xs = jnp.arange(8.) # Length of array matches number of devices
jax.pmap(f, axis_name='i')(xs)
ShardedDeviceArray([28., 28., 28., 28., 28., 28., 28., 28.], dtype=float32)
The psum collective aggregates the value of x on each device and synchronizes its value across the map i.e. out is 28. on each device. We're no longer performing a simple "map", but we're executing
an SPMD program where each device's computation can now interact with the same computation on other devices, albeit in a limited way using collectives. In this scenario, we can use out_axes = None,
because psum will synchronize the value.
def f(x):
out = lax.psum(x, axis_name='i')
return out
jax.pmap(f, axis_name='i', out_axes=None)(jnp.arange(8.))
ShardedDeviceArray(28., dtype=float32)
SPMD enables us to write one program that is run on every device in any TPU configuration simultaneously. The same code that is used to do machine learning on 8 TPU cores can be used on a TPU pod
that may have hundreds to thousands of cores! For a more detailed tutorial about jax.pmap and SPMD, you can refer to the the JAX 101 tutorial.
MCMC at scale
In this notebook, we focus on using Markov Chain Monte Carlo (MCMC) methods for Bayesian inference. There are may ways we utilize many devices for MCMC, but in this notebook, we'll focus on two:
1. Running independent Markov chains on different devices. This case is fairly simple and is possible to do with vanilla TFP.
2. Sharding a dataset across devices. This case is a bit more complex and requires recently added TFP machinery.
Independent Chains
Say we'd like to do Bayesian inference on a problem using MCMC and would like to run several chains in parallel across several devices (say 2 on each device). This turns out to be a program we can
just "map" across devices, i.e. one that needs no collectives. To make sure each program executes a different Markov chain (as opposed to running the same one), we pass in a different value for the
random seed to each device.
Let's try it on a toy problem of sampling from a 2-D Gaussian distribution. We can use TFP's existing MCMC functionality out of the box. In general, we try to put most of the logic inside of our
mapped function to more explicitly distinguish between what's running on all the devices versus just the first.
def run(seed):
target_log_prob = tfd.Sample(tfd.Normal(0., 1.), 2).log_prob
initial_state = jnp.zeros([2, 2]) # 2 chains
kernel = tfm.HamiltonianMonteCarlo(target_log_prob, 1e-1, 10)
def trace_fn(state, pkr):
return target_log_prob(state)
states, log_prob = tfm.sample_chain(
return states, log_prob
By itself, the run function takes in a stateless random seed (to see how stateless randomness work, you can read the TFP on JAX notebook or see the JAX 101 tutorial). Mapping run over different seeds
will result in running several independent Markov chains.
states, log_probs = jax.pmap(run)(random.split(random.PRNGKey(0), 8))
print(states.shape, log_probs.shape)
# states is (8 devices, 1000 samples, 2 chains, 2 dimensions)
# log_prob is (8 devices, 1000 samples, 2 chains)
(8, 1000, 2, 2) (8, 1000, 2)
Note how we now have an extra axis corresponding to each device. We can rearrange the dimensions and flatten them to get an axis for the 16 chains.
states = states.transpose([0, 2, 1, 3]).reshape([-1, 1000, 2])
log_probs = log_probs.transpose([0, 2, 1]).reshape([-1, 1000])
fig, ax = plt.subplots(1, 2, figsize=(10, 5))
ax[0].plot(log_probs.T, alpha=0.4)
ax[1].scatter(*states.reshape([-1, 2]).T, alpha=0.1)
When running independent chains on many devices, it's as easy as pmap-ing over a function that uses tfp.mcmc, ensuring we pass different values for the random seed to each device.
Sharding data
When we do MCMC, the target distribution is often a posterior distribution obtained by conditioning on a dataset, and computing an unnormalized log-density involves summing likelihoods for each
observed data.
With very large datasets, it can be prohibitively expensive to even run one chain on a single device. However, when we have access to multiple devices, we can split up the dataset across the devices
to better leverage the compute we have available.
If we'd like to do MCMC with a sharded dataset, we need to ensure the unnormalized log-density we compute on each device represents the total, i.e. the density over all data, otherwise each device
will be doing MCMC with their own incorrect target distribution. To this end, TFP now has new tools (i.e. tfp.experimental.distribute and tfp.experimental.mcmc) that enable computing "sharded" log
probabilities and doing MCMC with them.
Sharded distributions
The core abstraction TFP now provides for computing sharded log probabiliities is the Sharded meta-distribution, which takes a distribution as input and returns a new distribution that has specific
properties when executed in an SPMD context. Sharded lives in tfp.experimental.distribute.
Intuitively, a Sharded distribution corresponds to a set of random variables that have been "split" across devices. On each device, they will produce different samples, and can individually have
different log-densities. Alternatively, a Sharded distribution corresponds to a "plate" in graphical model parlance, where the plate size is the number of devices.
Sampling a Sharded distribution
If we sample from a Normal distribution in a program being pmap-ed using the same seed on each device, we will get the same sample on each device. We can think of the following function as sampling a
single random variable that is synchronized across devices.
# `pmap` expects at least one value to be mapped over, so we provide a dummy one
def f(seed, _):
return tfd.Normal(0., 1.).sample(seed=seed)
jax.pmap(f, in_axes=(None, 0))(random.PRNGKey(0), jnp.arange(8.))
ShardedDeviceArray([-0.20584236, -0.20584236, -0.20584236, -0.20584236,
-0.20584236, -0.20584236, -0.20584236, -0.20584236], dtype=float32)
If we wrap tfd.Normal(0., 1.) with a tfed.Sharded, we logically now have eight different random variables (one on each device) and will therefore produce a different sample for each one, despite
passing in the same seed.
def f(seed, _):
return tfed.Sharded(tfd.Normal(0., 1.), shard_axis_name='i').sample(seed=seed)
jax.pmap(f, in_axes=(None, 0), axis_name='i')(random.PRNGKey(0), jnp.arange(8.))
ShardedDeviceArray([ 1.2152631 , 0.7818249 , 0.32549605, 0.6828047 ,
1.3973192 , -0.57830244, 0.37862757, 2.7706041 ], dtype=float32)
An equivalent representation of this distribution on a single device is just a 8 independent normal samples. Even though the value of the sample will be different (tfed.Sharded does pseudo-random
number generation slightly differently), they both represent the same distribution.
dist = tfd.Sample(tfd.Normal(0., 1.), jax.device_count())
DeviceArray([ 0.08086783, -0.38624594, -0.3756545 , 1.668957 ,
-1.2758069 , 2.1192007 , -0.85821325, 1.1305912 ], dtype=float32)
Taking the log-density of a Sharded distribution
Let's see what happens when we compute the log-density of a sample from a regular distribution in an SPMD context.
def f(seed, _):
dist = tfd.Normal(0., 1.)
x = dist.sample(seed=seed)
return x, dist.log_prob(x)
jax.pmap(f, in_axes=(None, 0))(random.PRNGKey(0), jnp.arange(8.))
(ShardedDeviceArray([-0.20584236, -0.20584236, -0.20584236, -0.20584236,
-0.20584236, -0.20584236, -0.20584236, -0.20584236], dtype=float32),
ShardedDeviceArray([-0.94012403, -0.94012403, -0.94012403, -0.94012403,
-0.94012403, -0.94012403, -0.94012403, -0.94012403], dtype=float32))
Each sample is the same on each device, so we compute the same density on each device too. Intuitively, here we only have a distribution over a single normally distributed variable.
With a Sharded distribution, we have a distribution over 8 random variables, so when we compute the log_prob of a sample, we sum, across devices, over each of the individual log densities. (You might
notice that this total log_prob value is larger than the singleton log_prob computed above.)
def f(seed, _):
dist = tfed.Sharded(tfd.Normal(0., 1.), shard_axis_name='i')
x = dist.sample(seed=seed)
return x, dist.log_prob(x)
sample, log_prob = jax.pmap(f, in_axes=(None, 0), axis_name='i')(
random.PRNGKey(0), jnp.arange(8.))
print('Sample:', sample)
print('Log Prob:', log_prob)
Sample: [ 1.2152631 0.7818249 0.32549605 0.6828047 1.3973192 -0.57830244
0.37862757 2.7706041 ]
Log Prob: [-13.7349205 -13.7349205 -13.7349205 -13.7349205 -13.7349205 -13.7349205
-13.7349205 -13.7349205]
The equivalent, "unsharded" distribution produces the same log density.
dist = tfd.Sample(tfd.Normal(0., 1.), jax.device_count())
DeviceArray(-13.7349205, dtype=float32)
A Sharded distribution produces different values from sample on each device, but get the same value for log_prob on each device. What's happening here? A Sharded distribution does a psum internally
to ensure the log_prob values are in sync across devices. Why would we want this behavior? If we're running the same MCMC chain on each device, we'd like the target_log_prob to be the same across
each device, even if some random variables in the computation are sharded across devices.
Additionally, a Sharded distribution ensures that gradients across devices are the correct, to ensure that algorithms like HMC, which take gradients of the log-density function as part of the
transition function, produce proper samples.
Sharded JointDistributions
We can create models with multiple Sharded random variables by using JointDistributions (JDs). Unfortunately, Sharded distributions cannot be safely used with vanilla tfd.JointDistributions, but
tfp.experimental.distribute exports "patched" JDs that will behave like Sharded distributions.
def f(seed, _):
dist = tfed.JointDistributionSequential([
tfd.Normal(0., 1.),
tfed.Sharded(tfd.Normal(0., 1.), shard_axis_name='i'),
x = dist.sample(seed=seed)
return x, dist.log_prob(x)
jax.pmap(f, in_axes=(None, 0), axis_name='i')(random.PRNGKey(0), jnp.arange(8.))
([ShardedDeviceArray([1.6121525, 1.6121525, 1.6121525, 1.6121525, 1.6121525,
1.6121525, 1.6121525, 1.6121525], dtype=float32),
ShardedDeviceArray([ 0.8690128 , -0.83167845, 1.2209264 , 0.88412696,
0.76478404, -0.66208494, -0.0129658 , 0.7391483 ], dtype=float32)],
ShardedDeviceArray([-12.214451, -12.214451, -12.214451, -12.214451,
-12.214451, -12.214451, -12.214451, -12.214451], dtype=float32))
These sharded JDs can have both Sharded and vanilla TFP distributions as components. For the unsharded distributions, we obtain the same sample on each device, and for the sharded distributions, we
get different samples. The log_prob on each device is synchronized as well.
MCMC with Sharded distributions
How do we think about Sharded distributions in the context of MCMC? If we have a generative model that can be expressed as a JointDistribution, we can pick some axis of that model to "shard" across.
Typically, one random variable in the model will correspond to observed data, and if we have a large dataset that we'd like to shard across devices, we want the variables that are associated to data
points to be sharded as well. We also may have "local" random variables that are one-to-one with the observations we are sharding, so we will have to additionally shard those random variables.
We'll go over examples of the usage of Sharded distributions with TFP MCMC in this section. We'll start with a simpler Bayesian logistic regression example, and conclude with a matrix factorization
example, with the goal of demonstrating some use-cases for the distribute library.
Example: Bayesian logistic regression for MNIST
We'd like to do Bayesian logistic regression on a large dataset; the model has a prior \(p(\theta)\) over the regression weights, and a likelihood \(p(y_i | \theta, x_i)\) that is summed over all
data \(\{x_i, y_i\}_{i = 1}^N\) to obtain the total joint log density. If we shard our data, we'd shard the observed random variables \(x_i\) and \(y_i\) in our model.
We use the following Bayesian logistic regression model for MNIST classification:
\[ \begin{align*} w &\sim \mathcal{N}(0, 1) \\ b &\sim \mathcal{N}(0, 1) \\ y_i | w, b, x_i &\sim \textrm{Categorical}(w^T x_i + b) \end{align*} \]
Let's load MNIST using TensorFlow Datasets.
mnist = tfds.as_numpy(tfds.load('mnist', batch_size=-1))
raw_train_images, train_labels = mnist['train']['image'], mnist['train']['label']
train_images = raw_train_images.reshape([raw_train_images.shape[0], -1]) / 255.
raw_test_images, test_labels = mnist['test']['image'], mnist['test']['label']
test_images = raw_test_images.reshape([raw_test_images.shape[0], -1]) / 255.
Downloading and preparing dataset mnist/3.0.1 (download: 11.06 MiB, generated: 21.00 MiB, total: 32.06 MiB) to /root/tensorflow_datasets/mnist/3.0.1...
WARNING:absl:Dataset mnist is hosted on GCS. It will automatically be downloaded to your
local data directory. If you'd instead prefer to read directly from our public
GCS bucket (recommended if you're running on GCP), you can instead pass
`try_gcs=True` to `tfds.load` or set `data_dir=gs://tfds-data/datasets`.
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Dataset mnist downloaded and prepared to /root/tensorflow_datasets/mnist/3.0.1. Subsequent calls will reuse this data.
We have 60000 training images but let's take advantage of our 8 available cores and split it 8 ways. We'll use this handy shard utility function.
def shard_value(x):
x = x.reshape((jax.device_count(), -1, *x.shape[1:]))
return jax.pmap(lambda x: x)(x) # pmap will physically place values on devices
shard = functools.partial(jax.tree.map, shard_value)
sharded_train_images, sharded_train_labels = shard((train_images, train_labels))
print(sharded_train_images.shape, sharded_train_labels.shape)
(8, 7500, 784) (8, 7500)
Before we continue, let's quickly discuss precision on TPUs and its impact on HMC. TPUs execute matrix multiplications using low bfloat16 precision for speed. bfloat16 matrix multiplications are
often sufficient for many deep learning applications, but when used with HMC, we have empirically found the lower precision can lead to diverging trajectories, causing rejections. We can use higher
precision matrix multiplications, at the cost of some additional compute.
To increase our matmul precision, we can use the jax.default_matmul_precision decorator with "tensorfloat32" precision (for even higher precision we could use "float32" precision).
Let's now define our run function, which will take in a random seed (which will be the same on each device) and a shard of MNIST. The function will implement the aforementioned model and we will then
use TFP's vanilla MCMC functionality to run a single chain. We'll make sure to decorate run with the jax.default_matmul_precision decorator to make sure the matrix multiplication is run with higher
precision, though in the particular example below, we could just as well use jnp.dot(images, w, precision=lax.Precision.HIGH).
# We can use `out_axes=None` in the `pmap` because the results will be the same
# on every device.
@functools.partial(jax.pmap, axis_name='data', in_axes=(None, 0), out_axes=None)
def run(seed, data):
images, labels = data # a sharded dataset
num_examples, dim = images.shape
num_classes = 10
def model_fn():
w = yield Root(tfd.Sample(tfd.Normal(0., 1.), [dim, num_classes]))
b = yield Root(tfd.Sample(tfd.Normal(0., 1.), [num_classes]))
logits = jnp.dot(images, w) + b
yield tfed.Sharded(tfd.Independent(tfd.Categorical(logits=logits), 1),
model = tfed.JointDistributionCoroutine(model_fn)
init_seed, sample_seed = random.split(seed)
initial_state = model.sample(seed=init_seed)[:-1] # throw away `y`
def target_log_prob(*state):
return model.log_prob((*state, labels))
def accuracy(w, b):
logits = images.dot(w) + b
preds = logits.argmax(axis=-1)
# We take the average accuracy across devices by using `lax.pmean`
return lax.pmean((preds == labels).mean(), 'data')
kernel = tfm.HamiltonianMonteCarlo(target_log_prob, 1e-2, 100)
kernel = tfm.DualAveragingStepSizeAdaptation(kernel, 500)
def trace_fn(state, pkr):
return (
states, trace = tfm.sample_chain(
return states, trace
jax.pmap includes a JIT compile but the compiled function is cached after the first call. We'll call run and ignore the output to cache the compilation.
output = run(random.PRNGKey(0), (sharded_train_images, sharded_train_labels))
jax.tree.map(lambda x: x.block_until_ready(), output)
CPU times: user 24.5 s, sys: 48.2 s, total: 1min 12s
Wall time: 1min 54s
We'll now call run again to see how long the actual execution takes.
states, trace = run(random.PRNGKey(0), (sharded_train_images, sharded_train_labels))
jax.tree.map(lambda x: x.block_until_ready(), trace)
CPU times: user 13.1 s, sys: 45.2 s, total: 58.3 s
Wall time: 1min 43s
We're executing 200,000 leapfrog steps, each of which computes a gradient over the entire dataset. Splitting the computation over 8 cores enables us to compute the equivalent of 200,000 epochs of
training in about 95 seconds, about 2,100 epochs per second!
Let's plot the log-density of each sample and each sample's accuracy:
fig, ax = plt.subplots(1, 3, figsize=(15, 5))
ax[0].set_title('Log Prob')
ax[2].set_title('Step Size')
If we ensemble the samples, we can compute a Bayesian model average to improve our performance.
@functools.partial(jax.pmap, axis_name='data', in_axes=(0, None), out_axes=None)
def bayesian_model_average(data, states):
images, labels = data
logits = jax.vmap(lambda w, b: images.dot(w) + b)(*states)
probs = jax.nn.softmax(logits, axis=-1)
bma_accuracy = (probs.mean(axis=0).argmax(axis=-1) == labels).mean()
avg_accuracy = (probs.argmax(axis=-1) == labels).mean()
return lax.pmean(bma_accuracy, axis_name='data'), lax.pmean(avg_accuracy, axis_name='data')
sharded_test_images, sharded_test_labels = shard((test_images, test_labels))
bma_acc, avg_acc = bayesian_model_average((sharded_test_images, sharded_test_labels), states)
print(f'Average Accuracy: {avg_acc}')
print(f'BMA Accuracy: {bma_acc}')
print(f'Accuracy Improvement: {bma_acc - avg_acc}')
Average Accuracy: 0.9188529253005981
BMA Accuracy: 0.9264000058174133
Accuracy Improvement: 0.0075470805168151855
A Bayesian model average increases our accuracy by almost 1%!
Example: MovieLens recommendation system
Let's now try doing inference with the MovieLens recommendations dataset, which is a collection of users and their ratings of various movies. Specifically, we can represent MovieLens as an \(N \times
M\) watch matrix \(W\) where \(N\) is the number of users and \(M\) is the number of movies; we expect \(N > M\). The entries of \(W_{ij}\) are a boolean indicating whether or not user \(i\) watched
movie \(j\). Note that MovieLens provides user ratings, but we're ignoring them to simplify the problem.
First, we'll load the dataset. We'll use the version with 1 million ratings.
movielens = tfds.as_numpy(tfds.load('movielens/1m-ratings', batch_size=-1))
GENRES = ['Action', 'Adventure', 'Animation', 'Children', 'Comedy',
'Crime', 'Documentary', 'Drama', 'Fantasy', 'Film-Noir',
'Horror', 'IMAX', 'Musical', 'Mystery', 'Romance', 'Sci-Fi',
'Thriller', 'Unknown', 'War', 'Western', '(no genres listed)']
Downloading and preparing dataset movielens/1m-ratings/0.1.0 (download: Unknown size, generated: Unknown size, total: Unknown size) to /root/tensorflow_datasets/movielens/1m-ratings/0.1.0...
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Shuffling and writing examples to /root/tensorflow_datasets/movielens/1m-ratings/0.1.0.incompleteYKA3TG/movielens-train.tfrecord
HBox(children=(FloatProgress(value=0.0, max=1000209.0), HTML(value='')))
Dataset movielens downloaded and prepared to /root/tensorflow_datasets/movielens/1m-ratings/0.1.0. Subsequent calls will reuse this data.
We'll do some preprocessing of the dataset to obtain the watch matrix \(W\).
raw_movie_ids = movielens['train']['movie_id']
raw_user_ids = movielens['train']['user_id']
genres = movielens['train']['movie_genres']
movie_ids, movie_labels = pd.factorize(movielens['train']['movie_id'])
user_ids, user_labels = pd.factorize(movielens['train']['user_id'])
num_movies = movie_ids.max() + 1
num_users = user_ids.max() + 1
movie_titles = dict(zip(movielens['train']['movie_id'],
movie_genres = dict(zip(movielens['train']['movie_id'],
movie_id_to_title = [movie_titles[movie_labels[id]].decode('utf-8')
for id in range(num_movies)]
movie_id_to_genre = [GENRES[movie_genres[movie_labels[id]][0]] for id in range(num_movies)]
watch_matrix = np.zeros((num_users, num_movies), bool)
watch_matrix[user_ids, movie_ids] = True
(6040, 3706)
We can define a generative model for \(W\), using a simple probabilistic matrix factorization model. We assume a latent \(N \times D\) user matrix \(U\) and a latent \(M \times D\) movie matrix \(V
\), which when multiplied produce the logits of a Bernoulli for the watch matrix \(W\). We'll also include a bias vectors for users and movies, \(u\) and \(v\).
\[ \begin{align*} U &\sim \mathcal{N}(0, 1) \quad u \sim \mathcal{N}(0, 1)\\ V &\sim \mathcal{N}(0, 1) \quad v \sim \mathcal{N}(0, 1)\\ W_{ij} &\sim \textrm{Bernoulli}\left(\sigma\left(\left(UV^T\
right)_{ij} + u_i + v_j\right)\right) \end{align*} \]
This is a pretty big matrix; 6040 user and 3706 movies leads to a matrix with over 22 million entries in it. How do we approach sharding this model? Well, if we assume that \(N > M\) (i.e. there are
more users than movies), then it would make sense to shard the watch matrix across the user axis, so each device would have a chunk of watch matrix corresponding to a subset of users. Unlike the
previous example, however, we'll also have to shard up the \(U\) matrix, since it has an embedding for each user, so each device will be responsible for a shard of \(U\) and a shard of \(W\). On the
other hand, \(V\) will be unsharded and be synchronized across devices.
sharded_watch_matrix = shard(watch_matrix)
Before we write our run, let's quickly discuss the additional challenges with sharding the local random variable \(U\). When running HMC, the vanilla tfp.mcmc.HamiltonianMonteCarlo kernel will sample
momenta for each element of the chain's state. Previously, only unsharded random variables were part of that state, and the momenta were the same on each device. When we now have a sharded \(U\), we
need to sample different momenta on each device for \(U\), while sampling the same momenta for \(V\). To accomplish this, we can use tfp.experimental.mcmc.PreconditionedHamiltonianMonteCarlo with a
Sharded momentum distribution. As we continue to make parallel computation first-class, we may simplify this, e.g. by taking a shardedness indicator to the HMC kernel.
def make_run(*,
@functools.partial(jax.pmap, in_axes=(None, 0), axis_name=axis_name)
def run(key, watch_matrix):
num_users, num_movies = watch_matrix.shape
Sharded = functools.partial(tfed.Sharded, shard_axis_name=axis_name)
def prior_fn():
user_embeddings = yield Root(Sharded(tfd.Sample(tfd.Normal(0., 1.), [num_users, dim]), name='user_embeddings'))
user_bias = yield Root(Sharded(tfd.Sample(tfd.Normal(0., 1.), [num_users]), name='user_bias'))
movie_embeddings = yield Root(tfd.Sample(tfd.Normal(0., 1.), [num_movies, dim], name='movie_embeddings'))
movie_bias = yield Root(tfd.Sample(tfd.Normal(0., 1.), [num_movies], name='movie_bias'))
return (user_embeddings, user_bias, movie_embeddings, movie_bias)
prior = tfed.JointDistributionCoroutine(prior_fn)
def model_fn():
user_embeddings, user_bias, movie_embeddings, movie_bias = yield from prior_fn()
logits = (jnp.einsum('...nd,...md->...nm', user_embeddings, movie_embeddings)
+ user_bias[..., :, None] + movie_bias[..., None, :])
yield Sharded(tfd.Independent(tfd.Bernoulli(logits=logits), 2), name='watch')
model = tfed.JointDistributionCoroutine(model_fn)
init_key, sample_key = random.split(key)
initial_state = prior.sample(seed=init_key, sample_shape=num_chains)
def target_log_prob(*state):
return model.log_prob((*state, watch_matrix))
momentum_distribution = tfed.JointDistributionSequential([
Sharded(tfd.Independent(tfd.Normal(jnp.zeros([num_chains, num_users, dim]), 1.), 2)),
Sharded(tfd.Independent(tfd.Normal(jnp.zeros([num_chains, num_users]), 1.), 1)),
tfd.Independent(tfd.Normal(jnp.zeros([num_chains, num_movies, dim]), 1.), 2),
tfd.Independent(tfd.Normal(jnp.zeros([num_chains, num_movies]), 1.), 1),
# We pass in momentum_distribution here to ensure that the momenta for
# user_embeddings and user_bias are also sharded
kernel = tfem.PreconditionedHamiltonianMonteCarlo(target_log_prob, step_size,
num_adaptation_steps = int(0.8 * num_burnin_steps)
kernel = tfm.DualAveragingStepSizeAdaptation(kernel, num_adaptation_steps)
def trace_fn(state, pkr):
return {
'log_prob': target_log_prob(*state),
'log_accept_ratio': pkr.inner_results.log_accept_ratio,
return tfm.sample_chain(
num_results, initial_state,
return run
We'll again run it once to cache the compiled run.
run = make_run(axis_name='data')
output = run(random.PRNGKey(0), sharded_watch_matrix)
jax.tree.map(lambda x: x.block_until_ready(), output)
CPU times: user 56 s, sys: 1min 24s, total: 2min 20s
Wall time: 3min 35s
Now we'll run it again without the compilation overhead.
states, trace = run(random.PRNGKey(0), sharded_watch_matrix)
jax.tree.map(lambda x: x.block_until_ready(), trace)
CPU times: user 28.8 s, sys: 1min 16s, total: 1min 44s
Wall time: 3min 1s
Looks like we completed about 150,000 leapfrog steps in about 3 minutes, so about 83 leapfrog steps per second! Let's plot the accept ratio and log density of our samples.
fig, axs = plt.subplots(1, len(trace), figsize=(5 * len(trace), 5))
for ax, (key, val) in zip(axs, trace.items()):
ax.plot(val[0]) # Indexing into a sharded array, each element is the same
Now that we have some samples from our Markov chain, let's use them to make some predictions. First, let's extract each of the components. Remember that the user_embeddings and user_bias are split
across device, so we need to concatenate our ShardedArray to obtain them all. On the other hand, movie_embeddings and movie_bias are the same on every device, so we can just pick the value from the
first shard. We'll use regular numpy to copy the values from the TPUs back to CPU.
user_embeddings = np.concatenate(np.array(states.user_embeddings, np.float32), axis=2)
user_bias = np.concatenate(np.array(states.user_bias, np.float32), axis=2)
movie_embeddings = np.array(states.movie_embeddings[0], dtype=np.float32)
movie_bias = np.array(states.movie_bias[0], dtype=np.float32)
samples = (user_embeddings, user_bias, movie_embeddings, movie_bias)
print(f'User embeddings: {user_embeddings.shape}')
print(f'User bias: {user_bias.shape}')
print(f'Movie embeddings: {movie_embeddings.shape}')
print(f'Movie bias: {movie_bias.shape}')
User embeddings: (500, 2, 6040, 20)
User bias: (500, 2, 6040)
Movie embeddings: (500, 2, 3706, 20)
Movie bias: (500, 2, 3706)
Let's try to build a simple recommender system that utilizes the uncertainty captured in these samples. Let's first write a function that ranks movies according to the watch probability.
def recommend(sample, user_id):
user_embeddings, user_bias, movie_embeddings, movie_bias = sample
movie_logits = (
jnp.einsum('d,md->m', user_embeddings[user_id], movie_embeddings)
+ user_bias[user_id] + movie_bias)
return movie_logits.argsort()[::-1]
We can now write a function that loops over all the samples and for each one, picks the top ranked movie that the user hasn't watched already. We can then see the counts of all recommended movies
across the samples.
def get_recommendations(user_id):
movie_ids = []
already_watched = set(jnp.arange(num_movies)[watch_matrix[user_id] == 1])
for i in range(500):
for j in range(2):
sample = jax.tree.map(lambda x: x[i, j], samples)
ranking = recommend(sample, user_id)
for movie_id in ranking:
if int(movie_id) not in already_watched:
return movie_ids
def plot_recommendations(movie_ids, ax=None):
titles = collections.Counter([movie_id_to_title[i] for i in movie_ids])
ax = ax or plt.gca()
names, counts = zip(*sorted(titles.items(), key=lambda x: -x[1]))
ax.bar(names, counts)
ax.set_xticklabels(names, rotation=90)
Let's take the user who has seen the most movies versus the one who has seen the least.
user_watch_counts = watch_matrix.sum(axis=1)
user_most = user_watch_counts.argmax()
user_least = user_watch_counts.argmin()
print(user_watch_counts[user_most], user_watch_counts[user_least])
We hope our system has more certainty about user_most than user_least, given that we have more information about what sorts of movies user_most is more likely to watch.
fig, ax = plt.subplots(1, 2, figsize=(20, 10))
most_recommendations = get_recommendations(user_most)
plot_recommendations(most_recommendations, ax=ax[0])
ax[0].set_title('Recommendation for user_most')
least_recommendations = get_recommendations(user_least)
plot_recommendations(least_recommendations, ax=ax[1])
ax[1].set_title('Recommendation for user_least');
We see that there is more variance in our recommendations for user_least reflecting our additional uncertainty in their watch preferences.
We can also see look at the genres of the recommended movies.
most_genres = collections.Counter([movie_id_to_genre[i] for i in most_recommendations])
least_genres = collections.Counter([movie_id_to_genre[i] for i in least_recommendations])
fig, ax = plt.subplots(1, 2, figsize=(20, 10))
ax[0].bar(most_genres.keys(), most_genres.values())
ax[0].set_title('Genres recommended for user_most')
ax[1].bar(least_genres.keys(), least_genres.values())
ax[1].set_title('Genres recommended for user_least');
user_most has seen a lot of movies and has been recommended more niche genres like mystery and crime whereas user_least has not watched many movies and was recommended more mainstream movies, which
skew comedy and action.
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Creation Science
Viewed purely as a literary composition, the Bible is without parallel
• in its continuity – written over a 1600-year span by more than 40 authors drawn from every walk of life
• in its translation and circulation – published in more languages and read by more people than any other book
• in its survival – through persecution and criticism
• in its teachings, its frank portrayal of man, and its prophecies – many of which have already been accurately fulfilled
• in its influence on human behaviour, on reform, and on literature, art and music
But the uniqueness of the Bible goes far beyond these considerations; it claims for itself the added distinction of being divinely-inspired, for we read “All Scripture is given by inspiration of God,
and is profitable for doctrine, for reproof, for correction, for instruction in righteousness: That the man of God may be perfect, thoroughly furnished unto all good works.” (2Tm.3:16,17). These
words clearly imply that this Book has been specifically provided for our benefit – to impart reliable information that addresses the deepest issues of life: in particular, the character of our
Creator, the origin and purpose of life, the true nature of man, and what it is that follows this earthly existence.
The truth of divine inspiration has been remarkably confirmed in recent years by the discovery that the opening verse of the Bible – obviously numbered among the most widely read of all time – is
copiously watermarked with structures that spring from the heart of mathematics. Indeed, as expressed in the original Hebrew, Genesis 1:1 may be fairly claimed to be the most remarkable combination
of words ever written, and the evidence presented here, though by no means exhaustive, is sufficient to explain why. Supplemented by the other pages provided on this site, here is proof that the many
and varied numerical features associated with these strategically placed words are clearly present by design – but, at the same time, defying all natural explanation!
The object of our study takes the form of a powerful and fundamental assertion: “In the beginning God created the heavens and the earth.” – this information conveyed in a sequence of seven Hebrew
words comprising a total of 28 letters, thus:
There are two things to note here: (a) in accordance with standard practice, the words are written and read from right to left, and (b) the central word (necessary to sustain the grammar) is
Let us immediately observe that the number of Hebrew letters in Genesis 1:1 is four times the number of words.
#1: No. of letters = 28; No. of words = 7; letters = 4 x words.
Separating the letters and arranging them in rows – one in the first, two in the second, three in the third, and so on – a simple geometrical relationship between letters and words is revealed, thus:
#2: The 28 letters of the verse form a numerical triangle of side 7 units.
However, as the following figures reveal, two further triangles are found to occur on word boundaries:
#3: The 21 letters of words 1 to 5 form a numerical triangle of side 6 units – 6 itself being a numerical triangle.
#4: The 6 letters of word 1 form a numerical triangle of side 3 units – 3 itself being a numerical triangle.
These elements of numerical geometry are examples of what may conveniently be referred to as figurate numbers.
#5: A figurate number – as here understood – is one which, when represented as a set of uniform circular or spherical counters, completely fills a symmetrical polygonal or polyhedral frame.
In the examples given, the relevant frame has the form of an equilateral triangle – which, for the Christian, clearly has trinitarian implications.
The triangles of letters representing first word and first verse, viz 6 and 28, are further related in being first and second perfect numbers. To qualify for the title perfect a number is required to
equal the sum of those smaller numbers (including 1) which divide it exactly. Thus, 6 is perfect because 1+2+3 = 6; and 28 is perfect because 1+2+4+7+14 = 28. Such objects are exceedingly rare –
there being only 5 instances in the first 8 billion natural numbers; all known examples are even and triangular; they have interested the world’s foremost mathematicians from the earliest times.
#6: The numbers of letters forming the Bible’s first word and first verse are 6 and 28, respectively; these are first and second perfect numbers, and third and seventh triangular numbers
The number of words in this first verse may be identified with another symmetrical figure (one involving sixness rather than threeness), viz a numerical hexagon, thus:
#7: The 7 words of Genesis 1:1 form a numerical hexagon and thereby reveal a hidden relationship with both 6 and 10.
As can be seen, figures of this kind derive directly from a pair of identical triangles that possess a central counter – the so-called centroid- or generator- triangles. One in every three terms of
the infinite triangle series possesses this feature. In Fig.5 above the numerical triangle PQR – value 10 – is first copied in situ as P’Q’R’ and this then rotated by 180 degrees about the centroid
counter C to generate, in combination with PQR, two symmetrical figures: a hexagon of 7 (by overlap – or intersection), and a 6-pointed star, or hexagram, of 13 (by addition – or union).
#8: In any given sequence of numerical triangles, 1 in 3 will be built around a central counter, and thus will be capable of generating a related hexagon/hexagram pair by self intersection/union.
An examination of Fig.1 reveals that 28 is also a generator triangle; this – combined in like manner with a rotated copy of itself – gives rise to the hexagon/hexagram pair, 19/37, thus:
Ten, of course, is closely associated with the naming and writing of numbers, and features also as collective unit in metrication and decimalisation. Interestingly (since all numbers presented here
are expressed as denary – or base 10 – objects), the sum of each pair of digits representing the figurate numbers involved in Fig.7 is ten! Furthermore, as Fig.9 reveals, ten has an absolute presence
which is completely independent of all other considerations – a quality which it transfers to Genesis 1:1 by association.
In Fig.8, we have 10-as-triangle centred within 28-as-triangle – the outline of the latter being revealed as 18.
#9: The outline of the triangle of Genesis 1:1 letters is 18 – or 6+6+6.
When the inner triangle is rotated by 180 degrees about its central counter (c) a trio of 6-as-triangle is generated. The resulting symmetry reveals the close connection between this important
number, ten, and the two first perfect numbers.
#10: Ten-as-triangle (also known to the Pythagoreans as tetraktys) has a close affinity with the two first perfect numbers – hence with the lexical structure of Genesis 1:1.
A comparison of Figs.1 and 6 reveals that the letter occupying the triangle centroid is
#11: Occupying the centroid position in the triangle of Genesis 1:1 letters, we find
To summarise: the key themes revealed in this examination of the lexical features of Genesis 1:1 are the manifestations of numerical geometry, viz generator triangle, hexagon/hexagram pair, and
10-as-tetraktys in a setting of perfect numbers.
Section B:- The words read as numbers
All Hebrew words and word sequences have a cryptic numerical presence arising from the ancient practice (from c200 BC) of using the complete set of 22 alphabetic characters as numerals [details of
this scheme are provided here]. Numbers are formed by assembling a string of letters whose sum represents the required value. We shall find it convenient to use the term characteristic value
(hereafter “CV”) for a Hebrew word or phrase interpreted in this manner. On this basis, Genesis 1:1 may be fairly read as a sequence of seven numbers, thus:
In this presentation, letter CVs are shown above, and word CVs (sums of the corresponding letter CVs), below. Viewed naturalistically, the latter can have no information content: they appear to be
merely fortuitous appendages of passing interest. However, as we have seen, the Bible is no ordinary book. It claims that our Creator is its Author. If this indeed be the case then it would surely be
unwise to dismiss the possibility that these numbers – indelibly associated as they are with its opening words – represent an additional and valid tool of biblical exegesis. Let us therefore consider
what these numbers might have to tell us.
Observe that the first word is by far the largest numerically, and that Elohim (translated ‘God’) is the smallest. Note too that 37 is a factor of words 6 and 7 ( ‘and the earth’).
The CV of the complete verse (ie the sum of the 7 word CVs) is 2701 (column A). This factorises in an interesting way, viz 2701 = 37 x 73. Remarkably, these factors are symmetrically revealed when
the number is added to that formed by reversing its digits, ie 2701 + 1072 = 3773.
#12: Expressed as base 10 numbers, the prime factors of the verse CV, 37 and 73, involve the digits 3 and 7, and are reflective. They are also related absolutely in being 4th hexagon/hexagram pair –
the generator triangle involved being the 10th, ie 55.
But 2701 (representing Genesis 1:1) has further significant attributes. For example, like 28 it has a geometrical presence as a triangular number – one standing on a base of 73 and having an outline
of 216, or 6x6x6 (it may be remembered that the outline of 28-as-triangle was 18, or 6+6+6 – see #9).
Again, its digits combine in various ways to produce further instances of numerical geometry, thus: 2+7+0+1 = 10 (4th triangle); 2+701 = 703 (37th triangle); 27+01 = 28 (7th triangle); and 270+1 =
271 (10th hexagon). Further, 2701 is a generator triangle (ie it is centred – the centroid counter occupying the 25th position in the 49th row) and gives rise to the 25th hexagon/hexagram pair 1801/
3601 by self intersection/union.
#13: The verse CV, 2701, turns out to be a remarkable number possessing a significant presence when viewed as denary object or as geometrical absolute.
To continue our examination of the foregoing table: the sum of the two multiples of 37, words 6 and 7 (column B), is 703 – 37th triangular number – its factors being 19 and 37.
#14: The combined CV of words 6 and 7 is 703 – the prime factors of this number, 19/37, being the hexagon/hexagram pair generated by 28-as-triangle (see Fig.7).
Further, we find that this triangle fits precisely within 2701 – the verse triangle. It thus mirrors the structure of Fig.9 and, in the process, generates a triple of 666-as-triangle (anticipated by
the triple of 6-as-triangle in the earlier figure!). As is demonstrated elsewhere, 666 is uniquely triangular.
#15: 2701 (verse total) and 703 (sum of words 6 and 7) are coordinated geometrically and closely associated with the unique triangular number, 666.
Referring to the remaining columns of Table 2, note the manner in which the CVs generate 3-digit sums that are each a multiple of 111, or 3×37. Clearly, their complements are multiples of 37 also.
#16: Genesis 1:1 is saturated with the factor 37 which include a number of its eye-catching repeated-digit multiples.
The next table reveals the fact that the residual CVs are precisely bisected, (a) when the first word, and (b) the final two words, are omitted from the proceedings. In the first case, the sum
involved is 894; in the second, 999.
#17: When the first word, or words 6 and 7, are withdrawn, the residue in each case divides equally on word boundaries.
These particular verse divisions also have geometrical implications. Thus, as Fig.12 reveals, Genesis 1:1 may be represented as a trapezium in which 703-as-triangle again occupies centre stage, now
flanked by two numerical parallelograms of 999 (ie 27×37) counters apiece. The combination has 37 rows – the first of 55 counters and the 37th of 91; its outline is again 216, or 6x6x6.
The difference between 999 and 894 is 105 – 14th triangular number. As the following figures reveal, 913 (the CV of the first word of Genesis 1:1) may be viewed as the sum of the three triangular
numbers 105, 703 and 105.
Bringing these five components together, we now have yet another symmetrical view of the Bible’s first verse – one in which the first word appears as central element. Keep in mind that each figure in
the following composite represents the sum of word CVs – as defined in Table 3.
The entry of 894 into the proceedings focusses attention on the smaller, extrabiblical, set of four 3-digit numbers which satisfy the requirement that each is the sum of the cubes of its digits. I
refer to {153, 370, 371, 407}. The sum of the first three of these is 894; the 4th is the 6th of Genesis 1:1; and the sum of the four is 1301 (compare with 2701) – this appearing in column P of Table
3 above. Moreover, two members of this set are multiples of 37, viz 370 and 407, whose sum is 777 (and thus equal to the sum of the nouns God, heaven and earth that occur in Genesis 1:1).
#18: There is a close affinity between Genesis 1:1 and the secular set {153, 370, 371, 407}. In addition, the first, 153, appears as a New Testament ‘surface feature’ in John 21:11
• Genesis 1:1 as a tessellated trefoil
Another division of the 7 CVs into the sums of ‘odds’ and ‘evens’ leads to yet more symmetrical representations of the verse. The details may be found here.
• Sums of the digits of the word CVs
A further feature of interest concerns the digit sums of these 7 word CVs. These are presented in Table 4. Here the primary digit sums occupy column K. Apart from that derived from word 3, we see
that all are prime numbers – but that the sum of the two digits of this is also prime. Column L summarises the situation up to this point – every entry now a prime number. Finally, the secondary sums
are recorded in column M. Observe that the column totals are 82 (digit reverse of 28 – the letter count of Genesis 1:1), 73 and 37 (the factors of Genesis 1:1) – 37 being the numerical hexagram
representing the self-union of 28-as-triangle (see Fig.7).
• Products of the letter and word CVs
A very large whole number is generated when the letter CVs are multiplied together. Thus, proceeding word by word, we have:
Clearly, if these seven products are multiplied together we obtain the desired result, viz 2^15 x 3^6 x 10^27, or – since each of the exponents is a multiple of 3 – (2^5 x 3^2 x 10^9)^3.
#19: The product of the letter values is a perfect cube.
This number may be alternatively expressed as 2.3887872 x 10^34
Another large number is obtained by multiplying together the word CVs, thus:
Observe, in the second line, that if the six groups of three digits are added together, we obtain 2701 – the CV of the verse! Also that the sums of their digits (following the procedure of Table 4)
are 73 and 37, respectively, ie the factors of 2701.
The ratio of these products is 3.04153… / 2.38878… x 10^(-17), or 1.273255… x 10^(-17). This figure – multiplied by 10^17 – is within 0.0013% of the ratio of the perimeter of a square to the
circumference of its inscribed circle (or, alternatively, the ratio of their areas). In other words, the significant 5-digit string “12732” is common to both.
#20: The ratio of the word and letter products of Genesis 1:1 is related to the matter of squaring the circle.
The inversion of this ratio leads to a value for the universal constant “pi” in a very interesting and remarkable manner, thus:
#21: The clear evidence that the Bible’s first verse contains, within itself, all that is necessary to determine a value for ‘pi ‘ correct to 0.001% – and in such a straightforward manner – suggests
overwhelmingly that its Author, at the instant of its composition, was already fully aware of its later numerical implications; indeed, had designed it with these (and the other matters) in mind.
• Concatenations of word CVs
Observe that in base 10 representation all Genesis 1:1 word CVs have three digits (which involves writing the 3rd as ‘086’ rather than ’86’). The 21-digit number formed by concatenation these 7
values factorises thus:
913203086401395407296 = 2^6 x 7 x 37 x 131 x 1291 x 4159 x 7879 x 9941
This result is particularly interesting for two reasons: (a) 2368 (ie 64 x 37) is a factor – this being the CV of the Greek form of the name “Jesus Christ”, nominative case – and since the claim is
made (eg Jn.1:1-3, 10) that he is our Creator (ie the ‘God’ of Genesis 1:1), the matter is potentially significant; and (b) the fact that it has 13 prime factors – the largest having only 4 digits –
is indicative of low probability and hence of design.
Only 120 of the 5040 concatenated arrangements are exactly divisible by 2368 – all of which are required to end with words 6 and 7 in situ. One final observation concerns the concatenation of these,
viz 407296, which factorises thus: 2 x 86 x 2368. Since the related words are translated “and the earth”, and 86 happens to be the CV of word 3 (‘God’), the association of these numbers strongly
suggests that Jesus Christ is, indeed, God Incarnate!
#22: The CVs of Jesus (888) and of Christ (1480) each have 37 and 296 (ie 8 x 37, and 7th word of Genesis 1:1) as factors. Since this opening verse is saturated with multiples of 37 – an object
uniquely interesting per se – this particular number effectively binds together these several textually-related components.
Observe that 37 (along with all numerical hexagons) is the difference of two consecutive cubes, thus: 37 = 64 – 27 = 4^3 – 3^3; and that 73 (its digit reversal and other factor of 2701) is the
difference between hypercube and cube, thus: 73 = 81 – 8 = 3^4 – 2^3. But there are other associations of interest; for example:
#23: These multiples of 37 (defined by the cubes of the rows of the tetraktys) strengthen the association of Jesus Christ with Genesis 1:1 and also uphold the Bible’s claim to his being Creator. In
this connection it is also worth noting the comments of #11, viz that the central letter of Fig.1 is ‘yod’, ie the first of ‘Yeshua’ (Hebrew form of ‘Jesus’), and the central (untranslatable) word of
Fig.4 is formed from the first and last letters of the Hebrew alphabet – clearly, equivalent to Alpha and Omega – the First and the Last (see Revelation 1:8, 21:6 and 22:13).
The CVs of Genesis 1:1 are not independent. Each can be shown to be a function of the parameters 37 (unique number) and 6 (first perfect number), thus:
#24: The 7 word CVs of Genesis 1:1 are functions of the parameters 37 and 6, and are therefore related.
To summarise: many of the features found in this set of numbers are developments of those hinted at in the lexical structure – in particular, the structures of numerical geometry (specifically,
triangle, hexagon and hexagram), and the important roles played by 37 (derivative of 28 – second perfect number) and 6 (first perfect number); other significant associations involve the Creator’s
name, the cubes and the problem of squaring the circle.
Section C: – New Testament Associations
Attention has already been drawn to the close numerical affinity that exists between the Lord’s name, ‘Jesus Christ’, and Genesis 1:1 – the common factor 37 being the principal numerical binding
agent. However, there is another matter that warrants attention. It concerns the riddle of Revelation 13:18 in which the uniquely triangular number 666 is associated with a promise of wisdom to those
who, with understanding, read words as numbers. A detailed consideration of the issues is provided in the accompanying page 666 – and All That! – where two more triangular numbers, 153 (John 21:11)
and 276 (Acts 27:37), are shown to be closely involved with 666 in directing attention to the Bible’s first verse. Here is the nub of the matter expressed diagrammatically:
At (a), we have a repeat of Fig.9 where 10 – radix or base of our number system – is shown as tetraktys (ie its triangular form) in a setting of perfect numbers 6 (number of letters in the first word
of the Bible) and 28 (number of letters in the first verse). Clearly, this is a highly significant structure per se – and one that is completely independent of radix. It is also a matter of some
significance that, as denary objects, 6, 36 (or 6×6), 66 and 666 are all triangular. Observe now that if structure (a) is taken as a template, and satellites of 36, 66 and 666 be substituted around
the triangular core (which, must become a triangle of order one more than these), then the respective outcomes are as shown at (b), (c) and (d).
At (d), of course, we have the triangular form of Genesis 1:1 with the 37th triangle representing the sum of words 6 and 7 inset.
#25: The riddle of Revelation 13:18 specifies the numerical features of Genesis 1:1.
To summarise: it may be argued that Revelation 13:18 – taken together with John 21:11 and Acts 27:37 – effectively specifies the numerical structure of Genesis 1:1. This strongly suggest that both
have emanated from the same mind.
Section D: – Genesis 1:1 as a recreational object
It comes as something of a surprise to find that the word CVs display features that appear designed to catch the eye of those interested in recreational mathematics. Two examples will suffice.
• The resilience of word CVs 1 to 5
These words form what might be termed the ‘supernatural component’ of the statement. Their sum is 1998 which divides 999/999 on word boundaries (see Table 3). When written in reverse, the sum of
these 5 numbers is still 1998 – as it is when the same cyclic permutation is applied to the digits of the individual values (rows 2 and 3), or to the block of five (rows 4 and 5), thus:
• The deletion of zeros phenomena
Here again is the numeric reading of Genesis 1:1:
Observe that the 2nd, 4th and 6th word CVs have no ‘tens’ digit. Clearly, the central zeros are essential to maintain the integrity of these numbers, but as the foregoing analysis has demonstrated,
this set of 7 is full of surprises. Accordingly, we are interested to know what would be the effect of omitting these zeros.
#26: Clearly, to plan such wonders must require foreknowledge of the system of numeration we enjoy today!
To summarise: these phenomena suggest that the Lord is prepared to use the lowliest of means to reach the heart of the unbeliever!
Section E: – An extrabiblical perspective
There are secular voices which reveal numerical links with Genesis 1:1. These provide hard evidence of supernatural intelligent design and purpose. Two examples follow.
Earlier, in Section B, it was revealed that each of the word CVs of Genesis 1:1 could be expressed concisely in terms of the two parameters 37 and 6 – Table 7 defining the required coefficients. In
an alternative schema, the final six may be similarly defined, now using the triangular number 105 with 99 – difference between 105 and the first perfect number, 6. Here is a picture of the
background geometry:
Observe also that if the triangle of Hebrew letters (Fig.1) be superimposed on the triangle in which 10 appears in the context of perfect numbers 6 and 28 (Fig.9) then the sums of the letter CVs
within the four segments are as shown below:
Let us now observe that these companions of 913 (CV of the Bible’s first word) may all be simply expressed in terms of the same parameters, viz 105, 99 and 500, thus: 697 = 2.500 – 2.99 – 105; 604 =
2.500 – 4.99; and 487 = 2.500 – 2.99 – 3.105.
#27: The word CVs of Genesis 1:1 are echoed in the metric dimensions of an artefact that came into being in the 1960’s. Adding to the mystery is the fact that the standard governing this matter is
designated ISO 216 – 216 (or 6x6x6) being the outline of both Genesis 1:1 triangle (Fig.11) and trapezium (Fig.13).
The page Exceptional Measures describes these matters in greater detail.
The sequence of natural numbers 1 – 36 can be represented as a square 6×6 matrix, thus:
The sequence total is 666 (or 6 x 111) – the unique 36th triangular number. Summing the triangles and the squares indexed by these numbers (ie 1+3+6+…+630+666 and 1+4+9…+1225+1296) we obtain 8436 and
16206, respectively, ie 12 x 703 and 6 x 2701. Observe that 666, 703 and 2701 are the first three triangular multiples of 37 that define the geometry of Genesis 1:1 (see Fig.11 and Fig.16d). The grid
of Fig.19 has 6 rows and 6 columns. Since the three sums referred to have 6 as factor, we inquire whether it be possible – by rearranging the numbers within the matrix – to generate for each row the
totals 111, 2×703 and 2701 (for the numbers, and for the triangles and the squares they index, respectively). This is indeed possible. Here is a solution:
However, in addition to meeting these requirements, we now find, (a) the sum in each column and long diagonal is also 111 (or 3×37), (b) the four corner elements total 74 (or 2×37), (c) the
peripheral elements total 370 (or 10×37), and (d) the central 16 total 296 (or 8×37).
#28: The 6×6 square demonstrates that there exists an absolute and extrabiblical relationship between the numbers 666, 703, 2701 and 296 – all multiples of 37 – and Genesis 1:1; in addition, 111
figures in the verse as factor of 777, 888 and 999; again, a number of these relationships carry geometrical overtones. Also, it should not go unnoticed that 370 links the square with the smaller set
{153, 370, 371, 407} – already shown to be linked to be closely linked to the verse (see #19).
To summarise: here is further evidence of the miraculous; how does it come about that A4 – carefully specified by man as the most reasonable paper size for general use in a metric environment –
embodies close links with the word CVs of Genesis 1:1? Clearly, nothing was further from the minds of those who drafted ISO 216; and neither, presumably, was the reference number of the document
representing the outcome of their deliberations – 216, or 6x6x6, being the outline of 2701-as-triangle!! And as for the square array, the suggestion must be that its features played a significant
part in the divine planning of Genesis 1:1!
The vexed question of origins is clearly subsidiary to the two more fundamental questions: Is there a God? and, if so, Is He the Sovereign God of the Judeo-Christian Scriptures? It is surely
unreasonable to suppose that the defence of God’s Being and Sovereignty should depend entirely upon the ingenuity and zeal of well-meaning supporters, and it should come as no surprise, therefore, to
find that the Bible has within itself the means of delivering a crushing response to all who would challenge its veracity. Accordingly, we find integrated with the opening Hebrew words a remarkable
confluence of numerical symmetry with features of considerable rarity or uniqueness. Such attributes invite a consideration of following train of logic:
• the opening words of Holy Scripture must, logically,be ascribed to the only eye-witness, viz God the Creator
• circa 200 BC, each of these words – in common with all written Hebrew words – acquired an additional meaning as a number
• on close examination, these numbers are found to be associated with a rich and meaningful structure which encompasses other portions of the biblical text as well as certain extrabiblical matters;
this suggests, overwhelmingly, that they are not random accretions to the text – as might be supposed – but rather are features of design
• patently, such phenomena do not evolve
• they came into human view many centuries after these words were first recorded
• we can only conclude that these arbiters of truth are of supernatural origin and, because they elevate the significance of these controversial words, engender a high view of God, and are
consistent with the remainder of Scripture, they must be regarded as purposeful additions – planned and implemented by the Creator Himself!
• these conclusions – backed up as they are by self-evident truths – pose formidable problems for all unbelievers, encourage believers, and remove all obstacles to a literal understanding of the
Scriptures – the Creation narrative, in particular!
Vernon Jenkins MSc
Created: 2001-03-26
Last modified: 2001-10-24
email: vernon.jenkins@virgin.net
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Dynamics vs. Kinematics - What's the Difference? | This vs. That
Dynamics vs. Kinematics
What's the Difference?
Dynamics and kinematics are two branches of physics that study the motion of objects, but they approach the subject from different perspectives. Kinematics focuses on describing and analyzing the
motion of objects without considering the forces that cause the motion. It deals with concepts such as displacement, velocity, and acceleration. On the other hand, dynamics is concerned with the
forces that cause motion and how they affect the motion of objects. It explores concepts like Newton's laws of motion, momentum, and energy. While kinematics provides a mathematical description of
motion, dynamics delves deeper into the underlying causes and interactions that govern the motion of objects.
Attribute Dynamics Kinematics
Definition The study of forces and their effects on motion. The study of motion without considering its causes.
Focus Concerned with the relationship between motion and the forces acting upon it. Concerned with describing and analyzing motion without considering its
Variables Considers variables such as force, mass, acceleration, and momentum. Considers variables such as displacement, velocity, and time.
Equations Uses Newton's laws of motion to derive equations of motion. Uses mathematical equations to describe and analyze motion.
Applications Applied in engineering, physics, and other fields to understand and predict the behavior of objects in Applied in physics, robotics, and animation to describe and simulate motion.
Force Considers the effects of forces on motion. Does not consider forces or their effects.
Acceleration Studies the causes and effects of acceleration. Describes acceleration without considering its causes.
Momentum Considers the momentum of objects in motion. Does not consider momentum.
Further Detail
When studying the field of physics, two fundamental branches that often come up are dynamics and kinematics. While both deal with the motion of objects, they focus on different aspects and provide
unique insights into the behavior of physical systems. In this article, we will explore the attributes of dynamics and kinematics, highlighting their differences and similarities.
Dynamics is the branch of physics that describes the relationship between motion and the forces acting upon an object. It focuses on understanding how forces influence the motion and behavior of
objects. In dynamics, we analyze the causes of motion, such as the forces applied to an object, and how these forces affect its acceleration and velocity.
One key attribute of dynamics is the concept of Newton's laws of motion. These laws provide a framework for understanding the relationship between forces, mass, and acceleration. The first law states
that an object at rest will remain at rest, and an object in motion will continue moving at a constant velocity unless acted upon by an external force. The second law relates the force applied to an
object to its mass and acceleration, stating that the acceleration is directly proportional to the force and inversely proportional to the mass. The third law states that for every action, there is
an equal and opposite reaction.
Another important aspect of dynamics is the study of different types of forces. Forces can be categorized into contact forces, such as friction or normal force, and non-contact forces, such as
gravity or electromagnetic forces. Understanding these forces allows us to analyze and predict the motion of objects in various scenarios.
In addition to forces, dynamics also considers concepts like work, energy, and power. Work is defined as the product of the force applied to an object and the displacement it undergoes. Energy is the
ability to do work, and power is the rate at which work is done. These concepts provide a deeper understanding of the relationship between forces and motion.
Kinematics, on the other hand, is the branch of physics that focuses on describing motion without considering the forces causing it. It deals with the mathematical representation of motion, including
concepts like position, velocity, and acceleration. Kinematics provides a framework for analyzing and predicting the motion of objects based solely on their initial conditions and the laws of motion.
One of the key attributes of kinematics is the use of mathematical equations to describe motion. These equations include formulas for displacement, velocity, and acceleration. For example, the
equation for displacement is given by Δx = v[0]t + 0.5at^2, where Δx represents the change in position, v[0] is the initial velocity, t is the time, and a is the acceleration.
Kinematics also introduces the concept of motion graphs, such as position-time graphs, velocity-time graphs, and acceleration-time graphs. These graphs provide a visual representation of an object's
motion and allow us to analyze its behavior more intuitively. By examining the slope and shape of these graphs, we can gain insights into the object's velocity and acceleration at different points in
Furthermore, kinematics allows us to study the motion of objects in different dimensions. While dynamics primarily focuses on motion in one dimension, kinematics extends its analysis to
two-dimensional and three-dimensional motion. This enables us to describe the complex motion of objects in real-world scenarios, such as projectiles or objects moving in curved paths.
Lastly, kinematics plays a crucial role in the field of robotics and computer animation. By understanding the principles of kinematics, engineers and animators can create realistic and accurate
simulations of motion, allowing for the development of advanced technologies and virtual environments.
While dynamics and kinematics both deal with the motion of objects, they approach the subject from different perspectives. Dynamics focuses on the forces causing motion and their effects, while
kinematics focuses on the mathematical representation of motion without considering the forces involved.
One key difference between dynamics and kinematics is their level of complexity. Dynamics involves the analysis of forces, energy, and power, which can be more mathematically and conceptually
challenging. On the other hand, kinematics provides a more straightforward approach to understanding motion, relying on mathematical equations and graphs.
Another difference lies in their applications. Dynamics is particularly useful in engineering, as it allows engineers to design structures and machines that can withstand and utilize forces
effectively. It is also essential in fields like astrophysics, where the study of celestial bodies requires an understanding of gravitational forces. Kinematics, on the other hand, finds applications
in robotics, animation, and computer simulations, where precise motion planning and control are crucial.
Despite their differences, dynamics and kinematics are interconnected. Dynamics relies on the mathematical descriptions provided by kinematics to analyze the motion of objects under the influence of
forces. Kinematics, on the other hand, can benefit from the insights gained through dynamics, as understanding the forces acting on an object can help predict its motion more accurately.
In conclusion, dynamics and kinematics are two fundamental branches of physics that provide valuable insights into the behavior of objects in motion. While dynamics focuses on the forces causing
motion and their effects, kinematics deals with the mathematical representation of motion without considering the forces involved. Both branches have their unique attributes and applications, and
understanding their differences and similarities is essential for a comprehensive understanding of the principles of motion.
Comparisons may contain inaccurate information about people, places, or facts. Please report any issues.
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The Washing Monument is 555 feet high. If you stand one quarter of a mile, or 1320 feet, from the base of the monument and look to the top, find the angle of elevation to the nearest degree.? | Socratic
The Washing Monument is 555 feet high. If you stand one quarter of a mile, or 1320 feet, from the base of the monument and look to the top, find the angle of elevation to the nearest degree.?
1 Answer
You have that (from trigonometry):
$\tan \alpha = \frac{555}{1320} = 0.42$
$\alpha = {\tan}^{-} 1 \left(0.42\right) \approx {23}^{\circ}$
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An Intrinsic Theory of Quantum Mechanics: Progress in Field's Nominalistic Program, Part I
Chen, Eddy Keming (2017) An Intrinsic Theory of Quantum Mechanics: Progress in Field's Nominalistic Program, Part I. [Preprint]
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In this paper, I introduce an intrinsic account of the quantum state. This account contains three desirable features that the standard platonistic account lacks: (1) it does not refer to any abstract
mathematical objects such as complex numbers, (2) it is independent of the usual arbitrary conventions in the wave function representation, and (3) it explains why the quantum state has its amplitude
and phase degrees of freedom.
Consequently, this account extends Hartry Field’s program outlined in Science Without Numbers (1980), responds to David Malament’s long-standing impossibility conjecture (1982), and establishes an
important first step towards a genuinely intrinsic and nominalistic account of quantum mechanics. I will also compare the present account to Mark Balaguer’s (1996) nominalization of quantum mechanics
and discuss how it might bear on the debate about “wave function realism.” In closing, I will suggest some possible ways to extend this account to accommodate spinorial degrees of freedom and a
variable number of particles (e.g. for particle creation and annihilation).
Along the way, I axiomatize the quantum phase structure as what I shall call a “periodic difference structure” and prove a representation theorem as well as a uniqueness theorem. These formal results
could prove fruitful for further investigation into the metaphysics of phase and theoretical structure.
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Traverse Surveying - Objective, Method and Procedure
A traverse survey is a type of control survey that involves the establishment of a series of points that are linked together by lines to form a framework. The series of straight lines that connect
the successive points are called traverse lines.
The ends that defined each traverse line are called traverse stations or traverse points. The framework formed by connected survey lines of known length and direction is called a traverse.
Image Courtesy: Braincart
In the figure-1 below, A,B, C and D are traverse stations. AB, BC, and CD are traverse lines.
In traversing, the surveyor moves from one point to another by simultaneously measuring bearings and distances by "dead reckoning". Dead reckoning is the process of calculating the current position
of some moving object by using a previously determined position. Employed when the construction work is long and narrow ( tunnel or motorway construction).
Scope and Objective of Traverse Survey
A Traverse survey is conducted to establish horizontal control in land areas, especially in areas where the line of sight (LOS) is short due to heavily built-up areas, where the survey methods
are not applicable.
The main objective of the traverse survey are:
1. To locate or establish boundaries
2. To achieve horizontal control for topographic surveys
3. To locate and prepare construction layouts for highways, railways, and other private and public works
4. To conduct ground control surveys for photogrammetric surveys
Types of Traverse
The two types of traverse encountered while conducting surveys are:
1. Open Traverse
2. Closed Traverse
1. Open Traverse
Open traverse is a traverse that starts at a point of known position and terminates at a point of unknown position. An open traverse is suitable for surveying along a narrow strip of land. For
example, it is used for surveying roads, railways, canals, rivers, coastlines, pipelines, etc.
An open traverse can run from a few hundred meters to kilometers. Figure (b) below shows an open traverse ABCDEF.
Fig.2. Types of Traverse - Open and Closed Traverse
The consistency of angles and distance measured cannot be checked in an open traverse. So, in order to minimize the errors, the distances can be measured twice, angles turned by repetition, etc.
2. Closed Traverse
Closed traverse originates at a point of known position and closes on another point of known horizontal position. A closed traverse can be a closed link traverse ( where the position of A and D is
known) or a closed loop traverse ( where the traverse starts and ends at A, whose position is known). A closed traverse is suitable for locating the boundaries of lakes, houses, lawns, and gardens
and for large areas like towns, residential campus etc.
The figure-3 shows a closed traverse ABCDE.
Closed traverse lets have a computation check that allows the detection of systematic errors in both distance and direction.
How are Traverse Lines Measured?
The traverse lines are either determined by:
1. Direct measurement: Tapes, EDM
2. Indirect measurement: Tachometric Methods
3. Angular measurement: Theodolite
Whenever there is a change in direction of the traverse, an angular measurement is taken. The traverse survey is performed by a traverse party and traverse equipment.
Traverse Party: Traverse party consists of an instrument operator, a head tape man, and a rare tape man.
Traverse Equipment: The equipment used for the traverse survey includes: theodolites, compass, tapes, chains, tachometer, hand level, leveling staff, ranging pole, plumb bobs, EDM and reflector,
stakes and hubs, tacks, marking crayons, points, walkie talkies, & hammer, etc. The instrument used for traversing is dependent on the method of traversing employed
Different Methods of Traversing in Surveying
Traverse surveying is a control survey that involves a number of survey lines forming a framework. The survey lines and directions are measured using distance-measuring devices and angle-measuring
Usually, it is recommended to perform traversing using a theodolite and a tape to measure the angles and distances respectively. However, any combination of linear and angular measuring devices can
be used.
The different methods of traversing are mentioned below:
1. Chain Traversing
2. Free or Loose Needle Method of Traversing
3. Fast Needle Method
1. Chain Traversing
Here, the traverse lines are measured using chain and tape alone. This is a very crude method and cannot be completely relied on.
2. Chain and Compass Surveying - Free or Loose Needle Method
In this method, the magnetic bearings of the survey lines are measured using compass and the traverse lines are measured using, a tape or chain. The direction of the magnetic meridian of each
traverse line is determined independently.
3. Theodolite Surveying
Traversing conducted using theodolite can be conducted using the following methods:
3.1. Included Angle Method
3.2. Deflection Angle Method
3.3. Fast Needle Method
3.1. Included Angle Method
• Included angle method of traversing makes use of theodolite to determine the included angles.
• Hence, the method is suitable only for closed traversing.
The procedure can be explained by considering a closed traverse ABCD. The included angles that need to be determined are x, y, z, and w.
Initially, the theodolite is set-up at point A. The North direction is set using the compass in the theodolite.
• After setting the instrument at A, fore-bearing to line AB is determined (Fab) and AD (Fad) is also determined. Hence, the included angle, x = Fab-Fad.
• Now, set the instrument at B, and find the North direction. At B, determine the fore-bearing of line BC (Fbc) and back-bearing of line BA (Fba). Then the included angle, y = Fba-Fbc.
• Now, set the instrument at C, and find the North direction. At C, find the fore-bearing of line CB
Traverse Using Included Angle Method
(Fcb), and the fore-bearing of CD (Fcd). Hence, the included angle, y = 360 - ( Fcd-Fcb).
• Now, set the instrument at D, take the fore-bearing of line DA (Fda), and the back bearing of line DC (Fdc). Hence, the included angle, l = Fdc-Fda.
• The linear measurement AB,BC,CD, and DA are measured either using a tape or a chain.
Here, after determining the included angles, starting from A, it must end at A itself. To check whether we get a closed traverse, the following formula is used:
Sum of the interior angles = 180 x (n-2);
Here, 'n' is the number of traverse lines involved in the traverse. In the above examples, n = 4; Therefore, (180 x (n-2)) = (180 x (4-2)) = 360 degrees. The sum of the included angles measured using
theodolite must be equal to the 360 degrees as calculated above. If there is variation, the correction is provided. It is mainly performed using Bowditch Rule.
Fast needle method in traversing gives the direct measurement of included angle. The work can be checked during its progress and the errors can be detected and rectified immediately. Hence, the field
work become less cumbersome and the computations are simple compared to other method of measuring angles.
3.2. Fast Needle Method
In this method, the magnetic meridian is established only at the starting station. This method is used to measure the magnetic bearings and length of traverse lines.
In this method, the instrument is set at A, and the magnetic meridian of line AB (Fab) is only determined at first. We will lock this angle in the theodolite (by tightening the upper clamp of the
theodolite). Now take the instrument and place it on station B, loosen the upper clamp and sight C. Now after sighting, tighten the upper clamp and angle measured, and place the instrument at D. Now
loosen the upper clamp, sight A. Now, if the procedure is done correctly, the final angle shown on the scale will be 360 degrees.
If it does not close by 360 degrees, there exist closing error, that need checking.
3.3. Deflection Angle Traversing
• Mostly employed for open traversing
• Used for location survey of railways, pipelines, highways, etc..
The above open traverse is ABCD.
• Place instrument at A and find magnetic bearing of that line AB.
• Now set instrument at B, sight A and set the instrument reading to zero. Transit the telescope, (now the instrument is set along the direction of line AB). Now measure the deflection angle by
sighting to C.
• Now place the instrument at C, sight B and set the instrument to zero reading. Now transit the telescope and sight D. This gives the deflection angle at C.
Based on whether the angle is measured clockwise or anti-clockwise, positive and negative signs are given.
What is the Procedure to Perform Traverse Survey?
The steps involved in Traverse Survey are:
1. Reconnaissance
2. Selection of Traverse Stations
3. Linear and Angular Measurements
Step 1: Reconnaissance
Reconnaissance is defined as a preliminary field inspection of the entire area that needs to be surveyed. This involves:
1. The surveyor goes to the field and checks the entire area.
2. He decides the best plan of working.
3. He checks the intervisibility of the traverse stations
4. He decides the method of traversing to be adopted
5. Based on the method chosen, the instruments and accessories are selected accordingly.
Step 2: Selection of Traverse Stations
The basic principle followed in surveying is " working from whole to part "and it is adopted.
1. A minimum number of traverse stations should be selected.
2. Take the length of the traverse line as long as possible to reduce the time and centering effect of stations.
3. Try to select stations on a level and firm ground
4. After selecting the stations, mark them using pegs.
Step 3: Linear and Angular Measurements
The distances between the stations are measured using a tape or chain or the Tacheometric method or EDM instruments. The angular measurements are done using a compass or theodolite.
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There is zero correlation between the Fed printing and the money supply. Deal with it.
There is zero correlation between the Fed printing and the money supply. If you don’t believe this, you owe it to yourself to study up on monetary policy until you do.
This is an issue that brings them out of the bunker like no other in economics. But if you are an investor, trader or economist, understanding—and I mean really understanding, not just recycling
things you overheard on a trading desk or recall from econ 101—the mechanics of monetary policy should be at the top of your checklist. With the US, Japan, the UK and maybe soon Europe all with their
pedals to the monetary metal, more hinges on understanding this now than ever before.
And, as we saw this week, even many of the Titans of finance and economics have it wrong.
“Wrong? You’re saying they’re wrong? They have tons of money. They have long track records. I mean, they’ve seen it all. How can you say that? That’s just arrogant. Besides, did I mention they have
tons of money?”
Here’s why the Titans are wrong
Brad DeLong had an entertaining piece on whales, super whales and men who hate the Fed, but the answer is much simpler than the one he offers. In fact, if you’ve ever been in the belly of a hedge
fund, you know the answer to most everything is much simpler than it appears to the mere mortals on the outside.
The bottom line is the titans are working from the wrong playbook. We’re all, to varying degrees, slaves to our experiences. Their formative experiences, almost to a man, were in the early 80s. This
is when they built their knowledge and assembled their financial playbooks. They learned words like Milton Freidman, money multiplier, Paul Volcker, Ronald Reagan, and the superneutrality of money.
Above all, they internalized one dictum: real men have hard money.
This understanding implies that an increase in bank reserves deposited at the Fed (i.e. “printing”) eventually feeds credit growth and thereby inflationary pressures; in other words, no base money
increase, no credit growth. Only one problem: reality disagrees.
Here are the facts
From 1981 to 2006 total credit assets held by US financial institutions grew by $32.3 trillion (744%). How much do you think bank reserves at the Federal Reserve grew by over that same period? They
fell by $6.5 billion.
How is that possible? I thought in a fractional reserve system base money had to grow for credit to expand?
The answer is structural. The financial deregulation that began in the early 80s (significantly, the abolition of regulation Q) and the consequent development of repo markets fundamentally changed
the transmission mechanism of monetary policy. Collateral lending is now king. Today, length of collateral chains and haircut rates—neither of which are determined by the Fed—define the upper bounds
of the money supply, not base money and reserve requirements.
What about the relationship to inflation? Isn’t base money correlated to that? Here’s a graph, from this piece by central banking expert Peter Stella.
The X axis shows 5-yr growth rate of base money (loosely defined) and the Y axis shows annual yoy inflation. That’s right. Nobody home here, either.
Don’t confuse liquidity with credit
The Federal Reserve only provides liquidity. The amount of liquidity it puts in the reserve system has no direct impact on the issuance of credit by banks or shadow banks. Only banks and shadow banks
can create credit. And they lend either out of cash on hand or by repo-ing treasuries, mortgages, or deposits, if cash on hand is insufficient. And collateral that is pledged once can be pledged over
and over and over (collateral chain). So, even though credit increases, the total amount of banking reserves on deposit at the Fed remains unchanged (though composition across banks may change).
So if the banks and shadow banks can just as easily repo their Treasury and mortgage holdings to finance lending, and there is no link between base money and credit creation, why is the Fed doing QE
in the first place?
By keeping rates low well out the yield curve and providing comfort that the Fed will be there to fight the risk of recession and deflation, it creates an environment that enables, over time, a
normalization of risk taking in the real economy. Our revealed belief is that the Fed can chop these nastier outcomes off the left-hand side of the distribution. As a result we start feeling better
about putting our getting our money back out of the mattress and putting it back to work.
Risk taking always starts in financial markets, but eventually bleeds it way into the real economy. And, if you listen carefully, you can hear over the pitched squeals of fixed income investors, who
are suffering from sticker shock and low yields, that this is exactly what’s transpiring. The time bought with aggressive monetary policy is allowing household balance sheets to the labor market to
slowly heal. Heck, even the fiscal position is rapidly improving.
Again, it is important to underscore that it is the indirect psychological effects from Fed support and the low cost of capital—not the popularly imagined injection of Fed liquidity into stock
markets—that have gotten investors to mobilize their idle cash from money market accounts, increase margin, and take financial risk. It is our money, not the Fed’s, that’s driving this rally.
Ironically, if we all understood monetary policy better, the Fed’s policies would be working far less well. Thank God for small favors.
This is not a semantic point. I can hear traders saying “yeah, whatever, who cares, don’t fight the Fed, just buy”. But this concept has huge implications for the phase where the Fed decides to
remove the training wheels. If the Fed money is not directly propping up the stock market and the economy underneath has been healing, the much talked about wedge between “Fed-induced valuations” and
“the fundamentals” is likely considerably smaller than the consensus seems to think. It’s less “artificial”. In short, what all this means is the day the Fed lets up off the gas might give us a blip,
or maybe that long-awaited correction, but ultimately the Policy Bears will end up getting crushed, again.
The other, more mechanical, implication is that financial sector lending is neither nourished nor constrained by base money growth. The truth is the Fed’s monetary policy can influence only the price
at which lending transacts. The main determinant of credit growth, therefore, really just boils down to risk appetite: whether banks and shadow banks want to lend and whether others want to borrow.
Do they feel secure in their wealth and their jobs? Do they see others around them making money? Do they see other banks gaining market share?
These questions drive money growth more than the interest rate and base money. And the fact that it is less about the price of money and more about the mental state of borrowers and lenders is
something many people have a hard time wrapping their heads around–in large part because of what Econ 101 misguidedly taught us about the primacy of price, incentives and rational behavior. If you
answer the behavioral questions and ignore the endless misinformation about base money—even when it’s coming from the titans of finance—as an investor you’ll be much better off.
13 Replies to “There is zero correlation between the Fed printing and the money supply. Deal with it.”
1. Interesting article
2. The only rational behavior that is truly dependable is when given a choice between monetary discipline and juicing the economy, by whatever means, nobody picks monetary discipline.
1. What constitutes ‘monetary discipline” is subjective. And unfortunately, many are still trying to define what that means by looking in the rear view mirror. I think the best guide to what
makes sense wrt monetary policy is looking at how well the Fed is fulfilling its mandate. And on that score–whatever your subjective view of equity valuations is telling you–the Fed has been
very disciplined and pragmatic
3. “There is zero correlation between the Fed printing and the money supply.”
Is that really so? Really?
Does that match experience?
The graph shown shows rates of growth on the x and y axis, not levels. Have you graphed the levels? Please, try that and post it.
The graph given says nothing of the levels. If you graph the levels it will match experience. The money level and price levels are correlated over the long term.
Even if you do not graph the levels; look at where the points are, please. They are mostly in the first and second quadrant of the graph you gave. What does that mean?
If you drew axis at x=2 and y=4 where are those growth points. Most points would be in the first and third quadrants. But not in a line. What would that mean?
A graph of changes accentuates the quick movements and not the longer steady movements. And, a graph of levels shows longer term dynamics.
1. Log plots of levels MZM and CPI:
Time series:
2. You can’t use levels when dealing with non-stationary time series–unless of course you use cointegration techniques to effectively de-trend them. Otherwise you get spurrious correlations.
4. So you see the levels graphed above do correlate well.
But, here for the changes correlation is much less or near zero.
Here are the same graphs as above but for percent changes.
Time Series of Percent Change:
x-y percent changes:
The moral is to graph levels also.
5. This piece on vectors illustrates some of the reasoning behind what the data shown in the blogs graph and the ones I have linked to above.
Basically, economists should also look at levels, not just throw them away.
6. RE: Graph of Changes above (are akin to displacements from the origen)
If you took a step in the y direction only (North) and then later took a step in the x direction only (East) and repeated you would net move in a diagonal direction. The net x movement would
correlate with the net y movement. You would have moved North East.
Or, the level of x would correlate with the level of y.
Both the CPI level and the amount (level) of reserves at the central bank (money)
have increased over time.
7. Pingback: The Oracle Speaks - Deflation Market
8. Any half-way decent principles text will tell its readers that the growth rate of quantity of base money is but one of several determinants of broad money growth rates, the other being the real
demand for bank reserves; and any such text will also point out that broad money growth in turn is but one of two mutually exhaustive determinants (the other being the velocity of money) of an
economy’s inflation rate.
The same textbooks are also likely to point out the many policy and environmental developments that led to substantial changes in both the real demand for bank reserves and the velocity of broad
money starting in the early 80s, if not before: rising inflation rates; the appearance of money-market mutual funds; the eventual deregulation of rates on bank deposits; and changes in both the
structure and the manner of enforcing of minimum bank reserve requirements, are just the most obvious of these.
For all these reasons the claim that growth in bank reserves leads to corresponding growth in money and prices is properly understood, and has been understood by all competent economists, as a
comparative-statics proposition. Just as importantly, all the divergences of measured inflation from the rate of reserve growth are perfectly consistent with this comparative-static proposition,
once one allows for other such propositions pertaining to the determinants of velocity and the multiplier. So there’s absolutely nothing mysterious about the data pointed out here. (On the other
hand, it is very easy to show despite everything said above that broad money growth rates and inflation have been very much positively correlated, both in the U.S. across time, and across
countries, for the 1960s-2000s taken as a whole.)
What purports here to be a demolition of Econ 101 is, in short, actually nothing more than a demonstration of the fact that the author may wish to consider enrolling in that class one more time!
9. Maybe you’d like to make a wager on your statement.
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1,999 research outputs found
This note will use Hitchin's generalized geometry and a model of axionic gravity developed by Warren Siegel in the mid-nineties to show that the construction of Lagrangians based on the inner product
arising from the pairing of a vector and its dual can lead naturally to the low-energy Lagrangian of the bosonic string.Comment: Conclusions basically unchanged, but presentation streamlined
significantly. Published versio
Researchers described a new representation, the local geometry, for early visual processing which is motivated by results from biological vision. This representation is richer than is often used in
image processing. It extracts more of the local structure available at each pixel in the image by using receptive fields that can be continuously rotated and that go to third order spatial variation.
Early visual processing algorithms such as edge detectors and ridge detectors can be written in terms of various local geometries and are computationally tractable. For example, Canny's edge detector
has been implemented in terms of a local geometry of order two, and a ridge detector in terms of a local geometry of order three. The edge detector in local geometry was applied to synthetic and real
images and it was shown using simple interpolation schemes that sufficient information is available to locate edges with sub-pixel accuracy (to a resolution increase of at least a factor of five).
This is reasonable even for noisy images because the local geometry fits a smooth surface - the Taylor series - to the discrete image data. Only local processing was used in the implementation so it
can readily be implemented on parallel mesh machines such as the MPP. Researchers expect that other early visual algorithms, such as region growing, inflection point detection, and segmentation can
also be implemented in terms of the local geometry and will provide sufficiently rich and robust representations for subsequent visual processing
Particulate flows have been largely studied under the simplifying assumptions of one-way coupling regime where the disperse phase do not react-back on the carrier fluid. In the context of turbulent
flows, many non trivial phenomena such as small scales particles clustering or preferential spatial accumulation have been explained and understood. A more complete view of multiphase flows can be
gained calling into play two-way coupling effects, i.e. by accounting for the inter-phase momentum exchange between the carrier and the suspended phase, certainly relevant at increasing mass loading.
In such regime, partially investigated in the past by the so-called Particle In Cell (PIC) method, much is still to be learned about the dynamics of the disperse phase and the ensuing alteration of
the carrier flow. In this paper we present a new methodology rigorously designed to capture the inter-phase momentum exchange for particles smaller than the smallest hydrodynamical scale, e.g. the
Kolmogorov scale in a turbulent flow. In fact, the momentum coupling mechanism exploits the unsteady Stokes flow around a small rigid sphere where the transient disturbance produced by each particle
is evaluated in a closed form. The particles are described as lumped, point masses which would lead to the appearance of singularities. A rigorous regularization procedure is conceived to extract the
physically relevant interactions between particles and fluid which avoids any "ah hoc" assumption. The approach is suited for high efficiency implementation on massively parallel machines since the
transient disturbance produced by the particles is strongly localized in space around the actual particle position. As will be shown, hundred thousands particles can therefore be handled at an
affordable computational cost as demonstrated by a preliminary application to a particle laden turbulent shear flow.Comment: Submitted to Journal of Fluid Mechanics, 56 pages, 15 figure
We study non-radial oscillations of neutron stars with superfluid baryons, in a general relativistic framework, including finite temperature effects. Using a perturbative approach, we derive the
equations describing stellar oscillations, which we solve by numerical integration, employing different models of nucleon superfluidity, and determining frequencies and gravitational damping times of
the quasi-normal modes. As expected by previous results, we find two classes of modes, associated to superfluid and non-superfluid degrees of freedom, respectively. We study the temperature
dependence of the modes, finding that at specific values of the temperature, the frequencies of the two classes of quasi-normal modes show avoided crossings, and their damping times become
comparable. We also show that, when the temperature is not close to the avoided crossings, the frequencies of the modes can be accurately computed by neglecting the coupling between normal and
superfluid degrees of freedom. Our results have potential implications on the gravitational wave emission from neutron stars.Comment: 16 pages, 7 figures, 2 table
We analyze damping of oscillations of general relativistic superfluid neutron stars. To this aim we extend the method of decoupling of superfluid and normal oscillation modes first suggested in
[Gusakov & Kantor PRD 83, 081304(R) (2011)]. All calculations are made self-consistently within the finite temperature superfluid hydrodynamics. The general analytic formulas are derived for damping
times due to the shear and bulk viscosities. These formulas describe both normal and superfluid neutron stars and are valid for oscillation modes of arbitrary multipolarity. We show that: (i) use of
the ordinary one-fluid hydrodynamics is a good approximation, for most of the stellar temperatures, if one is interested in calculation of the damping times of normal f-modes; (ii) for radial and
p-modes such an approximation is poor; (iii) the temperature dependence of damping times undergoes a set of rapid changes associated with resonance coupling of neighboring oscillation modes. The
latter effect can substantially accelerate viscous damping of normal modes in certain stages of neutron-star thermal evolution.Comment: 25 pages, 9 figures, 1 table, accepted for publication in MNRA
The Exact Regularized Point Particle method (ERPP), which is a new inter-phase momentum coupling ap- proach, is extensively used for the first time to explore the response of homogeneous shear
turbulence in presence of different particle populations. Particle suspensions with different Stokes number and/or mass loading are considered. Particles with Kolmogorov Stokes number of order one
suppress turbulent kinetic energy when the mass loading is increased. In contrast, heavier particles leave this observable almost un- changed with respect to the reference uncoupled case. Turbulence
modulation is found to be anisotropic, leaving the streamwise velocity fluctuations less affected by unitary Stokes number particles whilst it is increased by heavier particles. The analysis of the
energy spectra shows that the turbulence modulation occurs throughout the entire range of resolved scales leading to non-trivial augmentation/depletion of the energy content among the different
velocity components at different length-scales. In this regard, the ERPP approach is able to provide convergent statistics up to the smallest dissipative scales of the flow, giving the opportunity to
trust the ensuing results. Indeed, a substantial modification of the turbu- lent fluctuations at the smallest-scales, i.e. at the level of the velocity gradients, is observed due to the particle
backreaction. Small scale anisotropies are enhanced and fluctuations show a greater level of in- termittency as measured by the probability distribution function of the longitudinal velocity
increments and by the corresponding flatness
Double field theory was developed by theoretical physicists as a way to encompass $T$-duality. In this paper, we express the basic notions of the theory in differential-geometric invariant terms, in
the framework of para-Kaehler manifolds. We define metric algebroids, which are vector bundles with a bracket of cross sections that has the same metric compatibility property as a Courant bracket.
We show that a double field gives rise to two canonical connections, whose scalar curvatures can be integrated to obtain actions. Finally, in analogy with Dirac structures, we define and study
para-Dirac structures on double manifolds.Comment: The paper will appear in J. Math. Phys., 201
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cptsvx.f - Linux Manuals (3)
cptsvx.f (3) - Linux Manuals
cptsvx.f -
subroutine cptsvx (FACT, N, NRHS, D, E, DF, EF, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, RWORK, INFO)
CPTSVX computes the solution to system of linear equations A * X = B for PT matrices
Function/Subroutine Documentation
subroutine cptsvx (characterFACT, integerN, integerNRHS, real, dimension( * )D, complex, dimension( * )E, real, dimension( * )DF, complex, dimension( * )EF, complex, dimension( ldb, * )B, integerLDB,
complex, dimension( ldx, * )X, integerLDX, realRCOND, real, dimension( * )FERR, real, dimension( * )BERR, complex, dimension( * )WORK, real, dimension( * )RWORK, integerINFO)
CPTSVX computes the solution to system of linear equations A * X = B for PT matrices
CPTSVX uses the factorization A = L*D*L**H to compute the solution
to a complex system of linear equations A*X = B, where A is an
N-by-N Hermitian positive definite tridiagonal matrix and X and B
are N-by-NRHS matrices.
Error bounds on the solution and a condition estimate are also
The following steps are performed:
1. If FACT = 'N', the matrix A is factored as A = L*D*L**H, where L
is a unit lower bidiagonal matrix and D is diagonal. The
factorization can also be regarded as having the form
A = U**H*D*U.
2. If the leading i-by-i principal minor is not positive definite,
then the routine returns with INFO = i. Otherwise, the factored
form of A is used to estimate the condition number of the matrix
A. If the reciprocal of the condition number is less than machine
precision, INFO = N+1 is returned as a warning, but the routine
still goes on to solve for X and compute error bounds as
described below.
3. The system of equations is solved for X using the factored form
of A.
4. Iterative refinement is applied to improve the computed solution
matrix and calculate error bounds and backward error estimates
for it.
FACT is CHARACTER*1
Specifies whether or not the factored form of the matrix
A is supplied on entry.
= 'F': On entry, DF and EF contain the factored form of A.
D, E, DF, and EF will not be modified.
= 'N': The matrix A will be copied to DF and EF and
N is INTEGER
The order of the matrix A. N >= 0.
NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrices B and X. NRHS >= 0.
D is REAL array, dimension (N)
The n diagonal elements of the tridiagonal matrix A.
E is COMPLEX array, dimension (N-1)
The (n-1) subdiagonal elements of the tridiagonal matrix A.
DF is REAL array, dimension (N)
If FACT = 'F', then DF is an input argument and on entry
contains the n diagonal elements of the diagonal matrix D
from the L*D*L**H factorization of A.
If FACT = 'N', then DF is an output argument and on exit
contains the n diagonal elements of the diagonal matrix D
from the L*D*L**H factorization of A.
EF is COMPLEX array, dimension (N-1)
If FACT = 'F', then EF is an input argument and on entry
contains the (n-1) subdiagonal elements of the unit
bidiagonal factor L from the L*D*L**H factorization of A.
If FACT = 'N', then EF is an output argument and on exit
contains the (n-1) subdiagonal elements of the unit
bidiagonal factor L from the L*D*L**H factorization of A.
B is COMPLEX array, dimension (LDB,NRHS)
The N-by-NRHS right hand side matrix B.
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
X is COMPLEX array, dimension (LDX,NRHS)
If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
LDX is INTEGER
The leading dimension of the array X. LDX >= max(1,N).
RCOND is REAL
The reciprocal condition number of the matrix A. If RCOND
is less than the machine precision (in particular, if
RCOND = 0), the matrix is singular to working precision.
This condition is indicated by a return code of INFO > 0.
FERR is REAL array, dimension (NRHS)
The forward error bound for each solution vector
X(j) (the j-th column of the solution matrix X).
If XTRUE is the true solution corresponding to X(j), FERR(j)
is an estimated upper bound for the magnitude of the largest
element in (X(j) - XTRUE) divided by the magnitude of the
largest element in X(j).
BERR is REAL array, dimension (NRHS)
The componentwise relative backward error of each solution
vector X(j) (i.e., the smallest relative change in any
element of A or B that makes X(j) an exact solution).
WORK is COMPLEX array, dimension (N)
RWORK is REAL array, dimension (N)
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, and i is
<= N: the leading minor of order i of A is
not positive definite, so the factorization
could not be completed, and the solution has not
been computed. RCOND = 0 is returned.
= N+1: U is nonsingular, but RCOND is less than machine
precision, meaning that the matrix is singular
to working precision. Nevertheless, the
solution and error bounds are computed because
there are a number of situations where the
computed solution can be more accurate than the
value of RCOND would suggest.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
September 2012
Definition at line 234 of file cptsvx.f.
Generated automatically by Doxygen for LAPACK from the source code.
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Visualizing Domain & Range From a Graph
I learned a nifty domain and range trick from an online workshop about using stickie notes to help “frame” the graph of a function. The idea is to use 4 notes so that all you see is the graph, which
can make identifying the domain and range a little easier.
Slide a stickie note from left to right until you “bump into” the function, and stick it on the paper. Likewise, slide a stickie from right to left, top to bottom, and bottom to top, until the graph
is framed, like so:
Now, students can see that the domain can be expressed as -5 ≤ x < 5 and the range can be expressed as 0 ≤ y < 6 (It’s easier to see on a graph whose axes are numbered a little better, but you get
the idea if you peek at the original graph above).
I like the strategy a lot – it’s tough for kids to visualize domain and range with the plethora of unusual and squiggly graphs out there. Since I’m guessing most of my students won’t walk into the
EOC with Post-Its in their pockets, I like using highlighters to color-code things a bit. Here are several work samples from students today showing their different interpretations of the strategy.
I like how they took the stickie strategy and made it more practical based on the writing tools they *will* have when taking the Algebra 1 EOC next week.
Check out this idea I found on Pinterest – another nifty way to help students visualize domain and range:
6 Responses to Visualizing Domain & Range From a Graph
1. Thank you, thank you, thank you! My Algebra 2 students really struggled with visualizing what domain and range looked like this year. I can’t wait to try this out next year!
□ Awesome! It was fun to watch my 8th graders pick this strategy right up, and even make it their own! Enjoy!
2. I think this is an awesome idea and I am trying to figure out how I might possibly be able to apply it to left and right hand limit problems in calculus. Thanks!
3. I love this idea, but I think kiddos get more confused when they have to deal with infinity and negative infinity. It’s like they don’t understand that the graph does not end when the square grid
on your paper ends. Those graphs with arrows on the end they want to stop where they see the graph end. How could you extend this highlighting technique to that concept?
□ Hi Stacey!
In eye experience, when students “slide” their sticky note or highlighter across an axis to establish when the graph “begins” or “ends”, they’re pretty good about seeing that they can’t draw
a highlighter reference or place the sticky because the graph doesn’t have a definitive starting/ending point due to the arrow(s).
4. To help my students understand about the continuous arrows on a graphed figure:
1) we first review what infinity means and identify the symbol,
2) we then write the symbol on the graph anywhere an arrow appears.
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This entry was posted in Algebra 1 and tagged Algebra, domain, function, range, strategy. Bookmark the permalink.
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Doubt with IOI'19: Split the Attractions - Codeforces
Hi! I was trying to solve that problem but it gave me WA on subtask 3. Reading the official solution, I realized that it was pretty similar to what I did, but verifying that the two components have a
size more than a. I verified that the remaining component (the one not used to assign vertices to A) had a size more than or equal to b.
Could someone tell me why the solution works? I just don't get it. Logically speaking (or writing lol), it would be impossible to assign b vertices from a subgraph of size less than b.
Update: Thinking very much, I realized why it works. Please correct me if I am wrong. If the size of the remaining component is greater or equal to a and lower than b, then we can use such component
to assign vertices to A and the "original" one to assign vertices to B, because the size of the original component is greater than (a+b+c)-b = (a+c) >= b (remember b <= c so b <= a+c).
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Jump to navigation Jump to search
Regression quantifies the relationship between an outcome variable and one or more predictor variables. It is primarily used to understand causal relationships. For example, how will sales increase
if price is dropped by $1? How is life expectancy changed by consumption of a new drug?
The above definition is very broad and encapsulates much of machine learning. However, most of the time when people refer to regression they are referring to Linear Regression.
Creating regression models in Displayr
Various models are available via Anything > Advanced Analysis > Regression.
Alternative meanings of "regression" in data science
There are two other meanings of regression within data science:
• Regression tests and regression testing refers to the process of ensuring that software gives consistent results over time. This is the main usage of regression within software engineering.
• Regression can refer to the tendency of data or behaviors to revert over time to some typical or base level (e.g., regression to the mean).
Pages in category 'Regression'
The following 20 pages are in this category, out of 20 total.
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VLOOKUP For Text | How to Use VLOOKUP For Text in Excel?
Table of Contents
Introduction to VLOOKUP For Text in Excel
While using VLOOKUP in Excel to find any particular value, it is necessary that the format of the data in both the lookup table and the lookup value is the same. Mostly, we use VLOOKUP in Excel
to search only numerical data, but if you are using both text and number formats, you can get an #N/A error.
It happens when the lookup value and the data table have different formats, i.e., one is in number, and the other is in text. Here, VLOOKUP shows a #N/A error because it cannot find a match. So, when
we use it to find text-based values, you can use VLOOKUP for text to fix the error.
Before we learn the methods to solve this error, let us first understand how this error occurs with the help of an example.
Example: #N/A Error due to Numbers Stored as Text
Here, we have two tables. Table 1 is the main table with city codes, and their specific pin codes, and Table 2 is our lookup array. Let us try to find the pin code for the city code “415930”.
• Enter the following VLOOKUP formula in cell E3.
• Press Enter and then drag the formula until E14.
When we use the
VLOOKUP Function
in Column E, it shows a “
” error. It happens because the lookup_value (Column D) is in text format, and the table_array (Columns A & B) is in numeric format.
Therefore, to avoid this error, make sure the lookup value and the table values are in the same format.
How to Use VLOOKUP for Text?
To solve this #N/A error, we must make sure that the data table and lookup values are in the same format, i.e., either in text or number format. The Excel functions to correct this error are as
1. Paste Special feature
2. VALUE function
3. TEXT function (for reverse conversion)
Example #1 – Convert Text to Numeric Values Using the Paste Special Method
The values in Column D look like numbers but are in text format. In this example, we will convert values in column D to number format by using the Paste Special method.
Step 1: Select “Cell G3” and enter number 1.
Step 2: Copy “Cell G3” and select the range from Cell “D3 to D14”, as shown below.
Step 3: Right-click the selected range and select the “Paste Special” option, as shown below. You can also use the keyboard shortcut “Ctrl +Alt +V” to open the “Paste Special” window.
Step 4: Select the “All” option under “Paste” and “Multiply” under the “Operation” section.
Note: The “Multiply” option from the paste special window multiplies the value 1 (the one we copied) with the values in column D. This way, by multiplying a number to the text format, we convert
column D into a numeric format.
Step 5: Click on the OK button.
The format of Column D will change from text to number after applying the “Paste Special” method. Therefore, the VLOOKUP Function will give the correct match instead of the #N/A error.
Example #2 – Convert Text to Numeric Values Using the VALUE Function
We use the VALUE() function within the VLOOKUP function to directly change the format of a text-based cell into a numeric value.
Let us take the same data as Example #1.
Step 1: Select “Cell E3” and add the VALUE Function within the VLOOKUP function as follows:
Step 2: Press “Enter”.
The formula will display “U_362” in Cell E3.
Step 3: Drag the cell downwards to copy the formula in cell E3 to E14.
The formula will now display the actual pincode instead of a #N/A error.
The VALUE function changed the text form of lookup_value (Column D) into the numeric form so that the VLOOKUP function could fetch accurate results.
Example #3 – Convert Numeric to Text Values Using the TEXT Function
In all the above examples, the lookup_value (Column D) was in text format, and the table_array (Columns A & B) was in a numeric format. Hence, we changed the format for the lookup_value (Column D)
from text to number.
Now, let us see how to change the data format from numeric to text format.
Consider the same data from the above example. But this time, your original data, i.e., Column A, is in text format, and the lookup_value (Column D) is in numeric format. So, we have to use the TEXT
function to convert the lookup_value from the number format into text format.
Let us change the data formatting by simply adding the TEXT Function.
Step 1: Select “Cell E3” and enter the formula:
Step 2: Press “Enter”.
The formula displays “U_362” in “Cell E3”.
Step 3: Drag “Cell E3” downwards. The formula will display the accurate result throughout Column E.
Here, the TEXT function has first changed the lookup_value (D3) number format into text format.
Frequently Asked Questions (FAQs)
Q1. Does VLOOKUP work with alphanumeric data?
Answer: Yes, VLOOKUP can handle alphanumeric data in Excel. It’s a function that finds a specific value from a huge data table. It doesn’t matter if the values are a mix of letters and numbers. Just
make sure the data formatting is consistent.
Q2. Why won’t VLOOKUP work with text?
Answer: VLOOKUP for text can give an error in the following situations:
• If data is present in both text and number format.
• When the value you are looking for does not match the values in the original data.
• If there are spelling mistakes or the data is in the wrong order.
• If the table is not in ascending order before applying the function.
• If there are extra spaces in the original data or lookup data.
Recommended Articles
This EDUCBA article explains how to use VLOOKUP for text using various Excel functions. Here, we have mentioned methods that you can use to solve the #N/A error using practical examples and an Excel
template. Read the following articles to learn more.
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CourseNana | COMP4500/7500 Advanced Algorithms and Data Structures - Assignment 2
This question has been solved
Advanced Algorithms and Data Structures
Assignment 2
Due at 3pm, Wednesday 19th of October 2022.
This assignment is worth 20% (COMP4500) or 15% (COMP7500) of your final grade.
This assignment is to be attempted individually. It aims to test your understanding of dynamic program- ming. Please read this entire handout before attempting any of the questions.
Submission. Answers to each of the written (not programming) questions (i.e. Q1(b), Q1(d), Q1(e)) should be clearly labeled and included in a pdf file called A2.pdf.
You need to submit (i) your written answers in A2.pdf, as well as (ii) your source code files Recursive.java and Dynamic.java electronically using Blackboard according to the exact instructions on
the Blackboard website: https://learn.uq.edu.au/
You can submit your assignment multiple times before the assignment deadline but only the last submission will be saved by the system and marked. Only submit the files listed above. You are
responsible for ensuring that you have submitted the files that you intended to submit in the way that we have requested them. You will be marked on the files that you submitted and not on those that
you intended to submit. Only files that are submitted according to the instructions on Blackboard will be marked - incorrect submissions will receive 0 marks.
Submitted work should be neat, legible and simple to understand – you may be penalised for work that is untidy or difficult to read and comprehend.
For the programming part, you will be penalised for submitting files that are not compatible with the assignment requirements. In particular, code that is submitted with compilation errors, or is not
compatible with the supplied testing framework will receive 0 marks.
Late submission. See section 5.3 of the course profile for details. If the assignment is submitted after the deadline, without an approved extension, a late penalty will apply. The late penalty shall
be 10% of the maximum possible mark for the assessment item will be deducted per calendar day (or part thereof), up to a maximum of seven (7) days. After seven days, no marks will be awarded for the
item. A day is considered to be a 24 hour block from the assessment item due time. Negative marks will not be awarded.
If there are medical or exceptional circumstances that will affect your ability to complete an assignment by the due date, then you can apply for an extension as per Section 5.3 of the electronic
course profile (ECP). Extensions to assignments must be requested via my.UQ (https://my.uq.edu.au/). You can find instructions on how to submit your request online (https://my.uq.edu.au/
information-and-services/ manage-my-program/exams-and-assessment/applying-extension). Your extension application must be submitted on or before the assessment item’s due date and time.
School Policy on Student Misconduct. You are required to read and understand the School Statement on Misconduct, available at the School’s website at: http://www.itee.uq.edu.au/
itee-student-misconduct-including-plagiarism This is an individual assignment. If you are found guilty of misconduct (plagiarism or collusion) then penalties will be applied.
COMP4500/7500 Assignment 2 (September 30, 2022) 2 Question 1 (100 marks total)
You are in charge of managing a venue for k consecutive days from day 0 to day k − 1 (inclusive), where k ≥ 1.
There are n different configurations, c0, c1, . . . cn−1, that the venue can be in, where n ≥ 1. The venue must be in exactly one of the n different configurations at any one time, and is in
configuration c0 on day 0. The cost of having the venue in a configuration c for day d is given by c.cost(d), where c.cost(d) ≥ 0. The cost of a configuration can be different on different days (e.g.
if d ̸= d′ then c.cost(d) does not necessarily equal c.cost(d′)), and different configurations may have different costs.
Each configuration c has a set-up time, c.setupTime(), and a tear-down time, c.teardownTime(), such that c.setupTime() ≥ 1 and c.teardownTime() ≥ 1. It is possible to change the venue from its
current config- uration cold to any different configuration cnew, however the reconfiguration takes cold.teardownTime() + cnew.setupTime() whole consecutive days to complete. For the first cold
.teardownTime() of those days the venue is still in configuration cold, and for the last cnew.setupTime() of those days, the venue is in the new configuration cnew. Once a reconfiguration is started
it must be completed without interruption, and it must be completed before day k (e.g. the last day of any reconfiguration must be less than or equal to day k − 1). Additionally, the venue cannot be
used to host any bookings for the duration of the reconfiguration. Other than this, there are no limits on the number of times the venue can be reconfigured. (The configuration of the venue can only
be changed by reconfiguring it in the way described above.)
In order to earn money, the venue can host events.
The payment received for hosting the event booking b depends on the configuration of the venue. The payment that would be received for hosting event booking b in a configuration c is given by
b.payment(c) where b.payment(c) ≥ 0.
In summary there are three different kinds of activities that can be taking place at the venue on any one of the k days you are in charge of it: either it can be idle in its current configuration,
being reconfigured from one configuration to another one, or it can be hosting an event (for a booking request) in its current configuration.
A schedule for the venue, assigns each of the k days that you are in charge of the venue to an activity. In a schedule, the activities must be assigned in such a way as to respect the constraints
described above (e.g. the venue is in configuration c0 on day 0; once a reconfiguration is commenced it must be completed without interruption before day k; if the event from a booking request is
hosted, then it must be hosted for all of the whole days specified in the booking request, etc.).
The profit of a schedule is the sum of the payments received by hosting the events in the schedule, minus the configuration costs for each of the k days.
Your task is to find a schedule with the maximum profit. (That is, you want to work out how to manage the venue so that you can get the best possible profit.)
COMP4500/7500 Assignment 2 (September 30, 2022) 3 As an example, consider the scenario where k = 11, there are n = 2 configurations:
c0: (setup=1, teardown=1, cost=[1, 0, 2, 1, 0, 0, 1, 1, 1, 5, 0]) c1: (setup=2, teardown=1, cost=[0, 6, 3, 1, 1, 1, 1, 1, 2, 0, 8])
and m = 4 booking requests:
b0: (start=0, end=1, payment= (c0 7→ 4), (c1 7→ 3) ) b1: (start=0, end=0, payment= (c0 7→ 2), (c1 7→ 7) ) b2: (start=3, end=3, payment= (c0 7→ 2), (c1 7→ 5) ) b3: (start=4, end=6, payment= (c0 7→ 3),
(c1 7→ 12))
A schedule with the maximum profit is:
day 0: HOSTING b1 in configuration c0 day 1: RECONFIGURING c0 to c1
day 2: RECONFIGURING c0 to c1
day 3: RECONFIGURING c0 to c1
day 4: HOSTING b3 in configuration c1 day 5: HOSTING b3 in configuration c1 day 6: HOSTING b3 in configuration c1 day 7: IDLE in configuration c1
day 8: IDLE in configuration c1
day 9: RECONFIGURING c1 to c0 day 10: RECONFIGURING c1 to c0
in which: (i) event booking b1 is hosted in c0 from day 0 to day 0; (ii) the venue is reconfigured from c0 to c1 from day 1 to 3; (iii) the event booking b3 is hosted in c1 from day 4 to day 6; (iv)
the venue is idle in c1 for day 7; (v) the venue is idle in c1 for day 8; and finally (vi) the venue is reconfigured from c1 to c0 from day 9 to day 10.
In this schedule the payments received from hosting the events in the schedule is b1.payment(c0) + b3.payment(c1) = 2 + 12 = 14
and the configuration costs for each of the k days are
(1 + 0) + (3 + 1 + 1 + 1 + 1 + 1 + 2 + 0) + (0) = 11
since the venue is in c0 on days 0 and 1 (inclusive); in c1 from days 2 to 9 (inclusive); and in c0 again on day 10. This means that the schedule has a profit of 14 − 11 = 3, which is the maximum
profit of any schedule.
Note that there are many other possible schedules, but none of them have a profit which is greater than 3. For example, another possible schedule is:
day 0: HOSTING b0 in configuration c0 day 1: HOSTING b0 in configuration c0 day 2: IDLE in configuration c0
day 3: HOSTING b2 in configuration c0 day 4: HOSTING b3 in configuration c0 day 5: HOSTING b3 in configuration c0 day 6: HOSTING b3 in configuration c0 day 7: IDLE in configuration c0
day 8: IDLE in configuration c0 day 9: IDLE in configuration c0 day 10: IDLE in configuration c0
COMP4500/7500 Assignment 2 (September 30, 2022) 4 however the payments received from hosting the events in this schedule is
b0.payment(c0) + b2.payment(c0) + b3.payment(c0) = 4 + 2 + 3 = 9 and the configuration costs for each of the k days are (since the venue is always in c0):
1 + 0 + 2 + 1 + 0 + 0 + 1 + 1 + 1 + 5 + 0 = 12
and so this schedule has a profit of 9 − 12 = −3, which is less than the maximum possible profit of 3.
[Note: In the zip file that accompanies this handout you will find the Activity class in the assignment2 package. If you need clarification of what an activity is, please refer to this class. In the
assignment2.test package you will also find a method checkSchedule in the DynamicTest class, which, for testing purposes, is used to check that a schedule is valid with respect to the algorithm
inputs, and calculates the profit of the schedule. Except for testing purposes, you should not be using method checkSchedule yourself, but it may help you to refer to it if you are having trouble
understanding the problem.]
1. (20 marks) Your first task is to identify the optimal substructure of the problem. You must implement the public static method optimalProfitRecursive from the Recursive class in the assignment2
package that is available in the zip file that accompanies this handout, to provide a naive recursive algorithm to determine the maximum profit of any schedule.
The recursive solution does NOT need to return a schedule that has the maximum profit – it just needs to return the maximum profit. Efficiency is not a great concern for this part (the
inefficiency will be expected to come from recomputing solutions to overlapping subproblems), so focus on an elegant solution that identifies the optimal substructure of the problem. (You must
not provide a dynamic programming solution to this question.)
2. (15 marks) It is expected that your recursive algorithm will not be polynomial-time in the worst case. For the case where the number of days you are managing the venue for is k, the number of
configurations is n, and the number of booking requests is m, give an asymptotic lower bound on the worst-case time complexity of your recursive algorithm in terms of parameters k, n and m, or a
relevant subset of those parameters. Make your bound as tight as possible. (We would like you to be able to show that your recursive algorithm, in the worst case, has a time complexity that is
worse than polynomial-time.)
[Make your answer as concise as possible – it should be no more than a page using minimum 11pt font. Longer answers will not be marked.]
3. (30 marks) Develop an efficient bottom-up dynamic programming solution to the problem (not mem- oised) by implementing the public static method optimalProfitDynamic in the Dynamic class from the
assignment2 package that accompanies this handout.
Your dynamic programming solution should run in polynomial time (in terms of k, n and m), and it should be as efficient as possible.
The dynamic solution does NOT need to return a schedule that would result in the maximum profit – it just needs to return the maximum profit.
COMP4500/7500 Assignment 2 (September 30, 2022) 5
4. (10 marks) Provide an asymptotic upper bound on the worst-case time complexity of your dynamic programming solution for part (c) in terms of the parameters k (the number of days), n (the number
of configurations) and m (the number of booking requests), or an appropriate subset of those parameters. Make your bounds as tight as possible and justify your solution.
[Make your answer as concise as possible – it should be no more than half a page using minimum 11pt font. Longer answers will not be marked.]
5. (5 marks) Provide an asymptotic upper bound on the worst-case space complexity of your dynamic programming solution for part (c) in terms of the parameters k (the number of days), n (the number
of configurations) and m (the number of booking requests), or an appropriate subset of those parameters. Make your bounds as tight as possible and justify your solution.
[Make your answer as concise as possible – it should be no more than half a page using minimum 11pt font. Longer answers will not be marked.]
6. (20marks)Extendyourbottom-updynamicprogrammingsolutionfrompart(c)tocalculateaschedule with the maximum profit, by implementing the public static method optimalScheduleDynamic in the Dynamic class
from the assignment2 package.
Like method optimalProfitDynamic, your implementation of this method should run in polynomial time (in terms of k, n and m), and it should be as efficient as possible. It must be a bottom-up
dynamic programming (not memoised) solution.
Do not change the class name of the Recursive or Dynamic classes or the package to which those files belong. You many not change the signatures of the methods that you have to implement in any way or
alter their specifications. (That means that you cannot change the method name, parameter types, return types or exceptions thrown by the those methods.) Do not modify any of the other classes or
interfaces or enumerated types defined in package assignment2.
You are encouraged to use Java 8 SE API classes, but no third party libraries should be used. (It is not necessary, and makes marking hard.) Don’t write any code that is operating-system specific
(e.g. by hard- coding in newline characters etc.), since we will batch test your code on a Unix machine. Your source file should be written using ASCII characters only.
The zip file for the assignment also some junit4 test classes to help you get started with testing your code. The JUnit4 test classes as provided in the package assignment2.test are not intended to
be an exhaustive test for your code. Part of your task will be to expand on these tests to ensure that your code behaves as required.
Your programming implementations will be tested by executing our own set of junit test cases. Code that is submitted with compilation errors, or is not compatible with the supplied testing framework
will receive 0 marks. A Java 8 compiler will be used to compile and test the code. The Recursive class will be tested in isolation from the Dynamic class.
Implementations that do not satisfy the assignment requirements will receive 0 marks even if they pass some of the test cases (e.g. if the solution given to Q1(c) is not an efficient bottom-up
dynamic programming solution, then it will receive 0 marks.)
You may lose marks for poorly structured, poorly documented or hard to comprehend code, or code that is
COMP4500/7500 Assignment 2 (September 30, 2022) 6
not compatible with the assignment requirements. Line length should be less than or equal to 100 characters so that it can be printed – please use spaces to indent your code instead of tabs to ensure
compatability with different machines. Don’t leave print statements in your submitted code.
Evaluation Criteria
Question 1
• Question 1 (a) (20 marks)
Given that your implementation satisfies the requirements of the question (i.e. the method must be implemented using a naive recursive programming solution that identifies the optimal substructure of
the problem), your implementation will be evaluated for correctness by executing our own set of junit test cases.
20 : All of our tests pass
16 : at least 80% of our tests pass 12 : at least 60% of our tests pass
8 : at least 40% of our tests pass
4 : at least 20% of our tests pass
0 : less than 20% of our test pass or work with little or no academic merit
Note: Code that is submitted with compilation errors, or is not compatible with the supplied testing framework will receive 0 marks. A Java 8 compiler will be used to compile and test the code.
Implementations that do not satisfy the assignment requirements will receive 0 marks even if they pass some of the test cases.
The Recursive class will be tested in isolation from the Dynamic class.
• Question 1 (b) (15 marks)
For this part of the question, the analysis should be no more than one page using minimum 11pt font. Longer solutions will receive 0 marks. Also, if a plausible, neat, legible and simple to
understand solution to Q1(a) has not been given, this question will receive 0 marks. Otherwise the following marking criteria applies.
15 : A correct asymptotic lower bound on the worst-case time complexity the recursive algorithm from Q1(a) is given in terms of the parameters specified in the question. The lower bound, which should
be exponential, should be reasonably tight for the algorithm at hand. The time- complexity given should be clearly justified by giving, justifying and solving a correct (lower bound) recurrence
derived from your algorithm. Any assumptions made in the analysis are reasonable and clearly stated. Asymptotic notation should be used correctly and the asymptotic time complexity given has been
simplified to remove lower order terms and unnecessary constant factors.
11 : A very good attempt has been made to give an asymptotic lower bound on the worst-case time complexity the recursive algorithm from Q1(a) is given in terms of the parameters specified in the
question. The lower bound should be exponential. The answer and justification may contain at most one or two minor mistakes or omissions. The time-complexity given should be mostly clearly justified
by giving, justifying and solving a (lower bound) recurrence derived from your algorithm. Any assumptions made in the analysis are mostly reasonable and clearly stated.
COMP4500/7500 Assignment 2 (September 30, 2022) 7
7 : A reasonable attempt has been made to give a tight asymptotic lower bound on the worst-case time complexity of the recursive algorithm from Q1(a) in terms of the parameters specified in the
question, and to justify it with respect to a recurrence derived from the algorithm, however the analysis or justification may contain minor mistakes or omissions or lack clarity.
3 : An attempt has been made to both give an asymptotic lower bound on the worst-case time complexity of the recursive algorithm from Q1(a) in terms of the parameters specified in the question, and
to justify it in terms of a recurrence derived from your algorithm, however it contains either a major mistake or many mistakes, gives an unreasonably loose lower bound, or is not clearly justified
by giving, justifying and solving a correct (lower bound) recurrence derived from your algorithm.
0 : Work with little or no academic merit.
• Question 1 (c) (30 marks)
Given that your implementation satisfies the requirements of the question (i.e. it is a efficient bottom- up dynamic programming (not memoised) solution that runs in polynomial time in terms of k, n
and m), your implementation will be evaluated for correctness and efficiency by executing our own set of junit test cases.
30 : All of our tests pass
24 : at least 80% of our tests pass 18 : at least 60% of our tests pass 12 : at least 40% of our tests pass
6 : at least 20% of our tests pass
0 : less than 20% of our test pass or work with little or no academic merit
Note: Code that is submitted with compilation errors, or is not compatible with the supplied testing framework will receive 0 marks. A Java 8 compiler will be used to compile and test the code.
Implementations that do not satisfy the assignment requirements will receive 0 marks even if they pass some of the test cases.
The Dynamic class will be tested in isolation from the Recursive class.
• Question 1 (d) (10 marks)
For this part of the question, the analysis should be no more than 1/2 of a page using minimum 11pt font. Longer solutions will receive 0 marks. Also, if a plausible, neat, legible and simple to
understand solution to Q1(c) has not been given, this question will receive 0 marks. Otherwise the following marking criteria applies.
10 : A correct asymptotic upper bound on the worst-case time complexity of the algorithm from Q1(c) is given in terms of the parameters specified in the question. The upper bound, which should be
polynomial in the parameters specified in the question, should be as tight as reasonably possible for the algorithm at hand. The time-complexity given should be clearly justified with respect to the
algorithm. Any assumptions made in the analysis are reasonable and clearly stated. Asymptotic notation should be used correctly and the asymptotic time complexity given has been simplified to remove
lower order terms and unnecessary constant factors.
7 : A very good attempt has been made to give an asymptotic upper bound on the worst-case time complexity the algorithm from Q1(c) in terms of the parameters specified in the question. The upper
bound should be polynomial in terms of those parameters. The answer and justification
COMP4500/7500 Assignment 2 (September 30, 2022) 8
may contain at most one or two minor mistakes or omissions. The time-complexity given should be mostly clearly justified with respect to the algorithm. Any assumptions made in the analysis are mostly
reasonable and clearly stated.
5 : A reasonable attempt has been made to give a tight asymptotic upper bound on the worst-case time complexity of the algorithm from Q1(c) in terms of the parameters specified in the question, and
to justify it, however the analysis or justification may contain minor mistakes or omissions or lack clarity.
2 : An attempt has been made to give an asymptotic upper bound on the worst-case time complexity of the algorithm from Q1(c) in terms of the parameters specified in the question, and justify it,
however it contains either a major mistake or many mistakes, gives an unreasonably loose upper bound, or is not clearly justified.
0 : Work with little or no academic merit.
• Question 1 (e) (5 marks)
For this part of the question, the analysis should be no more than 1/2 of a page using minimum 11pt font. Longer solutions will receive 0 marks. Also, if a plausible, neat, legible and simple to
understand solution to Q1(c) has not been given, this question will receive 0 marks. Otherwise the following marking criteria applies.
5 : A correct asymptotic upper bound on the worst-case space complexity of the algorithm from Q1(c) is given in terms of the parameters specified in the question. The upper bound, which should be
polynomial in the parameters specified in the question, should be as tight as reasonably possible for the algorithm at hand. The space-complexity given should be clearly justified with respect to the
algorithm. Any assumptions made in the analysis are reasonable and clearly stated. Asymptotic notation should be used correctly and the asymptotic space complexity given has been simplified to remove
lower order terms and unnecessary constant factors.
4 : A very good attempt has been made to give an asymptotic upper bound on the worst-case space complexity the algorithm from Q1(c) in terms of the parameters specified in the question. The upper
bound should be polynomial in terms of those parameters. The answer and justification may contain at most one or two minor mistakes or omissions. The space-complexity given should be mostly clearly
justified with respect to the algorithm. Any assumptions made in the analysis are mostly reasonable and clearly stated.
2 : A reasonable attempt has been made to give a tight asymptotic upper bound on the worst-case space complexity of the algorithm from Q1(c) in terms of the parameters specified in the question, and
to justify it, however the analysis or justification may contain minor mistakes or omissions or lack clarity.
1 : An attempt has been made to give an asymptotic upper bound on the worst-case space com- plexity of the algorithm from Q1(c) in terms of the parameters specified in the question, and justify it,
however it contains either a major mistake or many mistakes, gives an unreasonably loose upper bound, or is not clearly justified.
0 : Work with little or no academic merit.
• Question 1 (f) (20 marks)
Given that your implementation satisfies the requirements of the question (i.e. it is an efficient bottom- up dynamic programming (not memoised) solution that runs in polynomial time in terms of k, n
and m), your implementation will be evaluated for correctness and efficiency by executing our own set of junit test cases.
20 : All of our tests pass
COMP4500/7500 Assignment 2 (September 30, 2022) 9
16 : at least 80% of our tests pass 12 : at least 60% of our tests pass 8 : at least 40% of our tests pass 4 : at least 20% of our tests pass
0 : less than 20% of our test pass or work with little or no academic merit
Note: Code that is submitted with compilation errors, or is not compatible with the supplied testing framework will receive 0 marks. A Java 8 compiler will be used to compile and test the code.
Implementations that do not satisfy the assignment requirements will receive 0 marks even if they pass some of the test cases.
The Dynamic class will be tested in isolation from the Recursive class.
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Micro hydropower Input-output economic benefit analysis - DLLP Power-Hydroelectric Equipment Solutions
1. Input estimates for each technology model
Micro hydropower construction investment varies greatly depending on the natural conditions. In general, the independent operation of the micro-hydro power supply system per kilowatt of construction
investment in l million yuan, the use of funds to allocate the proportion of the relationship is: equipment (including units and metal components) accounted for 30%, hydraulic construction accounted
for 40%, wire erection accounted for 30%. If it is some demanding demonstration projects, all by the regular engineering and technology team construction, plus the work of technical experts, the user
inside the wire installation labor costs, etc., the cost will reach about 20,000 yuan per kilowatt.
Grid-connected micro-hydro power station, eliminating the need for transmission lines, the construction investment per kilowatt is about 7,000 yuan. If it is “electricity for irrigation” project,
most of the hydraulic construction by the farmland water conservancy construction, micro hydropower station construction input can be reduced to about 4,000 yuan per kilowatt.
2. Analysis of the benefits of each technology model
(1) independent operation of power supply micro-hydropower. Independent operation of micro hydropower annual power generation time calculated by 9 months should be 6,570 hours, that is, the annual
power generation capacity of each kilowatt installed capacity of 6570 kilowatt hours, effective electricity rate of 40%, calculated at 0.5 yuan per kilowatt hour, the annual power generation income
should be 1314 yuan. In the case of not calculating the management and maintenance expenses, the payback period is 7.6 years. In general, the payback time should be more than 10 years.
The social benefit is that each kilowatt of installed power can solve the problem of 4 households (12-15 people), but there are still two or three months of the year without water and electricity,
and other power sources are needed.
The energy benefit is 263 kWh of electricity per kilowatt per year for the installed machine.
(2) “Electricity for irrigation” grid-connected operation of micro-hydropower. A simple illustration is given as an example of a small type II reservoir in Yudian Township, Jinzhai County, Liuan
City, Anhui Province. The local power generation households to get 0.34 yuan per kilowatt hour of online electricity prices. This 0.34 yuan, according to the national small hydropower tax policy to
pay 6% tax, each kilowatt hour should be 0.02 yuan; to the Water Resources Bureau water resources use fee of 0.01 yuan per kilowatt hour; the relevant government departments to charge management fees
per kilowatt hour 0.03 yuan; power companies need to pay back the loan for the construction of the power grid for each kilowatt hour to charge 0.03 yuan for the construction of the power grid loan;
the actual income of power generation households for each kilowatt hour 0.25 yuan. Yuan.
The small type II reservoir in Yaodian Township irrigates 33 hectares of paddy fields and has a 100-kilowatt micro-hydro power station. The construction cost of the micro-hydro power station (at
current prices) is estimated at 400,000 yuan. According to the power generation households, the micro-hydro power station has an annual power generation time, which translates into 4,000 hours of
full-load power generation time. The annual revenue from the sale of the reservoir is 136,000 yuan, of which: 100,000 yuan of revenue from the power generation households; the state receives 8,000
yuan of tax revenue; the power company receives 1.2 million yuan of loan repayment fees for the construction of the network, in addition to the local power supply price of 0.56 yuan / kWh, the power
company has 0.22 yuan / kWh, that is, 8.8 million yuan of the difference between the purchase and sale; the Water Resources Bureau to collect 4,000 yuan of water resources use fees; relevant
government departments The management fee is RMB 1.2 million.
Social benefits: The power producer acts as the main body of farmland water conservation and undertakes the daily management and maintenance of reservoirs and canals. Since the irrigation charge of
33 hectares of paddy fields is only about 7,500 yuan if calculated at 225 yuan per hectare per year, which only accounts for a fraction of 13.6 million yuan of power generation revenue, the
irrigation charge for farmers is completely exempted.
Energy benefits: 400,000 kWh of electricity is obtained each year, improving the local rural electricity shortage.
Environmental benefits: A small type II reservoir has functioned properly and stabilized the water flow in the downstream section of the river. Due to the improvement of power supply, local farmers
generally use electric rice cookers, microwave ovens and other cooking utensils, reducing the amount of fuel wood.
3. Suitable promotion analysis From the analysis of the flow of products sold by micro hydropower equipment manufacturers in recent years, three types of user groups exist for micro hydropower.
The first category is the households without electricity in mountainous areas, using micro hydropower to solve the problem of living electricity for their own families. With the vigorous promotion of
China’s rural power grid transformation, the trend of this type of users is declining year by year, there are still about 20 million people without electricity in China, mostly living in the deep
mountains and forests, the promotion of micro-hydro technology is difficult and costly. In the international market, the proportion of people without electricity in developing countries is much
larger than in China, with a total population of about 1.6 billion, and the sales of micro-hydropower equipment in the international market have been on the rise in recent years. 2006, Dali Lida
Energy Institute of Practical Technology sold more than 2 million yuan of micro-hydropower equipment, with exports accounting for 90% of the total.
The second category is micro hydropower generation households that aim to increase their income. Most of these people are rural electricians or plumbers, with a certain professional knowledge base
and management experience, but lack of capital. Can only rely on the local water conservancy facilities or old station transformation, engage in micro-hydro power generation with little investment,
its installed capacity between 5 to 500 kilowatts. Revenue from the sale of electricity varies from a few thousand to several hundred thousand yuan per year. This is a group in great need of policy
support and technical guidance.
The third category is the long-term continuous drainage of water to the outside of the enterprise for hydroelectric power recovery. Such as sewage treatment plants, high terrain water plants, etc..
Through self-generation and self-consumption can save hundreds of thousands of yuan of electricity expenditure each year. These users are fewer, but because their affordability is stronger than
farmers, they need high-end equipment, which has a considerable drag on the technical development of the industry.
According to the statistics of the Ministry of Agriculture and the Ministry of Water Resources in recent years, the installed capacity of micro-hydro power in China is close to 8 million kilowatts,
of which about 200,000 micro-hydro power units below 10 kilowatts, with a total installed capacity of 220,000 kilowatts; 19,545 micro-hydro power stations of 10-100 kilowatts, with 21,620 units and a
total installed capacity of 690,610,000 kilowatts; 100 There are 19,107 micro-hydropower stations of 100-500 kilowatts, with 38,652 units and a total installed capacity of 7.18 million kilowatts, of
which the majority of those below 30 kilowatts are off-grid independent power supply, while most of those above 50 kilowatts are grid-connected.
If calculated according to the rural per capita possession of electricity 200 watts (Ministry of Water Resources rural electrification standards), the rural population involved in micro-hydropower
has now reached more than 40 million people with electricity.
(1)独立运行供电的微水电。独立运行微水电的年发电时间按9个月计算应为6 570小时，即每千瓦装机容量的年发电能力为6570千瓦时，有效用电率40％，以每千瓦时0．5元计算，则年发电收入应为1314元。在不计算管理和
根据农业部和水利部最近几年的统计，我国微水电的装机容量已经接近了800万千瓦，其中10千瓦以下微水电机组约20万台，总装机容量22万千瓦；10—100千瓦的微水电站19545座，机组21 620台，总装机容量69．61万千瓦
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How to Find Standard Deviation: A Step-by-Step Guide
🔍 Introduction
Are you struggling to find the standard deviation of your data? Don’t worry; you’re not alone. It’s a common problem that many people face. Standard deviation is a statistical measure that helps you
understand how much the data is spread out from the mean. It’s an essential concept in statistics and plays a vital role in many fields, including finance, engineering, and physics.
In this article, we’ll guide you through the step-by-step process of finding standard deviation. We’ll explain the concept and its importance, how to calculate it by hand, and the formulas used in
Excel. By the end of this guide, you’ll have a clear understanding of standard deviation and its calculations, and you’ll be ready to use it in your work.
👋 Greeting the Audience
Hello, and welcome to our guide on how to find standard deviation. Whether you’re a student, researcher, or anyone who needs to understand and analyze data, this guide will be a valuable resource for
you. We’ll cover all the basics and provide you with the knowledge and tools you need to calculate standard deviation with ease.
📚 Understanding Standard Deviation
Before we dive into calculations, let’s first understand what standard deviation is and why it’s important. Standard deviation is a measure of how much the data is spread out from the mean, or
average value. It gives you an idea of how much the data points deviate from the central value. Simply put, it tells you how much the data varies.
Standard deviation is used to describe the distribution of data in a set or population. It’s an essential concept in statistics because it allows you to compare different datasets and draw
conclusions from them. For example, you can use it to compare the performance of two different investment portfolios or the effectiveness of two different medications.
🔢 How to Calculate Standard Deviation by Hand
The formula for calculating standard deviation by hand is:
Step Action Formula
1 Find the mean of the data x̄ = (∑x) / n
2 Subtract the mean from each data point x – x̄
3 Square the differences (x – x̄)²
4 Find the sum of the squared differences ∑(x – x̄)²
5 Divide the sum by the number of data points minus one s² = ∑(x – x̄)² / (n – 1)
6 Find the square root of the result from step 5 s = √(∑(x – x̄)² / (n – 1))
Let’s break down this formula step by step.
Step 1: Find the Mean
The first step is to find the mean, or average value, of the data. To do this, you add up all the data points and divide by the number of data points.
Step 2: Subtract the Mean
The second step is to subtract the mean from each data point. This gives you the deviation of each data point from the mean.
Step 3: Square the Differences
The third step is to square each deviation. This step is necessary because the deviations can be positive or negative, and we want to eliminate the negative signs before we calculate the average
Step 4: Find the Sum
The fourth step is to find the sum of the squared deviations. This sum is a measure of the total spread of the data.
Step 5: Divide by n-1
The fifth step is to divide the sum of squared deviations by the number of data points minus one. This step is necessary because we’re dealing with a sample of data rather than the entire population.
Dividing by n-1 instead of n gives us an unbiased estimate of the population variance.
Step 6: Find the Square Root
The sixth and final step is to find the square root of the result from step 5. This gives us the standard deviation of the data.
🖥️ Using Excel to Calculate Standard Deviation
If you have a large dataset or want to save time, you can use Excel to calculate standard deviation. Excel has built-in functions that can do the calculations for you. The two most commonly used
functions are:
• =STDEV(range) – This function calculates the standard deviation of a sample.
• =STDEVP(range) – This function calculates the standard deviation of an entire population.
The range argument is the set of data you want to calculate the standard deviation for. You can enter it manually or select the cells containing the data.
🙋 Frequently Asked Questions
1. What is the difference between standard deviation and variance?
Standard deviation and variance are both measures of the spread of data. However, they measure different things. Standard deviation is the square root of variance. Variance is calculated by finding
the average of the squared differences from the mean. Standard deviation is easier to interpret because it’s in the same units as the original data.
2. What does a high standard deviation mean?
A high standard deviation means the data points are spread out over a wide range. This can be due to large individual differences or a lack of consistency in the data. It may indicate that the data
is not reliable or that there are outliers.
3. What does a low standard deviation mean?
A low standard deviation means the data points are closely clustered around the mean. This indicates that the data is consistent and reliable. However, it may also indicate that the data is too
narrow to draw meaningful conclusions.
4. What is a good standard deviation?
There is no fixed value for what constitutes a good standard deviation. It depends on the context of the data and the purpose of the analysis. In some cases, a high standard deviation may be
desirable, while in others, a low one may be preferred.
5. What is a standard deviation in probability?
In probability theory, standard deviation is a measure of the amount of variability or spread of a random variable. It’s a common measure used to describe the distribution of probability in a set of
6. Does standard deviation measure the average distance from the mean?
No, standard deviation measures the spread of the data around the mean. It tells you how much the data points deviate from the mean.
7. What is a population standard deviation?
Population standard deviation is the measure of the variability of a population. It’s the square root of the variance of the entire population. It’s used when the entire population is available for
8. How do you calculate standard deviation in Python?
You can use the NumPy library in Python to calculate standard deviation. The function to use is numpy.std().
9. What is the difference between standard deviation and range?
Standard deviation and range are both measures of the spread of data. However, range only gives you the difference between the largest and smallest values, while standard deviation takes into account
all the data points and their deviations from the mean.
10. Can you have negative standard deviation?
No, standard deviation is always a positive value. It’s the square root of the variance, which is also always positive.
11. What does standard deviation tell you about the data?
Standard deviation tells you how much the data points deviate from the mean. It gives you an idea of how much the data varies and how spread out it is. It’s a measure of the dispersion or spread of
the data around the central value.
12. What is a sample standard deviation?
Sample standard deviation is the measure of the variability of a sample. It’s the square root of the variance of the sample. It’s used when only a subset of the data is available for analysis.
13. What is the difference between standard deviation and error?
Standard deviation is a measure of the spread of data, while error is a measure of the difference between an estimated value and the true value. Standard deviation is a statistical concept, while
error is a concept used in experimental design and data analysis.
🎉 Conclusion
Congratulations! You’ve reached the end of our guide on how to find standard deviation. We hope you found this guide helpful and informative. Standard deviation is an essential concept in statistics,
and it’s crucial to understand how to calculate it. We’ve provided you with a step-by-step guide for both manual and Excel calculations.
Remember that standard deviation is just one of many statistical measures, and it’s essential to choose the right measure for your analysis. Don’t be afraid to seek help or advice if you’re unsure
about anything.
👉 Take Action
Now that you have a good understanding of how to find standard deviation, it’s time to put it into practice. Choose a dataset and try calculating the standard deviation by hand and in Excel. Compare
the results and see how they differ.
📜 Closing/Disclaimer
We hope you found our guide on how to find standard deviation helpful. While every effort has been made to ensure the accuracy of the information presented in this article, we make no guarantees or
warranties of any kind, express or implied, about the completeness, accuracy, reliability, suitability or availability with respect to the content.
The information presented in this article is intended for educational and informational purposes only. It should not be used as a substitute for professional advice or judgment. We are not liable for
any damages whatsoever arising from the use or inability to use this information.
Video:How to Find Standard Deviation: A Step-by-Step Guide
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Understanding Calculation Group Precedence - SQLBI
It is possible to define multiple calculation groups in one same data model. Moreover, it is possible to apply multiple calculation items to the same measure. Even though each calculation group can
only have one active calculation item, the presence of multiple calculation groups can activate multiple calculation items at the same time. This happens when a user uses multiple slicers over
different calculation groups, or when a CALCULATE function filters calculation items in different calculation groups. For example, in the first article about calculation groups we defined two
calculation groups: one to define the base measure and the other to define the time intelligence calculation to apply to the base measure. We obtained the following result, where the user selects
both the time intelligence calculation and the base measure for the report.
If there are multiple calculation items active in the current filter context, it is important to define which calculation item is applied first, by defining a set of precedence rules. DAX enforces
this by making it mandatory to set the Precedence property in a calculation group, in models that have more than one calculation group. This article describes how to correctly set the Precedence
property of a calculation group by showing several examples where the definition of the precedence changes the result of the calculations.
To prepare the demonstration, we created two different calculation groups, each one containing only one calculation item:
-- Calculation Group: 'Time Intelligence'[Time calc]
-- Calculation Item: YTD
CALCULATE (
SELECTEDMEASURE (),
DATESYTD ( 'Date'[Date] )
-- Calculation Group: 'Averages'[Averages]
-- Calculation Item: Daily AVG
DIVIDE (
SELECTEDMEASURE (),
COUNTROWS ( 'Date' )
YTD is a regular year-to-date calculation, whereas Daily AVG computes the daily average by dividing the selected measure by the number of days in the filter context. Based on the two calculation
items, we defined two measures:
YTD :=
[Sales Amount],
'Time Aggregation'[Time calc] = "YTD"
Daily AVG :=
[Sales Amount],
'Averages'[Averages] = "Daily AVG"
Both measures work just fine, as you can see in the following report.
The scenario suddenly becomes more complex when both calculation items are used at the same time. Look at the following Daily YTD AVG measure definition:
Daily YTD AVG :=
[Sales Amount],
'Time Intelligence'[Time calc] = "YTD",
'Averages'[Averages] = "Daily AVG"
The measure invokes both calculation items at the same time, but this raises the issue of precedence. Should the engine apply YTD first and Daily AVG later, or the other way around? In other words,
which of these two expressions should be evaluated?
-- DIVIDE (Daily AVG) is applied first, and then YTD
DIVIDE (
CALCULATE (
[Sales Amount],
DATESYTD ( 'Date'[Date] )
COUNTROWS ( 'Date' )
-- YTD is applied first, and then DIVIDE (Daily AVG)
DIVIDE (
[Sales Amount],
COUNTROWS ( 'Date' )
DATESYTD ( 'Date'[Date] )
It is likely that the second expression is the correct one. Nevertheless, without further information, DAX cannot choose between the two. Therefore, you must define the correct order of application
of the calculation groups.
The order of application depends on the Precedence property in the two calculation groups: The calculation group with the highest value is applied first; then the other calculation groups are applied
according to their Precedence value in a descending order. For example, you produce the wrong result with the following settings:
• Time Intelligence calculation group – Precedence: 0
• Averages calculation group – Precedence: 10
The value of the Daily YTD AVG is clearly wrong in all the months displayed but January. Let us analyze what happened in more depth. Averages has a precedence of 10; therefore, it is applied first.
The application of the Daily AVG calculation item leads to this expression corresponding to the Daily YTD AVG measure reference:
DIVIDE (
[Sales Amount],
COUNTROWS ( 'Date' )
'Time Intelligence'[Time calc] = "YTD"
At this point, DAX activates the YTD calculation item from the Time Intelligence calculation group. The application of YTD rewrites the only measure reference in the formula, which is Sales Amount.
Therefore, the final code corresponding to the Daily YTD AVG measure becomes the following:
DIVIDE (
CALCULATE (
[Sales Amount],
DATESYTD ( 'Date'[Date] )
COUNTROWS ( 'Date' )
Consequently, the number shown is obtained by dividing the Sales Amount measure computed using the YTD calculation item, by the number of days in the displayed month. For example, the value shown in
December is obtained by dividing 9,353,814,87 (YTD of Sales Amount) by 31 (the number of days in December). The number should be much lower because the YTD variation should be applied to both the
numerator and the denominator of the DIVIDE function used in the Daily AVG calculation item.
To solve the issue, the YTD calculation item must be applied before Daily AVG. This way, the transformation of the filter context for the Date column occurs before the evaluation of COUNTROWS over
the Date table. In order to obtain this, we modify the Precedence property of the Time Intelligence calculation group to 20, obtaining the following settings:
• Time Intelligence calculation group – Precedence: 20
• Averages calculation group – Precedence: 10
Using these settings, the Daily YTD AVG measure returns the correct values.
This time, the two application steps are the following: DAX first applies the YTD calculation from the Time Intelligence calculation group, changing the expression to the following:
CALCULATE (
[Sales Amount],
DATESYTD ( 'Date'[Date] )
'Averages'[Averages] = "Daily AVG"
Then, DAX applies the Daily AVG calculation item from the Averages calculation group, replacing the measure reference with the DIVIDE function and obtaining the following expression:
DIVIDE (
[Sales Amount],
COUNTROWS ( 'Date' )
DATESYTD ( 'Date'[Date] )
The value displayed in December now considers 365 days in the denominator of DIVIDE, thus obtaining the correct number. Before moving further, please consider that, in this example, we followed the
best practice of using calculation items with a single measure. Indeed, the first call comes from the visual of Power BI. However, one of the two calculation items rewrote the Sales Amount measure in
such a way that the problem arose. In this scenario, following the best practices is not enough. It is mandatory that you understand and define the precedence of application of calculation groups
very well.
All calculation items in a calculation group share the same precedence. It is impossible to define different precedence values for different calculation items within the same group.
The Precedence property is an integer value assigned to a calculation group. A higher value means a higher precedence of application; the calculation group with the higher precedence is applied
first. In other words, DAX applies the calculation groups according to their Precedence value sorted in a descending order. The absolute value assigned to Precedence does not mean anything. What
matters is how it compares with the Precedence of other calculation groups. There cannot be two calculation groups in a model with the same Precedence.
Because assigning different Precedence values to multiple calculation groups is mandatory, you must pay attention when making this choice at the model design stage.. Choosing the right Precedence
upfront is important because changing the Precedence of a calculation group might affect the existing reports of a model already deployed in production. When you have multiple calculation groups in a
model, you should always spend time verifying that the results of the calculations are the results expected with any combination of calculation items. The chances of making mistakes in the definition
of the precedence values is quite high without proper testing and validation.
Calculation items and calculation groups are extremely powerful. At the same time, they are quite complex if used in non-trivial scenarios. Specifically, using multiple calculation groups in the same
model requires you to define the precedence of application of calculation items. Using the wrong setting for the precedence produces incorrect results which, moreover, are extremely hard to test and
If you need to use multiple calculation groups in the same model, then spend time learning and understanding the details of this article before moving further with the development. It is time well
spent, as it is going to save you a lot of debug time later, when the model is in production.
Context transition
Evaluates an expression in a context modified by filters.
CALCULATE ( <Expression> [, <Filter> [, <Filter> [, … ] ] ] )
Safe Divide function with ability to handle divide by zero case.
DIVIDE ( <Numerator>, <Denominator> [, <AlternateResult>] )
Articles in the Calculation Groups series
• Understanding Calculation Group Precedence
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What Is The Importance Of Mathematics In Our Lives?
What is the importance of mathematics in our lives?
One of the most important fields from which we can benefit in our daily activities and lives is mathematics. Who among us does not use basic mathematical operations during business dealings like
buying and selling? In this article, we highlight the value of mathematics in our lives, and we do so by examining the topics and facets of life that mathematics helps us with.
What is mathematics?
The name “mathematics” comes from the Greek and means “tending to learn,” referring to the study of numbers, measurement, and space. Due to its importance, mathematics has drawn the attention of
scientists, and as mathematics is the foundation of modern science, it is constantly evolving today. While the future will be established on the past and present together,
our present day is based on its past and the discoveries that took place during it. It is seen as an abstract science that has changed over time as a result of human development. In order to
understand quantitative numerical and engineering links and their importance, mathematics is concerned with thinking processes. such as coming to understand the importance of parallelism. In addition
to providing one way to explain a lot of what goes on in people’s lives, perpendicularity also plays a significant role in our lives.
The importance of mathematics in our lives
There are many things that can be benefited from through mathematics, and we learn about these things in the following paragraphs. Mathematics or the important mathematical operations that we need on
a daily basis are not the only things that mathematics is important for
• Mathematics improves our ability to think clearly and sharpens our logic, which helps us in a variety of other ways.
• Our kids learn different skills in mathematics that help them be more intelligent every day.
• enhances our capacity for sound and logical thinking in daily life.
• By resolving mathematical puzzles and connecting these mathematical processes to things we encounter in daily life, such as purchasing and selling, mathematics aids with analysis and logical
• Math helps us to be more logical and intelligent in our explanations.
• Numerous options exist in mathematics to enter and study different professions.
• We utilize it to figure out money and amounts, comprehend fractions, and use ratios in our daily lives.
The importance of mathematics in business administration
Mathematics is used in fields other than the sciences that are taught in schools and colleges. Rather, the majority of daily life and business require mathematics. Multiple responsibilities for
various vocations demand a basic comprehension of mathematics and, in other circumstances, in-depth mathematical knowledge. Additionally, mathematics plays a significant role in business, such as
when it comes to keeping track of company activities through the use of complicated computations like matrix algebra, calculus, and linear programming. Payroll calculations, current accounts, price
breaks, profit margins, and price decreases are typical examples of practical uses.
Mathematics is more than just numbers and the mathematical processes that are taught in schools. Rather, it goes beyond this and tremendously helps people manage their businesses. Along with its
contribution to financial analysis and creating budget plans before beginning any activity, mathematical operations are also the foundation for insurances, wagers on real estate, and tax operations.
or a venture.
Why do we learn mathematics?
Some students find it challenging to learn arithmetic, and they talk about it with their teachers while posing the following query: How do I comprehend math and why do we study it in the first place?
Their inquiries can be addressed by pointing out how mathematics connects with and relates to a wide range of other academic disciplines in a way that aids in understanding them.
The use of mathematics in the creation of artistic works, such as oriental carpets, tiles, mosaics, and Gothic cathedrals, as well as the assistance it gave to Cubist art and abstract expression
through mathematical geometric shapes to create their artistic works, made the connection between mathematics and the arts obvious thousands of years ago.
As technology, machines, computers, and map design depend on its principles and genetics is founded on the field of mathematics statistics, mathematics plays an important role in natural sciences.
Some of the ideas found in nature are ones that originated in mathematics, such as consistency and symmetry, which are utilized, for example, to look at the phenomenon of changing seasons.
Mathematics had to be used to determine the time of year, night, and day.
Math is very helpful to authors. For instance, when discussing poetry, we discover that they use it to order poetic tone and divide verses into equal numbers of words and components. In terms of
writing, we learn that the logical thinking developed through mathematics helps in the logical representation of meanings. Additionally, it helps literature students in understanding how long
activities will take to complete.
How to become good at mathematics
There are many kinds of techniques that may be used to improve one’s mathematics abilities, some of which can also be applied to other topics, such as the following:
• Going to classes: Because the subject is cumulative, it is important to attend every class in order to understand the one behind that.
• Go to class on time: You shouldn’t be late for the class because the first section of the lesson frequently contains a review of several key concepts.
• Listen carefully during the explanation: When the teacher is explaining something, pay close attention so you understand all the points since sometimes the teacher does not write what he says.
• Ask questions: if you experience any difficulties with comprehension or application. It is possible to determine whether a question’s content and its response have been understood by others by
paying attention to their questions, or by determining whether a repeat of the response is necessary.
Ways to use mathematics in real life
Here are Some beneficial uses of mathematics in daily life, both directly and indirectly:
1. shopping: If you enter into a clothes store and notice a 20% off sign, you must use mathematics to figure out the item’s pricing. Whether you are interested in calculating the percentage of
discounts, the worth of shopping coupons, or the price of the goods in the store, you will undoubtedly recognize the importance of mathematics.
2. Currency exchange: Whether you are traveling to a foreign country or living in an area where there are multiple currencies in use, converting currencies needs mental calculations. If you know the
conversion rates, math gives you the power to quickly calculate and convert any currency to another, so you can find out how much US$100 is worth in Euros or how many dollars are worth £500.
3. Cooking: Cooking can also benefit from mathematics, as preparing food requires calculations to figure out the precise amount of ingredients to add in order to know the correct amounts for
recipes. If you are cooking for a large group, you should double the recipe’s serving size because the amount of food served should correspond to the number of persons at the table.
In conclusion, mathematics is fundamental to our existence and impacts on a wide range of topics, from the basic to the deep. It is an important instrument for making decisions, solving problems, and
understanding the environment. Mathematics is an integral aspect of our daily lives, whether it be for handling personal finances, pursuing jobs in science and technology, or just admiring the beauty
of symmetry and patterns.
Why is mathematics important in our daily lives?
When trying to solve problems, make decisions, and handle tasks like budgeting, cooking, and remodeling a house, mathematics plays an important role in everyday life.
How does mathematics contribute to technological advancements?
The language of science and technology is mathematics. It gives engineers and scientists the resources they need to design and create anything, from smartphones to spacecraft.
What are some practical applications of mathematics in everyday life?
Calculating finances, measuring elements in recipes, figuring out travel distances, and understanding interest rates for loans or investments are just a few examples of practical uses.
How does mathematics enhance financial literacy?
The basis of financial literacy is mathematics. It enables people to efficiently manage their finances, comprehend investment possibilities, and make wise financial decisions.
Can studying mathematics improve critical thinking skills?
Yes, studying mathematics improves your analytical and critical thinking abilities. It imparts the capacity for thorough information evaluation as well as logical reasoning.
Why is mathematics considered a universal language?
Because it is independent of culture or language, mathematics is a universal language. Global understanding and use of mathematical concepts and symbols makes it easier to communicate and work
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RUME Seminar
RUME Seminar
Fall 2024
December 5 Ryan Gueli
10:30am (CST) University of Oklahoma Critical Visualizations of Calculus 1 Student Success
November 21 James White
10:30am (CST) University of Oklahoma Actor-Oriented Transfer of Self-Efficacy
November 14 Steven Boyce
10:30am (CST) Portland State University Calculus Students' Conceptual Structures
October 31 Milos Savic
10:30am (CDT) University of Oklahoma Shifts in Math Teaching Beliefs and Values through a Focus on Creativity: The Case of Jo Parker
October 24 Kaitlyn Serbin
10:30am (CDT) U. of Texas Rio Grande Valley Secondary Teachers’ Guided Reinvention of Unique Factorization Domains with Connections to Teaching
October 17 Cameron Byerley
10:30am (CDT) U. of Georgia POSTPONED UNTIL NOV 14
October 10 Paul Dawkins & Dov Zazkis
10:30am (CDT) Texas State U. & Arizona State U. Leveraging Reading Psychology Methodology to Study Reading of Mathematical Proof
October 3 Jessica Gehrtz
10:30am (CDT) U. of Texas at San Antonio How are undergraduate STEM instructors leveraging student thinking in their teaching?
September 26 Zack Reed, Michael Tallman & Michael Oehrtman
10:30am (CDT) Embry-Riddle Aeronautical U, Oklahoma State U. & Oklahoma State U. Assessing Productive Meanings in Calculus
September 19 George Kuster, Sarah Hartman & Nick Fortune
10:30am (CDT) Christopher Newport U., Middle Tennessee State U. & Western Kentucky U. Lesson Planning Practices of Undergraduate Mathematics Instructors: What Do We Know?
September 12 Brian Rickard
10:30am (CDT) U. of Arkansas Quantitative Research Methods in RUME Literature
September 5 Megan Ryals & Morgan Sellers
10:30am (CDT) U. of Virginia & Colorado Mesa U. Qualitative Methods in Analyzing Students' Logical Reasoning in Probability
August 29
10:00am (CDT) Deborah Moore-Russo SIGMAA RUME Annual Business Meeting
Spring 2024
April 26
12:30pm Jessi Lajos An Updated Conceptualization of the Intuition Construct for Mathematics Education Research
(CDT) Utah State University
Room 430
April 19
12:30pm Cory Wilson A Review of Literature on Student Conceptions and Understandings of Equivalence
(CDT) Oklahoma State University
Room 430
April 12
12:30pm April Richardson Students’ reasoning about equivalence with functions and non-functions in Abstract Algebra
(CDT) Oklahoma State University
Room 430
March 29
12:30pm Stepan Paul (North Carolina State University), Dusty Grundmeier (The Ohio State University), & 3D Manipulatives in Integral Calculus: Student Achievement and Confidence in Solids of
(CDT) Deborah Moore-Russo (University of Oklahoma) Revolution Tasks
March 15
12:30pm Tomoya Tatsuno Teachers’ Mistakes as an Instructional Tool: Mistake Game
(CDT) University of Oklahoma
Room 430
March 8
12:30pm Lesson Planning Practices of Undergraduate Mathematics Instructors: What Do We Know?
Room 430
March 1
12:30pm RUME Conference Reflections
16 Jeff Meyer , Sepideh Stewart, Avery Madden Teaching Proofs in a Second Linear Algebra Course: A Mathematician’s Resources,
12:30pm CSU San Bernardino & OU Orientations, Goals, and Continual Decision Makings
9 Paul Regier, Ashley Berger, Allison Dorko
12:30pm USAO, OU, OSU Instructor and Coordinator Perspectives within First-Year Mathematics Courses
2 Ryan Peffer Using CAS to Promote Students’ Ways of Thinking Through Observation and Conjectures: The
12:30pm Washington State University Case of Eigenvalues and Eigenvectors
Room 430
26 Anna Mikulo & Caleb Judkins
12:30pm University of Oklahoma
19 Milos Savic
12:30pm University of Oklahoma Teaching and RUME with (or without) Generative AI
Room 430
Fall 2023
December 7
10:30am Jonathan Troup Visualizing Triple Integral Bounds via Embodied and Symbolic Reasoning in Virtual Reality
(CST) California State University Bakersfield (CSUB)
Room 430
November 30
10:30am Lucas Yong Overview of Sinclair & Gol Tabaghi’s Paper: Drawing space: mathematicians’ kinetic conceptions of
(CST) University of Oklahoma eigenvectors
PHSC 430
November 16
10:30am Ryan Peffer Incorporating Digital Interactive Figures: Facilitating Student Exploration Into Properties of Eigenvalues
(CST) Washington State University and Eigenvectors
November 9
10:30am Caleb Judkins
(CST) University of Oklahoma
PHSC 430
November 2
10:30am Anna Mikulo
(CDT) University of Oklahoma
Room 430
October 26
10:30am Paul Dawkins A research-based approach to teaching logic and proof techniques for undergraduate introduction to proofs
(CDT) Texas State University
October 19 Richard Velasco
10:30am University of Oklahoma, Department of Instructional Leadership and Academic Reimagining and Rehumanizing Mathematics for a Math Teacher Education Program
(CDT) Curriculum
Room 430
October 12
10:30am Maria Meehan Video recordings to complement, or substitute for, the first-year mathematics lecture: One lecturer’s journey
(CDT) University College Dublin, Ireland
Room 430
September 28
10:30am Jeff Meyer Productive and Impactful Meanings in Linear Algebra
(CDT) California State University, San Bernardino
Room 430
September 21
10:30am Jessi Lajos
(CDT) Utah State University
September 14
10:30am Sepideh Stewart Collaboration within the mathematics community
(CDT) University of Oklahoma
Room 430
September 7
10:30am Milos Savic An analysis of data analysis: An example of a RUME qualitative study
(CDT) University of Oklahoma
August 31
10:30am Deborah Moore-Russo
(CDT) University of Oklahoma
Room 430
August 24
10:30am Introduction to Math Education Seminars- Fall 2023
Room 430
Spring 2023
May 4
2:30pm Lucas Yong Overview of Zandieh et al.'s Paper: Exploring everyday examples to explain basis: Insights into student
(CDT) University of Oklahoma understanding from students in Germany
April 27
2:30pm Mae Glock Review of the paper "Analyzing the Nature of University Students’ Difficulties with Algebra in Calculus:
(CDT) University of Oklahoma Students’ Voices during Problem Solving"
April 20 Heather Johnson, Gary Olsen, Belin Tsinnajinnie, Livvia Bechtold
2:30pm University of Colorado-Denver, University of Colorado-Denver, West Ed, University Boundary Transitions Within, Across, and Beyond a Set of Digital Resources: Brokering in College Algebra
(CDT) of Colorado-Denver
April 13 Antonio Estevan Martinez, Jessica Gehrtz, Chris Rasmussen, Talia LaTona-Tequida,
2:30pm Kristen Vroom Course Coordinator Orientations Toward their Work and Opportunities for Professional Development
(CDT) San Diego State, University of Texas-San Antonio, San Diego State, San Diego
Zoom State, Oregon State
April 6 Sean Yee, Jessica Deshler, Kimberly Cervello Rogers, Robert Petrulis, Christopher
2:30pm Potvin, James Sweeney Bridging the gap between observation protocols and formative feedback
(CDT) University of South Carolina, West Virginia University, Bowling Green, EPRE
Zoom Consulting, Michigan State, Coker College
March 30
2:30pm Alisa Ediger Overview of Marzocchi & Soto's Paper: From the front lines of active learning: Lessons learned from those
(CDT) University of Oklahoma who are trying
March 23
2:30pm Sajal Halder Overview of Martsching's Paper: Students’ Mathematical Reasoning With and About Representations
(CDT) University of Oklahoma
March 9
2:30pm Allison Dorko and John Paul Cook A Case Study of Why Three Students Learned from Homework Instead of Lecture
(CST) Oklahoma State University
March 2
2:30pm Keith Gallagher & Nicole Engelke Infante The Role of Visual Representations in Identifying Key Ideas: A Case Study from Topology
(CST) University of Nebraska-Omaha
16 Rachel Funk, Karina Uhing, Molly Williams A Snapshot of Three Studies: Conceptualizing and Connecting Active Learning (AL) and Equitable and
2:30pm University of Nebraska-Lincoln, University of Nebraska-Omaha, Murray State Inclusive Teaching (EIT) in Undergraduate Mathematics
(CST) University
9 Sepideh Stewart
2:30pm University of Oklahoma Qualitative Research Methods in Mathematics Education
2 Milos Savic
2:30pm University of Oklahoma Discussion of Stinson (2020) and Reflection on Worldviews
26 Deborah Moore-Russo
2:30pm University of Oklahoma Recognizing the Rhythm of Writing and Presenting Research in Undergraduate Mathematics Education
Fall 2022
December 8 Milos Savic and Cory Wilson
1:30pm (CST) University of Oklahoma Comparing Student and Instructor Perspectives of Teaching Actions to Foster Creativity
November 17 Sajal Halder
1:30pm (CST) University of Oklahoma Counterexamples and Refutations in Undergraduate Mathematics
November 3 Jessica Lajos
1:30pm (CDT) Colorado State University Planting Seeds through Embodiment to Teach Formal Concepts of Abstract Algebra
October 27 Alisa Ediger
1:30pm (CDT) University of Oklahoma Can discussion boards disrupt gendered and racialized discussion patterns in math classes?
October 20 Deborah Moore-Russo
1:30pm (CDT) University of Oklahoma A Study of the Consistency in New York State First Year Math Exams
October 13
1:30pm (CDT) Making Mathematics Meaningful for All Students: An Exploration of Self-Efficacy in Teaching Mathematics
September 29
1:30pm (CDT) A Quantitative Critical Analysis of Instructional Practices and Math Identity
September 22
1:30pm (CDT) Confronting Abstraction: An Analysis of Mathematicians’ Concept Images and Definitions
September 15
1:30pm (CDT) How Students Learn Math Best: Tutors’ Beliefs about Themselves Versus Their Tutees
September 8 Meijun Zhu
1:30pm (CDT) University of Oklahoma How to prepare for college math
September 1
1:30pm (CDT) Department's Openness to Change. A Study from Calculus Instructors' Perceptions
August 25 Sepideh Stewart
1:30pm (CDT) University of Oklahoma Research in Undergraduate Mathematics Education Seminar - Introduction
Spring 2022
April 25
2:00pm Kate Raymond Preparing Pre-Service Teachers of Mathematics to Choose Tasks
(CDT) University of Oklahoma
April 18
2:00pm Ashley Berger Lessons learned from an exponential learning module
(CDT) University of Oklahoma
April 11 Milos Savic and Houssein El Turkey
2:00pm University of Oklahoma, University of New Designing Calculus Tasks to Foster Creative Mathematical Thinking
(CDT) Haven
March 28
2:00pm Michael Oehrtman Advanced Students’ Operationalization of Quantification in Analysis
(CDT) Oklahoma State University
March 21
2:00pm Josiah Ireland Investigating the pedagogical practices of a mathematics instructor participating in an inquiry-oriented professional development initiative: An
(CDT) Oklahoma State University exploratory case study
March 7
2:00pm Deborah Moore-Russo Framing Research: A Self-Study of Theoretical Frameworks Used in Recent Months
(CST) University of Oklahoma
28 Jeffrey Meyer
2:00pm California State University, San Dynamic Visualizations in Linear Algebraic Reasoning
(CST) Bernardino
2:00pm What’s the Norm? Instructors Justify their Active Learning Moves and Decisions
2:00pm Influences on Problem Solving Practices of Emerging Mathematicians
February 7
2:00pm A Preservice Teacher’s Experience of Mathematical Research
January 31
2:00pm Preservice secondary teachers’ reasoning about static and dynamic representations of function
January 24
Fall 2021
November 17 Thembinkosi (Peter) Mkhatshwa
2:30pm (CST) Miami University Ohio Investigating Students’ Thinking about Fundamental Concepts/Topics in Calculus Through the Lens of Quantitative and Covariational Reasoning
November 10 Dr. Mollee Shultz
2:30pm (CST) Texas State University Instructional Decision-Making around Inquiry-Oriented Instructional Practices and Culturally Relevant Pedagogy
November 3 Dr. Kaitlyn Serbin
2:30pm (CDT) University of Texas - Rio Grande Valley Prospective Teachers’ Understanding of Connections Between Inverses, Identities, and Binary Operations
October 27 V. Rani Satyam
2:30pm (CDT) Virginia Commonwealth University Affect Graphing: Leveraging Graphical Representations in the Study of Students' Affect in Mathematics
October 20 Amanda Lake Heath
2:30pm (CDT) Middle Tennessee State University Collaborative Creativity in Proving: Adapting a Measurement Tool for Group Use
October 13 Micah Godfrey
2:30pm (CDT) University of Oklahoma Proof and Problem Solving: an Article Review
September 29 Antonio Martinez
2:30pm (CDT) San Diego State University Exploring Factors that Contribute to the Development of One’s Mathematical Identity
September 22 Dr. Alison Mirin
2:30pm (CDT) University of Arizona Disability Accommodations in College: Alarming Discrimination in Mathematics
September 15 Dr. Estrella Johnson
2:30pm (CDT) Virginia Tech Undergraduate Math and Science Instructor’s Attitudes, Beliefs, and Views on Diversity, Inclusion, and Equity
September 1 Miloš Savić
2:30pm (CDT) University of Oklahoma RUME - PhD, MS, and Departmental Teaching Certificate
Spring 2021
May 7 Jessica Lajos
9:00am (CDT) University of Oklahoma Abstract Algebra Students' Representational Fluency during a Collapsing Structure Task
April 30 Anthony Cronin & Sepideh Stewart
9:00am (CDT) University College Dublin & OU An Analysis of Tutors’ Responses to Linear Algebra Students’ Queries in a Mathematics Center
April 23 Ashley Berger
9:00am (CDT) University of Oklahoma The "Knowing How" of Topology
April 16 Courtney Nagle & Pat Kelly
9:00am (CDT) Penn State Behrend Self-Assessment and Reflection in Precalculus and Calculus
April 9 Nicole Infante
8:30am (CDT) West Virginia University How Gesture Facilitates Diagram Construction and Problem Solving (Note the earlier start time)
March 26 Lora Park
9:00am (CDT) University at Buffalo, SUNY (Department of Psychology) Giving Feedback to College Students in Math: Insights from the Lab to the Real-World
March 19 Aaron Weinberg
9:00am (CDT) Ithaca College Student Learning from Instructional Calculus Videos
March 12 Nick Long and Steven Jones
9:00am (CST) Stephen F. Austin University and Brigham Young University Calculus in Virtual Reality: Studying VR Resources as Lessons and Manipulatives
February 26 Rafael Martinez Planell
9:00am (CST) University of Puerto Rico at Mayaguez Student Understanding of Exponential and Logarithmic Functions
February 19 Kaki Simmons
9:00am (CST) Arizona State Sign Language Variation and Iconicity in Undergraduate Mathematics
February 12 Deborah Moore-Russo
9:00am (CST) University of Oklahoma Theoretical Framing of Digital Resource Use in Mathematics Education
Fall 2020
December 11
8:30am Jessica Lajos A Methodology that Incorporates a New Survey Instrument to Characterize Non-creative versus Creative Forms of
(CST) University of Oklahoma Intuition
November 13
8:30am Doug Corey On A Knowledge Base for Teaching Undergraduate Mathematics
(CST) Brigham Young University
November 6
8:30am Ashley Berger Portfolio Assignments in First-Year Mathematics: An alternative to exams
(CST) University of Oklahoma
October 16
8:30am Deborah Moore-Russo University Calculus Students' Understanding of Slope
(CDT) University of Oklahoma
October 9
8:30am Jeffrey Meyer Learning Through the Triad: Cryptography, Number Theory, and Programming
(CDT) California State University, San Bernardino
18 Milos Savic
8:30am University of Oklahoma Calculus Students’ Definitions of Mathematical Creativity and its Association to Power
11 Sepideh Stewart
8:30am University of Oklahoma Linear Algebra Thinking in the Embodied, Symbolic and Formal Worlds
September 4 Željka Milin Šipuš
8:30am Department of Mathematics, Faculty of Science, University of Zagreb, Students’ understanding of geometrical objects (curves and surfaces) in multidimensional analysis
(CDT) Croatia, EU
Spring 2020
April 27
12:30pm Rosaura Uscanga Lomeli REMINDER: An Investigation of Students’ Thinking about Functions in Abstract Algebra
(CDT) Oklahoma State University
April 20 Erica R. Miller, Kimberly C. Rogers, Sean P. Yee
12:30pm Virginia Commonwealth University; Bowling Green State University; Analyzing Collegiate Mathematics Observation Protocols: Attending to the Instructional Triangle and Inquiry-Based
(CDT) University of South Carolina Mathematics Education Practices
April 6
12:30pm V. Rani Satyam Affective Pathways of Undergraduate Students While Engaged in Proof Construction Tasks
(CDT) Virginia Commonwealth University
March 23
12:30pm Math Department Checking In
(CDT) University of Oklahoma
March 9
12:35pm Milos Savic Shifting Pedagogical Beliefs into Action Through Teaching for Mathematical Creativity
(CDT) University of Oklahoma
PHSC 430
24 Sepideh Stewart Linear Algebra Thinking in the Embodied, Symbolic and Formal Worlds: Students' Reasoning behind Preferring
12:35pm University of Oklahoma certain Worlds
PHSC 430
17 Ashley Berger
12:30pm University of Oklahoma Examining the Qualities of Schema in Topology
PHSC 430
10 Jessica Lajos
12:35pm University of Oklahoma A Tour of Cognitive Transformations of Semiotic Representations in Advanced Mathematical Thinking
PHSC 430
February 3
12:35pm Milos Savic Undergraduate Learning Assistants and Mathematical Discourse in Active-Learning Precalculus
(CST) University of Oklahoma
PHSC 430
January 27
12:30pm Paul Regier The Impact of Creativity-Fostering Instruction on Student Self-efficacy in Upper-level Undergraduate Mathematics
PHSC 430
Fall 2019
December 4
4:00pm (CST) Sepideh Stewart Reflecting on Mathematics Education Research on Calculus
PHSC 1105
November 20 Deborah Moore-Russo
4:00pm (CST) University of Oklahoma Introduction to the APOS-Slope Framework
PHSC 1105
November 13 Paul Regier
4:00pm (CST) University of Oklahoma How problem posing can impact student motivation: a case study
PHSC 1105
November 6
4:00pm (CST) Discussion of two papers on calculus Transition-oriented pedagogies in university calculus; Designing textbooks
PHSC 1105
October 30
4:00pm (CDT) Discussion of two papers on calculus Calculus as a discursive bridge for Algebra, Geometry and Analysis; The dual nature of reasoning in Calculus
PHSC 1105
October 23
4:00pm (CDT) Discussion of Törner and Sangwin's papers
PHSC 1105
October 16 Sepideh Stewart
4:00pm (CDT) University of Oklahoma Examining unresolved difficulties with school algebra in calculus
PHSC 1105
October 9
4:00pm (CDT) Discussion of two papers on calculus
PHSC 1105
October 2 Discussion of Chris Rasmussen et al. paper
4:00pm (CDT) University of Oklahoma Research on Calculus: what do we know and where do we need to go?
PHSC 1105
September 25 Ashley Berger
4:00pm (CDT) University of Oklahoma Examining the Qualities of Schema in Topology
PHSC 1105
September 18
4:00pm (CDT) Discussion of Wangberg and Viirman's papers Raising Calculus to the Surface; What to do when there is no formula?
PHSC 1105
September 11
4:00pm (CDT) Discussion of Monaghan and Nilsen's papers The place of limits in elementary calculus courses; Reflections on the FTC
PHSC 1105
September 4
4:00pm (CDT) Discussion of Pat Thompson's paper Making the Fundamental Theorem of Calculus Fundamental to Students' Calculus
PHSC 1105
August 28 David Tall (via video)
4:00pm (CDT) University of Warwick Making Human Sense of Calculus
PHSC 1105
Spring 2019
April 29 Ben Gochanour
3:00 - 4:00 PM (CDT) University of Oklahoma Investigating Math Motivation and Math Anxiety in Undergraduates
PHSC 430
April 22 Allison Dorko
3:00 - 4:00 PM (CDT) Oklahoma State University Red X's and Green Checks: A Preliminary Model of Student Learning from Online Homework
PHSC 430
April 15 Andrew Lutz
3:00 - 4:00 PM (CDT) University of Oklahoma Reflection on Teaching Philosophy
PHSC 430
April 8 Milos Savic
3:00 - 4:00 PM (CDT) University of Oklahoma Future of RUME seminar - Talk by Allison Dorko CANCELLED
PHSC 430
April 2 Roger Howe
3:30 - 4:30 PM (CDT) Yale University Teachers' Institutes
LL123 Bizzell Library
March 11 RUME Conference Participants
3:00 - 4:00 PM (CDT) University of Oklahoma Reflections on the RUME conference
PHSC 430
March 4 Kerstin Pettersson
3:00 - 4:00 PM (CST) Stockholm University, Sweden Small-groups teaching in university mathematics – what did the students learn?
PHSC 430
February 25 John Paul Cook
3:00 PM - 4:00 PM (CST) Oklahoma State University Monster-barring as a Catalyst for Connecting Secondary Algebra to Abstract Algebra
PHSC 430
February 18 Sepideh Stewart
3:00-4:00 (CST) University of Oklahoma Examining unresolved difficulties with school algebra in calculus
PHSC 430
February 4 Katherine (Kaki) Simmons
3:00 PM - 4:00 PM (CST) University of Oklahoma Deaf and Hard of Hearing Students’ Perspectives on Undergraduate Mathematics Experience
PHSC 430
January 28 Paul Regier
3:00 PM (CST) University of Oklahoma Discussion of Problem Posing and Self-Determination Theory
PHSC 430
January 14 Milos Savic
3:00 - 4:00 PM (CST) University of Oklahoma The 3 dimensions of fostering creativity in the classroom
PHSC 1105
Fall 2018
November 19 Andrew Lutz & Deborah Moore-Russo
1:30 PM - 3:00 PM (CST) University of Oklahoma MAA's Instructional Practices & AMATYC's recent IMPACT document
PHSC 430
November 5 Andrew Lutz
1:30 PM - 3:00 PM (CST) University of Oklahoma MAA's Suggested Instructional Practices to Foster Engagement
PHSC 430
October 25 Andrew Lutz
1:00 PM - 3:00 PM (CDT) University of Oklahoma MAA Suggested Instructional Practices to Foster Engagement
PHSC 430
October 15 Milos Savic
1:30 PM - 3:00 PM (CDT) University of Oklahoma Insights and Recommendations from the MAA's National Calculus Study
PHSC 430
October 1 Andrew Lutz
1:30 PM - 3:00 PM (CDT) University of Oklahoma Overview of Standards for the First Two Years of College Math
PHSC 430
September 17 Deb Moore-Russo
1:30 PM - 3:00 PM (CDT) University of Oklahoma Recent History of Standards Movements in Math Education in the US
PHSC 430
August 20
1:30 PM - 3:20 PM (CDT) OU RUME group RUME Seminar Organization
PHSC 430
Spring 2018
April 23 Roundtable Session
12:30 - 1:20 PM (CDT) University of Oklahoma Best of 2018
PHSC 430
April 2 Andrew Lutz
12:30 PM - 1:20 PM (CDT) University of Oklahoma Discussion of Questions in the Classroom article
PHSC 430
March 26 Kyle McConnell
12:30 PM - 1:20 PM (CDT) University of Oklahoma Discussion of the Role of Evaluation in Research-Practice Integration
PHSC 430
March 12 Milos Savic
12:30 - 1:30 PM (CDT) University of Oklahoma Examining Intersections of Inquiry, Equity, and Creativity
PHSC 430
March 5 Milos Savic
12:30 PM - 1:20 PM (CST) University of Oklahoma Discussion of the RUME Paper "Didactical Disciplinary Literacy"
PHSC 430
February 26 RUME Conference Participants
12:30 PM - 1:20 PM (CST) University of Oklahoma Debriefing of the 21st Annual RUME Conference
PHSC 430
February 19 Ashley Berger
12:30 PM - 1:20 PM (CST) University of Oklahoma Schema Development in an Introductory Topology Proof
PHSC 1105
February 12 Sepideh Stewart and Jonathan Troup
12:30 PM - 1:20 PM (CST) University of Oklahoma Teaching Linear Algebra: Modeling One Instructor’s Decisions to Move between the Worlds of Mathematical Thinking
PHSC 430
February 5 Paul Regier
12:30 PM - 1:20 PM (CST) University of Oklahoma How may Fostering Creativity build Student Self-efficacy for Proving?
PHSC 430
January 29 Milos Savic
12:30 PM - 1:20 PM (CST) University of Oklahoma Productive Failures: From Pedagogical Requirement to Peer-Led Support Group
PHSC 430
January 22 Milos Savic
12:30 PM - 1:20 PM (CST) University of Oklahoma Interactive Discussion on Slightly Changing Tasks to Promote Mathematical Creativity
PHSC 430
Fall 2017
December 4 Round Table Discussion with Sepideh Stewart
2:30-3:30 (CST) University of Oklahoma Research on visualization in learning and teaching mathematics
PHSC 1105
November 20 Round Table Discussion with Sepideh Stewart
2:30-3:20 (CST) University of Oklahoma Research on visualization in learning and teaching mathematics
PHSC 1105
November 13 Round Table Discussion with Sepideh Stewart
2:30-3:20 (CST) University of Oklahoma Research on visualization in learning and teaching mathematics
PHSC 1105
November 6 Round Table Discussion with Sepideh Stewart
2:30-3:20 (CST) University of Oklahoma Research on visualization in learning and teaching mathematics
Room 430
October 30 Round Table Discussion with Sepideh Stewart
2:30-3:20 (CDT) University of Oklahoma Research on visualization in learning and teaching mathematics
PHSC 1105
October 23 Round Table Discussion with Sepideh Stewart
2:30-3:20 (CDT) University of Oklahoma Research on visualization in learning and teaching mathematics
PHSC 1105
October 16 Round Table Discussion with Sepideh Stewart
2:30-3:20 (CDT) University of Oklahoma Research on visualization in learning and teaching mathematics
PHSC 1105
October 9 Round Table Discussion with Sepideh Stewart
2:30-3:20 (CDT) University of Oklahoma Research on visualization in learning and teaching mathematics--Learning Styles: Concepts and Evidence
PHSC 1105
October 2 Round Table Discussion with Sepideh Stewart
2:30-3:20 (CDT) University of Oklahoma Research on visualization in learning and teaching mathematics: Reification as the Birth of Metaphor
PHSC 1105
September 25 Round Table Discussion with Sepideh Stewart
2:30-3:30 (CDT) University of Oklahoma Research on visualization in learning and teaching mathematics
PHSC 1105
September 18 Round Table Discussion with Sepideh Stewart
2:30-3:20 (CDT) University of Oklahoma Research on visualization in learning and teaching mathematics
PHSC 1105
September 11 Round Table Discussion with Sepideh Stewart
2:30-3:20 (CDT) University of Oklahoma Research on visualization in learning and teaching mathematics
PHSC 1105
August 28
2:30-3:20 (CDT) University of Oklahoma Research on visualization in learning and teaching mathematics
PHSC 1105
August 21 Sepideh Stewart
2:30-3:30 (CDT) University of Oklahoma RESEARCH ON VISUALIZATION IN LEARNING AND TEACHING MATHEMATICS
PHSC 1105
Spring 2017
May 1 Round Table Discussion-- Moderator: Kyle
12:30-1:30 (CDT) McConnell Mathematics Education Research at University Level
PHSC 1105 University of Oklahoma
April 24 Round Table Discussion -- Sepideh Stewart
12:30-1:30 (CDT) University of Oklahoma Mathematics Education Research at University Level
PHSC 1105
April 17 Round Table Discussion-- Moderator: Molly
12:30-1:30 (CDT) Beauchamp Mathematics Education Research at University Level
PHSC 1105 University of Oklahoma
April 10 Round Table Discussion-- Moderator: Mollie
12:30-1:30 (CDT) Mills-Weis Mathematics Education Research at University Level
PHSC 1105 University of Oklahoma
April 3 Round Table Discussion-- Moderator: Casey
12:30-1:30 (CDT) Haskins Mathematics Education Research at University Level
PHSC 1105 University of Oklahoma
March 27 Round Table Discussion-- Moderator: Ashley
12:30-1:30 (CDT) Berger Mathematics Education Research at University Level
PHSC 1105 University of Oklahoma
March 20 Round Table Discussion -- Sepideh Stewart
12:30-1:30 (CDT) University of Oklahoma Reflection on Mathematics Education Research at University Level
PHSC 1105
March 6 Round Table Discussion -- Sepideh Stewart
12:30-1:30 (CST) University of Oklahoma Mathematics Education Research at University Level -- Part 7
PHSC 1105
February 27 Round Table Discussion -- Sepideh Stewart
12:30-1:30 (CST) University of Oklahoma Mathematics Education Research at University Level -- Part 6
PHSC 1105
February 20 Round Table Discussion -- Sepideh Stewart
12:30-1:30 (CST) University of Oklahoma Mathematics Education Research at University Level -- Part 5
PHSC 1105
February 17
1:30 pm-2:20 pm (CST) Jonathan Troup Developing Students’ Reasoning about the Derivative of Complex-Valued Functions with the Aid of
HCLC Community Room, Rm. 118 , Lower Level 1, Bizzell University of Oklahoma Geometer’s Sketchpad (GSP)
Memorial Library
February 13 Round Table Discussion -- Sepideh Stewart
12:30-1:30 (CST) University of Oklahoma Mathematics Education Research at University Level -- Part 4
PHSC 1105
February 6 Round Table Discussion -- Moderator:
12:30-1:30 (CST) Sepideh Stewart The State of Mathematics Education Research at University Level -- Part 3
PHSC 1105 University of Oklahoma
January 30 Round Table Discussion-- Moderator: Ashley
12:30-1:30 (CST) Berger The State of Mathematics Education Research at University Level -- Part 2
PHSC 1105 University of Oklahoma
January 23 Sepideh Stewart
12:30-1:30 (CST) University of Oklahoma The State of Mathematics Education Research at University Level
PHSC 1105
Fall 2016
December 5 Rebecca Thomas
1:30 PM - 2:20 PM (CST) University of Oklahoma Discussion of Visualization paper
PHSC 1105
November 28 Milos Savic
1:30 PM - 2:20 PM (CST) University of Oklahoma Discussion of Blog post on Examples
PHSC 1105
November 21 Rebecca Thomas
1:30 PM - 2:20 PM (CST) University of Oklahoma Discussion of Blind Mathematicians Paper
PHSC 1105
November 16 Rebecca Thomas
4:00 PM - 5:00 PM (CST) University of Oklahoma Discussion of Blind Mathematicians Paper
PHSC 1105
November 14 Milos Savic
1:30 PM - 2:20 PM (CST) University of Oklahoma Discussion of Balance in instruction
PHSC 1105
November 7 Bryant Wilson
1:30 PM - 2:20 PM (CST) University of Oklahoma Discussion of Remedial Math and Quantitative Courses paper
PHSC 1105
October 31 Mahesh Sunkula
1:30 PM - 2:20 PM (CDT) University of Oklahoma Discussion of Teaching Methods Comparison of a Large Calculus Course
PHSC 1105
October 24 Ashley Berger
1:30 PM - 2:20 PM (CDT) University of Oklahoma Discussion of the Benny paper
PHSC 1105
October 17 Ore Adekoya
1:30 PM - 2:20 PM (CDT) University of Oklahoma Discussion of Proof and Problem Solving, Part 2
PHSC 1105
October 10 Ore Adekoya
1:30 PM - 2:20 PM (CDT) University of Oklahoma Discussion of Proof and Problem Solving at the University Level, Part 1
PHSC 1105
October 3 Kim Karlic
1:30 PM - 2:20 PM (CDT) University of Oklahoma Discussion of Textbooks and Continuity
PHSC 1105
September 26 Mahesh Sunkula
1:30 PM - 2:20 PM (CDT) University of Oklahoma Discussion of Time Constraints and Calculus Teaching paper
PHSC 1105
September 19 Kim Karlic
1:30 PM - 2:20 PM (CDT) University of Oklahoma Discussion of "Ways in which engaging in someone else's reasoning is productive"
PHSC 1105
September 12 Jieru Zhu
1:30 PM - 2:20 PM (CDT) University of Oklahoma Discussion of "Student connections among counting problems: an exploration using actor-oriented transfer"
PHSC 1105
August 29 Milos Savic
1:30 PM - 2:20 PM (CDT) University of Oklahoma Discussion of Two Implementations of Teaching Mathematical Creativity in Proving paper
PHSC 1105
August 22 Milos Savic
1:30 PM - 2:20 PM (CDT) University of Oklahoma Introduction and discussion of "Secret Mathematical Menu"
PHSC 1105
Spring 2016
May 2
12:30-1:30 Sepideh Stewart A time for reflection: Highlights from some of the papers we discussed this semester!!
(CDT) University of Oklahoma
PHSC 1105
April 25
12:30-1:30 Milos Savic Discussing the paper, titled: Ways in which engaging in someone elses reasoning is productive
(CDT) University of Oklahoma
PHSC 1105
April 11
12:30-1:30 Ashley Berger Resequencing Skills and Concepts in Applied Calculus Using the Computer as a Tool
(CDT) University of Oklahoma
PHSC 1105
April 4
12:30-1:30 Sepideh Stewart Contemplating visualization as an epistemological learning tool in mathematics
(CDT) University of Oklahoma
PHSC 1105
March 28 Jimmy Tran
12:30-1:30 University of Oklahoma, The four A's of Technology and Pedagogy
(CDT) Department of Psychology
PHSC 1105
March 21
12:30-1:30 Rebecca Thomas Types of Visual-Spatial Representations and Mathematical Problem Solving
(CDT) University of Oklahoma
PHSC 1105
March 7
12:30-1:30 Kim Do Discussion of the paper byJennifer E. Szydlik and Carol E. Seaman
(CST) University of Oklahoma
PHSC 1105
12:30-1:30 RUME Conference attendees Highlights from the 2016 RUME Conference
PHSC 1105
22 Sepideh Stewart
12:30-1:30 University of Oklahoma Physics: Bridging the embodied and symbolic worlds of mathematical thinking
PHSC 1105
15 David Plaxco
12:30-1:30 University of Oklahoma Re-claiming during proof production
PHSC 1105
February 8
12:30-1:30 Jieru Zhu With an Eye on the Mathematical Horizon: Dilemmas of Teaching Elementary School Mathematics. Jieru Zhu will lead the discussion of the paper by Deborah Ball.
(CST) University of Oklahoma The paper can be downloaded from: https://onedrive.live.com/redir?resid=88A2E1C9470B2911!111&authkey=!AL1SQIAlCJ1H4yI&ithint=file%2cpdf
PHSC 1105
February 1
12:30-1:30 Sepideh Stewart FROM PSYCHOLOGICAL IMPRISONMENT TO INTELLECTUAL FREEDOM – THE DIFFERENT ROLES THAT SCHOOL MATHEMATICS CAN TAKE IN STUDENT’S LIVES.
(CST) University of Oklahoma
PHSC 1105
January 25
12:30-1:30 Sepideh Stewart The Teaching for Robust Understanding (TRU) Framework
(CST) University of Oklahoma
PHSC 1105
Fall 2015
December 7 Rebecca Thomas
1:30 PM - 2:20 PM (CST) University of Oklahoma Presentation of "Two Proving Strategies of Highly Successful Math Majors" by Zazkis et al.
PHSC 1105
November 30 Milos Savic
1:30 PM - 2:20 PM (CST) University of Oklahoma How much time do students spend on homework in a proof-based inquiry-based learning course? PLUS BONUS TALK!
PHSC 1105
November 23 Mahesh Sunkula
1:30 PM - 2:20 PM (CST) University of Oklahoma Discussion of Tall's Visualization Paper
PHSC 1105
November 16 Oreoluwa Adekoya
1:30 PM - 2:20 PM (CST) University of Oklahoma Discussion of Pre-Calculus Teaching Paper
PHSC 1105
November 9 Milos Savic
1:30 PM - 2:20 PM (CST) University of Oklahoma TALK CANCELLED THIS WEEK
PHSC 1105
November 2 Sepideh Stewart
1:30 PM - 2:20 PM (CST) University of Oklahoma Discussion of a Meta-Analysis of Learning Styles Literature
PHSC 1105
October 26 Milos Savic
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OH level populations and accuracies of Einstein-A coefficients from hundreds of measured lines
Articles | Volume 20, issue 9
© Author(s) 2020. This work is distributed under the Creative Commons Attribution 4.0 License.
OH level populations and accuracies of Einstein-A coefficients from hundreds of measured lines
OH airglow is an important nocturnal emission of the Earth's mesopause region. As it is chemiluminescent radiation in a thin medium, the population distribution over the various roto-vibrational OH
energy levels of the electronic ground state is not in local thermodynamic equilibrium (LTE). In order to better understand these non-LTE effects, we studied hundreds of OH lines in a high-quality
mean spectrum based on observations with the high-resolution Ultraviolet and Visual Echelle Spectrograph at Cerro Paranal in Chile. Our derived populations cover vibrational levels between v=3 and 9,
rotational levels up to N=24, and individual Λ-doublet components when resolved. As the reliability of these results critically depends on the Einstein-A coefficients used, we tested six different
sets and found clear systematic errors in all of them, especially for Q-branch lines and individual Λ-doublet components. In order to minimise the deviations in the populations for the same upper
level, we used the most promising coefficients from Brooke et al. (2016) and further improved them with an empirical correction approach. The resulting rotational level populations show a clear
bimodality for each v, which is characterised by a probably fully thermalised cold component and a hot population where the rotational temperature increases between v=9 and 4 from about 700 to about
7000K, and the corresponding contribution to the total population at the lowest N decreases by an order of magnitude. The presence of the hot populations causes non-LTE contributions to rotational
temperatures at low N, which can be estimated quite robustly based on the two-temperature model. The bimodality is also clearly indicated by the dependence of the populations on changes in the
effective emission height of the OH emission layer. The degree of thermalisation decreases with increasing layer height due to a higher fraction of the hot component. Our high-quality population data
are promising with respect to a better understanding of the OH thermalisation process.
Received: 29 Nov 2019 – Discussion started: 18 Dec 2019 – Revised: 13 Feb 2020 – Accepted: 06 Apr 2020 – Published: 06 May 2020
The night-time emission of the Earth's atmosphere in the near-infrared is dominated by hydroxyl (OH) airglow (Meinel, 1950; Rousselot et al., 2000; Hanuschik, 2003; Noll et al., 2012, 2015), which
originates in the mesopause region in a layer with a width of about 8km and a typical peak height of 87km (Baker and Stair, 1988). The various bright roto-vibrational bands of the OH electronic
ground state, X^2Π, represent an important tracer for atmospheric dynamics (especially wave propagation), ambient temperatures, and chemical composition (especially atomic oxygen) at these high
altitudes, which are mostly probed by ground- and satellite-based remote sensing (e.g. Taylor et al., 1997; Beig et al., 2003; von Savigny et al., 2012; Mlynczak et al., 2013; Reisin et al., 2014;
Sedlak et al., 2016; Noll et al., 2017). For these applications, it is crucial to understand the physical mechanisms that lead to the observed line emission.
In the mesopause region, OH is mostly formed by the reaction of hydrogen and ozone (Bates and Nicolet, 1950; Xu et al., 2012), which excites the electronic ground state up to the ninth vibrational
level v (Charters et al., 1971; Llewellyn and Long, 1978; Adler-Golden, 1997). The nascent population distribution over the roto-vibrational levels is far from local thermodynamic equilibrium (LTE).
As the subsequent relaxation processes by collisions with other atmospheric species are relatively slow compared to the radiative lifetimes of the excited states (e.g. Adler-Golden, 1997; Xu et al.,
2012; Kalogerakis et al., 2018; Noll et al., 2018b), the OH emission bands (which contribute to the vibrational relaxation) reveal strong non-LTE effects. The vibrational level populations can be
fitted as a function of energy by an exponentially decreasing (i.e. Boltzmann-like) distribution with a pseudo-temperature of around 10000K (Khomich et al., 2008; Noll et al., 2015; Hart, 2019a).
Hence, OH bands with upper-state vibrational levels v^′ up to the highest nascent state can easily be measured. Moreover, the rotational level populations for the different v reveal high
overpopulation for high rotational states N compared to the lowest three or four levels under the assumption of a thermal distribution (Pendleton et al., 1989, 1993; Dodd et al., 1994; Cosby and
Slanger, 2007; Oliva et al., 2015; Noll et al., 2018b). The pseudo-temperatures for the high-N populations achieve values up to those found for the v levels (Oliva et al., 2015). The theoretical
explanation of these populations especially for low v is still uncertain (Dodd et al., 1994; Kalogerakis et al., 2018; Noll et al., 2018b) as their modelling suffers from limitations in the data sets
and uncertain input parameters (especially rate coefficients for collisional transitions). It is usually assumed that the ratios of lines related to the lowest N of a fixed v are sufficiently close
to LTE for a reliable estimate of the ambient temperature (e.g. Beig et al., 2003). However, this assumption appears to be insufficient at least for the highest v, where deviations of several kelvins
were found (Noll et al., 2016, 2018b). In addition, small modifications in the set of considered levels in terms of N can already significantly change the corresponding population temperature (Noll
et al., 2015).
A successful study of OH level populations requires accurate molecular parameters, i.e. line wavelengths, level energies, and Einstein-A coefficients. In particular, the latter suffer from relatively
high uncertainties despite numerous dedicated studies for their calculation (e.g. Mies, 1974; Langhoff et al., 1986; Turnbull and Lowe, 1989; Nelson et al., 1990; Goldman et al., 1998; van der Loo
and Groenenboom, 2007; Brooke et al., 2016) and evaluation (e.g. French et al., 2000; Pendleton and Taylor, 2002; Cosby and Slanger, 2007; Liu et al., 2015; Hart, 2019b). Apart from the derivation of
absolute OH level populations or densities (Noll et al., 2018b; Hart, 2019b), the quality of these transition probabilities especially affects OH-based temperature estimates (Liu et al., 2015; Noll
et al., 2015; Parihar et al., 2017; Hart, 2019b) and abundance retrievals for species like atomic oxygen (Mlynczak et al., 2013; Noll et al., 2018b). The persistent uncertainties in the Einstein-A
coefficients are obviously related to the molecular structure of OH and the lack of adequate data for the calculation of the molecular parameters (Nelson et al., 1990; Pendleton and Taylor, 2002;
Cosby and Slanger, 2007; van der Loo and Groenenboom, 2007; Brooke et al., 2016).
In order to improve our knowledge on OH level populations and Einstein-A coefficients, high-quality measurements of a large number of OH lines and a detailed analysis are required. We could perform
such a study based on high-resolution spectroscopic data taken with the Ultraviolet and Visual Echelle Spectrograph (UVES; Dekker et al., 2000) at the Very Large Telescope at Cerro Paranal in Chile
(24.6^∘S, 70.4^∘W). A mean spectrum of the highest-quality spectra (totalling 536h of exposure time) allowed us to investigate 723 lines with upper vibrational levels v^′ between 3 and 9 in the
optical and near-infrared regime in detail. In many cases, the small Λ doubling effect due to rotational–electronic perturbations between the ground and excited electronic states (Pendleton and
Taylor, 2002) was resolved.
In Sect. 2, we describe the UVES data set. Then, we discuss the data analysis involving the calculation of the mean spectrum, the measurement of line intensities, and a check of the line positions
(Sect. 3). Section 4 discusses the differences in the derived OH level populations for the Einstein-A coefficients of Mies (1974), Langhoff et al. (1986), Turnbull and Lowe (1989), van der Loo and
Groenenboom (2008), Rothman et al. (2013), and Brooke et al. (2016). Moreover, the Brooke et al. (2016) reference is used as the basis for an empirical improvement of the coefficients. The
corresponding OH level populations are then investigated in detail (Sect. 5). This involves population fitting, a study of the non-LTE contributions to rotational temperatures, and the investigation
of population differences caused by a change in the OH emission altitude. Finally, we draw our conclusions (Sect. 6).
This study is based on so-called Phase 3 products of the astronomical echelle spectrograph UVES (Dekker et al., 2000) provided by the European Southern Observatory. Noll et al. (2017) selected about
10400 archived spectra taken between April 2000 and March 2015, extracted the night-sky emission, and performed a complex flux calibration procedure in order to investigate long-term variations in
the mesopause region based on OH emission. The studied spectra comprise the wavelength range between 570 and 1040nm covered by two set-ups centred on 760 and 860nm. Depending on the width of the
entrance slit, the spectral resolving power varied between 20000 and 110000. Hence, these data are well suited for OH level population studies as they allow one to measure numerous resolved
emission lines.
As the exposure time (between 1 and 125min) and the contamination of the night-sky emission by the astronomical target (the slit length is only between 8 and 12arcsec) are also strongly varying, it
is important to focus on spectra of sufficient quality, especially if very weak lines are studied. The final sample of Noll et al. (2017), who studied relatively bright P-branch lines related to low
rotational levels, included 3113 suitable spectra. We use an even smaller subsample of 2299 spectra as the basis for this study. It is related to the investigation of the faint K(D[1]) potassium line
at 769.9nm with a mean intensity of about 1R (rayleigh) by Noll et al. (2019). For that sample, the selected spectra were carefully checked around the K(D[1]) line. In order to be able to measure
even fainter lines in the entire wavelength regime and to have a homogeneous data set for the calculation of a mean spectrum, we further reduced the sample. Forty-five spectra of the set-up centred
on 860nm were rejected as they showed severe flaws (wrong continuum levels) below 730nm. This was not a problem for the potassium study. Moreover, we increased the minimum exposure time from 10 to
45min and reduced the maximum continuum limit around K(D[1]) from 100 to 40Rnm^−1. Finally, we only considered spectra that were taken with the standard slit width of 1arcsec, which corresponds
to a resolving power of 42000. Thus, 63% of the sample of Noll et al. (2019) was taken with this slit width.
The resulting sample consists of 533 high-quality spectra with a total exposure time of 536h at a telescope with a diameter of the primary mirror of 8m.
3.1Mean spectrum
In order to calculate the probably best high-resolution airglow mean spectrum in the covered wavelength regime so far, we first mapped the 533 selected spectra (Sect. 2) with the flux calibration of
Noll et al. (2017) applied to a common wavelength grid from 560 to 1061nm with a step size of 1pm that well samples airglow lines, which have a full width at half maximum of about 20pm close to
800nm. The mapping is necessary since each UVES spectrum has its own wavelength grid. The set-up positioning appears to have an uncertainty of the order of 1nm. Moreover, the original step sizes
varied from 1.8 to 5.2pm, depending on the central wavelength of the set-up (760 or 860nm), the chip (two chips with spectra separated by a small gap at the central wavelength), and the pixel
binning. Pixel pairs in dispersion direction on the chips were merged for 85% of the sample. The rest of the data are unbinned. Before the mean calculation, the spectra were also scaled to be
representative of the zenith by using the van Rhijn correction (van Rhijn, 1921) for a thin layer at an altitude of 90km. As the zenith angles at the mid-exposure times vary from 3 to 64^∘, this is
a crucial correction with factors between 0.46 and 1.00. These factors do not significantly change across the entire OH emission layer; i.e. the choice of the reference altitude is not critical.
The mean spectrum was calculated by means of a pixel-dependent σ-clipping approach in order to avoid the contribution of strong sporadic outliers due to technical issues or the contamination by an
astronomical target. As the threshold was set to 10 standard deviations, statistical noise and natural variations of the airglow emission do not cause the rejection of a spectrum at a certain pixel.
The final number of considered spectra at each wavelength after three iterations of the σ-clipping approach is displayed in the lower panel of the upper plot in Fig. 1. The clipping only reduced the
numbers by a few spectra. There is a trend towards more rejections at longer wavelengths. The plot also reveals the impact of the combination of the two set-ups centred on 760 and 860nm with 231 and
302 spectra, respectively. The gap between the spectra of the two chips of each set-up and very narrow gaps between the spectral orders at long wavelengths can also be seen. The rounded edges of the
sample-related steps in the histogram reflect the variation in the wavelength positioning of a certain set-up.
The mean spectrum in Fig. 1a shows 15 OH bands marked by the upper and lower vibrational levels v^′ and v^′′. Bands with $\mathrm{\Delta }v={v}^{\prime }-{v}^{\prime \prime }$ between 3 and 6 are
covered. The band strength strongly increases from OH(5-0) to OH(4-1) as bands with higher v^′ and lower Δv tend to be stronger in the covered wavelength regime. Each band is split into the three
branches R, Q, and P, which are characterised by changes of the rotational quantum number N of −1, 0, and +1. While the R and Q branches at the short-wavelength side and central part of the band are
relatively compact, the P branch shows relatively wide spaces between the lines. P-branch lines with high upper rotational quantum numbers N^′ are located at distinctly longer wavelengths than those
with low N^′; i.e. they are found in regions that are dominated by other OH bands. For this reason, high-N^′ P-branch lines of the faint OH(7-1) band can also be detected in the UVES mean spectrum.
Figure 1b shows the narrow wavelength range between 727 and 745nm to demonstrate the good spectral resolution. The plotted range includes the full Q branch and the P branch up to lines with ${N}^{\
prime }=\mathrm{5}$ of OH(8-3), a band of intermediate strength. The plot clearly shows the splitting of each rotational state by spin–orbit coupling. The Q[1] and P[1] lines (F=1) related to the
electronic substate X^2Π[3∕2] are well separated from the fainter Q[2] and P[2] lines (F=2) of X^2Π[1∕2]. For the visible lines, the value of F does not change during the transition; i.e. ${F}^{\
prime \prime }={F}^{\prime }$. The inter-combination lines which show a change of F are much fainter and are therefore neglected in this study. The spectral resolving power of 42000 is sufficiently
high for seeing Λ doubling. The separation of both components can already be found for Q[1] and P[1] lines with relatively low N^′. In Fig. 1b, the largest separation is visible for Q[1](${N}^{\prime
}=\mathrm{4}$). It amounts to 55pm (Brooke et al., 2016); i.e. this Λ doublet is fully resolved. Separations of more than 200pm are measurable for P[1] lines with ${N}^{\prime }\ge \mathrm{11}$.
The faintest marked Λ doublets, Q[2](2) and Q[1](4), have intensities between 1 and 2R. They can easily be measured in the UVES mean spectrum, which allows one to also detect lines that are more
than 1 order of magnitude fainter (Sect. 3.2). For a general overview of lines (not only OH) that can be accessed with UVES data, see the catalogue of Cosby et al. (2006). It is based on the
night-sky atlas of Hanuschik (2003), which involves UVES observations with a total exposure time of 9h in the red and near-infrared wavelength range.
3.2Line intensities
As it is the most comprehensive list of calculated OH lines so far, we used the line wavelengths of Brooke et al. (2016) for the identification of lines in the UVES mean spectrum (Sect. 3.1) and the
derivation of their intensities. For this purpose, the calculated vacuum wavelengths were converted into air wavelengths by means of the formula of Edlén (1966) for standard air, which works well for
the UVES data. The default line integration range was set to a width of about 2 resolution elements of the spectrograph (Sect. 2), which is 40pm at 860nm, plus the separation of the two Λ-doublet
components. If the latter was wider than the 2 resolution elements, the components were measured independently. For an optimal continuum subtraction, the two continuum points for a linear
interpolation across the line were defined manually as this approach can better handle contaminations by nearby emissions and absorptions of other lines than an automatic procedure, which was also
tested. The wavelengths of the selected continuum points were also used as limits of the integration range. In particular, range modifications were necessary for lines at very long wavelengths around
1µm, where the spectrograph causes extended line wings. The resulting line intensities representing the zenith (Sect. 3.1) were also corrected for molecular absorption in the lower atmosphere. The
complex procedure involving high-resolution radiative transfer calculations and water vapour measurements in the astronomical target spectra is described by Noll et al. (2017). Further details are
given by Noll et al. (2015). For the correction of the measured intensities, the derived line transmission values for the 533 individual spectra were averaged. The resulting mean absorption of the
measured Λ doublets was 3%, and only 6% of the doublets was attenuated by more than 10%. Hence, the related intensity uncertainty after the correction, which can reduce the absorption by up to an
order of magnitude, is negligible for most lines.
As illustrated in Fig. 1, the mean spectrum is composed of UVES spectra of two different set-ups centred on 760 and 860nm. The wavelength shift between both set-ups causes changes in the data
properties, depending on wavelength. In order to minimise the impact of these changes on the measured line intensities, we investigated and corrected two effects: long-term variations in the OH line
intensity and flux calibration errors. The former are important since the two UVES set-ups cover very different parts of the sample-related period from May 2000 to July 2014. Before December 2004,
there were only observations with the 860nm set-up (Noll et al., 2017). On the other hand, spectra of this set-up are only present in the selected sample until May 2010. This results in mean 10.7cm
solar radio fluxes (Tapping, 2013) for an averaging period of 27d of 102sfu (solar flux units) for the 760nm set-up and 140sfu for the 860nm set-up. According to Noll et al. (2017) (also based
on UVES data), the mean solar cycle effect for v^′ between 5 and 9 is 16.1±1.9% per 100sfu. There is no significant change with v^′. We took this mean percentage, the set-up-specific mean solar
radio fluxes, and the wavelength-dependent fraction of 760 and 860nm spectra to correct the OH line intensities to be representative of the mean solar radio flux of the full sample of 123sfu. The
intensity corrections were up to a few per cent with line-dependent differences characterised by a standard deviation of 2.2%. We did not consider the impact of a possible linear long-term trend as
it is not significant for the UVES data (Noll et al., 2017). In order to test the flux calibration, we calculated mean spectra for each set-up and also measured line intensities. The latter were
performed automatically by using the same wavelengths for the line integration as in the case of the mean spectrum of the full sample. The continuum was measured in narrow intervals (about 0.25
resolution elements, i.e. 5pm wide at 860nm) around these limiting positions. The wavelength-dependent intensity ratios for the two set-ups were then used to derive correction factors depending on
set-up and chip. Taking the wavelength range between the set-up gaps around 760 and 860nm as the reference, we found correction factors close to 1 with a standard deviation of 2.7%, which is
consistent with the relative flux calibration uncertainty of about 2% for the UVES data set reported by Noll et al. (2017). Combining the solar activity and flux calibration correction, the
resulting standard deviation is only 1.7% as both effects partly cancel out.
The quality of the final line intensities was indicated by a flag consisting of a primary and a secondary classifier. Each classifier is represented by a digit between 0 and 3. A value of 3
corresponds to a reliable measurement of the entire Λ doublet, which requires symmetric line emission and a featureless underlying continuum, 1 and 2 refer to reliable measurements only for the Λ
-doublet component with e or f parity in the upper state, and 0 marks uncertainties for both components. For unresolved doublets, only 0 and 3 are possible digits. The introduction of the secondary
classifier allows for a finer classification scheme. For example, the combined classes 30 and 03 can be used for ambiguous cases. Reasons for measurement uncertainties are obvious or possible blends
with other emission lines, regions of significant absorptions in the continuum (often combined with very low transmission at the position of the line), and insufficient signal-to-noise ratio in the
case of very weak lines.
Figure 2 shows a histogram of the measured lines depending on the decadal logarithm of the intensity in rayleighs. In total, 723 Λ doublets are included. This neglects 13 measurements with digit 0
for the primary and secondary classifier. These lines were not further used in this study. Other potential lines could not be measured due to a blend with a stronger line, no detection, or a line
wavelength within the order gaps in the near-infrared (Fig. 1). The measured intensities of the 723 Λ doublets range from 0.01 to 600R; i.e. they comprise almost 5 orders of magnitude. Intensities
around 1R are most abundant. The median is 1.7R. There is a conspicuous drop in the occurrence frequency below about 0.2R, which suggests a strongly increasing incompleteness of detections for
fainter lines. The intensity of weaker Λ doublets also tends to be more uncertain as the intensity distributions for the different quality classes show. For the classes 3, 1+2, and 0, the median
intensities are 2.6, 0.83, and 0.086R. Then 546 doublets or 76% belong to class 3, where the two components were measured independently in 34% of the cases. There are 122 doublets (17%) with only
one reliable component (1+2). Finally, there are 55 cases (8%) with class 0 (75% of them with resolved doublets). Considering that detached components require independent line measurements (350
cases), the total number of measurements for the data in Fig. 2 amounts to 1073.
The 723 studied Λ doublets probe 236 different upper states characterised by v^′, N^′, and F^′. Up to nine doublets contribute to the population data for a certain level. The distribution of level
energies E^′ depending on v^′ is shown in Fig. 3. The energies range from 10211 to 28051cm^−1. Except for ${v}^{\prime }=\mathrm{3}$, where N^′ only up to 9 could be measured (mainly due to the
wavelength limitations of the UVES data), wide ranges of E^′ are covered by the data for the different v^′. A maximum energy range of 8861cm^−1 is achieved for ${v}^{\prime }=\mathrm{4}$. This is
possible due to N^′ up to 24. The energy ranges shrink for higher v^′ due to a steeper decrease of the line intensities with increasing N^′, which reduces the detectability of high-N^′ lines. An
important reason for this is certainly the closer exothermicity limit of the hydrogen–ozone reaction, which produces the excited OH. Nevertheless, there are nine levels above this limit if we assume
3.38eV (Cosby and Slanger, 2007), i.e. about 27260cm^−1 (Noll et al., 2018b). This suggests that the kinetic energy involved in the reaction is also important to populate the OH roto-vibrational
levels. For the excitation of the highest level found (${v}^{\prime }=\mathrm{9}$, ${N}^{\prime }=\mathrm{12}$, ${F}^{\prime }=\mathrm{1}$), about 800cm^−1 of additional energy would be needed.
Figure 3 also provides the primary quality classes for the lines related to the displayed states. Thus, 79% of the levels are covered by at least one line with class 3. An exclusive class 0
contribution is found for only 16 states. However, excluding uncertain lines can reduce the E^′ range for a given v^′. In particular, the maximum N^′ for ${v}^{\prime }=\mathrm{5}$ shows a decrease
from 23 to 20 in this case.
3.3Line positions
The high resolving power of 42000 of the UVES data used allows for a check of the quality of the input line positions, which were taken from Brooke et al. (2016) and were converted to standard air
using the formula of Edlén (1966). The default positioning of the integration ranges for the line intensity measurements described in Sect. 3.2 worked well in most cases. However, significant shifts
were necessary for some high-N^′ lines. For a systematic study of these offsets, we took the manually adapted integration windows to calculate the intensity-weighted centroid wavelength for each
In Fig. 4, we show the difference between observed and model wavelengths in picometres as a function of upper-state energy (neglecting the vibrational energy) for 406 individually measured Λ-doublet
components of the P branch. This selection rejects unresolved or only partly resolved Λ doublets and lines with uncertain central wavelengths due to blending with other lines. The remaining 66 Q
-branch and 67 R-branch Λ-doublet components are not plotted as they only probe ΔE^′ up to 5000cm^−1 and are essentially consistent with the P-branch lines, which tend to have higher signal-to-noise
ratios. The plot shows for all v^′ a very good agreement of observed and modelled wavelengths in the case of low energies. For ΔE^′ lower than 2000cm^−1, the mean value and standard deviation are
−0.4 and 0.7pm, respectively. The systematic offset is much less than the original pixel size in the UVES spectra (Sect. 3.1). Hence, it can be caused by uncertainties in the wavelength calibration.
Moreover, the assumption of standard air conditions (1013hPa, 288K, and no H[2]O) for the UVES instrument might cause a part of the offset. Beyond 3000cm^−1, the displayed wavelength offsets show
an increasing scatter. In part, this is caused by the higher measurement uncertainties for the fainter lines, but there are also clear trends depending on v^′. In general, the difference between the
measured and theoretical line wavelengths increases with ΔE^′. This increase appears to be stronger for higher v^′. While the change for ${v}^{\prime }=\mathrm{4}$ is only about 1pm at around
8000cm^−1, it is about 20pm for ${v}^{\prime }=\mathrm{5}$. For higher v^′ (at least for 6 and 7), the increase of the offsets appears to be even stronger. However, as the covered energy range
decreases as well, the maximum offsets only amount to a few picometres. Hence, only the measured shifts for ${v}^{\prime }=\mathrm{5}$ and N^′ of 22 and 23 are of the order of a spectral resolution
element. For all other detected lines, the quality of the theoretical line positions is much better.
The discussed results are for the line wavelengths published by Brooke et al. (2016). As the HITRAN line database (Gordon et al., 2017) is more frequently used, we also calculated the wavelength
shifts for those data. We took the version HITRAN2012 (Rothman et al., 2013), which does not differ from the more recent version HITRAN2016 (Gordon et al., 2017) in terms of the OH data. The results
are very similar to those in Fig. 4. Strong deviations above 10pm are found for the same small sample of lines, although OH(5-1)P[1](23) is missing in the HITRAN database. The mean difference
between the line wavelengths from Rothman et al. (2013) and Brooke et al. (2016) is 0.07pm. The standard deviation only amounts to 0.40pm.
Based on the UVES mean spectrum of Hanuschik (2003) (Sect. 3.1), the accuracy of OH line wavelengths was already investigated by Cosby et al. (2006). Their theoretical line positions originate from
Cosby et al. (2000) but should be similar to Goldman et al. (1998), the basis of the OH data in HITRAN, for low rotational levels. For higher rotational levels, the line wavelengths calculated by
Cosby et al. (2000) should be more precise. Indeed, although the spectrum of Hanuschik (2003) is noisier, the critical OH(5-1) lines do not show clear systematic offsets. However, all OH data
indicate a mean shift of the UVES-based wavelengths of about +1.0pm. This offset is similar to those for atomic and molecular oxygen in the same wavelength range, which were also measured by Cosby
et al. (2006). Thus, systematic errors in the wavelength calibration are the most likely explanation. In comparison, our measurements result in a negative mean offset. This could be caused by
differences in the UVES sample, data processing, and analysis.
4Einstein-A coefficients
4.1Full OH level populations
The intensities ${I}_{{i}^{\prime }{i}^{\prime \prime }}$ derived in Sect. 3.2, where i^′ and i^′′ are the upper and lower states of the roto-vibrational transition, can be converted into level
populations by dividing Einstein coefficients ${A}_{{i}^{\prime }{i}^{\prime \prime }}$. For a visualisation, these populations are usually normalised by dividing the statistical weight (i.e. the
degeneracy) of the upper state ${g}^{\prime }={g}_{{i}^{\prime }}$ and then converted to logarithms. Following Noll et al. (2015), we define
$\begin{array}{}\text{(1)}& y:=\mathrm{ln}\left(\frac{{I}_{{i}^{\prime }{i}^{\prime \prime }}}{{A}_{{i}^{\prime }{i}^{\prime \prime }}{g}^{\prime }}\right),\end{array}$
where the intensity is given in rayleighs and the Einstein-A coefficients are provided in inverse seconds, which is consistent with population column densities in units of 10^6cm^−2 (Noll et al.,
2018b). For Λ doublets, i is characterised by the vibrational level v, rotational level N, and electronic substate F. In the case of individual components, the parity p (i.e. e or f) is also a
parameter and g^′ is half as large as for the doublet.
Apart from the uncertainties in the line intensities, the quality of the resulting populations also depends on the reliability of the Einstein-A coefficients. As already mentioned in Sect. 1, the
latter is not satisfactory as the available sets differ quite significantly. With our large sample of energy levels, where the population of each state can be derived from up to nine different lines,
we can carry out a comprehensive comparison of Einstein-A coefficients. As a reference set, we take the coefficients calculated by Brooke et al. (2016) (B+16), who provide the most recent and largest
set of OH line parameters. Figure 5a shows the corresponding y for the 544 reliable Λ doublets of class 3 (neglecting OH(7-1), i.e. two doublets) as a function of the energy of the upper-state E^′.
The distribution of populations displays the well-known pattern of steep population decreases for low N^′ and weaker population gradients for higher N^′ (Pendleton et al., 1989, 1993; Cosby and
Slanger, 2007; Oliva et al., 2015; Kalogerakis et al., 2018; Noll et al., 2018b). Moreover, it indicates the expected decrease of populations for higher v^′ with a remarkable exception for ${v}^{\
prime }=\mathrm{8}$ (Cosby and Slanger, 2007; Noll et al., 2015). The latter is a signature of the nascent OH level population distribution, which mainly occupies ${v}^{\prime }=\mathrm{8}$ and 9.
The population properties will be discussed in more detail in Sect. 5.
It is now important to know how robust the observed pattern is with respect to changes in the set of Einstein-A coefficients. For this purpose, we also consider the HITRAN database with the version
from 2012 (Rothman et al., 2013) (see also Sect. 3.3), which is mainly based on the calculations of Goldman et al. (1998) for OH. Moreover, we use the coefficients from van der Loo and Groenenboom (
2008) (vdLG08), i.e. the corrected version of van der Loo and Groenenboom (2007), Turnbull and Lowe (1989) (TL89), Langhoff et al. (1986) (LWR86), and Mies (1974) (M74). We neglect the also still
popular data from Nelson et al. (1990) as their line list only focuses on low Δv and low N^′. Except for B+16, our selection of sets agrees with Liu et al. (2015) and Hart (2019b), who studied the
impact of Einstein-A coefficients on the populations of low rotational levels. Other comparisons used a smaller number of sets (French et al., 2000; Cosby and Slanger, 2007; Noll et al., 2015;
Parihar et al., 2017; Noll et al., 2018b). Note that the three oldest sets lack a significant number of the measured 723 Λ doublets. The set of Turnbull and Lowe (1989) only includes 634 doublets
with maximum N^′ between 13 (R[1] branch) and 15 (Q[2] and P[2] branches). In the case of LWR86 and M74, the N^′-related limits are higher by 1, but the bands with Δv=6, i.e. essentially OH(8-2) and
OH(9-3), are not covered. The number of doublets is therefore only 566 or 78% of the full sample.
^a Mean logarithmic level population y. ^b Mean difference of logarithmic level populations. ^c Mean absolute difference of logarithmic level populations. ^d Change in branch (Q−P and R−P). ^e Change
in v^′′ for all and individual branches (P, Q, R). ^f Number of selected Λ doublets (present in all sets of Einstein-A coefficients).
Figure 5 reveals clear discrepancies between the populations for the six investigated sets of Einstein-A coefficients. The general structure of the distribution is similar but the y values are
shifted. Taking 416 Λ doublets with ${N}^{\prime }\le \mathrm{12}$ and Δv≤5, which are present in all six sets, we find mean y between −1.43 for TL89 and −0.02 for LWR86 (Table 1). This corresponds
to an unsatisfactorily large population ratio of about 4.1. Substituting the extreme TL89 y value by the next highest one of −0.69 for HITRAN, the ratio is still about 1.9. The coefficients of M74,
B+16, and vdLG08 result in intermediate mean y values of −0.33, −0.18, and −0.13, respectively.
Any estimate of absolute OH level populations by means of OH line intensities will be highly uncertain with these results if the quality of the Einstein-A coefficients used cannot be evaluated. Tests
of the accuracy of the measured absolute populations require an alternative calculation which is less sensitive to the choice of the set of molecular parameters. Using a kinetic model for chemical OH
production and excitation relaxation via collisions and radiative transitions is a solution, although various required rate coefficients and molecular abundances (especially for atomic oxygen) are
quite uncertain. Noll et al. (2018b) found that the higher populations related to B+16 tend to be more reliable than the lower ones based on HITRAN; this was for v=9 based on OH line intensities from
UVES and pressure, temperature, and molecular abundance profiles from the satellite-based Sounding of the Atmosphere using Broadband Emission Radiometry (SABER) instrument (Russell et al., 1999) and
the empirical atmospheric NRLMISE-00 model (Picone et al., 2002). Hence, at least the very low populations related to TL89 appear to be quite unlikely.
4.2Detailed population comparisons
Figure 5 marks the populations derived from lines of the R, Q, and P branches by different symbols. A good set of Einstein-A coefficients should result in similar population distributions for the
three branches. However, the TL89 y values for the R branch are distinctly below those of the P branch. A similar but weaker effect can also be seen for the data related to HITRAN and M74. In order
to study these discrepancies in more detail, we calculated differences Δy for lines with the same upper state but different branches. We focus on Q versus P branch and R versus P branch. The
corresponding results for the six sets of Einstein-A coefficients and reliable Λ doublets of class 3 are shown as a function of E^′ relative to the lowest energy for a given v^′ in Fig. 6. For B+16
Einstein-A coefficients, 219 population ratios, Δy, are plotted.
All sets indicate unsatisfactory ratios for the comparison of Q- and P-related populations, especially for high ΔE^′ or N^′, where Δy can be lower than −1. The mean Δy for a subsample with ${N}^{\
prime }\le \mathrm{12}$ and Δv≤5 (cf. Sect. 4.1) are between −0.37 for TL89 and −0.25 for B+16, vdLG08, and LWR86 (Table 1); i.e. the different sets fail in a similar way. According to the
theoretical considerations of Pendleton and Taylor (2002) triggered by the OH(6-2) line intensity ratios measured by French et al. (2000), this can be explained by the general negligence of orbital
angular momentum uncoupling, which is related to rotational–electronic mixing of the electronic ground state X^2Π and the first excited state A^2Σ^+, for the calculation of the available Einstein-A
coefficients. OH line measurements in near-infrared spectroscopic data from the Nordic Optical Telescope at La Palma (Spain) by Franzen et al. (2019) indicate that too low Q-branch populations or too
high Einstein-A coefficients (based on HITRAN) are also an issue for OH bands with Δv=2 and 3 not covered by our study.
The comparison of populations based on R- and P-branch lines reveals a more complex situation than for the Q-branch data. All sets of Einstein-A coefficients show negative mean Δy, i.e. lower R
-related populations on average (Table 1). However, the range is relatively wide with values between −0.51 for TL89 and −0.02 for LWR86. The latter is the only satisfactory set for this comparison at
low ΔE^′ as it was already found by French et al. (2000) for OH(6-2) low-N^′ lines. For levels with high rotational energy, Δy tends to be positive. For the other sets, Fig. 6 shows a clear decrease
of Δy with increasing ΔE^′ at least below 1000cm^−1. For example, the most recent set, B+16, shows mean Δy of −0.04 and −0.14 below and above 400cm^−1, respectively. At high ΔE^′, the negative
trend appears to vanish for B+16, HITRAN, and vdLG08; the latter even indicating an increase at the highest energies as in the case of LWR86. The especially bad performance of the TL89, HITRAN, and
M74 coefficients is probably related to an underestimation of the vibration–rotation interaction (Pendleton and Taylor, 2002).
The differences in the dependence of the P- and R-branch-based populations on E^′ for the investigated sets of Einstein-A coefficients imply deviations in the related rotational temperatures,
$\begin{array}{}\text{(2)}& {T}_{\mathrm{rot}}=-\frac{\mathrm{1}}{{k}_{\mathrm{B}}\frac{\mathrm{d}y}{\mathrm{d}{E}^{\prime }}}\end{array}$
(Mies, 1974; Noll et al., 2018b), where k[B] is the Boltzmann constant and $\mathrm{d}y/\mathrm{d}{E}^{\prime }$ represents the slope of a regression line in a y(E^′) plot like Fig. 5 for the
included level populations. For a quantitative T[rot] comparison, we considered pairs of levels with a difference in N^′ of 1 where the populations were derived from reliable lines (class 3) of the
same OH band, F^′, and branch. Only those pairs that are available for the P and R branches were selected. This resulted in 35 pairs that are covered by all sets of Einstein-A coefficients up to ${N}
^{\prime }=\mathrm{12}$. The sample is relatively small since R-branch lines are often blended. The highest number of 12 pairs is found for the combination of the lowest N^′ of 2 and 3 (${N}^{\prime
}=\mathrm{1}$ does not exist for the R branch). Mean results for these 12 pairs (which minimise the measurement uncertainties) are shown in Fig. 7. The T[rot] differences based on higher N^′ agree
For the P branch, Fig. 7 reveals a wide range of mean temperatures between 195K for LWR86 and 208K for TL89; i.e. the selection of the set of Einstein-A coefficients strongly affects the derivation
of absolute T[rot]. The situation is better if the extremely high value for TL89 is neglected. In this case, the maximum difference (now limited by the HITRAN-related result) is only 6K instead of
13K. Moreover, the T[rot] for the two most recent sets, B+16 and vdLG08, agree well with the minimum related to LWR86. The temperature differences are consistent with those derived by Liu et al. (
2015) for low-N^′ P[1]-branch lines of the OH bands (3-0), (5-1), (6-2), (8-3), and (9-4) based on observations with a Czerny–Turner spectrometer at Xinglong in China. The differences between the
highest and lowest T[rot] related to TL89 and LWR86, respectively (B+16 was not published yet), were between 9K for OH(3-0) and 17K for OH(8-3) with the same mean of 13K. The trend of decreasing T
[rot] differences for OH bands with longer central wavelengths can also be observed in our data. Taking the differences between HITRAN and B+16 as an example, we find between 2K for OH(3-0)P[1] and
9K for OH(6-1)P[2] for the 12 selected line combinations. In this context, the result of Hart (2019b) for the P[1] branch of OH(4-2) is interesting. Based on data from an astronomical spectrograph
at Apache Point in the USA, he found a maximum difference of 3K for the same five sets investigated by Liu et al. (2015). If the minimum related to LWR86 is excluded, the variation is only about 1K
with the lowest T[rot] related to TL89.
Figure 7 also shows T[rot] based on R-branch lines, which were not used in the discussed studies. The set-dependent results are remarkable since they mirror those for the P-branch lines. Now, T[rot]
ranges from 179K for TL89 to 193K for LWR86; i.e. the maximum difference of 14K is very similar to the result for the P branch but the sign is reversed. Moreover, all T[rot] related to the R
branch are lower than those related to the P branch. Hence, the T[rot] difference between P and R branches is between 2K for LWR86 and 29K for TL89. For individual double pairs of lines, R
-branch-related T[rot] can also be higher than those for the P branch; i.e. LWR86 might not show the smallest differences. However, the large discrepancies for TL89 are obvious in any case.
As the P- and R-branch T[rot] values show an oppositional behaviour, we averaged the slopes, $\mathrm{d}y/\mathrm{d}{E}^{\prime }$, for both branches to derive more robust temperatures. As
demonstrated by Fig. 7, this was achieved. The mean value for all sets is 193.3K with a standard deviation of only 1.0K. The latter represents less than 20% of the variation for the individual
branches. Consequently, the combination of P- and R-branch data can significantly reduce the impact of the choice of the Einstein-A coefficients on the quality of the resulting T[rot]. However, in
practice, this will be difficult to apply due to the difficulties in measuring R-branch lines at moderate spectral resolution. Hence, it is more promising to improve the Einstein-A coefficients by a
better handling of the vibration–rotation interaction (Pendleton and Taylor, 2002), which appears to be the main reason for the set-dependent T[rot] discrepancies. Data as plotted in Fig. 7 can
provide important constraints for this purpose.
Another population-independent evaluation of Einstein-A coefficients is possible for transitions with the same upper and lower levels except for a different v^′′. For the comparison of the related y,
it was necessary to define a reference OH band for each v^′ between 4 and 9, where we have line measurements for two or more bands. We preferentially selected bands with good quality data in the
middle of the covered wavelength range: OH(4-0), OH(5-1), OH(6-2), OH(7-3), OH(8-3), and OH(9-4). The resulting Δy values are plotted in Fig. 8 as a function of line wavelength for the six sets of
Einstein-A coefficients. In the case of B+16, population ratios for 182 pairs of reliable Λ doublets (class 3) are shown. For LWR86 and M74, this number is only 136 due to the limitations in N^′ and
Δv (Sect. 4.1). The plots indicate a complex behaviour where the Δy values depend on band, branch, and N^′ in a different way for each set of Einstein-A coefficients. The data points for the lowest N
^′, which tend to cluster for each band, show a clear trend with wavelength (or Δv) for all sets except B+16. The data for HITRAN, vdLG08, and TL89 indicate an increase of Δy with wavelength, whereas
the M74 data show a decrease. For LWR86, Δy is mainly negative; i.e. the reference bands in the middle of the wavelength range with Δy=0 (not plotted) indicate the highest relative populations.
The overall performance of each set can be evaluated by measuring the mean absolute Δy for line pairs where Einstein-A coefficients are available in all sets. The corresponding results for 127 line
pairs fulfilling Δv≤5 and ${N}^{\prime }\le \mathrm{12}$ are provided in Table 1. The highest and hence worst $〈|\mathrm{\Delta }y|〉$ were found for M74 (0.41) and TL89 (0.38). Lower but still
unsatisfactory values of around 0.18 were obtained for HITRAN, vdLG08, and LWR86. B+16 clearly shows the best performance with a value of 0.11. Table 1 also contains $〈|\mathrm{\Delta }y|〉$
depending on branch. The best results are obtained for the P branch for all sets except M74, which is unsatisfactory for all branches.
The large Δy values and their trend with wavelength for M74 and TL89 shown in Fig. 8 were already found by Cosby and Slanger (2007). Also using UVES data, they compared the populations derived from
the P[1](1) line of the accessible OH bands with v^′ of 6, 8, and 9. Including the transition probabilities of M74, LWR86, TL89, and Goldman et al. (1998), their analysis favoured the latter, i.e.
the main input source for HITRAN. Cosby and Slanger (2007) explained the bad performance of the TL89 coefficients by the erroneous intensity calibration of data used for the applied empirical dipole
moment function (DMF), which is the basis for the calculation of the transition probabilities. Population comparisons for OH lines from near-infrared bands with low Δv, mostly not covered by UVES,
were performed by Oliva et al. (2013) based on observations between 0.95 and 2.4µm with the high-resolution echelle spectrograph GIANO at the Telescopio Nazionale Galileo at the Roque de los
Muchachos Observatory (La Palma) in Spain. The results show clear discrepancies between populations derived from lines of bands with Δv=2, 3, and 4 for the Einstein-A coefficients from van der Loo
and Groenenboom (2007). Interestingly, the corresponding trend with wavelength, displayed in Fig. 8, seems to be reversed for bands at longer wavelengths. In general, it can be expected that the
accuracy of Einstein-A coefficients for bands with high v^′ in the optical tends to be worse than in the case of bands with low v^′ in the near-infrared. Theoretical ab initio DMF calculations, as
used by Mies (1974), van der Loo and Groenenboom (2007), and van der Loo and Groenenboom (2008), are more uncertain for internuclear distances between the O and H atoms that are far from the
equilibrium. Moreover, the input data for empirical DMFs (Turnbull and Lowe, 1988, 1989; Nelson et al., 1990), theoretically extended empirical DMFs (Goldman et al., 1998), and modified ab initio
DMFs (Langhoff et al., 1986; Brooke et al., 2016) were mainly restricted to low v or low Δv.
As discussed in Sect. 3.2, about half of the measured Λ doublets are resolved due to the high spectral resolving power of UVES. This allowed us to systematically study deviations between the
Einstein-A coefficients of the e and f components. The older sets of transition probabilities (M74, LWR86, and TL89) do not provide information on the individual components. The HITRAN database (
Gordon et al., 2017) contains these components but the Einstein-A coefficients were just set to the value of the corresponding doublet. Finally, vdLG08 and B+16 consider Λ doubling but the
differences between the coefficients are very small. For B+16, the mean relative difference for our sample of 723 doublets is only 0.04%. The largest deviations are related to P- and Q-branch lines
with high N^′. The maximum in our sample of 0.25% is linked to OH(5-1)Q[2](6). As the corresponding values for vdLG08 are almost identical, it is sufficient to use only B+16 coefficients for the
comparison of the Λ-doublet components.
Figure 9 shows the results for 185 reliable (class 3) doublets with resolved components. The logarithmic population ratio Δy for f minus e for the upper-state parity p^′ is plotted as a function of
the corresponding difference in E^′. The latter is negative for ${F}^{\prime }=\mathrm{2}$ lines of the P and R branches according to the parity definition used by Brooke et al. (2016). If the
theoretically predicted equality of the transition probabilities was true, Δy should be close to 0. Small deviations in the populations are possible due to the small differences in E^′. Assuming a
Boltzmann-like distribution for a typical kinetic temperature of 190K at altitudes of the OH emission layer at Cerro Paranal (Noll et al., 2016), there would be $\mathrm{\Delta }y=+\mathrm{0.08}$
for $\mathrm{\Delta }{E}^{\prime }=-\mathrm{10}$cm^−1 and the same amount with negative sign for the corresponding positive energy difference. However, the true Δy values are about 1 order of
magnitude larger. The average absolute discrepancy in Δy is 0.22. In the case of large energy differences of at least 5cm^−1, it would even be 0.37, which corresponds to a ratio of 1.4. The clear
differences in the strengths of the Λ-doublet components are also illustrated by two example spectra. The weakly separated OH(6-2)P[1](4) doublet already shows slight differences in the intensity ($\
mathrm{\Delta }y=-\mathrm{0.04}$). For the widely separated OH(6-2)P[2](12) pair, the f component is about 1.7 times brighter than the e component (Δy=0.53). It is astonishing that it appears that
these large effects have not been recognised, so far. They can only be explained by inadequate Einstein-A coefficients since the P, Q, and R branches behave differently. The Δy related to P-branch
lines could be fitted by a non-LTE Boltzmann-like distribution with about 20K. However, the Δy distributions for the Q and R branches are less clear. A convincing regression line cannot be drawn,
and even if the fitting was performed the slopes would be very different. This rules out a significant impact of possible non-LTE-inducing propensity differences for the population of the e and f
states, even if the parity definitions are changed or more complex dependencies involving several level parameters are considered.
The discussion in the previous section has shown that the currently available sets of Einstein-A coefficients are not satisfactory, especially with respect to Q-branch lines and Λ-doublet components.
Overall, the B+16 set is the most promising since it is the most complete in terms of the included lines, shows the smallest band-to-band variations for constant v^′, and is only slightly worse than
LWR86 with respect to the population deviations between different branches. Hence, we focus on the B+16 set for the rest of this paper. However, as the remaining issues can still negatively affect
the evaluation of OH level population distributions as shown in Fig. 5, we tried to improve the coefficients to result in more consistent population ratios in the diagnostic plots discussed in Sect.
4.2. Our approach is fully empirical; i.e. it is based on regression lines and correction factors, and consists of three steps related to the correction of the discrepancies revealed by Figs. 6, 8,
and 9. Complex fitting approaches involving theoretical DMF-based calculations of the Einstein-A coefficients are out of the scope of this study.
We started with a correction of the discrepancies between the populations derived from different branches as shown in Fig. 6. For this purpose, we used 184 suitable, highly reliable Λ doublets
classified as 33, i.e. with a primary and secondary class of 3 (Sect. 3.2). It turned out that a linear fit of Δy for Q versus P and R versus P shows the best performance for N^′ as the independent
variable (instead of the plotted ΔE^′) and a separate fit for each OH band. The resulting slopes and intercepts and their uncertainties are provided in Fig. 10. There are no data points for OH(5-0),
OH(9-5), and OH(4-1) due to relatively high uncertainties caused by an insufficient number of reliable line measurements. For the remaining 12 bands, 5 to 16 (3 to 8) Λ doublets could be used for the
fit of the difference between R and P branch (Q and P branch). The slopes for the Q branch are clearly more negative than those for the R branch. The mean values are −0.15 and −0.02; i.e. the
apparent populations decrease by 14% and 2% compared to the P branch for an increase of N^′ by 1. The corresponding mean intercepts are +0.11 and +0.01, i.e. nearly zero for R versus P. The data
points are plotted as a function of the central wavelength of the Q[1](1) doublet of each band as especially the slope indicates a significant trend with wavelength, which suggests that the
branch-related errors in the B+16 Einstein-A coefficients depend on the energy difference between the upper and lower states. The discrepancies between the branches appear to be smaller for longer
wavelengths and might even vanish around 1µm. We used these correlations for a more robust correction approach involving error-weighted linear fits of slope and intercept as a function of Q[1](1)
wavelength. The error weighting allowed us to consider the strong dependence of the uncertainties on the band. The resulting fits are also shown in Fig. 10. The slopes change with $+\mathrm{0.45}±\
mathrm{0.05}$ (Q) and $+\mathrm{0.14}±\mathrm{0.03}$ (R) per micrometre. For the intercepts, we obtain changes of $-\mathrm{0.36}±\mathrm{0.08}$ (Q) and $+\mathrm{0.04}±\mathrm{0.11}$ (R) per
According to Eq. (1), the reciprocals of the population ratios Δy from our two-step fitting procedure indicate the systematic deviations of the Einstein-A coefficients. Thanks to the derived
regression lines, they can also be predicted for missing lines and bands, at least in the covered wavelength range. For the correction, we need to define a reference. While the Q-branch coefficients
do not appear to be reliable in general, it would be arbitrary to choose the P or the R branch. However, Fig. 7 suggests that the errors in the Einstein-A coefficients are minimised if both branches
are combined with the same weight. Hence, we corrected the transition probabilities for the three branches by using half the fitted deviation between P and R branches as the reference. Figure 11a
shows Δy for the 184 considered pairs of Λ doublets before and after the correction. While in the former case the mean value and standard deviation are −0.13 and 0.16, the corrected data reveal 0.00
and 0.08. Thus, the offsets vanished completely on average and the scatter was reduced by a factor of 2.
In the next step, we corrected Δy offsets between lines differing only in v^′′ as shown in Fig. 8. For this purpose, we focused on relatively bright lines with ${N}^{\prime }\le \mathrm{4}$ for the P
and R branches and ${N}^{\prime }=\mathrm{1}$ for the Q branch. As these Λ doublets indicate similar Δy (scatter of 0.03), differences in the selected line subsets do not critically affect the mean
values. For each considered OH band, 8 to 14 reliable pairs of Λ doublets of class 33 were available (89 in total). For the correction of Δy by changing the Einstein-A coefficients, we used the same
reference bands for each v^′ as discussed in Sect. 4.2. The choice is motivated by the accessibility and quality of the line measurements with UVES. It does not necessarily include the bands with the
most realistic transition probabilities. As discussed in Sect. 4.2, coefficients of bands with low v^′ and Δv tend to be more reliable as the DMF calculations are less challenging and the
experimental data are more abundant. Our reference bands have v^′ between 3 and 9 and Δv between 3 and 5. Hence, the most promising bands are beyond the UVES wavelength range. Nevertheless, there
does not appear to be a strong quality gradient with Δv or wavelength (v^′ cannot be tested) since the B+16 coefficients do not show such a dependence of Δy for the covered bands in Fig. 8 (in
contrast to the other investigated sets). Note that this is different from the situation for the branches illustrated in Fig. 10. In the end, we shifted the mean Δy for eight OH bands to zero by
multiplying the Einstein-A coefficients by factors between 0.85 for OH(7-2) and 1.06 for OH(8-4). This reduces the scatter in the measured populations for fixed v^′ in any case, even if the choice of
the reference bands might not be optimal. Figure 11b shows the corresponding results for 143 Λ doublets. The discussed corrections (also including the branch-related modifications) change the mean Δy
and standard deviation from −0.02 and 0.11 to −0.01 and 0.05.
Finally, we corrected the Δy between the Λ-doublet components as shown in Fig. 9. This is necessary in order to also use doublets with only one reliable component for the study of the OH level
populations discussed in Sect. 5. For the change of the Einstein-A coefficients, we assumed a natural population discrepancy between the two components consistent with a temperature of 190K as
illustrated in Fig. 9. We fitted the remaining Δy for each branch using N^′ as the independent variable due to a better performance with respect to linear regressions compared to ΔE^′. This approach
requires us to flip the sign for the data points with negative ΔE^′. This is reasonable since the amount of the deviations is very similar for Λ doublets with F^′ of 1 and 2. For the fits related to
P, Q, and R, we considered 99, 18, and 11 resolved doublets of class 33. The resulting slopes (and intercepts) are $-\mathrm{0.035}±\mathrm{0.003}$ ($+\mathrm{0.13}±\mathrm{0.03}$), $+\mathrm{0.065}±
\mathrm{0.019}$ ($-\mathrm{0.19}±\mathrm{0.08}$), and $-\mathrm{0.035}±\mathrm{0.035}$ ($+\mathrm{0.43}±\mathrm{0.48}$), respectively. Hence, the effect for the Q branch seems to be twice as large as
for the P branch and to also have a different sign. The R branch might behave similarly to the P branch but the uncertainties are high. Assuming that the e and f components equally contribute to the
fitted differences, we corrected the Einstein-A coefficients for the reliable N^′ range of the fits, i.e. we neglected doublets with low N^′ where both components are not sufficiently separated or
the fit crossed Δy=0. Figure 11c shows the resulting change in the mean Δy and scatter for the investigated 128 doublets. While the small mean value of +0.02 did not significantly change, the
standard deviation was clearly reduced from 0.25 to 0.11.
5.1Mean populations and rotational temperatures
With the correction of the B+16 Einstein-A coefficients in Sect. 4.3, we minimise the scatter in the OH level populations for upper states with different measured lines. Moreover, the change of the
populations with N^′ appears to be more reliable due to the promising combination of P- and R-branch data for T[rot] estimates (Fig. 7). The resulting population distributions for v^′ between 4 and 9
are shown in Fig. 12. We neglect ${v}^{\prime }=\mathrm{3}$ due to the lack of high-N^′ states in the UVES data (Fig. 3), which does not allow us to describe this population distribution in detail.
As already briefly discussed in Sects. 1 and 4.1, there is a characteristic pattern with a steep population decrease for low N^′ and a rather slow decrease for high N^′ for increasing level energy.
The difference between high and low N^′ tends to increase with decreasing v^′. Moreover, the change between the two extremes does not appear to happen continuously with increasing N^′. Instead, the
transition is mostly localised in a narrow ΔE^′ interval of a few hundred inverse centimetres. This known pattern (Cosby and Slanger, 2007; Oliva et al., 2015) suggests the definition of a cold and a
hot population for each v^′, which can be described by corresponding T[rot] and population ratios for fixed level energies (Oliva et al., 2015; Kalogerakis et al., 2018; Kalogerakis, 2019).
We applied this concept by fitting the natural logarithm of the sum of two exponential Boltzmann terms as a function of ΔE^′ (E^′ relative to the energy for ${N}^{\prime }=\mathrm{1}$ and ${F}^{\
prime }=\mathrm{1}$) to the corrected y for each v^′. We considered the populations from all Λ doublets with quality classes above 0. For classes 1 and 2, we derived the doublet-related populations
from the reliable components. An inspection of the change of the populations with increasing ΔE^′ resulted in the rejection of the highest N^′ levels for ${v}^{\prime }=\mathrm{8}$ and 9 as the
related populations cannot be reproduced satisfactorily by a two-component fit. In the case of ${v}^{\prime }=\mathrm{9}$, the seven measurements with ${N}^{\prime }\ge \mathrm{10}$ were neglected.
All corresponding E^′ values are between 250 and 790cm^−1 above the exothermicity limit of the hydrogen–ozone reaction (Sect. 3.2), which can explain the rapid population decrease in this energy
range, which could not clearly be constrained before due to a lack of data (Cosby and Slanger, 2007; Noll et al., 2018b). In the case of ${v}^{\prime }=\mathrm{8}$, a strong decrease of the
populations is found for eight measurements related to ${N}^{\prime }\ge \mathrm{14}$ with E^′ between 660cm^−1 below and 90cm^−1 above the exothermicity limit. This is an interesting result as it
provides valuable constraints on the nascent populations and the relaxation process from ${v}^{\prime }=\mathrm{9}$ to 8. The drop of the populations below the exothermicity limit also seems to be
present in the population distribution of Cosby and Slanger (2007), also based on UVES spectra (Hanuschik, 2003; Cosby et al., 2006). However, the authors do not discuss this phenomenon. Our ${v}^{\
prime }\le \mathrm{7}$ data, which are related to energies of more than 1200cm^−1 below the limit, do not show a cut in the populations. The data of Cosby and Slanger (2007) are not conclusive here
The remaining population measurements for each v^′, which varied between 83 for ${v}^{\prime }=\mathrm{4}$ and 124 for ${v}^{\prime }=\mathrm{6}$ and 8, were fitted with our two-component model by
means of robust least-squares minimisation, which resulted in the same best fits for a wide range of start values. Figure 12 shows the final best fits and also indicates the corresponding
temperatures T[cold] and T[hot] as well as the ratio of the hot and cold populations for $\mathrm{\Delta }{E}^{\prime }=\mathrm{0}$, r[pop,0]. Under consideration of the fit uncertainties, the
best-fit T[cold] values are very similar and consistent with a temperature of 190K, i.e. the typical ambient temperature at OH emission altitudes (Noll et al., 2016). Only ${v}^{\prime }=\mathrm{4}$
with 196±4K might slightly be higher. This could point to the weak influence of an intermediate population for low v^′. The second highest T[cold] value of 193±5K for ${v}^{\prime }=\mathrm{5}$
would be in agreement with this interpretation. Note that fixing the fit to a T[cold] of 190K did not significantly change the other parameters. The differences were much smaller than the
uncertainties. In contrast to T[cold], T[hot] shows a strong trend with v^′. The temperatures increase from about 700K for ${v}^{\prime }=\mathrm{9}$ to about 7000K for ${v}^{\prime }=\mathrm{4}$.
In parallel, r[pop,0] decreases from about 3% for ${v}^{\prime }=\mathrm{9}$ to about 0.3% for ${v}^{\prime }=\mathrm{4}$, i.e. hot populations with higher T[hot] show lower contributions to the
total population at low N^′. The strong change in r[pop,0] appears to be mainly caused by the decrease of the cold population with increasing v^′ since the $\mathrm{\Delta }{E}^{\prime }=\mathrm{0}$
intercepts of the lines describing the hot populations are located at similar y values of around −2 in Fig. 12. The fits for ${v}^{\prime }\le \mathrm{7}$ (no rejection of states) are convincing with
respect to the assumption of a homogeneous hot population, which can be described by a single temperature. Nevertheless, some fine structure might exist as the comparison of the individual
measurements and the fit lines suggest, although the possible population deviations appear to be not larger than 30%, which is small compared to population changes of the order of a magnitude in the
v^′-dependent energy ranges most contributing to T[hot]. Hence, the two-component fits are quite robust as the listed errors show. The highest uncertainties are related to ${v}^{\prime }=\mathrm{9}$
since the hot population is essentially constrained in an energy range of less than 300cm^−1, which includes 12 measurements with N^′ of 8 and 9.
Two-component fits were previously performed by Oliva et al. (2015) based on a near-infrared GIANO spectrum with a resolving power of 32000 taken during 2h with the spectrograph directly pointing
to the night sky at La Palma. The investigated lines belong to OH bands with low Δv and are complementary to those covered by our study. For the calculation of the populations, Oliva et al. (2015)
used the Einstein-A coefficients from van der Loo and Groenenboom (2007). For the fits, T[cold] was fixed at 200K. The resulting T[hot] and r[pop,0] varied from about 1300K and 1.8% for ${v}^{\
prime }=\mathrm{8}$ to about 7000K and 0.23% for ${v}^{\prime }=\mathrm{4}$. Although errors were not reported, these values are in good agreement with our results provided in Fig. 12. The fit
parameters for ${v}^{\prime }=\mathrm{9}$ are highly uncertain. However, Oliva et al. (2015) succeeded in fitting the populations for ${v}^{\prime }=\mathrm{2}$ and 3, which show an extension of the
trend found for the higher v^′. For ${v}^{\prime }=\mathrm{2}$, T[hot] and r[pop,0] resulted in 12000K and 0.14%, respectively. The GIANO data were refitted by Kalogerakis et al. (2018) with
unconstrained T[cold], which resulted in temperatures of about 190K but with larger scatter than in our case. For T[hot], the general trend was the same but with a large step from 900K for ${v}^{\
prime }=\mathrm{8}$ to 4000K for ${v}^{\prime }=\mathrm{7}$, which disagrees with our findings. Population ratios were not provided by Kalogerakis et al. (2018). Noll et al. (2018b) already
published populations related to ${v}^{\prime }=\mathrm{9}$ and P-branch lines based on the UVES data used in this study and B+16 Einstein-A coefficients. Kalogerakis (2019) fitted these populations
and found T[cold] and T[hot] of about 180 and 500K, respectively. Both temperatures are lower than our results, but the differences are less than 2 standard deviations. The fit of Kalogerakis (2019)
based on fewer data points seems to be related to a higher impact of the hot population at low N^′. Our results for T[hot] allow for an interesting comparison to the T[rot] of the nascent populations
of v^′ between 7 and 9, which were derived by Llewellyn and Long (1978) using laboratory data from Charters et al. (1971). Our best-fit T[hot] of 690±120, 1340±50, and 2180±50K agree well with their
760±20, 1230±30, and 1940±200K, which implies that the OH relaxation processes do not appear to significantly affect the hot populations of the highest v^′.
The previous discussion has shown that bimodality is a good concept for the description of the population distributions for each v^′. Moreover, the derived T[cold] are close to the expected effective
ambient temperatures for the v^′-dependent OH emission layers (Noll et al., 2016). The increasing trend of T[rot] derived from the lines with the lowest N^′ for increasing v^′ (Cosby and Slanger,
2007; Noll et al., 2015, 2017) is not found in the best-fit T[cold]. Hence, our fits could be used to estimate the non-LTE contributions to such T[rot], ΔT[NLTE], which are an issue for the use of T
[rot] as indicators of the temperatures in the mesopause region. Kalogerakis et al. (2018) and Kalogerakis (2019) compared T[cold] fits with T[rot] from linear regressions for levels with ΔE^′ lower
than 500 and 250cm^−1, respectively. The results indicate higher T[rot] than T[cold] at least for the highest v^′ (order of 20K). However, the uncertainties are large due to the strong impact of
the line selection (Noll et al., 2015), uncertainties in the line intensities (unclear for the GIANO data), and the choice of the Einstein-A coefficients (Fig. 7). Hence, we applied a different
approach by directly taking the two-component fit for the measurement of T[rot]. For this purpose, we derived the populations related to the first three P[1]-branch lines, which are often taken for T
[rot] determinations (e.g. Schmidt et al., 2013; Noll et al., 2016), from the fit curve at the corresponding v^′-dependent ΔE^′. The related T[rot] values were then calculated by a linear regression
of the three y for each v^′. Finally, the resulting ΔT[NLTE] is just the difference between T[rot] and T[cold]. This method is very robust as it is fully based on the two-component fit, which relies
on a high number of population measurements. Hence, uncertainties related to individual lines are negligible.
Our ΔT[NLTE] for v^′ between 4 and 9 are shown in Fig. 13. They increase relatively slowly between 4 and 6 from 1.2±0.1 to 1.8±0.1K and then faster to 6.0±1.6K for ${v}^{\prime }=\mathrm{9}$. As
the latter value is relatively uncertain, 4.6±0.3K for ${v}^{\prime }=\mathrm{8}$ might also be the maximum deviation. The errors were derived by displacing the hot component fit in both y
directions according to the uncertainty in r[pop,0] and refitting T[hot] as the only parameter to obtain modified ΔT[NLTE]. This approach considers that r[pop,0] and T[hot] are anticorrelated and
that the relative uncertainty in T[cold] is relatively small. Additional systematic uncertainties are caused by assuming only two components. It is required that the fit line for the hot component
can be linearly extrapolated to $\mathrm{\Delta }{E}^{\prime }=\mathrm{0}$. The good quality of the fits in the transition region between the dominance of the cold and hot components are promising.
Nevertheless, contributions of additional components of intermediate temperature cannot be excluded (as, for example, for ${v}^{\prime }=\mathrm{4}$ due to a possibly elevated T[cold]). Fits with
such an additional component and a fixed T[cold] of 190K, where the best-fit parameters of the intermediate and hot populations are not well constrained, showed possible ΔT[NLTE] increases by 10%
to 30%; i.e. the significance of positive non-LTE effects for all v^′ would remain high. It is hard to imagine situations where ΔT[NLTE] could significantly drop. A sharp cut of the hot population
for low N^′ would be inconsistent with a Boltzmann-like distribution as expected for relaxation processes. Another source of possible systematic errors are the Einstein-A coefficients, especially
with respect to their dependence on N^′. The latter was changed in Sect. 4.3 for the B+16 coefficients by the branch-specific corrections. Hence, we tested what happens if we consider the P- or R
-branch data as the standards instead of a combination of both. These modifications would change directly measured T[rot] to values as indicated in Fig. 7. However, the effect on the two-component
approach is much smaller. It is of the order of the already small fit errors for all v^′. As expected, non-LTE contributions related to R as the standard are lower than those for P. Furthermore, we
investigated the influence of the choice of the energy levels on ΔT[NLTE] by also simulating line sets consisting of the first two and first four P[1]-branch lines. For ${v}^{\prime }=\mathrm{8}$ as
an example, these changes cause ΔT[NLTE] of 3.3 and 6.9K, which clearly deviate from the plotted 4.6K. Hence, the non-LTE contributions are very sensitive to the selected energy levels, which is
consistent with the results from Noll et al. (2015, 2018b).
As shown in Fig. 13, we also calculated ΔT[NLTE] from the two-component fits of Oliva et al. (2015). Excluding their very uncertain fit parameters for ${v}^{\prime }=\mathrm{9}$, there is a very good
agreement with differences smaller than 0.4K. The only exception is ${v}^{\prime }=\mathrm{7}$, where our non-LTE contributions are about 1.6K lower. As the data basis and analysis were completely
different (including different Einstein-A coefficients), this convincing result demonstrates the robustness of the approach. The GIANO-related data for ${v}^{\prime }=\mathrm{2}$ and 3 suggest that Δ
T[NLTE] decreases only very slowly with decreasing v^′. The drop in r[pop,0] seems to be nearly compensated by the increase in T[hot].
Mean ΔT[NLTE] values for v^′ from 2 to 9 at Cerro Paranal were already derived by Noll et al. (2016) based on measurements of 25 OH bands and 2 O[2] bands (where non-LTE effects are less important)
in optical and near-infrared spectra from the echelle spectrograph X-shooter as well as from OH emission and kinetic temperature profile measurements with SABER. The ΔT[NLTE] from the complex
analysis for the first three P[1]-branch lines (derived from different band-specific line sets) are shown in Fig. 13. The values with a conspicuous maximum of 13.2±2.0K at ${v}^{\prime }=\mathrm{8}$
are clearly higher than those from the two-component fit. However, Noll et al. (2016) used HITRAN Einstein-A coefficients, which significantly deviate from our modified B+16 coefficients. As
demonstrated by Fig. 7, the impact on T[rot] can be large. Hence, we recalculated the ΔT[NLTE] of Noll et al. (2016) with the modified B+16 transition probabilities for the lines considered in Sect.
4.3 and the original ones (which result in about 2K higher T[rot] on average) in all other cases. Figure 13 indicates a clear reduction of ΔT[NLTE] for the relevant ${v}^{\prime }\ge \mathrm{4}$,
which better matches our results based on two-component population fits. Between v^′ of 4 and 7 the non-LTE contributions are almost constant with a mean of 2.5K. However, the absolute uncertainties
are larger. Only the maximum of 8.4±2.7K at ${v}^{\prime }=\mathrm{8}$ seems to be significant. It might also be present (but less pronounced) in the population fitting results. The high absolute
uncertainties from temperature comparisons (v^′-related differences are safer) are a critical drawback of that method and imply that two-component population fits provide the best constraints for ΔT
[NLTE], so far. The higher errors compared to the original Noll et al. (2016) data are partly related to the unavoidable mixture of corrected and uncorrected B+16 coefficients. However, B+16 line
parameters also appear to cause a larger scatter in T[rot] for different bands with the same v^′ compared to HITRAN data. The change in the ΔT[NLTE] differences between adjacent v^′ (especially
around ${v}^{\prime }=\mathrm{7}$) is mainly caused by a different calculation of T[rot] for the reference line set consisting of the first three P[1]-branch lines. Instead of using a constant
temperature offset for the conversion from the reference line set of Noll et al. (2015) including all P-branch lines up to ${N}^{\prime }=\mathrm{3}$ as discussed by Noll et al. (2016), we directly
corrected the band-specific T[rot] to be representative of the more recent reference line set.
5.2Population variability
The discussion of the roto-vibrational level populations of OH in Sect. 5.1 was only based on line intensity measurements in a single mean spectrum. We can learn more about these populations if we
also consider variations in the emission layer properties. In order to keep the signal-to-noise ratios high, we split the sample into two parts based on a characteristic layer parameter, calculated
the corresponding mean spectra, and derived level populations from the measured line intensities for a comparison. For the split, we selected the effective height of the OH emission layer h[eff],
i.e. the centroid altitude weighted by the volume emission rate, as it is positively correlated with the strength of the non-LTE effects (Noll et al., 2017, 2018a). There are fewer thermalising
collisions without v^′ change at higher altitudes due to lower air densities but higher atomic oxygen mixing ratios (Noll et al., 2018b). The impact of this effect is clearly reflected by the
observed higher h[eff] for higher v^′ (e.g. von Savigny et al., 2012), which are more affected by the hot nascent population and have lower effective lifetimes. According to the population modelling
of Noll et al. (2018b) for ${v}^{\prime }=\mathrm{9}$, h[eff] also increases for higher N^′.
In order to study the change of the population distribution for the different v^′ depending on the OH emission altitude, we need adequate space-based measurements of the emission profiles to be
linked with our ground-based OH level population data. This was already achieved by Noll et al. (2017) based on limb-sounding data for the Cerro Paranal region from the OH-specific channels of the
SABER radiometer (Russell et al., 1999). Here, we focus on the channel centred on 2.06µm, which covers OH(8-6) and OH(9-7). The effective v^′ is about 8.3 (Noll et al., 2016). Noll et al. (2017)
connected the resulting h[eff] for 4496 profiles to each UVES spectrum by a weighting procedure which involved temporal differences in day of year and local time measured in a two-dimensional
climatology. The approach also included the correction of differences in the solar activity, as measured by the solar radio flux (Sect. 3.2), for each UVES observation compared with the corresponding
weighted h[eff]. Finally, a most likely h[eff] was available for each UVES spectrum. We used these data to split our sample of 533 spectra (Sect. 2) at a median h[eff] of 89.2km for the 2.06µm OH
channel. The resulting subsamples show mean h[eff] of 88.7 and 89.6km; i.e. the height difference is almost 1km. The situation is very similar for the other OH channel at 1.64µm representing an
effective v^′ of about 4.6 (Noll et al., 2016), where the corresponding heights are 87.3 and 88.5km. Then, we performed the entire data analysis starting with the calculation of the mean spectra up
to the derivation of the final populations. In order to minimise systematic effects in the line measurement, the same wavelengths for the line integration and continuum derivation as for the full
sample spectrum were used (see also Sect. 3.2).
Figure 14 shows the resulting population ratios for the high and low h[eff] cases. For such a comparison, the choice of the Einstein-A coefficients does not matter. We only plot Δy related to the 257
best-quality Λ doublets (class 33) that are covered by all UVES spectra, i.e. reliable population ratios which are representative of the given h[eff]. We can identify three energy level regimes with
respect to the population change by a rise of the OH emission layer.
Up to about 600cm^−1, there is a general decrease of the populations, which strongly depends on v^′. From ${v}^{\prime }=\mathrm{4}$ to 9, the decrease shrinks from about 8% (−0.08) to about 1%;
i.e. the populations for higher v^′ are more stable. The largest difference in Δy is between the two lowest v^′. The decrease of the OH intensity for a rising OH emission layer is well known (Yee
et al., 1997; Melo et al., 1999; Liu and Shepherd, 2006). It is accompanied by a lower width of the layer (−0.5km for our subsamples) due to especially low OH production rates at the bottom side,
which are caused by a depletion of ozone. As band emissions with lower v^′ peak at lower altitudes (von Savigny et al., 2012; Noll et al., 2016), they seem to be more affected by this lack of fuel
for the OH production.
At ΔE^′ above 1200cm^−1, Fig. 14 shows a completely different behaviour. There is a general increase of the populations with a mean of +0.04 and no significant dependence on v^′. This finding
implies that the contribution of hot populations to the total population increases with h[eff]. The impact of non-LTE effects grows by a less efficient thermalisation process. The reduced
contributions from lower altitudes to the total emission certainly play an important role here since (similar to v^′) emission related to higher N^′ peaks higher in the atmosphere (Noll et al., 2018b
). There, collisional thermalisation of the rotational level population distributions is hampered by a relatively low density of nitrogen molecules and a relatively high volume mixing ratio of v^′
-deactivating (or even OH-destroying) atomic oxygen radicals (Noll et al., 2018b). Note that the location of the zero line in Fig. 14 is uncertain with respect to the degree of thermalisation as the
increase of 4% could also be caused by a change in the OH column density. If the hot populations define the zero line as they appear to be the most stable ones (which might be supported by the lack
of a v^′ dependence), the low-N^′ populations would further decrease.
Figure 14 is another good argument for the bimodality of rotational level population distributions. The change of y for low and high N^′ with increasing ΔE^′ seems to be very small. This suggests
that cold and hot populations are relatively homogeneous, which supports our two-temperature fit approach. Moreover, there is a quick transition between both populations in the relatively narrow ΔE^′
range between 600 and 1200cm^−1. In Fig. 12, this is the region where both fit components significantly contribute. Hence, a rise of the OH emission layer there should have the strongest impact on
the slope of the population distribution (i.e. T[rot]) by a change of the relative contribution of the cold and hot populations. As r[pop,0] increases, there is also an effect on ΔT[NLTE] as shown in
Fig. 13. Estimates of this quantity based on the populations for low and high OH layer are relatively uncertain due to the distinctly lower number of suitable lines and the high impact of increased
line intensity errors on the analysis of very small population differences. Nevertheless, we calculated ΔT[NLTE] changes of the order of a few tenths of a kelvin for the altitude difference of about
1km. This is clearly smaller than about 1K per kilometre, the order of magnitude from observational and modelling studies by Noll et al. (2017, 2018a, b), which might point to limitations in the
study of ΔT[NLTE] variations based on two-component population fits.
Based on averaged high-quality high-resolution spectra from the Ultraviolet and Visual Echelle Spectrograph at Cerro Paranal, we performed a detailed study of OH roto-vibrational level population
distributions. The mean populations for 723 Λ doublets with upper vibrational levels v^′ between 3 and 9 and upper rotational levels N^′ up to 24 were investigated. In about half the cases, the
doublet components were measured separately. The line wavelengths from literature (Rothman et al., 2013; Brooke et al., 2016) turned out to be sufficiently accurate in most cases. Only a small number
of lines with high N^′ and intermediate v^′ (especially ${v}^{\prime }=\mathrm{5}$) showed deviations by more than a few picometres.
The quality of population measurements is limited by uncertainties in the Einstein-A coefficients. We investigated this issue with comparisons of populations from different transitions with the same
upper state. We tested six sets of transition probabilities: Brooke et al. (2016), HITRAN (Rothman et al., 2013), van der Loo and Groenenboom (2008), Turnbull and Lowe (1989), Langhoff et al. (1986),
and Mies (1974). All sets fail in the case of Q-branch lines and the Λ-doublet components, where unexpectedly large intensity ratios are possible. The comparison of populations from P- and R-branch
lines indicated relatively small errors for the coefficients by Langhoff et al. (1986), van der Loo and Groenenboom (2008), and Brooke et al. (2016), whereas those from Turnbull and Lowe (1989) are
clearly the worst. The comparison of OH bands with the same v^′ showed a similar order of the different sets with respect to their quality. For this case, the coefficients of Brooke et al. (2016)
performed best. The widely used HITRAN data are only of intermediate and hence unsatisfactory quality.
For the population analysis, we focused on the Einstein-A coefficients from Brooke et al. (2016) due to their relatively good performance and the highest number of included lines. In order to
minimise the scatter in the populations, we further improved these coefficients by empirically correcting the found population discrepancies via regression lines related to N^′ and wavelength as well
as band-dependent correction factors. For the correction of the branch-related differences, we used P- and R-branch data combined with equal weights as the reference since this strongly reduced the
deviations between the different sets of Einstein-A coefficients with respect to rotational temperatures T[rot], i.e. the change of the populations with increasing N^′. The whole correction procedure
lowered the discrepancies in the coefficients by more than a factor of 2 for the measured lines. Nevertheless, the development of an improved set for all lines would need a more sophisticated
approach including modelling of the molecular parameters.
The resulting v^′-dependent population distributions show clearly bimodal structures, which were convincingly reproduced by two-temperature fits only excluding steep population decreases for ${v}^{\
prime }=\mathrm{8}$ and 9 at the highest N^′ with energies slightly below and above the exothermicity limit of the OH-producing hydrogen–ozone reaction, respectively. The fits show a cold population
with nearly ambient temperature of about 190K dominating at low N^′ and a hot population with temperatures between 700K for ${v}^{\prime }=\mathrm{9}$ and 7000K for ${v}^{\prime }=\mathrm{4}$ at
high N^′. In contrast, the ratio of the hot and cold populations at the level with the lowest energy of a given v^′ changes from 3 to 0.3% mainly due to a decrease of the cold component. The
significant contribution of a hot population to low N^′ causes deviations between T[rot] and ambient temperature, which we estimated by fitting our two-component model for the energy levels related
to the first three P[1]-branch lines. The results indicate non-LTE contributions that increase from about 1K for ${v}^{\prime }=\mathrm{4}$ to about 5K for ${v}^{\prime }=\mathrm{8}$. The best-fit
value for ${v}^{\prime }=\mathrm{9}$ is even higher (about 6K), but the fit uncertainties are by far the highest. In general, the applied approach is much more robust than the previously used method
based on comparisons of temperatures from different sources as it only weakly depends on uncertainties in the line intensities and Einstein-A coefficients. Our approach is mostly limited by the
reliability of the assumption of only two Boltzmann-like population distributions. There are hints of the existence of a more complex pattern, but the impact of these additional components appears to
be small.
This conclusion is supported by the change of the populations due to a rise of the OH emission layer, which we studied by the separation of the sample of spectra into two parts depending on the
effective emission height as obtained from height-resolved SABER OH volume emission rates. The energy regimes up to about 600cm^−1 and above about 1200cm^−1 relative to the lowest energy for a
given v^′ show clearly distinct variability in agreement with the energy ranges dominated by the cold and hot components in the derived population distributions. While the cold populations show a
decrease, which is stronger for lower N^′, the hot populations are relatively stable (or even increase) with increasing emission altitude. The largest measured effect is a 12% decrease of the cold
population at ${v}^{\prime }=\mathrm{4}$ relative to the hot population for a height difference of almost 1km.
The success of the two-component model for OH rotational level population distributions has implications for the thermalisation process of the highly non-thermal nascent populations. There are still
high uncertainties with respect to the rate coefficients for collisions with and without change of v^′. In particular, the modification of the rotational level population by v^′-changing collisions
is not known. Hence, the origin of the very hot populations at high N^′ of low v^′ is puzzling. Consequently, there is hope that the high-quality population data of this study can help to better
understand relaxation processes in OH by detailed modelling. This will be important knowledge with respect to the use of OH as an indicator of mesopause temperatures and for retrievals of atomic
abundances like those of oxygen.
This project made use of the ESO Science Archive Facility at http://archive.eso.org (ESO, last access: 2 May 2020). UVES Phase 3 spectra (release version 1) from different observing programmes of the
period from April 2000 to March 2015 were analysed. The v2.0 SABER data products used for this study were taken from http://saber.gats-inc.com (SABER Team, last access: 2 May 2020).
The supplement contains the data that are needed to reproduce Figs. 2 to 14. The supplement related to this article is available online at: https://doi.org/10.5194/acp-20-5269-2020-supplement.
SN has developed the project, processed the data, performed the analysis, produced the figures, and is the main author of the paper text, where all co-authors have made significant contributions. HW
and OG have also influenced the design of the study. In addition, HW has checked parts of the analysis and BP has been involved in the post-processing of the UVES Phase 3 products.
The authors declare that they have no conflict of interest.
We thank reviewer Ernesto Oliva and one anonymous referee for their positive and helpful reports.
Stefan Noll and this publication are financed by the German Research Foundation (DFG) (project NO 1328/1-1), and Holger Winkler is funded by the DFG (project NO 404/21-1).
This paper was edited by William Ward and reviewed by Ernesto Oliva and one anonymous referee.
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SPOJ.com - Problem TABLE
TABLE - Crash´s number table
In today's math lesson, Little Crash has just learnt Least Common Multiple (LCM). For two positive integers a and b, LCM(a, b) means the minimum positive integer which can be divisible by a and b.
After coming home, Crash is still thinking about what he learnt in the math lesson. Then he draw a table filled numbers in order to research LCM. The table has N rows and M columns. The number in the
ith row and jth column is LCM(i, j).
A table of 4*5 is just like this:
┃1│2│3 │4 │5 ┃
┃2│2│6 │4 │10 ┃
┃3│6│3 │12 │15 ┃
┃4│4│12 │4 │20 ┃
Now Little Crash wants to know the sum of all the numbers in the table. You just need to output the sum modulo 20101009.
Only two positive integers stand for N and M. (N, M <= 10^7)
A positive integer which means the sum modulo 20101009.
hide comments
pyy_official1: 2024-03-15 01:41:45
My solution runs about 0.2 second on my PC.But it gets TLE!
Last edit: 2024-03-15 01:42:11
Rohmad Raharjo: 2011-09-09 06:52:19
Would you like give me another test cases?
Nikmal Mursyidah: 2011-01-18 02:40:18
how to make a program runs faster?
my program always get TLE..
mike_nzk: 2010-12-25 06:14:36
My program runs about 1.4 seconds on my PC but gets TLE in SPOJ. It's so strange!
update:I finally AC the problem using a program running about 1.0 second on my PC.
Last edit: 2010-12-25 07:44:59
jiazhipeng: 2010-12-22 14:07:44
to xilinx
I didnt' do in that way because some parts of the program can run only once, but I want you to find the best solution for this part.
[Rampage] Blue.Mary: 2010-12-22 13:55:06
There's no need to set time limit 0.1 second for some test cases. I think a better way is to merge all test cases in 1 file and give out the total time limit. See problems added by me.
Last edit: 2010-12-22 13:56:09
jiazhipeng: 2010-12-22 02:47:08
to pratik:
My solution runs about 1 second on my PC.
.:: Pratik ::.: 2010-12-21 21:00:52
Is time limit strict the code in C++ runs on my machine in 5 seconds for worst case.
Added by: jiazhipeng
Date: 2010-12-20
Time limit: 0.100s-1.274s
Source limit: 50000B
Memory limit: 1536MB
Cluster: Cube (Intel G860)
Languages: C C++ 4.3.2 CPP C99 JAVA PAS-GPC PAS-FPC
Resource: Modified from task energy of NOI 2010
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CRP Toolbox
Removes thickening of diagonal lines in RP.
y = skel_crp(x) creates a new recurrence matrix y in which all recurrence points of the recurrence matrix x are removed which lead to a thickening of diagonal lines. Slubs, but also block structures
are removed in favour of the longest diagonal lines (skeletonization). Whenever a diagonal line (starting with the longest lines contained in the diagonal line length histogram) encounters an
adjacent diagonal line, this adjacent line and – recursively – all its consecutive adjacent lines, get deleted.
a = sin(linspace(0,5*2*pi,100));
X = crp(a,2,5,.5,'nonorm','nogui');
Y = skel_crp(X);
axis xy square
axis xy square
colormap([1 1 1; 0 0 0])
See Also
crp, dl,
Kraemer, K. H., Marwan, N.: Border effect corrections for diagonal line based recurrence quantification analysis measures, Phys. Lett. A, 383, 2019.
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ThmDex – An index of mathematical definitions, results, and conjectures.
Let $M = (X, \mathcal{F}, \mu)$ be a
D1158: Measure space
Let $E_1, \dots, E_N \in \mathcal{F}$ each be a
D1109: Measurable set
in $M$ such that
(i) $E_1, \dots, E_N$ is a D1681: Disjoint set collection
Then $$\mu \left( \bigcup_{n = 1}^N E_n \right) = \sum_{n = 1}^N \mu(E_n)$$
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e S
Case Studies
If you're interested in time series analysis and forecasting, this is the right place to be. The Time Series Lab (TSL) software platform makes time series analysis available to anyone with a basic
knowledge of statistics. Future versions will remove the need for a basic knowledge altogether by providing fully automated forecasting systems. The platform is designed and developed in a way such
that results can be obtained quickly and verified easily. At the same time, many advanced time series and forecasting operations are available for the experts. In our case studies, we often present
screenshots of the program so that you can easily replicate results.
Did you know you can make a screenshot of a TSL program window? Press Ctrl + p to open a window which allows you to save a screenshot of the program. The TSL window should be located on your main
Click on the buttons below to go to our case studies. At the beginning of each case study, the required TSL package is mentioned. Our first case study, about the Nile data, is meant to illustrate the
basic workings of the program and we advise you to start with that one.
Call center
Author: Rutger Lit
Date: July 05, 2022
Software: Time Series Lab - Home Edition
Topics: complex seasonal pattern
Batch program:
Call center
In this case study we analyse call center data to illustrate how to use TSL to model complex seasonal patterns. The series is used in the second application of the TBATS paper of De Livera, A.M.,
R.J. Hyndman, and R.D. Snyder (2011) and it can be downloaded from here. The call center time series consists of 10,140 observations on call arrivals per 5-minute interval between 7:00 AM and 9:00 PM
on weekdays. The series contains a daily seasonal pattern with period 169 (number of 5-minute intervals between 7:00 AM and 9:00 PM) and a weekly seasonal pattern with period 169 × 5 weekdays = 845.
Just as in De Livera, A.M., R.J. Hyndman, and R.D. Snyder (2011), we use 7605 observations (9 weeks) for our training sample which leaves 2535 observations (3 weeks) to analyse forecasting
performance. Note that in contrast to the data as shown in figure 1b and 5 of De Livera, A.M., R.J. Hyndman, and R.D. Snyder (2011), the data set we downloaded has two days of missing values (04/04/
2003 and 07/04/2003), see the figure below. On further inspection of figure 5 of De Livera, A.M., R.J. Hyndman, and R.D. Snyder (2011), we see that the corresponding days in their figure have a more
smooth pattern than the rest of the data so they might have used some fill-in values for the missing values. In TSL there is no need for this. Missing values are part of time series analysis, see
also Section 11.1.4 of the TSL manual.
Call center data set with two days of missing values
Building the model
We select a time-varying level, time-varying seasonal 1 with a period of 169 and 20 factors, and a time-varying seasonal 2 with a period of 845 and 10 factors.
Important: The number of factors, in combination with the time series length, strongly influences estimation times and higher numbers seldom lead to better forecasts, Therefore, it is strongly
advised to not choose the number of factors too high and it is almost never needed to go beyond 40.
Select a training sample of 7605 on the Estimation page and estimate the model. After TSL is done estimating, the graph page shows us the following figure:
Extracted level and two seasonals
From this figure we see that TSL has no problem with the missing data. The Kalman Filter / Smoother algorithm nicely interpolates all the selected components. Furthermore, we see from the top panel
that the Level is not smooth and it looks like the level itself picks up some dynamics. This is confirmed by the ACF of the predicted residuals which shows significant first order autocorrelation
among other lags. Before we fix this, let's first make some forecasts to compare this model to the following models. Go to the model comparison page and click the start loss calculation button in the
top right corner.
Our next modelling step is to add an ARMA(1,0) process to the existing model to do something about the first order autocorrelation that is still present in the residuals. Select the ARIMA(1,0)
component on the Build your own model page, leave everything else the same, and estimate the model. The result should be like the figure below. We see that the level is much smoother.
Extracted level and two seasonals + ARMA(1,0) errors
If we look at the ACF of the predicted residuals we see that the first order autocorrelation is still present. Again, let's make some forecasts to compare this model to the first and the following
models. Go to the model comparison page and click the start loss calculation button in the top right corner.
Seasonal variance extension
The default is to have one variance per seasonal component for all seasonal factors. In some situations, this is somewhat restrictive and estimating additional seasonal variances can improve model
fit and forecast performance. However, there are an extremely large number of factor combinations and machine learning needs to assist here since we cannot try all combinations. TSL has a machine
learning method that determines which seasonal factor gets its own variance parameter. To see this in action, go the top menu bar and click on Advanced settings and switch on Seasonal 1 variance
extension. Go to the Estimation page and make sure no parameters are set to fixed. Click on Estimate and wait till the algorithm is finished. This takes some time with an extensive model like this.
After the estimation is completed, go to the Model comparison page and start the loss calculation for this model as well. After that is completed you should see three check boxes in the top left
corner. When all are checked the resulting figure should look like the one below. The lowest loss line belongs to the last estimated model and is at least as good as the loss obtained from the TBATS
package as presented in De Livera, A.M., R.J. Hyndman, and R.D. Snyder (2011).
Keep in mind that the analysis performed by TSL in this case study is based on the call center time series with missing values.
Model comparison based on forecast performance
De Livera, A.M., R.J. Hyndman, and R.D. Snyder (2011). Forecasting Time Series With Complex Seasonal Patterns Using Exponential Smoothing. Journal of the American Statistical Association 106:496,
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Brainwallets: from the password to the address
Brainwallets are Bitcoin wallets generated uniquely from a passphrase that the users keeps in his mind so that it is required and sufficient to move the funds.
But what is actually the process that takes a password and spits a Bitcoin wallet address? Let’s dissect it.
1. From a password to a secret value
So, we have a password, but we need a fixed-size (256-bit) secret value to make our private key. This step can be done in a number of ways as it boils down to hashing the password but is crucial to
the strength of the resulting brainwallet.
Let’s have a look at how popular Brainwallet generators do it. (As of 20131204)
A lot of them just take the unsalted SHA256 hash of the password. This is wrong. Because SHA256 is fast and that means that an attacker can pregenerate huge tables of all possible brainwallets to
monitor and empty them (Spoiler: they do). This kind of thing – turning a human supplied password into a public hash – is exactly what password stretching are for, and not using them here is an
oversight as bad as not using them to store website user passwords, if not worse since here the hashes (the addresses) are public by default.
(Hint: use WarpWallet. It’s built by people who know what they are doing, and employs a proper KDF, making attacking your wallet really difficult.)
2. From the secret value to a private key
This is step is trivial. Actually, the output of the hashing above taken as a 256-bit unsigned number is already the private key, what is commonly called the secret exponent.
But we are used to see those pretty private keys beginning with a 5, so let’s see how it is encoded. That format is called WIF, Wallet import format, and it is pretty handy as it has checksumming
built in and employs a charset without confusing characters (Base58Check) – exactly like a Bitcoin address.
A snippet is worth a thousand words:
1 # Prepend the 0x80 version/application byte
2 private_key = b'\x80' + private_key
3 # Append the first 4 bytes of SHA256(SHA256(private_key)) as a checksum
4 private_key += sha256(sha256(private_key).digest()).digest()[:4]
5 # Convert to Base58 encoding
6 code_string = "123456789ABCDEFGHJKLMNPQRSTUVWXYZabcdefghijkmnopqrstuvwxyz"
7 value = int.from_bytes(private_key, byteorder='big')
8 output = ""
9 while value:
10 value, remainder = divmod(value, 58)
11 output = code_string[remainder] + output
3. From a private key to a public key
As Wikipedia tells us a ECDSA private key is just the scalar product of a private key (the secret exponent) and the curve – secp256k1 for Bitcoin – base point. How to do that is complex, but let’s
just take it for granted, as you’ll either use a librarty for this or research further by yourself.
What we get out of that operation is a pair (x, y) denoting a point on the curve, our public key.
4. From the public key to a Bitcoin address
We’re almost there! Now we just need to turn that ECDSA public key into a standard Bitcoin address.
The process is the same as point 4, executed on the SHA256+RIPEMD160 hash of the packed x and y values. Go go snippet:
1 # 1 byte 0x04, 32 bytes X, 32 bytes Y
2 public_key = b'\x04' + x.to_bytes(32, byteorder='big') + y.to_bytes(32, byteorder='big')
3 # Run SHA256 and RIPEMD-160 chained
4 address = ripemd160(sha256(public_key).digest())
5 # From now on it is point 4
6 # Prepend the 0x00 version/application byte for MainNet
7 address = b'\x00' + address
8 # Append the first 4 bytes of SHA256(SHA256(address)) as a checksum
9 address += sha256(sha256(address).digest()).digest()[:4]
10 # Convert to Base58 encoding
11 code_string = "123456789ABCDEFGHJKLMNPQRSTUVWXYZabcdefghijkmnopqrstuvwxyz"
12 value = int.from_bytes(address, byteorder='big')
13 output = ""
14 while value:
15 value, remainder = divmod(value, 58)
16 output = code_string[remainder] + output
17 # This wan not needed for the WIF format, but the encoding wants us to normalize the number
18 # (remove leading zeroes) and prepend a zero for each leading zero byte in the original
19 output = output.lstrip(code_string[0])
20 for ch in address:
21 if ch == 0: output = code_string[0] + output
22 else: break
And it’s done!
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Multiplication 6 Worksheet
Math, especially multiplication, forms the foundation of numerous scholastic disciplines and real-world applications. Yet, for many learners, grasping multiplication can present a challenge. To
resolve this obstacle, instructors and moms and dads have actually accepted a powerful device: Multiplication 6 Worksheet.
Intro to Multiplication 6 Worksheet
Multiplication 6 Worksheet
Multiplication 6 Worksheet -
These free 6 multiplication table worksheets for printing or downloading in PDF format are specially aimed at primary school students You can also make a multiplication worksheet yourself using the
worksheet generator These worksheets are randomly generated and therefore provide endless amounts of exercise material for at home or in class
These grade 6 math worksheets give additional computational practice particularly in column form multiplication and long division Sample Grade 6 Multiplication Worksheet More division worksheets
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Evolution of Multiplication 6 Worksheet
Multiplication 6 Worksheet
Multiplication 6 Worksheet
Free grade 6 multiplication worksheets to help your students improve their skills in Mathematics In grade 6 students must be able to multiply large numbers It is important for children to have a
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Basic worksheets for teaching kids to multiply by 6 Includes basic facts 0 through 6 as well as worksheets on 6s only and skip counting by 6 Multiplication by 6s Only Learn to Multiply by 6s After
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6 Times Table Multiplication Chart Exercise On 6 Times Table Table Of 6
6 Times Table Multiplication Chart Exercise On 6 Times Table Table Of 6
Multiply by 6 Samantha Jones Member for 3 years 5 months Age 7 10 Level 3 Language English en ID 521969 20 11 2020 Country code US Country United States School subject Math 1061955 Main content
Multiplication 2013181 Mulitply by 6 0 10 Share Print Worksheet Finish Mulitply by 6 0 10
Multiplication By 6 Worksheets helps your child learn and understand their tables in a slightly different way from learning about grouping to counting up and writing out their table facts Printable
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6 Times Table
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Multiplication Tables Check MTC Worksheets
6 Times Tables Worksheets
Multiplication Worksheets Numbers 1 6 PrintableMultiplication
6 Times Tables Worksheets
Grade 6 Multiplication Division Worksheets K5 Learning
These grade 6 math worksheets give additional computational practice particularly in column form multiplication and long division Sample Grade 6 Multiplication Worksheet More division worksheets
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6 Times Tables Worksheets
Kids Page 6 Times Tables Worksheets Maths Worksheets
Multiplication Worksheets Numbers 1 6 PrintableMultiplication
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6 Times Table Worksheets Activity Shelter
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How to Calculate an Elevation Gain for a Treadmill
Whenever we travel for work or when the weather goes against our training schedule, the trusty treadmill can be a real life saver to get the workout done. Here’s how you can get the elevation
training in using a treadmill!
Things You’ll Need: Smart Phone Calculator App
Step 1
Note the percent grade, or incline, setting of your treadmill. For example, 6 percent.
Step 2
Divide the percent grade by 100. For example, 6 / 100 = 0.06.
Step 3
Calculate the horizontal distance you have run on the treadmill. For example, you have run 2.5 kilometers. 2500m x cos [arctan (0.06)] = 2500m x cos [ 3.43º ] = 2500m x 0.998 = 2495m.
Step 4
Multiply your answer by 0.06. For example, 2495m x 0.06 = 149.7m. You have completed an elevation gain of approximately 150 meters (rounded up).
Using Arctangent
Work Out Slope Distance
Divide your target elevation gain by the percent grade and divide by the cosine of the arctangent of the slope ratio. For example, 150m / 0.06 = 2500m / 0.998 = 2505m.
Work Out Percent Grade
Divide your target elevation gain by the horizontal distance. For example, 150m / (2500m x 0.998)= 0.06 x 100 = 6.
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[Solved] CMSC 451 Project 2 Solved The second project involves comple
CMSC 451 Project 2 Solved The second project involves completing and extending the C++ program that evaluates statements of an expression language contained in the module 3 case study
Project 2
The second project involves completing and extending the C++ program that evaluates statements of an
expression language contained in the module 3 case study.
The statements of that expression language consist of an arithmetic expression followed by a list of
assignments. Assignments are separated from the expression and each other by commas. A semicolon
terminates the expression. The arithmetic expressions are fully parenthesized infix expressions
containing integer literals and variables. The valid arithmetic operators are +, –, *, /. Tokens can be
separated by any number of spaces. Variable names begin with an alphabetic character, followed by any
number of alphanumeric characters. Variable names are case sensitive. This syntax is described by BNF
and regular expressions in the case study.
The program reads in the arithmetic expression and encodes the expression as a binary tree. After the
expression has been read in, the variable assignments are read in and the variables and their values of
the variables are placed into the symbol table. Finally the expression is evaluated recursively.
Your first task is to complete the program provided by providing the three missing
classes, Minus, Times and Divide.
Next, you should extend the program so that it supports relational, logical and conditional expression
operators as defined by the following extension to the grammar:
-> '(' ')' |
'(' ':' '?' ')' |
'(' '!' ')'
-> '+' | '-' | '*' | '/' | '>' | '<' | '=' | '&' | '|'
Note that there are a few differences in the use of these operators compared to their customary use in
the C family of languages. Their differences are:
In the conditional expression operator, the symbols are reversed and the third operand
represents the condition. The first operand is the value when true and the second the value
when false
The logical operators use single symbols not double, for example the and operator is & not &&
The negation operator ! is a postfix operator, not a prefix one
There are only three relational operators not the usual six and the operator for equality
is = not ==
Like C and C++, any arithmetic expression can be interpreted as a logical value, taking 0 as false and
anything else as true
Your final task is to make the following two modifications to the program:
The program should accept input from a file, allowing for multiple expressions arranged one per
All results should be changed from double to int. In particular the evaluate function should
return an int.
You may assume that all input to the program is syntactically correct.
Deliverables for this project include the following:
1. Source code correctly implementing all required functionality. Your program must compile with
Microsoft Visual C++ or any modern C/C++ compiler on your O/S.
2. Word or PDF file providing screen shots of successfully compiling and executing the program.
3. Description of the process and lesson learned while completing this project (to be included in
the Word or PDF document).
4. A test plan that contains test cases that test all of the required operators. Each test case should
include the expression and its expected value (to be included in the Word or PDF document).
Grading rubric:
Attribute Meets Does not meet
Functionality 40 points
Completes the program provided in
Module 3 by providing the three
missing Classes:
Minus, Times and Divide.
0 points
Does not complete the program
provided in Module 3 by providing the
three missing Classes:
Minus, Times and Divide.
Extends Functionality 20 points
Extends the program so that it
supports relational, logical and
conditional expression operators.
All results should be changed
from double to int. In particular
the evaluate function should return
an int.
0 points
Does not extend the program so that
it supports relational, logical and
conditional expression operators.
All results should be changed
from double to int. In particular
the evaluate function should return
an int.
Input 20 points
Accepts input from a file, allowing
for multiple expressions arranged
one per line.
0 points
Does not accept input from a file,
allowing for multiple expressions
arranged one per line.
Documentation and
20 points
Includes source code correctly
implementing all required
0 points
Does not Include source code
correctly implementing all required
Program compiles with Microsoft
Visual C++ or any modern C/C++
compiler on your O/S.
Includes Word or PDF file providing
screen shots of successfully
compiling and executing the
Includes a description of the
process and lesson learned while
completing this project (to be
included in the Word or PDF
Includes a test plan that contains
test cases that test all of the
required operators. Each test case
should include the expression and
its expected value (to be included
in the Word or PDF document).
Program does not compile with
Microsoft Visual C++ or any modern
C/C++ compiler on your O/S.
Does not include Word or PDF file
providing screen shots of successfully
compiling and executing the program.
Does not include a description of the
process and lesson learned while
completing this project (to be included
in the Word or PDF document).
Does not include a test plan that
contains test cases that test all of the
required operators. Each test case
should include the expression and its
expected value (to be included in the
Word or PDF document)
Expert's Answer
2961 Times Downloaded
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1 Project 1 Introduction - the SeaPort Project series For this set of projects for the course, we wish to simulate some of the aspects of a number of
. Project 2 Introduction - the SeaPort Project series For this set of projects for the course, we wish to simulate some of the aspects of a number of Sea Ports. Here are the classes and their
instance variables we wish to define:
1 Project 2 Introduction - the SeaPort Project series For this set of projects for the course, we wish to simulate some of the aspects of a number of
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Isaac Newton Institute
The Isaac Newton Institute for Mathematical Sciences
is an international research institute for mathematics and its many applications at the
University of Cambridge
. It is named after one of the university's most illustrious figures, the mathematician and natural philosopher
Sir Isaac Newton
, and occupies one of the buildings in the Cambridge
Centre for Mathematical Sciences
Provided by Wikipedia
Published 2007
Cambridge University Press
...Isaac Newton Institute for Mathematical Sciences...
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EViews Help: graph
Create named graph object containing the results of a graph command, or created when merging multiple graphs into a single graph.
graph graph_name.graph_command(options) arg1 [arg2 arg3 ...]
graph graph_name.merge graph1 graph2 [graph3 ...]
Follow the keyword with a name for the graph, a period, and then a statement used to create a graph. There are two distinct forms of the command.
In the first form of the command, you create a graph using one of the graph commands, and then name the object using the specified name. The portion of the command given by,
graph_command(options) arg1 [arg2 arg3 ...]
should follow the form of one of the standard EViews graph commands:
Area graph (
area area
Area band graph (
band band
Bar graph (
bar bar
Boxplot graph (
boxplot boxplot
Distribution graph (
distplot distplot
Dot plot graph (
dot dot
Error bar graph (
errbar errbar
High-low(-open-close) graph (
hilo hilo
Line graph (
line line
Pie graph (
pie pie
Quantile-Quantile graph (
qqplot qqplot
Scatterplot—same as XY, but lines are initially turned off, symbols turned on, and a
scat scat
Matrix of scatterplots (
scatmat scatmat
Scatterplot pairs graph (
scatpair scatpair
Seasonal line graph (
seasplot seasplot
Spike graph (
spike spike
XY line-symbol graph with one X plotted against one or more Y’s using existing line-symbol settings (
xyarea xyarea
XY line-symbol graph with one X plotted against one or more Y’s using existing line-symbol settings (
xybar xybar
Same as XY, but symbols are initially turned off, lines turned on, and a
xyline xyline
Same as XY but sets XY settings to display pairs of X and Y plotted against each other (
xypair xypair
In the second form of the command, you instruct EViews to merge the listed graphs into a single graph, and then name the graph object using the specified name.
reset Resets all graph options to the global defaults. May be used to remove existing customization of the graph.
p Print the graph (for use when specified with a graph command).
Additional options will depend on the type of graph chosen. See the entry for each graph type for a list of the available options (for example, see
for details on bar graphs).
graph gra1.line(s, p) gdp m1 inf
creates and prints a stacked line graph object named GRA1. This command is equivalent to running the command:
line(s, p) gdp m1 inf
freezing the view, and naming the graph GRA1.
graph mygra.merge gr_line gr_scat gr_pie
creates a multiple graph object named MYGRA that merges three graph objects named GR_LINE, GR_SCAT, and GR_PIE.
“Graph Objects”
for a general discussion of graphs.
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GTM seminar
Speaker: Hsueh-Yung Lin (Kavli IPMU)
Title: Dynamical filtrations and applications
Date Thu, Jun 03, 2021, 15:30 - 17:00
Place: Zoom
(Joint with T.-C. Dinh, K. Oguiso, and D.-Q. Zhang) Let X be a compact Kähler manifold and let f be an automorphism of X (or more generally, a solvable group G acting on X). Given
Abstract: these data, we will introduce some filtrations on the space H^{1,1}(X) of (1,1)-classes of X, which capture the trade-off between the positivity of Kähler classes and the negativity
arising from (mixed) Hodge-Riemann relations. We will then explain how the fundamental properties of these filtrations lead to new upper bounds of various dynamical invariants, such as the
derived length of G among others, only in terms of the dimension of X.
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How to count the number of words in a cell in Excel - Free Excel Tutorial
This post explains that how to count the number of words in a cell using excel formula. How to count total words in a cell with User Defined Functions in excel VBA.
Count the number of words in a cell
If you want to count the number of words in a single cell, you can create an excel formula based on the IF function, the LEN function, the TRIM function and the SUBSTITUTE function. You need to use
the SUBSTITUTE function to remove all space strings from the text string in a cell, then the returned value goes into the LEN function to get the length of the text without space string. You still
need to get the length of the text with space strings and then subtract to the length of the text without space to get the number of spaces, then add 1 to get the number of words in a cell.
Assuming that you want to get the number of words in cell B1, you can write down an excel formula as follows:
=IF(LEN(TRIM(B1))=0, 0, LEN(TRIM(B1))-LEN(SUBSTITUTE(B1," ", ""))+1)
Let’s see how this formula works:
=LEN(TRIM(B1))-LEN(SUBSTITUTE(B1,” “, “”))+1
This formula returns the number of words in the text in Cell B1. The SUBSTITUTE function replaces all space strings with empty string to remove all spaces, and using LEN function to get the length of
the text without spaces.
The TRIM function will remove extra spaces from the text and just leave one space between words.
=IF(LEN(TRIM(B1))=0, 0, LEN(TRIM(B1))-LEN(SUBSTITUTE(B1,” “, “”))+1)
The IF function will check if it is an empty cell, if TRUE, then returns 0, otherwise, returns the number of words in the text in cell.
Count the number of words in a cell with User Defined Function
You can also create a new user defined function to count the number of words in a cell in Excel VBA:
1# click on “Visual Basic” command under DEVELOPER Tab.
2# then the “Visual Basic Editor” window will appear.
3# click “Insert” ->”Module” to create a new module
4# paste the below VBA code into the code window. Then clicking “Save” button.
Function countTotalWordCell(rng As Range) As Integer
countTotalWordCell = UBound(Split(rng.Value, " "), 1) + 1
End Function
5# back to the current worksheet, then enter the below formula in Cell C1:
Related Formulas
• count specific words in a cell or a range
If you want to count the number of a specific word in a single cell, you need to use the SUBSTITUTE function to remove all that certain word in text string, then using LEN function to calculate
the length of the substring that without that specific word.…
• Extract word that starting with a specific character
Assuming that you have a text string that contains email address in Cell B1, and if you want to extract word that begins with a specific character “@” sign, you can use a combination with the
TRIM function, the LEFT function, the SUBSTITUTE function, the MID function, the FIND function, the LEN function and the REPT function to create an excel formula.…
Related Functions
• Excel Substitute function
The Excel SUBSTITUTE function replaces a new text string for an old text string in a text string.The syntax of the SUBSTITUTE function is as below:= SUBSTITUTE (text, old_text, new_text,
• Excel IF function
The Excel IF function perform a logical test to return one value if the condition is TRUE and return another value if the condition is FALSE. The IF function is a build-in function in Microsoft
Excel and it is categorized as a Logical Function.The syntax of the IF function is as below:= IF (condition, [true_value], [false_value])….
• Excel LEN function
The Excel LEN function returns the length of a text string (the number of characters in a text string).The syntax of the LEN function is as below:= LEN(text)…
• Excel TRIM function
The Excel TRIM function removes all spaces from text string except for single spaces between words. The syntax of the TRIM function is as below:=TRIM(text)…
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McLaughlin, Richard - Department of Mathematics
McLaughlin, Richard M
Phillips Hall 320
Research Interests
Experimental, theoretical, and computational fluid dynamics, random phenomena, and stochastic partial differential equations
Professional Background
B.S Mathematics, the University of Arizona 1989; PhD Applied and Computational Mathematics, Princeton University, 1994; Wylie Instructor/NSF Postdoc, University of Utah, 1994-1996; Assistant
Professor, University of Utah 1996-1998; Associate Professor, University of North Carolina, 1998-2004; Full Professor, UNC, 2004-present; Chair of Mathematics, 2013-2023
Research Synopsis
My own work is in fundamental fluid dynamics. I use a blend of asymptotic and stochastic analysis, Monte-Carlo simulation, and experimental methods to uncover interesting fluid phenomena. Roberto
Camassa and I built a large-scale modern facility for exploring fundamental fluid dynamics, hosting a 120 foot long modular wave-tank, a tilting wind tunnel, a salt water processing center, as well a
huge array of instruments for making scientific measurements. Mathematics manages the facility which we share it with faculty in Marine Sciences, and have joint students and postdocs working in the
lab from math, physics, marine sciences, environmental science, computer science, and biology. Our scientific philosophy is to probe and unearth intriguing fluid phenomena and in turn to develop
predictive, first principled, mathematical theory to explain that phenomena. We’ve been fortunate to have made a number of exciting discoveries through this effort, including levitation phenomena
in settling particulates in stratified fluids, critical phenomena for the escape/trapping of fluid jets, blocking phenomena in shear flows past fixed bodies, paths of least time in potential flow,
discovering how geometry can be used to control asymmetries in solute delivery, and most recently a truly novel self-assembly mechanism by which particles suspended within a stratified fluid attract
seemingly to solve jig-saw like puzzles on its way to forming a large scale aggregate disc (this work appeared in Dec 2019 at Nature Communications, where it made its list of the top 50 most read
physics papers of 2019).
Representative Publications
Enhanced Diffusivity and Skewness of a Diffusing Tracer in the Presence of an Oscillating Wall
Lingyun Ding, Robert Hunt, Richard M. McLaughlin, and Hunter Woodie,
Research in the Mathematical Sciences, L. Ding et al. Res Math Sci, 2021, 8:34, 2021
Persisting Asymmetry in the Probability Distribution Function for a Random Advection–Diffusion Equation in Impermeable Channels
Roberto Camassa, Lingyun Ding, Zeliha Kilic, Richard M. McLaughlin,
Physica D, R. Camassa, L. Ding, Z. Kilic et al., Physica D 425, 2021, 132930, 2021
A First-Principle Mechanism for Particulate Aggregation and Self-Assembly in Stratified Fluids
R .Camassa, D. Harris, R. Hunt, Z. Kilic, and R. M. McLaughlin,
Nature Communications, 10, 5804, 2019
How Boundaries Shape Chemical Delivery in Microfluidics
M. Aminian, F. Bernardi, R. Camassa, R Harris, and R. M. McLaughlin,
Science, 354, 6317, 1252-1256, 2016
Squaring the Circle: Geometric Skewness and Symmetry Breaking for Passive Scalar Transport in Ducts and Pipes
M. Aminian, F. Bernardi, R. Camassa, and R. M. McLaughlin,
Physical Review Letters, 115, 154503, 2015
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MRI Fingerprinting with Philips | Jan Heiland Personal Homepage
MRI Fingerprinting with Philips
Apr 15, 2021 · 2 min read
Dr. Manuel Baumann (Philips Research), Jun-Prof. Jan Heiland FMA
A highly motivated Masters’ student of mathematics or statistics that feels comfortable with the problem below and that, at best, has good command of a programming language.
Problem Statement
MR Fingerprinting is a new, quantitative imaging technique in Magnetic Resonance Imaging (MRI)^1. In short,
1. MR Fingerprinting relies on the simulation of the Bloch equation, a parameter-dependent ODE for the magnetization $M$ of the form
$$\dot{M}(t) = f(M(t), B(t), T_1, T_2),$$
where the magnetic field $B$ is determined by the parameters of the acquisition.
2. Variation of the relaxation times $T_1$ and $T_2$ yield a series of trajectories forming a so-called dictionary.
3. Fingerprinting means the matching of the acquired under-sampled data with the dictionary entries for querying the relevant tissue-specific parameters.
The development and implementation opens a number of options for a Master’s project:
• System Theory: Re-formulation of the MRF dictionary computation as an input-state-output system and model order reduction of the underlying Bloch equation. While the reformulation enables the
matching by established system identification routines, a reduced order model can be used for a-posteriori checks of the selected parameters or for enriching the dictionary in the relevant
parameter range.
• Statistics: The estimation of parameters based on data is a common task in statistics. Apart from implementing and testing relevant routines for Fingerprinting, the inclusion of tailored
statistical approaches for the particular problem of Fingerprinting can be useful for improving the dictionary in general and for providing confidence estimates for the selection obtained from
classical matching.
• Optimization: Fingerprinting seeks for the best match of collected data with the precomputed dictionary entries. Tools from mathematical optimization will be used to improve the reliability of
the selected optimum and to enhance both the data aquisition and the dictionary through optimal design.
How to apply
Please see the job advertisement and apply by June 30.
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Model an Excavator Dipper Arm as a Flexible Body
The Reduced Order Flexible Solid block models a deformable body based on a reduced-order model that characterizes the geometric and mechanical properties of the body. The basic data imported from the
reduced-order model includes:
• A list of coordinate triples that specify the position of all interface frame origins relative to a common reference frame.
• A symmetric stiffness matrix that describes the elastic properties of the flexible body.
• A symmetric mass matrix that describes the inertial properties of the flexible body.
There are several ways to generate the reduced-order data required by this block. Typically, you generate a substructure (or superelement) by using finite-element analysis (FEA) tools.
This example uses the Partial Differential Equation Toolbox™ to create a reduced-order model for a flexible dipper arm, such as the arm for an excavator or a backhoe. You start with the CAD geometry
of the dipper arm, generate a finite-element mesh, apply the Craig-Bampton FEA substructuring method, and generate a reduced-order model. The model ReducedOrderFlexibleSolid uses the reduced-order
data from this example. In the model, the dipper arm is mounted on top of a rotating tower as part of a test rig. For more information, see Using the Reduced Order Flexible Solid Block - Flexible
Dipper Arm.
Add geometry file to the search path for the current MATLAB® session:
Step 1: Define the Geometry and Material Properties of the Dipper Arm
The file Dipper.STL contains a triangulation that defines the CAD geometry of the dipper arm. To view the geometry stored in this file, use the MATLAB® functions stlread and trisurf:
stlFile = 'Dipper.STL';
axis equal
The dipper arm is constructed from steel. To represent its material properties, set these values for Young's modulus, Poisson's ratio, and mass density:
E = 200e9; % Young's modulus in Pa
nu = 0.26; % Poisson's ratio (nondimensional)
rho = 7800; % Mass density in kg/m^3
Step 2: Specify the Locations of Interface Frames
The dipper arm has three interface frames where you can connect other Simscape™ Multibody™ elements, such as joints, constraints, forces, and sensors:
• The cylinder connection point, where the arm connects to a hydraulic cylinder that actuates the arm vertically.
• The bucket connection point, where the arm connects to the excavator bucket.
• The fulcrum point, where the arm connects to the excavator boom.
The positions of all interface frame origins are specified in meters relative to same common reference frame used by the CAD geometry.
origins = [-0.500 0 0 % Frame 1: Cylinder connection point
1.500 0 0 % Frame 2: Bucket connection point
0 -0.130 0]; % Frame 3: Fulcrum point
numFrames = size(origins,1);
Step 3: Create the Finite-Element Mesh
To generate the mesh for the dipper arm, first call the createpde (Partial Differential Equation Toolbox) function, which creates a structural model for modal analysis of a solid (3-D) problem. After
importing the geometry and material properties of the arm, the generateMesh (Partial Differential Equation Toolbox) function creates the mesh.
feModel = createpde('structural','modal-solid');
structuralProperties(feModel, ...
'YoungsModulus',E, ...
'PoissonsRatio',nu, ...
generateMesh(feModel, ...
'GeometricOrder','quadratic', ...
'Hmax',0.2, ...
Step 4: Set up the Multipoint Constraints for the Interface Frames
Each interface frame on the block corresponds to a boundary node that contributes six degrees of freedom to the reduced-order model. There are several ways to ensure that the FEA substructuring
method preserves the required degrees of freedom. For example, you can create a rigid constraint to connect the boundary node to a subset of finite-element nodes on the body. You can also use
structural elements, such as beam or shell elements, to introduce nodes with six degrees of freedom.
This example uses a multipoint constraint (MPC) to preserve the six degrees of freedom at each boundary node. To identify the geometric regions (such as faces, edges, or vertices) to associate with
each MPC, first plot the arm geometry by using the function pdegplot (Partial Differential Equation Toolbox):
You can zoom, rotate, and pan this image to determine the labels for the faces corresponding to the boundary nodes. These faces define the MPCs associated with the boundary nodes in the dipper arm:
• Cylinder connection point: face 1
• Bucket connection point: face 27
• Fulcrum point: face 23
faceIDs = [1,27,23]; % List in the same order as the interface frame origins
To verify these values, plot the mesh and highlight the selected faces:
hold on
colors = ['rgb' repmat('k',1,numFrames-3)];
assert(numel(faceIDs) == numFrames);
for k = 1:numFrames
nodeIdxs = findNodes(feModel.Mesh,'region','Face',faceIDs(k));
scatter3( ...
feModel.Mesh.Nodes(1,nodeIdxs), ...
feModel.Mesh.Nodes(2,nodeIdxs), ...
feModel.Mesh.Nodes(3,nodeIdxs), ...
scatter3( ...
origins(k,1), ...
origins(k,2), ...
origins(k,3), ...
hold off
Call the function structuralBC (Partial Differential Equation Toolbox) to define the MPCs for the boundary nodes in these faces:
for k = 1:numFrames
structuralBC(feModel, ...
'Face',faceIDs(k), ...
'Constraint','multipoint', ...
Step 5: Generate the Reduced-Order Model
The function reduce (Partial Differential Equation Toolbox) applies the Craig-Bampton order reduction method and retains all fixed-interface modes up to a frequency of $1{0}^{4}$ radians per second.
rom = reduce(feModel,'FrequencyRange',[0 1e4]);
Store the results of the reduction in a data structure arm. Transpose the ReferenceLocations matrix to account for the different layout conventions used by Partial Differential Equation Toolbox and
Simscape Multibody.
arm.P = rom.ReferenceLocations'; % Interface frame locations (n x 3 matrix)
arm.K = rom.K; % Reduced stiffness matrix
arm.M = rom.M; % Reduced mass matrix
The function computeModalDampingMatrix, which is defined at the bottom of this page, computes a reduced modal damping matrix with a damping ratio of 0.05:
dampingRatio = 0.05;
arm.C = computeModalDampingMatrix(dampingRatio,rom.K,rom.M);
The boundary nodes in the reduced-order model must be specified in the same order as the corresponding interface frames on the block. This order is given by the rows of the array origins. If the
order of the MPCs is different than the order specified by origins, permute the rows and columns of the various matrices so that they match the original order.
frmPerm = zeros(numFrames,1); % Frame permutation vector
dofPerm = 1:size(arm.K,1); % DOF permutation vector
assert(size(arm.P,1) == numFrames);
for i = 1:numFrames
for j = 1:numFrames
if isequal(arm.P(j,:),origins(i,:))
frmPerm(i) = j;
dofPerm(6*(i-1)+(1:6)) = 6*(j-1)+(1:6);
assert(numel(frmPerm) == numFrames);
assert(numel(dofPerm) == size(arm.K,1));
arm.P = arm.P(frmPerm,:);
arm.K = arm.K(dofPerm,:);
arm.K = arm.K(:,dofPerm);
arm.M = arm.M(dofPerm,:);
arm.M = arm.M(:,dofPerm);
arm.C = arm.C(dofPerm,:);
arm.C = arm.C(:,dofPerm);
Step 6: Import Reduced-Order Data
The model ReducedOrderFlexibleSolid uses the data structure arm to set up the parameters of the Reduced Order Flexible Solid block. In the block, these parameters import the reduced-order data:
• Origins: arm.P
• Stiffness Matrix: arm.K(1:24,1:24)
• Mass Matrix: arm.M(1:24,1:24)
• Damping Matrix: arm.C(1:24,1:24)
For more information, see Using the Reduced Order Flexible Solid Block - Flexible Dipper Arm.
Compute the Modal Damping Matrix
This function computes a modal damping matrix associated with the stiffness matrix K and mass matrix M. This function applies a single scalar damping ratio to all of the flexible (non-rigid-body)
normal modes associated with K and M.
function C = computeModalDampingMatrix(dampingRatio,K,M)
% To avoid numerical issues (such as complex eigenvalues with very small
% imaginary parts), make the matrices exactly symmetric.
K = (K+K')/2; % Stiffness matrix
M = (M+M')/2; % Mass matrix
% Compute the eigen-decomposition associated with the mass and stiffness
% matrices, sorting the eigenvalues in ascending order and permuting
% the corresponding eigenvectors.
[V,D] = eig(K,M);
[d,sortIdxs] = sort(diag(D));
V = V(:,sortIdxs);
% Due to small numerical errors, the six eigenvalues associated with the
% rigid-body modes may not be exactly zero. To avoid numerical issues,
% check that the first six eigenvalues are close enough to zero. Then
% replace them with exact 0 values.
assert(all(abs(d(1:6))/abs(d(7)) < 1e-9),'Error due to "zero" eigenvalues.');
d(1:6) = 0;
% Vectors of generalized masses and natural frequencies
MV = M*V;
generalizedMasses = diag(V'*MV);
naturalFrequencies = sqrt(d);
% Compute the modal damping matrix associated with K and M
C = MV * diag(2*dampingRatio*naturalFrequencies./generalizedMasses) * MV';
See Also
Reduced Order Flexible Solid | stlread | trisurf | createpde (Partial Differential Equation Toolbox) | importGeometry (Partial Differential Equation Toolbox) | structuralProperties (Partial
Differential Equation Toolbox) | generateMesh (Partial Differential Equation Toolbox) | pdegplot (Partial Differential Equation Toolbox) | structuralBC (Partial Differential Equation Toolbox) |
reduce (Partial Differential Equation Toolbox)
Related Topics
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A Practical Guide To Making Lineweaver-Burk Plots In Excel » The Tech Glance
A Practical Guide to Making Lineweaver-Burk Plots in Excel
The Lineweaver Burk Plot, also referred to as a Double Reciprocal Plot, is the output of the energy kinetics Lineweaver Burk Equation in the field of biochemistry. Users may therefore be unsure of
how to Making Lineweaver-Burk Plots in Excel.You’ve found the right spot if you’re looking for an easy way to make a Lineweaver Burk plot in Excel. This tutorial will walk you through Making
Lineweaver-Burk Plots in Excel so you can quickly see the connection between two sets of variables.
We’ll also go over some of the main characteristics of a Lineweaver Burk plot and discuss why this kind of plot is so helpful. You’ll get everything you need by the end of the article to create an
Excel Lineweaver Burk plot and make the most of your data. then let’s get going!
Step by Step Guide to Making Lineweaver-Burk Plots in Excel
What is a Lineweaver-Burk plot?
An illustration of the Michaelis-Menten equation for enzyme kinetics is called a Lineweaver-Burk plot. The reaction velocity (V) versus the reciprocal of the substrate concentration ([S]) are plotted
in a double reciprocal graph. The plot is a straight line with an x-intercept of -1/Km and a slope of Km/Vmax. 1/Vmax is the y-intercept value.
Determine the kinetic parameters of enzymes, such as Vmax and Km, using the Lineweaver-Burk plot. Additionally, it can be utilized to differentiate between various kinds of enzyme inhibition.
Excel Setup for Lineweaver-Burk Plots
Step by Step Excel Setup for Lineweaver-Burk Plots:
• Setting up the data correctly is the first step in making a Lineweaver-Burk display in Excel.
• Two columns should be used to organize the data in a table: the first column should hold the substrate concentrations, and the second should contain the reaction rates.
• Select the data once it has been properly formatted, and then click the “Insert” tab on the ribbon.
• Select the first scatter plot from the drop-down menu by selecting the “Scatter” button in the Charts section. The data will then be plotted as a scatter plot.
Calculations in Excel
These calculations are often used to determine the kinetic parameters of an enzyme-catalyzed reaction, such as the Michaelis-Menten constant (Km) and the maximum velocity (Vmax).
Step-by-step instructions on calculating the inverse of substrate concentration [1/[S]] and initial velocity [1/V0]:-
Step 1: Compile the required information.
You will require experimental data from an enzyme-catalyzed reaction in which the beginning velocity (V0) was recorded at various substrate concentrations ([S]). A variety of substrate concentrations
should ideally be represented in your data, so make sure to do this.
Step 2: Arrange the information
Make a table with two columns: one for the initial velocity (V0) and one for the substrate concentration ([S]). Make that the substrate concentrations and velocities are expressed in the same units
(for example, Molar and M/s).
Step 3: Inversely transform substrate concentrations ([1/[S]]).
Calculate the inverse of the concentration by taking the reciprocal (1/[S]) for each substrate concentration value ([S]) in your table.
Step 4: Convert initial velocities to the inverse ([1/V0])
Calculate the inverse of the velocity by getting the reciprocal (1/V0) for each beginning velocity (V0) value in your table.
Step 5: Visualize the data
Make a scatter plot with the beginning velocity ([1/V0]) and the inverse of substrate concentration ([1/[S]]) on the x- and y-axes, respectively. A data pair from your table is represented by each
point on the plot.
Step 6: Analyze the plot
The plot should show a straight line for enzyme-catalyzed reactions that follow Michaelis-Menten kinetics. This line’s equation is as follows:
y = (Km/Vmax) * x + (1/Vmax)
• y is the inverse of initial velocity ([1/V0])
• x is the inverse of substrate concentration ([1/[S]])
• Km is the Michaelis-Menten constant
• Vmax is the maximum velocity of the reaction
Step 7: Determine Km and Vmax
You can determine Km and Vmax by calculating the slope and intercept of the line you produced in Step 6. You may find the value of Km from the line’s slope (Km/Vmax) and the value of Vmax from the
line’s y-intercept (1/Vmax).
Step 8: Finalize the results
Now that you have calculated Km and Vmax, report these values as the kinetic parameters of the enzyme-catalyzed reaction.
NOTE:- Remember that the validity of the assumptions made for the kinetic model used—in this example, Michaelis-Menten kinetics—as well as the quality of the experimental data determine the
correctness and dependability of your results. Alternative kinetic models may need to be taken into account if your data does not fit a straight line on the plot and instead indicates departures from
straightforward Michaelis-Menten kinetics.
How to Making Lineweaver-Burk Plots in Excel
You’ll need the following information to make a Lineweaver-Burk plot:
The rate of the reaction (V) at various substrate concentrations ([S]). Following these steps will allow you to construct the plot after you have the data:
• Plot 1/[S] and 1/[V] on the x and y axes, respectively.
• A straight line should be drawn through the data points.
• Km/Vmax is the line’s slope.
• The line’s x-intercept is -1/Km.
• The line’s y-intercept is equal to 1/Vmax.
Lineweaver-Burk plot advantages
Comparing Lineweaver-Burk plots to other enzyme kinetic plots, like the Michaelis-Menten plot, reveals significant advantages. These benefits consist of:
• They are simpler to visually assess.
• They can be used to more precisely calculate the kinetic parameters of enzymes.
• They are useful for differentiating between various kinds of enzyme inhibition.
Disadvantages of Lineweaver-Burk plots
Additionally, there are some drawbacks to Lineweaver-Burk plots, such as:
• Compared to other forms of enzyme kinetic plots, they are less sensitive to variations in Vmax.
• Experimental flaws in the measurement of V and [S] may have an impact on them.
Troubleshooting Common Issues
If the data does not follow a linear trendline, the Michaelis-Menten equation may not be being followed by the enzyme or the data may not be in the proper format. Before attempting to generate a
Lineweaver-Burk plot in this situation, it is preferable to verify the data and ensure that it is in the proper format.
It is possible that the data is not exact enough if it fits a linear trendline yet the estimated values are inaccurate. In this situation, it is best to get more accurate data before retrying.
Read More : 5 Proven Methods for Fixing Black Spots on HP Laptop Screens
Clear PowerShell Previous command history
For those studying or researching enzyme kinetics, learning how to Making Lineweaver-Burk plot in Excel is a valuable skill. You now possess the skills necessary to create this potent graph, which
can help you determine important variables like Km and Vmax. Excel is a great tool for doing enzymatic analysis without the requirement for specialized software because of its usability and
This blog post explains how to Making Lineweaver-Burk Plots in Excel step-by-step. We hope that this article provides you with enough information about creating a Lineweaver Burk layout to allow you
to do so.
Hello, my name is Rishabh Kumar and I am the author of TheTechGlance.com. I am fond of writing and I have done engineering from NIT Hamirpur due to which I have good knowledge of technology, AI,
Crypto and network.
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The Stacks project
Example 39.5.3 (Additive group scheme). Consider the functor which associates to any scheme $T$ the group $\Gamma (T, \mathcal{O}_ T)$ of global sections of the structure sheaf. This is representable
by the scheme
\[ \mathbf{G}_ a = \mathop{\mathrm{Spec}}(\mathbf{Z}[x]) \]
The morphism giving the group structure is the morphism
\begin{eqnarray*} \mathbf{G}_ a \times \mathbf{G}_ a & \to & \mathbf{G}_ a \\ \mathop{\mathrm{Spec}}(\mathbf{Z}[x] \otimes _{\mathbf{Z}} \mathbf{Z}[x]) & \to & \mathop{\mathrm{Spec}}(\mathbf{Z}[x]) \
\ \mathbf{Z}[x] \otimes _{\mathbf{Z}} \mathbf{Z}[x] & \leftarrow & \mathbf{Z}[x] \\ x \otimes 1 + 1 \otimes x & \leftarrow & x \end{eqnarray*}
Hence we see that $\mathbf{G}_ a$ is a group scheme over $\mathbf{Z}$. For any scheme $S$ the base change $\mathbf{G}_{a, S}$ is a group scheme over $S$ whose functor of points is
\[ T/S \longmapsto \mathbf{G}_{a, S}(T) = \mathbf{G}_ a(T) = \Gamma (T, \mathcal{O}_ T) \]
as before.
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Re: st: Biprobit and clustering standard errors
Notice: On April 23, 2014, Statalist moved from an email list to a forum, based at statalist.org.
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Re: st: Biprobit and clustering standard errors
From Stas Kolenikov <[email protected]>
To [email protected]
Subject Re: st: Biprobit and clustering standard errors
Date Wed, 7 Sep 2011 09:09:57 -0500
Yes. You will have (at least) two issues.
1. Your variance-covariance matrix, -vce-, will not be of full rank.
Hence, you won't be able to estimate variances of certain combinations
of parameters (there is no telling which combinations will be affected
2. If you have but few clusters, the assumptions of the asymptotic
behavior may not be satisfied. The standard errors will suffer from
small sample biases, and the test statistics (z-statistics or
likelihood ratios) will have distributions different from their
asymptotic targets (normal or chi-squared distributions,
As a background, Stata (or any other statistical software) needs to
compute the likelihood scores, i.e., the derivatives of the likelihood
wrt to the parameters of the model. For variance estimation purposes,
you would need to have as many scores (represented by temporary
variables used in -robust- or -cluster- calculations) as you have
parameters. So this is not the number of variables, really, but the
number of parameters that matters.
On Wed, Sep 7, 2011 at 5:20 AM, Lina C <[email protected]> wrote:
> Hello everybody.
> I'm running a biprobit clustering the standard errors as follows:
> biprobit ( y1 = y2 x ) ( y2 = z x), robust cluster(area)
> The "x" vector of regressors is much below the number of clusters
> (areas), however Stata cannot calculate the chi_2. What I have noticed
> is that STATA use the sum of the X in both equations as the total
> number of regressors, and in this way the "x" of the first probit and
> the "x" of the second probit sum up a number that is above the number
> of clusters. Once I reduced the X to be, the sum in the first probit
> and in the second probit, below the number of clusters, the chi2
> appears..
> The problem is that I need to use more regressors..Is there a problem
> if I rely on that estimation with the missing estimation of the chi2?
> Thank you.
> Lina.
> *
> * For searches and help try:
> * http://www.stata.com/help.cgi?search
> * http://www.stata.com/support/statalist/faq
> * http://www.ats.ucla.edu/stat/stata/
Stas Kolenikov, also found at http://stas.kolenikov.name
Small print: I use this email account for mailing lists only.
* For searches and help try:
* http://www.stata.com/help.cgi?search
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/
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Using Python to Get All Combinations of Two Lists
In Python, we can get all combinations of two lists easily. The easiest way to obtain all combinations of two lists is with list comprehension.
list1 = ["a", "b", "c"]
list2 = [1, 2, 3]
combinations = [(x,y) for x in list1 for y in list2]
[('a', 1), ('a', 2), ('a', 3), ('b', 1), ('b', 2), ('b', 3), ('c', 1), ('c', 2), ('c', 3)]
You can also use a for loop to get all combinations of two list objects in Python.
list1 = ["a", "b"]
list2 = [1, 2]
def combinations(lst1, lst2):
result = []
for x in lst1:
for y in lst2:
return result
[('a', 1), ('a', 2), ('b', 1), ('b', 2)]
Finally, the Python itertools module has a function product() which finds the Cartesian product, or all combinations, for you.
from itertools import product
list1 = ["a", "b"]
list2 = [1, 2]
[('a', 1), ('a', 2), ('b', 1), ('b', 2)]
When working with collections of data in Python, the ability to manipulate them and create new collections is very valuable.
One such manipulation is the ability to get all combinations of two lists in a new list.
All combinations of two sets A and B is the Cartesian Product of these two sets.
The Cartesian Product of two sets A and B is the set of all possible ordered pairs (a, b), where a is in A and b is in B. We can get the Cartesian product between two lists easily with Python.
The easiest way to get the Cartesian product and all of the combinations of two lists is with list comprehension.
Below is a simple example of how to get all combinations of two lists in Python using list comprehension.
list1 = ["a", "b", "c"]
list2 = [1, 2, 3]
combinations = [(x,y) for x in list1 for y in list2]
[('a', 1), ('a', 2), ('a', 3), ('b', 1), ('b', 2), ('b', 3), ('c', 1), ('c', 2), ('c', 3)]
Using for Loop to Get Combinations of Two Lists in Python
We can also use a loop to get the different combinations of lists in Python.
We can define a loop easily which will loop over all possible combinations of our lists and create ordered pairs in the form of tuples.
Below is a simple example of how to get combinations of two lists in Python using iteration.
list1 = ["a", "b"]
list2 = [1, 2]
def combinations(lst1, lst2):
result = []
for x in lst1:
for y in lst2:
return result
[('a', 1), ('a', 2), ('b', 1), ('b', 2)]
Using itertools product() Function to Get Combinations of Two Lists in Python
The itertools module has many great functions which allow us to iterate over collections and perform complex tasks easily.
We can use the itertools product() function to get the Cartesian product of lists.
To get the all combinations of two lists in Python using product(), just pass the lists to the function.
Below is a simple example of how to get all combinations of two lists in Python using itertools and product().
from itertools import product
list1 = ["a", "b"]
list2 = [1, 2]
[('a', 1), ('a', 2), ('b', 1), ('b', 2)]
Hopefully this article has been useful for you to learn how to get combinations of lists in Python.
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RSA-210 has been factored!
RSA-210 has been factored! posted October 2013
The https://en.wikipedia.org/wiki/RSA_Factoring_Challenge" target="_blank">RSA Factoring Challenge has had one of its entry factored : RSA-210. More info here.
The RSA Factoring Challenge was a challenge put forward by RSA Laboratories on March 18, 1991 to encourage research into computational number theory and the practical difficulty of factoring
large integers and cracking RSA keys used in cryptography. They published a list of semiprimes (numbers with exactly two prime factors) known as the RSA numbers, with a cash prize for the
successful factorization of some of them. The smallest of them, a 100 decimal digit number called RSA-100 was factored by April 1, 1991, but many of the bigger numbers have still not been
factored and are expected to remain unfactored for quite some time.
The challenge is no longer active, this means no money for this brave Ryan P. And this doesn't mean RSA is less secure so no worries :)
leave a comment...
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QUT and All Star Manufacturing
Project 1 Calculations must be done in Excel – This question should be done using Method 1 as outlined in lecture 6 (i.e. Tax Effects, then Cash Flows then NPV).
QUT corporation projects their future unit sales for a new headphone. The projected unit sales are as below.
Unit sales 75000 88000 120000 95000 60000
To produce the headphones, the initial net working capital of $2,000,000 is required and additional net working capital is also required each year, which is 20% of the projected sales increase for
the following year. The net working capital will be recovered at the end of a project. In addition, the initial installation cost of the machine for production is $18,000,000. The machine will be
depreciated for tax purposes using straight-line depreciation with the useful life of 6 years. Also, costs and unit price are as below.
Fixed cost $2,800,000 per year
Variable cost $295 per unit
Price $420 per unit
In five years, the machine can be sold for about 30% of its acquisition cost. The tax rate is 30% and the required rate of return is 15%.
1. What is the NPV of the project?
2. Assuming that the project can be repeated indefinitely, what is the NPV∞ of the project?
Project 2 Calculations must be done in Excel – This question should be done using Method 1 as outlined in lecture 6 (i.e. Tax Effects, then Cash Flows then NPV).
As the financial advisor to All Star Manufacturing you are evaluating the following new investment in a manufacturing project: –
• The project has a useful life of 8 years.
• Land costs $10m and is estimated to have a resale value of $20m at the completion of the project.
• Buildings cost $12m, with allowable depreciation of 6% pa reducing balance and a salvage value of $10m.
• Equipment costs $5m, with allowable depreciation of 10% pa reducing balance and a salvage value of $1m. An investment allowance of 20% of the equipment cost is available.
• Revenues are expected to be $15m in year one and rise at 5% pa.
• Cash variable costs are estimated at 30% of revenue.
• Cash fixed costs are estimated at $3m pa.
• Managerial salaries of $800,000 will be allocated to the project, but these managerial positions will be unaffected by the acceptance of the project.
• An amount of $200,000 has been spent on a feasibility study for the new project.
• The project is to be partially financed with a loan of $13.5m to be repaid annually with equal instalments at a rate of 5% pa over 8 years.
• Except for initial outlays, assume cash flows occur at the end of each year.
• The tax rate is 30% and is payable in the year in which profit is earned.
• The after-tax required return for the project is 11% pa.
1. Calculate the NPV. Is the project acceptable? Why or why not?
2. Conduct a sensitivity analysis showing how sensitive the project is to revenues, fixed costs and to the required rate of return.
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Ensayo ingles | ramonpares
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DISTANCIAS PLANETARIAS Y LEY DE TITIUS-BODE, Historical essay
On the paradoxical and subsequent course of the Titius-Bode law
Johann Daniel Titius (1729-1796), professor of physics at the former University of Wittenberg (Saxony) translated into German the work Contemplation de la Nature, from the Swiss author Charles Bonnet
Without saying anything to anybody, Titius inserted two own paragraphs that are at the bottom of page 7 and at the beginning of the 8 in the German edition of 1766. In the preface, Bonnet warns
without accurate that Titius has interspersed some own notes, which suggests not only their knowledge but also their conformity. Of course, the new intercropping paragraph is not founded in the
original nor in translations of the Bonnet's work in Italian and English.
In the intercalated text we have refer there are to two parts, one after the other. In first part is exposed the succession of planetary distances from the Sun of historical planets, from Mercury to
Saturn, rounded to whole numbers as exposed: If we give 100 points to Saturn and 4 to Mercury, to Venus will correspond 4 + 3 = 7 points; to the Earth 4 + 6 = 10; to Mars, 4 + 12 = 16; to the next
would be 4 + 24 = 28, but with no planet; and will be 4 + 48 = 52 points and 4 + 96 = 100 points respectively, Jupiter for the and Saturn for the second.
In the second interleaved part is added as follows: If give to the Earth radius's orbit the value of 10, the other orbits radii are given by the formula Rn = 4 + (3 + 2^n), where n = -∞ for Mercury
and 0, 1, 2, 3, 4 and 5 for planets that follow.
These both statements, for all their particular typology and the radii of the orbits, seem to stem from an antique cossist[1]. In fact, many precedents have been finding up to the seventeenth
century. Titius was a disciple of the German philosopher Christian Freiherr von Wolf (1679-1754). The second part of the inserted text is also literally founded in a von Wolf's work dated in 1723,
and that's why, in twentieth century literature about Titius-Bode law, usually is assigned as authorship the German philosopher; In fact, Titius was a disciple of Wolf. Another reference, older than
before, is written by James Gregory in 1702, in his Astronomiae physicae et geometricae elementa, where the succession of planetary distances 4, 7, 10, 16, 52 and 100 becomes a geometric progression
of ratio 2. This is the nearest Newtonian formula, which is also contained in Benjamin Martin and Tomas Cerdà himself many years before the German publication of Bonnet's book.
As we have read (3), text interspersed by Titius in Bonnet's book was really transmitted in astronomy's work of Johann Elert Bode (1747-1826). In none of the issues appears Titius, and the authorship
of the law is not clearly assigned (Aleitung zur kenntnis des gestirnten Himmels, 1722). Only in a posthumous Bode's memoir can to be founded a reference to Titius with the clear recognition of their
priority. But in that moment, everyone knew what was Bode’s Law.
Titius and Bode hoped that the law would lead to the discovery of new planets. But it really was not. Those of the Uranus and Ceres rather contributed to the fame of the Titius-Bode law, but not
around Neptune and Pluto's discovery, just because both are excluded. However, it is applied to the satellites and even currently to the extrasolar planets (5).
Titius-Bode law remains a solid and convincing theoretical explanation of their physical meaning, and is not considered as a numerical device. Its history has always been linked as more soup than
substance. How can it be compared to the Hipparchus's work in respect to the planetary distances, those of Kepler regarding the orbit of Mars, the discovery of Neptune, the calculation of an event,
those of an orbit starting by only three positions, or the explanation about the Mercury's perihelion deviation? However, it is usually more cited.
In the nineteenth century, the Bode or Titius-Bode law: 1) Many authors or do not know, or never cite (9.4). 2) Others use it as if it were a basic law of the celestial mechanics, unrelated to
Newton. 3) There also are those who consider it an arithmetic casual approach, and 4) as established as law by Kepler, but with no demonstration.
It is interesting to mention the magnificent book titled The modern telescope from A.T. Arcemis, more than 1,500 pages in two great volumes, published in 1878 (Muntaner & Simon. Barcelona). That book
explains to us that Titius’s law has its origin in a French book written by that German author and called Contemplation of nature. It has been needed to reach till XXI century to clarify better this
issue, as described in (3).
[1] The cossists were experts in calculations of all kinds and were employed by merchants and businessmen to solve complex accounting problems. Their name derives from the Italian word cosa, meaning
“thing”, because they used symbols to represent an unknown quantity, similar to the way mathematicians use x today. All professional problem-solvers of this era invented their own clever methods for
performing calculations and would do their utmost to keep these methods secret in order to maintain their reputation as the only person capable of solving a particular problem.
An explanation of the Titius-Bode’s law that could be previous to its historical origin
The Jesuit Tomàs Cerdà (1715-1791) gave a famous astronomy course in Barcelona in 1760, at the Royal Chair of Mathematics of the College of Sant Jaume de Cordelles (Imperial and Royal Seminary of
Nobles of Cordellas). From the original manuscript preserved in the Royal Academy of History in Madrid, Lluís Gasiot remade Tratado de Astronomía from Cerdá, published in 1999, and which is based on
Astronomiae physicae from James Gregory (1702) and Philosophia Britannica from Benjamin Martin (1747). In the Cerdàs's Tratado appears the planetary distances obtained from the periodic times
applying the Kepler's third law, with an accuracy of 10^-3. Taking as reference the distance from Earth as 10 and rounding to whole, can be established the geometric progression [(Dn x 10) - 4] / [D
n-1 x 10) - 4] = 2, from n=2 to n=8. And using the circular uniform fictitious movement to the Kepler's Anomaly, it may be obtained Rn values corresponding to each planet's radios, which can be
obtained the reasons rn = (Rn - R1) / (Rn-1 - R1) resulting 1.82; 1.84; 1.86; 1.88 and 1.90, which rn = 2 - 0.02 (12 - n) that is the ratio between Keplerian succession and Titus-Bode Law, which
would be a casual numerical coincidence. The reason is close to 2, but really increases harmonically from 1.82.
The planet's average speed from n=1 to n=8 decreases when moving away the Sun and differs from uniform descent in n=2 to recover from n=7 (orbital resonance).
What announced the Titius-Bode’s law was previously established by Kepler
Among other authors, Comas Solà defined the Titius-Bode law half of the twentieth century as an arithmetic formula that establishes approximately a geometric progression of ratio 2 with the distances
of the planets from the Sun, as it had already been established by Kepler previously (4.5).
Like other astronomers of his time, Kepler already had the distances of the planets from the Sun relative to the Earth determined by trigonometric methods (4.1). He also knew the periodic times
directly seeing that ratio Pn / Pn-1~ 2, missing a planet between Mars and Jupiter (n = 5). Among n = 3 and n = 7 the distances from the Sun were an exponential function of the sequence of n.
Naturally, by the third law, DK = (Pn /P3)^2/3 also knew that (10 x DK)n – 4 / (10 x DK)n-1 – 4 = 2. If he calculates the abnormality he had the uniform circular movement to the real elliptical and
could calculate the corresponding radii Rn. This relationship is obtained
Rn – R1 / Rn-1 – R1 = 2 – 0,02 (12-n)
and perhaps it is the most important that exists between Titius-Bode’s law and what Kepler just established previously about it.
Titius-Bode’s law can only be applied to the historical planets of the Copernican solar system from n = 2, and it is not valid for asteroids nor comets
In Figure 17 we have an image, page 10 to the original facsimile of Copernicus’s manuscript (1539) of his famous De Revolutionibus Orbium coelestis. In Figure 16 we find the representation of N.
Winston included on page 5 of the Tratado de Astronomía from T. Cerdà T. (1760), comprising planets and comets with different periodic times and distances from the sun.
In Figures 14 and 15 we have the image of the manuscript named Tratado de arismética y geometría práctica from the author Juan de Área y Quiroga (1718, Figure 18), representing the Ptolemaic system
(s. II AD.) and Tychonic system (s. XVI AD.) as answer to the question 36 and which would not be applied to the Titius-Bode law.
As we have referred (6.1), each planetary orbit is defined by seven elements among which are the distance to the sun. With the mechanical of Newton it is possible to calculated all the elements from
three apparent positions of the planet, with its ascension, straight and decline. We know that this knowledge was historically achieved by successive approximations of many apparent positions,
followed by the corresponding adjustments. This is how Kepler and other observers already had much correct orbits of the historical planets.
Comets and asteroids do not follow the Titius-Bode law. Disregarding her, in 1850 Francisco Verdejo Paez, professor of geography at the University of Madrid, in his Geografía Histórica (Imprenta de
Repullés, Madrid), makes a unique sequence of distances from the sun, including asteroids known at this time: Mercury, 0.4; Venus, 0.7; Earth, 1.0; Mars, 1.6; Flora, 1.9; Vesta, 2.4; Iris, 2.4;
Metis, 2.4; Hebe, 2.4; Astrea, 2.4; Juno, 2.7; Ceres, 2.8; Shovels, 2.9; Higia, 3.0; Jupiter, 5.3; Saturn, 9.7; Uranus, 19.4; and Neptune, 40.5. The inclinations of the orbit from Flora to be Higia:
Flora, 5 ° 53 '; Vesta 7 ° 8 '; Iris 5 ° 28 '; Metis, 5 ° 35 '; Hebe, 14 ° 48 '; Astrea 5 ° 19 '; Juno, 13 ° 4 '; Ceres 10 ° 37 '; Palas, 34 ° 38 'and Higia, 3 ° 48'. The author J. Regueiro
Argüelles, cited above, in his Astronomia física written at the same year, leaves the distance corresponding to n = 5 with no sun planet, as the latest top authors, and only seven terms for the Bode
Interplanetary distances. Geometric mean ratio rn=1,86 from n=3 to n=7 of the Sun distances succession. The most paradoxical success of the Titius-Bode’s law. Influence of the periscope’s
The projected position of the planets in the sky from Earth changes as a result of the simultaneous movement of the Earth and the planet. For example, Earth gives twelve laps around the Sun an
Jupiter gives only one, and that’s why we see the planet running half year in one direction and another half on the other, the first faster than the second. There is a circle that is set with the
planet revolving twelve times, giving the full revolution to the Sun. This is called epicycles. Those of Saturn is the slowest and Mars the fastest. Those from Venus and Mercury are more complicated.
In addition, each planet comes to meet nearer of farther from Earth. For example, at certain points, Mars is four times farther away from us than in others. To explain these variations, it was
necessary to modify the circle, from which emerged the idea of a displaced centre of rotation, with the Earth farthest or nearest. This is the eccentric. With Kepler's laws all this was changed.
With its calculation of the anomaly it is obtained a circular uniform movement equivalent (4.5), between n = 1 and n = 7, with an average ratio (Rn - R1) / (Rn-1 - R1) = 2 - 0,02 (12, n) = 1.82;
1.84; 1.86; 1,88; 1,90 / 5 = 1.86 instead of 2 as an average value.
For the author of this essay, it is implausible to believe that Copernicus, Galileo, Kepler and Newton were able to interject an anonymous text in the translation of a book as apparently did Titius,
and also that they could appropriate this text inserted in an own book, as did Bode. Even today it seems paradoxical the continued success of this action, after three centuries, of what may simply be
a relic cosist from early eighteenth century. There are remote causes of these phenomena. In this case we could say that, without the introduction of the telescope by Galileo, the Titius-Bode law
would not have existed, but would still implicit in Kepler's work. It has also been said that Copernicus could pull the Earth from the centre of the world thanks to the discovery of America. However,
the catalogue of nebulae from Bode and its new constellations could have maligned anyone. But really, what it is happened is that no one remembers about that.
I do not think that rationality and, in particular, scientific knowledge has changed the average of the previous human nature, as it did, for example, passing from the Stone Age to the Age of metals.
Homo sapiens stage starts with the writing, which is a relatively recent fact, only about 5.000 to 6.000 years. In the remote darkness of time is Homo habilis, about two million years ago. Iron Age
changed as much as we have changed with science. Nothing to do with the small and slow changes that may come after knowing that the earth is a planet revolving around the sun, but eventually no one
knows what can happen in the future.
"After having spent more than half a century, the author wrote this essay to the memory of his former teachers."
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On The Stability Of The Solar System
, 7 min read
On The Stability Of The Solar System
Our solar system is not stable when considerung time ranges of several Gigayears. These are the results of simulations done by Laskar and Gastineau.
1. Nomenclature
Notions for ellipses, see below figure:
• center M
• semi-minor axis b
• semi-major axis a
• (linear) eccentricity e, numerical eccentricity $\varepsilon$
$$ \pmatrix{x(t)\cr y(t)} = \pmatrix{a \cos(t)\cr b \sin(t)}, \qquad \varepsilon = {e\over a} = {\sqrt{a^2-b^2}\over a} = \sqrt{1 - \left({b\over a}\right)^2} $$
The numerical eccentricity describes by "how much" the shape of the ellipse differs from a circle: a value of zero means it is a circle, anything large than zero becomes more deformed.
Below are the numerical eccentricities of the planets in our solar system. Semi-minor and semi-major axes length are given in AU.
Nr. Planet Eccentricity a b #moons
1 Mercury 0.206 0.38700 0.37870 0
2 Venus 0.007 0.72300 0.72298 0
3 Earth 0.017 1.00000 0.99986 1
4 Mars 0.093 1.52400 1.51740 2
5 Jupiter 0.049 5.20440 5.19820 95
6 Saturn 0.057 9.58250 9.56730 146
7 Uranus 0.046 19.21840 19.19770 28
8 Neptune 0.010 30.11000 30.10870 16
2. The Jovian Planets
Below text is from The Jovian Planets:
The four Jovian planets — Jupiter, Saturn, Uranus, and Neptune — are also called "giant planets". The Jovian planets occupy orbits in the outer solar system at distances ranging from 5 (Jupiter) to
30 (Neptune) times the Earth’s distance from the Sun. ...
Unlike the terrestrial planets that make up our inner solar system — Mercury, Venus, Earth, and Mars — the Jovian planets do not have solid surfaces. Instead, they are composed primarily of hydrogen
and helium, with traces of methane, ammonia, water, and other gases in their atmospheres. These gases make up a deep atmosphere and become tightly compressed around relatively tiny cores of rock. At
great depths within Jupiter, for example, the hydrogen gas is compacted so tightly that it exists in a rare metallic form.
3. Jovian problem
Below text is from N-body simulations: the performance of eleven integrators by P.W. Sharp.
The Jovian problem has the Sun, Jupiter, Saturn, Uranus and Neptune interacting through classical Newtonian gravitational forces. Let ${\bf r}_i$ denote the position of the i-th body, where the
bodies are ordered Sun to Neptune and the coordinate system is three-dimensional Cartesian with the origin at the barycenter of the bodies. G is the gravitational constant, $m_i$ is the i-th mass.
The differential equation is
$$ \ddot{\bf r}_i(t) = \sum_{j=1\atop j\ne i}^5 { G m_j ({\bf r}_j(t) - {\bf r}_i(t) \over \left\Vert {\bf r}_j(t) - {\bf r}_i(t) \right\Vert^2 }, \qquad i=1,\ldots,5. $$
Except for the emission of Pluto and a change in the coordinate system, above equation is problem C5 from Nonstiff DETEST.
This problem becomes particularly demanding when the integration interval is long, e.g., ten million years.
A simple test for correctness is to use the total energy:
$$ E = {1\over2} \left[ \sum_{i=1}^5 \left( m_i {\bf r}_i^2 - \sum_{j=1\atop j\ne i}^5 { G m_i m_j \over \left\Vert {\bf r}_j - {\bf r}_i \right\Vert } \right) \right]. $$
Total energy E must be constant over all time t. It is a not a very accurate measure for correctness.
Above paper gives more involved differential equations for:
• Nine planet problem
• Spin Axis problem
• DE102 problem
4. Evolution of planetary orbits
Below text is from Laskar: Stability of the solar system.
For all external planets, the maximum eccentricity is almost constant. That reflects the fact that these trajectories are very close to regular and quasiperiodic trajectories; possible instabilities
are insensitive with the scale of the drawing.
For Venus and the Earth, one observes moderated variations, but still significant. The maximum eccentricity of the Earth reached through chaotic diffusion reaches about 0.08, whereas its current
variations are approximately 0.06. It is about the same for Venus.
It should however be noted that to arrive at this possible collision between Mercury and Venus, the model was used beyond its rigorous field of validity, which does not includes the vicinity of
collisions. In addition, the solution was carefully chosen, so in any case, it is surely not a very probable one, and the majority of the solutions of close initial conditions will not lead to this
possible collision.
Concerning the system of the outer planets, the things are appreciably different, because the direct gravitational short period perturbations are more significant. The recent numerical simulations
show that particles placed among the outer planets do not remain beyond a few hundreds of million years, apart for some particular zones of stability or beyond Neptune, in the Kuiper belt, where
objects explicitly were found.
Finally, these observations also make it possible to have an idea of the general aspect of a planetary system around a star. Indeed, if the process of formation planetary from planetesimals is
correct, it becomes possible that the planetary systems will always be in a state of marginal stability, like our own Solar system. At the end of the phase of formation of the system, a large number
of bodies can remain, but in this case the system is strongly unstable, which will led to a collision or an ejection. After this event, the system becomes more stable, with constantly, a time of
stability comparable with its age.
5. Instability of solar system after 3 Gyr
Mogavero, Hoang and Laskar use below Hamiltonian
$$ \hat H = - \sum_{i=1}^8 \left[ \sum_{\ell=1}^{i-1} \left\langle { G m_i m_\ell \over \left\Vert {\bf r}_i - {\bf r}_\ell \right\Vert } \right\rangle + { 3 G^2 m_0^2 m_i \over c^2 a_i^2 \sqrt{1-\
varepsilon_i^2} } \right] . $$
• The $a_i$ are the semi-major axes.
• $m_0$ and $m_i$ are the masses of Sun and the planets.
• $\varepsilon_i$ are the eccentricities of the planets.
• The ${\bf r}_i$ are the heliocentric positions of the planets.
• c is the speed of light.
• The bracket operator represents the averaging over the mean longitudes resulting from the elimination of non-resonant Fourier harmonics of the N-body Hamiltonian.
See Mogavero+Laskar: Long-term dynamics of the inner planets in the Solar System.
The text Stability of the solar system shows below results for the eccentricity of Mercury after 1-5 Gyr.
Thanks to relativity the eccentricities of Mercury stay way lower than ignoring relativity. Nevertheless, after around 1Gyr the solar system becomes destabilized by Mercury crashing into Venus,
statistically speaking, i.e., in 1% of cases this can happen.
Beyond this spectacular aspect, these results also validated the methods of semi-analytical averaging developed for more than 20 years and which had allowed to show the possibility of collision
between Mercury and Venus (Laskar, 1994). These results also answer to the question raised more than 300 years ago by Newton, by showing that collisions among planets or ejections are actually
possible within the life expectancy of the Sun, that is, in less than 5 Gyr. The main surprise that comes from the numerical simulations of the recent years is that the probability for this
catastrophic events to occur is relatively high, of the order of 1%, and thus not just a mathematical curiosity with extremely low probability values. At the same time, 99% of the trajectories
will behave in a similar way as in the recent past millions of years, which is coherent with our common understanding that the Solar System has not much evolved in the past 4 Gyr. What is more
surprising is that if we consider a pure Newtonian world, starting with the present initial conditions, the probability of collisions within 5 Gyr grows to 60%, which can thus be considered as an
additional indirect confirmation of general relativity.
Also see:
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