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Publication dates of music in the British Library
If you go to the British Library online catalogue, search for music scores published in each year from 1650 to 1920, and plot the number of ‘hits’ by year, the result looks like this. Along the
bottom of the chart are years from 1650 to 1920, and up the side is the number of scores in the British Library attributed to each year. Note the logarithmic scale, where each gridline represents a
factor of 10. The blue line is the actual number of scores attributed to each year, and the red line is a five-year moving average, i.e. the average of each year and the two years either side of it.
There are two interesting things about this chart. The first is that the overall upward trend in the red line is remarkably straight. Over the period, the number of scores increased steadily by
roughly a factor of ten every 100 years.
The other interesting thing is that between 1700 and 1850 the blue line is very spiky, with regular spikes every five years. That is because a lot of printed music publication dates are unknown, but
have been estimated to the nearest round number – i.e. a year ending in ‘0’ or ‘5’. Judging by the movements in the red line, there are more ending in ‘0’ than ‘5’, as you would expect.
If you subtract the red line from the blue line you end up with the following chart.
The green line here represents the percentage by which the number of scores attributed to each date is above or below the five-year average. It is striking that the spikes are largest between 1700
and 1850. The small number of older scores are perhaps better studied, and more accurate date attributions are possible. After 1850, presumably it was more common for publishers to print a date on
the score. The pattern is perhaps also related to the large backlogs that accumulated in the early days of the British Library’s music department, so some newly published scores may not have been
logged when they arrived.^1
It is possible to estimate, as indicated by the little blue dashes, the proportion of total works from each five-year period that appear to have an approximate date. In the period from 1700 to 1850
these average out at about 40%. So between these dates, there is a 40% chance that we only have an approximate idea of a score’s actual publication date.
Of course, this is only one source. Other libraries might have different procedures in place for assigning publication dates, so we probably cannot generalise without some further investigation. In
many cases, it might be possible to work out more accurate dates for individual musical scores from other sources, but we cannot assume that a library catalogue reflects the best current scholarly
information on every item.
This is an example of where a bit of simple statistical analysis can give an interesting insight into a familiar dataset.
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• [Paused] Doors CE 9 (TI-84 Plus CE) Development thread
I can't count how many YouTube comments, emails, and Cemetech topics I've seen asked for a Doors CS/Doors CSE for the new
TI-84 Plus CE
, and finally, I've gotten around to start creating it. I won't deny that
by MateoConLechuga has pushed my competitive side to drop everything else and focus on this, and Epharius' work on PHASM has also motivated me. For about a month, I've been working to build the
original Doors CSE source code for the ez80 TI-84 Plus CE, and I've finally gotten it to the state where I can begin debugging the shell. For the unfamiliar, the stages in the process go something
1. Take the source, make it assemble by finding equivalents for all system calls and memory areas from the TI-84 Plus C Silver Edition on the TI-84 Plus CE.
2. Debug the shell, module, by module, modifying components to work properly. One needs to take into account things like 3-byte registers, the new memory format, and the fact that Doors CE is now
going to be a program rather than a shell.
3. Reconfigure for the TI-84 Plus CE. Like CesiumOS, Doors CE will divide itself into a RAM component (a launcher) and an Archived component (the shell itself) when it first runs, in order to
provide longer-lived hooks and resilience to RAM clears. This architecture will take time
4. Port libraries. Doors CE will provide ASM, C, and TI-BASIC libraries, but this will take time to get in place
I'm happy to present the first "screenshot" from Doors CE 9:
Great work Kerm! What stage in the porting process are you at now?
Ivoah wrote:
Great work Kerm! What stage in the porting process are you at now?
I have just started stage 2, and I expect it to be fairly time-consuming. The fact that we don't yet have a community TI-84 Plus CE emulator, including debugging tools, is a significant barrier to
moving fast with this.
So, the earlier request I made in the Cesium thread applies here too.
elfprince13 wrote:
MateoConLechuga wrote:
elfprince13 wrote:
Feature request: a standard argv mechanism (preferably for BASIC too, but definitely for C/Asm)
I would totally love to implement this, but I am not entirely sure what you are getting at. Could you please detail it a little more? Thanks!
Your standard entry point for programs should be akin to:
int main(int argc, char ** argv) {
return 0;
This allows for reasonable terminal interfaces, file opening mechanisms, and all sorts of goodies, for the same reasons it's a standard everywhere else.
The function called at program startup is named main. The implementation declares no prototype for this function. It shall be defined with a return type of int and with no parameters:
int main(void) { /* ... */ }
or with two parameters (referred to here as argc and argv, though any names may be used, as they are local to the function in which they are declared):
int main(int argc, char *argv[]) { /* ... */ }
or equivalent; or in some other implementation-defined manner.
If they are declared, the parameters to the main function shall obey the following constraints:
The value of argc shall be nonnegative.
argv[argc] shall be a null pointer.
If the value of argc is greater than zero, the array members argv[0] through argv[argc-1] inclusive shall contain pointers to strings, which are given implementation-defined values by the host
environment prior to program startup. The intent is to supply to the program information determined prior to program startup from elsewhere in the hosted environment. If the host environment is not
capable of supplying strings with letters in both uppercase and lowercase, the implementation shall ensure that the strings are received in lowercase.
If the value of argc is greater than zero, the string pointed to by argv[0] represents the program name; argv[0][0] shall be the null character if the program name is not available from the host
environment. If the value of argc is greater than one, the strings pointed to by argv[1] through argv[argc-1] represent the program parameters.
The parameters argc and argv and the strings pointed to by the argv array shall be modifiable by the program, and retain their last-stored values between program startup and program termination.
Yes, I definitely plan to do this (and I'll also be continuing to use the field-based shell program header from Doors CSE). Do you have any particular proposals for passing argc and argv to ASM
programs and getting them back from ASM programs? I guess for C programs, the API will be a little more obvious.
C programs and ASM programs should have the same format. ASM programs can handle the stack manipulations manually in the cases where they care, and ignore them in the cases where they don't.
I say celtic 3 this time
Ps: WOOOOT!!! Doors CSE 9.0!!!
Celtic 2 will be installed to help keep backwards compatibility of programs/games made on the CSE.
This looks great so far Kerm, let me know when there's a testing version to try!
tifreak8x wrote:
Celtic 2 will be installed to help keep backwards compatibility of programs/games made on the CSE.
Although this argument is perfectly logical and valid, I'd like to offer a counter argument
I'm not sure if all of the numbers used in Celtic 2 are also used in Celtic 3 for the same commands and some were added on, or if they were changed up for some reason, but assuming they weren't, then
CSE programs would be completely compatible with the CE. Of course the CE's programs wouldn't be compatible on the CSE, but that could be more of a long term fix with an eventual release of a new
version. Imagine a few years from now when ti will release a new calculator and the same type of projects will be launched. If these changes don't happen during development times such as these, I
don't think they will ever happen and people like Kerm will be porting outdated versions of libs until 2033 simply because they want to keep everything cross-compatible. However, If this "update" is
made right now, then of course it would take quite a bit of work, but it would be to everyone's advantage in the long run
Guess I better finish xLIBCE
tr1p1ea wrote:
Guess I better finish xLIBCE
That would be hugely helpful. Did we decided that 8bpp mode + using the two halves of the GRAM as the two buffers would be the right solution, albeit difficult to use to emulate the abuse of xLIBC in
full-resolution mode from the TI-84+CSE?
elfprince13 wrote:
C programs and ASM programs should have the same format. ASM programs can handle the stack manipulations manually in the cases where they care, and ignore them in the cases where they don't.
Okay, that's fair. Do we actually have a specific calling convention for C programs that we're using, or are we following whatever Zilog's compiler does? Mateo?
mr womp womp wrote:
tifreak8x wrote:
Celtic 2 will be installed to help keep backwards compatibility of programs/games made on the CSE.
Although this argument is perfectly logical and valid, I'd like to offer a counter argument
I'm not sure if all of the numbers used in Celtic 2 are also used in Celtic 3 for the same commands and some were added on, or if they were changed up for some reason, but assuming they weren't, then
CSE programs would be completely compatible with the CE. Of course the CE's programs wouldn't be compatible on the CSE, but that could be more of a long term fix with an eventual release of a new
version. Imagine a few years from now when ti will release a new calculator and the same type of projects will be launched. If these changes don't happen during development times such as these, I
don't think they will ever happen and people like Kerm will be porting outdated versions of libs until 2033 simply because they want to keep everything cross-compatible. However, If this "update" is
made right now, then of course it would take quite a bit of work, but it would be to everyone's advantage in the long run
Let's look at this another way: what Celtic III functions that aren't in Celtic 2 that you think are important? xLIBC and parts of Celtic 2 CSE cover all kinds of graphics functions, plus everything
that most people need to manipulate programs and AppVars. If there are specific functions that you think are important, I'd be more than happy to explore including them.
KermMartian wrote:
Okay, that's fair. Do we actually have a specific calling convention for C programs that we're using, or are we following whatever Zilog's compiler does? Mateo?
That's not the job of the compiler, that's the job of the startup routines, which also initialize the BSS and setup the vectors. These are written in assembly; here is the relevant standard C startup
portion, of which I have modified to support other things, and the relocation of errno. (I'm still working on the details, so the code below is the standard implementation). Implementing an input of
the standard arguments of main() is up to you.
The calling convention for C functions is pushing all arguments onto the stack in order of argument number, regardless of type. All items int and lower are passed as 24 bits, even though they aren't.
Return values are documented in the ZDS docs.
;* Copyright (c) 2007-2008 Zilog, Inc.
;* cstartup.asm
;* ZDS II C Runtime Startup for the eZ80 and eZ80Acclaim! C Compiler
xdef _errno
xdef __c_startup
xdef __cstartup
xref _main
xref __low_bss ; Low address of bss segment
xref __len_bss ; Length of bss segment
xdef _abort
__cstartup equ %1
;* Startup code. Reset entry point.
define .STARTUP, space = RAM
segment .STARTUP ; This should be placed properly
.assume ADL=1
; Initialize the .BSS section to zero
ld hl, __len_bss ; Check for non-zero length
ld bc, 0 ; *
or a, a ; clears carry bit
sbc hl, bc ; *
jr z, _c_bss_done ; .BSS is zero-length ...
ld hl, __low_bss ; [hl]=.bss
ld bc, __len_bss
ld (hl), 0
dec bc ; 1st byte's taken care of
ld hl, 0
sbc hl, bc
jr z, _c_bss_done ; Just 1 byte ...
ld hl, __low_bss ; reset hl
ld de, __low_bss+1 ; [de]=.bss+1
ldir ; Load Increment with Repeat
; prepare to go to the main system rountine
ld hl, 0 ; hl = NULL
push hl ; argv[0] = NULL
ld ix, 0
add ix, sp ; ix = &argv[0]
push ix ; &argv[0]
pop hl
ld de, 0 ; argc = 0
call _main ; int main( int argc, char *argv[]) )
pop de ; clean the stack
jr $ ; if we return from main loop forever here
;* Define global system var _errno. Used by floating point libraries
segment DATA
ds 3 ; extern int _errno
In addition, it is probably better if the shell initializes the BSS section in order to limit repeated code. C programs that run without a shell technically shouldn't be allowed. in my opinion,
mainly because of the requirements. There basically needs to be an overlaying layer. I haven't officially decided on a location for the BSS section, because it needs to be set statically by programs.
I imagine it will be at least 20kb though. I've also tried looking into the #nobss pragma, but it doesn't appear to function as desired.
Thanks, Mateo. is a bug, however.
Very nice work Kerm! As for names not displaying correctly, are you sure that you are looking at the name length byte when you display? You shouldn't be getting that many characters.
MateoConLechuga wrote:
As for names not displaying correctly, are you sure that you are looking at the name length byte when you display? You shouldn't be getting that many characters.
Haha, thanks for the suggestion.
KermMartian wrote:
Let's look at this another way: what Celtic III functions that aren't in Celtic 2 that you think are important? xLIBC and parts of Celtic 2 CSE cover all kinds of graphics functions, plus everything
that most people need to manipulate programs and AppVars. If there are specific functions that you think are important, I'd be more than happy to explore including them.
The particular commands I was looking at were:
NumToString, ExecHex, UngroupFile and MatToStr
mr womp womp wrote:
The particular commands I was looking at were:
NumToString, ExecHex, UngroupFile and MatToStr
The former two seem extremely useful, and I'll add them. Please remind me to do so. I think UngroupFile never really worked properly, from Iambian's notes, and I think MatToStr is too narrow (and can
be done with a combination of NumToString and simple TI-BASIC code) to include.
I've been working very hard, now that I have a fire lit under me, and here's what's new, judging from my commits:
• Repaired problems restoring folders from FLDSV AppVar after a crash
• Repaired 16->24 bit issues with PrevVATArray, the VFAT, routines like ldhlind, and so on.
• Fixed display of program icons, names, and property icons on the desktop
• Repaired the InfoPop box that appears when you hover over a program
• Made icons for TI-BASIC programs with :DCS-style headers display properly
• Attempted repairs for APD inside Doors CE
• Made the scrollbar work properly (by also fixing DivHLC and similar z80 Bits routines)
• Repaired initial display of Properties menu when you click on something
• Made creating new folders (and backing up those folders) work properly
• Repaired the string routine used for getting the name of new programs and folders
Looks like most (?) of the easier GUI part, and some of the VAT part, are done.
Have you given a close look to Mateo's work on the library and loader ?
I've been working very hard, now that I have a fire lit under me
This may be slightly off-topic (but hey, you brought the topic first yourself): what are you so afraid of ?
KermMartian wrote:
mr womp womp wrote:
The particular commands I was looking at were:
NumToString, ExecHex, UngroupFile and MatToStr
The former two seem extremely useful, and I'll add them. Please remind me to do so. I think UngroupFile never really worked properly, from Iambian's notes, and I think MatToStr is too narrow (and can
be done with a combination of NumToString and simple TI-BASIC code) to include.
This seems very reasonable and wonderful
I don't know how ExecHex is currently structured, but it seems it would be extremely useful to have an argument that specifies the address from which to copy + execute it. | {"url":"https://dev.cemetech.net/forum/viewtopic.php?p=240712","timestamp":"2024-11-04T21:37:19Z","content_type":"text/html","content_length":"100805","record_id":"<urn:uuid:833c2d93-31ae-49c9-9b0c-c18727486a0e>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00695.warc.gz"} |
How to find the resolution of a spherical harmonic field in GRIB - ecCodes GRIB FAQ
Some ECMWF data produced by the IFS is stored in GRIB with gridType=sh to indicate that the values are stored as spherical harmonic coefficients.
The resolution of such data is identified by the triangular truncation number which specifies the number of coefficients retained in the infinite sum. This number is stored in GRIB as the pentagonal
resolution parameters, J, K and M:
ecCodes key content GRIB edition 1 GRIB edition 2
J J pentagonal resolution parameter Section 2 Octets 7-8 Section 3 Octets 15-18
K K pentagonal resolution parameter Section 2 Octets 9-10 Section 3 Octets 19-22
M M pentagonal resolution parameter Section 2 Octets 11-12 Section 3 Octets 23-26
The pentagonal representation of resolution is general and the triangular representation used by ECMWF is a specific case with M = J = K. The truncation number, and hence the resolution of the
data, can be obtained by querying the value of any one of these keys. Most usually, the value of J - or its alias pentagonalResolutionParameterJ - is used:
% grib_get -p J t1000.grib
In this case, the value of J=1279 indicates the triangular truncation used is T1279.
How is the resolution of the spherical harmonic data related to the grid resolution ?
It is not possible to relate the resolution of the spherical harmonic data directly to the resolution of the corresponding grid as this depends on the specific relationship between the spectral
representation and the Gaussian grid used by the IFS. Three different relationships have been used by ECMWF over the years. For a triangular truncation of T, the corresponding Gaussian grid
resolutions are:
Relationship Gaussian grid resolution
linear N = (T + 1) / 2
quadratic N = 3 (T+1) / 4
cubic N = T + 1
where N corresponds to the number of latitude lines between the poles and the equator. However, to determine which relationship was used when the data were produced requires interrogation of a
corresponding grid point field.
See also:
Related articles | {"url":"https://confluence.ecmwf.int/display/UDOC/How+to+find+the+resolution+of+a+spherical+harmonic+field+in+GRIB+-+ecCodes+GRIB+FAQ","timestamp":"2024-11-10T09:37:03Z","content_type":"text/html","content_length":"63264","record_id":"<urn:uuid:7e53c95b-67df-4eca-b15d-003418a929da>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00871.warc.gz"} |
FK : in the same manner it may be demonstrated, that FL, FM, FG are each of them equal to FH, or FK : therefore the five straight lines FG, FH, FK, FL, FM are equal to one another : wherefore the
circle described from the centre F, at the distance of... Euclid's Elements of Geometry: The First Six, the Eleventh and Twelfth Books - Page 183by Euclid - 1765 - 464 pagesFull view
About this book
Euclid, John Keill - Geometry - 1723 - 436 pages
...Perpendicular F K. In the fame Manner we demonstrate, that FL, FM,- or FG, is equal to FH, or F K. Therefore the five Right Lines FG, FH, FK, FL, FM, are equal to each other. And fo a Circle
defcribed on the Center F, with either of the Diftances FG, FH, FK, FL,...
Euclid, John Keill - Geometry - 1733 - 446 pages
...Perpendicular FH equal to the Perpendicular F K. In the fame Manner we demonftrate, that FL, FM, or FG, is equal to FH, or FK. Therefore the five Right Lines FG, FH, FK, FL, FM, are equal to
each other. And fo a Circle defcribed oh the Center F, with either of the Diftances FG, FH, FK, FL,...
John Keill - Geometry - 1772 - 462 pages
...the Perpendicular F K. In the fame manner we demonftrate, that FL, FM, or FG, is equal to FH, or F K. Therefore the five Right • Lines FG, FH, FK, FL, FM, are equal to each other, and fo a
Circle defcribed on the Centre F, with either of the Diftances FG, FH, FK, FL,...
Euclid - 1781 - 552 pages
...perpendicular FH is equal to the perpendicular FK : In the fame manner it may be demonftrated that FL, FM, FG are each of them equal to FH or FK ; therefore the five ftraight lines FG, FH, FK,
FL, FM are equal to one another : Wherefore the cifcle defcribed from the...
Robert Simson - Trigonometry - 1781 - 534 pages
...perpendicular FH is equal to the perpendicular FK. in the fwne manner it may be demonftrated that FL, FM, FG are each of them equal to FH or FK; therefore the five ftraight lines FG, FH, FK,
FL, FM are equal to one another. wherefore the circle defcribe .'. from...
John Playfair - Euclid's Elements - 1795 - 462 pages
...perpendicular FH is equal to the perpendicular FK : in the fame manner it may be demonftrated, that FL, FM, FG are each of them equal to FH or FK : therefore the five ftraight lines FG, FH, FK,
FL, FM are equal to one another : wherefore the circle defcribed from the...
Robert Simson - Trigonometry - 1804 - 528 pages
...perpendicular FH is equal to the perpendicular FK. in the fame manner it may be demonftrated that FL, FM, FG are each of them equal to FH or FK; therefore the five ftraight lines FG, FH, FK,
FL, FM are equal to one another, wherefore the circle de~ fcribed from the...
Euclides - 1816 - 592 pages
...perpendicular FH is equal to the perpendicular FK : In the same manner it may be demonstrated ; that FL, FM, FG are each of them equal to FH, or FK : Therefore the five straight lines FG, FH,
FK, FL, FM are equal to one another : Wherefore the circle described from the...
John Playfair - Circle-squaring - 1819 - 348 pages
...perpendicular FH is equal to the perpendicular FK : in the same manner it may be demonstrated, that FL, FM, FG are each of them equal to FH or FK , therefore the five straight lines FG, FH, FK,
FL, FM are equal to one another : wherefore the circle described from the...
Peter Nicholson - Mathematics - 1825 - 1058 pages
...perpendicular FH is equal to the perpendicular FK : In the same manner it may be demonstrated, that FL, FM, FG are each of them equal to FH or FK : Therefore the uve straight linei FG, FH, FK ,
FL, FM are equal to on* another: Wherefore the circb described from... | {"url":"https://books.google.com.gi/books?id=kT44AAAAMAAJ&qtid=7e79d4&source=gbs_quotes_r&cad=4","timestamp":"2024-11-11T16:12:35Z","content_type":"text/html","content_length":"28240","record_id":"<urn:uuid:e2501356-4be1-44a9-b7dc-293c7566f614>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00405.warc.gz"} |
Getting Started with Data Structures in C
Learn Data Structures in C With Types
25 Oct 2024
3.24K Views
41 min read
What are Data Structures in C?
Have you begun to learn C Programming? Are you aware of the data structures used in C language? Data Structures in C provide a method for storing and organizing data in the computer memory. A data
structure is a fundamental building block for all critical operations performed on data. Effective utilization of the data structures leads to program efficiency.
In this C tutorial, we'll delve deep into the data structures used in the C language. We'll understand various types of data structures with examples. At the end of the tutorial, you'll differentiate
different data structures based on their characteristics. Let's get started.
Types of Data Structures in C
1. Primitive Data Structures: These data types are predefined in the C programming language. They store the data of only one type. Data types like short,integer, float, character, and double comes
in this category.
2. Non-Primitive Data Structures: They can store data of more than one type. They are derived from the built-in or the primitive data types and are called derived data types. Data types like arrays,
linked lists, trees, etc. come in this category. Non-Primitive Data Structures are also known as user-defined data types.
Based on the structure and arrangement of data, these data structures are further classified into two categories
1. Linear Data Structures: The data in this data structure is arranged in a sequence, one after the other i.e. each element appears to be connected linearly.
Based on the memory allocation, the Linear Data Structures are again classified into two types:
1. Static Data Structures: They have a fixed size. The memory is allocated at the compile time, and the user cannot change its size after being compiled; however, the data can be modified.
e.g. array
2. Dynamic Data Structures: They don't have a fixed size. The memory is allocated at the run time, and its size varies during the program execution. Moreover, the user can modify the size as
well as the data elements stored at the run time. e.g. linked list, stack, queue
2. Non-Linear Data Structures: The elements in this data structure are not arranged sequentially. They form a hierarchy i.e. the data elements have multiple ways to connect to other elements.
e.g. tree and graph
Types of Linear Data Structures in C
1. Arrays
An Array in the C programming language is a powerful data structure that allows users to store and manipulate a collection of elements, all of the same data types in a single variable. In simple
words, an array is a collection of elements of the same data type. The values get stored at contagious memory locations that can be accessed with their index number.
dataType arrayName[arraySize];
Example of Arrays in C
#include <stdio.h>
int main(){
int i=0;
int marks[5];//declaration of array
marks[0]=90;//initialization of array
//traversal of array
printf("%d \n",marks[i]);
}//end of for loop
return 0;
In this example, an integer array named "marks" of size 5 is declared and initialized with predetermined values, with each element's index denoting a different integer value.
There are two types of arrays in the C language:
1. Single-dimensional arrays: Here the elements are stored in a single row in contiguous memory locations. They are also known as 1D arrays.
2. Multi-dimensional arrays: They contain one or more arrays as their elements. They are known as arrays of arrays. Examples are 2D and 3D arrays.
2. Linked Lists
A Linked List is a linear data structure consisting of a series of connected nodes randomly stored in the memory. Here, each node consists of two parts, the first part is the data and the second part
contains the pointer to the address of the next node. The pointer of the last node of the linked list consists of a null pointer, as it points to nothing. The elements are stored dynamically in the
linked list.
Syntax to Declare Linked Lists in C
struct node
int data;
struct node *next;
Example of Linked Lists in C Compiler
#include <stdio.h>
#include <stdlib.h>
// Define the structure for a node
struct Node {
int data;
struct Node* next;
// Function to create a new node
struct Node* createNode(int data) {
struct Node* newNode = (struct Node*)malloc(sizeof(struct Node));
if (!newNode) {
printf("Memory error\n");
return NULL;
newNode->data = data;
newNode->next = NULL;
return newNode;
// Function to traverse and print the linked list
void traversal(struct Node* head) {
struct Node* itr = head;
while (itr != NULL) {
printf("%d\n", itr->data);
itr = itr->next;
// Function to insert a new node after a given previous node
void insertion(struct Node* prevNode, struct Node* newNode) {
if (prevNode == NULL) {
printf("The given previous node cannot be NULL\n");
newNode->next = prevNode->next;
prevNode->next = newNode;
int main() {
struct Node* head = createNode(1);
struct Node* second = createNode(2);
struct Node* third = createNode(3);
head->next = second;
second->next = third;
// Print the initial linked list
printf("Initial Linked List:\n");
// Insert a new node (4) after the second node
struct Node* newNode = createNode(4);
insertion(second, newNode);
// Print the linked list after insertion
printf("\nLinked List after Insertion:\n");
In the above code, a linked list is created and traversed. Afterward, we insert a new node in the linked list. Again it's traversed and printed with new values.
Initial Linked List:
Linked List after Insertion:
There are three types of Linked Lists used in C language:
1. Singly-linked list: Here, each node has dataand a pointer that contains a reference to the next node in the list. This means that we can traverse in only one direction, from the head to the tail.
It is the most common linked list.
2. Doubly linked list: In this type of linked list, each interconnected node contains three fields that contain references to both the next and previous nodes in the list along with the data. This
allows traversal in both directions, making it easier to insert or delete nodes at any point in the list.
3. Circular linked list: Here, the last node in the list contains a reference to the first node instead of NULL, creating a loop. This means that traversal can continue indefinitely, until the
program crashes.
3. Stack
A stack is an ordered list or a container in which insertion and deletion can be done from one end known as the top of the stack. The last inserted element is available first and is the first one to
be deleted. Hence, it is known as Last In, First Out LIFO, or First In, Last Out FILO. Stacks do not have a fixed size and their size can be increased or decreased depending on the number of
Basic Operations on Stack
• push: It is inserting an element at the top of the stack. If the stack is full, it is said to be an Overflow condition.
• pop: It removes the topmost element of the stack and returns its value.
• peek: It returns the value of the topmost element of the stack without modifying the stack.
• isFull: It determines if the stack is full or not.
• isEmpty: It checks if the stack is empty or not
Syntax to Define a Stack in C
struct stack
int myStack[capacity];
int top;
Example of Stack in C
#include <stdio.h>
#include <stdlib.h>
#define capacity 20
int count = 0;
struct STACK
int myStack[capacity];
int top;
void buildStack(struct STACK *stack)
stack->top = -1;
int isStackFull(struct STACK *stack)
if (stack->top == capacity - 1)
return 1;
return 0;
int isEmpty(struct STACK *stack)
if (stack->top == -1)
return 1;
return 0;
void push(struct STACK *stack, int dataValue)
if (isStackFull(stack))
printf("Overflow! The stack is full.");
stack->myStack[stack->top] = dataValue;
void pop(struct STACK *stack)
if (isEmpty(stack))
printf("\nUnderflow! The stack is empty. \n");
printf("The deleted value is: %d\n", stack->myStack[stack->top]);
void printValues(struct STACK *stack)
printf("The elements of the stack are:\n");
for (int i = 0; i < count; i++)
printf("%d \n", stack->myStack[i]);
int main()
struct STACK *stack = (struct STACK *)malloc(sizeof(struct STACK));
// create a stack.
// insert data values into the stack.
push(stack, 150);
push(stack, 270);
push(stack, 330);
push(stack, 400);
// print the stack elements.
// delete an item at the top.
// print the stack elements after performing the pop operation.
printf("\nThe stack elements after the pop operation: \n");
The above program in the C Editor demonstrates the creation of a Stack and its associated operations.
The elements of the stack are:
The deleted value is: 400
The stack elements after the pop operation:
The elements of the stack are:
3. Queue
A queue is an ordered list in which insertion is done at one end called REAR and deletion at another end called FRONT. The first inserted element is available first for the operations to be performed
and is the first one to be deleted. Hence, it is known as First In First Out, FIFO, or Last In Last Out, LILO.
Basic Operations on Queue
• enqueue: It is used to insert an element at the back of a queue or to the end or the rear end of the queue.
• dequeue: It removes and returns the element from the front of a queue.
• peek: It returns the value at the front end of the queue without removing it.
• isFull: It determines if the queue is full or not.
• isEmpty: check if the queue is empty or not. It returns a boolean value, true when the queue is empty, otherwise false.
Syntax to define a Queue in C
struct Queue {
int myQueue[capacity];
int front, rear;
Example of Queue in C
#include <stdio.h>
#include <stdlib.h>
#define capacity 15
struct QUEUE
int front, rear, size;
int myQueue[capacity];
void buildQueue(struct QUEUE *queue)
queue->front = -1;
queue->rear = -1;
int isFull(struct QUEUE *queue)
if (queue->front == 0 && queue->rear == capacity - 1)
return 1;
return 0;
int isEmpty(struct QUEUE *queue)
if (queue->front == -1)
return 1;
return 0;
void enqueue(struct QUEUE *queue, int item)
if (isFull(queue))
if (queue->front == -1)
queue->front = 0;
queue->myQueue[queue->rear] = item;
void dequeue(struct QUEUE *queue)
if (isEmpty(queue))
printf("The queue is empty.\n");
int deletedValue = queue->myQueue[queue->front];
if (queue->front >= queue->rear)
queue->front = -1;
queue->rear = -1;
printf("The deleted value is: %d\n", deletedValue);
void printValues(struct QUEUE *queue)
printf("The elements of the queue are:\n");
for (int i = queue->front; i <= queue->rear; i++)
printf("%d \n", queue->myQueue[i]);
int main()
struct QUEUE *queue = (struct QUEUE *)malloc(sizeof(struct QUEUE));
// create a queue.
enqueue(queue, 160);
enqueue(queue, 208);
enqueue(queue, 390);
enqueue(queue, 450);
// print the queue elements.
// delete an item from the queue.
// print the queue elements after performing the dequeue operation.
printf("\nThe queue elements after the queue operation: \n");
In the above code, we defined a queue. After that, we perform insertion, deletion, and print operations on the queue.
The elements of the queue are:
The deleted value is: 160
The queue elements after the queue operation:
The elements of the queue are:
Types of Non-Linear Data Structures in C
1. Graphs
A graph is a collection of nodes that consist of data and are connected to other nodes of the graph. Formally a Graph is composed of a set of vertices( V ) and a set of edges( E ). The graph is
denoted by G(V, E). In a graph, each entity is symbolized by a vertex, while edges denote the connections or associations between entities.
There are two ways to represent a graph in C:
1. Adjacency Matrix: It's a 2D array of size V x V where V is the number of nodes in a graph. It represents a finite graph, with 0's and 1's. It is a square matrix since it's a V x V matrix. The
elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph i.e. if there is any edge connecting a pair of nodes in the graph.
Example of Adjacency Matrix Representation of Graph in C
#include <stdio.h>
#include <stdlib.h>
#define MAX_VERTICES 10
void initializeMatrix(int matrix[MAX_VERTICES][MAX_VERTICES], int numVertices) {
for (int i = 0; i < numVertices; i++) {
for (int j = 0; j < numVertices; j++) {
matrix[i][j] = 0;
// Function to add an edge to the graph
void addEdge(int matrix[MAX_VERTICES][MAX_VERTICES], int start, int end) {
if (start >= MAX_VERTICES || end >= MAX_VERTICES || start < 0 || end < 0) {
printf("Invalid edge!\n");
} else {
matrix[start][end] = 1;
matrix[end][start] = 1; // For an undirected graph
// Function to print the adjacency matrix
void printMatrix(int matrix[MAX_VERTICES][MAX_VERTICES], int numVertices) {
for (int i = 0; i < numVertices; i++) {
for (int j = 0; j < numVertices; j++) {
printf("%d ", matrix[i][j]);
int main() {
int numVertices = 5;
int adjacencyMatrix[MAX_VERTICES][MAX_VERTICES];
initializeMatrix(adjacencyMatrix, numVertices);
addEdge(adjacencyMatrix, 0, 1);
addEdge(adjacencyMatrix, 0, 4);
addEdge(adjacencyMatrix, 1, 2);
addEdge(adjacencyMatrix, 1, 3);
addEdge(adjacencyMatrix, 1, 4);
addEdge(adjacencyMatrix, 2, 3);
addEdge(adjacencyMatrix, 3, 4);
printf("Adjacency Matrix:\n");
printMatrix(adjacencyMatrix, numVertices);
return 0;
The above program demonstrates the creation of a graph using an adjacency matrix. First of all, we initialize a matrix of size 10. Then we define a function to add an edge to the graph. At last,
we print the adjacency matrix.
Adjacency Matrix:
2. Adjacency List: It represents a graph as an array of linked lists. The index of the array represents a vertex and each element in its linked list represents the other vertices that form an edge
with the vertex.
Example of Adjacency List Representation of Graph in C
#include <stdio.h>
#include <stdlib.h>
struct AdjListNode {
int dest;
struct AdjListNode* next;
struct AdjList {
struct AdjListNode* head;
struct Graph {
int V;
struct AdjList* array;
struct AdjListNode* createAdjListNode(int dest) {
struct AdjListNode* newNode = (struct AdjListNode*)malloc(sizeof(struct AdjListNode));
newNode->dest = dest;
newNode->next = NULL;
return newNode;
struct Graph* createGraph(int V) {
struct Graph* graph = (struct Graph*)malloc(sizeof(struct Graph));
graph->V = V;
graph->array = (struct AdjList*)malloc(V * sizeof(struct AdjList));
for (int i = 0; i < V; ++i) {
graph->array[i].head = NULL;
return graph;
void addEdge(struct Graph* graph, int src, int dest) {
// Add an edge from src to dest. A new node is added to the adjacency list of src.
struct AdjListNode* newNode = createAdjListNode(dest);
newNode->next = graph->array[src].head;
graph->array[src].head = newNode;
newNode = createAdjListNode(src);
newNode->next = graph->array[dest].head;
graph->array[dest].head = newNode;
// Function to print the adjacency list representation of graph
void printGraph(struct Graph* graph) {
for (int v = 0; v < graph->V; ++v) {
struct AdjListNode* pCrawl = graph->array[v].head;
printf("\n Adjacency list of vertex %d\n head ", v);
while (pCrawl) {
printf("-> %d", pCrawl->dest);
pCrawl = pCrawl->next;
int main() {
int V = 5;
struct Graph* graph = createGraph(V);
addEdge(graph, 0, 1);
addEdge(graph, 0, 4);
addEdge(graph, 1, 2);
addEdge(graph, 1, 3);
addEdge(graph, 1, 4);
addEdge(graph, 2, 3);
addEdge(graph, 3, 4);
return 0;
The above code defines structures for nodes and lists. The createGraph() function initializes the graph, addEdge() adds edges between vertices and printGraph() displays the adjacency list of each
Adjacency list of vertex 0
head -> 4-> 1
Adjacency list of vertex 1
head -> 4-> 3-> 2-> 0
Adjacency list of vertex 2
head -> 3-> 1
Adjacency list of vertex 3
head -> 4-> 2-> 1
Adjacency list of vertex 4
head -> 3-> 1-> 0
2. Trees
Tree data structure forms a hierarchy containing a collection of nodes such that each node of the tree stores a value and a list of references to other nodes (the "children"). The topmost node of the
tree is called the root node from which the tree originates, and the nodes below it are called the child nodes.
Syntax to Declare a Tree in C
struct node {
int data;
struct node* left;
struct node* right;
There are some popular types of trees in data structures. Let's look at some of them
1. Binary Tree: In this according to the name, binary, each parent node can have at most two nodes or two children i.e. left and right child.
2. Binary Search Tree: They are special kinds of Binary Trees having some special characteristics. It is called a binary tree because each tree node has up to two children. It is called a search
tree because it can search a number in O(log(n)) time. The only difference between Binary Search and Binary Search Tree is that BST uses Binary Trees to represent the data set instead of an
Choosing the Right Data Structure in C
As a C developer, you need to know every data structure in C programming in complete detail. You must be careful enough while selecting the proper data structure for your project. We'll look at the
approach to selecting the data structure in C.
• Understand Requirements: Start by thoroughly understanding the requirements of your application like the type of data to be stored, the frequency and nature of data operations, memory
constraints, and expected performance metrics.
• Analyze Operations: Take a proper analysis of the operations that need to be performed on the data, including searching, sorting, updating, and traversing.
• Consider Time and Space Complexity: Evaluate the time and space complexity of the type of operations for various data structures.
• Size of Data Set: Different data structures and algorithms have different performance characteristics when operating on different size data sets.
Hence, in the above article, we saw all the prominent data structures in C with examples. We saw the two broad classifications of data structures as primitive and non-primitive. We got into the depth
of linear and non-linear non-primitive data structures. Enroll in our C Programming Course to learn C programming in complete detail.
Q1. What are data structures in C?
Data structures in C include arrays, linked lists, stacks, queues, trees, and graphs.
Q2. What is the function of data structures in C?
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Commutative operation | PHP
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PHP: Commutative operation
Do you remember the basic rule of arithmetic that changing the order of the numbers we're adding doesn't change the sum? It's one of the most basic and intuitive rules of arithmetics, it's called the
commutative law.
A binary operation is considered commutative if you get the same result after swapping operands. Obviously, addition is a commutative operation: 3 + 2 = 2 + 3.
But is subtraction a commutative operation? Of course not: 2 - 3 ≠ 3 - 2. In programming, this law applies in the same way as in arithmetic.
Moreover, most of the operations we face in real life aren't commutative. In conclusion, you should always pay attention to the order of things you're working with.
Write a program that counts and sequentially displays the values of the following mathematical expressions: "3 to the power of 5" and "-8 divided by -4".
Note that the results will appear stuck together, on one line, and without any spaces. We'll learn how to deal with this problem in future lessons.
The exercise doesn't pass checking. What to do? 😶
If you've reached a deadlock it's time to ask your question in the «Discussions». How ask a question correctly:
• Be sure to attach the test output, without it it's almost impossible to figure out what went wrong, even if you show your code. It's complicated for developers to execute code in their heads, but
having a mistake before their eyes most probably will be helpful.
In my environment the code works, but not here 🤨
Tests are designed so that they test the solution in different ways and against different data. Often the solution works with one kind of input data but doesn't work with others. Check the «Tests»
tab to figure this out, you can find hints at the error output.
My code is different from the teacher's one 🤔
It's fine. 🙆 One task in programming can be solved in many different ways. If your code passed all tests, it complies with the task conditions.
In some rare cases, the solution may be adjusted to the tests, but this can be seen immediately.
I've read the lessons but nothing is clear 🙄
It's hard to make educational materials that will suit everyone. We do our best but there is always something to improve. If you see a material that is not clear to you, describe the problem in
“Discussions”. It will be great if you'll write unclear points in the question form. Usually, we need a few days for corrections.
By the way, you can participate in courses improvement. There is a link below to the lessons course code which you can edit right in your browser.
• Commutativity is the property of an operation wherein changing the order of operands doesn't affect the result. For example, addition is a commutative operation; changing the order of the numbers
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Simulating quadratic dynamical systems is PSPACE-complete (preliminary version)
Quadratic Dynamical Systems (QDS), whose definition extends that of Markov chains, are used to model phenomena in a variety of fields like statistical physics and natural evolution. Such systems also
play a role in genetic algorithms, a widelyused class of heuristics that are notoriously hard to analyze. Recently Rabinovich et al. took an important step in the study of QDS'S by showing, under
some technical assumptions, that such systems converge to a stationary distribution (similar theorems for Markov Chains are well-known). We show, however, that the following sampling problem for
QDS'S is PSPACE-hard: Given an initial distribution, produce a random sample from the t'th generation. The hardness result continues to hold for very restricted classes of QDS'S with very simple
initial distributions, thus suggesting that QDS'S are intrinsically more complicated than Markov chains.
Original language English (US)
Title of host publication Proceedings of the 26th Annual ACM Symposium on Theory of Computing, STOC 1994
Publisher Association for Computing Machinery
Pages 459-467
Number of pages 9
ISBN (Electronic) 0897916638
State Published - May 23 1994
Externally published Yes
Event 26th Annual ACM Symposium on Theory of Computing, STOC 1994 - Montreal, Canada
Duration: May 23 1994 → May 25 1994
Publication series
Name Proceedings of the Annual ACM Symposium on Theory of Computing
Volume Part F129502
ISSN (Print) 0737-8017
Other 26th Annual ACM Symposium on Theory of Computing, STOC 1994
Country/Territory Canada
City Montreal
Period 5/23/94 → 5/25/94
All Science Journal Classification (ASJC) codes
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Sketcher of various interrelated fourfolds.
Don’t miss the grove for the tree(s).
August 15, 2015.
Latest substantive edit: October 16, 2024. More edits to come.
Table 1.
In short:
Major well-definable modes of inference,
distinctive heuristic merits,
typical disciplines.
Inference modes. Deduction Ampliative inference
= non-ampliative inference. = non-deductive inference (C.S.Peirce).
Premises entail the conclusion or more. Premises don’t entail the full conclusion.
Proposed name: A. C. By proposed definition as
Repletive inference Repletive-&-deductive: Repletive-&-ampliative:
= non-attenuative Reversible deduction Induction
inference. (i.e., through equivalence). from part to collective whole.
Conclusion entails the — Counterbalancingly, — Counterbalancingly,
premises or more. its conclusions gain worth by its conclusions gain worth by
nontriviality, depth. verisimilitude, likelihood
⇝ Typically formalized in PURE MATHS: structures of space, of counting & measure, of (in Peirce’s sense: resemblance of conclusion to premises).
group, ring, field, & of order, & their applications. ⇝ Typically formalized in STATISTICS & KINDRED DISCIPLINES: inverse
optimization, communication & control, maybe some of philosophy, & their
Proposed name: B. D. By proposed definition as
Attenuative inference Attenuative-&-deductive: Attenuative-&-ampliative:
= non-repletive Forward-only Surmise, conjecture, abductive inference.
inference. deduction. — Counterbalancingly,
Conclusion doesn’t — Counterbalancingly, its conclusions gain worth by
entail the full its conclusions gain worth by plausibility à la Peirce: instinctual simplicity & naturalness.
premises. newness of aspect. ⇝ Typically formalized in SCIENCES OF CONCRETE PHENOMENA of motion,
⇝ Typically formalized in MATHS OF STRUCTURES OF ALTERNATIVES IN TIMELIKE matter, life, & mind, & their applications.
PERSPECTIVES: optimal & feasible solutions, probabilities, information, logic, & their
Definitions of inference modes in terms of entailment or, just as well, truth/falsity-preservativeness.
Illustrative GIF 1.
Simplified proposed square of inference modes.
Deductive and ampliative inferences DON’T and DO — respectively — ADD, to their conclusions, things beyond that which their premises give. But I couldn’t find reasonably
simple names for inferences that DON’T and DO — respectively — OMIT, from their conclusions, things given by their premises.
So, I picked out a couple of words — repletive and attenuative — that people may find handy. “Repletive” is pronounced re‑PLEE‑tiv. It rhymes with “depletive” and
“completive”. I first discussed the words in a post “Inference terminology” to peirce-l 2015-04-07. (Under “Adjective choices” below, I discuss why the words seem better choices than others.)
Every inference is either deductive or ampliative (but not both) and is either repletive or attenuative (but not both), at least on assumptions of classical logic. Neither one of those two exclusive
exhaustive alternatives depends on the other.
The repletive-attenuative dichotomy, less prominent though it is, is worth a careful look, for it stands both analagous and crosswise to the deductive-ampliative dichotomy and adds some light in
completing a regular four-way division (a tetrachotomy), one that embraces both dichotomies. (Said tetrachotomy, displayed twice as a square above, contains a third clarifying dichotomy, consisting
of the two diagonals A.‑D. and B.‑C.)
Charles Sanders
Founder of the study
of abductive inference.
See Appendix for extensive quotes from his Century Dictionary definitions of: ampliation(3.); ampliative;
induction (5.);
hypothesis (4.);
and from his later
discussion of
simpler hypothesis.
Three notes.
1. Note on words. By standard definitions, all and only non-deductive inference is ampliative inference, concluding in something extra in the sense of its not following purely from the premises.
Still, many call it “inductive” rather than “ampliative”. (Meanwhile, none of that pertains to mathematical induction, which is deductive.) In the classification of inferences, the logician and
polymath C.S. Peirce typically made a trichotomy of deduction, induction, and abduction, and called non-deductive inference “ampliative”, which Peirce knew to be the adjective preferred over
“synthetic” for non-deductive judgment earlier by Sir W. Hamilton and before him by Kant’s translator Archbishop Thomson. See “Semantic discussion” far below. End of note.
2. Note on abductive inference. My proposed (and seemingly original) definition of abduction as both ampliative and attenuative results in classing as abductive, not inductive, the inference of a
hypothetical universal from one or more existential particulars in a populous enough universe. I’ve found Peirce classing inference of a rule from particulars as abductive only once, in 1903. Also,
abduction by the proposed definition is both ampliative and attenuative in both directions (between the premise set and the conclusion), ergo abductive in both directions, yet still tends to be
retroductive, inferring a cause, reason, explanation, in only one direction. See more below under “Inferring a rule: abductive vs. inductive”. End of note.
3. Note on entailment etc. For simplicity’s sake, I refer to just the same thing for this discussion’s purposes, when I say things like “entailment properties” and when I say things like
“truth-preservativeness and falsity-preservativeness”. End of note.
Table 2.
Elementary forms. Truth/falsity-preservative & entailmental properties.
▶ The forms (schemata) below show all & only pertinent logical structure. ◀
▶ “entail” ≡ “deductively imply” ◀
│ Inference Mode & │ Model-Theoretically. │ Proof-Theoretically. │ Informal Description. │
│ Elementary Forms. │ │ │ │
│ Deductive, e.g.: │ Truth-preservative. │ The premises │ The conclusion does not go │
│ p∴p.pq∴p. │ │ entail the conclusion. │ beyond the premises. │
│ Ampliative, e.g.: │ Not truth-preservative. │ The premises do not │ The conclusion goes │
│ p∴q.p∴pq. │ │ entail the conclusion. │ beyond the premises. │
│ Repletive, e.g.: │ Falsity-preservative. │ The premises are │ The premises do not go │
│ p∴p.p∴pq. │ │ entailed by the conclusion. │ beyond the conclusion. │
│ Attenuative, e.g.: │ Not falsity-preservative. │ The premises are not │ The premises go │
│ p∴q.pq∴p. │ │ entailed by the conclusion. │ beyond the conclusion. │
Each of the inference modes has its merits or virtues, as well as drawbacks or vices, in regard to the prospect of concluding in a truth or a falsehood:
Security (futility) vs. opportunity (risk)
as prospects seen apart from their being counterbalanced in some sense by heuristic perspectives given to premises by conclusions.
• Deductive inference does not decrease security / futility — i.e., does not increase opportunity / risk.
• Ampliative inference decreases security / futility — i.e., increases opportunity / risk — in some way.
• Repletive inference does not increase security / futility — i.e., does not decrease opportunity / risk.
• Attenuative inference increases security / futility — i.e., decreases opportunity / risk — in some way.
I use the phrase “in some way” twice, in the bulleted list just above, in order to allude to the fact that an inference can be both ampliative and attenuative, i.e., it can at one same time increase
risk (or whatever) in some way and decrease it in some other way. (Of course elementary inference, as a topic, does not exhaust the topic of security, opportunity, etc., and their increase and
interplay in inquiry generally. Even at an elementary level, the prospects can be counterbalanced in some sense by heuristic perspectives given by conclusions to premises.)
Each merit or virtue comes with a flipside drawback or vice. Risk managers sometimes say, “opportunity equals risk,” and call them two sides of the same coin. In that spirit: security, safeness,
equals futility — “nothing ventured, nothing gained.” Freud made much from the titanic fact that one tends to have less choice between pleasure and pain than between both and neither.
Each of the above four inference modes works as a trade-off. On classical assumptions of two-valued logic, four mutually exclusive intersections of the above trade-offs, the above inference modes,
necessarily arise and encompass all inferences:
Table 3.
Intersections of modes of inference,
focussing on changes & non-changes of opportunity / risk
as prospects apart from counterbalancement by heuristic
perspectives given to premises by conclusions.
Inference modes ⯆ ⯆
⯆ Deductive: ⯆ Ampliative (=non-deductive):
NO INCREASE of opportunity/risk. SOME INCREASE of opportunity/risk.
Repletive: ⯈⯈ A. Reversible (i.e., equative) C. Induction.
NO DECREASE of opportunity/risk. By proposed definition as:
Repletive-&-deductive: Repletive-&-ampliative:
NO decrease, NO increase, of opportunity/risk. NO decrease, SOME increase, of opportunity/risk.
Attenuative ⯈⯈ B. Forward-only D. Surmise, conjecture, abductive inference.
(=non-repletive): deduction.
By proposed definition as:
SOME DECREASE of opportunity/risk. Attenuative-&-deductive: Attenuative-&-ampliative:
SOME decrease, NO increase, of opportunity/risk. SOME decrease, SOME increase, in different ways, of opportunity/risk.
Any unilingual grab-bag of univocal propositions (or, more generally, terms, e.g., in calculations), no matter how one divides said bag’s contents into bag of conjoined premises and bag of conjoined
conclusions, belongs to exactly two of the above four simple modes of inference, and to exactly one of their four intersections.
• Some deductive conclusions, despite true premises and conformity to a standard definition of deduction, will be just too obvious, blatantly redundant, which is worthwhile and needful in elementary
rules of deduction but seldom elsewhere as far as I know. Conformity to the usual definitions of deduction is needful in deduction but does not guarantee a deductive conclusion worthwhile in that it
gives a non-obvious — new or nontrivial — aspect to the premises.
• Some ampliative conclusions, despite true premises and conformity to a standard definition of ampliative inference, will be arbitrary and capricious: all too unlikely or all too implausible.
Special forms are usually developed in order to help the practice of achieving worthwhile conclusions. Special forms can illuminate, but what common pattern, if any, guides the special forms of
disparate inference modes in their common truth-seeking mission?
Systematic opposition between definitive form and distinctive heuristics of each of the main four mutually exclusive modes of inference.
Circle Limit III,
1959, M.C. Escher
Inferences may be worth classifying in the above four-fold manner because then four major inference modes will be defined in a uniform hard-core formal manner that exhausts the possibilities, by
their basic internal entailment relations (or preservativeness or otherwise of truth and of falsity); meanwhile their various attempted heuristic merits — 
◆ abductive conclusion’s plausibility (natural simplicity, in C.S. Peirce’s sense),
◆ inductive conclusion’s verisimilitude / likelihood (in Peirce’s sense: resemblance of conclusion to premises),
◆ forward-only-deductive conclusion’s new aspect, and
◆ reversibly-deductive conclusion’s nontriviality / depth
 — can be treated as forming a systematic class of aspects of fruitfulness or promisingness of inference, with each of them related (as the compensatory opposite, in a sense) to its
respective inference mode’s definitive internal entailment relations.
Those heuristic merits are difficult to quantify usefully or even to define exactly; yet, together with the entailment relations, they illuminingly form a regular system in which each heuristic merit
helps to overcome, so to speak, the limitations of its inference mode’s definitive entailment relations. At any rate there is a fruitful tension between the heuristic merit and the entailment
relations in each inference mode.
Table 4.
Inference modes & counterbalancing heuristic merits
(on the assumption that induction is repletive & ampliative,
and that abduction is attenuative & ampliative).
The propositional schemata below, when taken to display all pertinent logical structure,
exemplify inferences lacking their modes’ normal heuristic merits.
Inferences Deductive: Ampliative:
Repletive: A. Reversible deduction, e.g.: C. Induction, e.g.:
pq ∴ pq. pq ∴ pqr.
Logically simple. Adds new claim(s).
Compensate with the Compensate with
nontrivial, complex, deep. verisimilitude, likelihood
(conclusion’s likeness to the old claims).
Attenuative: B. Forward-only deduction, e.g.: D. Abductive inference, e.g.:
pqr ∴ pq. pq ∴ qr.
Claims less, vaguer. Logically complicated.
Compensate with Compensate with
newness, by concision, of aspect. natural simplicity
(abductive plausibility).
Notes about Table 4 above:
• Systematic oppositions hold along the diagonals.
• All the heuristic merits considered here are those of aspects that conclusions give to premises, not those of the inferring or reasoning itself. The Pythagorean theorem is considered deep but its
proof is not considered especially deep or very nontrivial, especially in the sense of “difficult” that is often enough (and understandably) allied to the idea of the nontrivial.
• (i) Natural simplicity and (ii) verisimilitude / likelihood contribute, in variable degree, to inclining the reasoner to believe or suspect a conclusion, at least until it is
while, in systematic and symmetrical contrast to them,
(iii) novelty and (iv) nontriviality contribute, in variable degree, to inclining the reasoner to disbelieve or doubt a conclusion, at least until it becomes established.
Those things emerge along the deductive-ampliative axis, the strongest axis. The matter becomes more complex when one incorporates the risk-versus-doubt profile along the repletive-attenuative
Table 5.
Formally wrought decrease & increase of RISK versus
perspectivally courted increase & decrease of DOUBT.
(A) Deductive conclusion (B) Ampliative conclusion
does not increase risk increases risk
YET can invite increase YET can invite sameness
of doubt. or decrease of doubt.
(1) Repletive conclusion (A1) Repletively (B1) Inductive (defined as
does not decrease risk deductive repletively ampliative)
YET can invite decrease conclusion can have conclusion can have
of doubt. (2) Attenuative conclusion nontriviality, depth. likelihood, verisimilitude.
decreases risk (A2) Attenuatively (B2) Abductive (defined as
YET can invite sameness deductive attenuatively ampliative)
or increase of doubt. conclusion can have conclusion can have
newness of aspect. natural simplicity.
Now, an inference actually arising in a course of thought does not always present itself clearly. Hence, one may consider its seeming heuristic merit (such as plausibility), or its seeming purpose
(such as explanation), in order to help one decide what mode of inference it ought to be framed as instancing.
It may even seem that one can pair the four entailment-related definitions one-to-one with four definitions in terms of heuristics and/or function. Yet, for example, deduction is not defined as
explicative, making explicit the merely implicit, since an inference with the form “p∴p”, e.g., “all is well, ergo all is well”, is deductive but its conclusion extracts no new or nontrivial
perspective from its premises. Furthermore and as already said, the heuristic merits themselves resist exact definition.
On the other hand, if no deduction were ever to make explicit the merely implicit, then no mind would bother with deductive reasoning. The heuristic merits deserve attention because, in the pervasive
absence of all the heuristic merits, no mind would bother with any reasoning — explicit, consciously weighed inference — at all. Deduction, whether repletive or
attenuative, would lose as much in general justification and rationale as induction and abductive inference would lose. Little in general would remain of inference, conscious or unconscious, mainly
such activities as remembering, and free-associative supposing, which are degenerate inferences analogously as straight lines are degenerate conics.
Table 6.
Inferences, when sapped of heuristic merits
and associated with cognitive modes.
Even these in their seasons have other merits.
Inferences Deductive: Ampliative:
Repletive: Reversible deduction, e.g.: Induction, e.g.:
… ∴ p, ∴ p, ∴ p, ∴ … . … ∴ p∨q, ∴ q, ∴ qr, ∴ … .
Fixated remembrance. Swelling expectation.
Attenuative: Forward-only deduction, e.g.: Abductive inference, e.g.:
… ∴ pq, ∴ q, ∴ q∨r, ∴ … . … ∴ p, ∴ q, ∴ r, ∴ … .
Dwindling notice. Wild supposition.
Quite unpromising, and vacuous in that sense, is an ampliative reasoning without plausibility or verisimilitude, though it have “blackboard validity” (a phrase I reel in from
Jeff Downard
) as an ampliative inference (i.e., the premises don’t entail the conclusion).
Jeffrey Brian Downard
Source: ResearchGate
So, the proposed entailment-relational definitions of inductive and abductive inferences admit of pointless inferences. Yet this seeming problem may help solve a bigger problem. After all, the
standard definitions of deduction admit of pointless inferences as well —vacuous deductions. Definitions of deduction would be just as slippery as those of induction and abductive inference if
one were to try to define deduction generally in terms of purposive and heuristic properties rather than in terms of entailment relations (or, just as well, preservativeness and non-preservativeness
of truth and falsity).
In the study of arguments, most of the theoretical interest will be in schemata and procedures that both (A) bring “blackboard validity” — conformity with the entailment-relational
definition (and concomitant virtues) of the given inference mode — and also (B) ensure some modicum of the inference mode’s correlated mode of promise, such as natural simplicity (
Peirce’s forms), a new aspect (the categorical syllogistic forms), etc. Together, such conformity and such promise relate inference modes to distinctive aims at explanation, prediction, etc.
Let guesses seem guesses.
Nathan Houser
Source: Academia.edu
Editor, director, now retired,
at the Peirce Edition Project.
But now that abduction is taken seriously, and so much attention has turned to its examination, we find that it is indeed a very slippery conception.
— Peirce scholar Nathan Houser in “The Scent of Truth” in Semiotica 153—1/4 (2005), 455–466.
An unslippery definition of abductive inference is reached naturally in the course of defining all the broadest inference modes in a hard-core way systematically via entailment relations (or, just as
well, via truth/falsity-preservativeness), such that abductive inference is definable as both ampliative (not automatically preserving truth) and attenuative (not automatically preserving falsity)&
hairsp;— the premises neither entail, nor are entailed by, the conclusion.
Illustrative GIF 2. Simplified proposed square of inference modes.
The very idea of inference across both-ways non-entailment evokes the notion of leaps or guesses dauntingly to one who thinks of logic as properly strict and formal. The idea of course daunted me. I
thought that almost nobody would take it seriously. Yet, I thought it very plausible that Nature or the Logos or the like would indeed favor use of all four of the basic inferential forms that
suggest themselves to a systematic thinker. Then I came to read, gratefully, Peirce on abductive inference. Peirce never defined it as involving both-ways non-entailment, but he did call it guessing,
at least in later years. Still, the idea of guessing as a mode of inference dissatisfies quite a few even when they do read Peirce.
Yet, a guess — in the sense of a conjecture or surmise — is an inference insofar as it consists in acceptance of a proposition (or term), even if but tentatively, on the
basis of some proposition(s) (or term(s)). Now, a guess ought to be at least somewhat a leap, out of a box so to speak, just as a deductive conclusion ought to be technically redundant, staying in a
box. They are simply different trade-offs between opportunity (risk) and security (futility).
Ergo, let guessing seem guessing. Let the definition of abductive inference PLAINLY represent the potential wildness of abductive conclusions, ANALOGOUSLY as most definitions of deduction PLAINLY
represent the technical redundancy and potential vacuity of deductive conclusions.
Let the potential wildness of abductive conclusions be seen as counterbalanced by the practice, discussed richly by Peirce and exemplifiable in various particular forms, of finding plausibility
(natural simplicity), along with conceivably testable implications, analogously as the technical redundancy and potential vacuity of deductive conclusions are seen as counterbalanced by the practice,
exemplifiable in various particular forms, of finding a new or nontrivial aspect, also conceivable further testability. Analogous remarks can be made about induction, verisimilitude / 
likelihood, and testability.
Tomis Kapitan
Source: Academia.edu
Thusly defined as both ampliative and attenuative, — and thusly distinguished as attenuative, from induction as repletive, — abductive inference would plainly have the
autonomy found lacking by Tomis Kapitan (1949–2016) in “Peirce and the Autonomy of Abductive Reasoning” (PDF) in Erkenntnis 37 (1992), pages 1–26, an article discussed in Houser’s article linked
above. (Kapitan also wrote “Abduction as Practical Inference”, published online in 2000 in the Digital Encyclopedia of Charles S. Peirce, reproduced at Commens.org.)
In other words, abductive inference would not boil down, in such analysis, to some application, practical or theoretical, of deduction and/or induction. Worthwhile, pertinent ideas about natural
simplicity, explanatory power, pursuit-worthiness, etc., which contribute to the current slipperiness of definitions of abductive inference, would instead be further salient issues of abductive
inference, neither explicitly contemplated in its definition nor incorporated into the content of all abductive inferences. Such incorporation, besides the problems that Kapitan finds, would make one
abductive conclusion into many conflicting ones, a continuum of potential conflicting ones, just by people’s differing soever fuzzily in the amounts of plausibility, economy, pursuit-worthiness,
etc., that they assert in it. Define abductive inference by its form, not by its heuristics and functionality, just as with deduction, where the somewhat slippery but valuable and pertinent ideas of
novelty, nontriviality, predictive power, etc., are further salient issues of deduction, neither explicitly contemplated in its standard definitions nor incorporated into the content of all
deductions (and such couldn’t usefully be done deductively). Such spartanism at the elementary level need not and ought not go so far as to forbid explicitly qualifying the illative relation by
saying “therefore, abductively,” or “therefore, deductively,” or the like, which very arguably Peirce would have endorsed.
Yet, some slipperiness remains, which the proposed definitions of the inference modes do not entirely remedy. I will take this up in the next subsection “inferring a rule”.
Rules, cases, facts.
Inferring a rule: abductive vs. inductive.
Given this post’s proposed definitions of induction and abduction (pace tradition), logically attenuative inference from one or more particulars to a rule is classed among abductive forms rather than
For example: “some fruit is sweet, ergo (rule) no fruit is not sweet (i.e., in logicians’ lingo, all fruit is sweet)”.
That fruit example concludes with a potentially very long shot from an existential particular to a (Boolean) hypothetical universal. The example does feel like a surmising, a conjecturing but, under
the proposed definitions, we haven’t needed to guess its classification, since its classification is determined only by the overall relation of entailing and being entailed (or, just as well, truth/
falsity preservativeness) between the premises as a whole and the conclusion as a whole.
Now, the original idea of induction was inference from particulars to a rule (see Peirce below); but one typically assumed the rule to be an (Aristotelian) existential universal, such that “All fruit
is sweet” entails, deductively implies, its subaltern “Some fruit is sweet” (look up immediate inference). Even under the inference-mode definitions herein proposed, inference of an existential
universal from particulars in a populous enough universe will, in some cases, be inductive, not abductive.
A. In earlier years Peirce often portrayed abduction (calling it “hypothesis”) as inference from rule (all men are mortal) and fact (Socrates is mortal) to case (Socrates is a man), and as involving
no generalization (see below for a quote from 1889–91). Later, Peirce at least once countenanced the abducing of rules from particulars. In his 1903 Syllabus of his Lowell lecture series Some
Topics of Logic, he wrote (Essential Peirce 2, 287, in 1st paragraph, see at Commens):
The whole operation of reasoning begins with Abduction […]. Its occasion is a surprise. […]. The mind seeks to bring the facts, as modified by the new discovery, into order; that is, to form
a general conception embracing them. In some cases, it does this by an act of generalization. In other cases, no new law is suggested, but only a peculiar state of facts that will “explain”
the surprising phenomenon; and a law already known is recognized as applicable to the suggested hypothesis, so that the phenomenon, under that assumption, would not be surprising, but quite
likely, or even would be a necessary result. […].
B. One may wonder whether, in that passage, Peirce implicitly meant an INDUCTIVE generalization, not an abductive one, to a new law; but soon (in the 2nd paragraph on the same page 287 in EP 2, see
at Google Books), Peirce wrote regarding deduction’s taking up an abduced hypothesis:
[…] either in the hypothesis itself, or in the pre-known truths brought to bear upon it, there must be a universal proposition, or Rule.
So, again, Peirce held at least in 1903 that a hypothesis abduced from particulars can contain a newly guessed rule.
Feynman asked, how do we find new laws? and emphasized that first, we guess them. Induction on the other hand, if defined as ampliative and repletive, seems oftenest concerned with estimating about
one or another population or universe of discourse that requires not some universal quantification(s) “∀x ∀y Hxy” or the like that cuts through the crap, so to speak, but instead a
collective predicate (such as the term “gamut”) attributing a distribution of properties, a relation, a percentage, etc., across that entire population or universe as one entire polyadic instance (at
least implicitly), an ascription toward which an inductive inference seeks to expand the claims in its premises without squintingly narrowing its focus exclusively to rules up, up, and away, so to
speak, from a variegated densely woven landscape.
Table 7.
Inference from particulars to a rule:
universe-size axioms & inference modes
(with induction defined as both repletive & ampliative, and
with abduction defined as both attenuative & ampliative)
│ │ There is a horse; │ There is something both a horse & magnificent; │
│ │ ergo anything at all is a horse. │ ergo anything at all is, if a horse, then magnificent. │
│ Given a postulate or │ ∃H ∴ ∀H. │ ∃HM ∴ ∀(H→M). │
│ axiom of a universe of… │ The above inference to a rule is: │ The above inference to a rule is: │
│ …zero objects │ Attenuatively deductive but necessarily unsound. │ Attenuatively deductive but necessarily unsound. │
│ …one object │ Reversibly deductive. │ Attenuatively deductive. │
│ …zero or one object │ Attenuatively deductive. │ Attenuatively deductive. │
│ …one or more objects │ Inductive. │ Abductive. │
│ …zero or more objects │ Abductive. │ Abductive. │
To make the abductive equine example somewhat more realistic, suppose that the inference is something more like: There exists planet Earth with quite various countries crowded on it, among most of
which are known to be distributed all known horses, and every such horse is known to be magnificent. Ergo, all horses are magnificent.
Richard Feynman, chalking cursives
Source: BBC video via Cornell U. via YouTube.
[…]. In general, we look for a new law by the following process.
First, we guess it. Then we com- — w- – don’t laugh, that’s the - really true. Then we compute the consequences of the guess, to see what, if this is right, if this
law that we guessed is right, we see what it would imply and then we compare those computation results to nature or we say compare to experiment or experience, — compare it directly
with observations to see if it, if it works.
If it disagrees with experiment, — it’s wrong. In that simple statement is the key to science. It doesn’t make a difference how beautiful your guess is, it doesn’t matter how smart
you are, who made the guess, or what his name is. If it disagrees with experiment, it’s wrong. That’s all there is to it. It’s therefore not unscientific to take a guess although many people who
are not in science think it is. […].
— Richard Feynman, 11/19/1964, his Cornell U. Messenger Lecture series “The Character of Physical Law”, Lecture 7: “Seeking New Laws” (go to 16:29).
C.S. Peirce does at least once discuss a kind of abductive inference that concludes in a “generalization” to a new law (1903, see The Essential Peirce v. 2, p. 287, both paragraphs, see above).
Peirce in earlier years described generalization as selective of the characters generalized (decreasing the comprehension while increasing the extension, see “Upon Logical Comprehension and
Extension”, 1867, Collected Papers v. 2 ¶422, also in Writings v. 2, p. 84), and as casting out “sporadic” cases (“A Guess at the Riddle”, 1877–8 draft, see The Essential Peirce v. 1, p. 273). If, in
the 1903 passage, he is discussing abductive inference by selective generalization quite generically, then any crude induction from the mass of experience seems to count as abductive instead.
Induction from sample to definite total population, as people actually frame it in practice, is sometimes not only ampliative but also attenuative — the conclusions do not always
entail the premises, even though one usually thinks of induction as inferring from a part to a whole including the part.
There are ways to reframe the induction “half this [designated, indicated] actual sample is blue, so half the total is blue” so that its conclusion will entail its premise, ways that are perhaps to
be favored over the example if they reflect better the inquirial interest involved in induction:
—one could say “some subset” or “some randomly drawn subset” (or the like) instead of “this actual sample” (or the like); or
—one could characterize this actual sample in the conclusion as well as in the premise (like a concluding graph that represents the actual measurements with a heavier line); or
—one could define a certain designated total population such that the definition is a kind of census explicitly designating each of the individuals in the population, and
—of course there are likely other ways that I haven’t thought of.
Such perspectives suit checking consistency by the deducibility of the premise from the conclusion and, applying probability calculations, deducing what would be the probability of drawing a given
subset as a sample given alternate possible sets of parameters of the total population.
Abductive yet non-explanatory inference.
Take a famous example of abductive inference:
The lawn is wet this morning.
Whenever it rains at night, the lawn is wet the next morning.
Ergo [hypothetically] it rained last night.
Now, suppose that the example of abductive inference is instead:
It rained last night.
Whenever the lawn is wet in the morning, it has rained the previous night.
Ergo [hypothetically], the lawn is wet this morning.
That is not a usual abductive inference to a cause or reason, because it is hard to see how the hypothesized circumstance that the lawn is wet this morning would explain the fact that it rained last
Rather than try to see every individual effect in an individualized teleological light, it is simpler to note that abductive inference (in the sense of inference both ampliative and attenuative) can
go in either direction between its set of premises and its set of conclusions, likewise as reversible deduction can. This view does not vitiate the connection between abductive inference and
explanation, likewise as the analogous view about reversible deduction does not vitiate the connection between reversible deduction and proving a theorem or corollary.
Just as mathematical reversible deduction has a preferred direction given as its purpose of moving from things already postulated or proven (i.e., moving from reasons) to new conclusions, so likewise
abductive inference in the sciences of concrete phenomena has a preferred direction by its purpose of moving from effects to causes (i.e., moving to reasons). Thus both abductive inference and
reversible deduction can be both (A) defined in terms of entailment (or truth/falsity preservativeness) and (B) characterized by their respective purposes. It’s not a bad thing.
Mathematicians often enough move in the reverse direction in their “scratch work”, from thesis that needs proof, to something already postulated or proven — then, if every deductive
step has been through an equivalence, they can simply reverse the order of the scratch work to produce the proof. Mobility in both directions seems less dazzling in abductive inference, at least
insofar as abductive inferences (and non-mathematical inferences generally as Zadeh mentioned somewhere) tend to be less lengthy than reversible deductions which can become very long chains. End of
“Reverse inferences”: abduction vs. induction vs. deduction.
Abduction can seem deduction in reverse, e.g., using Peirce’s terms “rule”, “case”, and “fact”:
Rule: All food tastes good.
Case: This fish is food.
Ergo, deductively,
Fact: This fish tastes good.
Fact [oddly]: This fish tastes good.
Rule: All food tastes good.
Ergo, abductively,
Case [plausibly]: this fish is food.
Yet the reversed Barbara deduction is not the abductive syllogism above. Instead, it is the capricious:
Fact: This fish tastes good.
Ergo: Case: This fish is food; and Rule: All food tastes good.
—which is both ampliative and repletive (i.e., inductive, not abductive, under present text’s proposal).
Also, the reversed traditional abductive syllogism is the capricious:
Case: This fish is food.
Ergo: Fact: This fish tastes good; and Rule: All food tastes good.
However, suppose that one makes not just a premise but a postulate of the rule that all food tastes good. Then the rule will remain among the premises when the case and the fact switch places between
the premise set and the conclusion. Under the postulate that all food tastes good:
• from the fact that this fish tastes good, one becomes able to induce the case that this fish is food; and
• from the case that this fish is food, one becomes able to (attenuatively) deduce the fact that this fish tastes good.
So, as a result of the proposed definitions of induction and abduction:
• an inference will be abductive if it extends an unpostulated rule to cover something as a case to explain a fact, and will not be the reverse of deduction of same fact from rule and case; and, by
the same stroke,
• if the rule has been postulated, then the aforesaid inference will be inductive and its reverse will be (attenuatively) deductive.
Abductive inference and statistical syllogism.
“It has rained every day for a week, ergo tomorrow it will rain.” Thusly framed, that inference is both ampliative and attenuative, hence abductive under the proposed definition. But it is just as
natural to frame the thought as being, that it has rained every day for a week and that ergo tomorrow it will again rain, rain for the eighth consecutive day, etc., at which point the inference is
framed as inductive; that seems much of its spirit. It turns into a kind of statistical syllogism, that is, a statistical induction concluding in a premise for an attenuative deduction —&
hairsp;in this case, an induction from seven consecutive rainy days as of today to eight consecutive rainy days as of tomorrow, followed by an attenuative deduction (from the eight consecutive rainy
days as of tomorrow) to a rainy day tomorrow, full stop. The restatement does justice to both the expansion, and the subsequent contraction, of focus of the original inference, by framing them
separately in component inferences. When an inference, seemingly in a given mode, is so easily analyzed, reduced, into worthwhile component inferences in other modes, then it seems fairer to regard
it as basically such a composition.
Fields that aim toward: reversible deduction; forward-only deduction; induction; and abductive inference.
Edgar A. Poe’s grave
Source: Herald-Tribune
The highest order of the imaginative intellect is always pre-eminently mathematical, or analytical; and the converse of this proposition is equally true.
— E. A. Poe, “American Poetry”, 1845, first paragraph, link to text.
Municipality of Aristotle
Reciprocation of premises and conclusion is more frequent in mathematics, because mathematics takes definitions, but never an accident, for its premises — a second characteristic
distinguishing mathematical reasoning from dialectical disputations.
— Aristotle, Posterior Analytics, Bk. 1, Ch. 12, link to text.
Pure mathematics is marked by far-reaching networks of bridges of equivalences between sometimes the most disparate-seeming things. With good reason, popularizations often focus on examples of the
metamorphosic power of mathematics. A topologist once told me that the statement “These two statements are equivalent” is itself one of the most common statements in mathematics. In a mundane example
of a bridge by equivalence, in mathematical induction (actually a kind of deduction), one takes a thesis that is to be proved, and transforms it (in a fairly simple step) into the ancestral case and
the heredity, conjoined. Once they’ve been separately proved (such is the hard part, also, I’ve read, often done by deductions through equivalences), then the mathematical induction itself, the
induction step, consists in transforming (again, in a simple step) the conjunction of ancestral case with heredity back into the thesis, demonstrating the thesis. The reasoning in pure mathematics
tends to be transformative, from one proposition (or compound) to another proposition equivalent to it and already proved or postulated or just easier or more promising to work with for the purpose
at hand. When one’s scratch work proceeds through equivalences from a thesis to postulates or established theorems, then one can simply reverse the order of the scratch work for the proof of the
thesis. Reverse mathematics, a project born in mathematical logic, takes up the question of just which mathematical theorems entail which sets of postulates as premises. This shows again the
prominence of deduction through equivalences in pure mathematics. The reverse of the reasoning in pure mathematics is typically still reasoning by pure mathematics (even if with inquisitive guidance
from mathematical logic).
Jay L. Devore
Source: Columbia U
In an example contrasting to that, deduction of probabilities and statistical induction, two neighborly forms of quite different modes of inference, are seen as each other’s reverse or inverse,
deduction of probabilities inferring (through forward-only deduction) from a total population’s parameters to particular cases, and statistical induction inferring in the opposite direction (e.g., in
Jay L. Devore’s Probability and Statistics for Engineering and the Sciences, 8th Edition, 2011, beginning around “inverse manner” on page 5, into page 6). Such deductive fields as probability theory
seem to involve the development of applications of pure mathematics in order to address “forward problems” in general, the problems of deducing solutions, predicting data, etc. from the given
parameters of a universe of discourse, a total population, etc., with special attention to structures of alternatives and of implications. That description fits the deductive mathematics of
optimization, probability (and uncertainty in Zadeh’s sense), and information (including algebra of information), and at least some of mathematical logic.
Now, statistical probability should not be nicknamed “inverse probability”, an obsolete phrase that comes from DeMorgan’s discussion of LaPlace and refers to a more specific idea, involving the
method of Bayesian probability. On the other hand, the inverse of mathematics of optimization actually goes by such names as inverse optimization and inverse variations. On a third hand, inverse
problem theory seems to concern inferring from observed effects to unobserved causes governed by known rules, and this seems a kind of abductive inference, albeit with a special emphasis on knowing
the governing rules pretty comprehensively.
It is in the (comparatively) concrete sciences, the sciences of motion, matter, life, and mind, that abductive inference takes center stage. I’ll add some discussion here later.
Semantic discussion: “ampliative inference” ≡ “non-deductive inference”.
Question: does the phrase “ampliative inference” mean
• simply non-deductive inference (as I’ve taken it to mean) or, instead,
• inference that both (A) is non-deductive and (B) entails all its premises in its conclusion?
Peirce’s examples of abductive reasoning had premises that were not only far from entailing their conclusions, but also far (too far for a fair reframing to close the gap) from being entailed by
their conclusions. He often classified abductive inference (which he also called hypothesis, retroduction, and presumption) as ampliative. Hence it is safe to say that his “ampliative” inference
meant simply the non-deductive, not the both repletive and non-deductive. This was both (A) during the years that he treated abductive inference as based on sampling and as a rearrangement of the
Barbara syllogism, and (B) afterwards, in the 1900s.
⯈ 1883:
We are thus led to divide all probable reasoning into deductive and ampliative, and further to divide ampliative reasoning into induction and hypothesis. [….]
—Page 143 in “A Theory of Probable Inference” in Studies in Logic. ⯇
⯈ 1889–91 Century Dictionary (“CD”), first edition: Peirce’s definition of “ampliative” (see below) says that it and “explicative” are alternates for the translative words “synthetic” and “analytic”
in Kantian philosophy. Peirce also opposes “explicative inference” and “ampliative inference” to each other in his bountiful definition of inference (see below) in CD. Note that explicative inference
amounts to deduction of a not fully obvious conclusion; it doesn’t merely parrot a premise:
[….] Explicative inference, an inference which consists in the observation of new relations between the parts of a mental diagram (see above) constructed without addition to the facts contained
in the premises. It infers no more than is strictly involved in the facts contained in the premises, which it thus unfolds or explicates. This is the opposite of ampliative inference, in which,
in endeavoring to frame a representation, not merely of the facts contained in the premises, but also of the way in which they have come to present themselves, we are led to add to the facts
directly observed. [….]
What may remain unclear to a reader of that passage is whether no ampliative conclusion omits a fact from its premises. Yet, if ampliative inference includes abductive inference (a.k.a. hypothesis),
then an ampliative conclusion certainly can omit facts from the premises and remain ampliative, as one can see from Peirce’s many examples.
Meanwhile, the line between non-explicative and explicative deduction, i.e., between obvious and non-obvious deduction, seems not perfectly clear.
Also, the ampliative-deductive division is a dichotomy of all inferences. Explicative inferences, as forming a strict subset of deductions, are better opposed to an analogous strict subset of
ampliative inferences, a subset consisting of ampliative inferences of which each conclusion accepts at least some definite matter from its premises, rather than veering arbitrarily into an unrelated
matter. Non‑explicative deductions parrot too much, as opposed to capricious ampliative inferences that parrot too little.
Anyway, like most who contemplate inference modes, Peirce usually refers to deductive (rather than explicative) inference in discussing inference in mathematics, traditional categorical syllogisms,
and mathematical probabilities. ⯇
⯈ 1892: Peirce treats “non-deductive” and “ampliative” as alternate labels for the same kinds of inference, as follows in “The Doctrine of Necessity Examined”, § II, 2nd paragraph:
[….] Non-deductive or ampliative inference is of three kinds: induction, hypothesis, and analogy. If there be any other modes, they must be extremely unusual and highly complicated, and may be
assumed with little doubt to be of the same nature as those enumerated. For induction, hypothesis, and analogy, as far as their ampliative character goes, that is, so far as they conclude
something not implied in the premises, depend upon one principle and involve the same procedure. All are essentially inferences from sampling. [….]
(Throughout the years, he usually regarded analogy as a combination of induction and hypothetical inference.) ⯇
⯈ 1900s: Peirce ceased holding that hypothetical (a.k.a. abductive, a.k.a. retroductive) inference aims at a likely conclusion from parts considered as samples, and argued instead that abductive
inference aims at a plausible, natural, instinctually simple explanation as (provisional) conclusion and that it introduces an idea new to the given case, while induction merely extends to a larger
whole of cases an idea already asserted (not merely framed) in the premises. This does not mean that only abductive inference is ampliative; instead at most it means that only abductive inference is
ampliative with regard to ideas, while induction is ampliative of the extension of ideas. (I’m unsure whether Peirce regarded abduced ideas as being definable by comprehension a.k.a. intension (as
opposed to extension a.k.a. denotation). ⯇
⯈ 1902: In a draft, regarding his past treatment of abductive inference, Peirce wrote,
I was too much taken up in considering syllogistic forms and the doctrine of logical extension and comprehension, both of which I made more fundamental than they really are.
— Collected Papers v. 2, ¶ 102. ⯇
Adjective choices.
“Repletive” (re-PLEE-tiv) seems a better adjective than “retentive” for non-attenuative inference (although maybe it’s just me), because “retentive” suggests not just keeping the premises, but
restraining them or the conclusions in one sense or another. The word “preservative”, to convey the idea of preserving the premises into the conclusions, would lead to confusion with the more usual
use of “preservative” in logic’s context to pertain to truth-preservativeness (not to mention falsity-preservativeness). If people dislike the word “repletive” for the present purpose, then I suppose
that “transervative” would do.
“Attenuative” (a-TEN-yoo-ay-tiv) seems much better than “precisive” or “reductive” for non-repletive inference. The adjective “precisive” seems applicable only abstrusely to an apparently
dis-precisive inference in the form of “p, ergo p or q”. Attenuative inference generally narrows logical focus, but in doing this it renders vague (i.e., omits) some of that which had been in focus.
“Reductive” may be less bad than “precisive” in those respects but is rendered too slippery by irrelevant senses clinging from other contexts and debates.
Made-up names.
Table 8.
Here is a little system of handy made-up nouns that could be used to name four inference modes and,
while we’re at it, made-up adjectives for their respective desired heuristic properties, in case the
view should tend to prevail that the traditional definitions of induction and abduction ought not to
be altered more than minimally.
Inferences Deduction: Ampliative inference:
Repletive A. Equiduction C. Pluduction
inference: (ē-kwi-duk′ -sho̤n) (plo̅o̅-duk′ -sho̤n)
≡ reversible deduction. ≋ induction.
Conclusion tries to be Conclusion tries to be
basatile (bā′- sa̤-tīl) veteratile (ve′ -te̤-ra̤-tīl)
(i.e., nontrivial, deep, (i.e., verisimilar, likely, in
complex). Peirce’s sense).
Attenuative B. Minuduction D. Mercuduction
inference: (mī-nū-duk′ -sho̤n) (me̤r-kū-duk′ -sho̤n)
≡ forward-only deduction. ≋ abductive inference.
Conclusion tries to be Conclusion tries to be
novatile (nō′ -va̤-tīl) aviatile (ā′ -vi-a̤-tīl)
(i.e., new in aspect). (i.e., plausible à la Peirce:
instinctually simple-&-natural,
APPENDIX: C.S. Peirce’s definitions of ampliation 3. (in logic), ampliative, inference, induction 5. (in logic), and hypothesis 4., in the Century Dictionary; and his 1908 discussion elsewhere of
simpler hypothesis.
Here are two excerpts from 1889–91 (and identically in 1911) from the Century Dictionary’s definitions of “ampliation” and “ampliative”, of which Peirce was in charge of the definitions, per the list
of words compiled by the Peirce Edition Project’s former branch at the Université de Québec à Montréal (UQÁM).
ampliation (am-pli-ā′ sho̤n) […] — 3. In logic, such a modification of the verb of a proposition as makes the subject denote objects which without such modification it would not denote,
especially things existing in the past and future. Thus, in the proposition, “Some man may be Antichrist,” the modal auxiliary may enlarges the breadth of man, and makes it apply to future men as
well as to those who now exist.
ampliative (am′ pli-ạ̄-tiv) […] Enlarging; increasing; synthetic. Applied— (a) In logic, to a modal expression causing an ampliation (see ampliation, 3); thus, the word may in “Some man
may be Antichrist” is an ampliative term. (b) In the Kantian philosophy, to a judgment whose predicate is not contained in the definition of the subject: more commonly termed by Kant a synthetic
judgment. [“Ampliative judgment” in this sense is Archbishop Thomson’s translation of Kant’s word Erweiterungsurtheil, translated by Prof. Max Müller “expanding judgment.”]
No subject, perhaps, in modern speculation has excited an intenser interest or more vehement controversy than Kant’s famous distinction of analytic and synthetic judgments, or, as I think
they might with far less of ambiguity be denominated, explicative and ampliative judgments. Sir W. Hamilton.
— Century Dictionary, p. 187, in Part 1: A – Appet., 1889–91, of Volume 1 of 6, first edition, and identically on p. 187 in Volume 1 of 12, 1911 edition. The brackets around the sentence mentioning
Archbishop Thomson are in the originals.
Peirce for his own part focused on the deductiveness or ampliativeness of inference, not of ready-made judgments (he once said that a Kantian synthetic judgment is a “genuinely dyadic” judgment, see
Collected Papers v. 1 ¶ 475). Peirce argued that mathematics aims at theorematic deductions that require experimentation with diagrams, a.k.a. schemata, and that it concerns purely hypothetical
objects. (So much for Kant’s synthetic a priori.)
The following is from the first edition, 1889–91, CD volume 3, CD part 11 Ihleite – Juno, CD pages 3080–1. Peirce was in charge of the CD’s definition of inference, per the list of words compiled by
the Peirce Edition Project’s former branch at the Université de Québec à Montréal (UQÁM). I recently typed the text out from the Internet Archive, so I’d suggest checking it against the Internet
Archive version if you quote or copy from the resultant text below. Let me know if you find an error.
inference (in′ fėr-ens), n. [= F. inférence = Sp. Pg. inferencia, ˂ ML. inferentia, inference, ˂ L. inferre, infer: see infer.] 1. The formation of a belief or opinion, not as directly
observed, but as constrained by observations made of other matters or by beliefs already adopted; the system of propositions or judgments connected together by such an act in a syllogism —
 namely, the premises, or the judgment or judgments which act as causes, and the conclusion, or the judgment which results as an effect; also, the belief so produced. The act of inference
consists psychologically in constructing in the imagination a sort of diagram or skeleton image of the essentials of the state of things represented in the premises, in which, by mental
manipulation and contemplation, relations that had not been noticed in constructing it are discovered. In this respect inference is analogous to experiment, where, in place of a diagram, a
simplified state of things is used, and where the manipulation is real instead of mental. Unconscious inference is the determination of a cognition by previous cognitions without consciousness or
voluntary control. The lowest kind of conscious inference is where a proposition is recognized as inferred, but without distinct apprehension of the premises from which is has been inferred. The
next lowest is the simple consequence, where a belief is recognized as caused by another belief, according to some rule or psychical force, but where the nature of this rule or leading principle
is not recognized, and it is in truth some observed fact embodied in a habit of inference. Such, for example, is the celebrated inference of Descartes, Cogito, ergo sum (‘I think, therefore I
exist’). Higher forms of inference are the direct syllogism (see syllogism); apagogic inference, or the reductio ad absurdum, which involves the principle of contradiction; dilemmatic inference,
which involves the principle of exluded middle; simple inferences turning upon relations; inferences of transposed quantity (see below); and the Fermatian inference (see Fermatian). Scientific
inferences are either inductive or hypothetic. See induction, 5, and analogy, 3.
2. Reasoning from effect to cause; reasoning from signs; conjecture from premises or criteria; hypothesis.
An excellent discourse on . . . the inexpressible happiness and satisfaction of a holy life, with pertinent inferences to prepare us for death and a future state.
Evelyn, Diary, Nov. 21, 1703.
He has made not only illogical inferences, but false statements.
Macaulay, Mitford’s Hist. Greece.
Take, by contrast, the word inference, which I have been using: it may stand for the act of inferring, as I have used it; or for the connecting principle, or inferentia, between premises and
conclusions; or for the conclusion itself.
J. H. Newman, Gram. of Assent, p. 254.
Alternative inference. See alternative. — Ampliative inference. See explicative inference, below. — Analogical inference, the inference that a certain thing, which is
known to possess a certain number of characters belonging to a limited number of objects or to one only, also possesses another character common to those objects. Such would be the inference that
Mars is inhabited, owing to its general resemblance to the earth. Mill calls this inference from particulars to particulars, and makes it the basis of induction. — Apagogical
inference, an inference reposing on the principle of contradiction, that A and not-A cannot be predicated of the same subject; the inference that a proposition is false because it leads to a
false conclusion. Such is the example concerning mercury, under deductive inference, below. — Comparative inference. See comparative. — Complete inference, an
inference whose leading principle involves no matter of fact over and above what is implied in the conception of reasoning or inference: opposed to incomplete inference, or enthymeme. Thus, if a
little girl says to hereself, “It is naughty to do what mamma tells me not to do; but mamma tells me not to squint; therefore it is naughty to squint,” this is a complete inference; while if the
first premise does not completely and explicitly appear in her thought, although really operative in leading her to the conclusion, it ceases to be properly a premise, and the inference is
incomplete. — Correct inference, an inference which conforms to the rules of logic, whether the premises are true or not. — Deductive inference, inference from a
general principle, or the application of a precept or maxim to a particular case recognized as coming under it: a phrase loosely applied to all explicative inference. Example: Mercury is a metal,
and mercury is a liquid; hence, not all metals are solid. The general rule here is that all metals are solid, which is concluded to be false, because the necessary consequence that mercury would
be solid is false. — Direct deductive inference, the simple inference from an antecedent to a consequent, in virtue of a belief in their connection as such. Example: All men die;
Enoch and Elijah were men; therefore they must have died. — Disjunctive inference. Same as alternative inference. — Explicative inference, an inference which consists
in the observation of new relations between the parts of a mental diagram (see above) constructed without addition to the facts contained in the premises. It infers no more than is strictly
involved in the facts contained in the premises, which it thus unfolds or explicates. This is the opposite of ampliative inference, in which, in endeavoring to frame a representation, not merely
of the facts contained in the premises, but also of the way in which they have come to present themselves, we are led to add to the facts directly observed. Thus, if I see the full moon partly
risen above the horizon, it is absolutely out of my power not to imagine the entire disk as completed, and then partially hidden; and it will be an addition to and correction of this idea if I
then stop to reflect that since the moon rose last the hidden part may have been torn away; the inference that the disk of the moon is complete is an irresistible ampliative inference. All the
demonstrations of mathematics proceed by explicative inferences. — Fermatian inference. See Fermatian. — Hypothetic inference, the inference that a hypothesis, or
supposition, is true because its consequences, so far as tried, have been found to be true; in a wider sense, the inference that a hypothesis resembles the truth as much as its consequences have
been found to resemble the truth. Thus, Schliemann supposes the story of Troy to be historically true in some measure, on account of the agreement of Homer’s narrative with the findings in his
excavations, all of which would be natural results of the truth of the hypothesis. — Immediate inference. See immediate. — Incomplete inference. See complete inference
, above. — Indirect inference, any inference reposing on the principle that the consequence of a consequence is itself a consequence. The same inference will be regarded as direct
or indirect, according to the degree of importance attached to the part this principle plays in it. Example: All men die; but if Enoch and Elijah died, the Bible errs; hence, if Enoch and Elijah
were men, the Bible errs. — Inductive inference. See induction, 5. — Inference of transposed quantity, any inference which reposes on the fact that a certain lot of
things is infinite in number, so that the inference would lose its cogency were this not the case. The following is an example: Every Hottentot kills a Hottentot; but nobody is killed by more
than one person; consequently, every Hottentot is killed by a Hottentot. If the foolish first premise is supposed to hold good of the finite number of Hottentots who are living at any one time,
the inference is conclusive. But if the infinite succession of generations is taken into account, then each Hottentot might kill a Hottentot of the succeeding generation, say one of his sons, yet
many might escape being killed.Tetrast Note 1: Peirce previously discussed inference of transposed quantity in 1881: Google on "Peirce" "every Texan kills". End of note. — Leading
principle of inference, the formula of the mental habit governing an inference. — Necessary inference, an explicative inference in which it is logically impossible for the premises
to be true without the truth of the conclusion.Tetrast Note 2: Peirce had necessary inference and probable inference together forming a dichotomy of all inferences. Probable inference included
deduction of a mathematically probable conclusion, induction, & abduction. But, by the 1900s, he regarded abduction as seeking to be plausible rather than probable in the inductive sense. By that
standard, deductions of optimal and feasible solutions should also be classed as plausible inferences. Mathematics of optimization (longer known as linear and nonlinear programming) was not yet a
discipline while Peirce lived. Also in later years he ascribed likelihood (which he also called verisimilitude) to worthwhile inductions. On such distinctions in general, see my Plausibility,
likelihood, novelty, nontriviality, versus optima, probabilities, information, givens.. End of note. — Probable inference, a kind of inference embracing all ampliative and some
explicative inference, in which the premises are recognized as possibly true without the truth of the conclusion, but in which it is felt that the reasoner is following a rule which may be
trusted to lead him to the truth in the main and in the long run. — Ricardian inference, the mode of inference employed by Ricardo to establish his theory of rent. See Ricardian.&
hairsp;— Statistical inference, an inference in regard to the magnitude of a quantity, where it is concluded that a certain value is the most probable, and that other possible values
gradually fall off in probability as they depart from the most probable value. All of the inferences of those sciences which are dominated by mathematics are of this character. = Syn. Analysis,
Anticipation, Argument, Argumentation, Assay, Assent, Assumption, Conclusion, Conjecture, Conviction, Corollary, Criterion, Decision, Deduction, Demonstration, Dilemma, Discovery, Elench,
Enthymeme, Examination, Experiment, Experimentation, Finding, Forecast, Generalization, Guess, Hypothesis, Illation, Induction, Inquiry, Investigation, Judgment, Lemma, Moral, Persuasion, Porism,
Prediction, Prevision, Presumption, Probation, Prognostication, Proof, Ratiocination, Reasoning, Research, Sifting, Surmise, Test, Theorem, Verdict. Of these words, illation is a strict synonym
for inference in the first and principal meaning of the latter word, but is pedantic and little used. Reasoning has the same meaning but is not used as a relative noun with of; thus, we speak of
the inference of the conclusion from the premises, and of reasoning from the premises to the conclusion. A reasoning may consist of a series of acts of inferences. Ratiocination is abstract and
severe reasoning, involving only necessary inferences. Conclusion differs from inference mainly in being applied preferentially to the result of the act called inference; but conclusion would
further usually imply a stronger degree of persuasion than inference. Conviction and persuasion denote the belief attained, or its attainment, from a psychological point of view, while inference,
illation, reasoning, ratiocination, and conclusion direct attention to the logic of the procedure. Conviction is perhaps a stronger word than persuasion, and more confined to serious and moral
inferences. Discovery is the inferential or other attainment of a new truth. Analysis, assay, examination, experiment, experimentation, inquiry, investigation, and research are processes
analogous to inference, and also involving acts of inference. Anticipation, assent, assumption, and presumption express the attainment of belief either without inference or considered
independently of any inference. Presumption is used for a probable inference or for the ground of it. Argument, argumentation, demonstration, and proof set forth the logic of inferences already
drawn. Criterion and test are rules of inference. Elench is that relation between the premises which compels assent to the conclusion; it is translated “evidence” in Heb. xi. 1, where an
intellectual conclusion is meant. Corollary, deduction, dilemma, enthymeme, forecast, generalization, induction, lemma, moral, porism, prediction, prevision, prognostication, sifting, and theorem
are special kinds of inference. (See those words). Conjecture, guess, hypothesis, and surmise are synonyms of inference in its secondary sense. Guess and surmise are weaker words.
The following is “induction, 5.” in the first edition, 1889–91, CD volume 3, CD part 11 Ihleite – Juno, CD page 3068. Peirce was in charge of the CD’s definition of induction, per the list of words
compiled by the Peirce Edition Project’s former branch at the Université de Québec à Montréal (UQÁM).
induction […] 5. In logic, the process of drawing a general conclusion from particular cases; the inference from the character of a sample to that of the whole lot sampled. Aristotle’s example
is: Man, the horse, and the mule are animals lacking a gall-bladder; now, man, the horse, and the mule are long-lived animals; hence, all animals that lack the gall-bladder are long-lived.
Logicians usually make it essential to induction that it should be an inference from the possession of a character by all the individuals of the sample to its possession by the whole class; but
the meaning is to be extended so as to cover the case in which, from the fact that a character is found in a certain proportion of individuals of the sample, its possession by a like proportion
of individuals of the whole lot sampled is inferred. Thus, if one draws a handful of coffee from a bag, and, finding every bean of the handful to be a fine one, concludes that all the beans in
the bag are fine, he makes an induction; but the character of the inference is essentially the same if, instead of finding that all the beans are fine, he finds that two thirds of them are fine
and one third inferior, and thence concludes that about two thirds of all the beans in the bag are fine. On the other hand, induction, in the strict sense of the word, is to be distinguished from
such methods of scientific reasoning as, first, reasoning by signs, as, for example, the inference that because a certain lot of coffee has certain characters known to belong to coffee grown in
Arabia, therefore this lot grew in Arabia; and, second, reasoning by analogy, where, from the possession of certain characters by a certain small number of objects, it is inferred that the same
characters belong to another object, which considerably resembles those objects named, as the inference that Mars is inhabited because the earth is inhabited. But the term induction has a second
and wider sense, derived from the use of the term inductive philosophy by Bacon. In this second sense, namely, every kind of reasoning which is neither necessary nor a probable deduction, and
which, though it may fail in a given case, is sure to correct itself in the long run, is called an induction. Such inference is more properly called ampliative inference. Its character is that,
though the special conclusion drawn might not be verified in the long run, yet similar conclusions would be, and in the long run the premises would be so corrected as to change the conclusion and
make it correct. Thus, if, from the fact that female births are generally in excess among negroes, it is inferred that they will be so in the United States during any single year, a probable
deduction is drawn, which, even if it happens to fail in the special case, will generally be found true. But if, from the fact that female births are shown to be in excess among negroes in any
one census of the United States, it is inferred that they are generally so, an induction is made, and if it happens to be false, then on continuing that sort of investigation, new premises will
be obtained from other censuses, and thus a correct general conclusion will in the long run be reached. Induction, as above defined, is called philosophical or real induction, in
contradistinction to formal or logical induction, which rests on a complete enumeration of cases and is thus induction only in form. A real induction is never made with absolute confidence, but
the belief in the conclusion is always qualified and shaded down. Socratic induction is the formation of a definition from the consideration of single instances. Mathematical induction, so
called, is a peculiar kind of demonstration introduced by Fermat, and better termed Fermatian inference. This demonstration, which is indispensable in the theory of numbers, consists in showing
that a certain property, if possessed by any number whatever, is necessarily possessed by the number next greater than that number, and then in showing that the property in question is in fact
possessed by some number, N; whence it follows that the property is possessed by every number greater than N.
Socrates used a kind of induccion by asking many questions, the whiche when thei were graunted he broughte thereupon his confirmacion concerning the present controversie; which kinde of
argument hath his name of Socrates himself, called by the learned Socratic induction.
Sir T. Wilson, Rule of Reason.
Our memory, register of sense,
And mould of arts, as mother of induction.
Lord Brooke, Human Learning (1633), st. 14.
Inductions will be more sure, the larger the experience from which they are drawn.
Bancroft, Hist. Const. I. 5.
The following is “hypothesis, 4.” in the first edition, 1889–91, CD volume 3, CD part 10 Halve – Iguvine, CD page 2959. Peirce was in charge of the CD’s definition of hypothesis, per the list of
words compiled by the Peirce Edition Project’s former branch at the Université de Québec à Montréal (UQÁM).
hypothesis […]. 4. The conclusion of an argument from consequent and antecedent; a proposition held to be probably true because its consequences, according to known general principles, are found
to be true; the supposition that an object has a certain character, from which it would necessarily follow that it must possess other characters which it is observed to possess. The word has
always been applied in this sense to theories of the planetary system. Kepler held the hypothesis that Mars moves in an elliptical orbit with the sun in one focus, describing equal areas in equal
times, the ellipse having a certain size, shape, and situation, and the perihelion being reached at a certain epoch. Of the three coordinates of the planet’s position, two, determining its
apparent position, were directly observed, but the third, its varying distance from the earth, was the subject of hypothesis. The hypothesis of Kepler was adopted because it made the apparent
places just what they were observed to be. A hypothesis is of the general nature of an inductive conclusion, but it differs from an induction proper in that it involves no generalization, and in
that it affords an explanation of observed facts according to known general principles. The distinction between induction and hypothesis is illustrated by the process of deciphering a despatch
written in a secret alphabet. A statistical investigation will show that in English writing, in general, the letter e occurs far more frequently than any other; this general proposition is an
induction from the particular cases examined. If now the despatch to be deciphered is found to contain 26 characters or less, one of which occurs much more frequently than any of the others, the
probable explanation is that each character stands for a letter, and the most frequent one for e: this is hypothesis. At the outset, this is a hypothesis not only in the present sense, but also
in that of being a provisional theory insufficiently supported. As the process of deciphering proceeds, however, the inferences become more and more probable, until practical certainty is
attained. Still the nature of the evidence remains the same; the conclusion is held true for the sake of the explanation it affords of observed facts. Generally speaking, the conclusions of
hypothetic inference cannot be arrived at inductively, because their truth is not susceptible of direct observation in single cases; nor can the conclusions of inductions, on account of their
generality, be reached by hypothetic inference. For instance, any historical fact, as that Napoleon Bonapart once lived, is a hypothesis; for we believe the proposition because its effects&
hairsp;— current tradition, the histories, the monuments, etc. — are observed. No mere generalization of observed facts could ever teach us that Napoleon lived. Again, we
inductively infer that every particle of matter gravitates toward every other. Hypothesis might lead to this result for any given pair of particles, but could never show that the law is
universal. The chief precautions to be used in adopting hypotheses are two: first, we should take pains not to confine our verifications to certain orders of effects to which the supposed fact
would give rise, but to examine effects of every kind; secondly, before a hypothesis can be regarded as anything more than a suggestion, it must have produced successful predictions. For example,
hypotheses concerning the luminiferous ether have had the defect that they would necessitate certain longitudinal oscillations to which nothing in the phenomenon corresponds; and consequently
these theories ought not to be held as probably true, but only as analogues of the truth. As long as the kinetical theory of gases merely explained the laws of Boyle and Charles, which it was
constructed to explain, it had little importance; but when it was shown that diffusion, viscosity, and conductibility in gases were connected and subject to those laws which theory had predicted,
the probability of the hypothesis became very great.
I asked him what he thought of Locusts, and whether the History might not be better accounted for, supposing them to be the winged Creatures that fell so thick about the Camp of Israel? but
by his answer it appear’d that he had never heard of any such Hypothesis.
Maundrell, Aleppo to Jerusalem, p. 61.
We have explained the phænomena of the heavens and of our sea by the power of gravity. . . . . But hitherto I have not been able to discover the cause of those properties of gravity from
phænomena, and I frame no hypothesis; for whatever is not deduced from the phænomena is to be called an hypothesis; and hypotheses, whether metaphysical or physical, whether occult qualities
or mechanical, have no place in experimental philosophy.
Newton, Principia (tr. by Motte), iii.
The following is from Peirce’s 1908 discussion of the idea of simpler hypothesis, from near the end of IV in “A Neglected Argument for the Reality of God” (via Wikisource with added good notes),
originally published (incompletely) in Hibbert Journal v. 7 and less incompletely in Collected Papers v. 6 and Essential Peirce v. 2. For the whole essay go to https://gnusystems.ca/CSPgod.htm#na0.
By “il lume naturale” is understood “the natural light of reason”.
Modern science has been builded after the model of Galileo, who founded it on il lume naturale. That truly inspired prophet had said that, of two hypotheses, the simpler is to be preferred; but I
was formerly one of those who, in our dull self-conceit fancying ourselves more sly than he, twisted the maxim to mean the logically simpler, the one that adds the least to what has been
observed, in spite of three obvious objections: first, that so there was no support for any hypothesis; secondly, that by the same token we ought to content ourselves with simply formulating the
special observations actually made; and thirdly, that every advance of science that further opens the truth to our view discloses a world of unexpected complications. It was not until long
experience forced me to realise that subsequent discoveries were every time showing I had been wrong,— while those who understood the maxim as Galileo had done, early unlocked the secret,—
 that the scales fell from my eyes and my mind awoke to the broad and flaming daylight that it is the simpler Hypothesis in the sense of the more facile and natural, the one that instinct
suggests, that must be preferred; for the reason that unless man have a natural bent in accordance with nature’s, he has no chance of understanding nature at all. Many tests of this principal and
positive fact, relating as well to my own studies as to the researches of others, have confirmed me in this opinion; and when I shall come to set them forth in a book, their array will convince
everybody. Oh no! I am forgetting that armour, impenetrable by accurate thought, in which the rank and file of minds are clad! They may, for example, get the notion that my proposition involves a
denial of the rigidity of the laws of association: it would be quite on a par with much that is current. I do not mean that logical simplicity is a consideration of no value at all, but only that
its value is badly secondary to that of simplicity in the other sense.
HOME || Deductive vs. ampliative. Repletive vs. attenuative. Induction & abduction starkly defined thereby. || Plausibility, likelihood, novelty, nontriviality, versus optima, probabilities,
information, givens || Logical quantity & research scopes - universal, general, special, particular, individual, singular. || Telos, entelechy, Aristotle's Four Causes, pleasure, & happiness ||
Compare to Aristotle, Aquinas, & Peirce. || Semiotic triad versus tetrad. || Tetrachotomies of future-oriented virtues and vices. || What of these other fours? || Fantastic Four. || Why tetrastic? ||
The Four Causes, their principles, special relativity, Thomistic beauty. || Logical quantities, categories of research, and categories. || Semiotics: collaterally based recognition, the proxy, and
counting‑as. || A periodic table of aspects of humanity […] | {"url":"https://tetrast.blogspot.com/","timestamp":"2024-11-14T01:49:42Z","content_type":"text/html","content_length":"166624","record_id":"<urn:uuid:56f22f14-97c3-4793-80a6-1f7265328b19>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00461.warc.gz"} |
Christmas Tree Spread with Puts Option Strategy - #1 Options Strategies Center
A Christmas tree spread with puts is an advanced options strategy that consists of three legs and six total options. The option strategy involves buying one put at strike price D, skipping strike
price C, selling three puts at strike price B, and then buying two puts at strike price A. It is somewhat similar to a butterfly spread, where the desired outcome is a pin at the short middle
strikes, but given a more room for the stock to drop in price, thus a more bearish option strategy.
Costs are also higher versus a standard butterfly spread due to the long upper legged strike price that is further away from the short middle strikes. Because the position is opened at a higher
cost, the stock must move lower to become profitable. Although it does have a higher probability of a loss, in the event the stock does move against the trader, losses are capped at the opening
The maximum profit is calculated by taking the difference between the highest strike price and the three short middle put strike price, then subtracting the cost of the trade. For example, if the
distance between point D and B was $10 and the trade cost $2.50, then the max profit would be $7.50.
The maximum loss would be the cost of the trade, so with the same example, the maximum loss would be $2.50.
There are two breakeven points. The upper breakeven point would be calculated by taking the top strike price (point D) and subtracting the cost of the trade. So, if point D was a 110-strike price
option, and the trade cost $2.50, the upper breakeven would be $107.50.
The lower breakeven point would be calculated by taking the lowest strike price (point A) and adding one-half of the net debit. So, if the point A strike price was equal to $95 and the cost of the
trade was $2.50, then our lower breakeven would be $96.25 (95+2.50/2).
If XYZ is trading $110 and is expected to trade lower over the next three months, a trader could execute a 110/100/95 Christmas tree spread by buying one 110-strike price put, selling three
100-strike price puts, and buying two 95-strike price puts for the following prices:
• Buy 1 XYZ 110-strike price put for $8.00
• Sell 3 XYZ 100-strike price puts for $6.90 ($2.30 each)
• Buy 2 XYZ 95-strike price puts for $1.40 ($.70 each)
• Total cost = $2.50
If the stock trades up over the next three months, the investor will have lost his full $2.50, and all positions will be removed from his account.
If the stock trades down to $100 at expiration, then the trader will gain $10 on the stock movement, capitalizing on the 110-strike price option, while the rest of the options expire worthless.
Because they paid $2.50 for the trade, their net profit would be $7.50.
If the stock traded down to $90, the investor would make $20 on the $110 put. Lose $10 X3 on the short middle 100-puts for a total of $30. Gain $5 X2 on the long lower 95-puts for a total of $10.
The net result would be $20 – $30 + $10, which mean all the profits are a wash, but because the investor paid $2.50 for the trade, their net loss is $2.50.
The lower a trader sets the strike prices, the more bearish a Christmas tree spread becomes, while at the same time, reducing the cost of the trade. However, the lower the strike prices are set, the
lower the probability of success.
This type of spread is sensitive to changes in implied volatility. The net price of the spread drops when implied volatility rises and the price increases when implied volatility falls, meaning it
has an inverse relationship to implied volatility changes. The trader who executes this trade wants a drop to implied volatility.
This is not a strategy not recommended for a beginner, only experienced traders should execute a Christmas tree spread options strategy. | {"url":"https://optionstrategiesinsider.com/blog/christmas-tree-spread-with-puts-option-strategy/","timestamp":"2024-11-06T04:16:49Z","content_type":"text/html","content_length":"906041","record_id":"<urn:uuid:92b324c2-e3b6-4278-8119-5b4c2dc18717>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00017.warc.gz"} |
Explain the differences between maximum likelihood and method of moments estimation. | Hire Some To Take My Statistics Exam
Explain the differences between maximum likelihood and method of moments estimation. In-plot analysis of the fit summary statistics also used a likelihood approach. The length-frequency calculations
were calculated as the square root of the square of see here ratio of the standard deviation of the frequency at the *i*th peak (baseline peak) to the previous peak (baseline peak) of the same
dataset. Calculated model coefficients were used as parameters, and the standard errors were calculated as standard deviation of the frequencies. They were the average of the calculated model-free
coefficients. Mean square error was calculated as *R^2^*. The significance results were similar with the maximum likelihood approach, except for calculating three-way log likelihood. These features,
when combined with the goodness of fit and model’s properties, are referred to as these parameters. Because prior information about the models and the parameters is to be analyzed, the likelihood
method and estimation methods frequently take two (partial) least-squares fits of each parameter and then determine which one best describes the data better, if appropriate. Their differences were
analyzed for various models including no-, large-, and small-group. For the large cohort analysis, including in analyses of specific regions of interest (ROC). Based on a comparison of the respective
experimental and the statistical results, several separate goodness-of-fit tests were done. The range for *p* \< 0.01 and *p* \< 0.001 when comparing between two or different models ([Table 1](#
pone.0215953.t001){ref-type="table"}). Models with *p* \< 0.025, *p* \< 0.055, and *p* \< 0.
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1 were found to be associated with a superior fit. Models including a smaller and a larger number of peaks were associated with a superior fit, yet did not differ in their slope after a
standard-deviation analysis. As above, the slopes of maximum likelihood and method of moment estimators, using the number of peaks (*y*), and their variance divided by the standard deviation, were
not significantly different among any of the potential models. This can be seen as follows: Example 1. An interaction between the abundance data and the predicted abundance data {#sec016}
————————————————————————————— In model 1, no interaction could be detected (*p*\> 0.08; ROC is 0.935, SD non-parametric assessment). What is wanted to change the model accordingly is also dependent
on the results of the fit statistics in modeling certain types of parameters, for instance when using a method of moments estimation. In this example, when the *p*–*p*~0~ ratio is small (*p\> 0.05*),
the goodness-of-fit curves for maximum likelihood ([Fig 8](#pone.0215953.g008){ref-type=”fig”}-1), and method of moments measurement {HfExplain the differences between maximum likelihood and method
of moments estimation. In addition, they present several examples in the papers which contribute to their work and in fact have advantages and disadvantages, including the use of nonparametric
coefficients in the estimation of the mean rate of a process, check these guys out use of a power-law model of the exponent and the design of test frameworks for the estimation of the maximum
likelihood. This presentation is made for us providing examples of all of the proposed methods and considering a large amount of information. Section ii. Section iii. Section iv. Method 1
Conjecturing of maximum likelihood method of moments (CMLM-1) The method of moments was derived using the maximum likelihood equation after the change of basis. It assumes a power-law frequency fit
parameter of the underlying model to the parameters. The parameters of the model are: Power $\bar{f}(u):=\sum_{j=0}^{p-1}u^{j}t_{ij}$ This assumption is applied to the frequency data $C_{ij}^{(k)}$
that the fit parameter $t_{ij}$, while $\bar{f}(u)$ is the frequency value of $p-1$.
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After taking the factorial part $\mathcal{R}$ for this fit parameter $t_{ij}$, we compute the dispersion constant for the frequency $C_{\mathcal{R}}^{(k}$, given at that stage. Next we make the
change of basis. The model parameters (and $C_{\mathcal{R}}^{(k)}$) are: Power $\bar{f}(v:p):=\sum_{k=0}^{p-1}f_{k}v^{k}$ This assumption is applied to the frequency data $C_{\mathcal{R}}^{(k)}$
which are the power values for each $Explain the differences between maximum likelihood and method of moments estimation. This paper introduces the standard normal method of moments, which is a more
precise approximation of normal estimates of moments when (almost) continuous underlying data is non-discrete and non-moving. Second, given data points $x_t$ and points $x_N$ for which $p(x_t| x_N) \
leq 1$, then the second moment of the matrix $\mu$ is a classical measure of randomization as stated in Theorem \[thm:measpop\], with probability taking a value $p(\mu| e_1)$ if $|e_1| > 1/2$. We
call this measure an estimate of moments for randomization when $p(x_t| x_N) \to < 1$, as obtained when the randomization parameter $p(x_N| e_1)$ is replaced by $p(x_t| \mathbf{x})$ with a
probability $\omega$. We specify its value as $p(x_N| \mathbf{x})$ if $|e_1| > 1/2$ and $\omega$ if $|e_1| = |e_2| > 1/2$. We also give an explicit description of the relationship between our measure
and moments (see Theorem \[thm:moments\_estimate\] below). The two helpful site important of these, and therefore the following (and to be precise, actually appearing) more refined results are
contained in Proposition \[prp:thm\_moments\_estimate\]. \[prp:thm\_moments\_estimate\] Let $i$ be a non-negative integer, and let $\mu_i$ be the first eigenvalue of ${\mathbf{Q}}\!\!(\cdot,\cdot)$.
Then $$\sin{(\Pi f(i,j))} = \begin{cases} 1 & i=j, \\ -1 & i\neq j. \end{cases}$$ \[prp:moments\_estimate\_2\] (i) If $n$ has Extra resources least two eigenvalues with $|E_{ij}^{\rm odd}| \gtrsim
1$, then $$\begin{aligned} \sec{|\sum_{i=j}^2 {\mathbf{P}}(\cos i + \alpha \cos \beta |x_S,x_T) – \sum_{i=j}^{N-1}\alpha \cos \beta |x_S,x_T)| = \sec{|\mu_i|} {\rm erfc}\left({\mathbf{p}}(\cos i ) \
right) + \sec{|\omega|} {\rm erfc}\left({\mathbf{p}}(\cos \beta ) \right),\end{aligned}$$ where $\alpha \leq 2$. (ii) If click to investigate has at least three eigenvalues with $|E_{ij}^{\rm odd}| \
gtrsim 1$, then $$\begin{aligned} \min{\left\{ {|\mu_i|} {\rm erfc}\left({\mathbf{p}}(\cos i ) \right) : 1 \leq i \leq N \right\}}+ \frac{1}{N}+ \frac{2}{N} {\rm erfc}\left(\prod_{i=1}^{N-1} \mu_i \
right) + \frac{1}{N}- \frac{2}{N} | {"url":"https://hireforstatisticsexam.com/explain-the-differences-between-maximum-likelihood-and-method-of-moments-estimation","timestamp":"2024-11-07T03:07:45Z","content_type":"text/html","content_length":"169595","record_id":"<urn:uuid:abc6809f-686f-474d-9162-54da41ec89ea>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00414.warc.gz"} |
Percentage calculator Original price is discounted/raised by A%. The resulting sales price is B How much is A % of B? How many % is A of B? How many percentages bigger or smaller is the second
number? Number A is increased by B % Number A is decreased by B %
Is there some calculation missing or do you have language corrections? Send feedback! | {"url":"https://vat-calculator.net/percentage-calculator/","timestamp":"2024-11-11T01:30:26Z","content_type":"text/html","content_length":"15917","record_id":"<urn:uuid:26c35363-1613-48fe-b46a-e040217fbb3c>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00264.warc.gz"} |
Course Information
APSC 101, APSC 102, APSC 103 (Formerly APSC 100) - Engineering Practice I
This course provides the laboratory experience and professional skills fundamental to the practice of engineering. It consists of three modules: Module 1, Complex problem solving (Fall term); Module
2, Laboratory Skills (Fall term); Module 3, Engineering Design Project(Winter term). The course provides an introduction to personal learning styles, team dynamics, oral and written presentation
skills, laboratory data collection, analysis and presentation, design methodologies, project management, information literacy, and workplace safety.
An introduction to Newtonian mechanics - a subject that is applicable to everyday engineering problems. Lecture topics are vectors, motion of a particle, particle dynamics, work and energy, statics
and dynamics of rigid bodies, conservation of energy, momentum, and collisions.
3 lecture hours per week, 1 hour tutorial per week
This course provides an introduction to the chemistry of materials: thermochemistry, heat, work, internal energy, enthalpy and the first law of thermodynamics; gas laws in ideal and non-ideal
systems; phase equilibria in one-component systems; concepts of bonding in the classification of materials; the physical, electrical and mechanical properties of metals, polymers, semiconductors, and
ceramics; techniques of characterizing materials.
APSC 141 - Introduction to Computer Programming for Engineers 1- 4 week course (Week 2-5)
This course introduces concepts, theory, and practice of computer programming. The implementation uses microcomputers. The emphasis is on the design of correct and efficient algorithms and on
programming style. Applications are made to engineering problems. Part 2 is taken in the Winter term.
2 lecture hours per week, 1 two-hour lab per week
APSC 143 - Introduction to Computer Programming for Engineers- MREN students only
This course introduces concepts, theory, and practice of computer programming. The implementation uses microcomputers. The emphasis is on the design of correct and efficient algorithms and on
programming style. Applications are made to engineering problems.
This course provides an introduction to the complex Earth System (which encompasses the solid earth, hydrosphere, atmosphere, and biosphere), and our interactions with it. Using the Earth System as a
framework, the science behind our exploration and understanding of our planet is explored. Its ongoing evolution is explored in combination with the over-arching themes of engineering geology,
sustainability, and geo-materials. Key concepts/issues relevant to engineers are dealt with, including population demographics, geo-dynamics, geopolitics, resource usage, modeling of "fuzzy" systems,
and risk assessment. The connection between the Earth System, risk management, and local-human activity is explored in-depth, including local and global-scale impacts of engineering, geopolitics, and
resource issues. Examples of the terrestrial sources of geo-materials used in engineering activities are highlighted along with the government, technical, social, economic and long-term natural
environmental challenges associated with their life cycle. The evolution and anthropogenic changes (including global warming), and the moral complexities of the biosphere are introduced (at the
component and system scale), as well as examples of key sensitivities and ethical considerations/threats including contamination, biodiversity loss, and climate change.
The principal objectives of the course are (1) to develop the student's ability to visualize and communicate three-dimensional shapes and (2) to acquire the skills needed to use computer-aided design
software. Topics covered are orthographic projection, isometric sketching, auxiliary and section views as well as dimensioning and working drawings. Computer-aided design software is used to create
solid models of the parts and assemblies as well as to generate dimensioned drawings. Students apply their learning in a project where they design their own version of a consumer product. Students
learn by hands-on exercises in free-hand sketching and computer-based drawing.
1 lecture hour per week, 1 two-hour laboratory period per week
Functions, limits, Graphs and derivatives; optimization, rate problems, exponentials, logarithms, inverse trigonometric of vector-valued functions; exponential growth as an example of a differential
equation related applications. Implicit derivatives and related rate applications. Fundamental Theorem of Calculus, Riemann integral; applications to problems involving areas, volumes, mass, charge,
work, etc. Some integration techniques Integration by substitution, by parts, and partial fractions. Introduction to second-order differential equations and complex numbers.
1 lecture hour per week, 1 hour LDI per week, 1 hour tutorial per week
This course develops skills that are necessary to organize and present technical information in a professional context.
This course continues from APSC 111 to introduce electricity and further develop fundamental ideas of mechanics in the context of engineering applications. Lecture topics include oscillations and
waves, electric charge, electrical current and resistance, EMF, D.C. circuits and electrical measurements, electric field and potential, magnetic fields and their origin, and electromagnetic
This course combines the fundamentals of chemistry with the engineering issues associated with them. Areas of study are entropy and the second law of thermodynamics, thermodynamics, chemical
equilibrium, electrochemistry, chemical kinetics, and organic chemistry. Environmental issues associated with each of these topics will be incorporated into lectures when appropriate.
APSC 142 - Introduction to Computer Programming for Engineers 2
This course introduces concepts, theory, and practice of computer programming. The implementation uses microcomputers. The emphasis is on the design of correct and efficient algorithms and on
programming style. Applications are made to engineering problems. Part 1 is taken in the Fall term.
Functions of several variables, partial derivatives, differentials, gradient, maxima, and minima. Double and triple integrals, polar and cylindrical coordinates; applications to mass, center of mass,
moment, etc. Series, Ratio test, power series; Taylor polynomial approximations. 3 lecture hours per week, 1 hour tutorial per week
Systems of linear equations; real vectors spaces and subspaces; linear combinations and linear spans; linear dependence and linear independence; applications to systems of linear equations and their
solution via Gaussian elimination; bases and dimension of real vector spaces; linear transformations, range, kernel, and Rank-Nullity theorem; matrix representation of a linear transformation;
composition of linear transformations and matrix multiplication; invertible matrices and determinants; eigenvalues and eigenvectors of square matrices. Applications of the course material to
engineering systems are illustrated.
Identification, visualization and quantification of forces on elements and forces within statically determinate engineering structures and systems. Two- and three-dimensional force equilibrium of
rigid bodies; force distribution within engineering systems like simple trusses, frames and machines; internal shear forces and bending moments in force carrying elements; and engineering stress and
1 lecture hour per week, 1 hour tutorial per week | {"url":"https://smithengineering.queensu.ca/first-year/first-year-courses.html","timestamp":"2024-11-11T20:59:11Z","content_type":"application/xhtml+xml","content_length":"53751","record_id":"<urn:uuid:0e158825-2395-47a9-bffc-7af29c3e83cb>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00192.warc.gz"} |
Hilbert's Twentieth Problem | Solutions of Regular Problem
Hilbert’s Twentieth Problem: Do all variational problems with certain boundary conditions have solutions
Do solutions in general exist? The calculus of variations is a field concerned with optimizing certain types of functions called functionals. In his 19th and 20th problems, Hilbert asked whether
certain classes of problems in the calculus of variations have solutions (his 20th) and, if so, whether those solutions are particularly smooth (19th). | {"url":"https://abakcus.com/directory/hilberts-twentieth-problem/","timestamp":"2024-11-11T06:54:43Z","content_type":"text/html","content_length":"124085","record_id":"<urn:uuid:189c2615-cba9-4956-a005-3e1da54504ee>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00687.warc.gz"} |
- Who was Pythagoras and what are his contributions? - What is Pythagoras known for in philosophy? - What are some interesting facts about Pythagoras? - How did Pythagoras influence mathematics and philosophy? - Duongle entertainment - Breaking news
– Who was Pythagoras and what are his contributions? – What is Pythagoras known for in philosophy? – What are some interesting facts about Pythagoras? – How did Pythagoras influence mathematics and
Have you ever paused to consider the life of the man who formulated the renowned Pythagorean theorem? Pythagoras was far more than just a mathematician; he was a multifaceted individual who made
significant contributions as a philosopher, educator, and a groundbreaking thinker. His ideas and teachings laid the groundwork for many concepts in both mathematics and philosophy that continue to
influence our understanding today. Pythagoras founded a religious movement known as Pythagoreanism, which emphasized the importance of numbers and their relationships in understanding the universe.
His belief that mathematics was the key to unlocking the mysteries of existence has left an indelible mark on various fields of study. Join us as we explore the fascinating life, teachings, and
enduring legacy of this extraordinary figure who has shaped the intellectual landscape for centuries!
Who Was Pythagoras?
Pythagoras, a prominent figure in the history of mathematics and philosophy, was born around 570 BCE on the picturesque island of Samos, located in Greece. While he is most famously recognized for
the Pythagorean theorem, which establishes a fundamental relationship between the sides of a right triangle, his life and contributions extend far beyond this singular achievement.
### Early Life and Education
During his formative years on Samos, Pythagoras was immersed in a rich tapestry of philosophical thought and inquiry. At the young age of 20, he embarked on a journey to the island of Miletus, where
he had the privilege of studying under some of the most esteemed philosophers of his time, including Thales and Anaximander. This period of education was pivotal, as it ignited in him a profound
passion for both philosophy and mathematics, shaping the intellectual path he would follow for the rest of his life.
### Influences from Egypt and Babylon
Following his studies in Miletus, Pythagoras sought further knowledge and enlightenment by traveling to Egypt and Babylon. In these ancient civilizations, he encountered a wealth of advanced
mathematical concepts, as well as mystical traditions that would deeply influence his own teachings and beliefs. The insights he gained during these travels not only enriched his understanding of
mathematics but also contributed to the development of his philosophical ideas, which emphasized the importance of numbers and their relationship to the universe. Thus, Pythagoras emerged as a key
figure whose work laid the groundwork for future mathematical and philosophical exploration.
The Pythagorean Brotherhood
In approximately 532 BCE, the renowned philosopher Pythagoras made his way to southern Italy, where he established a community that would come to be known as the Pythagorean Brotherhood. This group
was far more than just a collective of mathematicians; it represented a comprehensive way of life that intricately wove together elements of philosophy, religion, and ethics.
### Core Beliefs of the Pythagoreans
The Pythagoreans embraced a set of beliefs that were remarkably distinctive for their era. Among these was the concept of **reincarnation**, which posited that the soul undergoes a cycle of rebirth,
transitioning into new bodies after death. This belief in the **transmigration of souls** underscored their understanding of life and existence. Additionally, they held a profound reverence for
**harmony in numbers**, viewing numerical relationships as the fundamental essence of all things. They believed that by comprehending these numerical principles, one could unlock the mysteries of the
universe itself. Furthermore, the Pythagorean way of life was characterized by **ethical living**, which included strict guidelines governing behavior and lifestyle choices. For instance, they
adhered to specific dietary restrictions, notably avoiding beans, which they considered to be impure.
### Religious Practices
The Pythagorean lifestyle was also deeply intertwined with various religious practices. Members of the Brotherhood engaged in rituals that reflected their spiritual beliefs, including entering
temples barefoot as a sign of respect and humility. They often donned white garments, symbolizing purity, and participated in sacrificial rites. These observances were not merely ceremonial; they
embodied the Pythagorean conviction in the sanctity of life and the interconnectedness of all existence within the universe. Through these practices, the Pythagoreans sought to cultivate a deeper
understanding of themselves and their place in the cosmos.
The Contributions of Pythagoras
Pythagoras is most famously associated with the Pythagorean theorem, but his influence and contributions to mathematics and philosophy extend far beyond this well-known equation. Let’s delve into
some of his most notable achievements and ideas that have shaped various fields of study.
### The Pythagorean Theorem
The Pythagorean theorem is a fundamental principle in geometry that asserts that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the
sum of the squares of the lengths of the other two sides. This can be expressed mathematically as \( a² + b² = c² \), where \( c \) represents the length of the hypotenuse. This theorem is not only a
cornerstone of geometric theory but also has a multitude of practical applications in areas such as architecture, engineering, and various fields of science.
| Triangle Sides | Formula |
| \( a² + b² \) | Where \( c \) is the hypotenuse |
### Musical Harmony and Mathematics
In addition to his work in geometry, Pythagoras made significant strides in understanding the relationship between music and mathematics. He discovered that musical intervals could be represented
through numerical ratios, which led to a deeper comprehension of harmony in music. This groundbreaking connection between numbers and sound transformed the way people perceived music, laying the
foundation for the study of acoustics and musical theory.
#### Incommensurability
Another pivotal concept attributed to Pythagoras is that of **incommensurability**. This idea posits that not all lengths can be expressed as a ratio of whole numbers, which was a revolutionary
realization at the time. It challenged the prevailing mathematical beliefs and opened the door to new avenues of thought in both mathematics and philosophy. Pythagoras’s insights into
incommensurability not only advanced mathematical theory but also had profound implications for the understanding of the nature of numbers and their relationships.
In summary, while the Pythagorean theorem is a significant part of Pythagoras’s legacy, his contributions to the fields of music, mathematics, and philosophy are equally important and continue to
influence our understanding of these disciplines today.
The Mystical Side of Pythagoras
While Pythagoras made substantial contributions to mathematics, he also delved into the mystical and philosophical realms. His teachings often blended science with spirituality.
Pythagoreanism: A Way of Life
Pythagoreanism was more than just a mathematical doctrine; it was a comprehensive worldview. Followers believed that living in harmony with the universe was essential for personal and communal
The Role of Numbers in Nature
Pythagoras posited that numbers were the key to understanding the natural world. He believed that everything could be quantified and that numerical relationships governed the universe. This idea laid
the groundwork for future scientific inquiry.
The Legacy of Pythagoras
Though Pythagoras left no written records, his influence is undeniable. His ideas shaped the thoughts of later philosophers like Plato and Aristotle, and his teachings continue to resonate in modern
mathematics and philosophy.
Impact on Mathematics and Philosophy
The principles established by Pythagoras and his followers contributed significantly to the development of Western rational thought. His emphasis on logic and reason paved the way for future
mathematicians and philosophers.
Modern Applications of Pythagorean Thought
Today, Pythagorean concepts are foundational in various fields, including architecture, engineering, and physics. The Pythagorean theorem is taught in schools worldwide, illustrating its enduring
Pythagoras was a multifaceted thinker whose contributions to mathematics, philosophy, and spirituality have left an indelible mark on history. His belief in the power of numbers and their connection
to the universe continues to inspire and challenge us today. So, the next time you use the Pythagorean theorem, remember the incredible journey of the man behind the numbers!
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Dividing Neighborhoods
Dividing Neighborhoods
1. Draw a Ray from Point B to Point A 2. Draw a Ray from Point B to Point C. 3. You have now formed ABC. Bisect this angle. (Hint: hover over each icon to see the name of the tool)
Question 1
What street does your angle bisector run along?
Question 2
Your angle bisector is dividing two of Baltimore’s neighborhoods. Use your phone/prior knowledge/take a look at a map of Baltimore to determine what two neighborhoods you have divided: Neighborhood #
1: ___________________________________ Neighborhood #2:__________________________________
Question 3
What prior information or knowledge do you have about these two neighborhoods? What is your impression about their differences? If you only know one of them, what do you know about it? | {"url":"https://stage.geogebra.org/m/yytrfg4t","timestamp":"2024-11-14T04:11:50Z","content_type":"text/html","content_length":"96474","record_id":"<urn:uuid:75ee5214-cec3-4dfb-a023-a5aff0304fcf>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00868.warc.gz"} |
seminars - Approximation Methods of Multivariate Functions for Homomorphic Data Ordering
Homomorphic Encryption (HE) is a cryptographic primitive that enables computations between encrypted data without decryption.
HE allows operations that use sensitive data to be delegated to others who service outsourced computations while data information is not exposed.
With these characteristics, HE is considered one of the important technologies for privacy preservation.
Most HE schemes, however, support few operations only, mainly multiplication and addition.
Thus, non-polynomial operations between word-wisely encrypted data require much more computational cost than between plain data.
Although many approximation algorithms have been suggested to solve this problem, these works mainly focus on the one-variable functions and cannot be directly generalized to the multivariable
functions because of the algorithmic limit or the growth of computational cost.
In this thesis, we propose new approximation methods of three fundamental multivariate functions: sorting, max index, and softmax.
First, We propose an efficient sorting method for encrypted data that works with approximate comparison.
Using our method as a building block, we exploit k-way sorting network algorithm to show the implementation result that sorting 5^6=15625 data using 5-way sorting network which is about 23.3% faster
than sorting 2^14=16384 data using the general 2-way method.
Second, we propose a polynomial approximation method of the multivariate max function that inherits the method of Cheon emph{et al.} (ASIACRYPT 2020).
Our algorithm is the generalization of the previous two-variable approach of approximating sign function, analyzing that our algorithm requires 30% less depths to find the largest element than using
a state-of-the-art two-variable comparison.
Lastly, we suggest the approximation method for softmax activation for a neural network model.
By exploiting the algorithm, we develop a secure multi-label tumor classification method using the CKKS scheme, the approximate homomorphic encryption scheme. | {"url":"https://www.math.snu.ac.kr/board/index.php?mid=seminars&page=89&sort_index=speaker&order_type=asc&l=en&document_srl=828279","timestamp":"2024-11-13T11:34:49Z","content_type":"text/html","content_length":"48924","record_id":"<urn:uuid:eeb5cea6-abb7-4bf9-b103-9eda478cce0c>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00701.warc.gz"} |
How many miles will £10 petrol get me?How many miles will £10 petrol get me? | Petrol Calculator
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Aligned Pair Exclusion
APE example 1
(requires Y-Wing unchecked) : Load Example or : From the Start
Aligned Pair Exclusion
can be succinctly stated:
Any two cells that can see each other CANNOT contain a pair of numbers than will empty a cell in an Almost Locked Set they both entirely see.
Remember - a
bi-value cell
(with two candidates) is the simplest
Almost Locked Set
since it is a set of size '1' with size+1 (ie two) candidates.
Let's consider the simplest possible example - two bi-value cell attacking the pair. I have also shown the Y-Wing in the diagram so we can see there is a simpler way to do the same job - but only in
some cases.
We consider ALL the possible pairs of numbers that will fit in [
]. These are for
[S:2 and 2:S]
2 and 5
2 and 8
4 and 2
4 and 5
4 and 8
Apart from the first being impossible (2 and 2) since
can see each other, we have problems with some of the other combinations. What if 2 and 8 were tried as the solutions? Well, that would duplicate and therefore empty
. Also 4 and 8 would empty
APE Example 3 : Load Example or : From the Start
Further into the same puzzle we come across a 3-cell ALS plus a bi-value in
. The ALS in [
] contains the four numbers {1,4,7,9} which the solver thinks of as a quadruple combination. The combinations of 'abcd' are ab, ac, ad, bc, bd and cd. Back to the base pair: We can list the
combinations for
4 and 3
[S:4 and 7:S]
- excluded by [
5 and 3
[S:5 and 7:S]
- excluded by
7 and 3
[S:7 and 7:S]
The tricky one with the 3-cell ALS is not the fact that the base pair will empty it (it can't since it is two cells and the ALS is 3 cells). It's the fact that a solution of 4 in
and 7 in
would mean there'd be only two candidates left to fill three cells. Thats enough to rule out the combination.
Aligned Pair Exclusion can also work even if the pair is not aligned. Sounds like a joke, but it's too late now to rename this strategy :) Perhaps 'Subset Exclusion' was a better idea. There is a
subtle logical different but I have found many examples and it boosts the usefulness of this strategy.
I'm very grateful to Joseph Aleardi for putting me on the scent of this elegant logic.
APE Example 4 (turn off Y-Wings) : Load Example or : From the Start
The simplest type of APE2 using just two bi-value cells duplicates the Y-Wing, but I include an example to illustrate how APE2 works.
The diagram here shows first the Y-Wing based on
(the pivot) -
. It's quite easy to see that 8 must occur in either
, thus removing it from
But let's follow the APE logic with the non-aligned pair
. (Note: We could also choose
and eliminate the 8 there also).
pairs with
using these combinations:
1 and 1 - POSSIBLE!
1 and 6
[S:1 and 8:S]
- excluded by
9 and 1
9 and 6
[S:9 and 8:S]
- excluded by
The only difference between APE 1 and APE 2 is that with non-aligned pairs the same candidate *could* be a solution in both cells. So 1 and 1 is definitely on the cards. Not that it is critical in
this case. The other exclusions mean we can't have an 8 in
, just as we thought.
... by: Pulsar
Tuesday 15-Feb-2022
It looks like all the given examples can also be solved using two Almost Locked Sets with restricted commons. Consider the most difficult example, the Eight-Cell Aligned Pair:
We can identify a 2-cell ALS {B5 C5} and a 4-cell ALS {D5 D6 E6 F4}. These two sets have a pair of restricted commons: 1 and 9. It is impossible for both sets to contain both digits, because within
these sets they can only appear in the cells B5, C5, D5 and F4. These four cells cannot contain two 1's and two 9's, because the first three cells all see each other. Therefore, at least one ALS has
to lose either a 1 or a 9 to become a locked set.
Now 2 becomes the 'pincer' digit: cell E5 can see all the 2's in both ALS's and can therefore not contain a 2; if it did, both ALS's would lose their 2's, but we already know that at least one ALS
must lose a 1 or a 9, and an ALS can only lose 1 digit.
You can apply the same strategy with the 2-cell ALS {D6 E6} and the 4-cell ALS {B5 C5 D5 H5}. In this case, 6 and 7 are the restricted commons.
Two Almost Locked Sets with restricted commons appear to cover a wide variety of cases. I'd love to see if they encompass all forms of aligned pair exclusion.
... by: Robert
Sunday 22-Aug-2021
Interesting strategy. I do have some thoughts though.
All of the examples, with the exception of the last, are cases of an AIC with almost locked sets. My own solver, which implements this without a size limitation on the ALS (the solver at this website
seems to consider only ALSs with two cells and three values) does not solve the Klaus Brener 8-cell example; however, my cell-forcing implementation does.
I think there is another way to view this sort of strategy. There are a number of ALSs, and also some AALSs shown. The logic of an ALS in an AIC (described on another page is this) - if a chain turns
a value in an ALS "off", by virtue of being weakly linked to something somewhere that has already been turned "on", then the other values in the ALS are all turned "on" - that is, the values in an
ALS are strongly linked (but not weakly linked) to each other. Or, put another way, once a value is turned off, the ALS becomes a locked set, and the values in the locked set can be eliminated from
other cells in the unit containing the ALS that was promoted to a locked set. The chain can then continue, and possibly allow us to draw an inference.
But this idea can be extended. Suppose you have an AALS - then two values need to be turned "off" in order to turn it into a locked set. This will happen quite a bit in a forcing net (the only
significant strategy I have implemented in my own solver, that is not described in detail at this web site). However, it could even happen in an AIC - the chain could turn "off" some candidate in the
AALS, then wander around for a while, and come back and turn off another candidate in the same AALS. Two is all that is needed - the AALS has now become a locked set, and the inference can continue.
I have been calling this a "dynamic locked set" - it isn't a locked set, but conditional on some assumption (an AIC or forcing net becomes simply by assuming some candidate is on or off), once all
the implications of that assumption are considered, the "dynamic locked set", which might have had one, two, or even more values than a locked set would, magically becomes a locked set. And we can
then turn "off" the remaining values in the dynamic locked set if they are found in other cells in the unit, and possibly continuing following the chain/net until it allows us to make some inference.
I have similarly implemented a forcing net with what I call "dynamic groups", "dynamic fish", and "dynamic finned fish". I tried "dynamic Y-wings" and "dynamic XYZ-wings", but in my small database of
245 advanced puzzles, but this resulted in exactly zero additional inference that the forcing net could not already make. In fact, I think among the strategies described at this web site in the
"Basic", "Bent Set", "Strategies", and "Forcing Chains" sections, the only ones which are possibly not special cases of "forcing net with dynamic groups/fish/finned fish" are AIC with unique
rectangles, and SK loops. Even the "exotic" strategies include some special cases - I'm not sure about pattern overly, but other than that and Franken Fish, all "exotic" strategies listed up to APE
are special cases.
So I'd say you could view this one is a particular case of "forcing net with dynamic locked sets". That is meant in no way to denigrate this quite elegant strategy; I just think it can be
Franken fish is my next project :)
... by: giuseppe rizza
Sunday 4-Oct-2020
if I hit in APE Example 3 , it will be changed in the solver, different from what I can see in this picture.
Best regards
Giuseppe Rizza
Andrew Stuart writes:
True. You have to turn off WXYZ-Wing to see it.
... by: pie314271
Saturday 30-Nov-2019
I've found a very unusual APE example -
try this one
The only other thing that happens here is Hidden Singles, and on the step with the APE 4 different ALSs are used to eliminate a total of five candidates from the two grey cells, which is definitely
really high.
Andrew Stuart writes:
Great example!
... by: David Anez
Saturday 23-Feb-2019
I need some help, because I thought I understood WXYZ-wing, but in example 2, JUST before the APE would kick in, wouldn't it be a valid WXYZ-wing with A3 as hinge, together with A1/A4/H3? Locked Set
of 1/4/5/7. It would clear 7 from B3, therefore solving it as 3.
* If A3 is 4, A4 solves to 1, which leads A1 to be solved as 7, which removes the 7 of B3
* If A3 is 5, H3 is solved as 7, which removes the 7 of B3
* If A3 is 7, directly removes the 7
The 7 in question is removed anyway by a second APE after the APE and Simple Colouring, but is removed because the combination 5/7 in A3/B3 would clear H3... which is part of my possible WXYZ-wing,
together with A3.
Is there an assumption of WXYZ-wings that I have somehow missed?
Andrew Stuart writes:
You are correct. In fact with example 2 the solver finds the WXYZ wing first. I think the solver strategy order has changed a little since 2018.
... by: TAU
Monday 20-Apr-2015
I don't know whether your solver implements it, but the article on APE Type 2 does not mention that the combination of the target value with the same value in a non-aligned pair cell CAN be excluded,
not by an ALS, but by a Locked Set, IF the target and pair cells, between them, can see all occurrences of the common value in the Locked Set.
... by: rRcCV
Tuesday 22-Jul-2014
Like your site very much.
But I would appreciate to get some help in understanding the last "Eight-Cell Aligned Pair" example.
As far as I can see eliminating candidate 2 from E5 just requires four of the eight cells:
B5, C5, D6 and E6.
And completing the exercise of checking all possible combinations in D5E5 against the impact of all the ALS highlighted in the diagram unveals a lot of other combinations as being impossible, but
they are not enough to eliminate any candiates from either D5 or E5, since I#m left with the remaining combinations as follows:
So what's the benefit of taking EF4 and HJ5 into consideration?
I just don't see what makes the statement true that "Each cell is necessary to produce all the pairs used to cross reference..."
Thanks for any explanation.
... by: Anton Delprado
Monday 10-Mar-2014
I think it is important to note that any ALS seen by both cells doesn't have to be entirely seen by both cells.
For example if we were excluding "3 and 5" then the first cell only has to see the cells in the ALS that can be 3 and the second cell only has to see the cells in the ALS that can be 5.
Andrew Stuart writes:
... by: LeProf
Sunday 16-Feb-2014
We are great fan of your site and tools, which we use for training, and I for some leisure. I am not an expert on sudoku, but I am something of an expert on training people to do complicated
procedures. In my work, is we fail at a the training, later, either something/someone blows up (quite literally, I'm afraid), or a 1M$+ piece of equipment is taken offline or ruined.
The following comments are offered, then, as a more general comment (one example), to challenge your process of explaining how one begins to apply advanced "strategies". It is communicated because
those students and I using your clearly thoughtful and effort-intensive explanations find it possible, even so, to apply them, in largest part, only if the explained strategy is already understood
beforehand/separately from the explanation you provide. It is offered in the spirit of good organizational practice—that it is better we be talkers (providing critical feedback, even if negative)
than walkers (departing the organization without comment, when needs are not met).
First, note, in your pedagogy of advanced strategies, you *do* explain superbly how the pattern you choose to act upon leads to elimination of possible numbers in cells, based on the new method being
applied—the examples are all, generally, excellently explained.
But, there is little or no explanation on how one engaging a puzzle at an advanced state of solving (that shows all possibles, and that is taken as far as preceding more basic techniques allow) can
discern either that no other advanced strategy is better and actionable, or that the currently presented strategy, here APE, is present and actionable, and if several starting points are possible,
how the specific one to act upon is chosen.
Hence, what we seek (elsewhere at present) are explanatory statements for various strategies, that indicate, for given points on the paths to a puzzle's solution, what patterns are to be looked for,
among which frequencies of remaining numbers/number groups, to indicate the best advanced strategy to attempt to apply.
In APE Example 1, perhaps what is sought is as statement like, "After scanning the puzzle for… [patterns] and seeing none further—note that the apparent starting points for strategies P and Q in
cells A and B are dead ends—it is then may be time to apply an Aligned Pair Exclusion. To see if this strategy is applicable, units with between 4 and 5 unsolved cells are examined for presence of 2
or more bi-value cells that can *see* the rest of the cells, and that *share candidates with them*. Of the possible starting points in Example 1, we choose row G and box GHJ123 because… and we focus
on cells G2 and G3 because… " [Note, no mention at all, yet, of Almost Locked Sets of greater complexity than bi-value.]
A related point here is that a very broad and general statement of the strategy is likely most suitable at the end, where it can be readily understood, e.g., here, where the extension from bi-value
to all other Almost Locked Sets can be explained without undue complexity.
Remarkably, we know that you have already thought through this logical, truly strategic decision-making—it is what your programming, in the solver, clearly already does, walking stepwise, in a
particular order, through the strategies, deciding which to apply or dismiss, based on looked-for patterns. It is this very process of logic, of what to apply when in the course of a puzzle, and
where (very specifically) to begin applying it within a puzzle matrix, at a given point of being solved, that is missing in the current "strategy" explanations.
Finally, we are aware that explaining "strategies" is not necessarily the same as explaining the process of developing an over-arching strategy of applying them, in orderly fashion to given
situations. In this regard, I would say you have laid out an excellent list of *tactics*, and that what is yet missing is an adequate discussion of strategy—the way in which "battlefield" conditions
are accessed in each situation, to decide which particular tactic best applies.
Hope this might be go some help. We will continue exploring and learning tactics, separately, and coming here to review your well-explained examples, and to sort on our own the best over-arching
strategies to apply. Best wishes for this continuing, clearly dedicated and effort-intesive endeavour. LeProf
Andrew Stuart writes:
Late to answer this but very appreciated nonetheless. You are correct that I separate the 'tactic' from the 'strategy' and use the later word where tactic might be a better word. To be honest I don't
know what the best search for a strategy might be if confronted with a board of candidates. I am in awe of the pen-and-paper solvers who do use these strategies. When I started on the advanced ones I
imagined I was only talking to the programming crowd since I consider them too difficult to manually search and use. Sudoku is interesting for having the necessity for extending the logic so far - it
will be a rare puzzle indeed that is published in a newspaper that requires them. I'm hoping the real mental solvers will chime on on this topic.
... by: ad.joe
Monday 25-Jun-2012
General structure of the APE
Maybe it's not easy to put this straight, but in fact there are eliminations in 2 boxes:
g k p | _ _ _ | Y Z _
_ _ _ | _ _ _ | _ _ _
_ _ X | _ _ _ | a b c
The "wings" are X=2#, Y=2*, Z=2+
Then XYZ eliminates 2 on the aligned cells abcgkp ...
(provided that these have "not much more than #*+ on it", so you have to "cross-check")
- Doesn't the APE-structure look like an xy-wing (y-wing)? Only the function does work backwards and there is no pivot necessary!!
So you could also call it a BACKWING or a BACKSWING!
- And of course, in an X-Sudoku the 2 aligned fields need not be in the same box
(as long as one of the two is on a diagonal).
... by: ad.joe
Monday 25-Jun-2012
Let's build a general structure for the APE
_ _ _ | g k p | y z _
_ _ _ | _ _ _ | _ _ _
_ _ _ | x _ _ | a b c
the "wings" xyz (x=2#, y=2*, z=2+) eliminate 2 on the aligned cells abcgkp
(of course also in box 2, not only in box 3)
- While seeing this structure we can say, the form looks the same as an xy-wing (y-wing) or an xyz-wing, but the function works backward.
So you could call it also a BACKWING or a BACKSWING
- And of course, in an X-Sudoku the aligned fields need not be in ONE box (as soon as one of the two is on a diagonal)
... by: ad.joe
Friday 22-Jun-2012
How to tease out the APE:
First, it's not our job to solve one of the examples in another way. But it's nice that all is so understandable.
(Matt, I'd add three words: duplicate the contents of any "ONE of the" two-candidate cells they both see. )
So, as we look for the third cell of an xy-wing (having seen two "similar" ones at first), what can we do here:
Let's solve it by cross-checking: We assume that the most FREQUENT number of the bivalue cells can be eliminated in the two cells in question.
We assume this number being correct there and quickly end in error, quicker than in looking up all combinations.
(Who'd think that we would not score more than 90 percent, he/she can feel free to try with the other numbers)
... by: Anon
Sunday 18-Dec-2011
I want this software. how I can have it?
... by: Phil Gooda
Monday 8-Feb-2010
I actually managed to understand all this, except for one thing......why bother, there is a far simpler soultion within that box? Let me demonstrate by considering ONLY that box and naming the cells
as follows:
A B C
D E F
G H I ..... which means that my C is the X mentioned, and my F is the Y mentioned.
Cells B, C and G contain 3 numbers and only 3 numbers between them. Therefore those 3 numbers cannot appear in the other cells. Which immediately gives you the paired 5/9 in F and I, which means that
E has to be 4. Why bother doing all the APE stuff?
... by: Patrick Barnaby
Monday 25-Jan-2010
An XY-Chain removes 7 from r8c8. A Bilocation Graph reveals a 2 at r7c2 and then you'll have a naked 7 and then a naked 8 and then quite a few hidden singles on the first and second rows. So this is
easier than APE except the Bilocational graph techinque.
... by: Matt Lala
Thursday 12-Nov-2009
I love the site but some of the explanations need help. Or I guess I do. I think this is one of the less clear ones.
The rule you have says:
Any two cells aligned on a row or column within the same box CANNOT duplicate the contents of any two-candidate cell they both see.
If I take that literally in the first example... X in the first example can see the bivalues 25 and 37. It cannot duplicate the contents of those cells. Therefore... it can't be 2,5,3 or 7? Obviously
that's not what was meant.
Or is it... if the two aligned cells see some bivalue cells, and they both mutually share a certain candidate [that's part of those bivalue cells] then that shared candidate can go? But no, that's
not it either.
If either if the cells see two bivalue cells, and those two bivalue cells both share a candidate with the cell under consideration... that shared one goes?
Does one actually have to enumerate the possibilities? This seems like something that can be spotted with a glance, if only it can be made clear the exact conditions needed.
... by: Bernard Gervais
Wednesday 30-Sep-2009
Excellent site, thank you.
for example 1, I use the unicity concept which pinpoints A4 = 1.
Best regards. | {"url":"https://www.sudokuwiki.org/Aligned_Pair_Exclusion","timestamp":"2024-11-04T05:13:52Z","content_type":"text/html","content_length":"53025","record_id":"<urn:uuid:4c404fb8-0de1-4727-91e8-04289186fc00>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00287.warc.gz"} |
Zero kinetic undercooling limit in the supercooled Stefan problem
We study the solutions of the one-phase supercooled Stefan problem with kinetic undercooling, which describes the freezing of a supercooled liquid, in one spatial dimension. Assuming that the initial
temperature lies between the equilibrium freezing point and the characteristic invariant temperature throughout the liquid our main theorem shows that, as the kinetic undercooling parameter tends to
zero, the free boundary converges to the (possibly irregular) free boundary in the supercooled Stefan problem without kinetic undercooling, whose uniqueness has been recently established in (Delarue,
Nadtochiy and Shkolnikov (2019), Ledger and Søjmark (2018)). The key tools in the proof are a Feynman-Kac formula, which expresses the free boundary in the problem with kinetic undercooling through a
local time of a reflected process, and a resulting comparison principle for the free boundaries with different kinetic undercooling parameters.
All Science Journal Classification (ASJC) codes
• Statistics and Probability
• Statistics, Probability and Uncertainty
• Feynman-Kac formula
• Free boundary problems
• Kinetic undercooling
• Local time
• Reflected processes
• Supercooled Stefan problem
Dive into the research topics of 'Zero kinetic undercooling limit in the supercooled Stefan problem'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/zero-kinetic-undercooling-limit-in-the-supercooled-stefan-problem","timestamp":"2024-11-06T11:21:14Z","content_type":"text/html","content_length":"50615","record_id":"<urn:uuid:6dbed885-6f06-4b9e-96b2-b305d1965f94>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00775.warc.gz"} |
Embracing firefly flash pattern variability with data-driven species classification
Acquisition of flash sequence data
To extract flash sequence data, we perform 3D reconstruction of firefly swarms based on stereoscopic video recordings. Recordings were conducted at specific locations across the country where certain
species were known to emerge^24. Field researchers placed two spherical (360) GoPro Max cameras at known distances from each other on a level surface (Fig. 1A). Recordings started at dusk when the
first flashes were seen, and filmographers performed a simultaneous catch-and-release identification process to acquire ground-truth labels from visual inspection of the individuals present in the
swarm. All recordings are made at a frame rate of 30 frames per second. The movies were subsequently processed as described in a previous work^22,24 to extract the 3D locations of flash occurrences.
From these locations, we apply a simple distance-based linkage method to concatenate flashes into streaks and streaks into trajectories. We consider flashes at consecutive timesteps within a small
radius to be part of the same streak; streaks occurring within both 1s and 1m of each other are assumed to come from the same individual and placed in a set of transitively connected streaks called a
trajectory. To eliminate noise effects from the trajectory extraction, we threshold the trajectories to eliminate those that only contain one flash. The dataset^24 includes ten total species before
the application of the thresholding process. Following the thresholding, we also remove any species from the dataset that have fewer than one hundred total trajectories, leaving us with seven species
total. Finally, from the trajectories, we extract a binary time sequence by considering the time coordinates of flashes, i.e. a sequence of ones and zeroes where ones represent flashing and zeroes
represent interflash gaps. Each element (or bit) of the time series represents a single frame of a video, such that 30 bits represents 1 full second of recording. We further clean the dataset by
recognizing that any interflash gaps 1 or 2 bits in length (less than 0.07s) are likely caused by an error in the tracking or trajectorization process, or the firefly briefly being obscured by brush
as it moves. These short gaps are replaced by ones to connect the interrupted flash.
This process enables the capture of individual behavior from simple footage of firefly swarms of any species, provided individuals of that species flash frequently enough to meet the threshold
standards of the trajectory generation. Our data acquisition procedure highlights the presence of intraspecies behavioral variability, and characterizes this variability by representing flash
patterns as distributions (Fig. 4A–C).
The result of this process is 124,503 flash trajectories from the seven species before thresholding, and 65,389 after those with only one flash have been removed. More than half of these are P.
carolinus sequences – the majority class. About 1 percent of these are P. forresti and P. bethaniensis sequences – the minority classes. The rest of the classes range between 4 and 14 percent of the
total distribution. The dataset comprises binary sequences of between 6 and 1366 bits (0.2s to 45.5s) in duration, each labeled with the corresponding firefly class.
We implemented a bespoke neural network architecture with PyTorch to solve our classification problem. For the curious, technical details about the implementation and data-wrangling practices follow.
Additionally, all code is open source and available as mentioned in Section “Data availability”.
Neural network architecture
RNNs are a class of neural networks suitable for sequence learning tasks. They are characterized by feedback connectivity and the consequent ability to encode long-range temporal dependencies, such
as those intrinsic to firefly flash patterns. The defining computational step of an RNN is the hidden state update, which is a function of the input at the current timestep, \(x^{
where f is a non-linear activation function, such as hyperbolic tangent, \(W_{hh}\) and \(W_{xh}\) are weight matrices that map hidden-to-hidden (i.e. the feedback connectivity) and input-to-hidden,
respectively, and b is a bias term.
Importantly, Eq. (1) enables a recurrent neural network to ingest variable length input sequences, as the update rule can be applied recurrently, which is suitable for firefly flash sequences that
are of variable temporal duration. However, if the input duration is sufficiently large–as is the case with some of flash pattern sequences–vanishing gradients will arise when computing weight
updates via backpropagation through time (BPTT)^29 due to the non-linearity in f, ultimately prohibiting learning.
To address this issue, we leverage an extension to RNNs–gated recurrent units (GRUs)–that introduces gating mechanisms that regulate which information is stored to and retrieved from the hidden state
^30. These gating mechanisms enable the model to more effectively regulate the temporal context encoded in the hidden state and enables the encoding of longer-range temporal dependencies.
Additionally, GRU RNNs are computationally more efficient than other kinds of RNNs like long short-term memory networks (LSTMs), and use fewer parameters, which was a consideration due to our plans
for eventual downstream application of the model in real-time population monitoring. Consequently, we implement the model in PyTorch^31 as a 2-layer GRU with 128-dimension hidden layers, no dropout,
and LeakyReLU activation layers with a negative slope of 0.1.
Data preprocessing
To evaluate our model’s predictive ability on unseen data, we perform 60-fold stratified cross validation to ensure that each sequence in the dataset is used at least once in training and at least
once in testing, but never simultaneously. Each fold divides the data into ninety percent training, and ten percent testing, preserving the same class ratios as the original dataset. Due to the
severe class imbalance (e.g. some species only comprise 1% of the dataset, whereas others comprise close to 50% of the dataset), we perform random undersampling on the training set of each fold to
equalize the class count in the training and validation sets for each fold. This takes the form of a secondary k-fold cross validation procedure to sample from each class until classes are equalized.
All the remaining data are used for testing. The reported results are thus the ensemble precision, recall, and accuracy of each model on its respective test set of approximately 30,000 sequences,
averaged over the 60 model folds. The ground truth targets are species names; we performed label encoding to transform the targets into machine-readable integers representing each class.
Training and evaluation
We trained the model with the Adam optimizer^32 and evaluated performance via cross-entropy loss. During training, we set an early stopping callback that monitored the validation loss with a patience
of 50 epochs to prevent overfitting on the training set. Additionally, to alleviate exploding gradients, we applied a gradient clipping of 0.1 to the gradient norms following the procedure
recommended in Ref.^33. We conducted a hyperparameter sweep over the batch size and learning rate, testing all combinations of batch size \(\in \{8,16,32\}\) and learning rate \(\in \{10^{-3},10^
{-4},10^{-5}\}\). We selected the combination that had the highest validation set accuracy on a four-species subset of the data, which resulted in the choice of a batch size of 8 and a learning rate
of \(10^{-5}\). No data augmentation was applied during training.
We evaluate the performance of the RNN, along with the signal processing methods described in the following section, on the test data by examining the receiver-operating characteristic (ROC) curves
for each species (Fig. 6A–E). Per-species precision and recall are tabulated in Fig. 6F.
Sympatric species experiments
To explore the capabilities of the model when faced with sympatric swarms, we first gave the model a different training regimen. All of the data except for sequences from five days – one day each for
B. wickershamorum, P. carolinus, P. frontalis, P. knulli, and P. obscurellus – serve as the training and validation set, and sequences from the five held-out days enter the test set. The five held
out days and their codes as referenced in^24 are as listed in Table 3. Holding out single days like this ensures that a) the sequences being tested do not occur in the test set and b) the model can
identify new sequences on a new day for a species it has already seen before.
Table 3 Metadata of held-out sequences for sympatry experiments.
We note that P. bethaniensis and P. forresti are excluded from these experiments. This is because for both of these species, holding out one day of data would reduce the total number of sequences in
the training set to below one hundred, which is against our recommendation for sufficiency. However, these species remain in the training set, as none of their dates are held out, and thus the model
can still predict them during these sympatry experiments.
The goal of these experiments is to vary the different densities of each possible pair of species to test whether the model as trained can capture the presence of each species. This tests whether the
model is applicable in future hypothetical scenarios where more than one species may be present in a recording, but each of the species present are already part of the training set in some form. For
each experiment, we generated a test set of 400 sequences comprising two species from the holdout days, mixed together at a particular density ratio to create an articifial instance of sympatry. This
means that for each iteration of the experiment, the number of sequences for each species ranged from two to 398. We ran 500 iterations of each density ratio for each pair, where each iteration was
randomly sampled from the set of sequences corresponding with the held out date. We recorded the true positive rate for each class at each density reported by the model. The results are shown in Fig
Signal processing methods
For the purposes of comparison with the RNN, we implemented four alternative classifiers which use standard signal processing algorithms to compare our dataset against ground truth references for
each species. We implement these classifiers using two types of ground truth references: “literature references”, which use flash patterns as previously published in the literature, and “population
references”, which are generated by aggregating sequences in our own dataset.
Literature references
“Characteristic” flash patterns for six out of the seven species analyzed in this paper, excluding B. wickershamorum, have been previously recorded and published in the literature. These recorded
flash patterns hence served as the primary reference for researchers in identifying signals observed in the field. These reference flash patterns are typically reported pictorially; thus, we convert
images to binary-valued time series by computing the relative sizes of flashes and gaps, in pixels. We determine the pixel-to-second conversion to then convert the sequence to a 30 frames per second
time series, matching the sampling frequency of our data. We have quantified these variables from published works to underscore the prevalent tendency toward qualitative approximations over
quantitative analyses. Flash signals are commonly documented in scholarly articles and monographs through visual representations, frequently drawing from multiple, and occasionally ambiguous, primary
and secondary information sources for individual species.
These six reference time series then form the ground-truth comparisons against which our dataset is compared, using the four signal processing techniques described below in Methods Section Signal
processing methods. We omit B. wickershamorum as there is currently no published reference pattern.
Population references
We also generate “population references” by aggregating sequences in our own dataset. For each species, we first perform an 80:20 train:test split, similar to the preprocessing procedure performed
for the RNN (see above in Methods Section”Data preprocessing”). The population references are obtained by averaging the sequences in each training set. The remaining test data is then classified
using the signal processing algorithms described below in Methods Section “Signal processing algorithms”. As with the RNN, we perform \(N = 100\) iterations and take the ensemble average of the
performance across all iterations.
Signal processing algorithms
Jaccard index
The Jaccard index compares the similarity between two sets by taking the ratio of the size of the intersection of the sets with the size of the union^34 and has found broad application, for example
in genomics^35. For two binary-valued sequences \((a_m)_{m=1}^{M}\) and \((b_n)_{n=1}^N\) of lengths M and N, respectively, with \(a_m, b_n \in \{0,1\}\) for all m and n, we define the size of the
intersection as \(\sum _{i=1}^{\text {min}(M,N)} a_i b_i\), the number of ‘on’ (flashing) bits that occur simultaneously in both sequences. We define the union as \(\sum _{m=1}^M a_m + \sum _{n=1}^N
b_n\), the number of on bits for both sequences combined. The Jaccard index can also be evaluated in the same manner for two sequences that are binary-valued. Generally speaking, the intersection can
also be thought of as the dot product between the two sequences. To classify a sequence using the Jaccard index, the Jaccard index between a sequence and each species reference is computed, and the
softmax of the vector of Jaccard index values is computed to determine a probability of the sequence being from each species. The predicted species is then the argument maximum (arg max) of the
softmax vector.
Dot product
The dot product between two sequences is given by the sum of the product of the sequences, i.e. \(\sum _{i=1}^{\text {min}(M,N)} a_i b_i\) for two sequences \((a_m)_{m=1}^{M}\) and \((b_n)_{n=1}^N\)
of lengths M and N, respectively. Sequences are then classified by taking the arg max of the softmax of the dot product with each reference.
Dynamic time warping
Dynamic time warping (DTW) is an algorithm that computes the distance between two time series by locally stretching or compressing the sequences to optimize the match. DTW is useful for comparing
time series that are qualitatively similar but vary locally in speed and has found significant application in speech recognition^36,37,38,39. We implement DTW in MATLAB^40 to compute the distance
between sequences and species references. Similarly to the other metrics, the predicted species is taken to be the arg max of the softmax of the distances, which is a vector of probabilities that
maps to the probability of each label.
Each flash sequence can be parametrized by 3 values (Fig. 1A): the number of flashes in the sequence, the average duration of each flash, and the average time between flashes (inter-flash gap). We
perform support vector machine (SVM) classification in this 3-dimensional space, using a radial basis kernel function.
Figure 5
Reference sequences for the firefly species examined in this paper, as previously published in the literature, with the exception of B. wickershamorum. (A) Reference pattern for P. frontalis from
Barber, 1951^41. (B) Reference patterns for P. obscurellus and P. bethaniensis from Faust, 2017^7. (C) Reference patterns for P. carolinus and P. knulli from Stanger-Hall and Lloyd, 2015^8. ( D)
Reference pattern for P. forresti from Fallon et al., 2022^6. (E) Illustration of extracting P. frontalis sequence from literature pattern (top) and converting to time series (bottom).
Figure 6
Per-species ensemble of classification results. (A–E) Receiver operating characteristic (ROC) curves representing the true positive rate (TPR) as a function of the false positive rate (FPR) across
all model thresholds of classification, labeled by method. All non-RNN classification methods are conducted using population references. (F) Table of per-species precision and recall across all
surveyed methods (N=100). Bold statistics in the table represent the highest performer for each metric and species. Precision values for P. bethaniensis and P. forresti are low because these two
species represent the classes with the fewest number of samples, and so there is a very small amount of true positive values. However, these still greatly exceed what would be expected by chance
(0.001 and 0.006, respectively). The high recall for these classes indicates that the true positives are correctly captured.
The data acquisition procedure is not without noise, so we perform filtering to produce accurate quantitative characterization of flash phrases that falls in alignment with previous literature
observations. We leveraged the ability of the RNN to distinguish between sequences by choosing the sequences which the RNN scored highest as the top one hundred most confident classifications for
each species. This subset acts as the dataset on which characterization exercises are performed for Fig. 1B and Fig. 4. The procedure is as follows:
1. 1.
Initialize the empty list c
2. 2.
for each i sequence in the test set D:
1. (a)
Run a forward model step
2. (b)
Let p = the maximum probability in the resulting vector of softmax predictions
3. (c)
If the index of p corresponds with the correct label, add the pair (p, index) to the list c
3. 3.
Sort c by probability p and choose the top 100
4. 4.
Index into the dataset D using the associated indices of the top 100 probabilities to produce the subset
Characterizing in this way leverages the variability in the entire dataset by training the predictive classifier, then asks the predictive classifier only for what it is most confident about in order
to filter out sequences that may be missing flashes or exhibiting patterns that are far from the statistical norms of the species.
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The Professor loved prime numbers - The Patient Poppy
Photo by Ryoji Iwata on Unsplash
Excerpt from The Housekeeper and the Professor by Yōko Ogawa
This is an excerpt from the book The Housekeeper and the Professor by Yōko Ogawa, translated by Stephen Snyder.
The Professor loved prime numbers more than anything in the world. I’d been vaguely aware of their existence, but it never occurred to me that they could be the object of someone’s deepest affection.
He was tender and attentive and respectful; by turns he would caress them or prostrate himself before them; he never strayed far from his prime numbers. Whether at his desk or at the dinner table,
when he talked about numbers, primes were most likely to make an appearance. At first, it was hard to see their appeal. They seemed so stubborn, resisting division by any number but one and
themselves. Still, as we were swept up in the Professor’s enthusiasm, we gradually came to understand his devotion, and the primes began to seem more real, as though we could reach out and touch
them. I’m sure they meant something different to each of us, but as soon as the Professor would mention prime numbers, we would look at each other with conspiratorial smiles. Just as the thought of a
caramel can cause your mouth to water, the mere mention of prime numbers made us anxious to know more about their secrets.
Evening was a precious time for the three of us. The vague tension around my morning arrival—which for the Professor was always our first encounter—had dissipated, and Root livened up our quiet days.
I suppose that’s why I’ll always remember the Professor’s face in the evening, in profile, lit by the setting sun.
Inevitably, the Professor repeated himself when he talked about prime numbers. But Root and I had promised each other that we would never tell him, even if we had heard the same thing several times
before—a promise we took as seriously as our agreement to hide the truth about Enatsu. No matter how weary we were of hearing a story, we always made an effort to listen attentively. We felt we owed
that to the Professor, who had put so much effort into treating the two of us as real mathematicians. But our main concern was to avoid confusing him. Any kind of uncertainty caused him pain, so we
were determined to hide the time that had passed and the memories he’d lost. Biting our tongues was the least we could do.
But the truth was, we were almost never bored when he spoke of mathematics. Though he often returned to the topic of prime numbers—the proof that there were an infinite number of them, or a code that
had been devised based on primes, or the most enormous known examples, or twin primes, or the Mersenne, primes—the slightest change in the shape of his argument could make you see something you had
never understood before. Even a difference in the weather or in his tone of voice seemed to cast these numbers in a different light.
To me, the appeal of prime numbers had something to do with the fact that you could never predict when one would appear. They seemed to be scattered along the number line at any place that took their
fancy. The farther you get from zero, the harder they are to find, and no theory or rule could predict where they will turn up next. It was this tantalizing puzzle that held the Professor captive.
“Let’s try finding the prime numbers up to 100,” the Professor said one day when Root had finished his homework. He took his pencil and began making a list: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31,
37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
It always amazed me how easily numbers seemed to flow from the Professor, at any time, under any circumstances. How could these trembling hands, which could barely turn on the microwave, make such
precise numbers of all shapes and sizes?
I also liked the way he wrote his numbers with his little stub of a pencil. The 4 was so round it looked like a knot of ribbon, and the 5 was leaning so far forward it seemed about to tip over. They
weren’t lined up very neatly, but they all had a certain personality. The Professor’s lifelong affection for numbers could be seen in every figure he wrote.
“So, what do you see?” He tended to begin with this sort of general question.
They’re scattered all over the place.” Root usually answered first. “And 2 is the only one that’s even.” For some reason, he always noticed the odd man out.
“You’re right. Two is the only even prime. It’s the leadoff batter for the infinite team of prime numbers after it.”
“That must be awfully lonely,” said Root.
“Don’t worry,” said the Professor. “If it gets lonely, it has lots of company with the other even numbers.”
“But some of them come in pairs, like 17 and 19, and 41 and 43,” I said, not wanting to be shown up by Root.
“A very astute observation,” said the Professor. “Those are known as ‘twin primes.’”
I wondered why ordinary words seemed so exotic when they were used in relation to numbers. Amicable numbers or twin primes had a precise quality about them, and yet they sounded as though they’d been
taken straight out of a poem. In my mind, the twins had matching outfits and stood holding hands as they waited in the number line.
“As the numbers get bigger, the distance between primes increases as well, and it becomes more difficult to find twins. So we don’t know yet whether twin primes are infinite the way prime numbers
themselves are.” As he spoke, the Professor circled the consecutive pairs.
Among the many things that made the Professor an excellent teacher was the fact that he wasn’t afraid to say “we don’t know.” For the Professor, there was no shame in admitting you didn’t have the
answer, it was a necessary step toward the truth. It was as important to teach us about the unknown or the unknowable as it was to teach us what had already been safely proven.
Have you read this book? I’d love to hear your thoughts in a comment below!
The Housekeeper and the Professor – Summary
Here is the book summary from Goodreads:
He is a brilliant math Professor with a peculiar problem–ever since a traumatic head injury, he has lived with only eighty minutes of short-term memory.
She is an astute young Housekeeper, with a ten-year-old son, who is hired to care for him.
And every morning, as the Professor and the Housekeeper are introduced to each other anew, a strange and beautiful relationship blossoms between them.
Though he cannot hold memories for long (his brain is like a tape that begins to erase itself every eighty minutes), the Professor’s mind is still alive with elegant equations from the past. And the
numbers, in all of their articulate order, reveal a sheltering and poetic world to both the Housekeeper and her young son. The Professor is capable of discovering connections between the simplest of
quantities–like the Housekeeper’s shoe size–and the universe at large, drawing their lives ever closer and more profoundly together, even as his memory slips away.
The Housekeeper and the Professor is an enchanting story about what it means to live in the present, and about the curious equations that can create a family.
Copyright © 2003 by Yōko Ogawa.
Translated by: Stephen Snyder
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Is There an Issue with the LegendreP function?
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Is There an Issue with the LegendreP function?
I am trying to evaluate a sum over a large number of terms, each of which includes a Legendre polynomial. At fairly low numbers of the order parameter, I get enormous values for the polynomial.
I have created a simple Notebook to illustrate the issue. You can see that I can plot the Legendre polynomial for orders between 0 and 100
0 and get seemingly sensible answers. If I then do a single call for the final value, I get an error.
I am using MMA 13.2 on a MacBook Pro with Apple silicon.
Any suggestions will be much appreciated.
Thanks, David
2 Replies
Many thanks for all of these ideas.
The issue, it turns out, is that LegendreP, by default, attempts to return an answer in polynomial form. As the order increases, the coefficients in the polynomial get enormous and all sorts of
numerical horrors appear.
Applying //N after the function call is not the solution, in fact the answer is to put the N wrapper around the argument as follows:
LegendreP [1000,N[x]]
which internally invokes a purely numerical solution using the recurrence relationship I imagine.
Once again, thank you for your prompt and helpful ideas. David
One way is:
n = 20000;
th = \[Pi]/7;
$MaxExtraPrecision = 10000;(* Increase this value if necessary.*)
N[LegendreP[n, th], 20]
Second way:
n = 2000000;
th = \[Pi]/7;
LegendrePPP[n_, z_] := (1/Pi) NIntegrate[(z - Sqrt[z^2 - 1] Cos[t])^n, {t, 0, Pi},
WorkingPrecision -> 30, MaxRecursion -> 20]
LegendrePPP[n, th]
Third way is the best it's very fast:
ClearAll["`*"]; Remove["`*"];
S[n_, z_] = (Series[LegendreP[n, z], {n, Infinity, 1}] // Normal // Simplify)
n = 10000000;(* Try: n = 10^100 *)
th = \[Pi]/7;
N[S[n, th], 20]
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Measuring One Year’s Growth
What is One Year Growth?
In simple terms, one year of growth is the amount of growth a student makes during a school year. Fastbridge assessments represent growth using the rate of improvement (ROI) metric. Depending on the
assessment, the ROI will measure the change in score over weeks or months. For example, the ROI for aReading represents the average monthly growth between two administrations. Therefore, if a student
scores 472 on aReading screening in August and 495 on the same assessment 9 months later, in May, then that student will have a growth rate of 2.56 scaled score points per month.
When considering the question, what is one year of growth in an educational setting, what is often of interest is the answer to the question, what is typical growth among the population of students?
That is, how much growth should I expect students to attain in one year?
This is a normative question that often anticipates a single value at or above which a student made one year of growth, and below which a student did not make one year of growth. Such an approach
neglects the variability in growth across the student population and the imprecision of test scores that will naturally vary from day to day. Furthermore, it neglects the instability of growth due to
other factors that vary throughout a year and across students such as the quality of instruction, social-emotional states, life events, and brain development as well as the phenomenon of regression
to the mean. As a result of such factors students who make lower-than-average growth during one interval, often make higher-than-average growth in the next. The amount of growth one should expect to
see from students will differ.
One Year’s Growth “Expected Annual Growth” From a Normative Perspective
Unlike proficiency standards derived from expert analysis of what students should know and be able to do, to date there is no external criterion to define growth standards. Rather, growth standards
are typically norm-referenced. That is, characterizations of growth are anchored on the distribution of growth scores in a national sample.
Fastbridge provides ROI growth norms that can be used to describe typical student growth rates within the student population. The ROI growth percentiles indicate how a student’s growth compares to
the population of students at each grade and for each assessment. FastBridge provides seasonal and annual growth percentiles: Fall to Winter, Winter to Spring, and Fall to Spring.
The table below presents the growth norms for grade 2 students on CBMreading. For every 5th percentile, the table shows the corresponding average weekly change. For example, a student with a Fall to
Spring ROI of 1.55 points per week, grew at a rate at or above 70% of the students in the norm group.
Statistically speaking, what is typical or expected growth is commonly understood as that part of a score range where most students fall. ROI distributions are generally normal, bell-shaped curves.
In a normal curve, the most common score is the mid-point or median. On the percentile metric, the median is the 50th percentile. Half of the population has a slower growth rate while the other half
has a faster growth rate. As such, the 50th percentile is the most reasonable approximation of one year’s growth. Thus, the 50th percentile is a reasonable standard for establishing growth
Limitations of Using a Single Cut-Score
All scores are affected by random factors. The standard error of measurement (SEM) indexes the magnitude of the random factors (random error) on a student’s score. A student’s true score at a given
point in time is best understood within a range of plus or minus one SEM. For CBMreading, the SEM is about 9 points. Thus, on a given administration, we can have confidence that the student’s true
ability score lies within a range from 9 points below to 9 points above the observed score.
Growth scores are also affected by random error. As such a student’s true growth score is also best approximated by a range of values. By estimating the reliability of the ROI scores, we can
approximate the range within which a student’s true ROI is likely to fall. On CBMreading in grade 2, a student’s true Fall to Spring ROI is likely to fall within a range of approximately plus or
minus 0.30 ROIs. Using the grade 2 CBMreading ROI norms, for a student with an observed Fall to Spring ROI of 1.27 (50th percentile), the true ROI falls within a range from about 0.97 to about 1.62.
This is roughly the range from the 30th to 75th national percentile.
Because scores contain random error and given that 50 percent of students fall within the range from the 25th through 75th national percentile, it would be appropriate to treat any ROI within that
range as approximating one year’s growth (see the table below).
Student Growth in FastBridge
Student growth information is available in the Group Growth Report and via the Data Download.
The Group Growth Report can be accessed by selecting the Reporting tab and choosing View Report on the Group Growth Report icon (see Figure 1 below) and selecting the assessment to create a report.
Once accessed, select Fall as the Start and Spring as the End intervals at the top of the page (see Figure 2). To obtain the ROI growth percentile, the other options at the top of the report can be
left as the defaults.
To see the national ROI percentile, select the + icon on the right side of the screen (see Figure 3). The growth percentiles for each student are displayed in the Growth %ile column of the table. The
Growth Score column presents the ROI value.
Growth data are also summarized in the bar graphs in the top center window. The categories are defined by percentile ranges and are the same percentile ranges used to define benchmarks or norm ranges
in the Group Screening Report. The key at the bottom of the page provides the percentile ranges for the normative view color coding scheme. Although there is not a range that breaks at the 50th
percentile, these categories can be helpful to quickly summarize the level of growth in a group.
Figure 1
Figure 2
Figure 3
Users can also download raw data on students to investigate student growth patterns through their own means. Data Download can be selected through the District Manager or Reporting tabs. Data
Download is located at the top of the District Manager tab (see Figure 4). On the Reporting tab, find Student Data Download and choose View Report. On the next page, select the school year, schools,
and assessment. Select submit and a .csv file will download in the window. The Growth Percentile from Fall to Spring column, found in the data file, provides the national growth percentile of each
student based on growth from Fall to Spring.
Figure 4 | {"url":"https://fastbridge.illuminateed.com/hc/en-us/articles/10741164677787-Measuring-One-Year-s-Growth","timestamp":"2024-11-08T15:29:27Z","content_type":"text/html","content_length":"39752","record_id":"<urn:uuid:d813cb4c-72fc-44ac-bde8-0e20ba5e8096>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00578.warc.gz"} |
DCF Analysis: Forecasting Cash Flows
• Writer
Adam Fish
• Revealed
July 14, 2011
• Phrase rely
So as to get began with a reduced money circulation evaluation, we forecast an organization’s free money flows after which low cost them to the current worth utilizing the corporate’s
weighted-average value of capital (WACC).
Forecasting free money flows, nonetheless, could be fairly sophisticated – it’s really an artwork. There are numerous issues that may affect money flows and as many as attainable needs to be taken
under consideration when making a forecast:
What’s the outlook for the corporate and its business?
What’s the outlook for the economic system as a complete?
Is there any elements that make the corporate kind of aggressive inside its business?
The solutions to those questions will assist you to to regulate income progress charges and EBIT margins for the corporate. Let’s assume a hypothetical instance during which we now have a standard
financial outlook for the long run, a optimistic outlook for the business and a mean outlook for our firm.
Given these assumptions, we are able to merely take a look at our firm’s historic efficiency and proceed this efficiency out into the long run. our hypothetical firm’s revenues for the previous three
years, we are able to calculate the compound annual progress charge (CAGR) and use it to forecast income for the subsequent 5 years. The method for calculating CAGR is:
(12 months 3 Income/12 months 1 Income)^(1/2 Years of Development)-1
Subsequent, let’s calculate the corporate’s EBIT margin in order that we are able to forecast earnings earlier than curiosity and taxes. The method for EBIT margin is solely EBIT over Revenues. To
forecast EBIT we merely multiply our forecasted revenues by our EBIT margin.
The Taxman Cometh
To get to free money flows, we now have to forecast taxes and make sure assumptions in regards to the firm’s wants for working capital and capital expenditures. We calculate our firm’s tax charge by
dividing the corporate’s historic tax bills by its historic earnings earlier than taxes (EBIT much less curiosity expense). We will then forecast tax bills by multiplying the tax charge by our
forecasted EBIT for every year.
As soon as we now have after-tax revenue forecasted (EBIT – taxes), we have to add again depreciation and amortization, subtract capital expenditures and subtract working capital investments. We will
forecast depreciation and amortization bills by calculated their share of historic revenues and multiplying that share by forecasted revenues.
Capital expenditures are made to improve depreciating tools and put money into new belongings and tools for progress. Though capital expenditure is usually increased than depreciation and
amortization for rising firms, we’ll make the easy assumption that capital expenditure is the same as depreciation and amortization in an effort to forecast capital expenditures sooner or later.
Lastly, we have to forecast working capital investments. So as to develop the enterprise, we would wish a rising quantity of working capital on the steadiness sheet in an effort to obtain increased
revenues. This addition of capital to the steadiness sheet would lead to a damaging money circulation. For our mannequin we’ll assume that working capital must develop by 1% of income, due to this
fact our working capital funding forecast would merely be 1% multiplied by our forecasted revenues.
We will now get to free money circulation by including depreciation and amortization to after-tax revenue and subtracting capital expenditure and dealing capital funding.
With these projected free money flows, we are able to now proceed with the remainder of a reduced money circulation evaluation by calculating a terminal worth, a weighted common value of capital
after which calculating the online current worth to find out the enterprise worth for the corporate.
This text has been considered 480 instances.
Leave a Comment | {"url":"https://www.recentlyheard.com/dcf-analysis-forecasting-cash-flows/","timestamp":"2024-11-09T10:49:32Z","content_type":"text/html","content_length":"78772","record_id":"<urn:uuid:a24c7977-4a2e-4a06-b456-69a486c07f1b>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00036.warc.gz"} |
Disintegration versus divergence: The EMU Experience - Flossbach von Storch
Aglietta, M. and Brand, T. (2013), Un New Deal pour l’Europe, Éditions Odile Jacob.
Blanchard, O., Katz, L.F. (1992), Regional evolutions, Brooking Papers on Economic Activity, 23(1): 1-75.
Duarte, P. (2020) „More money will not safe the EMU“, Flossbach von Storch Research Institute, available at www.flossbachvonstorch-researchinstitute.com/en/comments/more-money-will-not-save-the-emu/.
Eichengreen, B. (1991), Is Europe an optimum currency area?, NBER Working Paper No. 3579
Frankel, J.A., Rose, A.K. (1997), Is EMU more justifiable ex post than ex ante?, European Economic Review, 41(3): 753-760,
Frankel, J. A., & Rose, A. K. (1998). The endogenity of the optimum currency area criteria. The Economic Journal, 108(449), 1009–1025.
Franks, J., Barkbu, B., Blavy, R., Oman, W., Schoelermann, H. (2018), Economic convergence in the euro area: Coming together or drifting apart?, IMF Working Paper No. 18/10
Gehringer, A., König, J., Ohr, R. (2020) „European (Monetary) Union: Until death do us apart“, Flossbach von Storch Research Institute, Macroeconomics 04/09/2020.
Glick, R., & Rose, A. K. (2016). Currency unions and trade: A post-emu reassessment. European Economic Review, 87, 78–91.
Krugman, P. (1993). Lessons of Massachusetts for EMU in Adjustment and Growth in the European Monetary Union. F. Torres and F. Giavazzi. Cambridge: Cambridge University Press and CEPR, 241-61. Mody,
A. (2018). EuroTragedy: A drama in nine acts. Oxford University Press.
Mundell, R. A. (1961). A theory of optimum currency areas. The American Economic Review, 51(4), 657–665.
Polák, P. (2019). The euro’s trade effect: A meta-analysis. Journal of Economic Surveys, 33(1), 101–124.
Thimann, C. (2005). Real convergence, economic dynamics, and the adoption of the euro in the new european union member states. In S. Schadler (Ed.), Euro adoption in central and eastern europe:
Opportunities and challenges (pp. 24–32). International Monetary Fund.
Thirlwall, A.P., Pacheco-López, P. (2017), Economics of Development, 10th ed., Palgrave Macmillan Publishers: London. | {"url":"https://www.flossbachvonstorch-researchinstitute.com/en/studies/disintegration-versus-divergence-the-emu-experience/","timestamp":"2024-11-02T20:24:26Z","content_type":"text/html","content_length":"60871","record_id":"<urn:uuid:8a839ca8-4b07-4fd0-9ba2-22d499b5129d>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00523.warc.gz"} |
Generic parallel functional programming: Scan and FFT
February 2017
Appeared at ICFP 2017
Parallel programming, whether imperative or functional, has long focused on arrays as the central data type. Meanwhile, typed functional programming has explored a variety of data types,
including lists and various forms of trees. Generic functional programming decomposes these data types into a small set of fundamental building blocks: sum, product, composition, and their
associated identities. Definitions over these few fundamental type constructions then automatically assemble into algorithms for an infinite variety of data types—some familiar and some new.
This paper presents generic functional formulations for two important and well-known classes of parallel algorithms: parallel scan (generalized prefix sum) and fast Fourier transform (FFT).
Notably, arrays play no role in these formulations. Consequent benefits include a simpler and more compositional style, much use of common algebraic patterns and freedom from possibility of
run-time indexing errors. The functional generic style also clearly reveals deep commonality among what otherwise appears to be quite different algorithms. Instantiating the generic formulations,
two well-known algorithms for each of parallel scan and FFT naturally emerge, as well as two possibly new algorithms.
author = {Conal Elliott},
title = {Generic functional parallel programming: Scan and {FFT}},
journal = {Proc. ACM Program. Lang.},
volume = {1},
number = {ICFP},
articleno = {48},
numpages = {24},
month = sep,
year = {2017},
url = {http://conal.net/papers/generic-parallel-functional},
doi = {http://dx.doi.org/10.1145/3110251}, | {"url":"http://conal.net/papers/generic-parallel-functional/","timestamp":"2024-11-13T16:03:14Z","content_type":"application/xhtml+xml","content_length":"3921","record_id":"<urn:uuid:1f024abe-0ab8-423d-8691-bdea261f5eb5>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00496.warc.gz"} |
Please Note: If you wish to receive credit points for this class, please let us know soon which module this would be for you. The physics "Pflichtseminar" requires a written summary of your talk.
A organizational meeting will be held on Monday, July 22, at 2pm, in SR 00.200 MATHEMATIKON.
First meeting on October 15.
Each session will be devoted to a fairly broad topic that should be split among two (or more) responsible speakers. Each team will be tutored by one of the organizers. The current distribution of
topics and speakers is here. There are still a few open slots, please contact us or come to the first meeting for further information.
Meeting time and place: Tuesday 2:30p.m., MATHEMATIKON SR 2
Dr. Ingmar Saberi, saberi@mathi.uni-heidelberg.de
Prof. Dr. J. Walcher, walcher@uni-heidelberg.de
Mirror symmetry is one of the oldest and best understood dualities that have emerged from string theory and has had a profound impact in certain areas of pure mathematics. The original statements,
motivated by Conformal Field Theory, concerned certain enumerative questions in algebraic geometry and their solution in terms of Hodge theory. Soon after, two mathematical formulations were put
forward: Kontsevich's Homological Mirror Symmetry inteprets the duality as an equivalence of symplectic and algebraic categories. The study of torus fibrations initiated by Strominger, Yau and Zaslow
gives an explicit geometric correspondence. Today, mirror symmetry remains an extremely active research field, reaching in influence far beyond its original formulation as a duality between
Calabi-Yau manifolds, to such subjects as representation theory, singularity theory, and knot theory. This seminar will trace mirror symmetry from its origins to some modern developments. One of our
main goals will be to understand Nick Sheridan's 2011 proof of homological mirror symmetry for the quintic threefold.
Prerequisites: The seminar is aimed at Masters students in mathematics and mathematical physics. Participants should have a solid background in at least one of symplectic geometry, algebraic geometry
or quantum field and string theory, with nodding acquaintance of the other two.
Schedule of Talks
Schedule still subject to change.
Date Speakers TitleAbstractNotes
October 15 Fabian, Benjamin T-duality
October 22 Levin, Sebastian Kähler geometry
October 29 Tristan, Aarya Mirror symmetry for \(T^2\)
November 5 Tobias, David 2d \(\mathcal N=(2,2)\) SuSy
November 12 Niklas Toric geometry
November 19 Lukas, Michael Hori-Vafa mirror symmetry
November 26 N.N. T.B.A.
December 3 Ingmar, Johannes D-branes
December 10 Fabio, N.N. Derived categories
December 17 Menelaos, N.N. Fukaya categories
January 7 Johanna, Steffen HMS for elliptic curve
January 14 Adnan, N.N. HMS for projective space
January 21 N.N., N.N. HMS for quintic | {"url":"https://mathi.uni-heidelberg.de/~walcher/teaching/wise1920/mirrorsymmetry/?lang=en","timestamp":"2024-11-14T23:41:14Z","content_type":"text/html","content_length":"13613","record_id":"<urn:uuid:07a33b31-c735-4834-859f-0816f7b745eb>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00838.warc.gz"} |
Investigate Linear Infeasibilities
This example shows how to investigate the linear constraints that cause a problem to be infeasible. For further details about these techniques, see Chinneck [1] and [2].
If linear constraints cause a problem to be infeasible, you might want to find a subset of the constraints that is infeasible, but removing any member of the subset makes the rest of the subset
feasible. The name for such a subset is Irreducible Infeasible Subset of Constraints, abbreviated IIS. Conversely, you might want to find a maximum cardinality subset of constraints that is feasible.
This subset is called a Maximum Feasible Subset, abbreviated MaxFS. The two concepts are related, but not identical. A problem can have many different IISs, some with different cardinality.
This example shows two ways of finding an IIS, and two ways of obtaining a feasible set of constraints. The example assumes that all given bounds are correct, meaning the lb and ub arguments have no
Infeasible Example
Create a random matrix A representing linear inequalities of size 150-by-15. Set the corresponding vector b to a vector with entries of 10, and change 5% of those values to –10.
N = 15;
rng default
A = randn([10*N,N]);
b = 10*ones(size(A,1),1);
Aeq = [];
beq = [];
b(rand(size(b)) <= 0.05) = -10;
f = ones(N,1);
lb = -f;
ub = f;
Check that problem is infeasible.
[x,fval,exitflag,output,lambda] = linprog(f,A,b,Aeq,beq,lb,ub);
No feasible solution found.
Linprog stopped because no point satisfies the constraints.
Deletion Filter
To identify an IIS, perform the following steps. Given a set of linear constraints numbered 1 through N, where all problem constraints are infeasible:
For each i from 1 to N:
• Temporarily remove constraint i from the problem.
• Test the resulting problem for feasibility.
• If the problem is feasible without constraint i, return constraint i to the problem.
• If the problem is not feasible without constraint i, do not return constraint i to the problem
Continue to the next i (up to value N).
At the end of this procedure, the constraints that remain in the problem form an IIS.
For MATLAB® code that implements this procedure, see the deletionfilter helper function at the end of this example.
Note: If you use the live script file for this example, the deletionfilter function is already included at the end of the file. Otherwise, you need to create this function at the end of your .m file
or add it as a file on the MATLAB path. The same is true for the other helper functions used later in this example.
See the effect of deletionfilter on the example data.
[ineqs,eqs,ncall] = deletionfilter(A,b,Aeq,beq,lb,ub);
The problem has no equality constraints. Find the indices for the inequality constraints and the value of b(iis).
Only one inequality constraint causes the problem to be infeasible, along with the bound constraints. The constraint is
A(iis,:)*x <= b(iis).
Why is this constraint infeasible together with the bounds? Find the sum of the absolute values of that row of A.
Due to the bounds, the x vector has values between –1 and 1, and so A(iis,:)*x cannot be less than b(iis) = –10.
How many linprog calls did deletionfilter perform?
The problem has 150 linear constraints, so the function called linprog 150 times.
Elastic Filter
As an alternative to the deletion filter, which examines every constraint, try the elastic filter. This filter works as follows.
First, allow each constraint i to be violated by a positive amount e(i), where equality constraints have both additive and subtractive positive elastic values.
$\begin{array}{l}{A}_{ineq}x\le {b}_{ineq}+e\\ {A}_{eq}x={b}_{eq}+{e}_{1}-{e}_{2}\end{array}$
Next, solve the associated linear programming problem (LP)
$\underset{x,e}{\mathrm{min}}\sum {e}_{i}$
subject to the listed constraints and with ${e}_{i}\ge 0$.
• If the associated LP has a solution, remove all constraints that have a strictly positive associated ${e}_{i}$, and record those constraints in a list of indices (potential IIS members). Return
to the previous step to solve the new, reduced associated LP.
• If the associated LP has no solution (is infeasible) or has no strictly positive associated ${e}_{i}$, exit the filter.
The elastic filter can exit in many fewer iterations than the deletion filter, because it can bring many indices at once into the IIS, and can halt without going through the entire list of indices.
However, the problem has more variables than the original problem, and its resulting list of indices can be larger than an IIS. To find an IIS after running an elastic filter, run the deletion filter
on the result.
For MATLAB® code that implements this filter, see the elasticfilter helper function at the end of this example.
See the effect of elasticfilter on the example data.
[ineqselastic,eqselastic,ncall] = ...
The problem has no equality constraints. Find the indices for the inequality constraints.
iiselastic = find(ineqselastic)
iiselastic = 5×1
The elastic IIS lists five constraints, whereas the deletion filter found only one. Run the deletion filter on the returned set to find a genuine IIS.
A_1 = A(ineqselastic > 0,:);
b_1 = b(ineqselastic > 0);
[dineq_iis,deq_iis,ncall2] = deletionfilter(A_1,b_1,Aeq,beq,lb,ub);
iiselasticdeletion = find(dineq_iis)
The fifth constraint in the elastic filter result, inequality 114, is the IIS. This result agrees with the answer from the deletion filter. The difference between the approaches is that the combined
elastic and deletion filter approach uses many fewer linprog calls. Display the total number of linprog calls used by the elastic filter followed by the deletion filter.
Remove IIS in a Loop
Generally, obtaining a single IIS does not enable you to find all the reasons that your optimization problem fails. To correct an infeasible problem, you can repeatedly find an IIS and remove it from
the problem until the problem becomes feasible.
The following code shows how to remove one IIS at a time from a problem until the problem becomes feasible. The code uses an indexing technique to keep track of constraints in terms of their
positions in the original problem, before the algorithm removes any constraints.
The code keeps track of the original variables in the problem by using a Boolean vector activeA to represent the current constraints (rows) of the A matrix, and a Boolean vector activeAeq to
represent the current constraints of the Aeq matrix. When adding or removing constraints, the code indexes into A or Aeq so that the row numbers do not change, even though the number of constraints
Running this code returns idx2, a vector of the indices of the nonzero elements in activeA:
idx2 = find(activeA)
Suppose that var is a Boolean vector that has the same length as idx2. Then
expresses var as indices into the original problem variables. In this way, the indexing can take a subset of a subset of constraints, work with only the smaller subset, and still unambiguously refer
to the original problem variables.
opts = optimoptions('linprog','Display',"none");
activeA = true(size(b));
activeAeq = true(size(beq));
[~,~,exitflag] = linprog(f,A,b,Aeq,beq,lb,ub,opts);
ncl = 1;
while exitflag < 0
[ineqselastic,eqselastic,ncall] = ...
ncl = ncl + ncall;
idxaa = find(activeA);
idxae = find(activeAeq);
tmpa = idxaa(find(ineqselastic));
tmpae = idxae(find(eqselastic));
AA = A(tmpa,:);
bb = b(tmpa);
AE = Aeq(tmpae,:);
be = beq(tmpae);
[ineqs,eqs,ncall] = ...
ncl = ncl + ncall;
activeA(tmpa(ineqs)) = false;
activeAeq(tmpae(eqs)) = false;
disp(['Removed inequalities ',int2str((tmpa(ineqs))'),' and equalities ',int2str((tmpae(eqs))')])
[~,~,exitflag] = ...
ncl = ncl + 1;
Removed inequalities 114 and equalities
Removed inequalities 97 and equalities
Removed inequalities 64 82 and equalities
Removed inequalities 60 and equalities
fprintf('Number of linprog calls: %d\n',ncl)
Number of linprog calls: 28
Notice that the loop removes inequalities 64 and 82 simultaneously, which indicates that these two constraints form an IIS.
Find MaxFS
Another approach for obtaining a feasible set of constraints is to find a MaxFS directly. As Chinneck [1] explains, finding a MaxFS is an NP-complete problem, meaning the problem does not necessarily
have efficient algorithms for finding a MaxFS. However, Chinneck proposes some algorithms that can work efficiently.
Use Chinneck's Algorithm 7.3 to find a cover set of constraints that, when removed, gives a feasible set. The algorithm is implemented in the generatecover helper function at the end of this example.
[coversetineq,coverseteq,nlp] = generatecover(A,b,Aeq,beq,lb,ub)
coversetineq = 5×1
Remove these constraints and solve the LP.
usemeineq = true(size(b));
usemeineq(coversetineq) = false; % Remove inequality constraints
usemeeq = true(size(beq));
usemeeq(coverseteq) = false; % Remove equality constraints
[xs,fvals,exitflags] = ...
Notice that the cover set is exactly the same as the iiselastic set from Elastic Filter. In general, the elastic filter finds too large a cover set. Chinneck's Algorithm 7.3 starts with the elastic
filter result and then retains only the constraints that are necessary.
Chinneck's Algorithm 7.3 takes 40 calls to linprog to complete the calculation of a MaxFS. This number is a bit more than 28 calls used earlier in the process of deleting IIS in a loop.
Also, notice that the inequalities removed in the loop are not exactly the same as the inequalities removed by Algorithm 7.3. The loop removes inequalities 114, 97, 82, 60, and 64, while Algorithm
7.3 removes inequalities 114, 97, 82, 60, and 2. Check that inequalities 82 and 64 form an IIS (as indicated in Remove IIS in a Loop), and that inequalities 82 and 2 also form an IIS.
usemeineq = false(size(b));
usemeineq([82,64]) = true;
ineqs = deletionfilter(A(usemeineq,:),b(usemeineq),Aeq,beq,lb,ub);
usemeineq = false(size(b));
usemeineq([82,2]) = true;
ineqs = deletionfilter(A(usemeineq,:),b(usemeineq),Aeq,beq,lb,ub);
[1] Chinneck, J. W. Feasibility and Infeasibility in Optimization: Algorithms and Computational Methods. Springer, 2008.
[2] Chinneck, J. W. "Feasibility and Infeasibility in Optimization." Tutorial for CP-AI-OR-07, Brussels, Belgium. Available at https://www.sce.carleton.ca/faculty/chinneck/docs/
Helper Functions
This code creates the deletionfilter helper function.
function [ineq_iis,eq_iis,ncalls] = deletionfilter(Aineq,bineq,Aeq,beq,lb,ub)
ncalls = 0;
[mi,n] = size(Aineq); % Number of variables and linear inequality constraints
f = zeros(1,n);
me = size(Aeq,1); % Number of linear equality constraints
opts = optimoptions("linprog","Algorithm","dual-simplex","Display","none");
ineq_iis = true(mi,1); % Start with all inequalities in the problem
eq_iis = true(me,1); % Start with all equalities in the problem
for i=1:mi
ineq_iis(i) = 0; % Remove inequality i
[~,~,exitflag] = linprog(f,Aineq(ineq_iis,:),bineq(ineq_iis),...
ncalls = ncalls + 1;
if exitflag == 1 % If now feasible
ineq_iis(i) = 1; % Return i to the problem
for i=1:me
eq_iis(i) = 0; % Remove equality i
[~,~,exitflag] = linprog(f,Aineq,bineq,...
ncalls = ncalls + 1;
if exitflag == 1 % If now feasible
eq_iis(i) = 1; % Return i to the problem
This code creates the elasticfilter helper function.
function [ineq_iis,eq_iis,ncalls,fval0] = elasticfilter(Aineq,bineq,Aeq,beq,lb,ub)
ncalls = 0;
[mi,n] = size(Aineq); % Number of variables and linear inequality constraints
me = size(Aeq,1);
Aineq_r = [Aineq -1.0*eye(mi) zeros(mi,2*me)];
Aeq_r = [Aeq zeros(me,mi) eye(me) -1.0*eye(me)]; % Two slacks for each equality constraint
lb_r = [lb(:); zeros(mi+2*me,1)];
ub_r = [ub(:); inf(mi+2*me,1)];
ineq_slack_offset = n;
eq_pos_slack_offset = n + mi;
eq_neg_slack_offset = n + mi + me;
f = [zeros(1,n) ones(1,mi+2*me)];
opts = optimoptions("linprog","Algorithm","dual-simplex","Display","none");
tol = 1e-10;
ineq_iis = false(mi,1);
eq_iis = false(me,1);
[x,fval,exitflag] = linprog(f,Aineq_r,bineq,Aeq_r,beq,lb_r,ub_r,opts);
fval0 = fval;
ncalls = ncalls + 1;
while exitflag == 1 && fval > tol % Feasible and some slacks are nonzero
c = 0;
for i = 1:mi
j = ineq_slack_offset+i;
if x(j) > tol
ub_r(j) = 0.0;
ineq_iis(i) = true;
c = c+1;
for i = 1:me
j = eq_pos_slack_offset+i;
if x(j) > tol
ub_r(j) = 0.0;
eq_iis(i) = true;
c = c+1;
for i = 1:me
j = eq_neg_slack_offset+i;
if x(j) > tol
ub_r(j) = 0.0;
eq_iis(i) = true;
c = c+1;
[x,fval,exitflag] = linprog(f,Aineq_r,bineq,Aeq_r,beq,lb_r,ub_r,opts);
if fval > 0
fval0 = fval;
ncalls = ncalls + 1;
This code creates the generatecover helper function. The code uses the same indexing technique for keeping track of constraints as the Remove IIS in a Loop code.
function [coversetineq,coverseteq,nlp] = generatecover(Aineq,bineq,Aeq,beq,lb,ub)
% Returns the cover set of linear inequalities, the cover set of linear
% equalities, and the total number of calls to linprog.
% Adapted from Chinneck [1] Algorithm 7.3. Step numbers are from this book.
coversetineq = [];
coverseteq = [];
activeA = true(size(bineq));
activeAeq = true(size(beq));
% Step 1 of Algorithm 7.3
[ineq_iis,eq_iis,ncalls] = elasticfilter(Aineq,bineq,Aeq,beq,lb,ub);
nlp = ncalls;
ninf = sum(ineq_iis(:)) + sum(eq_iis(:));
if ninf == 1
coversetineq = ineq_iis;
coverseteq = eq_iis;
holdsetineq = find(ineq_iis);
holdseteq = find(eq_iis);
candidateineq = holdsetineq;
candidateeq = holdseteq;
% Step 2 of Algorithm 7.3
while sum(candidateineq(:)) + sum(candidateeq(:)) > 0
minsinf = inf;
ineqflag = 0;
for i = 1:length(candidateineq(:))
activeA(candidateineq(i)) = false;
idx2 = find(activeA);
idx2eq = find(activeAeq);
[ineq_iis,eq_iis,ncalls,fval] = elasticfilter(Aineq(activeA,:),bineq(activeA),Aeq(activeAeq,:),beq(activeAeq),lb,ub);
nlp = nlp + ncalls;
ineq_iis = idx2(find(ineq_iis));
eq_iis = idx2eq(find(eq_iis));
if fval == 0
coversetineq = [coversetineq;candidateineq(i)];
if fval < minsinf
ineqflag = 1;
winner = candidateineq(i);
minsinf = fval;
holdsetineq = ineq_iis;
if numel(ineq_iis(:)) + numel(eq_iis(:)) == 1
nextwinner = ineq_iis;
nextwinner2 = eq_iis;
nextwinner = [nextwinnner,nextwinner2];
nextwinner = [];
activeA(candidateineq(i)) = true;
for i = 1:length(candidateeq(:))
activeAeq(candidateeq(i)) = false;
idx2 = find(activeA);
idx2eq = find(activeAeq);
[ineq_iis,eq_iis,ncalls,fval] = elasticfilter(Aineq(activeA,:),bineq(activeA),Aeq(activeAeq,:),beq(activeAeq),lb,ub);
nlp = nlp + ncalls;
ineq_iis = idx2(find(ineq_iis));
eq_iis = idx2eq(find(eq_iis));
if fval == 0
coverseteq = [coverseteq;candidateeq(i)];
if fval < minsinf
ineqflag = -1;
winner = candidateeq(i);
minsinf = fval;
holdseteq = eq_iis;
if numel(ineq_iis(:)) + numel(eq_iis(:)) == 1
nextwinner = ineq_iis;
nextwinner2 = eq_iis;
nextwinner = [nextwinnner,nextwinner2];
nextwinner = [];
activeAeq(candidateeq(i)) = true;
% Step 3 of Algorithm 7.3
if ineqflag == 1
coversetineq = [coversetineq;winner];
activeA(winner) = false;
if nextwinner
coversetineq = [coversetineq;nextwinner];
if ineqflag == -1
coverseteq = [coverseteq;winner];
activeAeq(winner) = false;
if nextwinner
coverseteq = [coverseteq;nextwinner];
candidateineq = holdsetineq;
candidateeq = holdseteq;
See Also
Related Topics | {"url":"https://kr.mathworks.com/help/optim/ug/investigate-linear-infeasibilities.html","timestamp":"2024-11-13T18:06:49Z","content_type":"text/html","content_length":"101479","record_id":"<urn:uuid:17eb95b0-96d7-4384-a5cb-850328c5e4fd>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00541.warc.gz"} |
CBSE Physics Class 12th Board Question Paper 2020 | SET -2
SECTION — A
(1 Mark Questions)
1. The wavelength and intensity of light emitted by a LED depend upon
a) forward bias and energy gap of the semiconductor
b) energy gap of the semiconductor and reverse bias
c) energy gap only
d) forward bias only
2. The graph showing the correct variation of linear momentum (p) of a charge particle with its de-Broglie wavelength (λ) is —
3. The selectivity of a series LCR a.c. circuit is large, when
a) L is large and R is large
b) L is small and R is small
c) L is large and R is small
d) L=R
4. Photo diodes are used to detect
a) Radio Waves
b) Gamma Rays
c) IR Rays
d) Optical Signals
5. The relationship between Brewster angle ‘θ’and the speed of light ‘v’ in the denser medium is —
a) c cos θ = v
b) v tan θ = c
c) c = v tan θ
d) v css θ = c
6. A biconvex lens of focal length f is cut into two identical plano-convex lenses. The focal length of each part will be
a) f
b) f/2
c) 2f
d) 4f
7. The phase difference between the current and voltage in series LCR circuit at resonance is
a) π
b) π/2
c) π/3
d) zero
8. Photons of frequency ν are incident on the surfaces of metals A & B of threshold frequencies 3/4 ν and 2/3 ν, respectively. The ratio of the maximum kinetic energy of electrons emitted from A to
that from B is
a) 2:3
b) 4:3
c) 3:4
d) 3:2
9. The electric flux through a closed Gaussian surface depends upon
a) Net charge enclosed and permittivity of the medium
b) Net charge enclosed, permittivity of the medium and the size of the Gaussian surface.
c) Net charge enclosed only
d) Permittivity of the medium only
10. A charge particle after being accelerated through a potential difference ‘V’ enters in a uniform magnetic field and moves in a circle of radius r. If V is doubled, the radius of the circle will
a) 2r
b) √2 r
c) 4r
d) r/√2
Note: Fill in the blanks with appropriate answer.
11. A point charge is placed at the centre of a hollow conducting sphere of internal radius ‘r’ and outer radius ‘2r’. The ratio of the surface charge density of the inner surface to that of the
outer surface will be ______________.
12. The _______________, a property of material C, Si and Ge depends upon the energy gap between their conduction and valence bands.
13. The ability of a junction diode to _____________ an alternating voltage, is based on the fact that it allows current to pass only when it is forward biased.
14. The physical quantity having SI unit NC^-1m is ______________.
15. A copper wire of non-uniform area of cross-section is connected to d.c. battery. The physical quantity which remains constant along the wire is ____________________.
Note : Answer the following :
16. Write the condition on path difference under which (i) constructive (ii) destructive interference occurs in Young’s double-slit experiment.
17. Plot a graph showing the variation of induced e.m.f. with the rate of change of current flowing through a coil.
A series combination of inductor (L), capacitor (C) and a resistor (R) is connected across an ac source of emf of peak value E₀ and angular frequency (ω). Plot a graph to show the variation of
impedance of the circuit with angular frequency (ω).
18. Depict the fields diagram of an electromagnetic wave propagating along positive X-axis with its electric field along Y-axis.
19. An electron moves along +x direction. It enters into a region of uniform magnetic field B directed along -z direction as shown in fig. Draw the shape of trajectory followed by the electron after
entering the field.
A square shaped current carrying loop MNOP is placed near a straight long current carrying wire AB as shown in fig. The wire and the loop lie in the same plane. If the loop experiences a net force F
towards the wire, find the magnitude of the force on the side ‘NO’ of the loop.
20. Define the term “current sensitivity” of a moving coil galvanometer.
SECTION — B
(2 Mark Questions)
21. a) Define one Becquerel.
b) A radioactive substance disintegrates into two types of daughter nuclei, one type with disintegration constant λ[1] and the other type with disintegration constant λ[2]. Determine the half-life of
the radioactive substance.
22. In a single slit diffraction experiment, the width of the slit is increased. How will the (i) size and (ii) intensity of central bright band be affected? Justify your answer.
23. In case of photo electric effect experiment, explain the following facts, giving reasons.
(a) The wave theory of light could not explain the existence of the threshold frequency.
(b) The photo electric current increases with increase of intensity of incident light.
24. Gamma rays and radio waves travel with the same velocity in free space. Distinguish between them in terms of their origin and the main application.
25. Use Bohr’s model of hydrogen atom to obtain the relationship between the angular momentum and the magnetic moment of the revolving electron.
26. Light from sodium lamp (S) passes through two polaroid sheets P[1] and P[2] as shown in fig. What will be the effect on the intensity of light transmitted (i) by P[1] and (ii) by P[2] on rotating
polaroid P[1] about the direction of propagation of light? Justify your answer in both cases.
Define the term ‘wave front of light’. A plane wave front AB propagating from denser medium (1) into a rarer medium (2) is incident on the surface P[1]P[2] separating the two media as shown in fig.
Using Huygen’s principle, draw the secondary wavelets and obtain the refracted wave front in the diagram.
27. Derive the expression for the torque acting on an electric dipole, when it is held in a uniform electric field. Identify the orientation of the dipole in the electric field, in which it attains a
stable equilibrium.
Obtain the expression for the energy stored in a capacitor connected across a dc battery. Hence define the energy density of the capacitor.
SECTION — C
(3 Mark Questions)
28. (a) Two point charges q[1 ]and q[2] are kept at a distance r[12] in air. Deduce the expression for the electrostatic potential energy of this system.
(b) If an external electric field (E) is applied on the system, write the expression for the total energy of this system.
29. What is solar cell? Draw its V-I characteristics. Explain the three processes involved in its working.
Draw the circuit diagram of a full wave rectifier. Explain its working showing its input and output waveforms.
30. (a) Define internal resistance of a cell.
(b) A cell of emf E and internal resistance r is connected across a variable resistor R. Plot the shape of graphs showing the variation of terminal voltage V with (i) R and (ii) circuit current I.
31. Calculate the de-Broglie wavelength associated with the electron revolving in the first excited state of hydrogen atom. The ground state energy level of the hydrogen atom is – 13.6 eV.
32. When a conducting loop of resistance 10 Ω and area 10 cm^2 is removed from an external magnetic field acting normally, the variation of induced current in the loop with time is shown in the
Find the
(i) total charge passed through the loop.
(ii) change in magnetic flux through the loop.
(iii) magnitude of the magnetic field applied.
33. Draw the curve showing the variation of binding energy per nucleon with the mass number of nuclei. Using it explain the fusion of nuclei lying on ascending part and fission of nuclei lying on
descending part of this curve.
34. An optical instrument uses a lens of power 100 D for objective lens and 50 D for its eyepiece. When the tube length is kept at 20 cm. The final image is formed at infinity.
a) Identify the optical instrument.
b) Calculate the magnification produced by the instrument.
SECTION — D
(5 Mark Questions)
35. (a) Write two important characteristics of equipotential surfaces.
(b) A thin circular ring of radius r is charged uniformly so that its linear charge density becomes λ. Derive an expression for the electric field at a point P at a distance x from it along the axis
of the ring. Hence, prove that at large distance (x>>r), the ring behaves as a point charge.
(a) State Gauss’s law on electrostatics and derive an expression for the electric field due to a long straight thin uniformly charged wire (linear charge density λ) at a point lying at a distance r
from the wire.
(b) The magnitude of electric field (in NC-1) in a region varies with the distance r (in m) as E = 10r + 5
By how much does the electric potential increases in moving from point at r= 1 m to a point at r = 10m.
36. (a) Define the term ‘focal length of a mirror’. With the help of ray diagram, obtain the relation between its focal length and radius of curvature.
(b) Calculate the angle of emergence (e) of the ray of light incident normally on the face AC of a glass prism ABC of refractive index √3. How will the angle of emergence change qualitatively, if the
ray of light emerges from the prism into a liquid of refractive index 1.3 instead of air?
(a) Define the term ‘resolving power of a telescope’. How will the resolving power be affected with the increase in
(i) Wavelength of light used.
(ii) Diameter of the objective lens.
Justify your answers.
(b) A screen is placed 80 cm from an object. The image of the object on the screen is formed by a convex lens placed between them at two different locations separated by a distance of 20 cm.
Determine the focal length of the lens.
37. (a) Show that an ideal inductor does not dissipate power in an ac circuit.
(b) The variation of inductive reactance (XL) of an inductor with the frequency (f) of the ac source of 100 V and variable frequency is shown in fig.
(i) Calculate the self-inductance of the inductor.
(ii) When this inductor is used in series with a capacitor of unknown value and a resistor of 10 Ω at 300 s^-1, maximum power dissipation occurs in the circuit. Calculate the capacitance of the
(a) A conductor of length ‘l’ is rotated about one of its ends at a constant angular speed ‘ω’ in a plane perpendicular to a uniform magnetic field B. Plot graphs to show variation of the emf induced
across the ends of the conductor with (i) angular speed ω and (ii) length of the conductor l.
(b) Two concentric circular loops of radius 1 cm and 20 cm are placed coaxially.
(i) Find mutual inductance of the arrangement.
(ii) If the current passed through the outer loop is changed at a rate of 5 A/ms, find the emf induced in the inner loop. Assume the magnetic field on the inner loop to be uniform. | {"url":"https://letxy.com/cbse-physics-class-12th-board-question-paper-2020-2/","timestamp":"2024-11-05T10:52:04Z","content_type":"text/html","content_length":"63096","record_id":"<urn:uuid:32043b89-9c90-451e-b152-c41f719ab68d>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00302.warc.gz"} |
Unique & FUN Halloween Math Game to Review Any Math Skill | Math Geek Mama
Unique & FUN Halloween Math Game to Review Any Math Skill
Looking for an easy and yet super engaging way to review math skills before a test? This Halloween math game can be easily set up and adapted for any math skill, but the fun twist makes it engaging
for the whole class!
Are you tired of using the same old review games before math tests? Are your students sick and tired of study guides? Do you want to add a little fun and excitement to your test prep days? Well,
you’re in luck! In this blog post, I’m going to introduce you to “Ghosts in the Graveyard,” a game that can transform any boring test prep day into an interactive, collaborative math experience.
While it’s perfect for Halloween, I’ve created templates for every holiday, so you can use them year-round!
Test review days don’t have to be boring!
*This is a guest post from Asia at The Sassy Math Teacher.
A Halloween Math Game for All Ages:
I first discovered this game during my student teaching days, and since then, I’ve made it even easier to play with a digital template. But don’t worry; you can still play it the original way if you
What I love about this game is that it’s suitable for any math classroom that knows how to add and subtract, and I guarantee your students will LOVE IT!
An Overview of Ghosts in the Graveyard:
Students work in teams to answer a series of challenges.
Correct answers earn them ghosts, which they can place on tombstones.
Materials Needed:
1. 16-24 Review Questions (A study guide or worksheet you already have would be excellent!)
2. Choose one of the following “Ghosts in the Graveyard” game boards:
– Option A: 64 Miniature Ghosts Printed on 8 Different Colors of Paper, then Laminated and Cut Out (8 ghosts of each color; each team will be assigned a color).
– Option B: Ghosts In The Graveyard Digital Template (I’ll share more about this later in the blog post!)
Materials Setup:
1. Question Preparation: Transform the dull study guide into “Challenges.” Ideally, have 2-3 questions on each challenge. For example, questions #1 and #2 become Challenge #1, questions #3, #4, and
#6 become Challenge #2, and so on. You’ll need a total of 8 challenges, each on separate pieces of paper. Make 8 copies of each challenge.
2. Create a Graveyard: Draw 3 tombstones on black construction paper large enough to accommodate several ghosts. Set up a designated area in your classroom or learning space as the “graveyard” by
displaying the 3 tombstones horizontally.
3. Ghost Game Pieces: Prepare ghost cards – you’ll need enough for each team to have a total of 8, one for each challenge. Teams only receive a ghost AFTER they’ve correctly answered a challenge.
How to Play “Ghosts in the Graveyard”:
1. Split students into teams of 4.
2. Provide each student with a recording sheet to keep track of their answers.
3. Place the challenges within easy reach for students.
4. Once the game begins, teams work on completing one challenge at a time.
5. When they finish, one student from the team brings the group’s recording sheet to you to have their answers checked.
6. If all the answers are correct, give them one of their team’s ghosts and let them choose which tombstone to place it on. They can even put all their ghosts on the same tombstone if they prefer.
7. If answers are incorrect, provide a hint and have them return to their team to fix it.
8. Repeat until about 5 minutes before class ends.
How to Determine the Winner:
Each tombstone is worth a mysterious point value, which you pre-determine and keep secret until 5 minutes before class ends.
Once revealed, each ghost becomes worth a certain point value.
For instance, if Team #4 has 2 ghosts on Tombstone #1, and you reveal that Tombstone #1 is worth 3 points, then they have a total of 6 points.
The team with the most points wins!
“I like that you don’t have to be the smartest group to win!” – 7th Grader
Tips for a Successful Halloween Math Game:
• The line to get challenges checked may get long. Consider playing on a day when you have an extra teacher in the room to help.
• Group students with at least one friend in each group.
Ghosts in the Graveyard Digital Template for Busy Teachers:
If you’re short on prep time, you can skip the laminating and cutting by using the Ghosts in the Graveyard Digital template.
The digital template includes a digital graveyard in Google Slides, challenge templates, and recording sheets. All you need to do is add your questions, print, and make copies.
Find out more about the digital template set for Ghosts in the Graveyard here.
More Ways to Play “Ghosts in the Graveyard”:
• Play it all year long with other holiday themes, such as “Turkeys in the Oven” for Thanksgiving.
• Make one of the tombstones worth a negative point value for an extra twist.
Transforming a test prep day with a game like “Ghosts in the Graveyard” can make a world of difference in your classroom.
Give it a try, and watch your classroom engagement soar!
Are you ready to turn your next worksheet into a spine-tingling adventure?
Grab your FREE Ghosts in The Graveyard Bonus Kit here.
Hey! My name is Asia and I am the face behind thesassymathteacher.com. Bethany invited me to write this guest post today because she loves finding ways to engage students in math class just as much
as I do. Find more fun math resources on my website!
More Not-So-Scary Halloween Math Ideas: | {"url":"https://mathgeekmama.com/halloween-math-game/","timestamp":"2024-11-14T13:57:48Z","content_type":"text/html","content_length":"185604","record_id":"<urn:uuid:eb716216-6dad-4667-bacd-6c2bb5e80ef2>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00008.warc.gz"} |
A Generative Model for Punctuation in Dependency Trees
Treebanks traditionally treat punctuation marks as ordinary words, but linguists have suggested that a tree’s “true” punctuation marks are not observed (Nunberg, 1990). These latent “underlying”
marks serve to delimit or separate constituents in the syntax tree. When the tree’s yield is rendered as a written sentence, a string rewriting mechanism transduces the underlying marks into
“surface” marks, which are part of the observed (surface) string but should not be regarded as part of the tree. We formalize this idea in a generative model of punctuation that admits efficient
dynamic programming. We train it without observing the underlying marks, by locally maximizing the incomplete data likelihood (similarly to the EM algorithm). When we use the trained model to
reconstruct the tree’s underlying punctuation, the results appear plausible across 5 languages, and in particular are consistent with Nunberg’s analysis of English. We show that our generative model
can be used to beat baselines on punctuation restoration. Also, our reconstruction of a sentence’s underlying punctuation lets us appropriately render the surface punctuation (via our trained
underlying-to-surface mechanism) when we syntactically transform the sentence.
Punctuation enriches the expressiveness of written language. When converting from spoken to written language, punctuation indicates pauses or pitches; expresses propositional attitude; and is
conventionally associated with certain syntactic constructions such as apposition, parenthesis, quotation, and conjunction.
In this paper, we present a latent-variable model of punctuation usage, inspired by the rule-based approach to English punctuation of Nunberg (1990). Training our model on English data learns rules
that are consistent with Nunberg’s hand-crafted rules. Our system is automatic, so we use it to obtain rules for Arabic, Chinese, Spanish, and Hindi as well.
Moreover, our rules are stochastic, which allows us to reason probabilistically about ambiguous or missing punctuation. Across the 5 languages, our model predicts surface punctuation better than
baselines, as measured both by perplexity (§4) and by accuracy on a punctuation restoration task (§6.1). We also use our model to correct the punctuation of non-native writers of English (§6.2), and
to maintain natural punctuation style when syntactically transforming English sentences (§6.3). In principle, our model could also be used within a generative parser, allowing the parser to evaluate
whether a candidate tree truly explains the punctuation observed in the input sentence (§8).
Punctuation is interesting
In The Linguistics of Punctuation, Nunberg (1990) argues that punctuation (in English) is more than a visual counterpart of spoken-language prosody, but forms a linguistic system that involves
“interactions of point indicators (i.e. commas, semicolons, colons, periods and dashes).” He proposes that much as in phonology (Chomsky and Halle, 1968), a grammar generates underlying punctuation
which then transforms into the observed surface punctuation.
Consider generating a sentence from a syntactic grammar as follows:
Hail the king [, Arthur Pendragon ,][, who wields [ “ Excalibur ” ] ,] .
Although the full tree is not depicted here, some of the constituents are indicated with brackets. In this underlying generated tree, each appositive NP is surrounded by commas. On the surface,
however, the two adjacent commas after Pendragon will now be collapsed into one, and the final comma will be absorbed into the adjacent period. Furthermore, in American English, the typographic
convention is to move the final punctuation inside the quotation marks. Thus a reader sees only this modified surface form of the sentence:
Hail the king, Arthur Pendragon,who wields“Excalibur.”
Note that these modifications are string transformations that do not see or change the tree. The resulting surface punctuation marks may be clues to the parse tree, but (contrary to NLP convention)
they should not be included as nodes in the parse tree. Only the underlying marks play that role.
Punctuation is meaningful
Pang et al. (2002) use question and exclamation marks as clues to sentiment. Similarly, quotation marks may be used to mark titles, quotations, reported speech, or dubious terminology (University of
Chicago, 2010). Because of examples like this, methods for determining the similarity or meaning of syntax trees, such as a tree kernel (Agarwal et al., 2011) or a recursive neural network (Tai et
al., 2015), should ideally be able to consider where the underlying punctuation marks attach.
Punctuation is helpful
Surface punctuation remains correlated with syntactic phrase structure. NLP systems for generating or editing text must be able to deploy surface punctuation as human writers do. Parsers and grammar
induction systems benefit from the presence of surface punctuation marks (Jones, 1994; Spitkovsky et al., 2011). It is plausible that they could do better with a linguistically informed model that
explains exactly why the surface punctuation appears where it does. Patterns of punctuation usage can also help identify the writer’s native language (Markov et al., 2018).
Punctuation is neglected
Work on syntax and parsing tends to treat punctuation as an afterthought rather than a phenomenon governed by its own linguistic principles. Treebank annotation guidelines for punctuation tend to
adopt simple heuristics like “attach to the highest possible node that preserves projectivity” (Bies et al., 1995; Nivre et al., 2018).^^1 Many dependency parsing works exclude punctuation from
evaluation (Nivre et al., 2007b; Koo and Collins, 2010; Chen and Manning, 2014; Lei et al., 2014; Kiperwasser and Goldberg, 2016), although some others retain punctuation (Nivre et al., 2007a;
Goldberg and Elhadad, 2010; Dozat and Manning, 2017).
In tasks such as word embedding induction (Mikolov et al., 2013; Pennington et al., 2014) and machine translation (Zens et al., 2002), punctuation marks are usually either removed or treated as
ordinary words (Řehůřek and Sojka, 2010).
Yet to us, building a parse tree on a surface sentence seems as inappropriate as morphologically segmenting a surface word. In both cases, one should instead analyze the latent underlying form,
jointly with recovering that form. For example, the proper segmentation of English hoping is not hop-ing but hope-ing (with underlying e), and the proper segmentation of stopping is neither stopp-ing
nor stop-ping but stop-ing (with only one underlying p). Cotterell et al. (2015); Cotterell et al. (2016) get this right for morphology. We attempt to do the same for punctuation.
2Formal Model
We propose a probabilistic generative model of sentences (
Figure 1
First, an
dependency tree
is stochastically generated by some recursive process
(e.g., Eisner,
, Model C).
Second, each constituent (i.e., dependency subtree) sprouts optional underlying punctuation at its left and right edges, according to a probability distribution
that depends on the constituent’s syntactic role (e.g.,
for “direct object”). This
yields the underlying string
′) , which is edited by a finite-state noisy channel
to arrive at the surface sentence
This third step may alter the sequence of punctuation tokens at each slot between words—for example, in §1, collapsing the double comma , , between Pendragon and who. u and x denote just the
punctuation at the slots of ū and $x¯$ respectively, with u[i] and x[i] denoting the punctuation token sequences at the i^th slot. Thus, the transformation at the i^th slot is u[i]↦x[i].
Since this model is generative, we could train it without any supervision to explain the observed surface string $x¯$: maximize the likelihood $p(x¯)$ in (1), marginalizing out the possible T, T′
In the present paper, however, we exploit known
values (as observed in the “depunctuated” version of a treebank). Because
is observed, we can jointly train
to maximize just
That is, the
model that generated
becomes irrelevant, but we still try to predict what surface punctuation will be added to
. We still marginalize over the underlying punctuation marks u. These are
never observed
, but they must explain the surface punctuation marks x (§
), and they must be explained in turn by the syntax tree
). The trained generative model then lets us restore or correct punctuation in new trees
2.1Generating Underlying Punctuation
The A
model characterizes the probability of an underlying punctuated tree
given its corresponding unpunctuated tree
, which is given by
are the left and right
attaches to the tree node
. Each puncteme (Krahn,
) in the finite set
is a string of 0 or more underlying punctuation
The probability
) is given by a log-linear model
$pθ(l,r|w)∝expθ⊤f(l,r,w)if (l,r)∈Wd(w)0otherwise$
is the finite set of possible punctemes and
gives the possible puncteme pairs for a node
that has dependency relation
) to its parent.
are estimated heuristically from the tokenized surface data (§
is a sparse binary feature vector, and θ is the corresponding parameter vector of feature weights. The feature templates in Appendix A
consider the symmetry between
, and their compatibility with (a) the POS tag of
’s head word, (b) the dependency paths connecting
to its children and the root of
, (c) the POS tags of the words flanking the slots containing
, (d) surface punctuation already added to
’s subconstituents.
2.2From Underlying to Surface
From the tree T′, we can read off the sequence of underlying punctuation tokens u[i] at each slot i between words. Namely, u[i] concatenates the right punctemes of all constituents ending at i with
the left punctemes of all constituents starting at i (as illustrated by the examples in §1 and Figure 1). The NoisyChannel model then transduces u[i] to a surface token sequence x[i], for each i = 0,
…, n independently (where n is the sentence length).
Nunberg’s formalism
Much like Halle’s (1968) phonological grammar of English, Nunberg’s 1990 descriptive English punctuation grammar (Table 1) can be viewed computationally as a priority string rewriting system, or
Markov algorithm (Markov, 1960; Caracciolo di Forino, 1968). The system begins with a token string u. At each step it selects the highest-priority local rewrite rule that can apply, and applies it as
far left as possible. When no more rules can apply, the final state of the string is returned as x.
Table 1:
1. Point Absorption,,↦,,.↦.-,↦--;↦;;.↦. 3. Period Absorption.?↦?.!↦!abbv.↦abbv
2. Quote Transposition",↦,"".↦." 4. Bracket Absorptions,)↦)-)↦) (,↦(,"↦"“,↦“
1. Point Absorption,,↦,,.↦.-,↦--;↦;;.↦. 3. Period Absorption.?↦?.!↦!abbv.↦abbv
2. Quote Transposition",↦,"".↦." 4. Bracket Absorptions,)↦)-)↦) (,↦(,"↦"“,↦“
Simplifying the formalism
Markov algorithms are Turing complete. Fortunately, Johnson (1972) noted that in practice, phonological u ↦ x maps described in this formalism can usually be implemented with finite-state transducers
For computational simplicity, we will formulate our punctuation model as a probabilistic FST (PFST)—a locally normalized left-to-right rewrite model (Cotterell et al., 2014). The probabilities for
each language must be learned, using gradient descent. Normally we expect most probabilities to be near 0 or 1, making the PFST nearly deterministic (i.e., close to a subsequential FST). However,
permitting low-probability choices remains useful to account for typographical errors, dialectal differences, and free variation in the training corpus.
Our PFST generates a surface string, but the invertibility of FSTs will allow us to work backwards when analyzing a surface string (§3).
A sliding-window model
Instead of having rule priorities, we apply Nunberg-style rules within a 2-token window that slides over u in a single left-to-right pass (Figure 2). Conditioned on the current window contents ab, a
single edit is selected stochastically: either ab ↦ ab (no change), ab ↦ b (left absorption), ab ↦ a (right absorption), or ab ↦ ba (transposition). Then the window slides rightward to cover the next
input token, together with the token that is (now) to its left. a and b are always real tokens, never boundary symbols. ϕ specifies the conditional edit probabilities.^^5
These specific edit rules (like Nunberg’s) cannot insert new symbols, nor can they delete all of the underlying symbols. Thus, surface x[i] is a good clue to u[i]: all of its tokens must appear
underlyingly, and if x[i] = ϵ (the empty string) then u[i] = ϵ.
The model can be directly implemented as a PFST (Appendix D^4) using Cotterell et al.’s (2014) more general PFST construction.
Our single-pass formalism is less expressive than Nunberg’s. It greedily makes decisions based on at most one token of right context (“label bias”). It cannot rewrite ’”. ↦ .’” or “,. ↦ .’” because
the . is encountered too late to percolate leftward; luckily, though, we can handle such English examples by sliding the window right-to-left instead of left-to-right. We treat the sliding direction
as a language-specific parameter.^^6
2.3Training Objective
Building on (2), we train
to locally maximize the regularized conditional log-likelihood
where the sum is over a training treebank.
The expectation $E[⋯]$ is over $T′∼p(⋅∣T,x)$. This generalized expectation term provides posterior regularization (Mann and McCallum, 2010; Ganchev et al., 2010), by encouraging parameters that
reconstruct trees T′ that use symmetric punctuation marks in a “typical” way. The function c(T′) counts the nodes in T′ whose punctemes contain “unmatched” symmetric punctuation tokens: for example,
) is “matched” only when it appears in a right puncteme with ( at the comparable position in the same constituent’s left puncteme. The precise definition is given in Appendix B.^4
In our development experiments on English, the posterior regularization term was necessary to discover an aesthetically appealing theory of underlying punctuation. When we dropped this term (ξ = 0)
and simply maximized the ordinary regularized likelihood, we found that the optimization problem was underconstrained: different training runs would arrive at different, rather arbitrary underlying
punctemes. For example, one training run learned an Attach model that used underlying " to terminate sentences, along with a NoisyChannel model that absorbed the left quotation mark into the period.
By encouraging the underlying punctuation to be symmetric, we broke the ties. We also tried making this a hard constraint (ξ = ∞), but then the model was unable to explain some of the training
sentences at all, giving them probability of 0. For example, I went to the“ special place ” cannot be explained, because special place is not a constituent.^^8
In principle, working with the model (1) is straightforward, thanks to the closure properties of formal languages. Provided that p[syn] can be encoded as a weighted CFG, it can be composed with the
weighted tree transducer p[θ] and the weighted FST p[ϕ] to yield a new weighted CFG (similarly to Bar-Hillel et al., 1961; Nederhof and Satta, 2003). Under this new grammar, one can recover the
optimal T,T′ for $x¯$ by dynamic programming, or sum over T,T′ by the inside algorithm to get the likelihood $p(x¯)$. A similar approach was used by Levy (2008) with a different FST noisy channel.
In this paper we assume that T is observed, allowing us to work with (2). This cuts the computation time from O(n^3) to O(n).^^9 Whereas the inside algorithm for (1) must consider O(n^2) possible
constituents of $x¯$ and O(n) ways of building each, our algorithm for (2) only needs to iterate over the O(n) true constituents of T and the 1 true way of building each. However, it must still
consider the $|Wd|$ puncteme pairs for each constituent.
Given an input sentence $x¯$ of length n, our job is to sum over possible trees T′ that are consistent with T and $x¯$, or to find the best such T′. This is roughly a lattice parsing problem—made
easier by knowing T. However, the possible ū values are characterized not by a lattice but by a cyclic WFSA (as |u[i]| is unbounded whenever |x[i]| > 0).
For each slot 0 ≤ i ≤ n, transduce the surface punctuation string x[i] by the inverted PFST for p[ϕ] to obtain a weighted finite-state automaton (WFSA) that describes all possible underlying strings
u[i].^^10 This WFSA accepts each possible u[i] with weight p[ϕ](x[i]∣u[i]). If it has N[i] states, we can represent it (Berstel and Reutenauer, 1988) with a family of sparse weight matrices $Mi(υ)
∈RNi×Ni$, whose element at row s and column t is the weight of the s → t arc labeled with υ, or 0 if there is no such arc. Additional vectors $λi,ρi∈RNi$ specify the initial and final weights. (λ[i]
is one-hot if the PFST has a single initial state, of weight 1.)
For any puncteme l (or r) in $V$, we define M[i](l) = M[i](l[1])M[i](l[2])⋯M[i](l[|l|]), a product over the 0 or more tokens in l. This gives the total weight of all s →^*t WFSA paths labeled with l
The subprocedure in Algorithm 1 essentially extends this to obtain a new matrix $In(w)∈RNi×Nk$, where the subtree rooted at w stretches from slot i to slot k. Its element In(w)[st] gives the total
weight of all extended paths in the ū WFSA from state s at slot i to state t at slot k. An extended path is defined by a choice of underlying punctemes at w and all its descendants. These punctemes
determine an s-to-final path at i, then initial-to-final paths at i + 1 through k − 1, then an initial-to-t path at k. The weight of the extended path is the product of all the WFSA weights on these
paths (which correspond to transition probabilities in p[ϕ] PFST) times the probability of the choice of punctemes (from p[θ]).
This inside algorithm computes quantities needed for training (§2.3). Useful variants arise via well-known methods for weighted derivation forests (Berstel and Reutenauer, 1988; Goodman, 1999; Li and
Eisner, 2009; Eisner, 2016).
Specifically, to modify Algorithm 1 to maximize over T′ values (§§6.2–6.3) instead of summing over them, we switch to the derivation semiring (Goodman, 1999), as follows. Whereas In(w)[st] used to
store the total weight of all extended paths from state s at slot i to state t at slot j, now it will store the weight of the best such extended path. It will also store that extended path’s choice
of underlying punctemes, in the form of a puncteme-annotated version of the subtree of T that is rooted at w. This is a potential subtree of T′.
Thus, each element of I
) has the form (
) where
∈ ℝ and
is a tree. We define addition and multiplication over such pairs:
$(r,D)+(r′,D′)=(r,D)if r>r′(r′,D′)otherwise$
denotes an ordered combination of two trees. Matrix products
and scalar-matrix products
are defined in terms of element addition and multiplication as usual:
What is DD′? For presentational purposes, it is convenient to represent a punctuated dependency tree as a bracketed string. For example, the underlying tree T′ in Figure 1 would be [ [“ Dale ”] means
[“ [ river ] valley ”] ] where the words correspond to nodes of T. In this case, we can represent every D as a partial bracketed string and define DD′ by string concatenation. This presentation
ensures that multiplication (7) is a complete and associative (though not commutative) operation, as in any semiring. As base cases, each real-valued element of M[i](l) or M[k](r) is now paired with
the string [l or r] respectively, ^^11and the real number 1 at line 10 is paired with the string w. The real-valued elements of the λ[i] and ρ[i] vectors and the 0 matrix at line 11 are paired with
the empty string ϵ, as is the real number p at line 13.
In practice, the D strings that appear within the matrix M of Algorithm 1 will always represent complete punctuated trees. Thus, they can actually be represented in memory as such, and different
trees may share subtrees for efficiency (using pointers). The product in line 10 constructs a matrix of trees with root w and differing sequences of left/right children, while the product in line 14
annotates those trees with punctemes l,r.
To sample a possible
from the derivation forest in proportion to its probability (§
), we use the same algorithm but replace (
) with
$(r,D)+(r′,D′)=(r+r′,D)if u<rr+r′(r+r′,D′)otherwise$
∼Uniform(0,1) being a random number.
Having computed the objective (5), we find the gradient via automatic differentiation, and optimize $θ,ϕ$ via Adam (Kingma and Ba, 2014)—a variant of stochastic gradient decent—with learning rate
0.07, batchsize 5, sentence per epoch 400, and L2 regularization. (These hyperparameters, along with the regularization coefficients ς and ξ from (5), were tuned on dev data (§4) for each language
respectively.) We train the punctuation model for 30 epochs. The initial NoisyChannel parameters (ϕ) are drawn from $N(0,1)$, and the initial Attach parameters (θ) are drawn from $N(0,1)$ (with one
minor exception described in Appendix A).
4Intrinsic Evaluation of the Model
Throughout §§4–6, we will examine the punctuation model on a subset of the Universal Dependencies (UD) version 1.4 (Nivre et al., 2016)—a collection of dependency treebanks across 47 languages with
unified POS-tag and dependency label sets. Each treebank has designated training, development, and test portions. We experiment on Arabic, English, Chinese, Hindi, and Spanish (Table 2)—languages
with diverse punctuation vocabularies and punctuation interaction rules, not to mention script directionality. For each treebank, we use the tokenization provided by UD, and take the punctuation
tokens (which may be multi-character, such as …) to be the tokens with the PUNCT tag. We replace each straight double quotation mark " with either “ or ” as appropriate, and similarly for single
quotation marks.^^12 We split each non-punctuation token that ends in . (such as etc.) into a shorter non-punctuation token (etc) followed by a special punctuation token called the “abbreviation dot”
(which is distinct from a period). We prepend a special punctuation mark ∘ to every sentence $x¯$, which can serve to absorb an initial comma, for example.^^13 We then replace each token with the
special symbol UNK if its type appeared fewer than 5 times in the training portion. This gives the surface sentences.
Table 2:
Language . Treebank . #Token . %Punct . #Omit . #Type .
Arabic 𝖺𝗋 282K 7.9 255 18
Chinese 𝗓𝗁 123K 13.8 3 23
English 𝖾𝗇𝖾𝗇_𝖾𝗌𝗅 255K97.7K 11.79.8 402 3516
Hindi 𝗁𝗂 352K 6.7 21 15
Spanish 𝖾𝗌_𝖺𝗇𝖼𝗈𝗋𝖺 560K 11.7 25 16
Language . Treebank . #Token . %Punct . #Omit . #Type .
Arabic 𝖺𝗋 282K 7.9 255 18
Chinese 𝗓𝗁 123K 13.8 3 23
English 𝖾𝗇𝖾𝗇_𝖾𝗌𝗅 255K97.7K 11.79.8 402 3516
Hindi 𝗁𝗂 352K 6.7 21 15
Spanish 𝖾𝗌_𝖺𝗇𝖼𝗈𝗋𝖺 560K 11.7 25 16
To estimate the vocabulary $V$ of underlying punctemes, we simply collect all surface token sequences x[i] that appear at any slot in the training portion of the processed treebank. This is a
generous estimate. Similarly, we estimate $Wd$ (§2.1) as all pairs $(l,r)∈V2$ that flank any d constituent.
Recall that our model generates surface punctuation given an unpunctuated dependency tree. We train it on each of the 5 languages independently. We evaluate on conditional perplexity, which will be
low if the trained model successfully assigns a high probability to the actual surface punctuation in a held-out corpus of the same language.
We compare our model against three baselines to show that its complexity is necessary. Our first baseline is an ablation study that does not use latent underlying punctuation, but generates the
surface punctuation directly from the tree. (To implement this, we fix the parameters of the noisy channel so that the surface punctuation equals the underlying with probability 1.) If our full model
performs significantly better, it will demonstrate the importance of a distinct underlying layer.
Our other two baselines ignore the tree structure, so if our full model performs significantly better, it will demonstrate that conditioning on explicit syntactic structure is useful. These baselines
are based on previously published approaches that reduce the problem to tagging: Xu et al. (2016) use a BiLSTM-CRF tagger with bigram topology; Tilk and Alumäe (2016) use a BiGRU tagger with
attention. In both approaches, the model is trained to tag each slot i with the correct string $xi∈V*$ (possibly ϵ or ^∧). These are discriminative probabilistic models (in contrast to our generative
one). Each gives a probability distribution over the taggings (conditioned on the unpunctuated sentence), so we can evaluate their perplexity.^^14
As shown in Table 3, our full model beats the baselines in perplexity in all 5 languages. Also, in 4 of 5 languages, allowing a trained NoisyChannel (rather than the identity map) significantly
improves the perplexity.
Table 3:
. Attn. . CRF . Attach . +NC . Dir .
Arabic 1.4676 1.3016 1.2230 1.1526 L
Chinese 1.6850 1.4436 1.1921 1.1464 L
English 1.5737 1.5247 1.5636 1.4276 R
Hindi 1.1201 1.1032 1.0630 1.0598 L
Spanish 1.4397 1.3198 1.2364 1.2103 R
. Attn. . CRF . Attach . +NC . Dir .
Arabic 1.4676 1.3016 1.2230 1.1526 L
Chinese 1.6850 1.4436 1.1921 1.1464 L
English 1.5737 1.5247 1.5636 1.4276 R
Hindi 1.1201 1.1032 1.0630 1.0598 L
Spanish 1.4397 1.3198 1.2364 1.2103 R
5Analysis of the Learned Grammar
5.1Rules Learned from the Noisy Channel
We study our learned probability distribution over noisy channel rules (ab ↦ b, ab ↦ a, ab ↦ ab, ab ↦ ba) for English. The probability distributions corresponding to six of Nunberg’s English rules
are shown in Figure 3. By comparing the orange and blue bars, observe that the model trained on the 𝖾𝗇_𝖼𝖾𝗌𝗅 treebank learned different quotation rules from the one trained on the 𝖾𝗇 treebank. This is
because 𝖾𝗇_𝖼𝖾𝗌𝗅 follows British style, whereas 𝖾𝗇 has American-style quote transposition.^^15
We now focus on the model learned from the 𝖾𝗇 treebank. Nunberg’s rules are deterministic, and our noisy channel indeed learned low-entropy rules, in the sense that for an input ab with underlying
count ≥ 25,^^16 at least one of the possible outputs (a, b, ab or ba) always has probability >0.75. The one exception is ”.↦.” for which the argmax output has probability ≈ 0.5, because writers do
not apply this quote transposition rule consistently. As shown by the blue bars in Figure 3, the high-probability transduction rules are consistent with Nunberg’s hand-crafted deterministic grammar
in Table 1.
Our system has high precision when we look at the confident rules. Of the 24 learned edits with conditional probability >0.75, Nunberg lists 20.
Our system also has good recall. Nunberg’s hand-crafted schemata consider 16 punctuation types and generate a total of 192 edit rules, including the specimens in Table 1. That is, of the 16^2 = 256
possible underlying punctuation bigrams ab, $34$ are supposed to undergo absorption or transposition. Our method achieves fairly high recall, in the sense that when Nunberg proposes ab ↦ γ, our
learned p(γ∣ab) usually ranks highly among all probabilities of the form p(γ′∣ab). 75 of Nunberg’s rules got rank 1, 48 got rank 2, and the remaining 69 got rank >2. The mean reciprocal rank was
0.621. Recall is quite high when we restrict to those Nunberg rules ab ↦ γ for which our model is confident how to rewrite ab, in the sense that some p(γ′∣ab) > 0.5. (This tends to eliminate rare ab:
see §5.) Of these 55 Nunberg rules, 38 rules got rank 1, 15 got rank 2, and only 2 got rank worse than 2. The mean reciprocal rank was 0.836.
¿What about Spanish? Spanish uses inverted question marks ¿ and exclamation marks ¡, which form symmetric pairs with the regular question marks and exclamation marks. If we try to extrapolate to
Spanish from Nunberg’s English formalization, the English mark most analogous to ¿ is (. Our learned noisy channel for Spanish (not graphed here) includes the high-probability rules ,¿↦,¿ and :¿↦:¿
and ¿,↦¿ which match Nunberg’s treatment of ( in English.
5.2Attachment Model
What does our model learn about how dependency relations are marked by underlying punctuation?
The above example^^17 illustrates the use of specific puncteme pairs to set off the advmod, ccomp, and nmod relations. Notice that said takes a complement (ccomp) that is symmetrically quoted but
also left delimited by a comma, which is indeed how direct speech is punctuated in English. This example also illustrates quotation transposition. The top five relations that are most likely to
generate symmetric punctemes and their top (l, r) pairs are shown in Table 4.
Table 4:
parataxis2.38 . appos2.29 . list1.33 . advcl0.77 . ccomp0.53 .
, , 26.8 , , 18.8 εε 60.0 εε 73.8 εε 90.8
εε 20.1 : ε 18.1 , , 22.3 , , 21.2 “” 2.4
( ) 13.0 - ε 15.9 , ε 5.3 ε , 3.1 , , 2.4
- ε 9.7 εε 14.4 < > 3.0 ( ) 0.74 :“” 0.9
: ε 8.1 ( ) 13.1 ( ) 3.0 ε - 0.21 “ ,” 0.8
parataxis2.38 . appos2.29 . list1.33 . advcl0.77 . ccomp0.53 .
, , 26.8 , , 18.8 εε 60.0 εε 73.8 εε 90.8
εε 20.1 : ε 18.1 , , 22.3 , , 21.2 “” 2.4
( ) 13.0 - ε 15.9 , ε 5.3 ε , 3.1 , , 2.4
- ε 9.7 εε 14.4 < > 3.0 ( ) 0.74 :“” 0.9
: ε 8.1 ( ) 13.1 ( ) 3.0 ε - 0.21 “ ,” 0.8
The above example^^18 shows how our model handles commas in conjunctions of 2 or more phrases. UD format dictates that each conjunct after the first is attached by the conj relation. As shown above,
each such conjunct is surrounded by underlying commas (via the N.,.,.conj feature from Appendix A), except for the one that bears the conjunction and (via an even stronger weight on the C.ε.ε.$conj→$
.cc feature). Our learned feature weights indeed yield p(ℓ = ε, r = ε) > 0.5 for the final conjunct in this example. Some writers omit the “Oxford comma” before the conjunction: this style can be
achieved simply by changing “surrounded” to “preceded” (that is, changing the N feature to N.,.ε.conj).
6Performance on Extrinsic Tasks
We evaluate the trained punctuation model by using it in the following three tasks.
6.1Punctuation Restoration
In this task, we are given a depunctuated sentence $d¯$^^19 and must restore its (surface) punctuation. Our model supposes that the observed punctuated sentence $x¯$ would have arisen via the
generative process (1). Thus, we try to find T, T′, and $x¯$ that are consistent with $d¯$ (a partial observation of $x¯$).
The first step is to reconstruct T from $d¯$. This initial parsing step is intended to choose the T that maximizes $psyn(T∣d¯)$.^^20 This step depends only on p[syn] and not on our punctuation
model (p[θ], p[ϕ]). In practice, we choose T via a dependency parser that has been trained on an unpunctuated treebank with examples of the form $(d¯,T)$.^^21
Equation 2 now defines a distribution over (
, x) given this
. To obtain a single prediction for
, we adopt the minimum Bayes risk (MBR) approach of choosing surface punctuation
that minimizes the expected loss with respect to the unknown truth
. Our loss function is the total edit distance over all slots (where edits operate on punctuation tokens). Finding
exactly would be intractable, so we use a sampling-based approximation and draw
= 1000 samples from the posterior distribution over (
). We then define
) is the set of unique
values in the sample and
is the empirical distribution given by the sample. This can be evaluated in
) time.
We evaluate on Arabic, English, Chinese, Hindi, and Spanish. For each language, we train both the parser and the punctuation model on the training split of that UD treebank (§4), and evaluate on
held-out data. We compare to the BiLSTM-CRF baseline in §4 (Xu et al., 2016).^^22 We also compare to a “trivial” deterministic baseline, which merely places a period at the end of the sentence (or a
"|" in the case of Hindi) and adds no other punctuation. Because most slots do not in fact have punctuation, the trivial baseline already does very well; to improve on it, we must fix its errors
without introducing new ones.
Our final comparison on test data is shown in the table in Figure 4. On all 5 languages, our method beats (usually significantly) its 3 competitors: the trivial deterministic baseline, the
BiLSTM-CRF, and the ablated version of our model (Attach) that omits the noisy channel.
Of course, the success of our method depends on the quality of the parse trees T (which is particularly low for Chinese and Arabic). The graph in Figure 4 explores this relationship, by evaluating
(on dev data) with noisier trees obtained from parsers that were variously trained on only the first 10%, 20%, …of the training data. On all 5 languages, provided that the trees are at least 75%
correct, our punctuation model beats both the trivial baseline and the BiLSTM-CRF (which do not use trees). It also beats the Attach ablation baseline at all levels of tree accuracy (these curves are
omitted from the graph to avoid clutter). In all languages, better parses give better performance, and gold trees yield the best results.
6.2Punctuation Correction
Our next goal is to correct punctuation errors in a learner corpus. Each sentence is drawn from the Cambridge Learner Corpus treebanks, which provide original (𝖾𝗇_𝖾𝗌𝗅) and corrected (𝖾𝗇_𝖼𝖾𝗌𝗅)
sentences. All kinds of errors are corrected, such as syntax errors, but we use only the 30% of sentences whose depunctuated trees T are isomorphic between 𝖾𝗇_𝖾𝗌𝗅 and 𝖾𝗇_𝖼𝖾𝗌𝗅. These 𝖾𝗇_𝖼𝖾𝗌𝗅 trees may
correct word and/or punctuation errors in 𝖾𝗇_𝖾𝗌𝗅, as we wish to do automatically.
We assume that an English learner can make mistakes in both the attachment and the noisy channel steps. A common attachment mistake is the failure to surround a non-restrictive relative clause with
commas. In the noisy channel step, mistakes in quote transposition are common.
Correction model.
Based on the assumption about the two error sources, we develop a discriminative model for this task. Let $x¯e$ denote the full input sentence, and let x[e] and x[c] denote the input (possibly
errorful) and output (corrected) punctuation sequences. We model $p(xc∣x¯e)=∑T∑Tc′psyn(T∣x¯e)⋅pθ(Tc′∣T,xe)⋅pϕ(xc∣Tc′)$. Here T is the depunctuated parse tree, T[c]′ is the corrected underlying tree,
T[e]′ is the error underlying tree, and we assume $pθ(Tc′∣T,xe)=∑Te′p(Te′∣T,xe)⋅pθ(Tc′∣Te′)$.
In practice we use a 1-best pipeline rather than summing. Our first step is to reconstruct
from the error sentence
. We choose
that maximizes
from a dependency parser trained on 𝖾𝗇_𝖾𝗌𝗅 treebank examples (
). The second step is to reconstruct
based on our punctuation model trained on 𝖾𝗇_𝖾𝗌𝗅. We choose
that maximizes
). We then reconstruct
is the node in
, and
) is a similar log-linear model to (
) with additional features (Appendix C
) which look at
Finally, we reconstruct x[c] based on the noisy channel p[ϕ](x[c]∣T[c]′) in §2.2. During training, ϕ is regularized to be close to the noisy channel parameters in the punctuation model trained on
We use the same MBR decoder as in §6.1 to choose the best action. We evaluate using AED as in §6.1. As a second metric, we use the script from the CoNLL 2014 Shared Task on Grammatical Error
Correction (Ng et al., 2014): it computes the F[0.5]-measure of the set of edits found by the system, relative to the true set of edits.
As shown in Table 5, our method achieves better performance than the punctuation restoration baselines (which ignore input punctuation). On the other hand, it is soundly beaten by a new BiLSTM-CRF
that we trained specifically for the task of punctuation correction. This is the same as the BiLSTM-CRF in the previous section, except that the BiLSTM now reads a punctuated input sentence (with
possibly erroneous punctuation). To be precise, at step 0 ≤ i ≤ n, the BiLSTM reads a concatenation of the embedding of word i (or BOS if i = 0) with an embedding of the punctuation token sequence x[
i]. The BiLSTM-CRF wins because it is a discriminative model tailored for this task: the BiLSTM can extract arbitrary contextual features of slot i that are correlated with whether x[i] is correct in
Table 5:
. ♦ . ★ . ●-- . parsed . gold . ★-corr .
AED 0.052 0.051 0.047 0.034 0.033 0.005
F[0.5] 0.779 0.787 0.827 0.876 0.881 0.984
. ♦ . ★ . ●-- . parsed . gold . ★-corr .
AED 0.052 0.051 0.047 0.034 0.033 0.005
F[0.5] 0.779 0.787 0.827 0.876 0.881 0.984
6.3Sentential Rephrasing
We suspect that syntactic transformations on a sentence should often preserve the underlying punctuation attached to its tree. The surface punctuation can then be regenerated from the transformed
tree. Such transformations include edits that are suggested by a writing assistance tool (Heidorn, 2000), or subtree deletions in compressive summarization (Knight and Marcu, 2002).
For our experiment, we evaluate an interesting case of syntactic transformation. Wang and Eisner (2016) consider a systematic rephrasing procedure by rearranging the order of dependent subtrees
within a UD treebank, in order to synthesize new languages with different word order that can then be used to help train multi-lingual systems (i.e., data augmentation with synthetic data).
As Wang and Eisner acknowledge (2016, footnote 9), their permutations treat surface punctuation tokens like ordinary words, which can result in synthetic sentences whose punctuation is quite unlike
that of real languages.
In our experiment, we use Wang and Eisner (2016) “self-permutation” setting, where the dependents of each noun and verb are stochastically reordered, but according to a dependent ordering model that
has been trained on the same language. For example, rephrasing a English sentence
under an English ordering model may yield
which is still grammatical except that , and . are wrongly swapped (after all, they have the same POS tag and relation type). Worse, permutation may yield bizarre punctuation such as , , at the start
of a sentence.
Our punctuation model gives a straightforward remedy—instead of permuting the tree directly, we first discover its most likely underlying tree
by the maximizing variant of Algorithm 1 (§3.1). Then, we permute the underlying tree and sample the surface punctuation from the distribution modeled by the trained PFST, yielding
We leave the handling of capitalization to future work.
We test the naturalness of the permuted sentences by asking how well a word trigram language model trained on them could predict the original sentences.^^23 As shown in Figure 6, our permutation
approach reduces the perplexity over the baseline on 4 of the 5 languages, often dramatically.
Table 6:
. Punctuation . All .
. Base . Half . Full . Base . Half . Full .
Arabic 156.0 231.3 186.1 540.8 590.3 553.4
Chinese 165.2 110.0 61.4 205.0 174.4 78.7
English 98.4 74.5 51.0 140.9 131.4 75.4
Hindi 10.8 11.0 9.7 118.4 118.8 91.8
Spanish 266.2 259.2 194.5 346.3 343.4 239.3
. Punctuation . All .
. Base . Half . Full . Base . Half . Full .
Arabic 156.0 231.3 186.1 540.8 590.3 553.4
Chinese 165.2 110.0 61.4 205.0 174.4 78.7
English 98.4 74.5 51.0 140.9 131.4 75.4
Hindi 10.8 11.0 9.7 118.4 118.8 91.8
Spanish 266.2 259.2 194.5 346.3 343.4 239.3
7Related Work
Punctuation can aid syntactic analysis, since it signals phrase boundaries and sentence structure. Briscoe (1994) and White and Rajkumar (2008) parse punctuated sentences using hand-crafted
constraint-based grammars that implement Nunberg’s approach in a declarative way. These grammars treat surface punctuation symbols as ordinary words, but annotate the nonterminal categories so as to
effectively keep track of the underlying punctuation. This is tantamount to crafting a grammar for underlyingly punctuated sentences and composing it with a finite-state noisy channel.
The parser of Ma et al. (2014) takes a different approach and treats punctuation marks as features of their neighboring words. Zhang et al. (2013) use a generative model for punctuated sentences,
leting them restore punctuation marks during transition-based parsing of unpunctuated sentences. Li et al. (2005) use punctuation marks to segment a sentence: this “divide and rule” strategy reduces
ambiguity in parsing of long Chinese sentences. Punctuation can similarly be used to constrain syntactic structure during grammar induction (Spitkovsky et al., 2011).
Punctuation restoration (§6.1) is useful for transcribing text from unpunctuated speech. The task is usually treated by tagging each slot with zero or more punctuation tokens, using a traditional
sequence labeling method: conditional random fields (Lui and Wang, 2013; Lu and Ng, 2010), recurrent neural networks (Tilk and Alumäe, 2016), or transition-based systems (Ballesteros and Wanner, 2016
8Conclusion and Future Work
We have provided a new computational approach to modeling punctuation. In our model, syntactic constituents stochastically generate latent underlying left and right punctemes. Surface punctuation
marks are not directly attached to the syntax tree, but are generated from sequences of adjacent punctemes by a (stochastic) finite-state string rewriting process . Our model is inspired by Nunberg (
1990) formal grammar for English punctuation, but is probabilistic and trainable. We give exact algorithms for training and inference.
We trained Nunberg-like models for 5 languages and L2 English. We compared the English model to Nunberg’s, and showed how the trained models can be used across languages for punctuation restoration,
correction, and adjustment.
In the future, we would like to study the usefulness of the recovered underlying trees on tasks such as syntactically sensitive sentiment analysis (Tai et al., 2015), machine translation (Cowan et
al., 2006), relation extraction (Culotta and Sorensen, 2004), and coreference resolution (Kong et al., 2010). We would also like to investigate how underlying punctuation could aid parsing. For
discriminative parsing, features for scoring the tree could refer to the underlying punctuation, not just the surface punctuation. For generative parsing (§3), we could follow the scheme in (1). For
example, the p[syn] factor in (1) might be a standard recurrent neural network grammar (RNNG) (Dyer et al., 2016); when a subtree of T is completed by the Reduce operation of p[syn], the
punctuation-augmented RNNG (1) would stochastically attach subtree-external left and right punctemes with p[θ] and transduce the subtree-internal slots with p[ϕ].
In the future, we are also interested in enriching the T′ representation and making it more different from T, to underlyingly account for other phenomena in T such as capitalization, spacing,
morphology, and non-projectivity (via reordering).
This material is based upon work supported by the National Science Foundation under Grant Nos. 1423276 and 1718846, including a REU supplement to the first author. We are grateful to the state of
Maryland for the Maryland Advanced Research Computing Center, a crucial resource. We thank Xiaochen Li for early discussion, Argo lab members for further discussion, and the three reviewers for
quality comments.
Our model could be easily adapted to work on constituency trees instead.
Multi-token punctemes are occasionally useful. For example, the puncteme … might consist of either 1 or 3 tokens, depending on how the tokenizer works; similarly, the puncteme ?! might consist of 1
or 2 tokens. Also, if a single constituent of T gets surrounded by both parentheses and quotation marks, this gives rise to punctemes (“ and ”). (A better treatment would add the parentheses as a
separate puncteme pair at a unary node above the quotation marks, but that would have required T′ to introduce this extra node.)
Rather than learn a separate edit probability distribution for each bigram ab, one could share parameters across bigrams. For example, Table 1’s caption says that “stronger” tokens tend to absorb
“weaker” ones. A model that incorporated this insight would not have to learn O(|Σ|^2) separate absorption probabilities (two per bigram ab), but only O(|Σ|) strengths (one per unigram a, which may
be regarded as a 1-dimensional embedding of the punctuation token a). We figured that the punctuation vocabulary Σ was small enough (Table 2) that we could manage without the additional complexity of
embeddings or other featurization, although this does presumably hurt our generalization to rare bigrams.
We could have handled all languages uniformly by making ≥ 2 passes of the sliding window (via a composition of ≥ 2 PFSTs), with at least one pass in each direction.
In retrospect, there was no good reason to square the $ET′[c(T′)]$term. However, when we started redoing the experiments, we found the results essentially unchanged.
Recall that the NoisyChannel model family (§2.2) requires the surface “ before special to appear underlyingly, and also requires the surface ε after special to be empty underlyingly. These hard
constraints clash with the $ξ=∞$ hard constraint that the punctuation around special must be balanced. The surface ” after place causes a similar problem: no edge can generate the matching underlying
We do O(n) multiplications of N × N matrices where N = O(# of punc types ⋅max # of punc tokens per slot).
Constructively, compose the u-to-x PFST (from the end of §2.2) with a straight-line FSA accepting only x[i], and project the resulting WFST to its input tape (Pereira and Riley, 1996), as explained
at the end of Appendix D.
We still construct the real matrix M[i](l) by ordinary matrix multiplication before pairing its elements with strings. This involves summation of real numbers: each element of the resulting real
matrix is a marginal probability, which sums over possible PFST paths (edit sequences) that could map the underlying puncteme l to a certain substring of the surface slot x[i]. Similarly for M[k](r).
For 𝖾𝗇 and 𝖾𝗇_𝖾𝗌𝗅, “ and ” are distinguished by language-specific part-of-speech tags. For the other 4 languages, we identify two " dependents of the same head word, replacing the left one with “ and
the right one with ”.
For symmetry, we should also have added a final mark.
These methods learn word embeddings that optimize conditional log-likelihood on the punctuation restoration training data. They might do better if these embeddings were shared with other tasks, as
multi-task learning might lead them to discover syntactic categories of words.
American style places commas and periods inside the quotation marks, even if they are not logically in the quote. British style (more sensibly) places unquoted periods and commas in their logical
place, sometimes outside the quotation marks if they are not part of the quote.
For rarer underlying pairs ab, the estimated distributions sometimes have higher entropy due to undertraining.
[𝖾𝗇] Earlier, Kerry said,“Just because you get an honorable discharge does not, in fact, answer that question.”
[𝖾𝗇] Sections 1, 2, 5, 6, 7, and 8 will survive any termination of this License.
To depunctuate a treebank sentence, we remove all tokens with POS-tag PUNCT or dependency relation punct. These are almost always leaves; else we omit the sentence.
Ideally, rather than maximize, one would integrate over possible trees T, in practice by sampling many values T[k] from $psyn(⋅∣u¯)$ and replacing S(T) in (10) with $⋃kS(Tk)$.
Specifically, the Yara parser (Rasooli and Tetreault, 2015), a fast non-probabilistic transition-based parser that uses rich non-local features (Zhang and Nivre, 2011).
We copied their architecture exactly but re-tuned the hyperparameters on our data. We also tried tripling the amount of training data by adding unannotated sentences (provided along with the original
annotated sentences by Ginter et al. (2017)), taking advantage of the fact that the BiLSTM-CRF does not require its training sentences to be annotated with trees. However, this actually hurt
performance slightly, perhaps because the additional sentences were out-of-domain. We also tried the BiGRU-with-attention architecture of Tilk and Alumäe (2016), but it was also weaker than the
BiLSTM-CRF (just as in Table 3). We omit all these results from Figure 4 to reduce clutter.
So the two approaches to permutation yield different training data, but are compared fairly on the same test data.
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東京大学理学系研究科 上田研究室
2006/1/26 (Thu) 16:00
│ Lecturer │ Neil Mochan (M2) │
│ Title │ Spin Dynamics in an optically trapped, spatially unifrom F=1 Bose Einstein Condensate │
│ │ Optical trapping of a BEC removes the tendency for the atoms to exist in a single hyperfine magnetic state. This new degree of freedom allows spin dynamics to occur. For a spatially │
│ Abstract │ uniform condensate in the absence of a magnetic field, the dynamics between the m=1,0,-1 states is modelled over varying initial conditions and compared to recent experimental data. │
│ │ Also studied is the situation where the interaction is oscillated by varying the s-wave scattering lengths by sweeping over the Feshbach resonance. │
2006/1/17 (Tue)
│ Lecturer │ Yuki Kawaguchi (PD) │
│ Title │ Einstein-de Haas effect in dipolar Bose-Einstein condensates │
│ │ The magnetic dipole-dipole interaction couples the spin and orbital angular momenta so that spin relaxation causes the system to rotate mechanically (Einstein-de Haas effect) or, │
│ │ conversely, a solid-body rotation of the system leads to its magnetization (Barnett effect). We show that these effects also occur in a dipolar Bose-Einstein condensate in which atoms │
│ Abstract │ undergo scalar, spinor, and dipolar interactions. General properties of the order parameter for a dipolar spinor Bose-Einstein condensate is discussed based on symmetries of the │
│ │ interactions, and an initially spin-polarized dipolar condensate is shown to dynamically generate a new type of non-singular vortex via spin-orbit interactions. We also discuss the │
│ │ effect of the external magnetic field and that of the trap geometry on the properties of the condensate. │
2005/12/6 (Tue)
│ Lecturer │ Shuta Nakajima (B4) │
│ Title │ Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms │
│ │ In 2002, I. Bloch et al. observed a superfluid Mott-insulator phase transition in a Bose-Einstein condensate held in a three-dimensional optical lattice potential. As the potential │
│ Abstract │ depth of the lattice is increased, a transition is observed from a superfluid to a Mott insulator phase. In the superfluid phase, each atom is spread out over the entire lattice, with │
│ │ long-range phase coherence. But in the insulating phase, exact numbers of atoms are localized at individual lattice sites, with no phase coherence across the lattice. In this seminar, │
│ │ I provide brief explanations about optical lattice, Bose-Hubbard model and superfluid Mott-insulator transition. And then, I introduce the experiment fulfilled by Bloch et al. │
2005/11/29 (Tue)
│ Lecturer │ Tomoya Ono (B4) │
│ Title │ Coherent collisional spin dynamics in optical lattices │
│ │ Immanuel Bloch and his collaborators observed coherent, purely collisionally driven spin dynamics of rubidium87 atoms in an optical lattice. For high lattice depths, atom pairs │
│ Abstract │ confined to the same latteice site show weakly damped Rabi-type oscillations between two-particle Zeeman states of equal magnetization, induced by spin changing collisions. I perform a │
│ │ review about their paper. │
2005/11/22 (Tue)
│ Lecturer │ Yuji Kurotani (M1) │
│ Title │ Theoretical circuit analysis of quantum measurement processes │
│ │ Measurement is to obtain information about a measured system. von Neumann have proposed that we can decompose any quantum measurement processes into two parts. The first part is a │
│ │ unitary transformation to move the information from the measured system to the measuring apparatus. The second part is the projective measurement on the apparatus (non-unitary │
│ Abstract │ transformation) to obtain the information about the initial system state. If we obtain the information of the system precisely, it seems that the post-measurement state of the system │
│ │ must change into an eigenstate of the measured observable. However, it is not universally true when using the `contractive state measurement (CSM)' proposed in 1980's. The CSM is an │
│ │ epoch-making model, but there is a little problem. In this seminar, I explain `swapping state measurement' as a improvement model of the CSM by using quantum circuit. These are usually │
│ │ discussed in continuous variable system, but we can also discuss them in spin-1/2 system. │
2005/11/15 (Tue)
│ Lecturer │ Dr. David Roberts (Ecole Normale Superieur) │
│ Title │ Casimir-like drag in a slow-moving Bose-Einstein condensate │
│ │ It is widely accepted that a superfluid flow exhibits a critical velocity below which there is no dissipation. However, the often-neglected zero-temperature quantum fluctuations have │
│ Abstract │ implications for the existence of this critical velocity. The drag force on an object created by the scattering of these quantum fluctuations in a three-dimensional, weakly interacting │
│ │ Bose-Einstein condensate is discussed. A non-zero force at low velocities is found to exist for two specific experimentally realizable examples, which suggests that the effective │
│ │ critical velocity in these systems is zero. │
2005/11/8 (Tue)
│ Lecturer │ Hiromi Arai (M2) │
│ Title │ Protein stability analysis by statistical learning │
│ │ Proteins are by far most structurally complex and functionally sophisticated molecules known. A protein molecule is made from a long chain of amino acids. There are 20 types of amino │
│ │ acids. Each type of protein has a unique sequence of them and, surprisingly, has a unique fold: the protein chain FOLDS in its unique stable form and exists in the physiologic │
│ Abstract │ condition. The mechanism of protein folding is still remain unsolved and prediction of protein fold by its sequence is left unfulfilled because of complexity of the system. Moreover, │
│ │ not all protein folds: folding condition is sensitive and limited sequence folds stable. But recently statsitical learning method has been developed and performs modestly. In my study, │
│ │ I marked on the critical point whether one protein is stable or not amd have been made a statistically analysis. In this seminar I am going to give a brief review of bioinformatics │
│ │ around my study and introduce what I am studying. │
2005/10/25 (Tue)
│ Lecturer │ Keiji Murata (M2) │
│ Title │ Bogoliubov excitations on spontaneously-broken axisymmetry phase of f=1 spinor BEC │
│ │ Spinor BEC system in the magnetic field is interesting for the rich ground state structures depending on the linear and quadratic Zeeman effects. MFT approach shows the characteristic │
│ │ phase of the f=1 spinor BEC, called `mixed' phase in MIT paper, is the phase that has spontaneously-broken axisymmetry. That is, the magnetization in this phase is not parallel to the │
│ Abstract │ applied magnetic field but tilted against it. We formulated the Bogoliubov theory on spinor BEC and verified the existence of gapless Goldstone modes predicted by the Goldstone's │
│ │ theorem. The interpretations of them are also very characteristic because they are phonon-magnon coupled excitation modes (In other words, they are such modes that recover the │
│ │ spontaneously-broken U(1) and SO(2) symmetries "simultaneously"). In this seminar, I will explain how to get this conclusion analytically and display the numerical verification of │
│ │ them. │
2005/10/18 (Tue)
│ Lecturer │ Dr. Michikazu Kobayashi (Osaka City Univ.) │
│ Title │ Localization of Bose-Einstein Condensation and Disappearance of Superfluidity of Strongly Correlated Bose Fluid in a Confined Potential │
│ │ Recently, some experiments of liquid 4He confined in porous glass have found the signal of Bose-Einstein condensation without superfluidity, suggesting a new phase transition to │
│ Abstract │ localized BECs (Bose glass). Motivated by the recent observation of quantum phase transition at high pressures and near the zero temperatures, observed by Yamamoto et al., we develop │
│ │ the three-dimensional Bose fluid in a confined potential. By introducing the localization length of localized condensates, we make a new analytical criterion for the localization of │
│ │ the Bose condensate. The critical pressure of the transition from normal condensate to localized condensates is quantitatively consistent with observations without free parameters. │
2005/10/11 (Tue)
│ Lecturer │ Teppei Sekizawa (D2) │
│ Title │ Quantum Statistical Machanics of Spin-1 Hard-Sphere Bose Gas with Spin-dependent Interaction │
│ │ We discuss statistical mechanics of spin-1 hard-sphere Bose gas with spin-dependent interaction by using Lee-Yang work in 1959. Although they studied spin-1 hard-sphere Bose gas with │
│ Abstract │ spin-independent interaction, we first study spin-1 hard-sphere Bose gas with spin-dependent interaction by using Lee-Yang method. We discover the binary kernel in our system and │
│ │ calculate the pertition function when the system does not exhibit Bose-condensation. │
2005/9/6 (Tue) 13:30
│ Lecturer │ Dr. Miguel A. Cazalilla (Donostia International Physics Center) │
│ Title │ "Breaking up" the BEC │
│ │ In recent years, interest in strongly correlated systems is driving us beyond the study of phenomena related to Bose-Einstein condensates (BEC) in weakly interacting gases of ultracold │
│ │ atoms. In this seminar, we shall focus on two much investigated routes to "break up" the BEC: low-dimensional systems in optical lattices and fast rotating Bose gases. Whereas the │
│ │ former has already been realized in various kinds of experiments, the latter still remains an exciting playground for theorists (although experiments are getting closer and closer to │
│ │ the strongly correlated regime). In the first part of the seminar, we shall discuss our recent results on the phase diagram and excitation spectra of two-dimensional (or very │
│ Abstract │ anisotropic) optical lattices. These are described as arrays of one-dimensional Bose gases. We compare our results with the experiments of the Zurich group, and show that they are, at │
│ │ least, in good qualitative agreement. In the second part of the seminar, we shall discuss the low-energy excitations of rapidly rotating Bose gases. We consider the so-call quantum │
│ │ Hall regime, where this system exhibits a series of vortex liquid states and it is no longer a BEC. We use exact numerical diagonalization of small atom systems to study the excitation │
│ │ of the vortex liquids in a harmonic trap. The low-lying excited states turn out to be surface (edge) waves. Comparison with existing theories for the edge excitations of quantum Hall │
│ │ states will be also presented. We thus find that the Pfaffian (or Moore-Read) state presents a number of anomalies, which may have their origin in a reconstruction of the edge of this │
│ │ vortex liquid. │
2005/7/1 (Fri) 15:00-17:00 理学部第二会議室 (H3-45)
│ Lecturer │ Prof. Sandro Stringari (Trento Univ.) │
│ Title │ Bloch oscillations in ultra-cold atoms and test of the Casimir-Polder force │
│ │ After a brief summary of recent theoretical and experimental work on Bloch oscillations in ultracold gases, I will discuss possible applications to the inteferometic study of the │
│ Abstract │ Casimir-Polder force generated by the surface of a substrate on a single atom. Recent work on the surface-atom force at the micron distance will be reviewed and new perspectives opened │
│ │ by atomic physics in this area will be outlined. │
│ │ 詳細 http://www.phys.titech.ac.jp/coe21/seminar/index.html │ | {"url":"http://cat.phys.s.u-tokyo.ac.jp/seminar2005.html","timestamp":"2024-11-14T01:14:26Z","content_type":"text/html","content_length":"23258","record_id":"<urn:uuid:3f947aa4-ac8e-4d57-8cbe-452fdb33d520>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00592.warc.gz"} |
Deeper Neural Networks Lead To Simpler Embeddings | Experfy.com
A surprising explanation for generalization in neural networks
Recent research is increasingly investigating how neural networks, being as hyper-parametrized as they are, generalize. That is, according to traditional statistics, the more parameters, the more the
model overfits. This notion is directly contradicted by a fundamental axiom of deep learning:
Increased parametrization improves generalization.
Although it may not be explicitly stated anywhere, it’s the intuition behind why researchers continue to push models larger to make them more powerful.
There have been many efforts to explain exactly why this is so. Most are quite interesting; the recently proposed Lottery Ticket Hypothesis states that neural networks are just giant lotteries
finding the best subnetwork, and another paper suggests through theoretical proof that such phenomenon is built into the nature of deep learning.
Perhaps one of the most intriguing, though, is one proposing that deeper neural networks lead to simpler embeddings. Alternatively, this is known as the “simplicity bias” — neural network parameters
have a bias towards simpler mappings.
Minyoung Huh et al proposed in a recent paper, “The Low-Rank Simplicity Bias in Deep Networks”, that depth increases the proportion of simpler solutions in the parameter space. This makes neural
networks more likely — by chance — to find simple solutions rather than complex ones.
On terminology: the authors measure the “simplicity” of a matrix based on its rank — roughly speaking, a measurement of how linearly independent parts of the matrix are on other parts. A higher rank
can be considered more complex since its parts are highly independent and thus contain more “independent information”. On the other hand, a lower rank can be considered simpler.
Huh et al begin by analyzing the rank of linear networks — that is, networks without any nonlinearities, like activation functions.
The authors trained several linear networks of different depths on the MNIST dataset. For each network, they randomly drew 128 neural network weights and the associated kernel, plotting their ranks.
As the depth increases, the rank of the network’s parameters decreases.
This can be derived from the fact that the rank of the product of two matrices can only decrease or remain the same from the ranks of each of its constituents. If we get a little bit more abstract,
we can think of this intuitively as: each matrix contains its own independent information, but when they are combined, the information of one matrix can only get muddled and entangled with the
information in another matrix.
rank(AB) ≤ min (rank(A), rank(B))
What is more interesting, though, is that the same applies to nonlinear networks. When nonlinear activation functions like tanh or ReLU are applied, the pattern is repeated: higher depth, lower rank.
The authors also performed hierarchical cluster kernels for different depths of the network for the two nonlinearities (ReLU and tanh). As the depth increases, the presence of block structures shows
decreasing rank. The “independent information” each kernel carries decreases with depth.
Hence, although it may seem odd, over-parametrizing a network acts as implicit regularization. This is especially true with linearities; thus, a model’s generalization can be improved by increasing
In fact, the authors find that on both the CIFAR-10 and CIFAR-100 datasets, linearly expanding the network increases accuracy by 2.2% and 6.5% from a baseline simple CNN. On ImageNet, linear
over-parametrization of AlexNet increases accuracy by 1.8%; ResNet10 accuracy increases by 0.9%, and ResNet18 accuracy increases by 0.4%.
This linear over-parametrization — which is not adding any serious learning capacity to the network, only more linear transformations — performs even better than explicit regularizers, like
penalties. Moreover, this implicit regularization does not change the objective being minimized.
Perhaps what is most satisfying about this contribution is that it is in agreement with Occam’s razor — a statement many papers have questioned or outright refuted as being relevant to deep learning.
The simplest solution is usually the right one.
– Occam’s razor
Indeed, much of prior discourse has taken more parameters to mean more complex, leading many to assert that Occam’s razor, while being relevant in the regime of classical statistics, does not apply
to over parametrized spaces.
This paper’s fascinating contribution argues instead that simpler solutions are in fact better, and that more successful, highly parameterized neural networks arrive at those simpler solutions
because, not despite, their parametrization.
Still, as the authors term it, this is a “conjecture” — it still needs refinement and further investigation. But it’s a well-founded, and certainly interesting, one at that — perhaps with the
potential to shift the conversation on how we think about generalization in deep learning.
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Excel Formula to Calculate Average Without OutliersCalculating Average Excluding Outliers In Excel: A Formula Guide
Excel Formula to Calculate Average Without Outliers
When working with data in Microsoft Excel, calculating averages is a common task. However, sometimes your data may contain outliers – values that are significantly higher or lower than most of the
data points. These outliers can skew the average and make it less representative of the typical values in your dataset. In this article, we’ll show you how to calculate the average in Excel while
excluding outliers using various methods.
Understanding Averages and Outliers
Before we dive into the methods for calculating averages without outliers, let’s briefly review what these terms mean:
• An average (or arithmetic mean) is the sum of a set of values divided by the number of values. It represents the central tendency or typical value of the dataset.
• An outlier is a data point that differs significantly from other observations. Outliers can be much higher or lower than the majority of values in the dataset.
Outliers can occur for various reasons, such as data entry errors, measurement issues, or genuine extreme values. While outliers may be interesting to investigate, they can distort the average and
lead to misleading conclusions if not handled appropriately.
Why Exclude Outliers When Calculating the Average?
Outliers can have a significant impact on the average, pulling it towards the extreme value and making it less representative of the central tendency of the data. By excluding outliers, we can
calculate an average that better reflects the typical values in the dataset.
For example, let’s say you have the following dataset of salaries:
The last value, $500,000, is an outlier that is much higher than the other salaries. If we calculate the average salary using the standard AVERAGE function in Excel, we get:
The average salary of $146,000 is not a good representation of the typical salary in this dataset, as it is skewed upwards by the outlier. By excluding the outlier, we can calculate an average that
better captures the central tendency of the salaries.
Method 1: Using the TRIMMEAN Function
Excel provides a built-in function called TRIMMEAN that allows you to calculate the mean of a dataset while excluding a specified percentage of data points from the top and bottom of the sorted
values. Here’s how to use it:
1. Arrange your data in a single column or row.
2. In a cell where you want the result to appear, type =TRIMMEAN(
3. Select the range of cells containing your data.
4. Type a comma (,), and then enter the percentage of data points you want to exclude from each end of the sorted values (e.g., 0.1 for 10%).
5. Close the parentheses and press Enter.
For example, to calculate the average of the salaries while excluding the top and bottom 10% of values, you would use:
=TRIMMEAN(A2:A6, 0.1)
In this case, the TRIMMEAN function sorts the salaries, removes the top and bottom values (10% of 5 data points is 0.5, rounded up to 1), and calculates the mean of the remaining values.
The TRIMMEAN function is useful when you have a large dataset and want to exclude a fixed percentage of extreme values from both ends of the distribution. However, it may not always remove all
outliers, especially if they are not in the top or bottom percentiles.
Method 2: Using the AVERAGEIF Function with Criteria
Another way to exclude outliers when calculating the average is to use the AVERAGEIF function with criteria that define the range of acceptable values. Here’s how:
1. Arrange your data in a single column or row.
2. Determine the lower and upper bounds for your acceptable values (e.g., 1st and 99th percentile, or mean ± 2 standard deviations).
3. In a cell where you want the result to appear, type =AVERAGEIF(
4. Select the range of cells containing your data.
5. Type a comma (,), and then enter your criteria using cell references or values (e.g., "B2:B6",">="&C1,"<="&C2 where C1 contains the lower bound and C2 the upper bound).
6. Close the parentheses and press Enter.
For example, let’s say we want to calculate the average of salaries between $40,000 and $70,000. We could use:
This subtracts the average of salaries greater than $70,000 from the average of salaries greater than or equal to $40,000, effectively giving us the average of salaries between $40,000 and $70,000.
The AVERAGEIF function allows you to specify precise criteria for inclusion in the average calculation. However, it requires you to determine appropriate lower and upper bounds, which may involve
some data exploration and statistical analysis.
Method 3: Using an Array Formula
Array formulas allow you to perform calculations on multiple values and return a result. Here’s how to use an array formula to calculate the average excluding outliers:
1. Arrange your data in a single column or row.
2. Determine the lower and upper bounds for your acceptable values.
3. In a cell where you want the result to appear, type =AVERAGE(IF(
4. Select the range of cells containing your data.
5. Type >lower_bound,
6. Select the range of cells again.
7. Type ,IF(
8. Select the range a third time.
9. Type <upper_bound,
10. Select the range a fourth time.
11. Type )))
12. Instead of pressing Enter, hold down Ctrl+Shift and press Enter to enter the formula as an array.
For example, to calculate the average of salaries between $40,000 and $70,000, you would use:
The IF functions check each value in the range A2:A6 to see if it’s greater than $40,000 and less than $70,000. If both conditions are true, the value is included in the AVERAGE calculation. The
result is the average of salaries within the specified range.
Important: Remember to enter array formulas using Ctrl+Shift+Enter, not just Enter. Excel will automatically add curly braces {} around the formula to indicate it’s an array.
Array formulas offer flexibility in defining complex criteria for inclusion in the average. However, they can be more difficult to set up and modify compared to regular formulas.
Tips for Identifying and Handling Outliers
Deciding what constitutes an outlier and how to handle it depends on your data and analysis goals. Here are some tips:
• Create a box plot, histogram, or scatter plot of your data to visually identify potential outliers. These charts can help you spot values that are far from the main cluster of data points.
• Calculate statistical measures like z-scores, interquartile range (IQR), or Tukey fences to objectively define outliers. These methods compare each value to the center and spread of the dataset
to determine if it’s an unusual observation.
• Consider the context and subject area. Unusually large or small values may be legitimate data points that shouldn’t be discarded. For example, in a dataset of housing prices, a multi-million
dollar mansion may be a genuine outlier rather than a data entry error.
• If outliers are due to data entry errors, try to correct them if possible. Check the original data source and make necessary adjustments. If the outliers are due to measurement issues, consider
re-collecting the data to ensure accuracy.
• Document how you identified and handled outliers so others can understand and evaluate your analysis. Transparency is key to reproducibility and credibility in data analysis.
Final Thoughts
Outliers can have a significant impact on averages in Excel, making them less representative of typical values. By using functions like TRIMMEAN, AVERAGEIF with criteria, or array formulas, you can
calculate averages that exclude outliers and better capture the central tendency of your data.
By mastering these techniques for calculating averages without outliers, you’ll be able to analyze your data more effectively, draw meaningful insights, and make better-informed decisions. Excel
provides powerful tools for data analysis, and understanding how to handle outliers is a crucial skill for any data analyst or business professional.
What is an outlier in a dataset?
An outlier is a data point that differs significantly from other observations in a dataset. Outliers can be much higher or lower than the majority of values and may occur due to data entry errors,
measurement issues, or genuine extreme values.
Why should I exclude outliers when calculating the average in Excel?
Outliers can have a significant impact on the average, pulling it towards the extreme value and making it less representative of the typical values in the dataset. By excluding outliers, you can
calculate an average that better reflects the central tendency of the data.
What is the TRIMMEAN function in Excel?
The TRIMMEAN function is a built-in Excel function that allows you to calculate the mean of a dataset while excluding a specified percentage of data points from the top and bottom of the sorted
values. This function is useful for removing extreme values from both ends of the distribution.
How can I use the AVERAGEIF function to exclude outliers in Excel?
You can use the AVERAGEIF function with criteria that define the range of acceptable values. By specifying a lower and upper bound for your data, you can calculate the average of values that fall
within that range, effectively excluding outliers.
What should I consider when identifying and handling outliers in my data?
When identifying and handling outliers, consider creating visual representations of your data (e.g., box plots, histograms) to spot potential outliers, calculate statistical measures (e.g., z-scores,
IQR) to objectively define outliers, and take into account the context and subject area. If outliers are due to errors, try to correct them or consider re-collecting the data. Always document your
process for handling outliers to ensure transparency and reproducibility.
Vaishvi Desai is the founder of Excelsamurai and a passionate Excel enthusiast with years of experience in data analysis and spreadsheet management. With a mission to help others harness the power of
Excel, Vaishvi shares her expertise through concise, easy-to-follow tutorials on shortcuts, formulas, Pivot Tables, and VBA.
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Heat Flux Boundary Condition for 1D Line Element
Hello everyone,
I have a question regarding boundary conditions in FreeFEM++. Specifically, I would like to know if it is possible to apply a heat flux boundary condition to a one-dimensional line element at any of
its boundaries.
For context, I am working on a heat transfer problem where I need to specify a heat flux instead of a temperature at one of the boundaries. The problem statement and the .edp file are attached below.
I would appreciate any guidance on how to implement this in FreeFEM++, or if there are specific commands or examples you could point me to.
Thank you!!!
trial.edp (8.0 KB)
I think that you need to set non-homogeneous Neumann BC.
For your heat equation it means -K\frac{dT}{dx}n=g, with g a given boundary flux, n the external normal (+1 on the right, -1 on the left of the domain)
For that you juste delete the +on(1,T=1000)
and replace it by +int0d(Th)(g*v)
However if you want to put different conditions on the left and on the right, you need to define boundary labels, I don’t know how to do it in 1d.
1 Like
Thanks for the suggestion, it worked
I’m working on a simulation where a material surface is undergoing ablation (material is being “burned” or removed) over time, which causes the surface to shrink. In FreeFem++,
I need to do the following:
Update the mesh: As the material shrinks, I need to update the mesh at each time step to reflect this shrinkage and also remesh the surface.
Update the finite element space (Fespace): After the mesh is updated, I also need to update the finite element space to reflect the new geometry.
Transfer old solution fields: After the mesh is updated, I need to transfer my old solution fields (like temperature, density, etc.) to the new mesh.
My questions are:
How do I properly update the mesh and remesh during the simulation?
How do I reinitialize the finite element space after remeshing?
How do I transfer the old solution fields (temperature, density, etc.) to the new mesh? Should I interpolate them, or can I initialize them based on the new geometry?
It would be helpful if you can provide any guidance on how to implement this in FreeFEM++, or if there are specific commands or examples you could point me to.
Thank you!!! | {"url":"https://community.freefem.org/t/heat-flux-boundary-condition-for-1d-line-element/3568","timestamp":"2024-11-10T20:58:41Z","content_type":"text/html","content_length":"27391","record_id":"<urn:uuid:63b97d62-40b9-4ad4-80d8-da04f70f18fa>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00673.warc.gz"} |
pl/sql program for reverse of a number | SQL Queries
pl/sql program for reverse of a number
Here’s a PL/SQL program to find the reverse of a number:
num NUMBER;
reverse_num NUMBER := 0;
remainder NUMBER;
-- Prompt the user for input
num := #
-- Store the original number in a temporary variable
original_num NUMBER := num;
-- Calculate the reverse of the number
WHILE num > 0 LOOP
remainder := num MOD 10;
reverse_num := reverse_num * 10 + remainder;
num := FLOOR(num / 10);
END LOOP;
-- Display the reverse of the number
DBMS_OUTPUT.PUT_LINE('The reverse of ' || original_num || ' is: ' || reverse_num);
1. DECLARE: This keyword indicates the start of the declaration section where we define variables.
2. num NUMBER;: This line declares a variable named num of the NUMBER data type, which will store the original number.
3. reverse_num NUMBER := 0;: This line initializes the variable reverse_num to 0. It will store the reverse of the original number.
4. remainder NUMBER;: This line declares a variable named remainder of the NUMBER data type. It will store the remainder when dividing the number by 10.
5. BEGIN: This keyword signifies the beginning of the executable section.
6. num := #: This line prompts the user to enter the number and assigns the input value to the num variable. The & symbol is used to retrieve the value entered by the user.
7. DECLARE: This keyword indicates the start of the nested block where we define another variable to store the original number.
8. original_num NUMBER := num;: This line declares a variable named original_num of the NUMBER data type and assigns it the value of num. This variable will store the original number for display
9. BEGIN: This keyword signifies the beginning of the nested block’s executable section.
10. WHILE num > 0 LOOP: This loop will execute as long as the value of num is greater than 0.
11. remainder := num MOD 10;: This line calculates the remainder when num is divided by 10 and assigns it to the remainder variable.
12. reverse_num := reverse_num * 10 + remainder;: This line updates the reverse_num variable by multiplying it by 10 and adding the value of remainder. This effectively builds the reverse of the
13. num := FLOOR(num / 10);: This line updates the value of num by dividing it by 10 and taking the floor of the result. This effectively removes the rightmost digit from num in each iteration.
14. END LOOP;: This statement marks the end of the loop.
15. DBMS_OUTPUT.PUT_LINE('The reverse of ' || original_num || ' is: ' || reverse_num);: This line uses the DBMS_OUTPUT.PUT_LINE procedure to display the reverse of the original number. It
concatenates the values of original_num and reverse_num to form a descriptive message.
By running this PL/SQL program, you can find the reverse of a given number. The program prompts the user to enter the number, calculates the reverse of the number using a loop, and then displays the
reverse on the console. This allows you to quickly obtain the reverse of the given number. | {"url":"https://sqlqueries.in/pl-sql-program-for-reverse-of-a-number/","timestamp":"2024-11-03T07:47:02Z","content_type":"text/html","content_length":"99989","record_id":"<urn:uuid:29eb7e60-fa7e-4f57-aaf5-477a23783268>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00012.warc.gz"} |
60 Card Bingo house edge
The game Go-Go bingo has four cards of fifteen numbers each, with numbers 1-12 always in row 1 of each card, 13-24 in row 2, etc., with no repeating of the 60 numbers. An initial bet is $1 (which can
be changed, but payouts remain proportional) and the following payouts:
ANY LINE $.75
LINE UP $.75
LINE DOWN $.75
CROSS $1.50
ROMAN I $2.50
BULLSEYE $7.50
CHECKERED $7.50
ANY 2 LINES $20
TT $25
EYE $50
FRAMED $125
FULL HOUSE $375
30 balls are drawn, and extra balls can be bought for a price determined by the maximum potential payout of the next ball (if there is none, it is free) divided by the number of balls remaining, and
rounded up to the nearest quarter. There seems to be an edge to the player when multiple payouts are available (such as 2 ways to win $20 with 30 balls left--the player receives odds of 52.33:1 on a
29:1 event), but I am unable to calculate the house edge or optimal strategy for maximizing the return from the game.
The only "strategy" appears to be whether or not to buy an extra ball.
As for house edge, even if you limit the calculation to the original 30 balls, you need to describe (a) just what each winning layout looks like, (b) the difference between "any line," "line up," and
"line down," (c) whether the $1 bet is for all four cards or per card, and (d) can there be more than one payout, or is it just "highest value pays" (so, for example, a full house doesn't also pay
off all of the other results as well)?
To determine the extra ball strategy, you'll need to be more specific as to what the cost is. What do you mean by "number of balls remaining" - for example, on your first extra ball, would that be
30, since there are still 30 undrawn balls? | {"url":"https://wizardofvegas.com/forum/gambling/betting-systems/32210-60-card-bingo-house-edge/","timestamp":"2024-11-12T22:00:15Z","content_type":"text/html","content_length":"42601","record_id":"<urn:uuid:b0d1cc3a-0569-4f17-beb1-fc2add2076c5>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00189.warc.gz"} |
A method of immediate detection of objects with a near-zero apparent motion in series of CCD-frames
Issue A&A
Volume 609, January 2018
Article Number A54
Number of page(s) 11
Section Numerical methods and codes
DOI https://doi.org/10.1051/0004-6361/201630323
Published online 05 January 2018
A&A 609, A54 (2018)
A method of immediate detection of objects with a near-zero apparent motion in series of CCD-frames
^1 Western Radio Technical Surveillance Center, State Space Agency of Ukraine, Kosmonavtiv Street, 89600 Mukachevo, Ukraine
e-mail: vadym@savanevych.com
^2 Uzhhorod National University, Laboratory of space research, 2a Daleka Street, 88000 Uzhhorod, Ukraine
e-mail: sergii.khlamov@gmail.com
^3 Main Astronomical Observatory of the NAS of Ukraine, 27 Akademika Zabolotnogo Street, 03143 Kyiv, Ukraine
^4 Kharkiv National University of Radio Electronics, 14 Nauki Avenue, 61166 Kharkiv, Ukraine
^5 National Astronomical Research Institute of Thailand, 260 Moo 4, T. Donkaew, A. Maerim, 50180 Chiangmai, Thailand
^6 Institute of Physics, Faculty of Natural Sciences, University of P. J. Safarik, Park Angelinum 9, 04001 Kosice, Slovakia
^7 Scientific Research, Design and Technology Institute of Micrographs, 1/60 Akademika Pidgornogo Street, 61046 Kharkiv, Ukraine
^8 Department of Physics and Astronomy, University of North Carolina at Chapel Hill, Chapel Hill NC-27599, North Carolina, USA
Received: 22 December 2016
Accepted: 25 September 2017
The paper deals with a computational method for detection of the solar system minor bodies (SSOs), whose inter-frame shifts in series of CCD-frames during the observation are commensurate with the
errors in measuring their positions. These objects have velocities of apparent motion between CCD-frames not exceeding three rms errors (3σ) of measurements of their positions. About 15% of objects
have a near-zero apparent motion in CCD-frames, including the objects beyond the Jupiter’s orbit as well as the asteroids heading straight to the Earth. The proposed method for detection of the
object’s near-zero apparent motion in series of CCD-frames is based on the Fisher f-criterion instead of using the traditional decision rules that are based on the maximum likelihood criterion. We
analyzed the quality indicators of detection of the object’s near-zero apparent motion applying statistical and in situ modeling techniques in terms of the conditional probability of the true
detection of objects with a near-zero apparent motion. The efficiency of method being implemented as a plugin for the Collection Light Technology (CoLiTec) software for automated asteroids and comets
detection has been demonstrated. Among the objects discovered with this plugin, there was the sungrazing comet C/2012 S1 (ISON). Within 26 min of the observation, the comet’s image has been moved by
three pixels in a series of four CCD-frames (the velocity of its apparent motion at the moment of discovery was equal to 0.8 pixels per CCD-frame; the image size on the frame was about five pixels).
Next verification in observations of asteroids with a near-zero apparent motion conducted with small telescopes has confirmed an efficiency of the method even in bad conditions (strong backlight from
the full Moon). So, we recommend applying the proposed method for series of observations with four or more frames.
Key words: methods: numerical / methods: data analysis / techniques: image processing / minor planets, asteroids: general / comets: general / comets: individual: ISON
© ESO, 2018
1. Introduction
Different types of objects are detected in series of CCD-frames during observations: solar system minor bodies (SSOs); stars and large-scale diffuse sources (non-SSOs); charge transfer tails from
bright stars, bright streaks from satellites, and noise sources amongst others. The difference between the detected SSOs and non-SSOs is that the non-SSOs have a zero velocity apparent motion on a
set of frames, while the SSOs have a non-zero one. Wherein, a rapid detection of the objects with a near-zero velocity apparent motion both from the main belt of asteroids and beyond the Jupiter’s
orbit is very important for the asteroid-comet hazard problem as well as for the earliest recording new SSOs.
Over the past few decades, several powerful software tools and methods had been developed, allowing discovery and cataloging of thousands of SSOs (asteroids, comets, trans-Neptunians, Centaurs,
etc.). First of all, it was the Lincoln Near-Earth Asteroid Research (LINEAR) project (Stokes 1998), which outperformed all asteroid search programs acted until 1998. This project brought the number
of discovered SSOs to over 230000, including 2423 near-Earth objects (NEOs) and 279 comets (Stokes 2000). The second biggest asteroid survey, the Catalina Sky Survey (2016), started in 2005 as a
search program for any potentially hazardous minor planets and allowed to discover more than 6500 NEOs. The same program in the southern hemisphere, the Siding Spring Survey (SSS), was closed in
A successful operation of these programs has stimulated new instruments and advanced CCD-cameras manufacturing as well as the development of new methods and algorithms for image processing and
detecting faint SSOs. These methods of the automated search for very faint objects in a CCD-frame series were based, mostly, on the matched filter or the combined multiple frames along the typical
SSO’s motion (Yanagisawa et al. 2005). For example, the implementation of a multi-hypothesis velocity matched filter for LINEAR archive of images has produced about 25% new detections (mostly of
faint SSOs) that were missed at the stage of a primary processing of observations (Shucker 2008). Another algorithm, the interacting multiple model (IMM), was introduced as a modification of matched
filter and provided a new structure for effective management of multiple filter models, while the selected parameters must be considered for the IMM optimizing (Genovese 2001).
The Panoramic Survey Telescope and Rapid Response System (Pan-STARRS) for surveying the sky for moving objects on a continual basis was designed as an array of four telescopes. The first telescope,
PS1, is in a full operation since 2010 and is able to observe objects down to 22.5^m apparent magnitude. With the help of PS1 more than 2860 NEOs and many comets have already been discovered (see
Hsieh et al. 2013). PS1 uses the Moving Object Pipeline System, MOPS (Heasley et al. 2007), which includes some methods and techniques for searching for the extremely faint and distant Sedna-like
objects (Jedicke et al. 2009), such as for example the modified intra-nightly linking algorithm, which includes a partial Hough transform method for quickly identifying of the multiple detections and
post-processing step for intra-nightly linking (see, Parker et al. 2009; Myers et al. 2008).
These methods were successfully tested for simulations of processing the moving objects with MOPS on the Pan-STARRS and the Large Synoptic Survey Telescope, LSST (Barnard et al. 2006), the latter
will be provided by the same pipeline system as on the Pan-STARRS (Myers et al. 2008). It is important that PS1 is highly effective for discovering objects that could actually impact the Earth next
100 years (Jedicke et al. 2009) and was complemented with the infrared data of the former WISE orbital telescope (Dailey et al. 2010).
In 2009, the authors of this paper developed the CoLiTec (Collection Light Technology) software for the automated detection of the solar system minor bodies in CCD-frames series^1 (see, in detail,
Savanevych 1999, 2006; Savanevych et al. 2012, 2015a; Vavilova et al. 2012a,b, 2017; Vavilova 2016; Pohorelov et al. 2016). Since 2009 it has been installed at several observatories: Andrushivka
Astronomical Observatory (A50, Ukraine; Ivashchenko et al. 2013), ISON-NM Observatory (H15, the US; Elenin et al. 2013), ISON-Kislovodsk Observatory (D00, Russia; ISON-Kislovodsk 2016),
ISON-Ussuriysk Observatory (C15, Russia; Elenin et al. 2014), Odessa-Mayaki (583, Ukraine; Troianskyi et al. 2014), Vihorlat Observatory (Slovakia; Dubovsky et al. 2017).
The preliminary object’s detection with CoLiTec software is based on the accumulation of the energy of signals along possible object tracks in a series of CCD-frames. Such accumulation is reached by
the method of the multivalued transformation of the object coordinates that is equivalent to the Hough transformation (Savanevych 2006; Savanevych et al. 2012). In general, CoLiTec software allows
detecting of the objects with different velocities of the apparent motion by individual plugins for fast and slow objects, and objects with the near-zero apparent motion. CoLiTec software is widely
used in a number of observatories. In total, four comets (C/2011 X1 (Elenin), P/2011 NO1 (Elenin), C/2012 S1 (ISON) and P/2013 V3 (Nevski)) and more than 1560 asteroids including 5 NEOs, 21 Trojan
Jupiter asteroids and one Centaur were discovered using CoLiTec software as well as more than 700000 positional CCD-measurements were sent to the Minor Planet Center (Ivashchenko et al. 2013; Elenin
et al. 2013, 2014; Savanevych et al. 2015a). Our comparison of statistical characteristics of positional CCD-measurements with CoLiTec and Astrometrica^2 (Miller et al. 2008; Raab 2012) software in
the same set of test CCD-frames has demonstrated that the limits for reliable positional CCD-measurements with CoLiTec software are wider than those with Astrometrica one, in particular, for the area
of extremely low signal-to-noise ratio (S/N; Savanevych et al. 2015a).
Besides the requirement of large computational effort (see Shucker 2008), the main disadvantage of all the above mentioned methods implemented into software is a neglecting of near-zero apparent
motion of objects in CCD-frames that has yet to be described and tested. So, the aim of this paper is to introduce a new computational method for detection of SSOs with a near-zero velocity of
apparent motion in a series of CCD-frames. We propose considering these SSOs as a separate subclass, which includes objects whose inter-frame shifts during the observational session are commensurate
with the errors in measuring their positions. We call the maximum permissible velocity of a near-zero apparent motion as ”-velocity. Then, a subclass of SSOs with a near-zero apparent motion includes
such SSOs, which have velocities of apparent motion between CCD-frames that are not exceed three rms errors, 3σ, of measurements of their positions (”= 3σ). We will also use the notation of 3σ
-velocity instead of ”-velocity to describe a near-zero apparent motion of SSOs.
The economy in the observational search resource leads to a reduction in the time between CCD-frames. This, in turn, leads to the fact that a significant part of SSOs will have an ”-velocity apparent
motion, in other words, have a shift, which is commensurate with the errors in estimating of their position. In general, there are about 15% of SSOs with ”-velocity motion. They are the objects
beyond the Jupiter’s orbit as well as asteroids moving to the observer along the view axis (heading straight to the Earth). Of course, when such an object is close enough, a parallax from the Earth’s
rotation will introduce a significant transverse motion that can be detectable. The proposed method allows us to locate objects with a near-zero apparent motion, including the potentially dangerous
objects, at larger distances from the Earth than trivial methods. It gives more time to study such objects and to warn about their approach to the Earth in case of their hazardous behavior.
The structure of our paper is as follows. We describe a problem statement, a model of the apparent motion and hypothesis verification in Chapter 2. The task solution and new method are described in
Chapter 3. Analysis of quality indicators of near-zero motion detection is provided in Chapter 4. Concluding remarks and discussion are given in Chapter 5. A mathematical rationale of the method is
described in Appendices A−C.
2. Problem statement
The apparent motion of any object may be represented as the projection of its trajectory on the focal plane of a telescope. It is described by the model of rectilinear and uniform motion of an object
along each coordinate independently during the tracking and formation of the series of its CCD-measurements (see Appendix A).
Objects with significant apparent motion are easily detected by any methods of the trajectory determination, for example, the methods for inter-frame processing (Garcia et al. 2008; Gong et al. 2004;
Vavilova et al. 2012b). The problem arises when we would like to detect an object with a near-zero apparent motion in CCD-frame series. Such an object can be falsely identified as the object with a 3
The first step for solving this problem is a formation of the set of measurements Ω[set] (A.5) (no more than one measurement per frame) for the object, which was preliminarily assigned to the objects
with 3σ-velocities. In its turn, such objects should be registered in the internal catalog of objects that are motionless in the series of CCD-frames (Vavilova et al. 2012a). This catalog is also
helpful to reduce the number of false SSO detections in the software for automatic CCD-frame processing of asteroid surveys (Pohorelov et al. 2016).
In other words, the hypothesis H[0] that a certain set Ω[set] (A.5) of measurements complies to the objects with a 3σ-velocity is as follows: $H0:Vx2+Vy2=0,$(1)where V[x], V[y] are the apparent
velocities of object along each coordinate.
Then the more complicative alternative H[1] that the object with the set of measurements Ω[set] (A.5) has a 3σ-velocity will be written as: $H1:Vx2+Vy2>0.$(2)The false detection of the near-zero
apparent motion of the object is an error of the first kind α assuming the validity of H[0] hypothesis (1). The skipping of the object with a 3σ-velocity is an error of the second kind fi under
condition that the alternative hypothesis H[1] (2) is true. It is accepted in the community that the conditional probabilities of errors of the first α kind (conditional probability of the false
detection, CPFD) and the second fi kind (skipping of the object) are the indicators of a good quality detection (Kuzmyn 2000). We also used the conditional probability of the true detection (CPTD) as
a complement to the conditional probability of an error of the second fi kind to unity (1 −fi).
So, the task solution may be formulated as follows: 1) it is necessary to develop computational methods for detecting the near-zero apparent motion of the object based on the analysis of a set Ω[set]
of measurements (A.5) obtained from a series of CCD-frames; 2) computational methods have to check the competing hypotheses of zero H[0] (1) and near-zero H[1] (2) apparent motion of the object.
Maximum likelihood criterion. Usually, hypotheses such as H[0] (6) and H[1] (7) are tested according to a maximum likelihood criterion (Masson 2011; Myung 2003; Miura et al. 2005; Sanders-Reed 2005)
or any other criterion of the Bayesian group (Lee et al. 2014). The sufficient statistic for all the criteria is the likelihood ratio (LR), which is compared with critical values that are selected
according to the specific criteria (Morey et al. 2014). If there are no opportunities to justify the a priori probabilities of hypotheses and losses related to wrong decisions, the developer can use
either a maximum likelihood criteria or Neyman-Pearson approach (Lee et al. 2014). The unknown parameters of the likelihood function are evaluated by the same sample in which the hypotheses are
tested. In mathematical statistics, such rules are called “substitutional rules for hypothesis testing” (Lehman et al. 2010; Morey et al. 2014). In the technical literature, such rules are called
“detection-measurement” (Morey et al. 2014).
The “detection” procedure precedes the “measurement” procedure for the substitutional decision rule. And this is a general principle for solving the problem of mixed optimization with discrete and
continuous parameters (Arora et al. 1994). The decision statistics of hypotheses that correspond to different values of discrete parameters are compared with each other after the optimization of
conditional likelihood functions for the value of their continuous parameters. The software developers use the substitution rule of maximum likelihood despite the fact that the evidence is not proved
mathematically. It should be compared with any new methods of hypothesis testing with a priori parametric uncertainty (Gunawan 2006). The quality indicators of hypothesis testing can be examined only
by statistical modeling or on the training samples of large experimental datasets.
A likelihood function for detection of a near-zero apparent motion can be defined as the common density distribution of measurements of the object positions in a set of measurements (see Appendix B).
Ordinary least square (OLS) evaluation of the parameters of the object’s apparent motion as well as the variance of the object’s positional estimates in a set of measurements are described in
Appendix C. Using these parameters, one can obtain the maximum allowable (critical) value of the LR estimate for the detection of a near-zero apparent motion for the substitutional methods (C.11−
3. Task solution
Conversion of testing the hypothesis H[1] to the problem of validation of the statistical significance factor of the apparent motion. One of the disadvantages of substitutional methods based on
maximum likelihood criteria (Masson 2011; Myung 2003) is the insufficient justification of their application when some parameters of likelihood function are unknown. The second one leads to the
necessity of selecting the value of boundary decisive statistics (Miura et al. 2005; Sanders-Reed 2005). Moreover, in our case, the substitutional methods are inefficient when the object’s apparent
motion is near-zero.
Models (A.1) and (A.2) of the independent apparent motion along each coordinate are the classical models of linear regression with two parameters (start position and the velocity along each
coordinate). Thus, in our case, the alternative H[1] hypothesis (2) about the object to be the SSO with a near-zero apparent motion is identical to the hypothesis about the statistical significance
of the apparent motion. We propose to check the statistical significance of the entire velocity for detection of a 3σ-velocity, which is equivalent to check the hypothesis H[1].
A method for detection of the near-zero apparent motion using Fisher f-criterion. We propose to check the statistical significance of the entire velocity of the apparent motion of the object using f
-criterion. F-test should be applied, when variances of the positions in a set of measurements are unknown. It is based on the fact that the f-distribution does not depend on the distribution of
positional errors in a set of measurements (Phillips 1982; Johnson et al. 1995). Furthermore, there are also tabulated values of the Fisher distribution statistics (Burden et al. 2010; Melard 2014).
The f-criterion to check the statistical significance of the entire velocity of the apparent motion is represented as (Phillips 1982): $f(Ωset)=R02−R12R12Nmea−rw,$(3)where w = 1 is the number of
factors of the linear regression model that are verified by the hypothesis. In our case, the factor is the velocity of the apparent motion; r is a rank of the plan matrix F[x] (Burden et al. 2010,
rang (F[x] = r ≤ min(m,N[mea]))); $Fx=∥∥∥∥∥∥∥∥∥∥∥∥1Δτ1=(τ1−τ0)......1Δτk=(τk−τ0)......1ΔτNmea=(τNmea−τ0)∥∥∥∥∥∥∥∥∥∥∥∥.$(4)The rank of the F[x] matrix defined by (4) is equal to two for the linear
model of the motion along one coordinate because a number m of the estimated parameters of the motion is equal to two. As the apparent motion occurs along two coordinates, the number m of its
estimated parameters is equal to four. Accordingly, the rank r of the F[x] matrix is four because r = m.
The statistic (3) has a Fisher probability distribution with (w, N[mea]−r) degrees of freedom (Phillips 1982). Its distribution corresponds to the distribution of the ratio of two independent random
variables with a chi-square distribution (Park et al. 2011), degrees of freedom w, and N[mea]−r. For example, let the number N[fr] of CCD-frames in a series of frames to be N[fr] = 4, and each frame
contains the measurement of the object’s position. Hence, for two coordinates the number of measurements is 2N[mea] = 8, w = 1, and the rank r of the matrix F[x] (4) is r = 4. Therefore, statistic
(3) has a Fisher probability distribution with (1, 4) degrees of freedom.
To determine the maximum allowable (critical) tabulated value of the Fisher distribution statistics, we have to use the predefined significance level α. Its value is the conditional probability of
the false detection, CPFD, of the near-zero apparent motion. For example, if α = 10^-3, the maximum allowable f[cr] value of the Fisher distribution statistics with (1, 4) degrees of freedom is f[cr]
= 74.13 (Melard 2014).
After transformation, the method for detection of the near-zero apparent motion using Fisher f-criterion is represented as: $R02−R12R12≥wfcrNmea−r·$(5)
4. Indicators of quality of the near-zero apparent motion detection
Number of experiments for statistical modeling. Errors in statistical modeling are defined by estimates of conditional probabilities of the false detection fl[0] (validity of the H[0] hypothesis) and
true detection fl[1] (validity of the alternative H[1] using the critical values of the decision statistics after modeling the H[0] hypothesis).
In our research we assumed that the reasonable values of errors of experimental frequencies are equal to fl[0accept] = α/ 10, fl[1accept] = 10^-3. Their dependence on the number of experiments for
the statistical modeling (under the condition of a validity of the hypothesis H[0] and the alternative H[1]) is determined by the empirical formulas: $N0exp=102/fl0accept;N1exp=102/fl1accept=10-6.$
Preconditions and constants for the methods of the statistical and in situ modeling. To study the indicators of quality of the near-zero apparent motion detection using substitutional methods (see,
Appendix C and formulas C.11−C.13) in maximum likelihood approach, the appropriate maximum allowable values λ[cr] should be applied. These values are determined in accordance with the predefined
level of significance α in the modeling of the hypothesis H[0] (V = 0).
For the statistical and in situ modeling, where the method (5) was used, we applied the tabulated value f[cr] of the Fisher distribution statistics with (w, N[mea]−r) degrees of freedom (Phillips
1982). As an alternative, the critical value f[cr] is determined according to the predefined level of significance α in the modeling of the hypothesis H[0] (V = 0). Normally distributed random
variables were modeled using the Ziggurat method (Marsaglia et al. 2000). All the methods for detection of the near-zero apparent motion were analyzed on the same data set.
The following values of constants were used: the significance level is taken as α = 10^-3 and α = 10^-4; the number N[fr] of frames in a series is equal to N[fr] = (4,6,8,10,15). For modeling H[1] (V
> 0) hypothesis the velocity module V of the apparent motion was defined in relative terms, namely, rms error of measurement deviations of the object’s position (V = kσ).
Here the coefficient is equal to k = (0,0.5,1,1.25,1.5, 1.75,2,3,4,5,10). Mathematical expectation of external estimation of positional rms error is and its rms error is . If α = 10^-3, the maximum
allowable tabulated value of the Fisher distribution statistics with (1, 4) degrees of freedom is equal to f[cr] = 74.13 and if α = 10^-4, it is f[cr] = 241.62 (Melard 2014).
A method of statistical modeling for analysis of indicators of quality of the near-zero apparent motion detection in a series of CCD-frames. Conditional probability of the true detection (CPTD) is
calculated in terms of the frequency of LR estimates , or f(Ω[set]) exceeding the maximum allowable values λ[cr], or f[cr] for all methods of near-zero apparent motion detection: $Dtrue=Nexc/N1exp,$
(8)where N[exc] is the number of exceedings of the critical value λ[cr] for the substitutional methods of maximum likelihood or f[cr] for the method with f-criterion. CPTD estimation is determined
for the various number of frames N[fr] and various values of the apparent motion velocity module V.
Fig. 1
Curves of the near-zero apparent motion detection obtained by the method using Fisher f-criterion (1), substitutional methods with the known variance (2), with external estimations of rms error
0.15 (3) and rms error 0.25 (4).
Figure 1 (α = 10^-3) shows the curves of near-zero apparent motion detected by different methods: the Fisher f-criterion (5) method (curve 1); substitutional method for maximum likelihood detection
using the known variance of the position measurements (C.12) (curve 2); and substitutional method for maximum likelihood detection using external estimation of rms error (C.13) (curve 3) and (curve
Figure 2 (α = 10^-3) shows the curves of near-zero apparent motion detection obtained by the Fisher f-criterion method (5) with the critical tabulated value f[cr] of the Fisher distribution
statistics with (w, N[mea]−r) degrees of freedom (Phillips 1982) and the critical value f[cr] according to the predefined significance level α.
Fig. 2
Curves of the near-zero apparent motion detection obtained by the Fisher f-criterion method with the critical tabulated value (solid line) and the critical value according to the predefined
significance level α (dashed line).
A method of in situ modeling for analysis of indicators of quality of the near-zero apparent motion detection on a series of CCD-frames. In this case, it is impossible to restore the real law of the
errors’ distribution completely. The method of in situ modeling is, therefore, more appropriate (Kuzmyn 2000).
We compiled the set of objects with practically zero apparent motion in the framework of the CoLiTec project (Savanevych et al. 2015a,b) and used it as the internal catalog (IC) of motionless objects
in a series of frames (Vavilova et al. 2012a).
It is important to note that the objects exactly from the internal catalog were selected as in situ data. Because the positions of objects from this catalog are fixed, so deviations of their
estimated positions from their average value can be regarded as evaluations of their errors. These values can be used in the in situ modeling.
Further, these deviations should be added to the determined values of the object’s displacements according to their velocities of the apparent motion. Thereby, it is possible to use the real laws of
the positional errors distribution in the study of their motion by the in situ modeling method.
In situ data. Series of CCD-frames from observatories ISON-NM (MPC code – “H15”; Molotov et al. 2009) and ISON-Kislovodsk (MPC code – “D00”; ISON-Kislovodsk 2016) were selected as the in situ data.
The ISON-NM observatory is equipped with a 40 cm telescope SANTEL-400AN with CCD-camera FLI ML09000-65 (3056 × 3056 pixels, the pixel size is 12 microns). Exposure time was 150 s.
The ISON-Kislovodsk observatory is equipped with a 19.2 cm wide-field telescope GENON (VT-78) with CCD-camera FLI ML09000-65 (4008 × 2672 pixels, the pixel size is 9 microns). Exposure time was 180
s. Figures 3 and 4 show the curves of the near-zero apparent motion detection obtained by the Fisher f-criterion (5) and by the substitutional method of maximum likelihood with an external estimation
of rms error (C.13) for two sources of in situ data.
Fig. 3
Curves of the near-zero apparent motion detection with the SANTEL-400AN telescope obtained by the Fisher f-criterion method (solid line) and by the substitutional method with external estimation of
rms error 0.15 (dashed line).
Fig. 4
Curves of the near-zero apparent motion detection with the GENON (VT-78) telescope obtained by the Fisher f-criterion method (solid line) and by the substitutional method with external estimation
of rms error 0.15 (dashed line).
Analysis of indicators of quality of the near-zero apparent motion detection in a series of CCD-frames by the method of statistical modeling. Analyzing different approaches, we can note that the
substitutional methods of maximum likelihood detection with known variance of the object’s position (C.12) depicted by the curve 2 in Fig. 1, and the methods with external estimation of rms errors
(C.13) represented by the curve 3 in the same figure are the most sensitive to the object velocity changes. For example, CPTD of the near-zero apparent motion for these methods increases in the
series consisting of four frames and having V = 0.5σ. Here, σ is an rms error of the errors of estimated positions. For other methods the velocity module of the apparent motion is not less than V =
1.25σ, and if N[fr] = 6, not less than V = σ.
The curve 1 in Fig. 1 demonstrates that the near-zero apparent motion detection method with Fisher f-criterion (5) is not effective enough with the data of statistical modeling, when the number of
frames N[fr] is small. But if N[fr] is not less than eight, this method is not inferior to other ones by CPTD. In own turn, the substitutional method of maximum likelihood with the known variance of
the object’s position (C.12) exists only in theory and can not be applied in practice.
Hereby, the substitutional method of maximum likelihood with external estimation of rms error (C.13) described by curve 3 in Fig. 1 is the most effective and flexible. We remember that the external
estimation can be obtained from measurements of the other objects in CCD-frame.
On the other hand, the determination of critical values for all substitutional methods encounters formidable obstacles. First of all, it is not clear how to separate a set of stars (objects with a
zero rate motion) from the objects with a near-zero apparent motion to determine them. Also, this process is very time- and resource-consuming and difficult to apply in rapidly changing conditions of
observations in modern asteroid surveys.
In statistical modeling, the critical values f[cr] of the f-criterion determined according to the predefined significance levels are almost equal to the tabulated critical values of Fisher
distribution statistics with (w, N[mea]−r) degrees of freedom (Phillips 1982; Melard 2014) of the method (5). It is obviously seen in Fig. 2. Moreover, these figures demonstrate that the similarity
of these critical values of decisive statistic does not depend on the number of frames in the series.
Hence, it is not necessary to determine them for the different number of frames N[fr] and observation conditions. It is enough to use the maximum allowable tabulated value (Melard 2014).
Following from our statistical experiments, we can note that the method for the near-zero apparent motion detection with Fisher f-criterion (5) is more effective for the large number of CCD-frames
and the velocity module of the apparent motion V = 0.5σ as it’s seen in Fig. 2.
Analysis of indicators of quality of the near-zero apparent motion detection in a series of CCD-frames by the method of in situ modeling. It is found that the method for detection of the object’s
near-zero apparent motion using Fisher f-criterion (5) is the most sensitive to changes in the object’s velocity (Figs. 3, 4). As shown earlier, CPTD for this method increases when series includes
four frames or more and when V = 0.5σ. For other methods the velocity module of the apparent motion should be not less than V = 1.25σ.
In addition, the method of the near-zero apparent motion detection using Fisher f-criterion (5) is stable and does not depend on the kind of telescope (Fig. 5a). Therefore, there is no need to
undertake additional steps for determining the critical value of the decisive statistic after the equipment replacement or observational conditions change. Other methods of the apparent motion
detection encounter problems when determining the critical values as it is obvious from Fig. 5b.
Fig. 5
Curves of the near-zero apparent motion detection with the GENON (VT-78) (solid line) and SANTEL-400AN (dashed line) telescopes (α = 10^-3) obtained by the Fisher f-criterion method (a),
substitutional method for maximum likelihood detection with external estimation of rms error (b).
Examples of objects discovered by the method of near-zero apparent motion detection in a series of CCD-frames using significance criteria of the apparent motion. There are many of objects with
near-zero apparent motion that were detected by the CoLiTec software for automated asteroids and comets discoveries (Savanevych et al. 2015b). The plugin implements the method of detection using the
Fisher f-criterion (5). Table 1 gives information about several observatories at which the CoLiTec software is installed.
Table 1
Information about observatories and telescopes at which the CoLiTec software is installed.
The real-life examples of detection of asteroids 1917, 6063, 242211, 3288 and 1980, 20460, 138846, 166 with a near-zero apparent motion aredescribed in Tables 2 and 3, respectively.
The observations were conducted in 2017 in the period from 3 to 19 July with different small telescopes and confirmed an efficiency of the method even in bad conditions (strong backlight from the
full Moon).
Tables 2 and 3 contain the following apparent motion parameters of the aforementioned asteroids: date of observations; name of telescope; exposure time during the observation; apparent velocities of
object along each coordinate and in the rectangular coordinate system (CS; see, Appendix C, formulas (C.1), (C.2)); apparent velocities of objects and in the equatorial CS determined from the
observational data; apparent velocities of object and in the equatorial CS determined from the Horizons system (Giorgini et al. 2001) for the same times of observation; velocity module of the
apparent motion of object determined from the observational data ($V̂=V̂2x+V̂2y$); velocity module of the apparent motion of object determined from the Horizons system; average FWHM of object in five
frames; average S/N of object in five frames; rms error of stars positional estimates (C.7) from UCAC4 catalog (Zacharias et al. 2013) with S/N approximately equal to the object’s S/N; brightness Mag
[cat] of the object determined from the Horizons system; angular distance between the observed asteroid and the Moon; phase of the Moon, percentage illumination by the Sun; coefficient of the
velocity module of the apparent motion of object determined in relative terms, in other words, rms error of measurement deviations of the object’s position ().
Table 2
Examples of asteroids 1917, 6063, 242211, 3288 with a near-zero apparent motion that were detected by the proposed method using Fisher f-criterion (5).
Table 3
Examples of asteroids 1980, 20460, 138846, 166 with a near-zero apparent motion that were detected by the proposed method using Fisher f-criterion (5).
Discovery of the sungrazing comet C/2012 S1 (ISON). On September 21, 2012 the sungrazing comet C/2012 S1 (ISON) was discovered (Fig. 6) at the ISON-Kislovodsk Observatory (ISON-Kislovodsk 2016) of
the International Scientific Optical Network (ISON) project (Molotov et al. 2009; Minor Planet Center 2012). Information about observatory and telescope is available in Table 1.
Fig. 6
Sungrazing comet C/2012 S1 (ISON) at the moment of discovery in the center of crop of CCD-frame with field of view 20 × 20 arcmin (panel a), 8 × 8 arcminutes (panel b).
At the moment of discovery, the magnitude of the comet was equal to 18.8^m, and its coma had 10 arc seconds in diameter that corresponds to 50000 km at a heliocentric distance of 6.75 au. Its
apparent motion velocity at the moment of discovery was equal to 0.8 pixels per frame. The size of the comet image in the frame was about five pixels. In Fig. 7a the cell size corresponds to the size
of the pixel and is equal to 2 arc seconds. Within 26 min of the observation, the image of the comet has been moved by three pixels in the series of 4 CCD-frames (Fig. 7b).
Fig. 7
Panel a: images of C/2012 S1 (ISON) comet on CCD-frames: the image size is five pixels (a), the shift of comet image between the first and the fourth CCD-frames of series is three pixels (panel b).
C/2012 S1 (ISON) comet (Fig. 8) was detected using the CoLiTec software for automated asteroids and comets discoveries (Savanevych et al. 2015b) with the implemented method of detection using Fisher
f-criterion (5).
Fig. 8
Sungrazing comet C/2012 S1 (ISON) in a series of four CCD-frames.
C/2012 S1 (ISON) comet was disintegrated at an extremely small perihelion distance of about 1 million km on the day of perihelion passage, on November 28, 2013. Its disintegration was caused by the
Sun’s tidal forces and the significant mass loss due to the alterations in the moments of inertia of its nucleus. Despite having a short visible life time for our observations, this comet
supplemented our knowledge of cometary astronomy.
5. Conclusions
We proposed a computational method for the detection of objects with the near-zero apparent motion on a series of CCD-frames, which is based on the Fisher f-criterion (Phillips 1982) instead of using
the traditional decision rules that based on the maximum likelihood criterion (Myung 2003).
For the analysis of the indicators of quality of the apparent motion detection, we applied statistical and in situ modeling methods and determined their conditional probabilities of true detection
(CPTD) of the near-zero motion on a series of CCD-frames.
The statistical modeling showed that the most effective and adaptive method for the apparent motion detection is the substitutional method of maximum likelihood using the external estimation of rms
errors (C.13, Fig. 1). But the process of determining the critical values of decisive statistics is very time- and resource-consuming in the rapidly changing observational conditions. By this reason,
we recommended to apply the method of the near-zero apparent motion detection for the subclass of objects with 3σ-velocity using Fisher f-criterion (5) for series with the number of frames N[fr] = 4
or more (Fig. 1). The condition of a large number of frames in the series also makes the proposed method not inferior to other methods of apparent motion detection by CPTD.
When studying the indicators of quality of near-zero apparent motion detection by the in situ modeling method the objects from the internal catalog fixed on a series of CCD-frames were used as in
situ data. It was found that in the case when the velocity does not exceed 3 rms errors in object position per frame, the most effective method for near-zero apparent motion detection is the method
which uses Fisher f-criterion (Figs. 3, 4). When compared with other methods, this method is stable at the equipment replacement (Fig. 5).
The proposed method for detection of the objects with 3σ-velocity apparent motion using Fisher f-criterion was verified by authors and implemented in the embedded plugin developed in the CoLiTec
software for automated discovery of asteroids and comets (Savanevych et al. 2015b).
Among the other objects detected and discovered with this plugin, there was the sungrazing comet C/2012 S1 (ISON; Minor Planet Center 2012). The velocity of the comet apparent motion at the moment of
discovery was equal to 0.8 pixels per CCD-frame. Image size of the comet on the frame was about five pixels (Fig. 7a). Within 26 min of the observation, the image of the comet had moved by three
pixels in the series of four CCD-frames (Fig. 7b). So, it was considered to belong to the subclass of SSOs that have a velocity of apparent motion between CCD-frames not exceeding three rms errors σ
of measurements of its position (”= 3σ). In total, about 15% of SSO objects with ”-velocity apparent motion in the CCD-frames. These are the objects beyond the Jupiter’s orbit as well as asteroids
heading straight to the Earth.
The authors thank observatories that have implemented CoLiTec software for observations. We especially thank Vitaly Nevski and Artyom Novichonok for their discovery of ISON comet and others SSOs. We
are grateful to the reviewer for their helpful remarks that improved our paper and, in particular, for the suggestion “to add a few real-life examples, where the method provides a detection of motion
for an object that would otherwise be difficult to detect”. We express our gratitude to Mr. W. Thuillot, coordinator of the Gaia-FUN-SSO network (Thuillot et al. 2014), for the approval of CoLiTec as
a well-adapted software to the Gaia-FUN-SSO conditions of observation (https://gaiafunsso.imcce.fr). Research is supported by the APVV-15-0458 grant and the VVGS-2016-72608 internal grant of the
Faculty of Science, P. J. Safarik University in Kosice (Slovakia). The CoLiTec software is available on http://neoastrosoft.com.
Appendix A: Model of the motion parameters
The model of rectilinear and uniform motion of an object along each coordinate independently can be represented with the set of equations: $xk(θx)=x0+Vx(τk−τ0);yk(θy)=y0+Vy(τk−τ0),$where k(i,n) = k
is the index number of measurement in the set, namely, ith measurement of n[fr]th CCD-frame with the observed object; x[0], y[0] are the coordinates of object from the set of measurements at the time
τ[0] of the base frame timing; V[x], V[y] are the apparent velocities of object along each coordinate: $θx=(x0,Vx)T;θy=(y0,Vy)T;$are the vectors of the parameters of the apparent motion of the object
along each coordinate, respectively.
The measured coordinates x[k], y[k] at the time τ[k] are also determined by the parameters of the apparent motion of object in CCD-frame and can be calculated according to Eqs. (A.1) and (A.2).
So, the set of N[fr] measurements of n[fr]th frame timing at the time τ[n] is generated from observations of a certain area of the celestial sphere. One frame of the series is a base CCD-frame, and
time of its anchoring is the base frame timing τ[0]. The asteroid image on n[fr]th frame has no differences from the images of stars on the same frame. Results of intra-frame processing (one object
per CCD-frame) can be presented as the Y[in] measurement (ith measurement on the n[fr]th frame). In general, the ith measurement on the n[fr]th frame contains estimates of coordinates Y[Kin] = { x
[in];y[in] } and brightness A[in] of the object: Y[in] = { Y[Kin];A[in] }. We used a rectangular coordinate system (CS) with the center located in the upper left corner of CCD-frame. It is assumed
that all the positional measurements of the object are previously transformed into coordinate system of the base CCD-frame.
A set of measurements (no more than one in the frame), belonging to the object, has the form as follows: $Ωset=(YK1(i,1),...,YKk(i,n),...,YKNmea(i,Nfr))=((x1,y1),...,(xk,yk),...,(xNmea,yNmea)),$(A.5)
where N[mea] is the number of the position measurements of the object in N[fr] frames. Measurements Y[k] from the set Ω[set] (A.5) of measurements are selected by the rule of no more than one
measurement per frame. Measurements of the object positions can not be obtained in all CCD-frames. Therefore, the number of measurements which belong to the object in certain set of measurements will
generally be equal to N[mea] (N[mea] ≤ N[fr]).
It is supposed that the observational conditions are practically unchanged during observations of object with near-zero apparent motion. So, the rms errors of estimates of its coordinates in the
different CCD-frames are almost identical. Deviations of estimates of coordinates of this object, which belong to the same set Ω[set] of measurements, are independent of each other both inside the
one measurement and between measurements obtained in different frames. Deviations of coordinates are normally distributed (Kuzmyn 2000), have a zero mathematical expectation and unknown variances
(standard deviations) $σx2$, $σy2$.
Appendix B: Likelihood function for detection of a near-zero apparent motion
This common density distribution for H[0] hypothesis (1), assuming that the object is a star with zero rate apparent motion, is defined as follows: $f0(x̅,y̅,σ)=k=1Nmea[Nxk(x̅,σ2)Nyk(y̅,σ2)],$(B.1)where
, are the coordinates of the object; $Nz(mz,σ2)=12πσexp(−12σ2(z−mz)2)$ is the density of normal distribution with mathematical expectation m[z] and variance σ^2 in z point.
The common density distribution for H[1] hypothesis (2) is defined otherwise. Namely, the coordinates x[k](θ[x]), y[k](θ[y]) at the time τ[k], calculated from Eqs. (A.1) and (A.2), must be used
instead of the object’s position parameters , : $f1(θ,σ)=k=1Nmea[Nxk(xk(θx),σ2)Nyk(yk(θy),σ2)].$(B.2)Absence of information on the position of the object, its apparent motion and variance of
estimates of object position in a set of measurements leads to the necessity of using the substitutional decision rule (Lehman et al. 2010; Morey et al. 2014). In this case, the statistics for
distinguishing these hypotheses is the LR estimate (Morey et al. 2014).
Appendix C: Evaluation of parameters for substitutional methods of maximum likelihood detection of a near-zero apparent motion
OLS-evaluation of the parameters of the object’s apparent motion may be represented in the scalar form (Kuzmyn 2000): $x̂0=DAx−CBxNmeaD−C2;V̂x=NmeaBx−CAxNmeaD−C2;ŷ0=DAy−CByNmeaD−C2;V̂y=
NmeaBy−CAyNmeaD−C2,$where $Ax=∑k=1Nmeaxk$; $Ay=∑k=1Nmeayk$; $Bx=∑k=1NmeaΔτkxk$; $By=∑k=1NmeaΔτkyk$; $C=∑k=1NmeaΔτk$; $D=∑k=1NmeaΔτk2$; Δ[τk] = (τ[k]−τ[0]) is the difference between the time τ[0] of
the base frame and time τ[k] of the frame, in which the kth measurement is obtained.
The interpolated coordinates of the object in the kth frame are represented as $x̂k=x̂k(θ̂x)=x̂0(θ̂x)+V̂x(θ̂x)(τk−τ0);ŷk=ŷk(θ̂y)=ŷ0(θ̂y)+V̂y(θ̂y)(τk−τ0).$Thus, for each (kth) measurement from N[mea]
measurements of the set Ω[set] (A.5), we have:
• the unknown real position of the objectx[k](θ[x]), y[k](θ[y]);
• the measured object coordinates x[k], y[k] at the time τ[k] in the coordinate system of the base frame;
• the interpolated coordinates , defined by Eqs. (C.3) and (C.4).
The variance of the object’s positional estimates in a set of measurements. Using the measured x[k], y[k] (A.1), (A.2) and the interpolated , ŷ[k]) (C.3), (C.4) coordinates, the variance estimates
$σ̂2x$ and $σ̂2y$ (hereinafter – variances) of the object’s positions can be represented as: $σ̂2x=∑k=1Nmea(xk−x̂k(θ̂x))2/(Nmea−m);σ̂2y=∑k=1Nmea(yk−ŷk(θ̂y))2/(Nmea−m),$where m = 2 is the number of
parameters of the apparent motion along each coordinate in a set of measurements.
Assuming the validity of the hypothesis about zero (H[0]) and near-zero (H[1]) apparent motions, the conditional variances $σ̂20$, $σ̂21$ of the object’s position can be represented as: $σ̂20=R022
(Nmea−m);σ̂21=R122(Nmea−m),$where $R02=∑k=1Nmea((xk−x̂)2+(yk−ŷ)2);R12=∑k=1Nmea((xk−x̂k(θ̂x))2+(yk−ŷk(θ̂y))2),$are the residual sums of the squared deviations of object’s positions (Burden et al. 2010).
We note also that the variance of the positions in a set of measurements can be obtained by the external data, for example, from measurements of another objects on a series of CCD-frames. Hence, the
required estimate is a variance estimation of all position measurements of objects detected in CCD-frame and identified in any astrometric catalog.
Substitutional methods for maximum likelihood detection of a near-zero apparent motion may operate with unknown real position x[k](θ[x]), y[k](θ[y]) of the object at a time τ[k] and unknown variances
$σx2$, $σy2$ of the object’s position in CCD-frames.
It is easy to show that in the latter case the substitutional method can be represented as $R02−R12R02R12≥ln(λcr)ANmea,$(C.11)where λ[cr] is the maximum allowable (critical) value of the LR estimate
for the detection of a near-zero apparent motion; A = 2(N[mea]−m).
If the variance σ^2 of the object’s position is known, the substitutional method can be represented as $R02−R12≥2σ2ln(λcr).$(C.12)In that case, if the external variance estimation $σ̂2out$ of the
position is used, the substitutional method takes the form: $R02−R12σ̂2out≥2ln(λcr).$(C.13)
All Tables
Table 1
Information about observatories and telescopes at which the CoLiTec software is installed.
Table 2
Examples of asteroids 1917, 6063, 242211, 3288 with a near-zero apparent motion that were detected by the proposed method using Fisher f-criterion (5).
Table 3
Examples of asteroids 1980, 20460, 138846, 166 with a near-zero apparent motion that were detected by the proposed method using Fisher f-criterion (5).
All Figures
Fig. 1
Curves of the near-zero apparent motion detection obtained by the method using Fisher f-criterion (1), substitutional methods with the known variance (2), with external estimations of rms error
0.15 (3) and rms error 0.25 (4).
In the text
Fig. 2
Curves of the near-zero apparent motion detection obtained by the Fisher f-criterion method with the critical tabulated value (solid line) and the critical value according to the predefined
significance level α (dashed line).
In the text
Fig. 3
Curves of the near-zero apparent motion detection with the SANTEL-400AN telescope obtained by the Fisher f-criterion method (solid line) and by the substitutional method with external estimation of
rms error 0.15 (dashed line).
In the text
Fig. 4
Curves of the near-zero apparent motion detection with the GENON (VT-78) telescope obtained by the Fisher f-criterion method (solid line) and by the substitutional method with external estimation
of rms error 0.15 (dashed line).
In the text
Fig. 5
Curves of the near-zero apparent motion detection with the GENON (VT-78) (solid line) and SANTEL-400AN (dashed line) telescopes (α = 10^-3) obtained by the Fisher f-criterion method (a),
substitutional method for maximum likelihood detection with external estimation of rms error (b).
In the text
Fig. 6
Sungrazing comet C/2012 S1 (ISON) at the moment of discovery in the center of crop of CCD-frame with field of view 20 × 20 arcmin (panel a), 8 × 8 arcminutes (panel b).
In the text
Fig. 7
Panel a: images of C/2012 S1 (ISON) comet on CCD-frames: the image size is five pixels (a), the shift of comet image between the first and the fourth CCD-frames of series is three pixels (panel b).
In the text
Fig. 8
Sungrazing comet C/2012 S1 (ISON) in a series of four CCD-frames.
In the text
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C Program to Sort an Array in Ascending Order - Ccodelearner
• C Program to Sort an Array in Ascending Order
Welcome to this comprehensive guide on sorting an array in ascending order using the C programming language. Sorting an array is a fundamental operation in computer science and is used in various
applications. Whether you are new to programming or already familiar with the concept, understanding how to sort an array is essential.
In this blog post, we will cover the basics of sorting algorithms, explain how to implement a C program to sort an array in ascending order, and provide detailed insights and tips along the way.
So, let’s get started!
Sorting refers to the arrangement of elements in a particular order, such as ascending or descending. When it comes to sorting arrays in C, there are several well-known algorithms to choose from.
These sorting algorithms differ in terms of their efficiency, simplicity, and performance characteristics.
In this blog post, we will focus on the selection sort algorithm, which is a simple yet effective sorting technique. We will walk you through the step-by-step process of implementing a C program
that sorts an array in ascending order using the selection sort algorithm.
Basics of Sorting Algorithms
Before diving into the implementation details, let’s briefly discuss some basic concepts related to sorting algorithms. Sorting algorithms can be categorized based on their approach, such as
comparison-based sorting and non-comparison-based sorting.
Comparison-Based Sorting
Most sorting algorithms, including the selection sort algorithm, fall under the category of comparison-based sorting. These algorithms compare pairs of elements and reorder them based on certain
criteria. The criteria could be the values themselves or some derived key associated with each element.
Non-Comparison-Based Sorting
Non-comparison-based sorting algorithms, such as counting sort or radix sort, do not rely on pairwise comparisons. Instead, they exploit additional information about the elements to sort them
efficiently. While these algorithms can achieve faster sorting times in certain scenarios, they are often specialized for particular data types or constraints.
For the purpose of this blog post, we will focus on comparison-based sorting algorithms, as they are more general and widely applicable.
Selection Sort Algorithm
The selection sort algorithm is based on the idea of finding the minimum (or maximum) element in an array and placing it at the beginning (or end) of the array. By repeatedly performing this
operation on the remaining unsorted portion of the array, the entire array becomes sorted in ascending (or descending) order.
The selection sort algorithm proceeds as follows:
1. Find the minimum element in the unsorted portion of the array.
2. Swap the minimum element with the first element in the unsorted portion.
3. Move the boundary of the sorted portion one element to the right.
4. Repeat steps 1-3 until the entire array becomes sorted.
The time complexity of the selection sort algorithm is O(n^2), where n is the number of elements in the array. This makes the selection sort algorithm suitable for small to medium-sized arrays,
but it may become inefficient for larger arrays.
Implementation of C Program to Sort an Array
Now that we have a good understanding of the selection sort algorithm, let’s dive into the implementation details. We will walk you through the step-by-step process of writing a C program to sort
an array in ascending order using the selection sort algorithm.
Here is the C code for sorting an array using the selection sort algorithm:
#include <stdio.h>
// Function to swap two elements
void swap(int* a, int* b)
int temp = *a;
*a = *b;
*b = temp;
// Function to perform selection sort
void selectionSort(int arr[], int n)
int i, j, min_idx;
// One by one move the boundary of the unsorted subarray
for (i = 0; i < n - 1; i++) {
// Find the minimum element in the unsorted array
min_idx = i;
for (j = i + 1; j < n; j++)
if (arr[j] < arr[min_idx])
min_idx = j;
// Swap the found minimum element with the first element
swap(&arr[min_idx], &arr[i]);
// Function to print the array
void printArray(int arr[], int n)
int i;
for (i = 0; i < n; i++)
printf("%d ", arr[i]);
int main()
int arr[] = { 64, 25, 12, 22, 11 };
int n = sizeof(arr) / sizeof(arr[0]);
printf("Array before sorting: ");
printArray(arr, n);
selectionSort(arr, n);
printf("\nArray after sorting in ascending order: ");
printArray(arr, n);
return 0;
The above C program consists of four functions: swap(), selectionSort(), printArray(), and the main function.
The swap() function is used to swap the values of two elements in the array. It takes two pointers as arguments and uses a temporary variable to perform the swapping operation.
The selectionSort() function performs the selection sort algorithm on the given array. It iterates over the unsorted portion of the array, finds the minimum element, and swaps it with the first
element of the unsorted portion. This process is repeated until the entire array becomes sorted.
The printArray() function is responsible for printing the elements of the array in a formatted manner.
In the main() function, we create an example array, arr, and determine its size. We then print the array before sorting, call the selectionSort() function to sort the array in ascending order,
and finally print the sorted array.
Testing and Running the Program
Let’s test and run the C program to see the selection sort algorithm in action. Copy the above code into a C compiler, compile and execute it. You should see the following output:
Array before sorting: 64 25 12 22 11
Array after sorting in ascending order: 11 12 22 25 64
As you can see, the array is successfully sorted in ascending order using the selection sort algorithm.
You can also try running the program with your own array elements to get different results. This will help you understand the behavior of the selection sort algorithm in different scenarios.
Performance Analysis
Now that we have implemented the selection sort algorithm and tested our C program, let’s briefly analyze its performance characteristics.
The selection sort algorithm has a time complexity of O(n^2). This means that the time taken to sort an array using selection sort grows quadratically with the number of elements in the array. As
a result, the algorithm becomes slower as the size of the array increases.
Due to its nested iteration structure, the selection sort algorithm also has a space complexity of O(1). This means that the algorithm uses a constant amount of additional memory, regardless of
the size of the input array.
While the selection sort algorithm is simple to understand and implement, it is not the most efficient sorting algorithm available. If you are working with large arrays or require faster sorting
times, you may want to consider alternative sorting algorithms such as quicksort, mergesort, or heapsort.
In this blog post, we have covered the basics of sorting algorithms and provided a comprehensive guide on implementing a C program to sort an array in ascending order using the selection sort
We started by introducing the topic, discussing the basics of sorting algorithms, and specifically focusing on comparison-based sorting algorithms. We then explained the selection sort algorithm
in detail, providing a step-by-step breakdown of its implementation.
Following that, we presented the complete C program to sort an array using the selection sort algorithm and discussed the individual functions involved. We also provided instructions on how to
test and run the program, allowing you to see the sorting algorithm in action.
Lastly, we briefly analyzed the performance characteristics of the selection sort algorithm, highlighting its time and space complexities. We also mentioned alternative sorting algorithms for
more efficient sorting in certain scenarios.
Sorting an array is an important skill for any programmer, and understanding the underlying algorithms is crucial for developing efficient and optimized code. We hope this blog post has provided
you with a solid foundation and practical knowledge in sorting arrays using the selection sort algorithm in C.
If you want to further explore this topic, we encourage you to experiment with the code, try different input arrays, and explore alternative sorting algorithms. The more you practice and play
around with sorting algorithms, the better you will become at implementing efficient solutions.
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In formal terms, a directed graph is an ordered pair G = (V, A) where^[1]
• V is a set whose elements are called vertices, nodes, or points;
• A is a set of ordered pairs of vertices, called arcs, directed edges (sometimes simply edges with the corresponding set named E instead of A), arrows, or directed lines.
It differs from an ordinary or undirected graph, in that the latter is defined in terms of unordered pairs of vertices, which are usually called edges, links or lines.
The aforementioned definition does not allow a directed graph to have multiple arrows with the same source and target nodes, but some authors consider a broader definition that allows directed graphs
to have such multiple arcs (namely, they allow the arc set to be a multiset). Sometimes these entities are called directed multigraphs (or multidigraphs).
On the other hand, the aforementioned definition allows a directed graph to have loops (that is, arcs that directly connect nodes with themselves), but some authors consider a narrower definition
that does not allow directed graphs to have loops.^[2] Directed graphs without loops may be called simple directed graphs, while directed graphs with loops may be called loop-digraphs (see section
Types of directed graph).
Types of directed graphs
A simple directed acyclic graph
A tournament on 4 vertices
• Symmetric directed graphs are directed graphs where all edges appear twice, one in each direction (that is, for every arrow that belongs to the digraph, the corresponding inverse arrow also
belongs to it). (Such an edge is sometimes called "bidirected" and such graphs are sometimes called "bidirected", but this conflicts with the meaning for bidirected graphs.)
• Simple directed graphs are directed graphs that have no loops (arrows that directly connect vertices to themselves) and no multiple arrows with same source and target nodes. As already
introduced, in case of multiple arrows the entity is usually addressed as directed multigraph. Some authors describe digraphs with loops as loop-digraphs.^[2]
□ Complete directed graphs are simple directed graphs where each pair of vertices is joined by a symmetric pair of directed arcs (it is equivalent to an undirected complete graph with the edges
replaced by pairs of inverse arcs). It follows that a complete digraph is symmetric.
□ Semicomplete multipartite digraphs are simple digraphs in which the vertex set is partitioned into sets such that for every pair of vertices x and y in different sets, there is an arc between
x and y. There can be one arc between x and y or two arcs in opposite directions.^[3]
□ Semicomplete digraphs are simple digraphs where there is an arc between each pair of vertices. Every semicomplete digraph is a semicomplete multipartite digraph in a trivial way, with each
vertex constituting a set of the partition.^[4]
□ Quasi-transitive digraphs are simple digraphs where for every triple x, y, z of distinct vertices with arcs from x to y and from y to z, there is an arc between x and z. There can be just one
arc between x and z or two arcs in opposite directions. A semicomplete digraph is a quasi-transitive digraph. There are extensions of quasi-transitive digraphs called k-quasi-transitive
□ Oriented graphs are directed graphs having no opposite pairs of directed edges (i.e. at most one of (x, y) and (y, x) may be arrows of the graph). It follows that a directed graph is an
oriented graph if and only if it has no 2-cycle.^[6] Such a graph can be obtained by applying an orientation to an undirected graph.
☆ Tournaments are oriented graphs obtained by choosing a direction for each edge in undirected complete graphs. A tournament is a semicomplete digraph.^[4]
☆ A directed graph is acyclic if it has no directed cycles. The usual name for such a digraph is directed acyclic graph (DAG).^[7]
○ Multitrees are DAGs in which there are no two distinct directed paths from the same starting vertex to the same ending vertex.
○ Oriented trees or polytrees are DAGs formed by orienting the edges of trees (connected, acyclic undirected graphs).
■ Rooted trees are oriented trees in which all edges of the underlying undirected tree are directed either away from or towards the root (they are called, respectively,
arborescences or out-trees, and in-trees.
Digraphs with supplementary properties
• Weighted directed graphs (also known as directed networks) are (simple) directed graphs with weights assigned to their arrows, similarly to weighted graphs (which are also known as undirected
networks or weighted networks).^[2]
□ Flow networks are weighted directed graphs where two nodes are distinguished, a source and a sink.
• Rooted directed graphs (also known as flow graphs) are digraphs in which a vertex has been distinguished as the root.
□ Control-flow graphs are rooted digraphs used in computer science as a representation of the paths that might be traversed through a program during its execution.
• Signal-flow graphs are directed graphs in which nodes represent system variables and branches (edges, arcs, or arrows) represent functional connections between pairs of nodes.
• Flow graphs are digraphs associated with a set of linear algebraic or differential equations.
• State diagrams are directed multigraphs that represent finite state machines.
• Commutative diagrams are digraphs used in category theory, where the vertices represent (mathematical) objects and the arrows represent morphisms, with the property that all directed paths with
the same start and endpoints lead to the same result by composition.
• In the theory of Lie groups, a quiver Q is a directed graph serving as the domain of, and thus characterizing the shape of, a representation V defined as a functor, specifically an object of the
functor category FinVct[K]^F(Q) where F(Q) is the free category on Q consisting of paths in Q and FinVct[K] is the category of finite-dimensional vector spaces over a field K. Representations of
a quiver label its vertices with vector spaces and its edges (and hence paths) compatibly with linear transformations between them, and transform via natural transformations.
Basic terminology
Oriented graph with corresponding incidence matrix
An arc (x, y) is considered to be directed from x to y; y is called the head and x is called the tail of the arc; y is said to be a direct successor of x and x is said to be a direct predecessor of y
. If a path leads from x to y, then y is said to be a successor of x and reachable from x, and x is said to be a predecessor of y. The arc (y, x) is called the reversed arc of (x, y).
The adjacency matrix of a multidigraph with loops is the integer-valued matrix with rows and columns corresponding to the vertices, where a nondiagonal entry a[ij] is the number of arcs from vertex i
to vertex j, and the diagonal entry a[ii] is the number of loops at vertex i. The adjacency matrix of a directed graph is a logical matrix, and is unique up to permutation of rows and columns.
Another matrix representation for a directed graph is its incidence matrix.
See direction for more definitions.
Indegree and outdegree
A directed graph with vertices labeled (indegree, outdegree)
For a vertex, the number of head ends adjacent to a vertex is called the indegree of the vertex and the number of tail ends adjacent to a vertex is its outdegree (called branching factor in trees).
Let G = (V, E) and v ∈ V. The indegree of v is denoted deg^−(v) and its outdegree is denoted deg^+(v).
A vertex with deg^−(v) = 0 is called a source, as it is the origin of each of its outcoming arcs. Similarly, a vertex with deg^+(v) = 0 is called a sink, since it is the end of each of its incoming
The degree sum formula states that, for a directed graph,
${\displaystyle \sum _{v\in V}\deg ^{-}(v)=\sum _{v\in V}\deg ^{+}(v)=|E|.}$
If for every vertex v ∈ V, deg^+(v) = deg^−(v), the graph is called a balanced directed graph.^[8]
Degree sequence
The degree sequence of a directed graph is the list of its indegree and outdegree pairs; for the above example we have degree sequence ((2, 0), (2, 2), (0, 2), (1, 1)). The degree sequence is a
directed graph invariant so isomorphic directed graphs have the same degree sequence. However, the degree sequence does not, in general, uniquely identify a directed graph; in some cases,
non-isomorphic digraphs have the same degree sequence.
The directed graph realization problem is the problem of finding a directed graph with the degree sequence a given sequence of positive integer pairs. (Trailing pairs of zeros may be ignored since
they are trivially realized by adding an appropriate number of isolated vertices to the directed graph.) A sequence which is the degree sequence of some directed graph, i.e. for which the directed
graph realization problem has a solution, is called a directed graphic or directed graphical sequence. This problem can either be solved by the Kleitman–Wang algorithm or by the Fulkerson–Chen–Anstee
Directed graph connectivity
A directed graph is weakly connected (or just connected^[9]) if the undirected underlying graph obtained by replacing all directed edges of the graph with undirected edges is a connected graph.
A directed graph is strongly connected or strong if it contains a directed path from x to y (and from y to x) for every pair of vertices (x, y). The strong components are the maximal strongly
connected subgraphs.
A connected rooted graph (or flow graph) is one where there exists a directed path to every vertex from a distinguished root vertex.
See also
External links
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Skewness and Kurtosis Calculator | Online Tutorials Library List | Tutoraspire.com
Skewness and Kurtosis Calculator
by Tutor Aspire
Skewness is a measure of the asymmetry of a dataset or distribution. This value can be positive or negative. A negative skew typically indicates that the tail is on the left side of the distribution.
A positive value typically indicates that the tail is on the right.
Kurtosis is simply a measure of the “tailedness” of a dataset or distribution. The kurtosis formula used by this calculator is identical to the formula used in Excel, which finds what is known as
excess kurtosis.
To find the skewness and kurtosis of a dataset, simply enter the comma-separated values in the box below, then click the “Calculate” button.
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Combined actions resistance - Member capacity (Columns: AS 4100)
The combined actions member capacity check is performed according to AS 4100 clause 8.4
The higher tier equations will be used automatically if the conditions for their use are met.
In the higher tier equation for M[i], the ratio β[m] will be based on the relevant strut length; if the strut length has bending induced by transverse load within its length then β[m] will be taken
as -1.0, and the ratio of end moments otherwise.
In the higher tier equation for M[ox], the ratio β[m] will be based on the LTB segment length, and taken as the ratio of end moments, using real moments only.
In the member capacity check, the design forces are the maxima in the design length being considered, where the design lengths are based on the major and minor strut lengths within a loop of LTB
Therefore, since any one design length will comprise both major and minor strut lengths, the design axial force for each design length will be taken as the maximum axial compression or axial tension
force from the major and minor strut lengths considered together.
Note that if the design axial force exceeds the design axial member capacity then the check will automatically be set as Fail.
Since both axial compression and axial tension are to be considered, but make use of different equations, then in cases where both axial forces exist within a design length the compression equations
and tension equations will both be evaluated and the worst case of the two will be reported.
Note that in bi-axial bending cases, zero axial force will be treated as compression. | {"url":"https://support.tekla.com/doc/tekla-structural-designer/2024/ref_combinedactionsresistancemembercapacitycolumnsas4100","timestamp":"2024-11-04T17:50:07Z","content_type":"text/html","content_length":"52895","record_id":"<urn:uuid:accf8033-683c-4e3e-b765-6a75507140d6>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00105.warc.gz"} |
Properties of Circles Worksheets
Conic Sections
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Values the Radius and Center point may contain
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Algebra 2 - Conic Sections: Properties of Circles Tests with Answers
Interactive PDF worksheet: Printing is not necessary! | {"url":"https://k12xl.com/algebra-2/conic-sections/properties-of-circles-worksheets","timestamp":"2024-11-15T03:33:19Z","content_type":"text/html","content_length":"84848","record_id":"<urn:uuid:8e9843d1-45da-4a57-8469-8f47213a50d7>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00441.warc.gz"} |
Franklin Pezzuti Dyer
More Functional Equations
2017 Aug 15
NOTE: I will be using superscripts in three different ways in this post, so be careful not to get confused. The first way is as an exponent. The expression reads "f of x, raised to the nth power."
The second way is to represent iteration. The expression reads "the nth iterate of f of x." The last way is to represent multiple derivatives. The expression reads "the nth derivative of f of x.
It's time for some more functional equations!
In this previous post, I looked at a few interesting functional equations, and in retrospect, it doesn't seem like they were very difficult. The answers were mostly polynomial functions that could be
solved by differentiating a few times. This time, however, I have some very hard functional equations to try.
Find non-constant functions (if they exist) with the following properties:
Let's start with the first problem. It is rather easy, but it is a good problem to start with. This can be simplified with a technique that is analogous to substitution in normal equations. To
simplify the equation, we can define so that we have and making the substitution $\log_2 x\to x$, we have At this point, you should be able to recognize a similarity between this equation and another
functional equation involving the trigonometric functions: or Noticing this yields the obvious solution or
Now it's time for the hard problems!
I used differentiation in my previous post about functional equations, but only to solve functional equations with polynomial solutions. I would like to show how to use differentiation to solve other
types of functional equations. For example, the second problem: This one stumped me for a while, and I was rather annoyed to find out that there was no non-constant solution. Watch what happens when
we differentiate: And if we differentiate again: and so on: Which means that the nth derivative of $\beta$ at $x=0$ is and so, for all positive integer $n$, And, of course, the only function all of
whose derivatives are equal to zero is the constant function, meaning that there exists no non-constant $\beta$ satisfying the functional equation.
On to the third one: The solution to this one is rather complicated. You could try the method that we used on the first problem, but you would end up with Yuck.
As it turns out, the answer may not even have a closed-form representation, and the answe has to be expressed as a Taylor Series. In case you are not familiar with the idea, the Taylor Series of a
function $f$ is and Taylor's Theorem states that This can be very helpful with some functional equations. If we differentiate both sides of our functional equation $n$ times, we end up with and so
and so
The next functional equation is and it uses the same strategy, but required a bit of manipulation before iterated differentiation can be applies. First we must take the natural logarithm of both
sides: Now we must make a substitution. Define $\psi_1$ as $\ln\delta(x)=\psi(x)$ so that our functional equation is transformed into Now we can differentiate both sides $n$ times to get for $n\ge
1$. Then we have for $n\ge 1$. Then, since we have all but the first term of the Taylor Series (which is zero anyways since $\ln (1)=0$, so we don't need to worry about it) we can express $\psi_1$ as
and so, when we reverse our substitution, we have
Of course, this solution isn't perfect - since we used the natural logarithm and obtained $\ln(x+1)$ on the right side of one of our equations, we basically invalidated our answer for any values of
$x$ for which $\ln(x+1)$ is nonreal; namely, $x\le 0$.
Now let's try and tackle those last two functional equations.
Before we try the next one, I would like to revisit a problem from my previous post about functional equations. Here it is: The method that I used to solve it was to substitute $x\to \frac{2}{x}$ to
get or I then took the original equation and this new equation: and treated them like a system of equations to be solved. If the second equation is multiplied by two, we obtain and when we add this
to the original equation, we get
Notice now that if we have any functional equation in the form Where we are trying to solve for $f$, $g$ is given, $a$ is a constant, and $\theta$ has the property In fact, a formula can be obtained
for $f$ in this equation. Using the same method as we did with my example, through the substitution $x\to\theta(x)$ repeated $n-1$ times, we can obtain the system of equations Now we multiply the
second equation by $-a$, the third by $a^2$, the fourth by $-a^3$, and so on, until we multiply the last equation by $(-a)^{n-1}$, which leaves us with And now watch what happens when we add this
system together. When we add the first two, we have and when we add this to the third one: and if you keep adding them up until the very last one, you will eventually end up with and Which is our
We just need one more thing before we try the next functional equation. I'm not going to go over it again in this post, but if you haven't already, you should go back to this post and read the very
end of it.
You should be able to notice now when you look at the next functional equation That this fits the form of our formula perfectly, with $a=-1$, $g(x)=\ln(x)$, and and after careful observation using
the information found in my previous post, we have $n=100$. Additionally, we can determine that and so, by plugging into the formula, we have There's no need to evaluate this sum - this answer is
good enough.
The last functional equation is similar to the first in that it is best solved using substitution and intuition, but it is much more difficult. This can be rearranged to get Let us now make the
substitution $\psi_1(2^x)=\zeta(x)$. Then and by taking $2^x\to x$,
At this point, you may be wondering how I decided to make those substitutions. This is where the intuition part of the problem comes in.
Upon seeing the problem, it was immediately obvious to me that the answer would not end up being a non-constant polynomial. This can be easily determined by noticing that if it was a polynomial with
degree $d$, then the left side of the functional equation would also have degree $d$, whereas the right side would have degree $2d$.
Once I found this out, I began to think that the solution would involve trigonometric functions, since there are so many identities relating the trigonometric functions to their squares. And when I
factored the right side as I was reminded of the pythagorean identities almost immediately.
If you were unable to figure this one out before, stop reading now and let what I have revealed so far be a hint to you.
Because of my suspicions about the pythagorean identities, I made another substitution by letting $\sin^2 \psi_2(x)=\psi_1(x)$. Then I got At this point I realized that I could just let $\psi_2(x)=
x$, because my equation had turned into the double-angle identity. When I substituted back in to find $\zeta$, I ended up with Which I thought was a rather satisfying solution.
That's all that I am going to do with functional equations for now. | {"url":"https://franklin.dyer.me/post/80","timestamp":"2024-11-11T23:17:46Z","content_type":"text/html","content_length":"18729","record_id":"<urn:uuid:e331f68c-2674-4227-8026-3cce5e07fc1c>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00653.warc.gz"} |
Quantum Monte Carlo Simulation of the 3
SciPost Submission Page
Quantum Monte Carlo Simulation of the 3D Ising Transition on the Fuzzy Sphere
by Johannes S. Hofmann, Florian Goth, Wei Zhu, Yin-Chen He, Emilie Huffman
This is not the latest submitted version.
This Submission thread is now published as
Submission summary
Authors (as registered SciPost users): Johannes Stephan Hofmann · Emilie Huffman
Submission information
Preprint Link: scipost_202401_00004v1 (pdf)
Date submitted: 2024-01-05 20:52
Submitted by: Huffman, Emilie
Submitted to: SciPost Physics Core
Ontological classification
Academic field: Physics
• Condensed Matter Physics - Computational
Specialties: • High-Energy Physics - Theory
Approaches: Theoretical, Computational
We present a numerical quantum Monte Carlo (QMC) method for simulating the 3D phase transition on the recently proposed fuzzy sphere [Phys. Rev. X 13, 021009 (2023)]. By introducing an additional $SU
(2)$ layer degree of freedom, we reformulate the model into a form suitable for sign-problem-free QMC simulation. From the finite-size-scaling, we show that this QMC-friendly model undergoes a
quantum phase transition belonging to the 3D Ising universality class, and at the critical point we compute the scaling dimensions from the state-operator correspondence, which largely agrees with
the prediction from the conformal field theory. These results pave the way to construct sign-problem-free models for QMC simulations on the fuzzy sphere, which could advance the future study on more
sophisticated criticalities.
Current status:
Has been resubmitted
Reports on this Submission
Report #3 by Anonymous (Referee 3) on 2024-3-25 (Invited Report)
• Cite as: Anonymous, Report on arXiv:scipost_202401_00004v1, delivered 2024-03-25, doi: 10.21468/SciPost.Report.8765
1 -The new four-flavor fermions model overcomes the sign problem for QMC of the fuzzy-ball regularization approach to 3D Ising criticality.
2 - The paper is well-written and provides the necessary technical details of the analytical derivations.
3 - The numerical part provides a careful finite-size scaling analysis over several measures.
1 - The performance of the current model compared to the traditional MC approach, such as the lattice Ising model, remains unclear.
2 - The proposed model seems to lack an intuitive physical interpretation linking it to actual physical systems. While it serves as a numerical tool for determining critical exponents, its complexity
may hinder future analytical progress.
The paper presents a new four-flavor fermions model as an extension of the authors' recent work on the fuzzy-ball regularization approach to 3D Ising criticality. In contrast to the existing model,
which consists of two-flavor fermions and suffers from the sign problem in Quantum Monte Carlo (QMC) simulations, the new model introduces an extra layer of SU(2) symmetry that protects the model
from the sign problem. The paper is well-written, offering the necessary technical details for the analytical derivations.
The numerical section provides a thorough finite-size scaling analysis across several measures. However, the current model's performance relative to traditional Monte Carlo approaches, such as the
lattice Ising model, remains unclear. Including a computational complexity analysis, even if only numerical, would significantly enhance the model's validity.
Additionally, the proposed model appears overly complex when attempting to link it to actual physical systems. Although it serves as a numerical tool for determining critical exponents, its
complexity could impede future analytical advancements.
Requested changes
1 - Include a computational complexity analysis.
2 - The parameters V1 = 0.1 and V2 = 0.5564 appear to be fine-tuned, particularly for V2, which is chosen with four significant figures. Although the authors mention that this choice minimizes
finite-size effects, it would be valuable to examine the results for other parameters.
Requested changes:
1. We thank the reviewer for this suggestion and have added a discussion of the N^4 scaling under equation 16 in the QMC simulations section of Results.
2. At the Ising critical point, the finite-size effect can be significantly reduced by tuning away the irrelevant operator epsilon’ (with scaling dimension ~3.83). In the present work, we tune the
interaction parameters to reach this goal, i.e. g1=0.4 and g0=1.3 from the exact diagonalization. By changing these numbers to the conventions of QMC parameters, it gives the repeating decimals
V0=0.61818181... and V1=0.1111111... We then scaled them so that V1=0.1, giving the repeating decimal V0=0.55636363... In principle, the Ising transition and its typical critical exponents should
not depend on the specific parameters in the model setup in the UV limit. But the RG flow distances to the IR fixed point are distinct, if choosing the different parameters in the UV limit. So
when we tune away from the "sweet point" as shown above, the finite-size effect will become significant and one needs much heavier computation on larger system sizes to reach the critical
We have attached an updated version of the manuscript.
Report #2 by Anonymous (Referee 2) on 2024-3-22 (Invited Report)
• Cite as: Anonymous, Report on arXiv:scipost_202401_00004v1, delivered 2024-03-22, doi: 10.21468/SciPost.Report.8746
1 - The presented method shows promise to circumvent the fermion sign problem for a new class of models.
2 - The paper presents a QMC implementation of the recently introduced fuzzy sphere regularization and takes care of benchmarking of known results.
4 - The paper is clearly written and self-contained concerning the technical steps.
1 - It is not clear whether the paper meets the criterion of presenting new research results that significantly advance the current understanding of the field as it is limited to one benchmarking
2- It does not become clear whether the method can substantially advance open problems in the field.
3 - The benchmarking seems to only allow for a check of consistency with the Ising universality class, but not to make actual precision predictions.
This manuscript presents an interesting sign-free implementation of the quantum Monte Carlo method employing the recently proposed fuzzy sphere regularization. It studies the quantum phase transition
and critical behavior as a function of the transverse field and shows that the behavior is compatible with 3D Ising universality. The authors claim that this strategy can be used to generalize to
other models and simultaneously avoid the notorious sign problem.
The content of the paper is interesting, it is well written, self-contained, and the results are presented clearly. The benchmark for Ising criticality presents an interesting proof of concept for
the method. However, I'm missing a concrete example and actual results that go beyond the benchmark example.
If this is clearly beyond the scope of the current work, it should be argued more clearly why -- despite the limitation to the benchmarking result -- the presented result meets the acceptance
criteria, i.e., that the work significantly advances the current understanding of the field and that it not only reproduces known results. A better alternative would be to find and calculate an
actual example that goes beyond established results.
Requested changes
1 - Above Eq. (3), the authors mention that they consider the limit that the kinetic energy is much larger than the interaction energy. It should be discussed how restrictive this constraint and how
it affects the presented results.
2 - The authors should discuss more clearly, whether consistency is all that can be expected to be achieved within the new method or whether there is a realistic perspective to achieve quantitative
precision results, e.g., on critical exponents.
3 - If the answer to the previous question is that quantitative results can be achieved, the authors should discuss how expensive it would be to do this for their current model or -- even better --
provide such quantitative precision results.
4 - I'm worried that the fitting in Sec. 3.2 may be affected by some sort of overfitting due to the presence of many parameters. The authors should comment on that.
5 - The quantity chi^2 should be introduced/defined. It might not be obvious to everyone outside the numerical community.
6 - The authors should consider to discuss the possible sources/implications of corrections to scaling and how they might affects their results.
7 - An interesting non-trivial benchmark could be the critical behavior of Dirac fermions, where high precision estimates are available from the conformal bootstrap community, but the QMC results are
not fully settled yet. Maybe the authors could consider this.
We would like to thank the referee for taking the time to read our manuscript and for providing a report that helped us to improve this paper. We are happy to hear that the referee finds our work
interesting and for rating the originality of this study as 'High'.
The target of the present work is to introduce a new method, namely the fuzzy sphere QMC, and the feasibility of this approach is demonstrated via the benchmark on the Ising model. We believe this
approach will advance the coming studies on more challenging problems, which we leave for the future work. We do think, as a paper introducing new techniques, it should not influence the quality of
the current paper.
First, we agree that high-precision results for critical exponents have already been obtained for the Ising universality class by various methods such as the conformal bootstrap, and QMC for spin
lattice models. Instead of competing with these already very efficient methods for the Ising universality class, the fuzzy sphere setup allows us to examine the proposed emergent conformal symmetry
at the critical point, and further make use of the conformal symmetry to compute universal properties such as critical exponents.
Second, in a previous study introducing the fuzzy sphere setup, we have used both exact diagonalization and DMRG to identify the Ising critical point and analyze the Hamiltonian spectrum at
criticality to verify the emergent conformal symmetry. However, both methods struggle with increasing fermion degrees of freedom, due to the exponential compute cost. Here, we introduce a polynomial
algorithm to overcome these limitations and benchmark the new method using the Ising criticality.
Finally, the polynomial scaling allows us to study models with larger local Hilbert spaces and the plethora of interesting quantum critical phenomena therein. We agree with the referee that it would
be rewarding to study critical Dirac fermions, for example. However, this is left for future work.
Hence, we strongly believe that the presented research is already an important result and meets the acceptance criteria of SciPost. A detailed response to the list of requested changes can be found
below. With these changes we believe the manuscript is ready for publication.
Requested changes:
1. We agree that it is useful to clarify the projection. As given in the manuscript, we have H = H_{kin} + H_{int}, with H_{kin} giving rise to Landau levels and H_{int} the Hamiltonian of physical
interest. The level spacing of the Landau levels is controlled by the cyclotron frequency $\omega_c$, and we consider a half-filled lowest Landau level (LLL). In general, the interaction scale
could induce excitations to higher Landau levels, however, this can be avoided by taking the limit $\omega_c \rightarrow \infty$. Hence, the projection of H_int, encoding the physics of interest,
to the LLL is not posing any constraint. We have added a couple of sentences above equation (3) to clarify this.
2.,3. To reiterate the argument from above, the purpose of this work is not to produce high-precision critical exponents for the Ising universality class. Instead, we are introducing a sign-free QMC
algorithm with a polynomial computational effort. While in principle, one can produce relatively precise exponents, we would like to highlight that this setup also enables us to analyze the existence
of an emergent conformal symmetry at criticality, especially for models with larger local Hilbert spaces. However, applying the new algorithm to such models to study, e.g., critical gauge theories,
relevant for deconfined quantum critical points, is left for future studies. Here, we have shown that the algorithm actually works by benchmarking it using the Ising model. To address the referee's
comments, we have added a discussion of the numerical complexity below Eq. (16), and discuss future directions in the conclusion.
4.,5.,6. We have added the definition of the chi^2 to the paper, see new Eq. (20). While there are seven parameters, we also have thirty-two data points in our fit, so overfitting should not be a big
concern. Note that we are not trying to extract high-precision critical exponents from the scaling collapse, which are known to be prone to finite size scaling corrections, but we are rather giving
evidence that the critical point of the fuzzy sphere model belongs to the Ising universality class. We believe the current system sizes are sufficiently large for this statement as shown by the
quality of the scaling collapse in Fig. 1(c).
7., We thank the referee for this interesting comment and agree that this would be an interesting future direction. However, it is still an open nontrivial question how to properly describe Dirac
fermions on a fuzzy sphere. A more directly accessible follow-up is to study critical gauge theories, as already mentioned in the conclusion. However, we believe that this would be a sufficiently
different and new result to be left for a future work.
We have attached a revised version of the manuscript for the reviewer to examine.
Report #1 by Anonymous (Referee 1) on 2024-3-5 (Invited Report)
• Cite as: Anonymous, Report on arXiv:scipost_202401_00004v1, delivered 2024-03-05, doi: 10.21468/SciPost.Report.8662
1- the topic is very interesting
2- the mathematical steps are clear and well documented
1- quite specialised
2- not very clear how to generalise the method
The paper deals with a very interesting problem of simulating phase transitions on the fuzzy sphere using quantum Monte Carlo. Recently fuzzy sphere regularization has received attention in several
branches of theoretical physics so it is timely to address and resolve potential issues. The authors propose a novel approach to the simulation by introducing flavor index to the fermions.
Remarkably, this approach does not display any sign problem. The authors show it in the case of the 3D Ising model. While the paper is well written and provides evidences of the effectiveness of
quantum Monte Carlo in this setting, it would be nice if the authors will clarify in more details how general the method of adding flavour to fermions related to the sign problem. In particular how
it can be used in more general phase transitions.
Requested changes
1-explain the limitations of the approach
2-discuss the generality of the method
We thank the referee for taking the time to read our manuscript and provide a useful report. As we detail below, we have addressed the referee's comments appropriately and feel confident that the
modified manuscript can now be published by SciPost.
1. We have added a discussion of the numerical limitation, i.e., the scaling of compute time with system size, below Eq. [16]. Due to the rank of the operators that couple to the auxiliary fields,
the computational effort scales as N^4 instead of the usual N^3, which can be achieved for conventional lattice models. However, we stress that this method still scales polynomially with size, in
contrast to exact diagonalization and DMRG. Hence, the QMC remains the method of choice when many fermionic degrees of freedom are studied, e.g., for studying critical gauge theories with
multiple fermion flavors.
2. We have added a more detailed discussion of the guiding design principle for the additional flavor in the last paragraph of Sec. 2.3. It might indeed be interesting to study how generically one
can use this trick to study universal aspects of critical phenomena. However, the purpose of this paper is not to show that introducing an additional flavor can always provide a sign-free model
to study any universality class of interest. Instead, we introduce the possibility of using the QMC method on the fuzzy sphere, which has never been done before in a way that exploits the unique
features of it. For example, the state-operator correspondence on the fuzzy sphere gives rise to unambiguous fingerprints of conformal symmetry in the spectrum (Fig. 3). Note that this cannot be
achieved on a tight-binding Hamiltonian on a 2D lattice with periodic boundary conditions, nor by conformal bootstrap as the conformal symmetry that assumes conformal symmetry from the beginning.
Similarly, the conformal symmetry highly constrains the functional form of (equal-time) correlation functions which allow a fit-free extraction of critical exponents/operator dimensions (Fig. 2c)
We agree with the referee that this paper mostly focuses on the Ising universality class to benchmark the method. In response to their comments, we have also added a discussion regarding the
"generality of the method" to the conclusion. For the Ising model, we needed an additional flavor degree of freedom to avoid the sign problem. However, many interesting universality classes can
be studied using sign-free model Hamiltonians without the need to introduce artificial flavor degrees of freedom! Examples are the critical gauge theories that have been proposed to describe
deconfined quantum critical points. | {"url":"https://scipost.org/submissions/scipost_202401_00004v1/","timestamp":"2024-11-13T07:33:41Z","content_type":"text/html","content_length":"57942","record_id":"<urn:uuid:05c826f7-a471-4bb0-bf79-fcf0f714356b>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00268.warc.gz"} |
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1. Name: Simple Harmonic Motion - Force, Acceleration, and Velocity at 3 Positions Category: Oscillations Date Added: 2018-04-15 Submitter: Flipping Physics Identifying the spring force,
acceleration, and velocity at the end positions and equilibrium position of simple harmonic motion. Amplitude is also defined and shown. Want Lecture Notes? This is an AP Physics 1 topic. Content
Times: 0:01 Identifying the 3 positions 0:43 Velocity 1:43 Spring Force 2:14 Amplitude 2:30 Acceleration 3:22 Velocity at position 2 4:12 Is simple harmonic motion also uniformly accelerated
motion? Thank you to Anish, Kevin, and Olivia for being my “substitute students” in this video! Next Video: Horizontal vs. Vertical Mass-Spring System Multilingual? Please help translate Flipping
Physics videos! Previous Video: Simple Harmonic Motion Introduction via a Horizontal Mass-Spring System Please support me on Patreon! Thank you to Jonathan Everett, Sawdog, and Christopher Becke
for being my Quality Control Team for this video. Thank you to Youssef Nasr for transcribing the English subtitles of this video. Simple Harmonic Motion - Force, Acceleration, and Velocity at 3
2. I've been extremely curious on how much Physics Education professional dart players have on shooting? It's quite impressive to throw 3 darts in such a small group repeatedly without any fixed
sights. If you have any Physics, mathematics, knowledge,suggestion to this either by text, video, illustration would you be so kind to share? Im looking for anything and everything to do with
start to finish with throwing and standing also throwing a Steel Tip Dart (with a flight and its uses along with balance and it's shaft) The functions of each piece of the process compared to
it's closest similarities. Thank You So Much.
3. Name: AP Physics C: Rotational vs. Linear Review (Mechanics) Category: Rotational Motion Date Added: 2017-04-28 Submitter: Flipping Physics Calculus based review and comparison of the linear and
rotational equations which are in the AP Physics C mechanics curriculum. Topics include: displacement, velocity, acceleration, uniformly accelerated motion, uniformly angularly accelerated
motion, mass, momentum of inertia, kinetic energy, Newton’s second law, force, torque, power, and momentum. Want Lecture Notes? Content Times: 0:12 Displacement 038 Velocity 1:08 Acceleration
1:33 Uniformly Accelerated Motion 2:15 Uniformly Angularly Accelerated Motion 2:34 Mass 3:19 Kinetic Energy 3:44 Newton’s Second Law 4:18 Force and Torque 5:12 Power 5:45 Momentum Multilingual?
Please help translate Flipping Physics videos! AP Physics C Review Website Next Video: AP Physics C: Universal Gravitation Review (Mechanics) Previous Video: AP Physics C: Rotational Dynamics
Review - 2 of 2 (Mechanics) Please support me on Patreon! Thank you to Sawdog for being my Quality Control individual for this video. AP Physics C: Rotational vs. Linear Review (Mechanics)
4. Name: AP Physics C: Dynamics Review (Mechanics) Category: Dynamics Date Added: 2017-03-23 Submitter: Flipping Physics Calculus based review of Newton’s three laws, basic forces in dynamics such
as the force of gravity, force normal, force of tension, force applied, force of friction, free body diagrams, translational equilibrium, the drag or resistive force and terminal velocity. For
the calculus based AP Physics C mechanics exam. Want Lecture Notes? Content Times: 0:18 Newton’s First Law 1:30 Newton’s Second Law 1:55 Newton’s Third Law 2:29 Force of Gravity 3:36 Force Normal
3:58 Force of Tension 4:24 Force Applied 4:33 Force of Friction 5:46 Static Friction 6:17 Kinetic Friction 6:33 The Coefficient of Friction 7:26 Free Body Diagrams 10:41 Translational equilibrium
11:41 Drag Force or Resistive Force 13:25 Terminal Velocity Next Video: AP Physics C: Work, Energy, and Power Review (Mechanics) Multilingual? Please help translate Flipping Physics videos! AP
Physics C Review Website Previous Video: AP Physics C: Kinematics Review (Mechanics) Please support me on Patreon! Thank you to Aarti Sangwan for being my Quality Control help. AP Physics C:
Dynamics Review (Mechanics)
5. Name: Review of Momentum, Impact Force, and Impulse Category: Momentum and Collisions Date Added: 2017-01-26 Submitter: Flipping Physics An important review highlighting differences between the
equations for Conservation of Momentum, Impact Force and Impulse. Want lecture notes? This is an AP Physics 1 Topic. Content Times: 0:17 Conservation of Momentum 1:01 An explosion is a collision
in reverse 1:22 Impact Force 1:39 Impulse 2:16 Impulse equals 3 things 2:53 How many objects are in these equations? A big THANK YOU to Elle Konrad who let me borrow several of her old dance
costumes! Next Video: Using Impulse to Calculate Initial Height Multilingual? Please help translate Flipping Physics videos! Previous Video: Demonstrating How Helmets Affect Impulse and Impact
Force Please support me on Patreon! Thank you to my Quality Control help: Christopher Becke, Scott Carter and Jennifer Larsen Review of Momentum, Impact Force, and Impulse
6. Name: Demonstrating How Helmets Affect Impulse and Impact Force Category: Momentum and Collisions Date Added: 2016-12-08 Submitter: Flipping Physics Demonstrating and measuring how a helmet
changes impulse, impact force and change in time during a collision. Want lecture notes? This is an AP Physics 1 Topic. Content Times: 0:21 The demonstration without a helmet 1:15 The equation
for Impulse 1:55 How a helmet should affect the variables 2:36 The demonstration with a helmet 3:29 Comparing with and without a helmet Next Video: Review of Momentum, Impact Force, and Impulse
Multilingual? Please help translate Flipping Physics videos! Previous Video: Demonstrating Impulse is Area Under the Curve Please support me on Patreon! Thank you to my Quality Control help:
Christopher Becke, Scott Carter, and Jennifer Larsen Demonstrating How Helmets Affect Impulse and Impact Force
7. Name: Introductory Conservation of Momentum Explosion Problem Demonstration Category: Momentum and Collisions Date Added: 2016-10-13 Submitter: Flipping Physics Now that we have learned about
conservation of momentum, let’s apply what we have learned to an “explosion”. Okay, it’s really just the nerd-a-pult launching a ball while on momentum carts. Want lecture notes? This is an AP
Physics 1 Topic. Content Times: 0:38 The demonstration 1:16 The known values 2:07 Solving the problem using conservation of momentum 4:00 Measuring the final velocity of the nerd-a-pult 4:39
Determining relative error 5:09 What happens with a less massive projectile? Multilingual? Please help translate Flipping Physics videos! Previous Video: Introduction to Conservation of Momentum
with Demonstrations Please support me on Patreon! Introductory Conservation of Momentum Explosion Problem Demonstration
8. Name: Proving and Explaining Impulse Approximation Category: Momentum and Collisions Date Added: 2016-09-22 Submitter: Flipping Physics Know when and how to use the “Impulse Approximation”. Want
lecture notes? This is an AP Physics 1 Topic. Content Times: 0:12 Reviewing the examples 0:43 Defining Impulse Approximation 1:41 Determining the forces during the collision 2:27 Solving for the
Force Normal (or Force of Impact) 3:12 Determining our error Next Video: How to Wear A Helmet - A PSA from Flipping Physics Multilingual? Please help translate Flipping Physics videos! Previous
Video: Impulse Introduction or If You Don't Bend Your Knees When Stepping off a Wall Please support me on Patreon! Proving and Explaining Impulse Approximation
9. Name: Impulse Introduction or If You Don't Bend Your Knees When Stepping off a Wall Category: Momentum and Collisions Date Added: 2016-09-22 Submitter: Flipping Physics Now mr.p doesn’t bend his
knees when stepping off a wall. What is the new force of impact? Want lecture notes? This is an AP Physics 1 Topic. Content Times: 0:18 How much does mr.p bend his knees? 1:00 Reviewing the
previous problem 1:57 What changes if I don’t bend my knees? 2:41 Impulse introduction 3:36 The impulse during this collision 4:51 Why is it bad to not bend your knees? 5:22 Estimating time of
collision if I don’t bend my knees 6:09 Solving for the force of impact 6:51 Review 7:28 No tomatoes were wasted in the making of this video Next Video: Proving and Explaining Impulse
Approximation Multilingual? Please help translate Flipping Physics videos! Previous Video: Calculating the Force of Impact when Stepping off a Wall Please support me on Patreon! Impulse
Introduction or If You Don't Bend Your Knees When Stepping off a Wall
10. Name: Calculating the Force of Impact when Stepping off a Wall Category: Momentum and Collisions Date Added: 2016-09-08 Submitter: Flipping Physics A 73 kg mr.p steps off a 73.2 cm high wall. If
mr.p bends his knees such that he stops his downward motion and the time during the collision is 0.28 seconds, what is the force of impact caused by the ground on mr.p? Want lecture notes? This
is an AP Physics 1 Topic. Content Times: 0:21 Translating the problem 1:32 Splitting the problem into parts 3:07 Substituting in known variables 4:30 Finding the final velocity for part 1 6:21
Substituting back into Force of Impact equation 7:23 Converting to pounds Next Video: Impulse Introduction or If You Don't Bend Your Knees When Stepping off a Wall Multilingual? Please help
translate Flipping Physics videos! Previous Video: Instantaneous Power Delivered by a Car Engine - Example Problem Please support me on Patreon! A big thank you to Jean Gifford for donating the
money for Bo and Billy’s bathrobes! Calculating the Force of Impact when Stepping off a Wall
11. Name: Force of Impact Equation Derivation Category: Momentum and Collisions Date Added: 2017-01-12 Submitter: Flipping Physics Rearranging Newton’s Second Law to derive the force of impact
equation. Want lecture notes? This is an AP Physics 1 Topic. Content Times: 0:09 Newton’s Second Law 1:57 The Force of Impact equation 2:33 The paradigm shift Next Video: Calculating the Force of
Impact when Stepping off a Wall Multilingual? Please help translate Flipping Physics videos! Previous Video: You Can't Run From Momentum! (a momentum introduction) Please support me on Patreon!
Force of Impact Equation Derivation
12. Name: Calculating Average Drag Force on an Accelerating Car using an Integral Category: Dynamics Date Added: 2016-08-11 Submitter: Flipping Physics A vehicle uniformly accelerates from rest to
3.0 x 10^1 km/hr in 9.25 seconds and 42 meters. Determine the average drag force acting on the vehicle. Want lecture notes? This is an AP Physics C Topic. Content Times: 0:14 The Drag Force
equation 0:39 The density of air 1:33 The drag coefficient 1:59 The cross sectional area 3:11 Determining instantaneous speed 4:08 Instantaneous Drag Force 4:36 Graphing Drag Force as a function
of Time 5:17 The definite integral of drag force with respect to time 5:42 Average Drag Force times Total Change in Time Next Video: Instantaneous Power Delivered by a Car Engine - Example
Problem Multilingual? Please help translate Flipping Physics videos! Previous Video: Average Power Delivered by a Car Engine - Example Problem Please support me on Patreon! Calculating Average
Drag Force on an Accelerating Car using an Integral
13. Name: Instantaneous Power Delivered by a Car Engine - Example Problem Category: Work, Energy, Power Date Added: 2017-01-12 Submitter: Flipping Physics A Toyota Prius is traveling at a constant
velocity of 113 km/hr. If an average force of drag of 3.0 x 10^2 N acts on the car, what is the power developed by the engine in horsepower? Want Lecture Notes? This is an AP Physics 1 Topic.
Content Times: 0:15 The problem 1:18 Which equation to use and why 2:20 Billy solves the problem 3:59 What if the car is moving at 129 km/hr? Next Video: You Can't Run From Momentum! (a momentum
introduction) Multilingual? Please help translate Flipping Physics videos! Previous Video: Average Power Delivered by a Car Engine - Example Problem Please support me on Patreon! Instantaneous
Power Delivered by a Car Engine - Example Problem
14. Name: Average Power Delivered by a Car Engine - Example Problem Category: Work, Energy, Power Date Added: 2016-07-28 Submitter: Flipping Physics A 1400 kg Prius uniformly accelerates from rest to
30 km/hr in 9.25 seconds and 42 meters. If an average force of drag of 8.0 N acts on the car, what is the average power developed by the engine in horsepower? Want Lecture Notes? This is an AP
Physics 1 Topic. Content Times: 0:15 Translating the example to physics 2:13 The equation for power 3:37 Drawing the Free Body Diagram and summing the forces 4:47 Solving for acceleration and
Force Applied 5:43 Determining theta 6:01 Solving for Average Power 6:53 Understanding our answer 7:34 The Horse Pedal 9:13 Comparing to a larger acceleration example Next Video: Instantaneous
Power Delivered by a Car Engine - Example Problem Multilingual? Please help translate Flipping Physics videos! Previous Video: Graphing Instantaneous Power Please support me on Patreon! Average
Power Delivered by a Car Engine - Example Problem
15. Name: Introductory Kinetic Friction on an Incline Problem Category: Dynamics Date Added: 2016-06-16 Submitter: Flipping Physics You place a book on a 14° incline and then let go of the book. If
the book takes 2.05 seconds to travel 0.78 meters, what is the coefficient of kinetic friction between the book and the incline? Want Lecture Notes? This is an AP Physics 1 Topic. Content Times:
0:01 The example 0:13 Listing the known values 1:09 Drawing the free body diagram 1:58 Net force in the perpendicular direction 2:34 Net force in the parallel direction 4:03 Solving for
acceleration 5:07 Solving for Mu 5:40 We made a mistake Multilingual? Please help translate Flipping Physics videos! Previous Video: Introductory Static Friction on an Incline Problem Please
support me on Patreon! Introductory Kinetic Friction on an Incline Problem
16. Name: Physics "Magic Trick" on an Incline Category: Dynamics Date Added: 2016-06-06 Submitter: Flipping Physics Understand the forces acting on an object on an incline by analyzing the forces on
a “floating block”. Want Lecture Notes? This is an AP Physics 1 topic. Content Times: 0:28 Finding the incline angle 1:17 Drawing the Free Body Diagram 2:26 Summing the forces in the
perpendicular direction 3:49 Summing the forces in the parallel direction 5:04 Determining masses for the “Magic Trick” 6:11 Adding pulleys, strings and mass 7:34 Floating the block 8:18
Analyzing the forces on the floating block Next Video: Introductory Static Friction on an Incline Problem Multilingual? Please help translate Flipping Physics videos! Previous Video: Breaking the
Force of Gravity into its Components on an Incline Thanks to Nic3_one and Cyril Laurier for their Fire Sounds: Fire in a can! » constant spray fire 1 by Nic3_one Earth+Wind+Fire+Water » Fire.wav
by Cyril Laurier 1¢/minute Physics "Magic Trick" on an Incline
17. Name: Breaking the Force of Gravity into its Components on an Incline Category: Dynamics Date Added: 2015-10-16 Submitter: Flipping Physics Resolve the force of gravity into its parallel and
perpendicular components so you can sum the forces. Want Lecture Notes? This is an AP Physics 1 topic. Content Times: 0:12 Drawing the Free Body Diagram 1:04 Introducing the parallel and
perpendicular directions 2:19 Drawing the components of the force of gravity 2:49 Finding the angle used to resolve the force of gravity into its components 4:33 Solving for the force of gravity
parallel 5:15 Solving for the force of gravity perpendicular 5:53 Redrawing the Free Body Diagram Next Video: Physics "Magic Trick" on an Incline Multilingual? Please help translate Flipping
Physics videos! Previous Video: Determining the Static Coefficient of Friction between Tires and Snow 1¢/minute Breaking the Force of Gravity into its Components on an Incline
18. Name: Newton's Laws of Motion in Space: Force, Mass, and Acceleration Category: Dynamics Date Added: 2015-10-07 Submitter: FizziksGuy Uploaded on Apr 18, 2010ESA Science - Newton In Space (Part
2): Newton's Second Law of Motion - Force, Mass And Acceleration. Newton's laws of motion are three physical laws that form the basis for classical mechanics. They have been expressed in several
different ways over nearly three centuries. --- Please subscribe to Science & Reason: • http://www.youtube.com/Best0fScience • http://www.youtube.com/ScienceMagazine • http://www.youtube.com/
FFreeThinker --- The laws describe the relationship between the forces acting on a body and the motion of that body. They were first compiled by Sir Isaac Newton in his work "Philosophiæ
Naturalis Principia Mathematica", first published on July 5, 1687. Newton used them to explain and investigate the motion of many physical objects and systems. For example, in the third volume of
the text, Newton showed that these laws of motion, combined with his law of universal gravitation, explained Kepler's laws of planetary motion. --- Newton's Second Law of Motion: A body will
accelerate with acceleration proportional to the force and inversely proportional to the mass. Observed from an inertial reference frame, the net force on a particle is equal to the time rate of
change of its linear momentum: F = d(mv)/dt. Since by definition the mass of a particle is constant, this law is often stated as, "Force equals mass times acceleration (F = ma): the net force on
an object is equal to the mass of the object multiplied by its acceleration." History of the second law Newton's Latin wording for the second law is: "Lex II: Mutationem motus proportionalem esse
vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur." This was translated quite closely in Motte's 1729 translation as: "LAW II: The alteration of motion is ever
proportional to the motive force impress'd; and is made in the direction of the right line in which that force is impress'd." According to modern ideas of how Newton was using his terminology,
this is understood, in modern terms, as an equivalent of: "The change of momentum of a body is proportional to the impulse impressed on the body, and happens along the straight line on which that
impulse is impressed." Motte's 1729 translation of Newton's Latin continued with Newton's commentary on the second law of motion, reading: "If a force generates a motion, a double force will
generate double the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively. And this motion (being always directed the
same way with the generating force), if the body moved before, is added to or subtracted from the former motion, according as they directly conspire with or are directly contrary to each other;
or obliquely joined, when they are oblique, so as to produce a new motion compounded from the determination of both." The sense or senses in which Newton used his terminology, and how he
understood the second law and intended it to be understood, have been extensively discussed by historians of science, along with the relations between Newton's formulation and modern
formulations. Newton's Laws of Motion in Space: Force, Mass, and Acceleration
19. Name: Experimentally Graphing the Force of Friction Category: Dynamics Date Added: 2015-08-19 Submitter: Flipping Physics To help understand the force of friction, mr.p pulls on a wooden block
using a force sensor. Want Lecture Notes? This is an AP Physics 1 topic. Content Times: 0:17 Drawing the Free Body Diagram 0:43 Summing the forces in the x-direction 1:21 Graph when the block
doesn’t move 1:46 Graph with the block moving Next Video: Does the Book Move? An Introductory Friction Problem Multilingual? Please help translate Flipping Physics videos! Previous Video:
Understanding the Force of Friction Equation 1¢/minute Experimentally Graphing the Force of Friction
20. Name: Understanding the Force of Friction Equation Category: Dynamics Date Added: 2015-08-18 Submitter: Flipping Physics The Force of Friction Equation is actually three equations is one. Learn
why! Want Lecture Notes? This is an AP Physics 1 topic. Content Times: 0:00 The basic Force of Friction Equation 0:20 One Kinetic Friction Equation 0:39 The Two Static Friction Equations 1:40
Example Free Body Diagram 2:16 The direction of the Force of Friction 3:20 Determining the magnitude of the Force of Static Friction 4:09 Understanding the “less than or equal” sign 6:08 If the
“less than or equal” sign were not there Next Video: Experimentally Graphing the Force of Friction Multilingual? Please help translate Flipping Physics videos! Previous Video: Introduction to the
Coefficient of Friction 1¢/minute Understanding the Force of Friction Equation
21. Name: Introduction to the Coefficient of Friction Category: Dynamics Date Added: 2015-08-09 Submitter: Flipping Physics Please do not confuse the Coefficient of Friction with the Force of
Friction. This video will help you not fall into that Pit of Despair! Want Lecture Notes? This is an AP Physics 1 topic. Content Times: 0:00 The equation for the Force of Friction 0:17 Mu, the
symbol for the Coefficient of Friction 1:21 Tables of Coefficients of Friction 2:49 Comparing the values of static and kinetic coefficients of friction 3:54 A typical range of values Next Video:
Understanding the Force of Friction Equation Multilingual? Please help translate Flipping Physics videos! Previous Video: Introduction to Static and Kinetic Friction by Bobby 1¢/minute
Introduction to the Coefficient of Friction
22. Name: An Introductory Tension Force Problem Category: Dynamics Date Added: 2015-07-30 Submitter: Flipping Physics Learn how to solve a basic tension force problem with demonstration! Want Lecture
Notes? This is an AP Physics 1 topic. Content Times: 0:00 The Problem Demonstrated 0:29 5 Steps to Solve and Free Body Diagram Problem 0:50 Drawing the Free Body Diagram 2:03 Resolving Tension
Force 1 into its components (numbers dependency) 4:00 Introducing the Equation Holster! 5:11 Redraw the Free Body Diagram 5:32 Sum the forces in the y-direction 7:24 Sum the forces in the
x-direction 8:29 Demonstrating our solution is correct Multilingual? Please help translate Flipping Physics videos! Next Video: Introduction to Static and Kinetic Friction by Bobby Previous
Video: 5 Steps to Solve any Free Body Diagram Problem 1¢/minute An Introductory Tension Force Problem
23. Name: 5 Steps to Solve any Free Body Diagram Problem Category: Dynamics Date Added: 2015-07-30 Submitter: Flipping Physics Learn how to solve problems that have Free Body Diagrams! Want Lecture
Notes? This is an AP Physics 1 topic. Content Times: 0:15 Step 1) Draw the Free Body Diagram 0:50 Step 2) Break Forces into Components 1:37 Step 3) Redraw the Free Body Diagram 2:15 Step 4) Sum
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A Guide to Inferencing With Bayesian Network in Python
Bayesian networks can model nonlinear, multimodal interactions using noisy, inconsistent data. It has become a prominent tool in many domains despite the fact that recognizing the structure of these
networks from data is already common. For modelling the conditionally dependent data and inferencing out of it, Bayesian networks are the best tools used for this purpose. In this post, we will walk
through the fundamental principles of the Bayesian Network and the mathematics that goes with it. Also, we will also learn how to infer with it through a Python implementation. The key points to be
covered in this post are listed below.
Table of Contents
1. What is Bayesian Network?
2. What is Directed Acyclic Graph (DAG)?
3. The Maths Behind Bayesian Network
4. Inferencing with Bayesian Network in Python
Let’s start the discussion by understanding the what is Bayesian Network.
What is Bayesian Network?
A Bayesian network (also spelt Bayes network, Bayes net, belief network, or judgment network) is a probabilistic graphical model that depicts a set of variables and their conditional dependencies
using a directed acyclic graph (DAG).
Bayesian networks are perfect for taking an observed event and forecasting the likelihood that any of numerous known causes played a role. A Bayesian network, for example, could reflect the
probability correlations between diseases and symptoms. Given a set of symptoms, the network may be used to calculate the likelihood of the presence of certain diseases.
What is Directed Acyclic Graph (DAG)?
In graph theory and computer science, a directed acyclic graph (DAG) is a directed graph with no directed cycles. In other words, it’s made up of vertices and edges (also called arcs), with each edge
pointing from one vertex to the next in such a way that following those directions would never lead to a closed-loop as depicted in below picture.
A directed graph is one in which all edge directions are consistent and the vertices can be topologically arranged in a linear order. DAGs have various scientific and computing applications,
including biology evolution, family trees, and epidemiology, and sociology.
Let’s see quickly what are fundamental maths involved with Bayesian Network.
The Maths Behind the Bayesian Network
An acyclic directed graph is used to create a Bayesian network, which is a probability model. It’s factored by utilizing a single conditional probability distribution for each variable in the model,
whose distribution is based on the parents in the graph. The simple principle of probability underpins Bayesian models. So, first, let’s define conditional probability and joint probability
Conditional Probability
Conditional probability is a measure of the likelihood of an event occurring provided that another event has already occurred (through assumption, supposition, statement, or evidence). If A is the
event of interest and B is known or considered to have occurred, the conditional probability of A given B is generally stated as P(A|B) or, less frequently, P[B](A) if A is the event of interest and
B is known or thought to have occurred. This can also be expressed as a percentage of the likelihood of B crossing with A:
Joint Probability
The chance of two (or more) events together is known as the joint probability. The sum of the probabilities of two or more random variables is the joint probability distribution.
For example, the joint probability of events A and B is expressed formally as:
• The letter P is the first letter of the alphabet (A and B).
• The upside-down capital “U” operator or, in some situations, a comma “,” represents the “and” or conjunction.
• P(A ^ B)
• P(A, B)
By multiplying the chance of event A by the likelihood of event B, the combined probability for occurrences A and B is calculated.
Posterior Probability
In Bayesian statistics, the conditional probability of a random occurrence or an ambiguous assertion is the conditional probability given the relevant data or background. “After taking into account
the relevant evidence pertinent to the specific subject under consideration,” “posterior” means in this case.
The probability distribution of an unknown quantity interpreted as a random variable based on data from an experiment or survey is known as the posterior probability distribution.
Inferencing with Bayesian Network in Python
In this demonstration, we’ll use Bayesian Networks to solve the well-known Monty Hall Problem. Let me explain the Monty Hall problem to those of you who are unfamiliar with it:
This problem entails a competition in which a contestant must choose one of three doors, one of which conceals a price. The show’s host (Monty) unlocks an empty door and asks the contestant if he
wants to swap to the other door after the contestant has chosen one.
The decision is whether to keep the current door or replace it with a new one. It is preferable to enter by the other door because the price is more likely to be higher. To come out from this
ambiguity let’s model this with a Bayesian network.
For this demonstration, we are using a python-based package pgmpy is a Bayesian Networks implementation written entirely in Python with a focus on modularity and flexibility. Structure Learning,
Parameter Estimation, Approximate (Sampling-Based) and Exact inference, and Causal Inference are all available as implementations.
from pgmpy.models import BayesianNetwork
from pgmpy.factors.discrete import TabularCPD
import networkx as nx
import pylab as plt
# Defining Bayesian Structure
model = BayesianNetwork([('Guest', 'Host'), ('Price', 'Host')])
# Defining the CPDs:
cpd_guest = TabularCPD('Guest', 3, [[0.33], [0.33], [0.33]])
cpd_price = TabularCPD('Price', 3, [[0.33], [0.33], [0.33]])
cpd_host = TabularCPD('Host', 3, [[0, 0, 0, 0, 0.5, 1, 0, 1, 0.5],
[0.5, 0, 1, 0, 0, 0, 1, 0, 0.5],
[0.5, 1, 0, 1, 0.5, 0, 0, 0, 0]],
evidence=['Guest', 'Price'], evidence_card=[3, 3])
# Associating the CPDs with the network structure.
model.add_cpds(cpd_guest, cpd_price, cpd_host)
Now we will check the model structure and associated conditional probability distribution by the argument get_cpds() will return True if every this is fine else through an error msg.
Now let’s infer the network, if we want to check at the next step which door will the host open now. For that, we need access to the posterior probability from the network and while accessing we need
to pass the evidence to the function. Evidence is needed to be given when we are evaluating posterior probability, here in our task evidence is nothing but which door is Guest selected and where is
the Price.
# Infering the posterior probability
from pgmpy.inference import VariableElimination
infer = VariableElimination(model)
posterior_p = infer.query(['Host'], evidence={'Guest': 2, 'Price': 2})
The probability distribution of the Host is clearly satisfying the theme of the contest. In the reality also, in this situation host definitely not going to open the second door he will open either
of the first two and that’s what the above simulation tells.
Now, let’s plot our above model. This can be done with the help of Network and Pylab. NetworkX is a Python-based software package for constructing, altering, and researching the structure, dynamics,
and function of complex networks. PyLab is a procedural interface to the object-oriented charting toolkit Matplotlib, and it is used to examine large complex networks represented as graphs with nodes
and edges.
nx.draw(model, with_labels=True)
This gives us the Directed Acyclic Graph (DAG) as like below.
Through this post, we have discussed what a Bayesian Network is. In addition to that we have discussed how the Bayesian network can be represented using DAG and also we have discussed what are the
general and simple mathematical concepts are associated with the network. Lastly, we have seen the practical implementation of the Bayesian network with help of the python tool pgmpy, and also
plotted a DAG of our model using Netwrokx and pylab. | {"url":"https://aiws.net/practicing-principles/modern-causal-inference/augmenting/on-media-augmenting/a-guide-to-inferencing-with-bayesian-network-in-python/","timestamp":"2024-11-13T07:56:28Z","content_type":"text/html","content_length":"165237","record_id":"<urn:uuid:8c635210-a8a0-498e-859e-5486fa0720b9>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00566.warc.gz"} |
Content-Aware SLIC Super-Pixels for Semi-Dark Images (SLIC++)
High Performance Cloud Computing Center (HPC3), Department of Computer and Information Sciences, Universiti Teknologi PETRONAS, Seri Iskandar 32610, Malaysia
Faculty of Engineering Science and Technology, Iqra University, Karachi 75600, Pakistan
Department of Computer Science, Shaheed Zulfiqar Ali Bhutto Institute of Science and Technology, Karachi 75600, Pakistan
Author to whom correspondence should be addressed.
Submission received: 27 September 2021 / Revised: 16 November 2021 / Accepted: 17 November 2021 / Published: 25 January 2022
Super-pixels represent perceptually similar visual feature vectors of the image. Super-pixels are the meaningful group of pixels of the image, bunched together based on the color and proximity of
singular pixel. Computation of super-pixels is highly affected in terms of accuracy if the image has high pixel intensities, i.e., a semi-dark image is observed. For computation of super-pixels, a
widely used method is SLIC (Simple Linear Iterative Clustering), due to its simplistic approach. The SLIC is considerably faster than other state-of-the-art methods. However, it lacks in
functionality to retain the content-aware information of the image due to constrained underlying clustering technique. Moreover, the efficiency of SLIC on semi-dark images is lower than bright
images. We extend the functionality of SLIC to several computational distance measures to identify potential substitutes resulting in regular and accurate image segments. We propose a novel SLIC
extension, namely, SLIC++ based on hybrid distance measure to retain content-aware information (lacking in SLIC). This makes SLIC++ more efficient than SLIC. The proposed SLIC++ does not only hold
efficiency for normal images but also for semi-dark images. The hybrid content-aware distance measure effectively integrates the Euclidean super-pixel calculation features with Geodesic distance
calculations to retain the angular movements of the components present in the visual image exclusively targeting semi-dark images. The proposed method is quantitively and qualitatively analyzed using
the Berkeley dataset. We not only visually illustrate the benchmarking results, but also report on the associated accuracies against the ground-truth image segments in terms of boundary precision.
SLIC++ attains high accuracy and creates content-aware super-pixels even if the images are semi-dark in nature. Our findings show that SLIC++ achieves precision of 39.7%, outperforming the precision
of SLIC by a substantial margin of up to 8.1%.
1. Introduction
Image segmentation has potential to reduce the image complexities associated with processing of singular image primitives. Low-level segmentation of an image in non-overlapping set of regions called
super-pixels helps in pre-processing and speeding up further high-level computational tasks related to visual images. The coherence feature of super-pixels allows faster architectural functionalities
of many visual applications including object localization [
], tracking [
], posture estimation [
], recognition [
], semantic segmentation [
], instance segmentation [
], and segmentation of medical imagery [
]. These applications will be aided by super-pixels in terms of boosted performances, as the super-pixels put forward only the discriminating visual information [
Low-level segmentation tends to result in incorrect segmentations if the visual image has high pixel intensities; these high pixeled values are usually the biproduct of visual scenes captured in low
lighting conditions, i.e., semi-dark images and dark images. The obtained incorrect super-pixels are attributed to the underlying approach used for the final segment creation. The existing
super-pixel methods fail due to incorrect pixel manipulations for base operational functionality. The currently employed pixel manipulation relies on straight line differences for super-pixel
creation. These straight-line difference manipulations fail to retain the content-aware information of the image. The results are further degraded if the image has low contrasted values, which result
in no clear discrimination among the objects present. The existing super-pixel creation methods are divided into two categories based on the implemented workflow. The two categories are graph-based
and gradient ascent based [
]. The former, focuses on minimization of cost function by grouping and treating each pixel of the image as a graph node. The later, iteratively processes each image pixel using clustering techniques
until convergence [
]. One of the typical features observed for identification of super-pixel accuracy is regularity, i.e., to what extent the super-pixel is close to the actual object boundary of the image. All the
graph-based methods for super-pixel calculations suffer from poorly adhered super-pixels which result in irregularity of segments presented by super-pixels [
]. Additionally, graph-based methods are constrained by the excessive computational complexity and other initialization parameters. Whereas, the gradient-ascent methods are simplistic in nature and
are recommended in the literature due to resultant high performance and accuracy [
]. However, there are some issues associated with content irrelevant manipulation of singular pixels to form resultant super-pixels. Some of the key features of using super-pixel segmentations are:
• Super-pixels abstraction potentially decreases the overhead of processing each pixel at a time.
• Gray-level or color images can be processed by a single algorithm implementation.
• Integrated user control provided for performance tuning.
With these advantages of super-pixels, they are highly preferred. However, super-pixel abstraction methods backed by gradient-ascent workflows are also limited in their working functionality to
retain the contextual information of the given image [
]. The contextual information retainment is required to achieve the richer details of the visual image. This loss of contextual information is caused by flawed pixel clusters created based on
Euclidean distance measure [
]. As the Euclidean distance measures calculates straight line differences among pixels which ends up in irregular and lousy super-pixels. Moreover, further degradation can be expected to process the
semi-dark images where high pixel intensities along with no clear boundaries are observed. In such scenarios, the propagation of inaccurate super-pixels will affect the overall functionality of
automated solutions [
]. To overcome this problem of information loss for creation of compact and uniform super-pixels, we propose content-aware distance measures for image pixel cluster creation. The content-aware
distance measure as the core foundational component of gradient-ascent methods for super-pixel creation will not only help in alleviation of information loss, but it will also help in preserving less
observant/perceptually visible information of semi-dark images. The state-of-the-art methods for super-pixels creation have not been analyzed exclusively on the semi-dark images which further raises
the concerns related to segmentation accuracy. In nutshell, the problems in existing segmentation algorithms are:
• Absence of classification of state-of-the-art methods based on low level manipulations.
• Inherited discontinuity in super-pixel segmentation due to inconsistent manipulations.
• Unknown pixel grouping criteria in terms of distance measure to retain fine grained details.
• Unknown effect of semi-dark images on final super-pixel segmentations.
To resolve these issues, the presented research exclusively presents multifaceted study offering following features:
• Classification of Literary Studies w.r.t Singular Pixel Manipulation Strategies:
The categorization of the existing studies is based on entire image taken as one entity. The image entity can represent either graph or a feature space to be clustered, i.e., graph-based or
gradient-ascent based methodology for pixel grouping. To the best of our knowledge, there has been no study that categorizes existing studies based on the manipulation strategy performed over each
pixel. We present the detailed comparative analysis of existing research highlighting their core functionality as the basis for classification.
Investigation of Appropriate Pixel Grouping Scheme:
The grouping scheme backing the image segmentation module being crucial component can highly affect the accuracy of the entire model. For this reason, to propose a novel extension as a generalized
solution of all types of images including semi-dark images, we present a detailed qualitative investigation of up to seven distance measures for grouping pixels to create super-pixels. The
investigation resulted in shortlisted pixel grouping measures to retain fine grained details of the visual image.
Novel Hybrid Content-Aware Extension of SLIC—SLIC++:
SLIC, being the simplest and fastest solution for the pixel grouping, remains the inspiration, and we enhance the performance of SLIC by adding content-aware feature in its discourse. The proposed
extension holds the fundamental functionality with improved features to preserve content-aware information. The enhancement results in better segmentation accuracy by extracting regular and
continuous super-pixels for all scenarios including semi-dark scenarios.
Comprehensive Perceptual Results focusing Semi-dark Images:
To assess the performance of the proposed extension SLIC++ for extraction of the richer information of the visual scene, we conduct experiments over semi-dark images. The experimental analysis is
benchmarked against the standard super-pixel creation methods to verify that incorporating content-aware hybrid distance measure leads to improved performance. The perceptual results further conform
better performance, the scalability, and generalizability of results produced by SLIC++.
Paper Organization
The remainder of paper is organized as follows:
Section 2
presents the prior super-pixel creation research and its relevance to semi-dark technology. In this section we also present critical analysis of studies proposed over period of two decades and their
possible applicability for semi-dark imagery. We also critically analyze two closely related studies and highlight the difference among them w.r.t SLIC++.
Section 3
describes the extension hypothesis and the final detailed proposal for super-pixel segmentation of semi-dark images.
Section 4
presents the detailed quantitative and qualitative analysis of SLIC extension against state-of-the-art algorithms validating the proposal.
Section 5
discusses the applicability of the proposed algorithm in the domain of computer vision. Finally,
Section 6
concludes the presented research and points out some future directions of research.
2. Literature Review
2.1. Limited Semi-Dark Image Centric Research Focusing Gradient-Ascent Methods
The gradient ascent methods are also called clustering-based methods. These methods take the input image and rasterize the image. Then, based on the local image cues such as color and spatial
information the pixels are clustered iteratively. After each iteration, gradients are calculated to refine the new clusters from the previously created clusters [
]. The iterative process continues till the algorithm converges after the gradients stop changing, thus named gradient-ascent methods. A lot of research has been already done in the domain of
gradient-ascent methods; the list of these methods is presented in
Table 1
The gradient ascent methods for super-pixel creation seems to be a promising solution due to their simplicity of implementation, speed of processing and easy adaptation for handling the latest
demands of complex visual image scenarios. However, the concerns associated with underlying proper extraction strategies is one of the challenging aspects to cater the dynamic featural requirements
imposed by complex visual image scenarios such as semi-dark images.
2.2. Critical Analysis of Gradient-Ascent Super-Pixel Creation Algorithms Based on Manipulation Strategy
For the critical analysis we have considered gradient ascent based super-pixel algorithms presented over period of two decades ranging from 2001 through 2021. The studies are retrieved from Google
Scholar’s repository with keywords including super-pixel segmentation, pixel abstraction, content sensitive super-pixel creation, content-aware super-pixel segmentation. The search resulted in a lot
of segmentation related studies in domain of image processing including the basic image transformations along with related super-pixel segmentation studies. For the critical analysis, the studies
mentioning clustering based super-pixel creations were shortlisted due to their relevance with proposed algorithm. The key features of these studies are critically analyzed and comprehensively
presented in
Table 1
, along with the critiques for respective handling concerns associated with semi-dark imagery.
Key Takeaways
The critical analysis presented in
Table 1
uncovers the fact that recent research explicitly points towards the need of segmentation algorithm which considers the content relevant super-pixel segmentations. To accomplish this task several
techniques are proposed with incorporation of prior transformations of image via deep learning methods, simple image processing, and probabilistic methods. Mostly the research uses and conforms the
achievements of Simple Linear Iterative Clustering (SLIC). Furthermore, most of the research studies are using SLIC algorithm for super-pixel creation as base mechanisms with added features.
Generally, the algorithms proposed in last decade have computational complexity of
$O N$
, whereas if neural networks are employed for automation of required parameter initialization, then the complexity becomes
$O N N o . o f l a y e r s$
. All the proposed algorithms use two distance measures for final super-pixel creation, i.e., Euclidean, or geodesic distance measure. However, all these studies have not mentioned the occurrence of
semi-dark images and their impact on the overall performance. It is estimated that huge margin of the existing image dataset already includes the problem centric image data. The Berkeley dataset that
is substantially used for performance analysis of super-pixel algorithms contains up to 63% semi-dark images. The proposed study uses the semi-dark images of Berkeley dataset for benchmarking
analysis of the SLIC++ algorithm.
2.3. Exclusiveness of SLIC++ w.r.t Recent Developments
The recent studies substantially focus on super-pixels with the induced key features of content sensitivity and adherence of the final segmentations; consequently several related research studies
have been proposed. Generally, the desired features are good boundary adherence, compact super-pixel size and low complexity. The same features are required for super-pixel segmentation of semi-dark
images. In this section we briefly review the recent developments which are closely related to our proposed method for creating super-pixel in semi-dark images.
BASS (Boundary-Aware Super-pixel Segmentation) [
], is closely related to the methodology that we have chosen, i.e., incorporation of content relevant information in the final pixel labeling which ends up with the creation of super-pixels. However,
the major difference resides in the initialization of the super-pixel seeds/centers. BASS recommends the usage of forest classification method prior to the super-pixel creation. This forest
classification of image space results with the creation of a binary image with highlighted boundary information over the image space. This boundary information is then utilized to aid the
initialization process of the seed/ cluster centers. Theoretically, the problem with this entire configuration is additional complexity of boundary map creation which raise the complexity from
$O N$
$O N l o g N$
. This boundary map creation and its associated condition of addition and deletion of seeds is expected to further introduce undesired super-pixel feature under-segmentation. The under-segmentation
might take place due to easy seed deletion condition and difficult addition condition which means more seeds would be deleted and less would be added. This aspect of the study is not desired for the
super-pixel creation in semi-dark scenarios. On the contrary we propose regularly distributed seeds along with usage of both the recommended distance measures without prior image transformation which
further reduces the overall complexity. Finally, we also propose the usage of geodesic distance for color components of the pixel rather than only using it for spatial component.
Intrinsic manifold SLIC [
], is an extension of manifold SLIC which proposes usage of manifolds to map high dimensional image data on the manifolds resembling Euclidean space near each point. IMSLIC uses Geodesic Centroidal
Voronoi Tessellations (GCVT) this allows flexibility of skipping the post-processing heuristics. For computation of geodesic distance on image manifold weighted image graph is overlayed with the same
graph theory of edges, nodes, and weights. For this mapping 8-connected neighbors of pixel are considered. This entire process of mapping and calculation of geodesic distances seems complex. The
theoretical complexity is
$O N$
, however with the incorporation of image graph the computational complexity will increase. Moreover, the conducted study computes only geodesic distance between the pixels leaving behind the
Euclidean counterpart. With substantially less complexity we propose to implement both the distance measures for all the crucial pixel components.
2.4. Summary and Critiques
The comprehensive literature survey is conducted to benefit the readers and provide a kickstart review of advancements of super-pixel segmentation over the period of two decades. Moreover, the survey
resulted in critical analysis of existing segmentation techniques which steered the attention to studies conducted for adverse image scenarios such as semi-dark images. Arguably, with the increased
automated solutions the incoming image data will be of dynamic nature (including lighting conditions). To deal with this dynamic image data, there is a critical need of a super-pixel segmentation
technique that takes into account the aspect of semi-dark imagery and results in regular and content-aware super-pixel segmentation in semi-dark scenarios. The super-pixel segmentation techniques
currently employed for the task suffer from two major issues, i.e., high complexity, and information loss. The information loss associated with the gradient-ascent methods is attributed to
restrictions imposed due to usage of Euclidean image space which totally loses the context of the information present in the image by calculating straight line differences. Many attempts have been
made to incorporate CNN probabilistic methods in super-pixel creation methods to optimize and aid the final segmentation results. However, to the best of our knowledge there has been no method
proposed exclusively for semi-dark images scenarios keeping the simplicity and optimal performance intact.
In following sections, we describe the preliminaries which are the base for the proposed extension of SLIC namely SLIC++. We also present several distance measures incorporated in base SLIC algorithm
namely SLIC+ to analyze the performance for semi-dark images.
3. Materials and Methods
3.1. The Semi-Dark Dataset
For the analysis of the content-aware super-pixel segmentation algorithm, we have used the state-of-the-art dataset which has been used in the literature for years now. The Berkeley image dataset [
] has been used for the comprehensive analysis and benchmarking of the proposed SLIC++ algorithm with the state-of-the-art algorithms. The Berkeley image dataset namely BSDS 500 has got five hundred
images overall, whereas the problem under consideration is of semi-dark images. For this purpose, we have initially extracted semi-dark images using RPLC (Relative Perceived Luminance Classification)
algorithm. The labels are created based on the manipulation of color model information, i.e., Hue, Saturation, Lightness (HSL) [
]. The final semi-dark images extracted from the BSDS-500 dataset turned out to be 316 images. Each image has resolution of either 321 × 481 or 481 × 321 dimensions. The BSDS-500 image dataset
provides the basis for empirical analysis of segmentation algorithms. For the performance analysis and boundary detection, the BSDS-500 dataset provides ground-truth labels by at least five human
annotators on average. This raises questions about the selection of annotation provided by the subjects. To deal with this problem, we have performed a simple logic over the image ground truth
labels. All the image labels are iterated with ‘OR’ operation to generate singular ground truth image label. The ‘OR’ operation is performed to make sure that the final ground truth is suggested by
most of the human annotators. Finally, every image is segmented and benchmarked against this single ground truth labeled image.
3.2. Desiderata of Accurate Super-Pixels
Generally, for super-pixel algorithms there are no definite features for super-pixels to be accurate. The literary studies refer accurate super-pixels in terms of boundary adherence, connectivity of
super-pixels, super-pixel partitioning, compactness, regularity, efficiency, controllable number of super-pixels and so on [
]. As the proposed study is research focused on semi-dark images, we take into account features that are desired for conformation of accurate boundary extraction in semi-dark images.
Boundary Adherence
The boundary adherence is the measure to compute the accuracy to which the boundary has been extracted by super-pixels against boundary image or ground-truth images. The idea is to preserve
information as much as possible by creating super-pixels over the image. The boundary adherence feature is basically a measure that results how accurately the super-pixels have followed the
ground-truth boundaries. This can be easily calculated by segmentation quality metrics precision-recall.
Efficiency with Less Complexity
As super-pixel segmentation algorithms are now widely used as preprocessing step for further visually intelligent tasks. The second desired feature is efficiency with less complexity. The focus
should be creation of memory-efficient and optimal usage of processing resources so that more memory and computational resources can be used by subsequent process. We take into account this feature
and propose an algorithm that uses exactly same resources as of Basic SLIC with added distance measures in its discourse.
Controllable Number of Super-Pixels
The controllable number is super-pixels is a desired feature to ensure the optimal boundary is extracted using the computational resources ideally. The super-pixel algorithms are susceptible to this
feature that is number of super-pixels. The number of super-pixels to be created can directly impact the overall algorithm performance. The performance is degraded in terms of under-segmentation or
over-segmentation error. In the former one, the respective algorithm fails to retrieve most of the boundaries due to the smaller number of super-pixels to be created, whereas the latter one retrieves
maximum boundary portions of the ground-truth images but there is surplus of computational resources.
Nevertheless, as mentioned earlier there is a huge list of accuracy measures and all those measures refer to different segmentation aspects and features. The required features and subsequent accuracy
measures to be reported depend on the application of algorithm. For semi-dark image segmentation, it is mandatory to ensure that most of the optimal boundary is extracted and this requirement can be
related to precision-recall metrics.
3.3. SLIC Preliminaries
Before presenting the SLIC++, we first introduce base functionality of SLIC. The overall functionality is based on creation of restricted windows in which the user defined seeds are placed, and
clustering of image point is performed in this restricted window. This restricted window is called Voronoi Tessellations [
]. Voronoi Tessellations is all about partitioning the image plane into convex polygons. This polygon is square in case of SLIC initialization windows. The Voronoi tessellations are made such that
each partition has one generating point and all the point in the partition are close to the generating point or the mass center of that partition. As the generating point lies in the center these
partitions are also called Centroidal Voronoi Tessellation (CVT). The SLIC algorithm considers CIELAB color space for the processing, where every pixel
on image
is presented by color components and spatial components as
$c p = l p , a p , b p$
being colour components and
$p u , v$
being spatial components. For any two pixels SLIC measures straight line difference or Euclidean distance between the two pixels for the entire image space
$ℝ 5$
The spatial distance between two pixels is given by
$d s$
and color component distance
$d c$
are given in Equations (1) and (2).
$d s = u 1 − u 2 2 + v 1 − v 2 2$
$d c = l 1 − l 2 2 + a 1 − a 2 2 + b 1 − b 2 2$
$d s$
$d c$
represent Euclidean distance between pixel
$p 1$
$p 2$
. Instead of simple Euclidean, SLIC uses distance term infused with Euclidean norm given by Equation (3).
The final distance term is normalized using interval $S$ and $m$ provides control over the super-pixel compactness which results in perceptually meaningful distance with balanced aspect of spatial
and color components. Provided the number of super-pixels $K$ seeds $( s i ) i = 1 K$ are evenly distributed in over the image $I$ clusters are created in restricted regions of Voronoi Tessellations.
The initialization seed are placed in image space within a window of $2 S × 2 S$ having center $s i$. After that simple K-means is performed over the pixels residing in the window to its center. SLIC
computes the distance between pixels using Equation (3) and iteratively processes the pixels until convergence.
3.4. The Extension Hypothesis—Fusion Similarity Measure
The super-pixels created by the SLIC algorithm basically uses the Euclidean distance measure to create pixel clusters or the super-pixels based on the seed or cluster centers. The Euclidean distance
measure takes into account the similarity among pixels using straight line differences among cluster centers and the image pixels. This property of distance measure results in distortion of extracted
boundaries of image. The reason is measure remains same no matter if there is a path along the pixels. The path along the pixels will result in smoother and content relevant pixels [
]. The Euclidean distance overlays a segmentation map over the image without having relevance to the actual content present in the image. Moreover, large diversity in the image (light conditions/high
density portions) result in unavoidable distortion. Therefore, we hypothesize to use accurate distance measure which presents content relevant information of the visual scene. For this reason, we
extend the functionality of SLIC by replacing the Euclidean distance measure with four potential similarity measures including chessboard, cosine, Minkowski, and geodesic and named it as SLIC+. These
distance measures have been used in the literature integrated in clustering algorithms for synthetic textual data clustering where studies mentioned to render reasonable results for focused problem
solving [
]. However, we use these similarity measures to investigate the effects on visual images using SLIC approach. Prior to implementation, a brief introductory discussion will help understand the overall
integration and foundation for choosing these similarity measures. The distance measures are basically the distance transforms applied on different images, specifying the distance from each pixel to
the desired pixel. For uniformity and easy understanding, let pixel
$p 1$
$p 2$
have the coordinates
$( x 1 , y 1 )$
$( x 2 , y 2 ) ,$
• Chessboard:
This measure calculates the maximum distance between vectors. This can be referred to measuring path between the pixels based on eight connected neighborhood whose edges are one unit apart. The
chessboard distance along any co-ordinate is given by identifying maximum, as presented in Equation (4).
$D c h e s s = max x 2 − x 1 , y 2 − y 1$
Rationale of Consideration: Since the problem with existing similarity measures is loss of information, chessboard is one of the alternate to be incorporated in super-pixel creation base. This
measure is considered as it takes into account information of eight connected neighbors of pixels under consideration. However, it might add computational overhead due to the same.
• Cosine:
This measure calculates distance based on the angle between two vectors. The cosine angular dimensions counteract the problem of high dimensionality. The inner angular product between the vectors
turns out to be one if vectors were previously normalized. Cosine distance is based on cosine similarity which is then plugged-in distance equation. Equations (5) and (6) shows calculation of
cosine distance between pixels.
$c o s i n e s i m i l a r i t y = p 1 . p 2 p 1 2 p 2 2$
$D c o s i n e = 1 − c o s i n e s i m i l a r i t y$
Rationale of Consideration: One of the aspects of content aware similarity measure is to retain the angular information thus we attempted to incorporate this measure. The resulting super-pixels
are expected to retain the content relevant boundaries. However, this measure does not consider magnitude of the vectors/pixels due to which boundary performance might fall.
• Minkowski:
This measure is a bit more intricate. It can be used for normed vector spaces, where distance is represented as vector having some length. The measure multiplies a positive weight value which
changes the length whilst keeping its direction. Equation (7) presents distance formulation of Minkowski similarity measure.
$D m i n = p 2 − p 1 µ 1 / µ$
Here $µ$ is the weight, if its value is set to 1 the resultant measure corresponds to Manhattan distance measure. $µ = 2$, refers to euclidean and $µ = ∞$, refers to chessboard or Chebyshev
distance measure.
Rationale of Consideration: As user-control in respective application is desired, Minkowski similarity provides the functionality by replacing merely one parameter which changes the entire
operationality without changing the core equations. However, here we still have problems relating to the retainment of angular information.
• Geodesic:
This measure considers geometric movements along the pixel path in image space. This distance presents locally shortest path in the image plane. Geodesic distance computes distance between two
pixels which results in surface segmentation with minimum distortion. Efficient numerical implementation of geodesic distance is achieved using first order approximation. For approximation
parametric surfaces are considered with
number of points on the surface. Given an image mask, geodesic distance for image pixels can be calculated using Equation (8).
$D g e o = min P x i , x j ∫ 0 1 D ( P x i , x j t ‖ P ˙ x i , x j t ‖ d t$
$P x i , x j t$
is connected path between pixel
$x i , x j$
, provided
= 0,1. The density function
$D x$
increments the distance and can be computed using Equation (9).
$D x = e E x υ , E x = ‖ ▽ I ‖ G σ ∗ ‖ ▽ I ‖ + γ ′$
is scaling factor,
$E x$
is edge measurement also provides normalization of gradient magnitude of image
$‖ ▽ I ‖$
$G σ$
is the Gaussian function with its standard deviation being
minimizes the effect of weak intensity boundaries over density function.
) always produces constant distance, for homogeneous appearing regions if
) is zero
) becomes one.
Rationale of Consideration
: For shape analysis by computing distances geodesic has been the natural choice. However, computing geodesic distance is computationally expensive and is susceptible to noise [
]. Therefore, to overcome effect of noise geodesic distance should be used in amalgamation of Euclidean properties to retain maximum possible information in terms of minimum distance among pixels
and their relevant angles.
The mentioned distance measures for identification of similarity among pixels based on pixel proximity provides different functionality features including extraction of information based on the
4-connected and 8-connected pixel neighborhood, and incorporation of geometric flows to keep track of angular movements of image pixels. However, none of these similarity measures provide balanced
equation with integrated features of optimal boundary extraction based on connected neighbors and their angular movements. Thus, we hypothesize boundary extraction to be more accurate and intricate
in presence of a similarity measure which provides greater information of spatial component provided by neighborhood pixels along with geometric flows.
3.5. SLIC++ Proposal
3.5.1. Euclidean Geodesic—Content-Aware Similarity Measure
Considering the simplicity and fast computation as critical components for segmentation, the proposed algorithm uses fusion of Euclidean and geodesic distance measures. The depiction of Euclidean and
geodesic similarity is presented in
Figure 1
, where straight line shows Euclidean similarity while curved line shows geodesic similarity. Since using only Euclidean similarity loses the context information due to usage of straight-line
distance and geodesic similarity focuses more on the actual possible path along the pixels. We propose the fusion of both the similarities to extract accurate information of image pixels and their
Using the same logic as SLIC, we propose a normalized similarity measure. The normalization is based on the interval
between the pixel cluster centers. To provide the control over super-pixel same variable
is also used. Beforehand the contribution of Euclidean and geodesic distance in final similarity measure cannot be determined in terms of optimized performance. Hence, we have introduced two weight
parameters for proposal of final similarity measure which based on weighted combination of Euclidean and geodesic distance. The proposed similarity measure is presented in Equation (10).
$D c a = w 1 d 1 + m S d 2 + w 2 d 3 + m S d 4$
$D c a$
is content-aware distance measure,
$d 1$
$d 2$
are same as
$d s$
$d c$
(Equations (5) and (6)) calculating the Euclidean distance for spatial and color component of image pixels. Variables
$d 3$
$d 4$
presents color and spatial component distance calculation using geodesic distance equation 8. Specifically,
$d 3$
represents geodesic calculation of color components of image pixel and
$d 4$
represents geodesic calculation of spatial components of pixel. Here, again we introduce similar normalization as of SLIC using variable
to provide control super-pixel compactness using geometric flows. The weights
$w 1$
$w 2$
further provides user control to choose the contribution of Euclidean and geodesic distance in final segmentation. These weights provide user flexibility, and these values can be changed based on the
application. Moreover, these weights can be further tuned in future studies.
3.5.2. Proposal of Content-Aware Feature Infusion in SLIC
The SLIC++ is proposed to extract the optimal information from a visual scene captured in semi-dark scenarios. Nevertheless, the same algorithm holds for any type of image if the objective is to
retrieve maximum information from the image space. The steps involved in computing super-pixels are written in SLIC++ algorithm (refer Algorithm 1). Basically, super-pixels are perceptual cluster
computed based on pixel proximity and color intensities. Some of the parameters include:
$K —$
being the number of super-pixels,
$N —$
total number of pixels,
$A —$
approximate number of pixels also called area of super-pixel, and
$S —$
length of super-pixel.
Algorithm 1. SLIC++ Algorithm
1:Initialize $K$ cluster center with seed $( s i ) i = 1 K$ defined at regular intervals $S$
2:Move cluster centers in $n × n$ pixel neighborhood to lowest gradient
4:$F o r$ each cluster center $s i$ do
5:Assign the pixel from $2 S × 2 S$ in a square window or CVT using distance measure given by Equation (3).
6:Assign the pixel from $2 S × 2 S$ in a square window or CVT using distance measure given by Equation (10).
7:End for
8:Compute new cluster centers and residual error $ε r r$(distance between previous centers and recomputed centers).
9:$U n t i l ε r r$<= threshold
10: Enforce connectivity.
Keeping simplicity and fast computation intact we present SLIC++ algorithm, here only one of the steps mentioned on step 5 or 6 will be used. If step 5 is implemented, i.e., distance measure given by
Equation (3) is used entire functionality of SLIC algorithm is implemented. Whereas, if step 6 is implemented, i.e., distance measure given by Equation (10) is used entire functionality of SLIC
algorithm is implemented.
• Initialization and Termination
For initialization a grid of initial point is created separated by distance S in each direction as seen in
Figure 2
. The number of initial centers is given as parameter
. Placement of initial center in restricted squared grids can result in error if the initial center is placed on the edge of image content. This initial center is termed as confused center. To
overcome this error gradient of the image is computed and the cluster center is moved in the direction of minimum gradient. The gradient is computed with 4-neighboring pixels and the centroid is
moved. To solve this mathematically L2 Norm distance is computed among four connected neighbors of center pixel. The gradient calculation is given by Equation (11).
$G x , y = ‖ x + 1 , y − x − 1 , y ‖ 2 + ‖ x , y + 1 − x , y − 1 ‖ 2$
$G x , y$ is the gradient of center pixel under consideration.
The gradient of the image pixels is calculated until stability where pixels stop changing the clusters based on the initialized clusters. Overall, the termination and optimization is controlled by
parameter ‘n’ which represents number of iterations the overall SLIC algorithm goes through before finally resulting in super-pixel creation of the image. To keep the uniformity in presented research
we have selected ‘
’ as 10 which has been a common practice [
How it works?
The incoming image is converted to CIELAB space. The user provides information of all the initialization parameters including ‘$K$’, ‘$m$’, ‘$n$’. Referring to the algorithm steps presented in SLIC++
algorithm. Step 1, places $K$ number of super-pixels provided by user on an equidistant grid. This grid is created separated by $S$, where $S$ is given by $N K$, $N$ is total number of image pixels.
Step 2 performs reallocation of initial seed takes places subjected to the gradient condition to overcome the effect of initial centers placed over the edge pixels in image. Step 3 through step 7,
are iteratively executed till the image pixels stop changing the clusters based on the cluster centers/seeds. Steps 5 or 6 are chosen based for respective implementation of SLIC or SLIC++ vice versa.
Step 5 and 6 basically performs clustering over the image pixels based on different distance measures. If user opts for SLIC then Euclidean distance measure is used (base functionality). If user opts
for SLIC++ then proposed hybrid distance measure is used. Step 8 checks if the new cluster center after every iteration of clustering is different than the previous center (distance between previous
centers and recomputed centers). Step 9 keeps track of the threshold value for iterations as specified by the parameter ‘n’. Step 10 enforces connectivity among the created super-pixel/clusters of
image pixels.
The simple difference in the implementation of SLIC and SLIC++ lies in the usage of distance measure being used for the computation of image super-pixels. The presented research shows merely changing
the distance measure to content-aware computational distance measure leads to better accuracy of results against the ground-truth for semi-dark images.
Algorithm Complexity
The proposed algorithm follows the same steps as of Basic SLIC by introducing a new content-aware distance equation, thus the complexity of the proposed SLIC++ remains the same without any addition
of new parameters, except the weights associated to the Euclidean and geodesic distance. These weights are merely scaler values to be taken into account in the core implementation of content-aware
variant of SLIC, i.e., SLIC++. Hence, the complexity for the pixel manipulation is up to
$O N$
is the total number of image pixels. With the minimum possible imposed requirements of computation SLIC++ manages to find accurate balance of implementation with infused functionality of Euclidean
and geodesic distance. This fusion results in optimal boundary detection verified in terms of precision-recall in
Section 4
4. Validation of the Proposed Algorithm
4.1. Experimental Setup and Implementation Details
Following the proposed algorithm and details of implementation scheme, SLIC++ is implemented in MATLAB. The benchmarking analysis and experiments are conducted in MATLAB workspace version R2020a
using the core computer vision and machine learning toolboxes. For experiments, the semi-dark images of Berkeley dataset have been used. The reported experiments are conducted on processor with specs
core i7 10750H CPU, 16 GB RAM and 64-bit operating system.
The images are extracted form a folder using Fullfile method then incoming RGB images are converted in CIELAB space. After that parameter initialization takes place to get the algorithm started.
Based on the number of K seed are initialized on the CIELAB image space and the condition relating to the gradient is checked using several different built-in methods. After that each pixel is
processed using the proposed similarity measure and super-pixels are created until the threshold specified by user is reached. Similarly, the performance of reported state-of-the-arts is checked
using the same environmental setup using the relevant parameters. Finally, the reported boundary performance is reported in form of precision recall measure to check the boundary adherence of
super-pixel methods including Meanshift, SLIC and SLIC++. For analysis in terms of precision recall bfscore method is used which takes in the segmented image, ground-truth image and compares the
extracted boundary with the ground-truth boundary by returning parameters precision, recall and score.
4.2. Parameter Selection
In this section we introduce the parameter associated with Meanshift, SLIC and SLIC++. Starting off with the proposed algorithm, SLIC++ uses several parameters as of Basic SLIC. Scaling factor
is set to 10, threshold on the iteration is set to value 10 represented by variable
and parameter S is computed based on N number of image pixels divided by user defined number of super-pixels in terms of variable K. The variable K provides user control for the number of
super-pixels. Compact super-pixels are created as the value of K is increased but it increases the computational overhead. We have reported the performance using four different set of values of K,
i.e., 500, 1000, 1500 and 2000. All these parameters including m, n and K are kept same as for the basic SLIC experiments. However, there are some additional parameters associated with SLIC++ which
$w 1$
$w 2$
and their values are set to 0.3175 and 0.6825, respectively. The weights are cautiously picked based on trial-and-error experimentation procedure. The images were tested for a range of different
weights. The weight values were varied to have weight ratios including 10:90, 30:70, 50:50, 70:30, and 90:10 for Euclidean and Geodesic distance, respectively. The ratio of 30:70 retains empirically
maximum and perceptually meaningful super-pixels resulting in the optimal performance against the ground-truth. For Meanshift implementation the bandwidth parameter is set to 16 and 32, keeping rest
of the implementation parameter default.
Table 2
shows the averaged performance of the proposed SLIC++ algorithm acquired by varying different values of weights for random test cases.
Empirically optimized performance of SLIC++ over 30:70 weight ratio for Euclidean and Geodesic distance hybridization has been tabulated in
Table 2
row number 4, 9, 14 (formatted bold and italics). Moreover, the parameter values have been set as
= 500,
= 10 and
= 10 for the conducted experiments.
4.3. Performance Analysis
For performance analysis we considered two different experimental setups including qualitative analysis and quantitative analysis. Initially, we extended and analyzed the performance of SLIC with
different distance measures to propose the most relevant distance measure for optimal boundary extension in semi-dark images. Then we compare the proposed algorithm with state-of-the-art super-pixel
segmentation algorithms. The detail of the analysis is presented in following sub-sections.
4.3.1. Numeric Analysis of SLIC Extension with Different Distance Measures
For the detailed analysis of the proposed algorithm, we first compare the performance of basic SLIC with the variants of SLIC+ proposed in this study. The evaluation is presented in form of precision
recall. For the optimal boundary detection greater values of precision are required. High precision rates relate to low number of false positives eventually resulting in high chance of accurate
boundary retrieval, whereas high recall rates are relevant to matching of ground-truth boundaries to segmented boundary. Mathematically, precision is probability of valid results and recall is
probability of detected ground-truths data [
]. For analysis of image segmentation modules, both high precision and recall are required to ensure maximum information retrieval [
Table 3
shows performance analysis of basic SLIC and its variants over randomly picked semi-dark images.
Table 2
depicts all the extension of SLIC perform better in terms of precision-recall. The parameters are kept uniform for all the experiments specifically parameter
as in SLIC [
]. Moreover, there is up to 3–9% gain in precision percentage using SLIC++ as compared to the basic SLIC algorithm. The relevant scores based on precision and recall also shoot up by margin of 5–9%
using SLIC++ (row 1 vs. 6 and row 7 vs. 12). However, the performance of other variants of SLIC is subjective to dimensions of incoming data, magnitudes, and memory overload. There usually is no
defined consensus regarding best generalized performer in terms of similarity measure so far [
]. Thus, we propose an integration of two similarity measures which takes into account minimal processing resources and still provides optimal boundary detection.
For further detailed qualitative analysis using the same test cases by changing the number of super-pixels we extend the analysis of SLIC versus SLIC++. The precision recall and score graphs are
shown in
Figure 3
Figure 3
, solid lines represents performance of SLIC and SLIC++ for Test case 1 and dashed lines represents performance for Test case 2.
Figure 3
a shows precision curves of SLIC++ are substantially better than the SLIC presented by brown (dark and light) lines for test case 1 and 2, respectively.
Figure 3
b shows the SLIC++ recall is less the resulting recall of SLIC for the same images. Subsequently, based on the precision, recall and the final scores SLIC++ outperforms basic SLIC on semi-dark
images. For number of pixels set to 1000 there is a drop observed in precision and recall of SLIC++, this behavior can be attributed to accuracy measure’s intolerance, i.e., even mutual refinements
may result in low precision and recall values [
]. Nevertheless, performance for retrieval increases with increasing number of super-pixels and SLIC++ outperforms SLIC up to margin of 10%.
4.3.2. Comparative Analysis with State-of-the-Art
For the benchmarking of SLIC++ two different algorithms, i.e., SLIC and Meanshift are considered. To investigate the performance of SLIC and SLIC++ for the analysis over entire Berkeley dataset
(semi-dark images), we set the number of super-pixels to 1500. The number of super-pixels is chosen 1500 because the peak performance of both the algorithms in experiment for test case 1 and 2 (refer
Figure 3
) is achieved by setting this parameter to value 1500. For the comparative analysis we also used Meanshift algorithm with input parameter, i.e., bandwidth set to 32. The bandwidth of Meanshift
decides the complexity of the algorithm as this value is decreased the segmentation becomes more intricate with the overhead of computational complexity. To maintain computational resources
throughout the experiment and keeping it uniform the parameters are chosen. The summary statistics of the obtained super-pixel segmentation results are shown in
Table 4
. The numerals presented in table are averaged values of precision, recall, and scores obtained for 316 images separately. The average precision, recall, and scores are presented in
Table 4
Table 4
shows SLIC++ achieves average percentage score up to 54%, whereas SLIC maintains a score of 47%. Finally, Meanshift achieves a score of 55%, which is greater than SLIC++ but as stated earlier for
segmentation application greater values of precision and recall are required. So, comparing the recall of SLIC++ versus Meanshift a huge difference is observed. This difference is in terms of low
recall of Meanshift which means algorithm fails to capture salient image structure [
] which is not desired for semi-dark image segmentation.
4.3.3. Boundary Precision Visualization against Ground-Truth
To validate the point-of-view relating to high precision and high recall we present perceptual results of Meanshift, SLIC and SLIC++. Notice that, the high precision means the algorithm has retrieved
most of the boundary as presented by the ground-truth, whereas high recall means most of the salient structural information is retrieved from the visual scene. Meanshift resulted in a minimum recall,
which hypothetically means the structural information was lost.
Table 5
presents how Meanshift, SLIC, and SLIC++ performed in terms of perceptual results for visual information retrieval. The reported results are for parameters
= 1500 for SLIC and SLIC++ and bandwidth = 32 for Meanshift.
As super-pixels are not just about the boundary detection, resulting applications also expect the structural information present in the visual scene. Consequently, we are not just interested in the
object boundaries but also the small structural information present in the visual image specifically semi-dark images.
Table 5
shows SLIC and SLIC++ not only retrieves boundaries correctly with minimal computational power consumed but also retrieves the structural information. Column 4 shows the fact by mapping prediction
over ground-truth image. For test case 1, in column 4 row id 3 Meanshift fails to extract the structural information as few green lines are observed. Whereas, for the same image SLIC and SLIC++
perform better as a lot of green textured lines are observed (refer column 4 row id 1 and 2). Meanwhile for test case 2, all three algorithms perform equally likely. Similar performance is observed
with test case 3, SLIC and SLIC++ retains structural information better than Meanshift. Since Meanshift resulted in minimum recall over the entire semi-dark Berkeley dataset (refer
Table 4
) it does not qualify to be a good fit for super-pixel segmentation. The reason is less reliability of structural information fetching and its performance is highly subjective to the incoming input
4.3.4. Visualizing Super-Pixels on Images
For one more layer of subjective analysis of super-pixel performance we present super-pixel masks in this section. Initially, we present the input image in
Figure 4
with the highlighted boxes to look closely for retrieval of structural information from the image. Here, the red box shows the texture information present on the hill whereas the green box shows
water flowing in a very dark region of the semi-dark image.
Using the input image presented in
Figure 4
, we conducted experiments by changing the initialization parameters of all three algorithms.
Table 5
shows the perceptual analysis visualizing the retrieval of salient structural information.
Table 6
shows that Meanshift extracts the boundaries correctly, whereas it loses all the contextual information when the bandwidth parameter is set to 32. This loss of information is attributed to low recall
scores, whereas decreasing the value of the bandwidth increases the computational complexity and at the cost of additional complexity Meanshift now retrieves contextual information. SLIC and SLIC++
with minimal computational power retains structural information as seen in the red and green boxes in rows 1 and 2 of
Table 5
. Moreover, as the number of super-pixels ‘
’ increases, better and greater structural information retrieval is observed.
Figure 5
shows a zoomed in view of the super-pixels created by SLIC and SLIC++, residing in the red box. Here, we can see that SLIC++ retrieves content-aware information and SLIC ends up creating circular
super-pixels (
Figure 5
a) due to the content irrelevant distance measure being used in its operational discourse.
Key Takeaways
The benchmarking analysis shows that the proposed algorithm SLIC++ achieves robust performance over different cases. The results of SLIC++ are more predictable as compared to the state-of-the-art
methods Meanshift and SLIC. The performance of Meanshift is highly subjective as the recall keeps changing. Less recall values eventually result in less scores at the cost of information loss.
Whereas, SLIC achieves 7% less scores and 8% less precision values in terms of boundary retrieval. The results of SLIC++ indicate that the proposed content-aware distance measure integrated in base
SLIC demonstrates superior results. The significant improvement to the existing knowledge of super-pixel creation research is hybridization of proximity measures. Based on the comprehensive research
it is seen that the hybrid measure performs better than the singular proximity measure counterparts of the same algorithm. These measures substantially control the end results of super-pixel
segmentation in terms of accuracy. The proposed hybrid proximity measure carefully finds a balance between the two existing distance measure and performs clustering over image pixels making sure to
retain content-aware information.
5. Limitations of Content-Aware SLIC++ Super-Pixels
The super-pixel segmentation algorithms are considered pre-processing step for wide range of computer vision applications. To obtain the optimal performance of sophisticated applications, the base
super-pixel algorithm SLIC uses set of input parameters. These parameters allow the user control over different aspects of image segmentation. The idea is to extract uniform super-pixels throughout
the image grid to maintain reliable learning statistics throughout the process. To make this possible the SLIC initially allows user to choose number of pixels ‘K’ (values ranging from 500–2000),
parameter ‘m’ (where m = 10) which decides the extent of enforcement of color similarity over spatial similarity, number of iterations ‘N’ (where N = 10) which decides the convergence of the
algorithm, neighborhood window ‘w’ (where w = 3) for gradient calculation to relocate cluster centers (if placed on edge pixel). This makes four input parameters for the base SLIC, whereas the
proposed extension SLIC++ introduces two more weight parameters, $w 1$ and $w 2$ (0.3175 and 0.6825, respectively), to decide the impact of each distance measure in the hybrid distance measure. All
these parameters significantly control the accuracy of segmentation results. Incorrect selection of these parameters leads to overall poor performance. Hence, for diverse applications, initial
parameter search is necessary, which in turn requires several runs. For the reported research, using the state-of-the-art segmentation dataset, i.e., Berkeley dataset we chose the parameters as
selected by the base SLIC. These parameters offer good performance over the image size of 321 × 481 or 481 × 321, whereas, as we increased the resolution of images during the extended research we
observed that a higher value of ‘K’ is required for better segmentation accuracy.
For the existing research, we conducted experiments focused to identify the gains associated with usage of the proposed content-aware distance measure over the straight line distance measure. For the
extended research, the input parameters shall be considered for optimization.
6. Emerging Successes and Practical Implications
Several decades of research in computer vision for boosted implementations resulting in fast accurate decisions, super-pixels have been a topic of research for long time now. The super-pixel
segmentation is taken as entry stage providing pre-processing functionality for sophisticated intelligent workflows such as semantic image segmentation. To speed up the overall process of training
and testing of these intelligent systems super-pixels are probable to provide remedies. As the intelligent automated vision systems have critical applications in medicine [
], manufacturing [
], surveillance [
], tracking [
] and so on. For this reason, fast and accurate visual decision are required. As the environmental conditions in form of visual dynamicity is challenging task to tackle by pre-processing modules.
These modules are required to provide reliable visual results. Many super-pixel creation algorithms have been proposed over time to solve focused problems of image content sparsity [
], initialization optimization [
], and accurate edge detection [
]. However, the topic of the lightning condition in this domain remains untouched and needs attention. The dynamic lightning condition is a key component in autonomous vehicles, autonomous robots,
surgical robots. The Berkeley dataset is comprised of images of different objects, ranging from humans, flowers, mountains, animals and so on. The conducted research holds for applications of
autonomous robots and autonomous vehicles. However, the proposed algorithm is backed by the core concepts of image segmentation. For this reason, the presented work can be extended for variety of
applications. Depending on the nature of application, the ranges of input parameters would be changed based on the required sensitivity of the end results, such as for the segmentation application in
the medical domain compact where content-aware information is required. Consequently, the input values including the number of super-pixels to be created will be carefully selected. To handle the
pre-processing problems associated with dynamic lightning conditions focusing autonomous robotics, the proposed extension of SLIC is a good fit. SLIC++ imposing minimum prerequisite conditions
provides direct control over the working functionality and still manages to provide optimal information retrieval from the visual scenes for not only normal images but rather inclusive of semi-dark
7. Conclusions and Future Work
7.1. Conclusions
In this paper, we introduced a content-aware similarity measure which not only solved the problem of boundary retrieval in semi-dark images but is also applicable to other image types such as bright
and dark. The content-aware measure is integrated in SLIC to create content-aware super-pixels which can then be used by other automated applications for fast implementations of high-level vision
task. We also report results of integration of SLIC with existing similarity measures and describe their limitations of applicability for visual image data. To validate out proposed algorithm along
with the proposed similarity measure, we conduct qualitative and quantitative evaluations on semi-dark images extracted from Berkeley dataset. We also compare SLIC++ with state-of-the-art super-pixel
algorithms. Our comparisons show that the SLIC++ outperforms the existing super-pixel algorithms in terms of precision and score values by a margin of 8% and 7%, respectively. Perceptual experimental
results also confirm that the proposed extension of SLIC, i.e., SLIC++ backed by content-aware distance measure outperforms the existing super-pixel creation methods. Moreover, SLIC++ results in
consistent and reliable performance for different test image cases characterizing a generic workflow for super-pixel creation.
7.2. Future Work
For the extended research, density-based similarity measures integrated with content-aware nature may lead the future analysis. The density-based feature is expected to aid the overall all working
functionality with noise resistant properties against the noisy incoming image data. Moreover, another aspect is the creation of accurate super-pixels in the presence of non-linearly separable data
properties. Finally, the input parameter selection for the initialization of SLIC variants depending on the application domain and incoming image size remains an open area of research.
Author Contributions
Project administration, supervision, conceptualization, formal analysis, investigation, review, M.A.H.; conceptualization, methodology, analysis, visualization, investigation, writing—original draft
preparation, data preparation and curation, M.M.M.; project administration, resources, review, K.R.; funding acquisition, conceptualization, review, S.H.A.; methodology, software, analysis,
visualization, S.S.R.; validation, writing—review and editing, M.U. All authors have read and agreed to the published version of the manuscript.
This research study is a part of the funded project under a matching grant scheme supported by Iqra University, Pakistan, and Universiti Teknologi PETRONAS (UTP), Malaysia. Grant number: 015MEO-227.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
Conflicts of Interest
The authors declare no conflict of interest.
Figure 1. Irrelevance of Euclidean distance measure for super-pixel creation relating to image content.
Figure 2.
Restricted Image search area for super-pixel creation specified by input argument for image window under consideration [
Figure 3. SLIC v/s SLIC++ performance over different number of pixels: (a) precision values; (b) recall value; (c) score values.
Figure 5. Zoomed in view of test case image for content-aware super-pixel analysis created by SLIC++ (b) against SLIC (a).
Method Complexity Pixel Manipulation Strategy Distance Measure Dataset Image Year Ref.
Meanshift $O N 2$ Mode seeking to locate local maxima Euclidean Not mentioned ✘ 2002 [17]
Medoidshift $O N 2$ Approximates local gradient using weighted estimates of Euclidean Not mentioned ✘ 2007 [18]
Quickshift $O d N 2$ Parzen’s density estimation for pixel values Non-Euclidean Caltech-4 ✘ 2008 [19]
TurboPixel $O N$ Geometric flows for limited pixels Gradient calculation for Berkeley Dataset ✘ 2009 [20]
boundary pixels only
$O N 3 2 l o Probabilistic modeling Dynamic road scenes. No explicit
Scene Shape Super-pixel (SSP) g N$ Shortest path manipulation with prior information of boundary. plus Euclidean space mentions of semi-dark images but we ✓ 2009 [21]
manipulation suspect presence of semi
Compact Super-pixels (CS) - Approximation of distance between pixels and further Euclidean 3D images ✘ 2010 [22]
optimization with graph cut methods
Compact Intensity Super-pixels - Same as CS, With added color constant information. Euclidean 3D images ✘ 2010 [22]
SLIC $O N$ Gradient optimization after every iteration Euclidean Berkeley Dataset ✘ 2012 [11]
SEEDS - Energy optimization for super-pixel is based on hill-climbing. Euclidean Berkeley Dataset ✘ 2012 [23]
Structure Sensitive Super-pixels $O N$ Super-pixel densities are checked, and energy minimization is Geodesic Berkeley Dataset ✘ 2013 [16]
Depth-adaptive super-pixels $O N$ Super-pixel density identification, followed by sampling and Euclidean RGB-D dataset consisting of 11 images ✘ 2013 [24]
finally k-means to create final clusters
Contour Relaxed Super-pixels $O N$ Uses pre-segmentation technique to create homogeneity Not mentioned Not mentioned ✘ 2013 [25]
Saliency-based super-pixel $O N$ Super-pixel creation followed by merging operator based on Euclidean Not mentioned ✘ 2014 [26]
Linear Spectral Clustering $O N$ Two-fold pixel manipulation strategy of optimization based on Euclidean Berkeley Dataset ✘ 2015 [27]
graph and clustering based algorithms.
Manifold SLIC $O N$ Same as SLIC but with mapping over manifold. Euclidean Berkeley Dataset ✘ 2016 [15]
Berkeley Segmentation Dataset (BSD),
BASS (Boundary-Aware Super-pixel $O N l o g Extension of SLIC initially creates boundary then uses SLIC HorseSeg,
Segmentation) N$ with different distance measures, along with optimization of Euclidean + Geodesic DogSeg, MSRA Salient Object Database, ✓ 2016 [28]
initialization parameters. Complex Scene Saliency Dataset (CSSD)
and Extended CSSD
BSLIC $O 6 N$ Extension of SLIC initializes seed within a hexagonal space Euclidean Berkeley Dataset ✘ 2017 [29]
rather than square
Intrinsic Manifold SLIC $O N$ Extension of Manifold SLIC with geodesic distance measure Geodesic Berkeley Dataset ✘ 2017 [30]
Similarity Ratio based $O N$ Extension of SLIC. Proposes automatic scaling of coordinate Mahanlanobis SAR Image dataset ✓ 2017 [31]
Super-pixels axes.
Optimized initialization parameters such as ‘n’ number of cyrosection Visible Human Male
Scalable SLIC $O N$ super-pixels, focused research to parallelization of sequential Euclidean dataset ✘ 2018 [32]
Content adaptive super-pixel Work on prior transformation of image with highlighted edges Euclidean (with
segmentation $O N 2$ created by edge filters graph-based Berkeley Dataset ✘ 2019 [33]
BASS (Bayesian Adaptive $O N$ Uses probabilistic methods to intelligently initialize the Euclidean Berkeley Dataset ✘ 2019 [34]
Super-Pixel Segmentation) super-pixel seeds.
Super-pixel segmentation with $O N N o . o Attempts to use neural networks for automatic seed
fully convolutional networks f l a y e r initialization over grid. Euclidean Berkeley Dataset, SceneFlow Dataset ✘ 2020 [35]
Texture-aware and structure Three different distance
preserving Super-pixels $O N$ The seed initialization takes place in circular grid. measure (without explicit Berkeley Dataset ✘ 2021 [36]
Efficient Image-Warping Framework Warping transform is used along with SLIC for creation of
for Content-Adaptive Super-pixels $O N$ adaptive super-pixels. Euclidean Berkeley Dataset ✘ 2021 [37]
Edge aware super-pixel $O N N o . o Edges are detected using unsupervised convolutional neural
segmentation with unsupervised CNN f l a y e r networks then passed to super-pixel segmentation algorithms Entropy based clustering Berkeley Dataset ✘ 2021 [38]
Row Ratio $w 1$ (Euclidean) $w 2$ (Geodesic) Precision Recall Score
Test Case 1 (Image ID = 14037):
1 10:90 0.1123 0.8877 0.47882 0.88930 0.62248
2 70:30 0.6825 0.3175 0.38850 0.92210 0.54660
3 50:50 0.4863 0.5137 0.37780 0.93040 0.53740
4 30:70 0.3175 0.6825 0.48854 0.89124 0.63113
5 90:10 0.8877 0.1123 0.38840 0.87340 0.53770
Test Case 2 (Image ID = 26031):
6 10:90 0.1123 0.8877 0.21623 0.78808 0.33935
7 70:30 0.6825 0.3175 0.18370 0.82790 0.30070
8 50:50 0.4863 0.5137 0.18910 0.85000 0.31000
9 30:70 0.3175 0.6825 0.22661 0.86520 0.35920
10 90:10 0.8877 0.1123 0.18650 0.79000 0.30220
Test Case 3 (Image ID = 108082):
11 10:90 0.1123 0.8877 0.27023 0.89832 0.41548
12 70:30 0.6825 0.3175 0.21840 0.82160 0.34510
13 50:50 0.4863 0.5137 0.22640 0.86800 0.35920
14 30:70 0.3175 0.6825 0.28547 0.91629 0.43532
15 90:10 0.8877 0.1123 0.22360 0.79470 0.34900
Row K m n Parameters Score Precision Recall Distance Measure
Test Case 1 (Image ID = 14037):
1 500 10 10 0.54430 0.39120 0.89390 Euclidean—SLIC
2 500 10 10 0.61234 0.46563 0.89406 Chessboard—SLIC+
3 500 10 10 0.59713 0.44407 0.91118 Cosine—SLIC+
4 500 10 10 $µ = 4$ 0.62792 0.47345 0.93199 Min4—SLIC+
5 500 10 10 0.56128 0.43777 0.78186 Geodesic—SLIC+
6 500 10 10 $w 1 = 0.3175$ $w 2 = 0.6825$ 0.63113 0.48854 0.89124 Euclidean Geodesic—SLIC++
Test Case 2 (Image ID = 26031):
7 500 10 10 0.30420 0.18690 0.81740 Euclidean—SLIC
8 500 10 10 0.35454 0.22098 0.89623 Chessboard—SLIC+
9 500 10 10 0.35698 0.22329 0.88957 Cosine—SLIC+
10 500 10 10 $µ = 4$ 0.34057 0.20959 0.90798 Min4—SLIC+
11 500 10 10 0.33715 0.21369 0.79842 Geodesic—SLIC+
12 500 10 10 $w 1 = 0.3175$ $w 2 = 0.6825$ 0.3592 0.22661 0.86525 Euclidean Geodesic—SLIC++
Test Case 3 (Image ID = 108082):
13 500 10 10 0.35410 0.22720 0.80260 Euclidean—SLIC
14 500 10 10 0.42099 0.27720 0.87476 Chessboard—SLIC+
15 500 10 10 0.38368 0.24251 0.91811 Cosine—SLIC+
16 500 10 10 $µ = 4$ 0.42465 0.27694 0.91004 Min4—SLIC+
17 500 10 10 0.40382 0.26764 0.82216 Geodesic—SLIC+
18 500 10 10 $w 1 = 0.3175$ $w 2 = 0.6825$ 0.43532 0.28547 0.91629 Euclidean Geodesic—SLIC++
Algorithm Score Precision Recall
SLIC 0.47020 0.31604 0.97719
SLIC++ 0.54799 0.39776 0.93470
Meanshift-32 0.55705 0.57573 0.68416
Row ID Image Groundtruth Image Prediction Prediction Map Compared with Groundtruth
Test Case 1:
Test Case 2:
Test Case 3:
Number of Super-Pixels/Algorithm 500 1000 1500 2000
Bandwidth/ 16 32
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Hashmani, M.A.; Memon, M.M.; Raza, K.; Adil, S.H.; Rizvi, S.S.; Umair, M. Content-Aware SLIC Super-Pixels for Semi-Dark Images (SLIC++). Sensors 2022, 22, 906. https://doi.org/10.3390/s22030906
AMA Style
Hashmani MA, Memon MM, Raza K, Adil SH, Rizvi SS, Umair M. Content-Aware SLIC Super-Pixels for Semi-Dark Images (SLIC++). Sensors. 2022; 22(3):906. https://doi.org/10.3390/s22030906
Chicago/Turabian Style
Hashmani, Manzoor Ahmed, Mehak Maqbool Memon, Kamran Raza, Syed Hasan Adil, Syed Sajjad Rizvi, and Muhammad Umair. 2022. "Content-Aware SLIC Super-Pixels for Semi-Dark Images (SLIC++)" Sensors 22,
no. 3: 906. https://doi.org/10.3390/s22030906
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Reciprocal of a Number
Experienced Forum Member
• Feb 2005
• 499
I am creating a report and I need to calcualate the reciprocal of a negative number. Can anyone think back to their algerbra days and tell me how I can come up with the reciprocal of 10.00- which
would be 10.00.
Tags: None
Re: Reciprocal of a Number
I thought reciprocals only deal with positives to fractions and back again.
are you saying if the number is negative youd like to see it as positive
if so then
if number is less than *zeros
number *= -1
All my answers were extracted from the "Big Dummy's Guide to the As400"
and I take no responsibility for any of them.
Ex - Solutions Architect
• Aug 2005
• 453
Re: Reciprocal of a Number
I thought the reciprocal of -10 = -.1 = 1/-10?
If you are looking for the absolute use:
PHP Code:
number = %Abs(number) ;
Predictions are usually difficult, especially about the future. ~Yogi Berra
Vertical Software Systems
• Oct 2005
• 517
Re: Reciprocal of a Number
From Wikipedia,
Reciprocal (mathematics), the number 1/x, which multiplied by x gives the product 1
"Tis better to be thought a fool then to open one's mouth and remove all doubt." - Benjamin Franklin
Sr. Product Specialist
• Nov 2005
• 2614
Re: Reciprocal of a Number
Now class.... let's get back to basics...
1 + 1 = 2
2 + 2 = 4
3 + 3 = 6
My blood runneth Orange
• Mar 2005
• 1119
Re: Reciprocal of a Number
We've had this discussion before! You know the one. About how subtracting a negative from zero gives the same result as multiplying by negative one, but at a fraction of the cost!
if number is less than *zeros
number = 0 - number
"Time passes, but sometimes it beats the <crap> out of you as it goes."
• Aug 2006
• 568
Re: Reciprocal of a Number
Personally I would move all numbers into a large character field then search the character field for a minus sign. If you find one, change it to a space and then move the entire field back into a
Re: Reciprocal of a Number
I have a headache......anyone know what I should take
All my answers were extracted from the "Big Dummy's Guide to the As400"
and I take no responsibility for any of them.
• Oct 2005
• 221
Re: Reciprocal of a Number
You're right. That is the reciprocal of -10.00
You don't stop playing games because you get old, You get old because you stop playing games!
• Aug 2005
• 78
Re: Reciprocal of a Number
I thought you multiplied it by 100.000001... or was that something else... heh
Hail to the king, baby. | {"url":"https://code400.com/forum/forum/iseries-programming-languages/free-format/3707-reciprocal-of-a-number","timestamp":"2024-11-08T16:56:14Z","content_type":"application/xhtml+xml","content_length":"150536","record_id":"<urn:uuid:a4d36087-a5dd-493a-bf96-5167960fe10f>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00313.warc.gz"} |
Quick Start To set_cfa_layout
Shu Fai Cheung & Mark Hok Chio Lai
The package semptools (CRAN page) contains functions that post-process an output from semPlot::semPaths(), to help users to customize the appearance of the graphs generated by semPlot::semPaths().
For the introduction to functions for doing very specific tasks, such as moving the parameter estimate of a path or rotating the residual of a variable, please refer to vignette("semptools"). The
present guide focuses on how to use set_cfa_layout() to configure various aspects of an semPaths graph generated for a typical confirmatory factor analysis (CFA) model.
The Initial semPaths Graph
Let us consider a CFA model. We will use cfa_example, a sample CFA dataset from semptools with 14 variables for illustration.
head(round(cfa_example, 3), 3)
#> x01 x02 x03 x04 x05 x06 x07 x08 x09 x10 x11
#> 1 1.159 1.271 1.451 -0.691 -0.015 -0.212 -0.336 1.559 0.870 1.115 -1.251
#> 2 0.059 -0.496 -0.585 -1.800 -0.555 0.012 1.208 0.551 0.055 -0.365 -0.142
#> 3 -0.737 2.933 1.625 0.642 -1.218 -0.155 -0.861 0.862 0.738 2.443 -0.628
#> x12 x13 x14
#> 1 0.253 0.663 -1.049
#> 2 0.110 -0.207 -0.226
#> 3 1.604 -1.688 0.395
This is the CFA model to be fitted:
mod <-
'f1 =~ x01 + x02 + x03
f2 =~ x04 + x05 + x06 + x07
f3 =~ x08 + x09 + x10
f4 =~ x11 + x12 + x13 + x14
Fitting the model by lavaan::cfa()
This is the plot from semPlot::semPaths():
p <- semPaths(fit, whatLabels="est",
sizeMan = 3.25,
node.width = 1,
edge.label.cex = .75,
style = "ram",
mar = c(10, 5, 10, 5))
The default layout is sufficient to have a quick examination of the results. We will see how set_cfa_layout() can be used to do the following tasks to post-process the graph:
• Change the order of the indicators.
• Change the order of the factors.
• Change the curvature of the inter-factor covariances.
• Move the loadings along the paths from factors to indicators.
• Rotate the graph.
Order the Indicators and Factors
Suppose we want to do this:
• Order the factors this way, from the left to the right:
• Order the indicators this way, from the left to the right:
□ x04, x05, x06, x07, x01, x02, x03, x11, x12, x13, x14, x08, x09, x10
• We would like to place the factors this way:
□ f2 above the center of x04, x05, x06, and x07.
□ f1 above the center of x01, x02, and x03.
□ f4 above the center of x11, x12, x13, and x14.
□ f3 above the center of x08, x09, and x10.
To do this, we create two vectors, one for the argument indicator_order and the other for the argument indicator_factor.
• indicator_order is a string vector with length equal to the number of indicators, with the desired order. In this example, it will be like this:
indicator_order <- c("x04", "x05", "x06", "x07",
"x01", "x02", "x03",
"x11", "x12", "x13", "x14",
"x08", "x09", "x10")
• indicator_factor is a string vector with length equal to the number of indicators. The elements are the names of the latent factors, denoting which indicators will be used to compute the mean
positions to place the latent factors:
indicator_factor <- c( "f2", "f2", "f2", "f2",
"f1", "f1", "f1",
"f4", "f4", "f4", "f4",
"f3", "f3", "f3")
The set_cfa_layout() function needs at least three arguments:
• semPaths_plot: The semPaths plot.
• indicator_order: The vector for the order of indicators.
• indicator_factor: The vector for the placement of the latent factors.
They do not have to be named if they are in this order.
We now use set_cfa_layout() to post-process the graph:
Change the Curvatures of the Factor Covariances
The graph has the factors and indicators ordered as required. However, the inter-factor covariances are too close to the factors. To increases the curvatures of the covariances, we can use the
argument fcov_curve. The default is .4. Let us increase it to 1.75.
The covariances are now more readable. The exact effect of the values vary from graph to graph. Therefore, trial and error is required to find a value suitable for a graph.
Move the Loadings
We can also move all the factor loadings together using the argument loading_position. The default value is .5, at the middle of the paths. If we want to move the loadings closer to the indicators,
we increase this number. If we want to move the loadings closer to the indicators, we decrease this number. In the following example, we move the loadings closer to the indicators, and increase the
distance between them in the process.
p2 <- set_cfa_layout(p,
fcov_curve = 1.75,
loading_position = .8)
The factor loadings are now easier to read, and also closer to the corresponding indicators.
Rotate the Model
The default orientation is “pointing downwards”: latent factors on the top, pointing down to the indicators on the bottom. The orientation can be set to one of these four directions: down (default),
left, up, and right. This is done by the argument point_to.
Like other functions in semptools, the set_cfa_layout() function can be chained with other functions using the pipe operator, %>%, from the package magrittr, or the native pipe operator |> available
since R 4.1.x. Suppose we want to mark the significant test results for the free parameters using mark_sig():
# If R version >= 4.1.0
p2 <- set_cfa_layout(p,
fcov_curve = 1.75,
loading_position = .9,
point_to = "up") |>
#> Loading required package: magrittr
• Currently, if a function needs the SEM output, only lavaan output is supported. | {"url":"http://rsync.jp.gentoo.org/pub/CRAN/web/packages/semptools/vignettes/quick_start_cfa.html","timestamp":"2024-11-03T01:18:08Z","content_type":"text/html","content_length":"104797","record_id":"<urn:uuid:6ee2f93a-b5a0-43db-9947-27611a7ea4d0>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00158.warc.gz"} |
The volume of the blue block is four times the volume of the red block.
What is the volume of the green block?
A pig built a house of bricks.
The pig got a good deal on his phone bill.
It cost him $9 the first month, $19 the second month, and $29 the following months.
What was his bill in the first year?
The factors of a positive integer are the numbers that divide that integer exactly.
( Strictly speaking, 12345 is the largest factor of 12345, so we really mean the next largest factor! )
How many of them are big?
The photograph courtesy of Roland Sauter | {"url":"https://aplusclick.org/k/k6-algebra.htm","timestamp":"2024-11-04T15:39:20Z","content_type":"text/html","content_length":"23669","record_id":"<urn:uuid:f4b68d56-5af9-43ff-bebc-ac6bff04e2d4>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00617.warc.gz"} |
Rhythm Quest
Devlog 2 - Jump Arcs
Published: May 25th, 2021
I spent some time yesterday refactoring and refining the jump mechanics for Rhythm Quest and thought I'd write a bit about it, since it's more interesting than it might seem at first blush.
In Rhythm Quest, each (normal) jump is roughly one beat in length. The exact horizontal distance varies, though, since the scroll rate is different for each song (and even within a song).
The naive approach
Your first instinct when implementing something like this might be to use simple platformer-style tracking of vertical velocity and acceleration:
void Jump() {
yVelocity = kJumpVelocity;
void FrameUpdate(float deltaTime) {
yVelocity += kGravityConstant * deltaTime;
yPosition += yVelocity * deltaTime;
Here we're essentially using a "semi-implicit euler integration", where we modify the differential velocity and position at each timestep. Surprisingly, it's actually fairly important that we modify
the velocity before the position and not the other way around! See https://gafferongames.com/post/integration_basics/ for more on this.
There are a number of issues with this approach. Probably the biggest one is that the behavior is different depending on the framerate (deltaTime), which means that the jump will behave differently
depending on the FPS of the game! You can fix that by using a fixed timestep to do this physics calculation instead of a variable timestep (Unity and Godot both provide a FixedUpdate physics
processing step for this purpose).
The other problem is one that's specific to Rhythm Quest...
The problem
So after some algebra (or trial and error), we have our jump velocity and gravity figured out. The jump paths look something like this:
That's all fine and good, but when we add height changes into the mix, it's a different story:
The sloping ramps in Rhythm Quest aren't actually supposed to have any bearing on gameplay -- they're mostly there for visual interest and to accentuate the phrasing of the music (it looks a lot more
interesting than just running across an entirely flat plain). But they're actually causing gameplay issues now, since they throw off the duration of each jump. It might not seem like much, but it can
add up and cause mistakes, especially in sections like this:
The above gif looks perfectly-aligned though, because it's using a better and robust solution. How did I manage to dynamically alter the jumping behavior to match the changing platform heights?
A more robust solution
The first thing we need to do is throw out our ideas of having a predetermined/fixed jump arc, since that obviously didn't work. Instead, we're going to calculate each jump arc dynamically. No
trial-and-error here, we're going to use quadratics!
The fundamental equation for a parabolic / ballistic arc trajectory is given by y = ax^2 + bx + c. If we assume a start position of x = 0 and y = 0 (this would be the start of the jump), then we can
simplify that to y = ax^2 + bx.
In other words, if we know the two constants a and b, then we can just plug them into this equation and have a mapping between y and x which will trace out an arc. a here represents our "gravity"
term and b represents our initial velocity.
Since we have two unknowns (a and b), we can solve for them if we're given two nonzero points. In other words, we just need to pick two points along the path of the jump arc, and then we can solve
the algebra and have a formula that we can use to calculate our y-coordinates.
The whole idea of this exercise is to have the player land exactly on the target position at the end of the jump, so let's pencil that in as one of our points (shown here in green). To make our lives
easier, we'll say that the x-coordinate at this point is 1:
Of course, in order for this to work, we need to know exactly what end_y is. We could try to calculate this using a raycast, but that wouldn't work if your jump "landing" position isn't actually on
the ground (e.g. you mis-timed a jump and are going to fall into a pit!).
Instead the way that this works is that I have a series of "ground points" that are generated on level initialization. These form a simple graph of the "ground height" of the level, minus any
obstacles. This lets me easily query for the "ground height" at any x-coordinate by using linear interpolation. Conceptually it looks like this:
For our third point, let's just have that be in the middle of the jump horizontally.
There are a couple of different options we could use for determining mid_y, the height of this point. Here's what I ended up with after some twiddling around:
// Some constant base jump height.
float midHeight = GameController.JumpHeight;
if (yDiff > 0.0f) {
// The end of the jump is higher than the beginning.
// Bias towards jumping higher since that looks more natural.
midHeight += yDiff * 0.5f;
} else {
// The end of the jump is lower than the beginning.
// Here I bias towards jumping lower, but not as much as above.
// It ends up looking a bit more natural this way.
midHeight += yDiff * 0.25f;
Time for some math
We have all of our variables and knowns, so let's actually do the math now! We have two equations that we get from plugging in our two points into the basic formula y = ax^2 + bx:
mid_y = 0.25a + 0.5b // x = 0.5, y = mid_y
end_y = a + b // x = 1, y = end_y
This is extremely easy to solve -- just multiply the top equation by two and take the difference. In the end we get:
a = 2 * end_y - 4 * mid_y
b = end_y - a
Now that we know a and b, we can store them and then use them to trace out the path of the arc!
So to wrap it up, each time the player presses jump, we:
• Store the beginning x-coordinate and y-coordinate
• Calculate a and b as shown above
• On subsequent frames, set y = ax^2 + bx + starting_y
The end result, once more:
This is way better than our naive solution from the beginning of the article. Not only does it work with varying heights, but we derived an exact equation to trace out our jump arc (no
approximations!), which means we can just update the visual x and y coordinates in the rendering update instead of having to deal with physics-based timesteps.
A few extra things I also ended up doing to make the jumps feel better:
First, I made jumps actually last slightly shorter than one beat. This is because it looks more natural to have a short period where the character has definitively landed on the ground before the
next jump. This also allows for some more leeway for timing consecutive jumps, and ensures that timing errors don't compound unfairly.
I also allow for early jumps -- in other words, you can re-jump slightly before you actually touch the ground. This again helps with ensuring that timing errors don't compound, and is a nice
quality-of-life improvement for players. In this case I make sure to "snap" your y-coordinate downwards at the beginning of the jump, so it still looks as if you ended up touching the ground (even
though you didn't really).
<< Back: Devlog 1 - Welcome to Rhythm Quest
>> Next: Devlog 3 - Flying Mechanic and Level Generation | {"url":"https://rhythmquestgame.com/devlog/02.html","timestamp":"2024-11-06T19:02:11Z","content_type":"text/html","content_length":"13095","record_id":"<urn:uuid:63096c63-0164-4bf3-b474-f71c0f1159c2>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00225.warc.gz"} |
Mathematics Education
Up to the present, the more significant contributions of LTEE laboratory members in the international scientific and educational community of Mathematics Education could be categorised in the
following dimensions:
Α. Didactics of Mathematics, Teaching and Learning in Primary Education
• A.1. Teaching and learning of arithmetic
• A.2. Teaching and learning of algebra
• A.3. Teaching and learning of geometry and measurement
• A.4. Teaching and learning of stochastic concepts (statistics and probability)
• A.5. Problem solving
Β. Educational materials and games for mathematics in primary education
• B.1. Educational materials for early childhood mathematics
• B.2. Design, development, management and evaluation of educational materials for early childhood mathematics
• B.3. Games for mathematics in primary education
• B.4. Design, development, management and evaluation of games for early childhood mathematics
• B.5. Picture book in early childhood mathematics
• B.6. Designing interdisciplinary educational materials and games
C. Socio-cultural issues on mathematics education
• C1. Mathematics education and family
• C.2. Issues on social interaction in a mathematics classroom
• C.3. Performance in mathematics and school failure
• C.4. Issues on gender
• C.5. Issues on multiculturalism
• C.6. Democratic mathematics education
• C.7. Issues on mathematics school textbooks
D. Teachers’ professional development
E. Theoretical and methodological issues on mathematics education and didactics of mathematics
• E.1. Issues on research’ methodology for mathematics teaching
• E.2. Interdisciplinarity and complexity in mathematics education
• E.3. Inquiry based mathematics education
• E.4. Theoretical Issues in Didactics of Mathematics
F. Mathematics education for students with special needs
• F1. Differential and inclusive pedagogical approaches in the learning of mathematics
Α. Didactics of Mathematics, Teaching and Learning in Primary Education
2008 Kafoussi, S. & Skoumpourdi, C. Mathematics of children 4-6 years old. Numbers and Space.Patakis, Athens, Greece. [GR]
Teaching and learning of arithmetic
1994 Kafoussi, S. (1994). Errors in learning and teaching of arithmetical operations. Euclides γ΄, 39, 41-57 [GR]
1996 Kafoussi, S. & Ntziahristos, E. (1996). Children’s capabilities in counting, entering primary school. Proceedings of the 1st Hellenic Conference of Mathematics Education, 291-308. University of
Athens, Athens[GR]
1998 Kafoussi, S. & Ntziahristos, E. (1998). Mathematical knowledge of third grade students on place value, addition and subtraction of three-digit numbers. Mathematical Review, 49-50, 205-217 [GR]
2001 Kafoussi, S. & Ntziahristos, E. (2001). In-service teachers’ training: the case of fractions. Mathematics, Informatics & Physics, 1, 127-138.
2001 Sonia Kafoussi & Evagelos Ntziahristos. (2001). An assessment of Greek students’ knowledge on place value concepts and whole number operations. Proceedings of thirtieth Spring Conference of the
Union of Bulgarian Mathematicians, “Mathematics and Education in Mathematics”, 297-303. Borovets.
2002 Kalavassis, F. & Skoumpourdi, C. Critical alignments for the use of set theory in early childhood. Bridges, 2, 48-55.
2002 Kafoussi, S. (2002). Discussing about the history of mathematics in classroom. Proceedings of the 19th Hellenic Conference of Mathematics Education, 175-184. HMS, Komotini[GR]
2003 Kapelou, K. & Kafoussi, S. (2003). Approaching the concept of natural number as an operator in a kindergarten school. Proceedings of the 20th Hellenic Conference of Mathematics Education,
211-221. HMS , Veroia [GR]
2007 Kafoussi, S. & Skoumpourdi, C. Kindergarten children analyze numbers and pose word problems for addition and subtraction. 2^nd Conference of GARME Formal and Informal Mathematics, 326-335,
Alexandroupolis, Greece. [GR]
2007 Skoumpourdi, C. Kindergarten children represent quantities helping Snow White. 6^th Dialogue for Mathematics Teaching: Mathematics and Literature, 151-163, Thessalonica, Greece.
2008 Kafoussi, S. (2008). Subitizing of children 3-6 years old. Research in Didactics of Mathematics, 2, 9-28. [GR]
2008 Kafoussi, S. (2008). Research approaches for the development of arithmetical thinking of kindergarten children. In H. Athanasiadis (ed.), Dimensions of research in education and pedagogy, (pp.
240-246). Editions of New technologies, Athens. [GR]
2009 Skoumpourdi, C. Set Theory and object collection: two dimensions of a mathematics teaching. Euclides γ΄, 70, 65-86.
2012 Kafoussi, S. (2012). Informal mathematics of kindergarten children in multicultural classrooms: a pilot research. Euclides γ΄, 77, 1-15. [GR]
2014 Tsampouraki, A. & Kafoussi, S. (2014). The notion of infinity – Thoughts and approaches from sixth grade students. Proceedings of the 31st Hellenic Conference of Mathematics Education, 940-949.
HMS, Veroia[GR]
2015 Tsampouraki, A. & Kafoussi, S. (2015). Sixth grade students solve problems about infinity. 6th Conference of GARME, CD-ROM, Thessaloniki, Greece. [GR]
2016 Intzidou, G. & Kafoussi, S. (2016). Designing learning activities for Sciences and Mathematics Fractions & mixtures. Proceedings of the 2^nd Hellenic Conference “Educational material in
Mathematics and in Sciences”, 165-175. Eds. M. Skoumios & C. Skoumpourdi, Rhodes, Greece. [GR]
2017 Tsampouraki, A. & Kafoussi, S. (2017). Investigating sixth grade’s students’ beliefs about infinity. Research in Didactics of Mathematics, 9, 43-57. [GR]
Teaching and learning of algebra
2013 Skoumpourdi, C. Kindergartners’ performance levels on patterning. International Journal for Mathematics in Education, HMS i JME, 5, 108-131.
2014 Skoumpourdi, C. Patterns in everyday life and in kindergarten. 5^th GARME Conference, CD-ROM, Florina, Greece.
2016 Skoumpourdi, C. Patterns for kindergartners: a developmental framework. In B. Maj-Tatsis, M. Pytlak & E. Swoboda (Eds.) CME Inquiry Based Mathematical Education (pp. 107-116), University of
Rzeszow, Rzeszow, Poland.
Teaching and learning of geometry and measurement
2009 Skoumpourdi, C. & Kafoussi. S. (2009). Oscar and Lisa discuss about shapes with kindergarten children. Journal of Contemporary Education, 156, 125-135. [GR]
2011 Skoumpourdi, C. & Papaioannou_Stravolemou, D. Kindergarten children measure area with the use of auxiliary means. Research in Mathematics Education, 6(1), 39-59.
2015 Skoumpourdi, C. Kindergartners measuring length. In K. Krainer & N. Vondrová (Eds.) 9^th Congress of the European Society for Research in Mathematics Education (CERME 9) (pp. 1989-1995), Charles
University in Prague, Faculty of Education and ERME, Prague, Czech Republic.
2015 Vaitsidi, G. & Skoumpourdi, C. Surfaces’ comparison through estimation and measurement using educational materials. 6^th GARME Conference, 369-378. Thessalonika, Greece.
2016 Tsafou, D. & Skoumpourdi, C. Primary school children estimate lengths. 9^th HMS Conference, 1306-1319. Diadrasi, Athens, Greece.
2017 Malamateniou P.-K. & Skoumpourdi, C. Kindergarten children’ informal strategies for the measurement and comparison of surfaces using educational material. 10^th OMEP Conference, 55-63,
Alexandroupolis, Greece.
Teaching and learning of stochastic concepts (statistics and probability)
1999 Kafoussi, S. (1999). Fifth and sixth grade students’ ideas about the concept of probability. Mathematical Review, 52, 31-41 [GR]
2001 Skoumpourdi, C. & Kalavassis, F. Primary school children’ views about probability concept. 18^th HMS Conference: School’ Role in the Society of the Information and New Technologies, 186-197,
Rhodes, Greece.
2002 Athanassiadis, E., Skoumpourdi, C. & Kalavassis, F. A didactical classification of probability problems linked with their formulation. 2^nd International Conference on the Teaching of
Mathematics αt the Undergraduate Level (ICTM), Crete. (CD-ROM)
2002 Kafoussi, S. & Skoumpourdi, C. Students’ abilities in probability concept in kindergarten. Proceedings of the 15^th Greek Conference on Statistics, vol. A, 339-346. Ioannina, Greece. [GR]
2002 Kafoussi, S. (2002). Learning opportunities in a kindergarten about the concept of probability. Proceedings of the 26^th International Conference on the Psychology of Mathematics Education
(PME), “Learning from Learners”, vol. 3, 161-168. Eds. A. Cockburn & E. Nardi, UEA, Norwich.
2003 Kafoussi, S. (2003). Discussing about the concept of probability in a kindergarten classroom. Proceedings of the 3^rd Mediterranean Conference on Mathematical Education, 459- 465. Eds. A.
Gagatsis & S. Papastavridis, Athens, Hellas.
2003 Skoumpourdi, C. & Kalavassis, F. Development of primary school students’ thought about probability expressions. 20^th HMS Conference: Students’ Course in Mathematics from Pre-School Age to the
Adultness, 509-518, Veroia, Greece.
2004 Skoumpourdi, C. The teaching of probability theory as a new trend in Greek primary education. ICME 10 (10^th International Congress on Mathematical Education), Denmark.
2004 Skoumpourdi, C. & Kalavassis, F. Didactic material in 3^rd grade’ probability chapter: Understanding and manipulation from students and teachers. 3^rd Dialogue for Mathematics Teaching: Picture,
Shape, Speech in Mathematics Teaching, 117-124, Thessalonica, Greece.
2004 Kafoussi, S. (2004). Can kindergarten children be successfully involved in probabilistic tasks? Statistics Education Research Journal (SERJ), 3(1), 29-39.
2006 Kafoussi, S. & Skoumpourdi, C. (2006). Teachers’ evaluation about students’ errors in probability and their didactical interventions. Proceedings of CIEAEM 58, “Changes in society: A challenge
for Mathematics Education”, Quaderni di Ricerca in Didattica (Mathematica), 2009, 19(3), 1-7. Srni, Czech Republic.
2007 Skoumpourdi, C., Tatsis, K. & Kafoussi, S. (2007).Kindergarten children’s informal knowledge about probability. Proceedings of CIEAEM 59, Mathematical Activity in Classroom Practice and as
Research Object in Didactics: two Complementary Perspectives (pp. 59-63), Hungary.
2008 Fesakis, G. & Kafoussi, S. (2008). The development of combinatorial thinking of kindergarten children with the help of ICT. Proceedings of the 6th Hellenic Conference of ICT in education,
129-136, Cyprus. [GR]
2008 Fesakis, G., Kafoussi, S. & Skoumpourdi, C. (2008). Creating stochastic experiences for the development of kindergartner’s intuitive concepts with the use of internet microworlds. 6^th National
Conference of ICT in Education, 280-287, Cyprus. [GR]
2008 Tatsis, K., Kafoussi, S. & Skoumpourdi, C. (2008). Kindergarten children discussing the fairness of probabilistic games: The creation of a primary discursive community. Early Childhood Education
Journal, 36 (3), 221-226.
2009 Skoumpourdi, C., Kafoussi, S. & Tatsis, K. (2009). Designing Probabilistic Tasks for kindergartners. Journal of Early Childhood Research, 7(2), 155-174.
Fesakis, G. & Kafoussi, S. (2009). Kindergarten children capabilities in combinatorial problems using computer microworlds and manipulatives. Proceedings of the 33rd Conference of the
2009 International Group for the Psychology of Mathematics Education(PME), “In Search for Theories in Mathematics Education”, vol. 3, 41-48. Eds. M. Tzekaki, M. Kaldrimidou, H. Sakonidis,
Thessaloniki, Greece.
2011 Fesakis, G., Kafoussi, S. & Malisiova, Ε. (2011). Intuitive conceptions of children in kindergarten, primary and secondary schools about the two dice sum problem through the use of a microworld.
Research in Didactics of Mathematics, 6, 11-37. [GR]
2011 Fesakis, G., Kafoussi, S. & Malisiova, Ε. (2011). Intuitive conceptions of kindergarten children about the two dice sum problem, through the use of a microworld. International Journal for
Technology in Mathematics Education, 18(2), 61-70.
Fessakis, G. (2012). ICT enhanced Data and Graphs comprehension activities in the kindergarten. Preparing the citizens of modern democracies. In S. Kafoussi, C. Skoumpourdi, & F. Kalavasis
2012 (Eds.), Proceedings of the CIEAM64: Mathematics Education and Democracy: learning and teaching practices Conference, Rhodes, Greece, 23-27 July 2012, International Journal for Mathematics in
Education (HMS i JME), Special Issue, Vol. 4, 2012, ISSN: 1791-6321, pp.: 238-242.
2012 Skoumpourdi, C. & Kalavasis, F. (2012). Ways of presenting tasks and their effect on primary pupils’ estimations of probability. In E. Avgerinos & A. Gagatsis (Eds.) Research on mathematical
education and mathematics applications (pp. 196-206), Mathematics Education and Multimedia Laboratory Department of Education, University of the Aegean, Rhodes, Greece.
2012 Kafoussi, S. (2012). Mathematical values and design of activities: the case of stochastic mathematics. Proceedings of the Conference “Innovative approaches in Education”, 65-72. Eds. E Koleza,
Α. Garbis & c. Markopoulos, Patra[GR]
2016 Petrelli, P. & Kafoussi, S. (2016). Designing learning activities for Sciences and Mathematics in a kindergarten school: Weather & Statistics. Proceedings of the 2^nd Hellenic Conference
“Educational material in Mathematics and in Sciences”, 186-195. Eds. M. Skoumios & C. Skoumpourdi, Rhodes, Greece. [GR]
2016 Frantzeskaki, K., Fesakis, G. & Kafoussi, S. (2016). Designing learning activities for Mathematics and ICT for kindergarten children: combinatorics and microworlds. Proceedings of the 2^nd
Hellenic Conference “Educational material in Mathematics and in Sciences”, 176-185. Eds. M. Skoumios & C. Skoumpourdi, Rhodes, Greece. [GR]
2017 Frantzeskaki, K., Kafoussi, S., & Fesakis, G. (2017). The development of kindergarten’ s children combinatorial thinking with the help of ICT. Euclides γ΄ (in press) [GR]
Problem solving
1996 Kafoussi, S. & Ntziahristos, E. (1996). Solving addition and subtraction problems: children’s capabilities entering primary school. Mathematical Review, 45, 54-66 [GR]
2008 Mokos, E. & Kafoussi, S. (2008). Methodological approaches for the study of students’ metacognitive activities in mathematics. Proceedings of the 25th Hellenic Conference of Mathematics
Education, 419-433. HMS, Volos[GR]
2008 Mokos, E., Kafoussi,S. & Kalavasis, F. (2008). Research about metacognition and learning mathematics. Proceedings of the 5th International Meeting of Didactics of Mathematics, vol. 1,
379-391.Eds. M. Kourkoulos & K. Tzanakis, Rethymno [GR]
2009 Mokos, E., Chaviaris, P. & Kafoussi, S. (2009). Metacognitive skills and different kinds of mathematical problems at fourth grade. Proceedings of CIEAEM61, Quaderni di Ricerca in Didattica
(Matematica), 19 (2), 172-177. Montreal, Canada.
2011 Fessakis, G., Lappas, D. (2011). Developing kindergartener’s creativity through creative problem solving in computational learning environments, In proceedings of the European and the 8th of
OMEP: Creativity and learning in Early School-Age, Nicosia, Cyprus, 6-8 May 2011, ISBN: 978-9963-7377-0-3, pp. 859-869
2011 Mokos, E. & Kafoussi, S. (2011). Metacognitive functions during problem solving in pairs: A case study. Proceedings of CIEAEM63, Quaderni di Ricerca in Didattica (Matematica), 361-365.
Barcelona, Spain.
2012 Mokos, E. & Kafoussi, S. (2012). Linking democracy and metacognition: the case of open-ended problems. Proceedings of CIEAEM64, “Mathematics Education and Democracy”, Special Issue of the
Journal HMS i JME, 427-433. Eds. S. Kafoussi, C. Skoumpourdi, F. Kalavasis, Rhodes, Greece.
2013 Fessakis, G., Gouli, E., & Mavroudi, E., (2013). Problem solving by 5–6 years old kindergarten children in a computer programming environment: A case study. Computers & Education, 63, 87–97.
2013 Fessakis G. Lappas D., (2013). Cultivating Preschoolers Creativity Using Guided Interaction with Problem Solving Computer Games, In C. Carvallo and P. Escudeiro (eds.), Proceedings of the 7th
European Conference on Games Based Learning (ECGBL2013), Vol. 2, 2-4 October 2013, Porto, Portugal, pp.: 763-770. Academic Conferences International Limited.
2013 Mokos, E. & Kafoussi, S. (2013). Elementary students’ spontaneous metacognitive functions in different types of mathematical problems. Journal of Research in Mathematics Education (REDIMAT), 2
(2), 242-267.
2014 Mokos, E. & Kafoussi, S. (2014). Metacognitive functions and problem solving The case of authentic mathematical problems. Proceedings of the 5th Conference of Greek Association of Researchers
in Mathematics Education, Florina (CD-ROM) [GR]
2014 Skandalaki, E. & Skoumpourdi, C. Solving division algorithm by 7-year-old students. 31^st HMS Conference, 881-890, Beroia, Greece.
Skandalaki, E. & Skoumpourdi, C. Utilization pictorial representations in problem solving.
2016 Skandalaki, E. & Skoumpourdi, C. Two-step problem solution by 7-8 year-old students. In B. Maj-Tatsis, M. Pytlak & E. Swoboda (Eds.) CME Inquiry Based Mathematical Education (pp. 117-126),
University of Rzeszow, Rzeszow, Poland.
2016 Skandalaki, E. & Skoumpourdi, C. Students of 7-8 years solve comparison problems. 2^nd DEMMS Conference “Developing Educational Materials for Mathematics and Science Learning”, 289-298, Rhodes,
Logico-mathematical and epistemological foundations of the pre-school learning process
B. Educational materials and games for mathematics in primary education
Educational materials for early childhood mathematics
2000 Kafoussi, S. & Ntziahristos, E. (2000). The role of teaching aids for the learning and teaching of mathematics. Mathematical Review, 53, 62-72 [GR]
2003 Skoumpourdi, C. & Kalavassis, F. Didactic materials used in probabilistic activities. CIEAEM 55, The use of didactic materials for developing pupil’s mathematical activities (pp. 35-37), Poland.
2004 Skoumpourdi, C. Didactic Material’ Classification for the Teaching of Mathematics. 21^st HMS Conference: The Curriculum and the Teaching Approach in Primary and Secondary School, 383-393,
Trikala, Greece.
2006 Skoumpourdi, C. Teaching aids for set concept instruction in kindergarten teachers. ICTM 3 (International Conference on the Teaching of Mathematics), Istanbul. (CD-ROM)
2008 Skoumpourdi, C. The use of ‘counting board’ in kindergarten’ mathematics. Research in Mathematics Education, 2, 29-50.
2008 Skoumpourdi, C. The number line in the teaching and learning of mathematics. 25^th HMS Conference: Mathematics education and the reality of the 21^st century, 303-312, Bolos, Greece.
2009 Skoumpourdi, C., Desli, D., Stathopoulou, C. & Tatsis, K. Didactic material in the teaching and learning of mathematics (Thematic Session). 3^rd GARME Conference: Mathematics Education and
Family Practices, 465-466, NT Publications, Athens, Greece.
2010 Skoumpourdi, C. The number line: an auxiliary means or an obstacle? International Journal for Mathematics Teaching and Learning (Electronic Journal).
2011 Skoumpourdi, C. & Kossopoulou, A. The geoboard as an instrument for shapes construction by kindergartners. 4^th GARME Conference, 441-447, Ioannina, Greece.
2012 Skoumpourdi, C. Designing the integration of materials and means in young children’ mathematics education. Patakis, Athens, Greece.
2014 Skoumpourdi, C. The role of educational materials in primary school mathematics. 12^th Dialogue for Mathematics Teaching, 31-55, Athens, Greece. (Invited plenary)
2014 Skoumpourdi, C. The educational material in the relation between teaching mathematics and mathematics education. 5^th GARME Conference, CD-ROM, Florina, Greece.
2015 Skoumpourdi, C. & Goutzina, O. Montessori mathematics material for kindergarten. 1^st DEMMS Conference “Developing Educational Materials for Mathematics and Science Learning”, 227-237, Rhodes,
Design, development, management and evaluation of educational materials for early childhood mathematics
2002 Skoumpourdi, C. & Kalavassis, F. Didactic material for the learning of probability theory with the use of historical problems. 19^th HMS Conference: The Mathematics as Diachronic Factor of
Civilization, 141-150, Komotini, Greece.
2003 Skoumpourdi, C. Forms of teaching activities for the introduction of concept of probability in primary school. Department of Preschool Education Sciences and Educational Design, University of
the Aegean.
2004 Skoumpourdi, C. & Kalavassis, F. Probabilistic situations with the use of manipulatives and students’ heuristics. The Cognitive Psychology Today: Bridges for the study of understanding, 117-119,
Typothito publications, Alexandroupolis, Greece.
2006 Fessakis G., Tasoula E., (2006). Design of a computer controlled robotic device for the construction of mathematical and spatial concepts for preschoolers, “Astrolavos” Journal of Greek
Mathematical Society, Vol 6, July-December 2006, pp. 33-54 (in Greek)
2006 Skoumpourdi, C. Stochastic situations with manipulatives and students strategies. Issues in Education, 7(1), 67-82.
2007 Skoumpourdi, C. The grandfather’ quilt. The little Euclides, 11, 30.
2009 Skoumpourdi, C. Designing a ‘modern’ abacus for early childhood mathematics. Teaching Mathematics and Computer Science, 7(2), 187-199.
2009 Skoumpourdi, C., Kafoussi, S. & Tatsis, K. Designing probabilistic tasks for kindergartners. Journal of Early Childhood Research, 7(2), 153-172.
2010 Skoumpourdi, C. Kindergarten mathematics with ‘Pepe the Rabbit’: how kindergartners use auxiliary means to solve problems. European Early Childhood Education Research Journal, 18(3), 149-157. [3
2011 Skoumpourdi, C. Manipulative bulletin board for early categorization. Teaching Mathematics and Computer Science, 9(1), 1-12.
2012 Skoumpourdi, C. Maria’s collection. The little Euclides, 31, 2-3.
2012 Skoumpourdi, C. Pizza in the wooden oven. The little Euclides, 32, 6.
Skoumpourdi, C. Designing educational material for early childhood mathematics education. In H. Switzer & D. Foulke (Eds.) Kindergartens: Teaching Methods, Expectations and Current Challenges
2013 (pp. 1-82), Series Education in a Competitive and Globalizing World. New York: Nova Science Publishers. ISBN: 978-1-62417-787-3 https://www.novapublishers.com/catalog/product_info.php?
Skoumpourdi, C. Early childhood mathematics: Designing educational material. In R. V. Nata (Ed.) Progress in Education, Volume 31 (Chapter 7, pp. 117-160). Nova Science Publishers, Inc.
Games for mathematics in primary education
2005 Skoumpourdi, C. & Kalavassis, F. Classification of educational games: Connection with game theory. 22^nd HMS Conference: The Modern Applications of Mathematics and their Development, 504-514,
Lamia, Greece.
2007 Skoumpourdi, C. & Kalavassis, F. Designing the incorporation of play in early childhood mathematics education. In F. Kalavasis & A. Kodakos (Eds.) Themes of Education Design (pp. 137-156).
Atrapos, Athens, Greece.
2007 Skoumpourdi, C. & Kalavassis, F. Games as a mathematical activity: the coexistence of differing perceptions in the primary school community (teachers, students, parents). CIEAEM 59, Mathematical
Activity in Classroom Practice and as Research Object in Didactics: two Complementary Perspectives (pp. 92-95), Hungary.
2009 Skoumpourdi, C & Kalavasis, F. The role of play in mathematics education: competitive attitudes and illusion of consensus. Pedagogical Inspection, 47, 139-154.
2010 Skoumpourdi, C. Play as context for approaching early childhood mathematics: Designing board games. Sygchroni Ekpaidevsi, 162, 82-99.
2012 Skoumpourdi, C. The game’ rules. 10^th Dialogue for Mathematics Teaching, CDROM. Athens, Greece.
2015 Skoumpourdi, C. The game in young children’ mathematics education. Athens. [E-Book]
2015 Skoumpourdi, C. “Pleasure, fun, easy learning” … is this the (educational) role of play? 6^th GARME Conference, 608-617. Thessalonika, Greece.
Design, development, management and evaluation of games for early childhood mathematics
2004 Skoumpourdi, C. Find the way. The little Euclides, 2, 29.
2004 Skoumpourdi, C. A simple construction: making shapes with the rope (part 1). The little Euclides, 2, 30.
2005 Skoumpourdi, C. A simple construction: making shapes with the rope (part 2). The little Euclides, 4, 20.
2006 Skoumpourdi, C. Constructing geometrical shapes. The little Euclides, 9, 33.
2006 Skoumpourdi, C. Constructing numbers’ symbols. The little Euclides, 8, 5.
2007 Skoumpourdi, C. The tower of the triangle. The little Euclides, 14, 18-19.
2008 Tatsis, K., Kafoussi, S. & Skoumpourdi, C. Discussing on the fairness of probabilistic games: the creation of a discursive community with kindergarten children. In B. Maj, M. Pytlak, E. Swoboda
(Eds.) Supporting Independent Thinking Through Mathematical Education (pp. 167-173), Rzeszów University of Rzeszów, Poland.
2008 Tatsis, K., Kafoussi, S. & Skoumpourdi, C. Kindergarten children discussing the fairness of probabilistic games: the creation of a primary discursive community. Early Childhood Education
Journal, 36(3), 221-226.
2008 Fesakis, G., Kafoussi, S. & Skoumpourdi, C. Creating stochastic experiences for the development of kindergartner’s intuitive concepts with the use of internet microworlds. 6^th National
Conference of ICT in Education, 280-287, Cyprus.
2009 Skoumpourdi, C. Board games for mathematics (workshop). 3^rd GARME Conference: Mathematics Education and Family Practices, 837-839, NT Publications, Athens, Greece.
2009 Skoumpourdi, C. The cat, the mice and the cheese. The little Euclides, 21, 4-5.
2009 Skoumpourdi, C. & Sofikiti, D. Young children’s material manipulating strategies in division tasks. In Tzekaki, M., Kaldrimidou, M. & Sakonidis, H. (Eds.) 33^rd Conference of the International
Group for the Psychology of Mathematics Education (Vol. 5, pp. 137-144), Thessaloniki, Greece: PME.
2012 Skoumpourdi, C. Playing board games inside and outside the classroom. Quaderni di Ricerca in Didattica (Matematica), 22(1), 130-134.
2013 Skoumpourdi, C. Guess which: shapes’ descriptions by toddlers. Quaderni di Ricerca in Didattica (Matematica), 23(1), 400-409.
2015 Skoumpourdi, C. & Malamateniou, P. Surface coverage game for kindergarten. 1^st DEMMS Conference “Developing Educational Materials for Mathematics and Science Learning”, 238-246, Rhodes, Greece.
2015 Skoumpourdi, C., Pinnika, B., Androti, M., Karagkiozoglou, N.-E., Katsara, E., Mandylaki, E, Pardali, B. & Chatzikokolaki, I. Educational game’ laboratory: evaluation of educational games. 1^st
DEMMS Conference “Developing Educational Materials for Mathematics and Science Learning”, 108-114, Rhodes, Greece.
2016 Skoumpourdi, C. Kindergarten children communicate geometric shapes through game. Communication and Education Vol. 2 (pp. 385-414). Diadrasi, Athens.
2016 Skoumpourdi, C. Different modes of communicating geometric shapes, through a game, in kindergarten. International Journal for Mathematics Teaching and Learning, 17(2), ISSN: 1473 – 0111.
2017 Kokiopoulou, P. Tatsis, K. & Skoumpourdi, C. Farmer-tangles: The use of a board game for the study of 4th grade pupils strategies and mathematical skills. 7^th GARME Conference, 428-438, Athens,
Picture book in early childhood mathematics
2008 Skoumpourdi, C. Designing a picture book for approaching shapes by kindergarten children. 7^th Dialogue for Mathematics Teaching: Book in the teaching of mathematics, 321-332, Thessalonica,
2008 Skoumpourdi, C. & Kafoussi, S. Developing mathematical discussion through picture book reading. 8^th Conference of Mathematics and Human Sciences, 524-533, Athens, Greece.
2009 Skoumpourdi, C. & Kafoussi, S. Oscar and Liza discuss shapes with kindergarten children. Sygchroni Ekpaidevsi, 156, 125-135.
2011 Skoumpourdi, C. & Mpakopoulou, I. The prints: a picture book for pre-formal geometry. Early Childhood Education Journal, 39(3), 197-206.
2016 Matha, A. & Skoumpourdi, C. Epistemic validity of illustrated books with geometric shapes. 2^nd DEMMS Conference “Developing Educational Materials for Mathematics and Science Learning”, 279-288,
Rhodes, Greece.
2017 Skoumpourdi, C. & Chrysanthi, P.-T. Picture book with repetitive division situations and kindregarten children’ action. Research in Mathematics Education, 9, 59-82.
Designing interdisciplinary educational materials and games
2008 Skoumpourdi, C. & Stathopoulou, C. Designing learning activities for Science with the use of educational material. In B. Christidou (Ed.) Educating young children in Sciences (pp. 345-361).
Kyriakidis Publications, Thessalonika, Greece.
2015 Skoumios, M. & Skoumpourdi, C. Developing educational material for mathematics and science, 1^st DEMMS Conference “Developing Educational Materials for Mathematics and Science Learning”, 14-37,
Rhodes, Greece.
2016 Skoumpourdi, C. & Skoumios, M. Educational material for mathematics and educational material for Science: lonely pathways or interactions? 2^nd DEMMS Conference “Developing Educational Materials
for Mathematics and Science Learning”, 15-52, Rhodes, Greece.
C. Socio-cultural issues on mathematics education
Mathematics education and family
2003 Kafoussi, S. & Ntziahristos, E. (2003). Parents’ beliefs about mathematics in primary school. Proceedings of the 20th Hellenic Conference of Mathematics Education, 264-273. HMS, Veroia[GR]
2003 Ntziahristos, E. & Kafoussi, S. (2003). Parents, students, teachers and mathematics. Mathematical Review, 60, 63-84 [GR]
2005 Kafoussi, S. , Skoumpourdi, C. & Kalavassis, F. (2005). Parents’ and teachers’ beliefs about kindergarten children’s informal knowledge in mathematics. Proceedings of the First Conference of
Greek Association of Researchers in Mathematics Education, 313-321. Athens [GR]
2005 Kafoussi, S. (2005). Parents’ and teachers’ interaction concerning students’ mathematics learning. Proceedings of CIEAEM 57, “Changes in society: A challenge for Mathematics Education”, 221-225.
2006 Kafoussi, S. (2006). Parents’ and students’ interaction in mathematics: designing home mathematical activities. Proceedings of CIEAEM 58, “Changes in society: A challenge for Mathematics
Education”, Quaderni di Ricerca in Didattica (Mathematica), 2009, 19(3), 79-85. Srni, Czech Republic.
2005 Kafoussi, S. (2005). Parents, students and emotions about mathematics. In H. Papailiou, G. Xanthakou & S. Chatzihristou (Eds.), Educational School Psychology (pp. 72-78). Athens, Atrapos [GR]
2007 Kafoussi, S. (2007). Interaction between school and family in mathematics education. In F. Kalavasis & A. Kontakos (Eds.), Themes of Educational Design, (pp. 267- 281). Atrapos editions. Athens
2009 Chaviaris, P. & Kafoussi, S. (2009). Communication between school and family through worksheets in Mathematics: a case study. Proceedings of the 3rd Conference of Greek Association of
Researchers in Mathematics Education, 181-191. Eds. Kalavassis, F., Kafoussi, S., Chionidou, M., Skoumpourdi, C., & Fesakis G. , Rhodes. [GR]
2009 Skoumpourdi, C., Tatsis, K. & Kafoussi, S. The involvement of mathematics in everyday activities and games: Parents’ views. 3^rd GARME Conference: Mathematics Education and Family Practices,
131-139, NT Publications, Athens, Greece
2013 Kafoussi, S., Chaviaris, P. (2013). School classroom, family, society and mathematics education. Patakis, Athens [GR]
2014 Chaviaris, P., Kafoussi, S. & Chronaki, A. (2014). Students’ and parents’ practices as they communicate statistical information. Proceedings of the 31st Hellenic Conference of Mathematics
Education, 997-1007. HMS, Veroia[GR]
2014 Moutsios-Rentzos, A., Chaviaris, P. & Kafoussi, S. (2014). Primary school students’ perceptions about parental involvement in mathematics in different school communities. Proceedings of the 5^
th GARME Conference, CD-ROM, Florina, Greece. [GR]
2014 Kafoussi, S., Moutsios-Rentzos, A. & Chaviaris, P. (2014). An investigation on the school socio-cultural identity and the perceived parental involvement on mathematics learning in Greece
.Proceedings of CIEAEM66, Quaderni di Ricerca in Didattica Mathematics, 24(1), 308-311. Lyon, France.
2015 Moutsios-Rentzos, A., Chaviaris, P., & Kafoussi, S. (2015). School socio-cultural identity and perceived parental involvement about mathematics learning in Greece. REDIMAT, 4(3), 234-259. doi:
2017 Kafoussi, S., Moutsios-Rentzos, A. & Chaviaris, P. (2017). Investigating parental influences on sixth graders’ mathematical identity: the case of attainment. Proceedings of MES 9, vol. 2,
592-602. Ed. Anna Chronaki, Volos, Greece.
2017 Chaviaris, P., Kafoussi, S., Moutsios-Rentzos, A. (2017). Sixth graders’ perceived parental influences on their mathematical identity: Attainment and attitudes. Proceedings of the 7^th GARME
Conference, 691-700. Athens [GR]
Issues on social interaction in a mathematics classroom
1993 Boufi, A. & Kafoussi, S. (1993). Communication among students in classroom: its role on in-service teachers’ reflection about issues concerning mathematics teaching. Proceedings of the 9^th
Hellenic Conference of Mathematics Education, 140-155. HMS, Patras[GR]
2003 Chaviari, P. & Kafoussi, S. (2003). Small group students’ involvement in a mathematical activity and its relation to their beliefs about cooperation. Proceedings of CIEAEM 55, “The use of
didactic materials for developing pupil’s mathematical activities”. Poland.
2003 Chaviaris, P. , Kafoussi, S. & Kalavassis F. (2003). Methodological tools for the analysis of social interaction in mathematics classroom. Proceedings of the Second Conference for Mathematics in
Secondary Education, Athens (CD-ROM). [GR]
2005 Chaviaris, P. & Kafoussi, S. (2005). Students’ reflection on their sociomathematical small-group interaction: a case study. Proceedings of the 29th Conference of the International Group on the
Psychology of Mathematics Education (PME), Vol. 2, 241-248. Eds. H. L. Chick & J. L.
2007 Chaviaris, P., Kafoussi, S. & Kalavassis, F. (2007). Pupils’ meta-discursive reflection on mathematics: a case study. Teaching Mathematics and Computer Science, 5(1), 147-169.
2007 Chaviaris, P. & Kafoussi, S. (2007). Metacognitive processes and social interaction among students in mathematics: the case of school textbooks in fifth and sixth grades. Teaching of Sciences:
Research and Practice, 20-21, 59-65. [GR]
2009 Kafoussi, S., Chaviaris, P. & Dekker, R. (2009-10). Factors that influence the development of students’ regulating activities as they collaborate in mathematics. International Journal for
Mathematics in Education (HMS i JME), 2, 46-74.
2010 Chaviaris, P. & Kafoussi, S. (2010). Developing students’ collaboration in a mathematics classroom through dramatic activities. International Electronic Journal of Mathematics Education, 5(2),
2011 Mokos, E. & Kafoussi, S. (2011). Metacognition, mathematics and work in groups. Proceedings of the 4^thConference of Greek Association of Researchers in Mathematics Education, 341-350. Eds. M.
Kaldrimidou & X. Vamvakousi, Ioannnina[GR]
2011 Kafoussi, S. & Chaviaris, P. (2011). Students’ beliefs and practices as they collaborate in mathematics. In P. Fokialis, N. Abdreadakis & G. Xanthakou (Eds.), Investigation of thinking at
school and society, vol. Α΄, (pp 560-579). Editions Pedio, Athens. [GR]
2013 Kafoussi, S. (2013). Can preschool children collaborate in mathematical tasks productively? International Journal for Mathematics in Education (HMS i JME) , vol. 6, 106-119.
Public use of concepts and mathematical expressions and critical mathematical literacy
… ….
Performance in mathematics and school failure
2002 Kalavassis, F., Mitsoullis, C., Orfanos, S., Skoumpourdi, C. & Tzortzakakis, G. Error and stigma: Error’s evaluation in mathematics and school failure prevention. In N. Polemikos, M. Kaila & F.
Kalavassis (Eds) Educational, Family and Political Psychopathology (vol.C΄, pp. 120-155). Deviations in Education. Atrapos, Athens, Greece.
2005 Skoumpourdi, C. Mathematics nature, ability and teaching: Three factors which may cause negative attitudes for mathematics education. In C. Papailiou, G. Xanthakou & S. Chatzichristou (Eds)
Educational School Psychology (vol. Α΄ pp. 39-46). Atrapos, Athens, Greece.
2005 Kalavassis, F., Kokkalidis, S., Stathopoulou, C. & Skoumpourdi, C. Student’s School Route from Kindergarten to High School as a Parameter of Schooling Performance: Research Approach in
Dodecanese Islands for the project “Pythagoras”, 111. Abstract in the Proceedings of the Greek Conference School and Family, Ioannina, Greece.
2005 Chaviaris, P. & Skoumpourdi, C. The students’ personal school-routes as a variable for the study of the relation between mathematics education and school failure. CIEAEM 57, Changes in society:
A Challenge for Mathematics Education (pp. 298-299), Italy. (Poster)
2006 Kalavassis, F., Kafoussi, S., Skoumpourdi, C. Kapelou, K. Stathopoulou, C. Chaviaris, P. & Kokkalidis, S. Interaction between school failure and mathematics education: An approach of educational
mechanics in the case of multi-island cluster of dodecanese. University of the Aegean, Rhodes, Greece.
2006 Stathopoulou, C., Kafoussi, S. & Chaviaris, P. (2006).The relationship between school failure and mathematics education. Themes in Education, 7(2), 179-196. [GR]
2006 Skoumpourdi, C. & Kafoussi, S. Students’ attrition rate: gender and mathematical performance. 23^rd HMS Conference, 583-595, Patra, Greece.
2006 Skoumpourdi, C. & Kalavassis, F. How the performance in primary school mathematics influence students’ school-route in difficult territorial areas: the case of a 18 small-islands complex in
Greece. CIEAEM 58, Changes in society: A Challenge for Mathematics Education (pp. 226-231), Czech Republic.
2006 Skoumpourdi, C. & Kapelou, K. The attrition rate of students and its relation with performance in mathematics. ICTM 3 (International Conference on the Teaching of Mathematics), Istanbul.
2008 Kalavassis, F., Skoumpourdi, C. & Stathopoulou, C. Models of students’ school routes. The case of the Dodecanese Islands. Pedagogical Inspection, 45, 94-110.
2010 Skoumpourdi, C. & Kalavassis, F. How the performance in primary school mathematics influence student’s school-route in difficult territorial areas: the case of a 18 small-islands complex in
Greece. Quaderni di Ricerca in Didattica (Matematica), 20, 145-152.
2010 Kafoussi, S., Stathopoulou, C. & Skoumpourdi, C. Assessment in mathematics and the school failure: the case of the evening schools students. Mentor 12, 19-35.[GR]
Issues on gender
2005 Kalavassis, F. & Skoumpourdi, C. The effect of educational material to the gender discrimination in mathematics. Abstract in Symposium Gender and Education: Mathematics, Physical Sciences and
New Technologies, 4, Thessalonica, Greece.
2007 Skoumpourdi, C. & Kalavassis, F. The effect of educational material to the gender discrimination in mathematics. In E. Drenoyanni, F. Seroglou & E. Tressou (Eds.) Mathematics, Science and
Technology (pp. 63-81). Kaleidoskopio, Athens, Greece.
2008 Kafoussi, S., Skoumpourdi, C. & Kalavassis, F. Gender and mathematics: A need for a total renegotiation of mathematics in education. Euclides γ΄, 69, 23-39.
Skoumpourdi, C. & Kafoussi, S. Mathematics for women or mathematics for all?
Issues on multiculturalism
2002 Kalavassis, F., Kafoussi, S., Stathopoulou, H. & Kapelou, K. (2002). The teaching of mathematics in primary education in Egypt, Greece, Cyprus, and Lebanon. Themes in Education, 3, 2-3,155-166
2005 Skoumpourdi, C. & Stathopoulou, C. The role of mathematics education in contemporary multicultural society. IA΄ International Conference of Pedagogical Society, 821-828, Rhodes, Greece.
2009 Stathopoulou, C., Skoumpourdi, C. & Kafoussi, S. The teaching of mathematics in multilingual classrooms. In C. Govaris (Ed.) Texts for teaching and learning at multicultural schools (pp.
131-148). Atrapos, Athens, Greece.
Democratic mathematics education
2004 Skoumpourdi, C. Translation of ‘Manifesto 2000 for the Year of Mathematics CIEAEM’. Euclides γ΄, 60-61, 90-102.
2012 Fessakis, G., Thoma, R., (2012). Can digital games democratize access to mathematics learning? Tracing the relationship between learning potential and popularity. In S. Kafoussi, C. Skoumpourdi,
& F. Kalavasis (Eds.), Proceedings of the CIEAM64: Mathematics Education and Democracy: learning and teaching practices Conference, Rhodes, Greece, 23-27 July 2012, International Journal
2012 S. Kafoussi, C. Skoumpourdi & F. Kalavasis (Eds). Mathematics Education and Democracy: Learning and teaching practices. CIEAEM64.
2012 Koza, M. & Skoumpourdi, C. Democratic education for blind students. CIEAEM 64, Mathematics Education and Democracy: Learning and Teaching Strategies (pp. 275-280), Rhodes, Greece.
2012 Skoumpourdi, C. Factors in creating a democratic game play. CIEAEM 64, Mathematics Education and Democracy: Learning and Teaching Strategies (pp. 493-496), Rhodes, Greece. (Workshop)
2012 Skoumpourdi, C. Democratic game play: is it a matter of rules? CIEAEM 64, Mathematics Education and Democracy: Learning and Teaching Strategies (pp. 183-188), Rhodes, Greece.
2012 Koza, M. & Skoumpourdi, C. Teaching and methodological approaches to the mathematics of blind children. 29^th HMS Conference, 343-352, Kalamata, Greece.
2013 Koza, M. & Skoumpourdi, C. The contribution of technology in mathematics education of children with visual impairments. 30^th HMS Conference, 468-477, Karditsa, Greece.
Koza, M. & Skoumpourdi, C. Designing tactile educational material for teaching mathematics to blind students.
2015 Fokiali, P. & Skoumpourdi, C. Linking studies with the society and the labor market. Experiences from an internship program. In V. Oikonomidis & E. Kourkoutas (Eds.). University-School
partnership. Creating communities of learning (pp. 145-165). Rethymno: University of Crete.
2016 Koza, M. & Skoumpourdi, C. Educational materials and other resources for teaching the concept of area to blind students. 2^nd DEMMS Conference “Developing Educational Materials for Mathematics
and Science Learning”, 411-420, Rhodes, Greece.
Issues on mathematics school textbooks
2001 Kalavassis, F. & Skoumpourdi, C. The probability concept in primary school textbooks (from Greece, Cyprus and UK): Analysis and comparative elements. 5^th Conference of Didactics on Mathematics
and Informatics in Education, 84-86, Thessalonica, Greece.
2003 Kafoussi, S., Skoumpourdi, C. & Kalavassis, F. An analysis of Greek textbooks’ pictorial representations about multiplication. CIEAEM 55, The use of didactic materials for developing pupil’s
mathematical activities (pp. 99-100), Poland.
2004 Skoumpourdi, C. Types of visual representation of triangle concept in primary school’ mathematics textbooks. 3^rd Dialogue for Mathematics Teaching: Picture, Shape, Speech in Mathematics
Teaching, 105-116, Thessalonica, Greece.
2005 Skoumpourdi, C. Quantitative and qualitative changes in the contents of schoolbooks for mathematics. CIEAEM 57, Changes in society: A Challenge for Mathematics Education (pp. 72-76), Italy.
2009 Kafoussi, S., Skoumpourdi, C. & Tatsis, K. (2009). Analyzing the first grade mathematics school textbook. Euclides γ΄, 71, 42-62.[GR]
2009 Skoumpourdi, C. The representations of the number line in primary mathematics school textbooks. Research in Mathematics Education, 3, 67-87, Kedros, Athens, Greece.
2009 Tatsis, K. & Skoumpourdi, C. First grade school mathematical textbook activities’ context. 3^rd GARME Conference: Mathematics Education and Family Practices, 383-393, NT Publications, Athens,
2009 Skoumpourdi, C. Greek’ educational reformation and the mathematics curriculum. Sygchroni Ekpaidevsi, 159, 95-118.
2013 Skandalaki, E. & Skoumpourdi, C. Two-digit numbers’ addition and subtraction: School Books instructions. 30^th HMS Conference: Mathematics in Education, in Technology and in Society, 820-830,
Karditsa, Greece.
2013 Skoumpourdi, C. & Skandalaki, E. Historical overview of iconic representations of fractions problems. Greek Pedagogical Conference: Childhood: Sociological, Cultural, Historical and Educational
Dimensions, 1265-1275, Athens, Greece.
D. Teachers’ professional development
1991 Boufi, Α. & Kafoussi, S. (1991). Teachers’ conceptions of students’ mathematical errors and conceived treatment of them. Proceedings of the Fifteenth International Conference on the Psychology
of Mathematics Education (PME), vol. I, 176-183. Ed. Fulvia Furinghetti, Assisi, Italy.
1993 Boufi, Α. & Kafoussi, S. (1993). Learning opportunities in an in-service teacher development program. Proceedings of the Seventeenth International Conference on the Psychology of Mathematics
Education (PME), vol. II, 207-214. Eds. Ichiei Hirabayashi, Nobuhiko Nohda et al., Tsukuba, Ibaraki, Japan.
1995 Boufi, A. & Kafoussi, S. (1995). Helping pre-service teachers to teach mathematics: the role of practical exercises. Euclides γ΄, 44, 12, 15-33 [GR]
1999 Kafoussi, S. & Chaviaris, P. (1999). Teachers’ beliefs about the learning and teaching of mathematics in primary education. Proceedings of the 16th Hellenic Conference of Mathematics Education,
169-176. HMS, Larisa [GR]
1999 Kalavassis, F. & Kafoussi, S. (1999). Τhe influence of teachers’ beliefs of different levels of education on students’ mathematical culture. Proceedings of CIEAEM 51, “Cultural Diversity in
Mathematics (Education)”, 217-222. Eds. A. Ahmed, J.- M. Kraemer & H. Williams, Horwood Publishing, Chichester.
2000 Kafoussi, S. & Ntziahristos, E. (2000). Teachers’ beliefs about teaching aids in elementary mathematics. Euclides γ΄, 53-54, 17,102-114 [GR]
2003 Kafoussi,S. & Kaila, M. (2003). In-service teachers’ training in mathematics: balancing between theory and practice. In P. Fokiali, B. Triarhi –Herrmann & Μ. Kaila (Eds.), Issues on teachers’
in- service training and further education (pp. 621-633). Dillingen Academy, Munchen- University of the Aegean, Rhodes. Atrapos editions. Athens
2004 Skoumpourdi, C. Training material for mistakes evaluation and decision’s prevention concerning early childhood education. ICME 10 (p. 159), Denmark. (Poster)
2005 Kalavassis, F., Kafoussi, S. & Skoumpourdi, C. An instrument for errors’ evaluation and decision’s prevention in primary school. CIEAEM 57, Changes in society: A Challenge for Mathematics
Education (pp. 273-277), Italy. (Workshop).
2005 Kafoussi, S., Skoumpourdi, C. & Kalavassis, F. Parents’ and teachers’ beliefs about the informal knowledge of kindergarten children in mathematics. 1^st GARME: Mathematics Education as a
Research Field, 313-321, Athens, Greece.
2005 Kalavassis, F. & Skoumpourdi, C. Is it possible to produce mathematics in the classroom? In K. Bratsalis (Ed.) Texts for teachers training (pp. 157-170). Atrapos, Athens, Greece.
2006 Kafoussi, S. & Skoumpourdi, C. Teachers’ evaluation about students’ errors in probability and their didactical interventions. CIEAEM 58, Changes in society: A Challenge for Mathematics Education
(pp. 58-63), Czech Republic.
2009 Kafoussi, S. (2009). Designing intercultural instructional tasks for kindergarten classrooms with pre-service teachers. Proceedings of CIEAEM61, Quaderni di Ricerca in Didattica (Matematica), 19
(2), 394-399. Montreal, Canada.
2012 Kafoussi, S. & Chaviaris, P. (2012). The Greek primary school teachers’ values about mathematical thinking. Proceedings of CIEAEM64, “Mathematics Education and Democracy”, Special Issue of the
Journal HMS i JME, 331-336. Eds. S. Kafoussi, C. Skoumpourdi, F. Kalavasis, Rhodes, Greece.
2014 Skoumpourdi, C. The length measurement for kindergarten teachers. 31^st HMS Conference, 891-900, Beroia, Greece.
2015 Kafoussi, S. & Chaviaris, P. (2015). Investigating primary school teachers’ values in mathematics. Research in Didactics of Mathematics, 8, 37-54.[GR]
2016 Kalafata, M., Skoumpourdi, C. & Chrysanthi, P.-T. Educators’ views for the educational material in the teaching of mathematics. 2^nd DEMMS Conference “Developing Educational Materials for
Mathematics and Science Learning”, 391-400, Rhodes, Greece.
E. Theoretical and methodological issues on mathematics education and didactics of mathematics
Issues on research’ methodology for mathematics teaching
2008 Skoumpourdi, C., Fesakis, G., Kafoussi, S. & Kalavassis, F. Research in Mathematics Education: Increasing multiplicity of research tools and continuous broadening of theoretical approaches to
the study of a complex and complicated phenomenon. Symposium of Research in the field of Education and Pedagogy, 227-239, Rhodes, Greece.
2008 Kalavassis, F., Kafoussi, S. & Skoumpourdi, C. Connecting research for teaching and learning mathematics with school practice. In B. Sbolopoulos (Ed.), Connecting educational research with
practice (pp. 87-96). Atrapos, Athens, Greece.
2011 Skoumpourdi, C., Fesakis, G., Kafoussi, S. & Kalavassis, F. The increasing diversity of research tools in mathematics education. Euclides γ΄, 75, 57-76.[GR]
Interdisciplinarity and complexity in mathematics education
2010 Kalavasis, F., Kafoussi, S., Skoumpourdi, C. & Tatsis, K. Interdisciplinarity and Complexity (I-C) in Mathematics Education: A proposal for their systematic implementation and the role of an
international scientific community. Revue de l’Interdisciplinarité Didactique, 1(1), 31-40.
2010 Kalavasis, F., Kafoussi, S., Skoumpourdi, C. & Tatsis, K. (2010). Interdisciplinarity and complexity (I-C) in mathematics education: mathematical aspects, school practice and didactical
approach. CIEAEM 62, Mathematics as a living, growing discipline CIEAEM’s contribution to making this explicit (pp. 23-27), London.
Inquiry based mathematics education
2017 Skoumpourdi, C. A framework for designing inquiry based activities (FIBA) for early childhood mathematics. In T. Dooley, & G. Gueudet, (Eds.) 10^th Congress of the European Society for Research
in Mathematics Education (CERME 10) (pp. 1901-1908). Dublin, Ireland: DCU Institute of Education and ERME.
2017 Skoumpourdi, C. Inquiry based mathematical activities for young children: Considerations on their design. 7^th GARME Conference, 246-255, Athens, Greece.
Theoretical Issues in Didactics of Mathematics
2002 Kafoussi, S. (2002). The history of mathematics as a source of creating activities in primary education. In M. Kaila, F. Kalavassis & N. Polemikos (Eds.), Intercultural approach of Myths,
Mathematics and Learning Disabilities in the Information Society (pp.145-153). University of the Aegean, Rhodes.
2002 Kafoussi, S. (2002). Errors as a starting point for a reform in school mathematics. In N. Polemikos, M. Kaila & F. Kalavassis (Eds.), Educational, Parental and Political Psychopathology (pp.
156-170). Atrapos editions. Athens [GR]
2005 Kafoussi, S. (2005). The evolution of research problems in mathematics learning and teaching in primary school. Euclides γ΄, 63,16-30 [GR]
2014 Kafoussi, S. (2014). Teaching mathematics in early childhood: values and activities. 12^th Dialogue for Mathematics Teaching, 13-30. Ed. D. Chasapis, Athens. [GR]
F. Mathematics education for students with special needs
Differential and inclusive pedagogical approaches in the learning of mathematics
2011 Noulis, I. & Kafoussi, S. (2011). Multiplication strategies of children with asperger syndrome: a pilot research. Proceedings of the 28th Hellenic Conference of Mathematics Education, 523-537.
HMS, Athens [GR]
2011 Noulis, I. & Kafoussi, S. (2011). Mathematical difficulties of students with asperger syndrome in multiplication: a pilot research. Proceedings of CIEAEM63, Quaderni di Ricerca in Didattica
(Matematica), 429-433. Barcelona, Spain.
2012 Noulis, I. & Kafoussi, S. (2012). Greek school textbooks and children with asperger syndrome: the case of multiplication. Proceedings of CIEAEM64, “Mathematics Education and Democracy”,
Special Issue of the Journal HMS i JME, 286-291. Eds. S. Kafoussi, C. Skoumpourdi, F. Kalavasis, Rhodes, Greece.
2013 Noulis, I., Kafoussi, S., Papailiou, C. & Polemikos, Ν. (2013). The appropriateness of multiplicative tasks of third and fourth grade’s greek school textbooks for children with asperger
syndrome. Proceedings of 3^rd Hellenic Conference of Special Education, Athens. (CD-ROM). [GR]
2014 Noulis, I. & Kafoussi, S. (2014). The investigation of multiplicative thinking of children with asperger syndrome through verbal problems. Proceedings of the 5^th GARME Conference, CD-ROM,
Florina, Greece. [GR]
2014 Noulis, I. & Kafoussi, S. (2014). Manipulative material as a means for the multiplicative reasoning to children with asperger syndrome. Proceedings of the 1^st Hellenic Conference “Developing
educational material in Mathematics and in Sciences”, 295-312. Eds. C. Skoumpourdi & M. Skoumios, Rhodes, Greece. [GR]
2015 Noulis, I., Kafoussi, S. & Kalavasis, F. (2015). The treatment of multiplicative tasks for asperger syndrome: a case study. Research in Didactics of Mathematics, 8, 11-36. [GR]
Ma Mathematical activities and the reflexive construction of theories in cooperation projects
Sys Systemic interactions between school, family and society on the mathematical abilities and intelligences of children with disabilities | {"url":"http://ltee.aegean.gr/mathematics-education/","timestamp":"2024-11-14T08:19:08Z","content_type":"text/html","content_length":"223694","record_id":"<urn:uuid:278c1d8e-4860-4c3e-8fc1-40edd833880d>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00816.warc.gz"} |
Supporting electrolyte asymptotics and the electrochemical pickling of steel
Asymptotic methods are used to analyse a recent numerical model for the electrochemical pickling of steel. Although the process is characterized by an excess of supporting electrolyte, the asymptotic
structure of the solution turns out not to be the same as that given classically for such situations. In the original theory, the asymptotic expansions for the ionic concentrations and electric
potential are regular and uniformly valid; here, singular perturbation theory is required to take account of the bulk electrolyte and concentration boundary layers adjacent to reacting surfaces. The
reworked theory gives a leading-order problem that is solved numerically; the results are in excellent agreement with those of the earlier computations. Also, invoking the slenderness of the geometry
yields, in a combined quintuple asymptotic limit, an analytical estimate for the current density that captures well the qualitative trends and that enables a rapid assessment of the effect of
operating conditions on the process.
• Electrochemical pickling
• Stainless steel
• Supporting electrolyte
Dive into the research topics of 'Supporting electrolyte asymptotics and the electrochemical pickling of steel'. Together they form a unique fingerprint. | {"url":"https://pure.ul.ie/en/publications/supporting-electrolyte-asymptotics-and-the-electrochemical-pickli","timestamp":"2024-11-10T22:12:55Z","content_type":"text/html","content_length":"53755","record_id":"<urn:uuid:970917cc-d50e-4e30-a8c2-20f2f827caf9>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00283.warc.gz"} |
Unit 7
Lesson 9
Compare Capacity
Warm-up: Choral Count: Count by 10 (10 minutes)
The purpose of this warm-up is for students to count by 10 to 100. Although students see the written sequence of numbers, they are not required to identify numbers beyond 20 until Grade 1.
• Display numbers from 1 to 100.
• “Let’s count to 100.”
• Point to the numbers as students count to 100.
• “Now let’s count to 100 by 10.”
• Demonstrate counting by 10 to 100, pointing to each number as you count.
• “Let’s all count to 100 by 10.”
• Have students repeat the count multiple times.
Activity Synthesis
• “Take turns counting to 100 by 10 with your partner.”
Activity 1: Capacity of Cups (10 minutes)
The purpose of this activity is for students to think about and compare the capacities of containers. Students start by comparing two containers where it is visually obvious which one holds more.
They use comparison language such as “The pitcher holds more than the cup.” Then, students consider containers that have capacities that are not easy to compare visually and they brainstorm ways to
compare the capacities of the containers. Students may need to see the water poured between the two containers multiple times before they determine and can explain which container has more capacity.
As students make predictions and then discuss and justify their comparisons, they share a mathematical claim and the thinking behind it (MP3).
MLR8 Discussion Supports. Use multimodal examples to clarify what it means for a container to hold liquid. Use verbal descriptions along with gestures or drawings to show the meaning of the word hold
in this context.
Advances: Listening, Representing
Required Preparation
• Gather a larger pitcher and a small cup to display during the launch.
• Gather 2 cups with capacities that are not easy to compare visually, such as a tall stemmed glass and a short, wide cup for the activity.
• Groups of 2
• Display a pitcher and a small cup.
• “Diego’s class needs a lot of lemonade for a lemonade sale they are going to have at school. Which container do you think they should use to hold the lemonade? Why do you think that?” (The
pitcher holds more lemonade because it is bigger. The cup is small.)
• 30 seconds: quiet think time
• 1 minute: partner discussion
• Share responses.
• “We think that the pitcher will hold more lemonade.”
• Display 2 cups and give each student a sticky note.
• “Which of these cups do you think would hold more lemonade? Put your sticky note by the cup that you think would hold more lemonade.”
• 3 minutes: independent work time
• “People had different answers about which cup would hold more lemonade. What can we do to figure out which cup can hold more lemonade?”
• 1 minute: quiet think time
• 1 minute: partner discussion
• Share and record responses.
• Demonstrate filling one of the cups with water and then slowly pour that water into the other cup.
• “I filled up the red cup and poured the same water into the blue cup, but the blue cup overflowed. Which cup do you think can hold more lemonade?”
• 30 seconds: quiet think time
• 1 minute: partner discussion
• Share responses.
• “The red cup can hold more lemonade than the blue cup.”
Activity Synthesis
• “We can use water to help us figure out which cup can hold more lemonade.”
Activity 2: Which Cup Can Hold More Water? (15 minutes)
The purpose of this task is for students to compare the capacities of two containers where the comparison is not easy to see visually. Students experiment with filling containers with water to
determine which has a greater capacity. Each group of students needs two cups or containers that they can compare the capacities of, a container of water, and a plastic or foil tray to catch any
water that spills. Students can also use small paper cups to fill up the containers. This activity can also be completed outside or at a water table. There are multiple ways that students can compare
the capacities of the containers, including by pouring water from one container to the other and seeing if the water overflows or if there is room left over or by counting how many small cups it
takes to fill up each container (MP7).
Engagement: Develop Effort and Persistence. Invite students to generate a list of shared expectations for group work. Ask students to share explicit examples of what those expectations would look
like in this activity.
Supports accessibility for: Social-Emotional Functioning
Required Preparation
• Each group of 4 students needs 2 cups or containers that are not easy to compare the capacity visually, such as a short, wide container and a tall, thin container.
• Each group of 4 students needs 1 small paper cup, a container filled with water, and a plastic or foil tray.
• Groups of 4
• Give each group of students two cups or containers, a container of water, a small paper cup, and a plastic or foil tray.
• “Elena likes to drink lots of water after dance class. She’s trying to figure out which cup to use. Which cup holds more water? Work with your group to figure it out.”
• 8 minutes: small-group work time
• Monitor for students who compare the capacities in different ways.
Activity Synthesis
• Invite each group of students to share how they compared the capacities of the cups with the class. As each group shares, ask:
□ “Which cup holds more water?”
□ “Which cup holds less water?”
□ “Tell your partner about the cups using the word ‘less.’”
Activity 3: Centers: Choice Time (20 minutes)
The purpose of this activity is for students to choose from activities that offer practice with number and shape concepts.
Students choose from any stage of previously introduced centers.
• Counting Collections
• Match Mine
• Shake and Spill
Required Preparation
• Gather materials from:
□ Counting Collections, Stage 1
□ Match Mine, Stage 1
□ Shake and Spill, Stages 1-4
• “Today we are going to choose from centers we have already learned.”
• Display the center choices in the student book.
• “Think about what you would like to do first.”
• 30 seconds: quiet think time
• Invite students to work at the center of their choice.
• 8 minutes: center work time
• “Choose what you would like to do next.”
• 8 minutes: center work time
Student Facing
Choose a center.
Counting Collections
Match Mine
Shake and Spill
Activity Synthesis
• “Where can you find any materials that you need to play Match Mine? Where do the materials go when you are finished playing?”
Lesson Synthesis
“Today we figured out which container would hold more water.”
Display a variety of containers used throughout the lesson.
“Tell your partner about the shapes of the containers.” (The glass looks like a cylinder. The cup has a circle on top.)
Cool-down: Unit 7, Section B Checkpoint (0 minutes) | {"url":"https://curriculum.illustrativemathematics.org/k5/teachers/kindergarten/unit-7/lesson-9/lesson.html","timestamp":"2024-11-03T07:25:45Z","content_type":"text/html","content_length":"90031","record_id":"<urn:uuid:da2ba8fe-c67a-4b14-804e-f52f165431ac>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00441.warc.gz"} |
Mathematics Rising
By Joselle, on September 27th, 2017
I read about the sad passing of Maryam Mirzakhani in July, and the extraordinary trajectory of her career in mathematics. But I did not know much about what she was actually doing. A recent post in
Quanta Magazine, with the title: Why Mathematicians Like to Classify Things, caught my attention because the title suggested that the post was about one of the most important ways that mathematics
succeeds – namely by finding sameness among diversity. I found that the work discussed in this post addresses the mathematical world to which Mirzakhani has made significant contributions. Looking
further into the content of the post and Mirzakhani’s experience invigorated both my emotional and my intellectual responses to mathematics.
A Quanta article by Erica Klarreich was written in 2014, when Mirzakhani won the Fields Medal. There Klarreich tells us that when Mirzakhani began her graduate career at Harvard, she became
fascinated with hyperbolic surfaces and, it seems, that this fascination lit the road she would journey. These are surfaces with a hyperbolic geometry rather than a Euclidean one. They can only be
explored in the abstract. They cannot be constructed in ordinary space.
I find it worth noting that the ancestry of these objects can be traced back to the 19th century when, while investigating the necessity of Euclid’s postulate about parallel lines, mathematicians
brought forth a new world, a new geometry, known today as hyperbolic geometry. This new geometry is sometimes identified with the names of mathematicians János Bolyai and Nikolai Ivanovich
Lobachevsky. Bolyai and Lobachevsky independently confirmed its existence when they allowed Euclid’s postulate about parallel lines to be replaced by another. In hyperbolic geometry, given a line
and a point not on it, there are many lines going through the given point that are parallel to the given line. In Euclidean geometry there is only one. With this change, Bolyai and Lobachevsky
developed a consistent and meaningful non-Euclidean geometry axiomatically. Extensive work on the ideas is also attributed to Carl Friedrich Gauss. One of the consequences of the change is that the
sum of the angles of a hyperbolic triangle is strictly less than 180 degrees. The depth of this newly discovered world was ultimately investigated analytically. And Riemann’s famous lecture in 1854
brought definitive clarity to the notion of geometry itself.
With her doctoral thesis in 2004, Mirzakhani was able to answer some fundamental questions about hyperbolic surfaces and, at the same time, build a connection to another major research effort
concerning what is called moduli space. The value of moduli space is the other thing that captured my attention in these articles.
In his more extended piece for Quanta, Kevin Hartnett provides a very accessible description of moduli space that is reproduced here:
In mathematics, it’s often beneficial to study classes of objects rather than specific objects — to make statements about all squares rather than individual squares, or to corral an infinitude of
curves into one single object that represents them all.
“This is one of the key ideas of the last 50 years, that it is very convenient to not study objects individually, but to try to see them as a member of some continuous family of objects,” said
Anton Zorich, a mathematician at the Institute of Mathematics of Jussieu in Paris and a leading figure in dynamics.
Moduli space is a tidy way of doing just this, of tallying all objects of a given kind, so that all objects can be studied in relation to one another.
Imagine, for instance, that you wanted to study the family of lines on a plane that pass through a single point. That’s a lot of lines to keep track of, but you might realize that each line
pierces a circle drawn around that point in two opposite places. The points on the circle serve as a kind of catalog of all possible lines passing through the original point. Instead of trying to
work with more lines than you can hold in your hands, you can instead study points on a ring that fits around your finger.
“It’s often not so complicated to see this family as a geometric object, which has its own existence and own geometry. It’s not so abstract,” Zorich said.
This way of collapsing one world into another is particularly interesting. And one of the results in Mirzakhani’s doctoral thesis concerned a formula for the volume of the moduli space created by
the set of all possible hyperbolic structures on a given surface. Mirzakhani’s research has roots in all of these – hyperbolic geometry, Riemann’s manifold, and moduli space.
Her work, and the work of her colleagues, is often characterized as an analysis of the paths of imagined billiard balls inside a polygon. This is not for the sake of understanding the game of pool
better, it’s just one of the ways to see the task at hand. Their strategies are interesting and, I might say, provocative . With this in mind, Hartnett provides a simple statement of process:
Start with billiards in a polygon, reflect that polygon to create a translation surface, and encode that translation surface as a point in the moduli space of all translation surfaces. The
miracle of the whole operation is that the point in moduli space remembers something of the original polygon — so by studying how that point fits among all the other points in moduli space,
mathematicians can deduce properties of billiard paths in the original polygon. (emphasis added)
The ‘translation surface’ is just a series of reflections of the original polygon over its edges.
These are beautiful conceptual leaps and they have answered many questions that inevitably concern both mathematics and physics. In 2014, Klarreich’s article captured some of Mirzakhani’s
In a way, she said, mathematics research feels like writing a novel. “There are different characters, and you are getting to know them better,” she said. “Things evolve, and then you look back at
a character, and it’s completely different from your first impression.”
The Iranian mathematician follows her characters wherever they take her, along story lines that often take years to unfold.
In the article she was described as someone with a daring imagination. Reading about how she experienced mathematics made the nature of these efforts even more striking. There is a mysterious
reality in these abstract worlds that grow out of measuring the earth. The two and three dimensional worlds of our experience become represented by ideals which then, almost like an
Alice-in-Wonderland rabbit hole, lead the way to unimaginable depths. We find purely abstract spaces that have volume. We get there by looking further and looking longer. I feel a happy and eager
inquisitiveness when I ask myself the question: “What are we looking at?” And I would like to find a new way to begin an answer. It seems to me that Mirzakhani loved looking. A last little bit from
Unlike some mathematicians who solve problems with quicksilver brilliance, she gravitates toward deep problems that she can chew on for years. “Months or years later, you see very different
aspects” of a problem, she said. There are problems she has been thinking about for more than a decade. “And still there’s not much I can do about them,” she said.
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Recent Comments
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• Vincent Migliore on Infinity is not the problem
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September 27th, 2017 | Tags: abstraction, geometry, manifold, math history, mathematics, topology | Category: mathematics, philosophy of mathematics | Comments are closed | {"url":"https://mathrising.com/?p=1518","timestamp":"2024-11-10T15:38:02Z","content_type":"application/xhtml+xml","content_length":"132353","record_id":"<urn:uuid:42714f88-f67c-49c2-a243-7005d4010886>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00249.warc.gz"} |
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Do Your Own Encryption: A Simple Guide
Encrypting your own data is easier than you might think. This guide will show you how encryption works and how you can use it to protect your communications. No need to trust a third party like
Google, Facebook, Microsoft or Signal.
Why You Need Encryption
Your communication provider can easily monitor your activities. A small number of global infrastructure providers can monitor and read all internet traffic. The post office can read your snail mail,
telecom companies can listen to your calls, and your WhatsApp messages may be revealed by Facebook to comply with government requests. However, by creating your own encryption, you can ensure that
your messages remain private. Here's how you can do it.
Bob and Alice: An Introduction to Asymmetric Encryption
Asymmetric encryption involves each user having two keys: a private key and a public key. The private key must remain secret, while the public key can be shared openly.
Creating your own encryption requires writing some lines of code. Don’t worry, it’s simple. You’ll need to install Python on your computer. Search for "install Python on Windows/Mac/Linux" for
Step 1: Create RSA Keys
Start by generating a pair of RSA keys in Python:
from cryptography.hazmat.primitives.asymmetric import rsa, padding
from cryptography.hazmat.primitives import hashes, serialization
bob_private_key = rsa.generate_private_key(key_size=4096, public_exponent=65537)
bob_public_key = bob_private_key.public_key()
If you encounter an error with `from cryptography.hazmat...`, you may need to install the package with `pip install cryptography`. A key size of 4096 bits is currently (2024) considered secure. The
`public_exponent` is kind of a standard constant.
Next, generate keys for Alice also:
alice_private_key = rsa.generate_private_key(key_size=4096, public_exponent=65537)
alice_public_key = alice_private_key.public_key()
Step 2: Encrypt the Message
Now you can use these secret keys to establish an encrypted conversation. Let’s say Bob wants to schedule a secret meeting with Alice:
secret_appointment = 'Let’s meet at the theater at 8 p.m. next Saturday'
Bob will encrypt this message using Alice’s public key, remember that public keys are openly shared:
encrypted_msg = alice_public_key.encrypt(
The encrypted message will look unreadable, something like this:
# Output: b"w\x0f\x07\xa4\xe0>g[4Q\xb3\xa16\xeb3\xd5)..."
Step 3: Decrypt the Message
Bob sends this encrypted message to Alice. Alice can decrypt it because she owns her secret private key:
decrypted_secret_message = alice_private_key.decrypt(
# Output: b'Let\xe2\x80\x99s meet at the theater at 8 p.m. next Saturday'
That was easy, right? If you have a reliable communication channel, you're all set. Unfortunately, most of the time, such highly secure channels aren't available.
Improvement: Sign the Message
So far, we've covered encryption and decryption. But how can Alice be sure that the message was sent by Bob? Just like with snail mail and email, you can’t always trust the sender’s identity.
Fortunately, RSA provides a solution. Bob can add a digital signature to the secret_appointment using his bob_private_key:
bob_signature = bob_private_key.sign(
secret_appointment_signed = bob_signature + b"|||" + encrypted_msg
Bob now sends the secret_appointment_signed to Alice.
Alice to verify the Signature
Alice splits the message and verifies Bob's signature with Bob's public key:
bob_signature, encrypted_secret_message = secret_appointment_signed.split(b'|||')
bob_public_key = bob_private_key.public_key()
print("Signature is valid.")
print("Signature is invalid.")
When the signature is valid, Alice can be sure that Bob is the sender and now can decrypt the secret with her private key:
decrypted_secret_message = alice_private_key.decrypt(
# Output: b'Let\xe2\x80\x99s meet at the theater at 8 p.m. next Saturday'
This process ensures Alice that the message is indeed from Bob and that only she can read it.
Just a few lines of code are needed. Bob can send the encrypted message via email, phone, QR codes, images, CB radio or any other medium.
Final thoughts
• Public keys must be exchanged securely, such as through an in-person meeting.
• Parties must agree on some conditions like the padding, algorithm, split-sequence beforehand.
• With the advancement of quantum computing, RSA-4096 could become vulnerable to hacking within the next three years.
• Every encrypted communication on the internet leaves traces of metadata, use VPNs or public spots.
If you have any comments or questions, please reach out by nostr DM or via this Nostr note: nostr:note1vy5kf4em5670a79gxk5z5wsxle6wm4359d4kcauchqf83hqt2hhq7n54hg
If you found this guide useful, consider sending some sats to lud16 - cygus44@hubstr.org - or zap my profile nostr:npub1cygus44llmuj4m7w5yfpgqk83x9lvrtr2qavn26ugsyduzv7jc0qnjw7h8
Thanks, stay brave and private 🧡 | {"url":"https://hubstr.org/articles/how-to-encryption.html","timestamp":"2024-11-04T20:12:35Z","content_type":"text/html","content_length":"20372","record_id":"<urn:uuid:71447466-9eb0-4113-96ab-a89291377335>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00802.warc.gz"} |
What does K units stand for?
Kelvin (K), base unit of thermodynamic temperature measurement in the International System of Units (SI). This unit was originally defined as 100/27,316 of the triple point (equilibrium among the
solid, liquid, and gaseous phases) of pure water.
Why is Kelvin a SI unit of temperature?
The Kelvin scale fulfills Thomson’s requirements as an absolute thermodynamic temperature scale. It uses absolute zero as its null point (i.e. low entropy). The relation between kelvin and Celsius
scales is TK = t°C + 273.15….
Unit of Temperature
Symbol K
Named after William Thomson, 1st Baron Kelvin
What does K mean in weight?
Kilo is a decimal unit prefix in the metric system denoting multiplication by one thousand (103). It is used in the International System of Units, where it has the symbol k, in lower case.
What is the SI unit work?
The SI unit of work is the joule (J), the same unit as for energy.
How is the Kelvin of a SI unit defined?
SI Units – Temperature. The kelvin (K) is defined by taking the fixed numerical value of the Boltzmann constant k to be 1.380 649 ×10 −23 when expressed in the unit J K −1, which is equal to kg m 2 s
−2 K −1, where the kilogram, meter and second are defined in terms of h, c and ∆ν Cs.
How is the kilogram defined as a SI unit?
The kilogram, symbol kg, is the SI unit of mass. It is defined by taking the fixed numerical value of the Planck constant h to be 6.626 070 15 × 10 -34 when expressed in the unit J s, which is equal
to kg m 2 s -1, where the meter and the second are defined in terms of c and Δν Cs. Unit of time.
What are the different types of SI units?
Following are the commonly used System of Units in measurements: 1 CGS system 2 MKS system 3 SI system More
What are the physical quantities of the SI base unit?
The SI base units and their physical quantities are the metre for measurement of length, the kilogram for mass, the second for time, the ampere for electric current, the kelvin for temperature, the
candela for luminous intensity, and the mole for amount of substance. | {"url":"https://yycnewcentrallibrary.com/what-does-k-units-stand-for/","timestamp":"2024-11-03T09:53:05Z","content_type":"text/html","content_length":"40757","record_id":"<urn:uuid:5af8ba9f-6329-4ee9-b2c6-d2f9171d95d3>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00213.warc.gz"} |
How to make a vector in vpython?
In VPython, a vector can be created using the vector() function. The vector() function takes three arguments that represent the x, y, and z components of the vector. Here's an example of how to
create a vector in VPython:
1 from vpython import vector
3 # create a vector with components (1, 2, 3)
4 v = vector(1, 2, 3)
6 # print the vector
7 print(v)
This will output:
You can also create a vector using the vec() function, which is a shorthand for vector(). Here's an example:
1 from vpython import vec
3 # create a vector with components (4, 5, 6)
4 v = vec(4, 5, 6)
6 # print the vector
7 print(v)
This will output:
Once you have created a vector, you can perform operations on it, such as adding or subtracting it from another vector, multiplying it by a scalar, or taking its cross product or dot product with
another vector. | {"url":"https://devhubby.com/thread/how-to-make-a-vector-in-vpython","timestamp":"2024-11-08T20:51:58Z","content_type":"text/html","content_length":"124085","record_id":"<urn:uuid:d0e1a9c0-cf2e-4ce2-bf7e-51219a2187e1>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00198.warc.gz"} |
Area Formulas for various objects | Plumbing Help
Area Formulas for various objects
Area of a square / rectangle
Area = length x width
Area of a circle
Area = diameter x diameter x .7854
Surface area of a cone
Area = half of circumference of base x slant height + area of base
Area of ellipse
Area = short diameter x long diameter x 0.7854
Surface area of a cylinder
Area = diameter x 3.1416 x length x area of two bases
Area of a hexagon
Area = width of side x 2.598 x width of side
Area of a parallelogram
Area = length of base x distance between base and top
Surface area of a pyramid
Area = 1/2 the base perimeter x slant height + area of base
Surface area of a sphere
Area = diameter x diameter x 3.1416
Leave a Comment | {"url":"https://plumbinghelp.com/plumbing_math_area/","timestamp":"2024-11-01T20:31:06Z","content_type":"text/html","content_length":"218446","record_id":"<urn:uuid:c1695304-1dc3-4aa6-909b-026c7792aa2c>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00763.warc.gz"} |
ICEMAN Example of phylogenetic analysis
Ötzi the Iceman is a well-preserved natural mummy of a man frozen for about 5300 years and found in 1991 in a glacier of the Ötztal Alps, near the border between Austria and Italy. Recently
researchers found interesting informations about it from the exam of mitochondrial DNA taken from cells of the iceman intestine. In particular from phylogenetic analysis they made some hypotesis
about its mitochondrial haplogroup. A haplogroup is defined by set of characteristic mutations on the mitochondrial genome. Therefore it can be traced along a person’s maternal line and it can be
used to group populations by genetic features. The famous book “The Seven Daughters of Eve” by Bryan Sykes describes the classification of all modern humans into into mitochondrial haplogroups and
links each haplogroup to a specific prehistoric woman (“clan mothers”). In fact the branches of the mtDNA tree (composed of groups of people with related haplogroups) are continent-specific. In this
example, we analyze the statistical properties and perform phylogenetic analysis of the mtDNA of the iceman to investigate the relationship with modern humans of different geographical locations and
to determine useful information about its haplogroup.
The mtDNA sequence of the iceman can be obtained from the Genbank database (accession number S69989). Some other sequence of modern humans from different parts of the world are considered. They have
been extracted from the Max Ingman database (http://www.genpat.uu.se/mtDB/). All the sequences are stored in the same structure. The associated haplogrop is indicated.
fullset = {
'iceman' 'S69989' ''
'Aborigine' 'AF346963' 'A'
'Aborigine2' 'AF346965' 'A'
'Biaka Pygmy' 'AF346968' 'L1'
'Biaka Pygmy2' 'AF346969' 'L1'
'Buriat' 'AF346970' 'C'
'Chukchi' 'AF346971' 'A'
'Chinese' 'AF346972' 'F'
'Chinese2' 'AF346973' 'D'
'Crimean Tatar' 'AF346974' 'H'
'Dutch' 'AF346975' 'H'
'Effik' 'AF346976' 'L2'
'Effik2' 'AF346977' 'L2'
'English' 'AF346978' 'V'
'Evenki' 'AF346979' 'C'
'German' 'AF346983' 'J'
'Guarani' 'AF346984' 'D'
'Hausa' 'AF346985' 'L0'
'Ibo' 'AF346986' 'L1'
'Ibo2' 'AF346987' 'L1'
'Italian' 'AY882413' 'UK'
'Japanese' 'AF346989' 'D'
'Japanese2' 'AF346990' 'D'
'Kikuyu' 'AF346992' 'L1'
'Korean' 'AF346993' 'B'
'Mandenka' 'AF346995' 'L2'
'Mbenzele Pygmy' 'AF346996' 'L1'
'Mbenzele Pygmy2' 'AF346997' 'L1'
'Mbuti Pygmy' 'AF346998' 'L0'
'Mbuti Pygmy2' 'AF346999' 'L0'
'Piman' 'AF347001' 'B'
'PNG Coast' 'AF347002' 'N'
'PNG Highlands' 'AF347004' 'N'
'Samoan' 'AF347007' 'B'
'Saami' 'AF347006' 'V'
'San' 'AF347008' 'L0'
'San2' 'AF347009' 'L0'
'Siberian Inuit' 'AF347010' 'D'
'Warao' 'AF347012' 'C'
'Warao2' 'AF347013' 'C'
Since the mtDNA sequence of the Oetzi mummy is only partially available we consider the corresponding nucleotides in modern human sequences. To do that, we perform some local alignments. The scores
of local alignments are stored in a vector indicating various degrees of similarity between sequences.
for ind = 1:length(fullset)
fullPeople(ind).Header = [fullset{ind,1},' ', fullset{ind,3}];
fullPeople(1).Sequence = getgenbank(fullset{1,2},'sequenceonly','true');
for ind = 2:length(fullset)
temp = getgenbank(fullset{ind,2},'sequenceonly','true');
[score, localAlignment, Start] = swalign(fullPeople(1).Sequence,temp,'Alphabet', 'NT');
sci(ind-1) = score;
cutof = Start(2);
If you don’t have a live web connection, you can load the data from a MAT-file using the command.
% load bigPic % <== Uncomment this if no internet connection
Statistical analysis
Before performing phylogenic analysis, we analyse some statistical properties of the iceman sequence.
S = fullPeople(1).Sequence;
The MATLAB function basecount reports nucleotide counts and displays them in pie chart.
title('Distribution of nucleotide bases of Iceman mtDNA HVR');
We can also plot the density of each nucleotide.
The MATLAB function dimercount can be used to count dimers and plot them.
title('Dimer distribution for Iceman mtDNA')
The codons of each of reading frame of the positive strand are also counted and plotted. You can use the MATLAB function codoncount.
for f = 1 : 3,
codons{1,f} = codoncount(S,'frame',f,'figure',true);
title(sprintf('Codons for Frame %d of Iceman mtDNA ',f));
Phylogenetic analysis
The scores of local alignments between sequences previously computed are analyzed. We plot the associated histogram and we compute some statistics (mean, variance max, min).
title('Istogram of local alignment scores')
stats=[max(sci), min(sci), mean(sci) var(sci)]
stats =
490.8493 455.9075 474.6299 75.5072
Few sequences have high scores and then are strongly related to the iceman sequence. The multiple alignment gives some information about the relationship between the analysed sequences.
MultiAligned = multialign(fullPeople);
Then we calculate the pairwise distances using the Juckes Cantor correction. The MATLAB function seqpdist is used for that.
distances = seqpdist(fullPeople,'Method','Jukes-Cantor','Alphabet','NT');
The phylogenetic tree is built with the UPGMA algorithm an displayed.
UPGMAtree = seqlinkage(distances,'UPGMA',fullPeople);
h = plot(UPGMAtree,'orient','bottom');
title('UPGMA Distance Tree using Jukes-Cantor model');
ylabel('Evolutionary distance')
A neighbor-joining tree is also constructed using the seqneighjoin function. Neighbor-joining tree uses also the pairwise distance calculated above (using the seqpdist function).
NJtree = seqneighjoin(distances,'equivar',fullPeople);
h = plot(NJtree,'orient','left');
title('Neighbor-joining tree using Jukes-Cantor model');
The phylogenetic trees show that the iceman is more related to european population in particular to Italian belonging to the UK superhaplogroup. This is in accordance with the results presented in
the paper of Rollo et al. that associates Otzi to the K haplogroup. Both trees show also some infomation about the relationship and the evolution of haplogroups in specific regions of the world. The
neighbor joining algorithm seems more accurate. The tree depicts in a general manner also the relationship and the evolution of different populations. In Africa, the most ancient mtDNA haplogroups
(L0, L1, L2), make up macrohaplogroup L. It radiated to form the Eurasian macrohaplogroups M and N. Among Europeans, haplogroups H, I, J, N1b, T, U, V, W, and X make up about 98% of the mtDNAs. These
were derived primarily from macrohaplogroup N. In Asia, macrohaplogroups N and M contributed equally to mtDNA radiation. Here now the main haplogroups are A, C, D, G.
F. Rollo, L. Ermini, S. Luciani, I. Marota, C. Olivieri, D. Luiselli, Fine characterization of the Iceman’s mtDNA haplogroup, J Phys Anthropol. Jan 2006.
Torroni, K. Huoponen, P. Francalacci, M. Petrozzi, L. Morelli, R. Scozzari, D. Obinu, M. L. Savontaus, D.C. Wallace. Classification of European mtDNAs from an Analysis of Three European Populations,
Genetics, Vol 144, 18351850, 1996.
M.Ingman, H. Kaessmann, S. Paabo, U. Gyllensten, Mitochondrial genome variation and the origin of modern humans. Nature 408, 708-713, 2000.
D.Mishmar, E. Ruiz-Pesini, P. Golik, V. Macaulay, A. Clark, S. Hosseini, M.Brandon, K. Easley, M. Brown, R.I. Sukernik, A. Olckers, D. Wallace. Natural selection shaped regional mtDNA variation in
humans. Proc.Natl.Acad.Sci. 100, 171-176, 2003. | {"url":"https://computationalgenomics.blogs.bristol.ac.uk/case_studies/iceman_demo","timestamp":"2024-11-13T08:24:37Z","content_type":"text/html","content_length":"46445","record_id":"<urn:uuid:bbab562a-ce0c-40f4-9d0e-9c52a18b7cc6>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00590.warc.gz"} |
Section: Research Program
Dynamic non-regular systems
nonsmooth mechanical systems, impacts, friction, unilateral constraints, complementarity problems, modeling, analysis, simulation, control, convex analysis
Dynamical systems (we limit ourselves to finite-dimensional ones) are said to be non-regular whenever some nonsmoothness of the state arises. This nonsmoothness may have various roots: for example
some outer impulse, entailing so-called differential equations with measure. An important class of such systems can be described by the complementarity system
$\left\{\begin{array}{c}\stackrel{˙}{x}=f\left(x,u,\lambda \right)\phantom{\rule{0.166667em}{0ex}},\hfill \\ 0\le y\perp \lambda \ge 0\phantom{\rule{0.166667em}{0ex}},\hfill \\ g\left(y,\lambda
,x,u,t\right)=0\phantom{\rule{0.166667em}{0ex}},\hfill \\ \text{re-initialization}\phantom{\rule{4.pt}{0ex}}\text{law}\phantom{\rule{4.pt}{0ex}}\text{of}\phantom{\rule{4.pt}{0ex}}\text{the}\ (1)
phantom{\rule{4.pt}{0ex}}\text{state}\phantom{\rule{4.pt}{0ex}}x\left(·\right)\text{,}\hfill \end{array}\right\$
where $\perp$ denotes orthogonality; $u$ is a control input. Now (1) can be viewed from different angles.
• Hybrid systems: it is in fact natural to consider that (1) corresponds to different models, depending whether ${y}_{i}=0$ or ${y}_{i}>0$ (${y}_{i}$ being a component of the vector $y$). In some
cases, passing from one mode to the other implies a jump in the state $x$; then the continuous dynamics in (1) may contain distributions.
• Differential inclusions: $0\le y\perp \lambda \ge 0$ is equivalent to $-\lambda \in {\mathrm{N}}_{K}\left(y\right)$, where $K$ is the nonnegative orthant and ${\mathrm{N}}_{K}\left(y\right)$
denotes the normal cone to $K$ at $y$. Then it is not difficult to reformulate (1) as a differential inclusion.
• Dynamic variational inequalities: such a formalism reads as $〈\stackrel{˙}{x}\left(t\right)+F\left(x\left(t\right),t\right),v-x\left(t\right)〉\ge 0$ for all $v\in K$ and $x\left(t\right)\in K$,
where $K$ is a nonempty closed convex set. When $K$ is a polyhedron, then this can also be written as a complementarity system as in (1).
Thus, the 2nd and 3rd lines in (1) define the modes of the hybrid systems, as well as the conditions under which transitions occur from one mode to another. The 4th line defines how transitions are
performed by the state $x$. There are several other formalisms which are quite related to complementarity. See [7], [8], [15] for a survey on models and control issues in nonsmooth mechanical | {"url":"https://radar.inria.fr/report/2016/bipop/uid7.html","timestamp":"2024-11-04T08:08:23Z","content_type":"text/html","content_length":"50574","record_id":"<urn:uuid:d4ad7d64-3049-4df5-9806-fc3e7c72e41f>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00498.warc.gz"} |
How Much Does Your Geyser Cost?
How To Reduce My Geyser Expense
August 2, 2021
Geysers are often the single biggest consumer of electricity in the home, making up as much as 40% of the total bill. Naturally this does depend on how much hot water you consume. If you set your
thermostat to heat the water to 60 degrees and you want to heat up 150L from a starting temperature of 15 degrees.
150L x 4.16 (specific heat capacity of water) x 45degrees = 28, 080 Kilo joules of energy required
We are used to Kilo-watt hour, not kilo-joules so we need to divide by 3, 600 (number of seconds in an hour) to convert kilo-joule to kilo-watt.
28080/3600 = 7.8 kilo-watt hours to heat a geyser. At 2.45 per unit this geyser is costing around R19 per day to heat up.
Then we have to add the standing losses of 1.1kWh for a B rated 150L geyser.
In total we are using 8.9kWh which costs around R21.80 rand per day, or R654 per month. In some cases you will not be heating water up from 15^oC so this cost may be less, however, it is still a
significant part of your monthly bill.
As the element ages, performance also drops and the monthly cost of running the geyser increases. With XTEND elements we save money in many ways, our element is 25% more efficient at heating water
than conventional elements and our standing losses are reduced as we keep the element away from the base plate. Finally, XTEND elements are built to last with minimal drop in performance over time
as our element is protected from scaling with a marine grade stainless steel outer casing.
Source Article: https://blog.homebug.co.za/?p=34
Geysers are often the single biggest consumer of electricity in the home, making up as much as 40% of the total bill. Naturally this does depend on how much hot water you consume. If you set your
thermostat to heat the water to 60 degrees and you want to heat up 150L from a starting temperature of 15 degrees.
150L x 4.16 (specific heat capacity of water) x 45degrees = 28, 080 Kilo joules of energy required
We are used to Kilo-watt hour, not kilo-joules so we need to divide by 3, 600 (number of seconds in an hour) to convert kilo-joule to kilo-watt.
28080/3600 = 7.8 kilo-watt hours to heat a geyser. At 2.45 per unit this geyser is costing around R19 per day to heat up.
Then we have to add the standing losses of 1.1kWh for a B rated 150L geyser.
In total we are using 8.9kWh which costs around R21.80 rand per day, or R654 per month. In some cases you will not be heating water up from 15oC so this cost may be less, however, it is still a
significant part of your monthly bill.
As the element ages, performance also drops and the monthly cost of running the geyser increases. With XTEND elements we save money in many ways, our element is 25% more efficient at heating water
than conventional elements and our standing losses are reduced as we keep the element away from the base plate. Finally, XTEND elements are built to last with minimal drop in performance over time
as our element is protected from scaling with a marine grade stainless steel outer casing.
Source Article: https://blog.homebug.co.za/?p=34 | {"url":"https://xtendelements.co.za/how-much-does-your-geyser-cost/","timestamp":"2024-11-02T18:40:45Z","content_type":"text/html","content_length":"351243","record_id":"<urn:uuid:80e50469-0780-4c6a-b797-37dba14a01bb>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00737.warc.gz"} |
Free Orbit Insertion: Is it Possible?
• I
• Thread starter pasanta
• Start date
In summary, the conversation discussed the possibility of a mass from outer space orbiting the Earth without a change in energy. While one person believed this was mathematically feasible, the other
argued that it would require a change in kinetic energy. It was mentioned that objects within our solar system can be captured and start orbiting larger planets, but if the object originated from
outside the solar system, it is unlikely to be affected by gravitational interactions within our solar system.
TL;DR Summary
General querie about a recnt discussion with a collegue
Good evening,
This week I had a discussion/conversation with one of my coworkers about the posibility of a mass from outer spate that could start orbiting the earth. The issue is, I believe you cannot, whithout a
proper change of energy (velocity) start orbiting a celestial body just by coming whithin its sphere of influence. My friend, however, argued that although basically impossible it is mathematically
feasible to do just that and a body could start orbiting another indefinetely.
I would appreciate if you guys gave us some insight on whether this is possible or, indeed, you would need to temper with the bodys kynetic energy.
Thanks in advance!
Last edited by a moderator:
Vanadium 50
Staff Emeritus
Science Advisor
Education Advisor
2023 Award
If the body is coming ni from infinity, it has the energy needed to return to infinity. You need to remove energy, perhaps by an interaction with the moon.
Last edited:
If some other process can slow it down enough (likely the gravtational influence from another solar system body) then yes it could be captured and start orbiting.
When you say outer space do you mean originating from within our solar system? If so the larger gas giants have captured objects that now orbit them that originated within our solar system.
If you mean from outside the solar system, then any such objects are likely to have such a large velocity that no gravitation interactions taking place within the solar system are likely to be able
to slow it down enough, not unless the object is very small, probably too small for us to even detect.
FAQ: Free Orbit Insertion: Is it Possible?
1. Is it possible to insert a spacecraft into orbit without using any fuel?
No, it is not currently possible to insert a spacecraft into orbit without using any fuel. In order to achieve orbit, a spacecraft must reach a certain velocity and altitude, which requires energy.
However, there are ongoing research and development efforts to create more efficient propulsion systems that may reduce the amount of fuel needed for orbit insertion.
2. Can a spacecraft use a gravitational slingshot maneuver for free orbit insertion?
Yes, a spacecraft can use a gravitational slingshot maneuver to gain speed and change its trajectory without expending any fuel. This technique involves using the gravitational pull of a planet or
other celestial body to accelerate the spacecraft and change its direction.
3. What factors determine the feasibility of free orbit insertion?
The feasibility of free orbit insertion depends on several factors, including the spacecraft's mass, its initial velocity and altitude, the gravitational pull of the target planet or body, and the
presence of any atmospheric drag. Some of these factors can be controlled or optimized through careful mission planning and the use of advanced propulsion systems.
4. Are there any successful examples of free orbit insertion in space missions?
Yes, there have been several successful examples of free orbit insertion in space missions. For instance, the Cassini spacecraft used a gravitational slingshot maneuver to enter orbit around Saturn,
and the Juno spacecraft used a similar technique for its orbit insertion around Jupiter. These missions demonstrate the feasibility and effectiveness of using free orbit insertion techniques.
5. What are the potential benefits of free orbit insertion?
Free orbit insertion techniques have the potential to significantly reduce the amount of fuel needed for space missions, making them more cost-effective and sustainable. They also allow for more
flexibility in mission planning and can enable spacecraft to reach more distant or challenging destinations in the solar system. Additionally, the use of free orbit insertion techniques can reduce
the environmental impact of space missions by minimizing the amount of propellant and emissions released into the atmosphere. | {"url":"https://www.physicsforums.com/threads/free-orbit-insertion-is-it-possible.975140/","timestamp":"2024-11-05T19:58:43Z","content_type":"text/html","content_length":"89219","record_id":"<urn:uuid:b4dc34d5-b9b8-427b-8909-51ba8ab1c02a>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00565.warc.gz"} |
Lesson 9
Standard Form and Factored Form
Lesson Narrative
Previously, students used area diagrams to expand expressions of the form \((x+p)(x+q)\) and generalized that the expanded expressions take the form of \(x^2 + (p+q)x +pq\). In this lesson, they see
that the same generalization can be applied when the factored expression contains a sum and a difference (when \(p\) or \(q\) is negative) or two differences (when both \(p\) and \(q\) are negative).
Although they have encountered an algebraic approach, students still benefit from drawing diagrams to expand unfamiliar factored expressions. Area diagrams are intuitive for visualizing the product
of two sums, but they are less intuitive for visualizing the product of two differences (for example, \((x-5)^2\)) or of a sum and a difference (for example, \((x+3)(x-4)\)). Subtraction can be
represented by removing parts of a rectangle and finding the area of the remaining region, but this strategy can get complicated when both factors are differences.
At this point, students transition from thinking about rectangular diagrams concretely, in terms of area, to thinking about them more abstractly, as a way to organize the terms in each factor.
(Students made similar transitions from area diagrams to abstract diagrams in middle school, for example, when they learned to distribute the multiplication of a number or a variable—positive and
negative—over addition and subtraction.)
Students also learn to use the terms standard form and factored form. When classifying quadratic expressions by their form, students refine their language and thinking about quadratic expressions
(MP6). In an upcoming lesson, students will graph quadratic expressions of these forms and study how features of the graphs relate to the parts of the expressions.
Learning Goals
Teacher Facing
• Comprehend the terms “standard form” and “factored form” (in written and spoken language).
• Use rectangular diagrams to reason about the product of two differences or of a sum and difference and to write equivalent expressions.
• Use the distributive property to write quadratic expressions given in factored form in standard form.
Student Facing
• Let’s write quadratic expressions in different forms.
Student Facing
• I can rewrite quadratic expressions given in factored form in standard form using either the distributive property or a diagram.
• I know the difference between “factored form” and “standard form.”
CCSS Standards
Building Towards
Glossary Entries
• factored form (of a quadratic expression)
A quadratic expression that is written as the product of a constant times two linear factors is said to be in factored form. For example, \(2(x-1)(x+3)\) and \((5x + 2)(3x-1)\) are both in
factored form.
• standard form (of a quadratic expression)
The standard form of a quadratic expression in \(x\) is \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a\) is not 0.
Additional Resources
Google Slides For access, consult one of our IM Certified Partners.
PowerPoint Slides For access, consult one of our IM Certified Partners. | {"url":"https://curriculum.illustrativemathematics.org/HS/teachers/1/6/9/preparation.html","timestamp":"2024-11-02T21:17:54Z","content_type":"text/html","content_length":"79302","record_id":"<urn:uuid:2ae6c613-ea07-4939-b39e-29fc283153a1>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00135.warc.gz"} |
Stencil Area Ratio Calculator - Online Calculators
Enter values of aperture width, length, stencil thickness to calculate the stencil ratio with our basic and advanced Stencil Area Ratio Calculator
Stencil Area Ratio Calculator
Enter any 3 values to calculate the missing variable
The Stencil Area ratio calculator is important in the field of manufacturing and electronics where the accuracy of stencil area determines the quality of product.
The Stencil Area Ratio (SAR) Calculator uses a simple formula:
$\mathrm{ SAR}=\frac{2×\left(L+W\right)×T}{L×W}$
• SAR (Stencil Area Ratio): This is like a special number that tells us how the stencil area compares to the edges of the openings. We measure it in millimeters.
• L (Length of the Aperture): It’s how long the opening in the stencil is, measured in millimeters.
• W (Width of the Aperture): It’s how wide the opening in the stencil is, also measured in millimeters.
• T (Thickness of the Stencil): This is how thick the stencil material is, measured in millimeters.
How to Calculate ?
Following are the key steps;
1. Calculate Aperture Area: Multiply the length (L) by the width (W), then divide by
2. Calculate Perimeter: Add the length and width, then multiply by 2 (
3. Calculate Stencil Area Ratio: Divide the aperture area by the product from step 2.
Solved Calculations:
Example 1:
• Length of the aperture (L) = 5 mm
• Width of the aperture (W) = 3 mm
• Thickness of the stencil (T) = 0.2 mm
Calculation Instructions
Step 1: SAR = $\frac{(L \times W)}{2 \times (L + W) \times T}$ Start with the formula.
Step 2: SAR = $\frac{(5 \times 3)}{2 \times (5 + 3) \times 0.2}$ Replace L with 5 mm, W with 3 mm, and T with 0.2 mm.
Step 3: SAR = $\frac{15}{2 \times 8 \times 0.2}$ Multiply the length by the width and add L and W together.
Step 4: SAR = $\frac{15}{3.2}$ Multiply the sum by 2 and then by the thickness.
Step 5: SAR = 4.69 Divide the aperture area by the product to get the Stencil Area Ratio.
The Stencil Area Ratio is 4.69.
Example 2:
• Length of the aperture (L) = 8 mm
• Width of the aperture (W) = 4 mm
• Thickness of the stencil (T) = 0.3 mm
Calculation Instructions
Step 1: SAR = $\frac{(L \times W)}{2 \times (L + W) \times T}$ Start with the formula.
Step 2: SAR = $\frac{(8 \times 4)}{2 \times (8 + 4) \times 0.3}$ Replace L with 8 mm, W with 4 mm, and T with 0.3 mm.
Step 3: SAR = $\frac{32}{2 \times 12 \times 0.3}$ Multiply the length by the width and add L and W together.
Step 4: SAR = $\frac{32}{7.2}$ Multiply the sum by 2 and then by the thickness.
Step 5: SAR = 4.44 Divide the aperture area by the product to get the Stencil Area Ratio.
The Stencil Area Ratio is 4.44.
What is the Stencil Area Ratio Calculator?
The Stencil Area Ratio (SAR) Calculator helps to know the efficiency of a stencil in the printing process, especially in the field of electronics manufacturing. It is therefore an important factor
that influences the quality of solder paste deposition, which in turns, directly impacts the reliability of electronic assemblies.
A higher SAR generally shows the better paste release from the stencil, that leads to more accurate and reliable printing results. It is important for an optimized stencil design and to ensure the
quality of production. | {"url":"https://areacalculators.com/stencil-area-ratio-calculator/","timestamp":"2024-11-04T00:43:19Z","content_type":"text/html","content_length":"118303","record_id":"<urn:uuid:611f5474-7f70-41ea-83d0-64052a3b11a2>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00044.warc.gz"} |
Each Converter
Importance of Conversion Tools
Conversion tools are indispensable in various fields, such as education, engineering, and daily life activities. Whether you're looking to convert each degree measure into radians or transform a
mixed number into a fraction greater than 1, these tools simplify complex calculations and improve accuracy.
Convert Each Degree Measure into Radians
To accurately convert each degree measure into radians, conversion tools are essential. Degrees and radians are two units used to measure angles, with radians being more prevalent in higher
mathematics and physics. Conversion tools help bridge the gap, making it easier for students and professionals to switch between these units without confusion. By automating these conversions, one
can avoid manual errors and save significant time.
Convert Each Mixed Number to a Fraction Greater Than 1
When dealing with mixed numbers, the need to convert each mixed number to a fraction greater than 1 often arises in mathematical problems and practical applications. Conversion tools simplify this
process by providing quick, accurate results. These tools are particularly beneficial for students who are learning basic arithmetic and for professionals who need to perform these conversions
regularly in their work.
Convert Each Measurement
In fields like engineering, construction, and science, the ability to convert each measurement accurately is critical. Conversion tools facilitate the seamless transformation of measurements from one
unit to another, whether it’s converting inches to centimeters, gallons to liters, or pounds to kilograms. This ensures that projects are executed with precision, adhering to required specifications
and standards.
Convert Each Angle Measure to Decimal Degree Form
For tasks that require angle measurements, being able to convert each angle measure to decimal degree form is crucial. Decimal degrees are often used in various applications like navigation, mapping,
and 3D modeling. Conversion tools make this process straightforward, offering quick and precise results. This eliminates the need for manual calculations, reducing errors, and enhancing productivity.
Conversion tools are invaluable in modern-day tasks, ensuring that conversions are accurate and efficient. Whether you need to convert each degree measure into radians or transform other units, these
tools provide a reliable solution for accurate and timely results.
How to Convert Each Degree Measure into Radians
To convert each degree measure into radians, there are various tools that simplify the process. Each converter tool can be highly efficient and accurate, ensuring that your conversions are exact.
These tools are particularly useful when working with complex angles and measurements.
Online Calculators
Online calculators are one of the most accessible resources for converting each degree measure into radians. Simply input your degree measure, and the calculator will instantly provide the
corresponding radian value. These calculators often come with additional functionalities that allow you to convert each mixed number to a fraction greater than 1, facilitating various mathematical
Scientific Calculators
Most scientific calculators have built-in functions to convert each degree measure into radians. These calculators are particularly useful for students and professionals who need to perform
conversions on-the-go. Additionally, they can convert each angle measure to decimal degree form, providing a comprehensive tool for multiple types of measurements.
Spreadsheet Software
Spreadsheet software like Microsoft Excel and Google Sheets also offer functions that can convert each degree measure into radians. By using specific formulas, you can automate the conversion process
for large datasets. This functionality is especially useful for projects that require converting each measurement efficiently and accurately.
Mobile Apps
Several mobile apps are available that specialize in mathematical conversions. These apps not only convert each degree measure into radians but also offer features to convert each mixed number to a
fraction greater than 1 and convert each angle measure to decimal degree form. Mobile apps provide the convenience of performing conversions anytime, anywhere.
Manual Calculation
While tools are convenient, understanding the manual method to convert each degree measure into radians is essential. The formula to convert degrees to radians is to multiply the degree measure by π/
180. Knowing this formula aids in verifying the accuracy of your converter tools.
Steps to Convert Each Mixed Number to a Fraction Greater Than 1
When you need to convert each degree measure into radians, understanding the process of converting each mixed number to a fraction greater than 1 is crucial. This conversion is essential in various
mathematical and scientific applications, ensuring accuracy and consistency in calculations.
Step 1: Identify the Mixed Number
First, identify the mixed number you wish to convert. A mixed number consists of a whole number and a fraction, such as 3 1/2. This step is the foundation for converting each mixed number to a
fraction greater than 1.
Step 2: Convert the Whole Number to a Fraction
Next, convert the whole number part of the mixed number into a fraction. Multiply the whole number by the denominator of the fractional part. For example, with 3 1/2, you multiply 3 by 2, yielding 6/
Step 3: Add the Fractional Part
Now, add the fractional part of the mixed number to the fraction obtained from the whole number. Using our example, add 6/2 to 1/2, which results in 7/2. You have now converted each mixed number to a
fraction greater than 1.
Step 4: Simplify the Fraction
Finally, simplify the fraction if necessary. Ensure the fraction is in its simplest form to make further calculations easier. For instance, 7/2 is already in its simplest form, so no further
simplification is needed.
Practical Applications
These steps are beneficial when you need to convert each degree measure into radians or convert each measurement into a different unit. Similarly, converting each angle measure to decimal degree form
often requires handling fractions, making this knowledge indispensable.
Methods to Convert Each Measurement Accurately
Converting measurements accurately is crucial in many fields, from mathematics to engineering. To convert each degree measure into radians, you can use a variety of tools. Calculators, conversion
tables, and online converters can all help you convert each measurement precisely. One of the most common methods involves multiplying the degree measure by π/180. This simple formula ensures that
you convert each degree measure into radians consistently.
Using Calculators
Scientific calculators are excellent tools to convert each degree measure into radians. They often have built-in functions for this exact purpose. Simply enter the degree value, press the appropriate
key, and instantly convert each measurement.
Conversion Tables
If you prefer a manual approach, conversion tables are quite handy. These tables provide pre-calculated values that allow you to convert each mixed number to a fraction greater than 1 or convert each
degree measure into radians without complex calculations.
Online Converters
Online tools offer the simplest way to convert each angle measure to decimal degree form. With just a few clicks, you can convert any measurement. These tools are especially useful for converting
multiple values quickly and accurately.
Software Applications
Software like spreadsheets can be used to automate the conversion process. Custom formulas can convert each measurement with high precision, ensuring consistency across large datasets. In conclusion,
to convert each degree measure into radians, multiple methods are available. Whether you choose calculators, conversion tables, online converters, or software applications, understanding the tools at
your disposal will help you convert each measurement accurately.
Techniques to Convert Each Angle Measure to Decimal Degree Form
Using Online Calculators
Converting each degree measure into radians can be simplified with online calculators. These tools offer intuitive interfaces where you can input your angle measurement and receive an instant result.
First, enter the angle measure you want to convert. The calculator will automatically convert each angle measure to decimal degree form, making it easy to handle angles in various calculations.
Additionally, you can find features to convert each mixed number to a fraction greater than 1, ensuring all your measurements are in the correct format for further computations.
Software Applications
For those who frequently need to convert each measurement, specialized software applications can be a lifesaver. These applications not only convert each degree measure into radians but also come
with added functionalities.
Many software tools offer batch conversion capabilities, allowing you to convert each angle measure to decimal degree form in bulk. This is particularly useful in fields such as engineering and
Furthermore, these applications often include tutorials and guides on how to convert each mixed number to a fraction greater than 1, which can be very beneficial for beginners.
Graphing Calculators
Graphing calculators are another excellent tool for converting each degree measure into radians. These handheld devices often come pre-loaded with functions that simplify the process.
Simply input your angle measurement, and the calculator will effortlessly convert each angle measure to decimal degree form. This feature is particularly useful for students and professionals on the
In addition, many graphing calculators offer functions to convert each mixed number to a fraction greater than 1, aiding in various mathematical problems.
Manual Conversion
Though not as fast, manually converting each degree measure into radians can be a useful skill to master. This method provides a deeper understanding of the conversion process.
To manually convert each angle measure to decimal degree form, you can use mathematical formulas and conversion tables. This approach is beneficial in situations where digital tools are unavailable.
Manual conversion also includes the ability to convert each mixed number to a fraction greater than 1, ensuring your calculations remain accurate and consistent.
Common Challenges in Conversion and How to Overcome Them
Converting each degree measure into radians is a common task in mathematics and science, but it can present several challenges. One of the primary hurdles is understanding the relationship between
degrees and radians. A degree is a measure of angle, whereas a radian is the standard unit of angular measure used in many areas of mathematics. To overcome this, remember the conversion factor: π
radians = 180 degrees. Thus, to convert each degree measure into radians, multiply the number of degrees by π/180.
Understanding the Conversion Factor
A frequent mistake is miscalculation of the conversion factor. Always ensure your values are precise and correctly positioned. For instance, converting 45 degrees to radians involves multiplying 45
by π/180, which simplifies to π/4 radians. Familiarity with this simple step can significantly reduce errors.
Tools and Techniques
Using calculators or conversion tools can ease the conversion process. These tools are often equipped to convert each degree measure into radians accurately with minimal input. Ensure you are using
reliable and verified tools to avoid discrepancies in results.
Practice and Application
Regular practice can help solidify your understanding and reduce errors. Apply the conversion in various practical scenarios, such as converting each mixed number to a fraction greater than 1, which
aids in reinforcing the concept of unit conversions.
Converting Measurements
When converting measurements, whether it be lengths, weights, or temperatures, always double-check your units. For example, converting each measurement in a scientific experiment requires consistent
unit usage to maintain accuracy. This habit ensures that all derived results are reliable.
Decimal Degree Form
For tasks requiring you to convert each angle measure to decimal degree form, start by understanding the decimal representation. Decimal degrees make it easier to integrate with various mathematical
models and software. Ensure you are familiar with both the fractional and decimal forms to switch seamlessly between them.
Practical Applications of Conversion Tools
To convert each degree measure into radians is a fundamental task in various fields of science and engineering. For example, in physics, angular velocity often needs to be expressed in radians per
second. In such cases, conversion tools streamline the process, ensuring accuracy and efficiency.
Engineering Calculations
Engineers frequently convert each degree measure into radians when designing mechanical systems. Precise calculations of rotational movements are essential for developing machinery, robotics, and
aeronautical components. Automated conversion tools eliminate the risk of manual errors, leading to safer and more reliable designs.
Educational Use
Students and educators benefit from tools that convert each degree measure into radians during learning and teaching trigonometry. These tools enhance understanding and help in visualizing complex
concepts by simplifying the conversion process.
Construction and Architecture
Professionals in construction and architecture often need to convert each angle measure to decimal degree form for precise planning and building. Digital conversion tools help in translating design
blueprints into accurate physical structures, ensuring the integrity of buildings and other constructions.
Everyday Measurements
Everyday tasks, such as cooking or DIY projects, sometimes require you to convert each measurement from one unit to another. Conversion tools make this fast and easy, improving efficiency and
accuracy in daily activities.
Mathematical Conversions
Conversion tools also assist in converting each mixed number to a fraction greater than 1 which is useful in solving algebraic equations and in higher-level math courses. Accurate conversion is
crucial for simplifying complex problems and ensuring correct solutions. | {"url":"https://toolstalls.com/each-converter","timestamp":"2024-11-09T04:42:07Z","content_type":"text/html","content_length":"71598","record_id":"<urn:uuid:cb28a7c1-3f09-42e5-89c4-22220ad41ae0>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00417.warc.gz"} |
Merged leaves & optimistic rollup | zkopru
Merged leaves & optimistic rollup
Merged leaves is used for item append sequence validation for the merkle tree rollup of massive items.
Merged leaves concept is used for the sequence verification of the appended items when we want to prove Merkle tree updates through multiple transactions. Let us assume that what exactly we are going
to prove is the following result.
[Original merkle tree]
- Root: 0xabcd...
- Index: m
[Items to add]
- [leaf_1, leaf_2, ..., leaf_n]
[Updated merkle tree]
- Root: 0xfdec...
- Index: m + n
To prove above, we will run the following steps:
To add a massive number of items, we split the transactions and record the intermediate proof result.
Every item will be merged sequentially into the mergedLeaves value to ensure the sequence of the item appending.
mergedLeaves = 0
mergedLeaves = hash(mergedLeaves, items[0])
mergedLeaves = hash(mergedLeaves, items[1])
mergedLeaves = hash(mergedLeaves, items[n])
In the end, the result stored on the EVM will be
- start root
- start index
- result root
- result index
- mergedLeaves
And then we can compare the Merkle tree update result.
Let us see a more detailed example.
startRoot is 0x0001234...
itemsToAdd is [0x01, 0x02, 0x03, 0x04, 0x05, 0x06, 0x07, 0x08, 0x09]
we are now trying to prove resultRoot is 0xabcd1234... when after adding all items in the itemsToAdd list.
We're going to add three items at once, so after two times of transactions, we can have a proof for the Merkle tree transition on EVM. Note that we have used random values for the hash calculation
for this example.
[stored proof on the EVM]
startRoot = 0x0001234...;
startIndex = 38;
resultRoot = 0x0001234...;
resultIndex = 38;
mergedLeaves = 0x0
Add the first item
startRoot = 0x0001234...;
startIndex = 38;
resultRoot = 0xAFCA1234...;
resultIndex = 39;
mergedLeaves = keccak256(0x0, 0x01) = 0xA0...
Add the second item
startRoot = 0x0001234...;
startIndex = 38;
resultRoot = 0xBA891234...;
resultIndex = 40;
mergedLeaves = keccak256(0xA0..., 0x02); = 0xB1...
Add the third item & record to the storage
[stored proof on the EVM]
startRoot = 0x0001234...;
startIndex = 38;
resultRoot = 0xC9B31234...;
resultIndex = 41;
mergedLeaves = keccak256(0xB1..., 0x03); = 0xC3...
To add the fourth item, we will retrive the result from the storage and keep going to append items.
As a result, we now have the result on Ethereum storage, proving the valid Merkle tree transition by the EVM calculation.
[stored proof on the EVM]
startRoot = 0x0001234...;
startIndex = 38;
resultRoot = 0xF1F3A4B...;
resultIndex = 47;
mergedLeaves = 0xDEFEDFED...;
Using the stored proof, we can validate the following information is valid or not. To validate the information, it computes the mergedLeaves result of the itemsToAdd and compares it with the
stored mergedLeaves.
startRoot: 0x0001234...,
startIndex: 38,
itemsToAdd: [0x01, 0x02, 0x03, 0x04, 0x05, 0x06, 0x07, 0x08, 0x09],
resultRoot: 0xF1F3A4B,
resultIndex: 47
Here is how it computes the mergedLeaves of itemsToAdd
merged = bytes32(0);
for(uint i = 0; i < itemsToAdd.length; i++) {
merged = keccak256(merged, itemsToAdd[i]);
Finally, if the result merged value equals to the mergedLeaves value 0xDEFEDFED... of the stored proof, it returns true or will be reverted. | {"url":"https://docs.zkopru.network/develop/how-it-works/merged-leaves","timestamp":"2024-11-02T20:22:24Z","content_type":"text/html","content_length":"310792","record_id":"<urn:uuid:2dbdc26d-b8a7-4eaf-b6d2-f6eed222e94c>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00660.warc.gz"} |
Finding the Volume of a Cylinder in a Real-World Context
Question Video: Finding the Volume of a Cylinder in a Real-World Context Mathematics
A roll of paper towels has the given dimensions. Determine, to the nearest hundredth, the volume of the roll.
Video Transcript
A roll of paper towels has the given dimensions. Determine, to the nearest hundredth, the volume of the roll.
This roll of paper towels is in the shape of a cylinder. Now a roll of paper towels doesn’t have paper towels to fill up the entire cylinder. The center of it is hollow. That’s the holder of the
paper towels. So in order to find the volume, we need to find the volume of the large cylinder, the whole thing, and take away the small cylinder on the inside that’s hollow, that doesn’t have paper
The volume 𝑉 of a cylinder is equal to the area of the base 𝐵 times the height h. But since the base of a cylinder is a circle, we can replace 𝐵 with 𝜋𝑟 squared, where 𝑟 is the radius. So to find the
volume of the paper towels, as we said before, we need to take the volume of the large cylinder and subtract the volume of the small cylinder. Therefore, we need the radii of each cylinder and the
height of each cylinder. Let’s begin with the large cylinder.
The height of the large cylinder is thirty centimeters. However, we’re not given the radius. The sixteen centimeters is a diameter. That’s the complete width of a circle, from one end of the circle
to the other. The radius is from the centre to the outside of a circle. It’s exactly half of a diameter. So we need to take sixteen divided by two. So our radius is eight centimeters.
Now looking at our smaller cylinder, our height is still thirty, and our radius we need to find because they give us the diameter of the smaller cylinder. So we need the radius. So we need to take
four and divide it by two. This means our radius is two centimeters.
Now we need to simplify. First let’s square the eight and square the two. Now we need to multiply sixty-four and thirty together, and then we also need to multiply four and thirty together. Now we
subtract one thousand nine hundred and twenty 𝜋 minus one hundred and twenty 𝜋, resulting in one thousand eight hundred 𝜋.
Now it says determine to the nearest hundredth. That means we need to multiply one thousand eight hundred by 𝜋 and then round two decimal places. We get five thousand six hundred and fifty-four point
eight six six seven seven seven. Now to round two decimal places, we need to decide if the six should stay a six or if it should round up to a seven. So we look at the number to the right of it.
Since it is a six, which is five or larger, we will round our six, the first six, up to a seven.
Therefore, the volume of the roll of the paper towels is five thousand six hundred and fifty-four point eight seven centimeters cubed. | {"url":"https://www.nagwa.com/en/videos/789158976143/","timestamp":"2024-11-10T06:44:44Z","content_type":"text/html","content_length":"245034","record_id":"<urn:uuid:5e878227-8394-4999-a254-e38588425d0c>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00329.warc.gz"} |
Prescribing the Curvature of a Riemannian Manifoldsearch
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Prescribing the Curvature of a Riemannian Manifold
A co-publication of the AMS and CBMS
Softcover ISBN: 978-0-8218-0707-1
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Prescribing the Curvature of a Riemannian Manifold
A co-publication of the AMS and CBMS
Softcover ISBN: 978-0-8218-0707-1
Product Code: CBMS/57
List Price: $29.00
Individual Price: $23.20
eBook ISBN: 978-1-4704-2419-0
Product Code: CBMS/57.E
List Price: $27.00
Individual Price: $21.60
Softcover ISBN: 978-0-8218-0707-1
eBook ISBN: 978-1-4704-2419-0
Product Code: CBMS/57.B
List Price: $56.00 $42.50
• CBMS Regional Conference Series in Mathematics
Volume: 57; 1985; 55 pp
MSC: Primary 53; Secondary 58
These notes were the basis for a series of ten lectures given in January 1984 at Polytechnic Institute of New York under the sponsorship of the Conference Board of the Mathematical Sciences and
the National Science Foundation. The lectures were aimed at mathematicians who knew either some differential geometry or partial differential equations, although others could understand the
Author's Summary:Given a Riemannian Manifold \((M,g)\) one can compute the sectional, Ricci, and scalar curvatures. In other special circumstances one also has mean curvatures, holomorphic
curvatures, etc. The inverse problem is, given a candidate for some curvature, to determine if there is some metric \(g\) with that as its curvature. One may also restrict ones attention to a
special class of metrics, such as Kähler or conformal metrics, or those coming from an embedding. These problems lead one to (try to) solve nonlinear partial differential equations. However,
there may be topological or analytic obstructions to solving these equations. A discussion of these problems thus requires a balanced understanding between various existence and non-existence
The intent of this volume is to give an up-to-date survey of these questions, including enough background, so that the current research literature is accessible to mathematicians who are not
necessarily experts in PDE or differential geometry.
The intended audience is mathematicians and graduate students who know either PDE or differential geometry at roughly the level of an intermediate graduate course.
□ Chapters
□ Gaussian Curvature
□ Scalar Curvature
□ Ricci Curvature
□ Boundary Value Problems
□ Some Open Problems
• Book Details
• Table of Contents
• Requests
Volume: 57; 1985; 55 pp
MSC: Primary 53; Secondary 58
These notes were the basis for a series of ten lectures given in January 1984 at Polytechnic Institute of New York under the sponsorship of the Conference Board of the Mathematical Sciences and the
National Science Foundation. The lectures were aimed at mathematicians who knew either some differential geometry or partial differential equations, although others could understand the lectures.
Author's Summary:Given a Riemannian Manifold \((M,g)\) one can compute the sectional, Ricci, and scalar curvatures. In other special circumstances one also has mean curvatures, holomorphic
curvatures, etc. The inverse problem is, given a candidate for some curvature, to determine if there is some metric \(g\) with that as its curvature. One may also restrict ones attention to a special
class of metrics, such as Kähler or conformal metrics, or those coming from an embedding. These problems lead one to (try to) solve nonlinear partial differential equations. However, there may be
topological or analytic obstructions to solving these equations. A discussion of these problems thus requires a balanced understanding between various existence and non-existence results.
The intent of this volume is to give an up-to-date survey of these questions, including enough background, so that the current research literature is accessible to mathematicians who are not
necessarily experts in PDE or differential geometry.
The intended audience is mathematicians and graduate students who know either PDE or differential geometry at roughly the level of an intermediate graduate course.
• Chapters
• Gaussian Curvature
• Scalar Curvature
• Ricci Curvature
• Boundary Value Problems
• Some Open Problems
Please select which format for which you are requesting permissions. | {"url":"https://bookstore.ams.org/CBMS/57","timestamp":"2024-11-11T20:37:15Z","content_type":"text/html","content_length":"78808","record_id":"<urn:uuid:b0dd8c28-cfd9-44c7-a606-e067cfcc2979>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00800.warc.gz"} |
Explore Techniques to Estimate Battery State of Charge
This example explores different techniques to estimate the state of charge (SOC) of a battery, including the Kalman filter algorithm and the Coulomb counting method. The Kalman filter is an
estimation algorithm that infers the state of a linear dynamic system from incomplete and noisy measurements. This example illustrates the effectiveness of the Kalman filter in dynamically estimating
the SOC of a battery, even when starting with inaccurate initial conditions and when the measurements are noisy. The SOC of a battery measures the current capacity of the battery in relation to its
maximum capacity.
To highlight the accuracy of the Kalman filter, you also estimate the SOC of the battery by using the Coulomb counting algorithm. You then compare these results with the SOC values you previously
estimated using the Kalman filter.
For a more straightforward and compact version of this example, see the Battery State-of-Charge Estimation example.
Measuring State of Charge in Electric Vehicles
Electric vehicles (EVs) provide a cleaner alternative to traditional combustion engine vehicles by using electricity as power source. This alternative power source allows the EVs to reduce the
reliance on fossil fuels and cut down on greenhouse gas emissions and air pollution. EVs operate by using electric motors powered by batteries, which are the heart of any EV. These batteries
determine the EV range, performance, and environmental footprint. However, managing the battery inside an EV is complex due to the inherent characteristics of the battery itself, including the
battery energy density and weight, the thermal management, aging and degradation, and the estimation of the SOC and state of health (SOH).
The ability to track the SOC is crucial for managing battery systems efficiently and in applications where the battery performance and longevity are critical.
This figure shows a schematic of an electric vehicle (EV) that focuses on the battery management system (BMS) and its interaction with the battery, by highlighting how the BMS estimates the battery
state of charge (SOC) using sensor data and the Kalman filter algorithm. The EV is equipped with temperature, current, and voltage sensors that tell you the values of these physical parameters at any
given time. However, the EV lacks a dedicated SOC sensor because a direct SOC sensor as a single unit simply does not exist. To know the SOC value of your battery, you must estimate the SOC through
algorithms that interpret the data from various sensors.
In this example, you estimate the SOC of a battery by using the Coulomb counting method and the Kalman filter algorithm:
• Coulomb counting — Integrate current over time,
$\mathrm{SOC}=\mathrm{SOC}\left({\mathit{t}}_{0}\right)+\frac{1}{{\mathit{C}}_{\mathrm{rated}}}{\int }_{{\mathit{t}}_{0}}^{{\mathit{t}}_{0}+\tau }{\mathit{I}}_{\mathrm{batt}},$
where ${\mathit{C}}_{\mathrm{rated}}$ is the nominal battery capacity, in ampere-hours (Ah), and ${\mathit{I}}_{\mathrm{batt}}$ is the battery current, in amperes.
• Kalman filter — Predict the future state of charge of the battery and update this prediction based on new measurements.
State of Charge
The state of charge of a battery measures the current charge level of the battery as a percentage of its total capacity. The SOC defines how much more charge the battery can store or how much charge
you can use before recharging the battery. To calculate the SOC value as a percentage, you divide the current charge of the battery by its total capacity, and multiply this value by 100:
where ${\mathit{C}}_{\mathrm{releasable}}$ is the current charge level of the battery, in Ah, and ${\mathit{C}}_{\mathrm{rated}}$ is the battery rated capacity, in Ah. Manufacturers provide the value
of the rated capacity, which represents the maximum amount of charge in the battery. For example, if a battery has a capacity of 200 Ah and currently contains 100 Ah of charge, the SOC is equal to
The SOC is important for managing battery systems in many applications, including electric vehicles (EV) and renewable energy storage systems. If you know the SOC value of your battery, you can
manage energy efficiently, prolong the life of the battery by preventing deep discharge cycles, and ensure system reliability.
Open Model
Open the model EstimateBatterySOCUsingKF. The model automatically loads the required parameters inside the BatterySOCEstimationData.m script by using a PreLoadFcn model callback.
This model represents a simplified Simscape™ implementation of the EV schematic in the previous section. The real initial SOC of the battery is 0.5 (or 50%), but the Kalman filter estimator starts
with an initial condition that assumes the SOC to be 0.8 (or 80%). The battery undergoes a cycle of charging and discharging over a period of six hours. Despite the initial discrepancy between the
actual SOC and the initial condition of the estimator, the Kalman filter quickly adjusts to accurately reflect the real SOC value. The estimation converges to the actual value in less than 10 minutes
and then continues to accurately track the real SOC.
In this example, the extended Kalman filter is the algorithm that the SOC Estimator (Kalman Filter) block uses to estimate the battery SOC. To use a different Kalman filter implementation, in the SOC
Estimator (Kalman Filter) block, set the Filter type parameter to Extended Kalman-Bucy Filter, Unscented Kalman Filter, or Unscented Kalman-Bucy Filter.
The model comprises three major components:
• Thermal network — Model the heat exchange between a battery and its environment.
• Electrical network — Model the battery and the current profile that charges and discharges the battery during the simulation.
• SOC estimator — Estimate the SOC of the battery by using the Kalman filter algorithm.
Explore Thermal Network
The heat exchange between a battery and the environment affects the battery performance and its safety. Low or high temperatures reduce the performance of a battery, while excessive heat can lead to
thermal runaway. In thermal runaway, the battery generates more heat than it dissipates, leading to fires or explosions.
This exchange happens through both conduction and convection. Conduction transfers the heat through materials. A battery generates heat due to the internal resistance and through electrochemical
reactions during charging and discharge cycles. The materials of the battery then conduct this heat to its surface. Once the heat reaches the surface, the battery transfers this heat to the
surrounding air or cooling fluid through convection. The efficiency of convective heat transfer depends on multiple factors, including the temperature difference between the battery surface and the
environment, the surface area of the battery, and the velocity of the air or fluid. To learn more about the basics of conduction, convection, and thermal mass, see the Heat Conduction Through Iron
Rod example.
This figures shows how this example models the thermal network of the battery:
To capture the thermal dynamics of the battery system, this model comprises:
• Battery Equivalent Circuit block — Model the battery that generates and exchanges the heat.
• Thermal mass of the battery — Represent the mass of the battery and its ability to store the heat. To model this mass, in the Battery Equivalent Circuit block, set the Thermal model parameter to
Lumped thermal mass and, in the Thermal settings, specify a value for the Battery thermal mass parameter. This parameter represents the energy required to raise the temperature of the thermal
port by one Kelvin.
• Convective Heat Transfer block — Model the heat exchange between the battery and the environment.
• Controlled Temperature Source block — Represent an ideal energy source in a thermal network that maintains a controlled temperature difference regardless of the heat flow rate. This block models
the ambient temperature. A and B are thermal conserving ports associated with the source inlet and outlet, respectively. Port S is a physical signal port that applies the control signal driving
the source.
• Temperature Sensor block — Measure the temperature of the battery. The SOC Estimator (Kalman Filter) block requires this temperature value.
The Convective Heat Transfer block represents the heat transfer by convection between the battery and the environment by means of fluid motion. The Newton law of cooling describes the transfer,
$\mathit{Q}=\mathit{k}\cdot \mathit{A}\cdot \left({\mathit{T}}_{\mathit{A}}-{\mathit{T}}_{\mathit{B}}\right),$
• $\mathit{Q}$ is the heat flow, in watts (W).
• $\mathit{k}$ is the convection heat transfer coefficient. In this example, this coefficient is equal to 5 W/(K*${\mathrm{m}}^{2}$).
• $\mathit{A}$ is the surface area. In this example, the surface area is the area of the battery cell and it is equal to 0.1019 ${\mathrm{m}}^{2}$.
• ${\mathit{T}}_{\mathit{A}}$ and ${\mathit{T}}_{\mathit{B}}$ are the temperatures of the battery and the environment, in Kelvin.
A and B are thermal conserving ports associated with the points between which the heat transfer by convection takes place. Because the block positive direction is from port A to port B, the heat flow
is positive if it flows from A to B.
Explore Electrical Network
This figure shows the elements that comprise the electrical network of the model:
• Battery Equivalent Circuit block — Model the battery of which you want to estimate the SOC.
• Probe block — Output the SOC value of the battery as a Simulink® signal.
• Controlled Current Source block — Represent an ideal current source that is powerful enough to maintain the current you specify through it regardless of the voltage across it. The output current
${\mathit{I}}_{\mathit{s}}$ is the numerical value at the physical signal input port, iT.
• Current Profile subsystem — Model the profile of the current that you feed to the Controlled Current Source block through the input port, iT, which in turn generates the current flowing through
the battery.
• Sensor and noise — Measure the battery voltage that the SOC Estimator (Kalman Filter) block requires to estimate the SOC by using a Voltage Sensor block. The two Band-Limited White Noise blocks
add noise to the current and voltage measurements to more accurately represent real-world applications.
Battery Equivalent Circuit Block
The Battery Equivalent Circuit block models the battery terminal voltage by using a combination of electrical circuit elements arranged in a specific circuit topology. This figure shows the
equivalent circuit topology, which relies on variable resistances, variable capacitances, and a variable voltage source. Batteries do not respond instantaneously to load changes. They require some
time to achieve a steady state. This time-varying property is a result of the battery charge dynamics. The block models the charge dynamics by using parallel RC sections in the equivalent circuit.
The Battery Equivalent Circuit calculates the terminal voltage of the battery at every time step by solving Kirchhoff's voltage law,
${\mathit{V}}_{\mathit{t}}={\mathit{V}}_{0}\left(\mathrm{SOC},\mathit{T}\right)+\mathit{I}\cdot {\mathit{R}}_{0}\left(\mathrm{SOC},\mathit{T}\right)+{\mathit{V}}_{1},$
• ${\mathit{V}}_{0}\left(\mathrm{SOC},\mathit{T}\right)$ is the open-circuit voltage.
• $\mathit{I}\cdot {\mathit{R}}_{0}\left(\mathrm{SOC},\mathit{T}\right)$ is the instantaneous overpotential.
• ${\mathit{R}}_{0}\left(\mathrm{SOC},\mathit{T}\right)$ is the instantaneous resistance of the battery at the specified SOC and temperature values.
• ${\mathit{V}}_{1}=\Delta {\mathit{U}}_{{\mathrm{RC}}_{1}}\left(\mathrm{SOC},\mathit{T}\right)$ is the dynamic overpotential. This optional term depends on the number of parallel
resistor-capacitor pairs. As the Battery Equivalent Circuit block in this example only models one time-constant dynamics, the dynamic overpotential only comprises one term.
• $\Delta {\mathit{U}}_{{\mathrm{RC}}_{1}}$ is the voltage drop for parallel resistor-capacitor pair 1.
The software updates the battery state of charge SOC, the temperature T, and the corresponding variables that depend on them at every time step during the simulation. For the Battery Equivalent
Circuit block, $\mathit{I}$ has a negative sign during discharge and a positive sign during charge. During discharge, the block subtracts the battery overpotentials from the open-circuit voltage
value, lowering the terminal voltage of the battery. In the discharge and charge cases, the overpotentials dissipate energy as heat. This figure shows the evolution of the battery overpotentials
during a battery pulse discharge. The dynamic overpotential from the resistor-capacitor pairs increases slowly over the pulse, as opposed to the instantaneous overpotential, which increases during
the first few milliseconds.
The battery in this example does not model self discharge, hysteresis, current directionality, battery fade, or calendar aging.
The block determines the battery temperature by summing all the ohmic losses in the battery,
${\mathit{M}}_{\mathrm{th}}\stackrel{˙}{\mathit{T}}={\sum }_{\mathit{i}}\frac{{\mathit{V}}_{\mathit{T},\mathit{i}}^{2}}{{\mathit{R}}_{\mathit{T},\mathit{i}}}$,
• ${\mathit{M}}_{\mathrm{th}}$ is the battery thermal mass.
• $\mathit{i}$ corresponds to the ${\mathit{i}}^{\mathrm{th}}$ ohmic loss contributor. Since this block models only one time-constant dynamics, these losses include the series resistance and the
first charge dynamics segment.
• ${\mathit{V}}_{\mathit{T},\mathit{i}}$ is the voltage drop across the resistor $\mathit{i}$.
• ${\mathit{R}}_{\mathit{T},\mathit{i}}$ is the resistor $\mathit{i}$.
Battery Characterization
This example characterizes the battery by using the SOC, temperature, voltage, resistance, and cell capacity. The parameterization for the battery in this example corresponds to a lithium-ion
battery. If you do not possess meaningful values for these parameters, or if you want to represent components by specific suppliers, you can parameterize the Battery Equivalent Circuit block with
predefined parameterizations. These parameters match the manufacturer data sheets. To load a predefined parameterization, double-click the block, then click the <click to select> hyperlink of the
Selected part parameter. The Block Parameterization Manager window opens and you can then choose the desired parameterization. For more information about pre-parameterization and for a list of the
available components, see List of Pre-Parameterized Components (Simscape Electrical).
This figure shows the block parameters of the Battery Equivalent Circuit block.
Examine the vector of state of charge values, SOC_vec.
SOC_vec = 1×7
0 0.1000 0.2500 0.5000 0.7500 0.9000 1.0000
The SOC_vec variable represents the vector of SOC breakpoints that define the points at which you specify the lookup data. This vector must be strictly ascending. The block calculates the SOC value
with respect to the nominal battery capacity that you specify in the Cell capacity, AH parameter. This example characterizes the SOC of the battery with seven data points.
Examine the vector of temperatures, T_vec.
The T_vec variable represents the vector of temperature breakpoints that define the points at which you specify lookup data. This vector must be strictly ascending and greater than 0 Kelvin. This
example characterizes the battery temperature with three data points. These temperatures are in Kelvin.
Examine the open-circuit voltage, V0_mat.
V0_mat = 7×3
3.4900 3.5000 3.5100
3.5500 3.5700 3.5600
3.6200 3.6300 3.6400
3.7100 3.7100 3.7200
3.9100 3.9300 3.9400
4.0700 4.0800 4.0800
4.1900 4.1900 4.1900
The V0_mat variable represents the lookup data for the open-circuit voltages across the fundamental battery model at the SOC and temperature breakpoints you specified in the SOC_vec and T_vec
variables. V0_mat is a 7-by-3 matrix. Each column relates to a temperature, and the rows relate to the SOC.
Examine the terminal resistance, R0_mat.
R0_mat = 7×3
0.0117 0.0085 0.0090
0.0110 0.0085 0.0090
0.0114 0.0087 0.0092
0.0107 0.0082 0.0088
0.0107 0.0083 0.0091
0.0113 0.0085 0.0089
0.0116 0.0085 0.0089
The R0_mat variable represents the lookup data for the series resistance of the battery at the SOC and temperature breakpoints you specified in the SOC_vec and T_vec variables. R0_mat is a 7-by-3
matrix. Each column relates to a temperature, and the rows relate to the SOC.
Examine the cell capacity, AH.
The AH variable represents the capacity of the battery at full charge. The block calculates the SOC by dividing the accumulated charge by this value. The block calculates the accumulated charge by
integrating the battery current.
In this example, these matrices are small, but you can characterize these table-based models to the level of granularity you require.
Current Profile
Open the Current Profile subsystem.
open_system("EstimateBatterySOCUsingKF/Current Profile")
The Current profile subsystem implements a simplified version of a system load by defining the current that flows through the battery. This current charges and discharges the battery during the
simulation. During the charging process, the subsystem defines a current that charges the battery at a constant rate of 15 Amperes. During the discharging process, the subsystem introduces some
variability with white noise and with uniform random numbers. By providing the discharge with some variability, you can better test the accuracy of the estimation.
The model feeds the current profile directly into a Controlled Current Source block to drive the battery.
Introduce Noise in Measurements for Kalman Filter
To evaluate the performance and robustness of the estimator, your simulation must represent real-world applications as closely as possible. In real-world applications, the data from sensors and
measurements always contain noise due to sensor inaccuracies, environmental disturbances, or signal processing errors. A signal noise is the unwanted alteration in a signal that interferes with the
accurate transmission of information. By introducing noise in the simulations, the model reflects the real operational conditions, making these simulations more reliable.
Theoretically, continuous white noise has a correlation time of 0, a flat power spectral density (PSD), and a total energy of infinity. In practice, white noise never disturbs physical systems,
although white noise is a useful theoretical approximation when the noise disturbance has a correlation time that is very small relative to the natural bandwidth of the system. In Simulink, you can
simulate the effect of white noise by using a random sequence with a correlation time much smaller than the shortest time constant of the system. This example introduces noise to the current and
voltage measurements by using two Band-Limited White Noise blocks. The correlation time of the noise is the sample rate of the blocks. For accurate simulations, use a correlation time much smaller
than the fastest dynamics of the system. You can get good results by specifying:
$\mathrm{tc}\approx \frac{1}{100}\frac{2\pi }{{\mathit{f}}_{\mathrm{max}}},$
where ${\mathit{f}}_{\mathrm{max}}$ is the bandwidth of the system in rad/sec.
The SOC Estimator (Kalman Filter) block receives the noisy current and voltage measurements. Then the block estimates the battery SOC against such uncertainties and disturbances. By adding noise, you
can effectively assess how well the filter extracts the signal from the noise, which is essential for applications where the precision is critical.
Noisy measurements are also useful for tuning the parameters of the filter. The effectiveness of the Kalman filter depends on the correctness of its parameters, including the process noise covariance
and the measurement noise covariance matrices.
Kalman Filter for State of Charge Estimation
From a practical perspective, the direct measurement of the SOC of a battery is not straightforward because the SOC is not a directly observable quantity such as the voltage or the current. The SOC
is a derived parameter and reflects the internal chemical state of the battery. This chemical state varies with multiple factors including age, temperature, and discharge rate. Since direct
measurement is challenging, you must estimate the SOC value.
This example uses the Kalman filter algorithm to estimate the battery SOC. The Kalman filter algorithm estimates the state of a linear dynamic system from noisy measurements. To estimate the SOC of a
battery, the Kalman filter predicts the future state and then updates this prediction based on the new measurements. The Kalman filter provides accurate estimates even with noisy data, adapts to
behavioral changes in the battery, including aging effects, and incorporates different measurements to improve the SOC estimation, such as voltage, current, and temperature. To learn more about
Kalman filters, see the Understanding Kalman Filters introductory examples.
The SOC Estimator (Kalman Filter) block provides four different Kalman filter algorithms for SOC estimation. This example focuses on the extended Kalman filter (EKF) algorithm.
This figure show the equivalent circuit for a battery with one-time-constant dynamics:
The equations for the equivalent circuit with one time-constant dynamics are:
$\frac{\mathrm{dSOC}}{\mathrm{dt}}=-\frac{\mathit{i}}{3600\cdot \mathrm{AH}}$
$\frac{{\mathrm{dV}}_{1}}{\mathrm{dt}}=\frac{\mathit{i}}{{\mathit{C}}_{1}\left(\mathrm{SOC},\mathit{T}\right)}-\frac{{\mathit{V}}_{1}}{{\mathit{R}}_{1}\left(\mathrm{SOC},\mathit{T}\right)\cdot {\
In these equations:
• $\mathrm{SOC}$ is the state of charge.
• $\mathit{i}\text{\hspace{0.17em}}$is the current.
• ${\mathit{V}}_{0}$ is the no-load voltage.
• ${\mathit{V}}_{\mathit{t}}$ is the terminal voltage.
• $\mathrm{AH}$ is the ampere-hour rating.
• ${\mathit{R}}_{1}$ is the polarization resistance.
• ${\mathit{C}}_{1}$ is the parallel RC capacitance.
• $\mathit{T}$ is the temperature.
• ${\mathit{V}}_{1}$ is the polarization voltage over the RC network.
A time constant ${\tau }_{1}$ for the parallel section relates the polarization resistance ${\mathit{R}}_{1}$ and the parallel RC capacitance ${\mathit{C}}_{1}$ using the relationship ${\mathit{C}}_
{1}=\frac{{\tau }_{1}}{{\mathit{R}}_{1}}$.
For the Kalman filter algorithms, the block uses this state and these process and observation functions:
$\mathit{x}={\left[\begin{array}{cc}\mathrm{SOC}& {\mathit{V}}_{1}\end{array}\right]}^{\mathit{T}}$
$\mathit{f}\left(\mathit{x},\mathit{i}\right)=\left[\begin{array}{c}-\frac{\mathit{i}}{3600\cdot \mathrm{AH}}\\ \frac{\mathit{i}}{{\mathit{C}}_{1}\left(\mathrm{SOC},\mathit{T}\right)}-\frac{{\mathit
{V}}_{1}}{{\mathit{R}}_{1}\left(\mathrm{SOC},\mathit{T}\right)\cdot {\mathit{C}}_{1}\left(\mathrm{SOC},\mathit{T}\right)}\end{array}\right]$
This diagram shows the internal structure of the extended Kalman filter (EKF) in the SOC Estimator (Kalman Filter) block:
The EKF technique relies on a linearization at every time step to approximate the nonlinear system. To linearize the system at every time step, the algorithm computes these Jacobians:
$\mathit{F}=\frac{\partial \mathit{f}}{\partial \mathit{x}}$
$\mathit{H}=\frac{\partial \mathit{h}}{\partial \mathit{x}}$
The EKF is a discrete-time algorithm. After the discretization, the Jacobians for the SOC estimation of the battery are:
${\mathbf{F}}_{\mathit{d}}=\left[\begin{array}{cc}1& 0\\ 0& {\mathit{e}}^{\frac{-\mathrm{Ts}}{{\mathit{R}}_{1}\left(\mathrm{SOC},\mathit{T}\right)\cdot {\mathit{C}}_{1}\left(\mathrm{SOC},\mathit{T}\
${\mathit{H}}_{\mathit{d}}=\left[\begin{array}{cc}\frac{\partial {\mathit{V}}_{\mathrm{OC}}}{\partial \mathrm{SOC}}& -1\end{array}\right]$
${\mathit{G}}_{\mathit{d}}=\left[\begin{array}{cc}\frac{-\mathrm{Ts}}{3600\cdot \mathrm{AH}}& \left({\mathit{e}}^{\frac{-\mathrm{Ts}}{{\mathit{R}}_{1}\left(\mathrm{SOC},\mathit{T}\right)\cdot {\
mathit{C}}_{1}\left(\mathrm{SOC},\mathit{T}\right)}}+1\right){\cdot \mathit{R}}_{1}\left(\mathrm{SOC},\mathit{T}\right)\end{array}\right]$
where ${\mathit{T}}_{\mathit{S}}$ is the sample time and ${\mathit{V}}_{\mathrm{OC}}$ is the open-circuit voltage.
The EKF algorithm comprises three phases: initialization, prediction, and correction.
The EKF algorithm initializes the state estimate and the uncertainty (covariance) of this state at 0 seconds.
• $\stackrel{ˆ}{\mathit{x}}\left(0|0\right)$ — State estimate at time step 0 using measurements at time step 0.
• $\stackrel{ˆ}{\mathit{P}}\left(0|0\right)$ — State estimation error covariance matrix at time step 0 using measurements at time step 0.
${\mathbf{P}}_{0}$ is the covariance of the initial error. ${\mathbf{P}}_{0}$ is a square matrix that represents the uncertainty in the initial estimates of the state of the system and contains the
covariances between all pairs of the initial estimates of the state. The diagonal elements represent the uncertainties in each variable of the state. The off-diagonal elements represent the
covariances between the different variables of the state. This matrix depends on the knowledge of the initial state, desired convergence speed, and stability of the filter. If you have a good initial
estimate of the state, you can set this matrix to a small value. If the initial value is uncertain, set this matrix to higher values. The choice of the ${\mathbf{P}}_{0}$ matrix is important as it
affects the performance of the filter, especially at the beginning of the simulation. If you set ${\mathbf{P}}_{0}$ too low, you are representing an overconfidence in the initial estimates and the
filter might be too slow to adjust to the actual behavior of the system. If you set ${\mathbf{P}}_{0}$ too high, you are representing almost no confidence in the initial estimates and the filter
might rely too much on the first few measurements. This action might destabilize the estimation of the state if the measurements are noisy or inaccurate.
The EKF algorithm predicts the future SOC based on the behavior of the battery, such how it discharges or charges under certain conditions, and the control inputs, for example if the battery is
drawing or supplying current. This prediction results in a predicted state and a predicted error in the estimate. The prediction state moves the state estimate forward in time.
• Project the states ahead (a priori):
$\stackrel{ˆ}{\mathit{x}}\left(\mathit{k}+1|\mathit{k}\right)={\mathbf{F}}_{\mathit{d}}\cdot \stackrel{ˆ}{\mathit{x}}\left(\mathit{k}|\mathit{k}\right)+{\mathit{G}}_{\mathit{d}}\cdot \mathit{i}$
• Project the error covariance ahead:
where $\mathbf{Q}$ is the covariance of the process noise.
The covariance of the process noise, $\mathbf{Q}$, is a square matrix that represents the uncertainty in the state transition model and contains the covariances between all pairs of the components of
the process noise. The diagonal elements represent the uncertainties of each component in the process noise. The off-diagonal elements represent the covariances between the different components in
the process noise. You often determine it through a combination of system identification, experiments, and domain knowledge. Higher values of the elements in the $\mathbf{Q}$ matrix indicate greater
uncertainty and unpredictability in the dynamics of the system.
The EKF algorithm uses the information from new measurements to update the predictions of the state and error. The filter calculates the measurement residual, which is the difference between the
actual measurements and the predictions. This difference represents the inaccuracy in the prediction. The filter then updates the predicted state based on the measurement residual. This process
involves calculating the Kalman gain that determines how much the filter must correct the predictions based on the new measurements. To calculate the Kalman gain, the algorithm uses the covariance of
the measurement noise, $\mathbf{R}$.
The covariance of the measurement noise, $\mathbf{R}$, is a square matrix that represents the uncertainty in the observation model and contains the covariances between all pairs of measurement
errors. The diagonal elements represent the uncertainties in each measurement. The off-diagonal elements represent the covariances between different measurements, that indicate how much the noise in
one measurement correlates with the noise in another measurement. Higher values of the elements in the $\mathbf{R}$ matrix indicate higher measurement noise or less confidence in the sensor. This
matrix depends on the characteristics of the sensors and on their accuracy. For example, if the standard deviation of the sensor is equal to 0.01, then a good value for $\mathbf{R}$ is ${\left(0.01\
• Update the estimate with the measurement $\mathit{y}\left(\mathit{k}\right)$ (a posteriori):
$\stackrel{ˆ}{\mathit{x}}\left(\mathit{k}+1|\mathit{k}+1\right)=\stackrel{ˆ}{\mathit{x}}\left(\mathit{k}+1|\mathit{k}\right)+\mathbf{K}\left(\mathit{k}+1\right)\cdot \left({\mathit{V}}_{\mathit{t}}\
left(\mathit{k}\right)-\left({\mathit{V}}_{0}\left(\mathrm{SOC},\mathit{T}\right)-{\mathit{V}}_{1}-{\mathit{R}}_{0}\left(\mathrm{SOC},\mathit{T}\right)\cdot \mathit{i}\right)\right).$
• Update the error covariance:
This process repeats for each new measurement and the Kalman filter continuously improves the state estimate by balancing the predictions of the model with the real-world measurements.
Plot Real and Estimated State of Charge
Simulate the model.
Create a figure to plot the real and estimated SOC values.
FigureEstimateBatterySOCUsingKF = figure(Name="EstimateBatterySOCUsingKF");
Get the simulation time, real SOC, and estimated SOC values from the simulation output.
time = SimulationOutputKalmanFilter.time/3600;
SOCRealKF = SimulationOutputKalmanFilter.signals(1).values(:)*100;
SOCEstKF = SimulationOutputKalmanFilter.signals(2).values(:)*100;
Plot the real and estimated SOC values.
hold on
hold off
grid on
title('Real and Estimated State of Charge')
ylabel('SOC (%)')
xlabel('Time (hours)')
The initial SOC of the battery is equal to 0.5. The estimator uses an initial condition for the SOC equal to 0.8. The extended Kalman filter estimator converges to the real value of the SOC in less
than 10 minutes and then follows the real SOC value.
Compare Kalman Filter Estimation with Coulomb Counting Estimation
To highlight the accuracy of the Kalman filter algorithm, estimate the SOC of the same battery model by using the Coulomb counting method. The Coulomb counting method estimates the SOC of the battery
by integrating the current that flows through the battery over time. This section shows you how the Coulomb counting estimator fails to converge to the real value of the SOC if the initial estimate
is wrong.
Open Coulomb Counting Model
Open the model EstimateBatterySOCUsingCC. The real initial SOC of the battery is 0.5 (or 50%), but the Coulomb counting estimator starts with an initial condition that assumes the SOC to be 0.8 (or
80%). The battery undergoes a cycle of charging and discharging over a period of six hours.
The thermal and electrical network of this model are identical to those of the EstimateBatterySOCUsingKF model.
This diagram shows the structure of the SOC Estimator (Coulomb Counting) block:
To compute the SOC of the battery, the SOC Estimator (Coulomb Counting) block counts the Ampere hour and current integration.
Compare Estimated State of Charge with Coulomb Counting and Kalman Filter
Simulate the EstimateBatterySOCUsingCC model.
Create a figure to plot the real and estimated SOC values.
FigureEstimateBatterySOCUsingCC = figure(Name="EstimateBatterySOCUsingCC");
Get the simulation time, real SOC, and estimated SOC values from the simulation output.
time = SimulationOutputCoulombCounting.time/3600;
SOCRealCC = SimulationOutputCoulombCounting.signals(1).values(:)*100;
SOCEstCC = SimulationOutputCoulombCounting.signals(2).values(:)*100;
Plot the real SOC values and the SOC values that you estimated using both the Kalman filter and the Coulomb counting algorithms.
hold on
hold off
grid on
title('Real and Estimated SOC with Coulomb Counting and Kalman Filter')
ylabel('SOC (%)')
xlabel('Time (hours)')
legend({'Real','Estimated Coulomb Counting','Estimated Kalman Filter'},Location="Best");
The initial SOC of the battery is equal to 0.5. The estimator uses an initial condition for the SOC equal to 0.8. The Coulomb counting estimator fails to converge to the real value of the SOC due to
the error in the initial estimation. However, if you initialize the Coulomb counter correctly, and if you know the capacity of the battery with absolute certainty, the Coulomb counting method
actually outperforms the Kalman filter in accuracy for a specific load case of short duration. The main problem with the Coulomb counting method arises over longer periods of time, as the estimated
battery SOC slightly starts to drift away from the actual real value due to the self-discharge behavior of the battery. All batteries experience this self-discharge behavior during storage conditions
due to the internal chemical reactions or other processes that consume charge from the battery. When the battery is off, the battery management system (BMS) does not wake up often enough to
accurately recalculate the SOC. For example, if you leave an electric vehicle (EV) battery pack uncharged and in storage for a couple of weeks, a lithium-ion battery pack can lose around 2% of its
SOC. The BMS cannot detect this loss and it is not accounted for in the Coulomb counter. This self-discharge rate also changes over the lifetime of a battery. As such, with Coulomb counting, every
time the BMS wakes up after a long period of time (for example, after just hours or even a two-week storage period), it perceives its SOC to be different from its actual value and lacks the
capability to correct this discrepancy in the initial value. Over time, this difference significantly accumulates.
For these reasons, choosing the correct algorithm to estimate the SOC value of your battery depends on multiple factors, including the model dependency, computational complexity, accuracy, and
adaptability. In general, the Kalman filter offers higher accuracy and adaptability (if you properly set the $\mathbf{Q}$, $\mathbf{R}$, and ${\mathbf{P}}_{0}$ matrices), but requires more detailed
system models and is also more complex to implement and computationally more intensive. The Coulomb counting algorithm, instead, is easier to implement and does not require a model of the system, but
it is also less adaptable to changes in the battery characteristics.
[1] Plett, Gregory L. Battery Management Systems. Volume I, Battery Modeling. Artech House, 2015.
[2] Plett, Gregory L. Battery Management Systems. Volume II, Equivalent-Circuit Methods. Artech House, 2015.
See Also
SOC Estimator (Kalman Filter) | SOC Estimator (Coulomb Counting)
Related Topics | {"url":"https://jp.mathworks.com/help/simscape-battery/ug/estimate-battery-soc-using-kalman-filter-example.html","timestamp":"2024-11-09T04:40:18Z","content_type":"text/html","content_length":"157622","record_id":"<urn:uuid:a405c5aa-a1da-4df5-a44f-51352efc77ff>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00798.warc.gz"} |
Operators and separators in C programming - Codeforwin
Operators and separators in C programming
Quick links
Every C program is developed using five fundamental units keywords, identifiers, operators, separators and literals. In previous post, we already learned about keywords and identifiers. Here in this
post we will focus on operators and separators.
Operators in C
Operator is a symbol given to an operation that operates on some value. It tells the computer to perform some mathematical or logical manipulations. Such as + is an arithmetic operator used to add
two integers or real types.
C language provides a rich set of operators. Operators are classified into following categories based on their usage.
Various operators in C programming
Let us suppose a = 10, b = 5.
Operator Description Example
Arithmetic operator
Arithmetic operator are used to perform basic arithmetic operations.
+ Add two integer or real type. a + b gives 15
* Multiply two integer or real a * b gives 50
/ Divide two integer or real types. a / b gives 2
Modulus operator divide first
% operand from second and returns a % b gives 0 (As 10/5 will have 0 remainder)
Assignment operator
Assignment operator is used to assign value to a variable. The value is assigned from right to left.
= Assign value from right operand to a = 10 will assign 10 in a
left operand.
Relational operators
Relational operators are used to check relation between any two operands.
If value of left operand is
> greater than right, returns true (a > b) returns true
else returns false
If value of right operand is
< greater than left, returns true (a < b) returns false
else returns false
== If both operands are equal returns (a == b) returns false
true else false
!= If both operands are not equal (a != b) returns true
returns true else false.
If value of left operand is
>= greater or equal to right operand, (a >= b) returns true
returns true else false
If value of right operand is
<= greater or equal to left operand, (a <= b) will return false
returns true else false
Logical operators
Logical operators are used to combine two boolean expression together and results a single boolean value according to the operand and operator used.
Used to combine two expressions.
&& If both operands are true or ((a>=1) && (a<=10)) returns true since (a>=1) is true and also (a<=10) is true.
Non-Zero, returns true else false
|| If any of the operand is true or ((a>1) || (a<5)) will return true. As (a>1) is true. Since first operand is true hence there is no need to check for second operand.
Non-zero, returns true else false
Logical NOT operator is a unary
! operator. Returns the complement !(a>1) will return false. Since (a>1) is true hence its complement is false.
of the boolean value.
Bitwise operators
Bitwise operator performs operations on bit(Binary level). Lets suppose a = 10, b = 5
a = 0000 1010 (8-bit binary representation of 10)
b = 0000 0101 (8-bit binary representation of 5)
Bitwise AND performs anding <span class="token number">0000</span> <span class="token number">1010</span>
operation on two binary bits <span class="token operator">&</span> <span class="token number">0000</span> <span class="token number">0101</span>
& value. If both are 1 then will ____________
result is 1 otherwise 0. <span class="token number">0000</span> <span class="token number">0000</span>
<span class="token number">0000</span> <span class="token number">1010</span>
Bitwise OR returns 1 if any of the <span class="token operator">|</span> <span class="token number">0000</span> <span class="token number">0101</span>
| two binary bits are 1 otherwise 0. ___________
<span class="token number">0000</span> <span class="token number">1111</span>
<span class="token number">0000</span> <span class="token number">1010</span>
Bitwise XOR returns 1 if both the <span class="token operator">^</span> <span class="token number">0000</span> <span class="token number">0101</span>
^ binary bits are different else ___________
returns 0. <span class="token number">0000</span> <span class="token number">1111</span>
Bitwise COMPLEMENT is a unary <span class="token operator">~</span> <span class="token number">0000</span> <span class="token number">1010</span>
operator.It returns the complement ___________
~ of the binary value i.e. if the <span class="token number">1111</span> <span class="token number">0101</span>
binary bit is 0 returns 1 else
returns 0.
Bitwise LEFT SHIFT operator is
unary operator. It shift the <span class="token number">0000</span> <span class="token number">1010</span> <span class="token operator"><<</span> <span class="token number">2</span>
<< binary bits to the left. It <span class="token operator">=</span> <span class="token number">0010</span> <span class="token number">1000</span>
inserts a 0 bit value to the
extreme right of the binary value.
Bitwise RIGHT SHIFT operator is
unary operator. It shifts the <span class="token number">0000</span> <span class="token number">1010</span> <span class="token operator"><<</span> <span class="token number">2</span>
>> binary bits to the right. It <span class="token operator">=</span> <span class="token number">0000</span> <span class="token number">0010</span>
inserts a 0 bit value to the
extreme left of the binary value.
Increment/Decrement operator
Increment/Decrement operator is a unary operator used to increase an integer value by 1 or decrease it by 1. Increment/decrement operator are of two types Postfix and Prefix.
++ Increment operator will add 1 to a++ will result to 11
an integer value. ++a will result to 11
-- Decrement operator will subtract 1 a-- will result to 9
from an integer value. --a will result to 9
Conditional/Ternary operator
Ternary operator as a conditional operator and is similar to simple if-else. It takes three operand.
It is used as conditional
operator. Syntax of using ternary b = (a>1) ? a : b;
?: operator: will store the value 10 in b as (a>1) is true hence true part will execute, assigning the value of a in b.
(condition) ? (true part) : (false
Other operators
In addition to above mentioned operator, C supports many more operators.
Operator Name Description
. Member access operator Used to access the members of structures and unions
-> Member access operator Used to access the members of structures and unions
* Dereferencing operator Used to dereference the value of a pointer variable
& Address of operator Used to get the actual memory address of a variable
sizeof() Size of operator Used to get the size of a data type
At this point discussing these operators is not possible. I will introduce these operators later in this C programming tutorial series.
Read more – Operator precedence and associativity in C
Separators in C
Separators are used to separate one programming element from other. Such as separating keyword from keyword, keyword from identifier, identifier from other identifier etc. They are similar to
punctuation marks in English paragraphs.
In C programming every expression is separated using white space character/s, statements are separated from other using semicolon ;.
We can use any number of white space characters to separate two expressions. However we must use at least single white space character to separate one programming element from other. We can also use
number of semicolon to separate one statement from other.
Note: We can write entire C program in two lines if proper separators used. Take an example of below two programs.
#include <stdio.h>
int main(){int a=10;int b=20;int c=a+b;printf("Sum=%d",c);return 0;}
The above program displays sum of two numbers 10 and 20 using minimum number of separators. However, it is less readable and considered as poor programming practice. We must use proper separators
(spaces and indention) to make the program readable.
Consider the same program written with proper separators and is much more readable than previous program.
#include <stdio.h>
int main()
int a = 10;
int b = 20;
int c = a + b;
printf("Sum=%d", c);
return 0;
Hence, I always recommend you to maintain a good indention in your program. | {"url":"https://codeforwin.org/c-programming/operators-separators-c-programming","timestamp":"2024-11-05T07:25:03Z","content_type":"text/html","content_length":"148860","record_id":"<urn:uuid:28e0af5a-c0f5-4018-8d48-8f279eaeec61>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00731.warc.gz"} |
Matlab AND Operator | Learn Working And Uses of Matlab AND Operator
Updated March 23, 2023
Introduction to Matlab AND Operator
In this article, we will see an outline on Matlab AND Operator. Logical operators control the execution of program flow according to conditions that result from a set of expressions. They are very
easy to use and to understand the flow of any program. They can be used to check the number of zeroes in an array or any conditional statement if it matches a particular requirement. Three types of
logical operators are used in any programming language i.e. OR (C|D), AND (C & D), NOT(~C). They result in Boolean values i.e. either True/False or 0/1. If a particular condition is false, then it
results in 0 else 1.
Working of Matlab AND Operator
In Matlab, logical operators function in a similar way as in other programming languages. Logical AND operator results in 0/1 or True/False based on the type of signals that we provide to the input.
They are denoted by & operator (C&D). Please find the below truth table to see the output for different combinations of input signals.
Truth Table:
Input 1 Input 2 Output
(C) (D) (C&D)
According to the above table, when any of the operands i.e. C and D are 0 or false the resulting output is false or 0. Similarly, if the operands are true or 1 then the resulting output is true or 1.
In Matlab, we can use logical AND operator by defining as C&D. It can also be defined as ‘ and (C, D)’ but this syntax is used rarely because of operator overloading issues. So, it’s better to define
the operator using C&D format in Matlab. Please find the below example to understand how AND operator works:
Examples of Matlab AND Operator
Below are the examples of Matlab AND Operator:
Example #1
G = [0,1,0,0,0,1]
H= [ 0,1,0,0,1,1]
In the above example, the first and second array consists of an array of 0 and 1. If we use AND operator between two arrays then, if both the elements are true in the above two arrays, it results in
True or 1. The second and sixth element of both arrays has 1 so the resulting output is 1 while rest other combinations have 0, so the resulting output is 0. The inputs or operands can be vectors,
scalar, matrix or multi-dimensional array. They can be of the same size or a different size.
Like Logical OR operator, logical AND operator can also be used in the short-circuiting principle. They have a different working principle as compared to normal & operator in Matlab. They are defined
by the && operator. If there are two expressions, then the second part of the defined expression is not evaluated if the first part of the defined expression is false or 0. The resulting output from
the expressions is always scalar is nature if we are using short-circuiting principles. In short, the second part of the defined expression always depends on the first part whether we use logical &&
or || operator in Matlab defining its short-circuiting nature.
Example #2
C = 0
D= 18
Y= (C==1) && (C*D<0)
In the above expression, it evaluates the first part of the defined expression which is not true since we have assigned the values of C as 0. So, according to the short-circuiting behavior of AND
operator in Matlab, if the first part of the given expression is false then it doesn’t evaluate the second part of the defined expression and it results in logical 0 or false evaluating only the
first expression. The output is 0 which is scalar in nature. We should be careful while using the & and && operator in Matlab because both will give you different outputs.
Logical AND operator is also used to determine the condition satisfying a particular criterion by resulting in 0 and 1. If the result is 1 then it matches a particular condition else the result is 0.
Please find the below example demonstrating the above part:
Example #3
C = [3, 0 ,5 ; 8, 1 ,0 ; 4, 3, 0]
D= [8, 0, 6; 2, 1, 0; 5, 7, 0]
In the above two matrices, it checks both the elements of the matrix and results in 0 and 1 based on the values. If the values of the matrices are not zero, then then it results in 1 and 0 if both
the elements are 0.
Logical operators form a very important part in many programming languages like Java, Python, C, etc. So, it’s important to understand the working of these operators to use the program while
executing any code. We should be aware of the business requirements and use the operators as needed. For example, && and && or || and | operator will give different outputs when used in an
Recommended Articles
This is a guide to Matlab AND Operator. Here we discuss the Introduction to Matlab AND Operator with practical examples and different combinations of input signals. You can also go through our
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The Math - Connected Mathematics Project
The Math
This section provides several perspectives on the mathematics that is developed in Connected Mathematics including a detailed description of the development of the mathematics within each strand and
of the unifying mathematical themes across the curriculum. It also includes the mathematics, goals, and the correlations to the CCSSM for each unit. | {"url":"https://connectedmath.msu.edu/the-math/index.aspx","timestamp":"2024-11-13T07:33:20Z","content_type":"text/html","content_length":"48470","record_id":"<urn:uuid:6506b21c-a68a-4021-bfb5-5e8b2c2fd5cd>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00468.warc.gz"} |
How do you calculate optimal selling price?
How do you calculate optimal selling price?
The optimum selling price occurs where marginal cost = marginal revenue. Marginal cost is assumed to be the same as variable cost. From the data it can be seen that the costs of direct materials,
direct labour and variable overhead total $18.75 per unit.
What is optimal sale price?
The optimal price is the price at which a seller can make the most profit. In other words, the price point at which the seller’s total profit is maximized. We can refer to the optimal price as the
profit maximizing price. The word ‘price’ refers to how much the seller will accept for the sale of something.
How do you calculate optimal?
Calculate Your Optimal Order Quantity The formula you need to calculate optimal order quantity is: [2 * (Annual Usage in Units * Setup Cost) / Annual Carrying Cost per Unit]^(1/2). Substitute each
input with your own figures.
What is P A bQ?
P = a – bQ a = theoretical maximum price (if price is set at ‘a’ or above, demand will be zero), i.e. from the graph above, at a price of $200, demand is zero.
What is optimal pricing strategy?
Optimal pricing policy is also known as perfect price discrimination, which means that a company segments the market into distinct customer groups and charges each group exactly what it is willing to
pay. The optimal price and volume refer to the selling price and volume at which a company maximizes its profits.
How do you set optimal price?
Our formula for optimal pricing tells us that p* = c – q / (dq/dp). Here, marginal costs are a bit sneaky — they enter directly, through the c, but also indirectly because a change in marginal cost
will change prices which in turn changes both q and dq/dp.
What is the optimal profit?
The general rule is that the firm maximizes profit by producing that quantity of output where marginal revenue equals marginal cost. To maximize profit the firm should increase usage of the input “up
to the point where the input’s marginal revenue product equals its marginal costs”.
What is the optimal level of inventory?
Optimal inventory levels are the ideal quantities of products that you should have in a fulfillment center(s) at any given time. By optimizing inventory levels, you reduce the risk of common
inventory issues, from high storage costs to out-of-stock items.
How do you find optimal price elasticity?
Thus, when price elasticity is relatively low, the optimal price is much greater than marginal revenue. Conversely, when P = $8 and _P = –10, MR = $7.20. When the quantity demanded is highly elastic
with respect to price, the optimal price is close to marginal revenue. | {"url":"https://richardvigilantebooks.com/how-do-you-calculate-optimal-selling-price/","timestamp":"2024-11-07T06:54:55Z","content_type":"text/html","content_length":"47087","record_id":"<urn:uuid:62d66b55-f1e9-4065-8578-961ef5db106d>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00212.warc.gz"} |
Sensitivity analysis of experimental data
This paper examines how a number of sensitivity techniques may be adapted to deal with the analysis of real experiments. We show that the fundamental quantities in this context are the partial
derivatives of model input parameters with respect to experimental observables. They appear as a consequence of the procedure used to fit the model to the experiment. These so-called experimental
elementary sensitivities are then combined with the usual model elementary sensitivities to yield coefficients which relate experimental and model observables. The importance of such coefficients for
the planning of experiments is, in particular, discussed. Based on a linear analysis, we also derive simple expressions (containing the experimental elementary sensitivities) for the degree of
parameter deviation arising from uncertainties in and discrepancies between model and measured observables. Finally, an overview of the role of sensitivity theory in analyzing experimental data is
All Science Journal Classification (ASJC) codes
• Computational Mathematics
• Applied Mathematics
Dive into the research topics of 'Sensitivity analysis of experimental data'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/sensitivity-analysis-of-experimental-data","timestamp":"2024-11-02T02:21:31Z","content_type":"text/html","content_length":"49229","record_id":"<urn:uuid:a6f31c4c-6f2d-4548-b2bd-94812260649d>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00541.warc.gz"} |
Blue Heron Capital Partners - Case Solution
The Blue Heron Capital Partners case study deals with a socially-responsible hedge fund. Two firms, AstraZeneca plc and Medco Health Solutions, Inc., both practicing social responsibility, have
options valued written on them.
Kathleen Hevert
Harvard Business Review (BAB711-PDF-ENG)
February 01, 2012
Case questions answered:
1. What is distinctive about option security? Can calls and puts be valued using the traditional discounted cash flow model (DCF) model? Why or why not?
2. Consider the Medco call option with a $40 exercise price and an October expiration. Using the riskless hedge approach and monthly binomial trials, what is this option worth today?
3. Now consider the Medco call option with an exercise price of $40 and a January expiration. Using the riskless hedge approach and monthly binomial trials, what is the option worth today? How do
you explain the difference between values for the October and January calls?
4. What is the value of the Medco call option with a $40 exercise price and an October expiration using the Black Scholes option pricing model? Why does this value differ from the value using the
binomial approach?
5. Let us now turn to the AstraZeneca call option with a $40 exercise price and an October expiration. Using the riskless hedge binomial approach and the Black Sholes option pricing model, what is
this option worth? For now, disregard the dividend AZN is expected to pay in September. How does this value compare to the MHS call with the same contract terms? What accounts for the difference?
6. Returning to the AstraZeneca call option with a $40 exercise price and an October expiration, how do you expect the September dividend to affect the valuation? Why? Use the Black Sholes option
pricing model to validate your prediction. Assume that the 2008 dividend amount and timing will be identical to that of 2007. Are there any problems using the Black-Scholes model to value an
option on a dividend-paying stock?
Not the questions you were looking for? Submit your own questions & get answers.
Blue Heron Capital Partners Case Answers
This case solution includes an Excel file with calculations.
1. What is distinctive about option security? Can calls and puts be valued using the traditional discounted cash flow model (DCF) model? Why or why not?
An option is a right, usually obtained for a fee, to buy or sell an asset within a specified time at a set price, which comes in many forms, not only as stand-alone contracts but also as an embedded
element in securities (such as convertible bonds) and investments in real assets.
The payoff on the option diagram presented in Exhibit 01 can be used to illustrate the distinctive character of an option.
This diagram shows that the value of an option increases as the value of stock increases, and the limited losses and unlimited upside are faced by the option holder.
Basically, the risk-adjusted discount rate for the underlying asset can only be used to evaluate derivatives of that asset if the derivative has the same risk as the underlying asset.
Options are riskier because they are leveraged claims. Even small fluctuations in the value of the underlying asset can greatly affect the return on a small option investment (option premium).
Also, the holder may lose 100% of his investment in the call option, while the holder of the underlying asset almost never loses 100% of his investment.
For an option, the beneficial change in the value of the asset is limited, but for the underlying asset itself, the beneficial outcome is infinite.
These factors, combined with the fact that the risk profile of an option changes every time the value of the underlying asset changes, make estimating the appropriate risk-adjusted discount rate an
impossible task. Therefore, we cannot use DCF to assign options.
2. Consider the Medco call option with a $40 exercise price and an October expiration. Using the riskless hedge approach and monthly binomial trials, what is this option worth today? In order to
build the binomial tree, you need to estimate the volatility for Medco. Using the stock prices provided in the Excel file, follow these steps:
a. Compute daily return for each trading day using adjusted prices as follows:
b. Find the standard deviation of daily returns over the past year.
c. To annualize the standard deviation, take the standard deviation of daily return and multiply by the square root of 260 (the approximate number of trading days in one year).
For the year ended July 18, 2008, the daily volatility of MHS was approximately…
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The world’s most valuable resources are no longer oil, but data (Economist – May 6, 2017). Data visualization is becoming a rapidly evolving blend of science and art dramatically changing the
corporate landscape over the next few years.
Dromic combines expertize in creation of advanced graphics interfaces grounded on sophisticate mathematical models, Artificial Intelligence and data analytics, providing a bridge between business
users and data scientists.
Offer Dromic breakthrough multi-dimensional visualization toolkit for Big Data analytics and IoT (Internet of Things) application, and provide related design and development services
through commercial technological and scientific partnership with key players of these emerging markets.
OpenGL, WebGL
GPU shading languages (GLSL, HLSL, Cg) FFMPEG
DICOM (Digital Imaging and Communication in Medicine) format
Qt/C++, wsWidgets, CTK+, MS Web Form, Sencha ExtJS, React Babylonjs, Three.js, D3.js, PixiJS
Syntactic analysis, parsing and language design.
The world’s most valuable resources are no longer oil, but data (Economist – May 6, 2017). Data visualization is becoming a rapidly evolving blend of science and art dramatically changing the
corporate landscape over the next few years.
Dromic combines expertize in creation of advanced graphics interfaces grounded on sophisticate mathematical models, Artificial Intelligence and data analytics, providing a bridge between business
users and data scientists.
Offer Dromic breakthrough multi-dimensional visualization toolkit for Big Data analytics and IoT (Internet of Things) application, and provide related design and development services
through commercial technological and scientific partnership with key players of these emerging markets.
technological skills
OpenGL, WebGL
GPU shading languages (GLSL, HLSL, Cg) FFMPEG
DICOM (Digital Imaging and Communication in Medicine) format
Qt/C++, wsWidgets, CTK+, MS Web Form, Sencha ExtJS, React Babylonjs, Three.js, D3.js, PixiJS
Syntactic analysis, parsing and language design.
dromic toolkit
Multi-dimensional visualization toolkit for Big Data and IoT (Internet of Things).
dromic toolkit
Multi-dimensional visualization toolkit for Big Data and IoT (Internet of Things).
Hyperbolic tree viewer
Commonly, trees graph are displayed on a Euclidean plane with the root at the top and children below their parents and connected to their parents with edges. This solution is valid for small graphs
while large graphs are extremely difficult to lay out in a way that helps people understand them. Hyperbolic trees, which are a dynamic representation of hierarchical structure, are an effective way
to display complex trees clearly.
In the hyperbolic layout the root is placed at the center while the children are placed at an outer ring to their parents. The circumference jointly increases with the radius and more space becomes
available for the growing numbers of intermediate and leaf nodes.
• Carrying out both layout and drawing in 3D hyperbolic space lets us see a large amount of context around a focus point.
• Our layout is for a good balance between information density and clutter.
• Hyperbolic layout uses a nonlinear (distortion) technique to accommodate focus and context for a large number of nodes.
• Non Overlapping: to ensure that nodes do not overlap each other, hyperbolic layout algorithms assign an open angle for each node. All children of a node are laid out in this open angle.
• Refocusing: transformations are provided to allow fluent node re-positioning.
In this example it is represented the tree of life, kingdom-animalia. You can click on a node to move it to the center or to grab and reposition a single node.
Hyperbolic tree viewer
Commonly, trees graph are displayed on a Euclidean plane with the root at the top and children below their parents and connected to their parents with edges. This solution is valid for small graphs
while large graphs are extremely difficult to lay out in a way that helps people understand them. Hyperbolic trees, which are a dynamic representation of hierarchical structure, are an effective way
to display complex trees clearly.In the hyperbolic layout the root is placed at the center while the children are placed at an outer ring to their parents. The circumference jointly increases with
the radius and more space becomes available for the growing numbers of intermediate and leaf nodes.
• Carrying out both layout and drawing in 3D hyperbolic space lets us see a large amount of context around a focus point.
• Our layout is for a good balance between information density and clutter.
• Hyperbolic layout uses a nonlinear (distortion) technique to accommodate focus and context for a large number of nodes.
• Non Overlapping: to ensure that nodes do not overlap each other, hyperbolic layout algorithms assign an open angle for each node. All children of a node are laid out in this open angle.
• Refocusing: transformations are provided to allow fluent node re-positioning.
This tool is avilable in Virtual Reality environment too.
In this example it is represented the tree of life, kingdom-animalia.
Protein structures visualization and handling in virtual reality
Immersive collaborative and interactive visualization and manipulation are an extremely exciting trend for data analytics. Dealing with data in a tridimensional space through immersive VR headsets
and related interaction gear enable a better comprehension of distances and outliers and enables more natural interaction and an improved engagement and accuracy.
Dromic aims to make available an interactive XR environment for data interaction, visualization and sharing for data analyst and more generally for data consumers in different industrial and research
areas leveraging the enabling XR technologies.
The video on the right shows the three-dimensional structure of the spike protein of SARS-CoV2 (virus responsible for COVID19) in complex with the human ACE2 receptor, created in collaboration with
the engineering department of UBCM (Campus bio-medico university of Rome)
The structure model was downloaded from the Protein Databank website (rcsb.org), the code is 7A94 on the site.
Graph viewer 3D
A graph with many nodes often appears difficult to read because the nodes are overlapped. But what happens if we immerse the node space in a 3D environment? Nodes and connections may appear lighter
looking at the graph from a different point of view.
• Data visualization in 3D environment
• The connections of the nodes appear clearer
• Ability to rotate the graph in all directions
• Zoom and show only a portion of the graph
• Ability to move a node with automatic translation of child nodes
Here you can see an example of a graph drawn in a 3D environment. The data source is represented by the departments of a company. The nodes of the graph are represented by human resources, articles
and news of the company. Each node refers to a specific department.
By visualizing the graph in 3D and rotating the space, the division into departments that are shown as balloons is immediately clear. From the size of the balls one understands the extent of one
department compared to another. By turning the graph further, it is possible to easily identify the connections between the nodes of the same department and those of different departments.
Try to realize it!
Graph viewer 3D
A graph with many nodes often appears difficult to read because the nodes are overlapped. But what happens if we immerse the node space in a 3D environment? Nodes and connections may appear lighter
looking at the graph from a different point of view.
• Data visualization in 3D environment
• The connections of the nodes appear clearer
• Ability to rotate the graph in all directions
• Zoom and show only a portion of the graph
• Ability to move a node with automatic translation of child nodes
Here you can see an example of a graph drawn in a 3D environment. The data source is represented by the departments of a company. The nodes of the graph are represented by human resources, articles
and news of the company. Each node refers to a specific department.
By visualizing the graph in 3D and rotating the space, the division into departments that are shown as balloons is immediately clear. From the size of the balls one understands the extent of one
department compared to another. By turning the graph further, it is possible to easily identify the connections between the nodes of the same department and those of different departments.
Try to realize it!
Graph viewer
This is a tool for data exploration. Very often the items of a data set have a large number of properties associated with them. Starting from a big data source, this tool allows you to build a graph
of these items and show their properties and related connections in a simple and clear way. It’s an easy and fun way to explore data and find connections between them. These are the main features:
• Search Widget for a simple and fast search in the data sources
• Simple system to add nodes to the graph viewer by clicking on search results
• Card mode visualization of node data
• Easy graph navigation system with drag and zoom features
• Creation of automatic links between a new node and those already present on the graph
• Multiple graph drawing algorithms: force-directed, circle, hierarchical, grid, spread, etc.
• Ability to customize the displayed properties for each type of node (human, city, etc)
Here we show an example of the power of the tool. In this example the data source is represented by wikidata. Wikidata (link a wikidata
) is a free and open knowledge base. The Wikidata repository consists mainly of items, each one having a label, a description and any number of aliases. Through our tool you can search some wikidata
items and add them to graph viewer. You can have fun discovering the connections between the different entities. Try it.
Graph viewer
This is a tool for data exploration. Very often the items of a data set have a large number of properties associated with them. Starting from a big data source, this tool allows you to build a graph
of these items and show their properties and related connections in a simple and clear way. It’s an easy and fun way to explore data and find connections between them. These are the main features:
• Search Widget for a simple and fast search in the data sources
• Simple system to add nodes to the graph viewer by clicking on search results
• Card mode visualization of node data
• Easy graph navigation system with drag and zoom features
• Creation of automatic links between a new node and those already present on the graph
• Multiple graph drawing algorithms: force-directed, circle, hierarchical, grid, spread, etc.
• Ability to customize the displayed properties for each type of node (human, city, etc)
Here we show an example of the power of the tool. In this example the data source is represented by wikidata. Wikidata (link a wikidata
) is a free and open knowledge base. The Wikidata repository consists mainly of items, each one having a label, a description and any number of aliases. Through our tool you can search some wikidata
items and add them to graph viewer. You can have fun discovering the connections between the different entities. Try it.
Parallel lines
Parallel coordinates are a common way of visualizing high-dimensional geometry and analyzing multivariate data. It is a diagram composed by several entities visualized as points along a vertical
lines and links between them visualized as wavy lines.
The user is able to interact with the diagram. Basic operations are:
• Highlight families of wavy lines by hovering the mouse in the top part.
• Change the vertical lines order (e.g. swapping two adjacent vertical lines).
• By select a columns set change the mapping of points along the vertical lines.
• Automatic Scatter plot diagrams
• Diagram filters by clicking & dragging along vertical line.
• Scatter plot filters by clicking & dragging on a scatter plot.
• Multiple filters supported
The example shown here is based on a data model made of entities and relationships. Each entity is characterized by a number of properties. The relations are just connections or links between
This Example visualized data based on the SDG Index and Dashboards Report that provides a report card for country performance on the historic Agenda 2030 and the Sustainable Development Goals (SDGs).
Each vertical line of the diagram represents a set of same entity, in this way you can compare an arbitrary number of entities type by viewing the connections between the data.
You can see the relations between pair of adjacent vertical lines represented as curved arcs.
Parallel lines Tool is used by The Sustainable Development Solutions Network (SDSN): Twitter, LinkedIn, Facebook
Parallel lines
Parallel coordinates are a common way of visualizing high-dimensional geometry and analyzing multivariate data. It is a digram composed by several entities visualized as points along a vertical lines
and links between them visualized as wavy lines.
The user is able to interact with the diagram. Basic operations are:
• Highlight families of wavy lines by hovering the mouse in the top part.
• By SQL change the vertical lines order (e.g. swapping two adjacent vertical lines).
• By SQL change the mapping of points along the vertical lines.
The example shown here is based on a data model made of entities and relationships. Each entity is characterized by a number of properties. The relations are just connections or links between
This example visualizes a number of investments extracted from the demo dataset. For each investment we consider these entities:
• The invested amount
• The date of the investment
• The total amount of all the investments made by the investor
• The total amount of all the investments received by the company
• The number of company employees
• The number of articles mentioning the company
By simple SQL query it is possible to extract data from the database and show them directly in the form of a diagram. Each vertical line of the diagram represents a set of same entity, in this way
you can compare an arbitrary number of entities type by viewing the connections between the data.
You can see the relations between pair of adjacent vertical lines represented as curved arcs.
Dromic Voronoi Tool
Dromic Voronoi tool
Given a convex polygon and nested weighted data, it partitions the polygon in several inner tiles which represent the hierarchical structure of your data, such that the area of a cell represents the
weight of the underlying datum.
• Voronoi Layout
• Pan and zoom
• Customize colors and fonts
Dromic Voronoi tool
Given a convex polygon and nested weighted data, it partitions the polygon in several inner tiles which represent the hierarchical structure of your data, such that the area of a cell represents the
weight of the underlying datum.
• Voronoi Layout
• Pan and zoom
• Customize colors and fonts
Scatterplot matrix
A Scatterplot displays the value of 2 sets of data on 2 dimensions. Each observation (or point) in a scatterplot has two coordinates; the first corresponds to the first piece of data in the pair
(that’s the X coordinate; the amount that you go left or right). The second coordinate corresponds to the second piece of data in the pair (that’s the Y-coordinate; the amount that you go up or
down). The point representing that observation is placed at the intersection of the two coordinates. Scatterplots are useful for interpreting trends in statistical data.
Our tool compares a set of types of data. For each data pair, it creates a scatter plot and finally a matrix is drawn.
In the example is drawn a scatter plot matrix based on six types of data: investments, investors, investment amount, funding date, number of articles, number of employees.
Thus it showed a 6×6 square matrix with 30 scatter plots, which are the combination of all six types of data.
Scatterplot matrix
A Scatterplot displays the value of 2 sets of data on 2 dimensions. Each observation (or point) in a scatterplot has two coordinates; the first corresponds to the first piece of data in the pair
(that’s the X coordinate; the amount that you go left or right). The second coordinate corresponds to the second piece of data in the pair (that’s the Y-coordinate; the amount that you go up or
down). The point representing that observation is placed at the intersection of the two coordinates. Scatterplots are useful for interpreting trends in statistical data.
Our tool compares a set of types of data. For each data pair, it creates a scatter plot and finally a matrix is drawn.
In the example is drawn a scatter plot matrix based on six types of data: investments, investors, investment amount, funding date, number of articles, number of employees.
Thus it showed a 6×6 square matrix with 30 scatter plots, which are the combination of all six types of data.
Sankey Diagram
A Sankey diagram depicts flows of any kind, where the width of each flow pictured is based on its quantity.
Sankey diagrams are very good at showing particular kinds of complex information
• Where money came from & went to (budgets, contributions)
• Flows of energy from source to destination
• Flows of goods from place to place
Sankey diagram
A Sankey diagram depicts flows of any kind, where the width of each flow pictured is based on its quantity.
Sankey diagrams are very good at showing particular kinds of complex information
• Where money came from & went to (budgets, contributions)
• Flows of energy from source to destination
• Flows of goods from place to place
Geographic links
When dealing with data that has latitude and longitude between properties it becomes natural to represent them on the earth’s surface. This tool allows to visualize the connections among the nodes of
a graph placed on the terrestrial globe. You can rotate the globe and view connections from various points of view.
• Useful for viewing data that has a geographical position
• It highlights the relationships between the nodes and the geographical position
Here you can view Geolinks viewer in action. The example data set consists of two types of items: investors and companies. Each item has a defined by geographical position. The graph show links
between companies and related investors, in order to see where investors coming from in the world. You can rotate the space by click&drag or by arrows keys.
Geographic links
When dealing with data that has latitude and longitude between properties it becomes natural to represent them on the earth’s surface. This tool allows to visualize the connections among the nodes of
a graph placed on the terrestrial globe. You can rotate the globe and view connections from various points of view.
• Useful for viewing data that has a geographical position
• It highlights the relationships between the nodes and the geographical position
Here you can view Geolinks viewer in action. The example data set consists of two types of items: investors and companies. Each item has a defined by geografical position. The graph show links
between companies and related investors, in order to see where investors coming from in the world. You can rotate the space by click&drag or by arrows keys.
Map 2D
View geographic data on map 2D.
– Filter by single region or multiple regions
– Highlights connected region on mouse over
– Filter by amount
– Save chart as png image
Map 2D with directions
View geographic data on map 2D.
– Filter by single region or multiple regions
– Highlights connected region on mouse over
– Filter by amount
– Save chart as png image
Heat map 3D
A heat map is a two-dimensional representation of data in which values are represented by colors. A simple heat map provides an immediate visual summary of information. More elaborate heat maps allow
the viewer to understand complex data sets. There can be many ways to display heat maps, but they all share one thing in common they use color to communicate relationships between data values that
would be would be much harder to understand if presented numerically in a spreadsheet. In our tool we have co-linked 2 heat map graphs and drawn in a 3d environment.
• Heat maps 3D are a lot more visual than standard analytics reports, which can make them easier to analyze at a glance.
• It is easier to see the connections between 2 heat map charts.
In this example we have put in relation four data entities: investors, companies, years and amount of investment. The color variation indicates the amount of investment.
You can rotate and zoom the graph to display data from multiple points of view.
Heat map 3D
A heat map is a two-dimensional representation of data in which values are represented by colors. A simple heat map provides an immediate visual summary of information. More elaborate heat maps allow
the viewer to understand complex data sets. There can be many ways to display heat maps, but they all share one thing in common they use color to communicate relationships between data values that
would be would be much harder to understand if presented numerically in a spreadsheet. In our tool we have co-linked 2 heat map graphs and drawn in a 3d environment.
• Heat maps 3D are a lot more visual than standard analytics reports, which can make them easier to analyze at a glance.
• It is easier to see the connections between 2 heat map charts.
In this example we have put in relation four data entities: investors, companies, years and amount of investment. The color variation indicates the amount of investment.
You can rotate and zoom the graph to display data from multiple points of view.
Scatter 5D
Usually 3D Scatter plots are used to plot data points on three axes in the attempt to show the relationship between three variables. Each row in the data table is represented by a marker whose
position depends on its values in the columns set on the X, Y, and Z axes.
In this tool we added other 2 dimensions: color and size.
Thus we have five dimensions·
The relationship between different variables is called correlation. If the markers are close to making a straight line in any direction in the three-dimensional space of the 3D scatter plot, the
correlation between the corresponding variables is high. If the markers are equally distributed in the 3D scatter plot, the correlation is low, or zero. However, even though a correlation may seem to
be present, this might not always be the case. The variables could be related to some fourth variable, thus explaining their variation, or pure coincidence might cause an apparent correlation.
You can change how the 3D scatter plot is viewed by zooming in and out as well as rotating it by using the navigation controls located in the top right part of the visualization.
• Change data on five dimensions
• Filter by color and size
• Zoom and Rotation
Scatter 5D
Usually 3D Scatter plots are used to plot data points on three axes in the attempt to show the relationship between three variables. Each row in the data table is represented by a marker whose
position depends on its values in the columns set on the X, Y, and Z axes.
In this tool we added other 2 dimensions: color and size.
Thus we have five dimensions·
The relationship between different variables is called correlation. If the markers are close to making a straight line in any direction in the three-dimensional space of the 3D scatter plot, the
correlation between the corresponding variables is high. If the markers are equally distributed in the 3D scatter plot, the correlation is low, or zero. However, even though a correlation may seem to
be present, this might not always be the case. The variables could be related to some fourth variable, thus explaining their variation, or pure coincidence might cause an apparent correlation.
You can change how the 3D scatter plot is viewed by zooming in and out as well as rotating it by using the navigation controls located in the top right part of the visualization.
• Change data on five dimensions
• Filter by color and size
• Zoom and Rotation
Sunburst diagram
This type of visualisation shows hierarchy through a series of rings, that are sliced for each category node. Each ring corresponds to a level in the hierarchy, with the central circle representing
the root node and the hierarchy moving outwards from it.
Rings are sliced up and divided based on their hierarchical relationship to the parent slice. The angle of each slice is either divided equally under its parent node or can be made proportional to a
Colour can be used to highlight hierarchal groupings or specific categories.
Sunburst diagram
This type of visualisation shows hierarchy through a series of rings, that are sliced for each category node. Each ring corresponds to a level in the hierarchy, with the central circle representing
the root node and the hierarchy moving outwards from it.
Rings are sliced up and divided based on their hierarchical relationship to the parent slice. The angle of each slice is either divided equally under its parent node or can be made proportional to a
Colour can be used to highlight hierarchal groupings or specific categories.
Scatter Regression
Best regression line automatically calculated.
Scatter Regression
Linear regression attempts to model the relationship between two variables by fitting a linear equation to observed data. One variable is considered to be an explanatory variable, and the other is
considered to be a dependent variable. For example, a modeler might want to relate the weights of individuals to their heights using a linear regression model.
Regression analysis is an important tool for modelling and analyzing data. Here, we fit a curve / line to the data points, in such a manner that the differences between the distances of data points
from the curve or line is minimized.
There are innumerable forms of regressions, which can be performed. Each form has its own importance and a specific condition where they are best suited to apply
Regression tool compute the following regression types:
• Linear Regression
• Logarithmic Regression
• Exponential Regression
• Polynomial Regression
Regression Line tool automatically calculates the best regression type according to data.
Hyperbolic Matrix 2d/3d
This type of visualization shows multiple hierarchy data in a matrix through 2d/3d hyperbolic tree visualization.
In this visualization you can:
– Add multiple hyperbolic trees: at the top right we have placed a component that allows you to add a network by root node. Nodes are sorted by type.
– Remove hyperbolic tree: you can remove visualization pressing “Remove” button placed at the top right in each hyperbolic visualization
– Full Screen hyperbolic tree: you can view a hyperbolic tree in full screen pressing “Fullscreen” button placed at the top right in each hyperbolic visualization
– Click on single node to view label: as demo we randomly added images or only label text . Of course, you can view other metadata that you will provide.
– Zoom and Pan: hold the left mouse button to rotate or CTRL + left mouse button to pan, Wheel to zoom
– Colored network nodes: each network node is colored according to the node type
Title Address Description
DROMIC Via Sante Bargellini, 4, 00157 Roma RM, Italia Via Sante Bargellini, 4 00157 Rome Italy +39 06 43252306 | {"url":"https://www.dromic.com/","timestamp":"2024-11-11T17:08:06Z","content_type":"text/html","content_length":"211704","record_id":"<urn:uuid:4bd5801b-ab4a-4371-b6ee-d57aa7442418>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00519.warc.gz"} |
Sane's Monthly Algorithms Challenge: October 2008
Cow Daycare (Advanced Level)
Farmer John has just opened up a new daycare for cows of all sizes. The daycare consists of n (1 ≤ n ≤ 100, 000) cow pens. Each cow pen is built differently to accommodate cows of a unique size from
1 to n.
There is a mother cow inside every pen whose job is to take care of all the cows inside it. When there are no cows inside her pen, she does not need to do any work. For every new cow added to her
pen, the amount of work she has to do to take care of each individual cow is increased by 1.
For example, if she has 1 cow she does 1 piece of work. If she has 2 cows she does 4 pieces of work. 3 cows is 9 pieces of work. So on and so forth.
The daycare had become so popular that Farmer John is taking in more cows than the mother cows can handle. To fix this, he realizes that he can lessen the number of cows in some pens (and thus the
amount of work done by some mothers) by moving smaller cows to the pens built for larger cows.
Only smaller cows can be moved to larger pens (larger cows can not be moved to smaller pens). The size of each cow in a pen does not affect the amount of work done by a mother cow (only the quantity
does). Moreover, there is no limit to the number of cows that can be moved inside a pen.
Given the number of cow pens and the number of cows of each size, move the cows around so that the total amount of work done by all of the mother cows is minimal.
The first line of the input consists of a single integer, n (1 ≤ n ≤ 100,000), representing the number of uniquely sized cow pens.
The next n lines that follow will each contain a non-negative integer ≤ 100 on the i'th line, representing the number of cows that are size i. There will be one test case where cows rule the planet
and each integer may be as large as 100,000.
Output the sum of all work done by each mother such that it's minimal.
Note that it may be necessary to use a 64-bit integer (
long long
in C/C++,
in PAS) to compute your answer.
Sample Input
Sample Output
Explanation of Sample Data
Two of the size 1 cows are moved to pen 2 and another is moved to pen 3. The size 2 cow and one of the size 3 cows move to pen 4. The mothers in 2, 3, and 4 each do four pieces of work while the
mother in pen 1 does 1 piece of work. 1 + 4 + 4 + 4 = 13. While there are other equally optimal placements, there exist none better than 13.
Before (Total Work = 21)
After (Total Work = 13)
Mother cows have not been included in these diagrams.
All Submissions
Best Solutions
Point Value: 30 (partial)
Time Limit: 1.00s
Memory Limit: 64M
Added: Oct 20, 2008
Author: DrSane
Languages Allowed:
C++03, PAS, C, HASK, ASM, RUBY, PYTH2, JAVA, PHP, SCM, CAML, PERL, C#, C++11, PYTH3 | {"url":"https://wcipeg.com/problem/smac081p4","timestamp":"2024-11-06T07:46:15Z","content_type":"text/html","content_length":"12040","record_id":"<urn:uuid:92e1fd8e-bfb1-464e-951f-e1218095155c>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00370.warc.gz"} |
Documentation > Explore
Sensitivity analysis π
Sensitivity analysis correspond to a set of methods capturing how a model reacts to a change in its inputs. The goal of these statistical methods is to measure how variation propagates from the
inputs to the outputs. More specifically, sensitivity analysis is defined by (Saltelli et al., 2008) as describing the Β«relative importance of each input in determining [output] variabilityΒ». As a
consequence, typical result of such methods is an ordering of its inputs according to their sensitivity.
Sensitivity analysis generally involve an a priori sampling of the input space and a statistical method to analyse the co-variance of the inputs and outputs of the model.
Sensitivity analysis can be done at a global or local level. Global methods provide summary statistics of the effects of inputs variation in the complete input space, whereas local methods focus the
effect of inputs variation around a given point of the input space (think of a Jacobian matrix e.g.). The
one factor at a time method
can be viewed as a local sensitivity method, as only one factor vary, the other remaining fixed at their nominal value.
OpenMOLE implements two classical methods for global sensitivity analysis: Morris and Saltelli.
Morris method π
Principle π
Morris method
is a statistical method for global sensitivity analysis. This method is of the type "one-factor-at-a-time", and was conceived as a preliminary computational experiment, to grasp the relative
influence of each factor. In comparison to LHS screening, it has the advantage to provide information for each factor. The input space is considered as a grid and trajectories are sampled among these
points. The method captures output variation when one of the trajectory points is moved to one of its closest neighbors. This variation is called an elementary effect. A certain number of
are generated, in order to observe the consequence of elementary effects anywhere in the input space (trajectories are generated such that given a starting point, any point at fixed distance is
equiprobable - note that the method is still subject to the curse of dimensionality for trajectories to fill the input space). Finally, the method summarizes these elementary effect to estimate
global sensitivity in the output space. This method is computationally cheap, as each trajectory has
parameter points if
is the number of factors. The total number of model runs will thus be
, so the number of trajectory can be adjusted to the computational budget.
Results and Interpretation π
Morris' method computes three sensitivity indicators for each model input and each model output. An elementary effect for input
and output
is obtained when the factor
is changed during one trajectory by a step
from an point
, and computed as
= (y
* x
- y
* (x
+ delta
) / delta
). These are computed as summary statistics on simulated elementary effects and are:
• The overall sensitivity measure, i[j] is the average of the elementary effects epsilon[ij], computed on effects for which factor i was changed (by construction there are exactly R such effects,
one for each trajectory, and they are independent). It is interpreted as the average influence of the input i on the model output j variability. Note that it can aggregate very different
strengths of effects, and more dramatically will cancel opposite effects: an indicator which profile along a dimension is a squared function for example will be considered as unsensitive to the
input regarding this sensitivity index.
• A more robust version of the final sensitivity measure, mu^*[ij], is computed as the average of the absolute value of the elementary effects, ensuring robustness against non-monotonic models. It
is still an average and will miss non-linear effects.
• To account for non-linearities or interactions between factors, the measure sigma[ij] is computed as the standard deviation of the elementary effects. A low standard deviation means that effects
are constant, i.e. the output is linear on this factor. A high value will mean either that the indicator is non-linear on this factor (low variations at some places and high variations are
others) or that variations change a lot when changing other factor values, i.e. that this factor has interactions with others. Both are equivalent regarding a projection on the dimension of
factor considered.
Morris' method within OpenMOLE π
Specific constructor π
constructor is defined in OpenMOLE and takes the following parameters:
• evaluation is the task (or a composition of tasks) that uses your inputs, typically your model task.
• inputs is the list of your model's inputs.
• outputs is the list of your model's outputs, which behavior is evaluated by the method.
• sample is the number of trajectories sampled, which in practice will determine the accuracy of estimation of sensitivity indices but also the total number of runs.
• levels is the resolution of relative variations, i.e. the number of steps p for each dimension of the grid in which trajectories are sampled. In other words, any variation of a factor delta[i]
will be a multiple of 1/p. It should be adapted to the number of trajectories: a higher number of levels will be more suited to a high number of trajectories, to not miss parts of the space with
a small number of local trajectories.
Use example π
Here is how you can make use of this constructor in OpenMOLE:
val i1 = Val[Double]
val i2 = Val[Double]
val i3 = Val[Double]
val o1 = Val[Double]
val o2 = Val[Double]
evaluation = model,
inputs = Seq(
i1 in (0.0, 1.0),
i2 in (0.0, 1.0),
i3 in (0.0, 1.0)),
outputs = Seq(o1, o2),
sample = 10,
level = 10
) hook display
Additional material π
Paper describing method and its evaluation:
Campolongo F, Saltelli A, Cariboni, J, 2011, From screening to quantitative sensitivity analysis. A unified approach, Computer Physics Communication. 182 4, pp. 978-988.
The book on sensitivity analysis is also a good reference for the description of sensitivity analysis methods and case studies of their applications: Saltelli, A., Tarantola, S., Campolongo, F., &
Ratto, M. (2004). Sensitivity analysis in practice: a guide to assessing scientific models (Vol. 1). New York: Wiley.
OpenMOLE Market example
Saltelli's method π
Saltelli is a statistical method for global sensitivity analysis. It estimates sensitivity indices based on relative variances. More precisely, the first order sensitivity coefficient for factor
and output indicator
is computed by first conditionally to any
value, estimating the expectancy of
conditionally to the value of
with all other factors varying, and then considering the variance of these local conditional expectancies. In simpler words, it is the variance after projecting along the dimension of the factor. It
is written as
Var[~i]}[E[X[i]](y[j] x[i])] / Var[y[j]]} where X[~i] are all other factors but x[i]
. An other global sensitivity index does not consider a projection but the full behavior along the factor for all other possible parameter values. This corresponds to the total effect, i.e. first
order but also interactions with other factors. This index is written as
E[X[~i]][Var[x[i]](y[j]X[~i] / Var[y[j]]
. In practice, Sobol quasi-random sequences are used to estimate the indices. The computational budget for this method is fixed by the number of Sobol points drawn, so in practice the user controls
directly the number of model runs.
Saltelli's method within OpenMOLE π
Specific constructor π
constructor is defined in OpenMOLE and can take the following parameters:
• evaluation is the task (or a composition of tasks) that uses your inputs, typically your model task.
• inputs is the list of your model's inputs
• outputs is the list of your model's outputs for which the sensitivity indices will be computed.
• samples number of samples to draw for the estimation of the relative variances, which correspond exactly to the number of model runs. The higher the dimension, the poorer the estimation of
indices will be for low number of samples.
The @code{hook} keyword is used to save or display results generated during the execution of a workflow. The generic way to use it is to write either
hook(workDirectory / "path/of/a/file")
to save the results, or
hook display
to display the results in the standard output. In the output file contains for each index 2 parts: the
and the
, which contain each the matrices of indices for each factor and each indicator.
Use example π
Here is how you can make use of this constructor in OpenMOLE:
val i1 = Val[Double]
val i2 = Val[Double]
val i3 = Val[Double]
val o1 = Val[Double]
val o2 = Val[Double]
evaluation = model,
inputs = Seq(
i1 in (0.0, 1.0),
i2 in (0.0, 1.0),
i3 in (0.0, 1.0)),
outputs = Seq(o1, o2),
sample = 100
) hook display
Additional material π
The variance-based Saltelli method is well described in the paper: Saltelli, A., Annoni, P., Azzini, I., Campolongo, F., Ratto, M., & Tarantola, S. (2010). Variance based sensitivity analysis of
model output. Design and estimator for the total sensitivity index. Computer Physics Communications, 181(2), 259-270. Methods for checking the convergence of indices have been introduced in the
literature (see e.g. Sarrazin, F., Pianosi, F., & Wagener, T. (2016). Global Sensitivity Analysis of environmental models: Convergence and validation. Environmental Modelling & Software, 79,
135-152.), and will soon be introduced in OpenMOLE. | {"url":"https://next.openmole.org/Sensitivity.html","timestamp":"2024-11-08T06:05:07Z","content_type":"text/html","content_length":"28652","record_id":"<urn:uuid:c98fd344-5753-4082-b1a1-acbb1b3620ca>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00575.warc.gz"} |
• Kids math books
the mathematical cat
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Penrose, no ordinary cat, takes kids on adventurous tours of mathematical concepts and characters with his books.
Join Penrose as he continues on his mad cap tour of mathematical ideas.
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Quantitative Aspects of Electrolysis - UCalgary Chemistry Textbook
Electrical current is defined as the rate of flow for any charged species. Most relevant to this discussion is the flow of electrons. Current is measured in a composite unit called an ampere, defined
as one coulomb per second (A = 1 C/s). The charge transferred, Q, by passage of a constant current, I, over a specified time interval, t, is then given by the simple mathematical product
When electrons are transferred during a redox process, the stoichiometry of the reaction may be used to derive the total amount of (electronic) charge involved. For example, the generic reduction
$$M^{n+}(aq) + ne^- ⟶ M(s)$$
involves the transfer of n mole of electrons. The charge transferred is, therefore,
where F is Faraday’s constant, the charge in coulombs for one mole of electrons. If the reaction takes place in an electrochemical cell, the current flow is conveniently measured, and it may be used
to assist in stoichiometric calculations related to the cell reaction.
Converting Current to Moles of Electrons
In one process used for electroplating silver, a current of 10.23 A was passed through an electrolytic cell for exactly 1 hour. How many moles of electrons passed through the cell? What mass of
silver was deposited at the cathode from the silver nitrate solution?
Faraday’s constant can be used to convert the charge (Q) into moles of electrons (n). The charge is the current (I) multiplied by the time
$$n=\frac{Q}{F}=\frac{\frac{10.23\;C}{s}\times 1\;hr\times\frac{60\;min}{hr}\times\frac{60\;s}{min}}{96485\;C/mol\;e^-}=\frac{36830\;C}{96485\;C/mol\;e^-}=0.3817\;mol\;e^-$$
From the problem, the solution contains AgNO[3], so the reaction at the cathode involves 1 mole of electrons for each mole of silver
$$\text{cathode: }Ag^+(aq) + e^− ⟶ Ag(s)$$
The atomic mass of silver is 107.9 g/mol, so
$$\text{mass Ag}=0.3817\;mol\;e^− × \frac{1\;mol\;Ag}{1\;mol\;e^−} × \frac{107.9\;g\;Ag}{1\;mol\;Ag} = 41.19\;g\;Ag$$
Check Your Learning
Aluminum metal can be made from aluminum(III) ions by electrolysis. What is the half-reaction at the cathode? What mass of aluminum metal would be recovered if a current of 25.0 A passed through the
solution for 15.0 minutes?
$Al^{3+}(aq)+3e^−⟶Al(s)$; 0.0777 mol Al = 2.10 g Al.
Time Required for Deposition
In one application, a 0.010-mm layer of chromium must be deposited on a part with a total surface area of 3.3 m^2 from a solution of containing chromium(III) ions. How long would it take to deposit
the layer of chromium if the current was 33.46 A? The density of chromium (metal) is 7.19 g/cm^3.
First, compute the volume of chromium that must be produced (equal to the product of surface area and thickness):
Use the computed volume and the provided density to calculate the molar amount of chromium required:
$$\text{mass}=\text{volume}\times\text{density}=33\;\require{enclose}\enclose{horizontalstrike}{cm^3}\times\frac{7.19\;g}{\enclose{horizontalstrike}{cm^3}}=237\;g\;Cr$$ $$mol\;Cr=237\;g\;Cr\times\
The stoichiometry of the chromium(III) reduction process requires three moles of electrons for each mole of chromium(0) produced, and so the total charge required is:
Finally, if this charge is passed at a rate of 33.46 C/s, the required time is:
Check Your Learning
What mass of zinc is required to galvanize the top of a 3.00 m × 5.50 m sheet of iron to a thickness of 0.100 mm of zinc? If the zinc comes from a solution of Zn(NO[3])[2] and the current is 25.5 A,
how long will it take to galvanize the top of the iron? The density of zinc is 7.140 g/cm^3.
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How do you find m and b given -3x+y=6? | HIX Tutor
How do you find m and b given #-3x+y=6#?
Answer 1
Solve for $y$ to put in slope-intercept form:
To find #m# and #b# solve for #y# to put this equation in slope intercept form.
#color(red)(3x) - 3x + y = color(red)(3x) + 6#
#0 + y = color(red)(3x) + 6#
#y = 3x + 6#
The slope-intercept form of a linear equation is:
#y = color(red)(m)x + color(blue)(b)#
Where #color(red)(m)# is the slope and #color(blue)(b# is the y-intercept value.
#y = color(red)(3)x + color(blue)(6)#
The slope is #color(red)(m = 3)#
The y-intercept is #color(blue)(b = 6)# or (0, 6)
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Answer 2
To find the values of ( m ) and ( b ) given the equation ( -3x + y = 6 ), you need to rewrite the equation in slope-intercept form, ( y = mx + b ), where ( m ) represents the slope and ( b )
represents the y-intercept.
Given the equation ( -3x + y = 6 ), rearrange it to solve for ( y ):
[ y = 3x + 6 ]
Comparing this equation to the slope-intercept form ( y = mx + b ), you can see that ( m = 3 ) and ( b = 6 ).
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
Not the question you need?
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• Readily available 24/7 | {"url":"https://tutor.hix.ai/question/how-do-you-find-m-and-b-given-3x-y-6-8f9af92267","timestamp":"2024-11-04T11:11:19Z","content_type":"text/html","content_length":"570364","record_id":"<urn:uuid:38b6d34b-4eac-45d1-a84c-be0f2dcd7446>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00334.warc.gz"} |
Provides fundamental numerical algorithms including linear computations, Fourier transform, ordinary differential equations, finite element method, random numbers, digital filtering, and digital
image processing. Students will write programs on these topics to understand and to use practically the alogorithms. (Dept. Mechanical Eng. / Robotics / Micro System Tech., B3)
1st 4/ 8 Introduction: Analytical solution, Numerical solution
2nd 4/15 ODE: Canonical forms of ordinary differential equations, Euler/Heun/Runge-Kutta methods
3rd 4/22 ODE: Runge-Kutta-Fehlberg method, Holonomic constraints, Constraint stabilization method (CSM)
4th 4/29 Linear equations: Gaussian elimination, LU decomposition, Triangular matrices, Cholesky decomposition
5th 5/11 Projection: Minimum error solution, Projection matrix, Gram-Schmidt orthogonalization, QR decomposition
6th 5/13 (1st quiz) ODE, Linear equations
7th 5/20 Interpolation: Piecewise linear interpolation, Spline interpolation
8th 5/27 Variational principles: Lagrangian, Lagrangian under constraints, Lagrange eqs. of motion
9th 6/ 3 Variational principles: Open link mechanism, Closed link mechanism
10th 6/ 8 Variational principles: Rigid body rotation, Quaternions
11th 6/10 (2nd quiz) Projection, Interpolation, Variational principles
12th 6/17 FEM: Shape functions, Stiffness matrix, Static deformation of beam
13th 7/ 1 FEM: Inertia matrix, Dynamic deformation of beam
14th 7/ 8 Fourier Transform: Discrete fourier transform(DFT), Fast fourier transform (FFT)
15th 7/20 Fourier Transform: Matched filter, Phase-only correlation method
1st Introduction (updated 2013/4/2) reduced copy
2nd,3rd ODE (updated 2013/4/16) (updated 2013/4/14) reduced copy
4th Linear equations (updated 2013/4/16) reduced copy
5,6th Projection (updated 2013/4/29) reduced copy
7th Interpolation (updated 2013/5/15) reduced copy
8,9,10th Variational principles (updated 2013/5/27) (updated 2013/5/26) reduced copy
11,12,13th FEM (updated 2013/5/27) reduced copy
14,15th Fourier Transform (updated 2013/5/27) reduced copy
Numerical Methods for Mechanical Systems (2nd print) errata (updated 2013/5/27) (updated 2013/4/2) | {"url":"https://hirailab.com/edu/2013/algorithm/algorithm-e.html","timestamp":"2024-11-04T09:21:12Z","content_type":"text/html","content_length":"8531","record_id":"<urn:uuid:ca8c7271-4597-4d8f-a909-376a167b5d1f>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00544.warc.gz"} |
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Rounding the GPA can be done easily by using calculator. simply insert the values in the allocated fields and click the blue color calculate button and see the answer in fraction of seconds. | {"url":"https://roundingcalculator.guru/round-gpa-calculator/","timestamp":"2024-11-10T18:25:49Z","content_type":"text/html","content_length":"43342","record_id":"<urn:uuid:7382bf9c-5c43-4bb8-b81f-9840bc0a9104>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00567.warc.gz"} |
dpbsv.f - Linux Manuals (3)
dpbsv.f (3) - Linux Manuals
dpbsv.f -
subroutine dpbsv (UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO)
DPBSV computes the solution to system of linear equations A * X = B for OTHER matrices
Function/Subroutine Documentation
subroutine dpbsv (characterUPLO, integerN, integerKD, integerNRHS, double precision, dimension( ldab, * )AB, integerLDAB, double precision, dimension( ldb, * )B, integerLDB, integerINFO)
DPBSV computes the solution to system of linear equations A * X = B for OTHER matrices
DPBSV computes the solution to a real system of linear equations
A * X = B,
where A is an N-by-N symmetric positive definite band matrix and X
and B are N-by-NRHS matrices.
The Cholesky decomposition is used to factor A as
A = U**T * U, if UPLO = 'U', or
A = L * L**T, if UPLO = 'L',
where U is an upper triangular band matrix, and L is a lower
triangular band matrix, with the same number of superdiagonals or
subdiagonals as A. The factored form of A is then used to solve the
system of equations A * X = B.
UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N is INTEGER
The number of linear equations, i.e., the order of the
matrix A. N >= 0.
KD is INTEGER
The number of superdiagonals of the matrix A if UPLO = 'U',
or the number of subdiagonals if UPLO = 'L'. KD >= 0.
NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrix B. NRHS >= 0.
AB is DOUBLE PRECISION array, dimension (LDAB,N)
On entry, the upper or lower triangle of the symmetric band
matrix A, stored in the first KD+1 rows of the array. The
j-th column of A is stored in the j-th column of the array AB
as follows:
if UPLO = 'U', AB(KD+1+i-j,j) = A(i,j) for max(1,j-KD)<=i<=j;
if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(N,j+KD).
See below for further details.
On exit, if INFO = 0, the triangular factor U or L from the
Cholesky factorization A = U**T*U or A = L*L**T of the band
matrix A, in the same storage format as A.
LDAB is INTEGER
The leading dimension of the array AB. LDAB >= KD+1.
B is DOUBLE PRECISION array, dimension (LDB,NRHS)
On entry, the N-by-NRHS right hand side matrix B.
On exit, if INFO = 0, the N-by-NRHS solution matrix X.
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, 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.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
November 2011
Further Details:
The band storage scheme is illustrated by the following example, when
N = 6, KD = 2, and UPLO = 'U':
On entry: On exit:
* * a13 a24 a35 a46 * * u13 u24 u35 u46
* a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56
a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66
Similarly, if UPLO = 'L' the format of A is as follows:
On entry: On exit:
a11 a22 a33 a44 a55 a66 l11 l22 l33 l44 l55 l66
a21 a32 a43 a54 a65 * l21 l32 l43 l54 l65 *
a31 a42 a53 a64 * * l31 l42 l53 l64 * *
Array elements marked * are not used by the routine.
Definition at line 165 of file dpbsv.f.
Generated automatically by Doxygen for LAPACK from the source code. | {"url":"https://www.systutorials.com/docs/linux/man/3-dpbsv.f/","timestamp":"2024-11-10T20:37:00Z","content_type":"text/html","content_length":"10832","record_id":"<urn:uuid:8e756aa2-8023-4181-abca-3fc672796c36>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00651.warc.gz"} |
Add and Subtract 10 to Two-Digit Numbers: CCSS.MATH.CONTENT.1.NBT.C.5 - Common Core: 1st Grade Math
All Common Core: 1st Grade Math Resources
Example Questions
Example Question #1 : Add And Subtract 10 To Two Digit Numbers: Ccss.Math.Content.1.Nbt.C.5
Example Question #2 : Add And Subtract 10 To Two Digit Numbers: Ccss.Math.Content.1.Nbt.C.5
Example Question #1 : Add And Subtract 10 To Two Digit Numbers: Ccss.Math.Content.1.Nbt.C.5
Example Question #4 : Add And Subtract 10 To Two Digit Numbers: Ccss.Math.Content.1.Nbt.C.5
Example Question #12 : Place Value And Properties Of Operations To Add And Subtract
Example Question #281 : Number & Operations In Base Ten
Example Question #2 : Add And Subtract 10 To Two Digit Numbers: Ccss.Math.Content.1.Nbt.C.5
Example Question #8 : Add And Subtract 10 To Two Digit Numbers: Ccss.Math.Content.1.Nbt.C.5
Example Question #9 : Add And Subtract 10 To Two Digit Numbers: Ccss.Math.Content.1.Nbt.C.5
Example Question #2 : Add And Subtract 10 To Two Digit Numbers: Ccss.Math.Content.1.Nbt.C.5
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All Common Core: 1st Grade Math Resources | {"url":"https://cdn.varsitytutors.com/common_core_1st_grade_math-help/add-and-subtract-10-to-two-digit-numbers-ccss-math-content-1-nbt-c-5","timestamp":"2024-11-03T09:14:18Z","content_type":"application/xhtml+xml","content_length":"175750","record_id":"<urn:uuid:93f121eb-c518-4684-bfab-5173eafc25c4>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00343.warc.gz"} |
Exponential Dichotomy and Eberlein-Weak Almost Periodic Solutions
Exponential Dichotomy and Eberlein-Weak Almost Periodic Solutions ()
in a Banach space X, where
1. Introduction
The aim of this work is to investigate the existence and uniqueness of a weakly almost periodic solution in the sense of Eberlein for the following linear equation :
for all
Further, if
We also assume
1) For every fixed
The problem of the existence of almost periodic solutions has been extensively studies in the literature [1-6]. Eberlein-weak almost periodic functions are more general than almost periodic functions
and they were introduced by Eberlein [7], for more details about this topics we refer to [8-11] where the authors gave an important overview about the theory of Eberlein weak almost periodic
functions and their applications to differential equations. In the literature, many works are devoted to the existence of almost periodic and pseudo almost periodic solutions for differential
equations (a pseudo almost periodic function is the sum of an almost periodic function and of an ergodic perturbation), but results about Eberlein weak almost periodic solutions are rare [7,12-16].
In ([17], Chap. 3) the authors investigate the existence and uniqueness of an almost periodic solution for equation (1) when the corresponding homogeneous equation of (1) has an exponential dichotomy
and the function f is almost periodic. In ([17], Chap. 3) the authors showed that, if the corresponding homogeneous equation of (1) has an exponential dichotomy and the function f is almost periodic,
the equation (1) has a unique bounded integral solution on
2. Eberlein-Weak Almost Periodic Functions
In the sequel, we give some properties about weak almost periodic functions in the sense of Eberlein (Eberlein-weak almost periodic functions).
Let X and Y be two Banach spaces. Denote by
Definition 2.1 A bounded continuous function
is a relatively compact set in
We denote these functions by
Definition 2.2 A function J:
is relatively compact with respect to the weak topology of the sup-normed Banach space
For the sequel,
Theorem 2.3 Equipped with the norm
the vector space
In [18,19] Deleeuw and Glicksberg proved that if we consider the subspace of those Eberlein weakly almost periodic functions, which contain zero in the weak closure of the orbit (weak topology of
the following decomposition
holds. Moreover, if
uniformly in
For a more detailed information about the decomposition and the ergodic result we refer to the book of Krengel [20,21].
In order to prove the weak compactness of the translates, Ruess and Summers extended the double limits criterion of Grothendieck [22] to the following:
Proposition 2.4 A subset
1) H is bounded in
2) for all
whenever the iterated limits exist.
This result will be the main tool in verifying weak almost periodicity. For the other task we will use.
Proposition 2.5 For every Eberlein weakly almost periodic function f there exists a sequence
3. Statement of the Main Result
In this section, we state a result of the existence and uniqueness of an Eberlein-weakly almost periodic solution of the Periodic Inhomogeneous Linear Equation (1). The existence and uniqueness of an
almost periodic and bounded solution has been studied by M. N’Guérékata ([17]). More precisely, the author proved the following result.
Theorem 3.1 ([17]) Assume that
We propose to extend the above theorem to the case where f is Eberlein-weakly almost periodic.
Theorem 3.2 Assume that
For the proof of theorem (3.2), we use the following lemmas.
Lemma 3.3 Let
whenever the iterated limits exist.
Proof. Noting that only the equality of the iterated limits has to be proved, we may pass to subsequences. Therefore we assume that the following limits exists
here 1) and 2) can be obtained by a diagonalization argument. Since
Again by uniform continuity of f, and by the choice of subsequences we find for the interchanged limits.
Lemma 3.4 Let
Proof. We first prove that the set
whenever the iterated limits exist. Since
a standard trick of topology gives
Lemma 3.5 Let, for a Banach space
Proof. In Order to prove that
whenever the iterated limits exist. Assuming that the iterated limits exist and by the fact that we only have to prove the equality of them, we may pass to subsequences for the verification. Since g
is Eberlein weakly almost periodic with a relatively compact range, by a use of a diagonalization routine, we may assume that
for a suitable choice of subsequences
By hypothesis, we have
From the convergence of
Using the double limits condition of the sequence
This yields, by a standard estimate, that
The following example shows that the compactness assumption on the range of g is essential and that the periodicity of
Example 3.6 We let
Further, if
Using Lemma 2. 16 in ([13]), we obtain that g is Eberlein weakly almost periodic. Now, for the sequences
some calculations lead to the identity :
Proof. (of Theorem 3.2) Since f is Eberlein-weakly almost periodic, then f is continuous and bounded on
We claim that
In fact, for any
On the other hand, we have
which ends the claim.
Now, to complete the proof, it remains for us to prove that
whenever the iterated limits exist. Assuming that the iterated limits exist and by the fact that we only have to prove the equality of them, we may pass to subsequences.
Since by Lemma (3.5)
are Eberlein weakly almost periodic, we may assume that
Bringing the last estimate into play we obtain
The uniform boundedness of the sequences of linear functional
and the fact that Lemma (3.4) applies to
By going to appropriate subsequences, we can assume that the iterated double limits for
Starting with
which concludes the proof.
4. Acknowledgements
The authors would like to thank the I. R. D. (Institut de Recherche pour le Développement , UMI 209) for its hospitality and support. | {"url":"https://www.scirp.org/journal/paperinformation?paperid=22995","timestamp":"2024-11-05T22:49:14Z","content_type":"application/xhtml+xml","content_length":"134428","record_id":"<urn:uuid:584a6f57-5589-4fb5-9ab8-f38a98c2d956>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00777.warc.gz"} |
The Golden Ratio has many other names. You might hear it referred to as the Golden Section, Golden Proportion, Golden Mean, phi ratio, Sacred Cut, or Divine Proportion. They all mean the same thing.
In its simplest form, the Golden Ratio is 1:phi. This is not pi as in π or 3.14... and is not pronounced "pie." This is phi and is pronounced "fie."
Phi is represented by the lower-case Greek letter φ. Its numeric equivalent is 1.618...which means its decimal stretches to infinity and never repeats (much like pi). "The DaVinci Code" had it wrong
when the protagonist assigned an "exact" value of 1.618 to phi.
Phi also performs amazing feats of derring-do in trigonometry and quadratic equations. It can even be used to write a recursive algorithm when programming software. But let's get back to aesthetics.
What the Golden Ratio Looks Like
The easiest way to picture the Golden Ratio is by looking at a rectangle with a width of 1, and a length of 1.168... If you were to draw a line in this plane so that one square and one rectangle
resulted, the square's sides would have a ratio of 1:1. And the "leftover" rectangle? It would be exactly proportionate to the original rectangle: 1:1.618.
You could then draw another line in this smaller rectangle, again leaving a 1:1 square and a 1:1.618... rectangle. You can keep doing this until you're left with an indecipherable blob; the ratio
continues on in a downward pattern regardless. | {"url":"https://www.malekalqadi.com/post/the-golden-ratio","timestamp":"2024-11-01T19:51:10Z","content_type":"text/html","content_length":"1050379","record_id":"<urn:uuid:2505292b-8779-47cb-a24d-947abd0aa084>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00465.warc.gz"} |
Fully Polynomial-Time Algorithms Parameterized by Vertex Integrity Using Fast Matrix Multiplication
We study the computational complexity of several polynomial-time-solvable graph problems parameterized by vertex integrity, a measure of a graph’s vulnerability to vertex removal in terms of
connectivity. Vertex integrity is the smallest number ι such that there is a set S of ι^′ ≤ ι vertices such that every connected component of G−S contains at most ι−ι^′ vertices. It is known that the
vertex integrity lies between the well-studied parameters vertex cover number and tree-depth. Our work follows similar studies for vertex cover number [Alon and Yuster, ESA 2007] and tree-depth
[Iwata, Ogasawara, and Ohsaka, STACS 2018]. Alon and Yuster designed algorithms for graphs with small vertex cover number using fast matrix multiplications. We demonstrate that fast matrix
multiplication can also be effectively used when parameterizing by vertex integrity ι by developing efficient algorithms for problems including an O(ι^ω−1n)-time algorithm for Maximum Matching and an
O(ι^(ω−1)/^2n^2) ⊆ O(ι^0.687n^2)-time algorithm for All-Pairs Shortest Paths. These algorithms can be faster than previous algorithms parameterized by tree-depth, for which fast matrix multiplication
is not known to be effective.
Publication series
Name Leibniz International Proceedings in Informatics, LIPIcs
Volume 274
ISSN (Print) 1868-8969
Conference 31st Annual European Symposium on Algorithms, ESA 2023
Country/Territory Netherlands
City Amsterdam
Period 4/09/23 → 6/09/23
• Adaptive Algorithms
• Algebraic Algorithms
• APSP
• FPT in P
• Matching
• Subgraph Detection
ASJC Scopus subject areas
Dive into the research topics of 'Fully Polynomial-Time Algorithms Parameterized by Vertex Integrity Using Fast Matrix Multiplication'. Together they form a unique fingerprint. | {"url":"https://cris.bgu.ac.il/en/publications/fully-polynomial-time-algorithms-parameterized-by-vertex-integrit-2","timestamp":"2024-11-05T18:57:06Z","content_type":"text/html","content_length":"61159","record_id":"<urn:uuid:3adea0ff-7f77-491b-ae77-7d8580dab619>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00722.warc.gz"} |
Seasonal Indices for Quarterly Data
This online calculator calculates seasonal indices based on quarterly data using the method of simple averages
This calculator complements Seasonal fluctuations. Seasonal indices. Method of simple averages calculator, which works only on monthly data. Here you can enter your quarterly data, and the calculator
computes seasonal indices using the method of simple averages. There are several methods to calculate seasonal indices, and the method of simple averages is the simplest of them. Technically, you
calculate seasonal indices in three steps.
1. Calculate total average, that is, sum all data and divide by the number of periods (i.e., years) multiplied by the number of seasons (i.e., quarters). For example, for three years data, you have
to sum all entries and divide by 3(years)*4(quarters)=12.
2. Calculate the average for each season over periods, i.e., sum data for the first quarter, divide by the number of years, and repeat this to each quarter.
3. Calculate the seasonal index for each season by dividing seasonal average by total average and expressing the result in percents.
The sum of all indices should be 100%*(number of seasons). In case you have lost some precision during the calculation, you may need to normalize your data. For quarters, divide 400% by your sum,
then multiply each index by the obtained coefficient. After that, they will add up to 400%.
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Q The slope of the tangent to the cerve (y−x5)2=x(1+x2)2 at the... | Filo
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Question Text Q The slope of the tangent to the cerve at the point is
Updated On Sep 15, 2022
Topic Calculus
Subject Mathematics
Class Class 12
Answer Type Video solution: 1
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Primitive Type i16
1.0.0 ·
Expand description
The 16-bit signed integer type.
1.43.0 · source
pub const MIN: Self = -32_768i16
The smallest value that can be represented by this integer type (−2^15).
Basic usage:
assert_eq!(i16::MIN, -32768);
1.43.0 · source
pub const MAX: Self = 32_767i16
The largest value that can be represented by this integer type (2^15 − 1).
Basic usage:
assert_eq!(i16::MAX, 32767);
The size of this integer type in bits.
assert_eq!(i16::BITS, 16);
Converts a string slice in a given base to an integer.
The string is expected to be an optional + or - sign followed by digits. Leading and trailing whitespace represent an error. Digits are a subset of these characters, depending on radix:
This function panics if radix is not in the range from 2 to 36.
Basic usage:
assert_eq!(i16::from_str_radix("A", 16), Ok(10));
Returns the number of ones in the binary representation of self.
Basic usage:
let n = 0b100_0000i16;
assert_eq!(n.count_ones(), 1);
Returns the number of zeros in the binary representation of self.
Basic usage:
assert_eq!(i16::MAX.count_zeros(), 1);
Returns the number of leading zeros in the binary representation of self.
Depending on what you’re doing with the value, you might also be interested in the ilog2 function which returns a consistent number, even if the type widens.
Basic usage:
let n = -1i16;
assert_eq!(n.leading_zeros(), 0);
Returns the number of trailing zeros in the binary representation of self.
Basic usage:
let n = -4i16;
assert_eq!(n.trailing_zeros(), 2);
1.46.0 (const: 1.46.0) · source
Returns the number of leading ones in the binary representation of self.
Basic usage:
let n = -1i16;
assert_eq!(n.leading_ones(), 16);
1.46.0 (const: 1.46.0) · source
Returns the number of trailing ones in the binary representation of self.
Basic usage:
let n = 3i16;
assert_eq!(n.trailing_ones(), 2);
Shifts the bits to the left by a specified amount, n, wrapping the truncated bits to the end of the resulting integer.
Please note this isn’t the same operation as the << shifting operator!
Basic usage:
let n = -0x5ffdi16;
let m = 0x3a;
assert_eq!(n.rotate_left(4), m);
Shifts the bits to the right by a specified amount, n, wrapping the truncated bits to the beginning of the resulting integer.
Please note this isn’t the same operation as the >> shifting operator!
Basic usage:
let n = 0x3ai16;
let m = -0x5ffd;
assert_eq!(n.rotate_right(4), m);
Reverses the byte order of the integer.
Basic usage:
let n = 0x1234i16;
let m = n.swap_bytes();
assert_eq!(m, 0x3412);
Reverses the order of bits in the integer. The least significant bit becomes the most significant bit, second least-significant bit becomes second most-significant bit, etc.
Basic usage:
let n = 0x1234i16;
let m = n.reverse_bits();
assert_eq!(m, 0x2c48);
assert_eq!(0, 0i16.reverse_bits());
Converts an integer from big endian to the target’s endianness.
On big endian this is a no-op. On little endian the bytes are swapped.
Basic usage:
let n = 0x1Ai16;
if cfg!(target_endian = "big") {
assert_eq!(i16::from_be(n), n)
} else {
assert_eq!(i16::from_be(n), n.swap_bytes())
Converts an integer from little endian to the target’s endianness.
On little endian this is a no-op. On big endian the bytes are swapped.
Basic usage:
let n = 0x1Ai16;
if cfg!(target_endian = "little") {
assert_eq!(i16::from_le(n), n)
} else {
assert_eq!(i16::from_le(n), n.swap_bytes())
Converts self to big endian from the target’s endianness.
On big endian this is a no-op. On little endian the bytes are swapped.
Basic usage:
let n = 0x1Ai16;
if cfg!(target_endian = "big") {
assert_eq!(n.to_be(), n)
} else {
assert_eq!(n.to_be(), n.swap_bytes())
Converts self to little endian from the target’s endianness.
On little endian this is a no-op. On big endian the bytes are swapped.
Basic usage:
let n = 0x1Ai16;
if cfg!(target_endian = "little") {
assert_eq!(n.to_le(), n)
} else {
assert_eq!(n.to_le(), n.swap_bytes())
Checked integer addition. Computes self + rhs, returning None if overflow occurred.
Basic usage:
assert_eq!((i16::MAX - 2).checked_add(1), Some(i16::MAX - 1));
assert_eq!((i16::MAX - 2).checked_add(3), None);
Unchecked integer addition. Computes self + rhs, assuming overflow cannot occur.
This results in undefined behavior when self + rhs > i16::MAX or self + rhs < i16::MIN, i.e. when checked_add would return None.
1.66.0 (const: 1.66.0) · source
Checked addition with an unsigned integer. Computes self + rhs, returning None if overflow occurred.
Basic usage:
assert_eq!(1i16.checked_add_unsigned(2), Some(3));
assert_eq!((i16::MAX - 2).checked_add_unsigned(3), None);
Checked integer subtraction. Computes self - rhs, returning None if overflow occurred.
Basic usage:
assert_eq!((i16::MIN + 2).checked_sub(1), Some(i16::MIN + 1));
assert_eq!((i16::MIN + 2).checked_sub(3), None);
Unchecked integer subtraction. Computes self - rhs, assuming overflow cannot occur.
This results in undefined behavior when self - rhs > i16::MAX or self - rhs < i16::MIN, i.e. when checked_sub would return None.
1.66.0 (const: 1.66.0) · source
Checked subtraction with an unsigned integer. Computes self - rhs, returning None if overflow occurred.
Basic usage:
assert_eq!(1i16.checked_sub_unsigned(2), Some(-1));
assert_eq!((i16::MIN + 2).checked_sub_unsigned(3), None);
Checked integer multiplication. Computes self * rhs, returning None if overflow occurred.
Basic usage:
assert_eq!(i16::MAX.checked_mul(1), Some(i16::MAX));
assert_eq!(i16::MAX.checked_mul(2), None);
Unchecked integer multiplication. Computes self * rhs, assuming overflow cannot occur.
This results in undefined behavior when self * rhs > i16::MAX or self * rhs < i16::MIN, i.e. when checked_mul would return None.
Checked integer division. Computes self / rhs, returning None if rhs == 0 or the division results in overflow.
Basic usage:
assert_eq!((i16::MIN + 1).checked_div(-1), Some(32767));
assert_eq!(i16::MIN.checked_div(-1), None);
assert_eq!((1i16).checked_div(0), None);
1.38.0 (const: 1.52.0) · source
Checked Euclidean division. Computes self.div_euclid(rhs), returning None if rhs == 0 or the division results in overflow.
Basic usage:
assert_eq!((i16::MIN + 1).checked_div_euclid(-1), Some(32767));
assert_eq!(i16::MIN.checked_div_euclid(-1), None);
assert_eq!((1i16).checked_div_euclid(0), None);
Checked integer remainder. Computes self % rhs, returning None if rhs == 0 or the division results in overflow.
Basic usage:
assert_eq!(5i16.checked_rem(2), Some(1));
assert_eq!(5i16.checked_rem(0), None);
assert_eq!(i16::MIN.checked_rem(-1), None);
1.38.0 (const: 1.52.0) · source
Checked Euclidean remainder. Computes self.rem_euclid(rhs), returning None if rhs == 0 or the division results in overflow.
Basic usage:
assert_eq!(5i16.checked_rem_euclid(2), Some(1));
assert_eq!(5i16.checked_rem_euclid(0), None);
assert_eq!(i16::MIN.checked_rem_euclid(-1), None);
1.7.0 (const: 1.47.0) · source
Checked negation. Computes -self, returning None if self == MIN.
Basic usage:
assert_eq!(5i16.checked_neg(), Some(-5));
assert_eq!(i16::MIN.checked_neg(), None);
1.7.0 (const: 1.47.0) · source
Checked shift left. Computes self << rhs, returning None if rhs is larger than or equal to the number of bits in self.
Basic usage:
assert_eq!(0x1i16.checked_shl(4), Some(0x10));
assert_eq!(0x1i16.checked_shl(129), None);
Unchecked shift left. Computes self << rhs, assuming that rhs is less than the number of bits in self.
This results in undefined behavior if rhs is larger than or equal to the number of bits in self, i.e. when checked_shl would return None.
1.7.0 (const: 1.47.0) · source
Checked shift right. Computes self >> rhs, returning None if rhs is larger than or equal to the number of bits in self.
Basic usage:
assert_eq!(0x10i16.checked_shr(4), Some(0x1));
assert_eq!(0x10i16.checked_shr(128), None);
Unchecked shift right. Computes self >> rhs, assuming that rhs is less than the number of bits in self.
This results in undefined behavior if rhs is larger than or equal to the number of bits in self, i.e. when checked_shr would return None.
1.13.0 (const: 1.47.0) · source
Checked absolute value. Computes self.abs(), returning None if self == MIN.
Basic usage:
assert_eq!((-5i16).checked_abs(), Some(5));
assert_eq!(i16::MIN.checked_abs(), None);
1.34.0 (const: 1.50.0) · source
Checked exponentiation. Computes self.pow(exp), returning None if overflow occurred.
Basic usage:
assert_eq!(8i16.checked_pow(2), Some(64));
assert_eq!(i16::MAX.checked_pow(2), None);
🔬This is a nightly-only experimental API. (isqrt #116226)
Returns the square root of the number, rounded down.
Returns None if self is negative.
Basic usage:
assert_eq!(10i16.checked_isqrt(), Some(3));
Saturating integer addition. Computes self + rhs, saturating at the numeric bounds instead of overflowing.
Basic usage:
assert_eq!(100i16.saturating_add(1), 101);
assert_eq!(i16::MAX.saturating_add(100), i16::MAX);
assert_eq!(i16::MIN.saturating_add(-1), i16::MIN);
1.66.0 (const: 1.66.0) · source
Saturating addition with an unsigned integer. Computes self + rhs, saturating at the numeric bounds instead of overflowing.
Basic usage:
assert_eq!(1i16.saturating_add_unsigned(2), 3);
assert_eq!(i16::MAX.saturating_add_unsigned(100), i16::MAX);
Saturating integer subtraction. Computes self - rhs, saturating at the numeric bounds instead of overflowing.
Basic usage:
assert_eq!(100i16.saturating_sub(127), -27);
assert_eq!(i16::MIN.saturating_sub(100), i16::MIN);
assert_eq!(i16::MAX.saturating_sub(-1), i16::MAX);
1.66.0 (const: 1.66.0) · source
Saturating subtraction with an unsigned integer. Computes self - rhs, saturating at the numeric bounds instead of overflowing.
Basic usage:
assert_eq!(100i16.saturating_sub_unsigned(127), -27);
assert_eq!(i16::MIN.saturating_sub_unsigned(100), i16::MIN);
1.45.0 (const: 1.47.0) · source
Saturating integer negation. Computes -self, returning MAX if self == MIN instead of overflowing.
Basic usage:
assert_eq!(100i16.saturating_neg(), -100);
assert_eq!((-100i16).saturating_neg(), 100);
assert_eq!(i16::MIN.saturating_neg(), i16::MAX);
assert_eq!(i16::MAX.saturating_neg(), i16::MIN + 1);
1.45.0 (const: 1.47.0) · source
Saturating absolute value. Computes self.abs(), returning MAX if self == MIN instead of overflowing.
Basic usage:
assert_eq!(100i16.saturating_abs(), 100);
assert_eq!((-100i16).saturating_abs(), 100);
assert_eq!(i16::MIN.saturating_abs(), i16::MAX);
assert_eq!((i16::MIN + 1).saturating_abs(), i16::MAX);
Saturating integer multiplication. Computes self * rhs, saturating at the numeric bounds instead of overflowing.
Basic usage:
assert_eq!(10i16.saturating_mul(12), 120);
assert_eq!(i16::MAX.saturating_mul(10), i16::MAX);
assert_eq!(i16::MIN.saturating_mul(10), i16::MIN);
Saturating integer division. Computes self / rhs, saturating at the numeric bounds instead of overflowing.
Basic usage:
assert_eq!(5i16.saturating_div(2), 2);
assert_eq!(i16::MAX.saturating_div(-1), i16::MIN + 1);
assert_eq!(i16::MIN.saturating_div(-1), i16::MAX);
let _ = 1i16.saturating_div(0);
1.34.0 (const: 1.50.0) · source
Saturating integer exponentiation. Computes self.pow(exp), saturating at the numeric bounds instead of overflowing.
Basic usage:
assert_eq!((-4i16).saturating_pow(3), -64);
assert_eq!(i16::MIN.saturating_pow(2), i16::MAX);
assert_eq!(i16::MIN.saturating_pow(3), i16::MIN);
Wrapping (modular) addition. Computes self + rhs, wrapping around at the boundary of the type.
Basic usage:
assert_eq!(100i16.wrapping_add(27), 127);
assert_eq!(i16::MAX.wrapping_add(2), i16::MIN + 1);
1.66.0 (const: 1.66.0) · source
Wrapping (modular) addition with an unsigned integer. Computes self + rhs, wrapping around at the boundary of the type.
Basic usage:
assert_eq!(100i16.wrapping_add_unsigned(27), 127);
assert_eq!(i16::MAX.wrapping_add_unsigned(2), i16::MIN + 1);
Wrapping (modular) subtraction. Computes self - rhs, wrapping around at the boundary of the type.
Basic usage:
assert_eq!(0i16.wrapping_sub(127), -127);
assert_eq!((-2i16).wrapping_sub(i16::MAX), i16::MAX);
1.66.0 (const: 1.66.0) · source
Wrapping (modular) subtraction with an unsigned integer. Computes self - rhs, wrapping around at the boundary of the type.
Basic usage:
assert_eq!(0i16.wrapping_sub_unsigned(127), -127);
assert_eq!((-2i16).wrapping_sub_unsigned(u16::MAX), -1);
Wrapping (modular) multiplication. Computes self * rhs, wrapping around at the boundary of the type.
Basic usage:
assert_eq!(10i16.wrapping_mul(12), 120);
assert_eq!(11i8.wrapping_mul(12), -124);
Wrapping (modular) division. Computes self / rhs, wrapping around at the boundary of the type.
The only case where such wrapping can occur is when one divides MIN / -1 on a signed type (where MIN is the negative minimal value for the type); this is equivalent to -MIN, a positive value that is
too large to represent in the type. In such a case, this function returns MIN itself.
This function will panic if rhs is 0.
Basic usage:
assert_eq!(100i16.wrapping_div(10), 10);
assert_eq!((-128i8).wrapping_div(-1), -128);
1.38.0 (const: 1.52.0) · source
Wrapping Euclidean division. Computes self.div_euclid(rhs), wrapping around at the boundary of the type.
Wrapping will only occur in MIN / -1 on a signed type (where MIN is the negative minimal value for the type). This is equivalent to -MIN, a positive value that is too large to represent in the type.
In this case, this method returns MIN itself.
This function will panic if rhs is 0.
Basic usage:
assert_eq!(100i16.wrapping_div_euclid(10), 10);
assert_eq!((-128i8).wrapping_div_euclid(-1), -128);
Wrapping (modular) remainder. Computes self % rhs, wrapping around at the boundary of the type.
Such wrap-around never actually occurs mathematically; implementation artifacts make x % y invalid for MIN / -1 on a signed type (where MIN is the negative minimal value). In such a case, this
function returns 0.
This function will panic if rhs is 0.
Basic usage:
assert_eq!(100i16.wrapping_rem(10), 0);
assert_eq!((-128i8).wrapping_rem(-1), 0);
1.38.0 (const: 1.52.0) · source
Wrapping Euclidean remainder. Computes self.rem_euclid(rhs), wrapping around at the boundary of the type.
Wrapping will only occur in MIN % -1 on a signed type (where MIN is the negative minimal value for the type). In this case, this method returns 0.
This function will panic if rhs is 0.
Basic usage:
assert_eq!(100i16.wrapping_rem_euclid(10), 0);
assert_eq!((-128i8).wrapping_rem_euclid(-1), 0);
Wrapping (modular) negation. Computes -self, wrapping around at the boundary of the type.
The only case where such wrapping can occur is when one negates MIN on a signed type (where MIN is the negative minimal value for the type); this is a positive value that is too large to represent in
the type. In such a case, this function returns MIN itself.
Basic usage:
assert_eq!(100i16.wrapping_neg(), -100);
assert_eq!(i16::MIN.wrapping_neg(), i16::MIN);
Panic-free bitwise shift-left; yields self << mask(rhs), where mask removes any high-order bits of rhs that would cause the shift to exceed the bitwidth of the type.
Note that this is not the same as a rotate-left; the RHS of a wrapping shift-left is restricted to the range of the type, rather than the bits shifted out of the LHS being returned to the other end.
The primitive integer types all implement a rotate_left function, which may be what you want instead.
Basic usage:
assert_eq!((-1i16).wrapping_shl(7), -128);
assert_eq!((-1i16).wrapping_shl(128), -1);
Panic-free bitwise shift-right; yields self >> mask(rhs), where mask removes any high-order bits of rhs that would cause the shift to exceed the bitwidth of the type.
Note that this is not the same as a rotate-right; the RHS of a wrapping shift-right is restricted to the range of the type, rather than the bits shifted out of the LHS being returned to the other
end. The primitive integer types all implement a rotate_right function, which may be what you want instead.
Basic usage:
assert_eq!((-128i16).wrapping_shr(7), -1);
assert_eq!((-128i16).wrapping_shr(64), -128);
Wrapping (modular) absolute value. Computes self.abs(), wrapping around at the boundary of the type.
The only case where such wrapping can occur is when one takes the absolute value of the negative minimal value for the type; this is a positive value that is too large to represent in the type. In
such a case, this function returns MIN itself.
Basic usage:
assert_eq!(100i16.wrapping_abs(), 100);
assert_eq!((-100i16).wrapping_abs(), 100);
assert_eq!(i16::MIN.wrapping_abs(), i16::MIN);
assert_eq!((-128i8).wrapping_abs() as u8, 128);
1.51.0 (const: 1.51.0) · source
Computes the absolute value of self without any wrapping or panicking.
Basic usage:
assert_eq!(100i16.unsigned_abs(), 100u16);
assert_eq!((-100i16).unsigned_abs(), 100u16);
assert_eq!((-128i8).unsigned_abs(), 128u8);
Wrapping (modular) exponentiation. Computes self.pow(exp), wrapping around at the boundary of the type.
Basic usage:
assert_eq!(3i16.wrapping_pow(4), 81);
assert_eq!(3i8.wrapping_pow(5), -13);
assert_eq!(3i8.wrapping_pow(6), -39);
Calculates self + rhs
Returns a tuple of the addition along with a boolean indicating whether an arithmetic overflow would occur. If an overflow would have occurred then the wrapped value is returned.
Basic usage:
assert_eq!(5i16.overflowing_add(2), (7, false));
assert_eq!(i16::MAX.overflowing_add(1), (i16::MIN, true));
🔬This is a nightly-only experimental API. (bigint_helper_methods #85532)
Calculates self + rhs + carry and checks for overflow.
Performs “ternary addition” of two integer operands and a carry-in bit, and returns a tuple of the sum along with a boolean indicating whether an arithmetic overflow would occur. On overflow, the
wrapped value is returned.
This allows chaining together multiple additions to create a wider addition, and can be useful for bignum addition. This method should only be used for the most significant word; for the less
significant words the unsigned method u16::carrying_add should be used.
The output boolean returned by this method is not a carry flag, and should not be added to a more significant word.
If the input carry is false, this method is equivalent to overflowing_add.
// Only the most significant word is signed.
// 10 MAX (a = 10 × 2^16 + 2^16 - 1)
// + -5 9 (b = -5 × 2^16 + 9)
// ---------
// 6 8 (sum = 6 × 2^16 + 8)
let (a1, a0): (i16, u16) = (10, u16::MAX);
let (b1, b0): (i16, u16) = (-5, 9);
let carry0 = false;
// u16::carrying_add for the less significant words
let (sum0, carry1) = a0.carrying_add(b0, carry0);
assert_eq!(carry1, true);
// i16::carrying_add for the most significant word
let (sum1, overflow) = a1.carrying_add(b1, carry1);
assert_eq!(overflow, false);
assert_eq!((sum1, sum0), (6, 8));
1.66.0 (const: 1.66.0) · source
Calculates self + rhs with an unsigned rhs
Returns a tuple of the addition along with a boolean indicating whether an arithmetic overflow would occur. If an overflow would have occurred then the wrapped value is returned.
Basic usage:
assert_eq!(1i16.overflowing_add_unsigned(2), (3, false));
assert_eq!((i16::MIN).overflowing_add_unsigned(u16::MAX), (i16::MAX, false));
assert_eq!((i16::MAX - 2).overflowing_add_unsigned(3), (i16::MIN, true));
Calculates self - rhs
Returns a tuple of the subtraction along with a boolean indicating whether an arithmetic overflow would occur. If an overflow would have occurred then the wrapped value is returned.
Basic usage:
assert_eq!(5i16.overflowing_sub(2), (3, false));
assert_eq!(i16::MIN.overflowing_sub(1), (i16::MAX, true));
🔬This is a nightly-only experimental API. (bigint_helper_methods #85532)
Calculates self − rhs − borrow and checks for overflow.
Performs “ternary subtraction” by subtracting both an integer operand and a borrow-in bit from self, and returns a tuple of the difference along with a boolean indicating whether an arithmetic
overflow would occur. On overflow, the wrapped value is returned.
This allows chaining together multiple subtractions to create a wider subtraction, and can be useful for bignum subtraction. This method should only be used for the most significant word; for the
less significant words the unsigned method u16::borrowing_sub should be used.
The output boolean returned by this method is not a borrow flag, and should not be subtracted from a more significant word.
If the input borrow is false, this method is equivalent to overflowing_sub.
// Only the most significant word is signed.
// 6 8 (a = 6 × 2^16 + 8)
// - -5 9 (b = -5 × 2^16 + 9)
// ---------
// 10 MAX (diff = 10 × 2^16 + 2^16 - 1)
let (a1, a0): (i16, u16) = (6, 8);
let (b1, b0): (i16, u16) = (-5, 9);
let borrow0 = false;
// u16::borrowing_sub for the less significant words
let (diff0, borrow1) = a0.borrowing_sub(b0, borrow0);
assert_eq!(borrow1, true);
// i16::borrowing_sub for the most significant word
let (diff1, overflow) = a1.borrowing_sub(b1, borrow1);
assert_eq!(overflow, false);
assert_eq!((diff1, diff0), (10, u16::MAX));
1.66.0 (const: 1.66.0) · source
Calculates self - rhs with an unsigned rhs
Returns a tuple of the subtraction along with a boolean indicating whether an arithmetic overflow would occur. If an overflow would have occurred then the wrapped value is returned.
Basic usage:
assert_eq!(1i16.overflowing_sub_unsigned(2), (-1, false));
assert_eq!((i16::MAX).overflowing_sub_unsigned(u16::MAX), (i16::MIN, false));
assert_eq!((i16::MIN + 2).overflowing_sub_unsigned(3), (i16::MAX, true));
Calculates the multiplication of self and rhs.
Returns a tuple of the multiplication along with a boolean indicating whether an arithmetic overflow would occur. If an overflow would have occurred then the wrapped value is returned.
Basic usage:
assert_eq!(5i16.overflowing_mul(2), (10, false));
assert_eq!(1_000_000_000i32.overflowing_mul(10), (1410065408, true));
Calculates the divisor when self is divided by rhs.
Returns a tuple of the divisor along with a boolean indicating whether an arithmetic overflow would occur. If an overflow would occur then self is returned.
This function will panic if rhs is 0.
Basic usage:
assert_eq!(5i16.overflowing_div(2), (2, false));
assert_eq!(i16::MIN.overflowing_div(-1), (i16::MIN, true));
1.38.0 (const: 1.52.0) · source
Calculates the quotient of Euclidean division self.div_euclid(rhs).
Returns a tuple of the divisor along with a boolean indicating whether an arithmetic overflow would occur. If an overflow would occur then self is returned.
This function will panic if rhs is 0.
Basic usage:
assert_eq!(5i16.overflowing_div_euclid(2), (2, false));
assert_eq!(i16::MIN.overflowing_div_euclid(-1), (i16::MIN, true));
Calculates the remainder when self is divided by rhs.
Returns a tuple of the remainder after dividing along with a boolean indicating whether an arithmetic overflow would occur. If an overflow would occur then 0 is returned.
This function will panic if rhs is 0.
Basic usage:
assert_eq!(5i16.overflowing_rem(2), (1, false));
assert_eq!(i16::MIN.overflowing_rem(-1), (0, true));
1.38.0 (const: 1.52.0) · source
Overflowing Euclidean remainder. Calculates self.rem_euclid(rhs).
Returns a tuple of the remainder after dividing along with a boolean indicating whether an arithmetic overflow would occur. If an overflow would occur then 0 is returned.
This function will panic if rhs is 0.
Basic usage:
assert_eq!(5i16.overflowing_rem_euclid(2), (1, false));
assert_eq!(i16::MIN.overflowing_rem_euclid(-1), (0, true));
1.7.0 (const: 1.32.0) · source
Negates self, overflowing if this is equal to the minimum value.
Returns a tuple of the negated version of self along with a boolean indicating whether an overflow happened. If self is the minimum value (e.g., i32::MIN for values of type i32), then the minimum
value will be returned again and true will be returned for an overflow happening.
Basic usage:
assert_eq!(2i16.overflowing_neg(), (-2, false));
assert_eq!(i16::MIN.overflowing_neg(), (i16::MIN, true));
1.7.0 (const: 1.32.0) · source
Shifts self left by rhs bits.
Returns a tuple of the shifted version of self along with a boolean indicating whether the shift value was larger than or equal to the number of bits. If the shift value is too large, then value is
masked (N-1) where N is the number of bits, and this value is then used to perform the shift.
Basic usage:
assert_eq!(0x1i16.overflowing_shl(4), (0x10, false));
assert_eq!(0x1i32.overflowing_shl(36), (0x10, true));
1.7.0 (const: 1.32.0) · source
Shifts self right by rhs bits.
Returns a tuple of the shifted version of self along with a boolean indicating whether the shift value was larger than or equal to the number of bits. If the shift value is too large, then value is
masked (N-1) where N is the number of bits, and this value is then used to perform the shift.
Basic usage:
assert_eq!(0x10i16.overflowing_shr(4), (0x1, false));
assert_eq!(0x10i32.overflowing_shr(36), (0x1, true));
1.13.0 (const: 1.32.0) · source
Computes the absolute value of self.
Returns a tuple of the absolute version of self along with a boolean indicating whether an overflow happened. If self is the minimum value (e.g., i16::MIN for values of type i16), then the minimum
value will be returned again and true will be returned for an overflow happening.
Basic usage:
assert_eq!(10i16.overflowing_abs(), (10, false));
assert_eq!((-10i16).overflowing_abs(), (10, false));
assert_eq!((i16::MIN).overflowing_abs(), (i16::MIN, true));
1.34.0 (const: 1.50.0) · source
Raises self to the power of exp, using exponentiation by squaring.
Returns a tuple of the exponentiation along with a bool indicating whether an overflow happened.
Basic usage:
assert_eq!(3i16.overflowing_pow(4), (81, false));
assert_eq!(3i8.overflowing_pow(5), (-13, true));
const: 1.50.0 · source
pub const fn pow(self, exp: u32) -> Self
Raises self to the power of exp, using exponentiation by squaring.
Basic usage:
let x: i16 = 2; // or any other integer type
assert_eq!(x.pow(5), 32);
🔬This is a nightly-only experimental API. (isqrt #116226)
Returns the square root of the number, rounded down.
This function will panic if self is negative.
Basic usage:
assert_eq!(10i16.isqrt(), 3);
Calculates the quotient of Euclidean division of self by rhs.
This computes the integer q such that self = q * rhs + r, with r = self.rem_euclid(rhs) and 0 <= r < abs(rhs).
In other words, the result is self / rhs rounded to the integer q such that self >= q * rhs. If self > 0, this is equal to round towards zero (the default in Rust); if self < 0, this is equal to
round towards +/- infinity.
This function will panic if rhs is 0 or the division results in overflow.
Basic usage:
let a: i16 = 7; // or any other integer type
let b = 4;
assert_eq!(a.div_euclid(b), 1); // 7 >= 4 * 1
assert_eq!(a.div_euclid(-b), -1); // 7 >= -4 * -1
assert_eq!((-a).div_euclid(b), -2); // -7 >= 4 * -2
assert_eq!((-a).div_euclid(-b), 2); // -7 >= -4 * 2
Calculates the least nonnegative remainder of self (mod rhs).
This is done as if by the Euclidean division algorithm – given r = self.rem_euclid(rhs), self = rhs * self.div_euclid(rhs) + r, and 0 <= r < abs(rhs).
This function will panic if rhs is 0 or the division results in overflow.
Basic usage:
let a: i16 = 7; // or any other integer type
let b = 4;
assert_eq!(a.rem_euclid(b), 3);
assert_eq!((-a).rem_euclid(b), 1);
assert_eq!(a.rem_euclid(-b), 3);
assert_eq!((-a).rem_euclid(-b), 1);
Calculates the quotient of self and rhs, rounding the result towards negative infinity.
This function will panic if rhs is zero.
On overflow, this function will panic if overflow checks are enabled (default in debug mode) and wrap if overflow checks are disabled (default in release mode).
Basic usage:
let a: i16 = 8;
let b = 3;
assert_eq!(a.div_floor(b), 2);
assert_eq!(a.div_floor(-b), -3);
assert_eq!((-a).div_floor(b), -3);
assert_eq!((-a).div_floor(-b), 2);
Calculates the quotient of self and rhs, rounding the result towards positive infinity.
This function will panic if rhs is zero.
On overflow, this function will panic if overflow checks are enabled (default in debug mode) and wrap if overflow checks are disabled (default in release mode).
Basic usage:
let a: i16 = 8;
let b = 3;
assert_eq!(a.div_ceil(b), 3);
assert_eq!(a.div_ceil(-b), -2);
assert_eq!((-a).div_ceil(b), -2);
assert_eq!((-a).div_ceil(-b), 3);
If rhs is positive, calculates the smallest value greater than or equal to self that is a multiple of rhs. If rhs is negative, calculates the largest value less than or equal to self that is a
multiple of rhs.
This function will panic if rhs is zero.
On overflow, this function will panic if overflow checks are enabled (default in debug mode) and wrap if overflow checks are disabled (default in release mode).
Basic usage:
assert_eq!(16_i16.next_multiple_of(8), 16);
assert_eq!(23_i16.next_multiple_of(8), 24);
assert_eq!(16_i16.next_multiple_of(-8), 16);
assert_eq!(23_i16.next_multiple_of(-8), 16);
assert_eq!((-16_i16).next_multiple_of(8), -16);
assert_eq!((-23_i16).next_multiple_of(8), -16);
assert_eq!((-16_i16).next_multiple_of(-8), -16);
assert_eq!((-23_i16).next_multiple_of(-8), -24);
If rhs is positive, calculates the smallest value greater than or equal to self that is a multiple of rhs. If rhs is negative, calculates the largest value less than or equal to self that is a
multiple of rhs. Returns None if rhs is zero or the operation would result in overflow.
Basic usage:
assert_eq!(16_i16.checked_next_multiple_of(8), Some(16));
assert_eq!(23_i16.checked_next_multiple_of(8), Some(24));
assert_eq!(16_i16.checked_next_multiple_of(-8), Some(16));
assert_eq!(23_i16.checked_next_multiple_of(-8), Some(16));
assert_eq!((-16_i16).checked_next_multiple_of(8), Some(-16));
assert_eq!((-23_i16).checked_next_multiple_of(8), Some(-16));
assert_eq!((-16_i16).checked_next_multiple_of(-8), Some(-16));
assert_eq!((-23_i16).checked_next_multiple_of(-8), Some(-24));
assert_eq!(1_i16.checked_next_multiple_of(0), None);
assert_eq!(i16::MAX.checked_next_multiple_of(2), None);
🔬This is a nightly-only experimental API. (num_midpoint #110840)
Calculates the middle point of self and rhs.
midpoint(a, b) is (a + b) >> 1 as if it were performed in a sufficiently-large signed integral type. This implies that the result is always rounded towards negative infinity and that no overflow will
ever occur.
assert_eq!(0i16.midpoint(4), 2);
assert_eq!(0i16.midpoint(-1), -1);
assert_eq!((-1i16).midpoint(0), -1);
1.67.0 (const: 1.67.0) · source
pub const fn ilog(self, base: Self) -> u32
Returns the logarithm of the number with respect to an arbitrary base, rounded down.
This method might not be optimized owing to implementation details; ilog2 can produce results more efficiently for base 2, and ilog10 can produce results more efficiently for base 10.
This function will panic if self is less than or equal to zero, or if base is less than 2.
assert_eq!(5i16.ilog(5), 1);
Returns the base 2 logarithm of the number, rounded down.
This function will panic if self is less than or equal to zero.
assert_eq!(2i16.ilog2(), 1);
Returns the base 10 logarithm of the number, rounded down.
This function will panic if self is less than or equal to zero.
assert_eq!(10i16.ilog10(), 1);
1.67.0 (const: 1.67.0) · source
Returns the logarithm of the number with respect to an arbitrary base, rounded down.
Returns None if the number is negative or zero, or if the base is not at least 2.
This method might not be optimized owing to implementation details; checked_ilog2 can produce results more efficiently for base 2, and checked_ilog10 can produce results more efficiently for base 10.
assert_eq!(5i16.checked_ilog(5), Some(1));
1.67.0 (const: 1.67.0) · source
Returns the base 2 logarithm of the number, rounded down.
Returns None if the number is negative or zero.
assert_eq!(2i16.checked_ilog2(), Some(1));
1.67.0 (const: 1.67.0) · source
Returns the base 10 logarithm of the number, rounded down.
Returns None if the number is negative or zero.
assert_eq!(10i16.checked_ilog10(), Some(1));
const: 1.32.0 · source
pub const fn abs(self) -> Self
Computes the absolute value of self.
The absolute value of i16::MIN cannot be represented as an i16, and attempting to calculate it will cause an overflow. This means that code in debug mode will trigger a panic on this case and
optimized code will return i16::MIN without a panic.
Basic usage:
assert_eq!(10i16.abs(), 10);
assert_eq!((-10i16).abs(), 10);
Computes the absolute difference between self and other.
This function always returns the correct answer without overflow or panics by returning an unsigned integer.
Basic usage:
assert_eq!(100i16.abs_diff(80), 20u16);
assert_eq!(100i16.abs_diff(110), 10u16);
assert_eq!((-100i16).abs_diff(80), 180u16);
assert_eq!((-100i16).abs_diff(-120), 20u16);
assert_eq!(i16::MIN.abs_diff(i16::MAX), u16::MAX);
Returns a number representing sign of self.
• 0 if the number is zero
• 1 if the number is positive
• -1 if the number is negative
Basic usage:
assert_eq!(10i16.signum(), 1);
assert_eq!(0i16.signum(), 0);
assert_eq!((-10i16).signum(), -1);
Returns true if self is positive and false if the number is zero or negative.
Basic usage:
Returns true if self is negative and false if the number is zero or positive.
Basic usage:
1.32.0 (const: 1.44.0) · source
Return the memory representation of this integer as a byte array in big-endian (network) byte order.
let bytes = 0x1234i16.to_be_bytes();
assert_eq!(bytes, [0x12, 0x34]);
1.32.0 (const: 1.44.0) · source
Return the memory representation of this integer as a byte array in little-endian byte order.
let bytes = 0x1234i16.to_le_bytes();
assert_eq!(bytes, [0x34, 0x12]);
1.32.0 (const: 1.44.0) · source
Return the memory representation of this integer as a byte array in native byte order.
As the target platform’s native endianness is used, portable code should use to_be_bytes or to_le_bytes, as appropriate, instead.
let bytes = 0x1234i16.to_ne_bytes();
if cfg!(target_endian = "big") {
[0x12, 0x34]
} else {
[0x34, 0x12]
Create an integer value from its representation as a byte array in big endian.
let value = i16::from_be_bytes([0x12, 0x34]);
assert_eq!(value, 0x1234);
When starting from a slice rather than an array, fallible conversion APIs can be used:
fn read_be_i16(input: &mut &[u8]) -> i16 {
let (int_bytes, rest) = input.split_at(std::mem::size_of::<i16>());
*input = rest;
Create an integer value from its representation as a byte array in little endian.
let value = i16::from_le_bytes([0x34, 0x12]);
assert_eq!(value, 0x1234);
When starting from a slice rather than an array, fallible conversion APIs can be used:
fn read_le_i16(input: &mut &[u8]) -> i16 {
let (int_bytes, rest) = input.split_at(std::mem::size_of::<i16>());
*input = rest;
Create an integer value from its memory representation as a byte array in native endianness.
As the target platform’s native endianness is used, portable code likely wants to use from_be_bytes or from_le_bytes, as appropriate instead.
let value = i16::from_ne_bytes(if cfg!(target_endian = "big") {
[0x12, 0x34]
} else {
[0x34, 0x12]
assert_eq!(value, 0x1234);
When starting from a slice rather than an array, fallible conversion APIs can be used:
fn read_ne_i16(input: &mut &[u8]) -> i16 {
let (int_bytes, rest) = input.split_at(std::mem::size_of::<i16>());
*input = rest;
👎Deprecating in a future Rust version: replaced by the MIN associated constant on this type
New code should prefer to use i16::MIN instead.
Returns the smallest value that can be represented by this integer type.
👎Deprecating in a future Rust version: replaced by the MAX associated constant on this type
New code should prefer to use i16::MAX instead.
Returns the largest value that can be represented by this integer type.
Trait Implementations§
The resulting type after applying the + operator.
The resulting type after applying the + operator.
The resulting type after applying the + operator.
The resulting type after applying the + operator.
The resulting type after applying the & operator.
The resulting type after applying the & operator.
The resulting type after applying the & operator.
The resulting type after applying the & operator.
The resulting type after applying the | operator.
The resulting type after applying the | operator.
The resulting type after applying the | operator.
The resulting type after applying the | operator.
The resulting type after applying the | operator.
The resulting type after applying the | operator.
The resulting type after applying the ^ operator.
The resulting type after applying the ^ operator.
The resulting type after applying the ^ operator.
The resulting type after applying the ^ operator.
Returns the default value of 0
The resulting type after applying the / operator.
The resulting type after applying the / operator.
The resulting type after applying the / operator.
This operation rounds towards zero, truncating any fractional part of the exact result.
This operation will panic if other == 0 or the division results in overflow.
The resulting type after applying the / operator.
Converts a NonZeroI16 into an i16
Converts a bool to a i16. The resulting value is 0 for false and 1 for true values.
assert_eq!(i16::from(true), 1);
assert_eq!(i16::from(false), 0);
The associated error which can be returned from parsing.
Parses a string
to return a value of this type.
Read more
The resulting type after applying the * operator.
The resulting type after applying the * operator.
The resulting type after applying the * operator.
The resulting type after applying the * operator.
The resulting type after applying the - operator.
The resulting type after applying the - operator.
The resulting type after applying the ! operator.
The resulting type after applying the ! operator.
Compares and returns the maximum of two values.
Read more
Compares and returns the minimum of two values.
Read more
Restrict a value to a certain interval.
Read more
This method tests for self and other values to be equal, and is used by ==.
This method tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.
This method returns an ordering between
values if one exists.
Read more
This method tests less than (for
) and is used by the
Read more
This method tests less than or equal to (for
) and is used by the
Read more
This method tests greater than or equal to (for
) and is used by the
Read more
This method tests greater than (for
) and is used by the
Read more
Method which takes an iterator and generates Self from the elements by multiplying the items.
Method which takes an iterator and generates Self from the elements by multiplying the items.
The resulting type after applying the % operator.
The resulting type after applying the % operator.
The resulting type after applying the % operator.
This operation satisfies n % d == n - (n / d) * d. The result has the same sign as the left operand.
This operation will panic if other == 0 or if self / other results in overflow.
The resulting type after applying the % operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the << operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
The resulting type after applying the >> operator.
🔬This is a nightly-only experimental API. (portable_simd #86656)
The mask element type corresponding to this element type.
The resulting type after applying the - operator.
The resulting type after applying the - operator.
The resulting type after applying the - operator.
The resulting type after applying the - operator.
Method which takes an iterator and generates Self from the elements by “summing up” the items.
Method which takes an iterator and generates Self from the elements by “summing up” the items.
Try to create the target number type from a source number type. This returns an error if the source value is outside of the range of the target type.
The type returned in the event of a conversion error.
Attempts to convert i16 to NonZeroI16.
The type returned in the event of a conversion error.
Try to create the target number type from a source number type. This returns an error if the source value is outside of the range of the target type.
The type returned in the event of a conversion error.
Try to create the target number type from a source number type. This returns an error if the source value is outside of the range of the target type.
The type returned in the event of a conversion error.
Try to create the target number type from a source number type. This returns an error if the source value is outside of the range of the target type.
The type returned in the event of a conversion error.
Try to create the target number type from a source number type. This returns an error if the source value is outside of the range of the target type.
The type returned in the event of a conversion error.
Try to create the target number type from a source number type. This returns an error if the source value is outside of the range of the target type.
The type returned in the event of a conversion error.
Try to create the target number type from a source number type. This returns an error if the source value is outside of the range of the target type.
The type returned in the event of a conversion error.
Try to create the target number type from a source number type. This returns an error if the source value is outside of the range of the target type.
The type returned in the event of a conversion error.
Try to create the target number type from a source number type. This returns an error if the source value is outside of the range of the target type.
The type returned in the event of a conversion error.
Try to create the target number type from a source number type. This returns an error if the source value is outside of the range of the target type.
The type returned in the event of a conversion error.
Try to create the target number type from a source number type. This returns an error if the source value is outside of the range of the target type.
The type returned in the event of a conversion error.
Try to create the target number type from a source number type. This returns an error if the source value is outside of the range of the target type.
The type returned in the event of a conversion error.
Try to create the target number type from a source number type. This returns an error if the source value is outside of the range of the target type.
The type returned in the event of a conversion error.
Try to create the target number type from a source number type. This returns an error if the source value is outside of the range of the target type.
The type returned in the event of a conversion error.
Try to create the target number type from a source number type. This returns an error if the source value is outside of the range of the target type.
The type returned in the event of a conversion error.
Try to create the target number type from a source number type. This returns an error if the source value is outside of the range of the target type.
The type returned in the event of a conversion error.
Auto Trait Implementations§
Blanket Implementations§
Returns the argument unchanged.
Calls U::from(self).
That is, this conversion is whatever the implementation of From<T> for U chooses to do.
The type returned in the event of a conversion error.
Performs the conversion.
The type returned in the event of a conversion error.
Performs the conversion. | {"url":"https://rust-for-linux.github.io/docs/v6.8/core/primitive.i16.html","timestamp":"2024-11-06T12:12:23Z","content_type":"text/html","content_length":"961140","record_id":"<urn:uuid:df2a9e95-01ab-4bfd-95c0-aed7e762c7ce>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00004.warc.gz"} |
Circumference Calculator - Find the Circumference of a Circle
Our circumference calculator is here to help you find the circumference of a circle. With this tool, you can find the circumference using a diameter or a radius.
How to use the circumference calculator
1. Enter the radius or diameter of the circle you want to find the circumference of. Ensure the unit is cm.
2. Click on the Calculate button to get the result.
What is Circumference?
Circumference is a term used in geometry to describe the linear distance around a closed curve or circular shape. It represents the complete perimeter or outer boundary of a circle. The circumference
is an important measurement in understanding the properties of circles and applying them to real-world scenarios.
Formula for Circumference
The formula to calculate the circumference (C) of a circle is:
C = π × d
π (pi) is a mathematical constant approximately equal to 3.14159
d is the diameter of the circle
Alternatively, if you know the radius (r) of the circle, the circumference can be calculated as:
C = 2 × π × r
How to Find the Circumference of a Circle
To find the circumference of a circle, you need to know either the diameter or the radius of the circle. Then, simply plug the value into the appropriate formula mentioned above.
For example, if the diameter of a circle is 10 units, the circumference can be calculated as:
C = π × d
C = 3.14159 × 10
C = 31.4159 units
Circumference to Diameter Ratio
The ratio of the circumference to the diameter of any circle is always equal to π (pi). This means that if you divide the circumference of a circle by its diameter, you will always get the value of
C/d = π
This relationship is often used to estimate the value of π by measuring the circumference and diameter of a circular object.
Other Similar Calculators
Check out other calculators that are similar to this one.
Can the circumference of a circle be less than its diameter?
No, the circumference of a circle is always greater than its diameter. This is because the circumference represents the complete distance around the circle, while the diameter is a straight line
passing through the center.
Is the circumference formula the same for all closed curves?
No, the circumference formula C = π × d is specific to circles. Other closed curves, such as ellipses or irregular shapes, have different formulas or require numerical approximations to calculate
their perimeter.
How does the circumference change when the radius or diameter of a circle changes?
The circumference of a circle is directly proportional to its radius or diameter. If the radius or diameter increases, the circumference also increases proportionally. If the radius or diameter
decreases, the circumference decreases proportionally.
Can you have a circle with an infinite circumference?
No, a circle with an infinite circumference is not possible. The circumference is always a finite value determined by the diameter or radius of the circle.
Why is the circumference important in real-world applications?
The circumference of a circle finds applications in various fields, such as measuring the distance travelled by a wheel or gear, determining the length of circular pipes or cables, calculating the
area enclosed by a circular fence, and many more. Understanding the circumference is crucial in engineering, construction, and manufacturing processes involving circular components.
How do I find the diameter from the circumference?
To determine the diameter of a circle when you know its circumference, simply divide the circumference by π (pi), which is approximately equal to 3.14159. This calculation stems from the formula for
circumference: C = π × d, where C is the circumference and d is the diameter. By rearranging the formula, you can find the diameter as d = C/π.
How to find the area of a circle from the circumference?
The process to find the area of a circle from its circumference involves a few steps. First, divide the circumference by π to obtain the diameter. Then, divide the diameter by 2 to find the radius.
Once you have the radius, square it and multiply by π to get the area.
For example, if a circle has a circumference of 12.57 meters, its diameter would be 4 meters, and its radius would be 2 meters. Squaring 2 and multiplying by π gives an area of approximately 12.57
square meters.
How do I find the radius from the circumference?
To calculate the radius of a circle from its circumference, follow these steps: First, divide the circumference by π to find the diameter. Then, divide the diameter by 2 to obtain the radius. For
instance, if a circle has a circumference of 18.85 meters, its diameter would be 6 meters (18.85/π), and its radius would be 3 meters (6/2).
How to measure the circumference?
There are a few ways to measure the circumference of a circular object:
1. Use a flexible measuring tape or string to wrap around the object, and then measure the length of the tape or string.
2. If you know the diameter or radius, calculate the circumference using the appropriate formula (C = π × d or C = 2 × π × r).
3. Use a specialized tool or app designed to measure circumferences.
What is the formula for the circumference?
The formula for the circumference of a circle is:
C = π × d
Where C is the circumference, d is the diameter, and π (pi) is approximately equal to 3.14159.
Alternatively, if you know the radius (r) instead of the diameter, the formula is:
C = 2 × π × r
What is the circumference of a circle with a radius of 1 meter?
To find the circumference of a circle with a radius of 1 meter, we can use the formula C = 2 × π × r.
Plugging in the values, we get:
C = 2 × π × 1
C = 2 × 3.14159
C ≈ 6.28 meters
How do I find the circumference of a cylinder?
A cylinder's circumference is simply the circumference of its circular base. To find the circumference of a cylinder, you can use the same formula as for a circle:
C = π × d
Where d is the diameter of the circular base of the cylinder.
Alternatively, if you know the radius (r) of the cylinder's base, you can use the formula:
C = 2 × π × r
How do I find the area of a circle with a circumference of 1 meter?
To find the area of a circle with a circumference of 1 meter, follow these steps:
1. Divide the circumference by π to find the diameter: 1 meter / π ≈ 0.318 meters
2. Divide the diameter by 2 to find the radius: 0.318 meters / 2 ≈ 0.159 meters
3. Square the radius: (0.159 meters)^2 ≈ 0.0253 square meters
4. Multiply the squared radius by π: 0.0253 square meters × π ≈ 0.0795 square meters
Therefore, the area of a circle with a circumference of 1 meter is approximately 0.0795 square meters.
How to find the radius of a circle with a circumference of 10 centimeters?
To find the radius of a circle with a circumference of 10 centimeters, follow these steps:
1. Divide the circumference by π to find the diameter: 10 cm / π ≈ 3.18 cm
2. Divide the diameter by 2 to find the radius: 3.18 cm / 2 ≈ 1.59 cm
Therefore, the radius of a circle with a circumference of 10 centimeters is approximately 1.59 centimeters.
What is the unit of the circumference of a circle?
The circumference of a circle is a linear measurement, representing the distance around the circle's edge. As such, the unit of circumference is a unit of length. Common units used for circumference
• Metric units: millimeters (mm), centimeters (cm), meters (m), and kilometers (km)
• Imperial units: inches (in), feet (ft), yards (yd), and miles (mi)
The specific unit used depends on the context and the size of the circle being measured. For example, the circumference of a small circular object might be expressed in centimeters or inches, while
the circumference of a large circular structure could be given in meters or feet. | {"url":"https://thatcalculator.com/calculators/circumference-calculator/","timestamp":"2024-11-08T21:45:40Z","content_type":"text/html","content_length":"34772","record_id":"<urn:uuid:8a230212-d77a-4f26-858c-f71f6f064a82>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00442.warc.gz"} |
Within the indicated shell, the number of reflections that satisfy both resolution limits and observation criterion, and that were used pre-allocated as the cross-validation test reflections before a structure solution process. These data were not used in the structure solution and refinement process and were used to calculate the 'free' R factor | {"url":"https://mmcif.wwpdb.org/dictionaries/mmcif_pdbx_v50.dic/Items/_refine_ls_shell.number_reflns_R_free.html","timestamp":"2024-11-10T04:56:03Z","content_type":"text/html","content_length":"15732","record_id":"<urn:uuid:74dd7d72-7aab-4b02-85d1-5cde3bd26e6b>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00595.warc.gz"} |
How to convert e to decimal
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Deep Learning and Kurtosis-Controlled, Entropy-Based Framework for Human Gait Recognition Using Video Sequences
Department of Computer Science, HITEC University Taxila, Taxila 47080, Pakistan
College of Computer Engineering and Sciences, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
Faculty of Applied Mathematics, Silesian University of Technology, 44-100 Gliwice, Poland
Author to whom correspondence should be addressed.
Submission received: 28 December 2021 / Revised: 16 January 2022 / Accepted: 18 January 2022 / Published: 21 January 2022
Gait is commonly defined as the movement pattern of the limbs over a hard substrate, and it serves as a source of identification information for various computer-vision and image-understanding
techniques. A variety of parameters, such as human clothing, angle shift, walking style, occlusion, and so on, have a significant impact on gait-recognition systems, making the scene quite complex to
handle. In this article, we propose a system that effectively handles problems associated with viewing angle shifts and walking styles in a real-time environment. The following steps are included in
the proposed novel framework: (a) real-time video capture, (b) feature extraction using transfer learning on the ResNet101 deep model, and (c) feature selection using the proposed kurtosis-controlled
entropy (KcE) approach, followed by a correlation-based feature fusion step. The most discriminant features are then classified using the most advanced machine learning classifiers. The simulation
process is fed by the CASIA B dataset as well as a real-time captured dataset. On selected datasets, the accuracy is 95.26% and 96.60%, respectively. When compared to several known techniques, the
results show that our proposed framework outperforms them all.
1. Introduction
Human gait recognition (HGR) [
] is a biometric application used to solve human recognition’s behavioral characteristics from a distance. A few other biometrics, such as handwriting, face [
], iris [
], ear [
], electroencephalography (EEG) [
], etc., are also useful for identifying an individual in a defined vicinity [
]. Gait recognition is critical in security systems. In this modern technological era, we require an innovative and up-to-date biometric application; thus, gait is an ideal approach for identifying
individuals. The primary advantage of gait recognition over other biometric techniques is that it produces desirable results while avoiding identification from low-resolution videos [
]. Each individual has a few unique characteristics, such as walking style, speed, clothes variation, carrying conditions, and variation in angles [
Individual body gestures or walking styles are used to detect human gait features because each subject has a unique walking style. Each subject’s walking style varies depending on the situation and
the type of clothing he is wearing [
]. Additionally, when an individual holds a bag, the features are changed [
]. Gait recognition is divided into two approaches: model-based and model-free, also known as the holistic model [
]. The model-based method requires extravagant computing costs. However, in the free-model-based technique, we can detect suspicious activity through preprocessing and segmentation techniques [
]. The major part of the process is to detect the same person in a different environment because a person’s body language varies in different situations (for example, carrying a bag and moving fast)
Many gait-recognition techniques are presented in the literature using machine learning (ML) [
]. ML is an important research area; it is utilized in several applications, such as human action recognition [
], image processing [
], and recently, COVID-19 diagnostics [
]. A simple gait-recognition method involves a few essential steps, such as preprocessing original video frames, segmenting the region of interest (ROI), extracting features from ROI, and classifying
extracted features using classification algorithms [
]. Researchers use thresholding techniques to segment the ROI after enhancing the contrast of video sequences in the preprocessing step. This step is critical in traditional approaches because the
features are only extracted from these segmented regions. This procedure, however, is complicated and unauthentic.
Deep learning has had a lot of success with human gait recognition in recent years. The convolutional neural network (CNN) is a type of deep learning model that is used for several processes, such as
gait recognition [
], action recognition [
], medical imaging [
], and others [
]. A simple CNN model consists of a few important layers, such as convolutional, pooling, batch normalization, ReLu, GAP, fully connected, and classification layers [
Figure 1
depicts a simple architecture. A group of filters is adapted in the convolutional layer to extract some important features of an image, such as edges and shape. The non-linear conversion is carried
out by the ReLu layer, also known as the activation layer. The batch-normalization layer minimizes the number of training epochs, whereas the pooling layers solve the overflow problem. The fully
connected layer extracts the image’s deep features, also known as high-level features, and classifies them in the final step using the softmax function [
Our major contributions in this work are:
• A database captured in the real-time outdoor environment using more than 50 subjects. The captured videos include a high rate of noise and background complexity.
• Refinement of the contrast of extracted video frames using the 3D box filtering approach and then fine-tuning of the ResNet101 model. The transfer-learning-based model is trained on real-time
captured video frames and extracted features.
• A kurtosis-based heuristic approach is proposed to select the best features and fuse them in one vector using the correlation approach.
• Classification using multiclass one against all-SVM (OaA-SVM) and comparison of the performance of the proposed method on different feature sets.
This paper is organized as follows: previous techniques are discussed in
Section 2
Section 3
describes the proposed method, such as frame refinement, deep learning, feature selection, and classification. Results of the proposed technique are discussed in
Section 4
. Finally, the conclusion and future directions are presented in
Section 5
2. Related Work
There are several methods for HGR using deep learning [
]. The authors focused on deep-learning-based methods in these methods, but the majority of them focused on selecting important features and feature fusion. The experimental process used several gait
datasets, such as CASIA A, CASIA B, and CASIA C [
]. In this work, our focus is on the real-time recorded dataset and CASIA B datasets. CASIA B is a famous dataset and is mostly utilized for gait recognition.
Wang et al. [
] presented a novel gait-recognition method using a convolutional LSTM approach named Conv-LSTM. They performed a few steps to complete the gait-recognition process. At the start, they presented GEI
frame by frame and then expanded its volume to relax the constraint of the gait cycle. Later on, they performed an experiment to analyze the cross-covariance of one subject. After that, they design a
Conv-LSTM model for final gait recognition. The experiments were performed on the CASIA B dataset and OU-ISIR datasets. On the CASIA B dataset, they achieved an accuracy of 93%, and 95% on OU-ISIR,
Arshad et al. [
] presented a new approach for HGR. In this approach, two deep neural networks were used with the FEcS selection method. Two pre-trained models, VGG19 and AlexNet, were used for feature extraction.
The extracted features were refined in the later step using entropy and skewness vectors. Four datasets were used for the experimental process, CASIA A, CASIA B, CASIA C, and AVAMVG. On these
datasets, they achieved the accuracy of 99.8%, 99.7%, 93.3%, and 92.2%, respectively.
Mehmood et al. [
] presented an automated deep-learning-based system for HGR under multiple angles. Four key steps were performed, preprocessing, feature extraction, feature selection, and classification. They
extracted deep-learning features and applied the firefly algorithm for feature optimization. The experiments were performed on a widely available CASIA B dataset and achieved notable accuracy.
Anusha et al. [
] presented a technique for HGR based on multiple features. They extracted low-level features through spatial, texture, and gradient information. Five databases were used for the experimental
process, CASIA A, CASIA B, OU-ISIR D, CMU MoBo, and KTH video datasets. These all datasets were tested on different angles and achieved improved performance.
Sharif et al. [
] presented a method for HGR based on accurate ROI segmentation and multilevel features fusion. Several clothing and carrying conditions were considered for the experimental process and achieved an
accuracy of 98.6%, 93.5%, and 97.3%, respectively. A PoseGait model was presented by Liao et al. [
] for HGR. They considered the problem of drastic variations of individuals. The 3D models were used for capturing data from different angles. The 3D image was defined as 3D coordinates of the human
body joints.
Wu et al. [
] presented a graph-based approach for multiview HGR. The Spiderweb-graph-based approach was applied to capture the data in a single view and then connect to the other view of gait data concurrently.
Memory and capsule modules were used for the trajectory view of each gait as well as STF extraction. The experiments were performed on three challenging gait datasets, SDUgait, OU-MVLP, and CASIA B,
and achieved an accuracy of 98.54%, 96.91%, and 98.77%, respectively. Arshad et al. [
] focused on the feature-selection approach to improve the performance of HGR. They used the Bayesian model and quartile-deviation approaches for enhanced feature vectors.
In conclusion, the previous studies focused primarily on the selection of the most relevant features. They used CNN to extract features and then combined the results with data from a few other
channels. They did not, however, focus on real-time gait recognition due to the complexity of the system design. In this paper, we focus on real-time gait recognition in an outdoor setting.
3. Proposed Methodology
The proposed method for real-time human gait recognition is presented in this section with detailed mathematical modeling and visual results. The following steps are involved in the proposed
framework: video preprocessing, deep learning features extraction using transfer learning, kurtosis-based features selection, correlation-based features fusion, and finally, one-against-all-SVM
(OaA-SVM)-based classification. The main architecture diagram is shown in
Figure 2
. The proposed method is executed in a sequence, and at the end of the execution, it returns a labeled output and numerical results in the form of recall rate, precision, accuracy, etc. The details
of each step are given below in the following subsections.
3.1. Videos Preprocessing
One of the most prominent parts of digital image processing is to enhance the quality of visual information of an image. This technique helps to remove the noise and messy information from the image
and make it more readable [
]. In the area of machine learning, the algorithm requires better information about an object to learn a good model. In this work, we initially process with videos, and in a later stage, convert them
into frames and label them according to the actual class. Then, we resize images into a dimension of
$R × C × c h$
, where
denotes row pixels,
denotes column pixels, and
$c h$
represent number of channels, respectively. We set
$R = C = 224$
$c h = 3$
The real-time videos are recorded into four different angles, and each angle consists of two gait classes—walk while carrying a bag and normal walk without any bag. The four angles are 90°, 180°,
270°, and 360°. The original frames also captured some noise during the video recording. For this purpose, we used the 3D box filter, which is a perfect choice. The detail of 3D box filtering is
given in Algorithm 1.
Algorithm 1: Data Augmentation Process.
Input: Original video frame $← ϕ ( r , c )$.
Output: Improved video frame $← ϕ ˜ ( r , c )$.
Step 1: Load all video frames $← Δ ( ϕ )$.
for: $1 : t o ← N f r a m e s$
Step 2: Calculate filter size.
$F ˜ a b c h ˜ = ∑ r = 1 4 ∑ c = 1 4 ∑ c h ˜ = 1 3 ( F r c c h ˜ )$
Step 3: Perform padding.
$P a d d i n g = F ˜ a b c h ˜ − 1 2$
where horizontal padding is performed.
Step 4: Update padding window.
$U p d ( P a d d i n g ) = ∑ ( P a d d i n g ) t$
where $t = 2$
Step 5: Perform normalization.
$N o r m a l i z a t i o n = 1 ∏ ( U p d ( P a d d i n g ) )$
$∏ ←$ product of padding image.
end for
The visual effects are also shown in the main flow,
Figure 2
. Here, the frames are given before and after the filter processing. The improved frames are utilized in the next step for model learning.
3.2. Convolutional Neural Network
Deep learning emerged recently in the field of computer vision and has since spread to other fields. Convolutional Neural Network (CNN) is a deep learning approach that won a competition for image
classification using ImageNet in 2012 [
]. In deep learning, images are directly passed in the model without segmentation; therefore, it is called image-based ML. A simple CNN consists of convolutional layer, ReLu layer, dense layer,
pooling layer, and a softmax layer. Visually, simple architecture is shown in
Figure 1
. In a CNN, the input image is passed to the network in a fixed size. A convolutional operation is performed in the convolutional layer and in the output weights, and bias vectors are generated.
Back-propagation is used in the CNN for training the weights. Mathematically, convolutional operation is formulated as follows:
$Y ^ C = ∑ k = 1 C ψ k ′ k ∗ x k , ∀ C ′ ,$
$Y ^ C$
is convolutional layer output,
denotes the convolutional kernels,
is composed of channels
$c h$
denotes the convolutional operation,
represents convolutional kernel and is defined by
$ψ ∈ ℝ k × k × C$
. To increase the nonlinearity of a network, ReLu layer is applied and defined as follows:
$Y ^ r ˜ = m a x ( 0 , x ) ,$
Another layer named maxpooling is employed to reduce the dimension of extracted features. This layer is formulated as follows:
$Y ^ m a x ( i , j , c ) = max u , v ∈ N x + u , j + v , c , ∀ i , j , c ,$
$Y ^ m a x$
denotes the max pooling operation,
denotes the input weight matrix, and
$x ∈ Y ^ C$
$u , j , and v$
are static parameters. The filter of this layer is normally fixed as
$3 × 3$
, denoted by
, and stride is
, denoted by
. An important layer in a CNN is fully connected layer. In this layer, all neurons are connected to each other, and resultant output is obtained in a 1D matrix. Finally, the extracted features are
classified in the softmax layer.
3.3. Deep Features Extraction
Feature extraction is the process of reducing original images into small number of important information groups. These groups are called features. The features have few important characteristics such
as color, shape, texture, and location of an object. In this proposed technique, we utilized a pre-trained deep learning network named ResNet101. Originally, this model consists of total of 101
layers and was trained on ImageNet dataset. A visual flow is shown in
Figure 3
. Note that the filter size of first convolutional layer is
$7 × 7$
, stride is 2, and number of channels is
. In the next, a maxpooling layer is applied for filter size
$3 × 3 ,$
and stride is
. Five residual layers are included in this network, and lastly, global average pool layer and FC layer are added following a softmax layer.
Training Model using Transfer Learning: In this work, we use ResNet101 pre-trained CNN model for features extraction [
]. We trained this model using transfer learning, as visually shown in
Figure 4
]. Note that parameters of original ResNet101 pre-trained model are transferred and trained on the real-time captured gait database. We remove the last layer from modified ResNet101 and extract
features. The features are extracted from global average pool (GAP) and FC layers. Mathematically, TL can define as follows: given a source domain
$S D$
and learning task
$L t$
, a target domain
$T D ˜$
and learning task
$L t ˜$
, TL improves the learning of a target predictive function
$F t ( . )$
in the
$T D ˜$
using the knowledge in
$S D$
$L t$
, where
$S D ≠ T D ˜$
$L t ≠ L t ˜$
. After this process, we obtain two feature vectors from GAP and FC layers of dimensions
$N × 2048$
$N × 1000$
, respectively.
3.4. Kurtosis-Controlled, Entropy-Based Feature Selection
Feature selection (FS) is the process of selecting the most prominent features from the original feature vector. In the FS, the value of features is not updated, and features are selected in the
original form [
]. The main motivation behind this step is to obtain the most optimal information of an image and discard the redundant and irrelevant data. In this work, we proposed a new feature selection
technique named kurtosis-controlled OaA-SVM. The working of this technique is given in Algorithms 2 and 3 for both vectors.
Description: Consider that we have two extracted deep feature vectors denoted by $V 1$ and $V 2$, where the dimension of each vector is $N × 2048$ and $N × 1000$, respectively. In the first step,
features are initialized and processed in the system until all features are passed. In the next step, kurtosis is computed for all features in the pair, where window size and stride were $2 × 2$ and
1. A newly generated vector is obtained, which is evaluated in the next step by fitness function. In the fitness function, we select one vs. all SVM classifier that classifies the feature and, in the
output, returns an error rate.
This process continues until the error rate is minimized and stops when error increases for the next iteration. By following this, we obtain two feature vectors in the output of dimension $N × 600$,
as discussed in Algorithms 1 and 2.
Finally, we fused both selected vectors by employing correlation-based approach. In this approach, features of both vectors are paired as
$i ( 1 )$
$j ( 1 )$
. After that, the correlation coefficient is computed, and higher-correlated feature pairs are added to the fused vector. Mathematically, it is defined as follows:
$r v 1 v 2 = ∑ ( v 1 i − v 1 ¯ ) ( v 2 j − v 2 ¯ ) ( ∑ ( v 1 i − v 1 ¯ ) 2 ∑ ( v 2 j − v 2 ¯ ) 2 ) ,$
$v 1 ∈ V 1$
$v 2 ∈ V 2$
$r v 1 v 2$
is a correlation coefficient among two features
$v 1 i$
$v 2 j$
$v 1 i$
$i t h$
features of
$V 1$
, and
$v 2 j$
$j t h$
feature of
$V 2$
, respectively. The notation
$v 1 ¯$
is mean value of feature vector
$V 1$
$v 2 ¯$
is mean value of feature vector
$V 2$
In this approach, we choose those features for fused vector which have correlation value near to 1 or greater than 0. This means we only selected positively correlated features for final fused
vector. This procedure is performed for all features in both vectors and in the output; we obtained a resultant features vector of length $N × f k$, where $f k$ denotes the length of fused feature
Finally, the features in the fused vector are classified using OaA-SVM.
Algorithm 2: Features selection for deep learning model 1.
Input: Feature vector $V 1$ of dimension $N × 2048$.
Output: Selected feature $V s$ of dimension $N × 600$.
Step 1: Features initialization.
for $i ← 1 t o N$// $N = 2048$
Step 2: Compute kurtosis of each feature pair.
$K u r t o s i s = ∑ i = 1 N ( V i − V ¯ ) 4 N [ ∑ i = 1 N ( V i − V ¯ ) 2 N ] 2$// $V i$ is input features, $V ¯$ is mean feature value, and $N$ denotes the total features.
Window size is $2 × 2$// $σ$ is a standard deviation.
Stride is 1.
Step 3: Newly generated kurtosis vector $V ˜$// $V ˜$ is kurtosis vector.
Step 4: Perform fitness function.
OaA-SVM classifier.
Evaluate features.
Compute error rate.
Step 5: Repeat steps 2, 3, and 4 until error rate is minimized.
end for
Algorithm 3: Features selection for deep learning model 2.
Input: Feature vector $V 2$ of dimension$N × 1000$.
Output: Selected feature $V s$ of dimension$N × 600$.
Step 1: Features initialization.
for $i ← 1 t o N$// $N = 1000$
Step 2: Compute kurtosis of each feature pair.
$K u r t o s i s = ∑ i = 1 N ( V i − V ¯ ) 4 N [ ∑ i = 1 N ( V i − V ¯ ) 2 N ] 2$// $V i$ is input features, $V ¯$ is mean feature value, and $N$ denotes the total features.
Window size is $2 × 2$.
Stride is 1.
Step 3: Newly generated kurtosis vector $V ˜$// $V ˜ = random vector size$.
Step 4: Perform fitness function.
OaA-SVM classifier.
Evaluate features.
Compute error rate.
Step 5: Repeat steps 2, 3, and 4 until error rate is minimized.
end for
3.5. Recognition
The one against all SVM (OaA-SVM) is utilized in this work for features classification [
]. The OaA-SVM approach is usually to determine the separation of one gait action with other listed gait action classes. Mostly, this classification is used for unknown patterns to generate the
maximum results.
Here, we have N-class problem, and D represents the training samples:
${ f k 1 , l 1 } , … , { f k D , l D }$
. Here,
$f k i$$∈$
$R b$
represents b-dimensional feature vector, where
$l i ∈$
${ 1 , 2 , … , B }$
is the corresponding class label. This approach usually constructs the binary SVM classifiers. The
$i t h$
represent all the positive labels of training class and the other negative labels. Then,
$i t h$
SVM solves the problems with the following yield-decision function:
$f i ( s ) = h i T ϕ ( f k i ) + β i ,$
$f i ( s )$
is the yield-decision function,
$h i T$
represents weight values,
$f k 1$
denotes input features, and
$β i$
denotes the bias values. The minimization function is applied on weight values as:
$Minimize : M ( h , θ j i ) = 1 2 ‖ h i ‖ 2 + C ∑ l = 1 N θ j i ,$
$Subject to : l j ( h i T ϕ ) ( f k i ) + β i ≥ 1 − θ j i , θ j i ≥ O ,$
where M is a minimization parameter, C denotes the output class,
$θ j i$
is between-class distance,
$l j = 1$
$l j = i$
, and
$l j = − 1$
otherwise. A sample
is classified as in class
$i *$
$i * = i = 1 , … , B arg m a x f k i = i = 1 , … , B arg m a x ( h T ϕ ) ( f k i ) + β i ) ,$
Resultant visual results are also shown in
Figure 5
, in which the classifier returned a labeled output such as walking with bag and normal walk. The numerical results are also computed and described in
Section 4
4. Results
4.1. Datasets
The results of the proposed automated system for gait recognition are presented in this section. The proposed framework is tested on a real-time captured dataset and CASIA B dataset. In the real-time
dataset, a total of 50 students are included, and for each student, we recorded 8 videos in 4 angles, 90°, 180°, 270°, and 360°. For each angle, two videos are recorded—one while wearing a bag and
one without wearing a bag. A few samples are shown in
Figure 6
. For the CASIA B dataset [
], we only consider a 90° angle for evaluation results. In this dataset, three different covariant factors are included, normal walking, walking with a bag, and walking while wearing a coat, as shown
Figure 7
4.2. Experimental Setup
We utilized 70% video frames for training the proposed framework, and the remaining 30% were utilized for the testing. In the training process, a group of hyper parameters was employed, where the
learning rate was 0.001, epochs was 200, mini batch size was 64, the optimization algorithm was Adam, the activation function was sigmoid, the dropout factor was 0.5, momentum was 0.7, the loss
function was cross entropy, and the learning rate schedule was Piecewise. Several classifiers were utilized for validation, such as SVM, K-Nearest Neighbor, decision trees, and Naïve Bayes. The
following parameters were used for the analysis of selected classifiers: precision, recall, F1 Score, AUC, and classification time. All results were computed using K-Fold validation, whereas the
value of K was 10. The simulations of this work were conducted in MATLAB2020a using a dedicated personal desktop computer with 256 GB SSD, 16 GB RAM, and 16 GB memory graphics card.
4.3. Real-Time Dataset Results
In this section, we present our proposed method’s results on real-time video sequences. The results are computed in three different scenarios—normal walking results for all selected angles, walking
while carrying a bag for all three angles, and finally, classification of normal walking and walking while carrying a bag.
Table 1
shows the results of a normal walk for the four selected angles of 90°, 180°, 270°, and 360°, respectively. The OaA-SVM classifier achieved superior performance compared to other listed
classification algorithms. The achieved accuracy by OaA-SVM is 95.75%, whereas the precision rate is 95.25%, F1-Score is 95.98%, AUC is 1.00, and FPR is 0.0125, respectively. The Medium KNN also
gives better performance and achieves an accuracy of 95%. The recall and precision rates are 95% and 95.50%, respectively. The worst accuracy noted for this experiment is 48.75% on the Kernel Naïve
Bayes classifier. The change in accuracy among all classifiers shows the authenticity of this work. The performance of OaA-SVM can be further verified in
Figure 8
. The prediction accuracy for the 270° angle is maximum, whereas the correct prediction performance for 360° is 86%. In the latter, the recognition time is also tabulated in
Table 1
and shows that the Fine Tree classifier executed much faster compared to the other listed classifiers. The execution time of Fine Tree is 19.465 (s). The variation in execution time is also shown in
Figure 9
Table 2
shows results of walking while carrying a bag at 90°, 180°, 270°, and 360° angles, respectively. The OaA-SVM classifier attained better accuracy compared to other classification algorithms. For
OaA-SVM, the accuracy is 96.5%, whereas the precision rate is 97%, F1-Score is 96.7%, AUC is 1.00, and FPR is 0.01, respectively. The KNN series classifiers, such as Cubic KNN and Medium KNN, give
the second-best accuracy of 95.1%. Naïve Bayes classifier achieves an accuracy of 83.7%. The recall and precision rates of the Naïve Bayes classifier are 83.7% and 86.7%, respectively. The Kernel
Naïve Bayes, FG-SVM, and Coarse KNN classifiers do not perform well. They achieve accuracies of 48.6%, 61.3%, and 63%, respectively.
The performance of OaA-SVM can be further verified through
Figure 10
, which shows that the prediction accuracy for the 270° angle is the maximum (100%), whereas correct prediction performance for 360° is 89%. The execution time during the recognition process is also
tabulated in
Table 2
and shows that the Medium Tree classifier executed much faster compared to other listed classifiers. The execution time is also plotted in
Figure 11
, which shows a huge variation due to the classifier’s complexity.
Table 3
represents the classification results of a normal walk and walking while carrying a bag. The purpose of this experiment is to analyze the performance of the proposed algorithm for the binary class
classification problem.
Table 3
shows that the OaA-SVM attained an accuracy of 96.4%. The other calculated measures, such as recall rate, precision rate, f1-Score, AUC, and FPR values, are 96.5%, 96.5%, 96.5%, 1.00, and 0.03,
respectively. Medium KNN gives the second-best classification performance of 96.1%, whereas the recall and precision rates are 96% and 96%, respectively. The following classifiers, Naïve Bayes,
Coarse KNN, Kernel Naïve Bayes, and FG SVM, do not perform well and achieve an accuracy of 67.8%, 79.3%, 735, and 79.6%, respectively. The confusion matrix also provided and illustrated in
Figure 12
shows that the correct prediction rate of both classes is above 90%. In the latter, the execution time is also plotted for this experiment and illustrated in
Figure 13
, which shows that the Fine Tree execution time is better compared to other algorithms.
4.4. CASIA B Dataset Results at a 90° Angle
CASIA B dataset results are presented in this subsection. For experimental results, we only selected a 90° angle because most of the recent studies used this angle. Numerical results are given in
Table 4
. The OaA-SVM gives better results among all other implemented classifiers. The results are computed with different feature vectors, such as original global average pool (GAP) layer features, fully
connected layer features (FC), and the proposed approach. For GAP-layer features, the achieved accuracy is 90.22%, and the recall rate is 90.10%, whereas the FC layer gives an accuracy of 88.64%. The
proposed method gives an accuracy of 95.26%, where the execution time is 176.4450 (s). Using Cubic KNN, the attained accuracy is 93.60%, which is the second-best performance after the OaA-SVM
Table 5
presents the confusion matrix of the OaA-SVM classifier performance using the proposed scheme. In this table, it is illustrated that the corrected prediction accuracy of normal walking is 94%,
walking while wearing a coat (W-Coat) is 95%, and walking while carrying a bag is 97%. In addition, the computational time of all classifiers is plotted in
Figure 14
, which shows that Medium KNN efficiency is better than all other classifiers in terms of the computational cost.
4.5. Discussion
A detailed discussion on results and comparison is conducted in this section. As shown in
Figure 1
, the proposed method has a few important steps, such as video preprocessing, deep-learning feature extraction, feature selection through the kurtosis approach, the fusion of selected features
through the correlation approach, and finally, the OaA-SVM-based feature classification. The proposed method is evaluated on two datasets. One is recorded in the real-time environment, and the second
is CASIA B. The real-time captured dataset results are given in
Table 1
Table 2
Table 3
. In
Table 1
, results are presented for normal walking under four different angles, whereas in
Table 2
, results are given for walking while carrying a bag. The classification of the binary class problem is also conducted, and results are tabulated in
Table 3
. The OaA-SVM outperformed in all three experiments and can be verified through
Table 1
. Additionally, we used the CASIA B dataset for the validation of the proposed technique. The results are listed in
Table 4
and verified through
Table 5
We also performed experiments on different feature sets to confirm the authenticity of the proposed heuristic-feature-selection approach. For this purpose, we select several feature vectors, such as
300 features, 400 features, 500 features, 600 features, 700 features, and a complete feature vector. Results are given in
Table 6
for both datasets. This table shows that the results of 600 features are much better compared to other feature sets. The results increase initially, but after 600 features, accuracy is degraded by
approximately 1%. For all features, the accuracy difference is almost 4%. Based on this table, we can say that the proposed technique achieves a significant performance on 600 features.
The comparison of the proposed method is also listed in
Table 7
. In this table, we only added those techniques in which the CASIA B dataset was used. For a real-time dataset, the comparison with recent techniques is not fair. This table shows that the previous
best-reported accuracy on this dataset was 93.40% [
]. In this work, we have improved this accuracy by almost 2% and reached 95.26%.
Normally, researchers employ metaheuristic techniques, such as a genetic algorithm, PSO, ACO, and BCO [
]. These techniques have consumed too much time during the selection process. However, the proposed feature-selection approach is based on the single-kurtosis value activation function and is
executed fast compared to the GA, PSO, BCO, Whale, and ACO on both the selected dataset, as shown by time plotted in
Figure 15
Figure 16
. These figures show that the proposed feature-selection approach is more suitable for gait recognition in terms of computational time than the metaheuristic techniques.
A detailed analysis is also conducted through confidence intervals. For this purpose, the proposed framework is executed 500 times for both datasets. For the real-time dataset, we obtained three
accuracy values—maximum (96.8%), minimum (95.3%), and average (96.05%). Based on these values, the standard deviation and standard error mean (SEM) are computed and obtain values of 0.75 and 0.5303.
Similarly, for the CASIA B dataset, the standard deviation and standard error mean value are 0.56 and 0.3959, respectively. Using standard deviation and standard error mean values, the margin of
error is computed, as plotted in
Figure 17
Figure 18
. From these figures, it is shown that the proposed framework accuracy is consistent after the number of selected iterations. Moreover, the performance of the proposed framework (OvA-SVM) is also
analyzed with a few other classifiers, such as Softmax, ELM, and KELM, as illustrated in
Figure 19
. From this figure, it can be noted that the proposed framework shows better performance.
The reason for the selected hyper parameters is as follows. Normally, the researchers employed gradient descent [
] as an optimization function, but due to the complex nature of the selected pre-trained model Resnet101 (residual blocks), the ADAM optimizer [
] can work better. The initial learning rate is normally 0.010, but in this work, the selected dataset dimension is high; therefore, the learning rate of 0.001 is suitable. Mini batch size is always
selected based on the machine (desktop computer), but we do not have enough resources for the execution on a mini batch size of 128; therefore, we selected a value of 64. The maximum epochs are
normally 70–100, but in this work, due to the higher number of video frames, we obtained a better training accuracy after 150 epochs. A few training results that were noted during the training
process are given in
Table 8
Table 9
5. Conclusions
Human gait recognition is an active research domain based on an important biometric application. Through gait, the human walking style can be easily determined in the video sequences. The major use
of human gait recognition is in video surveillance, crime prevention, and biometrics. In this work, a deep-learning-based method is presented for human gait recognition in a real-time environment and
for offline publicly available datasets. Transfer learning is applied for feature extraction and then selects robust features using a heuristic approach. The correlation formulation is applied for
the best-selected feature fusion. In the end, multiclass OaA-SVM is applied for the final classification. The methodology is evaluated on a real-time captured database and CASIA B dataset, and it
achieves an accuracy of 96.60% and 95.26%, respectively.
We can conclude that the preprocessing step before the model’s learning reduces the error rate. This step further shows strength in selecting the best features, but some features are discarded, which
are essential for classification. The kurtosis-controlled entropy (KcE) is a new heuristic feature-selection technique that is executed in less time than the metaheuristic techniques. Another new
technique named correlation-formulation-based fusion is used in this work for the best feature fusion. We compare the results of this method to existing methods such as PCA and LDA, and our newly
proposed selection technique gives better results and has a shorter computational time for real-time video sequences. Moreover, the fusion process, through correlation formulation, increased the
information of the human walking style that later helped the improved gait-recognition accuracy.
The drawback of this work is as follows: in the correlation-formulation-based feature fusion step, the method added some redundant features that later increased the computational time and reduced the
recognition accuracy slightly. In the future, we shall improve the fusion approach and increase the database by adding data from more subjects.
Author Contributions
Conceptualization, M.I.S., M.A.K. and M.N.; methodology, M.I.S., M.A.K. and M.N.; software, M.I.S., M.A.K. and M.N.; validation, A.A., S.A. and A.B.; formal analysis, A.A., S.A. and A.B.;
investigation, A.A., S.A. and A.B.; resources, A.A., S.A. and A.B.; data curation, M.A.K. and R.D.; writing—original draft preparation, M.A.K., M.I.S. and M.N.; writing—review and editing, R.D. and
A.A.; visualization, R.D. and S.A.; supervision, R.D.; project administration, R.D. and A.B.; funding acquisition, R.D. All authors have read and agreed to the published version of the manuscript.
This research received no external funding.
Data Availability Statement
Conflicts of Interest
The authors declare no conflict of interest.
Figure 7.
Sample frames collected from CASIA B dataset [
Figure 10. Confusion matrix of OaA-SVM for walking while carrying a bag at different selected angles.
Figure 11. Variation in computational time for selected classifiers of 4 selected angles on Real time dataset.
Figure 12. Confusion matrix of OaA-SVM for normal walking and walking while carrying a bag classification.
Figure 13. Execution time of proposed method on real-time dataset (normal walking and walking while carrying a bag).
Figure 19. Comparison of OvS-SVM classifier with other classification algorithms for gait recognition.
Table 1. Proposed gait-recognition results using real-time captured dataset for normal walk on selected angles.
Performance Measures
Classifiers Recall Precision (%) FI Score AUC FPR Accuracy Time
(%) (%) (%) (s)
OaA-SVM 95.75 96.25 95.98 1.00 0.0125 96.0 204.050
Cubic KNN 94.25 94.75 94.48 1.00 0.0175 94.5 366.480
Medium KNN 95.00 95.50 95.24 1.00 0.0175 94.9 175.870
Bagged Trees 93.50 93.75 93.62 0.99 0.0200 93.5 345.750
CG-SVM 90.25 90.50 90.36 0.99 0.0325 90.1 189.300
Fine Tree 86.75 86.50 86.62 0.92 0.0425 86.7 19.565
Medium Tree 85.50 85.75 85.62 0.92 0.0475 85.4 39.907
Naïve Bayes 84.50 87.25 85.84 0.90 0.0500 84.3 34.035
Coarse KNN 61.50 72.50 66.54 0.92 0.1275 61.5 189.760
FG-SVM 61.00 84.75 70.92 0.91 0.1300 60.9 132.050
Kernel Bayes 48.75 63.50 50.54 0.73 0.1725 48.5 582.710
Table 2. Proposed gait-recognition results using real-time captured dataset for walking while carrying a bag at selected angles.
Classifiers Performance Measures
Recall (%) Precision (%) FI Score (%) AUC FPR Accuracy (%) Time (s)
OaA-SVM 96.5 97.0 96.7 1.00 0.01 96.6 189.850
Cubic KNN 95.0 95.2 95.1 1.00 0.01 95.1 274.850
Medium KNN 95.0 95.2 95.1 1.00 0.01 95.1 9.219
Bagged Trees 94.7 95.0 94.8 0.99 0.01 94.9 202.010
CG-SVM 90.5 91.0 90.7 0.99 0.03 90.5 137.480
Fine Tree 87.7 81.0 84.2 0.93 0.04 87.6 14.190
Medium Tree 86.7 87.0 86.8 0.93 0.04 86.8 13.590
Naïve Bayes 83.7 86.7 85.2 0.89 0.05 83.7 20.920
Coarse KNN 63.0 74.0 68.0 0.92 0.12 63.0 169.521
FG-SVM 61.0 84.7 70.9 0.91 0.13 61.3 27.030
Kernel Bayes 48.7 60.2 53.8 0.75 0.17 48.6 483.800
Table 3. Recognition results for all datasets with kurtosis function on 600 predictions on original deep-feature vector extracted from ResNet 101.
Performance Measures
Classifiers Recall Precision FI Score AUC FPR Accuracy Time
(%) (%) (%) (%) (s)
OaA-SVM 96.5 96.5 96.5 1.00 0.03 96.4 129.830
Cubic KNN 95.0 96.0 95.7 1.00 0.04 95.6 278.980
Medium KNN 96.0 96.0 96.0 1.00 0.04 96.1 106.850
Bagged Trees 90.5 90.5 90.5 0.97 0.09 90.3 209.610
CG-SVM 82.0 82.5 82.2 0.91 0.18 82.2 102.260
Fine Tree 80.0 80.0 80.0 0.81 0.20 79.8 11.954
Medium Tree 76.0 76.5 76.2 0.80 0.24 76.3 30.888
Naïve Bayes 68.0 69.0 68.4 0.78 0.32 67.8 18.767
Coarse KNN 79.0 79.5 79.2 0.88 0.21 79.3 117.990
FG-SVM 79.5 85.5 82.3 0.98 0.41 79.6 74.026
Kernel Bayes 73.0 74.5 73.7 0.84 0.27 73.0 254.580
Classifier Features Measures
GAP FC Proposed Recall (%) Accuracy (%) Time (s)
✓ 90.10 90.22 242.4426
OaA-SVM ✓ 88.52 88.64 176.4450
✓ 95.10 95.26 114.2004
✓ 84.42 84.54 165.5994
Cubic KNN ✓ 83.60 83.98 111.2011
✓ 93.60 93.60 82.1460
✓ 83.40 83.48 151.0014
Medium KNN ✓ 84.80 84.76 104.1446
✓ 93.40 93.46 64.2914
✓ 85.10 85.16 256.1130
Baggage Tree ✓ 84.14 84.33 201.0148
✓ 87.50 87.45 117.1106
✓ 71.10 71.04 171.2540
Naïve Bayes ✓ 74.94 74.82 104.3360
✓ 79.30 79.30 76.3114
Gait Name Gait Name
Normal Walk W-Coat W-Bag
Normal Walk 94% 4% 2%
W-Coat 3% 95% 2%
W-Bag 1% 2% 97%
Table 6. Analysis of proposed feature-selection framework on numerous feature sets. *The OvA-SVM is employed as a classifier for this table.
Dataset Accuracy (%) on Feature Sets
300 Features 400 Features 500 Features 600 Features 700 Features All Features
Real-time (normal walking) 93.70 94.24 95.35 96.00 95.70 93.04
Real-time (walking while carrying a bag) 94.10 94.90 95.80 96.60 96.32 92.10
Real-time (normal walking vs. walking while carrying a bag) 92.90 93.72 95.30 96.40 96.14 93.50
CASIA B Dataset 92.96 93.40 93.85 95.26 95.10 92.64
Reference Year Dataset Accuracy (%)
[45] 2015 CASIA B 86.30
[46] 2017 CASIA B 90.60
[37] 2019 CASIA B 87.7
[6] 2020 CASIA B 93.40
Proposed CASIA B 95.26
Proposed Real-time 96.60
Epochs Accuracy (%) Error (%) Time (min)
20 83.5 16.5 221.6784
40 87.9 12.1 375.7994
60 90.2 9.8 588.7834
80 92.6 5.4 792.5673
100 93.9 5.1 875.1247
150 96.8 1.2 988.0045
200 98.1 0.4 1105.5683
Epochs Accuracy (%) Error (%) Time (min)
20 81.4 18.6 174.8957
40 84.6 15.4 292.0645
60 88.0 12 411.4756
80 90.2 9.8 581.8322
100 91.6 8.4 695.4570
150 94.3 5.7 808.5334
200 97.5 2.5 981.6873
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Sharif, M.I.; Khan, M.A.; Alqahtani, A.; Nazir, M.; Alsubai, S.; Binbusayyis, A.; Damaševičius, R. Deep Learning and Kurtosis-Controlled, Entropy-Based Framework for Human Gait Recognition Using
Video Sequences. Electronics 2022, 11, 334. https://doi.org/10.3390/electronics11030334
AMA Style
Sharif MI, Khan MA, Alqahtani A, Nazir M, Alsubai S, Binbusayyis A, Damaševičius R. Deep Learning and Kurtosis-Controlled, Entropy-Based Framework for Human Gait Recognition Using Video Sequences.
Electronics. 2022; 11(3):334. https://doi.org/10.3390/electronics11030334
Chicago/Turabian Style
Sharif, Muhammad Imran, Muhammad Attique Khan, Abdullah Alqahtani, Muhammad Nazir, Shtwai Alsubai, Adel Binbusayyis, and Robertas Damaševičius. 2022. "Deep Learning and Kurtosis-Controlled,
Entropy-Based Framework for Human Gait Recognition Using Video Sequences" Electronics 11, no. 3: 334. https://doi.org/10.3390/electronics11030334
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